diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json
index 2ef99925..592dad10 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -13,12 +13,12 @@
             "state": {
               "type": "markdown",
               "state": {
-                "file": "AGENTS.md",
+                "file": "wiki/concepts/Abaqus Beam and Shell Section Definitions.md",
                 "mode": "source",
                 "source": false
               },
               "icon": "lucide-file",
-              "title": "AGENTS"
+              "title": "Abaqus Beam and Shell Section Definitions"
             }
           }
         ]
@@ -197,6 +197,13 @@
   },
   "active": "31299cd80f02d44f",
   "lastOpenFiles": [
+    "Clippings/260626_실밸개발자 바이브코딩 클럽.md",
+    "wiki/concepts/Abaqus Beam and Shell Section Definitions.md",
+    "Clippings",
+    "wiki/concepts/Abaqus Analysis Procedures.md",
+    "wiki/concepts/Abaqus Cavity Radiation Interactions.md",
+    "wiki/concepts/Abaqus Adaptivity and Mesh Replacement.md",
+    "AGENTS.md",
     "skills/fem-theory-query/vault-path.txt",
     "skills/fem-theory-query/agents/openai.yaml",
     "skills/fem-theory-query/agents",
@@ -221,11 +228,6 @@
     "wiki/entities/midas NFX.md",
     "wiki/sources/Midas-NFX-Analysis-Manual.md",
     "wiki/meta/lint-report-2026-06-02.md",
-    "wiki/sources/Midas-FEA-Analysis-Manual.md",
-    "wiki/concepts/Uniform Optimal Convergence.md",
-    "wiki/sources/On-the-Finite-Element-Analysis-of-Shell-Structures.md",
-    "wiki/concepts/Midas FEA Embedded Reinforcement Modeling.md",
-    "wiki/concepts/Midas Civil PSC and Prestress Loss Analysis.md",
     "wiki/Wiki Map.canvas",
     "wiki/canvases/main.canvas",
     "wiki/meta/dashboard.base",
@@ -233,8 +235,6 @@
     "wiki/새 폴더",
     "wiki/comparisons",
     "wiki/canvases",
-    "wiki/새 폴더 (2)",
-    "wiki/domains",
     "wiki/canvases/youtube-explainer.canvas",
     "wiki/canvases/welcome.canvas",
     "wiki/canvases/claude-obsidian-presentation.canvas"
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_001.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_001.md
new file mode 100644
index 00000000..953a55db
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_001.md
@@ -0,0 +1,319 @@
+<!-- source-page: 1 -->
+
+# ABAQUS 2016
+
+USER SUBROUTINES REFERENCE GUIDE
+
+![](images/page-001_ce57648d57b43b9a35b489df2a3253d8bc27ae54534ec5d3aa6f8535461a88b2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+3D
+i
+V+R
+</details>
+
+3DEXPERIENCE
+
+<!-- source-page: 2 -->
+
+<!-- source-page: 3 -->
+
+# Abaqus User Subroutines
+
+Reference Guide
+
+<!-- source-page: 4 -->
+
+# Legal Notices
+
+Abaqus, the 3DS logo, and SIMULIA are commercial trademarks or registered trademarks of Dassault Systèmes or its subsidiaries in the United States and/or other countries. Use of any Dassault Systèmes or its subsidiaries trademarks is subject to their express written approval.
+
+Abaqus and this documentation may be used or reproduced only in accordance with the terms of the software license agreement signed by the customer, or, absent such an agreement, the then current software license agreement to which the documentation relates.
+
+This documentation and the software described in this documentation are subject to change without prior notice.
+
+Dassault Systèmes and its subsidiaries shall not be responsible for the consequences of any errors or omissions that may appear in this documentation.
+
+© Dassault Systèmes, 2015
+
+Other company, product, and service names may be trademarks or service marks of their respective owners. For additional information concerning trademarks, copyrights, and licenses, see the Legal Notices in the Abaqus 2016 Installation and Licensing Guide.
+
+<!-- source-page: 5 -->
+
+# Preface
+
+This section lists various resources that are available for help with using Abaqus Unified FEA software.
+
+# Support
+
+Both technical software support (for problems with creating a model or performing an analysis) and systems support (for installation, licensing, and hardware-related problems) for Abaqus are offered through a global network of support offices, as well as through our online support system. Contact information for our regional offices is accessible from SIMULIA→Locations at www.3ds.com/simulia. The online support system is accessible by selecting the SUBMIT A REQUEST link at Support - Dassault Systèmes (http://www.3ds.com/support).
+
+# Online support
+
+Dassault Systèmes provides a knowledge base of questions and answers, solutions to questions that we have answered, and guidelines on how to use Abaqus, Engineering Process Composer, Isight, Tosca, fe-safe, and other SIMULIA products. The knowledge base is available by using the Search our Knowledge option on www.3ds.com/support (http://www.3ds.com/support).
+
+By using the online support system, you can also submit new requests for support. All support/service requests are tracked. If you contact us by means outside the system to discuss an existing support problem and you know the support request number, please mention it so that we can query the support system to see what the latest action has been.
+
+# Training
+
+All SIMULIA regional offices offer regularly scheduled public training classes. The courses are offered in a traditional classroom form and via the Web. We also provide training seminars at customer sites. All training classes and seminars include workshops to provide as much practical experience with Abaqus as possible. For a schedule and descriptions of available classes, see the Training link at www.3ds.com/productsservices/simulia (www.3ds.com/products-services/simulia) or call your support office.
+
+# Feedback
+
+We welcome any suggestions for improvements to Abaqus software, the support tool, or documentation. We will ensure that any enhancement requests you make are considered for future releases. If you wish to make a suggestion about the service or products, refer to www.3ds.com/simulia. Complaints should be made by contacting your support office or by visiting SIMULIA→Quality Assurance at www.3ds.com/simulia (www.3ds.com/simulia).
+
+<!-- source-page: 6 -->
+
+<!-- source-page: 7 -->
+
+# Contents
+
+# 1. User Subroutines
+
+# Abaqus/Standard subroutines
+
+CREEP: Define time-dependent, viscoplastic behavior (creep and swelling). 1.1.1
+
+DFLOW: Define nonuniform pore fluid velocity in a consolidation analysis. 1.1.2
+
+DFLUX: Define nonuniform distributed flux in a heat transfer or mass diffusion analysis. 1.1.3
+
+DISP: Specify prescribed boundary conditions. 1.1.4
+
+DLOAD: Specify nonuniform distributed loads. 1.1.5
+
+FILM: Define nonuniform film coefficient and associated sink temperatures for heat transfer analysis. 1.1.6
+
+FLOW: Define nonuniform seepage coefficient and associated sink pore pressure for consolidation analysis. 1.1.7
+
+FRIC: Define frictional behavior for contact surfaces. 1.1.8
+
+FRIC\_COEF: Define the frictional coefficient for contact surfaces. 1.1.9
+
+GAPCON: Define conductance between contact surfaces or nodes in a fully coupled temperature-displacement analysis, coupled thermal-electrical-structural analysis, or pure heat transfer analysis. 1.1.10
+
+GAPELECTR: Define electrical conductance between surfaces in a coupled thermal-electrical or a coupled thermal-electrical-structural analysis. 1.1.11
+
+HARDINI: Define initial equivalent plastic strain and initial backstress tensor. 1.1.12
+
+HETVAL: Provide internal heat generation in heat transfer analysis. 1.1.13
+
+MPC: Define multi-point constraints. 1.1.14
+
+ORIENT: Provide an orientation for defining local material directions or local directions for kinematic coupling constraints or local rigid body directions for inertia relief. 1.1.15
+
+RSURFU: Define a rigid surface. 1.1.16
+
+SDVINI: Define initial solution-dependent state variable fields. 1.1.17
+
+SIGINI: Define an initial stress field. 1.1.18
+
+UAMP: Specify amplitudes. 1.1.19
+
+UANISOHYPER\_INV: Define anisotropic hyperelastic material behavior using the invariant formulation. 1.1.20
+
+UANISOHYPER\_STRAIN: Define anisotropic hyperelastic material behavior based on Green strain. 1.1.21
+
+UCORR: Define cross-correlation properties for random response loading. 1.1.22
+
+UCREEPNETWORK: Define time-dependent behavior (creep) for models defined within the parallel rheological framework. 1.1.23
+
+UDECURRENT: Define nonuniform volume current density in an eddy current or magnetostatic analysis. 1.1.24
+
+<!-- source-page: 8 -->
+
+UDEMPOTENTIAL: Define nonuniform magnetic vector potential on a surface in an eddy current or magnetostatic analysis. 1.1.25
+
+UDMGINI: Define the damage initiation criterion. 1.1.26
+
+UDSECURRENT: Define nonuniform surface current density in an eddy current or magnetostatic analysis. 1.1.27
+
+UEL: Define an element. 1.1.28
+
+UELMAT: Define an element with access to materials. 1.1.29
+
+UEXPAN: Define incremental thermal strains. 1.1.30
+
+UEXTERNALDB: Manage user-defined external databases and calculate model-independent history information. 1.1.31
+
+UFIELD: Specify predefined field variables. 1.1.32
+
+UFLUID: Define fluid density and fluid compliance for hydrostatic fluid elements. 1.1.33
+
+UFLUIDCONNECTORLOSS: Define the loss coefficient for fluid flow in fluid pipe connector elements. 1.1.34
+
+UFLUIDCONNECTORVALVE: Define the valve opening to control flow in fluid pipe connector elements. 1.1.35
+
+UFLUIDLEAKOFF: Define the fluid leak-off coefficients for pore pressure cohesive elements. 1.1.36
+
+UFLUIDPIPEFRICTION: Define the frictional coefficient for fluid flow in fluid pipe elements. 1.1.37
+
+UGENS: Define the mechanical behavior of a shell section. 1.1.38
+
+UHARD: Define the yield surface size and hardening parameters for isotropic plasticity or combined hardening models. 1.1.39
+
+UHYPEL: Define a hypoelastic stress-strain relation. 1.1.40
+
+UHYPER: Define a hyperelastic material. 1.1.41
+
+UINTER: Define surface interaction behavior for contact surfaces. 1.1.42
+
+UMASFL: Specify prescribed mass flow rate conditions for a convection/diffusion heat transfer analysis. 1.1.43
+
+UMAT: Define a material’s mechanical behavior. 1.1.44
+
+UMATHT: Define a material’s thermal behavior. 1.1.45
+
+UMESHMOTION: Specify mesh motion constraints during adaptive meshing. 1.1.46
+
+UMOTION: Specify motions during cavity radiation heat transfer analysis or steady-state transport analysis. 1.1.47
+
+UMULLINS: Define damage variable for the Mullins effect material model. 1.1.48
+
+UPOREP: Define initial fluid pore pressure. 1.1.49
+
+UPRESS: Specify prescribed equivalent pressure stress conditions. 1.1.50
+
+UPSD: Define the frequency dependence for random response loading. 1.1.51
+
+URDFIL: Read the results file. 1.1.52
+
+USDFLD: Redefine field variables at a material point. 1.1.53
+
+UTEMP: Specify prescribed temperatures. 1.1.54
+
+UTRACLOAD: Specify nonuniform traction loads. 1.1.55
+
+UTRS: Define a reduced time shift function for a viscoelastic material. 1.1.56
+
+<!-- source-page: 9 -->
+
+UTRSNETWORK: Define a reduced time shift function for models defined within the parallel rheological framework. 1.1.57
+
+UVARM: Generate element output. 1.1.58
+
+UWAVE: Define wave kinematics for an analysis. 1.1.59
+
+UXFEMNONLOCALWEIGHT: Define the weight function used to compute the average stress/strain to determine the crack propagation direction. 1.1.60
+
+VOIDRI: Define initial void ratios. 1.1.61
+
+# Abaqus/Explicit subroutines
+
+VDFLUX: Specify nonuniform distributed fluxes in an explicit dynamic coupled temperature-displacement analysis. 1.2.1
+
+VDISP: Specify prescribed boundary conditions. 1.2.2
+
+VDLOAD: Specify nonuniform distributed loads. 1.2.3
+
+VEXTERNALDB: User subroutine that gives control to the user at key moments of the analysis so that data can be exchanged dynamically among user subroutines and with external programs or files. 1.2.4
+
+VFABRIC: Define fabric material behavior. 1.2.5
+
+VFRIC: Define frictional behavior for contact surfaces. 1.2.6
+
+VFRIC\_COEF: Define the frictional coefficient for contact surfaces. 1.2.7
+
+VFRICTION: Define frictional behavior for contact surfaces. 1.2.8
+
+VUAMP: Specify amplitudes. 1.2.9
+
+VUANISOHYPER\_INV: Define anisotropic hyperelastic material behavior using the invariant formulation. 1.2.10
+
+VUANISOHYPER\_STRAIN: Define anisotropic hyperelastic material behavior based on Green strain. 1.2.11
+
+VUCHARLENGTH: Define characteristic element length at a material point. 1.2.12
+
+VUCREEPNETWORK: Define time-dependent behavior (creep) for models defined within the parallel rheological framework. 1.2.13
+
+VUEL: Define an element. 1.2.14
+
+VUEOS: Define equation of state material model. 1.2.15
+
+VUFIELD: Specify predefined field variables. 1.2.16
+
+VUFLUIDEXCH: Define the mass flow rate/heat energy flow rate for fluid exchange. 1.2.17
+
+VUFLUIDEXCHEFFAREA: Define the effective area for fluid exchange. 1.2.18
+
+VUHARD: Define the yield surface size and hardening parameters for isotropic plasticity or combined hardening models. 1.2.19
+
+VUINTER: Define the interaction between contact surfaces. 1.2.20
+
+VUINTERACTION: Define the contact interaction between surfaces with the general contact algorithm. 1.2.21
+
+VUMAT: Define material behavior. 1.2.22
+
+VUMULLINS: Define damage variable for the Mullins effect material model. 1.2.23
+
+VUSDFLD: Redefine field variables at a material point. 1.2.24
+
+VUTRS: Define a reduced time shift function for a viscoelastic material. 1.2.25
+
+<!-- source-page: 10 -->
+
+VUVISCOSITY: Define the shear viscosity for equation of state models. 1.2.26
+
+VWAVE: Define wave kinematics for an analysis. 1.2.27
+
+# Abaqus/CFD subroutines
+
+SMACfdUserPressureBC: Specify prescribed pressure boundary conditions. 1.3.1
+
+SMACfdUserVelocityBC: Specify prescribed velocity boundary conditions. 1.3.2
+
+# 2. Utility Routines
+
+Obtaining Abaqus environment variables 2.1.1
+
+Obtaining the Abaqus job name 2.1.2
+
+Obtaining the Abaqus output directory name 2.1.3
+
+Obtaining parallel processes information 2.1.4
+
+Obtaining part information 2.1.5
+
+Obtaining material point information in an Abaqus/Standard analysis 2.1.6
+
+Obtaining material point information in an Abaqus/Explicit analysis 2.1.7
+
+Obtaining material point information averaged at a node 2.1.8
+
+Obtaining node point information 2.1.9
+
+Obtaining node to element connectivity 2.1.10
+
+Obtaining stress invariants, principal stress/strain values and directions, and rotating tensors in an Abaqus/Standard analysis 2.1.11
+
+Obtaining principal stress/strain values and directions in an Abaqus/Explicit analysis 2.1.12
+
+Obtaining wave kinematic data in an Abaqus/Aqua analysis 2.1.13
+
+Printing messages to the message or status file 2.1.14
+
+Terminating an analysis 2.1.15
+
+Obtaining sensor information 2.1.16
+
+Accessing Abaqus materials 2.1.17
+
+Accessing Abaqus thermal materials 2.1.18
+
+Obtaining scalar state information in an Abaqus/CFD analysis 2.1.19
+
+Obtaining vector state information in an Abaqus/CFD analysis 2.1.20
+
+Obtaining the MPI communicator in an Abaqus/CFD analysis 2.1.21
+
+Ensuring thread safety 2.1.22
+
+Allocatable arrays 2.1.23
+
+# A. Index
+
+User subroutines index A.1
+
+User subroutine functions listing A.2
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_002.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_002.md
new file mode 100644
index 00000000..d99c6c0a
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_002.md
@@ -0,0 +1,249 @@
+<!-- source-page: 11 -->
+
+This guide describes all of the user subroutines and utility routines available in Abaqus. The interface and requirements for each user subroutine are discussed in detail. References to practical examples of most subroutines are also provided. Utility routines can be used within user subroutines to perform a variety of common tasks. The interface for all available utility routines appears in a separate chapter. For information on incorporating a user subroutine into an Abaqus analysis, see “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide.
+
+Most user subroutine interfaces in this guide use the Fortran language, although user subroutines can be written using the C and C++ languages. Similarly, the utility routines can be invoked from within these C and C++ user subroutines. For more information, refer to “Writing a user subroutine” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide, and “Writing user subroutines in C++” in the Dassault Systèmes Knowledge Base at www.3ds.com/support/knowledge-base.
+
+This guide is divided into four main sections:
+
+• “Abaqus/Standard subroutines,” Section 1.1 covers all of the user subroutines available for use in an Abaqus/Standard analysis. Each section discusses a particular subroutine. The sections are organized alphabetically according to the subroutine name.   
+• “Abaqus/Explicit subroutines,” Section 1.2 covers all of the user subroutines available for use in an Abaqus/Explicit analysis. Each section discusses a particular subroutine. The sections are organized alphabetically according to the subroutine name.   
+• “Abaqus/CFD subroutines,” Section 1.3 covers all of the user subroutines available for use in an Abaqus/CFD analysis. Each section discusses a particular subroutine. The sections are organized alphabetically according to the subroutine name.   
+• “Utility routines,” Section 2.1 covers all of the utility routines available for use in coding user subroutines. Each section discusses a task that can be performed using a utility routine. All of the utility routines associated with a particular task appear in the same section.
+
+<!-- source-page: 12 -->
+
+<!-- source-page: 13 -->
+
+# 1. User Subroutines
+
+• “Abaqus/Standard subroutines,” Section 1.1   
+• “Abaqus/Explicit subroutines,” Section 1.2   
+• “Abaqus/CFD subroutines,” Section 1.3
+
+<!-- source-page: 14 -->
+
+<!-- source-page: 15 -->
+
+# 1.1 Abaqus/Standard subroutines
+
+• “CREEP,” Section 1.1.1   
+• “DFLOW,” Section 1.1.2   
+• “DFLUX,” Section 1.1.3   
+• “DISP,” Section 1.1.4   
+• “DLOAD,” Section 1.1.5   
+• “FILM,” Section 1.1.6   
+• “FLOW,” Section 1.1.7   
+• “FRIC,” Section 1.1.8   
+• “FRIC\_COEF,” Section 1.1.9   
+• “GAPCON,” Section 1.1.10   
+• “GAPELECTR,” Section 1.1.11   
+• “HARDINI,” Section 1.1.12   
+• “HETVAL,” Section 1.1.13   
+• “MPC,” Section 1.1.14   
+• “ORIENT,” Section 1.1.15   
+• “RSURFU,” Section 1.1.16   
+• “SDVINI,” Section 1.1.17   
+• “SIGINI,” Section 1.1.18   
+• “UAMP,” Section 1.1.19   
+• “UANISOHYPER\_INV,” Section 1.1.20   
+• “UANISOHYPER\_STRAIN,” Section 1.1.21   
+• “UCORR,” Section 1.1.22   
+• “UCREEPNETWORK,” Section 1.1.23   
+• “UDECURRENT,” Section 1.1.24   
+• “UDEMPOTENTIAL,” Section 1.1.25   
+• “UDMGINI,” Section 1.1.26   
+• “UDSECURRENT,” Section 1.1.27   
+• “UEL,” Section 1.1.28   
+• “UELMAT,” Section 1.1.29   
+• “UEXPAN,” Section 1.1.30   
+• “UEXTERNALDB,” Section 1.1.31   
+• “UFIELD,” Section 1.1.32
+
+<!-- source-page: 16 -->
+
+• “UFLUID,” Section 1.1.33   
+• “UFLUIDCONNECTORLOSS,” Section 1.1.34   
+• “UFLUIDCONNECTORVALVE,” Section 1.1.35   
+• “UFLUIDLEAKOFF,” Section 1.1.36   
+• “UGENS,” Section 1.1.38   
+• “UHARD,” Section 1.1.39   
+• “UHYPEL,” Section 1.1.40   
+• “UHYPER,” Section 1.1.41   
+• “UINTER,” Section 1.1.42   
+• “UMASFL,” Section 1.1.43   
+• “UMAT,” Section 1.1.44   
+• “UMATHT,” Section 1.1.45   
+• “UMESHMOTION,” Section 1.1.46   
+• “UMOTION,” Section 1.1.47   
+• “UMULLINS,” Section 1.1.48   
+• “UPOREP,” Section 1.1.49   
+• “UPRESS,” Section 1.1.50   
+• “UPSD,” Section 1.1.51   
+• “URDFIL,” Section 1.1.52   
+• “USDFLD,” Section 1.1.53   
+• “UTEMP,” Section 1.1.54   
+• “UTRACLOAD,” Section 1.1.55   
+• “UTRS,” Section 1.1.56   
+• “UTRSNETWORK,” Section 1.1.57   
+• “UVARM,” Section 1.1.58   
+• “UWAVE,” Section 1.1.59   
+• “UXFEMNONLOCALWEIGHT,” Section 1.1.60   
+• “VOIDRI,” Section 1.1.61
+
+<!-- source-page: 17 -->
+
+# 1.1.1 CREEP: User subroutine to define time-dependent, viscoplastic behavior (creep and swelling).
+
+Product: Abaqus/Standard
+
+# References
+
+• “Rate-dependent plasticity: creep and swelling,” Section 23.2.4 of the Abaqus Analysis User’s Guide   
+• “Extended Drucker-Prager models,” Section 23.3.1 of the Abaqus Analysis User’s Guide   
+• “Modified Drucker-Prager/Cap model,” Section 23.3.2 of the Abaqus Analysis User’s Guide   
+• “Defining the gasket behavior directly using a gasket behavior model,” Section 32.6.6 of the Abaqus Analysis User’s Guide   
+• \*CAP CREEP   
+• \*CREEP   
+• \*DRUCKER PRAGER CREEP   
+• \*SWELLING   
+• “Verification of creep integration,” Section 3.2.6 of the Abaqus Benchmarks Guide
+
+# Overview
+
+User subroutine CREEP will be called at all integration points of elements for which the material definition contains user-subroutine-defined metal creep, time-dependent volumetric swelling, Drucker-Prager creep, or cap creep behavior, during procedures that allow viscoplastic response of the above type to occur (such as the quasi-static procedure). This subroutine will also be called at all integration points of gasket elements for which the behavior definition contains user-subroutine-defined creep.
+
+If user subroutine CREEP is used to define a material behavior, the subroutine:
+
+• is intended to provide the “uniaxial” creep laws that are to be included in a general time-dependent, viscoplastic material formulation;   
+• can be used in the coupled-temperature displacement (“Fully coupled thermal-stress analysis,” Section 6.5.3 of the Abaqus Analysis User’s Guide), coupled thermal-electrical-structural (“Fully coupled thermal-electrical-structural analysis,” Section 6.7.4 of the Abaqus Analysis User’s Guide), soils (“Coupled pore fluid diffusion and stress analysis,” Section 6.8.1 of the Abaqus Analysis User’s Guide), and quasi-static (“Quasi-static analysis,” Section 6.2.5 of the Abaqus Analysis User’s Guide) procedures;   
+• allows for the definition of creep laws for which the meaning and internal use depend on the material model with which they are being used;
+
+<!-- source-page: 18 -->
+
+• allows creep and swelling to be combined with rate-independent plastic behavior in a coupled manner, or they may simply be the only inelastic behaviors of the material, in which case Mises behavior is assumed;   
+• can use and update solution-dependent state variables; and   
+• can be used in conjunction with user subroutine USDFLD to redefine any field variables before they are passed in.
+
+If user subroutine CREEP is used to define rate-dependent behavior in the thickness direction for a gasket, the subroutine:
+
+• is intended to provide the creep laws that are used to prescribe the thickness-direction behavior for a gasket;   
+• can be used only in a quasi-static (“Quasi-static analysis,” Section 6.2.5 of the Abaqus Analysis User’s Guide) procedure;   
+• is used in a coupled form with the elastic-plastic model used to define the rate-independent part of the thickness-direction behavior of the gasket; and   
+• can use and update solution-dependent variables.
+
+# Metals
+
+For metals whose material behavior includes metal creep and/or time-dependent volumetric swelling, the routine allows any “creep” and “swelling” laws (viscoplastic behavior) of the following general form to be defined:
+
+$$
+\bar {\varepsilon} ^ {\dot {c} r} = g ^ {c r} (p, \tilde {q}, \bar {\varepsilon} ^ {s w}, \bar {\varepsilon} ^ {c r}, \mathrm{time}, \dots),
+$$
+
+$$
+\bar {\varepsilon} ^ {\dot {s} w} = g ^ {s w} (p, \tilde {q}, \bar {\varepsilon} ^ {s w}, \bar {\varepsilon} ^ {c r}, \mathrm{time}, \dots),
+$$
+
+where
+
+$\bar { \varepsilon } ^ { c r }$ is the uniaxial equivalent “creep” strain, conjugate to $\tilde { q } ,$ the Mises or Hill equivalent stress; $\bar { \varepsilon } ^ { s w }$ is the volumetric swelling strain;
+
+$\pmb { p }$ is the equivalent pressure stress, $\begin{array} { r } { p = - \frac { 1 } { 3 } ( \sigma _ { 1 1 } + \sigma _ { 2 2 } + \sigma _ { 3 3 } ) } \end{array}$ ; and
+
+$\tilde { q }$ is the equivalent deviatoric stress (Mises’ or, if anisotropic creep behavior is defined, Hill’s definition).
+
+The user subroutine must define the increments of inelastic strain, $\bigtriangleup \bar { \varepsilon } ^ { c r }$ and $\triangle \bar { \varepsilon } ^ { s w }$ , as functions of $\pmb { p }$ and $\tilde { q }$ and any other variables used in the definitions of $\boldsymbol { g } ^ { c r }$ and $g ^ { s w }$ (such as solution-dependent state variables introduced by you) and of the time increment, $\triangle t .$ . If any solution-dependent state variables are included in the definitions of $\boldsymbol { g } ^ { c r }$ and $g ^ { s w }$ , they must also be integrated forward in time in this routine.
+
+Abaqus computes the incremental creep strain (or the incremental viscoplastic strain) components as
+
+<!-- source-page: 19 -->
+
+$$
+\Delta \varepsilon^ {c r} = \frac {1}{3} \Delta \bar {\varepsilon} ^ {s w} \mathbf {R} + \Delta \bar {\varepsilon} ^ {c r} \mathbf {n},
+$$
+
+where is the gradient of the deviatoric stress potential, defined as
+
+$$
+\mathbf {n} = \frac {\partial \tilde {q}}{\partial \pmb {\sigma}},
+$$
+
+and is a matrix with the anisotropic swelling ratios in the diagonal if anisotropic swelling is defined; otherwise, $\mathbf { R } = \mathbf { I }$ .
+
+# Drucker-Prager materials
+
+For materials that yield according to the extended Drucker-Prager plasticity models using Drucker-Prager creep, the routine allows any “creep” laws (viscoplastic behavior) of the following general form to be defined:
+
+$$
+\bar {\varepsilon} ^ {\dot {c} r} = g ^ {c r} (\bar {\sigma} ^ {c r}, \bar {\varepsilon} ^ {c r}, \mathrm{time}, \dots),
+$$
+
+where
+
+gcr $\bar { \boldsymbol { \sigma } } ^ { c r }$ is the equivalent creep stress defined as
+
+$$
+\begin{array}{l} \frac {q - p \tan \beta}{1 - \frac {1}{3} \tan \beta} \quad \text {   if   creep   is   defined   in   terms   of   uniaxial   compression,   } \\ \frac {q - p \tan \beta}{1 + \frac {1}{3} \tan \beta} \quad \text {   if   creep   is   defined   in   terms   of   uniaxial   tension,   and   } \\ q - p \tan \beta \quad \text {   if   creep   is   defined   in   terms   of   pure   shear,   } \\ \end{array}
+$$
+
+where q is the equivalent deviatoric Mises’ stress, p is the pressure stress, and $\beta$ is the friction angle, and
+
+mCr $\bar { \varepsilon } ^ { c r }$ is the uniaxial equivalent “creep” strain, conjugate to $\bar { \sigma } ^ { c r }$ such that $\bar { \sigma } ^ { c r } \Delta \bar { \varepsilon } ^ { c r } = \sigma _ { i j } \Delta \varepsilon _ { i j } ^ { c r }$
+
+The user subroutine must define the increment of inelastic strain, $\bigtriangleup \bar { \varepsilon } ^ { c r }$ , as a function of ${ \bar { \sigma } } ^ { c r }$ and any other variables used in the definitions of $\boldsymbol { g } ^ { c r }$ (such as solution-dependent state variables introduced by you) and of the time increment, $\triangle t$ . If any solution-dependent state variables are included in the definitions of $\boldsymbol { g } ^ { c r }$ , they must also be integrated forward in time in this routine.
+
+Abaqus computes the incremental creep strain (or the incremental viscoplastic strain) components as
+
+$$
+\Delta \varepsilon^ {c r} = \frac {\Delta \bar {\varepsilon} ^ {c r}}{f ^ {c r}} \left(\frac {q}{\sqrt {(\epsilon \bar {\sigma} | _ {0} \tan \psi) ^ {2} + q ^ {2}}} \mathbf {n} + \frac {1}{3} \tan \psi \mathbf {I}\right),
+$$
+
+where $\mathbf { n } = \partial \tilde { q } / \partial \pmb { \sigma }$ The variable $f ^ { c r }$ is determined in such a way that
+
+$$
+f ^ {c r} = \frac {1}{\bar {\sigma} ^ {c r}} \pmb {\sigma}: \frac {\partial G ^ {c r}}{\partial \pmb {\sigma}},
+$$
+
+<!-- source-page: 20 -->
+
+and
+
+$$
+G ^ {c r} = \sqrt {(\epsilon \bar {\sigma} | _ {0} \tan \psi) ^ {2} + q ^ {2}} - p \tan \psi
+$$
+
+is the hyperbolic creep potential, where $\psi ( \theta , f ^ { \alpha } )$ is the dilation angle measured in the p–q plane at high confining pressure, $\bar { \sigma } | _ { 0 } = \bar { \sigma } | _ { \bar { \varepsilon } ^ { p l } = 0 , \dot { \bar { \varepsilon } } ^ { p l } = 0 }$ is the initial yield stress, and is the eccentricity. See “Extended Drucker-Prager models,” Section 23.3.1 of the Abaqus Analysis User’s Guide, for a discussion of $\psi _ { : }$ , , and $\bar { \sigma } | _ { 0 }$ .
+
+# Capped Drucker-Prager materials
+
+For materials that yield according to the modified Drucker-Prager/Cap plasticity model using cap creep, the routine allows any “cohesion creep” and “consolidation creep” laws (viscoplastic behavior) of the following general form to be defined:
+
+$$
+\bar {\varepsilon} _ {s} ^ {\dot {c} r} = g _ {s} ^ {c r} (\bar {\sigma} ^ {c r}, \bar {\varepsilon} _ {s} ^ {c r}, \mathrm{time}, \dots),
+$$
+
+$$
+\bar {\varepsilon} _ {c} ^ {\dot {c} r} = g _ {c} ^ {c r} (\bar {p} ^ {c r}, \bar {\varepsilon} _ {c} ^ {c r}, \mathrm{time}, \dots),
+$$
+
+where
+
+gcr $\bar { \boldsymbol { \sigma } } ^ { c r }$ is the equivalent creep stress defined from uniaxial compression test data as
+
+$$
+\bar {\sigma} ^ {c r} = \frac {q - p \tan \beta}{(1 - \frac {1}{3} \tan \beta)},
+$$
+
+where q is the equivalent deviatoric Mises’ stress, p is the pressure stress, and $\beta$ is the friction angle;
+
+$\bar { \varepsilon } _ { s } ^ { c r }$ is the equivalent cohesion creep uniaxial strain, conjugate to $\bar { \sigma } ^ { c r }$ such that $\bar { \sigma } ^ { c r } \Delta \bar { \varepsilon } _ { s } ^ { c r } = \pmb { \sigma } : \Delta \varepsilon _ { s } ^ { c r }$ , where $\Delta \varepsilon _ { s } ^ { c r }$ is defined below;
+
+$\bar { p } ^ { c r } = p - p _ { a }$ is the effective creep pressure $\begin{array} { r } { ( p = - \frac { 1 } { 3 } ( \sigma _ { 1 1 } + \sigma _ { 2 2 } + \sigma _ { 3 3 } ) } \end{array}$ and $p _ { a }$ is the cap hardening parameter); and
+
+$\bar { \varepsilon } _ { c } ^ { c r }$ is the volumetric consolidation creep strain.
+
+The user subroutine must define the increments of inelastic strain, $\bigtriangleup \bar { \varepsilon } _ { s } ^ { c r }$ and/or $\triangle \bar { \varepsilon } _ { c } ^ { c r }$ , as functions of $\bar { \sigma } ^ { c r }$ and/or $\bar { p } ^ { c r }$ and any other variables used in the definitions of $g _ { s } ^ { c r }$ and $g _ { c } ^ { c r }$ (such as solution-dependent state variables introduced by you) and of the time increment, $\triangle t .$ . If any solution-dependent state variables are included in the definitions of $g _ { s } ^ { c r }$ and $g _ { c } ^ { c r }$ , they must also be integrated forward in time in this routine.
+
+# Calculation of incremental creep strains for the cohesion mechanism
+
+Abaqus computes the incremental creep strain (or the incremental viscoplastic strain) components of the cohesion mechanism as
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+<!-- source-page: 21 -->
+
+$$
+\Delta \varepsilon_ {s} ^ {c r} = \frac {\Delta \bar {\varepsilon} _ {s} ^ {c r}}{f ^ {c r}} \left(\frac {q}{\sqrt {(0 . 1 \frac {d}{(1 - \frac {1}{3} \tan \beta)} \tan \beta) ^ {2} + q ^ {2}}} \mathbf {n} + \frac {1}{3} \tan \beta \mathbf {I}\right),
+$$
+
+where $\mathbf { n } = \partial \tilde { q } / \partial \sigma$ d is the material cohesion, the variable $f ^ { c r }$ is determined in such a way that
+
+$$
+f ^ {c r} = \frac {1}{\bar {\sigma} ^ {c r}} \pmb {\sigma}: \frac {\partial G _ {s} ^ {c r}}{\partial \pmb {\sigma}},
+$$
+
+and $G _ { s } ^ { c r }$ is the cohesion creep potential
+
+$$
+G _ {s} ^ {c r} = \sqrt {(0 . 1 \frac {d}{(1 - \frac {1}{3} \tan \beta)} \tan \beta) ^ {2} + q ^ {2}} - p \tan \beta .
+$$
+
+# Calculation of incremental creep strains for the consolidation mechanism
+
+Abaqus computes the incremental creep strain (or the incremental viscoplastic strain) components of the consolidation mechanism as
+
+$$
+\Delta \varepsilon_ {c} ^ {c r} = \frac {\triangle \overline {{\varepsilon}} _ {c} ^ {c r}}{G _ {c} ^ {c r}} (R ^ {2} q \mathbf {n} - \frac {1}{3} (p - p _ {a}) \mathbf {I}),
+$$
+
+where R controls the shape of the cap, and $G _ { c } ^ { c r }$ is the consolidation creep potential
+
+$$
+G _ {c} ^ {c r} = \sqrt {(p - p _ {a}) ^ {2} + (R q) ^ {2}}.
+$$
+
+Cohesion material properties are determined with a uniaxial compression test in which $\begin{array} { r l } { d \mathcal { \bar { E } } _ { s } ^ { c r } = } \end{array}$ $\| d \epsilon _ { 1 1 } ^ { c r } \|$ , and consolidation material properties are determined with a volumetric compression test in which $d \bar { \varepsilon } _ { c } ^ { c r } = \| d \epsilon _ { v o l } ^ { c r } \|$ . Most likely, $g _ { s } ^ { c r }$ is a positive function of $\bar { \sigma } ^ { c r }$ , and $g _ { \mathrm { c } } ^ { c r }$ is a positive function of $\overline { { p } } ^ { c r }$ .
+
+# Gaskets
+
+For gaskets whose behavior includes creep, the routine allows any “creep” law of the following general form to be defined:
+
+$$
+\dot {\varepsilon} ^ {c r} = g ^ {c r} (\sigma , \varepsilon^ {c r}, \mathrm{time}, \ldots),
+$$
+
+where $\varepsilon ^ { c r }$ is the compressive creep strain, conjugate to $\sigma ,$ the compressive stress in the gasket.
+
+The user subroutine must define the increments of inelastic creep strain, $\triangle \varepsilon ^ { c r }$ , as functions of and any other variables used in the definitions of $\boldsymbol { g } ^ { c r }$ (such as solution-dependent state variables introduced by you) and of the time increment, $\triangle t .$ . If any solution-dependent state variables are included in the definitions of ${ \bf { \dot { g } } } ^ { c r }$ , they must also be integrated forward in time in this routine. Abaqus will automatically multiply this creep strain by the proper thickness (see “Defining the gasket behavior directly using a gasket behavior model,” Section 32.6.6 of the Abaqus Analysis User’s Guide) to obtain a creep closure.
+
+<!-- source-page: 22 -->
+
+Abaqus provides both explicit and implicit time integration of creep and swelling behavior defined in this routine. The choice of the time integration scheme depends on the procedure type, the procedure definition, and whether a geometric linear or nonlinear analysis is requested (see “Rate-dependent plasticity: creep and swelling,” Section 23.2.4 of the Abaqus Analysis User’s Guide).
+
+Implicit integration is generally more effective when the response period is long relative to typical relaxation times for the material. Simple high-temperature structural design applications usually do not need implicit integration, but more complicated problems (such as might arise in manufacturing processes), creep buckling applications, or nonstructural problems (such as geotechnical applications) often are integrated more efficiently by the implicit method provided in the program. If implicit integration is used with this subroutine, nonlinear equations must be solved at each time step and the variations of $\triangle \bar { \varepsilon } ^ { c r } , \triangle \bar { \varepsilon } ^ { s w } , \triangle \bar { \varepsilon } _ { s } ^ { c r }$ , or $\triangle \bar { \varepsilon } _ { c } ^ { c r }$ with respect to $\bar { \varepsilon } ^ { c r } , \bar { \varepsilon } ^ { s w } , \bar { \varepsilon } _ { s } ^ { c r } , \bar { \varepsilon } _ { c } ^ { c r } , { \pmb { p } } , \tilde { q } , \bar { p } ^ { c r }$ , or $\bar { \sigma } ^ { c r }$ must be defined in the subroutine. To obtain good convergence during implicit integration, it is essential to define these quantities accurately.
+
+At the start of a new increment the subroutine is called once for each integration point to calculate the estimated creep strain based on the state at the start of the increment. Subsequently, it is called twice for each iteration if explicit integration is used: once to calculate the creep strain increment at the start of the increment and once to calculate it at the end of the increment. This is needed to test the validity of the time increment with respect to the user-specified maximum allowable difference in the creep strain increment. The flag LEND indicates whether the routine is called at the start or the end of the increment. The subroutine must use the corresponding values of time, temperature, field variables, and solution-dependent state variables in the calculation of the creep strain increment.
+
+For implicit integration Abaqus uses a local iteration procedure to solve the nonlinear constitutive equations, and the subroutine is called multiple times. The exact number of calls depends on the convergence rate of the local iteration procedure and, hence, will vary from point to point. During these iterations it is possible for the values of the state variables to be far from their final values when the equations are solved. Therefore, the coding in the subroutine must adequately protect against arithmetic failures (such as floating point overflows) even when variables are passed in with physically unreasonable values. As in explicit integration, the variable LEND indicates whether the routine is called at the start or the end of the increment.
+
+# Constant stress assumption when defining creep and swelling
+
+When the creep and swelling behavior are defined by simple formulæ, it is often possible to calculate the increments of equivalent creep and swelling strain exactly if it is assumed that the stress is constant during the increment. This approach has the advantage that it provides very good accuracy within the constant stress assumption. It also avoids the problem that arises for some creep behavior definitions: that the creep strain rate becomes infinite at zero time (or strain). Otherwise, in such a case you must protect against causing arithmetic failures at the start of the solution.
+
+<!-- source-page: 23 -->
+
+# Defining both plasticity and creep
+
+If both plasticity and creep are defined for a material, Abaqus will calculate the creep strain before entering the plasticity routines. The stresses passed into the creep routine may, therefore, exceed the yield stress.
+
+# Interpretation of stress and strain variables
+
+In finite-strain applications strain variables should be interpreted as logarithmic strains and stresses as “true” stress.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE CREEP (DECRA, DESWA, STATEV, SERD, EC, ESW, P, QTILD,
+1 TEMP, DTEMP, PREDEF, DPRED, TIME, DTIME, CMNAME, LEXIMP, LEND,
+2 COORDS, NSTATV, NOEL, NPT, LAYER, KSPT, KSTEP, KINC)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME
+C
+DIMENSION DECRA(5), DESWA(5), STATEV(*), PREDEF(*), DPRED(*),
+1 TIME(3), EC(2), ESW(2), COORDS(*)
+user coding to define DECRA, DESWA
+RETURN
+END 
+```
+
+# Variables to be defined
+
+# In all cases
+
+# DECRA(1)
+
+The definition depends on the usage:
+
+• Metal creep: $\bigtriangleup \bar { \varepsilon } ^ { c r }$ , equivalent (uniaxial) deviatoric creep strain increment.   
+• Drucker-Prager creep: $\bigtriangleup \bar { \varepsilon } ^ { c r }$ , equivalent (uniaxial) creep strain increment.   
+• Capped Drucker-Prager creep: $\triangle \bar { \varepsilon } _ { s } ^ { c r }$ , equivalent (uniaxial) cohesion creep strain increment.   
+• Gasket creep: $\triangle \varepsilon ^ { c r }$ , uniaxial compressive creep strain increment.
+
+# DESWA(1)
+
+The definition depends on the usage:
+
+• Metal creep: $\bigtriangleup \bar { \varepsilon } ^ { s w }$ , volumetric swelling strain increment.
+
+<!-- source-page: 24 -->
+
+• Capped Drucker-Prager creep: $\triangle \bar { \varepsilon } _ { c } ^ { c r }$ , equivalent (volumetric) consolidation creep strain increment.   
+• Drucker-Prager and gasket creep: = 0.
+
+# For implicit creep integration (LEXIMP=1, see below)
+
+# DECRA(2)
+
+The definition depends on the usage:
+
+• Metal creep and Drucker-Prager creep: $\partial \triangle \bar { \varepsilon } ^ { c r } / \partial \bar { \varepsilon } ^ { c r }$ .   
+• Capped Drucker-Prager creep: $\partial \triangle \bar { \varepsilon } _ { s } ^ { c r } / \partial \bar { \varepsilon } _ { s } ^ { c r }$   
+• Gasket creep: $\partial \triangle \varepsilon ^ { c r } / \partial \varepsilon ^ { c r }$ .
+
+# DECRA(3)
+
+The definition depends on the usage:
+
+• Metal creep: $\partial \triangle \bar { \varepsilon } ^ { c r } / \partial \bar { \varepsilon } ^ { s w }$ .   
+• Drucker-Prager creep, gasket creep, and capped Drucker-Prager creep: = 0.
+
+# DECRA(4)
+
+The definition depends on the usage:
+
+• Metal creep: $\partial \triangle \bar { \varepsilon } ^ { c r } / \partial p$   
+• Drucker-Prager creep, gasket creep, and capped Drucker-Prager creep: = 0.
+
+# DECRA(5)
+
+The definition depends on the usage:
+
+• Metal creep: ${ \partial \triangle E ^ { c r } } / { \partial \tilde { q } } .$   
+• Drucker-Prager creep: $\partial \Delta \bar { \varepsilon } ^ { c r } / \partial \bar { \sigma } ^ { c r }$   
+• Capped Drucker-Prager creep: $\partial \Delta \bar { \varepsilon } _ { s } ^ { c r } / \partial \bar { \sigma } ^ { c r }$ .   
+• Gasket creep: $\partial \triangle \varepsilon ^ { c r } / \partial \sigma$
+
+# DESWA(2)
+
+The definition depends on the usage:
+
+• Metal creep: $\partial \triangle \bar { \varepsilon } ^ { s w } / \partial \bar { \varepsilon } ^ { c r }$ .   
+• Drucker-Prager creep, gasket creep, and capped Drucker-Prager creep: = 0.
+
+# DESWA(3)
+
+The definition depends on the usage:
+
+• Metal creep: $\partial \triangle \bar { \varepsilon } ^ { s w } / \partial \bar { \varepsilon } ^ { s w }$ .   
+• Capped Drucker-Prager creep: ${ \partial \bigtriangleup \bar { \varepsilon } _ { c } ^ { c r } } / { \partial \bar { \varepsilon } _ { c } ^ { c r } }$ .   
+• Drucker-Prager and gasket creep: = 0.
+
+# DESWA(4)
+
+The definition depends on the usage:
+
+<!-- source-page: 25 -->
+
+• Metal creep: $\partial \triangle \bar { \varepsilon } ^ { s w } / \partial p$ .   
+• Capped Drucker-Prager creep: ${ \partial \bigtriangleup \bar { \varepsilon } _ { c } ^ { c r } } / { \partial \bar { p } ^ { c r } }$ .   
+• Drucker-Prager and gasket creep: = 0.
+
+# DESWA(5)
+
+The definition depends on the usage:
+
+• Metal creep: $\partial \triangle \bar { \varepsilon } ^ { s w } / \partial \tilde { q } .$ .   
+• Drucker-Prager creep, gasket creep, and capped Drucker-Prager creep: = 0.
+
+# Variables that can be updated
+
+# STATEV
+
+An array containing the user-defined solution-dependent state variables at this point. This array will be passed in containing the values of these variables at the start of the increment unless they are updated in user subroutine USDFLD or UEXPAN, in which case the updated values are passed in. If any of the solution-dependent variables are being used in conjunction with the creep behavior and the routine was called at the end of the increment (LEND=1, see the definition of LEND below), they must be updated in this subroutine to their values at the end of the increment. Furthermore, if the solution-dependent state variables are defined as a function of the creep (swelling) strain increment, they must be updated based on the creep (swelling) strain increment computed as EC(2)-EC(1) (likewise ESW(2)-ESW(1)), where EC(1), EC(2), ESW(1), and ESW(2) are defined below. You define the size of this array by allocating space for it (see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide, for more information).
+
+# SERD
+
+Magnitude of the strain energy rate density, $\dot { W }$ (required only in -integral calculations). The strain energy rate density is defined as
+
+$$
+\dot {W} = \int_ {0} ^ {\dot {\bar {\varepsilon}} ^ {c r}} q d \dot {\bar {\varepsilon}} ^ {c r} + \int_ {0} ^ {\dot {\bar {\varepsilon}} ^ {s w}} p d \dot {\bar {\varepsilon}} ^ {s w} = \int_ {0} ^ {\dot {\bar {\varepsilon}}} \boldsymbol {\sigma}: (d \dot {\bar {\varepsilon}} ^ {c r} + d \dot {\bar {\varepsilon}} ^ {s w}).
+$$
+
+Elastic rates are ignored in the calculation of $\dot { W } .$ . The contour integral will, therefore, be path independent only for steady-state creep conditions; that is, when the creep straining dominates throughout the specimen.
+
+# Variables passed in for information
+
+# EC(1)
+
+The definition depends on the usage:
+
+• Metal creep and Drucker-Prager creep: $\bar { \varepsilon } ^ { c r }$ at the start of the increment.   
+• Capped Drucker-Prager creep: $\bar { \varepsilon } _ { s } ^ { c r }$ at the start of the increment.   
+• Gasket creep: $\varepsilon ^ { c r }$ at the start of the increment.
+
+<!-- source-page: 26 -->
+
+# EC(2)
+
+The definition depends on the usage:
+
+• Metal creep and Drucker-Prager creep: $\bar { \varepsilon } ^ { c r }$ at the end of the increment.   
+• Capped Drucker-Prager creep: $\bar { \varepsilon } _ { s } ^ { c r }$ at the end of the increment.   
+• Gasket creep: $\varepsilon ^ { c r }$ at the end of the increment.
+
+# ESW(1)
+
+The definition depends on the usage:
+
+• Metal creep: $\bar { \varepsilon } ^ { s w }$ at the start of the increment.   
+• Capped Drucker-Prager creep: $\bar { \varepsilon } _ { c } ^ { c r }$ at the start of the increment.   
+• Drucker-Prager and gasket creep: = 0.
+
+# ESW(2)
+
+The definition depends on the usage:
+
+• Metal creep: $\bar { \varepsilon } ^ { s w }$ at the end of the increment.   
+• Capped Drucker-Prager creep: $\bar { \varepsilon } _ { c } ^ { c r }$ at the end of the increment.   
+• Drucker-Prager and gasket creep: = 0.
+
+# P
+
+The definition depends on the usage:
+
+• Metal creep and Drucker-Prager creep: $\begin{array} { r } { p = - \frac { 1 } { 3 } ( \sigma _ { 1 1 } + \sigma _ { 2 2 } + \sigma _ { 3 3 } ) } \end{array}$ , equivalent pressure stress (in soils analysis this is the equivalent effective pressure stress).   
+• Capped Drucker-Prager creep: $\bar { p } ^ { c r } = p - p _ { a }$ , effective creep pressure (in soils analysis p is the effective pressure stress).   
+• Gasket creep: = 0.
+
+If LEND=0, the value is p or $\bar { p } ^ { c r }$ at the beginning of the increment. If LEND=1, the value is p or $\bar { p } ^ { c r }$ at the end of the increment.
+
+# QTILD
+
+The definition depends on the usage:
+
+• Metal creep: ${ \tilde { q } } ,$ Mises or Hill equivalent stress (the Hill formula is used if anisotropic creep is defined; see “Anisotropic creep” in “Rate-dependent plasticity: creep and swelling,” Section 23.2.4 of the Abaqus Analysis User’s Guide).   
+• Gasket creep: $\sigma ,$ the uniaxial compressive stress.   
+• Drucker-Prager creep: $\bar { \sigma } ^ { c r }$ , equivalent creep stress (in soils analysis this is based on effective stresses).   
+• Capped Drucker-Prager creep: $\bar { \sigma } ^ { c r }$ , equivalent creep stress (in soils analysis this is based on effective stresses).
+
+<!-- source-page: 27 -->
+
+If LEND=0, the value is $\tilde { q }$ or $\bar { \sigma } ^ { c r }$ at the beginning of the increment. If LEND=1, the value is $\tilde { q }$ or $\bar { \sigma } ^ { c r }$ at the end of the increment.
+
+# TEMP
+
+Temperature at the end of the increment.
+
+# DTEMP
+
+Increment of temperature during the time increment.
+
+# PREDEF
+
+An array containing the values of all of the user-specified predefined variables at this point at the end of the increment (initial values at the beginning of the analysis and current values during the analysis).
+
+# DPRED
+
+An array containing the increments of all of the predefined variables during the time increment.
+
+# TIME(1)
+
+Value of step time at the end of the increment.
+
+# TIME(2)
+
+Value of total time at the end of the increment.
+
+# TIME(3)
+
+Value of creep time at the end of the increment.
+
+# DTIME
+
+Time increment.
+
+# CMNAME
+
+User-specified material name or gasket behavior name, left justified. Some internal creep models are given names starting with the “ABQ\_” character string. To avoid conflict, you should not use “ABQ\_” as the leading string for CMNAME.
+
+# LEXIMP
+
+Explicit/implicit flag.
+
+If LEXIMP=0, explicit creep integration is being used and only DECRA(1) and DESWA(1) need be defined; DECRA(I) and DESWA(I), I=2,5, need not be defined.
+
+If LEXIMP=1, implicit creep integration is being used. The derivatives, DECRA(I) and DESWA(I), I=2,5, should be defined accurately to achieve rapid convergence of the solution.
+
+# LEND
+
+Start/end of increment flag.
+
+If LEND=0, the routine is being called at the start of the increment. In this case DECRA(1) and DESWA(1) must be defined as the equivalent creep and swelling rates calculated at the beginning of the increment, multiplied by the time increment.
+
+<!-- source-page: 28 -->
+
+If LEND=1, the routine is being called at the end of the increment. In this case DECRA(1) and DESWA(1) must be defined as the equivalent creep and swelling rates calculated at the end of the increment, multiplied by the time increment. If applicable, the solution-dependent state variables STATEV must be updated as well.
+
+# COORDS(3)
+
+An array containing the current coordinates of this point.
+
+# NSTATV
+
+Number of solution-dependent state variables associated with this material or gasket behavior type (specified when space is allocated for the array; see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# NOEL
+
+Element number.
+
+# NPT
+
+Integration point number.
+
+# LAYER
+
+Layer number (for composite shells and layered solids).
+
+# KSPT
+
+Section point number within the current layer.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# Example: Hyperbolic sine creep law
+
+Suppose that we wish to model a metal using the creep behavior
+
+$$
+\bar {\varepsilon} ^ {\dot {c} r} = A (\sinh \frac {q}{\sigma_ {0}}) ^ {n}, \quad \bar {\varepsilon} ^ {\dot {s} w} = 0,
+$$
+
+where A, , and n are constants.
+
+User subroutine CREEP can be coded as follows:
+```csv
+SUBROUTINE CREEP (DECRA, DESWA, STATEV, SERD, EC, ESW, P, QTILD,
+1 TEMP, DTEMP, PREDEF, DPRED, TIME, DTIME, CMNAME, LEXIMP, LEND,
+2 COORDS, NSTATV, NOEL, NPT, LAYER, KSPT, KSTEP, KINC)
+C
+INCLUDE 'ABA_PARAM.INC'
+C 
+```
+
+<!-- source-page: 29 -->
+
+```txt
+CHARACTER*80 CMNAME
+C
+DIMENSION DECRA(5), DESWA(5), STATEV(*), PREDEF(*), DPRED(*), 1 TIME(3), COORDS(*), EC(2), ESW(2)
+C
+C DEFINE CONSTANTS
+C
+A=
+SIG0=
+AN=
+C
+T1=EXP(QTILD/SIG0)
+T2=EXP(-QTILD/SIG0)
+DECRA(1) = A*(.5*(T1-T2))**AN*DTIME
+IF(LEXIMP.EQ.1) THEN
+DECRA(5) = AN*A*(.5*(T1-T2))**(AN-1.)*DTIME/
+1 SIG0*.5*(T1+T2)
+END IF
+C
+RETURN
+END 
+```
+
+The derivative
+
+$$
+\frac {\partial \triangle \overline {{\varepsilon}} ^ {c r}}{\partial q} = \frac {n A \Delta t}{\sigma_ {0}} (\sinh \frac {q}{\sigma_ {0}}) ^ {n - 1} \cosh \frac {q}{\sigma_ {0}}
+$$
+
+has been defined on the assumption that the subroutine will be used with implicit integration.
+
+<!-- source-page: 30 -->
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_004.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_004.md
new file mode 100644
index 00000000..8b679648
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_004.md
@@ -0,0 +1,288 @@
+<!-- source-page: 31 -->
+
+# 1.1.2 DFLOW: User subroutine to define nonuniform pore fluid velocity in a consolidation analysis.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Pore fluid flow,” Section 34.4.7 of the Abaqus Analysis User’s Guide   
+• \*DFLOW   
+• \*DSFLOW
+
+# Overview
+
+User subroutine DFLOW:
+
+• can be used to define the variation of the seepage magnitude as a function of position, time, pore pressure, etc. in a soils consolidation analysis;   
+• will be called at each flow integration point for each element-based or surface-based nonuniform flow definition in the analysis; and   
+• ignores any amplitude references that may appear with the associated nonuniform flow definition.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE DFLOW(FLOW,U,KSTEP,KINC,TIME,NOEL,NPT,COORDS,1 JLTYP,SNAME)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION TIME(2),COORDS(3)
+CHARACTER*80 SNAME
+user coding to define FLOW
+RETURN
+END 
+```
+
+<!-- source-page: 32 -->
+
+# Variable to be defined
+
+# FLOW
+
+Effective velocity of pore fluid crossing the surface at this point from the inside of the region modeled to the outside of the region modeled. Units are LT−1 . Effective velocity is the volumetric flow rate per unit area (refer to “Permeability,” Section 26.6.2 of the Abaqus Analysis User’s Guide).
+
+FLOW will be passed into the routine as the magnitude of the seepage specified as part of the element-based or surface-based flow definition. If the magnitude is not defined, FLOW will be passed in as zero.
+
+The effective velocity is not available for output purposes.
+
+# Variables passed in for information
+
+# U
+
+Estimated pore pressure at this time at this point.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# TIME(1)
+
+Current value of step time (defined only in transient analysis).
+
+# TIME(2)
+
+Current value of total time (defined only in transient analysis).
+
+# NOEL
+
+Element number.
+
+# NPT
+
+Integration point number on the element’s surface.
+
+# COORDS
+
+An array containing the coordinates of this point. These are the current coordinates if geometric nonlinearity is accounted for during the step (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide); otherwise, the array contains the original coordinates of the point.
+
+# JLTYP
+
+Identifies the element face for which this call to DFLOW is being made through the element-based flow definition. This information is useful when several different nonuniform distributed flows are being imposed on an element at the same time. See Part VI, “Elements,” of the Abaqus Analysis User’s Guide for identification of element faces. The key is as follows:
+
+<!-- source-page: 33 -->
+
+<table><tr><td>JLTYP</td><td>Flow type</td></tr><tr><td>0</td><td>Surface-based load</td></tr><tr><td>11</td><td>S1NU</td></tr><tr><td>12</td><td>S2NU</td></tr><tr><td>13</td><td>S3NU</td></tr><tr><td>14</td><td>S4NU</td></tr><tr><td>15</td><td>S5NU</td></tr><tr><td>16</td><td>S6NU</td></tr></table>
+
+# SNAME
+
+Surface name for which this call to DFLOW is being made through the surface-based flow definition (JLTYP=0). For an element-based flow definition the surface name is passed in as a blank.
+
+<!-- source-page: 34 -->
+
+<!-- source-page: 35 -->
+
+# 1.1.3 DFLUX: User subroutine to define nonuniform distributed flux in a heat transfer or mass diffusion analysis.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Thermal loads,” Section 34.4.4 of the Abaqus Analysis User’s Guide   
+• “Mass diffusion analysis,” Section 6.9.1 of the Abaqus Analysis User’s Guide   
+• \*DFLUX   
+• \*DSFLUX   
+• “DFLUX,” Section 4.1.1 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine DFLUX:
+
+• can be used to define a nonuniform distributed flux as a function of position, time, temperature, element number, integration point number, etc. in a heat transfer or mass diffusion analysis;   
+• will be called at each flux integration point for each element-based or surface-based (heat transfer only) nonuniform distributed flux definition in the analysis;   
+• ignores any amplitude references that may appear with the associated nonuniform distributed flux definition; and   
+• uses the nodes as flux integration points for first-order heat transfer, first-order coupled temperaturedisplacement, first-order coupled thermal-electrical-structural, and mass diffusion elements.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE DFLUX (FLUX, SOL, KSTEP, KINC, TIME, NOEL, NPT, COORDS, 1 JLTYP, TEMP, PRESS, SNAME)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION FLUX(2), TIME(2), COORDS(3)
+CHARACTER*80 SNAME
+user coding to define FLUX(1) and FLUX(2)
+RETURN
+END 
+```
+
+<!-- source-page: 36 -->
+
+# FLUX(1)
+
+Magnitude of flux flowing into the model at this point. In heat transfer cases the units are $\mathrm { J } \mathrm { T } ^ { - 1 } \mathrm { L } ^ { - 2 }$ for surface fluxes and $\mathrm { J } \mathrm { T } ^ { - 1 } \mathrm { L } ^ { - 3 }$ for body flux. In transient heat transfer cases where a non-default amplitude is used to vary the applied fluxes, the time average flux over the time increment must be defined rather than the value at the end of the time increment. In mass diffusion cases the units are $\mathrm { P L T ^ { - 1 } }$ for surface fluxes and $\mathrm { P T } ^ { - 1 }$ for body flux.
+
+FLUX(1) will be passed into the routine as the magnitude of the flux specified as part of the element-based or surface-based flux definition. If the magnitude is not defined, FLUX(1) will be passed in as zero.
+
+This flux is not available for output purposes.
+
+# FLUX(2)
+
+In heat transfer cases: $d q / d \theta$ , the rate of change of the flux with respect to the temperature at this point. The units are $\mathrm { J } \mathrm { T } ^ { - 1 } \mathrm { L } ^ { - 2 } \theta ^ { - 1 }$ for surface fluxes and $\mathrm { J } \mathrm { T } ^ { - 1 } \mathrm { L } ^ { - 3 } \theta ^ { - 1 }$ for body flux.
+
+In mass diffusion cases: $d q / d c ,$ the rate of change of the flux with respect to the mass concentration at this point. The units are $\mathrm { L T ^ { - 1 } }$ for surface fluxes and $\mathrm { T } ^ { - 1 }$ for body flux.
+
+The convergence rate during the solution of the nonlinear equations in an increment is improved by defining this value, especially when the flux is a strong function of temperature in heat transfer analysis or concentration in mass diffusion analysis.
+
+# Variables passed in for information
+
+# SOL
+
+Estimated value of the solution variable (temperature in a heat transfer analysis or concentration in a mass diffusion analysis) at this time at this point.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# TIME(1)
+
+Current value of step time (defined only in transient analysis).
+
+# TIME(2)
+
+Current value of total time (defined only in transient analysis).
+
+# NOEL
+
+Element number.
+
+<!-- source-page: 37 -->
+
+# NPT
+
+Integration point number in the element or on the element’s surface. The integration scheme depends on whether this is a surface or a body flux.
+
+# COORDS
+
+An array containing the coordinates of this point. These are the current coordinates if geometric nonlinearity is accounted for during the step (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide); otherwise, the array contains the original coordinates of the point.
+
+# JLTYP
+
+Identifies the flux type for which this call to DFLUX is being made. The flux type may be a body flux, a surface-based flux, or an element-based surface flux. For element-based surface fluxes, this variable identifies the element face for which this call to DFLUX is being made. This information is useful when several different nonuniform distributed fluxes are being imposed on an element at the same time. See Part VI, “Elements,” of the Abaqus Analysis User’s Guide for element face identification. The key is as follows:
+
+<table><tr><td>JLTYP</td><td>Flux type</td></tr><tr><td>0</td><td>Surface-based flux</td></tr><tr><td>1</td><td>BFNU</td></tr><tr><td>11</td><td>S1NU (SNEGNU for heat transfer shells)</td></tr><tr><td>12</td><td>S2NU (SPOSNU for heat transfer shells)</td></tr><tr><td>13</td><td>S3NU</td></tr><tr><td>14</td><td>S4NU</td></tr><tr><td>15</td><td>S5NU</td></tr><tr><td>16</td><td>S6NU</td></tr></table>
+
+# TEMP
+
+Current value of temperature at this integration point (defined only for a mass diffusion analysis). Temperature for a heat transfer analysis is passed in as variable SOL.
+
+# PRESS
+
+Current value of the equivalent pressure stress at this integration point (defined only for a mass diffusion analysis).
+
+# SNAME
+
+Surface name for a surface-based flux definition (JLTYP=0). For a body flux or an element-based surface flux the surface name is passed in as blank.
+
+<!-- source-page: 38 -->
+
+<!-- source-page: 39 -->
+
+# 1.1.4 DISP: User subroutine to specify prescribed boundary conditions.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Boundary conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.3.1 of the Abaqus Analysis User’s Guide   
+• “Connector actuation,” Section 31.1.3 of the Abaqus Analysis User’s Guide   
+• \*BOUNDARY   
+• \*CONNECTOR MOTION   
+• “Riser dynamics,” Section 12.1.2 of the Abaqus Example Problems Guide   
+• “DISP,” Section 4.1.2 of the Abaqus Verification Guide   
+• “Boundary conditions,” Section 5.1.5 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine DISP:
+
+• can be used to define the magnitudes of prescribed boundary conditions or connector motions;   
+• requires incremental values to be defined for prescribed rotation boundary conditions;   
+• is called for all degrees of freedom listed in a user-subroutine-defined boundary condition or connector motion definition;   
+• redefines any magnitudes that may be specified (and possibly modified by an amplitude) as part of the associated boundary condition or connector motion definition; and   
+• ignores the specified type, if any, of the associated boundary condition or connector motion definition.
+
+# User subroutine interface
+
+SUBROUTINE DISP(U,KSTEP,KINC,TIME,NODE,NOEL,JDOF,COORDS)
+
+```csv
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION U(3),TIME(3),COORDS(3)
+C 
+```
+
+user coding to define U
+
+<!-- source-page: 40 -->
+
+# RETURN END
+
+# Variable to be defined
+
+# U(1)
+
+All variable types except rotation: the total value of the prescribed variable at this point. The variable may be displacement, pore pressure, temperature, etc., depending on the degree of freedom constrained. U(1) will be passed into the user subroutine as the value defined by any magnitude and/or amplitude specification for the boundary condition or connector motion.
+
+Rotation variable type: the incremental value of the prescribed rotation at this point. The time increment, passed into the user subroutine through TIME(3), should be used to calculate the incremental value. In addition, U(1) will be passed into user subroutine DISP as the value defined by any magnitude or amplitude specification for the boundary condition.
+
+If the analysis procedure requires that the time derivatives of prescribed variables be defined (for example, in a dynamic analysis the velocity and acceleration, as well as the value of the variable, are needed), $d u / d t$ must be given in U(2) and $d ^ { 2 } u / d t ^ { 2 }$ in U(3). The total value of the variable (incremental value in the case of rotation) and its time derivatives must be given in user subroutine DISP, regardless of the type of boundary condition or connector motion.
+
+# Variables passed in for information
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# TIME(1)
+
+Current value of step time.
+
+# TIME(2)
+
+Current value of total time.
+
+# TIME(3)
+
+Current value of time increment.
+
+# NODE
+
+Node number. This variable cannot be used if user subroutine DISP is used to prescribe connector motions.
+
+# NOEL
+
+Element number. This variable cannot be used if user subroutine DISP is used to prescribe boundary conditions.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_005.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_005.md
new file mode 100644
index 00000000..4f6c3089
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_005.md
@@ -0,0 +1,235 @@
+<!-- source-page: 41 -->
+
+# JDOF
+
+Degree of freedom.
+
+# COORDS
+
+An array containing the current coordinates of this point. These are the coordinates at the end of the prior increment if geometric nonlinearity is accounted for during the step (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide); otherwise, the array contains the original coordinates of the node. This array cannot be used if user subroutine DISP is used to prescribe connector motions.
+
+<!-- source-page: 42 -->
+
+<!-- source-page: 43 -->
+
+# 1.1.5 DLOAD: User subroutine to specify nonuniform distributed loads.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Distributed loads,” Section 34.4.3 of the Abaqus Analysis User’s Guide   
+• \*DLOAD   
+• \*DSLOAD   
+• “Nonuniform crack-face loading and J -integrals,” Section 1.16.7 of the Abaqus Benchmarks Guide   
+• “Pure bending of a cylinder: CAXA elements,” Section 1.3.33 of the Abaqus Verification Guide   
+• “Cylinder subjected to asymmetric pressure loads: CAXA elements,” Section 1.3.35 of the Abaqus Verification Guide   
+• “Patch test for axisymmetric elements,” Section 1.5.4 of the Abaqus Verification Guide   
+• “Transient internal pressure loading of a viscoelastic cylinder,” Section 2.2.9 of the Abaqus Verification Guide   
+• “DLOAD,” Section 4.1.3 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine DLOAD:
+
+• can be used to define the variation of the distributed load magnitude as a function of position, time, element number, load integration point number, etc.;   
+• will be called at each load integration point for each element-based or surface-based nonuniform distributed load definition during stress analysis;   
+• will be called at each stiffness integration point for computing the effective axial force, ESF1, for pipe elements subjected to nonuniform load types PENU and PINU;   
+• cannot be used in mode-based procedures to describe the time variation of the load; and   
+• ignores any amplitude references that may appear with the associated step definition or nonuniform distributed load definition.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE DLOAD(F, KSTEP, KINC, TIME, NOEL, NPT, LAYER, KSPT, 1 COORDS, JLTYP, SNAME)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION TIME(2), COORDS (3) 
+```
+
+<!-- source-page: 44 -->
+
+# CHARACTER\*80 SNAME
+
+user coding to define F
+
+# RETURN
+
+# END
+
+# Variable to be defined
+
+F
+
+Magnitude of the distributed load. Units are $\mathrm { F L } ^ { - 2 }$ for surface loads and $\mathrm { F L } ^ { - 3 }$ for body forces. F will be passed into the routine as the magnitude of the load specified as part of the element-based or surfacebased distributed load definition. If the magnitude is not defined, F will be passed in as zero. For a static analysis that uses the modified Riks method (“Static stress analysis,” Section 6.2.2 of the Abaqus Analysis User’s Guide) F must be defined as a function of the load proportionality factor, . The distributed load magnitude is not available for output purposes.
+
+# Variables passed in for information
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# TIME(1)
+
+Current value of step time or current value of the load proportionality factor, , in a Riks step.
+
+# TIME(2)
+
+Current value of total time.
+
+# NOEL
+
+Element number.
+
+# NPT
+
+Load integration point number within the element or on the element’s surface, depending on the load type. (Stiffness integration point number while computing effective axial force, ESF1, for pipe elements subjected to load types PENU and PINU.)
+
+# LAYER
+
+Layer number (for body forces in layered solids).
+
+# KSPT
+
+Section point number within the current layer.
+
+<!-- source-page: 45 -->
+
+# COORDS
+
+An array containing the coordinates of the load integration point. These are the current coordinates if geometric nonlinearity is accounted for during the step (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide); otherwise, the array contains the original coordinates of the point. For axisymmetric elements that allow nonaxisymmetric deformation, COORDS(3) is the angular position of the integration point, in degrees.
+
+# JLTYP
+
+Identifies the load type for which this call to DLOAD is being made. The load type may be a body force, a surface-based load, or an element-based surface load. For element-based surface loads, this variable identifies the element face for which this call to DLOAD is being made. This information is useful when several different nonuniform distributed loads are being imposed on an element at the same time. See Part VI, “Elements,” of the Abaqus Analysis User’s Guide for element face identification. The key is as follows:
+
+<table><tr><td>JLTYP</td><td>Load type</td></tr><tr><td>0</td><td>Surface-based load</td></tr><tr><td>1</td><td>BXNU</td></tr><tr><td>1</td><td>BRNU</td></tr><tr><td>2</td><td>BYNU (except for axisymmetric elements)</td></tr><tr><td>2</td><td>BZNU (for axisymmetric elements only)</td></tr><tr><td>3</td><td>BZNU (for three-dimensional elements and asymmetric-axisymmetric elements)</td></tr><tr><td>20</td><td>PNU</td></tr><tr><td>21</td><td>P1NU</td></tr><tr><td>22</td><td>P2NU</td></tr><tr><td>23</td><td>P3NU</td></tr><tr><td>24</td><td>P4NU</td></tr><tr><td>25</td><td>P5NU</td></tr><tr><td>26</td><td>P6NU</td></tr><tr><td>27</td><td>PINU</td></tr><tr><td>28</td><td>PENU</td></tr><tr><td>41</td><td>PXNU</td></tr><tr><td>42</td><td>PYNU</td></tr><tr><td>43</td><td>PZNU</td></tr></table>
+
+<!-- source-page: 46 -->
+
+# SNAME
+
+Surface name for a surface-based load definition (JLTYP=0). For a body force or an element-based surface load the surface name is passed in as blank.
+
+<!-- source-page: 47 -->
+
+# 1.1.6 FILM: User subroutine to define nonuniform film coefficient and associated sink temperatures for heat transfer analysis.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Thermal loads,” Section 34.4.4 of the Abaqus Analysis User’s Guide   
+• \*CFILM   
+• \*FILM   
+• \*SFILM   
+• “Temperature-dependent film condition,” Section 1.3.42 of the Abaqus Verification Guide
+
+# Overview
+
+# User subroutine FILM:
+
+• can be used to define a node-based, element-based, or surface-based nonuniform film coefficient;   
+• can be used to define sink temperatures as functions of position, time, temperature, node number, element number, integration point number, etc.;   
+• will be called during procedures that allow heat transfer analysis at each node or surface integration point of those surfaces and elements for which node-based, element-based, or surface-based nonuniform film conditions are defined;   
+• ignores any amplitude references for the sink temperature or film coefficient that may appear with the associated nonuniform film definition; and   
+• uses the nodes for first-order heat transfer elements as surface integration points for both elementbased and surface-based films.
+
+# User subroutine interface
+
+```fortran
+SUBROUTINE FILM(H, SINK, TEMP, KSTEP, KINC, TIME, NOEL, NPT, 1 COORDS, JLTYP, FIELD, NFIELD, SNAME, NODE, AREA)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION H(2), TIME(2), COORDS(3), FIELD(NFIELD)
+CHARACTER*80 SNAME
+user coding to define H(1), H(2), and SINK
+RETURN
+END 
+```
+
+<!-- source-page: 48 -->
+
+# Variables to be defined
+
+# H(1)
+
+Film coefficient at this point. Units are $\mathrm { J } \mathrm { T } ^ { - 1 } \mathrm { L } ^ { - 2 } \theta ^ { - 1 }$ . H(1) will be passed into the routine as the magnitude of the film coefficient specified as part of the node-based, element-based, or surface-based film condition definition. If the magnitude is not defined, H(1) will be initialized to zero.
+
+# H(2)
+
+$d h / d \theta _ { ; }$ , rate of change of the film coefficient with respect to the surface temperature at this point. Units are $\mathrm { J } \mathrm { T } ^ { - 1 } \mathrm { L } ^ { - 2 } \theta ^ { - 2 }$ . The rate of convergence during the solution of the nonlinear equations in an increment is improved by defining this value, especially when the film coefficient is a strong function of surface temperature.
+
+# SINK
+
+Sink temperature. SINK will be passed into the routine as the sink temperature specified as part of the node-based, element-based, or surface-based film condition definition. If the sink temperature is not defined, SINK will be initialized to zero.
+
+# Variables passed in for information
+
+# TEMP
+
+Estimated surface temperature at this time at this point.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# TIME(1)
+
+Current value of step time.
+
+# TIME(2)
+
+Current value of total time.
+
+# NOEL
+
+Element number. This variable is passed in as zero for node-based films.
+
+# NPT
+
+Surface integration point number. This variable is passed in as zero for node-based films.
+
+# COORDS
+
+An array containing the coordinates of this point. These are the current coordinates if geometric nonlinearity is accounted for during the step (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide); otherwise, the array contains the original coordinates of the point.
+
+<!-- source-page: 49 -->
+
+# JLTYP
+
+Identifies the element face for which this call to FILM is being made for an element-based film coefficient specification. This information is useful when several different nonuniform film conditions are being imposed on an element at the same time. See Part VI, “Elements,” of the Abaqus Analysis User’s Guide for element face identification. The key is as follows:
+
+<table><tr><td>JLTYP</td><td>Film type</td></tr><tr><td>0</td><td>Node-based or surface-based loading</td></tr><tr><td>11</td><td>F1NU (FNEGNU for heat transfer shells)</td></tr><tr><td>12</td><td>F2NU (FPOSNU for heat transfer shells)</td></tr><tr><td>13</td><td>F3NU</td></tr><tr><td>14</td><td>F4NU</td></tr><tr><td>15</td><td>F5NU</td></tr><tr><td>16</td><td>F6NU</td></tr></table>
+
+# FIELD
+
+Interpolated values of field variables at this point.
+
+# NFIELD
+
+Number of field variables.
+
+# SNAME
+
+Surface name for which this call to FILM is being made for a surface-based film coefficient specification (JLTYP=0). This variable is passed in as blank for both node-based and element-based films.
+
+# NODE
+
+Node number. This variable is passed in as zero for both element-based and surface-based films.
+
+# AREA
+
+Nodal area for node-based films. AREA will be passed into the routine as the nodal area specified as part of the node-based film coefficient specification. This nodal area is not available for output purposes. This variable is passed in as zero for both element-based and surface-based films.
+
+<!-- source-page: 50 -->
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_006.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_006.md
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--- /dev/null
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+<!-- source-page: 51 -->
+
+# 1.1.7 FLOW: User subroutine to define nonuniform seepage coefficient and associated sink pore pressure for consolidation analysis.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Pore fluid flow,” Section 34.4.7 of the Abaqus Analysis User’s Guide   
+• \*FLOW   
+• \*SFLOW
+
+# Overview
+
+User subroutine FLOW:
+
+• can be used in a soils consolidation analysis to define the variation of the reference pore pressure and the seepage coefficient as functions of position, time, pore pressure, element number, integration point number, etc.;   
+• will be called at each integration point of element surfaces for which element-based or surface-based nonuniform surface seepage flow is defined; and   
+• ignores any amplitude references that may appear with the associated nonuniform flow definition.
+
+# User subroutine interface
+
+```fortran
+SUBROUTINE FLOW(H, SINK, U, KSTEP, KINC, TIME, NOEL, NPT, COORDS, 1 JLTYP, SNAME)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION TIME(2), COORDS(3)
+CHARACTER*80 SNAME
+user coding to define H and SINK
+RETURN
+END 
+```
+
+<!-- source-page: 52 -->
+
+# Variables to be defined
+
+# H
+
+Seepage coefficient at this point. Units are F−1 L3 T−1 . H will be passed into the routine as the reference seepage coefficient value specified as part of the element-based or surface-based flow definition. If the reference value is not defined, H will be passed in as zero.
+
+# SINK
+
+Sink pore pressure. SINK will be passed into the routine as the reference pore pressure value specified as part of the element-based or surface-based flow definition. If the reference value is not defined, SINK will be passed in as zero.
+
+# Variables passed in for information
+
+# U
+
+Estimated surface total pore pressure at this time and at this point.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# TIME(1)
+
+Current value of step time (defined only in transient analysis).
+
+# TIME(2)
+
+Current value of total time (defined only in transient analysis).
+
+# NOEL
+
+Element number.
+
+# NPT
+
+Surface integration point number.
+
+# COORDS
+
+An array containing the coordinates of this integration point. These are the current coordinates if geometric nonlinearity is accounted for during the step (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide); otherwise, the array contains the original coordinates of the point.
+
+# JLTYP
+
+Identifies the element face for which this call to FLOW is being made for an element-based flow. This information is useful when several nonuniform flow conditions are being imposed on an element at the same time. See Part VI, “Elements,” of the Abaqus Analysis User’s Guide for identification of the element faces. The key is as follows:
+
+<!-- source-page: 53 -->
+
+<table><tr><td>JLTYP</td><td>Flow type</td></tr><tr><td>0</td><td>Surface-based flow</td></tr><tr><td>61</td><td>Q1NU</td></tr><tr><td>62</td><td>Q2NU</td></tr><tr><td>63</td><td>Q3NU</td></tr><tr><td>64</td><td>Q4NU</td></tr><tr><td>65</td><td>Q5NU</td></tr><tr><td>66</td><td>Q6NU</td></tr></table>
+
+# SNAME
+
+Surface name for which this call to FLOW is being made for a surface-based flow (JLTYP=0). For an element-based flow the surface name is passed in as a blank.
+
+<!-- source-page: 54 -->
+
+<!-- source-page: 55 -->
+
+# 1.1.8 FRIC: User subroutine to define frictional behavior for contact surfaces.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Frictional behavior,” Section 37.1.5 of the Abaqus Analysis User’s Guide   
+• \*FRICTION   
+• “Thermal-stress analysis of a disc brake,” Section 5.1.1 of the Abaqus Example Problems Guide   
+• “FRIC,” Section 4.1.4 of the Abaqus Verification Guide
+
+# Overview
+
+# User subroutine FRIC:
+
+• can be used to define the frictional behavior between contacting surfaces;   
+• can be used when the extended versions of the classical Coulomb friction model provided in Abaqus are too restrictive and a more complex definition of shear transmission between contacting surfaces is required;   
+• will be called at points on the slave surface of a contact pair and at the integration points in a contact element (only when the contact point is closed) for which the contact interaction property model contains user-subroutine-defined friction;   
+• must provide the entire definition of shear interaction between the contacting surfaces; and   
+• can use and update solution-dependent state variables.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE FRIC(LM, TAU, DDTDDG, DDTDDP, DSLIP, SED, SFD,
+1 DDTDDT, PNEWDT, STATEV, DGAM, TAULM, PRESS, DPRESS, DDPDDH, SLIP,
+2 KSTEP, KINC, TIME, DTIME, NOEL, CINAME, SLNAME, MSNAME, NPT, NODE,
+3 NPATCH, COORDS, RCOORD, DROT, TEMP, PREDEF, NFDIR, MCRD, NPRED,
+4 NSTATV, CHRLNGTH, PROPS, NPROPS)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CINAME, SLNAME, MSNAME
+C
+DIMENSION TAU(NFDIR), DDTDDG(NFDIR, NFDIR), DDTDDP(NFDIR),
+1 DSLIP(NFDIR), DDTDDT(NFDIR, 2), STATEV(*), DGAM(NFDIR),
+2 TAULM(NFDIR), SLIP(NFDIR), TIME(2), COORDS(MCRD), 
+```
+
+<!-- source-page: 56 -->
+
+3 RCOORD(MCRD),DROT(2,2),TEMP(2),PREDEF(2,\*),PROPS(NPROPS)
+
+user coding to define LM, TAU, DDTDDG, DDTDDP, and, optionally, DSLIP, SED, SFD, DDTDDT, PNEWDT, STATEV
+
+RETURN END
+
+Variables to be defined
+
+# In all cases
+
+LM
+
+Relative motion flag. User subroutine FRIC is called only if the contact point is determined to be closed; that is, if the contact pressure is positive (the contact point was closed in the previous iteration) or if the contact point is overclosed (the contact point was open in the previous iteration).
+
+During iterations LM is passed into the subroutine as the value defined during the previous iteration. At the start of an increment or if the contact point opened during the previous iteration, this variable will be passed into the routine depending on the contact condition in the previous increment. If the contact point was slipping, LM is equal to 0; if the contact point was sticking, LM is equal to 1; and if the contact point was open, LM is equal to 2.
+
+Set LM equal to 0 if relative motion is allowed (either due to slip or elastic stick). In this case the subroutine must specify the frictional stress $\tau _ { 1 }$ (and $\tau _ { 2 }$ for three-dimensional analysis) as a function of the relative sliding motion $\gamma _ { 1 }$ (and $\gamma _ { 2 } )$ , the interface contact pressure p, and other predefined or user-defined state variables. In addition, the subroutine must define the derivatives of the frictional stress with respect to $\gamma _ { 1 } , ( \gamma _ { 2 } )$ , and p. For instance, in the case of isotropic elastic sticking, $\partial \tau _ { 1 } / \partial \gamma _ { 1 } =$ $\partial \tau _ { 2 } / \partial \gamma _ { 2 } = k _ { e l a s } , \partial \tau _ { 1 } / \partial \gamma _ { 2 } = \partial \tau _ { 2 } / \partial \gamma _ { 1 } = 0$ , where $k _ { e l a s }$ is the elastic stiffness of the interface.
+
+Set LM equal to 1 if no relative motion is allowed; a rigid sticking condition at the interface is enforced by a Lagrange multiplier method. In this case no further variables need to be updated. If LM is always set to 1, a “perfectly rough” interface is created. It is not advisable to set LM to 1 when the finite-sliding, surface-to-surface contact formulation is used.
+
+Set LM equal to 2 if friction is ignored (frictionless sliding is assumed). In this case no further variables need to be updated. If LM is always set to 2, a “perfectly smooth” interface is created.
+
+You can make decisions about the stick/slip condition based on incremental slip information and calculated frictional stresses. These quantities are passed in by Abaqus/Standard, as discussed below.
+
+To avoid convergence problems for the general class of frictional contact problems, set LM to 2 and exit this routine if the contact point was open at the end of the previous increment; that is, if Abaqus/Standard sets LM=2 when it calls this routine, simply exit the routine.
+
+<!-- source-page: 57 -->
+
+# If the return value of LM is 0
+
+# TAU(NFDIR)
+
+These values are passed in as the values of the frictional stress components, $\tau _ { \alpha }$ , at the beginning of the increment and must be updated to the values at the end of the increment. Here, and in the rest of this description, Greek subscripts ( , ) refer to frictional shear directions. The orientation of these directions on contact surfaces is defined in “Contact formulations in Abaqus/Standard,” Section 38.1.1 of the Abaqus Analysis User’s Guide.
+
+# DDTDDG(NFDIR,NFDIR)
+
+$\partial \Delta \tau _ { \alpha } / \partial \Delta \gamma _ { \beta }$ , partial derivative of the frictional stress in direction with respect to the relative motion in direction $\beta .$ .
+
+# DDTDDP(NFDIR)
+
+$\partial \Delta \tau _ { \alpha } / \partial \Delta p .$ , partial derivative of the frictional stress in direction with respect to the contact pressure. Since these terms yield an unsymmetric contribution to the stiffness matrix, they are used only if the unsymmetric equation solver is used (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide).
+
+# Variables that can be updated
+
+# DSLIP(NFDIR)
+
+$\Delta \gamma _ { \alpha } ^ { s l }$ , increment in nonrecoverable sliding motion (slip). If LM was 0 in the previous iteration, this array is passed in as the user-defined values during the previous iteration; otherwise, it will be zero. The array should be updated only if the return value of LM is 0.
+
+This array is useful to detect slip reversals between iterations. It is used by the output options to indicate whether this point is sticking or slipping. Upon convergence of an increment, the values in DSLIP(NFDIR) are accumulated in SLIP(NFDIR), which are stored as the plastic strains.
+
+# SED
+
+This variable is passed in as the value of the elastic energy density at the start of the increment and should be updated to the elastic energy density at the end of the increment. This variable is used for output only and has no effect on other solution variables.
+
+# SFD
+
+This variable should be defined as the incremental frictional dissipation. The units are energy per unit area if the contact element or contact pair calling FRIC uses stresses as opposed to forces. For regular stress analysis this variable is used for output only and has no effect on other solution variables. In coupled temperature-displacement and coupled thermal-electrical-structural analyses the dissipation is converted into heat if the gap heat generation model is used. If SFD is not defined, the heat generation is calculated based on the dissipation obtained as the product of the slip increment, DSLIP, and the frictional stress, TAU.
+
+<!-- source-page: 58 -->
+
+# DDTDDT(NFDIR,2)
+
+$\partial \Delta \tau _ { \alpha } / \partial \Delta \theta _ { 1 } , \partial \Delta \tau _ { \alpha } / \partial \Delta \theta _ { 2 }$ partial derivatives of the frictional stress in direction with respect to the temperatures of the two surfaces. This is required only for coupled temperature-displacement and coupled thermal-electrical-structural elements, in which the frictional stress is a function of the surface temperatures.
+
+# PNEWDT
+
+Ratio of suggested new time increment to the time increment currently being used (DTIME, see below). This variable allows you to provide input to the automatic time incrementation algorithms in Abaqus/Standard (if automatic time incrementation is chosen).
+
+PNEWDT is set to a large value before each call to FRIC.
+
+If PNEWDT is redefined to be less than 1.0, Abaqus/Standard must abandon the time increment and attempt it again with a smaller time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines that allow redefinition of PNEWDT for this iteration.
+
+If PNEWDT is given a value that is greater than 1.0 for all calls to user subroutines for this iteration and the increment converges in this iteration, Abaqus/Standard may increase the time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines for this iteration.
+
+If automatic time incrementation is not selected in the analysis procedure, values of PNEWDT greater than 1.0 will be ignored and values of PNEWDT less than 1.0 will cause the job to terminate.
+
+# STATEV(NSTATV)
+
+An array containing the user-defined solution-dependent state variables. You specify the number of available state variables; see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide, for details. This array will be passed in containing the values of these variables at the start of the increment. If any of the solution-dependent state variables is being used in conjunction with the friction behavior, they must be updated in this subroutine to their values at the end of the increment.
+
+# Variables passed in for information
+
+# DGAM(NFDIR)
+
+If LM was set to 0 in the previous iteration, this value is the increment of sliding motion in the current increment, $\Delta \gamma _ { \alpha }$ . Otherwise, it will be zero. Comparison with DSLIP(NFDIR) makes it possible to determine whether slip changes to stick at this point and/or if there is a slip direction reversal occurring at this point.
+
+# TAULM(NFDIR)
+
+If LM was set to 1 in the previous iteration, this value is the current value of the constraint stress at the end of the increment, $\tau _ { \alpha } ^ { L M }$ . Otherwise, it will be zero. Comparison with the critical shear stress makes it possible to determine whether stick changes to slip at this point.
+
+<!-- source-page: 59 -->
+
+# PRESS
+
+p, contact pressure at end of increment.
+
+# DPRESS
+
+, increment in contact pressure.
+
+# DDPDDH
+
+/ , current contact stiffness, in the case of soft contact (“Contact pressure-overclosure relationships,” Section 37.1.2 of the Abaqus Analysis User’s Guide).
+
+# SLIP(NFDIR)
+
+Total nonrecoverable sliding motion (slip) at the beginning of the increment, $\gamma _ { \alpha } ^ { s l }$ . This value is the accumulated value of DSLIP(NFDIR) from previous increments.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# TIME(1)
+
+Value of step time at the end of the increment.
+
+# TIME(2)
+
+Value of total time at the end of the increment.
+
+# DTIME
+
+Current increment in time.
+
+# NOEL
+
+Element label for contact elements. Passed in as zero if contact surfaces are defined.
+
+# CINAME
+
+User-specified surface interaction name associated with the friction definition, left justified. For contact elements it is the element set name given for the interface definition associated with the friction definition; if an optional name is assigned to the interface definition, CINAME is passed in as this name, left justified.
+
+# SLNAME
+
+Slave surface name. Passed in as blank if contact elements are used.
+
+# MSNAME
+
+Master surface name. Passed in as blank if contact elements are used.
+
+# NPT
+
+Integration point number for contact elements. Passed in as zero if contact surfaces are defined.
+
+<!-- source-page: 60 -->
+
+# NODE
+
+User-defined global slave node number (or internal node number for models defined in terms of an assembly of part instances) involved with this contact point. Corresponds to the predominant slave node of the constraint if the surface-to-surface contact formulation is used. Passed in as zero if called from a contact element.
+
+# NPATCH
+
+Not used.
+
+# COORDS(MCRD)
+
+An array containing the current coordinates of this point.
+
+# RCOORD(MCRD)
+
+If the master surface is defined as a rigid surface, this array is passed in containing the coordinates of the opposing point on the rigid surface in its current position and orientation.
+
+# DROT(2,2)
+
+Rotation increment matrix. For contact with a three-dimensional rigid surface, this matrix represents the incremental rotation of the surface directions relative to the rigid surface. It is provided so that vector- or tensor-valued state variables can be rotated appropriately in this subroutine. Stress and slip components are already rotated by this amount before FRIC is called. This matrix is passed in as a unit matrix for two-dimensional and axisymmetric contact problems.
+
+# TEMP(2)
+
+Current temperature at the slave node and the opposing master surface, respectively.
+
+# PREDEF(2,NPRED)
+
+An array containing pairs of values of all the user-specified field variables at the end of the current increment (initial values at the beginning of the analysis and current values during the analysis). If FRIC is called from a contact pair, the first value in a pair corresponds to the slave node and the second value corresponds to the nearest point on the master surface. If FRIC is called from a large-sliding contact element, PREDEF(1,NPRED) corresponds to the value at the integration point of the element and PFREDEF(2,NPRED) corresponds to the nearest point on the opposing surface. If FRIC is called from a small-sliding contact element, PREDEF(1,NPRED) corresponds to the value at the integration point of the first side and PFREDEF(2,NPRED) corresponds to the value at the integration point on the opposite face of the element.
+
+# NFDIR
+
+Number of friction directions.
+
+# MCRD
+
+Number of coordinate directions at the contact point.
+
+# NPRED
+
+Number of predefined field variables.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_007.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_007.md
new file mode 100644
index 00000000..baab4d5c
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@@ -0,0 +1,310 @@
+<!-- source-page: 61 -->
+
+# NSTATV
+
+Number of user-defined state variables.
+
+# CHRLNGTH
+
+Characteristic contact surface face dimension, which can be used to define the maximum allowable elastic slip.
+
+# PROPS(NPROPS)
+
+Array of user-specified property values that are used to define the frictional behavior between the contacting surfaces.
+
+# NPROPS
+
+User-specified number of property values associated with this friction model.
+
+<!-- source-page: 62 -->
+
+<!-- source-page: 63 -->
+
+# 1.1.9 FRIC\_COEF: User subroutine to define the frictional coefficient for contact surfaces.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Frictional behavior,” Section 37.1.5 of the Abaqus Analysis User’s Guide   
+• \*FRICTION   
+• “FRIC\_COEF,” Section 4.1.5 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine FRIC\_COEF:
+
+• can be used to define the isotropic frictional coefficient between contacting surfaces;   
+• corresponds to the classical Coulomb friction model; and   
+• can be used with the contact pair and general contact algorithms.
+
+# User subroutine interface
+
+```c
+subroutine fric_coef (
+C Write only -
+* fCoef, fCoefDeriv,
+C Read only -
+* nBlock, nProps, nTemp, nFields,
+* jFlags, rData,
+* surfInt, surfSlv, surfMst,
+* props, slipRate, pressure,
+* tempAvg, fieldAvg)
+C
+include 'aba_param.inc'
+C
+dimension fCoef(nBlock),
+* fCoefDeriv(nBlock,3),
+* props(nProps),
+* slipRate(nBlock),
+* pressure(nBlock),
+* tempAvg(nBlock),
+* fieldAvg(nBlock,nFields)
+C
+parameter( iKStep    = 1, 
+```
+
+<!-- source-page: 64 -->
+
+```matlab
+* iKInc = 2,
+* nFlags = 2 )
+C
+    parameter( iTimStep = 1,
+    * iTimGlb = 2,
+    * iDTimCur = 3,
+    * nData = 3 )
+C
+    dimension jFlags(nFlags), rData(nData)
+C
+    character*80 surfInt, surfSlv, surfMst
+C
+    user coding to define fCoef
+    return
+    end 
+```
+
+# Variables to be defined
+
+fCoef(nBlock)
+
+This array must be updated to the current values of the friction coefficient at the contact point.
+
+fCoefDeriv(nBlock,3)
+
+This array must be updated to the derivatives of the friction coefficient with respect to slip rate, pressure, and temperature at the contact point.
+
+# Variables passed in for information
+
+nBlock
+
+Equal to 1.
+
+nProps
+
+User-specified number of property values associated with this friction model.
+
+nTemp
+
+1 if the temperature is defined and 0 if the temperature is not defined.
+
+nFields
+
+Number of user-specified field variables.
+
+jFlag(1)
+
+Step number.
+
+jFlag(2)
+
+Increment number.
+
+<!-- source-page: 65 -->
+
+rData(1)
+
+Value of step time.
+
+rData(2)
+
+Value of total time.
+
+rData(3)
+
+Current increment in time from $t = t _ { c u r r } - \Delta t { \mathrm { t } } 0 t = t _ { c u r r } .$
+
+surfInt
+
+User-specified surface interaction name, left justified.
+
+surfSlv
+
+Slave surface name, left justified.
+
+surfMst
+
+Master surface name, left justified.
+
+props(nProps)
+
+User-specified vector of property values to define the frictional coefficient at the contact point.
+
+slipRate(nBlock)
+
+This array contains the rate of tangential slip at the contact point for the current time increment.
+
+pressure(nBlock)
+
+This array contains the pressure at the contact point projected at the end of the current time increment.
+
+tempAvg(nBlock)
+
+Average current temperature between the master and slave surfaces at the contact point.
+
+fieldAvg(nBlock,nFields)
+
+Average current value of all the user-specified field variables between the master and slave surfaces at the contact point.
+
+<!-- source-page: 66 -->
+
+<!-- source-page: 67 -->
+
+# 1.1.10 GAPCON: User subroutine to define conductance between contact surfaces or nodes in a fully coupled temperature-displacement analysis, coupled thermalelectrical-structural analysis, or pure heat transfer analysis.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Thermal contact properties,” Section 37.2.1 of the Abaqus Analysis User’s Guide   
+• \*GAP CONDUCTANCE   
+• “GAPCON,” Section 4.1.6 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine GAPCON:
+
+• assumes that the heat transfer between surfaces is modeled as $q = k ( \theta _ { A } - \theta _ { B } )$ , where q is the heat flux per unit area flowing between corresponding points A and B on the surfaces, k is the gap conductance, and $\theta _ { A }$ and $\theta _ { B }$ are the surface temperatures;   
+• is used to define k, providing greater flexibility than direct gap conductance definition in specifying the dependencies of k (for example, it is not necessary to define the gap conductance as a function of the average of the two surfaces’ temperatures, mass flow rates, or field variables);   
+• will be called at the slave nodes of a contact pair and at the integration points in a contact or a gap element for which the heat conductance definition contains a user-subroutine-defined gap conductance; and   
+• ignores any dependencies or data specified for the gap conductance outside the user subroutine.
+
+# Usage with contact pairs and gap elements
+
+When this subroutine is used with a contact pair, point A is on the slave surface and point B is on the master surface.
+
+When GAPCON is used with gap elements of type DGAP or GAPUNIT, point A is on the first node of the element and point B is the second node of the element.
+
+# User subroutine interface
+
+SUBROUTINE GAPCON(AK,D,FLOWM,TEMP,PREDEF,TIME,CINAME,SLNAME, 1 MSNAME,COORDS,NOEL,NODE,NPRED,KSTEP,KINC)
+
+```csv
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CINAME, SLNAME, MSNAME 
+```
+
+<!-- source-page: 68 -->
+
+C
+
+```txt
+DIMENSION AK(5), D(2), FLOWM(2), TEMP(2), PREDEF(2, *), 1 TIME(2), COORDS(3)
+user coding to define AK(1) -- AK(5)
+RETURN
+END 
+```
+
+# Variables to be defined
+
+# AK(1)
+
+Gap conductance, k. The units of k are energy per time (flux) per area per temperature $( \mathrm { J } \mathrm { T } ^ { - 1 } \mathrm { L } ^ { - 2 } \theta ^ { - 1 } )$ .
+
+# AK(2)
+
+$\partial k / \partial d ,$ derivative of the gap conductance with respect to the clearance between the bodies. If the gap conductance is not a function of gap clearance, AK(2)=0.0. This variable needs to be defined only for fully coupled temperature-displacement and coupled thermal-electrical-structural analyses.
+
+# AK(3)
+
+$\partial k / \partial p$ , derivative of the gap conductance with respect to the pressure between the bodies. If the gap conductance is not a function of the pressure, AK(3)=0.0. This variable needs to be defined only for fully coupled temperature-displacement and coupled thermal-electrical-structural analyses.
+
+# AK(4)
+
+$\partial k / \partial \theta _ { A }$ , derivative of the gap conductance with respect to the temperature of point A on the first surface of the interface.
+
+# AK(5)
+
+$\partial k / \partial \theta _ { B }$ , derivative of the gap conductance with respect to the temperature of point B on the second surface of the interface.
+
+# Variables passed in for information
+
+# D(1)
+
+Separation between the surfaces, d.
+
+# D(2)
+
+Pressure transmitted across the surfaces, p. This pressure is zero in pure heat transfer analysis.
+
+# FLOWM(2)
+
+${ \dot { m } } | _ { A } , { \dot { m } } | _ { B }$ , magnitudes of the mass flow rate per unit area at points A and B.
+
+# TEMP(2)
+
+Current temperature at points A and B.
+
+<!-- source-page: 69 -->
+
+# PREDEF(2,NPRED)
+
+An array containing pairs of values of all of the user-specified field variables at the end of the current increment at points A and B (initial values at the beginning of the analysis and current values during the analysis).
+
+# TIME(1)
+
+Value of step time at the end of the increment.
+
+# TIME(2)
+
+Value of total time at the end of the increment.
+
+# CINAME
+
+User-specified surface interaction name associated with the heat conductance definition, left justified. For contact elements it is the element set name given for the interface definition associated with the heat conductance definition; if an optional name is assigned to the interface definition, CINAME is passed in as this name, left justified. For gap elements it is the element set name for the element definition associated with the heat conductance definition.
+
+# SLNAME
+
+Slave surface name. Passed in as blank if contact or gap elements are used.
+
+# MSNAME
+
+Master surface name. Passed in as blank if contact or gap elements are used.
+
+# COORDS
+
+An array containing the coordinates of point A. These are the current coordinates if geometric nonlinearity is accounted for during the step (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide); otherwise, the array contains the original coordinates of the point.
+
+# NOEL
+
+Element label for contact or gap elements. Passed in as zero if contact surfaces are defined.
+
+# NODE
+
+Slave node number (point A) if GAPCON is called for a contact pair.
+
+# NPRED
+
+Number of predefined field variables.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+<!-- source-page: 70 -->
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_008.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_008.md
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+<!-- source-page: 71 -->
+
+# 1.1.11 GAPELECTR: User subroutine to define electrical conductance between surfaces in a coupled thermal-electrical or a coupled thermal-electrical-structural analysis.
+
+Product: Abaqus/Standard
+
+# References
+
+• “Electrical contact properties,” Section 37.3.1 of the Abaqus Analysis User’s Guide   
+• \*GAP ELECTRICAL CONDUCTANCE
+
+# Overview
+
+User subroutine GAPELECTR:
+
+• assumes that the electrical current flowing between the interface surfaces is modeled as $J = \sigma _ { g } ( \varphi _ { A } - \varphi _ { B } )$ where J is the electrical current density flowing across the interface from point A (the slave surface) to point B (the master surface), $\varphi _ { A }$ and $\varphi _ { B }$ are the electrical potential on opposite points of the surfaces, and $\sigma _ { g }$ is the surface electrical conductance;   
+• is used to define $\sigma _ { g }$ , providing much greater flexibility than direct gap electrical conductance definition in specifying the dependencies of $\sigma _ { g }$ (for instance, it is not necessary to define the gap electrical conductance as a function of the average of the two surfaces’ temperatures and/or field variables);   
+• will be called at the slave nodes of a contact pair (“Defining contact pairs in Abaqus/Standard,” Section 36.3.1 of the Abaqus Analysis User’s Guide) for which the gap electrical conductance is defined in a user subroutine; and   
+• ignores any dependencies or data specified for the gap electrical conductance outside the user subroutine.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE GAPELECTR(SIGMA, D, TEMP, PREDEF, TIME, CINAME, 1 SLNAME, MSNAME, COORDS, NODE, NPRED, KSTEP, KINC)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CINAME, SLNAME, MSNAME
+C
+DIMENSION SIGMA(5), D(2), TEMP(2), PREDEF(2, *), TIME(2), 1 COORDS(2, 3)
+user coding to define SIGMA(1) -- SIGMA(5)
+RETURN
+END 
+```
+
+<!-- source-page: 72 -->
+
+# SIGMA(1)
+
+Gap electrical conductance, $\sigma _ { g } .$
+
+# SIGMA(2)
+
+$\partial \sigma _ { g } / \partial \theta _ { A } ,$ , derivative of the gap electrical conductance with respect to the temperature of point A. If the gap electrical conductance is not a function of $\theta _ { A }$ , SIGMA(2) = 0.0.
+
+# SIGMA(3)
+
+$\partial \sigma _ { g } / \partial \theta _ { B }$ , derivative of the gap electrical conductance with respect to the temperature of point B. If the gap electrical conductance is not a function of $\theta _ { B }$ , SIGMA(3) = 0.0.
+
+# SIGMA(4)
+
+$\partial \sigma _ { g } / \partial d ,$ derivative of the gap electrical conductance with respect to the clearance between the bodies. If the gap electrical conductance is not a function of gap clearance, SIGMA(4)= 0.0. This variable needs to be defined only for a fully coupled thermal-electrical-structural analysis.
+
+# SIGMA(5)
+
+$\partial \sigma _ { g } / \partial p ,$ derivative of the gap electrical conductance with respect to the pressure between the bodies. If the gap electrical conductance is not a function of the pressure, SIGMA(5)= 0.0. This variable needs to be defined only for a fully coupled thermal-electrical-structural analysis.
+
+# Variables passed in for information
+
+# D(1)
+
+Separation between the interface surfaces, d.
+
+# D(2)
+
+Pressure transmitted across the surfaces, p.
+
+# TEMP(2)
+
+Current temperature at points A and B.
+
+# PREDEF(2,NPRED)
+
+An array containing pairs of values of all of the user-specified field variables at the end of the current increment at points A and B (initial values at the beginning of the analysis and current values during the analysis).
+
+# TIME(1)
+
+Value of step time at the end of the increment.
+
+# TIME(2)
+
+Value of total time at the end of the increment.
+
+<!-- source-page: 73 -->
+
+# CINAME
+
+User-specified surface interaction name, left justified.
+
+# SLNAME
+
+Slave surface name.
+
+# MSNAME
+
+Master surface name.
+
+# COORDS
+
+An array containing the current coordinates of points A and B. COORDS(1,K1) are the coordinates at point A, and COORDS(2,K1) are the coordinates at point B.
+
+# NODE
+
+Slave node number (point A).
+
+# NPRED
+
+Number of predefined field variables.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+<!-- source-page: 74 -->
+
+<!-- source-page: 75 -->
+
+# 1.1.12 HARDINI: User subroutine to define initial equivalent plastic strain and initial backstress tensor.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1 of the Abaqus Analysis User’s Guide   
+• “Classical metal plasticity,” Section 23.2.1 of the Abaqus Analysis User’s Guide   
+• “Models for metals subjected to cyclic loading,” Section 23.2.2 of the Abaqus Analysis User’s Guide   
+• “Extended Drucker-Prager models,” Section 23.3.1 of the Abaqus Analysis User’s Guide   
+• \*INITIAL CONDITIONS   
+• “HARDINI,” Section 4.1.8 of the Abaqus Verification Guide
+
+# Overview
+
+# User subroutine HARDINI:
+
+• can be used only for material models that use metal plasticity or Drucker-Prager plasticity;   
+• can be used to provide initial equivalent plastic strain values as a function of element number, material point number, and/or material point coordinates for isotropic and combined hardening;   
+• enables you to specify initial conditions for the backstress tensor as a function of element number, material point number, and/or material point coordinates for kinematic and combined hardening;   
+• will be called to define the initial equivalent plastic strain and, if relevant, the initial backstresses at material points for which user-subroutine-defined initial hardening conditions are specified; and   
+• is intended for use when the initial equivalent plastic strain and/or backstress distributions are too complicated to specify directly as initial hardening conditions.
+
+# Defining backstress components
+
+The number of backstress components that must be defined depends on the element type for which this routine is being called. Part VI, “Elements,” of the Abaqus Analysis User’s Guide describes the number of stress components for each element type; the number of backstress components is identical to the number of stress components. The order of the backstress components is the same as the order of the stress components. For example, in three-dimensional continuum elements six backstress components must be defined in the order $\alpha _ { 1 1 } , \alpha _ { 2 2 } , \alpha _ { 3 3 } , \alpha _ { 1 2 } , \alpha _ { 1 3 } , \alpha _ { 2 3 }$
+
+<!-- source-page: 76 -->
+
+User subroutine interface   
+```txt
+SUBROUTINE HARDINI (ALPHA, EQPS, COORDS, NTENS, NCRDS, NOEL, NPT, 1 LAYER, KSPT, LREBAR, REBARN)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION ALPHA (NTENS, *), COORDS (NCRDS)
+CHARACTER*80 REBARN
+user coding to define EQPS and, if relevant, ALPHA (NTENS)
+RETURN
+END 
+```
+
+Variables to be defined   
+```txt
+The variables described below are element-type dependent.
+
+EQPS
+Equivalent plastic strain.
+
+ALPHA (1,1)
+First backstress component of the first backstress.
+
+ALPHA (2,1)
+Second backstress component of the first backstress.
+
+ALPHA (3,1)
+Third backstress component of the first backstress.
+
+Etc.
+NTENS backstress component values should be defined for each backstress. 
+```  
+Variables passed in for information   
+COORDS
+
+An array containing the initial coordinates of this point.
+
+NTENS
+
+Number of backstress values to be defined. This number depends on the element type.
+
+<!-- source-page: 77 -->
+
+# NCRDS
+
+Number of coordinates.
+
+# NOEL
+
+Element number.
+
+# NPT
+
+Integration point number.
+
+# LAYER
+
+Layer number (for composite shells and layered solids).
+
+# KSPT
+
+Section point number within the current layer.
+
+# LREBAR
+
+Rebar flag. If LREBAR=1, the current integration point is associated with element rebar. Otherwise, LREBAR=0.
+
+# REBARN
+
+Name of the rebar to which the current integration point belongs, which is the name given in the rebar or rebar layer definition (“Defining reinforcement,” Section 2.2.3 of the Abaqus Analysis User’s Guide, or “Defining rebar as an element property,” Section 2.2.4 of the Abaqus Analysis User’s Guide). If no name was given in the rebar or rebar layer definition, this variable will be blank. This variable is relevant only when LREBAR=1.
+
+<!-- source-page: 78 -->
+
+<!-- source-page: 79 -->
+
+# 1.1.13 HETVAL: User subroutine to provide internal heat generation in heat transfer analysis.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Uncoupled heat transfer analysis,” Section 6.5.2 of the Abaqus Analysis User’s Guide   
+• “Fully coupled thermal-stress analysis,” Section 6.5.3 of the Abaqus Analysis User’s Guide   
+• “Fully coupled thermal-electrical-structural analysis,” Section 6.7.4 of the Abaqus Analysis User’s Guide   
+• \*HEAT GENERATION   
+• “HETVAL,” Section 4.1.9 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine HETVAL:
+
+• can be used to define a heat flux due to internal heat generation in a material, for example, as might be associated with phase changes occurring during the solution;   
+• allows for the dependence of internal heat generation on state variables (such as the fraction of material transformed) that themselves evolve with the solution and are stored as solution-dependent state variables;   
+• will be called at all material calculation points for which the material definition contains volumetric heat generation during heat transfer, coupled temperature-displacement, coupled thermal-electrical, or coupled thermal-electrical-structural analysis procedures;   
+• can be useful if it is necessary to include a kinetic theory for a phase change associated with latent heat release (for example, in the prediction of crystallization in a polymer casting process);   
+• can be used in conjunction with user subroutine USDFLD if it is desired to redefine any field variables before they are passed in; and   
+• cannot be used with user subroutine UMATHT.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE HETVAL (CMNAME, TEMP, TIME, DTIME, STATEV, FLUX, 1 PREDEF, DPRED)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME
+C 
+```
+
+<!-- source-page: 80 -->
+
+DIMENSION TEMP(2),STATEV(\*),PREDEF(\*),TIME(2),FLUX(2), 1 DPRED(\*)
+
+user coding to define FLUX and update STATEV
+
+RETURN END
+
+# Variables to be defined
+
+# FLUX(1)
+
+Heat flux, r (thermal energy per time per volume: $\mathrm { J } \mathrm { T } ^ { - 1 } \mathrm { L } ^ { - 3 } )$ , at this material calculation point.
+
+# FLUX(2)
+
+Rate of change of heat flux per temperature, . This variable is nonzero only if the heat flux depends on temperature. It is needed to define a correct Jacobian matrix.
+
+# Variable that can be updated
+
+# STATEV(\*)
+
+An array containing the user-defined solution-dependent state variables at this point.
+
+In an uncoupled heat transfer analysis STATEV is passed into subroutine HETVAL as the values of these variables at the beginning of the increment. However, any updating of STATEV in user subroutine USDFLD will be included in the values passed into subroutine HETVAL since this routine is called before HETVAL. In addition, if HETVAL is being used in a fully coupled temperature-displacement or coupled thermal-electrical-structural analysis and user subroutine UEXPAN, user subroutine CREEP, user subroutine UMAT, or user subroutine UTRS is used to define the mechanical behavior of the material, those routines are called before this routine; therefore, any updating of STATEV done in UEXPAN, CREEP, UMAT, or UTRS will be included in the values passed into this routine.
+
+In all cases STATEV should be passed back from user subroutine HETVAL containing the values of the state variables at the end of the current increment.
+
+# Variables passed in for information
+
+# CMNAME
+
+User-specified material name, left justified.
+
+# TEMP(1)
+
+Current temperature.
+
+# TEMP(2)
+
+Temperature increment.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_009.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_009.md
new file mode 100644
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@@ -0,0 +1,312 @@
+<!-- source-page: 81 -->
+
+# TIME(1)
+
+Step time at the end of the increment.
+
+# TIME(2)
+
+Total time at the end of the increment.
+
+# DTIME
+
+Time increment.
+
+# PREDEF(\*)
+
+An array containing the values of all of the user-specified field variables at this point (initial values at the beginning of the analysis and current values during the analysis).
+
+# DPRED(\*)
+
+Array of increments of predefined field variables.
+
+<!-- source-page: 82 -->
+
+<!-- source-page: 83 -->
+
+# 1.1.14 MPC: User subroutine to define multi-point constraints.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “General multi-point constraints,” Section 35.2.2 of the Abaqus Analysis User’s Guide   
+• \*MPC
+
+# Overview
+
+# User subroutine MPC:
+
+• is called to impose a user-defined multi-point constraint and is intended for use when general constraints cannot be defined with one of the MPC types provided by Abaqus/Standard;   
+• can use only degrees of freedom that also exist on an element somewhere in the same model (methods for overcoming this limitation are discussed below);   
+• can generate linear as well as nonlinear constraints;   
+• allows definition of constraints involving finite rotations; and   
+• makes it possible to switch constraints on and off during an analysis.
+
+# Coding methods
+
+There are two methods for coding this routine. By default, the subroutine operates in a degree of freedom mode. In this mode each call to this subroutine allows one individual degree of freedom to be constrained. Alternatively, you can specify that the subroutine operate in a nodal mode. In this mode each call to this subroutine allows a set of constraints to be imposed all at once; that is, on multiple degrees of freedom of the dependent node. In either case, the routine will be called for each user-subroutine-defined multi-point constraint or set of constraints. See “General multi-point constraints,” Section 35.2.2 of the Abaqus Analysis User’s Guide, for details.
+
+# Constraints that involve rotational degrees of freedom
+
+In geometrically nonlinear analyses Abaqus/Standard compounds three-dimensional rotations based on a finite-rotation formulation and not by simple addition of the individual rotation components (see “Conventions,” Section 1.2.2 of the Abaqus Analysis User’s Guide, and “Rotation variables,” Section 1.3.1 of the Abaqus Theory Guide). An incremental rotation involving one component usually results in changes in all three total rotation components. Therefore, any general constraint that involves large three-dimensional rotations should be implemented using the nodal mode of user subroutine MPC. The single degree of freedom version of user subroutine MPC can be used for geometrically linear problems, geometrically nonlinear problems with planar rotations, and constraints that do not involve rotation components.
+
+<!-- source-page: 84 -->
+
+The degrees of freedom involved in user MPCs must appear on some element or Abaqus/Standard MPC type in the model: user MPCs cannot use degrees of freedom that have not been introduced somewhere on an element. For example, a mesh that uses only continuum (solid) elements cannot have user MPCs that involve rotational degrees of freedom. The simplest way to overcome this limitation is to introduce an element somewhere in the model that uses the required degrees of freedom but does not affect the solution in any other way. Alternatively, if the degrees of freedom are rotations, they can be activated by the use of a library BEAM-type MPC somewhere in the model.
+
+# Use with nodal coordinate systems
+
+When a local coordinate system (“Transformed coordinate systems,” Section 2.1.5 of the Abaqus Analysis User’s Guide) and a user MPC are both used at a node, the variables at the node are first transformed before the MPC is imposed. Therefore, user-supplied MPCs must be based on the transformed degrees of freedom. The local-to-global transformation matrices $\mathbf { T } ^ { I }$ for the individual nodes:
+
+$$
+\mathbf {u} _ {g l o b a l} ^ {I} = \mathbf {T} ^ {I} \cdot \mathbf {u} _ {l o c a l} ^ {I}
+$$
+
+are passed in for information.
+
+# Degree of freedom version of user subroutine MPC
+
+This version of user subroutine MPC allows for one individual degree of freedom to be constrained and, thus, eliminated at a time. The constraint can be quite general and nonlinear of the form:
+
+$$
+f (u ^ {1}, u ^ {2}, u ^ {3}, \dots , u ^ {N}, \text {   geometry,   temperature,   field   variables   }) = 0.
+$$
+
+The first degree of freedom in this function, $u ^ { 1 }$ , is the degree of freedom that will be eliminated to impose the constraint. $u ^ { 2 } , u ^ { 3 }$ , etc. are any other degrees of freedom that are involved in the constraint. Since $u ^ { 1 }$ will be eliminated to impose the constraint, it cannot be used in subsequent kinematic constraints (multi-point constraints, linear equation constraints, or boundary conditions). Therefore, the user MPCs are imposed in the order given in the input.
+
+You must provide, at all times, two items of information in user subroutine MPC:
+
+1. A list of degree of freedom identifiers at the nodes that are listed in the corresponding multi-point constraint definition. This list corresponds to $u ^ { 1 } , u ^ { 2 } , u ^ { 3 }$ , etc. in the constraint as given above.   
+2. An array of the derivatives
+
+$$
+A ^ {1} = \frac {\partial f}{\partial u ^ {1}}, \quad A ^ {2} = \frac {\partial f}{\partial u ^ {2}}, \quad A ^ {3} = \frac {\partial f}{\partial u ^ {3}}, \quad \dots
+$$
+
+<!-- source-page: 85 -->
+
+of the constraint function with respect to the degrees of freedom involved. This array is needed for the redistribution of loads from degree of freedom $u ^ { 1 }$ to the other degrees of freedom and for the elimination of $u ^ { 1 }$ from the system matrices.
+
+In addition, you can provide the value of the dependent degree of freedom $u ^ { 1 }$ as a function of the independent degrees of freedom $u ^ { 2 } , u ^ { 3 }$ etc. If this value is not provided, Abaqus/Standard will update $u ^ { 1 }$ based on the linearized form of the constraint equation as
+
+$$
+u ^ {1} = - \frac {1}{A ^ {1}} \sum_ {i = 2} ^ {N} A ^ {i} u ^ {i}.
+$$
+
+Subroutine MPC should be coded and checked with care: if the array of derivatives $\partial f / \partial \boldsymbol { u } ^ { 1 }$ , etc. does not correspond to the definition of $\cdot _ { u ^ { 1 } }$ in terms of $u ^ { 2 } , u ^ { 3 }$ , etc., forces will be transmitted improperly by the MPC and violations of equilibrium may occur. In addition, convergence of the solution may be adversely affected.
+
+User subroutine interface   ```fortran
+SUBROUTINE MPC(UE,A,JDOF,MDOF,N,JTYPE,X,U,UINIT,MAXDOF, * LMPC,KSTEP,KINC,TIME,NT,NF,TEMP,FIELD,LTRAN,TRAN)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION A(N),JDOF(N),X(6,N),U(MAXDOF,N),UINIT(MAXDOF,N), * TIME(2),TEMP(NT,N),FIELD(NF,NT,N),LTRAN(N),TRAN(3,3,N)
+user coding to define UE, A, JDOF, and, optionally, LMPC
+RETURN
+END 
+```  
+Variables to be defined   
+A(N)
+
+An array containing the derivatives of the constraint function,
+
+$$
+A (1) = \frac {\partial f}{\partial u ^ {1}}, \quad A (2) = \frac {\partial f}{\partial u ^ {2}}, \quad \dots
+$$
+
+The coding in the subroutine must define N entries in A, where N is defined below.
+
+<!-- source-page: 86 -->
+
+# JDOF(N)
+
+An array containing the degree of freedom identifiers at the nodes that are involved in the constraint. For example, if $u ^ { 1 }$ is the z-displacement at a node, give JDOF(1) = 3; if $u ^ { 2 }$ is the x-displacement at a node, give $\pmb { \sigma } \pmb { \mathrm { D } } \pmb { 0 } \pmb { \mathrm { F } } ( 2 ) = 1$ . The coding in the subroutine must define N entries in JDOF, where N is defined below.
+
+# Variables that can be updated
+
+# UE
+
+This variable is passed in as the total value of the eliminated degree of freedom, $u ^ { 1 }$ . This variable will either be zero or have the current value of $\mathit { \Pi } _ { u } ^ { 1 }$ based on the linearized constraint equation, depending at which stage of the iteration the user subroutine is called. If the constraint is linear and is used in a smalldisplacement analysis (nonlinear geometric effects are not considered) or in a perturbation analysis, this variable need not be defined: Abaqus/Standard will compute $u ^ { 1 }$ as
+
+$$
+u ^ {1} = - \frac {1}{A ^ {1}} \sum_ {i = 2} ^ {N} A ^ {i} u ^ {i}.
+$$
+
+If the constraint is nonlinear, this variable should be updated to the value of $u ^ { 1 }$ at the end of the increment to satisfy the constraint exactly. If the return value is the same as the incoming value, Abaqus/Standard will update the eliminated degree of freedom based on the linearized form of the constraint equation. In this case the constraint is not likely to be satisfied exactly.
+
+# LMPC
+
+Set this variable to zero to avoid the application of the multi-point constraint. The MPC will be applied if the variable is not changed. This variable must be set to zero every time the subroutine is called if the user MPC is to remain deactivated. This MPC variable is useful for switching the MPC on and off during an analysis. However, the option should be used with care: switching off an MPC may cause a sudden disturbance in equilibrium, which can lead to convergence problems. If this variable is used to switch on an MPC during an analysis, the variable UE should be defined; otherwise, the constraint may not be satisfied properly.
+
+# Variables passed in for information
+
+# MDOF
+
+Maximum number of active degrees of freedom per node involved in the MPC. For the degree of freedom mode of user subroutine MPC, MDOF= 1.
+
+# N
+
+Number of degrees of freedom that are involved in the constraint, defined as the number of nodes given in the corresponding multi-point constraint definition. If more than one degree of freedom at a node is
+
+<!-- source-page: 87 -->
+
+involved in a constraint, the node must be repeated as needed or, alternatively, the nodal mode should be used.
+
+# JTYPE
+
+Constraint identifier given for the corresponding multi-point constraint definition.
+
+# X(6,N)
+
+An array containing the original coordinates of the nodes involved in the constraint.
+
+# U(MAXDOF,N)
+
+An array containing the values of the degrees of freedom at the nodes involved in the constraint. These values will be either the values at the end of the previous iteration or the current values based on the linearized constraint equation, depending at which stage of the iteration the user subroutine is called.
+
+# UINIT(MAXDOF,N)
+
+An array containing the values at the beginning of the current iteration of the degrees of freedom at the nodes involved in the constraint. This information is useful for decision-making purposes when you do not want the outcome of a decision to change during the course of an iteration. For example, there are constraints in which the degree of freedom to be eliminated changes during the course of the analysis, but it is necessary to prevent the choice of the dependent degree of freedom from changing during the course of an iteration.
+
+# MAXDOF
+
+Maximum degree of freedom number at any node in the analysis. For example, for a coupled temperature-displacement analysis with continuum elements, MAXDOF will be equal to 11.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number within the step.
+
+# TIME(1)
+
+Current value of step time.
+
+# TIME(2)
+
+Current value of total time.
+
+# NT
+
+Number of positions through a section where temperature or field variable values are stored at a node. In a mesh containing only continuum elements, NT=1. For a mesh containing shell or beam elements, NT is the largest of the values specified for the number of temperature points in the shell or beam section definition (or 2 for temperatures specified together with gradients for shells or two-dimensional beams, 3 for temperatures specified together with gradients for three-dimensional beams).
+
+<!-- source-page: 88 -->
+
+# NF
+
+Number of different predefined field variables requested for any node (including field variables defined as initial conditions).
+
+# TEMP(NT,N)
+
+An array containing the temperatures at the nodes involved in the constraint. This array is not used for a heat transfer, coupled temperature-displacement, coupled thermal-electrical, or coupled thermalelectrical-structural analysis since the temperatures are degrees of freedom of the problem.
+
+# FIELD(NF,NT,N)
+
+An array containing all field variables at the nodes involved in the constraint.
+
+# LTRAN(N)
+
+An integer array indicating whether the nodes in the MPC are transformed. If LTRAN(I)=1, a transformation is applied to node I; if LTRAN(I)=0, no transformation is applied.
+
+# TRAN(3,3,N)
+
+An array containing the local-to-global transformation matrices for the nodes used in the MPC. If no transformation is present at node I, TRAN(\*,\*,I) is the identity matrix.
+
+# Example: Nonlinear single degree of freedom MPC
+
+An example of a nonlinear single degree of freedom MPC is a geometrically nonlinear two-dimensional slider involving nodes a, b, and c. The constraint forces node a to be on the straight line connecting nodes b and c (see Figure 1.1.14–1).
+
+![](images/page-088_8f657df88c4514ae7b09613833ac3a286e1de527e185fd3e16beb725bfc277be.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+x
+a
+b
+c
+</details>
+
+Figure 1.1.14–1 Nonlinear MPC example: two-dimensional slider.
+
+<!-- source-page: 89 -->
+
+The constraint equation can be written in the form
+
+$$
+f (u ^ {a}, v ^ {a}, u ^ {b}, v ^ {b}, u ^ {c}, v ^ {c}) = (x ^ {a} - x ^ {b}) (y ^ {c} - y ^ {b}) - (y ^ {a} - y ^ {b}) (x ^ {c} - x ^ {b}) = 0,
+$$
+
+where $( x ^ { a } , y ^ { a } ) , ( x ^ { b } , y ^ { b } )$ , and $( x ^ { c } , y ^ { c } )$ are the current locations of ${ \pmb a } ,$ b, and c. The derivatives are readily obtained as
+
+$$
+\begin{array}{l} \frac {\partial f}{u ^ {a}} = y ^ {c} - y ^ {b}, \frac {\partial f}{u ^ {b}} = y ^ {a} - y ^ {c}, \frac {\partial f}{u ^ {c}} = y ^ {b} - y ^ {a}, \\ \frac {\partial f}{v ^ {a}} = x ^ {b} - x ^ {c}, \frac {\partial f}{v ^ {b}} = x ^ {c} - x ^ {a}, \frac {\partial f}{v ^ {c}} = x ^ {a} - x ^ {b}. \\ \end{array}
+$$
+
+Depending on the orientation of the segment $( b , c )$ we choose either $u ^ { a }$ or $v ^ { a }$ as the degree of freedom to be eliminated. If $\left| x ^ { c } - x ^ { b } \right| \geq \left| y ^ { \bar { c } } - y ^ { b } \right|$ , we choose $v ^ { a }$ as the dependent degree of freedom. If $\left| x ^ { c } - x ^ { b } \right| < \left| y ^ { c } - y ^ { b } \right|$ , we choose $u ^ { a }$ as the dependent degree of freedom. Moreover, if points b and c are coincident, the constraint is not applied.
+
+To prevent the choice of either $u ^ { a }$ or $v ^ { a }$ as the dependent degree of freedom from changing during the course of an iteration, the orientation of the segment $( b , c )$ is tested based on the geometry at the beginning of the iteration. The dependent degree of freedom is allowed to change from increment to increment.
+
+Suppose the above multi-point constraint is defined as type 1, with nodes a, a, b, b, c, c. The user subroutine MPC could be coded as follows:
+```fortran
+SUBROUTINE MPC(UE,A,JDOF,MDOF,N,JTYPE,X,U,UINIT,MAXDOF, *LMPC,KSTEP,KINC,TIME,NT,NF,TEMP,FIELD,LTRAN,TRAN)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION A(N),JDOF(N),X(6,N),U(MAXDOF,N),UINIT(MAXDOF,N), *TIME(2),TEMP(NT,N),FIELD(NF,NT,N),LTRAN(N),TRAN(3,3,N)
+PARAMETER( PRECIS = 1.D-15 )
+C
+IF (JTYPE .EQ. 1) THEN
+    DYBC0 = X(2,5) + UINIT(2,5) - X(2,3) - UINIT(2,3)
+    DXBC0 = X(1,3) + UINIT(1,3) - X(1,5) - UINIT(1,5)
+    DYBC = X(2,5) + U(2,5) - X(2,3) - U(2,3)
+    DXBC = X(1,3) + U(1,3) - X(1,5) - U(1,5)
+    A(3) = X(2,1) + U(2,1) - X(2,5) - U(2,5)
+    A(4) = X(1,5) + U(1,5) - X(1,1) - U(1,1)
+    A(5) = X(2,3) + U(2,3) - X(2,1) - U(2,1)
+    A(6) = X(1,1) + U(1,1) - X(1,3) - U(1,3)
+    JDOF(3) = 1 
+```
+
+<!-- source-page: 90 -->
+
+```prolog
+JDOF(4) = 2
+JDOF(5) = 1
+JDOF(6) = 2
+IF (ABS(DYBC0).LE.PRECIS .AND. ABS(DXBC0).LE.PRECIS) THEN
+POINTS B AND C HAVE COLLAPSED. DO NOT APPLY CONSTRAINT.
+LMPC = 0
+ELSE IF (ABS(DXBC0).LT. ABS(DYBC0)) THEN
+MAKE U_A DEPENDENT DOF.
+JDOF(1) = 1
+JDOF(2) = 2
+A(1) = DYBC
+A(2) = DXBC
+UE = A(5)A(2)/A(1) + X(1,3) + U(1,3) - X(1,1)
+ELSE
+MAKE V_A DEPENDENT DOF.
+JDOF(1) = 2
+JDOF(2) = 1
+A(1) = DXBC
+A(2) = DYBC
+UE = -A(6)A(2)/A(1) + X(2,3) + U(2,3) - X(2,1)
+END IF
+END IF
+RETURN
+END 
+```
+
+# Nodal version of user subroutine MPC
+
+The nodal version of user subroutine MPC allows for multiple degrees of freedom of a node to be eliminated simultaneously. The set of constraints can be quite general and nonlinear, of the form
+
+$$
+f _ {i} \left(\boldsymbol {u} ^ {1}, \boldsymbol {u} ^ {2}, \boldsymbol {u} ^ {3}, \dots , \boldsymbol {u} ^ {N}, \text {geometry, temperature, field variables}\right) = 0 \quad i = 1, 2, \dots , \text {NDEP}.
+$$
+
+NDEP is the number of dependent degrees of freedom that are involved in the constraint and should have a value between 1 and MDOF, which is the number of active degrees of freedom per node in the analysis. N is the number of nodes involved in the constraint. The scalar constraint functions $f _ { i }$ can also
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+<!-- source-page: 91 -->
+
+be considered as a vector function $f ,$ and the first set of degrees of freedom $\mathbf { \delta } _ { u } \mathbf { 1 }$ in the vector function $f$ will be eliminated to impose the constraint. The sets $u ^ { 2 } , u ^ { 3 }$ , etc. are the independent degrees of freedom at nodes 2, 3, etc. involved in the constraint. The set $u ^ { 1 }$ must be composed of NDEP degrees of freedom at the first node of the MPC definition. For example, if the dependent degrees of freedom are the x-displacement, the z-displacement, and the y-rotation at the first node, $\pmb { u } ^ { 1 } = \big ( \bar { u } _ { x } ^ { 1 } , u _ { z } ^ { 1 } , \phi _ { u } ^ { 1 } \big ) . \ \pmb { u } ^ { 2 } , \ \pmb { u } ^ { 3 }$ etc. can be composed of any number of degrees of freedom, depending on which ones play a role in the constraint, and need not be of the same size; for example, $u ^ { 2 } \overset { \cdot } { = } ( u _ { y } ^ { 2 } )$ and ${ \pmb u } ^ { 3 } = ( u _ { x } ^ { 3 } , u _ { y } ^ { 3 } , \bar { u _ { z } ^ { 3 } } )$ .
+
+The dependent node can also reappear as an independent node in the MPC. However, since the dependent degrees of freedom of this node will be eliminated, they cannot be used as independent degrees of freedom in this MPC. For example, if the rotations at node a are constrained by the MPC, the displacements of node a can still be used as independent degrees of freedom in the MPC, but the rotations themselves cannot. Similarly, the degrees of freedom that will be eliminated to impose the constraint cannot be used in subsequent kinematic constraints (multi-point constraints, linear equation constraints, or boundary conditions). The MPCs are imposed in the order given in the input for this purpose.
+
+The nodal version of user subroutine MPC was designed with the application of nonlinear constraints involving large three-dimensional rotations in mind. Due to the incremental nature of the solution procedure in Abaqus/Standard, a linearized set of constraints
+
+$$
+\delta f _ {i} = \boldsymbol {A} _ {i} ^ {1} \cdot \delta \boldsymbol {u} ^ {1} + \boldsymbol {A} _ {i} ^ {2} \cdot \delta \boldsymbol {u} ^ {2} + \boldsymbol {A} _ {i} ^ {3} \cdot \delta \boldsymbol {u} ^ {3} + \dots = 0 \quad i = 1, 2, \dots , \text {NDEP},
+$$
+
+where $A _ { i } ^ { 1 } \ = \ A _ { i } ^ { 1 } ( u ^ { 1 } , u ^ { 2 } , . . . ) , A _ { i } ^ { 2 } \ = \ A _ { i } ^ { 2 } ( u ^ { 1 } , u ^ { 2 } , . . . )$ , etc. is applied during each iteration. This linearized set of constraints is used for the calculation of equilibrium. For finite rotations the linearized equation is given in terms of the linearized rotations $\delta \pmb { \theta } ^ { 1 } , \delta \pmb { \theta } ^ { 2 } , \delta \pmb { \theta } ^ { 3 } , . . .$ , yielding
+
+$$
+\delta f _ {i} = \boldsymbol {A} _ {i} ^ {1} \cdot \delta \boldsymbol {\theta} ^ {1} + \boldsymbol {A} _ {i} ^ {2} \cdot \delta \boldsymbol {\theta} ^ {2} + \boldsymbol {A} _ {i} ^ {3} \cdot \delta \boldsymbol {\theta} ^ {3} + \dots = 0 \quad i = 1, 2, \dots , \mathrm{NDEP}.
+$$
+
+Since the linearized rotation field, , is not the variation of the total rotation vector, $\phi$ (see “Rotation variables,” Section 1.3.1 of the Abaqus Theory Guide), you cannot obtain the linearized constraint equation by simply taking derivatives of the vector function, $f ,$ with respect to the rotational degrees of freedom involved. The formulation of the linearized constraint in $\delta \pmb { \theta }$ is equivalent to the formulation of a geometrically linear constraint in the deformed configuration and is generally easier to formulate than the constraint in terms of $\delta \phi$ . For an exact formulation of the constraint, the dependent components of the total rotation vector $\phi ^ { 1 }$ must be defined exactly (see “Rotation variables,” Section 1.3.1 of the Abaqus Theory Guide).
+
+You must provide, at all times, two items of information in subroutine MPC:
+
+1. A matrix of degree of freedom identifiers at the nodes that are listed in the corresponding multipoint constraint definition. The columns of this matrix correspond to $\boldsymbol { u } ^ { 1 } , \boldsymbol { u } ^ { 2 } , \boldsymbol { u } ^ { 3 }$ , etc. in the set of constraints as given above, where unused entries are padded with zeros. The number of nonzero entries in $u ^ { 1 }$ will implicitly determine the number of dependent degrees of freedom, NDEP.   
+2. The matrices representing the linearized constraint function with respect to the degrees of freedom involved. These matrices are needed for the redistribution of loads from degrees of freedom $u ^ { 1 }$ to
+
+<!-- source-page: 92 -->
+
+the other degrees of freedom and for the elimination of $u ^ { 1 }$ from the system matrices. For constraints that do not involve three-dimensional rotations and constraints with planar rotations, these matrices can be readily obtained from the derivatives of the total constraint function with respect to the degrees of freedom involved:
+
+$$
+A _ {i j} ^ {1} = \frac {\partial f _ {i}}{\partial u _ {j} ^ {1}}, \quad A _ {i j} ^ {2} = \frac {\partial f _ {i}}{\partial u _ {j} ^ {2}}, \quad A _ {i j} ^ {3} = \frac {\partial f _ {i}}{\partial u _ {j} ^ {3}}, \quad \dots
+$$
+
+For constraints that involve finite rotations, the matrices follow from the linearized form:
+
+$$
+A _ {i j} ^ {1} \delta \theta_ {j} ^ {1} + A _ {i j} ^ {2} \delta \theta_ {j} ^ {2} + A _ {i j} ^ {3} \delta \theta_ {j} ^ {3} + \ldots = 0.
+$$
+
+In addition, you can provide the values of the dependent degrees of freedom $u ^ { 1 }$ , as a function of the independent degrees of freedom $u ^ { 2 } , u ^ { 3 }$ etc. For finite rotations, $\phi ^ { 1 }$ must be specified as a function of $\phi ^ { 2 } , \phi ^ { 2 }$ , etc. If these values are not provided, Abaqus/Standard will update $\boldsymbol { u } ^ { 1 }$ based on the linearized form of the constraint equations. Subroutine MPC should be coded and checked with care: if the matrices of derivatives $A _ { i j } ^ { 1 }$ , etc. do not correspond to the definition of $u ^ { 1 }$ in terms of $u ^ { 2 } , u ^ { 3 }$ etc., forces will be transmitted improperly by the MPC and violations of equilibrium may occur. In addition, convergence of the solution may be adversely affected.
+
+User subroutine interface   ```txt
+SUBROUTINE MPC(UE,A,JDOF,MDOF,N,JTYPE,X,U,UINIT,MAXDOF, * LMPC,KSTEP,KINC,TIME,NT,NF,TEMP,FIELD,LTRAN,TRAN)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION UE(MDOF),A(MDOF,MDOF,N),JDOF(MDOF,N),X(6,N), * U(MAXDOF,N),UINIT(MAXDOF,N),TIME(2),TEMP(NT,N), * FIELD(NF,NT,N),LTRAN(N),TRAN(3,3,N)
+user coding to define JDOF, UE, A and, optionally, LMPC
+RETURN
+END 
+```
+
+<!-- source-page: 93 -->
+
+# JDOF(MDOF,N)
+
+Matrix of degrees of freedom identifiers at the nodes involved in the constraint. Before each call to MPC, Abaqus/Standard will initialize all of the entries of JDOF to zero. All active degrees of freedom for a given column (first index ranging from 1 to MDOF) must be defined starting at the top of the column with no zeros in between. A zero will mark the end of the list for that column. The number of nonzero entries in the first column will implicitly determine the number of dependent degrees of freedom (NDEP). For example, if the dependent degrees of freedom are the z-displacement, the xrotation, and the z-rotation at the first node, NDEP and
+
+$$
+\mathrm{JDOF} (1, 1) = 3, \quad \mathrm{JDOF} (2, 1) = 4, \quad \mathrm{JDOF} (3, 1) = 6.
+$$
+
+If the degrees of freedom at the third node involved in the MPC are the x-displacement and the yrotation, define
+
+$$
+\mathrm{JDOF} (1, 3) = 1, \quad \mathrm{JDOF} (2, 3) = 5.
+$$
+
+# A(MDOF,MDOF,N)
+
+Submatrices of coefficients of the linearized constraint function,
+
+$$
+\mathsf {A} (\mathsf {I}, \mathsf {J}, 1) = A _ {i j} ^ {1}, \quad \mathsf {A} (\mathsf {I}, \mathsf {J}, 2) = A _ {i j} ^ {2}, \quad \dots
+$$
+
+Before each call to user subroutine MPC, Abaqus/Standard will initialize all of the entries of A to zero; therefore, only nonzero entries need to be defined. If the coding in the subroutine defines NDEP nonzero entries in the column JDOF(J,1), it should define NDEP × NDEP entries in the submatrix A(I,J,1). Since this submatrix will be inverted to impose the MPC, it must be nonsingular. A maximum of NDEP × MDOF entries can be defined for the remaining submatrices A(I,J,K), K = 2, , N. The number of columns in each submatrix A(I,J,K) must correspond to the number of nonzero entries in the corresponding column of the matrix JDOF(J,K).
+
+# Variables that can be updated
+
+# UE(NDEP)
+
+This array is passed in as the total value of the eliminated degrees of freedom, $u ^ { 1 }$ . This array will either be zero or contain the current values of $u ^ { 1 }$ based on the linearized constraint equations, depending at which stage of the iteration the user subroutine is called. For small-displacement analysis or perturbation analysis this array need not be defined: Abaqus/Standard will compute $u ^ { 1 }$ as
+
+$$
+u _ {i} ^ {1} = - \sum_ {r = 1} ^ {\mathrm{NDEP}} A _ {i r} ^ {1} {} ^ {- 1} \sum_ {s = 2} ^ {\mathrm{N}} \sum_ {t = 1} ^ {\mathrm{MDOF}} A _ {r t} ^ {s} u _ {\mathrm{JDOF} (t, s)} ^ {s} \qquad i = 1, \dots , \mathrm{NDEP}.
+$$
+
+<!-- source-page: 94 -->
+
+For large-displacement analysis this array can be updated to the value of $\boldsymbol { u } ^ { 1 }$ at the end of the increment to satisfy the constraint exactly. If the return values are the same as the incoming values, Abaqus/Standard will update the eliminated degrees of freedom based on the linearized form of the constraint equations. In this case the constraint is not likely to be satisfied exactly.
+
+# LMPC
+
+Set this variable to zero to avoid the application of the multi-point constraint. If the variable is not changed, the MPC will be applied. This variable must be set to zero every time the subroutine is called if the user MPC is to remain deactivated. This MPC variable is useful for switching the MPC on and off during an analysis. This option should be used with care: switching off an MPC may cause a sudden disturbance in equilibrium, which can lead to convergence problems.
+
+# Variables passed in for information
+
+# MDOF
+
+Number of active degrees of freedom per node in the analysis. For example, for a coupled temperaturedisplacement analysis with two-dimensional continuum elements, the active degrees of freedom are 1, 2, and 11 and, hence, MDOF will be equal to 3.
+
+# N
+
+Number of nodes involved in the constraint. The value of N is defined as the number of nodes given in the corresponding multi-point constraint definition.
+
+# JTYPE
+
+Constraint identifier given for the corresponding multi-point constraint definition.
+
+# X(6,N)
+
+An array containing the original coordinates of the nodes involved in the constraint.
+
+# U(MAXDOF,N)
+
+An array containing the values of the degrees of freedom at the nodes involved in the constraint. These values will either be the values at the end of the previous iteration or the current values based on the linearized constraint equation, depending at which stage of the iteration the user subroutine is called.
+
+# UINIT(MAXDOF,N)
+
+An array containing the values at the beginning of the current iteration of the degrees of freedom at the nodes involved in the constraint. This information is useful for decision-making purposes when you do not want the outcome of a decision to change during the course of an iteration. For example, there are constraints in which the degrees of freedom to be eliminated change during the course of the analysis, but it is necessary to prevent the choice of the dependent degrees of freedom from changing during the course of an iteration.
+
+<!-- source-page: 95 -->
+
+# MAXDOF
+
+Maximum degree of freedom number at any node in the analysis. For example, for a coupled temperature-displacement analysis with continuum elements, MAXDOF is equal to 11.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number within the step.
+
+# TIME(1)
+
+Current value of step time.
+
+# TIME(2)
+
+Current value of total time.
+
+# NT
+
+Number of positions through a section where temperature or field variable values are stored at a node. In a mesh containing only continuum elements, NT=1. For a mesh containing shell or beam elements, NT is the largest of the values specified for the number of temperature points in the shell or beam section definition (or 2 for temperatures specified together with gradients for shells or two-dimensional beams, 3 for temperatures specified together with gradients for three-dimensional beams).
+
+# NF
+
+Number of different predefined field variables requested for any node (including field variables defined as initial conditions).
+
+# TEMP(NT,N)
+
+An array containing the temperatures at the nodes involved in the constraint. This array is not used for a heat transfer, coupled temperature-displacement, coupled thermal-electrical, or coupled thermalelectrical-structural analysis since the temperatures are degrees of freedom of the problem.
+
+# FIELD(NF,NT,N)
+
+An array containing all field variables at the nodes involved in the constraint.
+
+# LTRAN(N)
+
+An integer array indicating whether the nodes in the MPC are transformed. If LTRAN(I)=1, a transformation is applied to node I; if LTRAN(I)=0, no transformation is applied.
+
+# TRAN(3,3,N)
+
+An array containing the local-to-global transformation matrices for the nodes used in the MPC. If no transformation is present at node I, TRAN(\*,\*,I) is the identity matrix.
+
+<!-- source-page: 96 -->
+
+As an example of a nonlinear MPC, consider the insertion of a rigid beam in a large-displacement, planar (two-dimensional) problem. This MPC is the two-dimensional version of library BEAM-type MPC. It can be implemented as a set of three different single degree of freedom MPCs or as a single nodal MPC. Here, the second method will be worked out because it is simpler and requires less data input.
+
+Let a and b (see Figure 1.1.14–2) be the ends of the beam, with a the dependent end.
+
+![](images/page-096_655003410d54524a381141dc8fedadb1043f04dd566e308a804672d81f22bf25.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+x
+L
+a
+b
+φ_b^b + φ_0
+</details>
+
+Figure 1.1.14–2 Nonlinear MPC example: rigid beam.
+
+The rigid beam will then define both components of displacement and the rotation at a in terms of the displacements and rotation at end b according to the set of equations:
+
+$$
+f _ {1} (\boldsymbol {u} ^ {a}, \boldsymbol {u} ^ {b}) = x ^ {a} - x ^ {b} - L \cos (\phi_ {z} ^ {b} + \phi_ {0}) = 0,
+$$
+
+$$
+f _ {2} (\pmb {u} ^ {a}, \pmb {u} ^ {b}) = y ^ {a} - y ^ {b} - L \sin (\phi_ {z} ^ {b} + \phi_ {0}) = 0,
+$$
+
+$$
+f _ {3} (\pmb {u} ^ {a}, \pmb {u} ^ {b}) = \phi_ {z} ^ {a} - \phi_ {z} ^ {b} = 0,
+$$
+
+where $\pmb { u } ^ { a } = ( u _ { x } ^ { a } , u _ { y } ^ { a } , \phi _ { z } ^ { a } )$ and $\boldsymbol { u } ^ { b } = ( u _ { x } ^ { b } , u _ { y } ^ { b } , \phi _ { z } ^ { b } ) , ( x ^ { a } , y ^ { a } )$ and $( x ^ { b } , y ^ { b } )$ are the current locations of a and $b , \phi _ { z } ^ { a }$ and $\phi _ { z } ^ { b }$ are the rotations at a and b about the z-axis, L is the length of the link, and $\phi _ { 0 }$ is the original orientation of the link.
+
+In terms of the original positions $( X ^ { a } , Y ^ { a } )$ and $( X ^ { b } , Y ^ { b } )$ of a and $\begin{array} { r } { \pmb { b } , } \end{array}$
+
+$$
+L = \sqrt {L _ {X} ^ {2} + L _ {Y} ^ {2}}
+$$
+
+and
+
+$$
+\cos \phi_ {0} = L _ {X} / L,
+$$
+
+$$
+\sin \phi_ {0} = L _ {Y} / L,
+$$
+
+where $L _ { X } = X ^ { a } - X ^ { b }$ and $L _ { Y } = Y ^ { a } - Y ^ { b }$ . Thus, the constraint equations can be expressed as
+
+<!-- source-page: 97 -->
+
+$$
+f _ {1} (\boldsymbol {u} ^ {a}, \boldsymbol {u} ^ {b}) = u _ {x} ^ {a} - u _ {x} ^ {b} + L _ {X} - L _ {X} \cos \phi_ {z} ^ {b} + L _ {Y} \sin \phi_ {z} ^ {b} = 0,
+$$
+
+$$
+f _ {2} (\boldsymbol {u} ^ {a}, \boldsymbol {u} ^ {b}) = u _ {y} ^ {a} - u _ {y} ^ {b} + L _ {Y} - L _ {X} \sin \phi_ {z} ^ {b} - L _ {Y} \cos \phi_ {z} ^ {b} = 0,
+$$
+
+$$
+f _ {3} (\pmb {u} ^ {a}, \pmb {u} ^ {b}) = \phi_ {z} ^ {a} - \phi_ {z} ^ {b} = 0.
+$$
+
+In light of the above formulation, the nontrivial portions of the matrices A and JDOF are
+
+$$
+\mathsf {A} (1: 3, 1: 3, 1) = \left( \begin{array}{c c c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right), \quad \mathsf {A} (1: 3, 1: 3, 2) = \left( \begin{array}{c c c} - 1 & 0 & L _ {X} \sin \phi_ {z} ^ {b} + L _ {Y} \cos \phi_ {z} ^ {b} \\ 0 & - 1 & - L _ {X} \cos \phi_ {z} ^ {b} + L _ {Y} \sin \phi_ {z} ^ {b} \\ 0 & 0 & - 1 \end{array} \right)
+$$
+
+and
+
+$$
+\mathrm{JDOF} (1: 3, 1) = \mathrm{JDOF} (1: 3, 2) = \left( \begin{array}{c} 1 \\ 2 \\ 6 \end{array} \right).
+$$
+
+Since degree of freedom 6 ( ) appears in this constraint, there must be an element in the mesh that uses that degree of freedom—a B21 beam element, for example.
+
+If the above multi-point constraint is defined as type 1 with nodes a and b, the user subroutine MPC could be coded as follows:
+```txt
+SUBROUTINE MPC(UE,A,JDOF,MDOF,N,JTYPE,X,U,UINIT,MAXDOF,LMPC,
+* KSTEP,KINC,TIME,NT,NF,TEMP,FIELD,LTRAN,TRAN)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION UE(MDOF),A(MDOF,MDOF,N),JDOF(MDOF,N),X(6,N),
+* U(MAXDOF,N),UINIT(MAXDOF,N),TIME(2),TEMP(NT,N),
+* FIELD(NF,NT,N),LTRAN(N),TRAN(3,3,N)
+C
+IF (JTYPE .EQ. 1) THEN
+COSFIB = COS(U(6,2))
+SINFIB = SIN(U(6,2))
+ALX = X(1,1) - X(1,2)
+ALY = X(2,1) - X(2,2)
+C
+UE(1) = U(1,2) + ALX*(COSFIB-1.) - ALY*SINFIB
+UE(2) = U(2,2) + ALY*(COSFIB-1.) + ALX*SINFIB
+UE(3) = U(6,2)
+C
+A(1,1,1) = 1.
+A(2,2,1) = 1.
+A(3,3,1) = 1. 
+```
+
+<!-- source-page: 98 -->
+
+```prolog
+A(1,1,2) = -1.
+A(1,3,2) = ALX*SINFIB + ALY*COSFIB
+A(2,2,2) = -1.
+A(2,3,2) = -ALX*COSFIB + ALY*SINFIB
+A(3,3,2) = -1.
+C
+JDOF(1,1) = 1
+JDOF(2,1) = 2
+JDOF(3,1) = 6
+JDOF(1,2) = 1
+JDOF(2,2) = 2
+JDOF(3,2) = 6
+END IF
+C
+RETURN
+END 
+```
+
+# Example: Nonlinear MPC involving finite rotations
+
+As an example of a nonlinear MPC involving finite rotations, consider a two-dimensional constant velocity joint that might be part of a robotics application. Let a, b, c (see Figure 1.1.14–3) be the nodes making up the joint, with a the dependent node.
+
+![](images/page-098_512737e1c62e4f7251984bccbaea6a0be28c00c8003060f74b1a2d139e1a281a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+x
+φ^c
+c
+b
+a
+φ^b
+</details>
+
+Figure 1.1.14–3 Nonlinear MPC example: constant velocity joint.
+
+The joint is operated by prescribing an axial rotation $\phi ^ { \mathrm { c } } = \phi ^ { \mathrm { c } } \mathbf { e } _ { x }$ at c and an out-of-plane rotation $\phi ^ { b } = $ $\phi ^ { b } \mathbf { e } _ { z }$ at b. The compounding of these two prescribed rotation fields will determine the total rotation at a. We can formally write this constraint as follows:
+
+<!-- source-page: 99 -->
+
+$$
+\boldsymbol {f} (\phi^ {a}, \phi^ {b}, \phi^ {c}) = \phi^ {a} - \phi^ {b} \circ \phi^ {c} = 0,
+$$
+
+where denotes the rotation product. The formulation of the linearized constraint can be readily achieved from geometrically linear considerations in the deformed state.
+
+In geometrically linear problems compound rotations are obtained simply as the linear superposition of individual rotation vectors. Consider the geometry depicted in Figure 1.1.14–3 and assume that the infinitesimal rotations $\delta \pmb { \theta } ^ { \mathrm { c } } = \delta \theta ^ { \mathrm { c } } \mathbf { e } _ { x }$ and $\delta \pmb { \theta } ^ { b } = \delta \theta ^ { b } \mathbf { e }$ are applied at c and b, respectively. The rotation $\delta \pmb { \theta } ^ { a }$ at a will simply be the sum of the vector $\delta \pmb { \theta } ^ { b }$ to the vector $\delta \pmb { \theta } ^ { c }$ rotated by an angle $\phi ^ { b }$ about the z-axis. Thus, the linearized constraint can be written directly as
+
+$$
+\delta f _ {1} \left(\phi^ {a}, \phi^ {b}, \phi^ {c}\right) = \delta \theta_ {x} ^ {a} - \cos \left(\phi^ {b}\right) \delta \theta^ {c} = 0,
+$$
+
+$$
+\delta f _ {2} (\phi^ {a}, \phi^ {b}, \phi^ {c}) = \delta \theta_ {y} ^ {a} - \sin (\phi^ {b}) \delta \theta^ {c} = 0,
+$$
+
+$$
+\delta f _ {3} (\phi^ {a}, \phi^ {b}, \phi^ {c}) = \delta \theta_ {z} ^ {a} - \delta \theta^ {b} = 0.
+$$
+
+In light of this formulation, the nontrivial portions of the matrices JDOF and A are
+
+$$
+\mathrm{JDOF} (1: 3, 1) = \left( \begin{array}{c} 4 \\ 5 \\ 6 \end{array} \right), \quad \mathrm{JDOF} (1, 2) = 6, \quad \mathrm{JDOF} (1, 3) = 4,
+$$
+
+and
+
+$$
+\mathsf {A} (1: 3, 1: 3, 1) = \left( \begin{array}{c c c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right), \quad \mathsf {A} (1: 3, 1, 2) = \left( \begin{array}{c} 0 \\ 0 \\ - 1 \end{array} \right), \quad \mathsf {A} (1: 3, 1, 3) = \left( \begin{array}{c} - \cos (\phi^ {b}) \\ - \sin (\phi^ {b}) \\ 0 \end{array} \right).
+$$
+
+Since degrees of freedom 4 $( \phi _ { x } ) , 5 ( \phi _ { y } )$ , and $6 ( \phi _ { z } )$ appear in this constraint, there must be an element in the mesh that uses these degrees of freedom—a B31 beam element, for example. The MPC subroutine has been coded with just this information. In this case Abaqus/Standard updates the dependent rotation field, $\phi ^ { a }$ , based on the linearized constraint equations. Although the constraint is not satisfied exactly, good results are obtained as long as the rotation increments are kept small enough. A more rigorous derivation of the linearized constraint and the exact nonlinear recovery of the dependent degrees of freedom is presented in “Rotation variables,” Section 1.3.1 of the Abaqus Theory Guide.
+
+If the above multi-point constraint is defined as type 1 with nodes $a , b ,$ and $c ,$ user subroutine MPC could be coded as follows:
+
+SUBROUTINE MPC(UE,A,JDOF,MDOF,N,JTYPE,X,U,UINIT,MAXDOF,LMPC,
+* KSTEP,KINC,TIME,NT,NF,TEMP,FIELD,LTRAN,TRAN)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION UE(MDOF),A(MDOF,MDOF,N),JDOF(MDOF,N),X(6,N),
+* U(MAXDOF,N),UINIT(MAXDOF,N),TIME(2),TEMP(NT,N),
+
+<!-- source-page: 100 -->
+
+\* FIELD(NF,NT,N),LTRAN(N),TRAN(3,3,N)
+
+```txt
+IF (JTYPE .EQ. 1) THEN
+A(1,1,1) = 1.
+A(2,2,1) = 1.
+A(3,3,1) = 1.
+A(3,1,2) = -1.
+A(1,1,3) = -COS(U(6,2))
+A(2,1,3) = -SIN(U(6,2)) 
+```
+
+C
+
+```txt
+JDOF(1,1) = 4
+JDOF(2,1) = 5
+JDOF(3,1) = 6
+JDOF(1,2) = 6
+JDOF(1,3) = 4
+END IF 
+```
+
+```txt
+RETURN
+END 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_011.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_011.md
new file mode 100644
index 00000000..54a310f6
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_011.md
@@ -0,0 +1,352 @@
+<!-- source-page: 101 -->
+
+# 1.1.15 ORIENT: User subroutine to provide an orientation for defining local material directions or local directions for kinematic coupling constraints or local rigid body directions for inertia relief.
+
+Product: Abaqus/Standard
+
+# References
+
+• “Orientations,” Section 2.2.5 of the Abaqus Analysis User’s Guide   
+• \*ORIENTATION   
+• “Eigenvalue analysis of a piezoelectric transducer,” Section 7.1.1 of the Abaqus Example Problems Guide
+
+# Overview
+
+# User subroutine ORIENT:
+
+• will be called at the start of the analysis at each location (material point, special-purpose element, coupling node, or reference point for inertia relief) for which local directions are defined with a user-subroutine-defined orientation;   
+• is used to define the direction cosines of a local system of (material) directions with respect to the default basis directions (default basis directions are defined as the global directions for continuum elements and as the default surface directions for shell, membrane, and surface elements, as described in “Conventions,” Section 1.2.2 of the Abaqus Analysis User’s Guide);   
+• can be used to define the direction cosines orienting the layer of reinforcing material in membrane, shell, or surface elements (see “Defining reinforcement,” Section 2.2.3 of the Abaqus Analysis User’s Guide);   
+• can be used to provide a local system for defining the direction of action of rotary inertia, spring, dashpot, flexible joint, and elastic-plastic joint elements;   
+• can be used with gasket elements to define the local in-plane directions for three-dimensional area and three-dimensional link elements that consider transverse shear and membrane deformations (see “Defining the gasket behavior directly using a gasket behavior model,” Section 32.6.6 of the Abaqus Analysis User’s Guide);   
+• can be used to define a local system in which coupling constraints are applied (see “Coupling constraints,” Section 35.3.2 of the Abaqus Analysis User’s Guide, and “Kinematic coupling constraints,” Section 35.2.3 of the Abaqus Analysis User’s Guide);   
+• can be used to define a local system at the reference point for the rigid body directions in which inertia relief loads are applied for the entire model (see “Inertia relief,” Section 11.1.1 of the Abaqus Analysis User’s Guide);   
+• will ignore rotation angles defined for layers of composite solids (see “Solid (continuum) elements,” Section 28.1.1 of the Abaqus Analysis User’s Guide) but will take into account rotation angles
+
+<!-- source-page: 102 -->
+
+defined for layers of composite shells (see “Using a shell section integrated during the analysis to define the section behavior,” Section 29.6.5 of the Abaqus Analysis User’s Guide, and “Using a general shell section to define the section behavior,” Section 29.6.6 of the Abaqus Analysis User’s Guide); and
+
+• ignores any data specified for the associated orientation definition outside the user subroutine.
+
+The local directions defined by user subroutine ORIENT must be specified relative to the default basis directions.
+
+User subroutine interface   
+```txt
+SUBROUTINE ORIENT(T, NOEL, NPT, LAYER, KSPT, COORDS, BASIS, 1 ORNAME, NNODES, CNODES, JNNUM)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 ORNAME
+C
+DIMENSION T(3, 3), COORDS(3), BASIS(3, 3), CNODES(3, NNODES)
+DIMENSION JNNUM(NNODES)
+user coding to define T
+RETURN
+END 
+```  
+Variable to be defined
+
+T
+
+An array containing the direction cosines of the preferred orientation in terms of the default basis directions. T(1,1), T(2,1), T(3,1) give the (1, 2, 3) components of the first direction; T(1,2), T(2,2), T(3,2) give the second direction; etc. For shell and membrane elements only the first and second directions are used. The directions do not have to be normalized. If the second direction is not orthogonal to the first direction, Abaqus/Standard will orthogonalize and normalize the second direction with respect to the first. The third direction is then determined by taking the cross product of the first and second directions. For planar elements the first two directions must lie in the plane of the element.
+
+For use with coupling constraints (“Coupling constraints,” Section 35.3.2 of the Abaqus Analysis User’s Guide), the local basis directions are used as the local constraint directions for application of the kinematic constraint.
+
+For use with inertia relief loads, the local basis directions are used as the rigid body direction vectors for computing the loads.
+
+<!-- source-page: 103 -->
+
+# NOEL
+
+Element number. This value is zero when the subroutine is called for use with coupling constraints or inertia relief loads.
+
+# NPT
+
+Integration point number. This variable is set only for relevant uses.
+
+# LAYER
+
+Layer number (for composite shells and layered solids). This variable is set only when relevant. It is equal to zero when it is irrelevant, such as in a regular solid element or in a shell element when transverse shear stiffness calculations are performed.
+
+# KSPT
+
+Section point number within the current layer. This variable is set only when relevant. It is equal to zero when it is irrelevant, such as in a regular solid element or in a shell element when transverse shear stiffness calculations are performed.
+
+# COORDS
+
+An array containing the initial coordinates of this point. This array contains the coordinates of the reference point for inertia relief loads.
+
+# BASIS
+
+An array containing the direction cosines of the normal material basis directions in terms of the global coordinates in the original configuration. BASIS(1,1), BASIS(2,1), BASIS(3,1) give the 1- direction, etc. This is useful only in shells or membranes since in all other cases the basis is the global coordinate system.
+
+# ORNAME
+
+User-specified orientation name, left justified, with one exception. When an overall section orientation is specified for a composite solid or shell section and the individual layer orientations are specified by an orientation angle, Abaqus defines an internal orientation name to represent the actual orientation of the layer. To avoid internal names, provide an orientation name rather than an orientation angle as part of the layer definition for each individual layer of a composite section.
+
+# NNODES
+
+Number of element nodes. This value is two when the subroutine is called for use with a kinematic coupling definition, where the two nodes are the reference and current coupling node. When used with a distributing coupling definition, this number is equal to the number of coupling nodes plus one for the reference node. It is one when used with inertia relief loads since the local basis is defined at the reference point.
+
+<!-- source-page: 104 -->
+
+# CNODES
+
+An array containing the original coordinates of the nodes. When used with a kinematic coupling definition, the first entry defines the reference node coordinates, and the second entry defines the coupling node coordinates. When used with a distributing coupling definition, the first entry defines the reference node coordinates, and the subsequent entries define the coupling node coordinates in the order defined by the JNNUM array. When used with inertia relief loads, this array is not used. For all other uses the entry order follows that of the element definition node ordering.
+
+# JNNUM
+
+An array containing the NNODES node numbers. When used with a kinematic coupling definition, the first entry is the reference node number, and the second entry is the node number for the current coupling node. When used with a distributing coupling definition, the first entry is the reference node number followed by the node numbers of all coupling nodes. When used with inertia relief loads, this array is not used. For all other uses the entry order follows that of the element definition node ordering.
+
+<!-- source-page: 105 -->
+
+# 1.1.16 RSURFU: User subroutine to define a rigid surface.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Analytical rigid surface definition,” Section 2.3.4 of the Abaqus Analysis User’s Guide   
+• \*SURFACE   
+• \*RIGID BODY   
+• “RSURFU,” Section 4.1.10 of the Abaqus Verification Guide
+
+# Overview
+
+# User subroutine RSURFU:
+
+• is used to define the surface of a rigid body for use in contact pairs;   
+• can be used to define a complex rigid surface if the various capabilities provided for defining a surface in Abaqus (see “Analytical rigid surface definition,” Section 2.3.4 of the Abaqus Analysis User’s Guide) are too restrictive;   
+• will be called at points on the slave surface of a contact pair or, if contact elements are used, at each integration point of each contact element with which the rigid surface is associated; and   
+• requires the definition of the closest point on the rigid surface, the normal and tangent directions, and the surface curvature.
+
+# Overpenetration constraint
+
+This routine must determine if a point on the slave surface has penetrated the rigid surface and define the local surface geometry. If the deforming and rigid surfaces are in contact at this point, Abaqus/Standard will impose a constraint at the point to prevent overpenetration. The local surface geometry must be defined to provide the necessary orientation for the constraint equations and friction directions and to allow Abaqus/Standard to compute the rate of change of these equations as the point moves around on the surface—the “tangent stiffness matrix” for the surface in the Newton algorithm. For the purpose of these calculations, it is best to define a smooth surface. If the surface is defined in a discontinuous manner, convergence may be adversely affected.
+
+# Calculations to be performed
+
+Each time RSURFU is called, Abaqus/Standard gives the current position of point A on the surface of the deforming structure, $\mathbf { x } _ { A }$ ; the current position of the rigid body reference point, $\mathbf { x } _ { C } \mathrm { ; }$ ; the total displacements of both of these points, $\mathbf { u } _ { A }$ and $\mathbf { u } _ { C }$ ; and the total rotation of the rigid body reference point, $\phi _ { C }$ .
+
+The routine should perform the following calculations:
+
+<!-- source-page: 106 -->
+
+1. A point, $A ^ { \prime } ,$ must be found on the rigid surface at which the normal to the surface passes through $\mathbf { x } _ { A }$ . If there is not a unique point $A ^ { \prime } ,$ the routine must choose the most suitable point (usually the closest A′ to A). The routine must pass back the coordinates of $A ^ { \prime }$ to Abaqus/Standard. For the surface-to-surface contact formulation, the slave normal, not the master normal, should be used.   
+2. RSURFU must define the distance, h, by which A has penetrated the surface below A′. A negative value of h means that A is outside the surface of the rigid body.   
+3. If the surfaces are in contact, which may sometimes be the case even if h is negative, RSURFU must define the local surface geometry.
+
+# Defining the local surface geometry
+
+There are two scenarios under which it is mandatory that the routine define the local surface geometry: if A has penetrated the surface— if the surface behavior is truly rigid, or h is greater than the maximum overclosure value specified for modified surface behavior using either contact controls (see “Adjusting contact controls in Abaqus/Standard,” Section 36.3.6 of the Abaqus Analysis User’s Guide) or a modified pressure-overclosure relationship (see “Contact pressure-overclosure relationships,” Section 37.1.2 of the Abaqus Analysis User’s Guide)—and if A was in contact at the beginning of the increment, in which case the flag LCLOSE=1 (see the variable list for the definition of LCLOSE). The variable LCLOSE is not relevant for the surface-to-surface contact formulation and is always passed in as 0. The routine can be coded so that local surface geometry definitions are always provided regardless of the scenario.
+
+The local surface geometry is specified by two orthogonal tangents to the rigid surface at $A ^ { \prime } ,$ as well as the rates of change of the outward pointing normal at $A ^ { \prime } ,$ , , with respect to local surface coordinates that are distance measuring along the tangents, $S ^ { 1 }$ and $S ^ { 2 }$ (see Figure 1.1.16–1).
+
+![](images/page-106_32d8871bb49521cac077e0a411d498a3f2dd8f21b9fae691fccd87ad51d6a9b7.jpg)
+
+<details>
+<summary>text_image</summary>
+
+n
+A
+t²
+S²
+A′
+S¹
+t¹
+</details>
+
+Figure 1.1.16–1 Local geometry on a rigid surface.
+
+The tangents to the surface at $A ^ { \prime }$ must be defined so that their positive, right-handed cross product is the outward normal to the surface. For two-dimensional cases Abaqus/Standard assumes that the second tangent is (0, 0, −1), so that when you give the direction cosines of the first tangent as $( t _ { 1 } , t _ { 2 } , 0 )$ , the
+
+<!-- source-page: 107 -->
+
+outward normal will be $( - t _ { 2 } , t _ { 1 } , 0 )$ . The rates of change of the normal with respect to $S ^ { 1 }$ and $S ^ { 2 }$ are required to define the local curvature of the surface.
+
+User subroutine interface   
+```txt
+SUBROUTINE RSURFU(H, P, TGT, DNDS, X, TIME, U, CINAME, SLNAME, 1 MSNAME, NOEL, NODE, LCLOSE)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CINAME, SLNAME, MSNAME
+C
+DIMENSION P(3), TGT(3, 2), DNDS(3, 2), X(3, 3), TIME(2), U(6, 2)
+user coding to define H, P, TGT, and DNDS
+RETURN
+END 
+```  
+Variables to be defined
+
+H
+
+Penetration of the point A on the deforming structure into the surface of the rigid body, measured down the outward normal to the rigid surface. A negative value of H indicates that A is outside the rigid surface. Even for a completely rigid surface, $A$ may appear to penetrate the surface during the iterations because the kinematic constraints are not fully satisfied until an increment converges.
+
+P(3)
+
+Position of the point $A ^ { \prime }$ on the surface of the rigid body closest to point A on the surface of the deforming structure.
+
+TGT(3,2)
+
+Direction cosines of the two unit tangents to the surface, $\mathbf { t } ^ { 1 }$ and $\mathbf { t } ^ { 2 } ,$ , at point A′. For two-dimensional cases only the first two components of $\mathbf { t } ^ { 1 }$ need be given since in this case Abaqus/Standard assumes that $\mathbf { t } ^ { 2 }$ is (0, 0, −1).
+
+DNDS(3,2)
+
+Rates of change of the surface normal, , at $A ^ { \prime } ,$ with respect to distance measuring coordinates, $S ^ { 1 }$ and $S ^ { 2 }$ , along $\mathbf { t } ^ { 1 }$ and $\mathbf { t } ^ { 2 }$ . For two-dimensional cases only the first two entries in the first column of DNDS $( \partial n _ { 1 } / \partial S ^ { 1 } , \partial n _ { 2 } / \partial S ^ { 1 } )$ are required. The array DNDS is not required to be assigned for the surface-tosurface contact formulation.
+
+<!-- source-page: 108 -->
+
+# Variables passed in for information
+
+X(K1,1)
+
+Current coordinates of point A on the surface of the deforming structure.
+
+X(K1,2)
+
+Current coordinates of the rigid body reference point.
+
+X(K1,3)
+
+Unit normal vector for point A; relevant only for the surface-to-surface contact formulation.
+
+TIME(1)
+
+Value of step time at the end of the increment.
+
+TIME(2)
+
+Value of total time at the end of the increment.
+
+U(K1,1)
+
+Total displacement of point A on the surface of the deforming structure.
+
+U(K1,2)
+
+Total displacement and rotation of the rigid body reference point; $k _ { 1 } = 1 , 2 , 3$ are the displacement components, $k _ { 1 } = 4 , 5 , 6$ are the rotation components. For two-dimensional cases the only nonzero rotation component is $k _ { 1 } = 6 \mathrm { : }$ : U(4,2) and U(5,2) are both zero.
+
+# CINAME
+
+User-specified surface interaction name, left justified. For user-defined contact elements it is either the element set name given for the interface definition or the optional name assigned to the interface definition.
+
+# SLNAME
+
+Slave surface name. Passed in as blank if RSURFU is called for contact elements.
+
+# MSNAME
+
+Master surface name. Passed in as blank if RSURFU is called for contact elements.
+
+NOEL
+
+Element label for contact elements. Passed in as zero if RSURFU is called for a contact pair.
+
+NODE
+
+Node number for point A. For the surface-to-surface contact formulation, this quantity is passed in as 0.
+
+LCLOSE
+
+Flag indicating contact status at the beginning of the increment. LCLOSE=1 indicates that A is in contact (closed) at the beginning of the increment. LCLOSE=0 indicates that A is not in contact (open)
+
+<!-- source-page: 109 -->
+
+at the beginning of the increment. If LCLOSE=1, P, TGT and DNDS must be defined even if A opens during this increment. LCLOSE is not used for the surface-to-surface contact formulation and is passed in as 0.
+
+# Example: Rigid punch
+
+The input files for the following examples can be found in “RSURFU,” Section 4.1.10 of the Abaqus Verification Guide. The following discussion pertains only to the node-to-surface contact formulation.
+
+Consider the punch shown in Figure 1.1.16–2.
+
+![](images/page-109_086fc60d83bb4dd3ca546f95e72b72562e5de88c25b26de1643aa118f2966d60.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z
+x₁
+A′
+t¹
+α
+Q
+α
+β
+b
+a
+x₁
+A′
+t¹
+r
+</details>
+
+Figure 1.1.16–2 Cross-section of a rigid punch.
+
+It consists of a spherical head of radius a, smoothly merging into a conical section with cone angle . The center of the sphere lies on the z-axis at Q. We assume that the punch is being driven down the z-axis by a prescribed displacement at the rigid body reference node defined as a boundary condition. (This same surface could be defined directly as a three-dimensional surface of revolution, as described in “Analytical rigid surface definition,” Section 2.3.4 of the Abaqus Analysis User’s Guide. We define it here in RSURFU as an illustration.)
+
+A point (slave node) on the surface of the deforming body will be associated with the spherical head or with the conical part of the punch, depending on whether it lies above or below the cone that passes through $Q$ and the circle of intersection of the sphere and cone. Thus, define
+
+$$
+r = \sqrt {x _ {1} ^ {2} + x _ {2} ^ {2}}, \quad z = x _ {3}
+$$
+
+<!-- source-page: 110 -->
+
+in the three-dimensional case or
+
+$$
+r = x _ {1}, \quad z = x _ {2}
+$$
+
+in the axisymmetric case. Then, if $\alpha < z _ { Q } - z _  $ , the point is associated with the spherical surface. Otherwise, it is associated with the cone (both cases are indicated in Figure 1.1.16–2).
+
+Consider first the axisymmetric case. Then, for $\alpha < z _ { Q } - z$ (the sphere) the overclosure is
+
+$$
+h = a - b,
+$$
+
+where
+
+$$
+b = \sqrt {r ^ {2} + (z - z _ {Q}) ^ {2}}.
+$$
+
+The position of the point A′ on the rigid surface is ( $\beta , z _ { Q } - a$ , 0), where
+
+$$
+\cos \beta = r / b, \quad \mathrm{and} \quad \sin \beta = (z _ {Q} - z) / b.
+$$
+
+The tangent to the rigid surface at A′ is $\mathbf { t } ^ { 1 } = \left( - \sin { \beta } , - \cos { \beta } , 0 \right)$ The positive direction for $\mathbf { t } ^ { 1 }$ must be chosen so that the normal satisfies the right-hand rule with respect to $\mathbf { t } ^ { 1 }$ and $\mathbf { t } ^ { 2 }$ and points out of the rigid body. Also, $d S ^ { 1 } = a d \beta$ , so that
+
+$$
+\frac {\partial \mathbf {n}}{\partial S ^ {1}} = (- \frac {1}{a} \sin \beta , - \frac {1}{a} \cos \beta , 0).
+$$
+
+For $\alpha > z _ { Q } - z$ (the conical surface) the clearance is
+
+$$
+h = - r \cos \alpha + (z - z _ {Q}) \sin \alpha + a,
+$$
+
+and the position of the point A′ on the rigid surface is $( r + h \cos \alpha , z - h \sin \alpha )$ The surface tangent is $\mathbf { t } ^ { 1 } = \left( - \sin \alpha , - \cos \alpha , 0 \right)$ and there is no change in with position, so that
+
+$$
+\frac {\partial \mathbf {n}}{\partial S ^ {1}} = (0, 0, 0).
+$$
+
+The routine can then be coded as follows:
+```csv
+SUBROUTINE RSURFU(H,P,TGT,DNDS,X,TIME,U,CINAME,SLNAME,1 MSNAME,NOEL,NODE,LCLOSE)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CINAME,SLNAME,MSNAME
+DIMENSION P(3),TGT(3,2),DNDS(3,2),X(3,2),TIME(2),U(6,2)
+C
+C DEFINE THE FOLLOWING QUANTITIES: 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_012.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_012.md
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+<!-- source-page: 111 -->
+
+```prolog
+C A = RADIUS 'A' OF THE SPHERICAL HEAD
+C SINA = SINE (CONE ANGLE ALPHA)
+C COSA = COSINE (CONE ANGLE ALPHA)
+C Z0 = ORIGINAL 'Z' COORDINATE OF POINT 'Q'
+C
+A=5.0
+SINA=0.5
+COSA=0.86602
+Z0=6.0
+ZQ=Z0 + U(2,2)
+C
+C TEST FOR SEGMENT
+C
+IF(X(1,1)*SINA/COSA.LT.ZQ-X(2,1)) THEN
+C
+C SPHERE
+C
+B=SQRT(X(1,1)**2 + (X(2,1)-ZQ)**2)
+H=A-B
+COSB=X(1,1)/B
+SINB=(ZQ-X(2,1))/B
+P(1)=A*COSB
+P(2)=ZQ-A*SINB
+TGT(1,1)=-SINB
+TGT(2,1)=-COSB
+DNDS(1,1)=-SINB/A
+DNDS(2,1)=-COSB/A
+ELSE
+C CONE
+H=-X(1,1)*COSA+(X(2,1)-ZQ)*SINA+A
+P(1)=X(1,1) + H*COSA
+P(2)=X(2,1) - H*SINA
+TGT(1,1)=-SINA
+TGT(2,1)=-COSA
+DNDS(1,1)=0.
+DNDS(2,1)=0.
+END IF
+RETURN
+END 
+```
+
+The above case can be directly extended to three dimensions. For this purpose we assume that the radial axis, r, is in the global (x–y) plane, so that
+
+<!-- source-page: 112 -->
+
+$$
+r = \sqrt {x _ {1} ^ {2} + x _ {2} ^ {2}}, \quad z = x _ {3}.
+$$
+
+For $\alpha < z _ { Q } - z$ (the sphere), the overclosure is $h = a - b$ , where again
+
+$$
+b = \sqrt {r ^ {2} + (z - z _ {Q}) ^ {2}}.
+$$
+
+The point $A ^ { \prime }$ on the rigid surface is ( , $\gamma , z _ { Q } - a \sin \beta )$ , where
+
+$$
+\cos \gamma = \frac {x _ {1}}{r}, \sin \gamma = \frac {x _ {2}}{r}.
+$$
+
+For $r = 0 , \gamma$ is not defined uniquely; in that case we arbitrarily choose $\gamma = 0$ . We now need two tangents to the surface. The tangent $\mathbf { t } ^ { 1 }$ used in the axisymmetric case is now
+
+$$
+\mathbf {t} ^ {1} = \left(- \sin \beta \cos \gamma , - \sin \beta \sin \gamma , - \cos \beta\right)
+$$
+
+and the orthogonal tangent is
+
+$$
+\mathbf {t} ^ {2} = (- \sin \gamma , \cos \gamma , 0).
+$$
+
+Again, the positive directions of $\mathbf { t } ^ { 1 }$ and $\mathbf { t } ^ { 2 }$ are chosen so that $\mathbf { t } ^ { 1 } \times \mathbf { t } ^ { 2 }$ defines an outward normal to the surface. The distance measures on the surface are
+
+$$
+d S ^ {1} = a d \beta , \quad d S ^ {2} = a \cos \beta d \gamma
+$$
+
+so that
+
+$$
+\frac {\partial \mathbf {n}}{\partial S ^ {1}} = \bigl (- \frac {1}{a} \sin \beta \cos \gamma , - \frac {1}{a} \sin \beta \sin \gamma , - \frac {1}{a} \cos \beta \bigr),
+$$
+
+$$
+\frac {\partial \mathbf {n}}{\partial S ^ {2}} = \bigl (- \frac {1}{a} \sin \gamma , \frac {1}{a} \cos \gamma , 0 \bigr).
+$$
+
+For the conical surface ( $\alpha \geq z _ { Q } - z \ )$ , the surface separation is
+
+$$
+h = - r \cos \alpha + (z - z _ {Q}) \sin \alpha + a.
+$$
+
+The point $A ^ { \prime }$ on the rigid surface is $\left( \left( r + h \cos \alpha \right) \cos \gamma , \left( r + h \cos \alpha \right) \sin \gamma , z - h \sin \alpha \right)$ and the surface tangents are
+
+$$
+\mathbf {t} ^ {1} = \left(- \sin \alpha \cos \gamma , - \sin \alpha \sin \gamma , - \cos \alpha\right)
+$$
+
+$$
+\mathbf {t} ^ {2} = (- \sin \gamma , \cos \gamma , 0).
+$$
+
+There is no change of with respect to $S ^ { 1 }$ , and, in this case $d S ^ { 2 } = c d \gamma$ , where $c = r + h$ so that
+
+<!-- source-page: 113 -->
+
+$$
+\frac {\partial \mathbf {n}}{\partial S ^ {2}} = \left(- \frac {1}{c} \cos \alpha \sin \gamma , + \frac {1}{c} \cos \alpha \cos \gamma , 0\right).
+$$
+
+The routine can then be coded as follows:
+```fortran
+SUBROUTINE RSURFU(H,P,TGT,DNDS,X,TIME,U,CINAME,SLNAME,
+1 MSNAME,NOEL,NODE,LCLOSE)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CINAME,SLNAME,MSNAME
+DIMENSION P(3), TGT(3,2),DNDS(3,2), X(3,2), TIME(2), U(6,2)
+C
+C DEFINE THE FOLLOWING QUANTITIES:
+C A = RADIUS 'A' OF THE SPHERICAL HEAD
+C SINA = SINE (CONE ANGLE ALPHA)
+C COSA = COSINE (CONE ANGLE ALPHA)
+C Z0 = ORIGINAL 'Z' COORDINATE OF POINT 'Q'
+C
+A=5.0
+SINA=0.5
+COSA=0.86603
+Z0=5.0
+ZQ= Z0 + U(3,2)
+C
+C TEST FOR SEGMENT
+C
+R = SQRT(X(1,1)*X(1,1)+X(2,1)*X(2,1))
+IF(R .GT. 0.0) THEN
+COSG = X(1,1)/R
+SING = X(2,1)/R
+ELSE
+COSG = 1.0
+SING = 0.0
+END IF
+IF(R*SINA/COSA .LT. ZQ -X(3,1)) THEN
+C
+C SPHERE
+C
+B=SQRT(R*R+(X(3,1)-ZQ)**2)
+H=A-B
+COSB=R/B
+SINB=(ZQ-X(3,1))/B 
+```
+
+<!-- source-page: 114 -->
+
+```prolog
+P(1)=A*COSB*COSG
+P(2)=A*COSB*SING
+P(3)=ZQ-A*SINB
+TGT(1,1)=-SINB*COSG
+TGT(2,1)=-SINB*SING
+TGT(3,1)=-COSB
+TGT(1,2)=-SING
+TGT(2,2)=COSG
+TGT(3,2)=0.0
+DNDS(1,1)=-SINB*COSG/A
+DNDS(2,1)=-SINB*SING/A
+DNDS(3,1)=-COSB/A
+DNDS(1,2)=-SING/A
+DNDS(2,2)=COSG/A
+DNDS(3,2)=0.0
+ELSE
+C
+C CONE
+C
+H=-R*COSA+(X(3,1)-ZQ)*SINA+A
+P(1)=(R+H*COSA)*COSG
+P(2)=(R+H*COSA)*SING
+P(3)=X(3,1)-H*SINA
+TGT(1,1)=-SINA*COSG
+TGT(2,1)=-SINA*SING
+TGT(3,1)=-COSA
+TGT(1,2)=-SING
+TGT(2,2)=COSG
+TGT(3,2)=0.0
+DNDS(1,1)=0.0
+DNDS(2,1)=0.0
+DNDS(3,1)=0.0
+C=R+H*COSA
+DNDS(1,2)=-COSA*SING/C
+DNDS(2,2)=COSA*COSG/C
+DNDS(3,2)=0.0
+END IF
+C
+RETURN
+END 
+```
+
+<!-- source-page: 115 -->
+
+# 1.1.17 SDVINI: User subroutine to define initial solution-dependent state variable fields.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide   
+• \*INITIAL CONDITIONS   
+• “SDVINI,” Section 4.1.11 of the Abaqus Verification Guide
+
+# Overview
+
+# User subroutine SDVINI:
+
+• will be called for user-subroutine-defined initial solution-dependent state variable fields at particular material points, shell section points, contact slave nodes, or for user elements (see “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1 of the Abaqus Analysis User’s Guide);   
+• can be used to initialize solution-dependent state variables allocated as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide; and   
+• returns a value of zero for any solution-dependent state variables that have no defined initial condition.
+
+# Use of solution-dependent state variables in other user subroutines
+
+Solution-dependent state variables initialized in SDVINI can be used and updated in the following user subroutines:
+
+• CREEP   
+• FRIC   
+• HETVAL   
+• UEL   
+• UEXPAN   
+• UGENS   
+• UHARD   
+• UMAT   
+• UMATHT   
+• USDFLD   
+• UTRS
+
+The solution-dependent state variables are passed into these routines in the order in which they are entered in SDVINI.
+
+<!-- source-page: 116 -->
+
+User subroutine interface   
+```prolog
+SUBROUTINE SDVINI (STATEV, COORDS, NSTATV, NCRDS, NOEL, NPT, 1 LAYER, KSPT)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION STATEV (NSTATV), COORDS (NCRDS)
+user coding to define STATEV (NSTATV)
+RETURN
+END 
+```
+
+Variables to be defined   
+```txt
+STATEV (1)
+First solution-dependent state variable.
+
+STATEV (2)
+Second solution-dependent state variable.
+
+STATEV (3)
+Third solution-dependent state variable.
+
+Etc.
+Only NSTATV solution-dependent state variable values should be defined. 
+```
+
+Variables passed in for information   
+```txt
+COORDS
+An array containing the initial coordinates of this point. Coordinates are not available for user elements.
+
+NSTATV
+User-defined number of solution-dependent state variables (see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+NCRDS
+Number of coordinates. This value is zero for user elements.
+
+NOEL
+Element number. 
+```
+
+<!-- source-page: 117 -->
+
+# NPT
+
+Integration point number in the element (not relevant for user elements).
+
+# LAYER
+
+Layer number (for composite shells and layered solids).
+
+# KSPT
+
+Section point number within the current layer or section. Section point 1 is used for all pure heat transfer, coupled temperature-displacement, and coupled thermal-electrical-structural analyses.
+
+<!-- source-page: 118 -->
+
+<!-- source-page: 119 -->
+
+# 1.1.18 SIGINI: User subroutine to define an initial stress field.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1 of the Abaqus Analysis User’s Guide   
+• \*INITIAL CONDITIONS
+
+# Overview
+
+User subroutine SIGINI:
+
+• will be called for user-subroutine-defined initial stress fields at particular material points (these are the effective stress values for soils analysis);   
+• is called at the start of the analysis for each applicable material calculation point in the model; and   
+• can be used to define all active initial stress components at material points as functions of coordinates, element number, integration point number, etc.
+
+# Stress components
+
+The number of stress components that must be defined depends on the element type for which this call is being made. Part VI, “Elements,” of the Abaqus Analysis User’s Guide,” describes the element stresses. The order in which the components must be defined is the same as in the element definition. For example, in three-dimensional continuum elements six stress components must be defined in the order 011,022,033,012,013,023.
+
+# Initial stress field equilibrium
+
+You should ensure that the initial stress field is in equilibrium with the applied forces and distributed loads by using a static step or a geostatic step to check the equilibrium of the initial stress field before starting the response history. See “Geostatic stress state,” Section 6.8.2 of the Abaqus Analysis User’s Guide, for a discussion of defining initial equilibrium conditions for problems that include pore fluid pressure.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE SIGINI (SIGMA, COORDS, NTENS, NCRDS, NOEL, NPT, LAYER, 1 KSPT, LREBAR, NAMES)
+C
+INCLUDE 'ABA_PARAM.INC'
+C 
+```
+
+<!-- source-page: 120 -->
+
+DIMENSION SIGMA(NTENS),COORDS(NCRDS) CHARACTER NAMES(2)\*80
+
+user coding to define SIGMA(NTENS)
+
+RETURN END
+
+# Variables to be defined
+
+# SIGMA(1)
+
+First stress component.
+
+# SIGMA(2)
+
+Second stress component.
+
+# SIGMA(3)
+
+Third stress component.
+
+# Etc.
+
+Only NTENS stress values should be defined, where NTENS depends on the element type.
+
+# Variables passed in for information
+
+# COORDS
+
+An array containing the initial coordinates of this point.
+
+# NTENS
+
+Number of stresses to be defined, which depends on the element type.
+
+# NCRDS
+
+Number of coordinates.
+
+# NOEL
+
+Element number.
+
+# NPT
+
+Integration point number in the element.
+
+# LAYER
+
+Layer number (for composite shells and layered solids).
+
+# KSPT
+
+Section point number within the current layer.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_013.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_013.md
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+<!-- source-page: 121 -->
+
+# LREBAR
+
+Rebar flag. If LREBAR=1, the current integration point is associated with element rebar. Otherwise, LREBAR=0.
+
+# NAMES(1)
+
+Name of the rebar to which the current integration point belongs, which is the name given in the rebar or rebar layer definition (“Defining reinforcement,” Section 2.2.3 of the Abaqus Analysis User’s Guide, or “Defining rebar as an element property,” Section 2.2.4 of the Abaqus Analysis User’s Guide). If no name was given in the rebar or rebar layer definition, this variable will be blank. This variable is relevant only when LREBAR=1.
+
+# NAMES(2)
+
+Element type name (see Section EI.1, “Abaqus/Standard Element Index,” of the Abaqus Analysis User’s Guide).
+
+<!-- source-page: 122 -->
+
+<!-- source-page: 123 -->
+
+# 1.1.19 UAMP: User subroutine to specify amplitudes.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Amplitude curves,” Section 34.1.2 of the Abaqus Analysis User’s Guide   
+• \*AMPLITUDE   
+• \*OUTPUT
+
+# Overview
+
+User subroutine UAMP:
+
+• allows you to define the current value of an amplitude definition as a function of time;   
+• can be used to model control engineering aspects of your system when sensors are used (sensor values are from the beginning of the increment);   
+• can use a predefined number of state variables in their definition; and   
+• can optionally compute the derivatives and integrals of the amplitude function.
+
+# Explicit solution dependence
+
+The solution dependence introduced in this user subroutine is explicit: all data passed in the subroutine for information or to be updated are values at the beginning of that increment.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE UAMP(
+*    ampName, time, ampValueOld, dt, nProps, props, nSvars,
+*    svars, lFlagsInfo,
+*    nSensor, sensorValues, sensorNames, jSensorLookUpTable,
+*    AmpValueNew,
+*    lFlagsDefine,
+*    AmpDerivative, AmpSecDerivative, AmpIncIntegral,
+*    AmpDoubleIntegral)
+C
+INCLUDE 'ABA_PARAM.INC'
+C time indices
+parameter (iStepTime = 1,
+*    iTotalTime = 2,
+*    nTime = 2) 
+```
+
+<!-- source-page: 124 -->
+
+```scala
+C flags passed in for information
+    parameter (iInitialization = 1,
+    * iRegularInc = 2,
+    * iCuts = 3,
+    * ikStep = 4,
+    * nFlagsInfo = 4)
+C optional flags to be defined
+    parameter (iComputeDeriv = 1,
+    * iComputeSecDeriv = 2,
+    * iComputeInteg = 3,
+    * iComputeDoubleInteg = 4,
+    * iStopAnalysis = 5,
+    * iConcludeStep = 6,
+    * nFlagsDefine = 6)
+    dimension time(nTime), lFlagsInfo(nFlagsInfo),
+    * lFlagsDefine(nFlagsDefine)
+    dimension jSensorLookUpTable(*)
+    dimension sensorValues(nSensor), svars(nSvars), props(nProps)
+    character*80 sensorNames(nSensor)
+    character*80 ampName
+
+    user coding to define AmpValueNew, and
+    optionally lFlagsDefine, AmpDerivative, AmpSecDerivative,
+    AmpIncIntegral, AmpDoubleIntegral
+
+    RETURN
+    END 
+```
+
+# Variable to be defined
+
+# AmpValueNew
+
+Current value of the amplitude.
+
+# Variables that can be updated
+
+# lFlagsDefine
+
+Integer flag array to determine whether the computation of additional quantities is necessary or to set step continuation requirements.
+
+<!-- source-page: 125 -->
+
+<table><tr><td>1FlagsDefine(iComputeDeriv)</td><td>If set to 1, you must provide the computation of the amplitude derivative. The default is 0, which means that Abaqus computes the derivative automatically.</td></tr><tr><td>1FlagsDefine(iComputeSecDeriv)</td><td>If set to 1, you must provide the computation of the amplitude second derivative. The default is 0, which means that Abaqus computes the second derivative automatically.</td></tr><tr><td>1FlagsDefine(iComputeInteg)</td><td>If set to 1, you must provide the computation of the amplitude incremental integral. The default is 0, which means that Abaqus computes the incremental integral automatically.</td></tr><tr><td>1FlagsDefine(iComputeDoubleInteg)</td><td>If set to 1, you must provide the computation of the amplitude incremental double integral. The default is 0, which means that Abaqus computes the incremental integral automatically.</td></tr><tr><td>1FlagsDefine(iStopAnalysis)</td><td>If set to 1, the analysis will be stopped and an error message will be issued. The default is 0, which means that Abaqus will not stop the analysis.</td></tr><tr><td>1FlagsDefine(iConcludeStep)</td><td>If set to 1, Abaqus will conclude the step execution and advance to the next step (if a next step exists). The default is 0.</td></tr></table>
+
+# svars
+
+An array containing the values of the solution-dependent state variables associated with this amplitude definition. The number of such variables is nsvars (see above). You define the meaning of these variables.
+
+This array is passed into UAMP containing the values of these variables at the start of the current increment. In most cases they should be updated to be the values at the end of the increment.
+
+# AmpDerivative
+
+Current value of the amplitude derivative.
+
+# AmpSecDerivative
+
+Current value of the amplitude second derivative.
+
+<!-- source-page: 126 -->
+
+# AmpIncIntegral
+
+Current value of the amplitude incremental integral.
+
+# AmpDoubleIntegral
+
+Current value of the amplitude incremental double integral.
+
+# Variables passed in for information
+
+# ampName
+
+User-specified amplitude name, left justified.
+
+# time(iStepTime)
+
+Current value of step time or frequency.
+
+# time(iTotalTime)
+
+Current value of total time.
+
+# ampValueOld
+
+Old value of the amplitude from the previous increment.
+
+# dt
+
+Time increment.
+
+# props
+
+User-specified array of material constants associated with this amplitude definition.
+
+# nProps
+
+User-defined number of material constants associated with this amplitude definition.
+
+# nSvars
+
+User-defined number of solution-dependent state variables associated with this amplitude definition.
+
+# lFlagsInfo
+
+Integer flag array with information regrading the current call to UAMP.
+
+# lFlagsInfo(iInitialization)
+
+This flag is equal to 1 if UAMP is called from the initialization phase of the first analysis step and is set to 0 otherwise.
+
+# lFlagsInfo(iRegularInc)
+
+This flag is equal to 1 if UAMP is called from a regular increment and is set to 0 if called from the initialization phase of the first analysis step.
+
+# lFlagsInfo(iCuts)
+
+Number of cutbacks in this increment.
+
+# lFlagsInfo(ikStep)
+
+Step number.
+
+<!-- source-page: 127 -->
+
+# nSensor
+
+Total number of sensors in the model.
+
+# sensorValues
+
+Array with sensor values at the end of the previous increment. Each sensor value corresponds to a history output variable associated with the output database request defining the sensor.
+
+# sensorNames
+
+Array with user-defined sensor names in the entire model, left justified. Each sensor name corresponds to a sensor value provided with the output database request. All names will be converted to uppercase characters if lowercase or mixed-case characters were used in their definition.
+
+# jSensorLookUpTable
+
+Variable that must be passed into the utility functions IGETSENSORID and GETSENSORVALUE.
+
+Example: Amplitude definition using sensor and state variables   
+```python
+c user amplitude subroutine
+Subroutine UAMP(
+C    passed in for information and state variables
+*    ampName, time, ampValueOld, dt, nProps, props, nSvars,
+*    svars, lFlagsInfo,
+*    nSensor, sensorValues, sensorNames,
+*    jSensorLookUpTable,
+C    to be defined
+*    ampValueNew,
+*    lFlagsDefine,
+*    AmpDerivative, AmpSecDerivative, AmpIncIntegral,
+*    AmpDoubleIntegral)
+include 'aba_param.inc'
+C    svars - additional state variables, similar to (V)UEL
+    dimension sensorValues(nSensor), svars(nSvars),
+*    props(nProps)
+    character*80 sensorNames(nSensor)
+    character*80 ampName
+C    time indices
+    parameter( iStepTime    = 1,
+*    iTotalTime    = 2,
+*    nTime    = 2)
+C    flags passed in for information 
+```
+
+<!-- source-page: 128 -->
+
+```txt
+parameter( iInitialization = 1,
+* iRegularInc = 2,
+* iCuts = 3,
+* ikStep = 4,
+* nFlagsInfo = 4)
+
+C optional flags to be defined
+parameter( iComputeDeriv = 1,
+* iComputeSecDeriv = 2,
+* iComputeInteg = 3,
+* iComputeDoubleInteg = 4,
+* iStopAnalysis = 5,
+* iConcludeStep = 6,
+* nFlagsDefine = 6)
+
+parameter( tStep=0.18d0, tAccelerateMotor = .00375d0,
+* omegaFinal=23.26d0,
+* zero=0.0d0, one=1.0d0, two=2.0d0, four=4.0d0)
+
+dimension time(nTime), lFlagsInfo(nFlagsInfo),
+* lFlagsDefine(nFlagsDefine)
+dimension jSensorLookUpTable(*)
+
+lFlagsDefine(iComputeDeriv) = 1
+lFlagsDefine(iComputeSecDeriv) = 1
+lFlagsDefine(iComputeInteg) = 1
+lFlagsDefine(iComputeDoubleInteg) = 1
+
+c get sensor value
+vTrans_CU1 = GetSensorValue('HORIZ_TRANSL_MOTION',
+* jSensorLookUpTable,
+* sensorValues)
+
+if (ampName(1:22).eq. 'MOTOR_WITH_STOP_SENSOR') then
+    if (lFlagsInfo(iInitialization).eq.1) then
+    AmpSecDerivative = zero
+    AmpDerivative = omegaFinal/tAccelerateMotor
+    ampValueNew = zero
+    AmpIncIntegral = zero
+    AmpDoubleIntegral = zero
+
+    svars(1) = zero
+    svars(2) = zero 
+```
+
+<!-- source-page: 129 -->
+
+```vba
+else
+    tim = time(iStepTime)
+
+c ramp up the angular rot velocity of the
+c electric motor
+c after which hold constant
+    if (tim .le. tAccelerateMotor) then
+    AmpSecDerivative = zero
+    AmpDerivative = omegaFinal/tAccelerateMotor
+    ampValueNew = omegaFinal*tim/tAccelerateMotor
+    AmpIncIntegral = dt*(ampValueOld+ampValueNew)/two
+    AmpDoubleIntegral = dt**2*(ampValueOld+ampValueNew)/four
+    else
+    AmpSecDerivative = zero
+    AmpDerivative = zero
+    ampValueNew = omegaFinal
+    AmpIncIntegral = dt*(ampValueOld+ampValueNew)/two
+    AmpDoubleIntegral = dt**2*(ampValueOld+ampValueNew)/four
+    end if
+
+c retrieve old sensor value
+    vTrans_CU1_old = svars(1)
+
+c detect a zero crossing and count the number of
+c crossings
+    if (vTrans_CU1_old*vTrans_CU1 .le. zero .and.
+    * tim .gt. tAccelerateMotor ) then
+    svars(2) = svars(2) + one
+    end if
+    nrCrossings = int(svars(2))
+
+c stop the motor if sensor crosses zero the second time
+    if (nrCrossings.eq.2) then
+    ampValueNew = zero
+    lFlagsDefine(iConcludeStep)=1
+    end if 
+```
+
+<!-- source-page: 130 -->
+
+```txt
+c store sensor value
+    svars(1) = vTrans_CU1
+    end if
+    end if
+    return
+    end 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_014.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_014.md
new file mode 100644
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--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_014.md
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+<!-- source-page: 131 -->
+
+# 1.1.20 UANISOHYPER\_INV: User subroutine to define anisotropic hyperelastic material behavior using the invariant formulation.
+
+Product: Abaqus/Standard
+
+# References
+
+• “Anisotropic hyperelastic behavior,” Section 22.5.3 of the Abaqus Analysis User’s Guide   
+• \*ANISOTROPIC HYPERELASTIC   
+• “UANISOHYPER\_INV and VUANISOHYPER\_INV,” Section 4.1.13 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine UANISOHYPER\_INV:
+
+• can be used to define the strain energy potential of anisotropic hyperelastic materials as a function of an irreducible set of scalar invariants;   
+• is called at all material calculation points of elements for which the material definition contains user-defined anisotropic hyperelastic behavior with an invariant-based formulation (“Anisotropic hyperelastic behavior,” Section 22.5.3 of the Abaqus Analysis User’s Guide);   
+• can include material behavior dependent on field variables or state variables;   
+• requires that the values of the derivatives of the strain energy density function of the anisotropic hyperelastic material be defined with respect to the scalar invariants; and   
+• is called twice per material point in each iteration.
+
+# Enumeration of invariants
+
+To facilitate coding and provide easy access to the array of invariants passed to user subroutine UANISOHYPER\_INV, an enumerated representation of each invariant is introduced. Any scalar invariant can, therefore, be represented uniquely by an enumerated invariant, $I _ { n } ^ { * }$ , where the subscript n denotes the order of the invariant according to the enumeration scheme in the following table:
+
+<table><tr><td>Invariant</td><td>Enumeration, n</td></tr><tr><td> $\overline{I}_{1}$ </td><td>1</td></tr><tr><td> $\overline{I}_{2}$ </td><td>2</td></tr><tr><td>J</td><td>3</td></tr><tr><td> $\overline{I}_{4(\alpha\beta)}$ </td><td> $4 + 2(\alpha - 1) + \beta(\beta - 1)$ ;  $\alpha \leq \beta$ </td></tr><tr><td> $\overline{I}_{5(\alpha\beta)}$ </td><td> $5 + 2(\alpha - 1) + \beta(\beta - 1)$ ;  $\alpha \leq \beta$ </td></tr></table>
+
+<!-- source-page: 132 -->
+
+For example, in the case of three families of fibers there are a total of 15 invariants: $\overline { { I } } _ { 1 } , \overline { { I } } _ { 2 } , J ,$ six invariants of type $\overline { { I } } _ { 4 ( \alpha \beta ) }$ , and six invariants of type $\overline { { I } } _ { 5 ( \alpha \beta ) }$ , with $\alpha , \beta = 1 , 2 , 3 \left( \alpha \leq \beta \right)$ . The following correspondence exists between each of these invariants and their enumerated counterpart:
+
+<table><tr><td>Enumerated invariant</td><td>Invariant</td></tr><tr><td> $I_{1}^{*}$ </td><td> $\overline{I}_{1}$ </td></tr><tr><td> $I_{2}^{*}$ </td><td> $\overline{I}_{2}$ </td></tr><tr><td> $I_{3}^{*}$ </td><td> $J$ </td></tr><tr><td> $I_{4}^{*}$ </td><td> $\overline{I}_{4(11)}$ </td></tr><tr><td> $I_{5}^{*}$ </td><td> $\overline{I}_{5(11)}$ </td></tr><tr><td> $I_{6}^{*}$ </td><td> $\overline{I}_{4(12)}$ </td></tr><tr><td> $I_{7}^{*}$ </td><td> $\overline{I}_{5(12)}$ </td></tr><tr><td> $I_{8}^{*}$ </td><td> $\overline{I}_{4(22)}$ </td></tr><tr><td> $I_{9}^{*}$ </td><td> $\overline{I}_{5(22)}$ </td></tr><tr><td> $I_{10}^{*}$ </td><td> $\overline{I}_{4(13)}$ </td></tr><tr><td> $I_{11}^{*}$ </td><td> $\overline{I}_{5(13)}$ </td></tr><tr><td> $I_{12}^{*}$ </td><td> $\overline{I}_{4(23)}$ </td></tr><tr><td> $I_{13}^{*}$ </td><td> $\overline{I}_{5(23)}$ </td></tr><tr><td> $I_{14}^{*}$ </td><td> $\overline{I}_{4(33)}$ </td></tr><tr><td> $I_{15}^{*}$ </td><td> $\overline{I}_{5(33)}$ </td></tr></table>
+
+A similar scheme is used for the array ZETA of terms $\zeta _ { \alpha \beta } = { \bf A } _ { \alpha } \cdot { \bf A } _ { \beta }$ . Each term can be represented uniquely by an enumerated counterpart $\zeta _ { m } ^ { * }$ , as shown below:
+
+<table><tr><td>Dot product</td><td>Enumeration, m</td></tr><tr><td> $\zeta_{\alpha\beta}$ </td><td> $\alpha + \frac{1}{2}(\beta - 2)(\beta - 1)$ ;  $\alpha < \beta$ </td></tr></table>
+
+As an example, for the case of three families of fibers there are three $\zeta _ { \alpha \beta }$ terms: $\zeta _ { 1 2 } , \zeta _ { 1 3 }$ , and $\zeta _ { 2 3 }$ . These are stored in the ZETA array as $\left( \zeta _ { 1 } ^ { * } , \zeta _ { 2 } ^ { * } , \zeta _ { 3 } ^ { * } \right)$ .
+
+# Storage of arrays of derivatives of the energy function
+
+The components of the array UI1 of first derivatives of the strain energy potential with respect to the
+
+<!-- source-page: 133 -->
+
+scalar invariants, ${ { \partial U } \mathord { \left/ { \vphantom { { \partial U } { \partial I } { \partial T } _ { i } ^ { * } } } \right. \kern - delimiterspace } { \partial T } } _ { i } ^ { * }$ , are stored using the enumeration scheme discussed above for the scalar invariants.
+
+The elements of the array UI2 of second derivatives of the strain energy function, $\partial ^ { 2 } U / \partial I _ { i } ^ { * } \partial I _ { j } ^ { * }$ , are laid out in memory using triangular storage: if denotes the component in this array corresponding to the term $\partial ^ { 2 } U / \partial I _ { i } ^ { * } \partial I _ { i } ^ { * }$ , then $k = i + j \times ( j - 1 ) / 2 ; ( i \leq j )$ . For example, the term $\partial ^ { 2 } U / \partial I _ { 2 } ^ { * } \partial I _ { 5 } ^ { * }$ is stored in component $k \overset { \circ } { = } 2 + ( 5 \times 4 ) / 2 = 1 2$ in the UI2 array.
+
+# Special considerations for various element types
+
+There are several special considerations that need to be noted.
+
+# Shells that calculate transverse shear energy
+
+When UANISOHYPER\_INV is used to define the material response of shell elements that calculate transverse shear energy, Abaqus/Standard cannot calculate a default value for the transverse shear stiffness of the element. Hence, you must define the element’s transverse shear stiffness. See “Shell section behavior,” Section 29.6.4 of the Abaqus Analysis User’s Guide, for guidelines on choosing this stiffness.
+
+# Elements with hourglassing modes
+
+When UANISOHYPER\_INV is used to define the material response of elements with hourglassing modes, you must define the hourglass stiffness for hourglass control based on the total stiffness approach. The hourglass stiffness is not required for enhanced hourglass control, but you can define a scaling factor for the stiffness associated with the drill degree of freedom (rotation about the surface normal). See “Section controls,” Section 27.1.4 of the Abaqus Analysis User’s Guide.
+
+User subroutine interface   
+```sql
+SUBROUTINE UANISOHYPER_INV (AINV, UA, ZETA, NFIBERS, NINV,
+1 UI1, UI2, UI3, TEMP, NOEL, CMNAME, INCMPFLAG, IHYBFLAG,
+2 NUMSTATEV, STATEV, NUMFIELDV, FIELDV, FIELDVINC,
+3 NUMPROPS, PROPS)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME
+DIMENSION AINV(NINV), UA(2),
+2 ZETA(NFIBERS*(NFIBERS-1)/2)), UI1(NINV),
+3 UI2(NINV*(NINV+1)/2), UI3(NINV*(NINV+1)/2),
+4 STATEV(NUMSTATEV), FIELDV(NUMFIELDV),
+5 FIELDVINC(NUMFIELDV), PROPS(NUMPROPS) 
+```
+
+user coding to define UA,UI1,UI2,UI3,STATEV
+
+<!-- source-page: 134 -->
+
+# RETURN
+
+# END
+
+# Variables to be defined
+
+# UA(1)
+
+U, strain energy density function. For a compressible material at least one derivative involving J should be nonzero. For an incompressible material all derivatives involving J are ignored.
+
+# UA(2)
+
+$\tilde { U } _ { d e v } ,$ the deviatoric part of the strain energy density of the primary material response. This quantity is needed only if the current material definition also includes Mullins effect (see “Mullins effect,” Section 22.6.1 of the Abaqus Analysis User’s Guide).
+
+# UI1(NINV)
+
+Array of derivatives of strain energy potential with respect to the scalar invariants, $\partial U / \partial I _ { i } ^ { * }$ , ordered using the enumeration scheme discussed above.
+
+# UI2(NINV\*(NINV+1)/2)
+
+Array of second derivatives of strain energy potential with respect to the scalar invariants (using triangular storage), $\partial ^ { 2 } U / \partial I _ { i } ^ { * } \partial I _ { j } ^ { * }$ .
+
+# UI3(NINV\*(NINV+1)/2)
+
+Array of derivatives with respect to J of the second derivatives of the strain energy potential (using triangular storage), ${ \partial ^ { 3 } } U / { \partial I _ { i } ^ { * } \partial I _ { j } ^ { * } \partial J }$ . This quantity is needed only for compressible materials with a hybrid formulation (when INCMPFLAG = 0 and IHYBFLAG = 1).
+
+# STATEV
+
+Array containing the user-defined solution-dependent state variables at this point. These are supplied as values at the start of the increment or as values updated by other user subroutines (see “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide) and must be returned as values at the end of the increment.
+
+# Variables passed in for information
+
+# NFIBERS
+
+Number of families of fibers defined for this material.
+
+# NINV
+
+Number of scalar invariants.
+
+# TEMP
+
+Current temperature at this point.
+
+<!-- source-page: 135 -->
+
+# NOEL
+
+Element number.
+
+# CMNAME
+
+User-specified material name, left justified.
+
+# INCMPFLAG
+
+Incompressibility flag defined to be 1 if the material is specified as incompressible or 0 if the material is specified as compressible.
+
+# IHYBFLAG
+
+Hybrid formulation flag defined to be 1 for hybrid elements; 0 otherwise.
+
+# NUMSTATEV
+
+User-defined number of solution-dependent state variables associated with this material (see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# NUMFIELDV
+
+Number of field variables.
+
+# FIELDV
+
+Array of interpolated values of predefined field variables at this material point at the end of the increment based on the values read in at the nodes (initial values at the beginning of the analysis and current values during the analysis).
+
+# FIELDVINC
+
+Array of increments of predefined field variables at this material point for this increment, including any values updated by user subroutine USDFLD.
+
+# NUMPROPS
+
+Number of material properties entered for this user-defined hyperelastic material.
+
+# PROPS
+
+Array of material properties entered for this user-defined hyperelastic material.
+
+# AINV(NINV)
+
+Array of scalar invariants, $I _ { i } ^ { * }$ , at each material point at the end of the increment. The invariants are ordered using the enumeration scheme discussed above.
+
+# ZETA(NFIBERS\*(NFIBERS-1)/2))
+
+Array of dot product between the directions of different families of fiber in the reference configuration, $\zeta _ { \alpha \beta } = { \bf A } _ { \alpha } \cdot { \bf A } _ { \beta }$ . The array contains the enumerated values $\zeta _ { m } ^ { * }$ using the scheme discussed above.
+
+<!-- source-page: 136 -->
+
+As an example of the coding of user subroutine UANISOHYPER\_INV, consider the model proposed by Kaliske and Schmidt (2005) for nonlinear anisotropic elasticity with two families of fibers. The strain energy function is given by a polynomial series expansion in the form
+
+$$
+\begin{array}{l} U = \frac {1}{D} (J - 1) ^ {2} + \sum_ {i = 1} ^ {3} a _ {i} (\overline {{I}} _ {1} - 3) ^ {i} + \sum_ {j = 1} ^ {3} b _ {j} (\overline {{I}} _ {2} - 3) ^ {j} + \sum_ {k = 2} ^ {6} c _ {k} (\overline {{I}} _ {4 (1 1)} - 1) ^ {k} + \sum_ {l = 2} ^ {6} d _ {l} (\overline {{I}} _ {5 (1 1)} - 1) ^ {l} \\ + \sum_ {m = 2} ^ {6} e _ {m} (\overline {{I}} _ {4 (2 2)} - 1) ^ {m} + \sum_ {n = 2} ^ {6} f _ {n} (\overline {{I}} _ {5 (2 2)} - 1) ^ {n} + \sum_ {p = 2} ^ {6} g _ {p} (\zeta_ {1 2} \overline {{I}} _ {4 (1 2)} - \zeta_ {1 2} ^ {2}) ^ {p}. \\ \end{array}
+$$
+
+The code in user subroutine UANISOHYPER\_INV must return the derivatives of the strain energy function with respect to the scalar invariants, which are readily computed from the above expression. In this example auxiliary functions are used to facilitate enumeration of pseudo-invariants of type $\overline { { I } } _ { 4 ( \alpha \beta ) }$ and $\overline { { I } } _ { 5 ( \alpha \beta ) }$ , as well as for indexing into the array of second derivatives using symmetric storage. The user subroutine would be coded as follows:
+```txt
+subroutine uanisohyper_inv (aInv, ua, zeta, nFibers, nInv,
+* ui1, ui2, ui3, temp, noel,
+* cmname, incmpFlag, ihybFlag,
+* numStatev, statev,
+* numFieldv, fieldv, fieldvInc,
+* numProps, props)
+C
+    include 'aba_param.inc'
+C
+    character *80 cmname
+    dimension aInv(nInv), ua(2), zeta(nFibers*(nFibers-1)/2)
+    dimension ui1(nInv), ui2(nInv*(nInv+1)/2)
+    dimension ui3(nInv*(nInv+1)/2), statev(numStatev)
+    dimension fieldv(numFieldv), fieldvInc(numFieldv)
+    dimension props(numProps)
+C
+    parameter ( zero = 0.d0,
+    * one = 1.d0,
+    * two = 2.d0,
+    * three = 3.d0,
+    * four = 4.d0,
+    * five = 5.d0,
+    * six = 6.d0 )
+C 
+```
+
+<!-- source-page: 137 -->
+
+```txt
+C Kaliske-Schmidt energy function (3D)
+C
+C Read material properties
+    d=props(1)
+    dInv = one / d
+    a1=props(2)
+    a2=props(3)
+    a3=props(4)
+    b1=props(5)
+    b2=props(6)
+    b3=props(7)
+    c2=props(8)
+    c3=props(9)
+    c4=props(10)
+    c5=props(11)
+    c6=props(12)
+    d2=props(13)
+    d3=props(14)
+    d4=props(15)
+    d5=props(16)
+    d6=props(17)
+    e2=props(18)
+    e3=props(19)
+    e4=props(20)
+    e5=props(21)
+    e6=props(22)
+    f2=props(23)
+    f3=props(24)
+    f4=props(25)
+    f5=props(26)
+    f6=props(27)
+    g2=props(28)
+    g3=props(29)
+    g4=props(30)
+    g5=props(31)
+    g6=props(32)
+C
+C Compute Udev and 1st and 2nd derivatives w.r.t invariants
+C - I1
+    bi1 = aInv(1)
+    term = bi1-three
+    ua(2) = a1*term + a2*term**2 + a3*term**3 
+```
+
+<!-- source-page: 138 -->
+
+```txt
+ui1(1) = a1 + two*a2*term + three*a3*term**2
+ui2(indx(1,1)) = two*a2 + three*two*a3*term
+C - I2
+bi2 = aInv(2)
+term = bi2-three
+ua(2) = ua(2) + b1*term + b2*term**2 + b3*term**3
+ui1(2) = b1 + two*b2*term + three*b3*term**2
+ui2(indx(2,2)) = two*b2 + three*two*b3*term
+C - I3 (=J)
+bi3 = aInv(3)
+term = bi3-one
+ui1(3) = two*dInv*term
+ui2(indx(3,3)) = two*dInv
+C - I4(11)
+nI411 = indxInv4(1,1)
+bi411 = aInv(nI411)
+term = bi411-one
+ua(2) = ua(2)
+* + c2*term**2 + c3*term**3 + c4*term**4
+* + c5*term**5 + c6*term**6
+ui1(nI411) =
+* two*c2*term
+* + three*c3*term**2
+* + four*c4*term**3
+* + five*c5*term**4
+* + six*c6*term**5
+ui2(indx(nI411,nI411)) =
+* two*c2
+* + three*two*c3*term
+* + four*three*c4*term**2
+* + five*four*c5*term**3
+* + six*five*c6*term**4
+C - I5(11)
+nI511 = indxInv5(1,1)
+bi511 = aInv(nI511)
+term = bi511-one
+ua(2) = ua(2)
+* + d2*term**2 + d3*term**3 + d4*term**4
+* + d5*term**5 + d6*term**6
+ui1(nI511) =
+* two*d2*term
+* + three*d3*term**2 
+```
+
+<!-- source-page: 139 -->
+
+```c
+* + four*d4*term**3
+* + five*d5*term**4
+* + six*d6*term**5
+ui2 (indx(nI511,nI511)) =
+* two*d2
+* + three*two*d3*term
+* + four*three*d4*term**2
+* + five*four*d5*term**3
+* + six*five*d6*term**4
+C - I4(22)
+nI422 = indxInv4(2,2)
+bi422 = aInv(nI422)
+term = bi422-one
+ua(2) = ua(2)
+* + e2*term**2 + e3*term**3 + e4*term**4
+* + e5*term**5 + e6*term**6
+ui1(nI422) =
+* two*e2*term
+* + three*e3*term**2
+* + four*e4*term**3
+* + five*e5*term**4
+* + six*e6*term**5
+ui2 (indx(nI422,nI422)) =
+* two*e2
+* + three*two*e3*term
+* + four*three*e4*term**2
+* + five*four*e5*term**3
+* + six*five*e6*term**4
+C - I5(22)
+nI522 = indxInv5(2,2)
+bi522 = aInv(nI522)
+term = bi522-one
+ua(2) = ua(2)
+* + f2*term**2 + f3*term**3 + f4*term**4
+* + f5*term**5 + f6*term**6
+ui1(nI522) =
+* two*f2*term
+* + three*f3*term**2
+* + four*f4*term**3
+* + five*f5*term**4
+* + six*f6*term**5
+ui2 (indx(nI522,nI522)) = 
+```
+
+<!-- source-page: 140 -->
+
+```matlab
+* two*f2
+* + three*two*f3*term
+* + four*three*f4*term**2
+* + five*four*f5*term**3
+* + six*five*f6*term**4
+C - I4(12)
+    nI412 = indxInv4(1,2)
+    bi412 = aInv(nI412)
+    term = zeta(1)*(bi412-zeta(1))
+    ua(2) = ua(2)
+    * + g2*term**2 + g3*term**3
+    * + g4*term**4 + g5*term**5
+    * + g6*term**6
+    ui1(nI412) = zeta(1) * (
+    * two*g2*term
+    * + three*g3*term**2
+    * + four*g4*term**3
+    * + five*g5*term**4
+    * + six*g6*term**5)
+    ui2(indx(nI412,nI412)) = zeta(1)**2 * (
+    * two*g2
+    * + three*two*g3*term
+    * + four*three*g4*term**2
+    * + five*four*g5*term**3
+    * + six*five*g6*term**4)
+C
+C Add volumetric energy
+C
+    term = aInv(3) - one
+    ua(1) = ua(2) + dInv*term*term
+C
+    return
+    end
+C
+C Maps index from Square to Triangular storage of symmetric
+C matrix
+C
+    integer function index( i, j )
+C
+    include 'aba_param.inc'
+C
+    ii = min(i,j) 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_015.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_015.md
new file mode 100644
index 00000000..fa86c9bc
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_015.md
@@ -0,0 +1,369 @@
+<!-- source-page: 141 -->
+
+```matlab
+jj = max(i, j)
+c
+    index = ii + jj*(jj-1)/2
+c
+    return
+    end
+c
+c
+C Generate enumeration of Anisotropic Pseudo Invariants of
+C type 4
+c
+    integer function indexInv4(i, j)
+c
+    include 'aba_param.inc'
+c
+    ii = min(i, j)
+    jj = max(i, j)
+c
+    indexInv4 = 4 + jj*(jj-1) + 2*(ii-1)
+c
+    return
+    end
+c
+c
+C Generate enumeration of Anisotropic Pseudo Invariants of
+C type 5
+c
+    integer function indexInv5(i, j)
+c
+    include 'aba_param.inc'
+c
+    ii = min(i, j)
+    jj = max(i, j)
+c
+    indexInv5 = 5 + jj*(jj-1) + 2*(ii-1)
+c
+    return
+    end 
+```
+
+<!-- source-page: 142 -->
+
+# Additional reference
+
+• Kaliske, M., and J. Schmidt, “Formulation of Finite Nonlinear Anisotropic Elasticity,” CADFEM GmbH Infoplaner 2/2005, vol. 2, pp. 22–23, 2005.
+
+<!-- source-page: 143 -->
+
+# 1.1.21 UANISOHYPER\_STRAIN: User subroutine to define anisotropic hyperelastic material behavior based on Green strain.
+
+Product: Abaqus/Standard
+
+# References
+
+• “Anisotropic hyperelastic behavior,” Section 22.5.3 of the Abaqus Analysis User’s Guide   
+• \*ANISOTROPIC HYPERELASTIC   
+• “UANISOHYPER\_INV and VUANISOHYPER\_INV,” Section 4.1.13 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine UANISOHYPER\_STRAIN:
+
+• can be used to define the strain energy potential of anisotropic hyperelastic materials as a function of the components of the Green strain tensor;   
+• is called at all material calculation points of elements for which the material definition contains user-defined anisotropic hyperelastic behavior with a Green strain-based formulation (“Anisotropic hyperelastic behavior,” Section 22.5.3 of the Abaqus Analysis User’s Guide);   
+• can include material behavior dependent on field variables or state variables;   
+• requires that the values of the derivatives of the strain energy density function of the anisotropic hyperelastic material be defined with respect to the components of the modified Green strain tensor; and   
+• is called twice per material point in each iteration.
+
+# Storage of strain components
+
+In the array of modified Green strain, EBAR, direct components are stored first, followed by shear components. There are NDI direct and NSHR tensor shear components. The order of the components is defined in “Conventions,” Section 1.2.2 of the Abaqus Analysis User’s Guide. Since the number of active stress and strain components varies between element types, the routine must be coded to provide for all element types with which it will be used.
+
+# Storage of arrays of derivatives of the energy function
+
+The array of first derivatives of the strain energy function, DU1, contains NTENS+1 components, with NTENS=NDI+NSHR. The first NTENS components correspond to the derivatives with respect to each component of the modified Green strain, ${ \partial U } / { \partial \overline { { \varepsilon } } _ { i j } ^ { G } }$ . The last component contains the derivative with respect to the volume ratio, .
+
+<!-- source-page: 144 -->
+
+The array of second derivatives of the strain energy function, DU2, contains (NTENS+1)\*(NTENS+2)/2 components. These components are ordered using the following triangular storage scheme:
+
+<table><tr><td>Component</td><td>2D Case</td><td>3D Case</td></tr><tr><td>1</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{11}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{11}^{G}$ </td></tr><tr><td>2</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{22}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{22}^{G}$ </td></tr><tr><td>3</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{22}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{22}^{G}$ </td></tr><tr><td>4</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{33}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{33}^{G}$ </td></tr><tr><td>5</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{33}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{33}^{G}$ </td></tr><tr><td>6</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{33}^{G}\partial\overline{\varepsilon}_{33}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{33}^{G}\partial\overline{\varepsilon}_{33}^{G}$ </td></tr><tr><td>7</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td></tr><tr><td>8</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td></tr><tr><td>9</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{33}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{33}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td></tr><tr><td>10</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{12}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{12}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td></tr><tr><td>11</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial J$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{13}^{G}$ </td></tr><tr><td>12</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial J$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{13}^{G}$ </td></tr><tr><td>13</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{33}^{G}\partial J$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{33}^{G}\partial\overline{\varepsilon}_{13}^{G}$ </td></tr><tr><td>14</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{12}^{G}\partial J$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{12}^{G}\partial\overline{\varepsilon}_{13}^{G}$ </td></tr><tr><td>15</td><td> $\partial^{2}U/\partial J^{2}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{13}^{G}\partial\overline{\varepsilon}_{13}^{G}$ </td></tr><tr><td>16</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{23}^{G}$ </td></tr><tr><td>17</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{23}^{G}$ </td></tr><tr><td>18</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{33}^{G}\partial\overline{\varepsilon}_{23}^{G}$ </td></tr><tr><td>19</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{12}^{G}\partial\overline{\varepsilon}_{23}^{G}$ </td></tr><tr><td>20</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{13}^{G}\partial\overline{\varepsilon}_{23}^{G}$ </td></tr><tr><td>21</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{23}^{G}\partial\overline{\varepsilon}_{23}^{G}$ </td></tr><tr><td>22</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial J$ </td></tr><tr><td>23</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial J$ </td></tr><tr><td>24</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{33}^{G}\partial J$ </td></tr></table>
+
+<!-- source-page: 145 -->
+
+<table><tr><td>Component</td><td>2D Case</td><td>3D Case</td></tr><tr><td>25</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{12}^{G}\partial J$ </td></tr><tr><td>26</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{13}^{G}\partial J$ </td></tr><tr><td>27</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{23}^{G}\partial J$ </td></tr><tr><td>28</td><td></td><td> $\partial^{2}U/\partial J^{2}$ </td></tr></table>
+
+Finally, the array of third derivatives of the strain energy function, DU3, also contains (NTENS+1)\*(NTENS+2)/2 components, each representing the derivative with respect to of the corresponding component of DU2. It follows the same triangular storage scheme as DU2.
+
+# Special considerations for various element types
+
+There are several special considerations that need to be noted.
+
+# Shells that calculate transverse shear energy
+
+When UANISOHYPER\_STRAIN is used to define the material response of shell elements that calculate transverse shear energy, Abaqus/Standard cannot calculate a default value for the transverse shear stiffness of the element. Hence, you must define the element’s transverse shear stiffness. See “Shell section behavior,” Section 29.6.4 of the Abaqus Analysis User’s Guide, for guidelines on choosing this stiffness.
+
+# Elements with hourglassing modes
+
+When UANISOHYPER\_STRAIN is used to define the material response of elements with hourglassing modes, you must define the hourglass stiffness for hourglass control based on the total stiffness approach. The hourglass stiffness is not required for enhanced hourglass control, but you can define a scaling factor for the stiffness associated with the drill degree of freedom (rotation about the surface normal). See “Section controls,” Section 27.1.4 of the Abaqus Analysis User’s Guide.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE UANISOHYPER_STRAIN (EBAR, AJ, UA, DU1, DU2, DU3,
+1 TEMP, NOEL, CMNAME, INCMPFLAG, IHYBFLAG, NDI, NSHR, NTENS,
+2 NUMSTATEV, STATEV, NUMFIELDV, FIELDV, FIELDVINC,
+3 NUMPROPS, PROPS)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME
+C
+DIMENSION EBAR(NTENS), UA(2), DU1(NTENS+1), 
+```
+
+<!-- source-page: 146 -->
+
+```txt
+2 DU2((NTENS+1)*(NTENS+2)/2),
+3 DU3((NTENS+1)*(NTENS+2)/2),
+4 STATEV(NUMSTATEV), FIELDV(NUMFIELDV),
+5 FIELDVINC(NUMFIELDV), PROPS(NUMPROPS) 
+```
+
+user coding to define UA,DU1,DU2,DU3,STATEV
+
+RETURN
+
+END
+
+# Variables to be defined
+
+# UA(1)
+
+U, strain energy density function. For a compressible material at least one derivative involving J should be nonzero. For an incompressible material all derivatives involving J are ignored.
+
+# UA(2)
+
+$\tilde { U } _ { d e v }$ , the deviatoric part of the strain energy density of the primary material response. This quantity is needed only if the current material definition also includes Mullins effect (see “Mullins effect,” Section 22.6.1 of the Abaqus Analysis User’s Guide).
+
+# DU1(NTENS+1)
+
+Derivatives of strain energy potential with respect to the components of the modified Green strain tensor, ${ \partial U } / { \partial \overline { { \varepsilon } } _ { i j } ^ { G } }$ , and with respect to the volume ratio, .
+
+# DU2((NTENS+1)\*(NTENS+2)/2)
+
+Second derivatives of strain energy potential with respect to the components of the modified Green strain tensor and the volume ratio (using triangular storage, as mentioned earlier).
+
+# DU3((NTENS+1)\*(NTENS+2)/2)
+
+Derivatives with respect to J of the second derivatives of the strain energy potential (using triangular storage, as mentioned earlier). This quantity is needed only for compressible materials with a hybrid formulation (when INCMPFLAG = 0 and IHYBFLAG = 1).
+
+# STATEV
+
+Array containing the user-defined solution-dependent state variables at this point. These are supplied as values at the start of the increment or as values updated by other user subroutines (see “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide) and must be returned as values at the end of the increment.
+
+<!-- source-page: 147 -->
+
+# Variables passed in for information
+
+# TEMP
+
+Current temperature at this point.
+
+# NOEL
+
+Element number.
+
+# CMNAME
+
+User-specified material name, left justified.
+
+# NDI
+
+Number of direct stress components at this point.
+
+# NSHR
+
+Number of shear components at this point.
+
+# NTENS
+
+Size of the stress or strain component array (NDI + NSHR).
+
+# INCMPFLAG
+
+Incompressibility flag defined to be 1 if the material is specified as incompressible or 0 if the material is specified as compressible.
+
+# IHYBFLAG
+
+Hybrid formulation flag defined to be 1 for hybrid elements; 0 otherwise.
+
+# NUMSTATEV
+
+User-defined number of solution-dependent state variables associated with this material (see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# NUMFIELDV
+
+Number of field variables.
+
+# FIELDV
+
+Array of interpolated values of predefined field variables at this material point at the end of the increment based on the values read in at the nodes (initial values at the beginning of the analysis and current values during the analysis).
+
+# FIELDVINC
+
+Array of increments of predefined field variables at this material point for this increment, including any values updated by user subroutine USDFLD.
+
+# NUMPROPS
+
+Number of material properties entered for this user-defined hyperelastic material.
+
+<!-- source-page: 148 -->
+
+# PROPS
+
+Array of material properties entered for this user-defined hyperelastic material.
+
+# EBAR(NTENS)
+
+Modified Green strain tensor, $\overline { { \varepsilon } } ^ { G }$ , at the material point at the end of the increment.
+
+# AJ
+
+J, determinant of deformation gradient (volume ratio) at the end of the increment.
+
+# Example: Orthotropic Saint-Venant Kirchhoff model
+
+As a simple example of the coding of user subroutine UANISOHYPER\_STRAIN, consider the generalization to anisotropic hyperelasticity of the Saint-Venant Kirchhoff model. The strain energy function of the Saint-Venant Kirchhoff model can be expressed as a quadratic function of the Green strain tensor, $\varepsilon ^ { G }$ , as
+
+$$
+U (\boldsymbol {\varepsilon} ^ {G}) = \frac {1}{2} \boldsymbol {\varepsilon} ^ {G}: \mathbf {D}: \boldsymbol {\varepsilon} ^ {G},
+$$
+
+where is the fourth-order elasticity tensor. The derivatives of the strain energy function with respect to the Green strain are given as
+
+$$
+\frac {\partial U}{\partial \varepsilon^ {G}} = \mathbf {D}: \varepsilon^ {G},
+$$
+
+$$
+\frac {\partial^ {2} U}{\partial \varepsilon^ {G} \partial \varepsilon^ {G}} = \mathbf {D}.
+$$
+
+However, user subroutine UANISOHYPER\_STRAIN must return the derivatives of the strain energy function with respect to the modified Green strain tensor, $\overline { { \varepsilon } } ^ { G }$ , and the volume ratio, J, which can be accomplished easily using the following relationship between $\varepsilon ^ { G } , \overline { { \varepsilon } } ^ { G }$ , and :
+
+$$
+\boldsymbol {\varepsilon} ^ {G} = J ^ {\frac {2}{3}} \overline {{\boldsymbol {\varepsilon}}} ^ {G} + \frac {1}{2} (J ^ {\frac {2}{3}} - 1) \mathbf {I},
+$$
+
+where is the second-order identity tensor. Thus, using the chain rule we find
+
+$$
+\frac {\partial U}{\partial \overline {{\varepsilon}} ^ {G}} = J ^ {\frac {2}{3}} \frac {\partial U}{\partial \varepsilon^ {G}},
+$$
+
+$$
+\frac {\partial U}{\partial J} = \frac {\partial \varepsilon^ {G}}{\partial J}: \frac {\partial U}{\partial \varepsilon^ {G}},
+$$
+
+$$
+\frac {\partial^ {2} U}{\partial \overline {{\varepsilon}} ^ {G} \partial \overline {{\varepsilon}} ^ {G}} = J ^ {\frac {4}{3}} \frac {\partial^ {2} U}{\partial \varepsilon^ {G} \partial \varepsilon^ {G}},
+$$
+
+<!-- source-page: 149 -->
+
+$$
+\begin{array}{l} \frac {\partial^ {2} U}{\partial J ^ {2}} = \frac {\partial^ {2} \varepsilon^ {G}}{\partial J ^ {2}}: \frac {\partial U}{\partial \varepsilon^ {G}} + \frac {\partial \varepsilon^ {G}}{\partial J}: \frac {\partial^ {2} U}{\partial \varepsilon^ {G} \partial \varepsilon^ {G}}: \frac {\partial \varepsilon^ {G}}{\partial J}, \\ \frac {\partial^ {2} U}{\partial \overline {{\varepsilon}} ^ {G} \partial J} = \frac {2}{3 J} J ^ {\frac {2}{3}} \frac {\partial U}{\partial \varepsilon^ {G}} + J ^ {\frac {2}{3}} \frac {\partial^ {2} U}{\partial \varepsilon^ {G} \partial \varepsilon^ {G}}: \frac {\partial \varepsilon^ {G}}{\partial J}, \\ \end{array}
+$$
+
+where
+
+$$
+\frac {\partial \pmb {\varepsilon} ^ {G}}{\partial J} = \frac {2}{3 J} J ^ {\frac {2}{3}} (\overline {{\pmb {\varepsilon}}} ^ {G} + \frac {1}{2} \mathbf {I}) = \frac {2}{3 J} (\pmb {\varepsilon} ^ {G} + \frac {1}{2} \mathbf {I})
+$$
+
+and
+
+$$
+\frac {\partial^ {2} \varepsilon^ {G}}{\partial J ^ {2}} = - \frac {1}{3 J} \frac {\partial \varepsilon^ {G}}{\partial J}.
+$$
+
+In this example an auxiliary function is used to facilitate indexing into a fourth-order symmetric tensor. The user subroutine would be coded as follows:
+```txt
+subroutine uanisohyper_strain (
+    *    ebar, aj, ua, du1, du2, du3, temp, noel, cmname,
+    *    incmpFlag, ihybFlag, ndi, nshr, ntens,
+    *    numStatev, statev, numFieldv, fieldv, fieldvInc,
+    *    numProps, props)
+c
+    include 'aba_param.inc'
+c
+    dimension ebar(ntens), ua(2), du1(ntens+1)
+    dimension du2((ntens+1)*(ntens+2)/2)
+    dimension du3((ntens+1)*(ntens+2)/2)
+    dimension statev(numStatev), fieldv(numFieldv)
+    dimension fieldvInc(numFieldv), props(numProps)
+c
+    character*80 cmname
+c
+    parameter ( half    = 0.5d0,
+    $    one    = 1.d0,
+    $    two    = 2.d0,
+    $    third    = 1.d0/3.d0,
+    $    twothds    = 2.d0/3.d0,
+    $    four    = 4.d0 )
+*
+* Orthotropic Saint-Venant Kirchhoff strain energy function (3D)
+* 
+```
+
+<!-- source-page: 150 -->
+
+```python
+D1111=props(1)
+D1122=props(2)
+D2222=props(3)
+D1133=props(4)
+D2233=props(5)
+D3333=props(6)
+D1212=props(7)
+D1313=props(8)
+D2323=props(9)
+
+*
+d2UdE11dE11 = D1111
+d2UdE11dE22 = D1122
+d2UdE11dE33 = D1133
+
+*
+d2UdE22dE11 = d2UdE11dE22
+d2UdE22dE22 = D2222
+d2UdE22dE33 = D2233
+
+*
+d2UdE33dE11 = d2UdE11dE33
+d2UdE33dE22 = d2UdE22dE33
+d2UdE33dE33 = D3333
+
+*
+d2UdE12dE12 = D1212
+
+*
+d2UdE13dE13 = D1313
+
+*
+d2UdE23dE23 = D2323
+
+*
+xpow = exp (log(aj) * twothds)
+detuInv = one / aj
+
+*
+E11 = xpow * ebar(1) + half * (xpow - one)
+E22 = xpow * ebar(2) + half * (xpow - one)
+E33 = xpow * ebar(3) + half * (xpow - one)
+E12 = xpow * ebar(4)
+E13 = xpow * ebar(5)
+E23 = xpow * ebar(6)
+
+*
+term1 = twothds * detuInv
+dE11Dj = term1 * (E11 + half)
+dE22Dj = term1 * (E22 + half)
+dE33Dj = term1 * (E33 + half) 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_016.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_016.md
new file mode 100644
index 00000000..1dc8c93f
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_016.md
@@ -0,0 +1,313 @@
+<!-- source-page: 151 -->
+
+```txt
+dE12Dj = term1 * E12
+dE13Dj = term1 * E13
+dE23Dj = term1 * E23
+term2 = - third * detuInv
+d2E11DjDj = term2 * dE11Dj
+d2E22DjDj = term2 * dE22Dj
+d2E33DjDj = term2 * dE33Dj
+d2E12DjDj = term2 * dE12Dj
+d2E13DjDj = term2 * dE13Dj
+d2E23DjDj = term2 * dE23Dj
+*
+dUdE11 = d2UdE11dE11 * E11
+* + d2UdE11dE22 * E22
+* + d2UdE11dE33 * E33
+dUdE22 = d2UdE22dE11 * E11
+* + d2UdE22dE22 * E22
+* + d2UdE22dE33 * E33
+dUdE33 = d2UdE33dE11 * E11
+* + d2UdE33dE22 * E22
+* + d2UdE33dE33 * E33
+dUdE12 = two * d2UdE12dE12 * E12
+dUdE13 = two * d2UdE13dE13 * E13
+dUdE23 = two * d2UdE23dE23 * E23
+*
+U = half * ( E11*dUdE11 + E22*dUdE22 + E33*dUdE33 )
+* + E12*dUdE12 + E13*dUdE13 + E23*dUdE23
+*
+ua(2) = U
+ua(1) = ua(2)
+*
+du1(1) = xpow * dUdE11
+du1(2) = xpow * dUdE22
+du1(3) = xpow * dUdE33
+du1(4) = xpow * dUdE12
+du1(5) = xpow * dUdE13
+du1(6) = xpow * dUdE23
+du1(7) = dUdE11*dE11Dj + dUdE22*dE22Dj + dUdE33*dE33Dj
+* + two * ( dUdE12*dE12Dj
+* + dUdE13*dE13Dj
+* + dUdE23*dE23Dj )
+*
+xpow2 = xpow * xpow
+```
+
+<!-- source-page: 152 -->
+
+\*
+
+```txt
+du2 (indx(1,1)) = xpow2 * d2UdE11dE11
+du2 (indx(1,2)) = xpow2 * d2UdE11dE22
+du2 (indx(2,2)) = xpow2 * d2UdE22dE22
+du2 (indx(1,3)) = xpow2 * d2UdE11dE33
+du2 (indx(2,3)) = xpow2 * d2UdE22dE33
+du2 (indx(3,3)) = xpow2 * d2UdE33dE33
+du2 (indx(1,4)) = zero
+du2 (indx(2,4)) = zero
+du2 (indx(3,4)) = zero
+du2 (indx(4,4)) = xpow2 * d2UdE12dE12
+du2 (indx(1,5)) = zero
+du2 (indx(2,5)) = zero
+du2 (indx(3,5)) = zero
+du2 (indx(4,5)) = zero
+du2 (indx(5,5)) = xpow2 * d2UdE13dE13
+du2 (indx(1,6)) = zero
+du2 (indx(2,6)) = zero
+du2 (indx(3,6)) = zero
+du2 (indx(4,6)) = zero
+du2 (indx(5,6)) = zero
+du2 (indx(6,6)) = xpow2 * d2UdE23dE23 
+```
+
+\*
+
+```txt
+du2(indx(1,7)) = xpow * (term1 * dUdE11
+* + d2UdE11dE11 * dE11Dj
+* + d2UdE11dE22 * dE22Dj
+* + d2UdE11dE33 * dE33Dj)
+du2(indx(2,7)) = xpow * (term1 * dUdE22
+* + d2UdE22dE11 * dE11Dj
+* + d2UdE22dE22 * dE22Dj
+* + d2UdE22dE33 * dE33Dj)
+du2(indx(3,7)) = xpow * (term1 * dUdE33
+* + d2UdE33dE11 * dE11Dj
+* + d2UdE33dE22 * dE22Dj
+* + d2UdE33dE33 * dE33Dj)
+du2(indx(4,7)) = xpow * (term1 * dUdE12
+* + two * d2UdE12dE12 * dE12Dj)
+du2(indx(5,7)) = xpow * (term1 * dUdE13
+* + two * d2UdE13dE13 * dE23Dj)
+du2(indx(6,7)) = xpow * (term1 * dUdE23
+* + two * d2UdE23dE23 * dE13Dj)
+du2(indx(7,7)) = dUdE11*d2E11DjDj
+```
+
+<!-- source-page: 153 -->
+
+```txt
+* +dUdE22*d2E22DjDj
+* +dUdE33*d2E33DjDj
+* + two*( dUdE12*d2E12DjDj
+* +dUdE13*d2E13DjDj
+* +dUdE23*d2E23DjDj)
+* + d2UdE11dE11 * dE11Dj * dE11Dj
+* + d2UdE22dE22 * dE22Dj * dE22Dj
+* + d2UdE33dE33 * dE33Dj * dE33Dj
+* + two * ( d2UdE11dE22 * dE11Dj * dE22Dj
+* +d2UdE11dE33 * dE11Dj * dE33Dj
+* +d2UdE22dE33 * dE22Dj * dE33Dj )
+* + four * ( d2UdE12dE12 * dE12Dj * dE12Dj
+* +d2UdE13dE13 * dE13Dj * dE13Dj
+* +d2UdE23dE23 * dE23Dj * dE23Dj )
+*
+return
+end
+*
+* Maps index from Square to Triangular storage
+* of symmetric matrix
+*
+integer function index( i, j )
+*
+include 'aba_param.inc'
+* 
+ii = min(i,j)
+jj = max(i,j)
+*
+indx = ii + jj*(jj-1)/2
+*
+return
+end 
+```
+
+<!-- source-page: 154 -->
+
+<!-- source-page: 155 -->
+
+# 1.1.22 UCORR: User subroutine to define cross-correlation properties for random response loading.
+
+Product: Abaqus/Standard
+
+# References
+
+• “Random response analysis,” Section 6.3.11 of the Abaqus Analysis User’s Guide   
+• \*CORRELATION   
+• “Random response to jet noise excitation,” Section 1.4.10 of the Abaqus Benchmarks Guide
+
+# Overview
+
+User subroutine UCORR:
+
+• can be used to define the coefficients for the cross-correlation matrix in a random response analysis;   
+• will be called once for the combination of any two degrees of freedom with nonzero prescribed loads for each load case specified as a concentrated or distributed load or once for the combination of any two excitation directions specified as a base motion;   
+• allows correlation coefficients to be defined as a function of nodal coordinates; and   
+• ignores any data specified outside the user subroutine for the associated cross-correlation matrix.
+
+# Cross-correlation for base motion excitation
+
+The spatial correlation matrix for base motion excitation is defined by the coefficients $\Psi _ { i j } ^ { I J }$ in user subroutine UCORR, where $i , j$ are excitation directions and J corresponds to the Jth frequency function referenced under load case I.
+
+# Cross-correlation for point loads and distributed loads
+
+The spatial correlation matrix of the load is defined as follows. Let $F _ { ( N , i ) } ^ { I }$ be the load applied to degree of freedom i at node N in load case I, through the use of a concentrated or distributed load. Let J correspond to the Jth frequency function referenced under load case I. The spatial correlation matrix used in the random response analysis for this load case is then
+
+$$
+\Psi_ {(N, i) (M, j)} ^ {I J} = C _ {(N, i) (M, j)} ^ {I J} F _ {(N, i)} ^ {I} F _ {(M, j)} ^ {I},
+$$
+
+where $C _ { ( N , i ) ( M , j ) } ^ { I J }$ are the coefficients defined in user subroutine UCORR. Typically the load magnitude is given as 1.0; therefore, the load definition is simply selecting the nonzero terms that will appear in $\Psi _ { ( N , i ) ( M , j ) } ^ { \bar { I } , J }$ .
+
+<!-- source-page: 156 -->
+
+User subroutine interface   
+```txt
+SUBROUTINE UCORR(PSD,CORRR,CORRI,KSTEP,LCASE,JNODE1,JDOF1,1 JNODE2,JDOF2,COOR1,COOR2)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION COOR1(3),COOR2(3)
+CHARACTER*80 PSD
+user coding to define CORRR and CORRI
+RETURN
+END 
+```
+
+Variables to be defined   
+```txt
+CORRR
+Real part of the cross-correlation scaling factor.
+CORRI
+Imaginary part of the cross-correlation scaling factor. 
+```
+
+Variables passed in for information   
+```txt
+PSD
+User-specified name for the frequency function that references this correlation, left justified.
+
+KSTEP
+Step number.
+
+LCASE
+Load case number, I.
+
+JNODE1
+First node involved, N (not used for base motion excitation).
+
+JDOF1
+Degree of freedom i at the first node (for concentrated or distributed load excitation) or global e direction i (for base motion excitation).
+
+JNODE2
+Second node involved, M (not used for base motion excitation). 
+```
+
+<!-- source-page: 157 -->
+
+# JDOF2
+
+Degree of freedom $j$ at the second node (for concentrated or distributed load excitation) or global excitation direction $j$ (for base motion excitation).
+
+# COOR1
+
+An array containing the coordinates of the first node (not used for base motion excitation).
+
+# COOR2
+
+An array containing the coordinates of the second node (not used for base motion excitation).
+
+<!-- source-page: 158 -->
+
+<!-- source-page: 159 -->
+
+# 1.1.23 UCREEPNETWORK: User subroutine to define time-dependent behavior (creep) for models defined within the parallel rheological framework.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Parallel rheological framework,” Section 22.8.2 of the Abaqus Analysis User’s Guide   
+• “Nonlinear large-strain viscoelasticity with hyperelasticity,” Section 2.2.8 of the Abaqus Verification Guide   
+• \*VISCOELASTIC
+
+# Overview
+
+User subroutine UCREEPNETWORK:
+
+• is intended to provide creep laws for nonlinear viscoelastic networks for models defined using the parallel rheological framework (see “Parallel rheological framework,” Section 22.8.2 of the Abaqus Analysis User’s Guide);   
+• can use and update solution-dependent state variables; and   
+• can be used in conjunction with user subroutine USDFLD to redefine any field variables before they are passed in.
+
+# Model description
+
+The user subroutine allows a creep law of the following general form to be defined:
+
+$$
+\dot {\bar {\varepsilon}} ^ {c r} = g ^ {c r} (\bar {\varepsilon} ^ {c r}, I _ {1} ^ {c r}, \bar {I} _ {1}, \bar {I} _ {2}, J, p, \tilde {q}, t, \theta , F V),
+$$
+
+where
+
+$$
+I _ {1} ^ {c r} = \mathbf {I}: \mathbf {C} ^ {c r},
+$$
+
+and
+
+I is the identity tensor, $\mathbf{C}^{cr}$ is the right Cauchy-Green creep strain tensor, $\dot{\bar{\varepsilon}}^{cr}$ is the equivalent creep strain rate, $\bar{\varepsilon}^{cr}$ is the equivalent creep strain, $\bar{I}_1$ is the first invariant of $\bar{\mathbf{B}}$ , $\bar{I}_2$ is the second invariant of $\bar{\mathbf{B}}$ , $J$ is the determinant of the deformation gradient, $\mathbf{F}$ ,
+
+<!-- source-page: 160 -->
+
+p is the Kirchhoff pressure, $\tilde{q}$ is the equivalent deviatoric Kirchhoff stress, $t$ is the time, $\theta$ is the temperature, and $FV$ are field variables.
+
+The left Cauchy-Green strain tensor, , is defined as
+
+$$
+\bar {\mathbf {B}} = \bar {\mathbf {F}} \bar {\mathbf {F}} ^ {T},
+$$
+
+where is the deformation gradient with volume change eliminated, which is computed using
+
+$$
+\bar {\mathbf {F}} = J ^ {- \frac {1}{3}} \mathbf {F}.
+$$
+
+The user subroutine must define the increment of creep equivalent strain, $\Delta \bar { \varepsilon } ^ { c r }$ , as a function of the time increment, $\Delta t ,$ and the variables used in the definition of $\boldsymbol { \cdot } \boldsymbol { g } ^ { c r }$ , as well as the derivatives of the equivalent creep strain increment with respect to those variables. If any solution-dependent state variables are included in the definition of $\boldsymbol { g } ^ { c r }$ , they must also be integrated forward in time in this user subroutine.
+
+User subroutine interface   ```csv
+subroutine ucreepnetwork (
+C Must be updated
+* outputData,
+C Can be updated
+* statev,
+C Information (Read only)
+* nOutput,
+* nstatv,
+* networkid,
+* coords,
+* temp,
+* dtemp,
+* nfield,
+* predef,
+* dpred,
+* nprops,
+* props,
+* i_array,
+* niarray,
+* r_array,
+* nrarray,
+* c_array, 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_017.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_017.md
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--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_017.md
@@ -0,0 +1,476 @@
+<!-- source-page: 161 -->
+
+```c
+* ncarray)
+C
+    include 'aba_param.inc'
+C
+    parameter( io_creep_equiv_creepinc    = 1,
+    *    io_creep_deqcreepinc_deqcreep    = 2,
+    *    io_creep_deqcreepinc_dqtild    = 3,
+    *    io_creep_deqcreepinc_dinv1crp    = 4,
+    *    io_creep_deqcreepinc_dinv1    = 5,
+    *    io_creep_deqcreepinc_dinv2    = 6,
+    *    io_creep_deqcreepinc_ddetf    = 7,
+    *    io_creep_deqcreepinc_dpress    = 8 )
+C
+    parameter( i_creep_kstep    = 1,
+    *    i_creep_kinc    = 2,
+    *    i_creep_noel    = 3,
+    *    i_creep_npt    = 4,
+    *    i_creep_layer    = 5,
+    *    i_creep_kspt    = 6,
+    *    i_creep_lend    = 7 )
+C
+    parameter( ir_creep_step_time    = 1,
+    *    ir_creep_total_time    = 2,
+    *    ir_creep_creep_time    = 3,
+    *    ir_creep_timeinc    = 4,
+    *    ir_creep_equiv_creep_strain    = 5,
+    *    ir_creep_qtild    = 6,
+    *    ir_creep_inv1crp    = 7,
+    *    ir_creep_inv1    = 8,
+    *    ir_creep_inv2    = 9,
+    *    ir_creep_detf    = 10,
+    *    ir_creep_press    = 11 )
+C
+    parameter( ic_creep_material_name    = 1 )
+C
+    dimension
+    *    statev(nstatv),
+    *    predef(nfield),
+    *    dpred(nfield),
+    *    coords(*),
+    *    props(nprops),
+    *    outputData(nOutput), 
+```
+
+<!-- source-page: 162 -->
+
+```txt
+* i_array(niarray),
+* r_array(nrarray)
+
+character*80 c_array(ncarray)
+
+C
+
+user coding to define outputData(io_creep_equiv_creepinc),
+    outputData(io_creep_deqcreepinc_deqcreep),
+    outputData(io_creep_deqcreepinc_dqtild),
+    outputData(io_creep_deqcreepinc_dinv1crp),
+    outputData(io_creep_deqcreepinc_dinv1),
+    outputData(io_creep_deqcreepinc_dinv2),
+    outputData(io_creep_deqcreepinc_ddetf) and
+    outputData(io_creep_deqcreepinc_dpress)
+
+return
+end 
+```
+
+Variables to be defined   
+outputData(io_creep_equiv_creepinc)
+Equivalent creep strain increment, $\Delta\bar{\varepsilon}^{cr}$ .
+
+outputData(io_creep_deqcreepinc_deqcreep)
+The derivative: $\partial\Delta\bar{\varepsilon}^{cr}/\partial\bar{\varepsilon}^{cr}$ .
+
+outputData(io_creep_deqcreepinc_dqtild)
+The derivative: $\partial\Delta\bar{\varepsilon}^{cr}/\partial\tilde{q}$ .
+
+outputData(io_creep_deqcreepinc_dinv1crp)
+The derivative: $\partial\Delta\bar{\varepsilon}^{cr}/\partial I_{1}^{cr}$ .
+
+outputData(io_creep_deqcreepinc_dinv1)
+The derivative: $\partial\Delta\bar{\varepsilon}^{cr}/\partial\bar{I}_{1}$ .
+
+outputData(io_creep_deqcreepinc_dinv2)
+The derivative: $\partial\Delta\bar{\varepsilon}^{cr}/\partial\bar{I}_{2}$ .
+
+outputData(io_creep_deqcreepinc_ddetf)
+The derivative: $\partial\Delta\bar{\varepsilon}^{cr}/\partial J$ .
+
+outputData(io_creep_deqcreepinc_dpress)
+The derivative: $\partial\Delta\bar{\varepsilon}^{cr}/\partial p$ .
+
+<!-- source-page: 163 -->
+
+# Variable that can be updated
+
+# statev
+
+An array containing the user-defined solution-dependent state variables at this point.
+
+# Variables passed in for information
+
+# nOutput
+
+Size of array outputData.
+
+# nstatv
+
+Number of solution-dependent state variables associated with this material.
+
+# networkid
+
+Network identification number, which identifies the network for which creep is defined.
+
+# coords
+
+An array containing the current coordinates at this point.
+
+# temp
+
+Temperature at the end of the increment.
+
+# dtemp
+
+Increment of temperature.
+
+# nfield
+
+Number of field variables.
+
+# predef
+
+An array of interpolated values of predefined field variables at this point at the end of the increment, based on the values read in at the nodes and, optionally, redefined in user subroutine USDFLD.
+
+# dpred
+
+An array of increments of predefined field variables.
+
+# nprops
+
+User-specified number of property values associated with this creep model.
+
+# props
+
+An array of user-specified property values that are used to define the creep model.
+
+# i\_array(i\_creep\_kstep)
+
+Step number.
+
+# i\_array(i\_creep\_kinc)
+
+Increment number.
+
+<!-- source-page: 164 -->
+
+i\_array(i\_creep\_noel)
+
+Element number.
+
+i\_array(i\_creep\_npt)
+
+Integration point.
+
+i\_array(i\_creep\_layer)
+
+Layer number (for layered solids).
+
+i\_array(i\_creep\_kspt)
+
+Section point number within the current layer.
+
+i\_array(i\_creep\_lend)
+
+Start/end of increment flag. The value of 0 denotes the beginning of the increment, and the value of 1 denotes the end of the increment.
+
+niarray
+
+Size of array i\_array.
+
+r\_array(ir\_creep\_step\_time)
+
+Value of step time at the end of the increment.
+
+r\_array(ir\_creep\_total\_time)
+
+Value of total time at the end of the increment.
+
+r\_array(ir\_creep\_creep\_time)
+
+Value of creep time at the end of the increment.
+
+r\_array(ir\_creep\_timeinc)
+
+Time increment.
+
+r\_array(ir\_creep\_equiv\_creep\_strain)
+
+Equivalent creep strain.
+
+r\_array(ir\_creep\_qtild)
+
+Equivalent deviatoric Kirchhoff stress.
+
+r\_array(ir\_creep\_inv1crp)
+
+The first invariant, , of the right Cauchy-Green creep strain tensor, .
+
+r\_array(ir\_creep\_inv1)
+
+The first invariant, , of the left Cauchy-Green strain tensor, .
+
+r\_array(ir\_creep\_inv2)
+
+The second invariant, , of the left Cauchy-Green strain tensor, .
+
+<!-- source-page: 165 -->
+
+r\_array(ir\_creep\_detf)
+
+The determinant of the deformation gradient, .
+
+r\_array(ir\_creep\_press)
+
+Kirchhoff pressure.
+
+nrarray
+
+Size of array r\_array.
+
+c\_array(ic\_creep\_material\_name)
+
+User-specified material name, left justified. Some internal material models are given names starting with the “ABQ\_” character string. To avoid conflict, you should not use “ABQ\_” as the leading string for the material name.
+
+ncarray
+
+Size of array c\_array.
+
+# Example: Bergstrom-Boyce model
+
+As an example of the coding of user subroutine UCREEPNETWORK, consider the Bergstrom-Boyce model. In this case the equivalent creep strain rate is expressed as (see “Parallel rheological framework,” Section 22.8.2 of the Abaqus Analysis User’s Guide)
+
+$$
+\dot {\bar {\varepsilon}} ^ {c r} = A (\lambda^ {c r} - 1 + E) ^ {C} (\tilde {q}) ^ {m},
+$$
+
+where
+
+$$
+\lambda^ {c r} = \sqrt {\frac {1}{3} \mathbf {I} : \mathbf {C} ^ {c r}}
+$$
+
+and
+
+$\mathbf{C}^{cr}$ is the right Cauchy-Green creep strain tensor, $\tilde{q}$ is the equivalent deviatoric Kirchhoff stress, and $A, m, C,$ and $E$ are material parameters.
+
+The user subroutine would be coded as follows:
+```c
+subroutine ucreepnetwork (
+C Must be updated
+* outputData,
+C Can be updated
+* statev,
+C Information (Read only)
+* nOutput,
+* nstatv,
+* networkid, 
+```
+
+<!-- source-page: 166 -->
+
+```txt
+* coords,
+* temp,
+* dtemp,
+* nfield,
+* predef,
+* dpred,
+* nprops,
+* props,
+* i_array,
+* niarray,
+* r_array,
+* nrarray,
+* c_array,
+* ncarray) 
+```
+
+include 'aba\_param.inc'   
+```txt
+parameter( io_creep_equiv_creepinc = 1,
+* io_creep_deqcreepinc_deqcreep = 2,
+* io_creep_deqcreepinc_dqtild = 3,
+* io_creep_deqcreepinc_dinv1crp = 4,
+* io_creep_deqcreepinc_dinv1 = 5,
+* io_creep_deqcreepinc_dinv2 = 6,
+* io_creep_deqcreepinc_ddetf = 7,
+* io_creep_deqcreepinc_dpress = 8 ) 
+```
+
+```txt
+parameter( i_creep_kstep = 1,
+* i_creep_kinc = 2,
+* i_creep_noel = 3,
+* i_creep_npt = 4,
+* i_creep_layer = 5,
+* i_creep_kspt = 6,
+* i_creep_lend = 7 
+```
+
+```c
+parameter( ir_creep_step_time = 1,
+* ir_creep_total_time = 2,
+* ir_creep_creep_time = 3,
+* ir_creep_timeinc = 4,
+* ir_creep_equiv_creep_strain = 5,
+* ir_creep_qtild = 6,
+* ir_creep_inv1crp = 7,
+* ir_creep_inv1 = 8, 
+```
+
+<!-- source-page: 167 -->
+
+```python
+* ir_creep_inv2 = 9,
+* ir_creep_detf = 10,
+* ir_creep_press = 11
+C
+    parameter(ic_creep_material_name = 1)
+C
+C model parameters
+    parameter ( zero=0.0d0, half=0.5d0, one=1.0d0, two=2.0d0, & three=3.0d0, five=5.0d0, six=6.0d0 )
+C
+    dimension
+    * statev(nstatv),
+    * predef(nfield),
+    * dpred(nfield),
+    * coords(*),
+    * props(nprops),
+    * outputData(nOutput),
+    * i_array(niarray),
+    * r_array(nrarray)
+
+    character*80 c_array(ncarray)
+C
+C Bergstrom-Boyce Model
+C
+    A = props(1)
+    dm = props(2)
+    C = props(3)
+    E = props(4)
+C
+    dI1 = r_array(ir_creep_inv1crp)
+    dLamb = (dI1/three)**half
+    sigmaB = r_array(ir_creep_qtild)
+    dt = r_array(ir_creep_timeinc)
+C
+C deq
+    deq = dt*A*(dLamb-one+E)**C*sigmaB**dm
+C
+C d(deq)/(dI1crp)
+    deqdi1 = deq*C/(dLamb-one+E)/dLamb/six
+C
+C d(eq)/d(eq)
+    deqeq = zero 
+```
+
+<!-- source-page: 168 -->
+
+```txt
+C
+C d(eq)/d(q)
+    deqdq = dm*dt*A*(dLamb-one+E)**C*sigmaB**(dm-one)
+C
+C set output
+    outputData(io_creep_equiv_creepinc) = deq
+    outputData(io_creep_deqcreepinc_deqcreep) = deqeq
+    outputData(io_creep_deqcreepinc_dqtild) = deqdq
+    outputData(io_creep_deqcreepinc_dinv1crp) = deqdi1
+    outputData(io_creep_deqcreepinc_dinv1) = zero
+    outputData(io_creep_deqcreepinc_dinv2) = zero
+    outputData(io_creep_deqcreepinc_ddetf) = zero
+    outputData(io_creep_deqcreepinc_dpress) = zero
+C
+    return
+    end 
+```
+
+<!-- source-page: 169 -->
+
+# 1.1.24 UDECURRENT: User subroutine to define nonuniform volume current density in an eddy current or magnetostatic analysis.
+
+Product: Abaqus/Standard
+
+# References
+
+• “Eddy current analysis,” Section 6.7.5 of the Abaqus Analysis User’s Guide   
+• “Magnetostatic analysis,” Section 6.7.6 of the Abaqus Analysis User’s Guide   
+• \*DECURRENT
+
+# Overview
+
+User subroutine UDECURRENT:
+
+• can be used to define the variation of volume current density vector as a function of position, time, element number, etc. for a transient eddy current or magnetostatic analysis or as a function of position, excitation frequency, phase, element number, etc. for a time-harmonic eddy current analysis;   
+• will be called at each load integration point for each element-based nonuniform volume current density definition during eddy current or magnetostatic analysis; and   
+• ignores any amplitude references that may appear with the associated step definition or nonuniform distributed volume current density definition.
+
+# User subroutine interface
+
+```txt
+subroutine udecurrent (
+C Write only -
+* bodycurrent,
+C Read only -
+* predef, coords, nBlock,
+* i_array, niarray,
+* r_array, nrarray,
+* c_array, narray )
+C
+include 'aba_param.inc'
+C
+dimension bodycurrent(nBlock,*),
+* predef(nBlock,2,*),
+* coords(nBlock,*),
+* i_array(*),
+* r_array(*) 
+```
+
+<!-- source-page: 170 -->
+
+```python
+c
+    character*80 c_array(*)
+
+    parameter( i_udecurr_kstep = 1,
+    * i_udecurr_kinc = 2,
+    * i_udecurr_noel = 3,
+    * i_udecurr_npt = 4,
+    * i_udecurr_jltyp = 5,
+    * i_udecurr_phase = 6,
+    * i_udecurr_proc = 7,
+    * i_udecurr_nfld = 8 )
+
+    parameter( ir_udecurr_time_1 = 1,
+    * ir_udecurr_time_2 = 2,
+    * ir_udecurr_time_3 = 3 )
+
+    parameter( i_jltyp_cj = 1 )
+
+    parameter( i_proc_lf_th = 1,
+    * i_proc_lf_td = 2,
+    * i_proc_ms = 3 )
+
+    parameter( i_udecurr_phase_real = 1,
+    * i_udecurr_phase_imag = 2 )
+
+    user coding to define bodycurrent
+    return
+    end 
+```
+
+# Variable to be defined
+
+bodycurrent(nBlock,\*)
+
+Components of the body current density vector for a block of load integration points. The units are $\mathrm { C L } ^ { - 2 } \mathrm { T } ^ { - 1 }$ . bodycurrent will be passed into the routine as the vector specified as part of the elementbased distributed volume current density definition. If the vector is not defined, bodycurrent will be passed in as zero.
+
+# Variables passed in for information
+
+predef(2,\*)
+
+An array containing values of temperature and all the predefined field variables at the current load integration point, based on interpolation from the values specified at the nodes. The first value in a pair, predef(1,\*), corresponds to initial values; the second value, predef(2,\*), corresponds to
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_018.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_018.md
new file mode 100644
index 00000000..b07e8aca
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_018.md
@@ -0,0 +1,445 @@
+<!-- source-page: 171 -->
+
+incremental values of the temperature and field variables. The first entry (for example, predef(1,1) or predef(2,1)) contains the temperature; the subsequent entries (for example, predef(1,2) or predef(2,2) onward) contain the field variables.
+
+coords(nBlock,\*)
+
+An array containing the coordinates of the load integration points.
+
+nBlock
+
+Number of load integration points in this block. Currently equal to 1.
+
+i\_array(i\_udecurr\_kstep)
+
+Step number.
+
+i\_array(i\_udecurr\_kinc)
+
+Increment number.
+
+i\_array(i\_udecurr\_noel)
+
+Element number.
+
+i\_array(i\_udecurr\_npt)
+
+Load integration point number.
+
+i\_array(i\_udecurr\_jltyp)
+
+Currently equal to 1.
+
+i\_array(i\_udecurr\_phase)
+
+This value is relevant only for a time-harmonic eddy current analysis and is either 1 (i\_udecurr\_phase\_real) or 2 (i\_udecurr\_phase\_imag), depending on whether the current call to the user subroutine defines the real (in-phase) or the imaginary (out-of-phase) part of the volume current density vector.
+
+i\_array(i\_udecurr\_proc)
+
+Equal to 1 for a time-harmonic eddy current procedure, 2 for a transient eddy current procedure, and 3 for a magnetostatic procedure.
+
+i\_array(i\_udecurr\_nfld)
+
+Total number of predefined field variables.
+
+niarray
+
+Size of array i\_array. Currently equal to 8.
+
+r\_array(ir\_udecurr\_time\_1)
+
+Excitation frequency in cycles/time for a time-harmonic eddy current analysis; alternatively, the value of step time at the beginning of the current increment for a transient eddy current or magnetostatic analysis.
+
+<!-- source-page: 172 -->
+
+r\_array(ir\_udecurr\_time\_2)
+
+Excitation frequency in radians/time for a time-harmonic eddy current analysis; alternatively, the value of total time at the beginning of the current increment for a transient eddy current or magnetostatic analysis.
+
+r\_array(ir\_udecurr\_time\_3)
+
+Time increment for a transient eddy current or magnetostatic analysis.
+
+nrarray
+
+Size of array r\_array. Currently equal to 3.
+
+c\_array(1)
+
+Not used.
+
+ncarray
+
+Size of array c\_array(1). Currently equal to 1.
+
+<!-- source-page: 173 -->
+
+# 1.1.25 UDEMPOTENTIAL: User subroutine to define nonuniform magnetic vector potential on a surface in an eddy current or magnetostatic analysis.
+
+Product: Abaqus/Standard
+
+# References
+
+• “Eddy current analysis,” Section 6.7.5 of the Abaqus Analysis User’s Guide   
+• “Magnetostatic analysis,” Section 6.7.6 of the Abaqus Analysis User’s Guide   
+• \*D EM POTENTIAL
+
+# Overview
+
+User subroutine UDEMPOTENTIAL:
+
+• can be used to define the variation of the magnetic vector potential as a function of position, time, element number, etc. for a transient eddy current or magnetostatic analysis or as a function of position, excitation frequency, phase, element number, etc. for a time-harmonic eddy current analysis;   
+• will be called for each surface-based nonuniform electromagnetic potential definition during eddy current or magnetostatic analysis; and   
+• ignores any amplitude references that may appear with the associated step definition or nonuniform distributed electromagnetic potential definition.
+
+# User subroutine interface
+
+```txt
+subroutine udempotential (
+C Write only -
+* vecPot,
+C Read only -
+* coords, nBlock,
+* i_array, niarray,
+* r_array, nrarray,
+* c_array, ncarray )
+C
+include 'aba_param.inc'
+C
+dimension vecPot(nBlock,*),
+* coords(nBlock,*),
+* i_array(*),
+* r_array(*)
+C 
+```
+
+<!-- source-page: 174 -->
+
+```lua
+character*80 c_array(*)
+
+parameter(i_udempot_kstep = 1,
+* i_udempot_kinc = 2,
+* i_udempot_noel = 3,
+* i_udempot_currtyp = 4,
+* i_udempot_phase = 5,
+* i_udempot_proc = 6)
+
+parameter(ir_udempot_time_1 = 1,
+* ir_udempot_time_2 = 2,
+* ir_udempot_time_3 = 3)
+
+parameter(ic_udempot_surf = 1)
+
+parameter(i_pottyp_mvp = 1)
+
+parameter(i_proc_lf_th = 1,
+* i_proc_lf_td = 2,
+* i_proc_ms = 3)
+
+parameter(i_udempot_phase_real = 1,
+* i_udempot_phase_imag = 2)
+
+user coding to define vecPot
+
+return
+end 
+```
+
+# Variable to be defined
+
+vecPot(nBlock,\*)
+
+Components of the magnetic vector potential at a block of surface points. vecPot will be passed into the routine as the vector specified as part of the surface-based nonuniform magnetic vector potential definition. If the vector is not defined, vecPot will be passed in as zero.
+
+# Variables passed in for information
+
+coords(nBlock,\*)
+
+An array containing the coordinates of a block of surface points.
+
+nBlock
+
+Number of surface points in this block. Currently equal to 1.
+
+<!-- source-page: 175 -->
+
+```txt
+i_array(i_udempot_kstep)
+Step number. 
+```
+
+```txt
+i_array(i_udempot_kinc)
+Increment number. 
+```
+
+```pickle
+i_array(i_udempot_noel)
+Element number. 
+```
+
+```txt
+i_array(i_udempot_pottyp)
+Currently equal to 1. 
+```
+
+```python
+i_array(i_udempot_phase) 
+```
+
+This value is relevant only for a time-harmonic eddy current analysis and is either 1 (i\_udempot\_phase\_real) or 2 (i\_udempot\_phase\_imag), depending on whether the current call to the user subroutine defines the real (in-phase) or the imaginary (out-of-phase) part of the magnetic vector potential.
+
+```txt
+i_array(i_udempot_proc) 
+```
+
+Equal to 1 for a time-harmonic eddy current procedure, 2 for a transient eddy current procedure, and 3 for a magnetostatic procedure.
+
+```txt
+niarray 
+```
+
+Size of array i\_array. Currently equal to 6.
+
+```txt
+r_array(ir_udempot_time_1)
+```
+
+Excitation frequency in cycles/time for a time-harmonic eddy current analysis; alternatively, the value of step time at the beginning of the current increment for a transient eddy current or magnetostatic analysis.
+
+```txt
+r_array(ir_udempot_time_2) 
+```
+
+Excitation frequency in radians/time for a time-harmonic eddy current analysis; alternatively, the value of total time at the beginning of the current increment for a transient eddy current or magnetostatic analysis.
+
+```txt
+r_array(ir_udempot_time_3) 
+```
+
+Time increment for a transient eddy current or magnetostatic analysis.
+
+```txt
+nrarray 
+```
+
+Size of array r\_array. Currently equal to 3.
+
+```python
+c_array(ic_udempot_surf) 
+```
+
+Surface name.
+
+<!-- source-page: 176 -->
+
+# ncarray
+
+Size of array c\_array. Currently equal to 1.
+
+<!-- source-page: 177 -->
+
+# 1.1.26 UDMGINI: User subroutine to define the damage initiation criterion.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Progressive damage and failure,” Section 24.1.1 of the Abaqus Analysis User’s Guide   
+• “Modeling discontinuities as an enriched feature using the extended finite element method,” Section 10.7.1 of the Abaqus Analysis User’s Guide   
+• \*DAMAGE INITIATION
+
+# Overview
+
+User subroutine UDMGINI:
+
+• can be used to specify a user-defined damage initiation criterion;   
+• allows the specification of more than one failure mechanism in an element, with the most severe one governing the actual failure;   
+• can be used in combination with several Abaqus built-in damage evolution models, with each model corresponding to a particular failure mechanism;   
+• will be called at all integration points of elements for which the material definition contains userdefined damage initiation criterion;   
+• can call utility routine GETVRM to access material point data; and   
+• is currently available only for enriched elements.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE UDMGINI (FINDEX, NFINDEX, FNORMAL, NDI, NSHR, NTENS, PROPS,
+    1 NPROPS, STATEV, NSTATEV, STRESS, STRAIN, STRAINEE, LXFEM, TIME,
+    2 DTIME, TEMP, DTEMP, PREDEF, DPRED, NFIELD, COORDS, NOEL, NPT, LAYER,
+    3 KSPT, KSTEP, KINC, KDIRCYC, KCYCLELCF, TIMECYC, SSE, SPD, SCD, SVD,
+    4 SMD, JMAC, JMATYP, MATLAYO, LACCFLA, CELENT, DROT, ORI)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION FINDEX (NFINDEX), FNORMAL (NDI, NFINDEX), COORDS (*),
+    1 STRESS (NTENS), STRAIN (NTENS), STRAINEE (NTENS), PROPS (NPROPS),
+    2 STATEV (NSTATV), PREDEF (NFIELD), DPRED (NFIELD), TIME (2), JMAC (*),
+    3 JMATYP (*), DROT (3, 3), ORI (3, 3) 
+```
+
+<!-- source-page: 178 -->
+
+user coding to define FINDEX, and FNORMAL
+
+RETURN
+
+END
+
+# Variables to be defined
+
+# FINDEX(NFINDEX)
+
+A Vector defining the indices for all the failure mechanisms.
+
+# FNORMAL(NDI, NFINDEX)
+
+An Array defining the normal direction to the fracture plane (three dimensions) or line (two dimensions) for each failure mechanism.
+
+# Variables that can be updated
+
+# STATEV
+
+An array containing the user-defined solution-dependent state variables at this point. This array will be passed in containing the values of these variables at the start of the increment unless the values are updated in user subroutine USDFLD. They can be updated in this subroutine to their values at the end of the increment. You define the size of this array by allocating space for it (see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide, for more information).
+
+# SSE,SPD,SCD,SVD,SMD
+
+Specific elastic strain energy, plastic dissipation, “creep” dissipation, viscous, and damage energy, respectively, passed in as the values at the start of the increment and should be updated to the corresponding specific energy values at the end of the increment. They have no effect on the solution, except that they are used for energy output.
+
+# Variables passed in for information
+
+# NFINDEX
+
+Number of indices for all failure mechanisms.
+
+# NDI
+
+Number of direct stress components at this point.
+
+# NSHR
+
+Number of engineering shear stress components at this point.
+
+# NTENS
+
+Size of the stress or strain component array (NRI + NSHR).
+
+# PROPS(NPROPS)
+
+User-specified array of material constants associated with this user-defined failure criterion.
+
+<!-- source-page: 179 -->
+
+# NPROPS
+
+User-defined number of material constants associated with this user-defined failure criterion.
+
+# NSTATV
+
+Number of solution-dependent state variables associated with this material (specified when space is allocated for the array; see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# STRESS(NTENS)
+
+An Array passed in as the current stress tensor. If a local orientation is used at the same point as user subroutine UDMGINI, the stress components will be in the local orientation; in the case of finite-strain analysis, the basis system in which stress components are stored rotates with the material.
+
+# STRAIN(NTENS)
+
+An Array containing the current total strains. If a local orientation is used at the same point as user subroutine UDMGINI, the strain components will be in the local orientation; in the case of finite-strain analysis, the basis system in which strain components are stored rotates with the material.
+
+# STRAINEE(NTENS)
+
+An Array containing the current elastic strains. If a local orientation is used at the same point as user subroutine UDMGINI, the elastic strain components will be in the local orientation; in the case of finitestrain analysis, the basis system in which elastic strain components are stored rotates with the material.
+
+# LXFEM
+
+An integer flag to indicate an enriched element.
+
+# TIME(1)
+
+Value of step time at the beginning of the current increment.
+
+# TIME(2)
+
+Value of total time at the beginning of the current increment.
+
+# DTIME
+
+Time increment.
+
+# TEMP
+
+Temperature at the start of the increment.
+
+# DTEMP
+
+Increment of temperature during the time increment.
+
+# PREDEF
+
+An array containing the values of all of the user-specified predefined variables at this point at the start of the increment.
+
+<!-- source-page: 180 -->
+
+# DPRED
+
+An array containing the increments of all of the predefined variables during the time increment.
+
+# NFIELD
+
+Number of user-specified predefined variables.
+
+# COORDS
+
+An array containing the current coordinates of this point.
+
+# NOEL
+
+Element number.
+
+# NPT
+
+Integration point number.
+
+# LAYER
+
+Layer number (for composite shells and layered solids).
+
+# KSPT
+
+Section point number within the current layer.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# KDIRCYC
+
+Iteration number in a direct cyclic analysis.
+
+# KCYCLELCF
+
+Cycle number in a direct cyclic low-cycle fatigue analysis.
+
+# TIMECYC
+
+Time period in one loading cycle in a direct cyclic analysis.
+
+# JMAC
+
+Variable that must be passed into the GETVRM utility routine to access a material point variable.
+
+# JMATYP
+
+Variable that must be passed into the GETVRM utility routine to access a material point variable.
+
+# MATLAYO
+
+Variable that must be passed into the GETVRM utility routine to access a material point variable.
+
+# LACCFLA
+
+Variable that must be passed into the GETVRM utility routine to access a material point variable.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_019.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_019.md
new file mode 100644
index 00000000..21465187
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_019.md
@@ -0,0 +1,339 @@
+<!-- source-page: 181 -->
+
+# CELENT
+
+Characteristic element length, which is a typical length of a line across an element for a first-order element; it is half of the same typical length for a second-order element. For beams and trusses it is a characteristic length along the element axis. For membranes and shells it is a characteristic length in the reference surface. For axisymmetric elements it is a characteristic length in the (r, z) plane only. For cohesive elements it is equal to the constitutive thickness.
+
+# DROT(3,3)
+
+Rotation increment matrix. This matrix represents the increment of rigid body rotation of the basis system in which the components of stress (STRESS) and strain (STRAIN) are stored. It is provided so that vector- or tensor-valued state variables can be rotated appropriately in this subroutine: stress and strain components are already rotated by this amount before UDMGINI is called. This matrix is passed in as a unit matrix for small-displacement analysis and for large-displacement analysis if the basis system for the material point rotates with the material (as in a shell element or when a local orientation is used).
+
+# ORI(3,3)
+
+Material orientation with respect to global basis.
+
+# Example: User-defined damage initiation criterion with two different failure mechanisms
+
+As a simple example of the coding of user subroutine UDMGINI, consider a damage initiation criterion based on two different failure mechanisms: the maximum principal stress and the quadratic tractioninteraction.
+
+```csv
+SUBROUTINE UDMGINI(FINDEX,NFINDEX,FNORMAL,NDI,NSHR,NTENS,PROPS,
+1 NPROPS,STATEV,NSTATEV,STRESS,STRAIN,STRAINEE,LXFEM,TIME,
+2 DTIME,TEMP,DTEMP,PREDEF,DPRED,NFIELD,COORDS,NOEL,NPT,
+3 KLAYER,KSPT,KSTEP,INC,KDIRCYC,KCYCLELCF,TIMECYC,SSE,SPD,
+4 SCD,SVD,SMD,JMAC,JMATYP,MATLAYO,LACCFLA,CELENT,DROT,ORI)
+C
+INCLUDE 'ABA_PARAM.INC'
+CC
+DIMENSION FINDEX(NFINDEX),FNORMAL(NDI,NFINDEX),COORDS(*),
+1 STRESS(NTENS),STRAIN(NTENS),STRAINEE(NTENS),PROPS(NPROPS),
+2 STATEV(NSTATEV),PREDEF(NFIELD),DPRED(NFIELD),TIME(2),
+3 JMAC(*),JMATYP(*),DROR(3,3),ORI(3,3)
+DIMENSION PS(3), AN(3,3), WT(6)
+PS(1)=0.0
+PS(2)=0.0
+PS(3)=0.0 
+```
+
+<!-- source-page: 182 -->
+
+```txt
+C
+C ROTATE THE STRESS TO GLOBAL SYSTEM IF THERE IS ORIENTATION
+C
+CALL ROTSIG(STRESS,ORI,WT,1,NDI,NSHR)
+C
+C MAXIMUM PRINCIPAL STRESS CRITERION
+C
+CALL SPRIND(WT,PS,AN,1,NDI,NSHR)
+SIG1 = PS(1)
+KMAX=1
+DO K1 = 2, NDI
+IF(PS(K1).GT.SIG1) THEN
+SIG1 = PS(K1)
+KMAX = K1
+END IF
+END DO
+FINDEX(1) = SIG1/PROPS(1)
+DO K1=1, NDI
+FNORMAL(K1,1) = AN(KMAX,K1)
+END DO
+C
+C QUADRATIC TRACTION-INTERACTION CRITERION
+C
+FINDEX(2)=(STRESS(1)/PROPS(2))**2.0+(STRESS(NDI+1)/
+$ PROPS(3))**2.0+(STRESS(NDI+2)/PROPS(4))**2.0
+C
+FINDEX(2)=sqrt(FINDEX(2))
+C
+DO K1=1, NDI
+FNORMAL(K1,2)=ORI(K1,1)
+END DO
+RETURN
+END 
+```
+
+<!-- source-page: 183 -->
+
+# 1.1.27 UDSECURRENT: User subroutine to define nonuniform surface current density in an eddy current or magnetostatic analysis.
+
+Product: Abaqus/Standard
+
+# References
+
+• “Eddy current analysis,” Section 6.7.5 of the Abaqus Analysis User’s Guide   
+• “Magnetostatic analysis,” Section 6.7.6 of the Abaqus Analysis User’s Guide   
+• \*DSECURRENT
+
+# Overview
+
+User subroutine UDSECURRENT:
+
+• can be used to define the variation of surface current density vector as a function of position, time, element number, load integration point number, etc. for a transient eddy current or magnetostatic analysis or as a function of position, excitation frequency, phase, element number, load integration point number, etc. for a time-harmonic eddy current analysis;   
+• will be called at each surface load integration point for each nonuniform surface current density definition during eddy current and magnetostatic analyses; and   
+• ignores any amplitude references that may appear with the associated step definition or nonuniform distributed surface current density definition.
+
+# User subroutine interface
+
+```txt
+subroutine udsecurrent (
+C Write only -
+* surfacecurrent,
+C Read only -
+* coords, nBlock,
+* i_array, niarray,
+* r_array, nrarray,
+* c_array, narray )
+C
+include 'aba_param.inc'
+C
+dimension surfacecurrent(nBlock,*),
+* coords(nBlock,*),
+* i_array(*),
+* r_array(*)
+C 
+```
+
+<!-- source-page: 184 -->
+
+```lua
+character*80 c_array(*)
+
+parameter(i_udsecurr_kstep = 1,
+* i_udsecurr_kinc = 2,
+* i_udsecurr_noel = 3,
+* i_udsecurr_currtyp = 4,
+* i_udsecurr_phase = 5,
+* i_udsecurr_proc = 6)
+
+parameter(ir_udsecurr_time_1 = 1,
+* ir_udsecurr_time_2 = 2,
+* ir_udsecurr_time_3 = 3)
+
+parameter(ic_udsecurr_surf = 1)
+
+parameter(i_currtyp_tangential = 1)
+
+parameter(i_proc_lf_th = 1,
+* i_proc_lf_td = 2,
+* i_proc_ms = 3)
+
+parameter(i_udsecurr_phase_real = 1,
+* i_udsecurr_phase_imag = 2)
+
+user coding to define surfacecurrent
+
+return
+end 
+```
+
+# Variable to be defined
+
+surfacecurrent(nBlock,\*)
+
+Components of the surface current density vector at a block of surface integration points. The units are $\mathrm { C L ^ { - 1 } T ^ { - 1 } }$ . surfacecurrent will be passed into the routine as the vector specified as part of the surface-based distributed surface current density definition. If the vector is not defined, surfacecurrent will be passed in as zero.
+
+# Variables passed in for information
+
+coords(nBlock,\*)
+
+An array containing the coordinates of a block of surface load integration points.
+
+<!-- source-page: 185 -->
+
+nBlock
+
+Number of surface integration points in this block. Currently equal to 1.
+
+i\_array(i\_udsecurr\_kstep)
+
+Step number.
+
+i\_array(i\_udsecurr\_kinc)
+
+Increment number.
+
+i\_array(i\_udsecurr\_noel)
+
+Element number.
+
+i\_array(i\_udsecurr\_currtyp)
+
+Currently equal to 1.
+
+i\_array(i\_udsecurr\_phase)
+
+This value is relevant only for a time-harmonic eddy current analysis and is either 1 (i\_udsecurr\_phase\_real) or 2 (i\_udsecurr\_phase\_imag), depending on whether the current call to the user subroutine defines the real (in-phase) or the imaginary (out-of-phase) part of the surface current density vector.
+
+i\_array(i\_udsecurr\_proc)
+
+Equal to 1 for a time-harmonic eddy current procedure, 2 for a transient eddy current procedure, and 3 for a magnetostatic procedure.
+
+niarray
+
+Size of array i\_array. Currently equal to 6.
+
+r\_array(ir\_udsecurr\_time\_1)
+
+Excitation frequency in cycles/time for a time-harmonic eddy current analysis; alternatively, the value of step time at the beginning of the current increment for a transient eddy current or magnetostatic analysis.
+
+r\_array(ir\_udsecurr\_time\_2)
+
+Excitation frequency in radians/time for a time-harmonic eddy current analysis; alternatively, the value of total time at the beginning of the current increment for a transient eddy current or magnetostatic analysis.
+
+r\_array(ir\_udsecurr\_time\_3)
+
+Time increment for a transient eddy current or magnetostatic analysis.
+
+nrarray
+
+Size of array r\_array. Currently equal to 3.
+
+c\_array(ic\_udsecurr\_surf)
+
+Surface name.
+
+<!-- source-page: 186 -->
+
+# ncarray
+
+Size of array c\_array. Currently equal to 1.
+
+<!-- source-page: 187 -->
+
+# 1.1.28 UEL: User subroutine to define an element.
+
+# Product: Abaqus/Standard
+
+WARNING: This feature is intended for advanced users only. Its use in all but the simplest test examples will require considerable coding by the user/developer. “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide, should be read before proceeding.
+
+# References
+
+• “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide   
+• \*UEL PROPERTY  
+• \*USER ELEMENT
+
+# Overview
+
+User subroutine UEL:
+
+• will be called for each element that is of a general user-defined element type (i.e., not defined by a linear stiffness or mass matrix read either directly or from results file data) each time element calculations are required; and   
+• (or subroutines called by user subroutine UEL) must perform all of the calculations for the element, appropriate to the current activity in the analysis.
+
+# Wave kinematic data
+
+For Abaqus/Aqua applications four utility routines—GETWAVE, GETWAVEVEL, GETWINDVEL, and GETCURRVEL—are provided to access the fluid kinematic data. These routines are used from within user subroutine UEL and are discussed in detail in “Obtaining wave kinematic data in an Abaqus/Aqua analysis,” Section 2.1.13.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE UEL (RHS, AMATRX, SVARS, ENERGY, NDOFEL, NRHS, NSVARS,
+1 PROPS, NPROPS, COORDS, MCRD, NNODE, U, DU, V, A, JTYPE, TIME, DTIME,
+2 KSTEP, KINC, JELEM, PARAMS, NDLOAD, JDLTYP, ADLMAG, PREDEF, NPREF,
+3 LFLAGS, MLVARX, DDLMAG, MDLOAD, PNEWDT, JPROPS, NJPROP, PERIOD)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION RHS (MLVARX, *), AMATRX (NDOFEL, NDOFEL), PROPS (*),
+1 SVARS (*), ENERGY (8), COORDS (MCRD, NNODE), U (NDOFEL), 
+```
+
+<!-- source-page: 188 -->
+
+2 DU(MLVARX,\*),V(NDOFEL),A(NDOFEL),TIME(2),PARAMS(\*),   
+3 JDLTYP(MDLOAD,\*),ADLMAG(MDLOAD,\*),DDLMAG(MDLOAD,\*),   
+4 PREDEF(2,NPREDF,NNODE),LFLAGS(\*),JPROPS(\*)
+
+user coding to define RHS, AMATRX, SVARS, ENERGY, and PNEWDT
+
+RETURN END
+
+# Variables to be defined
+
+These arrays depend on the value of the LFLAGS array.
+
+# RHS
+
+An array containing the contributions of this element to the right-hand-side vectors of the overall system of equations. For most nonlinear analysis procedures, NRHS=1 and RHS should contain the residual vector. The exception is the modified Riks static procedure (“Static stress analysis,” Section 6.2.2 of the Abaqus Analysis User’s Guide), for which NRHS=2 and the first column in RHS should contain the residual vector and the second column should contain the increments of external load on the element. RHS(K1,K2) is the entry for the K1th degree of freedom of the element in the K2th right-hand-side vector.
+
+# AMATRX
+
+An array containing the contribution of this element to the Jacobian (stiffness) or other matrix of the overall system of equations. The particular matrix required at any time depends on the entries in the LFLAGS array (see below).
+
+All nonzero entries in AMATRX should be defined, even if the matrix is symmetric. If you do not specify that the matrix is unsymmetric when you define the user element, Abaqus/Standard will use the symmetric matrix defined by ${ \mathsf { \Omega } } _ { 2 } ^ { 1 } ( [ A ] + [ \dot { A } ] ^ { T } )$ , where is the matrix defined as AMATRX in this subroutine. If you specify that the matrix is unsymmetric when you define the user element, Abaqus/Standard will use AMATRX directly.
+
+# SVARS
+
+An array containing the values of the solution-dependent state variables associated with this element. The number of such variables is NSVARS (see below). You define the meaning of these variables.
+
+For general nonlinear steps this array is passed into UEL containing the values of these variables at the start of the current increment. They should be updated to be the values at the end of the increment, unless the procedure during which UEL is being called does not require such an update. This depends on the entries in the LFLAGS array (see below). For linear perturbation steps this array is passed into UEL containing the values of these variables in the base state. They should be returned containing perturbation values if you wish to output such quantities.
+
+<!-- source-page: 189 -->
+
+When KINC is equal to zero, the call to UEL is made for zero increment output (see “Output,” Section 4.1.1 of the Abaqus Analysis User’s Guide). In this case the values returned will be used only for output purposes and are not updated permanently.
+
+# ENERGY
+
+For general nonlinear steps array ENERGY contains the values of the energy quantities associated with the element. The values in this array when UEL is called are the element energy quantities at the start of the current increment. They should be updated to the values at the end of the current increment. For linear perturbation steps the array is passed into UEL containing the energy in the base state. They should be returned containing perturbation values if you wish to output such quantities. The entries in the array are as follows:
+
+<table><tr><td>ENERGY (1)</td><td>Kinetic energy.</td></tr><tr><td>ENERGY (2)</td><td>Elastic strain energy.</td></tr><tr><td>ENERGY (3)</td><td>Creep dissipation.</td></tr><tr><td>ENERGY (4)</td><td>Plastic dissipation.</td></tr><tr><td>ENERGY (5)</td><td>Viscous dissipation.</td></tr><tr><td>ENERGY (6)</td><td>“Artificial strain energy” associated with such effects as artificial stiffness introduced to control hourglassing or other singular modes in the element.</td></tr><tr><td>ENERGY (7)</td><td>Electrostatic energy.</td></tr><tr><td>ENERGY (8)</td><td>Incremental work done by loads applied within the user element.</td></tr></table>
+
+When KINC is equal to zero, the call to UEL is made for zero increment output (see “Output,” Section 4.1.1 of the Abaqus Analysis User’s Guide). In this case the energy values returned will be used only for output purposes and are not updated permanently.
+
+# Variable that can be updated
+
+# PNEWDT
+
+Ratio of suggested new time increment to the time increment currently being used (DTIME, see below). This variable allows you to provide input to the automatic time incrementation algorithms in Abaqus/Standard (if automatic time incrementation is chosen). It is useful only during equilibrium iterations with the normal time incrementation, as indicated by LFLAGS(3)=1. During a severe discontinuity iteration (such as contact changes), PNEWDT is ignored unless CONVERT SDI=YES is specified for this step. The usage of PNEWDT is discussed below.
+
+PNEWDT is set to a large value before each call to UEL.
+
+If PNEWDT is redefined to be less than 1.0, Abaqus/Standard must abandon the time increment and attempt it again with a smaller time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines that allow redefinition of PNEWDT for this iteration.
+
+<!-- source-page: 190 -->
+
+If PNEWDT is given a value that is greater than 1.0 for all calls to user subroutines for this iteration and the increment converges in this iteration, Abaqus/Standard may increase the time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines for this iteration.
+
+If automatic time incrementation is not selected in the analysis procedure, values of PNEWDT that are greater than 1.0 will be ignored and values of PNEWDT that are less than 1.0 will cause the job to terminate.
+
+# Variables passed in for information
+
+# Arrays:
+
+# PROPS
+
+A floating point array containing the NPROPS real property values defined for use with this element. NPROPS is the user-specified number of real property values. See “Defining the element properties” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide.
+
+# JPROPS
+
+An integer array containing the NJPROP integer property values defined for use with this element. NJPROP is the user-specified number of integer property values. See “Defining the element properties” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide.
+
+# COORDS
+
+An array containing the original coordinates of the nodes of the element. COORDS(K1,K2) is the K1th coordinate of the K2th node of the element.
+
+# U, DU, V, A
+
+Arrays containing the current estimates of the basic solution variables (displacements, rotations, temperatures, depending on the degree of freedom) at the nodes of the element at the end of the current increment. Values are provided as follows:
+
+<table><tr><td>U (K1)</td><td>Total values of the variables. If this is a linear perturbation step, it is the value in the base state.</td></tr><tr><td>DU (K1, KRHS)</td><td>Incremental values of the variables for the current increment for right-hand-side KRHS. If this is an eigenvalue extraction step, this is the eigenvector magnitude for eigenvector KRHS. For steady-state dynamics, KRHS = 1 denotes real components of perturbation displacement and KRHS = 2 denotes imaginary components of perturbation displacement.</td></tr><tr><td>V (K1)</td><td>Time rate of change of the variables (velocities, rates of rotation). Defined for implicit dynamics only (LFLAGS (1) = 11 or 12).</td></tr><tr><td>A (K1)</td><td>Accelerations of the variables. Defined for implicit dynamics only (LFLAGS (1) = 11 or 12).</td></tr></table>
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+<!-- source-page: 191 -->
+
+# JDLTYP
+
+An array containing the integers used to define distributed load types for the element. Loads of type Un are identified by the integer value n in JDLTYP; loads of type UnNU are identified by the negative integer value in JDLTYP. JDLTYP(K1,K2) is the identifier of the K1th distributed load in the K2th load case. For general nonlinear steps K2 is always 1.
+
+# ADLMAG
+
+For general nonlinear steps ADLMAG(K1,1) is the total load magnitude of the K1th distributed load at the end of the current increment for distributed loads of type Un. For distributed loads of type UnNU, the load magnitude is defined in UEL; therefore, the corresponding entries in ADLMAG are zero. For linear perturbation steps ADLMAG(K1,1) contains the total load magnitude of the K1th distributed load of type Un applied in the base state. Base state loading of type UnNU must be dealt with inside UEL. ADLMAG(K1,2), ADLMAG(K1,3), etc. are currently not used.
+
+# DDLMAG
+
+For general nonlinear steps DDLMAG contains the increments in the magnitudes of the distributed loads that are currently active on this element for distributed loads of type Un. DDLMAG(K1,1) is the increment of magnitude of the load for the current time increment. The increment of load magnitude is needed to compute the external work contribution. For distributed loads of type UnNU, the load magnitude is defined in UEL; therefore, the corresponding entries in DDLMAG are zero. For linear perturbation steps DDLMAG(K1,K2) contains the perturbation in the magnitudes of the distributed loads that are currently active on this element for distributed loads of type Un. K1 denotes the K1th perturbation load active on the element. K2 is always 1, except for steady-state dynamics, where K2=1 for real loads and K2=2 for imaginary loads. Perturbation loads of type UnNU must be dealt with inside UEL.
+
+# PREDEF
+
+An array containing the values of predefined field variables, such as temperature in an uncoupled stress/displacement analysis, at the nodes of the element (“Predefined fields,” Section 34.6.1 of the Abaqus Analysis User’s Guide).
+
+The first index of the array, K1, is either 1 or 2, with 1 indicating the value of the field variable at the end of the increment and 2 indicating the increment in the field variable. The second index, K2, indicates the variable: the temperature corresponds to index 1, and the predefined field variables correspond to indices 2 and above. In cases where temperature is not defined, the predefined field variables begin with index 1. The third index, K3, indicates the local node number on the element.
+
+<table><tr><td>PREDEF (K1,1,K3)</td><td>Temperature.</td></tr><tr><td>PREDEF (K1,2,K3)</td><td>First predefined field variable.</td></tr><tr><td>PREDEF (K1,3,K3)</td><td>Second predefined field variable.</td></tr><tr><td>Etc.</td><td>Any other predefined field variable.</td></tr></table>
+
+<!-- source-page: 192 -->
+
+<table><tr><td>PREDEF (K1, K2, K3)</td><td>Total or incremental value of the K2th predefined field variable at the K3th node of the element.</td></tr><tr><td>PREDEF (1, K2, K3)</td><td>Values of the variables at the end of the current increment.</td></tr><tr><td>PREDEF (2, K2, K3)</td><td>Incremental values corresponding to the current time increment.</td></tr></table>
+
+# PARAMS
+
+An array containing the parameters associated with the solution procedure. The entries in this array depend on the solution procedure currently being used when UEL is called, as indicated by the entries in the LFLAGS array (see below).
+
+For implicit dynamics (LFLAGS(1) = 11 or 12) PARAMS contains the integration operator values, as:
+
+<table><tr><td>PARAMS (1)</td><td> $\alpha$ </td></tr><tr><td>PARAMS (2)</td><td> $\beta$ </td></tr><tr><td>PARAMS (3)</td><td> $\gamma$ </td></tr></table>
+
+# LFLAGS
+
+An array containing the flags that define the current solution procedure and requirements for element calculations. Detailed requirements for the various Abaqus/Standard procedures are defined earlier in this section.
+
+<table><tr><td>LFLAGS (1)</td><td>Defines the procedure type. See “Results file output format,” Section 5.1.2 of the Abaqus Analysis User’s Guide, for the key used for each procedure.</td></tr><tr><td>LFLAGS (2) =0</td><td>Small-displacement analysis.</td></tr><tr><td>LFLAGS (2) =1</td><td>Large-displacement analysis (nonlinear geometric effects included in the step; see “General and linear perturbation procedures,” Section 6.1.3 of the Abaqus Analysis User’s Guide).</td></tr><tr><td>LFLAGS (3) =1</td><td>Normal implicit time incrementation procedure. User subroutine UEL must define the residual vector in RHS and the Jacobian matrix in AMATRX.</td></tr><tr><td>LFLAGS (3) =2</td><td>Define the current stiffness matrix (AMATRX =  $K^{NM} = -\partial F^{N}/\partial u^{M}$  or  $-\partial G^{N}/\partial u^{M}$ ) only.</td></tr><tr><td>LFLAGS (3) =3</td><td>Define the current damping matrix (AMATRX =  $C^{NM} = -\partial F^{N}/\partial \dot{u}^{M}$  or  $-\partial G^{N}/\partial \dot{u}^{M}$ ) only.</td></tr></table>
+
+<!-- source-page: 193 -->
+
+<table><tr><td>LFLAGS (3) =4</td><td>Define the current mass matrix ( $\mathbf{AMATRX} = M^{NM} = -\partial F^{N} / \partial \ddot{u}^{M}$ ) only. Abaqus/Standard always requests an initial mass matrix at the start of the analysis.</td></tr><tr><td>LFLAGS (3) =5</td><td>Define the current residual or load vector ( $\mathbf{RHS} = F^{N}$ ) only.</td></tr><tr><td>LFLAGS (3) =6</td><td>Define the current mass matrix and the residual vector for the initial acceleration calculation (or the calculation of accelerations after impact).</td></tr><tr><td>LFLAGS (3) =100</td><td>Define perturbation quantities for output.</td></tr><tr><td>LFLAGS (4) =0</td><td>The step is a general step.</td></tr><tr><td>LFLAGS (4) =1</td><td>The step is a linear perturbation step.</td></tr><tr><td>LFLAGS (5) =0</td><td>The current approximations to  $u^{M}$ , etc. were based on Newton corrections.</td></tr><tr><td>LFLAGS (5) =1</td><td>The current approximations were found by extrapolation from the previous increment.</td></tr></table>
+
+# TIME(1)
+
+Current value of step time or frequency.
+
+# TIME(2)
+
+Current value of total time.
+
+# Scalar parameters:
+
+# DTIME
+
+Time increment.
+
+# PERIOD
+
+Time period of the current step.
+
+# NDOFEL
+
+Number of degrees of freedom in the element.
+
+# MLVARX
+
+Dimensioning parameter used when several displacement or right-hand-side vectors are used.
+
+# NRHS
+
+Number of load vectors. NRHS is 1 in most nonlinear problems: it is 2 for the modified Riks static procedure (“Static stress analysis,” Section 6.2.2 of the Abaqus Analysis User’s Guide), and it is greater than 1 in some linear analysis procedures and during substructure generation.
+
+<!-- source-page: 194 -->
+
+# NSVARS
+
+User-defined number of solution-dependent state variables associated with the element (“Defining the number of solution-dependent variables that must be stored within the element” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+# NPROPS
+
+User-defined number of real property values associated with the element (“Defining the element properties” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+# NJPROP
+
+User-defined number of integer property values associated with the element (“Defining the element properties” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+# MCRD
+
+MCRD is defined as the maximum of the user-defined maximum number of coordinates needed at any node point (“Defining the maximum number of coordinates needed at any nodal point” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide) and the value of the largest active degree of freedom of the user element that is less than or equal to 3. For example, if you specify that the maximum number of coordinates is 1 and the active degrees of freedom of the user element are 2, 3, and 6, MCRD will be 3. If you specify that the maximum number of coordinates is 2 and the active degrees of freedom of the user element are 11 and 12, MCRD will be 2.
+
+# NNODE
+
+User-defined number of nodes on the element (“Defining the number of nodes associated with the element” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+# JTYPE
+
+Integer defining the element type. This is the user-defined integer value n in element type Un (“Assigning an element type key to a user-defined element” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+# KSTEP
+
+Current step number.
+
+# KINC
+
+Current increment number.
+
+# JELEM
+
+User-assigned element number.
+
+# NDLOAD
+
+Identification number of the distributed load or flux currently active on this element.
+
+# MDLOAD
+
+Total number of distributed loads and/or fluxes defined on this element.
+
+<!-- source-page: 195 -->
+
+# NPREDF
+
+Number of predefined field variables, including temperature. For user elements Abaqus/Standard uses one value for each field variable per node.
+
+# UEL conventions
+
+The solution variables (displacement, velocity, etc.) are arranged on a node/degree of freedom basis. The degrees of freedom of the first node are first, followed by the degrees of freedom of the second node, etc.
+
+# Usage with general nonlinear procedures
+
+The values of $u ^ { N }$ (and, in direct-integration dynamic steps, $\dot { u } ^ { N }$ and $\ddot { u } ^ { N } )$ enter user subroutine UEL as their latest approximations at the end of the time increment; that is, at time $t + \Delta t$ .
+
+The values of $H ^ { \alpha }$ enter the subroutine as their values at the beginning of the time increment; that is, at time t. It is your responsibility to define suitable time integration schemes to update $H ^ { \alpha }$ . To ensure accurate, stable integration of internal state variables, you can control the time incrementation via PNEWDT.
+
+The values of $\cdot _ { p ^ { \beta } }$ enter the subroutine as the values of the total load magnitude for the th distributed load at the end of the increment. Increments in the load magnitudes are also available.
+
+In the following descriptions of the user element’s requirements, it will be assumed that LFLAGS(3)=1 unless otherwise stated.
+
+# Static analysis (LFLAGS(1)=1,2)
+
+• $F ^ { N } = F ^ { N } ( u ^ { M } , H ^ { \alpha } , p ^ { \beta } , t )$ .   
+• Automatic convergence checks are applied to the force residuals corresponding to degrees of freedom 1–7.   
+• You must define AMATRX $= K ^ { N M } = - \partial F ^ { N } / \partial u ^ { M }$ and $\mathbf { R H S } = F ^ { N }$ and update the state variables, $H ^ { \alpha }$ .
+
+# Modified Riks static analysis (LFLAGS(1)=1) and (NRHS=2)
+
+• $F ^ { N } = F ^ { N } ( u ^ { M } , H ^ { \alpha } , p ^ { \beta } )$ , where $p ^ { \beta } = p _ { 0 } ^ { \beta } + \lambda q ^ { \beta } , p _ { 0 } ^ { \beta }$ and $q ^ { \beta }$ are fixed load parameters, and is the Riks (scalar) load parameter.   
+• Automatic convergence checks are applied to the force residuals corresponding to degrees of freedom 1–7.   
+• You must define $\begin{array} { l l l } { { \tt A M A T R X } } & { = } & { K ^ { N M } } & { = } & { - \partial F ^ { N } / \partial u ^ { M } } \end{array}$ , RHS $\mathbf { \Psi } ( \mathbf { 1 } ) \ = \ \mathbf { \Psi } F ^ { N }$ , and RHS(2) $= \Delta \lambda ( \partial { \cal F } ^ { N } / \partial \lambda )$ and update the state variables, $H ^ { \alpha }$ . RHS(2) is the incremental load vector.
+
+# Direct-integration dynamic analysis (LFLAGS(1)=11, 12)
+
+• Automatic convergence checks are applied to the force residuals corresponding to degrees of freedom 1–7.
+
+<!-- source-page: 196 -->
+
+• LFLAGS(3)=1: Normal time increment. Either the Hilber-Hughes-Taylor or the backward Euler time integration scheme will be used. With set to zero for the backward Euler, both schemes imply
+
+$$
+F ^ {N} = - M ^ {N M} \ddot {u} _ {t + \Delta t} + (1 + \alpha) G _ {t + \Delta t} ^ {N} - \alpha G _ {t} ^ {N},
+$$
+
+where $M ^ { N M } = M ^ { N M } ( u ^ { M } , \dot { u } ^ { M } , H ^ { \alpha } , p ^ { \beta } , t , . . . )$ and $G ^ { N } = G ^ { N } ( u ^ { M } , \dot { u } ^ { M } , H ^ { \alpha } , p ^ { \beta } , t , . . . )$ ; that is, the highest time derivative of $u ^ { M }$ in $M ^ { N M }$ and $G ^ { N }$ is $\dot { u } ^ { M }$ , so that
+
+$$
+- \frac {\partial F ^ {N}}{\partial \ddot {u} ^ {M} {} _ {t + \Delta t}} = M ^ {N M}.
+$$
+
+Therefore, you must store $G _ { t } ^ { N }$ as an internal state vector. If half-increment residual calculations are required, you must also store $G _ { t ^ { - } } ^ { N }$ as an internal state vector, where indicates the time at the beginning of the previous increment. For $\alpha = 0 , F ^ { N } = - M ^ { N M } \ddot { u } _ { t + \Delta t } + G ^ { N } { } _ { t + \Delta t }$ and $G _ { t } ^ { N }$ is not needed. You must define $\mathtt { A M A T R X } = M ^ { N M } \left( d \ddot { u } / d u \right) + \left( 1 + \alpha \right) C ^ { N M } \left( d \dot { u } / d u \right) + \left( 1 + \alpha \right) K ^ { N M }$ where $C ^ { N M } = - \partial G ^ { N } { } _ { t + \Delta t } / \partial \dot { u } ^ { M }$ and $K ^ { N M } = - \partial G ^ { N } { } _ { t + \Delta t } / \partial u ^ { M }$ . $\mathbf { R H S } = \mathbf { \nabla } F ^ { N }$ must also be defined and the state variables, $H ^ { \alpha }$ , updated. Although the value of given in the dynamic step definition is passed into UEL, the value of can vary from element to element. For example, can be set to zero for some elements in the model where numerical dissipation is not desired.
+
+• LFLAGS(3)=5: Half-increment residual $( F _ { 1 / 2 } ^ { N } )$ calculation. Abaqus/Standard will adjust the time increment so that $| F _ { 1 / 2 } ^ { N } | ~ < ~ t o l e r a n c e$ (where is specified in the dynamic step definition). The half-increment residual is defined as
+
+$$
+F _ {1 / 2} ^ {N} = - M ^ {N M} \ddot {u} _ {t + \Delta t / 2} + (1 + \alpha) G _ {t + \Delta t / 2} ^ {N} - \frac {\alpha}{2} (G _ {t} ^ {N} + G _ {t -} ^ {N}),
+$$
+
+where $t ^ { - }$ indicates the time at the beginning of the previous increment ( is a parameter of the Hilber-Hughes-Taylor time integration operator and will be set to zero if the backward Euler time integration operator is used). You must define $\mathbf { R H S } = F _ { 1 / 2 } ^ { N }$ . To evaluate $M ^ { N M }$ and ${ G ^ { N } } _ { t + \Delta t / 2 } , \mathrm { y o u }$ must calculate ${ H ^ { \alpha } } _ { t + \Delta t / 2 }$ . These half-increment values will not be saved. DTIME will still contain $\Delta t \left( { \mathrm { n o t } } \Delta t / 2 \right)$ . The values contained in U, V, A, and DU are half-increment values.
+
+• LFLAGS(3)=4: Velocity jump calculation. Abaqus/Standard solves $- M ^ { N M } \Delta \dot { u } ^ { M } = 0 \mathrm { f o r } \Delta \dot { u } ^ { M }$ , so you must define AMATRX . $\tt A M A T R X = M ^ { N M }$   
+• LFLAGS(3)=6: Initial acceleration calculation. Abaqus/Standard solves $- M ^ { N M } \ddot { u } ^ { M } + G ^ { N } = 0$ for $\ddot { u } ^ { M }$ , so you must define $\mathtt { A M A T R X } = M ^ { N M }$ and $\mathbf { R } \mathbf { \bar { H } } \mathbf { S } = G ^ { N }$ .
+
+# Subspace-based dynamic analysis (LFLAGS(1)=13)
+
+• The requirements are identical to those of static analysis, except that the Jacobian (stiffness), AMATRX, is not needed. No convergence checks are performed in this case.
+
+# Quasi-static analysis (LFLAGS(1)=21)
+
+• The requirements are identical to those of static analysis.
+
+<!-- source-page: 197 -->
+
+# Steady-state heat transfer analysis (LFLAGS(1)=31)
+
+• The requirements are identical to those of static analysis, except that the automatic convergence checks are applied to the heat flux residuals corresponding to degrees of freedom 11, 12, …
+
+# Transient heat transfer analysis $\left( \phantom { - } \theta _ { m a x } \right) \left( \tt L F L A G S \left( 1 \right) = 3 2 , \phantom { - } 3 3 \right)$
+
+• Automatic convergence checks are applied to the heat flux residuals corresponding to degrees of freedom 11, 12, …   
+• The backward difference scheme is always used for time integration; that is, Abaqus/Standard assumes that $\dot { u } _ { t + \Delta t } = \Delta u / \Delta t .$ , where $\Delta u = u _ { t + \Delta t } - u _ { t }$ and so $d \dot { u } / d u = 1 / \Delta t$ always. For degrees of freedom 11, 12, …, $\lvert \Delta u \rvert$ will be compared against the user-prescribed maximum allowable nodal temperature change in an increment, $\Delta \theta _ { m a x }$ , for controlling the time integration accuracy.   
+• You need to define $\mathtt { A M A T R X } = K ^ { N M } + ( 1 / \Delta t ) \ C ^ { N M }$ , where $C ^ { N M }$ is the heat capacity matrix and $\mathbf { R } \mathbf { \bar { H } } \mathbf { S } = F ^ { N }$ , and must update the state variables, $H ^ { \alpha }$ .
+
+# Geostatic analysis (LFLAGS(1)=61)
+
+• Identical to static analysis, except that the automatic convergence checks are applied to the residuals corresponding to degrees of freedom 1–8.
+
+# Steady-state coupled pore fluid diffusion/stress analysis (LFLAGS(1)=62, 63)
+
+• Identical to static analysis, except that the automatic convergence checks are applied to the residuals corresponding to degrees of freedom 1–8.
+
+# Transient coupled pore fluid diffusion/stress (consolidation) analysis $( ~ { u } _ { w } ^ { m a x } )$ (LFLAGS(1)=64, 65)
+
+• Automatic convergence checks are applied to the residuals corresponding to degrees of freedom 1–8.   
+• The backward difference scheme is used for time integration; that is, $\dot { u } _ { t + \Delta t } ^ { M } = \Delta u ^ { M } / \Delta t$ , where $\Delta u ^ { M } = u _ { t + \Delta t } ^ { M } - u _ { t } ^ { M }$ .   
+• For degree of freedom 8, $| \Delta u ^ { M } |$ will be compared against the user-prescribed maximum wetting liquid pore pressure change, $\Delta u _ { w } ^ { m a x }$ , for automatic control of the time integration accuracy.   
+• You must define $\mathtt { A M A T R X } = K ^ { N M } + ( 1 / \Delta t ) \ C ^ { N M }$ , where $C ^ { N M }$ is the pore fluid capacity matrix and $\mathbf { R H S } = F ^ { N }$ , and must update the state variables, $H ^ { \alpha }$ .
+
+# Steady-state fully coupled thermal-stress analysis (LFLAGS(1)=71)
+
+• Identical to static analysis, except that the automatic convergence checks are applied to the residuals corresponding to degrees of freedom 1–7 and 11, 12, …
+
+<!-- source-page: 198 -->
+
+Transient fully coupled thermal-stress analysis $( \theta _ { m a x } )$ (LFLAGS(1)=72,73)
+
+• Automatic convergence checks are applied to the residuals corresponding to degrees of freedom 1–7 and 11, 12, …   
+• The backward difference scheme is used for time integration; that is, $\dot { u } _ { t + \Delta t } ^ { M } = \Delta u ^ { M } / \Delta t .$ where $\Delta u ^ { M } = u _ { t + \Delta t } ^ { M } - u _ { t } ^ { M }$ .   
+• For degrees of freedom 11, 12, …, $| \Delta u ^ { M } |$ will be compared against the user-prescribed maximum allowable nodal temperature change in an increment, $\Delta \theta _ { m a x }$ , for automatic control of the time integration accuracy.   
+• You must define $\mathtt { A M A T R X } = K ^ { N M } + ( 1 / \Delta t ) \ C ^ { N M }$ , where $C ^ { N M }$ is the heat capacity matrix and $\mathbf { R } \mathbf { \bar { H } } \mathbf { S } = F ^ { N }$ , and must update the state variables, $H ^ { \alpha }$ .
+
+# Steady-state coupled thermal-electrical analysis (LFLAGS(1)=75)
+
+• The requirements are identical to those of static analysis, except that the automatic convergence checks are applied to the current density residuals corresponding to degree of freedom 9, in addition to the heat flux residuals.
+
+Transient coupled thermal-electrical analysis $\left( \phantom { - } \theta _ { m a x } \right) \left( \tt L F L A G S \left( 1 \right) = 7 6 , \eta 7 7 \right)$
+
+• Automatic convergence checks are applied to the current density residuals corresponding to degree of freedom 9 and to the heat flux residuals corresponding to degree of freedom 11.   
+• The backward difference scheme is always used for time integration; that is, Abaqus/Standard assumes that $\dot { u } _ { t + \Delta t } = \Delta u / \Delta t .$ , where $\Delta u = u _ { t + \Delta t } - u _ { t }$ . Therefore, $d \dot { u } / d u = 1 / \Delta t$ always. For degree of freedom 11 $| \Delta u |$ will be compared against the user-prescribed maximum allowable nodal temperature change in an increment, $\Delta \theta _ { m a x }$ , for controlling the time integration accuracy.   
+• You must define $\mathtt { A M A T R X } = K ^ { N M } + ( 1 / \Delta t ) \ C ^ { N M }$ , where $C ^ { N M }$ is the heat capacity matrix and $\mathbf { R } \mathbf { \bar { H } } \mathbf { S } = F ^ { N }$ , and must update the state variables, $H ^ { \alpha }$ .
+
+Steady-state coupled thermal-electrical-structural analysis (LFLAGS(1)=102)
+
+• Identical to static analysis, except that the automatic convergence checks are applied to the residuals corresponding to degrees of freedom 1–7, 9, and 11.
+
+Transient coupled thermal-electrical-structural analysis $( \theta _ { m a x } )$ (LFLAGS(1)=103,104)
+
+• Automatic convergence checks are applied to the residuals corresponding to degrees of freedom 1–7, 9, and 11.   
+• The backward difference scheme is always used for time integration; that is, Abaqus/Standard assumes that $\dot { u } _ { t + \Delta t } = \Delta u / \Delta t .$ , where $\Delta u = u _ { t + \Delta t } - u _ { t }$ . Therefore, $d \dot { u } / d u = 1 / \Delta t$ always. For degree of freedom 11 $| \Delta u |$ will be compared against the user-prescribed maximum allowable nodal temperature change in an increment, $\Delta \theta _ { m a x }$ , for controlling the time integration accuracy.   
+• You must define $\mathtt { A M A T R X } = K ^ { N M } + ( 1 / \Delta t ) \ C ^ { N M }$ , where $C ^ { N M }$ is the heat capacity matrix and $\mathbf { R } \mathbf { \bar { H } } \mathbf { S } = F ^ { N }$ ; and you must update the state variables, $H ^ { \alpha }$ .
+
+<!-- source-page: 199 -->
+
+“General and linear perturbation procedures,” Section 6.1.3 of the Abaqus Analysis User’s Guide, describes the linear perturbation capabilities in Abaqus/Standard. Here, base state values of variables will be denoted by $u ^ { M } , H ^ { \alpha }$ , etc. Perturbation values will be denoted by $\tilde { u } ^ { M } , \tilde { H } ^ { \alpha }$ , etc.
+
+Abaqus/Standard will not call user subroutine UEL for the eigenvalue buckling prediction procedure.
+
+For response spectrum, random response, transient modal dynamic, and mode-based steady-state dynamic procedures, user subroutine UEL is called only in a prior natural frequency extraction analysis, and the mass and stiffness contributions are taken into account during modal superposition.
+
+For direct-solution and mode-based steady-state dynamic, complex eigenvalue extraction, matrix generation, and substructure generation procedures, Abaqus/Standard will call user subroutine UEL, but only mass and stiffness contributions will be taken into account. The damping contributions will be neglected.
+
+# Static analysis (LFLAGS(1)=1, 2)
+
+• Abaqus/Standard will solve $K ^ { N M } \tilde { u } ^ { M } = \tilde { P } ^ { N }$ for $\tilde { u } ^ { M }$ , where $K ^ { N M }$ is the base state stiffness matrix and the perturbation load vector, $\tilde { P } ^ { N }$ , is a linear function of the perturbation loads, $\tilde { p } ;$ that is, $\tilde { P } ^ { N } =$ $\left( { \partial F } / { \partial \tilde { p } } \right) \tilde { p } .$ .   
+• $\mathtt { L F L A G S } \left( 3 \right) = 1$ : You must define $\mathbf { A M A T R X } = K ^ { N M }$ and $\mathbf { R } \mathbf { \bar { H } } \mathbf { S } = \mathbf { \tilde { \Lambda } } \tilde { P } ^ { N }$ .   
+• $\mathtt { L F L A G S } \left( 3 \right) = 1 0 0 $ : You must compute perturbations of the internal variables, $\tilde { H } ^ { \alpha }$ , and define RHS $= \tilde { P } ^ { N } - K ^ { N M } \tilde { u } ^ { M } $ for output purposes.
+
+# Eigenfrequency extraction analysis (LFLAGS(1)=41)
+
+$\bullet F ^ { N } = - M ^ { N M } \ddot { \tilde { u } } + G ^ { N } ( u ^ { M } + \tilde { u } ^ { M } , \ldots ) = - M ^ { N M } \ddot { \tilde { u } } + \left( \partial G ^ { N } / \partial u ^ { M } \right) \tilde { u } ^ { M } .$   
+• Abaqus/Standard will solve $\begin{array} { r l r } { K ^ { N M } \phi _ { i } ^ { M } } & { { } = } & { \omega _ { i } ^ { 2 } M ^ { N M } \phi _ { i } ^ { M } } \end{array}$ for $\phi _ { i } ^ { N }$ and $\omega _ { i }$ , where $\begin{array} { r l } { K ^ { N M } } & { { } = } \end{array}$ $- \partial F ^ { N } / \partial u ^ { M }$ is the base state stiffness matrix and $M ^ { N M } = - \partial F ^ { N M } / \partial \ddot { u } ^ { M }$ is the base state mass matrix.   
+• LFLAGS(3)=2: Define $\mathbf { A M A T R X } = K ^ { N M }$ .   
+• LFLAGS(3)=4: Define $\mathtt { A M A T R X } = M ^ { N M }$ .
+
+# Example: Structural and heat transfer user element
+
+Both a structural and a heat transfer user element have been created to demonstrate the usage of subroutine UEL. These user-defined elements are applied in a number of analyses. The following excerpt is from the verification problem that invokes the structural user element in an implicit dynamics procedure:
+
+```python
+*USER ELEMENT, NODES=2, TYPE=U1, PROPERTIES=4, COORDINATES=3, VARIABLES=12
+1, 2, 3
+*ELEMENT, TYPE=U1
+101, 101, 102 
+```
+
+<!-- source-page: 200 -->
+
+```csv
+*ELGEN, ELSET=UTRUSS
+101, 5
+*UEL PROPERTY, ELSET=UTRUSS
+0.002, 2.1E11, 0.3, 7200.
+```
+
+The user element consists of two nodes that are assumed to lie parallel to the x-axis. The element behaves like a linear truss element. The supplied element properties are the cross-sectional area, Young’s modulus, Poisson’s ratio, and density, respectively.
+
+The next excerpt shows the listing of the subroutine. The user subroutine has been coded for use in a perturbation static analysis; general static analysis, including Riks analysis with load incrementation defined by the subroutine; eigenfrequency extraction analysis; and direct-integration dynamic analysis. The names of the verification input files associated with the subroutine and these procedures can be found in “UEL,” Section 4.1.14 of the Abaqus Verification Guide. The subroutine performs all calculations required for the relevant procedures as described earlier in this section. The flags passed in through the LFLAGS array are used to associate particular calculations with solution procedures.
+
+During a modified Riks analysis all force loads must be passed into UEL by means of distributed load definitions such that they are available for the definition of incremental load vectors; the load keys Un and UnNU must be used properly, as discussed in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide. The coding in subroutine UEL must distribute the loads into consistent equivalent nodal forces and account for them in the calculation of the RHS and ENERGY arrays.
+
+```csv
+SUBROUTINE UEL(RHS,AMATRX,SVARS,ENERGY,NDOFEL,NRHS,NSVARS,
+1 PROPS,NPROPS,COORDS,MCRD,NNODE,U,DU,V,A,JTYPE,TIME,
+2 DTIME,KSTEP,KINC,JELEM,PARAMS,NDLOAD,JDLTYP,ADLMAG,
+3 PREDEF,NPREDF,LFLAGS,MLVARX,DDLMAG,MDLOAD,PNEWDT,
+4 JPROPS,NJPROP,PERIOD)
+C
+INCLUDE 'ABA_PARAM.INC'
+PARAMETER ( ZERO = 0.D0, HALF = 0.5D0, ONE = 1.D0 )
+C
+DIMENSION RHS(MLVARX, * ),AMATRX(NDOFEL,NDOFEL),
+1 SVARS(NSVARS),ENERGY(8),PROPS(*),COORDS(MCRD,NNODE),
+2 U(NDOFEL),DU(MLVARX, * ),V(NDOFEL),A(NDOFEL),TIME(2),
+3 PARAMS(3),JDLTYP(MDLOAD, * ),ADLMAG(MDLOAD, * ),
+4 DDLMAG(MDLOAD, * ),PREDEF(2,NPREDF,NNODE),LFLAGS(*),
+5 JPROPS(*)
+DIMENSION SRESID(6)
+C
+C UEL SUBROUTINE FOR A HORIZONTAL TRUSS ELEMENT
+C
+C SRESID - stores the static residual at time t+dt
+C SVARS - In 1-6, contains the static residual at time t
+C upon entering the routine. SRESID is copied to 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_021.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_021.md
new file mode 100644
index 00000000..170585e8
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_021.md
@@ -0,0 +1,332 @@
+<!-- source-page: 201 -->
+
+```txt
+C    SVARS(1-6) after the dynamic residual has been
+C    calculated.
+C    - For half-increment residual calculations: In 7-12,
+C    contains the static residual at the beginning
+C    of the previous increment. SVARS(1-6) are copied
+C    into SVARS(7-12) after the dynamic residual has
+C    been calculated.
+C
+AREA = PROPS(1)
+E    = PROPS(2)
+ANU    = PROPS(3)
+RHO    = PROPS(4)
+C
+ALEN = ABS(COORDS(1,2)-COORDS(1,1))
+AK    = AREA*E/ALEN
+AM    = HALF*AREA*RHO*ALEN
+C
+DO K1 = 1, NDOFEL
+    SRESID(K1) = ZERO
+    DO KRHS = 1, NRHS
+    RHS(K1,KRHS) = ZERO
+    END DO
+    DO K2 = 1, NDOFEL
+    AMATRX(K2,K1) = ZERO
+    END DO
+    END DO
+C
+IF (LFLAGS(3).EQ.1) THEN
+C    Normal incrementation
+    IF (LFLAGS(1).EQ.1 .OR. LFLAGS(1).EQ.2) THEN
+C    *STATIC
+    AMATRX(1,1) = AK
+    AMATRX(4,4) = AK
+    AMATRX(1,4) = -AK
+    AMATRX(4,1) = -AK
+    IF (LFLAGS(4).NE.0) THEN
+    FORCE = AK*(U(4)-U(1))
+    DFORCE = AK*(DU(4,1)-DU(1,1))
+    SRESID(1) = -DFORCE
+    SRESID(4) = DFORCE
+    RHS(1,1) = RHS(1,1)-SRESID(1)
+    RHS(4,1) = RHS(4,1)-SRESID(4) 
+```
+
+<!-- source-page: 202 -->
+
+```matlab
+ENERGY(2) = HALF*FORCE*(DU(4,1)-DU(1,1))
+* + HALF*DFORCE*(U(4)-U(1))
+* + HALF*DFORCE*(DU(4,1)-DU(1,1))
+ELSE
+FORCE = AK*(U(4)-U(1))
+SRESID(1) = -FORCE
+SRESID(4) = FORCE
+RHS(1,1) = RHS(1,1)-SRESID(1)
+RHS(4,1) = RHS(4,1)-SRESID(4)
+DO KDLOAD = 1, NDLOAD
+IF (JDLTYP(KDLOAD,1).EQ.1001) THEN
+RHS(4,1) = RHS(4,1)+ADLMAG(KDLOAD,1)
+ENERGY(8) = ENERGY(8)+(ADLMAG(KDLOAD,1)
+* - HALF*DDLMAG(KDLOAD,1))*DU(4,1)
+IF (NRHS.EQ.2) THEN
+C
+Riks
+RHS(4,2) = RHS(4,2)+DDLMAG(KDLOAD,1)
+END IF
+END IF
+END DO
+ENERGY(2) = HALF*FORCE*(U(4)-U(1))
+END IF
+ELSE IF (LFLAGS(1).EQ.11 .OR. LFLAGS(1).EQ.12) THEN
+C
+*DYNAMIC
+ALPHA = PARAMS(1)
+BETA = PARAMS(2)
+GAMMA = PARAMS(3)
+C
+DADU = ONE/(BETA*DTIME**2)
+DVDU = GAMMA/(BETA*DTIME)
+C
+DO K1 = 1, NDOFEL
+AMATRX(K1,K1) = AM*DADU
+RHS(K1,1) = RHS(K1,1)-AM*A(K1)
+END DO
+AMATRX(1,1) = AMATRX(1,1)+(ONE+ALPHA)*AK
+AMATRX(4,4) = AMATRX(4,4)+(ONE+ALPHA)*AK
+AMATRX(1,4) = AMATRX(1,4)-(ONE+ALPHA)*AK
+AMATRX(4,1) = AMATRX(4,1)-(ONE+ALPHA)*AK
+FORCE = AK*(U(4)-U(1))
+SRESID(1) = -FORCE
+SRESID(4) = FORCE 
+```
+
+<!-- source-page: 203 -->
+
+```vba
+RHS(1,1) = RHS(1,1) -
+* ((ONE+ALPHA)*SRESID(1)-ALPHA*SVARS(1))
+RHS(4,1) = RHS(4,1) -
+* ((ONE+ALPHA)*SRESID(4)-ALPHA*SVARS(4))
+ENERGY(1) = ZERO
+DO K1 = 1, NDOFEL
+SVARS(K1+6) = SVARS(k1)
+SVARS(K1) = SRESID(K1)
+ENERGY(1) = ENERGY(1)+HALF*V(K1)*AM*V(K1)
+END DO
+ENERGY(2) = HALF*FORCE*(U(4)-U(1))
+END IF
+ELSE IF (LFLAGS(3).EQ.2) THEN
+C Stiffness matrix
+AMATRX(1,1) = AK
+AMATRX(4,4) = AK
+AMATRX(1,4) = -AK
+AMATRX(4,1) = -AK
+ELSE IF (LFLAGS(3).EQ.4) THEN
+C Mass matrix
+DO K1 = 1, NDOFEL
+AMATRX(K1,K1) = AM
+END DO
+ELSE IF (LFLAGS(3).EQ.5) THEN
+C Half-increment residual calculation
+ALPHA = PARAMS(1)
+FORCE = AK*(U(4)-U(1))
+SRESID(1) = -FORCE
+SRESID(4) = FORCE
+RHS(1,1) = RHS(1,1)-AM*A(1)-(ONE+ALPHA)*SRESID(1)
+* + HALF*ALPHA*(SVARS(1)+SVARS(7))
+RHS(4,1) = RHS(4,1)-AM*A(4)-(ONE+ALPHA)*SRESID(4)
+* + HALF*ALPHA*(SVARS(4)+SVARS(10))
+ELSE IF (LFLAGS(3).EQ.6) THEN
+C Initial acceleration calculation
+DO K1 = 1, NDOFEL
+AMATRX(K1,K1) = AM
+END DO
+FORCE = AK*(U(4)-U(1))
+SRESID(1) = -FORCE
+SRESID(4) = FORCE
+RHS(1,1) = RHS(1,1)-SRESID(1) 
+```
+
+<!-- source-page: 204 -->
+
+```matlab
+RHS(4,1) = RHS(4,1) - SRESID(4)
+ENERGY(1) = ZERO
+DO K1 = 1, NDOFEL
+SVARS(K1) = SRESID(K1)
+ENERGY(1) = ENERGY(1) + HALF*V(K1) * AM*V(K1)
+END DO
+ENERGY(2) = HALF*FORCE*(U(4) - U(1))
+ELSE IF (LFLAGS(3).EQ.100) THEN
+C Output for perturbations
+IF (LFLAGS(1).EQ.1 .OR. LFLAGS(1).EQ.2) THEN
+C *STATIC
+FORCE = AK*(U(4) - U(1))
+DFORCE = AK*(DU(4,1) - DU(1,1))
+SRESID(1) = -DFORCE
+SRESID(4) = DFORCE
+RHS(1,1) = RHS(1,1) - SRESID(1)
+RHS(4,1) = RHS(4,1) - SRESID(4)
+ENERGY(2) = HALF*FORCE*(DU(4,1) - DU(1,1))
+* + HALF*DFORCE*(U(4) - U(1))
+* + HALF*DFORCE*(DU(4,1) - DU(1,1))
+DO KVAR = 1, NSVARS
+SVARS(KVAR) = ZERO
+END DO
+SVARS(1) = RHS(1,1)
+SVARS(4) = RHS(4,1)
+ELSE IF (LFLAGS(1).EQ.41) THEN
+C *FREQUENCY
+DO KRHS = 1, NRHS
+DFORCE = AK*(DU(4, KRHS) - DU(1, KRHS))
+SRESID(1) = -DFORCE
+SRESID(4) = DFORCE
+RHS(1, KRHS) = RHS(1, KRHS) - SRESID(1)
+RHS(4, KRHS) = RHS(4, KRHS) - SRESID(4)
+END DO
+DO KVAR = 1, NSVARS
+SVARS(KVAR) = ZERO
+END DO
+SVARS(1) = RHS(1,1)
+SVARS(4) = RHS(4,1)
+END IF
+END IF
+C 
+```
+
+<!-- source-page: 205 -->
+
+RETURN
+END
+
+<!-- source-page: 206 -->
+
+<!-- source-page: 207 -->
+
+# 1.1.29 UELMAT: User subroutine to define an element with access to Abaqus materials.
+
+# Product: Abaqus/Standard
+
+WARNING: This feature is intended for advanced users only. Its use in all but the simplest test examples will require considerable coding by the user/developer. “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide, should be read before proceeding.
+
+# References
+
+• “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide   
+• \*UEL PROPERTY  
+• \*USER ELEMENT   
+• “Accessing Abaqus materials,” Section 2.1.17   
+• “Accessing Abaqus thermal materials,” Section 2.1.18
+
+# Overview
+
+# User subroutine UELMAT:
+
+• will be called for each element that is of a general user-defined element type (i.e., not defined by a linear stiffness or mass matrix read either directly or from results file data) each time element calculations are required;   
+• (or subroutines called by user subroutine UELMAT) must perform all of the calculations for the element, appropriate to the current activity in the analysis;   
+• can access some of the Abaqus materials through utility routines MATERIAL\_LIB\_MECH and MATERIAL\_LIB\_HT;   
+• is available for a subset of the procedures supported for user subroutine UEL (see “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide); and   
+• is available for plane stress and three-dimensional element types in a stress/displacement analysis and for two-dimensional and three-dimensional element types in a heat transfer analysis (see “Userdefined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+# User subroutine interface
+
+SUBROUTINE UELMAT(RHS,AMATRX,SVARS,ENERGY,NDOFEL,NRHS,NSVARS,  
+1 PROPS,NPROPS,COORDS,MCRD,NNODE,U,DU,V,A,JTYPE,TIME,DTIME,   
+2 KSTEP,KINC,JELEM,PARAMS,NDLOAD,JDLTYP,ADLMAG,PREDEF,NPREDF,  
+3 LFLAGS,MLVARX,DDLMAG,MDLOAD,PNEWDT,JPROPS,NJPROP,PERIOD,   
+4 MATERIALLIB)
+
+<!-- source-page: 208 -->
+
+C
+
+INCLUDE 'ABA\_PARAM.INC'
+
+C
+
+DIMENSION RHS(MLVARX,\*),AMATRX(NDOFEL,NDOFEL),PROPS(\*),
+
+1 SVARS(\*),ENERGY(8),COORDS(MCRD,NNODE),U(NDOFEL),
+
+2 DU(MLVARX,\*),V(NDOFEL),A(NDOFEL),TIME(2),PARAMS(\*),
+
+3 JDLTYP(MDLOAD,\*),ADLMAG(MDLOAD,\*),DDLMAG(MDLOAD,\*),
+
+4 PREDEF(2,NPREDF,NNODE),LFLAGS(\*),JPROPS(\*)
+
+user coding to define RHS, AMATRX, SVARS, ENERGY, and PNEWDT
+
+RETURN
+
+END
+
+# Variables to be defined
+
+These arrays depend on the value of the LFLAGS array.
+
+# RHS
+
+An array containing the contributions of this element to the right-hand-side vectors of the overall system of equations. For most nonlinear analysis procedures, NRHS=1 and RHS should contain the residual vector. The exception is the modified Riks static procedure (“Static stress analysis,” Section 6.2.2 of the Abaqus Analysis User’s Guide), for which NRHS=2 and the first column in RHS should contain the residual vector and the second column should contain the increments of external load on the element. RHS(K1,K2) is the entry for the K1th degree of freedom of the element in the K2th right-hand-side vector.
+
+# AMATRX
+
+An array containing the contribution of this element to the Jacobian (stiffness) or other matrix of the overall system of equations. The particular matrix required at any time depends on the entries in the LFLAGS array (see below).
+
+All nonzero entries in AMATRX should be defined, even if the matrix is symmetric. If you do not specify that the matrix is unsymmetric when you define the user element, Abaqus/Standard will use the symmetric matrix defined by ${ \mathsf { \Omega } } _ { 2 } ^ { 1 } ( [ A ] + [ A ] ^ { T } )$ , where is the matrix defined as AMATRX in this subroutine. If you specify that the matrix is unsymmetric when you define the user element, Abaqus/Standard will use AMATRX directly.
+
+# SVARS
+
+An array containing the values of the solution-dependent state variables associated with this element. The number of such variables is NSVARS (see below). You define the meaning of these variables.
+
+For general nonlinear steps this array is passed into UELMAT containing the values of these variables at the start of the current increment. They should be updated to be the values at the end
+
+<!-- source-page: 209 -->
+
+of the increment, unless the procedure during which UELMAT is being called does not require such an update; this requirement depends on the entries in the LFLAGS array (see below). For linear perturbation steps this array is passed into UELMAT containing the values of these variables in the base state. They should be returned containing perturbation values if you wish to output such quantities.
+
+When KINC is equal to zero, the call to UELMAT is made for zero increment output (see “Output,” Section 4.1.1 of the Abaqus Analysis User’s Guide). In this case the values returned will be used only for output purposes and are not updated permanently.
+
+# ENERGY
+
+For general nonlinear steps array ENERGY contains the values of the energy quantities associated with the element. The values in this array when UELMAT is called are the element energy quantities at the start of the current increment. They should be updated to the values at the end of the current increment. For linear perturbation steps the array is passed into UELMAT containing the energy in the base state. They should be returned containing perturbation values if you wish to output such quantities. The entries in the array are as follows:
+
+<table><tr><td>ENERGY (1)</td><td>Kinetic energy.</td></tr><tr><td>ENERGY (2)</td><td>Elastic strain energy.</td></tr><tr><td>ENERGY (3)</td><td>Creep dissipation.</td></tr><tr><td>ENERGY (4)</td><td>Plastic dissipation.</td></tr><tr><td>ENERGY (5)</td><td>Viscous dissipation.</td></tr><tr><td>ENERGY (6)</td><td>“Artificial strain energy” associated with such effects as artificial stiffness introduced to control hourglassing or other singular modes in the element.</td></tr><tr><td>ENERGY (7)</td><td>Electrostatic energy.</td></tr><tr><td>ENERGY (8)</td><td>Incremental work done by loads applied within the user element.</td></tr></table>
+
+When KINC is equal to zero, the call to UELMAT is made for zero increment output (see “Output,” Section 4.1.1 of the Abaqus Analysis User’s Guide). In this case the energy values returned will be used only for output purposes and are not updated permanently.
+
+# Variable that can be updated
+
+# PNEWDT
+
+Ratio of suggested new time increment to the time increment currently being used (DTIME, see below). This variable allows you to provide input to the automatic time incrementation algorithms in Abaqus/Standard (if automatic time incrementation is chosen). It is useful only during equilibrium iterations with the normal time incrementation, as indicated by LFLAGS(3)=1. During a severe discontinuity iteration (such as contact changes), PNEWDT is ignored unless CONVERT SDI=YES is specified for this step. The usage of PNEWDT is discussed below.
+
+PNEWDT is set to a large value before each call to UELMAT.
+
+<!-- source-page: 210 -->
+
+If PNEWDT is redefined to be less than 1.0, Abaqus/Standard must abandon the time increment and attempt it again with a smaller time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines that allow redefinition of PNEWDT for this iteration.
+
+If PNEWDT is given a value that is greater than 1.0 for all calls to user subroutines for this iteration and the increment converges in this iteration, Abaqus/Standard may increase the time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines for this iteration.
+
+If automatic time incrementation is not selected in the analysis procedure, values of PNEWDT that are greater than 1.0 will be ignored and values of PNEWDT that are less than 1.0 will cause the job to terminate.
+
+# Variables passed in for information
+
+# Arrays:
+
+# PROPS
+
+A floating point array containing the NPROPS real property values defined for use with this element. NPROPS is the user-specified number of real property values. See “Defining the element properties” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide.
+
+# JPROPS
+
+An integer array containing the NJPROP integer property values defined for use with this element. NJPROP is the user-specified number of integer property values. See “Defining the element properties” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide.
+
+# COORDS
+
+An array containing the original coordinates of the nodes of the element. COORDS(K1,K2) is the K1th coordinate of the K2th node of the element.
+
+# U, DU, V, A
+
+Arrays containing the current estimates of the basic solution variables (displacements, rotations, temperatures, depending on the degree of freedom) at the nodes of the element at the end of the current increment. Values are provided as follows:
+
+U(K1)
+
+Total values of the variables. If this is a linear perturbation step, it is the value in the base state.
+
+DU(K1,KRHS)
+
+Incremental values of the variables for the current increment for right-hand-side KRHS. If this is an eigenvalue extraction step, this is the eigenvector magnitude for eigenvector KRHS. For steady-state dynamics KRHS denotes real components of perturbation displacement and KRHS denotes imaginary components of perturbation displacement.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_022.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_022.md
new file mode 100644
index 00000000..071ad1d4
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_022.md
@@ -0,0 +1,385 @@
+<!-- source-page: 211 -->
+
+V(K1) Time rate of change of the variables (velocities, rates of rotation). Defined for implicit dynamics only (LFLAGS(1) 11 or 12).
+
+A(K1) Accelerations of the variables. Defined for implicit dynamics only (LFLAGS(1) 11 or 12).
+
+# JDLTYP
+
+An array containing the integers used to define distributed load types for the element. Loads of type Un are identified by the integer value n in JDLTYP; loads of type UnNU are identified by the negative integer value in JDLTYP. JDLTYP(K1,K2) is the identifier of the K1th distributed load in the K2th load case. For general nonlinear steps K2 is always 1.
+
+# ADLMAG
+
+For general nonlinear steps ADLMAG(K1,1) is the total load magnitude of the K1th distributed load at the end of the current increment for distributed loads of type Un. For distributed loads of type UnNU, the load magnitude is defined in UELMAT; therefore, the corresponding entries in ADLMAG are zero. For linear perturbation steps ADLMAG(K1,1) contains the total load magnitude of the K1th distributed load of type Un applied in the base state. Base state loading of type UnNU must be dealt with inside UELMAT. ADLMAG(K1,2), ADLMAG(K1,3), etc. are currently not used.
+
+# DDLMAG
+
+For general nonlinear steps DDLMAG contains the increments in the magnitudes of the distributed loads that are currently active on this element for distributed loads of type Un. DDLMAG(K1,1) is the increment of magnitude of the load for the current time increment. The increment of load magnitude is needed to compute the external work contribution. For distributed loads of type UnNU the load magnitude is defined in UELMAT; therefore, the corresponding entries in DDLMAG are zero. For linear perturbation steps DDLMAG(K1,K2) contains the perturbation in the magnitudes of the distributed loads that are currently active on this element for distributed loads of type Un. K1 denotes the K1th perturbation load active on the element. K2 is always 1, except for steady-state dynamics, where K2=1 for real loads and K2=2 for imaginary loads. Perturbation loads of type UnNU must be dealt with inside UELMAT.
+
+# PREDEF
+
+An array containing the values of predefined field variables, such as temperature in an uncoupled stress/displacement analysis, at the nodes of the element (“Predefined fields,” Section 34.6.1 of the Abaqus Analysis User’s Guide).
+
+The first index of the array, K1, is either 1 or 2, with 1 indicating the value of the field variable at the end of the increment and 2 indicating the increment in the field variable. The second index, K2, indicates the variable: the temperature corresponds to index 1, and the predefined field variables correspond to indices 2 and above. In cases where temperature is not defined, the predefined field variables begin with index 1. The third index, K3, indicates the local node number on the element.
+
+<!-- source-page: 212 -->
+
+<table><tr><td>PREDEF (K1,1,K3)</td><td>Temperature.</td></tr><tr><td>PREDEF (K1,2,K3)</td><td>First predefined field variable.</td></tr><tr><td>PREDEF (K1,3,K3)</td><td>Second predefined field variable.</td></tr><tr><td>Etc.</td><td>Any other predefined field variable.</td></tr><tr><td>PREDEF (K1,K2,K3)</td><td>Total or incremental value of the K2th predefined field variable at the K3th node of the element.</td></tr><tr><td>PREDEF (1,K2,K3)</td><td>Values of the variables at the end of the current increment.</td></tr><tr><td>PREDEF (2,K2,K3)</td><td>Incremental values corresponding to the current time increment.</td></tr></table>
+
+# PARAMS
+
+An array containing the parameters associated with the solution procedure. The entries in this array depend on the solution procedure currently being used when UELMAT is called, as indicated by the entries in the LFLAGS array (see below).
+
+For implicit dynamics (LFLAGS(1) = 11 or 12) PARAMS contains the integration operator values, as:
+
+<table><tr><td>PARAMS (1)</td><td> $\alpha$ </td></tr><tr><td>PARAMS (2)</td><td> $\beta$ </td></tr><tr><td>PARAMS (3)</td><td> $\gamma$ </td></tr></table>
+
+# LFLAGS
+
+An array containing the flags that define the current solution procedure and requirements for element calculations. Detailed requirements for the various Abaqus/Standard procedures are defined earlier in this section.
+
+LFLAGS(1) Defines the procedure type. See “Results file output format,” Section 5.1.2 of the Abaqus Analysis User’s Guide, for the key used for each procedure.
+
+LFLAGS(2)=0 Small-displacement analysis.
+
+LFLAGS(2)=1 Large-displacement analysis (nonlinear geometric effects included in the step; see “General and linear perturbation procedures,” Section 6.1.3 of the Abaqus Analysis User’s Guide).
+
+LFLAGS(3)=1 Normal implicit time incrementation procedure. User subroutine UELMAT must define the residual vector in RHS and the Jacobian matrix in AMATRX.
+
+<!-- source-page: 213 -->
+
+$\begin{array} { r l } { \mathtt { L F L A G S \ ( 3 ) } = 2 \qquad } & { \mathtt { D e f i n e \ t h e \ c u r r e n t \ s t i f f n e s s \ m a t r i x \ ( a M A T R X } } \\ & { = K ^ { N M } = - \partial F ^ { N } / \partial u ^ { M } \ \mathrm { o r } \ - \partial G ^ { N } / \partial u ^ { M } \ \mathrm { o n l y } . } \end{array}$
+
+$\begin{array} { r l } { \mathtt { L F L A G S } \left( 3 \right) = 3 \qquad } & { \mathrm { D e f n e ~ t h e ~ c u r r e n t ~ d a m p i n g ~ m a t r i x ~ ( a M a T R X } } \\ & { = C ^ { N M } = - \partial F ^ { N } / \partial \dot { u } ^ { M } \ \mathrm { o r } \ - \partial G ^ { N } / \partial \dot { u } ^ { M } \ \mathrm { o n l y } . } \end{array}$
+
+LFLAGS(3)=4 Define the current mass matrix $( \tt { a M A T R X } = M ^ { N M } = \tt { \frac { \partial } { \partial } }$ $- \partial F ^ { N } / \partial \ddot { u } ^ { M } )$ only. Abaqus/Standard always requests an initial mass matrix at the start of the analysis.
+
+LFLAGS(3)=5 Define the current residual or load vector $( \mathtt { R H S } = F ^ { N } )$ only.
+
+LFLAGS(3)=6 Define the current mass matrix and the residual vector for the initial acceleration calculation (or the calculation of accelerations after impact).
+
+LFLAGS(3)=100 Define perturbation quantities for output.
+
+LFLAGS(4)=0 The step is a general step.
+
+LFLAGS(4)=1 The step is a linear perturbation step.
+
+LFLAGS(5)=0 The current approximations to $u ^ { M }$ , etc. were based on Newton corrections.
+
+LFLAGS(5)=1 The current approximations were found by extrapolation from the previous increment.
+
+# TIME(1)
+
+Current value of step time or frequency.
+
+# TIME(2)
+
+Current value of total time.
+
+# Scalar parameters:
+
+# DTIME
+
+Time increment.
+
+# PERIOD
+
+Time period of the current step.
+
+# NDOFEL
+
+Number of degrees of freedom in the element.
+
+# MLVARX
+
+Dimensioning parameter used when several displacement or right-hand-side vectors are used.
+
+<!-- source-page: 214 -->
+
+# NRHS
+
+Number of load vectors. NRHS is 1 in most nonlinear problems: it is 2 for the modified Riks static procedure (“Static stress analysis,” Section 6.2.2 of the Abaqus Analysis User’s Guide), and it is greater than 1 in some linear analysis procedures and during substructure generation.
+
+# NSVARS
+
+User-defined number of solution-dependent state variables associated with the element (“Defining the number of solution-dependent variables that must be stored within the element” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+# NPROPS
+
+User-defined number of real property values associated with the element (“Defining the element properties” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+# NJPROP
+
+User-defined number of integer property values associated with the element (“Defining the element properties” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+# MCRD
+
+MCRD is defined as the maximum of the user-defined maximum number of coordinates needed at any node point (“Defining the maximum number of coordinates needed at any nodal point” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide) and the value of the largest active degree of freedom of the user element that is less than or equal to 3. For example, if you specify that the maximum number of coordinates is 1 and the active degrees of freedom of the user element are 2, 3, and 6, MCRD will be 3. If you specify that the maximum number of coordinates is 2 and the active degrees of freedom of the user element are 11 and 12, MCRD will be 2.
+
+# NNODE
+
+User-defined number of nodes on the element (“Defining the number of nodes associated with the element” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+# JTYPE
+
+Integer defining the element type. This is the user-defined integer value n in element type Un (“Assigning an element type key to a user-defined element” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+# KSTEP
+
+Current step number.
+
+# KINC
+
+Current increment number.
+
+# JELEM
+
+User-assigned element number.
+
+<!-- source-page: 215 -->
+
+# NDLOAD
+
+Identification number of the distributed load or flux currently active on this element.
+
+# MDLOAD
+
+Total number of distributed loads and/or fluxes defined on this element.
+
+# NPREDF
+
+Number of predefined field variables, including temperature. For user elements Abaqus/Standard uses one value for each field variable per node.
+
+# MATERIALLIB
+
+A variable that must be passed to the utility routines performing material point computations.
+
+# UELMAT conventions
+
+The solution variables (displacement, velocity, etc.) are arranged on a node/degree of freedom basis. The degrees of freedom of the first node are first, followed by the degrees of freedom of the second node, etc.
+
+# Usage with general nonlinear procedures
+
+The values of $u ^ { N }$ (and, in direct-integration dynamic steps, $\dot { u } ^ { N }$ and $\ddot { u } ^ { N } )$ enter user subroutine UELMAT as their latest approximations at the end of the time increment; that is, at time $t + \Delta t$ .
+
+The values of $H ^ { \alpha }$ enter the subroutine as their values at the beginning of the time increment; that is, at time t. It is your responsibility to define suitable time integration schemes to update $H ^ { \alpha }$ . To ensure accurate, stable integration of internal state variables, you can control the time incrementation via PNEWDT.
+
+The values of $\cdot _ { p ^ { \beta } }$ enter the subroutine as the values of the total load magnitude for the th distributed load at the end of the increment. Increments in the load magnitudes are also available.
+
+In the following descriptions of the user element’s requirements, it will be assumed that LFLAGS(3)=1 unless otherwise stated.
+
+# Static analysis (LFLAGS(1)=1,2)
+
+• $F ^ { N } = F ^ { N } ( u ^ { M } , H ^ { \alpha } , p ^ { \beta } , t )$   
+• Automatic convergence checks are applied to the force residuals corresponding to degrees of freedom 1–7.   
+• You must define AMATRX $= K ^ { N M } = - \partial F ^ { N } / \partial u ^ { M }$ and $\mathbf { R H S } = F ^ { N }$ and update the state variables, $H ^ { \alpha }$ .
+
+# Direct-integration dynamic analysis (LFLAGS(1)=11, 12)
+
+• Automatic convergence checks are applied to the force residuals corresponding to degrees of freedom 1–7.
+
+<!-- source-page: 216 -->
+
+• LFLAGS(3)=1: Normal time increment. Either the Hilber-Hughes-Taylor or the backward Euler time integration scheme will be used. With set to zero for the backward Euler, both schemes imply
+
+$$
+F ^ {N} = - M ^ {N M} \ddot {u} _ {t + \Delta t} + (1 + \alpha) G ^ {N} - \alpha G _ {t} ^ {N},
+$$
+
+where $M ^ { N M } = M ^ { N M } ( u ^ { M } , \dot { u } ^ { M } , H ^ { \alpha } , p ^ { \beta } , t , . . . )$ and $G ^ { N } = G ^ { N } ( u ^ { M } , \dot { u } ^ { M } , H ^ { \alpha } , p ^ { \beta } , t , . . . )$ ; that is, the highest time derivative of $u ^ { M }$ in $M ^ { N M }$ and $G ^ { N }$ is $\dot { u } ^ { M }$ , so that
+
+$$
+- \frac {\partial F ^ {N}}{\partial \ddot {u} ^ {M} {} _ {t + \Delta t}} = M ^ {N M}.
+$$
+
+Therefore, you must store $G _ { t } ^ { N }$ as an internal state vector. If half-increment residual calculations are required, you must also store $G _ { t ^ { - } } ^ { N }$ as an internal state vector, where indicates the time at the beginning of the previous increment. For $\alpha = 0 , F ^ { N } = - M ^ { N M } \ddot { u } _ { t + \Delta t } + G ^ { N } { } _ { t + \Delta t }$ and $G _ { t } ^ { N }$ is not needed. You must define $\mathtt { A M A T R X } = M ^ { N M } \left( d \ddot { u } / d u \right) + \left( 1 + \alpha \right) C ^ { N M } \left( d \dot { u } / d u \right) + \left( 1 + \alpha \right) K ^ { N M }$ where $C ^ { N M } = - \partial G ^ { N } { } _ { t + \Delta t } / \partial \dot { u } ^ { M }$ and $K ^ { N M } = - \partial G ^ { N } { } _ { t + \Delta t } / \partial u ^ { M }$ . $\mathbf { R H S } = \mathbf { \nabla } F ^ { N }$ must also be defined and the state variables, $H ^ { \alpha }$ , updated. Although the value of given in the dynamic step definition is passed into UELMAT, the value of can vary from element to element. For example, can be set to zero for some elements in the model where numerical dissipation is not desired.
+
+• LFLAGS(3)=5: Half-increment residual $( F _ { 1 / 2 } ^ { N } )$ calculation. Abaqus/Standard will adjust the time increment so that $| F _ { 1 / 2 } ^ { N } | ~ < ~ t o l e r a n c e$ (where is specified in the dynamic step definition). The half-increment residual is defined as
+
+$$
+F _ {1 / 2} ^ {N} = - M ^ {N M} \ddot {u} _ {t + \Delta t / 2} + (1 + \alpha) G _ {t + \Delta t / 2} ^ {N} - \frac {\alpha}{2} (G _ {t} ^ {N} + G _ {t -} ^ {N}),
+$$
+
+where indicates the time at the beginning of the previous increment ( is a parameter of the Hilber-Hughes-Taylor time integration operator and will be set to zero if the backward Euler time integration operator is used). You must define $\mathbf { R H S } = F _ { 1 / 2 } ^ { N }$ . To evaluate $M ^ { N M }$ and ${ G ^ { N } } _ { t + \Delta t / 2 } , \mathrm { y o u }$ must calculate ${ H ^ { \alpha } } _ { t + \Delta t / 2 }$ . These half-increment values will not be saved. DTIME will still contain $\Delta t \left( { \mathrm { n o t } } \Delta t / 2 \right)$ . The values contained in U, V, A, and DU are half-increment values.
+
+• LFLAGS(3)=4: Velocity jump calculation. Abaqus/Standard solves $- M ^ { N M } \Delta \dot { u } ^ { M } = 0$ for $\Delta \dot { u } ^ { M }$ , so you must define AMATRX . $\tt A M A T R X = M ^ { N M }$   
+• LFLAGS(3)=6: Initial acceleration calculation. Abaqus/Standard solves $- M ^ { N M } \ddot { u } ^ { M } + G ^ { N } = 0$ for $\ddot { u } ^ { M }$ , so you must define $\mathtt { A M A T R X } = M ^ { N M }$ and $\mathbf { R } \mathbf { \bar { H } } \mathbf { S } = G ^ { N }$ .
+
+# Quasi-static analysis (LFLAGS(1)=21)
+
+• The requirements are identical to those of static analysis.
+
+# Steady-state heat transfer analysis (LFLAGS(1)=31)
+
+• The requirements are identical to those of static analysis, except that the automatic convergence checks are applied to the heat flux residuals corresponding to degrees of freedom 11, 12, …
+
+<!-- source-page: 217 -->
+
+# Transient heat transfer analysis $\left( \phantom { - } \theta _ { m a x } \right) \left( \tt L F L A G S \left( 1 \right) = 3 2 , \phantom { - } 3 3 \right)$
+
+• Automatic convergence checks are applied to the heat flux residuals corresponding to degrees of freedom 11, 12, …   
+• The backward difference scheme is always used for time integration; that is, Abaqus/Standard assumes that $\dot { u } _ { t + \Delta t } = \Delta u / \Delta t$ , where $\Delta u = u _ { t + \Delta t } - u _ { t }$ and so $d \dot { u } / d u = 1 / \Delta t$ always. For degrees of freedom 11, 12, …, $\lvert \Delta u \rvert$ will be compared against the user-prescribed maximum allowable nodal temperature change in an increment, $\Delta \theta _ { m a x }$ , for controlling the time integration accuracy.   
+• You need to define AMATRX $= K ^ { N M } + ( 1 / \Delta t ) \ C ^ { N M }$ , where $C ^ { N M }$ is the heat capacity matrix and $\mathbf { R } \mathbf { \bar { H } } \mathbf { S } = F ^ { N }$ , and must update the state variables, $H ^ { \alpha }$ .
+
+# Usage with linear perturbation procedures
+
+“General and linear perturbation procedures,” Section 6.1.3 of the Abaqus Analysis User’s Guide, describes the linear perturbation capabilities in Abaqus/Standard. Here, base state values of variables will be denoted by $u ^ { M } , H ^ { \alpha }$ , etc. Perturbation values will be denoted by $\tilde { u } ^ { M } , \tilde { H } ^ { \alpha }$ , etc.
+
+Abaqus/Standard will not call user subroutine UELMAT for the following procedures: eigenvalue buckling prediction, response spectrum, transient modal dynamic, steady-state dynamic (modal and direct), and random response.
+
+# Static analysis (LFLAGS(1)=1, 2)
+
+• Abaqus/Standard will solve $K ^ { N M } \tilde { u } ^ { M } = \tilde { P } ^ { N }$ for $\tilde { u } ^ { M }$ , where $K ^ { N M }$ is the base state stiffness matrix and the perturbation load vector, $\tilde { P } ^ { N }$ , is a linear function of the perturbation loads, $\tilde { p } ;$ that is, $\tilde { P } ^ { N } =$ $\left( { \partial F } / { \partial \tilde { p } } \right) \tilde { p } .$ .   
+• $\mathtt { L F L A G S } \left( 3 \right) = 1$ : You must define AMATRX $K ^ { N M }$ and $\mathbf { R } \mathbf { \tilde { H } S } = \mathbf { \tilde { \rho } } \tilde { P } ^ { N }$ .   
+• $\mathtt { L F L A G S } \left( 3 \right) = 1 0 0 $ : You must compute perturbations of the internal variables, $\tilde { H } ^ { \alpha }$ , and define RHS $= \tilde { P } ^ { N } - K ^ { N M } \tilde { u } ^ { M } $ for output purposes.
+
+# Eigenfrequency extraction analysis (LFLAGS(1)=41)
+
+$\bullet F ^ { N } = - M ^ { N M } \ddot { \tilde { u } } + G ^ { N } ( u ^ { M } + \tilde { u } ^ { M } , \ldots ) = - M ^ { N M } \ddot { \tilde { u } } + \left( \partial G ^ { N } / \partial u ^ { M } \right) \tilde { u } ^ { M } .$   
+• Abaqus/Standard will solve $\begin{array} { r l r } { K ^ { N M } \phi _ { i } ^ { M } } & { { } = } & { \omega _ { i } ^ { 2 } M ^ { N M } \phi _ { i } ^ { M } } \end{array}$ for $\phi _ { i } ^ { N }$ and $\omega _ { i }$ , where $\begin{array} { r l } { K ^ { N M } } & { { } = } \end{array}$ $- \partial \bar { F ^ { N } } / \partial u ^ { M }$ is the base state stiffness matrix and $M ^ { N M } = - \partial F ^ { \dot { N } \dot { M } } / \partial \ddot { u } ^ { M }$ is the base state mass matrix.   
+• LFLAGS(3)=2: Define $\mathbf { a } \mathbf { M } \mathbf { A } \mathbf { T } \mathbf { R } \mathbf { X } = K ^ { N M }$ .   
+• LFLAGS(3)=4: Define $\mathtt { A M A T R X } = M ^ { N M }$ .
+
+# Example: Structural user element with Abaqus isotropic linearly elastic material
+
+Both a structural and a heat transfer user element have been created to demonstrate the usage of subroutine UELMAT. These user-defined elements are applied in a number of analyses. The following excerpt
+
+<!-- source-page: 218 -->
+
+illustrates how the linearly elastic isotropic material available in Abaqus can be accessed from user subroutine UELMAT:
+
+```txt
+...
+*USER ELEMENT, TYPE=U1, NODES=4, COORDINATES=2, VAR=16,
+    INTEGRATION=4, TENSOR=PSTRAIN
+    1,2
+*ELEMENT, TYPE=U1, ELSET=SOLID
+    1, 1,2,3,4
+...
+*UEL PROPERTY, ELSET=SOLID, MATERIAL=MAT
+...
+*MATERIAL, NAME=MAT
+*ELASTIC
+7.00E+010, 0.33
+```
+
+The user element defined above is a 4-node, fully integrated plane strain element, similar to the Abaqus CPE4 element.
+
+The next excerpt shows the listing of the user subroutine. Inside the subroutine, a loop over the integration points is performed. For each integration point the utility routine MATERIAL\_LIB\_MECH is called, which returns stress and Jacobian at the integration point. These quantities are used to compute the right-hand-side vector and the element Jacobian.
+
+```csv
+c*****
+subroutine uelmat(rhs,amatrix,svars,energy,ndofel,nrhs,
+1 nsvars,props,nprops,coords,mcrd,nnode,u,du,
+2 v,a,jtype,time,dtime,kstep,kinc,jelem,params,
+3 ndload,jdltyp,adlmag,predef,npredf,lflags,mlvarx,
+4 ddlmag,mdload,pnewdt,jprops,njpro,period,
+5 materiallib)
+c
+include 'aba_param.inc'
+c
+dimension rhs(mlvarx,*, amatrix(ndofel, ndofel), props(*),
+1 svars(*), energy(*), coords(mcrd, nnode), u(ndofel),
+2 du(mlvarx,*, v(ndofel), a(ndofel), time(2), params(*),
+3 jdltyp(mdload,*, adlmag(mdload,*, ddlmag(mdload,*),
+4 predef(2, npredf, nnode), lflags(*), jprops(*)
+parameter (zero=0.d0, dmone=-1.0d0, one=1.d0, four=4.0d0,
+1 fourth=0.25d0, gaussCoord=0.577350269d0)
+parameter (ndim=2, ndof=2, nshr=1, nnodemax=4,
+1 ntens=4, ninpt=4, nsvint=4)
+c 
+```
+
+<!-- source-page: 219 -->
+
+```fortran
+c ndim ... number of spatial dimensions
+c ndof ... number of degrees of freedom per node
+c nshr ... number of shear stress component
+c ntens ... total number of stress tensor components
+c (=ndi+nshr)
+c ninpt ... number of integration points
+c nsvint... number of state variables per integration pt
+c (strain)
+c
+    dimension stiff(ndof*nnodemax,ndof*nnodemax),
+    1 force(ndof*nnodemax), shape(nnodemax), dshape(ndim,nnodemax),
+    2 xjac(ndim,ndim),xjaci(ndim,ndim), bmat(nnodemax*ndim),
+    3 statevLocal(nsvint),stress(ntens), ddsdde(ntens, ntens),
+    4 stran(ntens), dstran(ntens), wght(ninpt)
+c
+    dimension predef_loc(npredf), dpredef_loc(npredf),
+    1 defGrad(3,3),utmp(3),xdu(3),stiff_p(3,3),force_p(3)
+    dimension coord24(2,4),coords_ip(3)
+    data coord24 /dmone, dmone,
+    2 one, dmone,
+    3 one, one,
+    4 dmone, one/
+c
+    data wght /one, one, one, one/
+c
+c**************************
+c
+c U1 = first-order, plane strain, full integration
+c
+c State variables: each integration point has nsvint SDVs
+c
+c isvinc=(npt-1)*nsvint ... integration point counter
+c statev(1+isvinc ) ... strain
+c
+c**************************
+if (lflags(3).eq.4) then
+    do i=1, ndofel
+    do j=1, ndofel
+    amatrix(i,j) = zero
+    end do
+    amatrix(i,i) = one
+    end do 
+```
+
+<!-- source-page: 220 -->
+
+```matlab
+goto 999
+end if
+c
+c PRELIMINARIES
+c
+pnewdtLocal = pnewdt
+if(jtype .ne. 1) then
+    write(7,*)'Incorrect element type'
+    call xit
+endif
+if(nvars .lt. ninpt*nsvint) then
+    write(7,*)'Increase the number of SDVs to', ninpt*nsvint
+    call xit
+endif
+thickness = 0.1d0
+c
+c INITIALIZE RHS AND LHS
+c
+do k1=1, ndof*nnode
+    rhs(k1, 1)= zero
+    do k2=1, ndof*nnode
+    amatrix(k1, k2)= zero
+    end do
+end do
+c
+c LOOP OVER INTEGRATION POINTS
+c
+do kintk = 1, ninpt
+c
+c EVALUATE SHAPE FUNCTIONS AND THEIR DERIVATIVES
+c
+c determine (g,h)
+c
+g = coord24(1,kintk)*gaussCoord
+h = coord24(2,kintk)*gaussCoord
+c
+c shape functions
+    shape(1) = (one - g)*(one - h)/four;
+    shape(2) = (one + g)*(one - h)/four;
+    shape(3) = (one + g)*(one + h)/four;
+    shape(4) = (one - g)*(one + h)/four;
+c 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_023.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_023.md
new file mode 100644
index 00000000..e388b2d9
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_023.md
@@ -0,0 +1,335 @@
+<!-- source-page: 221 -->
+
+```txt
+c derivative d(Ni)/d(g)
+dshape(1,1) = -(one - h)/four;
+dshape(1,2) = (one - h)/four;
+dshape(1,3) = (one + h)/four;
+dshape(1,4) = -(one + h)/four;
+c
+c derivative d(Ni)/d(h)
+dshape(2,1) = -(one - g)/four;
+dshape(2,2) = -(one + g)/four;
+dshape(2,3) = (one + g)/four;
+dshape(2,4) = (one - g)/four;
+c
+c compute coordinates at the integration point
+c
+do k1=1, 3
+    coords_ip(k1) = zero
+end do
+do k1=1, nnode
+    do k2=1, mcrd
+    coords_ip(k2)=coords_ip(k2)+shape(k1)*coords(k2,k1)
+    end do
+end do
+c
+c INTERPOLATE FIELD VARIABLES
+c
+if (npredf.gt.0) then
+    do k1=1, npredf
+    predef_loc(k1) = zero
+    dpredef_loc(k1) = zero
+    do k2=1, nnode
+    predef_loc(k1) =
+    &    predef_loc(k1)+
+    &    (predef(1,k1,k2)-predef(2,k1,k2))*shape(k2)
+    dpredef_loc(k1) =
+    &    dpredef_loc(k1)+predef(2,k1,k2)*shape(k2)
+    end do
+    end do
+    end if
+c
+c FORM B-MATRIX
+c 
+```
+
+<!-- source-page: 222 -->
+
+```txt
+djac = one
+
+do i = 1, ndim
+    do j = 1, ndim
+    xjac(i,j) = zero
+    xjac(i,j) = zero
+    end do
+end do
+
+do inod = 1, nnode
+    do idim = 1, ndim
+    do jdim = 1, ndim
+    xjac(jdim,idim) = xjac(jdim,idim) +
+    dshape(jdim,inod)*coords(idim,inod)
+    end do
+    end do
+end do
+
+djac = xjac(1,1)*xjac(2,2) - xjac(1,2)*xjac(2,1)
+    if (djac .gt. zero) then
+    ! jacobian is positive - o.k.
+    xjac(1,1) = xjac(2,2)/djac
+    xjac(2,2) = xjac(1,1)/djac
+    xjac(1,2) = -xjac(1,2)/djac
+    xjac(2,1) = -xjac(2,1)/djac
+    else
+    ! negative or zero jacobian
+    write(7,*)'WARNING: element',jelem,'has neg.
+    Jacobian'
+    pnewdt = fourth
+    endif
+
+if (pnewdt .lt. pnewdtLocal) pnewdtLocal = pnewdt
+    do i = 1, nnode*ndim
+    bmat(i) = zero
+    end do
+
+do inod = 1, nnode
+    do ider = 1, ndim
+    do idim = 1, ndim
+    irow = idim + (inod - 1)*ndim 
+```
+
+<!-- source-page: 223 -->
+
+```fortran
+bmat(irow) = bmat(irow) +
+1 xjaci(idim,ider)*dshape(ider,inod)
+    end do
+    end do
+    end do
+c
+c CALCULATE INCREMENTAL STRAINS
+c
+do i = 1, ntens
+    dstran(i) = zero
+end do
+!
+! set deformation gradient to Identity matrix
+do k1=1,3
+    do k2=1,3
+    defGrad(k1,k2) = zero
+    end do
+    defGrad(k1,k1) = one
+end do
+c
+c COMPUTE INCREMENTAL STRAINS
+c
+do nodi = 1, nnode
+    incr_row = (nodi - 1)*ndof
+    do i = 1, ndof
+    xdu(i) = du(i + incr_row,1)
+    utmp(i) = u(i + incr_row)
+    end do
+    dNidx = bmat(1 + (nodi-1)*ndim)
+    dNidy = bmat(2 + (nodi-1)*ndim)
+    dstran(1) = dstran(1) + dNidx*xdu(1)
+    dstran(2) = dstran(2) + dNidy*xdu(2)
+    dstran(4) = dstran(4) +
+1    dNidy*xdu(1) +
+2    dNidx*xdu(2)
+c deformation gradient 
+```
+
+<!-- source-page: 224 -->
+
+```fortran
+defGrad(1,1) = defGrad(1,1) + dNidx*utmp(1)
+defGrad(1,2) = defGrad(1,2) + dNidy*utmp(1)
+defGrad(2,1) = defGrad(2,1) + dNidx*utmp(2)
+defGrad(2,2) = defGrad(2,2) + dNidy*utmp(2)
+end do
+
+c
+c CALL CONSTITUTIVE ROUTINE
+c
+isvinc= (kintk-1)*nsvint ! integration point increment
+c
+c prepare arrays for entry into material routines
+c
+do i = 1, nsvint
+statevLocal(i)=svars(i+isvinc)
+end do
+c
+c state variables
+c
+!DEC$ NOVECTOR
+do k1=1,ntens
+stran(k1) = statevLocal(k1)
+stress(k1) = zero
+end do
+c
+do i=1, ntens
+!DEC$ NOVECTOR
+do j=1, ntens
+ddsdde(i,j) = zero
+end do
+ddsdde(i,j) = one
+enddo
+c
+c compute characteristic element length
+c
+celent = sqrt(djac*dble(ninpt))
+dvmat = djac*thickness
+c
+dvdv0 = one
+call material_lib_mech(materialllib,stress,ddsdde, 
+```
+
+<!-- source-page: 225 -->
+
+```txt
+1 stran,dstran,kintk,dvdv0,dvmat,defGrad,
+2 predef_loc,dpredef_loc,npredf,celent,coords_ip)
+do k1=1,ntens
+statevLocal(k1) = stran(k1) + dstran(k1)
+end do
+isvinc= (kintk-1)*nsvint ! integration point increment
+update element state variables
+do i = 1, nsvint
+    svars(i+isvinc)=statevLocal(i)
+end do
+form stiffness matrix and internal force vector
+dNjdx = zero
+dNjdy = zero
+do i = 1, ndof*nnode
+    force(i) = zero
+    do j = 1, ndof*nnode
+    stiff(j,i) = zero
+    end do
+end do
+dvol= wght(kintk)*djac
+do nodj = 1, nnode
+incr_col = (nodj - 1)*ndof
+dNjdx = bmat(1+(nodj-1)*ndim)
+dNjdy = bmat(2+(nodj-1)*ndim)
+force_p(1) = dNjdx*stress(1) + dNjdy*stress(4)
+force_p(2) = dNjdy*stress(2) + dNjdx*stress(4)
+do jdof = 1, ndof
+jcol = jdof + incr_col
+force(jcol) = force(jcol) +
+& force_p(jdof)*dvol 
+```
+
+<!-- source-page: 226 -->
+
+```txt
+end do
+do nodi = 1, nnode
+incr_row = (nodi -1)*ndof
+dNidx = bmat(1+(nodi-1)*ndim)
+dNidy = bmat(2+(nodi-1)*ndim)
+stiff_p(1,1) = dNidx*ddsdde(1,1)*dNjdx
+& + dNidy*ddsdde(4,4)*dNjdy
+& + dNidx*ddsdde(1,4)*dNjdy
+& + dNidy*ddsdde(4,1)*dNjdx
+stiff_p(1,2) = dNidx*ddsdde(1,2)*dNjdy
+& + dNidy*ddsdde(4,4)*dNjdx
+& + dNidx*ddsdde(1,4)*dNjdx
+& + dNidy*ddsdde(4,2)*dNjdy
+stiff_p(2,1) = dNidy*ddsdde(2,1)*dNjdx
+& + dNidx*ddsdde(4,4)*dNjdy
+& + dNidy*ddsdde(2,4)*dNjdy
+& + dNidx*ddsdde(4,1)*dNjdx
+stiff_p(2,2) = dNidy*ddsdde(2,2)*dNjdy
+& + dNidx*ddsdde(4,4)*dNjdx
+& + dNidy*ddsdde(2,4)*dNjdx
+& + dNidx*ddsdde(4,2)*dNjdy
+do jdof = 1, ndof
+icol = jdof + incr_col
+do idof = 1, ndof
+irow = idof + incr_row
+stiff(irow,icol) = stiff(irow,icol) +
+& stiff_p(idof,jdof)*dvol
+end do
+end do
+end do
+end do
+end do
+c
+c assemble rhs and lhs
+c 
+```
+
+<!-- source-page: 227 -->
+
+```fortran
+do k1=1, ndof*nnode
+    rhs(k1, 1) = rhs(k1, 1) - force(k1)
+    do k2=1, ndof*nnode
+    amatrix(k1, k2) = amatrix(k1, k2) + stiff(k1,k2)
+    end do
+    end do
+    end do    ! end loop on material integration points
+    pnewdt = pnewdtLocal
+c
+999 continue
+c
+return
+end 
+```
+
+<!-- source-page: 228 -->
+
+<!-- source-page: 229 -->
+
+# 1.1.30 UEXPAN: User subroutine to define incremental thermal strains.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Thermal expansion,” Section 26.1.2 of the Abaqus Analysis User’s Guide   
+• \*EXPANSION   
+• “UEXPAN,” Section 4.1.16 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine UEXPAN:
+
+• can be used to define incremental thermal strains as functions of temperature, predefined field variables, and state variables;   
+• is intended for models in which the thermal strains depend on temperature and/or predefined field variables in complex ways or depend on state variables, which can be used and updated in this routine;   
+• is called at all integration points of elements for which the material or gasket behavior definition contains user-subroutine-defined thermal expansion; and   
+• is called twice per material point in each iteration during coupled temperature-displacement and coupled thermal-electrical-structural analyses.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE UEXPAN(EXPAN, DEXPANDT, TEMP, TIME, DTIME, PREDEF, 1 DPRED, STATEV, CMNAME, NSTATV, NOEL)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME
+C
+DIMENSION EXPAN(*), DEXPANDT(*), TEMP(2), TIME(2), PREDEF(*), 1 DPRED(*), STATEV(NSTATV)
+user coding to define EXPAN, DEXPANDT and update
+STATEV if necessary.
+RETURN
+END 
+```
+
+<!-- source-page: 230 -->
+
+# EXPAN(\*)
+
+Increments of thermal strain. The number of values to be defined and the order in which they are arranged depend on the type of thermal expansion being defined.
+
+• For isotropic expansion give the isotropic thermal strain increment as the first and only component of the matrix.   
+• For orthotropic expansion give $\Delta \epsilon _ { 1 1 } ^ { t h } , \Delta \epsilon _ { 2 2 } ^ { t h }$ , and $\Delta \epsilon _ { 3 3 } ^ { t h }$ as the first, second, and third components of the matrix, respectively.   
+• For anisotropic expansion give $\Delta \epsilon _ { 1 1 } ^ { t h } , \Delta \epsilon _ { 2 2 } ^ { t h } , \Delta \epsilon _ { 3 3 } ^ { t h } , \Delta \epsilon _ { 1 2 } ^ { t h } , \Delta \epsilon _ { 1 3 } ^ { t h }$ , and $\Delta \epsilon _ { 2 3 } ^ { t h }$ . Direct components are stored first, followed by shear components in the order presented here. For plane stress only three components of the matrix are needed; give $\Delta \epsilon _ { 1 1 } ^ { t h } , \Delta \epsilon _ { 2 2 } ^ { t h }$ , and $\Delta \epsilon _ { 1 2 } ^ { t h }$ , as the first, second, and third components, respectively.
+
+# DEXPANDT(\*)
+
+Variation of thermal strains with respect to temperature, $\partial \epsilon ^ { t h } / \partial \theta$ . The number of values and the order in which they are arranged depend on the type of thermal expansion being defined.
+
+• For isotropic expansion give the variation of the isotropic thermal strain with respect to temperature as the first and only component of the matrix.   
+• For orthotropic expansion give $\partial \epsilon _ { 1 1 } ^ { t h } / \partial \theta , \partial \epsilon _ { 2 2 } ^ { t h } / \partial \theta$ , and $\partial \epsilon _ { 3 3 } ^ { t h } / \partial \theta$ as the first, second, and third components of the matrix, respectively.   
+• For anisotropic expansion give $\partial \epsilon _ { 1 1 } ^ { t h } / \partial \theta , \partial \epsilon _ { 2 2 } ^ { t h } / \partial \theta , \partial \epsilon _ { 3 3 } ^ { t h } / \partial \theta , \partial \epsilon _ { 1 2 } ^ { t h } / \partial \theta , \partial \epsilon _ { 1 3 } ^ { t h } / \partial \theta$ , and $\partial \epsilon _ { 2 3 } ^ { t h } / \partial \theta$ . Direct components are stored first, followed by shear components in the order presented here. For plane stress only three components of the matrix are needed; give $\partial \epsilon _ { 1 1 } ^ { t h } / \partial \theta , \partial \epsilon _ { 2 2 } ^ { t h } / \partial \theta$ , and $\partial \epsilon _ { 1 2 } ^ { t h } / \partial \theta$ , as the first, second, and third components, respectively.
+
+# Variable that can be updated
+
+# STATEV(NSTATV)
+
+Array containing the user-defined solution-dependent state variables at this point. Except for coupled temperature-displacement and coupled thermal-electrical-structural analyses, these are supplied as values at the start of the increment and can be updated to their values at the end of the increment. For coupled temperature-displacement and coupled thermal-electrical-structural analyses, UEXPAN is called twice per material point per iteration. In the first call for a given material point and iteration, the values supplied are those at the start of the increment and can be updated. In the second call for the same material point and iteration, the values supplied are those returned from the first call, and they can be updated again to their values at the end of the increment.
+
+User subroutine UEXPAN allows for the incremental thermal strains to be only weakly dependent on the state variables. The Jacobian terms arising from the derivatives of the thermal strains with respect to the state variables are not taken into account.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_024.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_024.md
new file mode 100644
index 00000000..587e934f
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_024.md
@@ -0,0 +1,331 @@
+<!-- source-page: 231 -->
+
+# Variables passed in for information
+
+# TEMP(1)
+
+Current temperature (at the end of the increment).
+
+# TEMP(2)
+
+Temperature increment.
+
+# TIME(1)
+
+Step time at the end of the increment.
+
+# TIME(2)
+
+Total time at the end of the increment.
+
+# DTIME
+
+Time increment.
+
+# PREDEF(\*)
+
+Array containing the values of all the user-specified predefined field variables at this point (initial values at the beginning of the analysis and current values during the analysis).
+
+# DPRED(\*)
+
+Array of increments of predefined field variables.
+
+# CMNAME
+
+User-specified material name or gasket behavior name, left justified.
+
+# NSTATV
+
+Number of solution-dependent state variables associated with this material or gasket behavior type (specified when space is allocated for the array; see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# NOEL
+
+User-defined element number.
+
+<!-- source-page: 232 -->
+
+<!-- source-page: 233 -->
+
+# 1.1.31 UEXTERNALDB: User subroutine to manage user-defined external databases and calculate model-independent history information.
+
+Product: Abaqus/Standard
+
+# Reference
+
+• “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide
+
+# Overview
+
+User subroutine UEXTERNALDB:
+
+• is called once each at the beginning of the analysis, at the beginning of each increment, at the end of each increment, and at the end of the analysis (in addition, the user subroutine is also called once at the beginning of a restart analysis);   
+• can be used to communicate between other software and user subroutines within Abaqus/Standard;   
+• can be used to open external files needed for other user subroutines at the beginning of the analysis and to close those files at the end of the analysis;   
+• can be used to calculate or read history information at the beginning of each increment. This information can be written to user-defined COMMON block variables or external files for use during the analysis by other user subroutines; and   
+• can be used to write the current values of the user-calculated history information to external files.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE UEXTERNALDB (LOP, LRESTART, TIME, DTIME, KSTEP, KINC)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION TIME(2)
+C
+user coding to set up the Fortran environment, open files, close files,
+calculate user-defined model-independent history information,
+write history information to external files,
+recover history information during restart analyses, etc.
+do not include calls to utility routine XIT 
+```
+
+```txt
+RETURN
+END 
+```
+
+<!-- source-page: 234 -->
+
+None.
+
+# Variables passed in for information
+
+# LOP
+
+LOP=0 indicates that the user subroutine is being called at the start of the analysis.
+
+LOP=1 indicates that the user subroutine is being called at the start of the current analysis increment. The user subroutine can be called multiple times at the beginning of an analysis increment if the increment fails to converge and a smaller time increment is required.
+
+LOP=2 indicates that the user subroutine is being called at the end of the current analysis increment. When LOP=2, all information that you need to restart the analysis should be written to external files.
+
+LOP=3 indicates that the user subroutine is being called at the end of the analysis.
+
+LOP=4 indicates that the user subroutine is being called at the beginning of a restart analysis. When LOP=4, all necessary external files should be opened and properly positioned and all information required for the restart should be read from the external files.
+
+LOP=5 indicates that the user subroutine is being called at the start of a step. The KSTEP argument contains the current step number.
+
+LOP=6 indicates that the user subroutine is being called at the end of a step. The KSTEP argument contains the current step number.
+
+# LRESTART
+
+LRESTART=0 indicates that an analysis restart file is not being written for this increment.
+
+LRESTART=1 indicates that an analysis restart file is being written for this increment.
+
+LRESTART=2 indicates that an analysis restart file is being written for this increment and that only one increment is being retained per step so that the current increment overwrites the previous increment in the restart file (see “Restarting an analysis,” Section 9.1.1 of the Abaqus Analysis User’s Guide).
+
+# TIME(1)
+
+Value of current step time.
+
+# TIME(2)
+
+Value of current total time.
+
+# DTIME
+
+Time increment.
+
+# KSTEP
+
+Current step number. When LOP=4, KSTEP gives the restart step number.
+
+# KINC
+
+Current increment number. When LOP=4, KINC gives the restart increment number.
+
+<!-- source-page: 235 -->
+
+# 1.1.32 UFIELD: User subroutine to specify predefined field variables.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “USDFLD,” Section 1.1.53   
+• “Predefined fields,” Section 34.6.1 of the Abaqus Analysis User’s Guide   
+• \*FIELD   
+• “UTEMP, UFIELD, UMASFL, and UPRESS,” Section 4.1.25 of the Abaqus Verification Guide
+
+# Overview
+
+# User subroutine UFIELD:
+
+• allows you to prescribe predefined field variables at the nodes of a model—the predefined field variables at a node can be updated individually, or a number of field variables at the node can be updated simultaneously;   
+• is called whenever a user-subroutine-defined field appears;   
+• ignores any field variable values specified directly;   
+• can be used to modify field variable values read from a results file; and   
+• can be used in conjunction with user subroutine USDFLD such that the field variables that are passed in from UFIELD and interpolated to the material points can be modified (such changes are local to material point values, and nodal field variable values remain unaffected).
+
+# Updating field variables
+
+Two different methods are provided for updating field variables.
+
+# Individual variable updates
+
+By default, only one field variable at a time can be updated in user subroutine UFIELD. In this case the user subroutine will be called whenever a current value of a field variable is needed for a node that is listed in the specified field variable definition. This method can be used only for cases in which the field variables are independent of each other.
+
+# Simultaneous variable updates
+
+For cases in which the field variables depend on each other, multiple (possibly all) field variables at a point can be updated simultaneously in user subroutine UFIELD. In this case you must specify the number of field variables to be updated simultaneously at a point, and the user subroutine will be called each time the current field variable values are needed.
+
+<!-- source-page: 236 -->
+
+User subroutine interface   
+```fortran
+SUBROUTINE UFIELD(FIELD,KFIELD,NSECPT,KSTEP,KINC,TIME,NODE,1 COORDS,TEMP,DTEMP,NFIELD)
+INCLUDE 'ABA_PARAM.INC'
+DIMENSION FIELD(NSECPT,NFIELD), TIME(2), COORDS(3), 1 TEMP(NSECPT), DTEMP(NSECPT)
+user coding to define FIELD
+RETURN
+END 
+```  
+Variable to be defined   
+FIELD(NSECPT,NFIELD)
+
+Array of predefined field variable values at node number NODE. When updating only one field variable at a time, only the value of the specified field variable (see KFIELD below) must be returned. In this case NFIELD is passed into user subroutine UFIELD with a value of 1, and FIELD is thus dimensioned as FIELD(NSECPT,1). When updating all field variables simultaneously, the values of the specified number of field variables at the point must be returned. In this case FIELD is dimensioned as FIELD(NSECPT,NFIELD), where NFIELD is the number of field variables specified and KFIELD has no meaning.
+
+If NODE is part of any element other than a beam or shell, only one value of each field variable must be returned (NSECPT=1). Otherwise, the number of values to be returned depends on the mode of temperature and field variable input selected for the beam or shell section. The following cases are possible:
+
+1. Temperatures and field variables for a beam section are given as values at the points shown in the beam section descriptions. The number of values required, NSECPT, is determined by the particular section type specified, as described in “Beam cross-section library,” Section 29.3.9 of the Abaqus Analysis User’s Guide.   
+2. Temperatures and field variables are given as values at n equally spaced points through each layer of a shell section. The number of values required, NSECPT, is equal to n.   
+3. Temperatures and field variables for a beam section are given as values at the origin of the crosssection together with gradients with respect to the 2-direction and, for three-dimensional beams, the 1-direction of the section; or temperatures and field variables for a shell section are given
+
+<!-- source-page: 237 -->
+
+as values at the reference surface together with gradients through the thickness. The number of values required, NSECPT, is 3 for three-dimensional beams, 2 for two-dimensional beams, and 2 for shells. Give the midsurface value first, followed by the first and (if necessary) second gradients, as described in “Beam elements,” Section 29.3 of the Abaqus Analysis User’s Guide, and “Shell elements,” Section 29.6 of the Abaqus Analysis User’s Guide.
+
+Since field variables can also be defined directly, it is important to understand the hierarchy used in situations of conflicting information (see “Predefined fields,” Section 34.6.1 of the Abaqus Analysis User’s Guide).
+
+When the array FIELD is passed into user subroutine UFIELD, it will contain either the field variable values from the previous increment or those values obtained from the results file if this method was used. You are then free to modify these values within this subroutine.
+
+# Variables passed in for information
+
+# KFIELD
+
+User-specified field variable number. This variable is meaningful only when updating individual field variables at a time.
+
+# NFIELD
+
+User-specified number of field variables to be updated. This variable is meaningful only when updating multiple field variables simultaneously.
+
+# NSECPT
+
+Maximum number of section values required for any node in the model. The NSECPT can be 2 when only one field variable is specified at some non-beam or non-shell nodes in the model with contact.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# TIME(1)
+
+Current value of step time.
+
+# TIME(2)
+
+Current total time.
+
+# NODE
+
+Node number.
+
+# COORDS
+
+An array containing the coordinates of this node. These are the current coordinates if geometric nonlinearity is accounted for during the step (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide); otherwise, the array contains the original coordinates of the node.
+
+<!-- source-page: 238 -->
+
+# TEMP(NSECPT)
+
+Current temperature at the node. If user subroutines UTEMP and UFIELD are both used, user subroutine UTEMP is processed before user subroutine UFIELD.
+
+# DTEMP(NSECPT)
+
+Temperature increment at the node.
+
+<!-- source-page: 239 -->
+
+# 1.1.33 UFLUID: User subroutine to define fluid density and fluid compliance for hydrostatic fluid elements.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Fluid cavity definition,” Section 11.5.2 of the Abaqus Analysis User’s Guide   
+• \*FLUID BEHAVIOR   
+• “UFLUID,” Section 4.1.17 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine UFLUID:
+
+• is called for each cavity for which a user-defined fluid constitutive model is being specified;   
+• is called for every fluid element (“Surface-based fluid cavities: overview,” Section 11.5.1 of the Abaqus Analysis User’s Guide) and for every fluid exchange definition (“Fluid exchange definition,” Section 11.5.3 of the Abaqus Analysis User’s Guide) connected to a cavity reference node;   
+• requires that the fluid density, $\rho ( p , \theta )$ , and the fluid pressure compliance, $C _ { p }$ , be defined;   
+• requires that the fluid temperature compliance, $C _ { \theta }$ , be defined if the routine is to be used in a linear perturbation step and the fluid is subjected to a temperature excursion; and   
+• ignores any data specified for the fluid constitutive model outside the user subroutine.
+
+# Density and fluid mass
+
+At the start of the analysis (prior to the first iteration) the density calculated in user subroutine UFLUID (for the initial pressure, $p _ { I } .$ , and temperature, $\theta _ { I } )$ is used to calculate the fluid mass from the initial cavity volume. During the analysis the expected cavity volume is calculated from the fluid mass and the density.
+
+# User subroutine interface
+
+```fortran
+SUBROUTINE UFLUID(RHO, CP, CT, PNEWDT, ENER, PRESS, DPRESS, PRESSI, 1 TEMP, DTEMP, TEMPI, TIME, DTIME, KSTEP, KINC, NONUM, FLNAME, LFLAG)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 FLNAME
+DIMENSION TIME(2)
+user coding to define RHO, CP, and CT 
+```
+
+<!-- source-page: 240 -->
+
+# RETURN
+
+# END
+
+# Variables to be defined
+
+# RHO
+
+Fluid density, , at the end of the increment.
+
+# CP
+
+Fluid pressure compliance, $C _ { p } ,$ , at the end of the increment. For a linear perturbation step this is the base state compliance. Fluid pressure compliance is defined as
+
+$$
+C _ {p} = \frac {d \rho^ {- 1}}{d p} = - \rho^ {- 2} \frac {d \rho}{d p},
+$$
+
+where p is the fluid cavity pressure.
+
+# CT
+
+Fluid temperature compliance, $C _ { \theta }$ . This variable is needed only if a fluid temperature excursion occurs in a linear perturbation step and is the base state compliance. Fluid temperature compliance is defined as
+
+$$
+C _ {\theta} = \frac {d \rho^ {- 1}}{d \theta} = - \rho^ {- 2} \frac {d \rho}{d \theta},
+$$
+
+where is the fluid cavity temperature.
+
+# Variables that can be updated
+
+# PNEWDT
+
+Ratio of suggested new time increment to the time increment being used (DTIME, see below). This variable allows you to provide input to the automatic time incrementation algorithms in Abaqus/Standard (if automatic time incrementation is chosen).
+
+PNEWDT is set to a large value before each call to UFLUID.
+
+If PNEWDT is redefined to be less than 1.0, Abaqus/Standard must abandon the time increment and attempt it again with a smaller time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines that allow redefinition of PNEWDT for this iteration.
+
+If PNEWDT is given a value that is greater than 1.0 for all calls to user subroutines for this iteration and the increment converges in this iteration, Abaqus/Standard may increase the time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines for this iteration.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_025.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_025.md
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--- /dev/null
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+<!-- source-page: 241 -->
+
+If automatic time incrementation is not selected in the analysis procedure, values of PNEWDT that are greater than 1.0 will be ignored and values of PNEWDT that are less than 1.0 will cause the job to terminate.
+
+# ENER
+
+Energy per unit mass stored in the fluid. This variable is used for energy output only and has no effect on the solution.
+
+# Variables passed in for information
+
+# PRESS
+
+Fluid cavity pressure at the end of the increment. For a linear perturbation step this is the base state pressure.
+
+# DPRESS
+
+Fluid cavity pressure increment. For a linear perturbation step this value is zero.
+
+# PRESSI
+
+Fluid cavity pressure at the beginning of the analysis.
+
+# TEMP
+
+Fluid cavity temperature at the end of the increment. For a linear perturbation step this is the base state temperature.
+
+# DTEMP
+
+Fluid cavity temperature increment. For a linear perturbation step this value is zero.
+
+# TEMPI
+
+Fluid cavity temperature at the beginning of the analysis.
+
+# TIME(1)
+
+Current value of step time at the start of the increment.
+
+# TIME(2)
+
+Current value of total time at the start of the increment.
+
+# DTIME
+
+Time increment.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# NONUM
+
+Cavity reference node number.
+
+<!-- source-page: 242 -->
+
+# FLNAME
+
+User-specified fluid property name, left justified.
+
+# LFLAG
+
+Linear perturbation flag for the step. If this is a linear perturbation step, LFLAG=1. For a general analysis step LFLAG=0.
+
+<!-- source-page: 243 -->
+
+# 1.1.34 UFLUIDCONNECTORLOSS: User subroutine to define the loss coefficient for fluid flow in fluid pipe connector elements.
+
+Product: Abaqus/Standard
+
+# Reference
+
+• \*FLUID PIPE CONNECTOR LOSS
+
+# Overview
+
+User subroutine UFLUIDCONNECTORLOSS:
+
+• can be used to define the loss coefficient in a fluid pipe connector element;   
+• corresponds to the Darcy-Weisbach equation for pressure loss; and   
+• can be used with the fluid pipe connector elements.
+
+# User subroutine interface
+
+```txt
+subroutine ufluidconnectorloss (
+C Write only -
+* ak1, ak2,
+C Read only -
+* coords, flow, rho, visc,
+* dia, area,
+* ndim, jelno, kStep, kInc,
+* time,
+* nIarray,
+* i_array,
+* nRarray,
+* r_array,
+* ncarray,
+* c_array)
+C
+include 'aba_param.inc'
+C
+dimension time(2),
+* coords(2*ndim),
+* i_array(nIarray),
+* r_array(nRarray)
+C
+character*80 c_array(ncarray) 
+```
+
+<!-- source-page: 244 -->
+
+C
+
+```lua
+user coding to define connector friction values
+return
+end 
+```
+
+# Variables to be defined
+
+ak1
+
+This connector loss coefficient must be updated and is used when the flow is from node 1 to node 2.
+
+ak2
+
+This connector loss coefficient must be updated and is used when the flow is from node 2 to node 1.
+
+# Variables passed in for information
+
+coords(2\*ndim)
+
+Array containing original coordinates of the element. coords(1:ndim) is the coordinate of the first node, and coords(ndim+1:2\*ndim) is the coordinate of the second node.
+
+flow
+
+Current flow rate through the element.
+
+rho
+
+Current density of fluid flowing through the pipe.
+
+visc
+
+Current viscosity of fluid flowing through the pipe.
+
+dia
+
+User-specified hydraulic diameter.
+
+area
+
+User-specified hydraulic area.
+
+ndim
+
+Dimension of the element.
+
+jelno
+
+User element number for which friction coefficient is required.
+
+kStep
+
+Step number.
+
+kInc
+
+Increment number.
+
+<!-- source-page: 245 -->
+
+time(1)
+
+Current step time.
+
+time(2)
+
+Total time.
+
+nIarray
+
+Size of array i\_array.
+
+i\_array
+
+Integer array for future expansion.
+
+nRarray
+
+Size of array r\_array.
+
+r\_array
+
+Real array for future expansion.
+
+ncarray
+
+Size of array c\_array.
+
+c\_array
+
+Character array for future expansion.
+
+<!-- source-page: 246 -->
+
+<!-- source-page: 247 -->
+
+# 1.1.35 UFLUIDCONNECTORVALVE: User subroutine to define the valve opening to control flow in fluid pipe connector elements.
+
+Product: Abaqus/Standard
+
+# Reference
+
+• \*FLUID PIPE CONNECTOR LOSS
+
+# Overview
+
+User subroutine UFLUIDCONNECTORVALVE:
+
+• can be used to control the valve opening to turn off or turn on fluid flow; and   
+• can be used with fluid pipe connector elements.
+
+# User subroutine interface
+
+```txt
+subroutine ufluidconnectorvalve (
+C Write only -
+* valveOpening,
+C Read only -
+* coords, flow, rho, visc,
+* dia, area,
+* ndim, jelno, kStep, kInc,
+* time,
+* nIarray,
+* i_array,
+* nRarray,
+* r_array,
+* ncarray,
+* c_array)
+C
+    include 'aba_param.inc'
+C
+    dimension time(2),
+* coords(2*ndim),
+* i_array(nIarray),
+* r_array(nRarray)
+C
+    character*80 c_array(ncarray)
+C 
+```
+
+<!-- source-page: 248 -->
+
+```lua
+user coding to define control valve opening
+return
+end 
+```
+
+# Variable to be defined
+
+valveOpening
+
+The value of this variable must be set between 0.0 (closed/shut-off) and 1.0 (fully open) to determine whether the valve is fully or partially open or closed.
+
+# Variables passed in for information
+
+coords(2\*ndim)
+
+Array containing the original coordinates of the element. acoords(1:ndim) is the coordinate of the first node, and acoords(ndim+1:2\*ndim) is the coordinate of the second node.
+
+flow
+
+Current flow rate through the element.
+
+rho
+
+Current density of fluid flowing through the pipe.
+
+visc
+
+Current viscosity of fluid flowing through the pipe.
+
+dia
+
+User-specified hydraulic diameter.
+
+area
+
+User-specified hydraulic area.
+
+ndim
+
+Dimension of the element.
+
+jelno
+
+User element number for which a friction coefficient is required.
+
+kStep
+
+Step number.
+
+kInc
+
+Increment number.
+
+time(1)
+
+Current step time.
+
+<!-- source-page: 249 -->
+
+time(2)
+
+Total time.
+
+nIarray
+
+Size of array i\_array.
+
+i\_array
+
+Integer array for future expansion.
+
+nRarray
+
+Size of array r\_array.
+
+r\_array
+
+Real array for future expansion.
+
+ncarray
+
+Size of array c\_array.
+
+c\_array
+
+Character array for future expansion.
+
+<!-- source-page: 250 -->
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_026.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_026.md
new file mode 100644
index 00000000..3db3755c
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_026.md
@@ -0,0 +1,352 @@
+<!-- source-page: 251 -->
+
+# 1.1.36 UFLUIDLEAKOFF: User subroutine to define the fluid leak-off coefficients for pore pressure cohesive elements.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Defining the constitutive response of fluid within the cohesive element gap,” Section 32.5.7 of the Abaqus Analysis User’s Guide   
+• \*FLUID LEAKOFF   
+• “Propagation of hydraulically driven fracture,” Section 3.3.2 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine UFLUIDLEAKOFF:
+
+• can be used to define the fluid leak-off coefficients for pore pressure cohesive elements;   
+• is called at all material calculation points of elements for which the material definition contains user-defined leak-off coefficients; and   
+• can include material behavior dependent on field variables or state variables.
+
+# User subroutine interface
+
+```fortran
+SUBROUTINE UFLUIDLEAKOFF (PERM, PGRAD, DN, P_INT, P_BOT, P_TOP, 1 ANM, TANG, TIME, DTIME, TEMP, DTEMP, PREDEF, DPRED, C_BOT, C_TOP, 2 DC_BOT, DC_TOP, STATEV, NSTATV, NOEL, NPT, KSTEP, KINC)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME
+DIMENSION PERM(2), PGRAD(2), ANM(3), TANG(3, 2), TIME(2), PREDEF(1), 1 DPRED(1), DC_BOT(3), DC_TOP(3), STATEV(NSTATV)
+user coding to define C_BOT, C_TOP, DC_BOT, and DC_TOP
+RETURN
+END 
+```
+
+<!-- source-page: 252 -->
+
+Variables to be defined   
+C_BOT $C_{bot}$ , fluid leak-off coefficient on the bottom side of a pore pressure cohesive element.
+
+C_TOP $C_{top}$ , fluid leak-off coefficient on the top side of a pore pressure cohesive element.
+
+DC_BOT(1) $\partial C_{bot}/\partial d$ , where d=DN.
+
+DC_BOT(2) $\partial C_{bot}/\partial p_{int}$ , where $p_{int}=P\_INT$ .
+
+DC_BOT(3) $\partial C_{bot}/\partial p_{bot}$ , where $p_{bot}=P\_BOT$ .
+
+DC_TOP(1) $\partial C_{top}/\partial d$ , where d=DN.
+
+DC_TOP(2) $\partial C_{top}/\partial p_{int}$ , where $p_{int}=P\_INT$ .
+
+DC_TOP(3) $\partial C_{top}/\partial p_{top}$ , where $p_{top}=P\_TOP$ .
+
+# STATEV(NSTATV)
+
+An array containing the values of the solution-dependent state variables. You define the meaning of these variables. These are passed in as the values at the beginning of the increment and must be returned as the values at the end of the increment. The size of the array is defined as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide.
+
+Variables passed in for information   
+PERM (1)
+Fluid permeability.
+
+PERM (2)
+The derivative of fluid permeability with regard to the opening.
+
+PGRAD (1)
+The first component of internal pressure gradient.
+
+PGRAD (2)
+The second component of internal pressure gradient.
+
+<!-- source-page: 253 -->
+
+# DN
+
+The relative opening of the element.
+
+# P\_INT
+
+Internal pressure.
+
+# P\_BOT
+
+Bottom pressure.
+
+# P\_TOP
+
+Top pressure.
+
+# ANM
+
+Normal vector directed from the bottom face toward the top face.
+
+# TANG
+
+Tangent direction vectors.
+
+# TIME(1)
+
+Value of step time at the beginning of the current increment.
+
+# TIME(2)
+
+Value of total time at the beginning of the current increment.
+
+# DTIME
+
+Time increment.
+
+# TEMP
+
+Temperature at the start of the increment.
+
+# DTEMP
+
+Increment of temperature.
+
+# PREDEF
+
+Array of interpolated values of predefined field variables at this point at the start of the increment, based on the values read in at the nodes.
+
+# DPRED
+
+Array of increments of predefined field variables.
+
+# NSTATV
+
+Number of solution-dependent state variables that are associated with this material type (defined as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+<!-- source-page: 254 -->
+
+# UFLUIDLEAKOFF
+
+# NOEL
+
+Element number.
+
+# NPT
+
+Integration point number.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+<!-- source-page: 255 -->
+
+# 1.1.37 UFLUIDPIPEFRICTION: User subroutine to define the frictional coefficient for fluid flow in fluid pipe elements.
+
+Product: Abaqus/Standard
+
+# Reference
+
+• \*FLUID PIPE FLOW LOSS
+
+# Overview
+
+User subroutine UFLUIDPIPEFRICTION:
+
+• can be used to define the frictional coefficient for fluid flow to determine the pipe loss;   
+• corresponds to the Darcy-Weisbach equation for pressure loss; and   
+• can be used with the fluid pipe elements.
+
+# User subroutine interface
+
+```txt
+subroutine ufluidpipefriction (
+C Write only -
+* friction
+C Read only -
+* flow, rho, visc, rough,
+* dia, area,
+* ndim, jelno, kstep, kinc,
+* time, coords,
+* niarray,
+* i_array,
+* nrarray,
+* r_array,
+* ncarray,
+* c_array)
+include 'aba_param.inc'
+C
+    dimension time(2),
+* coords(2*ndim),
+* i_array(niarray),
+* r_array(nrarray)
+C
+    character*80 c_array(ncarray)
+C 
+```
+
+<!-- source-page: 256 -->
+
+```lua
+user coding to define friction
+return
+end 
+```
+
+# Variable to be defined
+
+# friction
+
+This value must be updated to the current value of the friction coefficient.
+
+# Variables passed in for information
+
+# flow
+
+Current flow rate through the element.
+
+# rho
+
+Current density of fluid flowing through the pipe.
+
+# visc
+
+Current viscosity of fluid flowing through the pipe.
+
+# rough
+
+User-specified pipe roughness.
+
+# dia
+
+User-specified hydraulic diameter.
+
+# area
+
+User-specified hydraulic area.
+
+# ndim
+
+Dimension of the element.
+
+# jelno
+
+User element number for which friction coefficient is required.
+
+# kstep
+
+Step number.
+
+# kinc
+
+Increment number.
+
+# time(1)
+
+Current step time.
+
+# time(2)
+
+Total time.
+
+<!-- source-page: 257 -->
+
+coords(2\*ndim)
+
+Array containing original coordinates of the element. coords(1:ndim) is the coordinate of the first node, and coords(ndim+1:2\*ndim) is the coordinate of the second node.
+
+niarray
+
+Size of array i\_array.
+
+i\_array
+
+Integer array for future expansion.
+
+nrarray
+
+Size of array r\_array.
+
+r\_array
+
+Real array for future expansion.
+
+ncarray
+
+Size of array c\_array.
+
+c\_array
+
+Character array for future expansion.
+
+<!-- source-page: 258 -->
+
+<!-- source-page: 259 -->
+
+# 1.1.38 UGENS: User subroutine to define the mechanical behavior of a shell section.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Using a general shell section to define the section behavior,” Section 29.6.6 of the Abaqus Analysis User’s Guide   
+• \*SHELL GENERAL SECTION
+
+# Overview
+
+User subroutine UGENS:
+
+• is used to define the (nonlinear) mechanical behavior of a shell section directly in terms of generalized section quantities;   
+• requires you to define the section behavior of the shell directly in terms of membrane stresses and forces, curvature changes, and bending moments;   
+• will be called at all integration points in all shell elements with a general, arbitrary, elastic shell section and a user-subroutine-defined shell section stiffness; and   
+• can be used with all static or dynamic procedures other than the quasi-static procedure, since that procedure uses automatic time stepping based on the techniques used by Abaqus/Standard to integrate standard creep laws.
+
+# Storage of membrane and bending components
+
+In the force and strain arrays and in the matrix DDNDDE, direct membrane terms are stored first, followed by the shear membrane term, and then the direct and shear bending terms. Only active components are stored, so the number of entries depends on the element type (see Table 1.1.38–1).
+
+Table 1.1.38–1 Active section force/moment components. 
+
+<table><tr><td>Element type</td><td>Force and moment components</td></tr><tr><td>Three-dimensional shells (S4R, S8R, S8R5, etc.) and axisymmetric shells with asymmetric deformation (SAXA1N, SAXA2N)</td><td> $N_{11}, N_{22}, N_{12}, M_{11}, M_{22}, M_{12}$ </td></tr><tr><td>Axisymmetric shells (SAX1, SAX2, etc)</td><td> $N_{11}, N_{22}, M_{11}, M_{22}$ </td></tr></table>
+
+There are NDI direct membrane and NSHR shear membrane components and NDI direct bending and NSHR shear bending components: a total of NSECV components. The order of the components is defined in “Using a general shell section to define the section behavior,” Section 29.6.6 of the Abaqus Analysis User’s Guide.
+
+Engineering measures of shear membrane strain $( \gamma _ { 1 2 } )$ and twist $\left( \mathrm { K } _ { 1 2 } \right)$ are used.
+
+<!-- source-page: 260 -->
+
+# Increments for which only the section stiffness can be defined
+
+Abaqus/Standard passes zero strain increments into user subroutine UGENS to start the first increment of all the steps and all increments of steps for which you have suppressed extrapolation in time from the previous incremental solution (“Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide). In this case you can define only the section stiffness (DDNDDE).
+
+# Stability
+
+You should ensure that the integration scheme coded in this routine is stable—no direct provision is made to include a stability limit in the time stepping scheme based on the calculations in UGENS.
+
+# Convergence rate
+
+DDNDDE must be defined accurately if rapid convergence of the overall Newton scheme is to be achieved. In most cases the accuracy of this definition is the most important factor governing the convergence rate. Unsymmetric equation solution is as much as four times as expensive as the corresponding symmetric system. Therefore, if the section stiffness matrix (DDNDDE) is only slightly unsymmetric, it may be computationally less expensive to use a symmetric approximation and accept a slightly slower rate of convergence.
+
+# Use with shells that have transverse shear and/or hourglass stiffness
+
+If user subroutine UGENS is used to describe the section behavior of shells with transverse shear, you must define the transverse shear stiffness (see “Defining the transverse shear stiffness” in “Using a general shell section to define the section behavior,” Section 29.6.6 of the Abaqus Analysis User’s Guide).
+
+If user subroutine UGENS is used to describe the section behavior of shells with hourglass stiffness, you must define the hourglass stiffness parameter for hourglass control based on total stiffness (see “Specifying nondefault hourglass control parameters for reduced-integration shell elements” in “Using a general shell section to define the section behavior,” Section 29.6.6 of the Abaqus Analysis User’s Guide). The hourglass stiffness parameter is not required for enhanced hourglass control, but you can define a scaling factor for the stiffness associated with the drill degree of freedom (rotation about the surface normal).
+
+# Use with continuum shell elements
+
+User subroutine UGENS cannot be used to describe the section behavior of continuum shell elements.
+
+# User subroutine interface
+
+SUBROUTINE UGENS(DDNDDE,FORCE,STATEV,SSE,SPD,PNEWDT,STRAN,   
+1 DSTRAN,TSS,TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED,CENAME,NDI,   
+2 NSHR,NSECV,NSTATV,PROPS,JPROPS,NPROPS,NJPROP,COORDS,CELENT,  
+3 THICK,DFGRD,CURV,BASIS,NOEL,NPT,KSTEP,KINC,NIT,LINPER)
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_027.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_027.md
new file mode 100644
index 00000000..ab3f9e70
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_027.md
@@ -0,0 +1,337 @@
+<!-- source-page: 261 -->
+
+```prolog
+C
+INCLUDE 'ABA_PARAM.INC'
+CHARACTER*80 CENAME
+DIMENSION DDNDDE(NSECV,NSECV),FORCE(NSECV),STATEV(NSTATV),
+1 STRAN(NSECV),DSTRAN(NSECV),TSS(2),TIME(2),PREDEF(*),
+2 DPRED(*),PROPS(*),JPROPS(*),COORDS(3),DFGRD(3,3),
+3 CURV(2,2),BASIS(3,3)
+user coding to define DDNDDE, FORCE, STATEV, SSE, PNEWDT
+RETURN
+END 
+```
+
+# Variables to be defined
+
+# DDNDDE(NSECV,NSECV)
+
+Section stiffness matrix of the shell section, , where are the section forces and moments on the shell section and are the generalized section strains in the shell. DDNDDE(I,J) defines the change in the Ith force component at the end of the time increment caused by an infinitesimal perturbation of the Jth component of the section strain increment array. The size of this matrix depends on the values of NSECV (see below for details).
+
+Unless you invoke the unsymmetric equation solution capability in the general shell section definition (“Defining whether or not the section stiffness matrices are symmetric” in “Using a general shell section to define the section behavior,” Section 29.6.6 of the Abaqus Analysis User’s Guide), Abaqus/Standard will use only the symmetric part of DDNDDE. The symmetric part of the matrix is calculated by taking one half the sum of the matrix and its transpose.
+
+# FORCE(NSECV)
+
+This array is passed in as the forces and moments per unit length on the shell surface at the beginning of the increment and must be updated in this routine to be the forces and moments at the end of the increment.
+
+# STATEV(NSTATV)
+
+An array containing the solution-dependent state variables. These are passed in as the values at the beginning of the increment and must be returned as the values at the end of the increment.
+
+# SSE, SPD
+
+Elastic strain energy and plastic dissipation, respectively. These are passed in as the values at the beginning of the increment and should be updated to the corresponding energy values at the end of the increment. These values have no effect on the solution; they are used for the energy output.
+
+<!-- source-page: 262 -->
+
+# PNEWDT
+
+Ratio of suggested new time increment to the time increment being used (DTIME, see below). This variable allows you to provide input to the automatic time incrementation algorithms in Abaqus/Standard (if automatic time incrementation is chosen).
+
+PNEWDT is set to a large value before each call to UGENS.
+
+If PNEWDT is redefined to be less than 1.0, Abaqus/Standard must abandon the time increment and attempt it again with a smaller time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines that allow redefinition of PNEWDT for this iteration.
+
+If PNEWDT is given a value that is greater than 1.0 for all calls to user subroutines for this iteration and the increment converges in this iteration, Abaqus/Standard may increase the time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines for this iteration.
+
+If automatic time incrementation is not selected in the analysis procedure, values of PNEWDT that are greater than 1.0 will be ignored and values of PNEWDT that are less than 1.0 will cause the job to terminate.
+
+# Variables passed in for information
+
+# STRAN(NSECV)
+
+An array containing the generalized section strains (membrane strains and curvature changes) at the beginning of the increment. The size of this array depends on the value of NSECV (see below for details).
+
+# DSTRAN(NSECV)
+
+Array of generalized section strain increments.
+
+# TSS(2)
+
+Array containing the transverse shear strains.
+
+# TIME(1)
+
+Value of step time at the beginning of the current increment.
+
+# TIME(2)
+
+Value of total time at the beginning of the current increment.
+
+# DTIME
+
+Time increment.
+
+# TEMP
+
+Temperature at the start of the increment.
+
+<!-- source-page: 263 -->
+
+# DTEMP
+
+Increment of temperature.
+
+# PREDEF
+
+Array of interpolated values of predefined field variables at this point at the start of the increment, based on the values read in at the nodes.
+
+# DPRED
+
+Array of increments of predefined field variables.
+
+# CENAME
+
+User-specified element set name associated with this section, left justified.
+
+# NDI
+
+Number of direct force components at this point.
+
+# NSHR
+
+Number of shear force components at this point.
+
+# NSECV
+
+Size of the force and strain component arrays.
+
+# NSTATV
+
+User-defined number of solution-dependent state variables associated with this section (“Defining the number of solution-dependent variables that must be stored for the section” in “Using a general shell section to define the section behavior,” Section 29.6.6 of the Abaqus Analysis User’s Guide).
+
+# PROPS(NPROPS)
+
+A floating point array containing the NPROPS real property values defined for use with this section.
+
+# JPROPS(NJPROP)
+
+An integer array containing the NJPROP integer property values defined for use with this section.
+
+# NPROPS
+
+User-defined number of real property values associated with this section (“Defining the section properties” in “Using a general shell section to define the section behavior,” Section 29.6.6 of the Abaqus Analysis User’s Guide).
+
+# NJPROP
+
+User-defined number of integer property values associated with the element (“Defining the section properties” in “Using a general shell section to define the section behavior,” Section 29.6.6 of the Abaqus Analysis User’s Guide).
+
+# COORDS
+
+An array containing the current coordinates of this integration point.
+
+<!-- source-page: 264 -->
+
+# CELENT
+
+Characteristic element length in the reference surface.
+
+# THICK
+
+Original section thickness.
+
+# DFGRD(3,3)
+
+An array containing the components of the midsurface deformation gradient, $\bar { f } _ { i j }$ . The deformation gradient curvature tensor is available for finite-strain shells (S3/S3R, S4, S4R, SAXs, and SAXAs); it is not available for small-strain shells.
+
+The deformation gradient is stored as a 3 × 3 matrix with component equivalence DFGRD $( \pmb { \tau } , \pmb { \ J } ) \Leftrightarrow$ $\bar { f } _ { i j } . \bar { f } _ { \alpha \beta }$ (Greek subscripts range from 1 to 2) are the in-plane components of the deformation gradient, and $\bar { f } _ { 3 3 }$ is the thickness change component. The components, $\bar { \bar { f } } _ { \alpha 3 }$ , are the transverse shear strains scaled by $\bar { f } _ { 3 3 }$ . The remaining components, $\bar { f } _ { 3 \beta }$ , are all zero.
+
+The tensor is provided in the local shell coordinate system.
+
+# CURV(2,2)
+
+An array containing the midsurface curvature tensor, $b _ { \alpha \beta }$ . The curvature tensor is available for finitestrain shells (S3/S3R, S4, S4R, SAXs, and SAXAs); it is not available for small-strain shells.
+
+The curvature tensor is stored as a 2 × 2 matrix with component equivalence CURV $( \pmb { \tau } , \pmb { \sigma } ) \Leftrightarrow b _ { \alpha \beta }$
+
+The tensor is provided in the local shell coordinate system.
+
+# BASIS(3,3)
+
+An array containing the direction cosines of the shell local surface coordinate system. BASIS(1,1), BASIS(2,1), and BASIS(3,1) give the (1, 2, 3) components of the first local direction, etc. The first two directions are in the plane of the element surface, and the third direction is the normal. The conventions for local directions on shell surfaces are defined in “Conventions,” Section 1.2.2 of the Abaqus Analysis User’s Guide. You can redefine the local system; see “Orientations,” Section 2.2.5 of the Abaqus Analysis User’s Guide.
+
+# NOEL
+
+Element number.
+
+# NPT
+
+Integration point number.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# NIT
+
+Iteration number. NIT=0 during the first assembly of the system matrix in any increment.
+
+<!-- source-page: 265 -->
+
+# LINPER
+
+Linear perturbation flag. LINPER=1 if the step is a linear perturbation step. LINPER=0 if the step is a general step.
+
+<!-- source-page: 266 -->
+
+<!-- source-page: 267 -->
+
+# 1.1.39 UHARD: User subroutine to define the yield surface size and hardening parameters for isotropic plasticity or combined hardening models.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Classical metal plasticity,” Section 23.2.1 of the Abaqus Analysis User’s Guide   
+• “Models for metals subjected to cyclic loading,” Section 23.2.2 of the Abaqus Analysis User’s Guide   
+• \*CYCLIC HARDENING   
+• \*PLASTIC
+
+# Overview
+
+User subroutine UHARD:
+
+• is called at all material calculation points of elements for which the material definition includes user-defined isotropic hardening or cyclic hardening for metal plasticity;   
+• can be used to define a material’s isotropic yield behavior;   
+• can be used to define the size of the yield surface in a combined hardening model;   
+• can include material behavior dependent on field variables or state variables; and   
+• requires, when appropriate, that the values of the derivatives of the yield stress (or yield surface size in combined hardening models) be defined with respect to the strain, strain rate, and temperature.
+
+# User subroutine interface
+
+```prolog
+SUBROUTINE UHARD (SYIELD, HARD, EQPLAS, EQPLASRT, TIME, DTIME, TEMP, 1 DTEMP, NOEL, NPT, LAYER, KSPT, KSTEP, KINC, CMNAME, NSTATV, 2 STATEV, NUMFIELDV, PREDEF, DPRED, NUMPROPS, PROPS)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME
+DIMENSION HARD(3), STATEV(NSTATV), TIME(*), $ PREDEF(NUMFIELDV), DPRED(*), PROPS(*)
+user coding to define SYIELD, HARD(1), HARD(2), HARD(3)
+RETURN
+END 
+```
+
+<!-- source-page: 268 -->
+
+# Variables to be defined
+
+# SYIELD
+
+. Yield stress for isotropic plasticity. Yield surface size for combined hardening.
+
+# HARD(1)
+
+Variation of SYIELD with respect to the equivalent plastic strain,
+
+# HARD(2)
+
+Variation of SYIELD with respect to the equivalent plastic strain rate,
+
+# HARD(3)
+
+Variation of SYIELD with respect to temperature, This quantity is required only in adiabatic, fully coupled temperature-displacement, and thermal-electrical-structural analyses.
+
+# STATEV(NSTATV)
+
+Array containing the user-defined solution-dependent state variables at this point. These are supplied as values at the beginning of the increment or as values updated by other user subroutines (see “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide) and must be returned as values at the end of the increment.
+
+# Variables passed in for information
+
+# EQPLAS
+
+Equivalent plastic strain,
+
+# EQPLASRT
+
+Equivalent plastic strain rate,
+
+# TIME(1)
+
+Value of step time at the beginning of the current increment.
+
+# TIME(2)
+
+Value of total time at the beginning of the current increment.
+
+# DTIME
+
+Time increment.
+
+# TEMP
+
+Temperature at the beginning of the increment.
+
+# DTEMP
+
+Increment of temperature.
+
+# NOEL
+
+Element number.
+
+<!-- source-page: 269 -->
+
+# NPT
+
+Integration point number.
+
+# LAYER
+
+Layer number (for composite shells and layered solids).
+
+# KSPT
+
+Section point number within the current layer.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# CMNAME
+
+User-specified material name, left justified.
+
+# NSTATV
+
+User-specified number of solution-dependent state variables associated with this material (“Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# NUMFIELDV
+
+Number of field variables.
+
+# PREDEF(NUMFIELDV)
+
+Array of interpolated values of predefined field variables at this material point at the start of the increment based on the values read in at the nodes (initial values at the beginning of the analysis and current values during the analysis).
+
+# DPRED(NUMFIELDV)
+
+Array of increments of predefined field variables at this material point for this increment; this includes any values updated by user subroutine USDFLD.
+
+# NPROPS
+
+Number of hardening properties entered for this user-defined hardening definition.
+
+# PROPS(NPROPS)
+
+Array of hardening properties entered for this user-defined hardening definition.
+
+<!-- source-page: 270 -->
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_028.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_028.md
new file mode 100644
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--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_028.md
@@ -0,0 +1,302 @@
+<!-- source-page: 271 -->
+
+# 1.1.40 UHYPEL: User subroutine to define a hypoelastic stress-strain relation.
+
+Product: Abaqus/Standard
+
+# References
+
+• “Hypoelastic behavior,” Section 22.4.1 of the Abaqus Analysis User’s Guide   
+• \*HYPOELASTIC
+
+# Overview
+
+User subroutine UHYPEL:
+
+• can be used to define isotropic hypoelastic material behavior, thus requiring the definition of Young’s modulus, E, and Poisson’s ratio, ;   
+• is called at all material calculation points of elements for which the material definition contains user-defined hypoelastic behavior;   
+• can be used in conjunction with user subroutine USDFLD to redefine any field variables that are passed in (see “USDFLD,” Section 1.1.53); and   
+• ignores any data specified outside the user subroutine for the associated hypoelastic material definition.
+
+# Special considerations for various element types
+
+There are several special considerations that need to be noted.
+
+# Beams and shells that calculate transverse shear energy
+
+When UHYPEL is used to define the material response of shell or beam elements that calculate transverse shear energy, Abaqus/Standard cannot calculate a default value for the transverse shear stiffness of the element. Hence, you must define the element’s transverse shear stiffness. See “Shell section behavior,” Section 29.6.4 of the Abaqus Analysis User’s Guide, and “Choosing a beam element,” Section 29.3.3 of the Abaqus Analysis User’s Guide, for guidelines on choosing this stiffness.
+
+# Elements with hourglassing modes
+
+If this capability is used to describe the material of elements with hourglassing modes, you must define the hourglass stiffness for hourglass control based on the total stiffness approach. The hourglass stiffness is not required for enhanced hourglass control, but you can define a scaling factor for the stiffness associated with the drill degree of freedom (rotation about the surface normal). See “Section controls,” Section 27.1.4 of the Abaqus Analysis User’s Guide.
+
+<!-- source-page: 272 -->
+
+User subroutine interface   
+```txt
+SUBROUTINE UHYPEL(E, GNU, STRAIN, NDI, NSHR, EINV1, EINV2, EINV3, 1 COORDS, NOEL, TEMP, PREDEF, CMNAME)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME
+C
+DIMENSION STRAIN(*), COORDS(3), PREDEF(*)
+user coding to define E and GNU
+RETURN
+END 
+```
+
+Variables to be defined   
+```txt
+E Young's modulus.  
+GNU Poisson's ratio. 
+```
+
+Variables passed in for information   
+STRAIN
+Array containing the total (elastic) strains, ( $\varepsilon$ ).
+NDI
+Number of direct strain components at this point.
+NSHR
+Number of shear strain components at this point.
+
+EINV1 $I_{1} = \mathrm{trace}(\varepsilon)$ , the first strain invariant.
+
+EINV2 $I_{2} = 1 / 2(\varepsilon :\varepsilon -I_{1}^{2})$ , the second strain invariant.
+
+EINV3 $I_{3} = \operatorname{det}(\varepsilon)$ , the third strain invariant.
+
+<!-- source-page: 273 -->
+
+# COORDS
+
+An array containing the coordinates of the material point. These are the current coordinates if geometric nonlinearity is accounted for during the step (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide); otherwise, the array contains the original coordinates of the point.
+
+# NOEL
+
+Element number.
+
+# TEMP
+
+Current temperature at this point.
+
+# PREDEF
+
+An array containing current values of the predefined field variables at this point (initial values at the beginning of the analysis and current values during the analysis).
+
+# CMNAME
+
+User-specified material name, left justified.
+
+<!-- source-page: 274 -->
+
+<!-- source-page: 275 -->
+
+# 1.1.41 UHYPER: User subroutine to define a hyperelastic material.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Hyperelastic behavior of rubberlike materials,” Section 22.5.1 of the Abaqus Analysis User’s Guide   
+• \*HYPERELASTIC   
+• “UMAT and UHYPER,” Section 4.1.21 of the Abaqus Verification Guide
+
+# Overview
+
+# User subroutine UHYPER:
+
+• can be used to define the strain energy potential for isotropic hyperelastic material behavior;   
+• is called at all material calculation points of elements for which the material definition contains user-defined hyperelastic behavior;   
+• can include material behavior dependent on field variables or state variables; and   
+• requires that the values of the derivatives of the strain energy density function of the hyperelastic material be defined with respect to the strain invariants.
+
+# Special considerations for various element types
+
+There are several special considerations that need to be noted.
+
+# Shells that calculate transverse shear energy
+
+When UHYPER is used to define the material response of shell elements that calculate transverse shear energy, Abaqus/Standard cannot calculate a default value for the transverse shear stiffness of the element. Hence, you must define the element’s transverse shear stiffness. See “Shell section behavior,” Section 29.6.4 of the Abaqus Analysis User’s Guide, for guidelines on choosing this stiffness.
+
+# Elements with hourglassing modes
+
+If this capability is used to describe the material of elements with hourglassing modes, you must define the hourglass stiffness for hourglass control based on the total stiffness approach. The hourglass stiffness is not required for enhanced hourglass control, but you can define a scaling factor for the stiffness associated with the drill degree of freedom (rotation about the surface normal). See “Section controls,” Section 27.1.4 of the Abaqus Analysis User’s Guide.
+
+# User subroutine interface
+
+SUBROUTINE UHYPER(BI1,BI2,AJ,U,UI1,UI2,UI3,TEMP,NOEL, 1 CMNAME,INCMPFLAG,NUMSTATEV,STATEV,NUMFIELDV,FIELDV,
+
+<!-- source-page: 276 -->
+
+UHYPER   
+```txt
+2 FIELDVINC, NUMPROPS, PROPS)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME
+DIMENSION U(2), UI1(3), UI2(6), UI3(6), STATEV(*), FIELDV(*), 2 FIELDVINC(*), PROPS(*)
+user coding to define U, UI1, UI2, UI3, STATEV 
+```  
+RETURN END
+
+Variables to be defined   
+U(1)
+U, strain energy density function. For a compressible material, at least one derivative involving J should be nonzero. For an incompressible material, all derivatives involving J will be ignored. The strain invariants— $\overline{I}_{1}$ , $\overline{I}_{2}$ , and J—are defined in “Hyperelastic behavior of rubberlike materials,” Section 22.5.1 of the Abaqus Analysis User’s Guide.
+
+U(2) $\tilde{U}_{dev}$ , the deviatoric part of the strain energy density of the primary material response. This quantity is needed only if the current material definition also includes Mullins effect (see “Mullins effect,” Section 22.6.1 of the Abaqus Analysis User’s Guide).
+
+UI1(1) $\partial U/\partial\overline{I}_{1}$ .
+
+UI1(2) $\partial U/\partial\overline{I}_{2}$ .
+
+UI1(3) $\partial U/\partial J$ .
+
+UI2(1) $\partial^{2}U/\partial\overline{I}_{1}^{2}$ .
+
+UI2(2) $\partial^{2}U/\partial\overline{I}_{2}^{2}$ .
+
+UI2(3) $\partial^{2}U/\partial J^{2}$ .
+
+<!-- source-page: 277 -->
+
+UI2 (4) $\partial^{2}U/\partial\overline{I}_{1}\partial\overline{I}_{2}.$
+
+UI2 (5) $\partial^{2}U/\partial\overline{I}_{1}\partial J.$
+
+UI2 (6) $\partial^{2}U/\partial\overline{I}_{2}\partial J.$
+
+UI3 (1) $\partial^{3}U/\partial\overline{I}_{1}^{2}\partial J.$
+
+UI3 (2) $\partial^{3}U/\partial\overline{I}_{2}^{2}\partial J.$
+
+UI3 (3) $\partial^{3}U/\partial\overline{I}_{1}\partial\overline{I}_{2}\partial J.$
+
+UI3 (4) $\partial^{3}U/\partial\overline{I}_{1}\partial J^{2}.$
+
+UI3 (5) $\partial^{3}U/\partial\overline{I}_{2}\partial J^{2}.$
+
+UI3 (6) $\partial^{3}U/\partial J^{3}.$
+
+# STATEV
+
+Array containing the user-defined solution-dependent state variables at this point. These are supplied as values at the start of the increment or as values updated by other user subroutines (see “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide) and must be returned as values at the end of the increment.
+
+# Variables passed in for information
+
+BI1 $\overline{I}_1$
+
+BI2 $\overline{I}_2$
+
+```txt
+AJ J. 
+```
+
+# TEMP
+
+Current temperature at this point.
+
+<!-- source-page: 278 -->
+
+# NOEL
+
+Element number.
+
+# CMNAME
+
+User-specified material name, left justified.
+
+# INCMPFLAG
+
+Incompressibility flag defined to be 1 if the material is specified as incompressible or 0 if the material is specified as compressible.
+
+# NUMSTATEV
+
+User-defined number of solution-dependent state variables associated with this material (see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# NUMFIELDV
+
+Number of field variables.
+
+# FIELDV
+
+Array of interpolated values of predefined field variables at this material point at the end of the increment based on the values read in at the nodes (initial values at the beginning of the analysis and current values during the analysis).
+
+# FIELDVINC
+
+Array of increments of predefined field variables at this material point for this increment; this includes any values updated by the user subroutine USDFLD.
+
+# NUMPROPS
+
+Number of material properties entered for this user-defined hyperelastic material.
+
+# PROPS
+
+Array of material properties entered for this user-defined hyperelastic material.
+
+<!-- source-page: 279 -->
+
+# 1.1.42 UINTER: User subroutine to define surface interaction behavior for contact surfaces.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “User-defined interfacial constitutive behavior,” Section 37.1.6 of the Abaqus Analysis User’s Guide   
+• \*SURFACE INTERACTION   
+• “UINTER,” Section 4.1.20 of the Abaqus Verification Guide
+
+# Overview
+
+# User subroutine UINTER:
+
+• is called at points on the slave surface of a contact pair with a user-defined constitutive model defining the interaction between the surfaces;   
+• can be used to define the mechanical (normal and shear) and thermal (heat flux) interactions between surfaces;   
+• can be used when the normal surface behavior (contact pressure versus overclosure) models (“Contact pressure-overclosure relationships,” Section 37.1.2 of the Abaqus Analysis User’s Guide) or the extended versions of the classical Coulomb friction model (“Frictional behavior,” Section 37.1.5 of the Abaqus Analysis User’s Guide) are too restrictive and a more complex definition of normal and shear transmission between contacting surfaces, including damping properties, are required;   
+• must provide the entire definition of the mechanical and the thermal interaction between the contacting surfaces (hence, no additional surface behaviors can be specified in conjunction with this capability);   
+• can provide the entire definition of viscous and structural damping for interactions between the contacting surfaces for direct and mode-based steady-state dynamic analysis (including the subspace projection method), transient mode-based analysis, complex eigenvalue extraction, matrix generation, and substructure generation;   
+• only accounts for element damping in mode-based procedures if the SIM architecture is used;   
+• can use and update solution-dependent state variables; and   
+• is not available for contact elements.
+
+# User subroutine interface
+
+SUBROUTINE UINTER(STRESS,DDSDDR,DVISCOUS,DSTRUCTURAL,FLUX,DDFDDT, 1 DDSDDT,DDFDDR,STATEV,SED,SFD,SPD,SVD,SCD,PNEWDT,RDISP,
+
+<!-- source-page: 280 -->
+
+```prolog
+2 DRDISP,
+3 TEMP, DTEMP, PREDEF, DPRED, TIME, DTIME, FREQR, CINAME, SLNAME,
+4 MSNAME,
+5 PROPS, COORDS, ALOCALDIR, DROT, AREA, CHRLNGTH, NODE, NDIR, NSTATV,
+6 NPRED, NPROPS, MCRD, KSTEP, KINC, KIT, LINPER, LOPENCLOSE, LSTATE,
+7 LSDI, LPRINT)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CINAME, SLNAME, MSNAME
+DIMENSION STRESS (NDIR), DDSDDR (NDIR, NDIR), FLUX (2), DDFDDT (2, 2),
+1 DDSDDT (NDIR, 2), DDFDDR (2, NDIR), STATEV (NSTATV),
+2 RDISP (NDIR), DRDISP (NDIR), TEMP (2), DTEMP (2), PREDEF (2, NPRED),
+3 DPRED (2, NPRED), TIME (2), PROPS (NPROPS), COORDS (MCRD),
+4 ALOCALDIR (3, 3), DROT (2, 2), DVISCOUS (NDIR, NDIR),
+5 DSTRUCTURAL (NDIR, NDIR)
+user coding to define STRESS, DDSDDR, FLUX, DDFDDT,
+DDSDDT, DDFDDR,
+and, optionally, STATEV, SED, SFD, SPD, SVD, SCD, PNEWDT,
+LOPENCLOSE, LSTATE, LSDI, DVISCOUS, DSTRUCTURAL
+RETURN
+END 
+```
+
+# Variables to be defined
+
+# STRESS(NDIR)
+
+This array is passed in as the stress between the slave and master surfaces at the beginning of the increment and must be updated in this routine to be the stress at the end of the increment. The stress must be defined in a local coordinate system (see ALOCDIR). This variable must be defined for a stress/displacement, a fully coupled temperature-displacement, or a coupled thermal-electrical-structural analysis. The sign convention for stresses is that a positive stress indicates compression across contact surfaces, while a negative stress indicates tension.
+
+# DDSDDR(NDIR,NDIR)
+
+Interface stiffness matrix. DDSDDR(I,J) defines the change in the Ith stress component at the end of the time increment caused by an infinitesimal perturbation of the Jth component of the relative displacement increment array. Unless you invoke the unsymmetric equation solution capability in the contact property model definition (“Use with the unsymmetric equation solver in Abaqus/Standard” in
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_029.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_029.md
new file mode 100644
index 00000000..bc35c1d2
--- /dev/null
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@@ -0,0 +1,299 @@
+<!-- source-page: 281 -->
+
+“User-defined interfacial constitutive behavior,” Section 37.1.6 of the Abaqus Analysis User’s Guide), Abaqus/Standard will use only the symmetric part of DDSDDR. For a particular off-diagonal (I,J) entry, the symmetrization is done by halving the sum of (I,J) and (J,I) components. DDSDDR must be defined for a stress/displacement, a fully coupled temperature-displacement, or a coupled thermal-electrical-structural analysis to ensure proper convergence characteristics.
+
+# FLUX(2)
+
+Magnitude of the heat flux flowing into the slave and master surfaces, respectively. This array is passed in as the value at the beginning of the increment and must be updated to the flux at the end of the increment. The convention for defining the flux is that a positive flux indicates heat flowing into a surface, while a negative flux indicates heat flowing out of the surface. This variable must be defined for a heat transfer, a fully coupled temperature-displacement, or a coupled thermal-electrical-structural analysis. The sum of these two flux terms represents the heat generated in the interface, and the difference in these flux terms represents the heat conducted through the interface.
+
+# DDFDDT(2,2)
+
+The negative of the variation of the flux at the two surfaces with respect to their respective temperatures, for a fixed relative displacement. This variable must be defined for a heat transfer, a fully coupled temperature-displacement, or a coupled thermal-electrical-structural analysis to ensure proper convergence characteristics. The entries in the first row contain the negatives of the derivatives of FLUX(1) with respect to TEMP(1) and TEMP(2), respectively. The entries in the second row contain the negatives of the corresponding derivatives of FLUX(2).
+
+# DDSDDT(NDIR,2)
+
+Variation of the stress with respect to the temperatures of the two surfaces for a fixed relative displacement. This variable is required only for thermally coupled elements (in a fully coupled temperature-displacement or a coupled thermal-electrical-structural analysis), in which the stress is a function of the surface temperatures. DDSDDT(NDIR,1) corresponds to the slave surface, and DDSDDT(NDIR,2) corresponds to the master surface.
+
+# DDFDDR(2,NDIR)
+
+Variation of the flux with respect to the relative displacement between the two surfaces. This variable is required only for thermally coupled elements (in a fully coupled temperature-displacement or a coupled thermal-electrical-structural analysis), in which the flux is a function of the relative displacement. DDFDDR(1,NDIR) corresponds to the slave surface, and DDFDDR(2,NDIR) corresponds to the master surface.
+
+# Variables that can be updated
+
+# DVISCOUS(NDIR,NDIR)
+
+Interface viscous damping matrix that can be used in direct steady-state dynamic analysis and transient and steady-state mode-based dynamic analysis (including the subspace projection method), as well as in complex eigenvalue extraction, matrix generation, and substructure generation. DVISCOUS(I,J)
+
+<!-- source-page: 282 -->
+
+defines an element in the material viscous damping matrix at the current frequency. Abaqus/Standard requires that this element is defined as a damping value for each (I,J) entry in the damping matrix.
+
+Unless you invoke the unsymmetric equation solution capability in the contact property model definition (“Use with the unsymmetric equation solver in Abaqus/Standard” in “User-defined interfacial constitutive behavior,” Section 37.1.6 of the Abaqus Analysis User’s Guide), Abaqus/Standard uses only the symmetric part of DVISCOUS. For a particular off-diagonal (I,J) entry the symmetrization is done by halving the sum of the (I,J) and (J,I) components.
+
+# DSTRUCTURAL(NDIR,NDIR)
+
+Interface structural damping matrix that can be used in direct steady-state dynamic analysis and steadystate mode-based dynamic analysis (including the subspace projection method), as well as in complex eigenvalue extraction, matrix generation, and substructure generation. DSTRUCTURAL(I,J) defines an element in the material structural damping matrix.
+
+Unless you invoke the unsymmetric equation solution capability in the contact property model definition (“Use with the unsymmetric equation solver in Abaqus/Standard” in “User-defined interfacial constitutive behavior,” Section 37.1.6 of the Abaqus Analysis User’s Guide), Abaqus/Standard uses only the symmetric part of DSTRUCTURAL. For a particular off-diagonal (I,J) entry the symmetrization is done by halving the sum of the (I,J) and (J,I) components.
+
+# STATEV(NSTATV)
+
+An array containing the solution-dependent state variables. These are passed in as values at the beginning of the increment and must be returned as values at the end of the increment. You define the number of available state variables as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide.
+
+# SED
+
+This variable is passed in as the value of the elastic energy density at the start of the increment and should be updated to the elastic energy density at the end of the increment. This variable is used for output only and has no effect on other solution variables. It contributes to the output variable ALLSE.
+
+# SFD
+
+This variable should be defined as the incremental frictional dissipation. The units are energy per unit area. This variable is used for output only and has no effect on other solution variables. It contributes to the output variables ALLFD and SFDR (and related variables). For computing its contribution to SFDR, SFD is divided by the time increment.
+
+# SPD
+
+This variable should be defined as the incremental dissipation due to plasticity effects in the interfacial constitutive behavior. The units are energy per unit area. This variable is used for output only and has no effect on other solution variables. It contributes to the output variable ALLPD.
+
+<!-- source-page: 283 -->
+
+# SVD
+
+This variable should be defined as the incremental dissipation due to viscous effects in the interfacial constitutive behavior. The units are energy per unit area. This variable is used for output only and has no effect on other solution variables. It contributes to the output variable ALLVD.
+
+# SCD
+
+This variable should be defined as the incremental dissipation due to creep effects in the interfacial constitutive behavior. The units are energy per unit area. This variable is used for output only and has no effect on other solution variables. It contributes to the output variable ALLCD.
+
+# PNEWDT
+
+Ratio of suggested new time increment to the time increment currently being used (DTIME, see below). This variable allows you to provide input to the automatic time incrementation algorithms in Abaqus/Standard (if automatic time incrementation is chosen).
+
+PNEWDT is set to a large value before each call to UINTER.
+
+If PNEWDT is redefined to be less than 1.0, Abaqus/Standard must abandon the time increment and attempt it again with a smaller time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines that allow redefinition of PNEWDT for this iteration.
+
+If PNEWDT is given a value that is greater than 1.0 for all calls to user subroutines for this iteration and the increment converges in this iteration, Abaqus/Standard may increase the time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines for this iteration.
+
+If automatic time incrementation is not selected in the analysis procedure, values of PNEWDT greater than 1.0 will be ignored and values of PNEWDT less than 1.0 will cause the job to terminate.
+
+# LOPENCLOSE
+
+An integer flag that is used to track the contact status in situations where user subroutine UINTER is used to model standard contact between two surfaces, like the default hard contact model in Abaqus/Standard. It comes in as the value at the beginning of the current iteration and should be set to the value at the end of the current iteration. It is set to −1 at the beginning of the analysis before UINTER is called. You should set it to 0 to indicate an open status and to 1 to indicate a closed status. A change in this flag from one iteration to the next will have two effects. It will result in output related to a change in contact status if you request a detailed contact printout in the message file (“The Abaqus/Standard message file” in “Output,” Section 4.1.1 of the Abaqus Analysis User’s Guide). In addition, it will also trigger a severe discontinuity iteration. Any time this flag is reset to a value of −1, Abaqus/Standard assumes that the flag is not being used. A change in this flag from −1 to another value or vice versa will not have any of the above effects.
+
+# LSTATE
+
+An integer flag that should be used in non-standard contact situations where a simple open/close status is not appropriate or enough to describe the state. It comes in as the value at the beginning of the current iteration and should be set to the value at the end of the current iteration. It is set to −1 at
+
+<!-- source-page: 284 -->
+
+the beginning of the analysis before UINTER is called. It can be assigned any user-defined integer value, each corresponding to a different state. You can track changes in the value of this flag and use it to output appropriate diagnostic messages to the message file (unit 7). You may choose to output diagnostic messages only when a detailed contact printout is requested (“The Abaqus/Standard message file” in “Output,” Section 4.1.1 of the Abaqus Analysis User’s Guide). In the latter case, the LPRINT parameter is useful. In conjunction with the LSTATE flag, you may also utilize the LSDI flag to trigger a severe discontinuity iteration any time the state changes from one iteration to the next. Any time this flag is reset to a value of −1, Abaqus/Standard assumes that the flag is not being used.
+
+# LSDI
+
+This flag is set to 0 before each call to UINTER and should be set to 1 if the current iteration should be treated as a severe discontinuity iteration. This would typically be done in non-standard contact situations based on a change in the value of the LSTATE flag from one iteration to the next. The use of this flag has no effect when the LOPENCLOSE flag is also used. In that case, severe discontinuity iterations are determined based on changes in the value of LOPENCLOSE alone.
+
+# Variables passed in for information
+
+# RDISP(NDIR)
+
+An array containing the current relative positions between the two surfaces at the end of the increment. The first component is the relative position of the point on the slave surface, with respect to the master surface, in the normal direction. The second and third components, if applicable, are the accumulated incremental relative tangential displacements, measured from the beginning of the analysis. For the relative position in the normal direction a negative quantity represents an open status, while a positive quantity indicates penetration into the master surface. For open points on the slave surface for which no pairing master is found, the first component is a very large negative number $( - 1 \times 1 0 ^ { 3 6 } )$ . The local directions in which the relative displacements are defined are stored in ALOCALDIR.
+
+# DRDISP(NDIR)
+
+An array containing the increments in relative positions between the two surfaces.
+
+# TEMP(2)
+
+Temperature at the end of the increment at a point on the slave surface and the opposing master surface, respectively.
+
+# DTEMP(2)
+
+Increment in temperature at the point on the slave surface and the opposing master surface, respectively.
+
+# PREDEF(2,NPRED)
+
+An array containing pairs of values of all the predefined field variables at the end of the current increment (initial values at the beginning of the analysis and current values during the analysis). The first value in a pair, PREDEF(1,NPRED), corresponds to the value at the point on the slave surface, and the second value, PFREDEF(2,NPRED), corresponds to the value of the field variable at the nearest point on the opposing surface.
+
+<!-- source-page: 285 -->
+
+# DPRED(2,NPRED)
+
+Array of increments in predefined field variables.
+
+# TIME(1)
+
+Value of step time at the end of the increment.
+
+# TIME(2)
+
+Value of total time at the end of the increment.
+
+# DTIME
+
+Current increment in time.
+
+# FREQR
+
+Current frequency for steady-state dynamic analysis in rad/time.
+
+# CINAME
+
+User-specified surface interaction name, left justified.
+
+# SLNAME
+
+Slave surface name.
+
+# MSNAME
+
+Master surface name.
+
+# PROPS(NPROPS)
+
+User-specified array of property values to define the interfacial constitutive behavior between the contacting surfaces.
+
+# COORDS(MCRD)
+
+An array containing the current coordinates of this point.
+
+# ALOCALDIR(3,3)
+
+An array containing the direction cosines of the local surface coordinate system. The directions are stored in columns. For example, ALOCALDIR(1,1), ALOCALDIR(2,1), and ALOCALDIR(3,1) give the (1, 2, 3) components of the normal direction. Thus, the first direction is the normal direction to the surface, and the remaining two directions are the local tangent directions in the plane of the surface. The local system is defined by the geometry of the master surface. The convention for the local directions is the same as the convention in situations where the model uses the built-in contact capabilities in Abaqus/Standard (described in “Contact formulations in Abaqus/Standard,” Section 38.1.1 of the Abaqus Analysis User’s Guide, for the tangential directions).
+
+# DROT(2,2)
+
+Rotation increment matrix. For contact with a three-dimensional rigid surface, this matrix represents the incremental rotation of the surface directions relative to the rigid surface. It is provided so that vector- or tensor-valued state variables can be rotated appropriately in this subroutine. Relative
+
+<!-- source-page: 286 -->
+
+# UINTER
+
+displacement components are already rotated by this amount before UINTER is called. This matrix is passed in as a unit matrix for two-dimensional and axisymmetric contact problems.
+
+# AREA
+
+Surface area associated with the contact point.
+
+# CHRLNGTH
+
+Characteristic contact surface face dimension.
+
+# NODE
+
+User-defined global slave node number (or internal node number for models defined in terms of an assembly of part instances) involved with this contact point. Corresponds to the predominant slave node of the constraint if the surface-to-surface contact formulation is used.
+
+# NDIR
+
+Number of force components at this point.
+
+# NSTATV
+
+Number of solution-dependent state variables.
+
+# NPRED
+
+Number of predefined field variables.
+
+# NPROPS
+
+User-defined number of property values associated with this interfacial constitutive model (“Interfacial constants” in “User-defined interfacial constitutive behavior,” Section 37.1.6 of the Abaqus Analysis User’s Guide).
+
+# MCRD
+
+Number of coordinate directions at the contact point.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# KIT
+
+Iteration number. KIT=0 for the first assembly, KIT=1 for the first recovery/second assembly, KIT=2 for the second recovery/third assembly, and so on.
+
+# LINPER
+
+Linear perturbation flag. LINPER=1 if the step is a linear perturbation step. LINPER=0 if the step is a general step. For a linear perturbation step, the inputs to user subroutine UINTER represent perturbation quantities about the base state. The user-defined quantities in UINTER are also perturbation quantities.
+
+<!-- source-page: 287 -->
+
+The Jacobian terms should be based on the base state. No change in contact status should occur during a linear perturbation step.
+
+# LPRINT
+
+This flag is equal to 1 if a detailed contact printout to the message file is requested and 0 otherwise (“The Abaqus/Standard message file” in “Output,” Section 4.1.1 of the Abaqus Analysis User’s Guide). This flag can be used to print out diagnostic messages regarding changes in contact status selectively only when a detailed contact printout is requested.
+
+<!-- source-page: 288 -->
+
+<!-- source-page: 289 -->
+
+# 1.1.43 UMASFL: User subroutine to specify prescribed mass flow rate conditions for a convection/diffusion heat transfer analysis.
+
+Product: Abaqus/Standard
+
+# References
+
+• “Uncoupled heat transfer analysis,” Section 6.5.2 of the Abaqus Analysis User’s Guide   
+• \*MASS FLOW RATE   
+• “UTEMP, UFIELD, UMASFL, and UPRESS,” Section 4.1.25 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine UMASFL:
+
+• can be used to prescribe the mass flow rate vector at the nodes of a model as a function of position and time;   
+• will be called whenever a current value of mass flow rate (per unit area) is needed for a node listed in a user-subroutine-defined mass flow rate definition (the node should belong to one or more convection/diffusion elements); and   
+• will overwrite any flow rate data specified for the associated mass flow rate definition outside the user subroutine.
+
+# User subroutine interface
+
+```fortran
+SUBROUTINE UMASFL (FLOW, KFLOW, KSTEP, KINC, TIME, NODE, COORDS)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION FLOW (KFLOW), TIME(2), COORDS(3)
+C
+user coding to define FLOW
+RETURN
+END 
+```
+
+<!-- source-page: 290 -->
+
+# Variable to be defined
+
+# FLOW
+
+Total value of the mass flow rate vector at this point. The number of components in this vector is KFLOW. If KFLOW=1, give the total mass flow rate through the cross-section (for one-dimensional elements). If KFLOW=2, give the x-component and y-component of the flow rate vector as FLOW(1) and FLOW(2). If KFLOW=3, give the x-component, y-component, and z-component as FLOW(1), FLOW(2), and FLOW(3).
+
+# Variables passed in for information
+
+# KFLOW
+
+The number of components in the mass flow rate vector. UMASFL will be called only once per node.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# TIME(1)
+
+Current value of step time.
+
+# TIME(2)
+
+Current value of total time.
+
+# NODE
+
+Node number.
+
+# COORDS
+
+An array containing the coordinates of this node.
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+<!-- source-page: 291 -->
+
+# 1.1.44 UMAT: User subroutine to define a material’s mechanical behavior.
+
+# Product: Abaqus/Standard
+
+WARNING: The use of this subroutine generally requires considerable expertise. You are cautioned that the implementation of any realistic constitutive model requires extensive development and testing. Initial testing on a single-element model with prescribed traction loading is strongly recommended.
+
+# References
+
+• “User-defined mechanical material behavior,” Section 26.7.1 of the Abaqus Analysis User’s Guide   
+• “User-defined thermal material behavior,” Section 26.7.2 of the Abaqus Analysis User’s Guide   
+• \*USER MATERIAL   
+• “SDVINI,” Section 4.1.11 of the Abaqus Verification Guide   
+• “UMAT and UHYPER,” Section 4.1.21 of the Abaqus Verification Guide
+
+# Overview
+
+# User subroutine UMAT:
+
+• can be used to define the mechanical constitutive behavior of a material;   
+• will be called at all material calculation points of elements for which the material definition includes a user-defined material behavior;   
+• can be used with any procedure that includes mechanical behavior;   
+• can use solution-dependent state variables;   
+• must update the stresses and solution-dependent state variables to their values at the end of the increment for which it is called;   
+• must provide the material Jacobian matrix, $\partial \triangle \sigma / \partial \triangle \varepsilon$ , for the mechanical constitutive model;   
+• can be used in conjunction with user subroutine USDFLD to redefine any field variables before they are passed in; and   
+• is described further in “User-defined mechanical material behavior,” Section 26.7.1 of the Abaqus Analysis User’s Guide.
+
+# Storage of stress and strain components
+
+In the stress and strain arrays and in the matrices DDSDDE, DDSDDT, and DRPLDE, direct components are stored first, followed by shear components. There are NDI direct and NSHR engineering shear components. The order of the components is defined in “Conventions,” Section 1.2.2 of the Abaqus Analysis User’s Guide. Since the number of active stress and strain components varies between element types, the routine must be coded to provide for all element types with which it will be used.
+
+<!-- source-page: 292 -->
+
+# Defining local orientations
+
+If a local orientation (“Orientations,” Section 2.2.5 of the Abaqus Analysis User’s Guide) is used at the same point as user subroutine UMAT, the stress and strain components will be in the local orientation; and, in the case of finite-strain analysis, the basis system in which stress and strain components are stored rotates with the material.
+
+# Stability
+
+You should ensure that the integration scheme coded in this routine is stable—no direct provision is made to include a stability limit in the time stepping scheme based on the calculations in UMAT.
+
+# Convergence rate
+
+DDSDDE and—for coupled temperature-displacement and coupled thermal-electrical-structural analyses—DDSDDT, DRPLDE, and DRPLDT must be defined accurately if rapid convergence of the overall Newton scheme is to be achieved. In most cases the accuracy of this definition is the most important factor governing the convergence rate. Since nonsymmetric equation solution is as much as four times as expensive as the corresponding symmetric system, if the constitutive Jacobian (DDSDDE) is only slightly nonsymmetric (for example, a frictional material with a small friction angle), it may be less expensive computationally to use a symmetric approximation and accept a slower convergence rate.
+
+An incorrect definition of the material Jacobian affects only the convergence rate; the results (if obtained) are unaffected.
+
+# Viscoelastic behavior in frequency domain
+
+The constitutive Jacobian (DDSDDE) must provide both the stiffness (storage modulus) and damping (loss modulus) for modeling frequency domain viscoelastic behavior.
+
+# Special considerations for various element types
+
+There are several special considerations that need to be noted.
+
+# Deformation gradient
+
+The deformation gradient is available for solid (continuum) elements, membranes, and finite-strain shells (S3/S3R, S4, S4R, SAXs, and SAXAs). It is not available for beams or small-strain shells. It is stored as a 3 × 3 matrix with component equivalence DFGRD0 $( \pmb { \tau } , \pmb { \sigma } ) \Leftrightarrow F _ { I J }$ . For fully integrated firstorder isoparametric elements (4-node quadrilaterals in two dimensions and 8-node hexahedra in three dimensions) the selectively reduced integration technique is used (also known as the technique). Thus, a modified deformation gradient
+
+$$
+\overline {{\mathbf {F}}} = \mathbf {F} \left(\frac {\overline {{J}}}{\overline {{J}}}\right) ^ {\frac {1}{n}}
+$$
+
+<!-- source-page: 293 -->
+
+is passed into user subroutine UMAT. For more details, see “Solid isoparametric quadrilaterals and hexahedra,” Section 3.2.4 of the Abaqus Theory Guide.
+
+The deformation gradient, which is passed to the user subroutine, is computed with respect to the initial configuration. If a local orientation is not specified, the components of the deformation gradient are expressed in the global coordinate system. If a local orientation is used, the components of the same deformation gradient are expressed in the local coordinate system; in the case of finite-strain analysis, the basis system rotates with the material.
+
+# Beams and shells that calculate transverse shear energy
+
+If user subroutine UMAT is used to describe the material of beams or shells that calculate transverse shear energy, you must specify the transverse shear stiffness as part of the beam or shell section definition to define the transverse shear behavior. See “Shell section behavior,” Section 29.6.4 of the Abaqus Analysis User’s Guide, and “Choosing a beam element,” Section 29.3.3 of the Abaqus Analysis User’s Guide, for information on specifying this stiffness.
+
+# Open-section beam elements
+
+When user subroutine UMAT is used to describe the material response of beams with open sections (for example, an I-section), the torsional stiffness is obtained as
+
+$$
+G J = \frac {(K _ {1 3} + K _ {2 3}) J}{2 k A},
+$$
+
+where J is the torsional constant, A is the section area, k is a shear factor, and $K _ { \alpha 3 }$ is the user-specified transverse shear stiffness (see “Transverse shear stiffness definition” in “Choosing a beam element,” Section 29.3.3 of the Abaqus Analysis User’s Guide).
+
+# Elements with hourglassing modes
+
+If this capability is used to describe the material of elements with hourglassing modes, you must define the hourglass stiffness factor for hourglass control based on the total stiffness approach as part of the element section definition. The hourglass stiffness factor is not required for enhanced hourglass control, but you can define a scaling factor for the stiffness associated with the drill degree of freedom (rotation about the surface normal). See “Section controls,” Section 27.1.4 of the Abaqus Analysis User’s Guide, for information on specifying the stiffness factor.
+
+# Pipe-soil interaction elements
+
+The constitutive behavior of the pipe-soil interaction elements (see “Pipe-soil interaction elements,” Section 32.12.1 of the Abaqus Analysis User’s Guide) is defined by the force per unit length caused by relative displacement between two edges of the element. The relative-displacements are available as “strains” (STRAN and DSTRAN). The corresponding forces per unit length must be defined in the STRESS array. The Jacobian matrix defines the variation of force per unit length with respect to relative displacement.
+
+<!-- source-page: 294 -->
+
+For two-dimensional elements two in-plane components of “stress” and “strain” exist (NTENS=NDI=2, and NSHR=0). For three-dimensional elements three components of “stress” and “strain” exist (NTENS=NDI=3, and NSHR=0).
+
+# Large volume changes with geometric nonlinearity
+
+If the material model allows large volume changes and geometric nonlinearity is considered, the exact definition of the consistent Jacobian should be used to ensure rapid convergence. These conditions are most commonly encountered when considering either large elastic strains or pressure-dependent plasticity. In the former case, total-form constitutive equations relating the Cauchy stress to the deformation gradient are commonly used; in the latter case, rate-form constitutive laws are generally used.
+
+For total-form constitutive laws, the exact consistent Jacobian is defined through the variation in Kirchhoff stress:
+
+$$
+\delta (J \pmb {\sigma}) = J (\mathbf {C}: \delta \mathbf {D} + \delta \mathbf {W} \cdot \pmb {\sigma} - \pmb {\sigma} \cdot \delta \mathbf {W})
+$$
+
+Here, J is the determinant of the deformation gradient, is the Cauchy stress, is the virtual rate of deformation, and is the virtual spin tensor, defined as
+
+$$
+\delta \mathbf {D} \stackrel {\mathrm{def}} {=} \mathrm{sym} (\delta \mathbf {F} \cdot \mathbf {F} ^ {- 1})
+$$
+
+and
+
+$$
+\delta \mathbf {W} \stackrel {\mathrm{def}} {=} \operatorname{asym} (\delta \mathbf {F} \cdot \mathbf {F} ^ {- 1}).
+$$
+
+For rate-form constitutive laws, the exact consistent Jacobian is given by
+
+$$
+\mathbf {C} = \frac {1}{J} \frac {\partial \Delta (J \pmb {\sigma})}{\partial \Delta \pmb {\varepsilon}}.
+$$
+
+# Use with almost incompressible or fully incompressible elastic materials
+
+For user-defined almost incompressible or incompressible elastic materials, a few different options are available depending on whether hybrid or nonhybrid elements are used. For all cases the first option should be to use user subroutine UHYPER instead of user subroutine UMAT when it is possible to do so. In user subroutine UMAT incompressible materials can be modeled via a penalty method; that is, you ensure that a finite bulk modulus is used. The bulk modulus should be large enough to model incompressibility sufficiently but small enough to avoid loss of precision. As a general guideline, the bulk modulus should be about $1 0 ^ { 4 } – 1 0 ^ { 6 }$ times the shear modulus. The tangent bulk modulus $K ^ { t }$ can be calculated from
+
+$$
+K ^ {t} = \frac {1}{9} \sum_ {\mathrm{I} = 1} ^ {3} \sum_ {\mathrm{J} = 1} ^ {3} \mathrm{DDSDDE} (\mathrm{I}, \mathrm{J}).
+$$
+
+<!-- source-page: 295 -->
+
+If a hybrid element is used with user subroutine UMAT, Abaqus/Standard, by default, replaces the pressure stress calculated from your definition of STRESS with that derived from the Lagrange multiplier and modifies the Jacobian appropriately (“Hybrid incompressible solid element formulation,” Section 3.2.3 of the Abaqus Theory Guide). This approach is suitable for material models that use an incremental formulation (for example, metal plasticity) but is not consistent with a total formulation that is commonly used for hyperelastic materials. In the latter situation, the default formulation may lead to convergence problems. Such convergence problems may be observed, for example, when an almost incompressible nonlinear elastic user material is subjected to large deformations. Abaqus/Standard provides an alternate total formulation when user materials are used with hybrid elements (see “User-defined mechanical material behavior,” Section 26.7.1 of the Abaqus Analysis User’s Guide). This formulation is consistent with the native almost incompressible formulation used by Abaqus for hyperelastic materials (“Hyperelastic material behavior,” Section 4.6.1 of the Abaqus Theory Guide) and works better than the default formulation for such cases.
+
+The total hybrid formulation assumes that the response of the material can be written as the sum of its deviatoric and volumetric parts and that these parts are decoupled from each other. In particular, the volumetric response is assumed to be defined in terms of a strain energy potential, $U ( \hat { J } )$ , which is a function of an alternate variable, $\hat { J }$ in place of the actual volume change . The alternate variable is made available inside user subroutine UMAT by extending the STRESS array beyond NTENS, with the NTENS+1 entry providing read access to ${ \hat { J } } .$ You must define the hydrostatic part of the stress tensor as $\begin{array} { r } { \hat { p } = - \frac { \partial U } { \partial \hat { J } } } \end{array}$ a . The formulation also requires the additional derivatives e these additional derivatives inside user subroutine UMAT $\begin{array} { r } { \hat { K } = J \frac { \partial ^ { 2 } \hat { \boldsymbol { U } } } { \partial \hat { J } ^ { 2 } } } \end{array}$ andk and S+1 $\begin{array} { r } { \frac { \partial \hat { K } } { \partial \hat { J } } = J \frac { \partial ^ { 3 } U } { \partial \hat { J } ^ { 3 } } } \end{array}$ a3 . YouNS+2 entry, respectively, of the STRESS array. In addition, the bulk modulus of the material (contributes toward the material Jacobian matrix, DDSDDE) must be defined as $\hat { K }$ .
+
+Abaqus/Standard also provides a fully incompressible user material formulation for use with hybrid elements to define a fully incompressible user material response. This formulation is consistent with the native formulation used by Abaqus for incompressible hyperelastic materials and assumes that the deviatoric stress can be derived from a strain energy potential function. You need define only the deviatoric stress and Jacobian to define a fully incompressible material response through user subroutine UMAT.
+
+For incompressible pressure-sensitive materials the element choice is particularly important when using user subroutine UMAT. In particular, first-order wedge elements should be avoided. For these elements the $\bar { B }$ technique is not used to alter the deformation gradient that is passed into user subroutine UMAT, which increases the risk of volumetric locking.
+
+# Increments for which only the Jacobian can be defined
+
+Abaqus/Standard passes zero strain increments into user subroutine UMAT to start the first increment of all the steps and all increments of steps for which you have suppressed extrapolation (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide). In this case you can define only the Jacobian (DDSDDE).
+
+<!-- source-page: 296 -->
+
+# Utility routines
+
+Several utility routines may help in coding user subroutine UMAT. Their functions include determining stress invariants for a stress tensor and calculating principal values and directions for stress or strain tensors. These utility routines are discussed in detail in “Obtaining stress invariants, principal stress/strain values and directions, and rotating tensors in an Abaqus/Standard analysis,” Section 2.1.11.
+
+User subroutine interface   
+SUBROUTINE UMAT(STRESS,STATEV,DDSDDE,SSE,SPD,SCD,
+1 RPL,DDSDDT,DRPLDE,DRPLDT,
+2 STRAN,DSTRAN,TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED,CMNAME,
+3 NDI,NSHR,NTENS,NSTATV,PROPS,NPROPS,COORDS,DROT,PNEWDT,
+4 CELENT,DFGRD0,DFGRD1,NOEL,NPT,LAYER,KSPT,JSTEP,KINC)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME
+DIMENSION STRESS(NTENS),STATEV(NSTATV),
+1 DDSDDE(NTENS,NTENS),DDSDDT(NTENS),DRPLDE(NTENS),
+2 STRAN(NTENS),DSTRAN(NTENS),TIME(2),PREDEF(1),DPRED(1),
+3 PROPS(NPROPS),COORDS(3),DROT(3,3),DFGRD0(3,3),DFGRD1(3,3),
+4 JSTEP(4)
+user coding to define DDSDDE, STRESS, STATEV, SSE, SPD, SCD
+and, if necessary, RPL, DDSDDT, DRPLDE, DRPLDT, PNEWDT
+RETURN
+END
+
+# Variables to be defined
+
+# In all situations
+
+DDSDDE(NTENS,NTENS)
+
+Jacobian matrix of the constitutive model, $\partial \Delta \sigma / \partial \Delta \varepsilon$ , where $\Delta \sigma$ are the stress increments and $\Delta \varepsilon$ are the strain increments. DDSDDE(I,J) defines the change in the Ith stress component at the end of the time increment caused by an infinitesimal perturbation of the Jth component of the strain increment array. Unless you invoke the unsymmetric equation solution capability for the user-defined material, Abaqus/Standard will use only the symmetric part of DDSDDE. The symmetric part of the matrix is calculated by taking one half the sum of the matrix and its transpose.
+
+For viscoelastic behavior in the frequency domain, the Jacobian matrix must be dimensioned as DDSDDE(NTENS,NTENS,2). The stiffness contribution (storage modulus) must be provided in
+
+<!-- source-page: 297 -->
+
+DDSDDE(NTENS,NTENS,1), while the damping contribution (loss modulus) must be provided in DDSDDE(NTENS,NTENS,2).
+
+# STRESS(NTENS)
+
+This array is passed in as the stress tensor at the beginning of the increment and must be updated in this routine to be the stress tensor at the end of the increment. If you specified initial stresses (“Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1 of the Abaqus Analysis User’s Guide), this array will contain the initial stresses at the start of the analysis. The size of this array depends on the value of NTENS as defined below. In finite-strain problems the stress tensor has already been rotated to account for rigid body motion in the increment before UMAT is called, so that only the corotational part of the stress integration should be done in UMAT. The measure of stress used is “true” (Cauchy) stress.
+
+If the UMAT utilizes a hybrid formulation that is total (as opposed to the default incremental behavior), the stress array is extended beyond NTENS. The first NTENS entries of the array contain the stresses, as described above. The additional quantities are as follows:
+
+STRESS(NTENS+1)
+
+Read only: ,
+
+STRESS(NTENS+2)
+
+Write only: , $\begin{array} { r } { \hat { K } = J \frac { \partial ^ { 2 } U } { \partial \hat { J } ^ { 2 } } } \end{array}$ and
+
+STRESS(NTENS+3)
+
+Write only: $\begin{array} { r } { \frac { \partial \hat { K } } { \partial \hat { J } } = J \frac { \partial ^ { 3 } U } { \partial \hat { J } ^ { 3 } } } \end{array}$ =Jou, , where is the volumetric part of the strain energy density potential.
+
+# STATEV(NSTATV)
+
+An array containing the solution-dependent state variables. These are passed in as the values at the beginning of the increment unless they are updated in user subroutines USDFLD or UEXPAN, in which case the updated values are passed in. In all cases STATEV must be returned as the values at the end of the increment. The size of the array is defined as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide.
+
+In finite-strain problems any vector-valued or tensor-valued state variables must be rotated to account for rigid body motion of the material, in addition to any update in the values associated with constitutive behavior. The rotation increment matrix, DROT, is provided for this purpose.
+
+# SSE, SPD, SCD
+
+Specific elastic strain energy, plastic dissipation, and “creep” dissipation, respectively. These are passed in as the values at the start of the increment and should be updated to the corresponding specific energy values at the end of the increment. They have no effect on the solution, except that they are used for energy output.
+
+# Only in a fully coupled thermal-stress or a coupled thermal-electrical-structural analysis
+
+# RPL
+
+Volumetric heat generation per unit time at the end of the increment caused by mechanical working of the material.
+
+<!-- source-page: 298 -->
+
+# DDSDDT(NTENS)
+
+Variation of the stress increments with respect to the temperature.
+
+# DRPLDE(NTENS)
+
+Variation of RPL with respect to the strain increments.
+
+# DRPLDT
+
+Variation of RPL with respect to the temperature.
+
+# Only in a geostatic stress procedure or a coupled pore fluid diffusion/stress analysis for pore pressure cohesive elements
+
+# RPL
+
+RPL is used to indicate whether or not a cohesive element is open to the tangential flow of pore fluid. Set RPL equal to 0 if there is no tangential flow; otherwise, assign a nonzero value to RPL if an element is open. Once opened, a cohesive element will remain open to the fluid flow.
+
+# Variable that can be updated
+
+# PNEWDT
+
+Ratio of suggested new time increment to the time increment being used (DTIME, see discussion later in this section). This variable allows you to provide input to the automatic time incrementation algorithms in Abaqus/Standard (if automatic time incrementation is chosen). For a quasi-static procedure the automatic time stepping that Abaqus/Standard uses, which is based on techniques for integrating standard creep laws (see “Quasi-static analysis,” Section 6.2.5 of the Abaqus Analysis User’s Guide), cannot be controlled from within the UMAT subroutine.
+
+PNEWDT is set to a large value before each call to UMAT.
+
+If PNEWDT is redefined to be less than 1.0, Abaqus/Standard must abandon the time increment and attempt it again with a smaller time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines that allow redefinition of PNEWDT for this iteration.
+
+If PNEWDT is given a value that is greater than 1.0 for all calls to user subroutines for this iteration and the increment converges in this iteration, Abaqus/Standard may increase the time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines for this iteration.
+
+If automatic time incrementation is not selected in the analysis procedure, values of PNEWDT that are greater than 1.0 will be ignored and values of PNEWDT that are less than 1.0 will cause the job to terminate.
+
+# Variables passed in for information
+
+# STRAN(NTENS)
+
+An array containing the total strains at the beginning of the increment. If thermal expansion is included in the same material definition, the strains passed into UMAT are the mechanical strains only (that is,
+
+<!-- source-page: 299 -->
+
+the thermal strains computed based upon the thermal expansion coefficient have been subtracted from the total strains). These strains are available for output as the “elastic” strains.
+
+In finite-strain problems the strain components have been rotated to account for rigid body motion in the increment before UMAT is called and are approximations to logarithmic strain.
+
+# DSTRAN(NTENS)
+
+Array of strain increments. If thermal expansion is included in the same material definition, these are the mechanical strain increments (the total strain increments minus the thermal strain increments).
+
+# TIME(1)
+
+Value of step time at the beginning of the current increment or frequency.
+
+# TIME(2)
+
+Value of total time at the beginning of the current increment.
+
+# DTIME
+
+Time increment.
+
+# TEMP
+
+Temperature at the start of the increment.
+
+# DTEMP
+
+Increment of temperature.
+
+# PREDEF
+
+Array of interpolated values of predefined field variables at this point at the start of the increment, based on the values read in at the nodes.
+
+# DPRED
+
+Array of increments of predefined field variables.
+
+# CMNAME
+
+User-defined material name, left justified. Some internal material models are given names starting with the “ABQ\_” character string. To avoid conflict, you should not use “ABQ\_” as the leading string for CMNAME.
+
+# NDI
+
+Number of direct stress components at this point.
+
+# NSHR
+
+Number of engineering shear stress components at this point.
+
+# NTENS
+
+Size of the stress or strain component array (NDI + NSHR).
+
+<!-- source-page: 300 -->
+
+# NSTATV
+
+Number of solution-dependent state variables that are associated with this material type (defined as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# PROPS(NPROPS)
+
+User-specified array of material constants associated with this user material.
+
+# NPROPS
+
+User-defined number of material constants associated with this user material.
+
+# COORDS
+
+An array containing the coordinates of this point. These are the current coordinates if geometric nonlinearity is accounted for during the step (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide); otherwise, the array contains the original coordinates of the point.
+
+# DROT(3,3)
+
+Rotation increment matrix. This matrix represents the increment of rigid body rotation of the basis system in which the components of stress (STRESS) and strain (STRAN) are stored. It is provided so that vector- or tensor-valued state variables can be rotated appropriately in this subroutine: stress and strain components are already rotated by this amount before UMAT is called. This matrix is passed in as a unit matrix for small-displacement analysis and for large-displacement analysis if the basis system for the material point rotates with the material (as in a shell element or when a local orientation is used).
+
+# CELENT
+
+Characteristic element length, which is a typical length of a line across an element for a first-order element; it is half of the same typical length for a second-order element. For beams and trusses it is a characteristic length along the element axis. For membranes and shells it is a characteristic length in the reference surface. For axisymmetric elements it is a characteristic length in the plane only. For cohesive elements it is equal to the constitutive thickness.
+
+# DFGRD0(3,3)
+
+Array containing the deformation gradient at the beginning of the increment. If a local orientation is defined at the material point, the deformation gradient components are expressed in the local coordinate system defined by the orientation at the beginning of the increment. For a discussion regarding the availability of the deformation gradient for various element types, see “Deformation gradient.”
+
+# DFGRD1(3,3)
+
+Array containing the deformation gradient at the end of the increment. If a local orientation is defined at the material point, the deformation gradient components are expressed in the local coordinate system defined by the orientation. This array is set to the identity matrix if nonlinear geometric effects are not included in the step definition associated with this increment. For a discussion regarding the availability of the deformation gradient for various element types, see “Deformation gradient.”
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_031.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_031.md
new file mode 100644
index 00000000..c42179e7
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_031.md
@@ -0,0 +1,381 @@
+<!-- source-page: 301 -->
+
+# NOEL
+
+Element number.
+
+# NPT
+
+Integration point number.
+
+# LAYER
+
+Layer number (for composite shells and layered solids).
+
+# KSPT
+
+Section point number within the current layer.
+
+# JSTEP(1)
+
+Step number.
+
+# JSTEP(2)
+
+Procedure type key (see “Results file output format,” Section 5.1.2 of the Abaqus Analysis User’s Guide).
+
+# JSTEP(3)
+
+1 if NLGEOM=YES for the current step; 0 otherwise.
+
+# JSTEP(4)
+
+1 if current step is a linear perturbation procedure; 0 otherwise.
+
+# KINC
+
+Increment number.
+
+# Example: Using more than one user-defined mechanical material model
+
+To use more than one user-defined mechanical material model, the variable CMNAME can be tested for different material names inside user subroutine UMAT as illustrated below:
+
+```txt
+IF (CMNAME(1:4) .EQ. 'MAT1') THEN
+    CALL UMAT_MAT1(argument_list)
+ELSE IF(CMNAME(1:4) .EQ. 'MAT2') THEN
+    CALL UMAT_MAT2(argument_list)
+END IF 
+```
+
+UMAT\_MAT1 and UMAT\_MAT2 are the actual user material subroutines containing the constitutive material models for each material MAT1 and MAT2, respectively. Subroutine UMAT merely acts as a directory here. The argument list may be the same as that used in subroutine UMAT.
+
+<!-- source-page: 302 -->
+
+As a simple example of the coding of user subroutine UMAT, consider the linear, viscoelastic model shown in Figure 1.1.44–1. Although this is not a very useful model for real materials, it serves to illustrate how to code the routine.
+
+The behavior of the one-dimensional model shown in the figure is
+
+$$
+\sigma + \frac {\mu_ {1}}{(E _ {1} + E _ {2})} \dot {\sigma} = \frac {\mu_ {1}}{(1 + E _ {1} / E _ {2})} \dot {\varepsilon} + \frac {1}{(1 / E _ {1} + 1 / E _ {2})} \varepsilon ,
+$$
+
+where $\dot { \sigma }$ and $\dot { \varepsilon }$ are the time rates of change of stress and strain. This can be generalized for small straining of an isotropic solid as
+
+$$
+\sigma_ {x x} + \tilde {\nu} \dot {\sigma} _ {x x} = \lambda \varepsilon_ {V} + 2 \mu \varepsilon_ {x x} + \tilde {\lambda} \dot {\varepsilon} _ {V} + 2 \tilde {\mu} \dot {\varepsilon} _ {x x}, \quad \mathrm{etc.},
+$$
+
+and
+
+$$
+\sigma_ {x y} + \tilde {\nu} \dot {\sigma} _ {x y} = \mu \gamma_ {x y} + \tilde {\mu} \dot {\gamma} _ {x y}, \quad \mathrm{etc.},
+$$
+
+where
+
+$$
+\varepsilon_ {V} = \varepsilon_ {x x} + \varepsilon_ {y y} + \varepsilon_ {z z},
+$$
+
+and $\tilde { \nu } , \lambda , \mu , \tilde { \lambda } ,$ , and $\tilde { \mu }$ are material constants ( and $\mu$ are the Lamé constants).
+
+A simple, stable integration operator for this equation is the central difference operator:
+
+$$
+\dot {f} _ {t + \frac {1}{2} \Delta t} = \frac {\Delta f}{\Delta t},
+$$
+
+$$
+f _ {t + \frac {1}{2} \Delta t} = f _ {t} + \frac {\Delta f}{2},
+$$
+
+where $f \mathrm { i }$ is some function, $f _ { t }$ is its value at the beginning of the increment, $\Delta f$ is the change in the function over the increment, and $\Delta t$ is the time increment.
+
+Applying this to the rate constitutive equations above gives
+
+$$
+(\frac {\Delta t}{2} + \tilde {\nu}) \Delta \sigma_ {x x} = (\Delta t \frac {\lambda}{2} + \tilde {\lambda}) \Delta \varepsilon_ {V} + (\Delta t \mu + 2 \tilde {\mu}) \Delta \varepsilon_ {x x} + \Delta t (\lambda \varepsilon_ {V} + 2 \mu \varepsilon_ {x x} - \sigma_ {x x}) _ {t}, \quad \mathrm{etc.,}
+$$
+
+and
+
+$$
+(\frac {\Delta t}{2} + \tilde {\nu}) \Delta \sigma_ {x y} = (\Delta t \frac {\mu}{2} + \tilde {\mu}) \Delta \gamma_ {x y} + \Delta t (\mu \gamma_ {x y} - \sigma_ {x y}) _ {t}, \quad \mathrm{etc.},
+$$
+
+so that the Jacobian matrix has the terms
+
+<!-- source-page: 303 -->
+
+![](images/page-303_26d520c0d5ad043c85fe843b76bda61a030fba18d68611d2dc64a52188ccef5e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+σ
+E₂
+μ₁
+E₁
+ε
+σ
+</details>
+
+Figure 1.1.44–1 Simple linear viscoelastic model.
+
+$$
+\frac {\partial \Delta \sigma_ {x x}}{\partial \Delta \varepsilon_ {x x}} = \frac {1}{(\frac {\Delta t}{2} + \tilde {\nu})} [ \Delta t (\frac {\lambda}{2} + \mu) + \tilde {\lambda} + 2 \tilde {\mu} ],
+$$
+
+$$
+\frac {\partial \Delta \sigma_ {x x}}{\partial \Delta \varepsilon_ {y y}} = \frac {1}{(\frac {\Delta t}{2} + \tilde {\nu})} [ \Delta t \frac {\lambda}{2} + \tilde {\lambda} ],
+$$
+
+and
+
+$$
+\frac {\partial \Delta \sigma_ {x y}}{\partial \Delta \gamma_ {x y}} = \frac {1}{(\frac {\Delta t}{2} + \tilde {\nu})} [ \Delta t \frac {\mu}{2} + \tilde {\mu} ].
+$$
+
+The total change in specific energy in an increment for this material is
+
+<!-- source-page: 304 -->
+
+$$
+(\sigma_ {i j} + \frac {1}{2} \Delta \sigma_ {i j}) \Delta \varepsilon_ {i j},
+$$
+
+while the change in specific elastic strain energy is
+
+$$
+(\varepsilon_ {i j} + \frac {1}{2} \Delta \varepsilon_ {i j}) D _ {i j k l} \Delta \varepsilon_ {k l},
+$$
+
+where D is the elasticity matrix:
+
+$$
+\left[ \begin{array}{c c c c c c} \lambda + 2 \mu & \lambda & \lambda & 0 & 0 & 0 \\ \lambda & \lambda + 2 \mu & \lambda & 0 & 0 & 0 \\ \lambda & \lambda & \lambda + 2 \mu & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu \end{array} \right].
+$$
+
+No state variables are needed for this material, so the allocation of space for them is not necessary. In a more realistic case a set of parallel models of this type might be used, and the stress components in each model might be stored as state variables.
+
+For our simple case a user material definition can be used to read in the five constants in the order $\lambda , \mu , { \tilde { \lambda } } , { \tilde { \mu } } ,$ and so that
+
+$$
+\text { PROPS } (1) = \lambda ,
+$$
+
+$$
+\text { PROPS } (2) = \mu ,
+$$
+
+$$
+\operatorname{PROPS} (3) = \tilde {\lambda},
+$$
+
+$$
+\mathrm{PROPS} (4) = \tilde {\mu},
+$$
+
+$$
+\mathrm{PROPS} (5) = \tilde {\nu}.
+$$
+
+The routine can then be coded as follows:
+```csv
+SUBROUTINE UMAT(STRESS,STATEV,DDSDDE,SSE,SPD,SCD,
+1 RPL,DDSDDT,DRPLDE,DRPLDT,
+2 STRAN,DSTRAN,TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED,CMNAME,
+3 NDI,NSHR,NTENS,NSTATV,PROPS,NPROPS,COORDS,DROT,PNEWDT,
+4 CELENT,DFGRD0,DFGRD1,NOEL,NPT,LAYER,KSPT,JSTEP,KINC)
+```
+
+C
+
+INCLUDE 'ABA\_PARAM.INC'
+
+<!-- source-page: 305 -->
+
+```csv
+C CHARACTER*80 CMNAME
+DIMENSION STRESS(NTENS),STATEV(NSTATV),
+1 DDSDDE(NTENS,NTENS),
+2 DDSDDT(NTENS),DRPLDE(NTENS),
+3 STRAN(NTENS),DSTRAN(NTENS),TIME(2),PREDEF(1),DPRED(1),
+4 PROPS(NPROPS),COORDS(3),DROT(3,3),DFGRD0(3,3),DFGRD1(3,3),
+5 JSTEP(4)
+DIMENSION DSTRES(6),D(3,3)
+
+C EVALUATE NEW STRESS TENSOR
+C
+EV = 0.
+DEV = 0.
+DO K1=1,NDI
+EV = EV + STRAN(K1)
+DEV = DEV + DSTRAN(K1)
+END DO
+C
+TERM1 = .5*DTIME + PROPS(5)
+TERM1I = 1./TERM1
+TERM2 = (.5*DTIME*PROPS(1)+PROPS(3))*TERM1I*DEV
+TERM3 = (DTIME*PROPS(2)+2.*PROPS(4))*TERM1I
+C
+DO K1=1,NDI
+DSTRES(K1) = TERM2+TERM3*DSTRAN(K1)
+1 +DTIME*TERM1I*(PROPS(1)*EV
+2 +2.*PROPS(2)*STRAN(K1)-STRESS(K1))
+STRESS(K1) = STRESS(K1) + DSTRES(K1)
+END DO
+C
+TERM2 = (.5*DTIME*PROPS(2) + PROPS(4))*TERM1I
+I1 = NDI
+DO K1=1,NSHR
+I1 = I1+1
+DSTRES(I1) = TERM2*DSTRAN(I1)+
+1 DTIME*TERM1I*(PROPS(2)*STRAN(I1)-STRESS(I1))
+STRESS(I1) = STRESS(I1)+DSTRES(I1)
+END DO
+C
+CREATE NEW JACOBIAN
+C 
+```
+
+<!-- source-page: 306 -->
+
+```asm
+TERM2 = (DTIME*(.5*PROPS(1)+PROPS(2))+PROPS(3)+1 2.*PROPS(4))*TERM1I
+TERM3 = (.5*DTIME*PROPS(1)+PROPS(3))*TERM1I
+DO K1=1,NTENS
+DO K2=1,NTENS
+DDSDDE(K2,K1) = 0.
+END DO
+END DO
+C
+DO K1=1,NDI
+DDSDDE(K1,K1) = TERM2
+END DO
+C
+DO K1=2,NDI
+N2 = K1-1
+DO K2=1,N2
+DDSDDE(K2,K1) = TERM3
+DDSDDE(K1,K2) = TERM3
+END DO
+END DO
+TERM2 = (.5*DTIME*PROPS(2)+PROPS(4))*TERM1I
+I1 = NDI
+DO K1=1,NSHR
+I1 = I1+1
+DDSDDE(I1,I1) = TERM2
+END DO
+C
+C TOTAL CHANGE IN SPECIFIC ENERGY
+C
+TDE = 0.
+DO K1=1,NTENS
+TDE = TDE + (STRESS(K1)-.5*DSTRES(K1))*DSTRAN(K1)
+END DO
+C
+C CHANGE IN SPECIFIC ELASTIC STRAIN ENERGY
+C
+TERM1 = PROPS(1) + 2.*PROPS(2)
+DO K1=1,NDI
+D(K1,K1) = TERM1
+END DO
+DO K1=2,NDI
+N2 = K1-1 
+```
+
+<!-- source-page: 307 -->
+
+```txt
+DO K2=1,N2
+    D(K1,K2) = PROPS(1)
+    D(K2,K1) = PROPS(1)
+END DO
+END DO
+DEE = 0.
+DO K1=1,NDI
+    TERM1 = 0.
+    TERM2 = 0.
+    DO K2=1,NDI
+    TERM1 = TERM1 + D(K1,K2)*STRAN(K2)
+    TERM2 = TERM2 + D(K1,K2)*DSTRAN(K2)
+END DO
+DEE = DEE + (TERM1+.5*TERM2)*DSTRAN(K1)
+END DO
+I1 = NDI
+DO K1=1,NSHR
+    I1 = I1+1
+    DEE = DEE + PROPS(2)*(STRAN(I1).5*DSTRAN(I1))*DSTRAN(I1)
+END DO
+SSE = SSE + DEE
+SCD = SCD + TDE - DEE
+RETURN
+END 
+```
+
+<!-- source-page: 308 -->
+
+<!-- source-page: 309 -->
+
+# 1.1.45 UMATHT: User subroutine to define a material’s thermal behavior.
+
+# Product: Abaqus/Standard
+
+WARNING: The use of this subroutine generally requires considerable expertise. You are cautioned that the implementation of any realistic thermal model requires significant development and testing. Initial testing on models with few elements under a variety of boundary conditions is strongly recommended.
+
+# References
+
+• “User-defined thermal material behavior,” Section 26.7.2 of the Abaqus Analysis User’s Guide   
+• \*USER MATERIAL   
+• “Freezing of a square solid: the two-dimensional Stefan problem,” Section 1.6.2 of the Abaqus Benchmarks Guide   
+• “UMATHT,” Section 4.1.22 of the Abaqus Verification Guide
+
+# Overview
+
+# User subroutine UMATHT:
+
+• can be used to define the thermal constitutive behavior of the material as well as internal heat generation during heat transfer processes;   
+• will be called at all material calculation points of elements for which the material definition includes a user-defined thermal material behavior;   
+• can be used with the procedures discussed in “Heat transfer analysis procedures: overview,” Section 6.5.1 of the Abaqus Analysis User’s Guide;   
+• can use solution-dependent state variables;   
+• must define the internal energy per unit mass and its variation with respect to temperature and to spatial gradients of temperature;   
+• must define the heat flux vector and its variation with respect to temperature and to gradients of temperature;   
+• must update the solution-dependent state variables to their values at the end of the increment;   
+• can be used in conjunction with user subroutine USDFLD to redefine any field variables before they are passed in; and   
+• is described further in “User-defined thermal material behavior,” Section 26.7.2 of the Abaqus Analysis User’s Guide.
+
+<!-- source-page: 310 -->
+
+# Use of subroutine UMATHT with coupled temperature-displacement and coupled thermal-electrical-structural elements
+
+User subroutine UMATHT should be used only with reduced-integration or modified coupled temperaturedisplacement and coupled thermal-electrical-structural elements if the mechanical and thermal fields are not coupled through plastic dissipation. No such restriction exists with fully integrated coupled temperature-displacement and coupled thermal-electrical-structural elements.
+
+User subroutine interface   
+```fortran
+SUBROUTINE UMATHT(U, DUDT, DUDG, FLUX, DFDT, DFDG,
+    1 STATEV, TEMP, DTEMP, DTEMDX, TIME, DTIME, PREDEF, DPRED,
+    2 CMNAME, NTGRD, NSTATV, PROPS, NPROPS, COORDS, PNEWDT,
+    3 NOEL, NPT, LAYER, KSPT, KSTEP, KINC)
+C
+INCLUDE 'ABA_PARAM.INC'
+CHARACTER*80 CMNAME
+DIMENSION DUDG(NTGRD), FLUX(NTGRD), DFDT(NTGRD),
+1 DFDG(NTGRD, NTGRD), STATEV(NSTATV), DTEMDX(NTGRD),
+2 TIME(2), PREDEF(1), DPRED(1), PROPS(NPROPS), COORDS(3)
+user coding to define U, DUDT, DUDG, FLUX, DFDT, DFDG,
+and possibly update STATEV, PNEWDT
+RETURN
+END 
+```
+
+# Variables to be defined
+
+U
+
+Internal thermal energy per unit mass, U, at the end of increment. This variable is passed in as the value at the start of the increment and must be updated to its value at the end of the increment.
+
+DUDT
+
+Variation of internal thermal energy per unit mass with respect to temperature, , evaluated at the end of the increment.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_032.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_032.md
new file mode 100644
index 00000000..27fb509f
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_032.md
@@ -0,0 +1,451 @@
+<!-- source-page: 311 -->
+
+# DUDG(NTGRD)
+
+Variation of internal thermal energy per unit mass with respect to the spatial gradients of temperature, $\partial U / \partial ( \partial \theta / \partial \mathbf { x } )$ , at the end of the increment. The size of this array depends on the value of NTGRD as defined below. This term is typically zero in classical heat transfer analysis.
+
+# FLUX(NTGRD)
+
+Heat flux vector, , at the end of the increment. This variable is passed in with the values at the beginning of the increment and must be updated to the values at the end of the increment.
+
+# DFDT(NTGRD)
+
+Variation of the heat flux vector with respect to temperature, , evaluated at the end of the increment.
+
+# DFDG(NTGRD,NTGRD)
+
+Variation of the heat flux vector with respect to the spatial gradients of temperature, , at the end of the increment. The size of this array depends on the value of NTGRD as defined below.
+
+# Variables that can be updated
+
+# STATEV(NSTATV)
+
+An array containing the solution-dependent state variables.
+
+In an uncoupled heat transfer analysis STATEV is passed into UMATHT with the values of these variables at the beginning of the increment. However, any changes in STATEV made in user subroutine USDFLD will be included in the values passed into UMATHT, since USDFLD is called before UMATHT. In addition, if UMATHT is being used in a fully coupled temperature-displacement or coupled thermalelectrical-structural analysis and user subroutine CREEP, user subroutine UEXPAN, user subroutine UMAT, or user subroutine UTRS is used to define the mechanical behavior of the material, those routines are called before this routine; therefore, any updating of STATEV done in CREEP, UEXPAN, UMAT, or UTRS will be included in the values passed into UMATHT.
+
+In all cases STATEV should be passed back from UMATHT as the values of the state variables at the end of the current increment.
+
+# PNEWDT
+
+Ratio of suggested new time increment to the time increment being used (DTIME, see below). This variable allows you to provide input to the automatic time incrementation algorithms in Abaqus/Standard (if automatic time incrementation is chosen).
+
+PNEWDT is set to a large value before each call to UMATHT.
+
+If PNEWDT is redefined to be less than 1.0, Abaqus/Standard must abandon the time increment and attempt it again with a smaller time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines that allow redefinition of PNEWDT for this iteration.
+
+If PNEWDT is given a value that is greater than 1.0 for all calls to user subroutines for this iteration and the increment converges in this iteration, Abaqus/Standard may increase the time increment. The
+
+<!-- source-page: 312 -->
+
+suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines for this iteration.
+
+If automatic time incrementation is not selected in the analysis procedure, values of PNEWDT that are greater than 1.0 will be ignored and values of PNEWDT that are less than 1.0 will cause the job to terminate.
+
+# Variables passed in for information
+
+# TEMP
+
+Temperature at the start of the increment.
+
+# DTEMP
+
+Increment of temperature.
+
+# DTEMDX(NTGRD)
+
+Current values of the spatial gradients of temperature,
+
+# TIME(1)
+
+Value of step time at the beginning of the current increment.
+
+# TIME(2)
+
+Value of total time at the beginning of the current increment.
+
+# DTIME
+
+Time increment.
+
+# PREDEF
+
+Array of interpolated values of predefined field variables at this point at the start of the increment, based on the values read in at the nodes.
+
+# DPRED
+
+Array of increments of predefined field variables.
+
+# CMNAME
+
+User-defined material name, left justified.
+
+# NTGRD
+
+Number of spatial gradients of temperature.
+
+# NSTATV
+
+Number of solution-dependent state variables associated with this material type (defined as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# PROPS(NPROPS)
+
+User-specified array of material constants associated with this user material.
+
+<!-- source-page: 313 -->
+
+# NPROPS
+
+User-defined number of material constants associated with this user material.
+
+# COORDS
+
+An array containing the coordinates of this point. These are the current coordinates in a fully coupled temperature-displacement or coupled thermal-electrical-structural analysis if geometric nonlinearity is accounted for during the step (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide); otherwise, the array contains the original coordinates of the point.
+
+# NOEL
+
+Element number.
+
+# NPT
+
+Integration point number.
+
+# LAYER
+
+Layer number (for composite shells and layered solids).
+
+# KSPT
+
+Section point number within the current layer.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# Example: Using more than one user-defined thermal material model
+
+To use more than one user-defined thermal material model, the variable CMNAME can be tested for different material names inside user subroutine UMATHT, as illustrated below:
+
+```sql
+IF (CMNAME(1:4) .EQ. 'MAT1') THEN
+CALL UMATHT_MAT1 (argument_list)
+ELSE IF(CMNAME(1:4) .EQ. 'MAT2') THEN
+CALL UMATHT_MAT2 (argument_list)
+END IF 
+```
+
+UMATHT\_MAT1 and UMATHT\_MAT2 are the actual user material subroutines containing the constitutive material models for each material MAT1 and MAT2, respectively. Subroutine UMATHT merely acts as a directory here. The argument list can be the same as that used in subroutine UMATHT.
+
+# Example: Uncoupled heat transfer
+
+As a simple example of the coding of user subroutine UMATHT, consider uncoupled heat transfer analysis in a material. The equations for this case are developed here, and the corresponding UMATHT is given.
+
+<!-- source-page: 314 -->
+
+This problem can also be solved by specifying thermal conductivity, specific heat, density, and internal heat generation directly.
+
+First, the equations for an uncoupled heat transfer analysis are outlined.
+
+The basic energy balance is
+
+$$
+\int_ {V} \rho \dot {U} d V = \int_ {S} q d S + \int_ {V} r d V,
+$$
+
+where V is the volume of solid material with surface area $s , \rho$ is the density of the material, $\dot { U }$ is the material time rate of the internal thermal energy, q is the heat flux per unit area of the body flowing into the body, and r is the heat supplied externally into the body per unit volume.
+
+A heat flux vector is defined such that
+
+$$
+q = - \mathbf {f} \cdot \mathbf {n},
+$$
+
+where is the unit outward normal to the surface S. Introducing the above relation into the energy balance equation and using the divergence theorem, the following relation is obtained:
+
+$$
+\int_ {V} \rho \dot {U} d V = - \int_ {V} \frac {\partial}{\partial \mathbf {x}} \cdot \mathbf {f} d V + \int_ {V} r d V.
+$$
+
+The corresponding weak form is given by
+
+$$
+\int_ {V} \delta \theta \rho \dot {U} d V - \int_ {V} \delta \mathbf {g} \cdot \mathbf {f} d V = \int_ {V} \delta \theta r d V + \int_ {S} \delta \theta q d S,
+$$
+
+where
+
+$$
+\mathbf {g} = \frac {\partial \theta}{\partial \mathbf {x}}
+$$
+
+is the temperature gradient and is an arbitrary variational field satisfying the essential boundary conditions.
+
+Introducing the backward difference integration algorithm:
+
+$$
+\dot {U} _ {t + \Delta t} = (U _ {t + \Delta t} - U _ {t}) (1 / \Delta t),
+$$
+
+the weak form of the energy balance equation becomes
+
+$$
+\frac {1}{\Delta t} \int_ {v} \delta \theta \rho (U _ {t + \Delta t} - U _ {t}) d V = \int_ {V} \delta \mathbf {g} \cdot \mathbf {f} d V + \int_ {V} \delta \theta r d V + \int_ {S} \delta \theta q d S.
+$$
+
+This nonlinear system is solved using Newton’s method.
+
+In the above equations the thermal constitutive behavior of the material is given by
+
+$$
+U = U (\theta , t, \partial \theta / \partial \mathbf {x}, s ^ {i}, \dots) U = U (\theta , \mathbf {g}, t, s ^ {i}, \dots)
+$$
+
+<!-- source-page: 315 -->
+
+and
+
+$$
+\mathbf {f} = \mathbf {f} (\theta , t, \partial \theta / \partial \mathbf {x}, s ^ {i}, \dots), \mathbf {f} = \mathbf {f} (\theta , \mathbf {g}, t, s ^ {i}, \dots),
+$$
+
+where $s ^ { i }$ are state variables.
+
+The Jacobian for Newton’s method is given by (after dropping the subscripts $t + \Delta t$ on U)
+
+$$
+\begin{array}{l} \frac {1}{\Delta t} \int_ {V} \delta \theta \rho \frac {\partial U}{\partial \theta} d \theta d V + \frac {1}{\Delta t} \int_ {V} \delta \theta \rho \frac {\partial U}{\partial \mathbf {g}} \cdot d \mathbf {g} d V \\ - \int_ {V} \delta \mathbf {g} \cdot \frac {\partial \mathbf {f}}{\partial \theta} d \theta d V - \int_ {V} \delta \mathbf {g} \cdot \frac {\partial \mathbf {f}}{\partial \mathbf {g}} \cdot d \mathbf {g} d V \\ - \int_ {V} \delta \theta \frac {\partial r}{\partial \theta} d \theta d V - \int_ {S} \delta \theta \frac {\partial q}{\partial \theta} d \theta d S. \\ \end{array}
+$$
+
+The thermal constitutive behavior for this example is now defined. We assume a constant specific heat for the material. The heat conduction in the material is assumed to be governed by Fourier’s law.
+
+The internal thermal energy per unit mass is defined as
+
+$$
+U = U (\theta),
+$$
+
+with
+
+$$
+\frac {\partial U}{\partial \theta} = c,
+$$
+
+where c is the specific heat of the material and
+
+$$
+\frac {\partial U}{\partial \mathbf {g}} = 0.
+$$
+
+Fourier’s law for heat conduction is given as
+
+$$
+\mathbf {f} = - \mathbf {k} \cdot \mathbf {g},
+$$
+
+where is the thermal conductivity matrix and is position, so that
+
+$$
+\frac {\partial \mathbf {f}}{\partial \mathbf {g}} = - \mathbf {k}
+$$
+
+and
+
+$$
+\frac {\partial \mathbf {f}}{\partial \theta} = - \frac {\partial \mathbf {k}}{\partial \theta} \cdot \mathbf {g}.
+$$
+
+The assumption of conductivity without any temperature dependence implies that
+
+<!-- source-page: 316 -->
+
+$$
+\frac {\partial \mathbf {f}}{\partial \theta} = 0.
+$$
+
+No state variables are needed for this material, so the allocation of space for them is not necessary.
+
+A thermal user material definition can be used to read in the two constants for our simple case, namely the specific heat, c, and the coefficient of thermal conductivity, k, so that
+
+$$
+\operatorname{PROPS} (1) = k,
+$$
+
+$$
+\operatorname{PROPS} (2) = c.
+$$```fortran
+SUBROUTINE UMATHT(U, DUDT, DUDG, FLUX, DFT, DFDG,
+    1 STATEV, TEMP, DTEMP, DTEMDX, TIME, DTIME, PREDEF, DPRED,
+    2 CMNAME, NTGRD, NSTATV, PROPS, NPROPS, COORDS, PNEWDT,
+    3 NOEL, NPT, LAYER, KSPT, KSTEP, KINC)
+C
+    INCLUDE 'ABA_PARAM.INC'
+C
+    CHARACTER*80 CMNAME
+    DIMENSION DUDG(NTGRD), FLUX(NTGRD), DFT(NTGRD),
+    1 DFDG(NTGRD, NTGRD), STATEV(NSTATV), DTEMDX(NTGRD),
+    2 TIME(2), PREDEF(1), DPRED(1), PROPS(NPROPS), COORDS(3)
+C
+    COND = PROPS(1)
+    SPECHT = PROPS(2)
+C
+    DUDT = SPECHT
+    DU = DUDT*DTEMP
+    U = U+DU
+C
+    DO I=1, NTGRD
+    FLUX(I) = -COND*DTEMDX(I)
+    DFDG(I, I) = -COND
+    END DO
+C
+    RETURN
+    END 
+```
+
+<!-- source-page: 317 -->
+
+# 1.1.46 UMESHMOTION: User subroutine to specify mesh motion constraints during adaptive meshing.
+
+Product: Abaqus/Standard
+
+# References
+
+• “Defining ALE adaptive mesh domains in Abaqus/Standard,” Section 12.2.6 of the Abaqus Analysis User’s Guide   
+• \*ADAPTIVE MESH   
+• \*ADAPTIVE MESH CONSTRAINT
+
+# Overview
+
+User subroutine UMESHMOTION:
+
+• is called at the end of any increment where adaptive meshing is performed (as specified by the frequency in increments);   
+• can be used to define the motion of nodes in an adaptive mesh constraint node set; and   
+• can call utility routines GETVRN, GETNODETOELEMCONN, and GETVRMAVGATNODE to access results data at the node.
+
+# Accessing node point data
+
+You are provided with access to the values of the node point quantities at the end of the increment through the utility routine GETVRN described in “Obtaining node point information,” Section 2.1.9. You can also access values of material point quantities extrapolated to, and averaged, at nodes at the end of the increment through the utility routine GETVRMAVGATNODE described in “Obtaining material point information averaged at a node,” Section 2.1.8. GETVRMAVGATNODE requires the list of elements attached to the node, which is obtained by calling the utility routine GETNODETOELEMCONN described in “Obtaining node to element connectivity,” Section 2.1.10.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE UMESHMOTION(UREF, ULOCAL, NODE, NNDOF,
+* LNODETYPE, ALOCAL, NDIM, TIME, DTIME, PNEWDT,
+* KSTEP, KINC, KMESHSWEEP, JMATYP, JGVBLOCK, LSMOOTH)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION ULOCAL(NDIM), JELEMLIST(*)
+DIMENSION ALOCAL(NDIM, *), TIME(2) 
+```
+
+<!-- source-page: 318 -->
+
+DIMENSION JMATYP(\*),JGVBLOCK(\*) C
+
+user coding to define ULOCAL and, optionally PNEWDT
+
+RETURN END
+
+# Variable to be defined
+
+# ULOCAL
+
+Components of the mesh displacement or velocity of the adaptive mesh constraint node, described in the coordinate system ALOCAL. ULOCAL will be passed into the routine as values determined by the mesh smoothing algorithm. All components of the mesh displacement or velocity will be applied; i.e., you do not have the ability to select the directions in which the mesh displacement should be applied.
+
+# Variables that can be updated
+
+# PNEWDT
+
+Ratio of suggested new time increment to the time increment currently being used (DTIME, see below). This variable allows you to provide input to the automatic time incrementation algorithms in Abaqus/Standard (if automatic time incrementation is chosen).
+
+PNEWDT is set to a large value before each call to UMESHMOTION.
+
+The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines that allow redefinition of PNEWDT for this increment.
+
+If automatic time incrementation is not selected in the analysis procedure, values of PNEWDT greater than 1.0 will be ignored and values of PNEWDT less than 1.0 will cause the job to terminate.
+
+# LSMOOTH
+
+Flag specifying that surface smoothing be applied after application of the mesh motion constraint. Set LSMOOTH to 1 to enable surface smoothing. When this flag is set, the constraint defined in ULOCAL will be modified by the smoothing algorithm. In cases where ULOCAL describes mesh motion normal to a surface, the smoothing will have a minor impact on this normal component of mesh motion.
+
+# Variables passed in for information
+
+# UREF
+
+The value of the user-specified displacement or velocity provided as part of the adaptive mesh constraint definition. This value is updated based on any amplitude definitions used with the adaptive mesh constraint or default ramp amplitude variations associated with the current step.
+
+<!-- source-page: 319 -->
+
+# NODE
+
+Node number.
+
+# NNDOF
+
+Number of degrees of freedom at the node.
+
+# LNODETYPE
+
+Node type flag.
+
+LNODETYPE=1 indicates that the node is on the interior of the adaptive mesh region.
+
+LNODETYPE=2 indicates that the node is involved in a tied constraint.
+
+LNODETYPE=3 indicates that the node is at the corner of the boundary of an adaptive mesh region.
+
+LNODETYPE=4 indicates that the node lies on the edge of a boundary of an adaptive mesh region.
+
+LNODETYPE=5 indicates that the node lies on a flat surface on a boundary of the adaptive mesh region.
+
+LNODETYPE=6 indicates that the node participates in a constraint (other than a tied constraint) as a master node.
+
+LNODETYPE=7 indicates that the node participates in a constraint (other than a tied constraint) as a slave node.
+
+LNODETYPE=10 indicates that a concentrated load is applied to the node.
+
+# ALOCAL
+
+Local coordinate system aligned with the tangent to the adaptive mesh domain at the node. If the node is on the interior of the adaptive mesh domain, ALOCAL will be set to the identity matrix. In other cases the 1-direction is along an edge or in the plane of a flat surface. When NDIM=2, the 2-direction is normal to the surface. When NDIM=3, the 2-direction also lies in the plane of a flat surface or is arbitrary if the node is on an edge. When NDIM=3 the 3-direction is normal to the surface or is arbitrary if the node is on an edge.
+
+# NDIM
+
+Number of coordinate dimensions.
+
+# TIME(1)
+
+Current value of step time.
+
+# TIME(2)
+
+Current value of total time.
+
+# DTIME
+
+Time increment.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+<!-- source-page: 320 -->
+
+# KMESHSWEEP
+
+Mesh sweep number.
+
+# JMATYP
+
+Variable that must be passed into the GETVRMAVGATNODE utility routine to access local results at the node.
+
+# JGVBLOCK
+
+Variable that must be passed into the GETVRN, GETNODETOELEMCONN, and GETVRMAVGATNODE utility routines to access local results at the node.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_033.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_033.md
new file mode 100644
index 00000000..96a7fc6c
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_033.md
@@ -0,0 +1,315 @@
+<!-- source-page: 321 -->
+
+# 1.1.47 UMOTION: User subroutine to specify motions during cavity radiation heat transfer analysis or steady-state transport analysis.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Cavity radiation,” Section 41.1.1 of the Abaqus Analysis User’s Guide   
+• “Steady-state transport analysis,” Section 6.4.1 of the Abaqus Analysis User’s Guide   
+• \*MOTION   
+• \*TRANSPORT VELOCITY
+
+# Overview
+
+User subroutine UMOTION:
+
+• can be used either to define the magnitude of the translational motion for degrees of freedom specified as a predefined field in a cavity radiation heat transfer analysis or to define the magnitude of the rotational velocity in a steady-state transport step; and   
+• will overwrite any motion or transport velocity magnitudes if they are defined directly (and possibly modified by including an amplitude reference) outside the user subroutine.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE UMOTION(U,KSTEP,KINC,TIME,NODE,JDOF)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION U,TIME(2)
+C
+user coding to define U
+RETURN
+END 
+```
+
+<!-- source-page: 322 -->
+
+# Variable to be defined
+
+U
+
+Total value of the component of the translation due to prescribed motion for the degree of freedom specified by JDOF. U will be passed into the routine as the value defined by any magnitude and/or amplitude specification in the motion definition for the degree of freedom JDOF. The total value of the translation must be given in user subroutine UMOTION, regardless of the type of motion defined (displacement or velocity).
+
+When used in conjunction with a steady-state transport analysis, U defines the magnitude of the rotational velocity. In such a case JDOF is passed in as 0.
+
+# Variables passed in for information
+
+KSTEP
+
+Step number.
+
+KINC
+
+Increment number.
+
+TIME(1)
+
+Current value of step time.
+
+TIME(2)
+
+Current value of total time.
+
+NODE
+
+Node number.
+
+JDOF
+
+Degree of freedom. When used in a steady-state transport analysis, JDOF is passed in as 0.
+
+<!-- source-page: 323 -->
+
+# 1.1.48 UMULLINS: User subroutine to define damage variable for the Mullins effect material model.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Mullins effect,” Section 22.6.1 of the Abaqus Analysis User’s Guide   
+• \*MULLINS EFFECT   
+• “Mullins effect and permanent set,” Section 2.2.3 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine UMULLINS:
+
+• can be used to define the damage variable for the Mullins effect material model, including the use of the Mullins effect approach to model energy dissipation in elastomeric foams;   
+• will be called at all material calculation points of elements for which the material definition contains a user-defined Mullins effect; and   
+• should be used when you do not want to use the Ogden and Roxburgh form of the damage variable, , that is used by Abaqus/Standard.
+
+# User subroutine interface
+
+```fortran
+SUBROUTINE UMULLINS (NUMPROPS, PROPS, UMAXNEW, UMAXOLD, SEDDEV, 1 ETA, DETADW, DMGDISSOLD, DMGDISSNEW, SENERNEW, NUMSTATEV, STATEV, 2 TEMP, DTEMP, NUMFIELDV, FIELDV, FIELDVINC, CMNAME, LINPER)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME
+DIMENSION PROPS (*), STATEV (*), FIELDV (*), FIELDVINC (*)
+user coding to define ETA, DETADW,
+and, optionally, DMGDISSNEW, SENERNEW, STATEV
+RETURN
+END 
+```
+
+<!-- source-page: 324 -->
+
+# Variables to be defined
+
+# ETA
+
+The damage variable, .
+
+# DETADW
+
+The derivative of the damage variable with respect to the elastic strain energy density of the undamaged material, $\begin{array} { l } { \frac { d \eta } { d \tilde { U } } } \end{array}$ . This quantity is needed for the Jacobian of the overall system of equations and needs to be defined accurately to ensure good convergence characteristics.
+
+# Variables that can be updated
+
+# DMGDISSNEW
+
+The energy dissipation density at the end of the increment. This quantity can be defined either in total form or in an incremental manner using the old value of the damage dissipation DMGDISSOLD and the increment in damage dissipation. This quantity is used for output purposes only.
+
+# SENERNEW
+
+The recoverable strain energy density at the end of the increment. This quantity is used for output purposes only.
+
+# STATEV
+
+Array containing the user-defined solution-dependent state variables at this point. These are supplied as values at the start of the increment or as values updated by other user subroutines (see “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide) and must be returned as values at the end of the increment.
+
+# Variables passed in for information
+
+# UMAXNEW
+
+The value, at the end of the increment, of the maximum primary strain energy density over its entire deformation history.
+
+# UMAXOLD
+
+The value, at the beginning of the increment, of the maximum primary strain energy density over its entire deformation history.
+
+# SEDDEV
+
+The value, at the end of the increment, of the deviatoric primary strain energy density when the primary material behavior is hyperelastic. The value, at the end of the increment, of the total primary strain energy density when the primary material behavior is hyperfoam.
+
+# DMGDISSOLD
+
+The value of energy dissipated at the beginning of the increment.
+
+<!-- source-page: 325 -->
+
+# CMNAME
+
+User-specified material name, left justified.
+
+# NUMSTATEV
+
+Number of solution-dependent state variables associated with this material (defined as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# NUMPROPS
+
+Number of material properties entered for this user-defined hyperelastic material.
+
+# PROPS
+
+Array of material properties entered for this user-defined hyperelastic material.
+
+# TEMP
+
+Temperature at the start of the increment.
+
+# DTEMP
+
+Increment of temperature.
+
+# NUMFIELDV
+
+Number of field variables.
+
+# FIELDV
+
+Array of interpolated values of predefined field variables at this material point at the beginning of the increment based on the values read in at the nodes (initial values at the beginning of the analysis and current values during the analysis).
+
+# FIELDVINC
+
+Array of increments of predefined field variables at this material point for this increment; this includes any values updated by user subroutine USDFLD.
+
+# LINPER
+
+Linear perturbation flag. LINPER=1 if the step is a linear perturbation step. LINPER=0 if the step is a general step.
+
+<!-- source-page: 326 -->
+
+<!-- source-page: 327 -->
+
+# 1.1.49 UPOREP: User subroutine to define initial fluid pore pressure.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1 of the Abaqus Analysis User’s Guide   
+• “Coupled pore fluid diffusion and stress analysis,” Section 6.8.1 of the Abaqus Analysis User’s Guide   
+• \*INITIAL CONDITIONS
+
+# Overview
+
+User subroutine UPOREP:
+
+• allows for the specification of the initial pore pressure values of a porous medium;   
+• can be used to define initial pore pressure values as functions of nodal coordinates and/or node numbers; and   
+• will be called to define initial fluid pore pressure values at all nodes of a coupled pore fluid diffusion and stress analysis whenever user-defined initial pore pressure conditions are specified.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE UPOREP(UW0,COORDS,NODE)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION COORDS(3)
+C
+user coding to define UW0
+RETURN
+END
+```
+
+# Variable to be defined
+
+UW0
+
+Initial fluid pore pressure.
+
+<!-- source-page: 328 -->
+
+# Variables passed in for information
+
+# COORDS
+
+An array containing the current coordinates of this node.
+
+# NODE
+
+Node number.
+
+<!-- source-page: 329 -->
+
+# 1.1.50 UPRESS: User subroutine to specify prescribed equivalent pressure stress conditions.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Mass diffusion analysis,” Section 6.9.1 of the Abaqus Analysis User’s Guide   
+• \*PRESSURE STRESS   
+• “UTEMP, UFIELD, UMASFL, and UPRESS,” Section 4.1.25 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine UPRESS:
+
+• allows you to prescribe equivalent pressure stress values at the nodes of a model;   
+• will be called in a mass diffusion analysis whenever a current value of equivalent pressure stress is needed for a node that has user-defined pressure stress conditions;   
+• can be used to modify any pressure stresses read in from a results file; and   
+• ignores any equivalent pressure stresses provided for the associated pressure stress definition outside the user subroutine.
+
+# User subroutine interface
+
+```fortran
+SUBROUTINE UPRESS(PRESS,KSTEP,KINC,TIME,NODE,COORDS)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION TIME(2), COORDS(3)
+C
+user coding to define PRESS
+RETURN
+END
+```
+
+# Variable to be defined
+
+# PRESS
+
+Total value of the equivalent pressure stress at the node.
+
+<!-- source-page: 330 -->
+
+You may have also requested equivalent pressure stress to be set in one of two other ways: from a previously generated results file or via direct data input. When PRESS is passed into user subroutine UPRESS, it will contain equivalent pressure stresses obtained from the results file only. You can modify these values within this routine. Any values given as direct data input will be ignored.
+
+# Variables passed in for information
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# TIME(1)
+
+Current value of step time.
+
+# TIME(2)
+
+Current value of total time.
+
+# NODE
+
+Node number.
+
+# COORDS
+
+An array containing the coordinates of this node.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_034.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_034.md
new file mode 100644
index 00000000..970b67c2
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_034.md
@@ -0,0 +1,300 @@
+<!-- source-page: 331 -->
+
+# 1.1.51 UPSD: User subroutine to define the frequency dependence for random response loading.
+
+Product: Abaqus/Standard
+
+# References
+
+• “Random response analysis,” Section 6.3.11 of the Abaqus Analysis User’s Guide   
+• \*RANDOM RESPONSE   
+• \*PSD-DEFINITION   
+• “Random response to jet noise excitation,” Section 1.4.10 of the Abaqus Benchmarks Guide
+
+# Overview
+
+User subroutine UPSD:
+
+• will be called once for each frequency at which calculations will be made during a random response analysis if the frequency function is defined in a user subroutine;   
+• is used to define complicated frequency dependencies for the cross-spectral density matrix of the random loading; and   
+• ignores any data given for the associated frequency function outside the user subroutine.
+
+# User subroutine interface
+
+```fortran
+SUBROUTINE UPSD(PSD,PSDR,PSDI,FREQ,KSTEP)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 PSD
+user coding to define PSDR and PSDI
+RETURN
+END 
+```
+
+# Variables to be defined
+
+# PSDR
+
+Real part of the frequency function at this frequency.
+
+<!-- source-page: 332 -->
+
+# PSDI
+
+Imaginary part of the frequency function at this frequency.
+
+# Variables passed in for information
+
+# PSD
+
+User-specified name for this frequency function definition, left justified.
+
+# FREQ
+
+Frequency, in radians per time.
+
+# KSTEP
+
+Step number.
+
+<!-- source-page: 333 -->
+
+# 1.1.52 URDFIL: User subroutine to read the results file.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Results file output format,” Section 5.1.2 of the Abaqus Analysis User’s Guide   
+• “Accessing the results file information,” Section 5.1.3 of the Abaqus Analysis User’s Guide   
+• “Utility routines for accessing the results file,” Section 5.1.4 of the Abaqus Analysis User’s Guide
+
+# Overview
+
+User subroutine URDFIL:
+
+• can be used to access the results file during an analysis;   
+• is called at the end of any increment in which new information is written to the results file;   
+• must call the utility routine DBFILE to read records from the results file (see “Utility routines for accessing the results file,” Section 5.1.4 of the Abaqus Analysis User’s Guide);   
+• can call the utility routine POSFIL to read from the results file starting at a specified step and increment as opposed to the beginning of the file, which would otherwise be done (see “Utility routines for accessing the results file,” Section 5.1.4 of the Abaqus Analysis User’s Guide);   
+• can force an analysis to terminate upon completion of a call by means of the variable LSTOP;   
+• allows the last increment written to the results file to be overwritten by means of the variable LOVRWRT; and   
+• allows access to the complete results file in a restarted job if the new results file is being appended to the old results file (see the description of the execution option fil in “Abaqus/Standard, Abaqus/Explicit, and Abaqus/CFD execution,” Section 3.2.2 of the Abaqus Analysis User’s Guide).
+
+# User subroutine interface
+
+```txt
+SUBROUTINE URDFIL(LSTOP,LOVRWRT,KSTEP,KINC,DTIME,TIME)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION ARRAY(513),JRRAY(NPRECD,513),TIME(2)
+EQUIVALENCE (ARRAY(1),JRRAY(1,1))
+user coding to read the results file
+RETURN
+END 
+```
+
+<!-- source-page: 334 -->
+
+# In all cases
+
+# LSTOP
+
+Flag to indicate whether an analysis should continue. The analysis will be terminated if LSTOP is set to 1. Otherwise, the analysis will continue.
+
+# LOVRWRT
+
+Flag to indicate that the information written to the results file for the increment can be overwritten. If LOVRWRT is set to 1, information for the current increment will be overwritten by information written to the results file in a subsequent increment unless the current increment is the final increment written to the results file. The purpose of this flag is to reduce the size of the results file by allowing information for an increment to be overwritten by information for a subsequent increment.
+
+# DTIME
+
+Time increment. This variable allows you to provide input to the automatic time incrementation algorithms in Abaqus (if automatic time incrementation is chosen). It is passed in as the value of the next time increment to be taken and can be updated to increase or reduce the time increment. If automatic time incrementation is not selected in the analysis procedure, updated values of DTIME are ignored.
+
+# Only if utility routine POSFIL is called
+
+# NSTEP
+
+Desired step at which file reading will begin via utility routine DBFILE. If NSTEP is set to 0, the first available step will be read.
+
+# NINC
+
+Desired increment at which file reading will begin via utility routine DBFILE. If NINC is set to 0, the first available increment of the specified step will be read.
+
+# Variables passed in for information
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# TIME(1)
+
+Value of the step time at the end of the increment.
+
+# TIME(2)
+
+Value of the total time at the end of the increment.
+
+<!-- source-page: 335 -->
+
+Example: Terminating an analysis upon exceeding a Mises stress limit
+
+The example below reads the values of Mises stress for the current increment from record 12 in the results file and terminates the analysis if any of the values of Mises stress written to the results file exceed 2.09 $\times 1 0 ^ { 8 }$ . Here, POSFIL is used to position you to read from the current increment.
+
+```fortran
+SUBROUTINE URDFIL(LSTOP,LOVRWRT,KSTEP,KINC,DTIME,TIME)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION ARRAY(513),JRRAY(NPRECD,513),TIME(2)
+EQUIVALENCE (ARRAY(1),JRRAY(1,1))
+PARAMETER(TOL=2.09D8)
+C
+C FIND CURRENT INCREMENT.
+C
+CALL POSFIL(KSTEP,KINC,ARRAY,JRCD)
+DO K1=1,999999
+CALL DBFILE(0,ARRAY,JRCD)
+IF (JRCD .NE. 0) GO TO 110
+KEY=JRRAY(1,2)
+C
+C RECORD 12 CONTAINS VALUES FOR SINV
+C
+IF (KEY.EQ.12) THEN
+IF (ARRAY(3).GT.TOL) THEN
+LSTOP=1
+GO TO 110
+END IF
+END IF
+END DO
+110 CONTINUE
+C
+RETURN
+END 
+```  
+Example: Terminating an analysis when the maximum Mises stress value stops increasing
+
+This example demonstrates the use of URDFIL and POSFIL to stop an analysis when the maximum value of Mises stress in the model does not increase from one increment in the results file to the next. A data statement is used to save the maximum Mises stress value from the last increment. LOVRWRT is also used in this case to overwrite an increment in the results file once it has been read in URDFIL.
+
+<!-- source-page: 336 -->
+
+The subroutine shown below must be modified to define the maximum Mises stress in the data statement each time a job is restarted. This can be avoided by removing the LOVRWRT=1 statement and recoding the routine to read both the previous and the current increment to check that the Mises stress increases from one increment to the next (in this case you must correctly handle the first increment written to the results file as there will be no previous increment). The results file must also be properly appended on restart if you wish to compare the values of Mises stress between the first increment of a restart and the final increment of the job being restarted. This approach has the disadvantage that the results file may become quite large, as no information in the file will be overwritten.
+
+```csv
+SUBROUTINE URDFIL(LSTOP,LOVRWRT,KSTEP,KINC,DTIME,TIME)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION ARRAY(513),JRRAY(NPRECD,513),TIME(2)
+EQUIVALENCE (ARRAY(1),JRRAY(1,1))
+C
+C INITIALIZE THE OLD MAXIMUM. FOR A JOB THAT IS BEING RESTARTED
+C THIS VALUE SHOULD BE SET TO THE MAXIMUM MISES STRESS IN THE
+C ORIGINAL ANALYSIS.
+C
+DATA OLDMAX/-1.D0/
+C
+CURRMAX = 0.D0
+C
+C FIND CURRENT INCREMENT.
+C
+CALL POSFIL(KSTEP,KINC,ARRAY,JRCD)
+C
+C SEARCH FOR THE HIGHEST VALUE OF MISES STRESS
+C AND STORE THIS IN CURRMAX
+C
+DO K1=1,999999
+CALL DBFILE(0,ARRAY,JRCD)
+IF (JRCD.NE.0) GO TO 110
+KEY=JRRAY(1,2)
+IF (KEY.EQ.12) THEN
+IF (ARRAY(3).GT.CURRMAX) CURRMAX=ARRAY(3)
+END IF
+END DO
+110 CONTINUE
+C
+C COMPLETED READING OF CURRENT INCREMENT. NOW CHECK TO
+C SEE IF VALUE OF MISES STRESS HAS INCREASED SINCE 
+```
+
+<!-- source-page: 337 -->
+
+```txt
+C LAST INCREMENT
+C
+IF (CURRMAX.LE.OLDMAX) LSTOP=1
+OLDMAX=CURRMAX
+LOVRWRT=1
+C
+RETURN
+END
+```
+
+<!-- source-page: 338 -->
+
+<!-- source-page: 339 -->
+
+# 1.1.53 USDFLD: User subroutine to redefine field variables at a material point.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Obtaining material point information in an Abaqus/Standard analysis,” Section 2.1.6   
+• “Material data definition,” Section 21.1.2 of the Abaqus Analysis User’s Guide   
+• \*USER DEFINED FIELD   
+• “Damage and failure of a laminated composite plate,” Section 1.1.14 of the Abaqus Example Problems Guide   
+• “USDFLD,” Section 4.1.24 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine USDFLD:
+
+• allows you to define field variables at a material point as functions of time or of any of the available material point quantities listed in the Output Variable Identifiers table (“Abaqus/Standard output variable identifiers,” Section 4.2.1 of the Abaqus Analysis User’s Guide) except the user-defined output variables UVARM and UVARMn;   
+• can be used to introduce solution-dependent material properties since such properties can easily be defined as functions of field variables;   
+• will be called at all material points of elements for which the material definition includes userdefined field variables;   
+• must call utility routine GETVRM to access material point data;   
+• can use and update state variables; and   
+• can be used in conjunction with user subroutine UFIELD to prescribe predefined field variables.
+
+# Explicit solution dependence
+
+Since this routine provides access to material point quantities only at the start of the increment, the solution dependence introduced in this way is explicit: the material properties for a given increment are not influenced by the results obtained during the increment. Hence, the accuracy of the results depends on the size of the time increment. Therefore, you can control the time increment in this routine by means of the variable PNEWDT.
+
+# Defining field variables
+
+Before user subroutine USDFLD is called, the values of the field variables at the material point are calculated by interpolation from the values defined at the nodes. Any changes to the field variables in the user subroutine are local to the material point: the nodal field variables retain the values defined
+
+<!-- source-page: 340 -->
+
+as initial conditions, predefined field variables, or in user subroutine UFIELD. The values of the field variables defined in this routine are used to calculate values of material properties that are defined to depend on field variables and are passed into other user subroutines that are called at the material point, such as the following:
+
+• CREEP   
+• HETVAL   
+• UEXPAN   
+• UHARD   
+• UHYPEL   
+• UMAT   
+• UMATHT   
+• UTRS
+
+Output of the user-defined field variables at the material points can be obtained with the element integration point output variable FV (see “Abaqus/Standard output variable identifiers,” Section 4.2.1 of the Abaqus Analysis User’s Guide).
+
+# Accessing material point data
+
+You are provided with access to the values of the material point quantities at the start of the increment (or in the base state in a linear perturbation step) through the utility routine GETVRM described in “Obtaining material point information in an Abaqus/Standard analysis,” Section 2.1.6. The values of the material point quantities are obtained by calling GETVRM with the appropriate output variable keys. The values of the material point data are recovered in the arrays ARRAY, JARRAY, and FLGRAY for floating point, integer, and character data, respectively. You may not get values of some material point quantities that have not been defined at the start of the increment; e.g., ER.
+
+# State variables
+
+Since the redefinition of field variables in USDFLD is local to the current increment (field variables are restored to the values interpolated from the nodal values at the start of each increment), any history dependence required to update material properties by using this subroutine must be introduced with userdefined state variables.
+
+The state variables can be updated in USDFLD and then passed into other user subroutines that can be called at this material point, such as those listed above. You specify the number of such state variables, as shown in the example at the end of this section (see also “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# User subroutine interface
+
+SUBROUTINE USDFLD(FIELD,STATEV,PNEWDT,DIRECT,T,CELENT,1 TIME,DTIME,CMNAME,ORNAME,NFIELD,NSTATV,NOEL,NPT,LAYER,
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_035.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_035.md
new file mode 100644
index 00000000..514bd620
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_035.md
@@ -0,0 +1,319 @@
+<!-- source-page: 341 -->
+
+```txt
+2 KSPT, KSTEP, KINC, NDI, NSHR, COORD, JMAC, JMATYP, MATLABO, LACCFLA)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME, ORNAME
+CHARACTER*3 FLGRAY(15)
+DIMENSION FIELD(NFIELD), STATEV(NSTATV), DIRECT(3,3),
+1 T(3,3), TIME(2)
+DIMENSION ARRAY(15), JARRAY(15), JMAC(*), JMATYP(*), COORD(*)
+user coding to define FIELD and, if necessary, STATEV and PNEWDT
+RETURN
+END 
+```
+
+# Variable to be defined
+
+# FIELD(NFIELD)
+
+An array containing the field variables at the current material point. These are passed in with the values interpolated from the nodes at the end of the current increment, as specified with initial condition definitions, predefined field variable definitions, or user subroutine UFIELD. The interpolation is performed using the same scheme used to interpolate temperatures: an average value is used for linear elements; an approximate linear variation is used for quadratic elements (also see “Solid (continuum) elements,” Section 28.1.1 of the Abaqus Analysis User’s Guide). The updated values are used to calculate the values of material properties that are defined to depend on field variables and are passed into other user subroutines (CREEP, HETVAL, UEXPAN, UHARD, UHYPEL, UMAT, UMATHT, and UTRS) that are called at this material point.
+
+# Variables that can be updated
+
+# STATEV(NSTATV)
+
+An array containing the solution-dependent state variables. These are passed in as the values at the beginning of the increment. In all cases STATEV can be updated in this subroutine, and the updated values are passed into other user subroutines (CREEP, HETVAL, UEXPAN, UMAT, UMATHT, and UTRS) that are called at this material point. The number of state variables associated with this material point is defined as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide.
+
+# PNEWDT
+
+Ratio of suggested new time increment to the time increment being used (DTIME, see below). This variable allows you to provide input to the automatic time incrementation algorithms in Abaqus/Standard (if automatic time incrementation is chosen).
+
+<!-- source-page: 342 -->
+
+PNEWDT is set to a large value before each call to USDFLD.
+
+If PNEWDT is redefined to be less than 1.0, Abaqus/Standard must abandon the time increment and attempt it again with a smaller time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines that allow redefinition of PNEWDT for this iteration.
+
+If PNEWDT is given a value that is greater than 1.0 for all calls to user subroutines for this iteration and the increment converges in this iteration, Abaqus/Standard may increase the time increment. The suggested new time increment provided to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user subroutines for this iteration.
+
+If automatic time incrementation is not selected in the analysis procedure, values of PNEWDT that are greater than 1.0 will be ignored and values of PNEWDT that are less than 1.0 will cause the job to terminate.
+
+# Variables passed in for information
+
+# DIRECT(3,3)
+
+An array containing the direction cosines of the material directions in terms of the global basis directions. DIRECT(1,1), DIRECT(2,1), DIRECT(3,1) give the (1, 2, 3) components of the first material direction; DIRECT(1,2), DIRECT(2,2), DIRECT(3,2) give the second material direction, etc. For shell and membrane elements, the first two directions are in the plane of the element and the third direction is the normal. This information is not available for beam elements.
+
+# T(3,3)
+
+An array containing the direction cosines of the material orientation components relative to the element basis directions. This is the orientation that defines the material directions (DIRECT) in terms of the element basis directions. For continuum elements T and DIRECT are identical. For shell and membrane elements T(1,1) , T(1,2) , T(2,1) , T(2,2) , T(3,3) , and all other components are zero, where is the counterclockwise rotation around the normal vector that defines the orientation. If no orientation is used, T is an identity matrix. Orientation is not available for beam elements.
+
+# CELENT
+
+Characteristic element length. This is a typical length of a line across an element for a first-order element; it is half of the same typical length for a second-order element. For beams and trusses it is a characteristic length along the element axis. For membranes and shells it is a characteristic length in the reference surface. For axisymmetric elements it is a characteristic length in the plane only.
+
+# TIME(1)
+
+Value of step time at the beginning of the current increment.
+
+# TIME(2)
+
+Value of total time at the beginning of the current increment.
+
+<!-- source-page: 343 -->
+
+# DTIME
+
+Time increment.
+
+# CMNAME
+
+User-specified material name, left justified.
+
+# ORNAME
+
+User-specified local orientation name, left justified.
+
+# NFIELD
+
+Number of field variables defined at this material point.
+
+# NSTATV
+
+User-defined number of solution-dependent state variables (see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# NOEL
+
+Element number.
+
+# NPT
+
+Integration point number.
+
+# LAYER
+
+Layer number (for composite shells and layered solids).
+
+# KSPT
+
+Section point number within the current layer.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# NDI
+
+Number of direct stress components at this point.
+
+# NSHR
+
+Number of shear stress components at this point.
+
+# COORD
+
+Coordinates at this material point.
+
+# JMAC
+
+Variable that must be passed into the GETVRM utility routine to access an output variable.
+
+<!-- source-page: 344 -->
+
+# JMATYP
+
+Variable that must be passed into the GETVRM utility routine to access an output variable.
+
+# MATLAYO
+
+Variable that must be passed into the GETVRM utility routine to access an output variable.
+
+# LACCFLA
+
+Variable that must be passed into the GETVRM utility routine to access an output variable.
+
+# Example: Damaged elasticity model
+
+Included below is an example of user subroutine USDFLD. In this example a truss element is loaded in tension. A damaged elasticity model is introduced: the modulus decreases as a function of the maximum tensile strain that occurred during the loading history. The maximum tensile strain is stored as a solution-dependent state variable—see “Defining solution-dependent field variables” in “Predefined fields,” Section 34.6.1 of the Abaqus Analysis User’s Guide.
+
+Input file   
+```csv
+*HEADING
+DAMAGED ELASTICITY MODEL WITH USER SUBROUTINE USDFLD
+*ELEMENT, TYPE=T2D2, ELSET=ONE
+1, 1, 2
+*NODE
+1, 0., 0.
+2, 10., 0.
+*SOLID SECTION, ELSET=ONE, MATERIAL=ELASTIC
+1.
+*MATERIAL, NAME=ELASTIC
+*ELASTIC, DEPENDENCIES=1
+** Table of modulus values decreasing as a function
+** of field variable 1.
+2000., 0.3, 0., 0.00
+1500., 0.3, 0., 0.01
+1200., 0.3, 0., 0.02
+1000., 0.3, 0., 0.04
+*USER DEFINED FIELD
+*DEPVAR
+1
+*BOUNDARY
+1, 1, 2
+2, 2
+*STEP
+*STATIC 
+```
+
+<!-- source-page: 345 -->
+
+```csv
+0.1, 1.0, 0.0, 0.1
+*CLOAD
+2, 1, 20.
+*END STEP
+*STEP
+*STATIC
+0.1, 1.0, 0.0, 0.1
+*CLOAD
+2, 1, 0.
+*END STEP
+*STEP, INC=20
+*STATIC
+0.1, 2.0, 0.0, 0.1
+*CLOAD
+2, 1, 40.
+*END STEP 
+```
+
+User subroutine   
+```fortran
+SUBROUTINE USDFLD(FIELD,STATEV,PNEWDT,DIRECT,T,CELENT,
+1 TIME,DTIME,CMNAME,ORNAME,NFIELD,NSTATV,NOEL,NPT,LAYER,
+2 KSPT,KSTEP,KINC,NDI,NSHR,COORD,JMAC,JMATYP,MATLAYO,
+3 LACCFLA)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME,ORNAME
+CHARACTER*3 FLGRAY(15)
+DIMENSION FIELD(NFIELD),STATEV(NSTATV),DIRECT(3,3),
+1 T(3,3),TIME(2)
+DIMENSION ARRAY(15),JARRAY(15),JMAC(*),JMATYP(*),
+1 COORD(*)
+C
+C Absolute value of current strain:
+CALL GETVRM('E',ARRAY,JARRAY,FLGRAY,JRCD,JMAC,JMATYP,
+MATLAYO,LACCFLA)
+EPS = ABS( ARRAY(1) )
+C Maximum value of strain up to this point in time:
+CALL GETVRM('SDV',ARRAY,JARRAY,FLGRAY,JRCD,JMAC,JMATYP,
+MATLAYO,LACCFLA)
+EPSMAX = ARRAY(1)
+C Use the maximum strain as a field variable
+FIELD(1) = MAX( EPS , EPSMAX ) 
+```
+
+<!-- source-page: 346 -->
+
+```txt
+C Store the maximum strain as a solution dependent state
+C variable
+    STATEV(1) = FIELD(1)
+C If error, write comment to .DAT file:
+    IF(JRCD.NE.0) THEN
+    WRITE(6,*) 'REQUEST ERROR IN USDFLD FOR ELEMENT NUMBER ', 1    NOEL,'INTEGRATION POINT NUMBER ',NPT
+    ENDIF
+C
+    RETURN
+    END 
+```
+
+<!-- source-page: 347 -->
+
+# 1.1.54 UTEMP: User subroutine to specify prescribed temperatures.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Predefined fields,” Section 34.6.1 of the Abaqus Analysis User’s Guide   
+• \*TEMPERATURE  
+• “LE11: Solid cylinder/taper/sphere—temperature loading,” Section 4.2.11 of the Abaqus Benchmarks Guide   
+• “UTEMP, UFIELD, UMASFL, and UPRESS,” Section 4.1.25 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine UTEMP:
+
+• allows you to prescribe temperatures at the nodes of a model;   
+• will be called whenever a current value of temperature is needed for a node that is listed under a user-defined temperature field definition;   
+• ignores any temperatures provided for the associated temperature field definition outside the user subroutine; and   
+• can be used to modify any temperatures read in from a results file.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE UTEMP(TEMP,NSECPT,KSTEP,KINC,TIME,NODE,COORDS)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION TEMP(NSECPT), TIME(2), COORDS(3)
+C
+user coding to define TEMP
+RETURN
+END 
+```
+
+<!-- source-page: 348 -->
+
+# TEMP(NSECPT)
+
+Array of temperature values at node number NODE. If the node is not connected to a beam or shell element, only one value of temperature must be returned (NSECPT=1). Otherwise, the number of temperatures to be returned depends on the mode of temperature and field variable input selected for the beam or shell section. The following cases are possible:
+
+1. Temperatures and field variables for a beam section are given as values at the points shown in the beam section descriptions. The number of values required, NSECPT, is determined by the particular section type specified, as described in “Beam cross-section library,” Section 29.3.9 of the Abaqus Analysis User’s Guide.   
+2. Temperatures and field variables are given as values at n equally spaced points through each layer of a shell section. The number of values required, NSECPT, is equal to n.   
+3. Temperatures and field variables for a beam section are given as values at the origin of the crosssection together with gradients with respect to the 2-direction and, for three-dimensional beams, the 1-direction of the section; or temperatures and field variables for a shell section are given as values at the reference surface together with gradients with respect to the thickness. The number of values required, NSECPT, is 3 for three-dimensional beams, 2 for two-dimensional beams, and 2 for shells. Give the midsurface value first, followed by the first and (if necessary) second gradients, as described in “Beam elements,” Section 29.3 of the Abaqus Analysis User’s Guide, and “Shell elements,” Section 29.6 of the Abaqus Analysis User’s Guide.
+
+You can also request temperatures to be set in one of two other ways: from a previously generated results file or via direct data input. When array TEMP is passed into user subroutine UTEMP, it will contain temperatures obtained from the results file only. You can modify these values within this routine. Any values given as direct data input will be ignored.
+
+# Variables passed in for information
+
+# NSECPT
+
+Maximum number of section values required for any node in the model.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# TIME(1)
+
+Current value of step time.
+
+# TIME(2)
+
+Current value of total time.
+
+<!-- source-page: 349 -->
+
+# NODE
+
+Node number.
+
+# COORDS
+
+An array containing the current coordinates of this point. These are the current coordinates if geometric nonlinearity is accounted for during the step (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide); otherwise, the array contains the original coordinates of the node.
+
+<!-- source-page: 350 -->
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_036.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_036.md
new file mode 100644
index 00000000..cceb41f2
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_036.md
@@ -0,0 +1,289 @@
+<!-- source-page: 351 -->
+
+# 1.1.55 UTRACLOAD: User subroutine to specify nonuniform traction loads.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Distributed loads,” Section 34.4.3 of the Abaqus Analysis User’s Guide   
+• \*DLOAD   
+• \*DSLOAD   
+• “Distributed traction and edge loads,” Section 1.4.18 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine UTRACLOAD:
+
+• can be used to define the variation of the distributed traction load magnitude as a function of position, time, element number, load integration point number, etc.;   
+• if needed, can be used to define the initial loading direction for the distributed traction load as a function of position, element number, load integration point number, etc.;   
+• will be called at each load integration point for each element-based, edge-based, or surface-based nonuniform distributed traction load definition during stress analysis;   
+• cannot be used in mode-based procedures to describe the time variation of the load; and   
+• ignores any amplitude references that may appear with the associated step definition or nonuniform distributed traction load definition.
+
+# User subroutine interface
+
+```csv
+SUBROUTINE UTRACLOAD(ALPHA,T_USER,KSTEP,KINC,TIME,NOEL,NPT,1 COORDS,DIRCOS,JLTYP,SNAME)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION T_USER(3), TIME(2), COORDS(3), DIRCOS(3,3)
+CHARACTER*80 SNAME
+user coding to define ALPHA and T_USER
+RETURN
+END 
+```
+
+<!-- source-page: 352 -->
+
+# ALPHA
+
+Magnitude of the distributed traction load. Units are $\mathrm { F L } ^ { - 2 }$ for surface loads, $\mathrm { F L } ^ { - 1 }$ for edge loads, and F for edge moments. ALPHA is passed into the routine as the magnitude of the load specified as part of the element-based or surface-based distributed load definition. If the magnitude is not defined, ALPHA is passed in as zero. For a static analysis that uses the modified Riks method (“Unstable collapse and postbuckling analysis,” Section 6.2.4 of the Abaqus Analysis User’s Guide) ALPHA must be defined as a function of the load proportionality factor, . The distributed load magnitude is not available for output purposes.
+
+# T\_USER
+
+Loading direction of the distributed traction load. T\_USER is passed into the routine as the load direction specified as part of the element-based or surface-based distributed load definition. The vector T\_USER passed out of the subroutine is used as the initial loading direction $\mathbf { t } _ { u s e r }$ discussed in “Distributed loads,” Section 34.4.3 of the Abaqus Analysis User’s Guide. The direction of T\_USER as defined by the subroutine should not change during a step. If it does, convergence difficulties might arise. Load directions are needed only for a nonuniform general surface traction, shear surface traction, and general edge traction. If a direction is defined for the nonuniform normal edge traction, shear edge traction, transverse edge traction, or edge moment, it will be ignored. See “Distributed loads,” Section 34.4.3 of the Abaqus Analysis User’s Guide, for details.
+
+# Variables passed in for information
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# TIME(1)
+
+Current value of step time or current value of the load proportionality factor, , in a Riks step.
+
+# TIME(2)
+
+Current value of total time.
+
+# NOEL
+
+User-defined element number.
+
+# NPT
+
+Load integration point number within the element or on the element’s surface, depending on the load type.
+
+<!-- source-page: 353 -->
+
+# COORDS
+
+An array containing the coordinates of the load integration point. These are the current coordinates if geometric nonlinearity is accounted for during the step (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide); otherwise, the array contains the original coordinates of the point.
+
+# DIRCOS
+
+Orientation of the face or edge in the reference configuration. For three-dimensional facets the first and second columns are the normalized local directions in the plane of the surface, and the third column is the normal to the face. For solid elements the normal points inward, which is the negative of what is defined in “Conventions,” Section 1.2.2 of the Abaqus Analysis User’s Guide; for shell elements the normal definition is consistent with the convention. For two-dimensional facets the first column is the normalized tangent, the second column is the facet normal, and the third column is not used. For three-dimensional shell edges the first column is the tangent to the shell edge (shear direction), the second column is the in-plane normal (normal direction), and the third column is the normal to the plane of the shell (transverse direction).
+
+# JLTYP
+
+Identifies the load type for which this call to UTRACLOAD is being made. The load type may be an element-based surface load, an edge-based load, or a surface-based load. This variable identifies the element face or edge for which this call to UTRACLOAD is being made. This information is useful when several different nonuniform distributed loads are being imposed on an element at the same time. See Part VI, “Elements,” of the Abaqus Analysis User’s Guide for element face and edge identification. The load labels are shown in Table 1.1.55–1. For surface- or edge-based loading (TRSHRNU, TRVECNU, EDLDNU, EDNORNU, EDSHRNU, EDTRANU, EDMOMNU), j in the load type identifies the face or edge of the element underlying the surface.
+
+Table 1.1.55–1 JLTYP values for surface traction and edge load labels. 
+
+<table><tr><td>Load Label</td><td>JLTYP</td><td>Load Label</td><td>JLTYP</td><td>Load Label</td><td>JLTYP</td></tr><tr><td>TRSHRNU</td><td>510+j</td><td>EDLDNU</td><td>540+j</td><td>EDTRANU</td><td>570+j</td></tr><tr><td>TRSHR1NU</td><td>511</td><td>EDLD1NU</td><td>543</td><td>EDTRANU</td><td>573</td></tr><tr><td>TRSHR2NU</td><td>512</td><td>EDLD2NU</td><td>544</td><td>EDTRANU</td><td>574</td></tr><tr><td>TRSHR3NU</td><td>513</td><td>EDLD3NU</td><td>545</td><td>EDTRANU</td><td>575</td></tr><tr><td>TRSHR4NU</td><td>514</td><td>EDLD4NU</td><td>546</td><td>EDTRANU</td><td>576</td></tr><tr><td>TRSHR5NU</td><td>515</td><td>EDNORNU</td><td>550+j</td><td>EDMOMNU</td><td>580+j</td></tr><tr><td>TRSHR6NU</td><td>516</td><td>EDNOR1NU</td><td>553</td><td>EDMOM1NU</td><td>583</td></tr><tr><td>TRVECNU</td><td>520+j</td><td>EDNOR2NU</td><td>554</td><td>EDMOM2NU</td><td>584</td></tr><tr><td>TRVEC1NU</td><td>521</td><td>EDNOR3NU</td><td>555</td><td>EDMOM3NU</td><td>585</td></tr></table>
+
+<!-- source-page: 354 -->
+
+<table><tr><td>Load Label</td><td>JLTYP</td><td>Load Label</td><td>JLTYP</td><td>Load Label</td><td>JLTYP</td></tr><tr><td>TRVEC2NU</td><td>522</td><td>EDNOR4NU</td><td>556</td><td>EDMOM4NU</td><td>586</td></tr><tr><td>TRVEC3NU</td><td>523</td><td>EDSHRNU</td><td>560+j</td><td></td><td></td></tr><tr><td>TRVEC4NU</td><td>524</td><td>EDSHRNU</td><td>563</td><td></td><td></td></tr><tr><td>TRVEC5NU</td><td>525</td><td>EDSHRNU</td><td>564</td><td></td><td></td></tr><tr><td>TRVEC6NU</td><td>526</td><td>EDSHRNU</td><td>565</td><td></td><td></td></tr><tr><td></td><td></td><td>EDSHRNU</td><td>566</td><td></td><td></td></tr></table>
+
+# SNAME
+
+Surface name for a surface-based load definition. For an element-based or edge-based load the surface name is passed in as blank.
+
+<!-- source-page: 355 -->
+
+# 1.1.56 UTRS: User subroutine to define a reduced time shift function for a viscoelastic material.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Time domain viscoelasticity,” Section 22.7.1 of the Abaqus Analysis User’s Guide   
+• \*TRS   
+• \*VISCOELASTIC   
+• “Transient thermal loading of a viscoelastic slab,” Section 3.1.2 of the Abaqus Benchmarks Guide
+
+# Overview
+
+User subroutine UTRS:
+
+• can be used to define a temperature-time shift for a time domain viscoelastic analysis;   
+• will be called for all material points of elements for which a user-defined shift function is specified to define the time-temperature correspondence as part of the viscoelastic material definition;   
+• will be called before user subroutine UMATHT and/or user subroutine HETVAL if either or both are to be used with UTRS in a fully coupled temperature-displacement or a coupled thermal-electricalstructural analysis;   
+• can use and update solution-dependent state variables; and   
+• can be used in conjunction with user subroutine USDFLD to redefine any field variables before they are passed in.
+
+# User subroutine interface
+
+```fortran
+SUBROUTINE UTRS (SHIFT,TEMP,DTEMP,TIME,DTIME,PREDEF,DPRED,1 STATEV,CMNAME,COORDS)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME
+DIMENSION SHIFT(2),TIME(2),PREDEF(1),DPRED(1),STATEV(1),1 COORDS(1)
+C
+user coding to define SHIFT(1) and SHIFT(2)
+RETURN
+END 
+```
+
+<!-- source-page: 356 -->
+
+# Variable to be defined
+
+# SHIFT
+
+An array of length two that defines the shift function, A $( A > 0 )$ , at this point. SHIFT(1) defines the shift function at the beginning of the increment, and SHIFT(2) defines the shift function at the end of the increment. Abaqus/Standard will apply an averaging scheme to these values that assumes that the natural logarithm of the shift function can be approximated by a linear function over the increment.
+
+If either element of SHIFT is found to be less than or equal to zero, the analysis will terminate with an error message.
+
+# Variable that can be updated
+
+# STATEV
+
+An array containing the solution-dependent state variables at this point. This array will be passed in containing the values of these variables at the start of the increment unless they are updated in user subroutines USDFLD or UEXPAN, in which case the updated values are passed in. If any of the solutiondependent state variables are being used in conjunction with the viscoelastic behavior, they must be updated in this subroutine to their values at the end of the increment.
+
+# Variables passed in for information
+
+# TEMP
+
+Temperature at the end of the increment.
+
+# DTEMP
+
+Increment of temperature during the time increment.
+
+# PREDEF
+
+An array containing the values of all of the user-specified field variables at this point at the end of the increment (initial values at the beginning of the analysis and current values during the analysis).
+
+# DPRED
+
+An array containing the increments of all of the predefined field variables during the time increment.
+
+# TIME(1)
+
+Value of step time at the end of the current increment.
+
+# TIME(2)
+
+Value of total time at the end of the current increment.
+
+# DTIME
+
+Time increment. If this subroutine is called during a procedure such as a static analysis in which the viscoelastic effects will not be taken into account, this variable is passed in as zero.
+
+# CMNAME
+
+User-specified material name, left justified.
+
+<!-- source-page: 357 -->
+
+# COORDS
+
+An array containing the coordinates of the material point. These are the current coordinates if geometric nonlinearity is accounted for during the step (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide); otherwise, the array contains the original coordinates of the point.
+
+<!-- source-page: 358 -->
+
+<!-- source-page: 359 -->
+
+# 1.1.57 UTRSNETWORK: User subroutine to define a reduced time shift function for models defined within the parallel rheological framework.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Parallel rheological framework,” Section 22.8.2 of the Abaqus Analysis User’s Guide   
+• “Nonlinear large-strain viscoelasticity with hyperelasticity,” Section 2.2.8 of the Abaqus Verification Guide   
+• \*VISCOELASTIC
+
+# Overview
+
+User subroutine UTRSNETWORK:
+
+• can be used to define a time-temperature shift for a nonlinear viscoelastic network for models defined using the parallel rheological framework;   
+• will be called for all material points of elements for which a user-defined shift function is specified to define the time-temperature correspondence as part of the viscoelastic material definition;   
+• can use and update solution-dependent state variables; and   
+• can be used in conjunction with user subroutine USDFLD to redefine any field variables before they are passed in.
+
+# User subroutine interface
+
+```c
+subroutine utrsnetwork (
+C Must be updated
+* outputData,
+C Can be updated
+* statev,
+C Information (Read only)
+* nOutput,
+* nstatv,
+* networkid,
+* coords,
+* temp,
+* dtemp,
+* nfield,
+* predef,
+* dpred,
+* nprops, 
+```
+
+<!-- source-page: 360 -->
+
+```python
+* props,
+* i_array,
+* niarray,
+* r_array,
+* nrarray,
+* c_array,
+* ncarray)
+
+C
+    include 'aba_param.inc'
+
+C
+    parameter( io_trs_shift_begin = 1,
+    *    io_trs_shift_end = 2 )
+
+C
+    parameter( i_trs_kstep = 1,
+    *    i_trs_kinc = 2,
+    *    i_trs_noel = 3,
+    *    i_trs_npt = 4,
+    *    i_trs_layer = 5,
+    *    i_trs_kspt = 6 )
+
+C
+    parameter( ir_trs_step_time = 1,
+    *    ir_trs_total_time = 2,
+    *    ir_trs_crep_time = 3,
+    *    ir_trs_timeinc = 4 )
+
+C
+    parameter( ic_trs_material_name = 1 )
+
+C
+    dimension
+    *    statev(nstatv),
+    *    predef(nfield),
+    *    dpred(nfield),
+    *    props(nprops),
+    *    coords(*),
+    *    outputData(nOutput),
+    *    i_array(niarray),
+    *    r_array(nrarray)
+
+    character*80 c_array(ncarray)
+
+C 
+```
+
+user coding to define outputData(io\_trs\_shift\_begin)
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_037.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_037.md
new file mode 100644
index 00000000..49778131
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_037.md
@@ -0,0 +1,428 @@
+<!-- source-page: 361 -->
+
+and outputData(io\_trs\_shift\_end)
+
+return
+
+end
+
+# Variables to be defined
+
+outputData(io\_trs\_shift\_begin)
+
+The shift function at the beginning of the increment.
+
+outputData(io\_trs\_shift\_end)
+
+The shift function at the end of the increment.
+
+# Variable that can be updated
+
+statev
+
+An array containing the user-defined solution-dependent state variables at this point.
+
+# Variables passed in for information
+
+nOutput
+
+Size of array outputData. Currently equal to 2.
+
+nstatv
+
+Number of solution-dependent state variables associated with this material.
+
+networkid
+
+Network identification number, which identifies the network for which creep is defined.
+
+coords
+
+An array containing the current coordinates at this point.
+
+temp
+
+Temperature at the end of the increment.
+
+dtemp
+
+Increment of temperature.
+
+nfield
+
+Number of field variables.
+
+predef
+
+An array of interpolated values of predefined field variables at this point at the end of the increment, based on the values read in at the nodes and, optionally, redefined in user subroutine USDFLD.
+
+dpred
+
+An array of increments of predefined field variables.
+
+<!-- source-page: 362 -->
+
+nprops
+
+User-specified number of user-defined material properties.
+
+props
+
+An array of user-specified property values.
+
+```cmake
+i_array(i_trs_kstep) 
+```
+
+Step number.
+
+```bazel
+i_array(i_trs_kinc) 
+```
+
+Increment number.
+
+```txt
+i_array(i_trs_noel) 
+```
+
+Element number.
+
+```python
+i_array(i_trs_npt) 
+```
+
+Integration point.
+
+```txt
+i_array(i_trs_layer) 
+```
+
+Layer number (for layered solids).
+
+```txt
+i_array(i_trs_kspt) 
+```
+
+Section point number within the current layer.
+
+niarray
+
+Size of array i\_array. Currently equal to 6.
+
+```txt
+r_array(ir_trs_step_time) 
+```
+
+Value of step time at the end of the increment.
+
+```txt
+r_array(ir_trs_total_time) 
+```
+
+Value of total time at the end of the increment.
+
+```txt
+r_array(ir_trs_creep_time) 
+```
+
+Value of creep time at the end of the increment.
+
+```python
+r_array(ir_trs_timeinc) 
+```
+
+Time increment.
+
+nrarray
+
+Size of array r\_array. Currently equal to 4.
+
+c\_array(ic\_trs\_material\_name)
+
+User-specified material name, left justified. Some internal material models are given names starting with the “ABQ\_” character string. To avoid conflict, you should not use “ABQ\_” as the leading string for the material name.
+
+<!-- source-page: 363 -->
+
+# ncarray
+
+Size of array c\_array. Currently equal to 1.
+
+# Example: Williams-Landel-Ferry shift function
+
+As an example of the coding of user subroutine UTRSNETWORK, consider the William-Landel-Ferry model to define the shift function. In this case the shift function is expressed as (see “Thermorheologically simple temperature effects” in “Time domain viscoelasticity,” Section 22.7.1 of the Abaqus Analysis User’s Guide)
+
+$$
+\log_ {1 0} (A) = - \frac {C _ {1} (\theta - \theta_ {0})}{C _ {2} + (\theta - \theta_ {0})},
+$$
+
+where
+
+$\theta$ is the temperature,
+
+00 $\theta _ { 0 }$ is the reference temperature, and
+
+$C _ { 1 }$ and $C _ { 2 }$ are constants.
+
+The user subroutine would be coded as follows:
+```txt
+subroutine utrsnetwork (
+C Must be updated
+* outputData,
+C Can be updated
+* statev,
+C Information (Read only)
+* nOutput,
+* nstatv,
+* networkid,
+* coords,
+* temp,
+* dtemp,
+* nfield,
+* predef,
+* dpred,
+* nprops,
+* props,
+* i_array,
+* niarray,
+* r_array,
+* nrarray,
+* c_array,
+* ncarray) 
+```
+
+<!-- source-page: 364 -->
+
+```python
+c
+    include 'aba_param.inc'
+
+c
+    parameter( io_trs_shift_begin = 1,
+    *    io_trs_shift_end = 2 )
+
+c
+    parameter( i_trs_kstep = 1,
+    *    i_trs_kinc = 2,
+    *    i_trs_noel = 3,
+    *    i_trs_npt = 4,
+    *    i_trs_layer = 5,
+    *    i_trs_kspt = 6 )
+
+c
+    parameter( ir_trs_step_time = 1,
+    *    ir_trs_total_time = 2,
+    *    ir_trs_creep_time = 3,
+    *    ir_trs_timeinc = 4 )
+
+c
+    parameter( ic_trs_material_name = 1 )
+
+c
+    parameter( zero=0.0d0, one=1.0d0, dln10=2.30258509299d0)
+
+c
+    dimension
+    *    statev(nstatv),
+    *    predef(nfield),
+    *    dpred(nfield),
+    *    props(nprops),
+    *    coords(*),
+    *    outputData(nOutput),
+    *    i_array(niarray),
+    *    r_array(nrarray)
+
+    character*80 c_array(ncarray)
+
+c
+    outputData(io_trs_shift_begin) = zero
+    outputData(io_trs_shift_end) = zero
+    temp0 = temp-dtemp
+
+c
+c WLF
+
+c 
+```
+
+<!-- source-page: 365 -->
+
+```julia
+theta0 = props(1)
+C1 = props(2)
+C2 = props(3)
+outputData(io_trs_shift_begin) =
+& exp(-dln10*C1*(temp0-theta0)/(C2+(temp0-theta0)))
+outputData(io_trs_shift_end) =
+& exp(-dln10*C1*(temp-theta0)/(C2+(temp-theta0)))
+return
+end 
+```
+
+<!-- source-page: 366 -->
+
+<!-- source-page: 367 -->
+
+# 1.1.58 UVARM: User subroutine to generate element output.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Obtaining material point information in an Abaqus/Standard analysis,” Section 2.1.6   
+• \*USER OUTPUT VARIABLES   
+• “UVARM,” Section 4.1.26 of the Abaqus Verification Guide
+
+# Overview
+
+# User subroutine UVARM:
+
+• will be called at all material calculation points of elements for which the material definition includes the specification of user-defined output variables;   
+• may be called multiple times for each material point in an increment, as Abaqus/Standard iterates to a converged solution;   
+• will be called for each increment in a step;   
+• allows you to define output quantities that are functions of any of the available integration point quantities listed in the Output Variable Identifiers table (“Abaqus/Standard output variable identifiers,” Section 4.2.1 of the Abaqus Analysis User’s Guide);   
+• allows you to define the material directions as output variables;   
+• can be used for gasket elements;   
+• can call utility routine GETVRM to access material point data;   
+• cannot be used with linear perturbation procedures; and   
+• cannot be updated in the zero increment.
+
+# Accessing material point data
+
+You are provided with access to the values of the material point quantities through the utility routine GETVRM described in “Obtaining material point information in an Abaqus/Standard analysis,” Section 2.1.6. In a nonlinear analysis values returned will correspond to the current solution iteration, representing a converged solution only at the final iteration for each increment. The values of the material point data are recovered in the arrays ARRAY, JARRAY, and FLGRAY for floating point, integer, and character data, respectively. Floating point data are recovered as double-precision data.
+
+# Using user-defined output variables
+
+The output identifier for the user-defined output quantities is UVARM. Individual components are accessed with UVARMn, where , NUVARM. You must specify the number of user-defined output variables, NUVARM, for a given material to allocate space at each material calculation point for
+
+<!-- source-page: 368 -->
+
+each variable. The user-defined output variables are available for both printed and results file output and are written to the output database and restart files for contouring, printing, and X–Y plotting in Abaqus/CAE. Any number of user-defined output variables can be used.
+
+# Output precision
+
+The data are provided in double precision for output to the data (.dat) and results (.fil) files and are written to the output database (.odb) file in single precision. Because the user provides UVARM output variables in double precision, numeric overflow errors related to output to the output database file may occur in cases where the output results exceed the capacity for single-precision representation even when no overflow errors occur in UVARM.
+
+User subroutine interface   
+```txt
+SUBROUTINE UVARM(UVAR, DIRECT, T, TIME, DTIME, CMNAME, ORNAME, 1 NUVARM, NOEL, NPT, LAYER, KSPT, KSTEP, KINC, NDI, NSHR, COORD, 2 JMAC, JMATYP, MATLABO, LACCFLA)
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME, ORNAME
+CHARACTER*3 FLGRAY(15)
+DIMENSION UVAR (NUVARM), DIRECT(3, 3), T(3, 3), TIME(2)
+DIMENSION ARRAY(15), JARRAY(15), JMAC(*), JMATYP(*), COORD(*)
+C The dimensions of the variables FLGRAY, ARRAY and JARRAY
+C must be set equal to or greater than 15.
+user coding to define UVAR
+RETURN
+END 
+```
+
+# Variable to be defined
+
+# UVAR(NUVARM)
+
+An array containing the user-defined output variables. These are passed in as the values at the beginning of the increment and must be returned as the values at the end of the increment.
+
+<!-- source-page: 369 -->
+
+# DIRECT(3,3)
+
+An array containing the direction cosines of the material directions in terms of the global basis directions. DIRECT(1,1), DIRECT(2,1), DIRECT(3,1) give the (1, 2, 3) components of the first material direction; DIRECT(1,2), DIRECT(2,2), DIRECT(3,2) give the second material direction, etc. For shell and membrane elements the first two directions are in the plane of the element and the third direction is the normal. This information is not available for beam and truss elements.
+
+# T(3,3)
+
+An array containing the direction cosines of the material orientation components relative to the element basis directions. This is the orientation that defines the material directions (DIRECT) in terms of the element basis directions. For continuum elements T and DIRECT are identical. For shell and membrane elements T(1,1) , T(1,2) , T(2,1) , T(2,2) , T(3,3) , and all other components are zero, where is the counterclockwise rotation around the normal vector that defines the orientation. If no orientation is used, T is an identity matrix. Orientation is not available for beam and truss elements.
+
+# TIME(1)
+
+Value of step time at the end of the current increment.
+
+# TIME(2)
+
+Value of total time at the end of the current increment.
+
+# DTIME
+
+Time increment.
+
+# CMNAME
+
+User-specified material name, left justified.
+
+# ORNAME
+
+User-specified local orientation name, left justified.
+
+# NUVARM
+
+User-specified number of user-defined output variables.
+
+# NOEL
+
+Element number.
+
+# NPT
+
+Integration point number.
+
+# LAYER
+
+Layer number (for composite shells and layered solids).
+
+<!-- source-page: 370 -->
+
+# KSPT
+
+Section point number within the current layer.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# NDI
+
+Number of direct stress components at this point.
+
+# NSHR
+
+Number of shear stress components at this point.
+
+# COORD
+
+Coordinates at this material point.
+
+# JMAC
+
+Variable that must be passed into the GETVRM utility routine to access an output variable.
+
+# JMATYP
+
+Variable that must be passed into the GETVRM utility routine to access an output variable.
+
+# MATLAYO
+
+Variable that must be passed into the GETVRM utility routine to access an output variable.
+
+# LACCFLA
+
+Variable that must be passed into the GETVRM utility routine to access an output variable.
+
+# Example: Calculation of stress relative to shift tensor
+
+Below is an example of user subroutine UVARM. The subroutine calculates the position of the current state of stress relative to the center of the yield surface for the kinematic hardening plasticity model by subtracting the kinematic shift tensor, , from the stress tensor, . See “Metal plasticity models,” Section 4.3.1 of the Abaqus Theory Guide, for additional details.
+
+```txt
+SUBROUTINE UVARM(UVAR, DIRECT, T, TIME, DTIME, CMNAME, ORNAME, 1 NUVARM, NOEL, NPT, LAYER, KSPT, KSTEP, KINC, NDI, NSHR, COORD, 2 JMAC, JMATYP, MATLAYO, LACCFLA)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+CHARACTER*80 CMNAME, ORNAME
+CHARACTER*3 FLGRAY(15)
+DIMENSION UVAR (NUVARM), DIRECT(3, 3), T(3, 3), TIME(2) 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_038.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_038.md
new file mode 100644
index 00000000..2af76da6
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_038.md
@@ -0,0 +1,335 @@
+<!-- source-page: 371 -->
+
+```txt
+DIMENSION ARRAY(15),JARRAY(15),JMAC(*),JMATYP(*),COORD(*)
+C
+C Error counter:
+JERROR = 0
+C Stress tensor:
+CALL GETVRM('S',ARRAY,JARRAY,FLGRAY,JRCD,JMAC,JMATYP,
+1 MATLAYO,LACCFLA)
+JERROR = JERROR + JRCD
+UVAR(1) = ARRAY(1)
+UVAR(2) = ARRAY(2)
+UVAR(3) = ARRAY(3)
+UVAR(4) = ARRAY(4)
+UVAR(5) = ARRAY(5)
+UVAR(6) = ARRAY(6)
+C Kinematic shift tensor:
+CALL GETVRM('ALPHA',ARRAY,JARRAY,FLGRAY,JRCD,JMAC,JMATYP,
+1 MATLAYO,LACCFLA)
+JERROR = JERROR + JRCD
+C Calculate the position relative to the center of the
+C yield surface:
+UVAR(1) = UVAR(1) - ARRAY(1)
+UVAR(2) = UVAR(2) - ARRAY(2)
+UVAR(3) = UVAR(3) - ARRAY(3)
+UVAR(4) = UVAR(4) - ARRAY(4)
+UVAR(5) = UVAR(5) - ARRAY(5)
+UVAR(6) = UVAR(6) - ARRAY(6)
+C If error, write comment to .DAT file:
+IF(JERROR.NE.0) THEN
+WRITE(6,*) 'REQUEST ERROR IN UVARM FOR ELEMENT NUMBER',
+1 NOEL,'INTEGRATION POINT NUMBER ',NPT
+ENDIF
+RETURN
+END 
+```
+
+<!-- source-page: 372 -->
+
+<!-- source-page: 373 -->
+
+# 1.1.59 UWAVE: User subroutine to define wave kinematics for an Abaqus/Aqua analysis.
+
+Products: Abaqus/Standard Abaqus/Aqua
+
+# References
+
+• “Abaqus/Aqua analysis,” Section 6.11.1 of the Abaqus Analysis User’s Guide   
+• \*WAVE
+
+# Overview
+
+User subroutine UWAVE:
+
+• will be called at each load integration point for which an Abaqus/Aqua load is specified and a userdefined gravity wave is specified;   
+• can be used to define the wave kinematics (fluid velocity and acceleration, dynamic pressure, vertical gradient of the dynamic pressure, and the instantaneous fluid surface elevation) as a function of time and space; and   
+• for stochastic analysis, can be used to determine when during the analysis the current configuration should be retained as the intermediate configuration upon which the wave kinematics are based.
+
+# User subroutine interface
+
+```prolog
+SUBROUTINE UWAVE(V, A, PDYN, DPDYNDZ, SURF, LPDYN
+1 LRECOMPUTE, LUPLOCAL, LUPGLOBAL,
+2 LSURF, NDIM, XCUR, XINTERMED,
+3 GRAV, DENSITY, ELEVB, ELEVS,
+4 SEED, NSPECTRUM, FREQWAMP,
+5 TIME, DTIME, NOEL, NPT, KSTEP, KINC)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION V(NDIM), A(NDIM), XCUR(NDIM), XINTERMED(NDIM),
+1 FREQWAMP(2, NSPECTRUM), TIME(2)
+user coding to define V, A, PDYN, DPDYNDZ, SURF
+and, if necessary, LUPGLOBAL and LUPLOCAL
+RETURN
+END 
+```
+
+<!-- source-page: 374 -->
+
+# When LSURF=0
+
+# V(NDIM)
+
+The total fluid velocity at the current load integration location. This array is passed into UWAVE as the steady current velocity. The array should be updated as the sum of the steady current velocity and the velocity contribution from the user-defined wave theory.
+
+# A(NDIM)
+
+The fluid acceleration at the current load integration location.
+
+# PDYN
+
+The dynamic pressure contribution to the total pressure. This variable is needed only for buoyancy loads. The total pressure at a location below the instantaneous surface elevation is the sum of the atmospheric pressure, the hydrostatic pressure measured to the mean fluid elevation, and the dynamic pressure. See “Airy wave theory,” Section 6.2.2 of the Abaqus Theory Guide, and “Stokes wave theory,” Section 6.2.3 of the Abaqus Theory Guide, for definitions of the dynamic pressure for Airy and Stokes waves, respectively.
+
+# DPDYNDZ
+
+The gradient of the dynamic pressure in the vertical direction. This variable is needed only for buoyancy loads.
+
+# When LSURF=1
+
+# SURF
+
+The vertical coordinate of the instantaneous fluid surface corresponding to the horizontal position of the load integration point (given in XCUR). If the current location of the load integration point is above the instantaneous surface elevation, no fluid loads will be applied.
+
+# Only in an analysis with stochastic wave kinematics based on an intermediate configuration
+
+# LUPLOCAL
+
+Flag to determine if the intermediate configuration will be updated for this element. This flag can be set only when LRECOMPUTE=1. Return LUPLOCAL as 0 (default) to indicate that the intermediate configuration should not be updated. Return LUPLOCAL as 1 if the intermediate configuration should be updated for this element. The intermediate configuration is stored on an element-by-element basis. Therefore, all integration points for a given element will have their intermediate configuration updated if an update is requested at any one integration point on the element.
+
+# LUPGLOBAL
+
+Flag to determine if the intermediate configuration will be updated for all elements. This flag can be set only when LRECOMPUTE=1. Return LUPGLOBAL as 0 (default) to indicate that the intermediate configuration should not be updated. Return LUPGLOBAL as 1 if the intermediate configuration should be updated for all elements with Abaqus/Aqua loads.
+
+<!-- source-page: 375 -->
+
+# LRECOMPUTE
+
+For stochastic analysis LRECOMPUTE=1 indicates that an update to the intermediate configuration is permitted during this call to user subroutine UWAVE. The local and global update flags must be set accordingly. If the intermediate configuration is to be updated, the local update flag LUPLOCAL or the global update flag LUPGLOBAL must be set to 1. When LRECOMPUTE=1 and the intermediate configuration needs to be updated, the user subroutine should recompute all wave kinematics information based on the new intermediate configuration. For nonstochastic analysis this flag is always set to 0.
+
+# LPDYN
+
+LPDYN=1 indicates that only the dynamic pressure and its gradient need to be calculated (i.e., buoyancy loads). LPDYN=0 indicates that only the fluid velocity and acceleration need to be calculated (i.e., drag or inertia loads).
+
+# LSURF
+
+LSURF=1 indicates that subroutine UWAVE only needs to return the instantaneous fluid surface elevation. When LSURF=1, no velocity, acceleration, or dynamic pressure needs to be calculated. LSURF=0 indicates that the instantaneous fluid surface elevation SURF is not needed.
+
+# NDIM
+
+Two or three, indicating that the analysis is in two or three dimensions. The vertical direction is the global y-direction in two-dimensional analysis and the global z-direction in three-dimensional analysis.
+
+# XCUR(NDIM)
+
+An array containing the current coordinates of the load integration point.
+
+# XINTERMED(NDIM)
+
+An array containing the intermediate configuration coordinates of the load integration point. For nonstochastic analysis this array is not used. In a stochastic analysis the wave field is based upon this configuration. At the beginning of each load increment the LRECOMPUTE flag is set to 1 to prompt you for update action. If the intermediate configuration should be replaced by the current configuration, the flag LUPLOCAL should be set to 1 to update the intermediate configuration for this element only or the flag LUPGLOBAL should be set to 1 to update the intermediate configuration for all elements that have Abaqus/Aqua loading. At the beginning of the analysis the intermediate configuration is the reference configuration.
+
+# GRAV
+
+The user-specified gravitational constant in the fluid variable definition.
+
+# DENSITY
+
+The user-specified fluid mass density in the fluid variable definition.
+
+<!-- source-page: 376 -->
+
+# ELEVB
+
+The user-specified elevation of the seabed in the fluid variable definition.
+
+# ELEVS
+
+The user-specified elevation of the still fluid level in the fluid variable definition.
+
+# SEED
+
+For stochastic analysis the user-specified random number seed in the gravity wave definition.
+
+# NSPECTRUM
+
+For stochastic analysis the number of user-specified frequency versus wave amplitude pairs in the gravity wave definition, used to define the wave spectrum.
+
+# FREQWAMP(1,NSPECTRUM)
+
+For stochastic analysis the frequency values used to define the wave spectrum.
+
+# FREQWAMP(2,NSPECTRUM)
+
+For stochastic analysis the wave amplitude values used to define the wave spectrum.
+
+# TIME(1)
+
+Value of step time at the end of the current increment.
+
+# TIME(2)
+
+Value of total time at the end of the current increment.
+
+# DTIME
+
+Time increment.
+
+# NOEL
+
+Element number.
+
+# NPT
+
+Load integration point number. All line elements use full integration for the application of external loads. For distributed loads applied to the ends of the element, NPT corresponds to the end number of the element.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+<!-- source-page: 377 -->
+
+# 1.1.60 UXFEMNONLOCALWEIGHT: User subroutine to define the weight function used to compute the average stress/strain to determine the crack propagation direction.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Modeling discontinuities as an enriched feature using the extended finite element method,” Section 10.7.1 of the Abaqus Analysis User’s Guide   
+• “Progressive damage and failure,” Section 24.1.1 of the Abaqus Analysis User’s Guide   
+• \*DAMAGE INITIATION
+
+# Overview
+
+User subroutine UXFEMNONLOCALWEIGHT:
+
+• can be used to specify a user-defined weight function; and   
+• is currently available only for enriched elements.
+
+# User subroutine interface
+
+```fortran
+SUBROUTINE UXFEMNONLOCALWEIGHT (WEIGHT, JELNO, NPT, COORDS, & CRACKTIPCOORD, NNCRD, RADIUS, KSTEP, KINC, TIME)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION TIME(2), COORDS(NNCRD), CRACKTIPCOORD(NNCRD)
+user coding to define weight
+RETURN
+END 
+```
+
+# Variable to be defined
+
+weight
+
+A scalar weight function used to compute the average stress/strain at the crack tip.
+
+# Variables passed in for information
+
+JELNO
+
+Element number.
+
+<!-- source-page: 378 -->
+
+# NPT
+
+Integration point number.
+
+# COORDS
+
+An array containing the current coordinates of this integration point.
+
+# CRACKTIPCOORDS
+
+An array containing the current coordinates of the crack tip.
+
+# NNCRD
+
+Dimension of the model.
+
+# RADIUS
+
+Influence radius in which the elements are included for averaging.
+
+# KSTEP
+
+Step number.
+
+# KINC
+
+Increment number.
+
+# TIME(1)
+
+Value of step time at the beginning of the current increment.
+
+# TIME(2)
+
+Value of total time at the beginning of the current increment.
+
+<!-- source-page: 379 -->
+
+# 1.1.61 VOIDRI: User subroutine to define initial void ratios.
+
+# Product: Abaqus/Standard
+
+# References
+
+• “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1 of the Abaqus Analysis User’s Guide   
+• “Coupled pore fluid diffusion and stress analysis,” Section 6.8.1 of the Abaqus Analysis User’s Guide   
+• \*INITIAL CONDITIONS
+
+# Overview
+
+User subroutine VOIDRI:
+
+• will be called to define initial void ratio values at material calculation points of continuum elements (see Part VI, “Elements,” of the Abaqus Analysis User’s Guide) in a porous medium whenever a user-defined initial condition on void ratio is specified; and   
+• can be used to define initial void ratio values as functions of material point coordinates and/or element numbers.
+
+# User subroutine interface
+
+```txt
+SUBROUTINE VOIDRI (EZERO, COORDS, NOEL)
+C
+INCLUDE 'ABA_PARAM.INC'
+C
+DIMENSION COORDS (3)
+C
+user coding to define EZERO
+RETURN
+END 
+```
+
+# Variable to be defined
+
+# EZERO
+
+Initial void ratio.
+
+<!-- source-page: 380 -->
+
+# Variables passed in for information
+
+# COORDS
+
+An array containing the current coordinates of this point.
+
+# NOEL
+
+Element number.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_039.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_039.md
new file mode 100644
index 00000000..789f2edb
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_039.md
@@ -0,0 +1,319 @@
+<!-- source-page: 381 -->
+
+# 1.2 Abaqus/Explicit subroutines
+
+• “VDFLUX,” Section 1.2.1   
+• “VDISP,” Section 1.2.2   
+• “VDLOAD,” Section 1.2.3   
+• “VEXTERNALDB,” Section 1.2.4   
+• “VFABRIC,” Section 1.2.5   
+• “VFRIC,” Section 1.2.6   
+• “VFRIC\_COEF,” Section 1.2.7   
+• “VFRICTION,” Section 1.2.8   
+• “VUAMP,” Section 1.2.9   
+• “VUANISOHYPER\_INV,” Section 1.2.10   
+• “VUANISOHYPER\_STRAIN,” Section 1.2.11   
+• “VUCHARLENGTH,” Section 1.2.12   
+• “VUCREEPNETWORK,” Section 1.2.13   
+• “VUEL,” Section 1.2.14   
+• “VUEOS,” Section 1.2.15   
+• “VUFIELD,” Section 1.2.16   
+• “VUFLUIDEXCH,” Section 1.2.17   
+• “VUFLUIDEXCHEFFAREA,” Section 1.2.18   
+• “VUHARD,” Section 1.2.19   
+• “VUINTER,” Section 1.2.20   
+• “VUINTERACTION,” Section 1.2.21   
+• “VUMAT,” Section 1.2.22   
+• “VUMULLINS,” Section 1.2.23   
+• “VUSDFLD,” Section 1.2.24   
+• “VUTRS,” Section 1.2.25   
+• “VUVISCOSITY,” Section 1.2.26   
+• “VWAVE,” Section 1.2.27
+
+<!-- source-page: 382 -->
+
+<!-- source-page: 383 -->
+
+# 1.2.1 VDFLUX: User subroutine to specify nonuniform distributed fluxes in an explicit dynamic coupled temperature-displacement analysis.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Thermal loads,” Section 34.4.4 of the Abaqus Analysis User’s Guide   
+• \*DFLUX   
+• \*DSFLUX
+
+# Overview
+
+User subroutine VDFLUX:
+
+• can be used to define the variation of the distributed flux as a function of position, temperature, time, velocity, element number, etc. for a group of points in a dynamic coupled thermal-stress analysis using explicit integration (for more information, see “Fully coupled thermal-stress analysis,” Section 6.5.3 of the Abaqus Analysis User’s Guide);   
+• will be called at each flux integration point associated with each element-based or surface-based nonuniform distributed flux definition in the analysis; and   
+• recognizes an amplitude reference (“Amplitude curves,” Section 34.1.2 of the Abaqus Analysis User’s Guide) if it appears with the associated nonuniform flux definition.
+
+# User subroutine interface
+
+```fortran
+subroutine vdflux (
+C Read only (unmodifiable)variables -
+    1 nblock, ndim, kStep, kIncr, stepTime, totalTime, jUid,
+    2 amplitude, temp, curCoords, velocity, dirCos, jltyp, sname,
+C Write only (modifiable) variable -
+    1 value )
+C
+    include 'vaba_param.inc'
+C
+    dimension curCoords(nblock,ndim), velocity(nblock,ndim),
+    1 jUid(nblock), dirCos(nblock,ndim,ndim), temp(nblock),
+    2 value(nblock)
+    character*80 sname
+C
+    do 100 km = 1, nblock
+    user coding to define value 
+```
+
+<!-- source-page: 384 -->
+
+100 continue
+
+return end
+
+# Variable to be defined
+
+value(nblock)
+
+Magnitude of the distributed flux. Units are $\mathrm { J } \mathrm { T } ^ { - 1 } \mathrm { L } ^ { - 2 }$ for surface fluxes and $\mathrm { J } \mathrm { T } ^ { - 1 } \mathrm { L } ^ { - 3 }$ for body fluxes.
+
+# Variables passed in for information
+
+nblock
+
+Number of points to be processed in this call to VDFLUX.
+
+ndim
+
+Number of coordinate directions: 2 for two-dimensional models, 3 for three-dimensional models. The model is considered three-dimensional if any three-dimensional elements are defined.
+
+kStep
+
+Step number.
+
+kIncr
+
+Increment number.
+
+stepTime
+
+Value of time since the step began.
+
+totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime − stepTime.
+
+jUid
+
+User-defined element numbers.
+
+amplitude
+
+Current value of the amplitude referenced for this flux (set to unity if no amplitude is referenced). You must multiply the flux by the current amplitude value within the user subroutine if the amplitude is required.
+
+TEMP
+
+Current value of temperature at this integration point.
+
+curCoords(nblock, ndim)
+
+Current coordinates of each point for which the flux is to be prescribed.
+
+<!-- source-page: 385 -->
+
+# velocity(nblock, ndim)
+
+Current velocity of each point for which the flux is to be prescribed.
+
+# dirCos(nblock, ndim, ndim)
+
+Current orientation of the face or edge (not applicable for body flux type loads). The second dimension indicates the vector, and the third dimension indicates the components of that vector. For faces (surface fluxes on three-dimensional continuum and shell elements) the first and second vectors are the local directions in the plane of the surface and the third vector is the normal to the face, as defined in “Conventions,” Section 1.2.2 of the Abaqus Analysis User’s Guide. For solid elements the normal points inward, which is the opposite of what is defined in the conventions; for shell elements the normal definition is consistent with the defined conventions. For edges (fluxes on two-dimensional continuum elements) the first vector is the normal to the edge; the second vector is the tangent to the edge; and, if ndim=3, the third vector is a unit normal in the out-of-plane direction.
+
+# jltyp
+
+Key that identifies the distributed flux type. The load type may be a body flux, a surface-based flux, or an element-based surface flux. For element-based surface fluxes this variable identifies the element face for which this call to VDFLUX is being made. See Part VI, “Elements,” of the Abaqus Analysis User’s Guide, for element load type identification. This information is useful when several different nonuniform distributed loads are being imposed on an element at the same time. The key is as follows:
+
+<table><tr><td>jltyp</td><td>Load type</td></tr><tr><td>0</td><td>Surface-based load</td></tr><tr><td>1</td><td>BFNU</td></tr><tr><td>11</td><td>S1NU or SNEGNU</td></tr><tr><td>12</td><td>S2NU or SPOSNU</td></tr><tr><td>13</td><td>S3NU</td></tr><tr><td>14</td><td>S4NU</td></tr><tr><td>15</td><td>S5NU</td></tr><tr><td>16</td><td>S6NU</td></tr></table>
+
+# sname
+
+Surface name for a surface-based flux load definition (JLTYP=0). For a body flux or an element-based face load the surface name is passed in as a blank.
+
+<!-- source-page: 386 -->
+
+<!-- source-page: 387 -->
+
+# 1.2.2 VDISP: User subroutine to specify prescribed boundary conditions.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Boundary conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.3.1 of the Abaqus Analysis User’s Guide   
+• \*BOUNDARY   
+• “VDISP,” Section 4.1.28 of the Abaqus Verification Guide
+
+# Overview
+
+# User subroutine VDISP:
+
+• can be used to prescribe translational and rotational boundary conditions;   
+• is called for all degrees of freedom listed in the associated boundary condition;   
+• allows user to specify values for either the degree of freedom or its time derivatives such as velocity and acceleration;   
+• releases the boundary condition by default if the user does not specify a value for the boundary condition;   
+• can be used to apply a concentrated load, instead, by adjusting the default motion of the node;   
+• can be called for blocks of nodes for which the boundary conditions are defined in the subroutine.
+
+# Initial velocity
+
+At the beginning of each step user subroutine VDISP is called once to establish the initial velocity; and then, it is called once on each configuration, including the initial configuration, to establish the nodal acceleration.
+
+The first call to user subroutine VDISP is made to establish the initial velocity, which is indicated by the passing of a step time value of into the subroutine, where is the current time increment. If displacement is prescribed, the returned variable, rval, corresponds to $\boldsymbol { u _ { o } } \mathrm { ~ - ~ } \boldsymbol { v _ { o } } d t$ , where $u _ { o }$ and $v _ { o }$ are the initial displacement and velocity respectively. If velocity is prescribed, the returned variable corresponds to the initial velocity $v _ { o }$ . If acceleration is prescribed, the returned variable corresponds to $\frac { v _ { o } } { d t }$ where $v _ { o }$ is the initial velocity.
+
+The default value of rval is consistent with the velocity at the end of previous step or that specified as an initial condition in case of the first step. You only need to reset the rval if a different initial velocity is desired. The arrays u and v stand for the default initial displacement and velocity, respectively. The array a contains a zero value.
+
+<!-- source-page: 388 -->
+
+# Acceleration
+
+During time incrementation user subroutine VDISP is called once for each configuration, including the initial configuration, to establish the nodal acceleration.
+
+If displacement is prescribed, the returned variable should be set equal to the displacement at stepTime+dtNext, where stepTime is the step time and dtNext is the next time increment. If velocity is prescribed, the returned variable should be set equal to the mean velocity at stepTime+dtNext/2. If acceleration is prescribed, the returned variable should be set equal to the acceleration at stepTime. Note that stepTime is zero for the initial configuration.
+
+The variable rval has a default value that is computed as if the boundary condition is released. You only need to reset the rval if the boundary condition is active. The variable u contains values at stepTime. Whereas, the variable v contains initial velocity when stepTime is zero and, otherwise, velocity at stepTime—dt/2. The variable a contains values at stepTime computed as if the bondary condition is released.
+
+Tip: If you wish to apply a concentrated load, instead of the boundary condition, you can compute the change in acceleration due to this load and modify the rval value to account for that change. Note that the nodal mass and the rotary inertia are available in VDISP for computing the change in acceleration. Also, note that the default value of rval already reflects all other forces acting at the node.
+
+User subroutine interface   
+```fortran
+subroutine vdisp(
+c Read only variables -
+    1 nblock, nDof, nCoord, kstep, kinc,
+    2 time, totalTime, dtNext, dt,
+    3 cbname, jBCType, jDof, jNodeUid, amp,
+    4 coordNp, u, v, a, rf, rmass, rotaryI,
+c Write only variable -
+    5 rval)
+c
+    include 'vaba_param.inc'
+c
+    character*80 cbname
+    dimension jDof(nDof), jNodeUid(nblock),
+    1 amp(nblock), coordNp(nCoord,nblock),
+    2 u(nDof,nblock), v(nDof,nblock), a(nDof,nblock),
+    3 rf(nDof,nblock), rmass(nblock), rotaryI(3,3,nblock),
+    4 rval(nDof,nblock)
+c
+    do 100 k = 1, nblock 
+```
+
+<!-- source-page: 389 -->
+
+```txt
+do 100 j = 1, nDof
+    if( jDof(j) .gt. 0 ) then
+    user coding to define rval(j, k)
+    end if
+100 continue
+c
+    return
+    end 
+```
+
+# Variable to be defined
+
+rval(nDof, nblock)
+
+Values of the prescribed variable for degrees of freedom 1–6 (translation and rotation) at the nodes. The variable can be displacement, velocity, or acceleration, depending on the type specified in the associated boundary condition. The variable type is indicated by jBCType. The variable rval has a default value that is computed as if the boundary condition is released. You only need to reset the rval if the boundary condition is active.
+
+# Variables passed in for information
+
+nblock
+
+Number of nodal points to be processed in this call to VDISP.
+
+nDof
+
+Number of degrees of freedom (equals 6).
+
+nCoord
+
+Number of coordinate components (equals 3).
+
+kstep
+
+Step number.
+
+kinc
+
+Increment number.
+
+stepTime
+
+Value of time since the step began.
+
+totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime-stepTime.
+
+dtNext
+
+Next time increment size.
+
+dt
+
+Current time increment size.
+
+<!-- source-page: 390 -->
+
+# cbname
+
+User-specified name corresponding to the associated boundary condition.
+
+# jBCType
+
+Indicator for type of prescribed variable: 0 for displacement, 1 for velocity, and 2 for acceleration.
+
+# jDof(nDof)
+
+Indicator for prescribed degrees of freedom. The values given by rval(j,k) are prescribed only if jDof(j) equals 1.
+
+# jNodeUid(nblock)
+
+Node numbers.
+
+# amp(nblock)
+
+Amplitude values corresponding to the associated amplitude functions. These values are passed in for information only and will not contribute to the values of the prescribed variable automatically.
+
+# coordNp(nCoord, nblock)
+
+Nodal point coordinates.
+
+# u(nDof, nblock)
+
+Initial displacements when stepTime is negative, and, otherwise, displacement at stepTime. All translations are included if one or more translational degrees of freedom are prescribed. All rotations are included if one or more rotational degrees of freedom are prescribed.
+
+# v(nDof, nblock)
+
+Initial nodal velocities when stepTime is non-positive and, otherwise, mean velocities at stepTime-dt/2 during time incrementation. All translational velocities are included if one or more translational degrees of freedom are prescribed. All angular velocities are included if one or more rotational degrees of freedom are prescribed.
+
+# a(nDof, nblock)
+
+Contains a zero value when stepTime is negative and, otherwise, the accelerations, computed without accounting for the boundary condition, at stepTime. All translational accelerations are included if one or more translational degrees of freedom are prescribed. All angular accelerations are included if one or more rotational degrees of freedom are prescribed.
+
+# rf(nDof, nblock)
+
+Nodal point reaction at stepTime-dt. All reaction forces are included if one or more translational degrees of freedom are prescribed. All reaction moments are included if one or more rotational degrees of freedom are prescribed.
+
+# rmass(nblock)
+
+Nodal point masses.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_040.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_040.md
new file mode 100644
index 00000000..2fcaf3fa
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_040.md
@@ -0,0 +1,289 @@
+<!-- source-page: 391 -->
+
+rotaryI(3, 3, nblock)
+
+Nodal point rotary inertia.
+
+# Example: Imposition of acceleration on a rigid body with nonzero initial velocity
+
+In this example a sinusoidal acceleration is imposed on the reference node of a rigid body. Nonzero initial velocity is also specified for the rigid body. User subroutine VDISP given below illustrates how the return value array is to be computed for different phases of the solution. The analysis results show that both the initial velocity and acceleration are correctly specified.
+
+Input file   
+```csv
+*HEADING
+Test VDISP with S4R element
+*NODE, NSET=NALL
+1,
+2, 2., 0.
+3, 0., 2.
+4, 2., 2.
+9, 1., 1., 0.
+*ELEMENT, TYPE=S4R, ELSET=SHELL
+10, 1,2,4,3
+*SHELL SECTION, ELSET=SHELL, MATERIAL=ELSHELL
+2.0000000e-02, 3
+*MATERIAL, NAME=ELSHELL
+*DENSITY
+7850.0,
+*ELASTIC
+2.5000000e+11, 3.0000000e-01
+*RIGID BODY, REF NODE=9, ELSET=SHELL
+*INITIAL CONDITIONS, Type=VELOCITY
+9, 1, 0.4
+*STEP
+*DYNAMIC, EXPLICIT, DIRECT USER CONTROL
+0.01, 0.8
+*BOUNDARY, USER, TYPE=ACCELERATION
+9, 1
+*OUTPUT, HISTORY, TIME INTERVAL=0.01, OP=NEW
+*NODE OUTPUT, NSET=NALL
+U, V, A
+*END STEP 
+```
+
+<!-- source-page: 392 -->
+
+User subroutine   
+```fortran
+subroutine vdisp(
+c Read only variables -
+    * nblock, nDof, nCoord, kstep, kinc,
+    * steppTime, totalTime, dtNext, dt,
+    * cbname, jBCType, jDof, jNodeUid, amp,
+    * coordNp, u, v, a, rf, rmass, rotaryI,
+c Write only variable -
+    * rval )
+c
+    include 'vaba_param.inc'
+    parameter( zero = 0.d0, half = 0.5d0, one = 1.d0 )
+c
+    character*80 cbname
+    dimension jDof(nDof), jNodeUid(nblock),
+    * amp(nblock), coordNp(nCoord,nblock),
+    * u(nDof,nblock), v(nDof,nblock), a(nDof,nblock),
+    * rf(nDof,nblock), rmass(nblock),
+    * rotaryI(3,3,nblock), rval(nDof,nblock)
+c
+c    Impose acceleration
+c
+    if( jBCType .eq. 2 ) then
+c
+    if( steppTime .lt. zero ) then
+c
+c    Initialization 1
+c
+    do 310 k=1, nblock
+    do 310 j=1, nDof
+    if ( jDof(j) .gt. 0 ) then
+    v0 = v(j,k)
+    rval(j,k) = v0/dt
+    end if
+310    continue
+c
+    else
+c
+c    Time incrementation
+c
+    amplitude = 2.0 
+```
+
+<!-- source-page: 393 -->
+
+```fortran
+period = 0.8
+twopi = 6.2831853d0
+c
+do 350 k=1, nblock
+do 350 j=1, nDof
+    if (jDof(j).gt.0) then
+    rval(j,k) = amplitude*
+*    sin(twopi*stepTime / period)
+    end if
+350 continue
+    end if
+    end if
+c
+    return
+    end 
+```
+
+<!-- source-page: 394 -->
+
+<!-- source-page: 395 -->
+
+# 1.2.3 VDLOAD: User subroutine to specify nonuniform distributed loads.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Applying loads: overview,” Section 34.4.1 of the Abaqus Analysis User’s Guide   
+• “Distributed loads,” Section 34.4.3 of the Abaqus Analysis User’s Guide   
+• \*DLOAD   
+• \*DSLOAD   
+• “Deformation of a sandwich plate under CONWEP blast loading,” Section 9.1.9 of the Abaqus Example Problems Guide
+
+# Overview
+
+User subroutine VDLOAD:
+
+• can be used to define the variation of the distributed load magnitude as a function of position, time, velocity, etc. for a group of points, each of which appears in an element-based or surface-based nonuniform load definition;   
+• will be called for load integration points associated with each nonuniform load definition including PENU and PINU loads applicable for pipe elements;   
+• does not make available the current value of the nonuniform distributed loads for file output purposes; and   
+• recognizes an amplitude reference (“Amplitude curves,” Section 34.1.2 of the Abaqus Analysis User’s Guide) if it appears with the associated nonuniform load definition.
+
+# User subroutine interface
+
+```python
+subroutine vdload (
+C Read only (unmodifiable)variables -
+    1 nBlock, ndim, stepTime, totalTime,
+    2 amplitude, curCoords, velocity, dirCos, jltyp, sname,
+C Write only (modifiable) variable -
+    1 value )
+C
+    include 'vaba_param.inc'
+C
+    dimension curCoords(nBlock,ndim), velocity(nBlock,ndim),
+    1 dirCos(nBlock,ndim,ndim), value(nBlock)
+    character*80 sname
+C 
+```
+
+<!-- source-page: 396 -->
+
+do 100 km = 1, nBlock
+
+user coding to define value
+
+100 continue
+
+return
+
+end
+
+# Variable to be defined
+
+value (nBlock)
+
+Magnitude of the distributed load. Units are FL−2 for surface loads, FL−3 for body forces.
+
+# Variables passed in for information
+
+nBlock
+
+Number of points to be processed in this call to VDLOAD.
+
+ndim
+
+Number of coordinate directions: 2 for two-dimensional models, 3 for three-dimensional models. The model will be considered three-dimensional if any three-dimensional elements are defined (including SPRINGA elements).
+
+stepTime
+
+Value of time since the step began.
+
+totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime − stepTime.
+
+amplitude
+
+Current value of the amplitude referenced for this load (set to unity if no amplitude is referenced). You must multiply the load by the current amplitude value within the user subroutine if the amplitude is required.
+
+curCoords (nBlock, ndim)
+
+Current coordinates of each point for which the load is to be calculated.
+
+velocity (nBlock, ndim)
+
+Current velocity of each point for which the load is to be calculated.
+
+dirCos (nBlock, ndim, ndim)
+
+Current orientation of the face, edge, pipe, or beam for pressure type loads (not applicable for body force type loads). The second dimension indicates the vector, and the third dimension indicates the components of that vector. For faces (pressures on three-dimensional continuum, shell, and membrane elements), the first and second vectors are the local directions in the plane of the surface and the third
+
+<!-- source-page: 397 -->
+
+vector is the normal to the face, as defined in “Conventions,” Section 1.2.2 of the Abaqus Analysis User’s Guide. For solid elements the normal points inward, which is the opposite of what is defined in the conventions; for shell elements the normal definition is consistent with the defined conventions. For edges (pressures on two-dimensional continuum elements and two-dimensional beams and pipes), the first vector is the normal to the edge, the second vector is the tangent to the edge, and, if ndim=3, the third vector will be a unit normal in the out-of-plane direction. For three-dimensional beam and pipe elements, the first and second vectors are the local axes ( , ) and the third vector is the tangent vector ( ), as defined in “Beam element cross-section orientation,” Section 29.3.4 of the Abaqus Analysis User’s Guide.
+
+For a discrete element analysis using PD3D elements, the first column of the array is the radius of the element.
+
+# jltyp
+
+Key that identifies the distributed load type. The load type may be a body force, a surface-based load, or an element-based surface load. For element-based surface loads, this variable identifies the element face for which this call to VDLOAD is being made. See Part VI, “Elements,” of the Abaqus Analysis User’s Guide for element load type identification. This information is useful when several different nonuniform distributed loads are being imposed on an element at the same time. The key is as follows:
+
+<table><tr><td>Jltype</td><td>Load type</td></tr><tr><td>0</td><td>Surface-based load</td></tr><tr><td>1</td><td>BXNU</td></tr><tr><td>2</td><td>BYNU</td></tr><tr><td>3</td><td>BZNU</td></tr><tr><td>20</td><td>PNU</td></tr><tr><td>21</td><td>P1NU</td></tr><tr><td>22</td><td>P2NU</td></tr><tr><td>23</td><td>P3NU</td></tr><tr><td>24</td><td>P4NU</td></tr><tr><td>25</td><td>P5NU</td></tr><tr><td>26</td><td>P6NU</td></tr><tr><td>27</td><td>PINU</td></tr><tr><td>28</td><td>PENU</td></tr><tr><td>41</td><td>PXNU</td></tr></table>
+
+<!-- source-page: 398 -->
+
+<table><tr><td>Jltype</td><td>Load type</td></tr><tr><td>42</td><td>PYNU</td></tr><tr><td>43</td><td>PZNU</td></tr></table>
+
+# sname
+
+Surface name for a surface-based load definition (JLTYP=0). For a body force or an element-based load the surface name is passed in as a blank.
+
+<!-- source-page: 399 -->
+
+# 1.2.4 VEXTERNALDB: User subroutine that gives control to the user at key moments of the analysis so that data can be exchanged dynamically among Abaqus user subroutines and with external programs or files.
+
+Product: Abaqus/Explicit
+
+# Reference
+
+• “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide
+
+# Overview
+
+User subroutine VEXTERNALDB:
+
+• is called once at the beginning of the analysis, at the beginning of each step, before each increment, at the start of each increment, at the end of each increment, at the end of each step, and finally at the end of the analysis;   
+• can be used to communicate data between external programs and user subroutines within Abaqus/Explicit;   
+• can be used to control the time incrementation of the Abaqus/Explicit analysis;   
+• can be used to control the output of the restart data for the analysis;   
+• can be used either to skip the remainder of an Abaqus step or to terminate the analysis;   
+• can be used to open and close external files as needed for exchange of data with the Abaqus analysis;   
+• can be used to exchange data with other user subroutines via user-allocated global and thread-local arrays (see “Allocatable arrays,” Section 2.1.23) and;   
+• can be used to exchange data with other Abaqus processes via an MPI mechanism (see “Obtaining parallel processes information,” Section 2.1.4) in domain-parallel analyses.
+
+# Dynamic exchange of data with other Abaqus user subroutines and external programs
+
+Typically, Abaqus user subroutines are called with the context data limited to a specific material point, a specific element, etc. Rarely, you need to know some nonlocal information such as the state of the neighboring material points or elements. In other situations you want to specify the behavior in the user subroutines to depend dynamically on the external programs. Both these complex scenarios can be addressed using user subroutine VEXTERNALDB.
+
+VEXTERNALDB is called once at the beginning of the analysis, at the beginning of each step, before each increment, at the start of each increment, at the end of each increment, at the end of each step, and finally at the end of the analysis. Other Abaqus subroutines are called after the call to user subroutine VEXTERNALDB at the start of the increment but before the next call at the end of that increment.
+
+<!-- source-page: 400 -->
+
+User subroutine interface   
+```fortran
+subroutine vexternaldb(lOp, i_Array, niArray, r_Array, nrArray)
+C
+    include 'vaba_param.inc'
+C
+C Contents of i_Array
+    parameter( i_int_nTotalNodes    = 1,
+    *    i_int_nTotalElements    = 2,
+    *    i_int_kStep    = 3,
+    *    i_int_kInc    = 4,
+    *    i_int_iStatus    = 5,
+    *    i_int_lWriteRestart    = 6 )
+C Possible values for the lOp argument
+    parameter( j_int_StartAnalysis    = 0,
+    *    j_int_StartStep    = 1,
+    *    j_int_SetupIncrement    = 2,
+    *    j_int_StartIncrement    = 3,
+    *    j_int_EndIncrement    = 4,
+    *    j_int_EndStep    = 5,
+    *    j_int_EndAnalysis    = 6 )
+C Possible values for i_Array(i_int_iStatus)
+    parameter( j_int_Continue    = 0,
+    *    j_int_TerminateStep    = 1,
+    *    j_int_TerminateAnalysis    = 2 )
+C Contents of r_Array
+    parameter( i_flt_TotalTime    = 1,
+    *    i_flt_StepTime    = 2,
+    *    i_flt_dTime    = 3 )
+C dimension i_Array(niArray),
+    *    r_Array(nrArray)
+    kStep = i_Array(i_int_kStep)
+    kInc = i_Array(i_int_kInc) 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_041.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_041.md
new file mode 100644
index 00000000..3c5c4ff5
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_041.md
@@ -0,0 +1,339 @@
+<!-- source-page: 401 -->
+
+Note that you can use the MPI communication between parallel Abaqus processes to gather and scatter the data.
+
+C Start of the analysis if (lOp .eq. j\_int\_StartAnalysis) then
+
+User coding to set up the environment, open files, launch/connect to the external programs, etc.
+
+C continuation from a previous analysis (restart) if (kStep .ne. 0) then end if
+
+C Start of the step else if (lOp .eq. j\_int\_StartStep) then
+
+Set up or exchange (import and export) initial values with external programs.
+
+C The initial values may need to match those at the point of restart. if ( kInc .ne. 0) then end if
+
+C Setup the increment else if (lOp .eq. j\_int\_SetupIncrement) then Change i\_Array(i\_int\_lWriteRestart) and i\_Array(i\_int\_iStatus) if desired. Change r\_Array(i\_flt\_dTime) if desired.
+
+C Start of the increment else if (lOp .eq. j\_int\_StartIncrement) then
+
+The time increment is finalized. Use r\_Array(i\_flt\_dTime) if desired. If needed, gather and export data from the configuration at the end of the previous increment to external programs. Import and scatter data from external program to influence the current Abaqus increment.
+
+C End of the increment else if (lOp .eq. j\_int\_EndIncrement) then
+
+Change i\_Array(i\_int\_iStatus) if desired. Gather and export data from the configuration at the end of the current increment
+
+<!-- source-page: 402 -->
+
+to external programs.
+
+C End of the step else if (lOp .eq. j\_int\_EndStep) then
+
+In the case of multiple steps, prepare the transition to the next step. For example, these data can serve as initial values for the next step.
+
+C End of the analysis else if (lOp .eq. j\_int\_EndAnalysis) then
+
+User coding to close files and disconnect any external programs, etc.
+
+```lua
+end if
+return
+end 
+```
+
+# Variables to be defined
+
+None.
+
+# Variables that can be updated
+
+i\_Array(i\_int\_lWriteRestart)
+
+i\_Array(i\_int\_lWriteRestart) indicates whether restart data are currently scheduled to be written. When lOp=j\_int\_SetupIncrement, you can optionally modify it either to write restart data or to skip it. A value of 1 would capture the data for a possible future restart of the analysis from the current time point; whereas 0 would forego such restart from the current time point.
+
+i\_Array(i\_int\_iStatus)
+
+i\_Array(i\_int\_iStatus) indicates the status of the analysis and has a default value of j\_int\_Continue. When lOp=j\_int\_SetupIncrement or j\_int\_EndIncrement, you can optionally modify it either to a value of j\_int\_TerminateStep to skip the remainder of the current step or to a value of j\_int\_TerminateAnalysis to terminate the analysis. If you request to terminate the analysis, the analysis will go through one additional increment with a zero time increment size to generate the field output that reflects the state at termination, as described in “Abaqus/Explicit output as a result of analysis termination” in “Output to the output database,” Section 4.1.3 of the Abaqus Analysis User’s Guide. When the passed in value is not equal to j\_int\_Continue, you can coordinate the necessary events with any external program.
+
+r\_Array(i\_flt\_dTime)
+
+Time increment. When lOp=j\_int\_SetupIncrement, it is the time increment proposed for the current increment and it can be modified to control the incrementation. When
+
+<!-- source-page: 403 -->
+
+lOp=j\_int\_StartIncrement, it is the finalized time increment for the increment to be taken; whereas when lOp=j\_int\_EndIncrement, it is the time increment just taken.
+
+Variables passed in for information   
+i_Array(i_int_nTotalNodes)
+    Total number of nodes in the model.
+
+i_Array(i_int_nTotalElements)
+    Total number of elements in the model.
+
+i_Array(i_int_kStep)
+    Current step number. When lOp=j_int_StartAnalysis, i_Array(i_int_kStep) gives the restart step number.
+
+i_Array(i_int_kInc)
+    Current increment number. When lOp=j_int_StartStep, i_Array(i_int_kInc) gives the restart increment number.
+
+lOp
+    lOp=j_int_StartAnalysis indicates that the user subroutine is being called at the start of the analysis. A nonzero i_Array(i_int_kStep) indicates that the analysis is starting from a prior analysis (restart).
+    lOp=j_int_StartStep indicates that the user subroutine is being called at the start of a step. A nonzero i_Array(i_int_kInc) indicates a continuation of the step from a prior analysis (restart).
+    lOp=j_int_SetupIncrement indicates that the user subroutine is being called to set up an increment and r_Array(i_flt_dTime) can be modified. In addition, i_Array(i_int_lWriteRestart) can be modified to control output of restart data at the end of the current increment. You can also control the continuation of the analysis via i_Array(i_int_iStatus).
+    lOp=j_int_StartIncrement indicates that the user subroutine is being called at the start of the agreed increment. You need to import or compute the data necessary for starting the increment.
+    lOp=j_int_EndIncrement indicates that the user subroutine is being called at the end of the increment. If you have results to export, this is a good time to do so. You can also control the continuation of the analysis via i_Array(i_int_iStatus).
+    lOp=j_int_EndStep indicates that the user subroutine is being called at the end of the step.
+    lOp=j_int_EndAnalysis indicates that the user subroutine is being called at the end of the analysis.
+
+r_Array(i_flt_StepTime)
+    Value of current step time. When lOp=j_int_SetupIncrement or j_int_StartIncrement, the step time is at the start of the increment. When lOp=j_int_EndIncrement, the step time is at the end of the increment.
+
+<!-- source-page: 404 -->
+
+r\_Array(i\_flt\_TotalTime)   
+```txt
+Value of current total time. When lOp =j_int_SetupIncrement or j_int_StartIncrement, the total time is at the start of the increment. When lOp =j_int_EndIncrement, the total time is at the end of the increment. 
+```
+
+<!-- source-page: 405 -->
+
+# 1.2.5 VFABRIC: User subroutine to define fabric material behavior.
+
+# Product: Abaqus/Explicit
+
+WARNING: The use of this user subroutine generally requires considerable expertise. You are cautioned that the implementation of any realistic constitutive model requires extensive development and testing. Initial testing on a single-element model with prescribed traction loading is strongly recommended.
+
+# References
+
+• “Fabric material behavior,” Section 23.4.1 of the Abaqus Analysis User’s Guide   
+• \*FABRIC
+
+# Overview
+
+User subroutine VFABRIC:
+
+• is used to define the mechanical constitutive behavior of a fabric material in the plane of the fabric;   
+• is valid for materials that exhibit two “structural” directions that may not be orthogonal to each other with deformation;   
+• is used to update the nominal fabric stresses for a given nominal fabric strain where the direct strains are defined as the nominal strain measured along the two yarn directions of the fabric and the engineering shear strain is defined as the drop in the angle between the two yarn directions going from the reference configuration to the current configuration;   
+• can be used with elements under plane stress conditions;   
+• will be called for blocks of material calculation points for which the material is defined in a user subroutine (“Material data definition,” Section 21.1.2 of the Abaqus Analysis User’s Guide);   
+• can use and update solution-dependent state variables;   
+• can use any field variables that are passed in; and   
+• cannot be used in an adiabatic analysis.
+
+# Component ordering in tensors
+
+The component ordering depends upon whether the tensor is a “strain” variable or a “stress” variable.
+
+# Symmetric tensors
+
+Tensors such as the strain and strain increment have four components, and tensors such as stress have three components, with the difference between the two sets of variables arising from the assumed plane stress condition. The component order with the arrays for these variables is listed in the table below:
+
+<!-- source-page: 406 -->
+
+<table><tr><td>Component</td><td>Strain</td><td>Stress</td></tr><tr><td>1</td><td> $\varepsilon_{11}$ </td><td> $\sigma_{11}$ </td></tr><tr><td>2</td><td> $\varepsilon_{22}$ </td><td> $\sigma_{22}$ </td></tr><tr><td>3</td><td> $\varepsilon_{33}$ </td><td> $\sigma_{12}$ </td></tr><tr><td>4</td><td> $\varepsilon_{12}$ </td><td></td></tr></table>
+
+The shear strain components in user subroutine VFABRIC are stored as tensor components and not as engineering components.
+
+# Initial calculations and checks
+
+In the datacheck phase of the analysis Abaqus/Explicit calls user subroutine VFABRIC with a set of fictitious strains and a totalTime and stepTime that are both equal to 0.0. This step serves as a check on your constitutive relation and calculates the equivalent initial material properties, upon which the initial elastic wave speeds are computed.
+
+# Orientation of the fabric yarn
+
+In general, the yarn directions may not be orthogonal to each other in the reference configuration. You can specify these local directions with respect to the in-plane axes of an orthogonal orientation system at a material point. Both the local directions and the orthogonal system are defined together as a single orientation definition. If the local directions are not specified, these directions are assumed to match the in-plane axes of the orthogonal system. The local direction may not remain orthogonal with deformation. Abaqus updates the local directions with deformation and computes the nominal strains along these directions and the drop in angle between them (the fabric engineering shear strain). The constitutive behavior for the fabric defines the fabric nominal stresses as a function of the fabric strains. Abaqus converts these fabric stresses into the Cauchy stress and the resulting internal forces.
+
+# Material point deletion
+
+Material points that satisfy a user-defined failure criterion can be deleted from the model (see “Userdefined mechanical material behavior,” Section 26.7.1 of the Abaqus Analysis User’s Guide). You must specify the state variable number controlling the element deletion flag when you allocate space for the solution-dependent state variables, as explained in “Fabric material behavior,” Section 23.4.1 of the Abaqus Analysis User’s Guide. The deletion state variable should be set to a value of one or zero in user subroutine VFABRIC. A value of one indicates that the material point is active, while a value of zero indicates that Abaqus/Explicit should delete the material point from the model by setting the stresses to zero. The structure of the block of material points passed to user subroutine VFABRIC remains unchanged during the analysis; deleted material points are not removed from the block. Abaqus/Explicit will pass zero stresses and strain increments for all deleted material points. Once a material point has been flagged as deleted, it cannot be reactivated.
+
+<!-- source-page: 407 -->
+
+```fortran
+subroutine vfabric(
+C Read only (unmodifiable)variables -
+    1    nblock, ndim, npt, layer, kspt, kstep, kinc,
+    2    nstatev, nfieldv, nprops,
+    3    lOp, jElem, stepTime, totalTime, dt, cmname, coordMp,
+    4    charLength, props, density, braidAngle, fabricStrain,
+    5    fabricStrainInc,
+    6    tempOld, fieldOld, fabricStressOld, stateOld,
+    7    tempNew, fieldNew, enerIntern,
+C Write only (modifiable)variables -
+    8    fabricStressNew, stateNew, enerInelas )
+C
+C NOTE: In addition to the above "Write only" variables,
+C the thickness direction component of fabricStrainInc
+C i.e, fabricStrainInc(*,ndirStrain) may also be set by
+C the user for changing thickness as a function
+C of material in-plane state.
+C
+    include 'vaba_param.inc'
+C
+    parameter (ndirStrain = 3, nshr = 1, ndirStress = 2)
+C
+C NOTE: The constants defined above are used for array
+C dimensions below.
+C
+    dimension
+    *    jElem(nblock),
+    *    coordMp(nblock,ndim),
+    *    charLength(nblock),
+    *    props(nprops),
+    *    density(nblock),
+    *    braidAngle(nblock),
+    *    fabricStrain(nblock,ndirStrain+nshr),
+    *    strainFabricInc(nblock,ndirStrain+nshr),
+    *    tempOld(nblock),
+    *    fieldOld(nblock,nfieldv),
+    *    fabricStressOld(nblock,ndirStress+nshr),
+    *    stateOld(nblock,nstatev),
+    *    tempNew(nblock), 
+```
+
+<!-- source-page: 408 -->
+
+```txt
+* fieldNew(nblock, nfieldv),
+* fabricStressNew(nblock, ndirStress+nshr),
+* stateNew(nblock, nstatev),
+* enerIntern(nblock),
+* enerInelas(nblock)
+*
+character*80 cmname
+C
+do 100 km = 1, nblock
+user coding
+100 continue
+return
+end 
+```
+
+# Variables to be defined
+
+fabricStressNew(nblock,ndirStress+nshr)
+
+Nominal fabric stress at each material point at the end of the increment. This nominal fabric stress can be requested as output variable SFABRIC.
+
+stateNew(nblock,nstatev)
+
+State variables at each material point at the end of the increment. You define the size of this array by allocating space for it (see “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide, for more information). This variable can be requested as output variable SDV.
+
+enerInelas(nblock)
+
+Total inelastic energy density at material points at the end of the increment. This variable can be requested as output variable ENER.
+
+# Variable that can be updated
+
+fabricStrainInc(\*,ndirStrain)
+
+Thickness direction strain increment. The thickness can be requested as output variable STH.
+
+# Variables passed in for information
+
+nblock
+
+Number of material points to be processed in this call to VFABRIC.
+
+ndim
+
+Two for a two-dimensional model and three for a three-dimensional model.
+
+Current integration point number.
+
+<!-- source-page: 409 -->
+
+# layer
+
+Current layer number in the case of a composite section.
+
+# kspt
+
+Current material point number within the section.
+
+# kStep
+
+Current Abaqus step number.
+
+# kInc
+
+Increment number of the current Abaqus step.
+
+# nstatev
+
+Number of user-defined state variables that are associated with this material type (you define this as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# nfieldv
+
+Number of user-defined external field variables.
+
+# nprops
+
+User-specified number of user-defined material properties.
+
+# lOp
+
+Integer flag indicating the computation that is expected. lOp = −2 indicates that the routine is being called to initialize the stresses corresponding to the initial strains, which can be large. lOp = −1 indicates that the routine is being called to update the stresses based on the instantaneous elastic response for a small “artificial” strain increment given. lOp = 0 indicates that this is an annealing process and you should reinitialize the internal state variables, stateNew, if necessary. The stresses will be set to zero by Abaqus. lOp = 1 indicates that the routine is being called to update the stresses and the state for a given strain increment.
+
+# jElem(nblock)
+
+Array of element numbers.
+
+# stepTime
+
+Value of time since the step began.
+
+# totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime - stepTime.
+
+# dt
+
+Time increment size.
+
+<!-- source-page: 410 -->
+
+# cmname
+
+User-specified material name, left justified. It is passed in as an uppercase character string. Some internal material models are given names starting with the character string “ABQ\_”. To avoid conflict, you should not use “ABQ\_” as the leading string for cmname.
+
+# coordMp(nblock,\*)
+
+Material point coordinates. It is the midplane material point for shell elements and the centroid for beam elements.
+
+# charLength(nblock)
+
+Characteristic element length, which is either the default value based on the geometric mean or the user-defined characteristic element length defined in user subroutine VUCHARLENGTH. The default value is a typical length of a line across an element for a first-order element; it is half of the same typical length for a second-order element. For membranes and shells the default value is a characteristic length in the reference surface.
+
+# props(nprops)
+
+User-supplied material properties.
+
+# density(nblock)
+
+Current density at the material points in the midstep configuration. This value may be inaccurate in problems where the volumetric strain increment is very small. If an accurate value of the density is required in such cases, the analysis should be run in double precision. This value of the density is not affected by mass scaling.
+
+# braidAngle(nblock)
+
+Angle in radians between the two yarn directions at the end of the increment.
+
+# fabricStrain(nblock,ndirStrain+nshr)
+
+Total nominal strain in the fabric at the end of increment. This variable can be requested as output variable EFABRIC.
+
+# fabricStrainInc(nblock,ndirStrain+nshr)
+
+Incremental nominal strain in the fabric.
+
+# tempOld(nblock)
+
+Temperatures at each material point at the beginning of the increment.
+
+# fieldOld(nblock,nfieldv)
+
+Values of the user-defined field variables at each material point at the beginning of the increment.
+
+# fabricStressOld(nblock,ndirStress+nshr)
+
+Nominal fabric stress at each material point at the beginning of the increment.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_042.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_042.md
new file mode 100644
index 00000000..c76f92f1
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_042.md
@@ -0,0 +1,385 @@
+<!-- source-page: 411 -->
+
+stateOld(nblock,nstatev)
+
+State variables at each material point at the beginning of the increment.
+
+tempNew(nblock)
+
+Temperatures at each material point at the end of the increment.
+
+fieldNew(nblock,nfieldv)
+
+Values of the user-defined field variables at each material point at the end of the increment.
+
+enerIntern(nblock)
+
+Internal energy per unit mass at each material point at the beginning of the increment.
+
+# Example: Using more than one user-defined material model
+
+To use more than one user-defined fabric material model, the variable cmname can be tested for different fabric material names inside user subroutine VFABRIC, as illustrated below:
+
+```txt
+if (cmname(1:4) .eq. 'MAT1') then
+    call VFABRIC_MAT1(argument_list)
+else if (cmname(1:4) .eq. 'MAT2') then
+    call VFABRIC_MAT2(argument_list)
+end if 
+```
+
+VFABRIC\_MAT1 and VFABRIC\_MAT2 are the actual fabric material user subroutines containing the constitutive material models for each material MAT1 and MAT2, respectively. User subroutine VFABRIC merely acts as a directory here. The argument list can be the same as that used in subroutine VFABRIC. The material names must be in uppercase characters since cmname is passed in as an uppercase character string.
+
+# Example: Influence of nonorthogonal material directions in highly anisotropic elastic material
+
+As an example of the coding of user subroutine VFABRIC, consider a simple elastic lamina material with highly anisotropic properties. For a fabric the material definitions need not remain orthogonal with deformation, whereas the directions do remain orthogonal for a built-in elastic material. The simple VFABRIC routine given below defines an elastic fabric and can be used to compare the fabric and the built-in elastic materials under different loading conditions.
+
+The user subroutine would be coded as follows:
+
+```csv
+subroutine vfabric(
+C Read only (unmodifiable)variables -
+1 nblock, ndim, npt, layer, kspt, kstep, kinc,
+2 nstatev, nfieldv, nprops,
+3 lOp, jElem, stepTime, totalTime, dt, cmname, coordMp,
+4 charLength, props, density, braidAngle, fabricStrain,
+5 fabricStrainInc, 
+```
+
+<!-- source-page: 412 -->
+
+```txt
+6 tempOld, fieldOld, fabricStressOld, stateOld,
+7 tempNew, fieldNew, enerIntern,
+C Write only (modifiable) variables -
+8 fabricStressNew, stateNew, enerInelas )
+C
+C NOTE: In addition to the above "Write only" variables,
+C the thickness direction component of fabricStrainInc
+C i.e, fabricStrainInc(*,ndirStrain) may also be set by the user
+C for changing thickness as a function of material in-plane
+C state.
+C
+    include 'vaba_param.inc'
+C
+    parameter( ndirStrain = 3, nshr = 1, ndirStress = 2,
+    one = 1.d0, two = 2.d0 )
+C
+C NOTE: The constants defined above are used for array
+C dimensions and computation below.
+C
+    dimension
+    * jElem(nblock),
+    * coordMp(nblock,ndim),
+    * charLength(nblock),
+    * props(nprops),
+    * density(nblock),
+    * braidAngle(nblock),
+    * fabricStrain(nblock,ndirStrain+nshr),
+    * fabricStrainInc(nblock,ndirStrain+nshr),
+    * tempOld(nblock),
+    * fieldOld(nblock,nfieldv),
+    * fabricStressOld(nblock,ndirStress+nshr),
+    * stateOld(nblock,nstatev),
+    * tempNew(nblock),
+    * fieldNew(nblock,nfieldv),
+    * fabricStressNew(nblock,ndirStress+nshr),
+    * stateNew(nblock,nstatev),
+    * enerIntern(nblock),
+    * enerInelas(nblock)
+C
+    character*80 cmname
+C 
+```
+
+<!-- source-page: 413 -->
+
+```python
+C
+C Read properties
+    E1 = props(1)
+    E2 = props(2)
+    xnu12 = props(3)
+    twiceG12 = two * props(4)
+C
+    xnu21 = E2 * xnu12 / E1
+C
+C Let us assume:
+    xnu13 = xnu12
+    xnu23 = xnu21
+C
+    xnu13OverE1 = xnu13/E1
+    xnu23OverE2 = xnu23/E2
+C
+    fr = one / (one - xnu12 * xnu21)
+    D11 = E1 * fr
+    D22 = E2 * fr
+    D12 = E2 * xnu12 * fr
+C
+    do k = 1, nblock
+C
+C Update the stress
+    stressInc11 = D11 * fabricStrainInc(k,1)
+    * + D12 * fabricStrainInc(k,2)
+    stressInc22 = D22 * fabricStrainInc(k,2)
+    * + D12 * fabricStrainInc(k,1)
+    stressInc12 = twiceG12 *
+    * fabricStrainInc(k,ndirStrain + 1)
+C
+    fabricStressNew(k,1) = fabricStressOld(k,1)
+    * + stressInc11
+    fabricStressNew(k,2) = fabricStressOld(k,2)
+    * + stressInc22
+C
+C    shear stress
+    fabricStressNew(k,ndirStress+1) =
+    * fabricStressOld(k,ndirStress+1) + stressInc12
+C
+C    Thickness direction strain 
+```
+
+<!-- source-page: 414 -->
+
+```prolog
+c
+fabricStrainInc(k,ndirStrain) =
+C * - ( xnu13OverE1 * stressInc11
+C * + xnu23OverE2 * stressInc22
+C *
+C
+end do
+return
+end 
+```
+
+<!-- source-page: 415 -->
+
+# 1.2.6 VFRIC: User subroutine to define frictional behavior for contact surfaces.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Frictional behavior,” Section 37.1.5 of the Abaqus Analysis User’s Guide   
+• \*FRICTION   
+• “VFRIC, VFRIC\_COEF, and VFRICTION,” Section 4.1.30 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine VFRIC:
+
+• can be used to define the frictional behavior between contact pair surfaces;   
+• can be used when the classical Coulomb friction model is too restrictive and a more complex definition of shear transmission between contacting surfaces is required;   
+• must provide the entire definition of shear interaction between the contacting surfaces;   
+• can use and update solution-dependent state variables;   
+• cannot be used in conjunction with softened tangential surface behavior; and   
+• cannot be used with the general contact algorithm.
+
+# Terminology
+
+The use of user subroutine VFRIC requires familiarity with the following terminology.
+
+# Surface node numbers
+
+The “surface node number” refers to the position of a particular node in the list of nodes on the surface. For example, there are nSlvNod nodes on the slave surface. Number nSlvNod, is the surface node number of the nth node in this list; jSlvUid is the user-defined global number of this node. An Abaqus/Explicit model can be defined in terms of an assembly of part instances (see “Defining an assembly,” Section 2.10.1 of the Abaqus Analysis User’s Guide). In such models a node number in jSlvUid is an internally generated node number. If the original node number and part instance name are required, call the utility routine VGETPARTINFO (see “Obtaining part information,” Section 2.1.5).
+
+# Contact points
+
+The nodes on the slave surface that are in contact in the current time increment are defined as “contact points.” The number of contact points is passed into this subroutine as nContact. The array jConSlvid(nContact) gives the surface node numbers for the contact points.
+
+<!-- source-page: 416 -->
+
+A local coordinate system is defined for each contact point to facilitate specification of frictional forces and incremental slips. The local 1-direction for both two-dimensional and three-dimensional contact is tangential to the master surface, and it is defined by $\mathbf { t } _ { 1 } ~ = ~ d \mathbf { s } / | d \mathbf { s } |$ , where is the incremental slip vector. The incremental slip vector used to define $\mathbf { t } _ { 1 }$ corresponds to the incremental slip in the current time increment for penalty contact and the predicted incremental slip for kinematic contact. The master surface normal direction, $\mathbf { n } ,$ is the local 2-direction for two-dimensional contact and the local 3-direction for three-dimensional contact. The local 2-direction for three-dimensional contact is given by $\mathbf { t } _ { 2 } = \mathbf { n } \times \mathbf { t } _ { 1 }$ , which is also tangent to the master surface. The vectors are shown in Figure 1.2.6–1 and Figure 1.2.6–2. The direction cosines for $\mathbf { t } _ { 1 }$ and with respect to the global coordinate system are available in dirCosT1 and dirCosN, respectively. In the case of zero incremental slip $( d \mathbf { s } | = 0 )$ we choose an arbitrary direction for $\mathbf { t } _ { 1 }$ that is orthogonal to the normal direction, .
+
+![](images/page-416_1b073b703d46fac241a31506484f9ae0fc24cd05a0531fdd51f8530f17fa3a07.jpg)
+
+<details>
+<summary>text_image</summary>
+
+surface normal
+slave node
+master surface
+n
+ds
+t₁
+slip direction
+incremental frictional slip
+</details>
+
+Figure 1.2.6–1 Local coordinate system for two-dimensional contact with VFRIC.
+
+# Frictional forces
+
+You specify the frictional force, fTangential, at each contact point in local coordinates in this subroutine. The array fTangential is dimensioned such that only the tangential components can be specified. Any components of the frictional force that are not specified will remain equal to zero. For three-dimensional contact with isotropic friction, only the first component of the frictional force need be specified since the second component should be zero. A “stick force” at each contact point is provided in the array fStickForce to assist you in setting the appropriate frictional force values. The stick force is the force required to prevent additional “plastic” slipping. The stick force at each contact point is provided as a scalar value as it would act in the direction opposite to $\mathbf { t } _ { 1 }$ . The stick force is computed prior to calling user subroutine VFRIC by either the kinematic or the penalty contact algorithm. See “Contact constraint enforcement methods in Abaqus/Explicit,” Section 38.2.3 of the Abaqus Analysis User’s Guide, for descriptions of the kinematic and penalty contact algorithms and the user interface for choosing between them. The first component of the frictional force should be in the range between
+
+<!-- source-page: 417 -->
+
+![](images/page-417_ecbb54cd0740af82f1db19d47d13d68d75f3d07909b05f56047ad61469d075ac.jpg)
+
+<details>
+<summary>text_image</summary>
+
+surface normal
+slave node
+master surface
+n
+ts
+ds
+t1
+slip directions
+incremental frictional slip
+</details>
+
+Figure 1.2.6–2 Local coordinate system for three-dimensional contact with VFRIC.
+
+zero and minus the stick force value. Typically, the stick force will be positive and the first component of the applied frictional force will be negative, opposing the incremental slip. Penalty contact includes an elastic slip regime due to finite penalty stiffness, so occasionally, during recovery of elastic slip, the stick force will be negative, indicating that it is appropriate for the first component of the frictional force to be positive (i.e., acting in the same direction as the incremental slip). A noisy or unstable solution is likely to result if the first component of fTangential is set outside of the range between zero and negative the value of the stick force.
+
+After user subroutine VFRIC is called, frictional forces that oppose the forces specified at the contact points are distributed to the master nodes. For balanced master-slave contact we then compute weighted averages of the frictional forces for both master-slave orientations. These forces are directly applied if the penalty contact algorithm is being used. If the kinematic contact algorithm is being used, the frictional forces are converted to acceleration corrections by dividing by the nodal masses.
+
+User subroutine interface   
+```txt
+subroutine vfric(
+C Write only -
+    1 fTangential,
+C Read/Write -
+    2 statev,
+C Read only -
+    3 kStep, kInc, nContact, nFacNod, nSlvNod, nMstNod,
+    4 nFricDir, nDir, nStateVar, nProps, nTemp, nPred, numDefTfv,
+    5 jSlvUid, jMstUid, jConSlvid, jConMstid, timStep, timGlb, 
+```
+
+<!-- source-page: 418 -->
+
+```txt
+6 dTimCur, surfInt, surfSlv, surfMst, lContType,
+7 dSlipFric, fStickForce, fTangPrev, fNormal, frictionWork,
+8 shape, coordSlv, coordMst, dirCosSl, dircosN, props,
+9 areaSlv, tempSlv, preDefSlv, tempMst, preDefMst)
+C
+    include `vaba_param.inc'
+C
+    character*80 surfInt, surfSlv, surfMst
+C
+    dimension props(nProps), statev(nStateVar,nSlvNod),
+    1 dSlipFric(nDir,nContact),
+    2 fTangential(nFricDir,nContact),
+    3 fTangPrev(nDir,nContact),
+    4 fStickForce(nContact), areaSlv(nSlvNod),
+    5 fNormal(nContact), shape(nFacNod,nContact),
+    6 coordSlv(nDir,nSlvNod), coordMst(nDir,nMstNod),
+    7 dirCosSl(nDir,nContact), dircosN(nDir,nContact),
+    8 jSlvUid(nSlvNod), jMstUid(nMstNod),
+    9 jConSlvid(nContact), jConMstid(nFacNod,nContact)
+    1 tempSlv(nContact), preDefSlv(nContact,nPred),
+    2 tempMst(numDefTfv), preDefMst(numDefTfv,nPred)
+
+    user coding to define fTangential
+    and, optionally, statev
+
+    return
+    end 
+```
+
+# Variable to be defined
+
+# fTangential(nFricDir, nContact)
+
+This array must be updated to the current values of the frictional force components for all contact points in the local tangent directions. See Figure 1.2.6–1 and Figure 1.2.6–2 for definition of the local coordinate system. This array will be zero (no friction force) until you reset it.
+
+# Variable that can be updated
+
+# statev(nstateVar, nSlvNod)
+
+This array contains the user-defined solution-dependent state variables for all the nodes on the slave surface. You define the size of this array (see “Frictional behavior,” Section 37.1.5 of the Abaqus Analysis User’s Guide, for more information). This array will be passed in containing the values of these variables prior to the call to user subroutine VFRIC. If any of the solution-dependent state variables is being used in conjunction with the friction behavior, it must be updated in this subroutine.
+
+<!-- source-page: 419 -->
+
+The state variables are available even for slave nodes that are not in contact. This may be useful when, for example, the state variables need to be reset for slave nodes that are not in contact.
+
+Variables passed in for information   
+kStep
+Step number.
+
+kInc
+Increment number.
+
+nContact
+Number of contacting slave nodes.
+
+nFacNod
+Number of nodes on each master surface facet (nFacNod is 2 for two-dimensional surfaces, nFacNod is 4 for three-dimensional surfaces). If the master surface is an analytical rigid surface, this variable is passed in as 0.
+
+nSlvNod
+Number of slave nodes.
+
+nMstNod
+Number of master surface nodes, if the master surface is made up of facets. If the master surface is an analytical rigid surface, this variable is passed in as 0.
+
+nFricDir
+Number of tangent directions at the contact points (nFricDir = nDir - 1).
+
+nDir
+Number of coordinate directions at the contact points. (In a three-dimensional model nDir will be two if the surfaces in the contact pair are two-dimensional analytical rigid surfaces or are formed by two-dimensional elements.)
+
+nStateVar
+Number of user-defined state variables.
+
+nProps
+User-specified number of property values associated with this friction model.
+
+nTemp
+1 if the temperature is defined and 0 if the temperature is not defined.
+
+nPred
+Number of predefined field variables.
+
+<!-- source-page: 420 -->
+
+# numDefTfv
+
+Equal to nContact if the master surface is made up of facets. If the master surface is an analytical rigid surface, this variable is passed in as 1.
+
+# jSlvUid(nSlvNod)
+
+This array lists the user-defined global node numbers (or internal node numbers for models defined in terms of an assembly of part instances) of the nodes on the slave surface.
+
+# jMstUid(nMstNod)
+
+This array lists the user-defined global node numbers (or internal node numbers for models defined in terms of an assembly of part instances) of the nodes on the master surface. If the master surface is an analytical rigid surface, this array is passed in as a dummy array.
+
+# jConSlvid(nContact)
+
+This array lists the surface node numbers of the slave surface nodes that are in contact.
+
+# jConMstid(nFacNod, nContact)
+
+This array lists the surface node numbers of the master surface nodes that make up the facet with which each contact point is in contact. If the master surface is an analytical rigid surface, this array is passed in as a dummy array.
+
+# timStep
+
+Value of step time.
+
+# timGlb
+
+Value of total time.
+
+# dtimCur
+
+Current increment in time from $t = t _ { c u r r } - \Delta t$ to .
+
+# surfInt
+
+User-specified surface interaction name, left justified.
+
+# surfSlv
+
+Slave surface name.
+
+# surfMst
+
+Master surface name.
+
+# lContType
+
+Contact type flag. This flag is set based on the type of constraint enforcement method (see “Contact constraint enforcement methods in Abaqus/Explicit,” Section 38.2.3 of the Abaqus Analysis User’s Guide) being used: 1 for kinematic contact and 2 for penalty contact. Stick conditions are satisfied exactly with the kinematic contact algorithm; they are satisfied only approximately (subject to an automatically chosen penalty stiffness value) with the penalty contact algorithm.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_043.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_043.md
new file mode 100644
index 00000000..9381f5a3
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_043.md
@@ -0,0 +1,364 @@
+<!-- source-page: 421 -->
+
+# dSlipFric(nDir, nContact)
+
+This array contains the incremental frictional slip during the current time increment for each contact point in the current local coordinate system. These incremental slips correspond to tangential motion in the time increment from $t = t _ { c u r r } - \Delta t$ to $t = t _ { c u r r }$ . For penalty contact this incremental slip is used to define the local coordinate system at each contact point (see Figure 1.2.6–1 and Figure 1.2.6–2) so that only the first component of dSlipFric can be nonzero in the local system. The contact points for kinematic contact are determined based on penetrations detected in the predicted configuration (at $t = t _ { c u r r } + \Delta t )$ , and the predicted incremental slip direction is used to define the local coordinate system at each contact point. If the slip direction changes between increments, dSlipFric may have a nonzero component in the local 2-direction and, if the surface is faceted and the contact point moves from one facet to another, in the local 3-direction.
+
+# fStickForce(nContact)
+
+This array contains the magnitude of frictional force required to enforce stick conditions at each contact point. For kinematic contact this force corresponds to no slip; for penalty contact this force depends on the previous frictional force, the value of the penalty stiffness, and the previous incremental slip. The penalty stiffness is assigned automatically. Occasionally, during recovery of elastic slip associated with the penalty method, the stick force will be assigned a negative value.
+
+# fTangPrev(nDir, nContact)
+
+This array contains the values of the frictional force components calculated in the previous increment but provided in the current local coordinate system (zero for nodes that were not in contact).
+
+# fNormal(nContact)
+
+This array contains the magnitude of the normal force for the contact points applied at the end of current time increment; i.e., at time $t = t _ { c u r r }$ .
+
+# frictionWork
+
+This variable contains the value of the total frictional dissipation in the entire model from the beginning of the analysis. The units are energy per unit area.
+
+# shape(nFacNod, nContact)
+
+For each contact point this array contains the shape functions of the nodes of its master surface facet, evaluated at the location of the contact point. If the master surface is an analytical rigid surface, this array is passed in as a dummy array.
+
+# coordSlv(nDir, nSlvNod)
+
+Array containing the nDir components of the current coordinates of the slave nodes.
+
+# coordMst(nDir, nMstNod)
+
+Array containing the nDir components of the current coordinates of the master nodes. If the master surface is an analytical rigid surface, this array is passed in as a dummy array.
+
+<!-- source-page: 422 -->
+
+dirCosSl(nDir, nContact)
+
+Direction cosines of the incremental slip at the contact points.
+
+dircosN(nDir, nContact)
+
+Direction cosines of the normals to the master surface at the contact points.
+
+props(nProps)
+
+User-specified vector of property values to define the frictional behavior between the contacting surfaces.
+
+areaSlv(nSlvNod)
+
+Area associated with the slave nodes (equal to 1 for node-based surface nodes).
+
+tempSlv(nContact)
+
+Current temperature at the slave nodes.
+
+preDefSlv(nContact,nPred)
+
+Current user-specified predefined field variables at the slave nodes (initial values at the beginning of the analysis and current values during the analysis).
+
+tempMst(numDefTfv)
+
+Current temperature at the nearest points on the master surface.
+
+preDefMst(numDefTfv,nPred)
+
+Current user-specified predefined field variables at the nearest points on the master surface (initial values at the beginning of the analysis and current values during the analysis).
+
+<!-- source-page: 423 -->
+
+# 1.2.7 VFRIC\_COEF: User subroutine to define the frictional coefficient for contact surfaces.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Frictional behavior,” Section 37.1.5 of the Abaqus Analysis User’s Guide   
+• \*FRICTION   
+• “VFRIC, VFRIC\_COEF, and VFRICTION,” Section 4.1.30 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine VFRIC\_COEF:
+
+• can be used to define the isotropic frictional coefficient between contacting surfaces;   
+• corresponds to the classical Coulomb friction model; and   
+• can be used only with the general contact algorithm.
+
+# User subroutine interface
+
+```txt
+subroutine vfric_coef (
+C Write only -
+* fCoef, fCoefDeriv,
+C Read only -
+* nBlock, nProps, nTemp, nFields,
+* jFlags, rData,
+* surfInt, surfSlv, surfMst,
+* props, slipRate, pressure,
+* tempAvg, fieldAvg)
+C
+include 'vaba_param.inc'
+C
+dimension fCoef(nBlock),
+* fCoefDeriv(nBlock,3),
+* props(nProps),
+* slipRate(nBlock),
+* pressure(nBlock),
+* tempAvg(nBlock),
+* fieldAvg(nBlock,nFields)
+C
+parameter( iKStep    = 1, 
+```
+
+<!-- source-page: 424 -->
+
+```matlab
+* iKInc = 2,
+* nFlags = 2 )
+C
+    parameter( iTimStep = 1,
+    * iTimGlb = 2,
+    * iDTimCur = 3,
+    * nData = 3 )
+C
+    dimension jFlags(nFlags), rData(nData)
+C
+    character*80 surfInt, surfSlv, surfMst
+C
+    user coding to define fCoef
+C
+    return
+    end 
+```
+
+# Variables to be defined
+
+fCoef(nBlock)
+
+This array must be updated to the current values of the friction coefficient for all contacting points.
+
+fCoefDeriv(nBlock,3)
+
+This array is not applicable to Abaqus/Explicit analyses.
+
+# Variables passed in for information
+
+nBlock
+
+Number of contacting points to be processed in this call to VFRIC\_COEF.
+
+nProps
+
+User-specified number of property values associated with this friction model.
+
+nTemp
+
+1 if the temperature is defined and 0 if the temperature is not defined.
+
+nFields
+
+Number of user-specified field variables.
+
+jFlag(1)
+
+Step number.
+
+jFlag(2)
+
+Increment number.
+
+<!-- source-page: 425 -->
+
+rData(1)
+
+Value of step time.
+
+rData(2)
+
+Value of total time.
+
+rData(3)
+
+Current increment in time from $t = t _ { c u r r } - \Delta t$ to .
+
+surfInt
+
+User-specified surface interaction name, left justified.
+
+surfSlv
+
+Slave surface name, not applicable to general contact.
+
+surfMst
+
+Master surface name, not applicable to general contact.
+
+props(nProps)
+
+User-specified vector of property values to define the frictional coefficient at contacting points.
+
+slipRate(nBlock)
+
+This array contains the rate of tangential slip at the contacting points for the current time increment.
+
+pressure(nBlock)
+
+This array contains the pressure at the contacting points applied at the end of the current time increment.
+
+tempAvg(nBlock)
+
+Average current temperature between the master and slave surfaces at the contacting points.
+
+fieldAvg(nBlock,nFields)
+
+Average current value of all the user-specified field variables between the master and slave surfaces at the contacting points.
+
+<!-- source-page: 426 -->
+
+<!-- source-page: 427 -->
+
+# 1.2.8 VFRICTION: User subroutine to define frictional behavior for contact surfaces.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Frictional behavior,” Section 37.1.5 of the Abaqus Analysis User’s Guide   
+• \*FRICTION   
+• “VFRIC, VFRIC\_COEF, and VFRICTION,” Section 4.1.30 of the Abaqus Verification Guide
+
+# Overview
+
+# User subroutine VFRICTION:
+
+• can be used to define the frictional behavior between contacting surfaces;   
+• can be used when the classical Coulomb friction model is too restrictive and a more complex definition of shear transmission between contacting surfaces is required;   
+• must provide the entire definition of shear interaction between the contacting surfaces;   
+• can use and update solution-dependent state variables for node-to-face and node-to-analytical rigid surface contact;   
+• cannot be used in conjunction with softened tangential surface behavior; and   
+• can be used only with the general contact algorithm.
+
+# Contact points
+
+The points considered in user subroutine VFRICTION are called contact points. Each contact point is primarily associated with a slave node or a point along a slave edge; the contact point also references the corresponding master surface that it contacts. A contact point exists for each pairing of slave node and master surface. Therefore, more than one contact point may reference the same slave node but different master surfaces, such as with contact at a corner.
+
+The number of contact points currently being passed into user subroutine VFRICTION is nBlock. The array jConSlvUid(nNodSlv,nBlock) gives the slave surface node numbers associated with the contact points. The variable nNodSlv indicates whether a single slave node (for node-to-face contact) or two slave nodes of an edge (for edge-to-edge contact) are associated with each contact point. Similarly, the array jConMstUid(nNodMst,nBlockAnal) gives the master surface node numbers associated with each contact point; the nodes can belong to a facet, an edge, or an analytical surface. The variable nNodMst indicates the number of master nodes associated with each contact point.
+
+An Abaqus/Explicit model can be defined in terms of an assembly of part instances (see “Defining an assembly,” Section 2.10.1 of the Abaqus Analysis User’s Guide). In such models a node number is an internally generated node number. If the original node number and part instance name are required, call the utility routine VGETPARTINFO (see “Obtaining part information,” Section 2.1.5).
+
+<!-- source-page: 428 -->
+
+# Local coordinate system
+
+A local coordinate system is defined for each contact point to facilitate specification of frictional forces and incremental slip. The local 1-direction is tangential to the master surface; it is defined by $\mathbf { t } _ { 1 } ~ =$ $d \mathbf { s } / | d \mathbf { s } |$ , where is the incremental slip vector. The incremental slip vector used to define $\mathbf { t } _ { 1 }$ corresponds to the incremental slip in the current time increment. The master surface normal direction, $\mathbf { n } ,$ is the local 3-direction. The local 2-direction is given by $\mathbf { t } _ { 2 } = \mathbf { n } \times \mathbf { t } _ { 1 }$ , which is also tangent to the master surface. The vectors are shown in Figure 1.2.8–1. The direction cosines for $\mathbf { t } _ { 1 }$ and with respect to the global coordinate system are available in dirCosS1 and dirCosN, respectively. In the case of zero incremental slip ( ) we choose an arbitrary direction for $\mathbf { t } _ { 1 }$ that is orthogonal to the normal direction, .
+
+![](images/page-428_ee0f0a9fc878b4b8608e793a1eac6039ee5e9782ac0f06804a5ced007f8deb3f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+surface normal
+slave node
+master surface
+n
+ts
+ds
+t1
+slip directions
+incremental frictional slip
+</details>
+
+Figure 1.2.8–1 Local coordinate system for three-dimensional contact with VFRICTION.
+
+# Frictional forces
+
+You specify the frictional force, fTangential, at each contact point in local coordinates in this subroutine. The array fTangential is dimensioned such that only the tangential components can be specified. Any components of the frictional force that are not specified will remain equal to zero. For isotropic friction, only the first component of the frictional force need be specified since the second component should be zero. A “stick force” at each contact point is provided in the array fStickForce to assist you in setting appropriate frictional force values. The stick force is the force required to prevent additional “plastic” slipping. The stick force at each contact point is provided as a scalar value as it would act in the direction opposite to $\mathbf { t } _ { 1 }$ . The stick force is computed prior to calling user subroutine VFRICTION. The first component of the frictional force should be in the range between zero and the
+
+<!-- source-page: 429 -->
+
+negative of the stick force value. Typically, the stick force will be positive and the first component of the applied frictional force will be negative, opposing the incremental slip. Penalty contact includes an elastic slip regime due to finite penalty stiffness; so occasionally the stick force will be negative during recovery of elastic slip, indicating that it is appropriate for the first component of the frictional force to be positive (i.e., acting in the same direction as the incremental slip). A noisy or unstable solution is likely to result if the first component of fTangential is set outside the range between zero and the negative of the stick force value.
+
+After user subroutine VFRICTION is called, frictional forces that oppose the forces specified at the contact points are distributed to the master nodes.
+
+User subroutine interface   
+```txt
+subroutine vfriction (
+C Write only -
+    * fTangential,
+C Read/Write -
+    * state,
+C Read only -
+    * nBlock, nBlockAnal, nBlockEdge,
+    * nNodState, nNodSlv, nNodMst,
+    * nFricDir, nDir,
+    * nStates, nProps, nTemp, nFields,
+    * jFlags, rData,
+    * surfInt, surfSlv, surfMst,
+    * jConSlvUid, jConMstUid, props,
+    * dSlipFric, fStickForce, fTangPrev, fNormal,
+    * areaCont, dircosN, dircosS1,
+    * shapeSlv, shapeMst,
+    * coordSlv, coordMst,
+    * velSlv, velMst,
+    * tempSlv, tempMst,
+    * fieldSlv, fieldMst )
+C
+    include `vaba_param.inc'
+C
+    dimension fTangential(nFricDir,nBlock),
+    * state(nStates,nNodState,nBlock),
+    * jConSlvUid(nNodSlv,nBlock),
+    * jConMstUid(nNodMst,nBlockAnal),
+    * props(nProps),
+    * dSlipFric(nDir,nBlock),
+    * fStickForce(nBlock), 
+```
+
+<!-- source-page: 430 -->
+
+```c
+* fTangPrev(nDir,nBlock),
+* fNormal(nBlock),
+* areaCont(nBlock),
+* dircosN(nDir,nBlock),
+* dircosS1(nDir,nBlock),
+* shapeSlv(nNodSlv,nBlockEdge),
+* shapeMst(nNodMst,nBlockAnal),
+* coordSlv(nDir,nNodSlv,nBlock),
+* coordMst(nDir,nNodMst,nBlockAnal),
+* velSlv(nDir,nNodSlv,nBlock),
+* velMst(nDir,nNodMst,nBlockAnal),
+* tempSlv(nBlock),
+* tempMst(nBlockAnal),
+* fieldSlv(nFields,nBlock),
+* fieldMst(nFields,nBlockAnal)
+C
+    parameter( iKStep    = 1,
+    *    iKInc    = 2,
+    *    iLConType    = 3,
+    *    nFlags    = 3 )
+C
+    parameter( iTimStep    = 1,
+    *    iTimGlb    = 2,
+    *    iDTimCur    = 3,
+    *    iFrictionWork    = 4,
+    *    nData    = 4 )
+C
+    dimension jFlags(nFlags), rData(nData)
+C
+    character*80 surfInt, surfSlv, surfMst
+C
+    user coding to define fTangential
+    and, optionally, state
+C
+    return
+    end 
+```
+
+# Variable to be defined
+
+# fTangential(nFricDir,nBlock)
+
+This array must be updated to the current values of the frictional force components for all contact points in the local tangent directions. See Figure 1.2.8–1 for a definition of the local coordinate system. This array will be zero (no friction force) until it is set.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_044.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_044.md
new file mode 100644
index 00000000..e48a5654
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_044.md
@@ -0,0 +1,448 @@
+<!-- source-page: 431 -->
+
+state(nStates,nNodState,nBlock)
+
+This array contains the user-defined, solution-dependent state variables for all the nodes on the slave surface. The use of state variables is applicable for node-to-face and node-to-analytical rigid surface contact. See “Frictional behavior,” Section 37.1.5 of the Abaqus Analysis User’s Guide, for more information on the size of this array. This array will be passed in containing the values of these variables prior to the call to user subroutine VFRICTION.
+
+If any of the solution-dependent state variables are being used in conjunction with the friction behavior, they must be updated in this subroutine. These state variables need to be updated with care: outside the user subroutine these state variables are single-valued per slave node, but multiple contact points may refer to the same slave node (if it contacts a master surface at more than one point). Each contact point may be passed into the user subroutine independently in a given increment, possibly on separate calls to the user subroutine; therefore, you may end up advancing the state variables for the associated node multiple times for a single increment. To keep track of whether or not a node state is advanced, you may want to use one of the state variables exclusively for this purpose. You could set that selected state variable to the current increment number and update the state only if it is not already set to the current increment number.
+
+Variables passed in for information   
+```txt
+nBlock
+Number of contact points to be processed in this call to VFRICCTION.
+
+nBlockAnal
+1 for analytical rigid master surface; nBlock otherwise.
+
+nBlockEdge
+nBlock for edge-type slave surface; 1 otherwise.
+
+nNodState
+1 for node-to-face contact and node-to-analytical rigid surface contact.
+
+nNodSlv
+1 for node-to-face and node-to-analytical rigid surface contact; 2 for edge-to-edge contact.
+
+nNodMst
+1 for analytical rigid master surface; 2 for edge-type master surface; 4 for facet-type master surface.
+
+nFricDir
+Number of tangent directions at the contact points (nFricDir = nDir - 1).
+
+nDir
+Number of coordinate directions at the contact points (equal to 3). 
+```
+
+<!-- source-page: 432 -->
+
+nStates
+
+Number of user-defined state variables.
+
+nProps
+
+User-specified number of property values associated with this friction model.
+
+nTemp
+
+1 if the temperature is defined and 0 if the temperature is not defined.
+
+nFields
+
+Number of predefined field variables.
+
+jFlag(1)
+
+Step number.
+
+jFlag(2)
+
+Increment number.
+
+jFlag(3)
+
+1 for node-to-face contact, 2 for edge-to-edge contact, and 3 for node-to-analytical rigid surface contact.
+
+rData(1)
+
+Value of step time.
+
+rData(2)
+
+Value of total time.
+
+rData(3)
+
+Current increment in time from to .
+
+rData(4)
+
+This variable contains the value of the total frictional dissipation in the entire model from the beginning of the analysis. The units are energy per unit area.
+
+surfInt
+
+User-specified surface interaction name, left justified.
+
+surfSlv
+
+Slave surface name, currently set to a blank.
+
+surfMst
+
+Master surface name, currently set to a blank.
+
+jConSlvUid(nNodSlv,nBlock)
+
+This array lists the surface node numbers of the slave surface nodes associated with each contact point.
+
+<!-- source-page: 433 -->
+
+# jConMstUid(nNodMst,nBlockAnal)
+
+This array lists the surface node numbers of the master surface nodes that make up the facet, edge, or analytical rigid surface associated with each contact point.
+
+# props(nProps)
+
+User-specified vector of property values to define the frictional behavior between the contacting surfaces.
+
+# dSlipFric(nDir,nBlock)
+
+This array contains the incremental frictional slip during the current time increment for each contact point in the current local coordinate system. These incremental slips correspond to tangential motion in the time increment from $t = t _ { c u r r } - \Delta t$ to $t = t _ { c u r r }$ . This incremental slip is used to define the local coordinate system at each contact point (see Figure 1.2.8–1) so that only the first component of dSlipFric can be nonzero in the local system.
+
+# fStickForce(nBlock)
+
+This array contains the magnitude of frictional force required to enforce stick conditions at each contact point. This force depends on the previous frictional force, the value of the penalty stiffness, and the previous incremental slip. The penalty stiffness is assigned automatically. Occasionally, during recovery of elastic slip associated with the penalty method, the stick force will be assigned a negative value.
+
+# fTangPrev(nDir,nBlock)
+
+This array contains the values of the frictional force components calculated in the previous increment but provided in the current local coordinate system (zero for nodes that were not in contact).
+
+# fNormal(nBlock)
+
+This array contains the magnitude of the normal force for the contact points applied at the end of current time increment; i.e., at time $t = t _ { c u r r }$ .
+
+# areaCont(nBlock)
+
+Area associated with the contact points. The sum of the contact areas among all contact points associated with a single slave node equals the surface area associated with that slave node (equal to 1 for node-based surface nodes). Therefore, the contact area at a contact point depends on the number of contact points currently associated with the same slave node. A contact point contributes a frictional stress to the associated slave node that is equal to fTangential(1,k) divided by areaCont(k).
+
+# dircosN(nDir,nBlock)
+
+Direction cosines of the normals to the master surface at the contact points.
+
+# dirCosS1(nDir,nBlock)
+
+Direction cosines of the incremental slip at the contact points. The direction cosines are undefined (all components zero) if the incremental frictional slip is zero.
+
+<!-- source-page: 434 -->
+
+shapeSlv(nNodSlv,nBlockEdge)
+
+For edge-to-edge contact this array contains the shape functions of the nodes of its slave edge, evaluated at the location of the contact point. If the contact is not edge-to-edge, this array is passed in as a dummy array.
+
+shapeMst(nNodMst,nBlockAnal)
+
+For node-to-face and edge-to-edge contact this array contains the shape functions of the nodes of its master surface, evaluated at the location of the contact point. If the master surface is an analytical rigid surface, this array is passed in as a dummy array.
+
+coordSlv(nDir,nNodSlv,nBlock)
+
+Array containing the nDir components of the current coordinates of the contact points.
+
+coordMst(nDir,nNodMst,nBlockAnal)
+
+Array containing the nDir components of the current coordinates of the master nodes associated with the contact points. If the master surface is an analytical rigid surface, this array is passed in as a dummy array.
+
+velSlv(nDir,nNodSlv,nBlock)
+
+Array containing the nDir components of the current velocity of the contact points.
+
+velMst(nDir,nNodMst,nBlockAnal)
+
+Array containing the nDir components of the current velocity of the master nodes associated with the contact points. If the master surface is an analytical rigid surface, this array is passed in as a dummy array.
+
+tempSlv(nBlock)
+
+Current temperature of the slave surface at the contact points.
+
+tempMst(nBlockAnal)
+
+Current temperature at the points on the master surface associated with the contact points.
+
+fieldSlv(nFields,nBlock)
+
+Current user-specified predefined field variables on the slave surface at the contact points (initial values at the beginning of the analysis and current values during the analysis).
+
+fieldMst(nFields,nBlockAnal)
+
+Current user-specified predefined field variables at the points on the master surface associated with the contact points (initial values at the beginning of the analysis and current values during the analysis).
+
+<!-- source-page: 435 -->
+
+# 1.2.9 VUAMP: User subroutine to specify amplitudes.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Amplitude curves,” Section 34.1.2 of the Abaqus Analysis User’s Guide   
+• \*AMPLITUDE   
+• \*OUTPUT
+
+# Overview
+
+User subroutine VUAMP:
+
+• allows you to define the current value of an amplitude definition as a function of time;   
+• can be used to model control engineering aspects of your system when sensors are used (sensor values are from the beginning of the increment);   
+• can use a predefined number of state variables in its definition; and   
+• can optionally compute the derivatives and integrals of the amplitude function.
+
+# Explicit solution dependence
+
+The solution dependence introduced in this user subroutine is explicit: all data passed in the subroutine for information or to be updated are values at the beginning of that increment.
+
+# User subroutine interface
+
+SUBROUTINE VUAMP(   
+```javascript
+* ampName, time, ampValueOld, dt, nprops, props, nSvars,
+* svars, lFlagsInfo, nSensor, sensorValues, sensorNames,
+* jSensorLookUpTable,
+* AmpValueNew,
+* lFlagsDefine,
+* AmpDerivative, AmpSecDerivative, AmpIncIntegral) 
+```  
+INCLUDE 'VABA\_PARAM.INC'
+
+```txt
+C time indices
+parameter (iStepTime = 1,
+* iTotalTime = 2,
+* nTime = 2) 
+```  
+C flags passed in for information
+
+<!-- source-page: 436 -->
+
+```prolog
+parameter (iInitialization = 1,
+* iRegularInc = 2,
+* ikStep = 3,
+* nFlagsInfo = 3)
+C optional flags to be defined
+parameter (iComputeDeriv = 1,
+* iComputeSecDeriv = 2,
+* iComputeInteg = 3,
+* iStopAnalysis = 4,
+* iConcludeStep = 5,
+* nFlagsDefine = 5)
+dimension time(nTime), lFlagsInfo(nFlagsInfo),
+* lFlagsDefine(nFlagsDefine),
+* sensorValues(nSensor),
+* props(nprops),
+* sVars(nSvars)
+
+character*80 sensorNames(nSensor)
+character*80 ampName
+dimension jSensorLookUpTable(*)
+
+user coding to define AmpValueNew, and
+optionally lFlagsDefine, AmpDerivative, AmpSecDerivative,
+AmpIncIntegral
+
+RETURN
+END 
+```
+
+# Variable to be defined
+
+# AmpValueNew
+
+Current value of the amplitude.
+
+# Variables that can be updated
+
+# lFlagsDefine
+
+Integer flag array to determine whether the computation of additional quantities is necessary or to set step continuation requirements.
+
+lFlagsDefine(iComputeDeriv)
+
+If set to 1, you must provide the computation of the amplitude derivative. The default is 0, which means that Abaqus computes the derivative automatically.
+
+<!-- source-page: 437 -->
+
+<table><tr><td>lFlagsDefine (iComputeSecDeriv)</td><td>If set to 1, you must provide the computation of the amplitude second derivative. The default is 0, which means that Abaqus computes the second derivative automatically.</td></tr><tr><td>lFlagsDefine (iComputeInteg)</td><td>If set to 1, you must provide the computation of the amplitude incremental integral. The default is 0, which means that Abaqus computes the incremental integral automatically.</td></tr><tr><td>lFlagsDefine (iStopAnalysis)</td><td>If set to 1, the analysis will be stopped and an error message will be issued. The default is 0, which means that Abaqus will not stop the analysis.</td></tr><tr><td>lFlagsDefine (iConcludeStep)</td><td>If set to 1, Abaqus will conclude the step execution and advance to the next step (if a next step is available). The default is 0.</td></tr></table>
+
+# svars
+
+An array containing the values of the solution-dependent state variables associated with this amplitude definition. The number of such variables is nsvars (see above). You define the meaning of these variables.
+
+This array is passed into VUAMP containing the values of these variables at the start of the current increment. In most cases they should be updated to be the values at the end of the increment.
+
+# AmpDerivative
+
+Current value of the amplitude derivative.
+
+# AmpSecDerivative
+
+Current value of the amplitude second derivative.
+
+# AmpIncIntegral
+
+Current value of the amplitude incremental integral.
+
+# Variables passed in for information
+
+# ampName
+
+User-specified amplitude name, left justified.
+
+# time(iStepTime)
+
+Current value of step time.
+
+<!-- source-page: 438 -->
+
+time(iTotalTime)
+
+Current value of total time.
+
+ampValueOld
+
+Old value of the amplitude from the previous increment.
+
+dt
+
+Current stable time increment.
+
+nprops
+
+User-defined number of properties associated with this amplitude definition.
+
+props(nprops)
+
+User-supplied amplitude properties.
+
+nSvars
+
+User-defined number of solution-dependent state variables associated with this amplitude definition.
+
+lFlagsInfo
+
+Integer flag array with information regrading the current call to VUAMP:
+
+lFlagsInfo(iInitialization)
+
+This flag is equal to 1 if VUAMP is called from the initialization phase of each step and is set to 0 otherwise.
+
+lFlagsInfo(iRegularInc)
+
+This flag is equal to 1 if VUAMP is called from a regular increment and is set to 0 otherwise.
+
+lFlagsInfo(ikStep)
+
+Step number.
+
+nSensor
+
+Total number of sensors in the model.
+
+sensorValues
+
+Array with sensor values at the end of the previous increment. Each sensor value corresponds to a history output variable associated with the output database request defining the sensor.
+
+sensorNames
+
+Array with user-defined sensor names in the entire model, left justified. Each sensor name corresponds to a sensor value provided with the output database request. All names will be converted to uppercase characters if lowercase or mixed-case characters were used in their definition.
+
+jSensorLookUpTable
+
+Variable that must be passed into the utility functions IVGETSENSORID and VGETSENSORVALUE.
+
+<!-- source-page: 439 -->
+
+Example: Amplitude definition using sensor and state variables   
+```txt
+c user amplitude subroutine
+Subroutine VUAMP(
+C    passed in for information and state variables
+*    ampName, time, ampValueOld, dt, nprops, props, nSvars,
+*    svars, lFlagsInfo, nSensor, sensorValues, sensorNames,
+*    jSensorLookUpTable,
+C    to be defined
+*    ampValueNew,
+*    lFlagsDefine,
+*    AmpDerivative, AmpSecDerivative, AmpIncIntegral)
+include 'vaba_param.inc'
+C    svars - additional state variables, similar to (V)UEL
+    dimension sensorValues(nSensor), props(nprops),
+*    svars(nSvars)
+    character*80 sensorNames(nSensor)
+    character*80 ampName
+C    time indices
+    parameter( iStepTime    = 1,
+*    iTotalTime    = 2,
+*    nTime    = 2)
+C    flags passed in for information
+    parameter( iInitialization    = 1,
+*    iRegularInc    = 2,
+*    ikStep    = 3,
+*    nFlagsInfo    = 3)
+C    optional flags to be defined
+    parameter( iComputeDeriv    = 1,
+*    iComputeSecDeriv    = 2,
+*    iComputeInteg    = 3,
+*    iStopAnalysis    = 4,
+*    iConcludeStep    = 5,
+*    nFlagsDefine    = 5)
+    parameter( tStep=0.18, tAccelerateMotor = .00375,
+*    omegaFinal=23.26) 
+```  
+c Alternatively, assign the user-defined amplitude
+
+<!-- source-page: 440 -->
+
+```txt
+c properties on the data lines rather than using a parameter
+c definition above.
+c tStep = props(1)
+c tAccelerateMotor = props(2)
+c omegaFinal = props(3)
+
+    dimension time(nTime), lFlagsInfo(nFlagsInfo),
+    *    lFlagsDefine(nFlagsDefine)
+    dimension jSensorLookUpTable(*)
+
+    lFlagsDefine(iComputeDeriv)    = 1
+    lFlagsDefine(iComputeSecDeriv) = 1
+
+c get sensor value
+    vTrans_CU1    = vGetSensorValue('HORIZ_TRANSL_MOTION',
+    *    jSensorLookUpTable,
+    *    sensorValues)
+
+    if (ampName(1:22) .eq. 'MOTOR_WITH_STOP_SENSOR') then
+    if (lFlagsInfo(iInitialization).eq.1) then
+    ampValueNew    = ampValueOld
+
+    svars(1) = 0.0
+    svars(2) = 0.0
+    else
+    tim = time(iStepTime)
+
+c ramp up the angular rot velocity of the electric
+c motor after which hold constant
+    if (tim .le. tAccelerateMotor) then
+    ampValueNew = omegaFinal*tim/tAccelerateMotor
+
+    else
+    ampValueNew = omegaFinal
+    end if
+
+c retrieve old sensor value
+    vTrans_CU1_old = svars(1)
+
+c detect a zero crossing and count the number of
+c crossings
+    if (vTrans_CU1_old*vTrans_CU1 .le. 0.0 .and. 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_045.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_045.md
new file mode 100644
index 00000000..2bfb596b
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_045.md
@@ -0,0 +1,307 @@
+<!-- source-page: 441 -->
+
+```txt
+* tim .gt. tAccelerateMotor ) then
+    svars(2) = svars(2) + 1.0
+    end if
+    nrCrossings = int(svars(2))
+
+c stop the motor if sensor crosses zero the second
+c time
+    if (nrCrossings.eq.2) then
+    ampValueNew = 0.0
+    lFlagsDefine(iConcludeStep)=1
+    end if
+
+c store sensor value
+    svars(1) = vTrans_CU1
+    end if
+    end if
+
+return
+end 
+```
+
+<!-- source-page: 442 -->
+
+<!-- source-page: 443 -->
+
+# 1.2.10 VUANISOHYPER\_INV: User subroutine to define anisotropic hyperelastic material behavior using the invariant formulation.
+
+Product: Abaqus/Explicit
+
+# References
+
+• “Anisotropic hyperelastic behavior,” Section 22.5.3 of the Abaqus Analysis User’s Guide   
+• \*ANISOTROPIC HYPERELASTIC   
+• “UANISOHYPER\_INV and VUANISOHYPER\_INV,” Section 4.1.13 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine VUANISOHYPER\_INV:
+
+• can be used to define the strain energy potential of anisotropic hyperelastic materials as a function of an irreducible set of scalar invariants;   
+• will be called for blocks of material calculation points for which the material definition contains user-defined anisotropic hyperelastic behavior with invariant-based formulation (“Anisotropic hyperelastic behavior,” Section 22.5.3 of the Abaqus Analysis User’s Guide);   
+• can use and update solution-dependent state variables;   
+• can use any field variables that are passed in; and   
+• requires that the values of the derivatives of the strain energy density function be defined with respect to the scalar invariants.
+
+# Enumeration of invariants
+
+To facilitate coding and provide easy access to the array of invariants passed to user subroutine VUANISOHYPER\_INV, an enumerated representation of each invariant is introduced. Any scalar invariant can, therefore, be represented uniquely by an enumerated invariant, $I _ { n } ^ { * }$ , where the subscript n denotes the order of the invariant according to the enumeration scheme in the following table:
+
+<table><tr><td>Invariant</td><td>Enumeration, n</td></tr><tr><td> $\overline{I}_{1}$ </td><td>1</td></tr><tr><td> $\overline{I}_{2}$ </td><td>2</td></tr><tr><td>J</td><td>3</td></tr><tr><td> $\overline{I}_{4(\alpha\beta)}$ </td><td> $4 + 2(\alpha - 1) + \beta(\beta - 1)$ ;  $\alpha \leq \beta$ </td></tr><tr><td> $\overline{I}_{5(\alpha\beta)}$ </td><td> $5 + 2(\alpha - 1) + \beta(\beta - 1)$ ;  $\alpha \leq \beta$ </td></tr></table>
+
+<!-- source-page: 444 -->
+
+For example, in the case of three families of fibers there are a total of 15 invariants: $\overline { { I } } _ { 1 } , \overline { { I } } _ { 2 } , J ,$ six invariants of type $\overline { { I } } _ { 4 ( \alpha \beta ) }$ , and six invariants of type $\overline { { I } } _ { 5 ( \alpha \beta ) }$ , with $\alpha , \beta = 1 , 2 , 3 \left( \alpha \leq \beta \right)$ . The following correspondence exists between each of these invariants and their enumerated counterpart:
+
+<table><tr><td>Enumerated invariant</td><td>Invariant</td></tr><tr><td> $I_1^*$ </td><td> $\overline{I}_1$ </td></tr><tr><td> $I_2^*$ </td><td> $\overline{I}_2$ </td></tr><tr><td> $I_3^*$ </td><td> $J$ </td></tr><tr><td> $I_4^*$ </td><td> $\overline{I}_{4(11)}$ </td></tr><tr><td> $I_5^*$ </td><td> $\overline{I}_{5(11)}$ </td></tr><tr><td> $I_6^*$ </td><td> $\overline{I}_{4(12)}$ </td></tr><tr><td> $I_7^*$ </td><td> $\overline{I}_{5(12)}$ </td></tr><tr><td> $I_8^*$ </td><td> $\overline{I}_{4(22)}$ </td></tr><tr><td> $I_9^*$ </td><td> $\overline{I}_{5(22)}$ </td></tr><tr><td> $I_{10}^*$ </td><td> $\overline{I}_{4(13)}$ </td></tr><tr><td> $I_{11}^*$ </td><td> $\overline{I}_{5(13)}$ </td></tr><tr><td> $I_{12}^*$ </td><td> $\overline{I}_{4(23)}$ </td></tr><tr><td> $I_{13}^*$ </td><td> $\overline{I}_{5(23)}$ </td></tr><tr><td> $I_{14}^*$ </td><td> $\overline{I}_{4(33)}$ </td></tr><tr><td> $I_{15}^*$ </td><td> $\overline{I}_{5(33)}$ </td></tr></table>
+
+A similar scheme is used for the array zeta of terms $\zeta _ { \alpha \beta } = { \bf A } _ { \alpha } \cdot { \bf A } _ { \beta }$ . Each term can be represented uniquely by an enumerated counterpart $\zeta _ { m } ^ { * }$ , as shown below:
+
+<table><tr><td>Dot product</td><td>Enumeration, m</td></tr><tr><td> $\zeta_{\alpha\beta}$ </td><td> $\alpha + \frac{1}{2}(\beta - 2)(\beta - 1)$ ;  $\alpha < \beta$ </td></tr></table>
+
+As an example, for the case of three families of fibers there are three $\zeta _ { \alpha \beta }$ terms: $\zeta _ { 1 2 } , \zeta _ { 1 3 }$ , and $\zeta _ { 2 3 }$ . These are stored in the zeta array as $\left( \zeta _ { 1 } ^ { * } , \zeta _ { 2 } ^ { * } , \zeta _ { 3 } ^ { * } \right)$ .
+
+<!-- source-page: 445 -->
+
+# Storage of arrays of derivatives of energy function
+
+The components of the array duDi of first derivatives of the strain energy potential with respect to the scalar invariants, ${ \partial U } / { \partial I _ { i } ^ { * } }$ , are stored using the enumeration scheme discussed above for the scalar invariants.
+
+The elements of the array d2uDiDi of second derivatives of the strain energy function, $\partial ^ { 2 } U / \partial I _ { i } ^ { * } \partial I _ { j } ^ { * }$ , are laid out in memory using triangular storage: if denotes the component in this array corresponding to the term $\partial ^ { 2 } U / \partial I _ { i } ^ { * } \partial I _ { i } ^ { * }$ , then $k = i + j \times ( j - 1 ) / 2 ; ( i \leq j )$ . For example, the term $\partial ^ { 2 } U / \partial I _ { 2 } ^ { * } \partial I _ { 5 } ^ { * }$ is stored in component $\overset { \cdot } { k } = 2 + ( 5 \times 4 ) / 2 = 1 2$ in the d2uDiDi array.
+
+# Special considerations for shell elements
+
+When VUANISOHYPER\_INV is used to define the material response of shell elements, Abaqus/Explicit cannot calculate a default value for the transverse shear stiffness of the element. Hence, you must define the element’s transverse shear stiffness. See “Shell section behavior,” Section 29.6.4 of the Abaqus Analysis User’s Guide, for guidelines on choosing this stiffness.
+
+# Material point deletion
+
+Material points that satisfy a user-defined failure criterion can be deleted from the model (see “User-defined mechanical material behavior,” Section 26.7.1 of the Abaqus Analysis User’s Guide). You must specify the state variable number controlling the element deletion flag when you allocate space for the solution-dependent state variables, as explained in “User-defined mechanical material behavior,” Section 26.7.1 of the Abaqus Analysis User’s Guide. The deletion state variable should be set to a value of one or zero in VUANISOHYPER\_INV. A value of one indicates that the material point is active, and a value of zero indicates that Abaqus/Explicit should delete the material point from the model by setting the stresses to zero. The structure of the block of material points passed to user subroutine VUANISOHYPER\_INV remains unchanged during the analysis; deleted material points are not removed from the block. Abaqus/Explicit will “freeze” the values of the invariants passed to VUANISOHYPER\_INV for all deleted material points; that is, the values remain constant after deletion is triggered. Once a material point has been flagged as deleted, it cannot be reactivated.
+
+# User subroutine interface
+
+```csv
+subroutine vuanisohyper_inv (
+C Read only (unmodifiable) variables -
+1 nblock, nFiber, nInv,
+2 jElem, kIntPt, kLayer, kSecPt,
+3 cmname,
+4 nstatev, nfieldv, nprops,
+5 props, tempOld, tempNew, fieldOld, fieldNew,
+6 stateOld, sInvariant, zeta,
+C Write only (modifiable) variables - 
+```
+
+<!-- source-page: 446 -->
+
+```txt
+7 uDev, duDi, d2uDiDi,
+8 stateNew)
+C
+    include 'vaba_param.inc'
+C
+    dimension props(nprops),
+    1 tempOld(nblock),
+    2 fieldOld(nblock,nfieldv),
+    3 stateOld(nblock,nstatev),
+    4 tempNew(nblock),
+    5 fieldNew(nblock,nfieldv),
+    6 sInvariant(nblock,nInv),
+    7 zeta(nblock,nFiber*(nFiber-1)/2),
+    8 uDev(nblock), duDi(nblock,nInv),
+    9 d2uDiDi(nblock,nInv*(nInv+1)/2),
+    * stateNew(nblock,nstatev)
+C
+    character*80 cmname
+C
+    do 100 km = 1,nblock
+    user coding
+100 continue
+    return
+    end 
+```
+
+# Variables to be defined
+
+udev(nblock)
+
+$\tilde { U } _ { d e v ; }$ , the deviatoric part of the strain energy density of the primary material response. This quantity is needed only if the current material definition also includes Mullins effect (see “Mullins effect,” Section 22.6.1 of the Abaqus Analysis User’s Guide).
+
+duDi(nblock,nInv)
+
+Array of derivatives of strain energy potential with respect to the scalar invariants, ${ \partial U } / { \partial I _ { i } ^ { * } }$ , ordered using the enumeration scheme discussed above.
+
+d2uDiDi(nblock,nInv\*(nInv+1)/2)
+
+Arrays of second derivatives of strain energy potential with respect to the scalar invariants (using triangular storage), $\partial ^ { 2 } U / \partial I _ { i } ^ { * } \partial I _ { j } ^ { * }$ .
+
+<!-- source-page: 447 -->
+
+stateNew(nblock,nstatev)
+
+State variables at each material point at the end of the increment. You define the size of this array by allocating space for it (see “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide, for more information).
+
+Variables passed in for information   
+nblock
+Number of material points to be processed in this call to VUANISOHYPER_STRAIN.
+nFiber
+Number of families of fibers defined for this material.
+nInv
+Number of scalar invariants.
+jElem(nblock)
+Array of element numbers.
+kIntPt
+Integration point number.
+kLayer
+Layer number (for composite shells).
+kSecPt
+Section point number within the current layer.
+cmname
+User-specified material name, left justified. It is passed in as an uppercase character
+internal material models are given names starting with the “ABQ_” character string. To
+you should not use “ABQ_” as the leading string for cmname.
+nstatev
+Number of user-defined state variables that are associated with this material type (as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1
+Analysis User’s Guide).
+nfieldv
+Number of user-defined external field variables.
+nprops
+User-specified number of user-defined material properties.
+props(nprops)
+User-supplied material properties.
+
+<!-- source-page: 448 -->
+
+tempOld(nblock)
+
+Temperatures at each material point at the beginning of the increment.
+
+tempNew(nblock)
+
+Temperatures at each material point at the end of the increment.
+
+fieldOld(nblock,nfieldv)
+
+Values of the user-defined field variables at each material point at the beginning of the increment.
+
+fieldNew(nblock,nfieldv)
+
+Values of the user-defined field variables at each material point at the end of the increment.
+
+stateOld(nblock,nstatev)
+
+State variables at each material point at the beginning of the increment.
+
+sInvariant(nblock,nInv)
+
+Array of scalar invariants, , at each material point at the end of the increment. The invariants are ordered using the enumeration scheme discussed above.
+
+zeta(nblock,nFiber\*(nFiber-1)/2) )
+
+Array of dot product between the directions of different families of fiber in the reference configuration, . The array contains the enumerated values using the scheme discussed above.
+
+# Example: Using more than one user-defined anisotropic hyperelastic material model
+
+To use more than one user-defined anisotropic hyperelastic material model, the variable cmname can be tested for different material names inside user subroutine VUANISOHYPER\_INV, as illustrated below:
+
+```txt
+if (cmname(1:4) .eq. 'MAT1') then
+    call VUANISOHYPER_INV1(argument_list)
+else if (cmname(1:4) .eq. 'MAT2') then
+    call VUANISOHYPER_INV2(argument_list)
+end if 
+```
+
+VUANISOHYPER\_INV1 and VUANISOHYPER\_INV2 are the actual subroutines containing the anisotropic hyperelastic models for each material MAT1 and MAT2, respectively. Subroutine VUANISOHYPER\_INV merely acts as a directory here. The argument list can be the same as that used in subroutine VUANISOHYPER\_INV. The material names must be in uppercase characters since cmname is passed in as an uppercase character string.
+
+# Example: Anisotropic hyperelastic model of Kaliske and Schmidt
+
+As an example of the coding of subroutine VUANISOHYPER\_INV, consider the model proposed by Kaliske and Schmidt (2005) for nonlinear anisotropic elasticity with two families of fibers. The strain energy function is given by a polynomial series expansion in the form
+
+<!-- source-page: 449 -->
+
+$$
+\begin{array}{l} U = \frac {1}{D} (J - 1) ^ {2} + \sum_ {i = 1} ^ {3} a _ {i} (\overline {{I}} _ {1} - 3) ^ {i} + \sum_ {j = 1} ^ {3} b _ {j} (\overline {{I}} _ {2} - 3) ^ {j} + \sum_ {k = 2} ^ {6} c _ {k} (\overline {{I}} _ {4 (1 1)} - 1) ^ {k} + \sum_ {l = 2} ^ {6} d _ {l} (\overline {{I}} _ {5 (1 1)} - 1) ^ {l} \\ + \sum_ {m = 2} ^ {6} e _ {m} (\overline {{I}} _ {4 (2 2)} - 1) ^ {m} + \sum_ {n = 2} ^ {6} f _ {n} (\overline {{I}} _ {5 (2 2)} - 1) ^ {n} + \sum_ {p = 2} ^ {6} g _ {p} (\zeta_ {1 2} \overline {{I}} _ {4 (1 2)} - \zeta_ {1 2} ^ {2}) ^ {p}. \\ \end{array}
+$$
+
+The code in subroutine VUANISOHYPER\_INV must return the derivatives of the strain energy function with respect to the scalar invariants, which are readily computed from the above expression. In this example auxiliary functions are used to facilitate enumeration of pseudo-invarinats of type $\overline { { I } } _ { 4 ( \alpha \beta ) }$ and $\overline { { I } } _ { 5 ( \alpha \beta ) }$ , as well as for indexing into the array of second derivatives using symmetric storage. The subroutine would be coded as follows:
+```txt
+subroutine vuanisohyper_inv (
+C Read only -
+* nblock, nFiber, nInv,
+* jElem, kIntPt, kLayer, kSecPt,
+* cmname,
+* nstatev, nfieldv, nprops,
+* props, tempOld, tempNew, fieldOld, fieldNew,
+* stateOld, sInvariant, zeta,
+C Write only -
+* uDev, duDi, d2uDiDi,
+* stateNew )
+C
+include 'vaba_param.inc'
+C
+dimension props(nprops),
+* tempOld(nblock),
+* fieldOld(nblock,nfieldv),
+* stateOld(nblock,nstatev),
+* tempNew(nblock),
+* fieldNew(nblock,nfieldv),
+* sInvariant(nblock,nInv),
+* zeta(nblock,nFiber*(nFiber-1)/2),
+* uDev(nblock), duDi(nblock,*),
+* d2uDiDi(nblock,*),
+* stateNew(nblock,nstatev)
+C
+character*80 cmname
+C
+parameter ( zero = 0.d0, one = 1.d0, two = 2.d0,
+* three = 3.d0, four = 4.d0, five = 5.d0, six = 6.d0 ) 
+```
+
+<!-- source-page: 450 -->
+
+```txt
+C Kaliske energy function (3D)
+C
+C Read material properties
+d=props(1)
+dinv = one / d
+a1=props(2)
+a2=props(3)
+a3=props(4)
+b1=props(5)
+b2=props(6)
+b3=props(7)
+c2=props(8)
+c3=props(9)
+c4=props(10)
+c5=props(11)
+c6=props(12)
+d2=props(13)
+d3=props(14)
+d4=props(15)
+d5=props(16)
+d6=props(17)
+e2=props(18)
+e3=props(19)
+e4=props(20)
+e5=props(21)
+e6=props(22)
+f2=props(23)
+f3=props(24)
+f4=props(25)
+f5=props(26)
+f6=props(27)
+g2=props(28)
+g3=props(29)
+g4=props(30)
+g5=props(31)
+g6=props(32)
+C
+do k = 1, nblock
+Udev(k) = zero
+C Compute Udev and 1st and 2nd derivatives w.r.t invariants
+C - I1 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_046.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_046.md
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+<!-- source-page: 451 -->
+
+```txt
+bi1 = sInvariant(k,1)
+term = bi1-three
+Udev(k) = Udev(k)
+* + a1*term + a2*term**2 + a3*term**3
+duDi(k,1) = a1 + two*a2*term + three*a3*term**2
+d2uDiDi(k,indx(1,1)) = two*a2 + three*two*a3*term
+C - I2
+bi2 = sInvariant(k,2)
+term = bi2-three
+Udev(k) = Udev(k)
+* + b1*term + b2*term**2 + b3*term**3
+duDi(k,2) = b1 + two*b2*term + three*b3*term**2
+d2uDiDi(k,indx(2,2)) = two*b2 + three*two*b3*term
+C - I3 (=J)
+bi3 = sInvariant(k,3)
+term = bi3-one
+duDi(k,3) = two*dinv*term
+d2uDiDi(k,indx(3,3)) = two*dinv
+C - I4(11)
+nI411 = indxInv4(1,1)
+bi411 = sInvariant(k,nI411)
+term = bi411-one
+Udev(k) = Udev(k)
+* + c2*term**2 + c3*term**3 + c4*term**4
+* + c5*term**5 + c6*term**6
+duDi(k,nI411) =
+* two*c2*term
+* + three*c3*term**2
+* + four*c4*term**3
+* + five*c5*term**4
+* + six*c6*term**5
+d2uDiDi(k,indx(nI411,nI411)) =
+* two*c2
+* + three*two*c3*term
+* + four*three*c4*term**2
+* + five*four*c5*term**3
+* + six*five*c6*term**4
+C - I5(11)
+nI511 = indxInv5(1,1)
+bi511 = sInvariant(k,nI511)
+term = bi511-one
+Udev(k) = Udev(k) 
+```
+
+<!-- source-page: 452 -->
+
+```txt
+*    + d2*term**2 + d3*term**3 + d4*term**4
+*    + d5*term**5 + d6*term**6
+duDi(k,nI511) =
+*    two*d2*term
+*    + three*d3*term**2
+*    + four*d4*term**3
+*    + five*d5*term**4
+*    + six*d6*term**5
+d2uDiDi(k,indx(nI511,nI511)) =
+*    two*d2
+*    + three*two*d3*term
+*    + four*three*d4*term**2
+*    + five*four*d5*term**3
+*    + six*five*d6*term**4
+C - I4(22)
+    nI422 = indxInv4(2,2)
+    bi422 = sInvariant(k,nI422)
+    term = bi422-one
+    Udev(k) = Udev(k)
+*    + e2*term**2 + e3*term**3 + e4*term**4
+*    + e5*term**5 + e6*term**6
+duDi(k,nI422) =
+*    two*e2*term
+*    + three*e3*term**2
+*    + four*e4*term**3
+*    + five*e5*term**4
+*    + six*e6*term**5
+d2uDiDi(k,indx(nI422,nI422)) =
+*    two*e2
+*    + three*two*e3*term
+*    + four*three*e4*term**2
+*    + five*four*e5*term**3
+*    + six*five*e6*term**4
+C - I5(22)
+    nI522 = indxInv5(2,2)
+    bi522 = sInvariant(k,nI522)
+    term = bi522-one
+    Udev(k) = Udev(k)
+*    + f2*term**2 + f3*term**3 + f4*term**4
+*    + f5*term**5 + f6*term**6
+duDi(k,nI522) =
+*    two*f2*term 
+```
+
+<!-- source-page: 453 -->
+
+```txt
+* + three*f3*term**2
+* + four*f4*term**3
+* + five*f5*term**4
+* + six*f6*term**5
+d2uDiDi(k, index(nI522, nI522)) =
+* two*f2
+* + three*two*f3*term
+* + four*three*f4*term**2
+* + five*four*f5*term**3
+* + six*five*f6*term**4
+C - I4(12)
+nI412 = indexInv4(1, 2)
+bi412 = sInvariant(k, nI412)
+term = zeta(k, 1) * (bi412 - zeta(k, 1))
+Udev(k) = Udev(k)
+* + g2*term**2 + g3*term**3
+* + g4*term**4 + g5*term**5
+* + g6*term**6
+duDi(k, nI412) = zeta(k, 1) * (
+* two*g2*term
+* + three*g3*term**2
+* + four*g4*term**3
+* + five*g5*term**4
+* + six*g6*term**5 )
+d2uDiDi(k, index(nI412, nI412)) = zeta(k, 1) ** 2 * (
+* two*g2
+* + three*two*g3*term
+* + four*three*g4*term**2
+* + five*four*g5*term**3
+* + six*five*g6*term**4 )
+C
+end do
+C
+return
+end
+C
+Function to map index from Square to Triangular storage
+C of symmetric matrix
+C
+integer function index(i, j)
+include 'vaba_param.inc'
+ii = min(i, j) 
+```
+
+<!-- source-page: 454 -->
+
+```matlab
+jj = max(i, j)
+    index = ii + jj*(jj-1)/2
+    return
+    end
+C
+C Function to generate enumeration of scalar
+C Pseudo-Invariants of type 4
+C integer function indexInv4(i, j)
+    include 'vaba_param.inc'
+    ii = min(i, j)
+    jj = max(i, j)
+    indexInv4 = 4 + jj*(jj-1) + 2*(ii-1)
+    return
+    end
+C
+C Function to generate enumeration of scalar
+C Pseudo-Invariants of type 5
+C integer function indexInv5(i, j)
+    include 'vaba_param.inc'
+    ii = min(i, j)
+    jj = max(i, j)
+    indexInv5 = 5 + jj*(jj-1) + 2*(ii-1)
+    return
+    end 
+```
+
+# Additional reference
+
+• Kaliske, M., and J. Schmidt, “Formulation of Finite Nonlinear Anisotropic Elasticity,” CADFEM GmbH Infoplaner 2/2005, vol. 2, pp. 22–23, 2005.
+
+<!-- source-page: 455 -->
+
+# 1.2.11 VUANISOHYPER\_STRAIN: User subroutine to define anisotropic hyperelastic material behavior based on Green strain.
+
+Product: Abaqus/Explicit
+
+# References
+
+• “Anisotropic hyperelastic behavior,” Section 22.5.3 of the Abaqus Analysis User’s Guide   
+• \*ANISOTROPIC HYPERELASTIC   
+• “UANISOHYPER\_INV and VUANISOHYPER\_INV,” Section 4.1.13 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine VUANISOHYPER\_STRAIN:
+
+• can be used to define the strain energy potential of anisotropic hyperelastic materials as a function of the components of the Green strain tensor;   
+• will be called for blocks of material calculation points for which the material definition contains user-defined anisotropic hyperelastic behavior with Green strain-based formulation (“Anisotropic hyperelastic behavior,” Section 22.5.3 of the Abaqus Analysis User’s Guide);   
+• can use and update solution-dependent state variables;   
+• can use any field variables that are passed in; and   
+• requires that the values of the derivatives of the strain energy density function be defined with respect to the components of the modified Green strain tensor and the volume ratio.
+
+# Component ordering in tensors
+
+The component ordering depends upon whether the tensor is second or fourth order.
+
+# Symmetric second-order tensors
+
+For symmetric second-order tensors, such as the modified Green strain tensor, there are ndir+nshr components; the component order is given as a natural permutation of the indices of the tensor. The direct components are first and then the indirect components, beginning with the 12-component. For example, a stress tensor contains ndir direct stress components and nshr shear stress components, which are passed in as
+
+<table><tr><td>Component</td><td>2D Case</td><td>3D Case</td></tr><tr><td>1</td><td> $\overline{\varepsilon}_{11}^{G}$ </td><td> $\overline{\varepsilon}_{11}^{G}$ </td></tr><tr><td>2</td><td> $\overline{\varepsilon}_{22}^{G}$ </td><td> $\overline{\varepsilon}_{22}^{G}$ </td></tr></table>
+
+<!-- source-page: 456 -->
+
+<table><tr><td>Component</td><td>2D Case</td><td>3D Case</td></tr><tr><td>3</td><td> $\overline{\varepsilon}_{33}^{G}$ </td><td> $\overline{\varepsilon}_{33}^{G}$ </td></tr><tr><td>4</td><td> $\overline{\varepsilon}_{12}^{G}$ </td><td> $\overline{\varepsilon}_{12}^{G}$ </td></tr><tr><td>5</td><td></td><td> $\overline{\varepsilon}_{23}^{G}$ </td></tr><tr><td>6</td><td></td><td> $\overline{\varepsilon}_{31}^{G}$ </td></tr></table>
+
+The shear strain components are stored as tensor components and not as engineering components.
+
+# Symmetric fourth-order tensors
+
+For symmetric fourth-order tensors, such as the deviatoric elasticity tensor $\partial ^ { 2 } U / \partial \overline { { \varepsilon } } _ { i j } ^ { G } \partial \overline { { \varepsilon } } _ { k l } ^ { G }$ , there are (ndir+nshr)\*(ndir+nshr+1)/2 independent components. These components are ordered using the following triangular storage scheme:
+
+<table><tr><td>Component</td><td>2D Case</td><td>3D Case</td></tr><tr><td>1</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{11}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{11}^{G}$ </td></tr><tr><td>2</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{22}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{22}^{G}$ </td></tr><tr><td>3</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{22}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{22}^{G}$ </td></tr><tr><td>4</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{33}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{33}^{G}$ </td></tr><tr><td>5</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{33}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{33}^{G}$ </td></tr><tr><td>6</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{33}^{G}\partial\overline{\varepsilon}_{33}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{33}^{G}\partial\overline{\varepsilon}_{33}^{G}$ </td></tr><tr><td>7</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td></tr><tr><td>8</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td></tr><tr><td>9</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{33}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{33}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td></tr><tr><td>10</td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{12}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{12}^{G}\partial\overline{\varepsilon}_{12}^{G}$ </td></tr><tr><td>11</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{23}^{G}$ </td></tr><tr><td>12</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{23}^{G}$ </td></tr><tr><td>13</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{33}^{G}\partial\overline{\varepsilon}_{23}^{G}$ </td></tr><tr><td>14</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{12}^{G}\partial\overline{\varepsilon}_{23}^{G}$ </td></tr><tr><td>15</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{23}^{G}\partial\overline{\varepsilon}_{23}^{G}$ </td></tr><tr><td>16</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{11}^{G}\partial\overline{\varepsilon}_{31}^{G}$ </td></tr></table>
+
+<!-- source-page: 457 -->
+
+<table><tr><td>Component</td><td>2D Case</td><td>3D Case</td></tr><tr><td>17</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{22}^{G}\partial\overline{\varepsilon}_{31}^{G}$ </td></tr><tr><td>18</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{33}^{G}\partial\overline{\varepsilon}_{31}^{G}$ </td></tr><tr><td>19</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{12}^{G}\partial\overline{\varepsilon}_{31}^{G}$ </td></tr><tr><td>20</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{23}^{G}\partial\overline{\varepsilon}_{31}^{G}$ </td></tr><tr><td>21</td><td></td><td> $\partial^{2}U/\partial\overline{\varepsilon}_{31}^{G}\partial\overline{\varepsilon}_{31}^{G}$ </td></tr></table>
+
+If Q denotes the component number of term $\partial ^ { 2 } U / \partial \overline { { \varepsilon } } _ { i j } ^ { G } \partial \overline { { \varepsilon } } _ { k l } ^ { G }$ in the above table and M and N (with $M \ \leq \ N )$ denote the component numbers of $\overline { { \varepsilon } } _ { i j } ^ { G }$ and $\overline { { \varepsilon } } _ { k l } ^ { G }$ , respectively, in the table for second-order tensors, Q is given by the relationship $Q = M ^ { ' } + N \times ( N - 1 ) / 2$ . For example, consider the term $\partial ^ { 2 } U / \partial \overline { { \varepsilon } } _ { 1 1 } ^ { G } \partial \overline { { \varepsilon } } _ { 2 3 } ^ { G }$ . The component numbers for $\overline { { \varepsilon } } _ { 1 1 } ^ { G }$ and $\overline { { \varepsilon } } _ { 2 3 } ^ { G }$ are $M = 1$ and $N = 5$ , respectively, giving $Q = 1 + ( 5 \times 4 ) / 2 = 1 1$ .
+
+# Special consideration for shell elements
+
+When VUANISOHYPER\_STRAIN is used to define the material response of shell elements, Abaqus/Explicit cannot calculate a default value for the transverse shear stiffness of the element. Hence, you must define the element’s transverse shear stiffness. See “Shell section behavior,” Section 29.6.4 of the Abaqus Analysis User’s Guide, for guidelines on choosing this stiffness.
+
+# Material point deletion
+
+Material points that satisfy a user-defined failure criterion can be deleted from the model (see “User-defined mechanical material behavior,” Section 26.7.1 of the Abaqus Analysis User’s Guide). You must specify the state variable number controlling the element deletion flag when you allocate space for the solution-dependent state variables, as explained in “User-defined mechanical material behavior,” Section 26.7.1 of the Abaqus Analysis User’s Guide. The deletion state variable should be set to a value of one or zero in VUANISOHYPER\_STRAIN. A value of one indicates that the material point is active, and a value of zero indicates that Abaqus/Explicit should delete the material point from the model by setting the stresses to zero. The structure of the block of material points passed to user subroutine VUANISOHYPER\_STRAIN remains unchanged during the analysis; deleted material points are not removed from the block. Abaqus/Explicit will “freeze” the values of the strains passed to VUANISOHYPER\_STRAIN for all deleted material points; that is, the strain values remain constant after deletion is triggered. Once a material point has been flagged as deleted, it cannot be reactivated.
+
+# User subroutine interface
+
+subroutine vuanisohyper\_strain( C Read only (unmodifiable) variables – 1 nblock,jElem,kIntPt,kLayer,kSecPt,cmname,
+
+<!-- source-page: 458 -->
+
+```csv
+2 ndir,nshr,nstatev,nfieldv,nprops,
+3 props,tempOld,tempNew,fieldOld,fieldNew,
+4 stateOld, ebar,detu,
+C Write only (modifiable) variables -
+4 udev,duDe,duDj,
+5 d2uDeDe,d2uDjDj,d2uDeDj,
+6 stateNew)
+C
+    include 'vaba_param.inc'
+C
+    dimension jElem(nblock),
+    1 props(nprops),
+    2 tempOld(nblock),
+    3 fieldOld(nblock,nfieldv),
+    4 stateOld(nblock,nstatev),
+    5 tempNew(nblock),
+    6 fieldNew(nblock,nfieldv),
+    7 ebar(nblock,ndir+nshr), detu(nblock),
+    8 uDev(nblock),
+    9 duDe(nblock,ndir+nshr), duDj(nblock),
+    * d2uDeDe(nblock,(ndir+nshr)*(ndir+nshr+1)/2),
+    1 d2uDjDj(nblock),
+    2 d2uDeDj(nblock,ndir+nshr),
+    3 stateNew(nblock,nstatev)
+C
+    character*80 cmname
+C
+    do 100 km = 1,nblock
+    user coding
+100 continue
+    return
+    end 
+```
+
+# Variables to be defined
+
+udev(nblock)
+
+$\tilde { U } _ { d e v ; }$ , the deviatoric part of the strain energy density of the primary material response. This quantity is needed only if the current material definition also includes Mullins effect (see “Mullins effect,” Section 22.6.1 of the Abaqus Analysis User’s Guide).
+
+<!-- source-page: 459 -->
+
+duDe(nblock,ndir+nshr)
+
+Derivatives of strain energy potential with respect to the components of the modified Green strain tensor, ${ \partial U } / { \partial \overline { { \varepsilon } } _ { i j } ^ { G } }$ .
+
+duDj(nblock,ndir+nshr)
+
+Derivatives of strain energy potential with respect to volume ratio, $\partial U / \partial J$
+
+d2uDeDe(nblock,(ndir+nshr)\*(ndir+nshr+1)/2)
+
+Second derivatives of strain energy potential with respect to the components of the modified Green strain tensor (using triangular storage), $\partial ^ { 2 } U / \partial \overline { { \varepsilon } } _ { i j } ^ { G } \partial \overline { { \varepsilon } } _ { k l } ^ { G }$ .
+
+d2uDjDj(nblock)
+
+Second derivatives of strain energy potential with respect to volume ratio, $\partial ^ { 2 } U / \partial J ^ { 2 }$ .
+
+d2uDeDj(nblock,ndir+nshr)
+
+Cross derivatives of strain energy potential with respect to components of the modified Green strain tensor and volume ratio, $\partial ^ { 2 } U / \bar { \partial \varepsilon } _ { i j } ^ { G } \partial J ^ { 2 }$ .
+
+stateNew(nblock,nstatev)
+
+State variables at each material point at the end of the increment. You define the size of this array by allocating space for it (see “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide, for more information).
+
+# Variables passed in for information
+
+nblock
+
+Number of material points to be processed in this call to VUANISOHYPER\_STRAIN.
+
+jElem(nblock)
+
+Array of element numbers.
+
+kIntPt
+
+Integration point number.
+
+kLayer
+
+Layer number (for composite shells).
+
+kSecPt
+
+Section point number within the current layer.
+
+cmname
+
+User-specified material name, left justified. It is passed in as an uppercase character string. Some internal material models are given names starting with the “ABQ\_” character string. To avoid conflict, you should not use “ABQ\_” as the leading string for cmname.
+
+<!-- source-page: 460 -->
+
+ndir
+
+Number of direct components in a symmetric tensor.
+
+nshr
+
+Number of indirect components in a symmetric tensor.
+
+nstatev
+
+Number of user-defined state variables that are associated with this material type (you define this as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+nfieldv
+
+Number of user-defined external field variables.
+
+nprops
+
+User-specified number of user-defined material properties.
+
+props(nprops)
+
+User-supplied material properties.
+
+tempOld(nblock)
+
+Temperatures at each material point at the beginning of the increment.
+
+tempNew(nblock)
+
+Temperatures at each material point at the end of the increment.
+
+fieldOld(nblock,nfieldv)
+
+Values of the user-defined field variables at each material point at the beginning of the increment.
+
+fieldNew(nblock,nfieldv)
+
+Values of the user-defined field variables at each material point at the end of the increment.
+
+stateOld(nblock,nstatev)
+
+State variables at each material point at the beginning of the increment.
+
+ebar(nblock,ndir+nshr)
+
+Modified Green strain tensor, $\overline { { \varepsilon } } ^ { G }$ , at each material point at the end of the increment.
+
+detu(nblock)
+
+J, determinant of deformation gradient (volume ratio) at the end of the increment.
+
+# Example: Using more than one user-defined anisotropic hyperelastic material model
+
+To use more than one user-defined anisotropic hyperelastic material model, the variable cmname can be tested for different material names inside user subroutine VUANISOHYPER\_STRAIN, as illustrated below:
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_047.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_047.md
new file mode 100644
index 00000000..e82a247a
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_047.md
@@ -0,0 +1,437 @@
+<!-- source-page: 461 -->
+
+```txt
+if (cmname(1:4) .eq. 'MAT1') then
+    call VUANISOHYPER_STRAIN1(argument_list)
+else if (cmname(1:4) .eq. 'MAT2') then
+    call VUANISOHYPER_STRAIN2(argument_list)
+end if 
+```
+
+VUANISOHYPER\_STRAIN1 and VUANISOHYPER\_STRAIN2 are the actual subroutines containing the anisotropic hyperelastic models for each material MAT1 and MAT2, respectively. Subroutine VUANISOHYPER\_STRAIN merely acts as a directory here. The argument list can be the same as that used in subroutine VUANISOHYPER\_STRAIN. The material names must be in uppercase characters since cmname is passed in as an uppercase character string.
+
+# Example: Orthotropic Saint-Venant Kirchhoff model
+
+As a simple example of the coding of subroutine VUANISOHYPER\_STRAIN, consider the generalization to anisotropic hyperelasticity of the Saint-Venant Kirchhoff model. The strain energy function of the Saint-Venant Kirchhoff model can be expressed as a quadratic function of the Green strain tensor, $\varepsilon ^ { G }$ , as
+
+$$
+U (\boldsymbol {\varepsilon} ^ {G}) = \frac {1}{2} \boldsymbol {\varepsilon} ^ {G}: \mathbf {D}: \boldsymbol {\varepsilon} ^ {G},
+$$
+
+where is the fourth-order elasticity tensor. The derivatives of the strain energy function with respect to the Green strain are given as
+
+$$
+\frac {\partial U}{\partial \varepsilon^ {G}} = \mathbf {D}: \varepsilon^ {G},
+$$
+
+$$
+\frac {\partial^ {2} U}{\partial \varepsilon^ {G} \partial \varepsilon^ {G}} = \mathbf {D}.
+$$
+
+However, subroutine VUANISOHYPER\_STRAIN must return the derivatives of the strain energy function with respect to the modified Green strain tensor, $\overline { { \varepsilon } } ^ { G }$ , and the volume ratio, J, which can be accomplished easily using the following relationship between $\varepsilon ^ { G } , \overline { { \varepsilon } } ^ { G }$ , and :
+
+$$
+\boldsymbol {\varepsilon} ^ {G} = J ^ {\frac {2}{3}} \overline {{\boldsymbol {\varepsilon}}} ^ {G} + \frac {1}{2} (J ^ {\frac {2}{3}} - 1) \mathbf {I},
+$$
+
+where is the second-order identity tensor. Thus, using the chain rule we find
+
+$$
+\frac {\partial U}{\partial \overline {{\varepsilon}} ^ {G}} = J ^ {\frac {2}{3}} \frac {\partial U}{\partial \varepsilon^ {G}},
+$$
+
+$$
+\frac {\partial U}{\partial J} = \frac {\partial \varepsilon^ {G}}{\partial J}: \frac {\partial U}{\partial \varepsilon^ {G}},
+$$
+
+<!-- source-page: 462 -->
+
+$$
+\frac {\partial^ {2} U}{\partial \overline {{\varepsilon}} ^ {G} \partial \overline {{\varepsilon}} ^ {G}} = J ^ {\frac {4}{3}} \frac {\partial^ {2} U}{\partial \varepsilon^ {G} \partial \varepsilon^ {G}},
+$$
+
+$$
+\frac {\partial^ {2} U}{\partial J ^ {2}} = \frac {\partial^ {2} \varepsilon^ {G}}{\partial J ^ {2}}: \frac {\partial U}{\partial \varepsilon^ {G}} + \frac {\partial \varepsilon^ {G}}{\partial J}: \frac {\partial^ {2} U}{\partial \varepsilon^ {G} \partial \varepsilon^ {G}}: \frac {\partial \varepsilon^ {G}}{\partial J},
+$$
+
+$$
+\frac {\partial^ {2} U}{\partial \overline {{\varepsilon}} ^ {G} \partial J} = \frac {2}{3 J} J ^ {\frac {2}{3}} \frac {\partial U}{\partial \varepsilon^ {G}} + J ^ {\frac {2}{3}} \frac {\partial^ {2} U}{\partial \varepsilon^ {G} \partial \varepsilon^ {G}}: \frac {\partial \varepsilon^ {G}}{\partial J},
+$$
+
+where
+
+$$
+\frac {\partial \pmb {\varepsilon} ^ {G}}{\partial J} = \frac {2}{3 J} J ^ {\frac {2}{3}} (\overline {{\pmb {\varepsilon}}} ^ {G} + \frac {1}{2} \mathbf {I}) = \frac {2}{3 J} (\pmb {\varepsilon} ^ {G} + \frac {1}{2} \mathbf {I})
+$$
+
+and
+
+$$
+\frac {\partial^ {2} \varepsilon^ {G}}{\partial J ^ {2}} = - \frac {1}{3 J} \frac {\partial \varepsilon^ {G}}{\partial J}.
+$$
+
+In this example an auxiliary function is used to facilitate indexing into a fourth-order symmetric tensor. The subroutine would be coded as follows:
+```prolog
+subroutine vuanisohyper_strain (
+C Read only -
+* nblock,
+* jElem, kIntPt, kLayer, kSecPt,
+* cmname,
+* ndir, nshr, nstatev, nfieldv, nprops,
+* props, tempOld, tempNew, fieldOld, fieldNew,
+* stateOld, ebar, detu,
+C Write only -
+* uDev, duDe, duDj,
+* d2uDeDe, d2uDjDj, d2uDeDj,
+* stateNew )
+C
+include 'vaba_param.inc'
+C
+dimension props(nprops),
+* tempOld(nblock),
+* fieldOld(nblock,nfieldv),
+* stateOld(nblock,nstatev),
+* tempNew(nblock),
+* fieldNew(nblock,nfieldv),
+* ebar(nblock,ndir+nshr), detu(nblock), 
+```
+
+<!-- source-page: 463 -->
+
+```txt
+* uDev(nblock), duDe(nblock,ndir+nshr), duDj(nblock),
+* d2uDeDe(nblock, *), d2uDjDj(nblock),
+* d2uDeDj(nblock,ndir+nshr),
+* stateNew(nblock,nstatev)
+
+character*80 cmname
+
+parameter( half = 0.5d0, one = 1.d0, two = 2.d0,
+* third = 1.d0/3.d0, twoths = 2.d0/3.d0, four = 4.d0,
+* dinv = 0.d0 )
+
+Orthotropic Saint-Venant Kirchhoff strain energy function
+(3D)
+
+D1111 = props(1)
+D1122 = props(2)
+D2222 = props(3)
+D1133 = props(4)
+D2233 = props(5)
+D3333 = props(6)
+D1212 = props(7)
+D1313 = props(8)
+D2323 = props(9)
+
+do k = 1, nblock
+
+d2UdE11dE11 = D1111
+d2UdE11dE22 = D1122
+d2UdE11dE33 = D1133
+d2UdE22dE11 = d2UdE11dE22
+d2UdE22dE22 = D2222
+d2UdE22dE33 = D2233
+d2UdE33dE11 = d2UdE11dE33
+d2UdE33dE22 = d2UdE22dE33
+d2UdE33dE33 = D3333
+d2UdE12dE12 = D1212
+d2UdE13dE13 = D1313
+d2UdE23dE23 = D2323
+
+xpow = exp ( log(detu(k)) * twoths )
+detuInv = one / detu(k)
+
+C 
+```
+
+<!-- source-page: 464 -->
+
+C
+
+```txt
+E11 = xpow * ebar(k,1) + half * ( xpow - one )
+E22 = xpow * ebar(k,2) + half * ( xpow - one )
+E33 = xpow * ebar(k,3) + half * ( xpow - one )
+E12 = xpow * ebar(k,4)
+E23 = xpow * ebar(k,5)
+E13 = xpow * ebar(k,6) 
+```
+
+```txt
+term1 = twothds * detuInv
+dE11Dj = term1 * (E11 + half)
+dE22Dj = term1 * (E22 + half)
+dE33Dj = term1 * (E33 + half)
+dE12Dj = term1 * E12
+dE23Dj = term1 * E23
+dE13Dj = term1 * E13
+term2 = - third * detuInv
+d2E11DjDj = term2 * dE11Dj
+d2E22DjDj = term2 * dE22Dj
+d2E33DjDj = term2 * dE33Dj
+d2E12DjDj = term2 * dE12Dj
+d2E23DjDj = term2 * dE23Dj
+d2E13DjDj = term2 * dE13Dj 
+```
+
+C
+
+```txt
+dUdE11 = d2UdE11dE11 * E11
+*    + d2UdE11dE22 * E22
+*    + d2UdE11dE33 * E33
+dUdE22 = d2UdE22dE11 * E11
+*    + d2UdE22dE22 * E22
+*    + d2UdE22dE33 * E33
+dUdE33 = d2UdE33dE11 * E11
+*    + d2UdE33dE22 * E22
+*    + d2UdE33dE33 * E33
+dUdE12 = two * d2UdE12dE12 * E12
+dUdE23 = two * d2UdE23dE23 * E23
+dUdE13 = two * d2UdE13dE13 * E13
+U = half * ( E11*dUdE11 + E22*dU
+*    + E12*dUdE12 + E13*dUdE13 +
+uDev(k) = U
+duDe(k,1) = xpow * dUdE11
+duDe(k,2) = xpow * dUdE22
+duDe(k,3) = xpow * dUdE33
+```
+
+C
+
+C
+
+<!-- source-page: 465 -->
+
+```fortran
+duDe(k,4) = xpow * dUdE12
+duDe(k,5) = xpow * dUdE23
+duDe(k,6) = xpow * dUdE13
+C
+xpow2 = xpow * xpow
+C Only update nonzero components
+d2uDeDe(k,indx(1,1)) = xpow2 * d2UdE11dE11
+d2uDeDe(k,indx(1,2)) = xpow2 * d2UdE11dE22
+d2uDeDe(k,indx(2,2)) = xpow2 * d2UdE22dE22
+d2uDeDe(k,indx(1,3)) = xpow2 * d2UdE11dE33
+d2uDeDe(k,indx(2,3)) = xpow2 * d2UdE22dE33
+d2uDeDe(k,indx(3,3)) = xpow2 * d2UdE33dE33
+d2uDeDe(k,indx(4,4)) = xpow2 * d2UdE12dE12
+d2uDeDe(k,indx(5,5)) = xpow2 * d2UdE23dE23
+d2uDeDe(k,indx(6,6)) = xpow2 * d2UdE13dE13
+C
+duDj(k) = dUdE11*dE11Dj + dUdE22*dE22Dj + dUdE33*dE33Dj
+* + two * (dUdE12*dE12Dj + dUdE13*dE13Dj
+* + dUdE23*dE23Dj)
+d2uDjDj(k) = dUdE11*d2E11DjDj + dUdE22*d2E22DjDj
+* + dUdE33*d2E33DjDj
+* + two * (dUdE12*d2E12DjDj + dUdE13*d2E13DjDj
+* + dUdE23*d2E23DjDj)
+* + d2UdE11dE11 * dE11Dj * dE11Dj
+* + d2UdE22dE22 * dE22Dj * dE22Dj
+* + d2UdE33dE33 * dE33Dj * dE33Dj
+* + two * (d2UdE11dE22 * dE11Dj * dE22Dj
+* + d2UdE11dE33 * dE11Dj * dE33Dj
+* + d2UdE22dE33 * dE22Dj * dE33Dj)
+* + four * (d2UdE12dE12 * dE12Dj * dE12Dj
+* d2UdE13dE13 * dE13Dj * dE13Dj
+* d2UdE23dE23 * dE23Dj * dE23Dj)
+C
+d2uDeDj(k,1) = xpow * (term1 * dUdE11
+* + d2UdE11dE11 * dE11Dj
+* + d2UdE11dE22 * dE22Dj
+* + d2UdE11dE33 * dE33Dj)
+d2uDeDj(k,2) = xpow * (term1 * dUdE22
+* + d2UdE22dE11 * dE11Dj
+* + d2UdE22dE22 * dE22Dj
+* + d2UdE22dE33 * dE33Dj)
+d2uDeDj(k,3) = xpow * (term1 * dUdE33
+```
+
+<!-- source-page: 466 -->
+
+```matlab
+*    + d2UdE33dE11 * dE11Dj
+*    + d2UdE33dE22 * dE22Dj
+*    + d2UdE33dE33 * dE33Dj )
+    d2uDeDj(k,4) = xpow * ( term1 * dUdE12
+*    + two * d2UdE12dE12 * dE12Dj )
+    d2uDeDj(k,5) = xpow * ( term1 * dUdE23
+*    + two * d2UdE23dE23 * dE23Dj )
+    d2uDeDj(k,6) = xpow * ( term1 * dUdE13
+*    + two * d2UdE13dE13 * dE13Dj )
+    end do
+C
+    return
+    end
+C
+    integer function index( i, j )
+C
+    include 'vaba_param.inc'
+C
+C
+Function to map index from Square to Triangular storage
+C of symmetric matrix
+C
+    ii = min(i,j)
+    jj = max(i,j)
+C
+    index = ii + jj*(jj-1)/2
+C
+    return
+    end
+```
+
+<!-- source-page: 467 -->
+
+# 1.2.12 VUCHARLENGTH: User subroutine to define characteristic element length at a material point.
+
+Product: Abaqus/Explicit
+
+# References
+
+• \*CHARACTERISTIC LENGTH   
+• “VUCHARLENGTH,” Section 4.1.32 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine VUCHARLENGTH:
+
+• is called at all material points of elements for which the material definition includes a user-defined characteristic element length and the constitutive model requires a characteristic length;   
+• allows the definition of characteristic element length at a material point as a function of element topology, nodal and material point coordinates, and material orientation;   
+• can use field variables that are passed in; and   
+• can use solution-dependent state variables that are passed in.
+
+# Defining characteristic element length
+
+The characteristic element length defined in user subroutine VUCHARLENGTH is used by Abaqus in regularization schemes needed to mitigate mesh dependency in constitutive models that include strain-softening, such as damage models (“Damage evolution and element removal for ductile metals,” Section 24.2.3 of the Abaqus Analysis User’s Guide), and concrete (“Concrete smeared cracking,” Section 23.6.1 of the Abaqus Analysis User’s Guide). It could be used with built-in Abaqus material models as well as user subroutine–based material models.
+
+The characteristic element length coming in user subroutine VUCHARLENGTH has a default value based on the geometric mean. This default value is a typical length of a line across an element for a first-order element and is half of the same typical length for a second-order element. For trusses the default value is a characteristic length along the element axis. For membranes and shells the default value is a characteristic length in the reference surface. For axisymmetric elements the default value is a characteristic length in the r–z plane only.
+
+Inside user subroutine VUCHARLENGTH you can redefine the value of the characteristic element length based on the element topology and geometry. The characteristic element length defined in user subroutine VUCHARLENGTH at a particular material point is passed to other user subroutines that are called at the same material point, such as user subroutines VFABRIC, VUMAT, VUSDFLD, and VUEOS. The characteristic element length calculated in user subroutine VUCHARLENGTH at a particular material point is also used in built-in Abaqus material models that require characteristic length and are called at the same material point.
+
+<!-- source-page: 468 -->
+
+# Array of element type and geometric properties
+
+jElType contains information about the element type and the geometry. jElType(1) provides information about the shape of the element.
+
+<table><tr><td>jElType(1)</td><td>Shape</td></tr><tr><td>1</td><td>line</td></tr><tr><td>2</td><td>triangle</td></tr><tr><td>3</td><td>quadrilateral</td></tr><tr><td>4</td><td>tetrahedron</td></tr><tr><td>5</td><td>wedge</td></tr><tr><td>6</td><td>hexahedron</td></tr></table>
+
+jElType(2) provides information about the element’s dimensionality.
+
+<table><tr><td>jElType (2)</td><td>Space</td></tr><tr><td>1</td><td>2D and plane strain</td></tr><tr><td>2</td><td>3D</td></tr><tr><td>3</td><td>axisymmetric</td></tr><tr><td>4</td><td>plane stress</td></tr></table>
+
+jElType(3) provides information about the section of the element.
+
+<table><tr><td>jElType(3)</td><td>Section</td></tr><tr><td>1</td><td>solid</td></tr><tr><td>2</td><td>shell</td></tr><tr><td>3</td><td>truss</td></tr><tr><td>4</td><td>membrane</td></tr></table>
+
+# Elements
+
+User subroutine VUCHARLENGTH can be used with membrane elements; shell elements; truss elements; and plane stress, plane strain, axisymmetric, and three-dimensional solid elements.
+
+# Special consideration for 8-node continuum shell elements
+
+For 8-node hexahedron continuum shell elements (SC8R and SC8RT), the order of nodes in the array of nodal coordinates (coordNode) passed to user subroutine VUCHARLENGTH depends on the stacking
+
+<!-- source-page: 469 -->
+
+direction. For a single element nodes 1–4 correspond to the bottom face and nodes 5–6 correspond to the top face.
+
+User subroutine interface   
+```fortran
+subroutine vucharlength(
+c Read only variables-
+    1    nblock, nfieldv, nprops, ncomp, ndim, nnode, nstatev,
+    2    kSecPt, kLayer, kIntPt, jElType, jElem,
+    3    totalTime, stepTime, dt,
+    4    cmname, coordMp, coordNode, direct, T, props,
+    5    field, stateOld,
+c Write only variables-
+    6    charLength )
+c
+    include 'vaba_param.inc'
+c
+    dimension jElType(3), jElem(nblock), coordMp(nblock,ndim),
+    1    coordNode(nblock, nnode, ndim),
+    2    direct(nblock,3,3), T(nblock,3,3), props(nprops),
+    3    stateOld(nblock, nstatev), charLength(nblock, ncomp),
+    4    field(nblock, nfieldv)
+c
+    character*80 cmname
+c
+    do 100 k = 1, nblock
+    user coding to define charLength(nblock, ncomp)
+    100 continue
+c
+    return
+    end 
+```  
+Variable to be defined   
+charLength(nblock,ncomp)
+
+Characteristic element length.
+
+Variables passed in for information
+
+nblock
+
+Number of material points to be processed in this call to user subroutine VUCHARLENGTH.
+
+<!-- source-page: 470 -->
+
+# nfieldv
+
+Number of user-defined external field variables.
+
+# nprops
+
+User-specified number of user-defined material properties.
+
+# ncomp
+
+User-specified number of components of characteristic element length. If ncomp is greater than 1, only the first component of characteristic element length would be used in the built-in Abaqus material models. However, all the components could be used in the above-mentioned user subroutines.
+
+# ndim
+
+Number of coordinate directions: 2 for two-dimensional models and 3 for three-dimensional models.
+
+# nnode
+
+Number of nodes of the element.
+
+# nstatev
+
+Number of user-defined state variables that are associated with this material type (define this as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# kSecPt
+
+Section point number within the current layer.
+
+# kLayer
+
+Layer number (for composite shells).
+
+# kIntPt
+
+Integration point number.
+
+# jElType(3)
+
+Array containing information about the element type and geometry.
+
+# jElem(nblock)
+
+Array of element numbers.
+
+# totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime-stepTime.
+
+# stepTime
+
+Value of time since the step began.
+
+# dt
+
+Time increment size.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_048.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_048.md
new file mode 100644
index 00000000..13dfd3a4
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_048.md
@@ -0,0 +1,416 @@
+<!-- source-page: 471 -->
+
+# cmname
+
+User-specified material name, left justified. It is passed in as an uppercase character string.
+
+# coordMp(nblock,ndim)
+
+Material point coordinates. It is the midplane material point for shell elements.
+
+# coordNode(nblock,nnode,ndim)
+
+Nodal coordinates.
+
+# direct(nblock,3,3)
+
+An array containing the direction cosines of the material directions in terms of the global basis directions. For material point k direct(k,1,1), direct(k,2,1), and direct(k,3,1) give the (1, 2, 3) components of the first material direction; direct(k,1,2), direct(k,2,2), and direct(k,3,2) give the second material direction and so on. For shell and membrane elements the first two directions are in the plane of the element and the third direction is the normal.
+
+# T(nblock,3,3)
+
+An array containing the direction cosines of the material orientation components relative to the element basis directions. For material point k this is the orientation that defines the material directions (direct) in terms of the element basis directions. For continuum elements T and direct are identical. For shell and membrane elements T(k,1,1) , T(k,1,2) , T(k,2,1) , T(k,2,2) , T(k,3,3) , and all other components are zero, where is the counterclockwise rotation around the normal vector that defines the orientation. If no orientation is used, T is an identity matrix.
+
+# props(nprops)
+
+User-supplied material properties.
+
+# field(nblock,nfieldv)
+
+Values of the user-defined field variables at each material point at the beginning of the increment. User subroutine VUCHARLENGTH is called before user subroutine VUSDFLD. Thus, any changes to the field variables made in user subroutine VUSDFLD are not available in user subroutine VUCHARLENGTH.
+
+# stateOld (nblock, nstatev)
+
+State variables at each material point at the beginning of the increment.
+
+<!-- source-page: 472 -->
+
+<!-- source-page: 473 -->
+
+# 1.2.13 VUCREEPNETWORK: User subroutine to define time-dependent behavior (creep) for models defined within the parallel rheological framework.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Parallel rheological framework,” Section 22.8.2 of the Abaqus Analysis User’s Guide   
+• “Nonlinear large-strain viscoelasticity with hyperelasticity,” Section 2.2.8 of the Abaqus Verification Guide   
+• \*VISCOELASTIC
+
+# Overview
+
+User subroutine VUCREEPNETWORK:
+
+• is intended to provide creep laws for nonlinear viscoelastic networks for models defined using the parallel rheological framework;   
+• can use and update solution-dependent state variables; and   
+• can be used in conjunction with user subroutine VUSDFLD to redefine any field variables before they are passed in.
+
+# Model description
+
+The user subroutine allows a creep law of the following general form to be defined:
+
+$$
+\dot {\bar {\varepsilon}} ^ {c r} = g ^ {c r} (\bar {\varepsilon} ^ {c r}, I _ {1} ^ {c r}, \bar {I} _ {1}, \bar {I} _ {2}, J, p, q, t, \theta , F V),
+$$
+
+where
+
+$$
+I _ {1} ^ {c r} = \mathbf {I}: \mathbf {C} ^ {c r},
+$$
+
+and
+
+I is the identity tensor, $\mathbf{C}^{cr}$ is the right Cauchy-Green creep strain tensor, $\dot{\bar{\varepsilon}}^{cr}$ is the equivalent creep strain rate, $\bar{\varepsilon}^{cr}$ is the equivalent creep strain, $\bar{I}_1$ is the first invariant of $\bar{\mathbf{B}}$ , $\bar{I}_2$ is the second invariant of $\bar{\mathbf{B}}$ , $J$ is the determinant of the deformation gradient, $\mathbf{F}$ , $p$ is the Kirchhoff pressure,
+
+<!-- source-page: 474 -->
+
+$q$ is the equivalent deviatoric Kirchhoff stress, $t$ is the time, $\theta$ is the temperature, and $FV$ are field variables.
+
+The left Cauchy-Green strain tensor, $\bar { \mathbf { B } } ,$ is defined as
+
+$$
+\bar {\mathbf {B}} = \bar {\mathbf {F}} \bar {\mathbf {F}} ^ {T},
+$$
+
+where is the deformation gradient with volume change eliminated, which is computed using
+
+$$
+\bar {\mathbf {F}} = J ^ {- \frac {1}{3}} \mathbf {F}.
+$$
+
+The user subroutine must define the increment of creep equivalent strain, $\Delta \bar { \varepsilon } ^ { c r }$ , as a function of the time increment, $\Delta t ,$ and the variables used in the definition of $\cdot g ^ { c r }$ , as well as the derivatives of the equivalent creep strain increment with respect to those variables. If any solution-dependent state variables are included in the definition of $\boldsymbol { g } ^ { c r }$ , they must also be integrated forward in time in this user subroutine.
+
+User subroutine interface   ```txt
+subroutine vucreepnetwork (
+C Read only -
+* nblock, networkid, nstatev, nfieldv,
+* nprops, nDg, stepTime, totalTime, dt,
+* jElem, kIntPt, kLayer, kSecPt, cmname,
+* props, coordMp, tempOld, fieldOld,
+* stateOld, tempNew, fieldNew,
+* nIarray, i_array, nRarray, r_array,
+* q, p, eqcs, TrCc,
+C Write only -
+* dg, stateNew )
+C
+include 'vaba_param.inc'
+C
+C indices for equivalent creep strain and its derivatives
+parameter( i_deqcs = 1,
+* i_DdeqcsDq = 2,
+* i_DdeqcsDeqcs = 3,
+* i_DdeqcsDilc = 4 )
+C
+C indices for strain invariants
+parameter( i_I1 = 1,
+* i_I2 = 2, 
+```
+
+<!-- source-page: 475 -->
+
+```txt
+* i_J = 3)
+C
+    dimension props(nprops),
+    * tempOld(nblock),
+    * fieldOld(nblock,nfieldv),
+    * stateOld(nblock,nstatev),
+    * tempNew(nblock),
+    * fieldNew(nblock,nfieldv),
+    * coordMp(nblock,*),
+    * jElem(nblock),
+    * i_array(nblock,nIarray),
+    * r_array(nblock,nRarray),
+    * q(nblock),
+    * p(nblock),
+    * eqcs(nblock),
+    * TrCc(nblock),
+    * stateNew(nblock,nstatev),
+    * dg(nblock,nDg)
+
+    character*80 cmname
+C
+    do 100 km = 1,nblock
+    user coding
+100 continue
+
+    return
+    end 
+```  
+Variables to be defined
+
+dg (nblock, i_deqcs)
+Equivalent creep strain increment, $\Delta\bar{\varepsilon}^{cr}$ .
+
+dg (nblock, i_DdeqcsDq)
+The derivative: $\partial\Delta\bar{\varepsilon}^{cr}/\partial q$ .
+
+dg (nblock, i_DdeqcsDeqcs)
+The derivative: $\partial\Delta\bar{\varepsilon}^{cr}/\partial\bar{\varepsilon}^{cr}$
+
+dg(nblock,i_DdeqcsDilc)
+The first invariant, $I_1^{cr}$ , of the right Cauchy-Green creep strain tensor, $\mathbf{C}^{\mathrm{cr}}$ .
+
+<!-- source-page: 476 -->
+
+# Variable that can be updated
+
+stateNew(nblock,nstatev)
+
+Array containing the user-defined solution-dependent state variables at this point.
+
+# Variables passed in for information
+
+nblock
+
+Number of material points to be processed in this call to user subroutine VUCREEPNETWORK.
+
+networkid
+
+Network identification number, which identifies the network for which creep is defined.
+
+nstatev
+
+Number of user-defined state variables that are associated with this material type (see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+nfieldv
+
+Number of user-defined external field variables.
+
+nprops
+
+User-specified number of user-defined material properties.
+
+nDg
+
+Size of array dg.
+
+stepTime
+
+Value of time since the step began.
+
+totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime-stepTime.
+
+dt
+
+Time increment size.
+
+jElem(nblock)
+
+Array of element numbers.
+
+kIntPt
+
+Integration point number.
+
+kLayer
+
+Layer number (for composite shells).
+
+kSecPt
+
+Section point number within the current layer.
+
+<!-- source-page: 477 -->
+
+# cmname
+
+Material name, left justified. It is passed in as an uppercase character string. Some internal material models are given names starting with the “ABQ\_” character string. To avoid conflict, “ABQ\_” should not be used as the leading string for cmname.
+
+# props(nprops)
+
+User-supplied material properties.
+
+# coordMp(nblock,\*)
+
+Material point coordinates. It is the midplane material point for shell elements and the centroid for beam elements.
+
+# tempOld(nblock)
+
+Temperatures at the material points at the beginning of the increment.
+
+# fieldOld(nblock,nfieldv)
+
+Values of the user-defined field variables at the material points at the beginning of the increment.
+
+# stateOld(nblock,nstatev)
+
+State variables at the material points at the beginning of the increment.
+
+# tempNew(nblock)
+
+Temperatures at the material points at the end of the increment.
+
+# fieldNew(nblock)
+
+Values of the user-defined field variables at the material points at the end of the increment.
+
+# nIarray
+
+Size of array i\_array.
+
+# i\_array(nblock,nIarray)
+
+Array containing integer arguments. Currently it is not used.
+
+# nRarray
+
+Size of array r\_array.
+
+# r\_array(nblock,i\_I1)
+
+The first invariant, , of the left Cauchy-Green strain tensor, .
+
+# r\_array(nblock,i\_I2)
+
+The second invariant, , of the left Cauchy-Green strain tensor, .
+
+# r\_array(nblock,i\_J)
+
+The determinant of the deformation gradient, .
+
+<!-- source-page: 478 -->
+
+q(nblock)
+
+Array containing equivalent deviatoric Kirchhoff stresses.
+
+p(nblock)
+
+Array containing Kirchhoff pressures.
+
+eqcs(nblock)
+
+Array containing equivalent creep strains.
+
+TrCc(nblock)
+
+Array containing the first invariants, , of the right Cauchy-Green creep strain tensor, .
+
+# Example: Power-law strain hardening model
+
+As an example of the coding of user subroutine VUCREEPNETWORK, consider the power-law strain hardening model. In this case the equivalent creep strain rate is expressed as
+
+$$
+\dot {\bar {\varepsilon}} ^ {c r} = \left(A q ^ {n} [ (m + 1) \bar {\varepsilon} ^ {c r} ] ^ {m}\right) ^ {\frac {1}{m + 1}},
+$$
+
+where
+
+$\bar{\varepsilon}^{cr}$ is the equivalent creep strain, $q$ is the equivalent deviatoric Kirchhoff stress, and $A, m,$ and $n$ are material parameters.
+
+The user subroutine would be coded as follows:
+```txt
+subroutine vucreepnetwork (
+C Read only -
+* nblock, networkid, nstatev, nfieldv,
+* nprops, nDg, stepTime, totalTime, dt,
+* jElem, kIntPt, kLayer, kSecPt, cmname,
+* props, coordMp, tempOld, fieldOld,
+* stateOld, tempNew, fieldNew,
+* nIarray, i_array, nRarray, r_array,
+* q, p, eqcs, TrCc,
+C Write only -
+* dg, stateNew )
+C
+include 'vaba_param.inc'
+C
+parameter ( one = 1.d0, half = 0.5d0 )
+parameter ( eqcsSmall = 1.d-8 )
+parameter ( rMinVal = 1.d-12 ) 
+```
+
+<!-- source-page: 479 -->
+
+C
+
+```txt
+parameter( i_deqcs = 1,
+* i_DdeqcsDq = 2,
+* i_DdeqcsDeqcs = 3,
+* i_DdeqcsDilc = 4 ) 
+```
+
+C
+
+```javascript
+dimension props(nprops),
+* tempOld(nblock),
+* fieldOld(nblock,nfieldv),
+* stateOld(nblock,nstatev),
+* tempNew(nblock),
+* fieldNew(nblock,nfieldv),
+* coordMp(nblock,*),
+* jElem(nblock),
+* i_array(nblock,nIarray),
+* r_array(nblock,nRarray),
+* q(nblock),
+* p(nblock),
+* eqcs(nblock),
+* TrCc(nblock),
+* stateNew(nblock,nstatev),
+* dg(nblock,nDg) 
+```
+
+C
+
+```txt
+character*80 cmname 
+```
+
+C
+
+C Read properties
+
+C
+
+```txt
+rA = props(1)
+rN = props(2)
+rM = props(3) 
+```
+
+C
+
+C Update equivalent creep strain and its derivatives
+
+C
+
+```txt
+do k = 1, nblock
+om1 = one / (one + rM)
+test = half - sign( half, q(k) - rMinVal )
+qInv = (one - test) / (q(k) + test)
+eqcs_t = eqcs(k)
+if (eqcs_t .le. eqcsSmall .and. q(k).gt.rMinVal ) then 
+```
+
+C Initial guess based on constant creep strain rate during increment eqcs\_t = dt\*(exp(log(rA)+rN\*log(q(k)))\* \* ((one+rM)\*dt)\*\*rM)
+
+<!-- source-page: 480 -->
+
+```txt
+end if
+test2 = half - sign( half, eqcs_t - rMinVal )
+eqcsInv = ( one - test2 ) / ( eqcs_t + test2 )
+g = dt*(exp(log(rA)+rN*log(q(k)))*
+* ((one+rM)*(test2+eqcs_t))**rM)**om1
+dg(k,i_deqcs) = g
+dg(k,i_DdeqcsDq) = qInv * rN * om1 * g
+dg(k,i_DdeqcsDeqcs) = eqcsInv * rM * om1 * g
+end do
+return
+end 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_049.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_049.md
new file mode 100644
index 00000000..ff30b44d
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_049.md
@@ -0,0 +1,421 @@
+<!-- source-page: 481 -->
+
+# 1.2.14 VUEL: User subroutine to define an element.
+
+# Product: Abaqus/Explicit
+
+WARNING: This feature is intended for advanced users only. Its use in all but the simplest test examples will require considerable coding by the user/developer. “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide, should be read before proceeding.
+
+# References
+
+• “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide   
+• “User-defined element library,” Section 32.17.2 of the Abaqus Analysis User’s Guide   
+• “UEL,” Section 1.1.28   
+• \*UEL PROPERTY  
+• \*USER ELEMENT
+
+# Overview
+
+User subroutine VUEL:
+
+• will be called for each element that is of a general user-defined element type each time element calculations are required; and   
+• (or subroutines called by user subroutine VUEL) must perform all of the calculations for the element, appropriate to the current activity in the analysis.
+
+# User subroutine interface
+
+```csv
+SUBROUTINE VUEL(nblock,rhs,amass,dtimeStable,svars,nsvars,
+1 energy,
+2 nnode,ndofel,props,nprops,jprops,njprops,
+3 coords,mcrd,u,du,v,a,
+4 jtype,jElem,
+5 time,period,dtimeCur,dtimePrev,kstep,kinc,
+6 lflags,
+7 dMassScaleFactor,
+8 predef,npredef,
+9 jdltyp,adImag)
+C
+include 'vaba_param.inc'
+C operational code keys 
+```
+
+<!-- source-page: 482 -->
+
+```javascript
+parameter ( jMassCalc = 1,
+*    jIntForceAndDtStable = 2,
+*    jExternForce = 3) 
+```
+
+```txt
+C flag indices
+parameter (iProcedure = 1,
+* iNlgeom = 2,
+* iOpCode = 3,
+* nFlags = 3) 
+```
+
+```txt
+C energy array indices
+parameter ( iElPd = 1,
+* iElCd = 2,
+* iElIe = 3,
+* iElTs = 4,
+* iElDd = 5,
+* iElBv = 6,
+* iElDe = 7,
+* iElHe = 8,
+* iElKe = 9,
+* iElTh = 10,
+* iElDmd = 11,
+* iElDc = 12,
+* nElEnergy = 12) 
+```
+
+```txt
+C predefined variables indices
+parameter ( iPredValueNew = 1,
+*    iPredValueOld = 2,
+*    nPred = 2) 
+```
+
+```txt
+C time indices
+parameter (iStepTime = 1,
+* iTotalTime = 2,
+* nTime = 2) 
+```
+
+```txt
+dimension rhs(nblock,ndofel), amass(nblock,ndofel,ndofel),
+1    dtimeStable(nblock),
+2    svars(nblock,nsvars), energy(nblock,nElEnergy),
+3    props(nprops), jprops(njprops),
+4    jElem(nblock), time(nTime), lflags(nFlags),
+5    coords(nblock, nnode, mcrd),
+6    u(nblock,ndofel), du(nblock,ndofel), 
+```
+
+<!-- source-page: 483 -->
+
+```prolog
+7 v(nblock,ndofel), a(nblock, ndofel),
+8 dMassScaleFactor(nblock),
+9 predef(nblock, nnode, npred, nPred),
+* adlmag(nblock)
+
+do kblock = 1, nblock
+    user coding to define rhs, amass, dtimeStable, svars and energy
+    end do
+
+RETURN
+END 
+```
+
+# Variables to be defined
+
+Some of the following arrays depend on the value of the lflags array.
+
+# rhs
+
+An array containing the contributions of each element to the right-hand-side vector of the overall system of equations. Depending on the settings of the lflags array, it contains either the internal force from the element or the external load calculated from the specified distributed loads.
+
+# amass
+
+An array containing the contribution of each element to the mass matrix of the overall system of equations.
+
+All nonzero entries in amass should be defined. Moreover, the mass matrix must be symmetric. There are several other requirements that apply depending on the active degrees of freedom specified. These requirements are explained in detail below.
+
+# dtimeStable
+
+A scalar value defining, for each element, the upper limit of the time increment for stability considerations. This would be the maximum time increment to be used in the subsequent increment for this element to be stable (to satisfy the Courant condition). This value depends strongly on the element formulation, and it is important that is computed appropriately.
+
+# svars
+
+An array containing the values of the solution-dependent state variables associated with each element. The number of such variables is nsvars (see below). You define the meaning of these variables.
+
+This array is passed into VUEL containing the values of these variables at the start of the current increment. In most cases they should be updated to be the values at the end of the increment. In rare cases such an update is not required.
+
+# energy
+
+The array energy contains the values of the energy quantities associated with each element. The values in this array when VUEL is called are the element energy quantities at the start of the current
+
+<!-- source-page: 484 -->
+
+increment. They should be updated to the correct values at the end of the current increment; otherwise, plots of the energy balance for the entire model will not be accurate. Depending on the element formulation, many of these energies could be zero at all times. The entries in the array are as follows:
+
+<table><tr><td>energy (nblock,iElPd)</td><td>Plastic dissipation.</td></tr><tr><td>energy (nblock,iElCd)</td><td>Creep dissipation.</td></tr><tr><td>energy (nblock,iElIe)</td><td>Internal energy.</td></tr><tr><td>energy (nblock,iElTs)</td><td>Transverse shear energy.</td></tr><tr><td>energy (nblock,iElDd)</td><td>Material damping dissipation.</td></tr><tr><td>energy (nblock,iElBv)</td><td>Bulk viscosity dissipation.</td></tr><tr><td>energy (nblock,iElDe)</td><td>Drill energy.</td></tr><tr><td>energy (nblock,iElHe)</td><td>Hourglass energy.</td></tr><tr><td>energy (nblock,iElKe)</td><td>Kinetic energy.</td></tr><tr><td>energy (nblock,iElTh)</td><td>Heat energy.</td></tr><tr><td>energy (nblock,iElDmd)</td><td>Damage dissipation.</td></tr><tr><td>energy (nblock,iElDc)</td><td>Distortion control energy.</td></tr></table>
+
+# Variables passed in for information
+
+# Arrays:
+
+# props
+
+A floating point array containing the nprops real property values defined for use with each element processed. nprops is the user-specified number of real property values. See “Defining the element properties” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide.
+
+# jprops
+
+An integer array containing the njprops integer property values defined for use with each element processed. njprops is the user-specified number of integer property values. See “Defining the element properties” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide.
+
+# coords
+
+An array containing the original coordinates of the nodes of the element. coords(kblock,k1,k2) is the k2th coordinate of the k1th node of the kblock element.
+
+<!-- source-page: 485 -->
+
+u, du, v, a
+
+Arrays containing the basic solution variables (displacements, rotations, temperatures, pressures, depending on the degree of freedom) at the nodes of the element. Values are provided as follows, illustrated below for the k1th degree of freedom of the kblock element:
+
+<table><tr><td>u(kblock,k1)</td><td>Total value of the variables (such as displacements or rotations) at the end of the current increment.</td></tr><tr><td>du(kblock,k1)</td><td>Incremental values of the variables in the current increment.</td></tr><tr><td>v(kblock,k1)</td><td>Time rate of change of the variables (velocities, rates of rotation) at the midpoint of the increment.</td></tr><tr><td>a(kblock,k1)</td><td>Accelerations of the variables at the end of the current increment.</td></tr></table>
+
+# jElem
+
+jElem(kblock) contains the element number for the kblock element.
+
+# adlmag
+
+adlmag(kblock) is the total load magnitude of the load type jdltyp (integer identifying the load number for distributed load type Un) distributed load at the end of the current increment for distributed loads of type Un.
+
+# predef
+
+An array containing the values of predefined field variables, such as temperature in an uncoupled stress/displacement analysis, at the nodes of the element (“Predefined fields,” Section 34.6.1 of the Abaqus Analysis User’s Guide).
+
+The second index, k2, indicates the local node number on the kblock element. The third index, k3, indicates the variable: the temperature is stored if the index is 1, and the predefined field variables are stored if the indices are greater than or equal to 2. The fourth index of the array, k4, is either 1 or 2, with 1 indicating the value of the field variable at the end of the increment and 2 indicating the value of the field variable at the beginning of the increment.
+
+<table><tr><td>predef (kblock, k2, 1, k4)</td><td>Temperature.</td></tr><tr><td>predef (kblock, k2, 2, k4)</td><td>First predefined field variable.</td></tr><tr><td>predef (kblock, k2, 3, k4)</td><td>Second predefined field variable.</td></tr><tr><td>Etc.</td><td>Any other predefined field variable.</td></tr></table>
+
+<!-- source-page: 486 -->
+
+predef(kblock,k2,k3,k4)
+
+Value of the (k3–1)th predefined field variable at the k2th node of the element at the beginning or the end of the increment.
+
+predef(kblock,k2,k3,1)
+
+Values of the variables at the end of the current increment.
+
+predef(kblock,k2,k3,2)
+
+Values of the variables at the beginning of the current increment.
+
+# lflags
+
+An array containing the flags that define the current solution procedure and requirements for element calculations.
+
+lflags(iProcedure)
+
+Defines the procedure type. See
+
+“Results file output format,”
+
+Section 5.1.2 of the Abaqus Analysis
+
+User’s Guide, for the key used for each procedure.
+
+lflags(iNlgeom)=0
+
+Small-displacement analysis.
+
+lflags(iNlgeom)=1
+
+Large-displacement analysis (nonlinear geometric effects included in the step; see “General and linear perturbation procedures,” Section 6.1.3 of the Abaqus Analysis User’s Guide).
+
+lflags(iOpCode)=jMassCalc
+
+Define the mass matrix amass in the beginning of the analysis.
+
+lflags(iOpCode)=jIntForceAnd-DtStable
+
+Define the element internal force.
+
+Define the stable time increment as well.
+
+lflags(iOpCode)=jExternForce
+
+Define the distributed load effect on the external force associated with the element.
+
+# dMassScaleFactor
+
+An array containing the mass scale factors for each element.
+
+# time(iStepTime)
+
+Current value of step time.
+
+<!-- source-page: 487 -->
+
+time(iTotalTime)
+
+Current value of total time.
+
+# Scalar parameters:
+
+nblock
+
+Number of user elements to be processed in this call to VUEL.
+
+dtimeCur
+
+Current time increment.
+
+dtimePrev
+
+Previous time increment.
+
+period
+
+Time period of the current step.
+
+ndofel
+
+Number of degrees of freedom in the elements processed.
+
+nsvars
+
+User-defined number of solution-dependent state variables associated with the element (“Defining the number of solution-dependent variables that must be stored within the element” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+nprops
+
+User-defined number of real property values associated with the elements processed (“Defining the element properties” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+njprops
+
+User-defined number of integer property values associated with the elements processed (“Defining the element properties” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+mcrd
+
+mcrd is defined as the maximum of the user-defined maximum number of coordinates needed at any node point (“Defining the maximum number of coordinates needed at any nodal point” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide) and the value of the largest active degree of freedom of the user element that is less than or equal to 3. For example, if you specify that the maximum number of coordinates is 1 and the active degrees of freedom of the user element are 2, 3, and 6 mcrd will be 3. If you specify that the maximum number of coordinates is 2 and the active degree of freedom of the user element is 11, mcrd will be 2.
+
+nnode
+
+User-defined number of nodes on the elements (“Defining the number of nodes associated with the element” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+<!-- source-page: 488 -->
+
+# jtype
+
+Integer defining the element type. This is the user-defined integer value n in element type VUn (“Assigning an element type key to a user-defined element” in “User-defined elements,” Section 32.17.1 of the Abaqus Analysis User’s Guide).
+
+# kstep
+
+Current step number.
+
+# kinc
+
+Current increment number.
+
+# npredef
+
+Number of predefined field variables, including temperature. For user elements Abaqus/Explicit uses one value for each field variable per node.
+
+# VUEL conventions
+
+The solution variables (displacement, velocity, etc.) are arranged on a node/degree of freedom basis. The degrees of freedom of the first node are first, followed by the degrees of freedom of the second node, etc. The degrees of freedom that will be updated automatically in Abaqus/Explicit are: 1–3 (displacements), 4–6 (rotations), 8 (pressure), and 11 (temperature). Depending on the procedure type (see below), only some of the degrees of freedom listed above will be updated. Other degrees of freedom will not be updated by the time integration procedure in Abaqus/Explicit and, hence, should not be used.
+
+The mass matrix defined in user subroutine VUEL must be symmetric. In addition, the following requirements apply:
+
+• The mass matrix entries associated with the translational degrees of freedom for a particular node must be diagonal. Moreover, these diagonal entries must be equal to each other.   
+• There must be no coupling (off-diagonal) entries specified between degrees of freedom of different kinds. For example, you cannot specify nonzero mass matrix entries to couple the translational degrees of freedom to the rotational degrees of freedom.   
+• There must be no coupling (off-diagonal) entries specified between degrees of freedom belonging to different nodes.
+
+You must be using appropriate lumping techniques to provide a mass matrix that follows these requirements. For the rotational degrees of freedom at a particular node in three-dimensional analyses, you can specify a fully populated symmetric 3 × 3 inertia tensor.
+
+# Usage with general nonlinear procedures
+
+The following illustrates the use in explicit dynamic procedures:
+
+# Direct-integration explicit dynamic analysis (lflags(iProcedure)=17)
+
+• Automatic updates for degrees of freedom 1–6, 8, and 11.
+
+<!-- source-page: 489 -->
+
+• The governing equations are as described in “Explicit dynamic analysis,” Section 6.3.3 of the Abaqus Analysis User’s Guide.   
+• Coding for the operational code lflags(iOpCode)=jExternForceis optional.
+
+# Transient fully coupled thermal-stress analysis (lflags(iProcedure)=74)
+
+• Automatic updates for degrees of freedom 1–6 and 11.   
+• The governing equations are as described in “Fully coupled thermal-stress analysis in Abaqus/Explicit” in “Fully coupled thermal-stress analysis,” Section 6.5.3 of the Abaqus Analysis User’s Guide.   
+• Coding for the operational code lflags(iOpCode)=jExternForce is optional.
+
+# Example: Structural user element
+
+A structural user element has been created to demonstrate the usage of subroutine VUEL. These userdefined elements are applied in a number of analyses. The following excerpt is from the verification problem that invokes the structural user element in an explicit dynamic procedure:
+
+```txt
+*USER ELEMENT, NODES=2, TYPE=VU1, PROPERTIES=4, COORDINATES=3, VARIABLES=12
+1, 2, 3
+*ELEMENT, TYPE=VU1
+101, 101, 102
+*ELGEN, ELSET=VUTRUSS
+101, 5
+*UEL PROPERTY, ELSET=VUTRUSS
+0.002, 2.1E11, 0.3, 7200. 
+```
+
+The user element consists of two nodes that are assumed to lie parallel to the x-axis. The element behaves similarly to a linear truss element. The supplied element properties are the cross-sectional area, Young’s modulus, Poisson’s ratio, and density, respectively.
+
+The next excerpt shows the listing of the subroutine. The user subroutine has been coded for use in an explicit dynamic analysis. The names of the verification input files associated with the subroutine and these procedures can be found in “VUEL,” Section 4.1.33 of the Abaqus Verification Guide.
+
+```csv
+subroutine vuel(
+* nblock,
+* rhs, amass, dtimeStable,
+* svars, nsvars,
+* energy,
+* nnode, ndofel,
+* props, nprops,
+* jprops, njprops,
+* coords, ncrd,
+* u, du, v, a, 
+```
+
+<!-- source-page: 490 -->
+
+```txt
+* jtype, jElem,
+* time, period, dtimeCur, dtimePrev, kstep, kinc, lflags,
+* dMassScaleFactor,
+* predef, npredef,
+* ndload, adImag)
+
+include 'vaba_param.inc'
+
+c operation code
+parameter ( jMassCalc = 1,
+* jIntForceAndDtStable = 4)
+
+c flags
+parameter (iProcedure = 1,
+* iNlgeom = 2,
+* iOpCode = 3,
+* nFlags = 3)
+
+c procedure flags
+parameter ( jDynExplicit = 17 )
+
+c time
+parameter (iStepTime = 1,
+* iTotalTime = 2,
+* nTime = 2)
+
+c energies
+parameter ( iElPd = 1,
+* iElCd = 2,
+* iElIe = 3,
+* iElTs = 4,
+* iElDd = 5,
+* iElBv = 6,
+* iElDe = 7,
+* iElHe = 8,
+* iElKe = 9,
+* iElTh = 10,
+* iElDmd = 11,
+* iElDc = 12,
+* nElEnergy = 12)
+
+parameter (factorStable = 0.99d0) 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_050.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_050.md
new file mode 100644
index 00000000..09141b04
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_050.md
@@ -0,0 +1,374 @@
+<!-- source-page: 491 -->
+
+```python
+parameter (zero = 0.d0, half = 0.5d0, one = 1.d0, two=2.d0)
+c
+    dimension rhs(nblock,ndofel), amass(nblock,ndofel,ndofel),
+    *    dtimeStable(nblock),
+    *    svars(nblock,nsvars), energy(nblock,nElEnergy),
+    *    props(nprops), jprops(njprops),
+    *    jElem(nblock), time(nTime), l(nFlags),
+    *    coords(nblock,nnode,ncrd), u(nblock,ndofel),
+    *    du(nblock,ndofel), v(nblock,ndofel), a(nblock, ndofel),
+    *    predef(nblock, nnode, npred, nPred), adlmag(nblock),
+    *    dMassScaleFactor(nblock)
+
+c    Notes:
+c    Define only nonzero entries; the arrays to be defined
+c    have been zeroed out just before this call
+
+if (jtype .eq. 1001 .and.
+    *    lflags(iProcedure).eq.jDynExplicit) then
+
+    area0 = props(1)
+    eMod = props(2)
+    anu = props(3)
+    rho = props(4)
+
+    eDampTra = zero
+    amassFact0 = half*area0*rho
+
+    if ( lflags(iOpCode).eq.jMassCalc ) then
+    do kblock = 1, nblock
+
+c    use original distance to compute mass
+    alenX0 = (coords(kblock,2,1) - coords(kblock,1,1))
+    alenY0 = (coords(kblock,2,2) - coords(kblock,1,2))
+    alenZ0 = (coords(kblock,2,3) - coords(kblock,1,3))
+    alen0 = sqrt(alenX0*alenX0 + alenY0*alenY0 +
+    *    alenZ0*alenZ0)
+    am0 = amassFact0*alen0
+    amass(kblock,1,1) = am0
+    amass(kblock,2,2) = am0
+    amass(kblock,3,3) = am0
+    amass(kblock,4,4) = am0
+    amass(kblock,5,5) = am0 
+```
+
+<!-- source-page: 492 -->
+
+```txt
+amass(kblock,6,6) = am0
+end do
+else if ( lflags(iOpCode) .eq.
+*    jIntForceAndDtStable) then
+do kblock = 1, nblock
+    alenX0 = (coords(kblock,2,1) - coords(kblock,1,1))
+    alenY0 = (coords(kblock,2,2) - coords(kblock,1,2))
+    alenZ0 = (coords(kblock,2,3) - coords(kblock,1,3))
+    alen0 = sqrt(alenX0*alenX0 + alenY0*alenY0 +
+*    alenZ0*alenZ0)
+    vol0 = area0*alen0
+    amElem0 = two*amassFact0*alen0
+    alenX = alenX0
+*    + (u(kblock,4) - u(kblock,1))
+    alenY = alenY0
+*    + (u(kblock,5) - u(kblock,2))
+    alenZ = alenZ0
+*    + (u(kblock,6) - u(kblock,3))
+    alen = sqrt(alenX*alenX + alenY*alenY + alenZ*alenZ)
+    area = vol0/alen
+    ak = area*eMod/alen
+c    stable time increment for translations
+    dtimeStable(kblock) = factorStable*sqrt(amElem0/ak)
+c    force = E * logarithmic strain *current area
+    strainLog = log(alen/alen0)
+    fElasTra = eMod*strainLog*area
+    forceTra = fElasTra
+c    assemble internal load in RHS
+    rhs(kblock,1) = -forceTra
+    rhs(kblock,4) = forceTra
+c    internal energy calculation
+    alenOld = svars(kblock,1)
+    fElasTraOld = svars(kblock,2)
+    energy(kblock, iElIe) = energy(kblock, iElIe) + 
+```
+
+<!-- source-page: 493 -->
+
+```fortran
+* half*(fElasTra+fElasTraOld)*(alen - alenOld)
+c update state variables
+    svars(kblock,1) = alen
+    svars(kblock,2) = fElasTra
+    end do
+    end if
+    end if
+c
+    return
+    end 
+```
+
+<!-- source-page: 494 -->
+
+<!-- source-page: 495 -->
+
+# 1.2.15 VUEOS: User subroutine to define equation of state material model.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Equation of state,” Section 25.2.1 of the Abaqus Analysis User’s Guide   
+• \*EOS   
+“Equation of state material,” Section 2.2.20 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine VUEOS:
+
+• can be used to define the hydrodynamic material model in which the material’s volumetric response is determined by the user-defined equation of state;   
+• will be called for blocks of material calculation points for which the material definition contains a user-defined equation of state;   
+• can use and update solution-dependent state variables; and   
+• can use any field variables that are passed in.
+
+# User subroutine interface
+
+```python
+subroutine vueos (
+C Read only (unmodifiable) variables -
+    1    nblock,
+    2    jElem, kIntPt, kLayer, kSecPt,
+    3    steTime, totalTime, dt, cmname,
+    4    nstatev, nfieldv, nprops,
+    5    props, tempOld, tempNew, fieldOld, fieldNew,
+    6    stateOld, charLength, coordMp,
+    7    densityMean, refDensity, densityNew,
+    8    dkk, Em,
+C Write only (modifiable) variables -
+    8    press, dPdRho, dPdEm,
+    9    stateNew )
+C
+    include 'vaba_param.inc'
+C
+    dimension props(nprops),
+    1    tempOld(nblock),
+    2    fieldOld(nblock, nfieldv), 
+```
+
+<!-- source-page: 496 -->
+
+```matlab
+3 stateOld(nblock, nstatev),
+4 tempNew(nblock),
+5 fieldNew(nblock, nfieldv),
+6 charLength(nblock), coordMp(nblock, *),
+7 densityMean(nblock), refDensity(nblock),
+8 densityNew(nblock),
+9 dkk(nblock), Em(nblock),
+1 press(nblock), dPdRho(nblock), dPdEm(nblock),
+2 stateNew(nblock)
+C
+    character*80 cmname
+C
+    do 100 km = 1, nblock
+    user coding to define/update press, dPdRho, dPdEm
+100 continue
+    return
+    end 
+```
+
+# Variables to be defined
+
+press(nblock)
+
+The material point pressure stress, p.
+
+dPdRho(nblock)
+
+The derivative of the pressure with respect to the density, $\partial p / \partial \rho .$ This quantity is needed for the evaluation of the effective moduli of the material, which enters the stable time increment calculation.
+
+dPdEm(nblock)
+
+The derivative of the pressure with respect to the internal energy, $\partial p / \partial E _ { m }$ . This quantity is needed for the iterative Newton loop used outside of the user subroutine to solve for pressure.
+
+# Variable that can be updated
+
+stateNew(nblock,nstatev)
+
+State variables at each material point at the end of the increment. You define the size of this array by allocating space for it (see “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide, for more information).
+
+<!-- source-page: 497 -->
+
+# Variables passed in for information
+
+nblock
+
+Number of material points to be processed in this call to VUEOS.
+
+jElem(nblock)
+
+Array of element numbers.
+
+kIntPt
+
+Integration point number.
+
+kLayer
+
+Layer number.
+
+kSecPt
+
+Section point number within the current layer.
+
+stepTime
+
+Value of time since the step began.
+
+totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime - stepTime.
+
+dt
+
+Time increment size.
+
+cmname
+
+User-specified material name, left justified. It is passed in as an uppercase character string. Some internal material models are given names starting with the “ABQ\_” character string. To avoid conflict, you should not use “ABQ\_” as the leading string for cmname.
+
+nstatev
+
+Number of user-defined state variables that are associated with this material type (you define the number as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+nfieldv
+
+Number of user-defined external field variables.
+
+nprops
+
+User-specified number of user-defined material properties.
+
+props(nprops)
+
+User-supplied material properties.
+
+<!-- source-page: 498 -->
+
+tempOld(nblock)
+
+Temperatures at each material point at the beginning of the increment.
+
+tempNew(nblock)
+
+Temperatures at each material point at the end of the increment.
+
+fieldOld(nblock,nfieldv)
+
+Values of the user-defined field variables at each material point at the beginning of the increment.
+
+fieldNew(nblock,nfieldv)
+
+Values of the user-defined field variables at each material point at the end of the increment.
+
+stateOld(nblock,nstatev)
+
+State variables at each material point at the beginning of the increment.
+
+charLength(nblock)
+
+Characteristic element length, which is either the default value based on the geometric mean or the user-defined characteristic element length defined in user subroutine VUCHARLENGTH. The default value is a typical length of a line across an element for a first-order element; it is half the same typical length for a second-order element. For beams, pipes, and trusses, the default value is a characteristic length along the element axis. For membranes and shells it is a characteristic length in the reference surface. For axisymmetric elements it is a characteristic length in the r–z plane only. For cohesive elements it is equal to the constitutive thickness.
+
+coordMp(nblock,\*)
+
+Material point coordinates.
+
+densityMean(nblock)
+
+The mean density.
+
+refDensity(nblock)
+
+The reference density.
+
+densityNew(nblock)
+
+The current density for this increment.
+
+dkk(nblock)
+
+The volumetric strain increment.
+
+Em(nblock)
+
+The element specific internal energy (per unit mass)
+
+Example: User subroutine VUEOS to reproduce results obtained with \*EOS, TYPE=USUP
+
+As a simple example of coding of user subroutine VUEOS, consider the following form of the Mie-Grüneisen equation of state with 0.0 and a linear dependency between pressure and internal energy:
+
+<!-- source-page: 499 -->
+
+$$
+p = \rho_ {0} c _ {0} ^ {2} \eta (1 - \frac {\Gamma_ {0} \eta}{2}) + \Gamma_ {0} \rho_ {0} E _ {m},
+$$
+
+where $\eta = 1 - \rho _ { 0 } / \rho .$ . Therefore, the results obtained with user subroutine VUEOS should be the same as the results obtained with the linear $U _ { s } - U _ { p }$ type of EOS already available in “Equation of state material,” Section 2.2.20 of the Abaqus Verification Guide.
+
+The code in user subroutine VUEOS must return the pressure, , as in the above equation; the derivative of the pressure with respect to the density, $\partial p / \partial \rho ;$ and the derivative of the pressure with respect to the energy, $\partial p / \partial E _ { m }$ . For the case considered here, these values are
+
+$$
+\frac {\partial p}{\partial \rho} = \frac {\rho_ {0} ^ {2} c _ {0} ^ {2}}{\rho^ {2}} (1 - \Gamma_ {0} \eta),
+$$
+
+$$
+\frac {\partial p}{\partial E _ {m}} = \Gamma_ {0} \rho_ {0}.
+$$```fortran
+subroutine vueos (
+C Read only (unmodifiable) variables -
+    1    nblock,
+    2    jElem, kIntPt, kLayer, kSecPt,
+    3    steTime, totalTime, dt, cmname,
+    4    nstatev, nfieldv, nprops,
+    5    props, tempOld, tempNew, fieldOld, fieldNew,
+    6    stateOld, charLength, coordMp,
+    7    densityMean, refDensity, densityNew,
+    8    dkk, Em,
+C Write only (modifiable) variables -
+    8    press, dPdRho, dPdEm,
+    9    stateNew)
+C
+    include 'vaba_param.inc'
+C
+    dimension props(nprops),
+    1    tempOld(nblock),
+    2    fieldOld(nblock,nfieldv),
+    3    stateOld(nblock,nstatev),
+    4    tempNew(nblock),
+    5    fieldNew(nblock,nfieldv),
+    6    charLength(nblock), coordMp(nblock,*),
+    7    densityMean(nblock), refDensity(nblock),
+    8    densityNew(nblock),
+    9    dkk(nblock), Em(nblock),
+    1    press(nblock), dPdRho(nblock), dPdEm(nblock), 
+```
+
+<!-- source-page: 500 -->
+
+```txt
+2 stateNew(nblock)
+C
+    character*80 cmname
+C
+    parameter ( zero = 0.d0, one = 1.d0, half = 0.5d0 )
+C
+    c0 = props(1)
+    gamma0 = props(2)
+    c02 = c0*c0
+C
+    do k=1, nblock
+    rho0 = refDensity(k)
+    eta = one - rho0/densityNew(k)
+    f1 = rho0*c02*eta*(one-half*gamma0*eta)
+    f2 = gamma0*rho0
+    press(k) = f1 + f2*Em(k)
+C dP/dEm
+    dPdEm(k) = f2
+C dP/dRho
+    dPdRho(k) = c02*(rho0/densityNew(k))**2*(one-gamma0*eta)
+    end do
+C
+    return
+    end 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_051.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_051.md
new file mode 100644
index 00000000..d852eaad
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_051.md
@@ -0,0 +1,379 @@
+<!-- source-page: 501 -->
+
+# 1.2.16 VUFIELD: User subroutine to specify predefined field variables.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Predefined fields,” Section 34.6.1 of the Abaqus Analysis User’s Guide   
+• \*FIELD
+
+# Overview
+
+User subroutine VUFIELD:
+
+• allows you to prescribe predefined field variables at the nodes of a model—the predefined field variables at a node can be updated individually, or a number of field variables at the nodes can be updated simultaneously;   
+• can be called for blocks of nodes for which the field variable values are defined in the subroutine;   
+• ignores any field variable values specified directly;   
+• can be used to modify field variable values read from a results file; and   
+• can be used in conjunction with user subroutine VUSDFLD such that the field variables that are passed in from VUFIELD and interpolated to the material points can be modified (such changes are local to material point values, and nodal field variable values remain unaffected).
+
+# Updating field variables
+
+Two different methods are provided for updating field variables.
+
+# Individual variable updates
+
+By default, only one field variable is updated at a time for given nodes or a given node set in user subroutine VUFIELD. The user subroutine is called whenever a current value of a field variable is needed for the nodes that are listed in the field variable definition. This method is ideal for cases in which the field variables are independent of each other.
+
+# Simultaneous variable updates
+
+User subroutine VUFIELD can also be used to update multiple field variables simultaneously for given nodes or a given node set. This method is well-suited for cases in which there are dependencies between some of the field variables. In this case you must specify the number of field variables to be updated simultaneously, and the user subroutine will be called each time the field variable values are needed.
+
+<!-- source-page: 502 -->
+
+User subroutine interface   
+```txt
+SUBROUTINE VUFIELD(FIELD, NBLOCK, NFIELD, KFIELD, NCOMP, 1 KSTEP, JFLAGS, JNODEID, TIME, 2 COORDS, U, V, A)
+
+C
+INCLUDE 'VABA_PARAM.INC'
+
+C indices for the time array TIME
+PARAMETER( i_ufld_Current = 1,
+* i_ufld_Increment = 2,
+* i_ufld_Period = 3,
+* i_ufld_Total = 4 )
+
+C indices for the coordinate array COORDS
+PARAMETER( i_ufld_CoordX = 1,
+* i_ufld_CoordY = 2,
+* i_ufld_CoordZ = 3 )
+
+C indices for the displacement array U
+PARAMETER( i_ufld_SpaDisplX = 1,
+* i_ufld_SpaDisplY = 2,
+* i_ufld_SpaDisplZ = 3,
+* i_ufld_RotDisplX = 4,
+* i_ufld_RotDisplY = 5,
+* i_ufld_RotDisplZ = 6,
+* i_ufld_AcoPress = 7,
+* i_ufld_Temp = 8 )
+
+C indices for the velocity array V
+PARAMETER( i_ufld_SpaVelX = 1,
+* i_ufld_SpaVelY = 2,
+* i_ufld_SpaVelZ = 3,
+* i_ufld_RotVelX = 4,
+* i_ufld_RotVelY = 5,
+* i_ufld_RotVelZ = 6,
+* i_ufld_DAcoPress = 7,
+* i_ufld_DTemp = 8 )
+
+C indices for the acceleration array A
+PARAMETER( i_ufld_SpaAccelX = 1, 
+```
+
+<!-- source-page: 503 -->
+
+```c
+* i_ufld_SpaAccelY = 2,
+* i_ufld_SpaAccelZ = 3,
+* i_ufld_RotAccelX = 4,
+* i_ufld_RotAccelY = 5,
+* i_ufld_RotAccelZ = 6,
+* i_ufld_DDAcoPress = 7,
+* i_ufld_DDTemp = 8)
+
+C indices for JFLAGS
+PARAMETER( i_ufld_kInc = 1,
+* i_ufld_kPass = 2 )
+
+C
+DIMENSION FIELD(NBLOCK, NCOMP, NFIELD)
+DIMENSION JFLAGS(2), JNODEID(NBLOCK), TIME(4),
+* COORDS(3, NBLOCK)
+DIMENSION U(8, NBLOCK), V(8, NBLOCK), A(8, NBLOCK)
+
+C
+user coding to define FIELD
+
+RETURN
+END 
+```
+
+# Variable to be defined
+
+# FIELD(NBLOCK,NCOMP,NFIELD)
+
+Array of field variable values at a collective number of nodes NBLOCK (see NBLOCK below). When updating one field variable at a time, only the value of the specified field variable KFIELD must be returned. In this case NFIELD is passed into user subroutine VUFIELD with a value of 1, and FIELD is thus dimensioned as FIELD(NBLOCK,NCOMP,1). When updating all field variables simultaneously, the values of the specified number of field variables must be returned. In this case FIELD is dimensioned as FIELD(NBLOCK,NCOMP,NFIELD), where NFIELD is the number of field variables specified and KFIELD, which is set to −1, has no meaning.
+
+If fields are applied to nodes that are not part of pipe, beam, or shell elements, only one value of each field variable is required (NCOMP=1), and the user subroutine is invoked in a single pass. For nodes that are part of pipe, beam, or shell elements, VUFIELD is invoked in two passes per increment for these elements, and the number of values to be returned depends on the mode of temperature and field variable input selected for the beam or shell section. The following cases are possible:
+
+1. Field variables are given as values at the points on the shell or beam section. For a beam section the number of values required is determined by the particular section type specified, as described in “Beam cross-section library,” Section 29.3.9 of the Abaqus Analysis User’s Guide. For a shell section temperatures and field variables are given as values at n equally spaced points through
+
+<!-- source-page: 504 -->
+
+each layer of a shell section. In the first pass NCOMP is passed in with the value of 1 to define the field variable values at the first point. The second pass is used to define field variables at the remaining points.
+
+2. Field variables for the shell or beam section are given as values at the origin of the cross-section together with gradients along the cross-section. The number of gradient values required is 2 for three-dimensional beams, 1 for two-dimensional beams, and 1 for shells. In the first pass NCOMP is passed in with the value of 1 to define the field variable values at the origin of the cross-section. The gradients are defined in the second pass.
+
+Because field variables can also be defined directly, it is important to understand the hierarchy used in situations with conflicting information (see “Predefined fields,” Section 34.6.1 of the Abaqus Analysis User’s Guide).
+
+When the array FIELD is passed into user subroutine VUFIELD, it will contain either the field variable values from the previous increment or those values obtained from the results file if this method was used. You can then modify these values within this subroutine.
+
+# Variables passed in for information
+
+# NBLOCK
+
+User-specified number of nodes to be processed as a block in this call to VUFIELD. The value is equal to the total number of nodes given in a node set when blocking is disabled. When blocking is enabled, NBLOCK is equal to a predefined number set in Abaqus/Explicit. You can also modify NBLOCK by specifying a blocking size in the Abaqus/Explicit analysis.
+
+# NFIELD
+
+User-specified number of field variables to be updated. The default value is 1.
+
+# KFIELD
+
+User-specified field variable number. This variable is meaningful only when updating individual field variables at a time; otherwise, the value is set to −1.
+
+# NCOMP
+
+Maximum number of section values to be defined for any node in the model in the current pass. The first pass to user subroutine VUFIELD has NCOMP passed in with the value of 1.
+
+# KSTEP
+
+Current step number.
+
+# JFLAGS(i\_ufld\_kInc)
+
+Increment number for step KSTEP.
+
+# JFLAGS(i\_ufld\_kPass)
+
+This flag is equal to 1 for the first pass to user subroutine VUFIELD and is equal to 2 for the second pass.
+
+<!-- source-page: 505 -->
+
+# JNODEUID(NBLOCK)
+
+Array for user-defined node numbers. This array is dimensioned based on the size of NBLOCK, and the contained node numbers are identical to those defined in the input file. You can perform additional interdependent field variable operations by using nodal indices stored in this array.
+
+# TIME(4)
+
+Array for information of analysis time. You can retrieve any time information from this array by using the parameters given above. TIME(i\_ufld\_Current) stores the current analysis time, TIME(i\_ufld\_Increment) gives the time increment at this instance, TIME(i\_ufld\_Period) is the time period of the current step, and TIME(i\_ufld\_Total) is the total analysis time up to this point. You can use this time information to perform possible time-dependent field variable operations.
+
+# COORDS(3,NBLOCK)
+
+Coordinates for nodes in the array JNODEUID. This array stores current coordinates of nodes in which the order of coordinates stored corresponds to the order of nodes listed in the array JNODEUID. The coordinates can be retrieved by using the parameters given above. You can make use of COORDS to define possible position-dependent field variable operations.
+
+# U(8,NBLOCK), V(8,NBLOCK), and A(8,NBLOCK)
+
+Arrays containing solution variables of displacements, rotations, temperatures, and pressures and their corresponding temporal derivatives. The order in which these solutions are stored follows the order defined in the array JNODEUID. For a specific node its solution variables can be retrieved by using the parameter indices given above. Depending on the degrees of freedom, some solution variables are not valid for a given node. The displacement values correspond to the current increment. However, the acceleration is from a configuration that is one increment behind, and the velocity is such that it is consistent with the displacement increment and the time increment between the two successive configurations.
+
+<!-- source-page: 506 -->
+
+<!-- source-page: 507 -->
+
+# 1.2.17 VUFLUIDEXCH: User subroutine to define the mass flow rate/heat energy flow rate for fluid exchange.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Fluid exchange definition,” Section 11.5.3 of the Abaqus Analysis User’s Guide   
+• \*FLUID EXCHANGE   
+• \*FLUID EXCHANGE ACTIVATION   
+• \*FLUID EXCHANGE PROPERTY
+
+# Overview
+
+User subroutine VUFLUIDEXCH:
+
+• can be used to define mass flow rate and/or heat energy flow rate for fluid exchange;   
+• can be used when built-in fluid exchange property types cannot satisfactorily model the mass/heat energy flow;   
+• can use and update solution-dependent state variables;   
+• can use any field variables that are passed in; and   
+• requires that the derivatives of mass/heat energy flow rates be defined with respect to pressure and temperature in the primary and secondary fluid cavities.
+
+# Conventions for defining mass flow/heat energy flow rate
+
+A positive mass/heat energy flow rate indicates flow from the primary fluid cavity to the secondary fluid cavity. A negative value for mass flow rate will be ignored if the fluid exchange is between a cavity and its environment.
+
+# User subroutine interface
+
+```txt
+subroutine vufluidexch(
+C Read only (unmodifiable)variables -
+1    nstatev, nfieldv, nprops,
+2    steppTime, totalTime, dt,
+3    jCavType, fluExchName, effArea, amplitude,
+4    props, lExchEnv, pcavNew, pcavOld,
+5    ctempNew, ctempOld, cvol, cmass,
+6    rMix, CpMix, DCpDtemp,
+7    field, stateOld, 
+```
+
+<!-- source-page: 508 -->
+
+```python
+C Write only (modifiable) variables
+    8 stateNew, rMassRate, rEneRate,
+    9 DMassRateDPcav, DMassRateDTemp,
+    * DEneRateDPcav, DEneRateDTemp)
+c
+    include 'vaba_param.inc'
+c
+    dimension props(nprops),
+    1 pcavNew(2), pcavOld(2),
+    2 ctempNew(2), ctempOld(2), cvol(2), cmass(2),
+    3 rMix(2), CpMix(2), dCpDtemp(2),
+    4 field(nfieldv),
+    5 stateOld(nstatev), stateNew(nstatev),
+    6 DMassRateDPcav(2), DMassRateDTemp(2),
+    7 DEneRateDPcav(2), DEneRateDTemp(2)
+
+c Fluid cavity type
+    parameter( iHydraulic = 1,
+    * iAdiabaticGas = 2,
+    * iIsothermalGas = 3)
+
+    character*80 fluExchName
+
+c User coding to calculate mass flow rate,
+c heat energy flow rate and its derivatives with respect
+c to fluid cavity pressure and temperature.
+
+    return
+    end 
+```
+
+# Variables to be defined
+
+# rMassRate
+
+Mass flow rate. The mass flow rate is negative if the flow is into the primary cavity.
+
+# DMassRateDPcav(2)
+
+Derivative of mass flow rate with respect to pressure in primary and secondary fluid cavities.
+
+# DMassRateDTemp(2)
+
+Derivative of mass flow rate with respect to temperature in primary and secondary fluid cavities.
+
+# rEneRate
+
+Heat energy flow rate. The energy flow rate is negative if the flow is into the primary cavity.
+
+<!-- source-page: 509 -->
+
+# DEneRateDPcav(2)
+
+Derivative of heat energy flow rate with respect to pressure in primary and secondary fluid cavities.
+
+# DEneRateDTemp(2)
+
+Derivative of heat energy flow rate with respect to temperature in primary and secondary fluid cavities.
+
+# Variable that can be updated
+
+# stateNew(nstatev)
+
+State variable for fluid exchange at the end of the increment. You define the size of this array by allocating space for it (see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide, for more information).
+
+# Variables passed in for information
+
+# nstatev
+
+Number of user-defined state variables that are associated with this fluid exchange (you define this as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# nfieldv
+
+Number of user-defined external field variables.
+
+# nprops
+
+User-specified number of user-defined fluid exchange properties required to define mass/heat energy flow rate.
+
+# stepTime
+
+Value of time since the step began.
+
+# totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime−stepTime.
+
+# dt
+
+Time increment size.
+
+# jCavType
+
+Indicator of fluid cavity type: 1 for fluid cavity with hydraulic fluids, 2 for fluid cavity with adiabatic gases, and 3 for fluid cavity with isothermal gases.
+
+# fluExchName
+
+User-specified fluid exchange name.
+
+# effArea
+
+Effective area for fluid exchange.
+
+<!-- source-page: 510 -->
+
+amplitude
+
+Current value of the amplitude referenced for this fluid exchange. You must multiply the flow rates by the current amplitude value within the user subroutine if the amplitude is required.
+
+props(nprop)
+
+User-defined fluid exchange properties.
+
+lExchEnv
+
+The fluid exchange is to the environment if lExchEnv=1 and to another fluid cavity if lExchEnv=0.
+
+pcavNew(2)
+
+Pressure in primary and secondary fluid cavities at the end of the increment.
+
+pcavOld(2)
+
+Pressure in primary and secondary fluid cavities at the beginning of the increment.
+
+ctempNew(2)
+
+Temperature in primary and secondary fluid cavities at the end of the increment.
+
+ctempOld(2)
+
+Temperature in primary and secondary fluid cavities at the beginning of the increment.
+
+cvol(2)
+
+Volume of primary and secondary fluid cavities.
+
+cmass(2)
+
+Mass of fluid in primary and secondary fluid cavities.
+
+rMix(2)
+
+Gas constant of mixture in primary and secondary fluid cavities.
+
+CpMix(2)
+
+Specific heat of mixture in primary and secondary fluid cavities.
+
+DCpDtemp(2)
+
+Derivative of specific heat with respect to temperature for primary and secondary fluid cavities.
+
+field(nfieldv)
+
+Field variables at orifice.
+
+stateOld(nstatev)
+
+State variables for fluid exchange at the beginning of the increment.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_052.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_052.md
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+<!-- source-page: 511 -->
+
+# 1.2.18 VUFLUIDEXCHEFFAREA: User subroutine to define the effective area for fluid exchange.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Fluid exchange definition,” Section 11.5.3 of the Abaqus Analysis User’s Guide   
+• \*FLUID EXCHANGE
+
+# Overview
+
+User subroutine VUFLUIDEXCHEFFAREA:
+
+• can be used to define an effective area for fluid exchange that depends on the material state in the underlying elements on the fluid exchange surface;   
+• will be called for blocks of material calculation points on the fluid exchange surface;   
+• can be used only if the specified surface over which the fluid exchange occurs is a surface defined over membrane elements; and   
+• can be used with any fluid exchange property type.
+
+# Defining effective area
+
+The contribution of each material point can be defined as a function of:
+
+• the original area associated with the material point;   
+• the current material state in the underlying elements; and   
+• the temperature and pressure in the primary fluid cavity and the secondary fluid cavity or environment.
+
+The effective area for fabric materials can depend on the nominal strain in the yarn directions and the change in angle between the two yarn directions, as well as the current angle between the two yarn directions. For nonfabric materials the effective area can depend on the material point strain.
+
+# User subroutine interface
+
+```csv
+subroutine vufluidexcheffarea(
+C Read only (unmodifiable) variables -
+1 nblock, nprop, props,
+2 steppTime, totalTime, fluExchName,
+3 cMatName, lFabric, braidAngle,
+4 strain, origArea, 
+```
+
+<!-- source-page: 512 -->
+
+```txt
+5 pcav, ctemp,
+C Write only (modifiable) variables
+6 effArea)
+c
+    include 'vaba_param.inc'
+c
+    parameter (ndir = 3, nshr=1)
+c
+c    pointers for retrieving fabric constitutive strains
+    parameter( iFiberStrain1    = 1,
+    *    iFiberStrain2    = 2,
+    *    iFiberChangeAng    = 4)
+c
+    dimension props(nprop),
+    1    braidAngle(nblock),
+    2    strain(nblock, ndir+nshr),
+    3    origArea(nblock),
+    4    pcav(2),ctemp(2),
+    5    effArea(nblock)
+
+    character*80 fluExchName, cMatName
+
+c    do k = 1, nblock
+c    User coding to update effArea(k) = area associated with
+c    material point contributing to area for fluid exchange
+c    (leakage).
+c    end do
+
+    return
+    end 
+```
+
+# Variable to be defined
+
+# effArea(nblock)
+
+Area associated with the material point contributing to the total effective area for fluid exchange. The subroutine is called with effArea set to the current area associated with the material point and should be updated to reflect the area that contributes to fluid exchange.
+
+# Variables passed in for information
+
+# nBlock
+
+Number of material points to be processed in this call to VUFLUIDEXCHEFFAREA.
+
+<!-- source-page: 513 -->
+
+# nprop
+
+User-specified number of user-defined fluid exchange properties required to define the effective area.
+
+# props(nprop)
+
+User-defined fluid exchange properties.
+
+# stepTime
+
+Value of time since the step began.
+
+# totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime−stepTime.
+
+# fluExchName
+
+User-specified fluid exchange name.
+
+# cMatName
+
+User-specified material name associated with material points processed in this call.
+
+# lFabric
+
+Flag indicating whether the subroutine is called for material points on the fluid exchange surface with a fabric material (lFabric=1 if fabric material, lFabric=0 otherwise).
+
+# braidAngle(nblock)
+
+Angle in radians between the two yarn directions for fabric materials.
+
+# strain(nblock, ndir+nshr)
+
+Fabric constitutive strains (nominal strain in the yarn directions and change in angle between the two yarn directions) or strains for nonfabric materials at current location.
+
+# origArea(nblock)
+
+Original area associated with current material point.
+
+# pcav(2)
+
+Absolute pressure in primary and secondary (or ambient) fluid cavities at the start of the increment.
+
+# ctemp(2)
+
+Temperature in primary and secondary (or ambient) fluid cavities at the start of the increment.
+
+<!-- source-page: 514 -->
+
+<!-- source-page: 515 -->
+
+# 1.2.19 VUHARD: User subroutine to define the yield surface size and hardening parameters for isotropic plasticity or combined hardening models.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Classical metal plasticity,” Section 23.2.1 of the Abaqus Analysis User’s Guide   
+• “Models for metals subjected to cyclic loading,” Section 23.2.2 of the Abaqus Analysis User’s Guide   
+• \*CYCLIC HARDENING   
+• \*PLASTIC   
+• “Deformation of a sandwich plate under CONWEP blast loading,” Section 9.1.9 of the Abaqus Example Problems Guide   
+• “VUHARD,” Section 4.1.35 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine VUHARD:
+
+• is called at all material points of elements for which the material definition includes user-defined isotropic hardening or cyclic hardening for metal plasticity;   
+• can be used to define a material’s isotropic yield behavior;   
+• can be used to define the size of the yield surface in a combined hardening model;   
+• can include material behavior dependent on field variables or state variables; and   
+• requires that the derivatives of the yield stress (or yield surface size in combined hardening models) be defined with respect to the appropriate independent variables, such as strain, strain rate, and temperature.
+
+# User subroutine interface
+
+```c
+subroutine vuhard(
+C Read only -
+* nblock,
+* jElem, kIntPt, kLayer, kSecPt,
+* lAnneal, stepTime, totalTime, dt, cmname,
+* nstatev, nfieldv, nprops,
+* props, tempOld, tempNew, fieldOld, fieldNew,
+* stateOld,
+* eqps, eqpsRate,
+C Write only - 
+```
+
+<!-- source-page: 516 -->
+
+```txt
+* yield, dyieldDtemp, dyieldDeqps,
+* stateNew)
+C
+    include 'vaba_param.inc'
+C
+    dimension props(nprops), tempOld(nblock), tempNew(nblock),
+    1 fieldOld(nblock, nfieldv), fieldNew(nblock, nfieldv),
+    2 stateOld(nblock, nstatev), eqps(nblock), eqpsRate(nblock),
+    3 yield(nblock), dyieldDtemp(nblock), dyieldDeqps(nblock, 2),
+    4 stateNew(nblock, nstatev), jElem(nblock)
+C
+    character*80 cmname
+C
+    do 100 km = 1, nblock
+    user coding
+100 continue
+C
+    return
+    end 
+```
+
+# Variables to be defined
+
+# yield(nblock)
+
+Array containing the yield stress (for isotropic plasticity) or yield surface size (for combined hardening) at the material points.
+
+# dyieldDeqps(nblock,1)
+
+Array containing the derivative of the yield stress or yield surface size with respect to the equivalent plastic strain at the material points.
+
+# dyieldDeqps(nblock,2)
+
+Array containing the derivative of the yield stress with respect to the equivalent plastic strain rate at the material points.
+
+# dyieldDtemp(nblock)
+
+Array containing the derivative of the yield stress or yield surface size with respect to temperature at the material points. This quantity is required only in adiabatic and fully coupled temperature-displacement analyses.
+
+# stateNew(nblock,nstatev)
+
+Array containing the state variables at the material points at the end of the increment. The allocation of this array is described in “Solution-dependent state variables” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide.
+
+<!-- source-page: 517 -->
+
+nblock
+
+Number of material points to be processed in this call to VUHARD.
+
+jElem(nblock)
+
+Array of element numbers.
+
+kIntPt
+
+Integration point number.
+
+kLayer
+
+Layer number (for composite shells).
+
+kSecPt
+
+Section point number within the current layer.
+
+lanneal
+
+Flag indicating whether the routine is being called during an annealing process. lanneal=0 indicates that the routine is being called during a normal mechanics increment. lanneal=1 indicates that this is an annealing process and the internal state variables, stateNew, should be reinitialized if necessary. Abaqus/Explicit will automatically set the stresses, stretches, and state to a value of zero during the annealing process.
+
+stepTime
+
+Value of time since the step began.
+
+totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime-stepTime.
+
+dt
+
+Time increment size.
+
+cmname
+
+Material name, left justified. It is passed in as an uppercase character string. Some internal material models are given names starting with the “ABQ\_” character string. To avoid conflict, “ABQ\_” should not be used as the leading string for cmname.
+
+nstatev
+
+Number of user-defined state variables that are associated with this material type (see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+nfieldv
+
+Number of user-defined external field variables.
+
+<!-- source-page: 518 -->
+
+nprops
+
+User-specified number of user-defined material properties.
+
+tempOld(nblock)
+
+Temperatures at the material points at the beginning of the increment.
+
+tempNew(nblock)
+
+Temperatures at the material points at the end of the increment.
+
+fieldOld(nblock,nfieldv)
+
+Values of the user-defined field variables at the material points at the beginning of the increment.
+
+fieldNew(nblock,nfieldv)
+
+Values of the user-defined field variables at the material points at the end of the increment.
+
+stateOld(nblock,nstatev)
+
+State variables at the material points at the beginning of the increment.
+
+eqps(nblock)
+
+Equivalent plastic strain at the material points.
+
+eqpsRate(nblock)
+
+Equivalent plastic strain rate at the material points.
+
+<!-- source-page: 519 -->
+
+# 1.2.20 VUINTER: User subroutine to define the interaction between contact surfaces.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “User-defined interfacial constitutive behavior,” Section 37.1.6 of the Abaqus Analysis User’s Guide   
+• \*SURFACE INTERACTION   
+• “VUINTER,” Section 4.1.36 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine VUINTER:
+
+• can be used to define the mechanical and thermal interaction between contacting surfaces;   
+• must provide the entire definition of the interaction between the contacting surfaces;   
+• can use and update solution-dependent state variables; and   
+• must be used with the penalty contact constraint algorithm.
+
+# Terminology
+
+The use of user subroutine VUINTER requires familiarity with the following terminology.
+
+# Surface node numbers
+
+The “surface node number” refers to the position of a particular node in the list of nodes on the surface. For example, there are nSlvNod nodes on the slave surface. Number nSlvNod, is the surface node number of the nth node in this list; jSlvUid is the user-defined global number of this node. An Abaqus/Explicit model can be defined in terms of an assembly of part instances (see “Defining an assembly,” Section 2.10.1 of the Abaqus Analysis User’s Guide). In such models a node number in jSlvUid is an internally generated node number. If the original node number and part instance name are required, call the utility routine VGETPARTINFO (see “Obtaining part information,” Section 2.1.5).
+
+# Local coordinate system
+
+The array alocaldir defines the direction cosines of a local coordinate system for each slave node. The first local direction corresponds to the contact normal direction from the perspective of the slave node. For a two-dimensional VUINTER model the second local direction is the tangent direction defined by the cross product of the vector into the plane of the model (0., 0., −1.0) and the slave normal. For a three-dimensional VUINTER model the second and third local directions correspond to two orthogonal tangent directions $\mathbf { t } _ { 1 }$ and $\mathbf { t } _ { 2 }$ , which are set as follows:
+
+<!-- source-page: 520 -->
+
+• If the master surface is a cylindrical analytical surface, the second local direction corresponds to the generator direction (see “Analytical rigid surface definition,” Section 2.3.4 of the Abaqus Analysis User’s Guide), and the third local direction is the cross product of the first and second local directions.   
+• If the master surface is an analytical surface of revolution, the third local direction corresponds to the hoop direction, and the second local direction is the cross product of the third and first local directions.   
+• If the master surface is a three-dimensional, element-based surface, the tangent directions are based on the slave normal, using the standard convention for calculating surface tangents (see “Conventions,” Section 1.2.2 of the Abaqus Analysis User’s Guide).
+
+For the two cases listed above involving three-dimensional analytical surfaces, the local tangent directions will reflect a rotation of the master surface. For the last case (three-dimensional, element-based master surface) the tangent directions may not follow the rotation of either the master or slave surfaces; for example, the local system would remain fixed with respect to the global system if a slave node and its surrounding facets rotate about an axes parallel to the slave normal.
+
+The 2 × 2 array stored in drot for each slave node represents the incremental rotation of the tangent directions within the tangent plane corresponding to the tracked point of a three-dimensional master surface. (This incremental rotation array is equal to a unit matrix if nDir is equal to 2.) This incremental rotation matrix is provided so that vector- or tensor-valued state variables defined within the tangent plane can be rotated in this subroutine. For example, the second and third components of the rdisp array (i.e., the relative slip components) are rotated by this amount before VUINTER is called. However, as already mentioned, the rotation of the tangent directions may not reflect a physical rotation of the master or slave surface.
+
+# Conventions for heat flux and stress
+
+A positive flux indicates heat flowing into a surface, and a negative flux denotes heat leaving the surface. Flux must be specified for both surfaces, and they need not be equal and opposite so that effects such as frictional dissipation and differential surface heating can be modeled.
+
+A positive normal stress denotes a pressure directed into the surface (opposite the local normal direction). Positive shear stresses denote shear tractions in the direction of the local surface tangents.
+
+# User subroutine interface
+
+```txt
+subroutine vuinter(
+C Write only
+    1 sfd, scd, spd, svd,
+C Read/Write -
+    2 stress, fluxSlv, fluxMst, sed, statev,
+C Read only -
+    3 kStep, kInc, nFacNod, nSlvNod, nMstNod, nSurfDir,
+    4 nDir, nStateVar, nProps, nTemp, nPred, numDefTfv, 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_053.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_053.md
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+<!-- source-page: 521 -->
+
+```julia
+5 jSlvUid, jMstUid, jConMstid, timStep, timGlb,
+6 dTimCur, surfInt, surfSlv, surfMst,
+7 rdisp, drdisp, drot, stiffDflt, condDflt,
+8 shape, coordSlv, coordMst, alocaldir, props,
+9 areaSlv, tempSlv, dtempSlv, preDefSlv, dpreDefSlv,
+1 tempMst, dtempMst, preDefMst, dpreDefMst)
+C
+    include `vaba_param.inc'
+C
+    character*80 surfInt, surfSlv, surfMst
+C
+    dimension props(nProps), statev(nStateVar,nSlvNod),
+    1 drot(2,2,nSlvNod), sed(nSlvNod), sfd(nSlvNod),
+    2 scd(nSlvNod), spd(nSlvNod), svd(nSlvNod),
+    3 rdisp(nDir,nSlvNod), drdisp(nDir,nSlvNod),
+    4 stress(nDir,nSlvNod), fluxSlv(nSlvNod),
+    5 fluxMst(nSlvNod), areaSlv(nSlvNod),
+    6 stiffDflt(nSlvNod), condDflt(nSlvNod),
+    7 alocaldir(nDir,nDir,nSlvNod), shape(nFacNod,nSlvNod),
+    8 coordSlv(nDir,nSlvNod), coordMst(nDir,nMstNod),
+    9 jSlvUid(nSlvNod), jMstUid(nMstNod),
+    1 jConMstid(nFacNod,nSlvNod), tempSlv(nSlvNod),
+    2 dtempSlv(nSlvNod), preDefSlv(nPred,nSlvNod),
+    3 dpreDefSlv(nPred,nSlvNod), tempMst(numDefTfv),
+    4 dtempMst(numDefTfv), preDefMst(nPred,numDefTfv),
+    5 dpreDefMst(nPred,numDefTfv)
+
+    user coding to define stress,
+    and, optionally, fluxSlv, fluxMst, statev, sed, sfd, scd, spd,
+    and svd
+
+    return
+    end 
+```
+
+# Variable to be defined
+
+stress(nDir, nSlvNod)
+
+On entry this array contains the stress at the interface during the previous time increment. It must be updated to the stress at the interface in the current time increment.
+
+<!-- source-page: 522 -->
+
+# fluxSlv(nSlvNod)
+
+On entry this array contains the flux entering the slave surface during the previous time increment. It must be updated to the flux entering the slave surface during the current increment.
+
+# fluxMst(nSlvNod)
+
+On entry this array contains the flux entering the master surface during the previous time increment. It must be updated to the flux entering the master surface during the current time increment.
+
+# sfd(nSlvNod)
+
+This array can be updated to contain the increment in frictional dissipation at each node (units of energy per unit area). These values contribute to the output variables SFDR and ALLFD and have no effect on other solution variables.
+
+# scd(nSlvNod)
+
+This array can be updated to contain the increment in creep dissipation at each node (units of energy per unit area). These values contribute to the output variables SFDR and ALLCD and have no effect on other solution variables.
+
+# spd(nSlvNod)
+
+This array can be updated to contain the increment in plastic dissipation at each node (units of energy per unit area). These values contribute to the output variables SFDR and ALLPD and have no effect on other solution variables.
+
+# svd(nSlvNod)
+
+This array can be updated to contain the increment in viscous dissipation at each node (units of energy per unit area). These values contribute to the output variables SFDR and ALLVD and have no effect on other solution variables.
+
+# sed(nSlvNod)
+
+On entry this array contains the elastic energy density at the slave nodes at the beginning of the increment. It can be updated to contain the elastic energy density at the end of the current time increment. These values contribute to the output variable ALLSE and have no effect on other solution variables.
+
+# statev(nstateVar, nSlvNod)
+
+This array contains the user-defined solution-dependent state variables for all the nodes on the slave surface. You define the size of this array (see “User-defined interfacial constitutive behavior,” Section 37.1.6 of the Abaqus Analysis User’s Guide, for more information). This array will be passed in containing the values of these variables prior to the call to user subroutine VUINTER. If any of the solution-dependent state variables is being used in conjunction with the surface interaction, it must be updated in this subroutine.
+
+<!-- source-page: 523 -->
+
+kStep
+
+Step number.
+
+kInc
+
+Increment number.
+
+nFacNod
+
+Number of nodes on each master surface facet. nFacNod is 2 for two-dimensional surfaces, and nFacNod is 4 for three-dimensional surfaces (the first and last nodes are the same for triangular facets). If the master surface is an analytical rigid surface, this variable is passed in as 0.
+
+nSlvNod
+
+Number of slave nodes.
+
+nMstNod
+
+Number of master surface nodes, if the master surface is made up of facets. If the master surface is an analytical rigid surface, this variable is passed in as 0.
+
+nSurfDir
+
+Number of tangent directions at the contact points (nSurfDir = nDir - 1).
+
+nDir
+
+Number of coordinate directions at the contact points. (In a three-dimensional model nDir will be 2 if the surfaces in the contact pair are two-dimensional analytical rigid surfaces or are formed by two-dimensional elements.)
+
+nStateVar
+
+Number of user-defined state variables.
+
+nProps
+
+User-specified number of property values associated with this surface interaction model.
+
+nTemp
+
+1 if the temperature is defined and 0 if the temperature is not defined.
+
+nPred
+
+Number of predefined field variables.
+
+numDefTfv
+
+Equal to nSlvNod if the master surface is made up of facets. If the master surface is an analytical rigid surface, this variable is passed in as 1.
+
+<!-- source-page: 524 -->
+
+jSlvUid(nSlvNod)
+
+This array lists the user-defined global node numbers (or internal node numbers for models defined in terms of an assembly of part instances) of the nodes on the slave surface.
+
+jMstUid(nMstNod)
+
+This array lists the user-defined global node numbers (or internal node numbers for models defined in terms of an assembly of part instances) of the nodes on the master surface. If the master surface is an analytical rigid surface, this array is passed in as a dummy array.
+
+jConMstid(nFacNod, nSlvNod)
+
+This array lists the surface node numbers of the master surface nodes that make up the facet onto which each slave node projects. If the master surface is an analytical rigid surface, this array is passed in as a dummy array.
+
+timStep
+
+Value of step time.
+
+timGlb
+
+Value of total time.
+
+dtimCur
+
+Current increment in time from $t = t _ { c u r r } - \Delta t { \mathrm { t } } 0 t = t _ { c u r r } .$
+
+surfInt
+
+User-specified surface interaction name, left justified.
+
+surfSlv
+
+Slave surface name.
+
+surfMst
+
+Master surface name.
+
+rdisp(nDir, nSlvNod)
+
+An array containing the relative positions between the two surfaces. The first component is the relative position of the slave node, with respect to the master surface, in the normal direction (a positive value indicates a penetration, and a negative value indicates a gap). The second and third components, if applicable, are the accumulated incremental relative tangential displacements of the slave node, measured from the beginning of the step in which the contact pair is defined. The local directions in which the relative displacements are defined are stored in alocaldir. If the master surface is an analytical surface, the elements in rdisp are set to r\_MaxVal for the slave nodes that are far from the master surface.
+
+<!-- source-page: 525 -->
+
+drdisp(nDir, nSlvNod)
+
+An array containing the increments in relative positions between the two surfaces during the current time increment. If the master surface is an analytical surface, the elements in drdisp are set to r\_MaxVal for the slave nodes that are far from the master surface.
+
+drot(2, 2, nSlvNod)
+
+Rotation increment matrix. This matrix represents the incremental rotation of the local surface tangent directions for a three-dimensional surface. This rotation matrix for each slave node is defined as a unit matrix for two-dimensional surfaces. If the master surface is an analytical surface, the elements in drot are set to r\_MaxVal for the slave nodes that are far from the master surface.
+
+stiffDflt(nSlvNod)
+
+Values of the default penalty stiffnesses for each slave node (units of FL ).
+
+condDflt(nSlvNod)
+
+Values of the default penalty conductances for each slave node (units of $J \theta ^ { - 1 } \mathrm { T } ^ { - 1 } )$ .
+
+shape(nFacNod, nSlvNod)
+
+For each contact point this array contains the shape functions of the nodes of its master surface facet, evaluated at the location of the contact point. If the master surface is an analytical rigid surface, this array is passed in as a dummy array.
+
+coordSlv(nDir, nSlvNod)
+
+Array containing the nDir components of the current coordinates of the slave nodes.
+
+coordMst(nDir, nMstNod)
+
+Array containing the nDir components of the current coordinates of the master nodes. If the master surface is an analytical rigid surface, this array is passed in as the coordinates of the contact points on the master surface.
+
+alocaldir(nDir, nDir, nSlvNod)
+
+Direction cosines of the local surface coordinate system. The first array index corresponds to the components of the local directions, and the second array index corresponds to the local direction number. The first direction (alocaldir(1..nDir,1,...)) is the normal to the surface. The second direction (alocaldir(1..nDir,2,...)) is the first surface tangent. For a three-dimensional surface, the third direction (alocaldir(1..3,3,...)) is the second surface tangent. If the master surface is an analytical rigid surface, the numbers in alocaldir are valid only if the corresponding parts in rdisp are valid (i.e., not equal to r\_MaxVal).
+
+props(nProps)
+
+User-specified vector of property values to define the behavior between the contacting surfaces.
+
+areaSlv(nSlvNod)
+
+Area associated with the slave nodes (equal to 1 for node-based surface nodes).
+
+<!-- source-page: 526 -->
+
+tempSlv(nSlvNod)
+
+Current temperature at the slave nodes.
+
+dtempSlv(nSlvNod)
+
+Increment in temperature during the previous time increment at the slave nodes.
+
+preDefSlv(nPred, nSlvNod)
+
+Current user-specified predefined field variables at the slave nodes (initial values at the beginning of the analysis and current values during the analysis).
+
+dpreDefSlv(nPred, nSlvNod)
+
+Increment in the predefined field variables at the slave nodes during the previous time increment.
+
+tempMst(numDefTfv)
+
+Current temperature at the nearest points on the master surface.
+
+dtempMst(numDefTfv)
+
+Increment in temperature during the previous time increment at the nearest points on the master surface.
+
+preDefMst(nPred, numDefTfv)
+
+Current user-specified predefined field variables at the nearest points on the master surface (initial values at the beginning of the analysis and current values during the analysis).
+
+dpreDefMst(nPred, numDefTfv)
+
+Increment in the predefined field variables during the previous time increment at the nearest points on the master surface.
+
+<!-- source-page: 527 -->
+
+# 1.2.21 VUINTERACTION: User subroutine to define the contact interaction between surfaces with the general contact algorithm.
+
+Product: Abaqus/Explicit
+
+# References
+
+• “User-defined interfacial constitutive behavior,” Section 37.1.6 of the Abaqus Analysis User’s Guide   
+• \*SURFACE INTERACTION   
+• “VUINTERACTION,” Section 4.1.37 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine VUINTERACTION:
+
+• can be used to define the mechanical and thermal interaction between contact surfaces;   
+• must provide the entire definition of the interaction between the contact surfaces;   
+• can utilize a user-specified tracking thickness to determine potential points of interaction on a surface (and thus which nodes should be passed into the subroutine);   
+• can use and update solution-dependent state variables for node-to-face contact and node-toanalytical rigid surface contact; and   
+• must be used with the general contact algorithm.
+
+# Terminology
+
+The use of user subroutine VUINTERACTION requires familiarity with the following terminology.
+
+# Tracking thickness
+
+For efficiency, user subroutine VUINTERACTION considers only regions of two surfaces that are likely to be in contact or come into contact in a given increment. This likelihood is defined by a tracking thickness: only portions of surfaces separated by less than the tracking thickness in a given increment are passed into the subroutine; portions of the surfaces with a separation larger than the tracking thickness are ignored for the current increment. Surface thicknesses are accounted for in the separation calculations.
+
+Abaqus/Explicit provides an internal default value for the tracking thickness, but a nondefault value can be specified; see “Tracking thickness when VUINTER or VUINTERACTION is used” in “Userdefined interfacial constitutive behavior,” Section 37.1.6 of the Abaqus Analysis User’s Guide. The tracking thickness is passed into VUINTERACTION using the variable rData(4).
+
+# Proximity points
+
+A proximity point is a potential point of interaction for you to consider in user subroutine VUINTERACTION. Each proximity point is primarily associated with a slave node or a point along
+
+<!-- source-page: 528 -->
+
+a slave edge; the proximity point also references a corresponding, locally nearest point on the master surface within the tracking thickness. A proximity point exists for each pairing of slave node and proximal master surface point. Therefore, more than one proximity point may reference the same node on the slave surface but different points on the master surface if multiple local minimum distances to the slave node exist on the master surface; this phenomenon commonly occurs near the corners of a master surface. No proximity points exist for a slave node that is separated from the master surface by more than the tracking thickness. A two-dimensional representation for contact between portions of shell surfaces is shown in Figure 1.2.21–1.
+
+![](images/page-528_d67aa33ffe88f079716c883afaefbecc9ec3b146925363501935690a1e52fef4.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Gap is greater than
+tracking thickness
+(no proximity point)
+4 proximity points for these local
+minima (within tracking thickness)
+Slave surface
+Master surface
+</details>
+
+Figure 1.2.21–1 Four proximity points are associated with three slave nodes in this surface pairing.
+
+The number of proximity points currently being passed into user subroutine VUINTERACTION is nBlock. The array jSlvUid(nNodSlv,nBlock) gives the slave surface node numbers associated with the proximity points. The variable nNodSlv indicates whether a single slave node (for node-to-face contact) or two slave nodes of an edge (for edge-to-edge contact) are associated with each proximity point. Similarly, the array jMstUid(nNodMst,nBlockAnal) gives the master surface node numbers associated with the proximity points; the nodes can belong to a facet, an edge, or an analytical surface. The variable nNodMst indicates the number of master nodes associated with each proximity point.
+
+An Abaqus/Explicit model can be defined in terms of an assembly of part instances (see “Defining an assembly,” Section 2.10.1 of the Abaqus Analysis User’s Guide). In such models a node number is an internally generated node number. If the original node number and part instance name are required, call the utility routine VGETPARTINFO (see “Obtaining part information,” Section 2.1.5).
+
+# Local coordinate system
+
+The array dircos defines the direction cosines of a local coordinate system for each proximity point. The first local direction corresponds to the contact normal direction from the perspective of the slave node. The second and third local directions correspond to two orthogonal tangent directions $\mathbf { t } _ { 1 }$ and $\mathbf { t } _ { 2 }$ , which are set as follows:
+
+<!-- source-page: 529 -->
+
+• If the master surface is a cylindrical analytical surface, the second local direction corresponds to the generator direction (see “Analytical rigid surface definition,” Section 2.3.4 of the Abaqus Analysis User’s Guide), and the third local direction is the cross product of the first and second local directions.   
+• If the master surface is an analytical surface of revolution, the third local direction corresponds to the hoop direction, and the second local direction is the cross product of the third and first local directions.   
+• If the master surface is element-based, the tangential directions are based on the slave normal and the line connecting the first and third nodes on the master facet.
+
+For the two cases listed above involving analytical surfaces, the local tangential directions will reflect a rotation of the master surface. For the last case (element-based master surface) the tangential directions follow the rotation of the master surface only approximately. The second tangential direction is constructed such that it is perpendicular to the slave normal and the line going from the first to the third node on the master facet. The slave normal, the first tangent, and the second tangent form a right-handed system.
+
+# Conventions for stress and heat flux
+
+A positive normal stress denotes a pressure directed into the surface (opposite the local normal direction). Positive shear stresses denote shear tractions in the direction of the local surface tangents.
+
+A positive flux indicates heat flowing into a surface, and a negative flux denotes heat leaving the surface. Flux must be specified for both surfaces, and they need not be equal and opposite so that effects such as frictional dissipation and differential surface heating can be modeled.
+
+# User subroutine interface
+
+```csv
+subroutine vinteraction (
+C Read/Write -
+* stress, fluxSlv, fluxMst,
+* state, sed,
+C Write only -
+* sfd, scd, spd, svd,
+C Read only -
+* nBlock, nBlockAnal, nBlockEdge,
+* nNodState, nNodSlv, nNodMst, nDir,
+* nStates, nProps, nTemp, nFields,
+* jFlags, rData,
+* surfInt, surfSlv, surfMst,
+* jSlvUid, jMstUid, props,
+* penetration, drDisp, dRot, dircos, stiffDef, conductDef,
+* coordSlv, coordMst, areaProx, shapeSlv, shapeMst, 
+```
+
+<!-- source-page: 530 -->
+
+```txt
+* tempSlv, tempMst, dTempSlv, dTempMst,
+* fieldSlv, fieldMst, dFieldSlv, dFieldMst)
+C
+    include `vaba_param.inc'
+C
+    dimension stress(nDir, nBlock),
+    *    fluxSlv(nBlock),
+    *    fluxMst(nBlock),
+    *    state(nStates, nNodState, nBlock),
+    *    sed(nBlock),
+    *    sfd(nBlock),
+    *    scd(nBlock),
+    *    spd(nBlock),
+    *    svd(nBlock),
+    *    jSlvUid(nNodSlv, nBlock),
+    *    jMstUid(nNodMst, nBlockAnal),
+    *    props(nProps),
+    *    penetration(nBlock),
+    *    drDisp(nDir, nBlock),
+    *    dRot(2, 2, nBlock),
+    *    stiffDef(nBlock),
+    *    conductDef(nBlock),
+    *    dircos(nDir, nDir, nBlock),
+    *    coordSlv(nDir, nNodSlv, nBlock),
+    *    coordMst(nDir, nNodMst, nBlockAnal),
+    *    areaProx(nBlock),
+    *    shapeSlv(nNodSlv, nBlockEdge),
+    *    shapeMst(nNodMst, nBlockAnal),
+    *    tempSlv(nBlock),
+    *    tempMst(nBlockAnal),
+    *    dTempSlv(nBlock),
+    *    dTempMst(nBlockAnal),
+    *    fieldSlv(nFields, nBlock),
+    *    fieldMst(nFields, nBlockAnal)
+    *    dFieldSlv(nFields, nBlock),
+    *    dFieldMst(nFields, nBlockAnal)
+C
+    parameter( iKStep    = 1,
+    *    iKInc    = 2,
+    *    iLConType    = 3,
+    *    nFlags    = 3 )
+C 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_054.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_054.md
new file mode 100644
index 00000000..df7cbb99
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+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_054.md
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+<!-- source-page: 531 -->
+
+```prolog
+parameter( iTimStep = 1,
+* iTimGlb = 2,
+* iDTimCur = 3,
+* iTrackThic = 4,
+* nData = 4 )
+C
+dimension jFlags(nFlags), rData(nData)
+C
+character*80 surfInt, surfSlv, surfMst
+C
+user coding to define stress,
+and, optionally, fluxSlv, fluxMst, state, sed, sfd, scd, spd,
+and svd
+C
+return
+end 
+```
+
+# Variable to be defined
+
+stress(nDir, nBlock)
+
+On entry this array contains the stress defined in the local system at the proximity points during the previous time increment. It must be updated to the stress at the interface in the current time increment.
+
+# Variables that can be updated
+
+fluxSlv(nBlock)
+
+On entry this array contains the flux entering the slave surface during the previous time increment. It must be updated to the flux entering the slave surface during the current increment.
+
+fluxMst(nBlock)
+
+On entry this array contains the flux entering the master surface during the previous time increment. It must be updated to the flux entering the master surface during the current time increment.
+
+state(nStates,nNodState,nBlock)
+
+This array contains the user-defined solution-dependent state variables for the proximity points. The use of the state variables is applicable only for node-to-face contact. See “User-defined interfacial constitutive behavior,” Section 37.1.6 of the Abaqus Analysis User’s Guide, for more information on the size of this array. This array will be passed in containing the values of these variables prior to the call to user subroutine VUINTERACTION.
+
+If any of the solution-dependent state variables is being used in conjunction with the interaction, it must be updated in this subroutine. These state variables need to be updated with care: outside the user subroutine these state variables are single-valued per slave node, but multiple proximity points may refer to the same slave node. Each proximity point may be passed into the user subroutine independently in a given increment, possibly on separate calls to the user subroutine; therefore, you may end up
+
+<!-- source-page: 532 -->
+
+advancing the state variables for the associated node multiple times for a single time increment. To keep track of whether or not a node state is advanced, you may want to use one of the state variables exclusively for this purpose. You could set that selected state variable to the current increment number and update the state only if it is not already set to the current increment number.
+
+# sed(nBlock)
+
+On entry this array contains the elastic energy density at the proximity points at the beginning of the increment. It can be updated to contain the elastic energy density at the end of the current time increment. These values contribute to the output variable ALLSE and have no effect on other solution variables. The use of this variable is applicable only for node-to-face contact.
+
+# sfd(nBlock)
+
+This array can be updated to contain the increment in frictional dissipation at each proximity point (units of energy per unit area). These values contribute to the output variables SFDR and ALLFD and have no effect on other solution variables. The use of this variable is applicable only for node-to-face contact.
+
+# scd(nBlock)
+
+This array can be updated to contain the increment in creep dissipation at each proximity point (units of energy per unit area). These values contribute to the output variables SFDR and ALLCD and have no effect on other solution variables. The use of this variable is applicable only for node-to-face contact.
+
+# spd(nBlock)
+
+This array can be updated to contain the increment in plastic dissipation at each proximity point (units of energy per unit area). These values contribute to the output variables SFDR and ALLPD and have no effect on other solution variables. The use of this variable is applicable only for node-to-face contact.
+
+# svd(nBlock)
+
+This array can be updated to contain the increment in viscous dissipation at each proximity point (units of energy per unit area). These values contribute to the output variables SFDR and ALLVD and have no effect on other solution variables. The use of this variable is applicable only for node-to-face contact.
+
+# Variables passed in for information
+
+# nBlock
+
+Number of proximity points to be processed in this call to VUINTERACTION.
+
+# nBlockAnal
+
+1 for analytical rigid master surface; nBlock otherwise.
+
+# nBlockEdge
+
+nBlock for edge type slave surface; 1 otherwise.
+
+# nNodState
+
+1 for node-to-face and node-to-analytical rigid surface contact; not applicable for edge-to-edge contact.
+
+<!-- source-page: 533 -->
+
+nNodSlv   
+1 for node-to-face and node-to-analytical rigid surface contact; 2 for edge-to-edge contact.
+
+nNodMst
+1 for analytical rigid master surface; 2 for edge-type master surface; 4 for facet-type master surface.
+
+nDir
+Number of coordinate directions at the proximity points (equal to 3).
+
+nStates
+Number of user-defined state variables.
+
+nProps
+User-specified number of property values associated with this interaction model.
+
+nTemp
+1 if the temperature is defined and 0 if the temperature is not defined.
+
+nFields
+Number of predefined field variables.
+
+jFlag(1)
+Step number.
+
+jFlag(2)
+Increment number.
+
+jFlag(3)
+1 for node-to-face contact, 2 for edge-to-edge contact, and 3 for node-to-analytical rigid surface contact.
+
+rData(1)
+Value of step time.
+
+rData(2)
+Value of total time.
+
+rData(3)
+Current increment in time from $t = t_{curr} - \Delta t$ to $t = t_{curr}$ .
+
+rData(4)
+This variable contains the value of the tracking thickness specified for the surface interaction.
+
+surfInt
+User-specified surface interaction name, left justified.
+
+surfSlv
+Slave surface name, currently set to a blank.
+
+<!-- source-page: 534 -->
+
+# surfMst
+
+Master surface name, currently set to a blank.
+
+# jSlvUid(nNodSlv,nBlock)
+
+This array lists the surface node numbers of the slave surface nodes associated with each proximity point.
+
+# jMstUid(nNodMst,nBlockAnal)
+
+This array lists the surface node numbers of the master surface nodes that make up the facet, edge, or analytical rigid surface associated with each proximity point.
+
+# props(nProps)
+
+User-specified vector of property values to define the interaction between the tracking surfaces.
+
+# penetration(nBlock)
+
+The relative position of the proximity points, with respect to the master surface, in the normal direction (a positive value indicates a penetration, and a negative value indicates a gap) during the current time increment.
+
+# drDisp(nDir,nBlock)
+
+An array containing the increments in relative positions of the proximity points with respect to the associated master surfaces during the current time increment.
+
+# dRot(2,2,nBlock)
+
+This argument is currently undefined.
+
+# stiffDef(nBlock)
+
+Values of the default penalty stiffnesses (stress per unit penetration, units of $\mathrm { F L } ^ { - 3 } )$ .
+
+# conductDef(nBlock)
+
+Values of the default penalty conductances (units of $\boldsymbol {  J } \boldsymbol { \theta } ^ { - 1 } \boldsymbol { \mathrm { T } } ^ { - 1 } )$ .
+
+# dircos(nDir,nDir,nBlock)
+
+Direction cosines of the local surface coordinate system. The first array index corresponds to the components of the local directions, and the second array index corresponds to the local direction number. The first direction (dircos(1..nDir,1,...)) is the normal to the surface. The second direction (dircos(1..nDir,2,...)) is the first surface tangent. For a three-dimensional surface, the third direction (dircos(1..3,3,...)) is the second surface tangent. If the master surface is an analytical rigid surface, the numbers in dircos are valid only if the corresponding parts in penetration are valid (i.e., not equal to r\_MaxVal).
+
+# coordSlv(nDir,nNodSlv,nBlock)
+
+Array containing the nDir components of the current coordinates of the proximity points.
+
+<!-- source-page: 535 -->
+
+# coordMst(nDir,nNodMst,nBlockAnal)
+
+Array containing the nDir components of the current coordinates of the nodes on the master surface.
+
+If the master surface is an analytical rigid surface, this array is passed in as a dummy array.
+
+# areaProx(nBlock)
+
+Contact area associated with a proximity point. The sum of the contact areas among all proximity points associated with a single slave node equals the surface area associated with that slave node (equal to 1 for node-based surface nodes). Therefore, the contact area at a proximity point depends on the number of other proximity points currently associated with the same slave node. A proximity point contributes a contact normal force to the associated slave node that is equal to stress(1,k) multiplied by areaProx(k).
+
+# shapeSlv(nNodSlv,nBlockEdge)
+
+For edge-to-edge contact this array contains the shape functions of the nodes of its slave edge, evaluated at the location of the contact point. If the contact is not edge-to-edge, this array is passed in as a dummy array.
+
+# shapeMst(nNodMst,nBlockAnal)
+
+For node-to-face and edge-to-edge contact this array contains the shape functions of the nodes of its master surface, evaluated at the location of the contact point. If the master surface is an analytical rigid surface, this array is passed in as a dummy array.
+
+# tempSlv(nBlock)
+
+Current temperature at the proximity points on the slave surface.
+
+# tempMst(nBlockAnal)
+
+Current temperature at the points on the master surface closest to the proximity points.
+
+# dTempSlv(nBlock)
+
+Increment in the temperature during the previous time increment at the proximity points on the slave surface.
+
+# dTempMst(nBlockAnal)
+
+Increment in the temperature during the previous time increment at the points on the master surface closest to the proximity points.
+
+# fieldSlv(nFields,nBlock)
+
+Current user-specified predefined field variables at the proximity points on the slave surface (initial values at the beginning of the analysis and current values during the analysis).
+
+# fieldMst(nFields,nBlockAnal)
+
+Current user-specified predefined field variables at the points on the master surface closest to the proximity points (initial values at the beginning of the analysis and current values during the analysis).
+
+<!-- source-page: 536 -->
+
+dFieldSlv(nFields,nBlock)
+
+Increment in the user-specified predefined field variables during the previous time increment at the proximity points on the slave surface.
+
+dFieldMst(nFields,nBlockAnal)
+
+Increment in the user-specified predefined field variables during the previous time increment at the points on the master surface closest to the proximity points.
+
+<!-- source-page: 537 -->
+
+# 1.2.22 VUMAT: User subroutine to define material behavior.
+
+# Product: Abaqus/Explicit
+
+WARNING: The use of this user subroutine generally requires considerable expertise. You are cautioned that the implementation of any realistic constitutive model requires extensive development and testing. Initial testing on a single-element model with prescribed traction loading is strongly recommended. The component ordering of the symmetric and nonsymmetric tensors for the three-dimensional case using C3D8R elements is different from the ordering specified in “Three-dimensional solid element library,” Section 28.1.4 of the Abaqus Analysis User’s Guide, and the ordering used in Abaqus/Standard.
+
+# References
+
+• “User-defined mechanical material behavior,” Section 26.7.1 of the Abaqus Analysis User’s Guide   
+• \*USER MATERIAL
+
+# Overview
+
+# User subroutine VUMAT:
+
+• is used to define the mechanical constitutive behavior of a material;   
+• will be called for blocks of material calculation points for which the material is defined in a user subroutine (“Material data definition,” Section 21.1.2 of the Abaqus Analysis User’s Guide);   
+• can use and update solution-dependent state variables;   
+• can use any field variables that are passed in; and   
+• can be used in an adiabatic analysis, provided you define both the inelastic heat fraction and the specific heat for the appropriate material definitions and you store the temperatures and integrate them as user-defined state variables.
+
+# Component ordering in tensors
+
+The component ordering depends upon whether the tensor is symmetric or nonsymmetric.
+
+# Symmetric tensors
+
+For symmetric tensors such as the stress and strain tensors, there are ndir+nshr components, and the component order is given as a natural permutation of the indices of the tensor. The direct components are first and then the indirect components, beginning with the 12-component. For example, a stress tensor contains ndir direct stress components and nshr shear stress components, which are passed in as
+
+<!-- source-page: 538 -->
+
+<table><tr><td>Component</td><td>2D Case</td><td>3D Case</td></tr><tr><td>1</td><td> $\sigma_{11}$ </td><td> $\sigma_{11}$ </td></tr><tr><td>2</td><td> $\sigma_{22}$ </td><td> $\sigma_{22}$ </td></tr><tr><td>3</td><td> $\sigma_{33}$ </td><td> $\sigma_{33}$ </td></tr><tr><td>4</td><td> $\sigma_{12}$ </td><td> $\sigma_{12}$ </td></tr><tr><td>5</td><td></td><td> $\sigma_{23}$ </td></tr><tr><td>6</td><td></td><td> $\sigma_{31}$ </td></tr></table>
+
+The shear strain components in user subroutine VUMAT are stored as tensor components and not as engineering components; this is different from user subroutine UMAT in Abaqus/Standard, which uses engineering components.
+
+# Nonsymmetric tensors
+
+For nonsymmetric tensors there are ndir+2\*nshr components, and the component order is given as a natural permutation of the indices of the tensor. The direct components are first and then the indirect components, beginning with the 12-component. For example, the deformation gradient is passed as
+
+<table><tr><td>Component</td><td>2D Case</td><td>3D Case</td></tr><tr><td>1</td><td> $F_{11}$ </td><td> $F_{11}$ </td></tr><tr><td>2</td><td> $F_{22}$ </td><td> $F_{22}$ </td></tr><tr><td>3</td><td> $F_{33}$ </td><td> $F_{33}$ </td></tr><tr><td>4</td><td> $F_{12}$ </td><td> $F_{12}$ </td></tr><tr><td>5</td><td> $F_{21}$ </td><td> $F_{23}$ </td></tr><tr><td>6</td><td></td><td> $F_{31}$ </td></tr><tr><td>7</td><td></td><td> $F_{21}$ </td></tr><tr><td>8</td><td></td><td> $F_{32}$ </td></tr><tr><td>9</td><td></td><td> $F_{13}$ </td></tr></table>
+
+# Initial calculations and checks
+
+In the data check phase of the analysis Abaqus/Explicit calls user subroutine VUMAT with a set of fictitious strains and a totalTime and stepTime both equal to 0.0. This is done as a check on your constitutive relation and to calculate the equivalent initial material properties, based upon which the initial elastic wave speeds are computed.
+
+<!-- source-page: 539 -->
+
+# Defining local orientations
+
+All stresses, strains, stretches, and state variables are in the orientation of the local material axes. These local material axes form a basis system in which stress and strain components are stored. This represents a corotational coordinate system in which the basis system rotates with the material. If a user-specified coordinate system (“Orientations,” Section 2.2.5 of the Abaqus Analysis User’s Guide) is used, it defines the local material axes in the undeformed configuration.
+
+# Special considerations for various element types
+
+The use of user subroutine VUMAT requires special consideration for various element types.
+
+# Shell and plane stress elements
+
+You must define the stresses and internal state variables. In the case of shell or plane stress elements, NDIR=3 and NSHR=1; you must define strainInc(\*,3), the thickness strain increment. The internal energies can be defined if desired. If they are not defined, the energy balance provided by Abaqus/Explicit will not be meaningful.
+
+# Shell elements
+
+When VUMAT is used to define the material response of shell elements, Abaqus/Explicit cannot calculate a default value for the transverse shear stiffness of the element. Hence, you must define the element’s transverse shear stiffness. See “Shell section behavior,” Section 29.6.4 of the Abaqus Analysis User’s Guide, for guidelines on choosing this stiffness.
+
+# Beam elements
+
+For beam elements the stretch tensor and the deformation gradient tensor are not available. For beams in space you must define the thickness strains, strainInc(\*,2) and strainInc(\*,3). strainInc(\*,4) is the shear strain associated with twist. Thickness stresses, stressNew(\*,2) and stressNew(\*,3), are assumed to be zero, and any values you assign are ignored.
+
+# Pipe elements
+
+For pipe elements the stretch tensor and the deformation gradient tensor are not available. The axial strain, strainInc(\*,1), and the shear strain, strainInc(\*,4), associated with twist are provided along with the hoop stress, stressNew(\*,2). The hoop stress is predefined based on your pipe internal and external pressure load definitions (PE, PI, HPE, HPI, PENU, and PINU), and it should not be modified here. The thickness stress, stressNew(\*,3), is assumed to be zero and any value you assign is ignored. You must define the axial stress, stressNew(\*,1), and the shear stress, stressNew(\*,4). You must also define hoop strain, strainInc(\*,2), and the pipe thickness strain, strainInc(\*,3).
+
+<!-- source-page: 540 -->
+
+# Deformation gradient
+
+The polar decomposition of the deformation gradient is written as , where and are the right and left symmetric stretch tensors, respectively. The constitutive model is defined in a corotational coordinate system in which the basis system rotates with the material. All stress and strain tensor quantities are defined with respect to the corotational basis system. The right stretch tensor, , is used. The relative spin tensor represents the spin (the antisymmetric part of the velocity gradient) defined with respect to the corotational basis system.
+
+# Special considerations for hyperelasticity
+
+Hyperelastic constitutive models in VUMAT should be defined in a corotational coordinate system in which the basis system rotates with the material. This is most effectively accomplished by formulating the hyperelastic constitutive model in terms of the stretch tensor, , instead of in terms of the deformation gradient, . Using the deformation gradient can present some difficulties because the deformation gradient includes the rotation tensor and the resulting stresses would need to be rotated back to the corotational basis.
+
+# Objective stress rates
+
+The Green-Naghdi stress rate is used when the mechanical behavior of the material is defined using user subroutine VUMAT. The stress rate obtained with user subroutine VUMAT may differ from that obtained with a built-in Abaqus material model. For example, most material models used with solid (continuum) elements in Abaqus/Explicit employ the Jaumann stress rate. This difference in the formulation will cause significant differences in the results only if finite rotation of a material point is accompanied by finite shear. For a discussion of the objective stress rates used in Abaqus, see “Stress rates,” Section 1.5.3 of the Abaqus Theory Guide.
+
+# Material point deletion
+
+Material points that satisfy a user-defined failure criterion can be deleted from the model (see “Userdefined mechanical material behavior,” Section 26.7.1 of the Abaqus Analysis User’s Guide). You must specify the state variable number controlling the element deletion flag when you allocate space for the solution-dependent state variables, as explained in “User-defined mechanical material behavior,” Section 26.7.1 of the Abaqus Analysis User’s Guide. The deletion state variable should be set to a value of one or zero in VUMAT. A value of one indicates that the material point is active, while a value of zero indicates that Abaqus/Explicit should delete the material point from the model by setting the stresses to zero. The structure of the block of material points passed to user subroutine VUMAT remains unchanged during the analysis; deleted material points are not removed from the block. Abaqus/Explicit will pass zero stresses and strain increments for all deleted material points. Once a material point has been flagged as deleted, it cannot be reactivated.
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+<!-- source-page: 541 -->
+
+```fortran
+subroutine vumat(
+C Read only (unmodifiable)variables -
+    1 nblock, ndir, nshr, nstatev, nfieldv, nprops, lanneal,
+    2 stepTime, totalTime, dt, cmname, coordMp, charLength,
+    3 props, density, strainInc, relSpinInc,
+    4 tempOld, stretchOld, defgradOld, fieldOld,
+    5 stressOld, stateOld, enerInternOld, enerInelasOld,
+    6 tempNew, stretchNew, defgradNew, fieldNew,
+C Write only (modifiable)variables -
+    7 stressNew, stateNew, enerInternNew, enerInelasNew )
+C
+    include 'vaba_param.inc'
+C
+    dimension props(nprops), density(nblock), coordMp(nblock,*),
+    1 charLength(nblock), strainInc(nblock,ndir+nshr),
+    2 relSpinInc(nblock,nshr), tempOld(nblock),
+    3 stretchOld(nblock,ndir+nshr),
+    4 defgradOld(nblock,ndir+nshr+nshr),
+    5 fieldOld(nblock,nfieldv), stressOld(nblock,ndir+nshr),
+    6 stateOld(nblock,nstatev), enerInternOld(nblock),
+    7 enerInelasOld(nblock), tempNew(nblock),
+    8 stretchNew(nblock,ndir+nshr),
+    8 defgradNew(nblock,ndir+nshr+nshr),
+    9 fieldNew(nblock,nfieldv),
+    1 stressNew(nblock,ndir+nshr), stateNew(nblock,nstatev),
+    2 enerInternNew(nblock), enerInelasNew(nblock)
+C
+    character*80 cmname
+C
+    do 100 km = 1,nblock
+    user coding
+100 continue
+    return
+    end 
+```
+
+<!-- source-page: 542 -->
+
+# Variables to be defined
+
+stressNew (nblock, ndir+nshr)
+
+Stress tensor at each material point at the end of the increment.
+
+stateNew (nblock, nstatev)
+
+State variables at each material point at the end of the increment. You define the size of this array by allocating space for it (see “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide, for more information).
+
+# Variables that can be updated
+
+enerInternNew (nblock)
+
+Internal energy per unit mass at each material point at the end of the increment.
+
+enerInelasNew (nblock)
+
+Dissipated inelastic energy per unit mass at each material point at the end of the increment.
+
+# Variables passed in for information
+
+nblock
+
+Number of material points to be processed in this call to VUMAT.
+
+ndir
+
+Number of direct components in a symmetric tensor.
+
+nshr
+
+Number of indirect components in a symmetric tensor.
+
+nstatev
+
+Number of user-defined state variables that are associated with this material type (you define this as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+nfieldv
+
+Number of user-defined external field variables.
+
+nprops
+
+User-specified number of user-defined material properties.
+
+lanneal
+
+Flag indicating whether the routine is being called during an annealing process. lanneal=0 indicates that the routine is being called during a normal mechanics increment. lanneal=1 indicates that this is an annealing process and you should re-initialize the internal state variables, stateNew, if necessary. Abaqus/Explicit will automatically set the stresses, stretches, and state to a value of zero during the annealing process.
+
+<!-- source-page: 543 -->
+
+# stepTime
+
+Value of time since the step began.
+
+# totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime - stepTime.
+
+# dt
+
+Time increment size.
+
+# cmname
+
+User-specified material name, left justified. It is passed in as an uppercase character string. Some internal material models are given names starting with the “ABQ\_” character string. To avoid conflict, you should not use “ABQ\_” as the leading string for cmname.
+
+# coordMp(nblock,\*)
+
+Material point coordinates. It is the midplane material point for shell elements and the centroid for beam and pipe elements.
+
+# charLength(nblock)
+
+Characteristic element length, which is either the default value based on the geometric mean or the user-defined characteristic element length defined in user subroutine VUCHARLENGTH. The default value is a typical length of a line across an element for a first-order element; it is half of the same typical length for a second-order element. For beams, pipes, and trusses, the default value is a characteristic length along the element axis. For membranes and shells it is a characteristic length in the reference surface. For axisymmetric elements it is a characteristic length in the r–z plane only. For cohesive elements it is equal to the constitutive thickness.
+
+# props(nprops)
+
+User-supplied material properties.
+
+# density(nblock)
+
+Current density at the material points in the midstep configuration. This value may be inaccurate in problems where the volumetric strain increment is very small. If an accurate value of the density is required in such cases, the analysis should be run in double precision. This value of the density is not affected by mass scaling.
+
+# strainInc (nblock, ndir+nshr)
+
+Strain increment tensor at each material point.
+
+# relSpinInc (nblock, nshr)
+
+Incremental relative rotation vector at each material point defined in the corotational system. Defined as , where is the antisymmetric part of the velocity gradient, , and $\pmb { \Omega } = \dot { \mathbf { R } } \cdot \mathbf { R ^ { T } }$ . Stored in 3D as and in 2D as .
+
+<!-- source-page: 544 -->
+
+tempOld(nblock)
+
+Temperatures at each material point at the beginning of the increment.
+
+stretchOld (nblock, ndir+nshr)
+
+Stretch tensor, , at each material point at the beginning of the increment defined from the polar decomposition of the deformation gradient by $\mathbf { F } = \mathbf { R } \cdot \mathbf { U }$ .
+
+defgradOld (nblock,ndir+2\*nshr)
+
+Deformation gradient tensor at each material point at the beginning of the increment. Stored in 3D as $( F _ { 1 1 } , F _ { 2 2 } , F _ { 3 3 } , F _ { 1 2 } , F _ { 2 3 } , F _ { 3 1 } , F _ { 2 1 } , F _ { 3 2 } , F _ { 1 3 } )$ and in 2D as $( F _ { 1 1 } , F _ { 2 2 } , F _ { 3 3 } , F _ { 1 2 } , F _ { 2 1 } )$ .
+
+fieldOld (nblock, nfieldv)
+
+Values of the user-defined field variables at each material point at the beginning of the increment.
+
+stressOld (nblock, ndir+nshr)
+
+Stress tensor at each material point at the beginning of the increment.
+
+stateOld (nblock, nstatev)
+
+State variables at each material point at the beginning of the increment.
+
+enerInternOld (nblock)
+
+Internal energy per unit mass at each material point at the beginning of the increment.
+
+enerInelasOld (nblock)
+
+Dissipated inelastic energy per unit mass at each material point at the beginning of the increment.
+
+tempNew(nblock)
+
+Temperatures at each material point at the end of the increment.
+
+stretchNew (nblock, ndir+nshr)
+
+Stretch tensor, , at each material point at the end of the increment defined from the polar decomposition of the deformation gradient by .
+
+defgradNew (nblock,ndir+2\*nshr)
+
+Deformation gradient tensor at each material point at the end of the increment. Stored in 3D as $( F _ { 1 1 }$ $F _ { 2 2 } , F _ { 3 3 } , F _ { 1 2 } , F _ { 2 3 } , F _ { 3 1 } , F _ { 2 1 } , F _ { 3 2 } , F _ { 1 3 } )$ and in 2D as $( F _ { 1 1 } , F _ { 2 2 } , F _ { 3 3 } , F _ { 1 2 } , F _ { 2 1 } )$ .
+
+fieldNew (nblock, nfieldv)
+
+Values of the user-defined field variables at each material point at the end of the increment.
+
+# Example: Using more than one user-defined material model
+
+To use more than one user-defined material model, the variable cmname can be tested for different material names inside user subroutine VUMAT, as illustrated below:
+
+```txt
+if (cmname(1:4) .eq. 'MAT1') then
+call VUMAT_MAT1(argument_list) 
+```
+
+<!-- source-page: 545 -->
+
+```txt
+else if (cmname(1:4) .eq. 'MAT2') then
+    call VUMAT_MAT2 (argument_list)
+end if 
+```
+
+VUMAT\_MAT1 and VUMAT\_MAT2 are the actual user material subroutines containing the constitutive material models for each material MAT1 and MAT2, respectively. Subroutine VUMAT merely acts as a directory here. The argument list can be the same as that used in subroutine VUMAT. The material names must be in uppercase characters since cmname is passed in as an uppercase character string.
+
+# Example: Elastic/plastic material with kinematic hardening
+
+As a simple example of the coding of subroutine VUMAT, consider the generalized plane strain case for an elastic/plastic material with kinematic hardening. The basic assumptions and definitions of the model are as follows.
+
+Let be the current value of the stress, and define to be the deviatoric part of the stress. The center of the yield surface in deviatoric stress space is given by the tensor , which has initial values of zero. The stress difference, , is the stress measured from the center of the yield surface and is given by
+
+$$
+\boldsymbol {\xi} = \mathbf {S} - \boldsymbol {\alpha}.
+$$
+
+The von Mises yield surface is defined as
+
+$$
+f (\pmb {\sigma}) = \frac {1}{2} \pmb {\xi}: \pmb {\xi} - \frac {1}{3} \sigma_ {0} ^ {2},
+$$
+
+where $\sigma _ { 0 }$ is the uniaxial equivalent yield stress. The von Mises yield surface is a cylinder in deviatoric stress space with a radius of
+
+$$
+R = \sqrt {\frac {2}{3}} \sigma_ {0}.
+$$
+
+For the kinematic hardening model, R is a constant. The normal to the Mises yield surface can be written as
+
+$$
+\mathbf {Q} = \sqrt {\frac {3}{2}} \frac {\boldsymbol {\xi}}{\sigma_ {0}}.
+$$
+
+We decompose the strain rate into an elastic and plastic part using an additive decomposition:
+
+$$
+\dot {\epsilon} = \dot {\epsilon} ^ {e l} + \dot {\epsilon} ^ {p l}.
+$$
+
+The plastic part of the strain rate is given by a normality condition
+
+$$
+\dot {\epsilon} ^ {p l} = \dot {\gamma} \mathbf {Q},
+$$
+
+<!-- source-page: 546 -->
+
+where the scalar multiplier $\dot { \gamma }$ must be determined. A scalar measure of equivalent plastic strain rate is defined by
+
+$$
+\dot {\bar {\epsilon}} ^ {p l} = \sqrt {\frac {2}{3} \dot {\epsilon} ^ {p l} : \dot {\epsilon} ^ {p l}}.
+$$
+
+The stress rate is assumed to be purely due to the elastic part of the strain rate and is expressed in terms of Hooke’s law by
+
+$$
+\dot {\pmb {\sigma}} = \lambda \mathrm{trace} (\dot {\pmb {\epsilon}} ^ {e l}) \mathbf {I} + 2 \mu \dot {\pmb {\epsilon}} ^ {e l},
+$$
+
+where and $2 \mu$ are the Lamés constants for the material.
+
+The evolution law for is given as
+
+$$
+\dot {\alpha} = \frac {2}{3} \dot {\gamma} H \mathbf {Q},
+$$
+
+where H is the slope of the uniaxial yield stress versus plastic strain curve.
+
+During active plastic loading the stress must remain on the yield surface, so that
+
+$$
+\sqrt {\mathbf {Q} : \mathbf {Q}} = 1.
+$$
+
+The equivalent plastic strain rate is related to $\dot { \gamma }$ by
+
+$$
+\dot {\bar {\epsilon}} ^ {p l} = \sqrt {\frac {2}{3}} \dot {\gamma}.
+$$
+
+The kinematic hardening constitutive model is integrated in a rate form as follows. A trial elastic stress is computed as
+
+$$
+\pmb {\sigma} _ {n e w} ^ {t r i a l} = \pmb {\sigma} _ {o l d} + \lambda \mathrm{trace} (\Delta \pmb {\epsilon}) \mathbf {I} + 2 \mu \Delta \pmb {\epsilon},
+$$
+
+where the subscripts and refer to the beginning and end of the increment, respectively. If the trial stress does not exceed the yield stress, the new stress is set equal to the trial stress. If the yield stress is exceeded, plasticity occurs in the increment. We then write the incremental analogs of the rate equations as
+
+$$
+\pmb {\sigma} _ {n e w} = \pmb {\sigma} _ {n e w} ^ {t r i a l} - 2 \mu \pmb {\Delta} \pmb {\epsilon} ^ {p l} = \pmb {\sigma} _ {n e w} ^ {t r i a l} - 2 \mu \Delta \gamma \mathbf {Q},
+$$
+
+$$
+\boldsymbol {\alpha} _ {n e w} = \boldsymbol {\alpha} _ {o l d} + \frac {2}{3} H \Delta \gamma \mathbf {Q},
+$$
+
+$$
+\bar {\epsilon} _ {n e w} ^ {p l} = \bar {\epsilon} _ {o l d} ^ {p l} + \sqrt {\frac {2}{3}} \Delta \gamma ,
+$$
+
+<!-- source-page: 547 -->
+
+where
+
+$$
+\Delta \gamma = \dot {\gamma} \Delta t.
+$$
+
+From the definition of the normal to the yield surface at the end of the increment, ,
+
+$$
+\alpha_ {n e w} + \sqrt {\frac {2}{3}} \sigma_ {0} \mathbf {Q} = \mathbf {S} _ {n e w}.
+$$
+
+This can be expanded using the incremental equations as
+
+$$
+\pmb {\alpha} _ {o l d} + \frac {2}{3} H \Delta \gamma \mathbf {Q} + \sqrt {\frac {2}{3}} \sigma_ {0} \mathbf {Q} = \mathbf {S} _ {n e w} ^ {t r i a l} - \Delta \gamma 2 \mu \mathbf {Q}.
+$$
+
+Taking the tensor product of this equation with , using the yield condition at the end of the increment, and solving for $\Delta \gamma \mathrm { : }$ :
+
+$$
+\Delta \gamma = \frac {1}{2 \mu (1 + H / 3 \mu)} \left(\left(\pmb {\xi} _ {n e w} ^ {t r i a l}: \pmb {\xi} _ {n e w} ^ {t r i a l}\right) ^ {1 / 2} - \sqrt {\frac {2}{3}} \sigma_ {0}\right).
+$$
+
+The value for $\Delta \gamma$ is used in the incremental equations to determine $\sigma _ { n e w } , \alpha _ { n e w } ,$ , and $\overline { { \epsilon } } _ { n e w } ^ { p l }$
+
+This algorithm is often referred to as an elastic predictor, radial return algorithm because the correction to the trial stress under the active plastic loading condition returns the stress state to the yield surface along the direction defined by the vector from the center of the yield surface to the elastic trial stress. The subroutine would be coded as follows:
+```txt
+subroutine vumat(
+C Read only -
+    1 nblock, ndir, nshr, nstatev, nfieldv, nprops, lanneal,
+    2 stepTime, totalTime, dt, cmname, coordMp, charLength,
+    3 props, density, strainInc, relSpinInc,
+    4 tempOld, stretchOld, defgradOld, fieldOld,
+    3 stressOld, stateOld, enerInternOld, enerInelasOld,
+    6 tempNew, stretchNew, defgradNew, fieldNew,
+C Write only -
+    5 stressNew, stateNew, enerInternNew, enerInelasNew )
+C
+    include 'vaba_param.inc'
+C
+C J2 Mises Plasticity with kinematic hardening for plane
+C strain case.
+C Elastic predictor, radial corrector algorithm.
+C
+C The state variables are stored as: 
+```
+
+<!-- source-page: 548 -->
+
+```txt
+C STATE(*,1) = back stress component 11
+C STATE(*,2) = back stress component 22
+C STATE(*,3) = back stress component 33
+C STATE(*,4) = back stress component 12
+C STATE(*,5) = equivalent plastic strain
+C
+C
+C All arrays dimensioned by (*) are not used in this algorithm
+    dimension props(nprops), density(nblock),
+    1 coordMp(nblock,*),
+    2 charLength(*), strainInc(nblock,ndir+nshr),
+    3 relSpinInc(*), tempOld(*),
+    4 stretchOld(*), defgradOld(*),
+    5 fieldOld(*), stressOld(nblock,ndir+nshr),
+    6 stateOld(nblock,nstatev), enerInternOld(nblock),
+    7 enerInelasOld(nblock), tempNew(*),
+    8 stretchNew(*), defgradNew(*), fieldNew(*),
+    9 stressNew(nblock,ndir+nshr), stateNew(nblock,nstatev),
+    1 enerInternNew(nblock), enerInelasNew(nblock)
+C
+    character*80 cmname
+C
+    parameter( zero = 0., one = 1., two = 2., three = 3.,
+    1 third = one/three, half = .5, twoThirds = two/three,
+    2 threeHalfs = 1.5 )
+C
+    e = props(1)
+    xnu = props(2)
+    yield = props(3)
+    hard = props(4)
+C
+    twomu = e / ( one + xnu )
+    thremu = threeHalfs * twomu
+    sixmu = three * twomu
+    alamda = twomu * ( e - twomu ) / ( sixmu - two * e )
+    term = one / ( twomu * ( one + hard/thremu ) )
+    con1 = sqrt( twoThirds )
+C
+    do 100 i = 1,nblock
+C
+C Trial stress
+    trace = strainInc(i,1) + strainInc(i,2) + strainInc(i,3) 
+```
+
+<!-- source-page: 549 -->
+
+```txt
+sig1 = stressOld(i,1) + alamda*trace + twomu*strainInc(i,1)
+sig2 = stressOld(i,2) + alamda*trace + twomu*strainInc(i,2)
+sig3 = stressOld(i,3) + alamda*trace + twomu*strainInc(i,3)
+sig4 = stressOld(i,4) + twomu*strainInc(i,4)
+
+C
+C Trial stress measured from the back stress
+s1 = sig1 - stateOld(i,1)
+s2 = sig2 - stateOld(i,2)
+s3 = sig3 - stateOld(i,3)
+s4 = sig4 - stateOld(i,4)
+
+C
+C Deviatoric part of trial stress measured from the back stress
+smean = third * (s1 + s2 + s3)
+ds1 = s1 - smean
+ds2 = s2 - smean
+ds3 = s3 - smean
+
+C
+C Magnitude of the deviatoric trial stress difference
+dsmag = sqrt(ds1**2 + ds2**2 + ds3**2 + 2.*s4**2)
+
+C
+C Check for yield by determining the factor for plasticity,
+C zero for elastic, one for yield
+radius = con1 * yield
+facyld = zero
+if( dsmag - radius .ge. zero ) facyld = one
+
+C
+C Add a protective addition factor to prevent a divide by zero
+C when dsmag is zero. If dsmag is zero, we will not have exceeded
+C the yield stress and facyld will be zero.
+dsmag = dsmag + (one - facyld)
+
+C
+C Calculated increment in gamma (this explicitly includes the
+C time step)
+diff = dsmag - radius
+dgamma = facyld * term * diff
+
+C
+C Update equivalent plastic strain
+deqps = con1 * dgamma
+stateNew(i,5) = stateOld(i,5) + deqps
+
+C
+C Divide dgamma by dsmag so that the deviatoric stresses are
+C explicitly converted to tensors of unit magnitude in the 
+```
+
+<!-- source-page: 550 -->
+
+```fortran
+C following calculations
+    dgamma = dgamma / dsmag
+C
+C Update back stress
+    factor = hard * dgamma * twoThirds
+    stateNew(i,1) = stateOld(i,1) + factor * ds1
+    stateNew(i,2) = stateOld(i,2) + factor * ds2
+    stateNew(i,3) = stateOld(i,3) + factor * ds3
+    stateNew(i,4) = stateOld(i,4) + factor * s4
+C
+C Update the stress
+    factor = twomu * dgamma
+    stressNew(i,1) = sig1 - factor * ds1
+    stressNew(i,2) = sig2 - factor * ds2
+    stressNew(i,3) = sig3 - factor * ds3
+    stressNew(i,4) = sig4 - factor * s4
+C
+C Update the specific internal energy -
+    stressPower = half * (
+    1    ( stressOld(i,1)+stressNew(i,1) ) *strainInc(i,1)
+    1 + ( stressOld(i,2)+stressNew(i,2) ) *strainInc(i,2)
+    1 + ( stressOld(i,3)+stressNew(i,3) ) *strainInc(i,3)
+    1 + two * ( stressOld(i,4)+stressNew(i,4) ) *strainInc(i,4) )
+C
+    enerInternNew(i) = enerInternOld(i)
+    1 + stressPower / density(i)
+C
+C Update the dissipated inelastic specific energy -
+    plasticWorkInc = dgamma * half * (
+    1    ( stressOld(i,1)+stressNew(i,1) ) *ds1
+    1 + ( stressOld(i,2)+stressNew(i,2) ) *ds2
+    1 + ( stressOld(i,3)+stressNew(i,3) ) *ds3
+    1 + two * ( stressOld(i,4)+stressNew(i,4) ) *s4 )
+    enerInelasNew(i) = enerInelasOld(i)
+    1 + plasticWorkInc / density(i)
+    100 continue
+C
+    return
+    end 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_056.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_056.md
new file mode 100644
index 00000000..255a6a73
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_056.md
@@ -0,0 +1,402 @@
+<!-- source-page: 551 -->
+
+# 1.2.23 VUMULLINS: User subroutine to define damage variable for the Mullins effect material model.
+
+Product: Abaqus/Explicit
+
+# References
+
+• “Mullins effect,” Section 22.6.1 of the Abaqus Analysis User’s Guide   
+• “Energy dissipation in elastomeric foams,” Section 22.6.2 of the Abaqus Analysis User’s Guide   
+• \*MULLINS EFFECT   
+• “Mullins effect and permanent set,” Section 2.2.3 of the Abaqus Verification Guide
+
+# Overview
+
+# User subroutine VUMULLINS:
+
+• can be used to define the damage variable for the Mullins effect material model (“Mullins effect,” Section 22.6.1 of the Abaqus Analysis User’s Guide), including the use of the Mullins effect approach to model energy dissipation in elastomeric foams (“Energy dissipation in elastomeric foams,” Section 22.6.2 of the Abaqus Analysis User’s Guide);   
+• will be called for blocks of material calculation points for which the material definition contains a user-defined Mullins effect;   
+• can be used to define a failure criterion based on the strain energy density of the material;   
+• can use and update solution-dependent state variables;   
+• can use any field variables that are passed in; and   
+• should be used when you do not want to use the Ogden and Roxburgh form of the damage variable, , that is used by Abaqus/Explicit.
+
+# Material point deletion
+
+Material points that satisfy a user-defined failure criterion can be deleted from the model (see “Userdefined mechanical material behavior,” Section 26.7.1 of the Abaqus Analysis User’s Guide). You must specify the state variable number controlling the element deletion flag when you allocate space for the solution-dependent state variables, as explained in “User-defined mechanical material behavior,” Section 26.7.1 of the Abaqus Analysis User’s Guide. The deletion state variable can be set to a value of one or zero inside user subroutine VUMULLINS. A value of one indicates that the material point is active, and a value of zero indicates that Abaqus/Explicit should delete the material point from the model by setting the stresses to zero. The structure of the block of material points passed to user subroutine VUMULLINS remains unchanged during the analysis; deleted material points are not removed from the block. Abaqus/Explicit will “freeze” the values of the strain energy density passed to user subroutine VUMULLINS for all deleted material points; that is, the values remain constant after deletion is triggered. Once a material point has been flagged as deleted, it cannot be reactivated.
+
+<!-- source-page: 552 -->
+
+```fortran
+subroutine vumullins (
+C Read only (unmodifiable) variables -
+    1    nblock,
+    2    jElem, kIntPt, kLayer, kSecPt,
+    3    cmname,
+    4    nstatev, nfieldv, nprops,
+    5    props, tempOld, tempNew, fieldOld, fieldNew,
+    6    stateOld, enerDamageOld,
+    7    uMaxOld, uMaxNew, uDev,
+C Write only (modifiable) variables -
+    8    eta, detaDuDev,
+    9    stateNew, enerDamageNew )
+C
+    include 'vaba_param.inc'
+C
+    dimension props(nprops),
+    1    tempOld(nblock),
+    2    fieldOld(nblock,nfieldv),
+    3    stateOld(nblock,nstatev),
+    4    tempNew(nblock),
+    5    fieldNew(nblock,nfieldv),
+    6    enerDamageOld(nblock),
+    7    uMaxOld(nblock), uMaxNew(nblock),
+    8    uDev(nblock),
+    9    eta(nblock), detaDuDev(nblock),
+    1    stateNew(nblock,nstatev),
+    2    enerDamageNew(nblock)
+C
+    character*80 cmname
+C
+    do 100 km = 1,nblock
+    user coding
+100 continue
+    return
+    end 
+```
+
+<!-- source-page: 553 -->
+
+# Variables to be defined
+
+eta(nblock)
+
+The damage variable, .
+
+detaDuDev(nblock)
+
+The derivative of the damage variable with respect to the deviatoric elastic strain energy density of the undamaged material, $d \eta / d \tilde { U } _ { d e v }$ , when the primary material behavior is hyperelastic. The derivative of the damage variable with respect to the total elastic strain energy density of the undamaged material, $d \eta / d { \tilde { U } }$ , when the primary material behavior is hyperfoam. This quantity is needed for the evaluation of the effective moduli of the material, which enters the stable time increment calculation.
+
+# Variables that can be updated
+
+stateNew(nblock,nstatev)
+
+State variables at each material point at the end of the increment. You define the size of this array by allocating space for it (see “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide, for more information).
+
+enerDamageNew(nblock)
+
+The energy dissipation density at the end of the increment. This quantity can be defined either in total form or in an incremental manner using the old value of the damage dissipation enerDamageOld and the increment in damage dissipation. This quantity is used for output purposes only.
+
+# Variables passed in for information
+
+nblock
+
+Number of material points to be processed in this call to VUMULLINS.
+
+jElem(nblock)
+
+Array of element numbers.
+
+kIntPt
+
+Integration point number.
+
+kLayer
+
+Layer number (for composite shells).
+
+kSecPt
+
+Section point number within the current layer.
+
+cmname
+
+User-specified material name, left justified. It is passed in as an uppercase character string. Some internal material models are given names starting with the “ABQ\_” character string. To avoid conflict, you should not use “ABQ\_” as the leading string for cmname.
+
+<!-- source-page: 554 -->
+
+# nstatev
+
+Number of user-defined state variables that are associated with this material type (you define the number as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# nfieldv
+
+Number of user-defined external field variables.
+
+# nprops
+
+User-specified number of user-defined material properties.
+
+# props(nprops)
+
+User-supplied material properties.
+
+# tempOld(nblock)
+
+Temperatures at each material point at the beginning of the increment.
+
+# tempNew(nblock)
+
+Temperatures at each material point at the end of the increment.
+
+# fieldOld(nblock,nfieldv)
+
+Values of the user-defined field variables at each material point at the beginning of the increment.
+
+# fieldNew(nblock,nfieldv)
+
+Values of the user-defined field variables at each material point at the end of the increment.
+
+# stateOld(nblock,nstatev)
+
+State variables at each material point at the beginning of the increment.
+
+# enerDamageOld(nblock)
+
+The value of energy dissipated at the beginning of the increment.
+
+# uMaxOld(nblock)
+
+The value, at the beginning of the increment, of the maximum primary strain energy density over its entire deformation history.
+
+# uMaxNew(nblock)
+
+The value, at the end of the increment, of the maximum primary strain energy density over its entire deformation history.
+
+# uDev(nblock)
+
+The value, at the end of the increment, of the deviatoric primary strain energy density, $\tilde { U } _ { d e v }$ , when the primary material behavior is hyperelastic. The value, at the end of the increment, of the total primary strain energy density, $\tilde { U }$ , when the primary material behavior is hyperfoam.
+
+<!-- source-page: 555 -->
+
+As a simple example of the coding of user subroutine VUMULLINS, consider the following damage model based on the softening hyperelasticity approach proposed by Volokh (2007). The damage variable is assumed to vary with the deformation according to
+
+$$
+\eta = \exp \left(\frac {- U _ {d e v} ^ {m}}{U _ {0}}\right),
+$$
+
+where $U _ { d e v } ^ { m }$ is the maximum value of $\tilde { U } _ { d e v }$ at a material point during its deformation history, $\tilde { U } _ { d e v }$ is the deviatoric part of the strain energy density of the undamaged hyperelastic behavior, and $U _ { 0 }$ is a material parameter with units of strain energy density. The energy dissipation function for this model takes the form
+
+$$
+\phi (\eta) = U _ {0} (1 - \eta + \eta \ln (\eta)).
+$$
+
+It can be shown that the functions and satisfy the following condition:
+
+$$
+\left(\tilde {U} _ {d e v} + \phi^ {\prime} (\eta)\right) \dot {\eta} = 0.
+$$
+
+The code in user subroutine VUMULLINS must return the damage variable, ; the derivative of the damage variable with respect to the elastic strain energy density of the undamaged material, $d \eta / d \tilde { U } _ { d e v }$ ; and the energy dissipation . The user subroutine would be coded as follows:
+```fortran
+subroutine vumullins (
+C Read only (unmodifiable) variables -
+    1    nblock,
+    2    jElem, kIntPt, kLayer, kSecPt,
+    3    cmname,
+    4    nstatev, nfieldv, nprops,
+    5    props, tempOld, tempNew, fieldOld, fieldNew,
+    6    stateOld, enerDamageOld,
+    7    uMaxOld, uMaxNew, uDev,
+C Write only (modifiable) variables -
+    8    eta, detaDuDev,
+    9    stateNew, enerDamageNew )
+C
+    include 'vaba_param.inc'
+C
+    dimension props(nprops),
+    1    tempOld(nblock),
+    2    fieldOld(nblock, nfieldv),
+    3    stateOld(nblock, nstatev), 
+```
+
+<!-- source-page: 556 -->
+
+```txt
+4 tempNew(nblock),
+5 fieldNew(nblock,nfieldv),
+6 enerDamageOld(nblock),
+7 uMaxOld(nblock), uMaxNew(nblock),
+8 uDev(nblock),
+9 eta(nblock), detaDuDev(nblock),
+1 stateNew(nblock,nstatev),
+2 enerDamageNew(nblock)
+
+C
+    character*80 cmname
+
+C
+    parameter ( zero = 0.d0, one = 1.d0 )
+
+C
+    u0 = props(1)
+    u0Inv = zero
+    if ( u0 .gt. zero ) u0Inv = one / u0
+
+C
+    do k=1, nblock
+    eta(k) = exp(-uMaxNew(k) * u0Inv)
+    detaDUdev(k) = zero
+    if ( uMaxNew(k) .gt. uMaxOld(k) )
+
+1    detaDUdev(k) = -u0Inv * eta(k)
+    enerDamageNew(k) = u0*(one-eta(k)+eta(k)*log(eta(k)))
+    end do
+
+C
+    return
+    end 
+```
+
+# Additional reference
+
+• Volokh, K. Y., “Hyperelasticity with Softening for Modeling Materials Failure,” Journal of the Mechanics and Physics of Solids, vol. 55, pp. 2237–2264, 2007.
+
+<!-- source-page: 557 -->
+
+# 1.2.24 VUSDFLD: User subroutine to redefine field variables at a material point.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Obtaining material point information in an Abaqus/Standard analysis,” Section 2.1.6   
+• “Material data definition,” Section 21.1.2 of the Abaqus Analysis User’s Guide   
+• \*USER DEFINED FIELD   
+• “Damage and failure of a laminated composite plate,” Section 1.1.14 of the Abaqus Example Problems Guide   
+• “VUSDFLD,” Section 4.1.39 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine VUSDFLD:
+
+• allows the redefinition of field variables at a material point as functions of time or of any of the available material point quantities listed in “Available output variable keys” in “Obtaining material point information in an Abaqus/Explicit analysis,” Section 2.1.7;   
+• can be used to introduce solution-dependent material properties since such properties can be easily defined as functions of field variables;   
+• will be called at all material points of elements for which the material definition includes userdefined field variables;   
+• can call utility routine VGETVRM to access material point data; and   
+• can use and update solution-dependent state variables.
+
+# Explicit solution dependence
+
+Since this routine provides access to material point quantities only at the start of the increment, the material properties for a given increment are not influenced by the results obtained during the increment. Hence, the accuracy of the results depends on the size of the time increment. However, in most situations this is not a concern for explicit dynamic analysis because the stable time increment is usually sufficiently small to ensure good accuracy.
+
+# Defining field variables
+
+Before user subroutine VUSDFLD is called, the values of the field variables at the material point are calculated by interpolation from the values defined at the nodes. Any changes to the field variables in the user subroutine are local to the material point: the nodal field variables retain the values defined as initial conditions or predefined field variables or the values defined in user subroutine VUFIELD. The values of the field variables defined in this routine are used to calculate values of material properties that
+
+<!-- source-page: 558 -->
+
+are defined to depend on field variables and are passed into other user subroutines that are called at the material point, such as the following:
+
+• VUANISOHYPER\_INV   
+• VUANISOHYPER\_STRAIN   
+• VUHARD   
+• VUMAT   
+• VUTRS   
+• VUVISCOSITY
+
+Output of the user-defined field variables at the material points can be obtained with the element integration point output variable FV (see “Abaqus/Explicit output variable identifiers,” Section 4.2.2 of the Abaqus Analysis User’s Guide).
+
+# State variables
+
+Since the redefinition of field variables in VUSDFLD is local to the current increment (field variables are restored to the values interpolated from the nodal values at the start of each increment), any history dependence required to update material properties by using this subroutine must be introduced with userdefined state variables.
+
+The state variables can be updated in VUSDFLD and then passed into other user subroutines that can be called at this material point, such as those listed above. The number of such state variables can be specified as shown in the example at the end of this section (see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# Accessing material point data
+
+The values of the material point quantities at the start of the increment can be accessed through the utility routine VGETVRM described in “Obtaining material point information in an Abaqus/Explicit analysis,” Section 2.1.7. The values of the material point quantities are obtained by calling VGETVRM with the appropriate output variable keys.
+
+# Component ordering in symmetric tensors
+
+For symmetric tensors such as the stress and strain tensors there are ndir+nshr components, and the component order is given as a natural permutation of the indices of the tensor. The direct components are first and then the indirect components, beginning with the 12-component. For example, a stress tensor contains ndir direct stress components and nshr shear stress components, which are returned as:
+
+<table><tr><td>Component</td><td>2D Case</td><td>3D Case</td></tr><tr><td>1</td><td> $\sigma_{11}$ </td><td> $\sigma_{11}$ </td></tr><tr><td>2</td><td> $\sigma_{22}$ </td><td> $\sigma_{22}$ </td></tr><tr><td>3</td><td> $\sigma_{33}$ </td><td> $\sigma_{33}$ </td></tr></table>
+
+<!-- source-page: 559 -->
+
+<table><tr><td>Component</td><td>2D Case</td><td>3D Case</td></tr><tr><td>4</td><td> $\sigma_{12}$ </td><td> $\sigma_{12}$ </td></tr><tr><td>5</td><td></td><td> $\sigma_{23}$ </td></tr><tr><td>6</td><td></td><td> $\sigma_{31}$ </td></tr></table>
+
+The shear strain components in user subroutine VUSDFLD are stored as tensor components and not as engineering components; unlike user subroutine USDFLD in Abaqus/Standard, which uses engineering components.
+
+User subroutine interface   
+```fortran
+subroutine vusdfld(
+c Read only variables -
+    1 nblock, nstatev, nfieldv, nprops, ndir, nshr,
+    2 jElem, kIntPt, kLayer, kSecPt,
+    3 stepTime, totalTime, dt, cmname,
+    4 coordMp, direct, T, charLength, props,
+    5 stateOld,
+c Write only variables -
+    6 stateNew, field )
+c
+    include 'vaba_param.inc'
+c
+    dimension jElem(nblock), coordMp(nblock,*),
+    1 direct(nblock,3,3), T(nblock,3,3),
+    2 charLength(nblock), props(nprops),
+    3 stateOld(nblock,nstatev),
+    4 stateNew(nblock,nstatev),
+    5 field(nblock,nfieldv)
+    character*80 cmname
+c
+c Local arrays from vgetvm are dimensioned to
+c maximum block size (maxblk)
+c
+    parameter( nrData=6 )
+    character*3 cData(maxblk*nrData)
+    dimension rData(maxblk*nrData), jData(maxblk*nrData)
+c
+    do 100 k = 1, nblock
+    user coding to define field(nblock,nfieldv)
+    and, if necessary, stateNew(nblock,nstatev) 
+```
+
+<!-- source-page: 560 -->
+
+```lua
+100 continue
+c
+return
+end 
+```
+
+# Variable to be defined
+
+# field(nblock,nfieldv)
+
+An array containing the field variables at the material points. These are passed in with the values interpolated from the nodes at the end of the current increment, as specified with initial condition definitions, predefined field variable definitions, or user subroutine VUFIELD. The updated values are used to calculate the values of material properties that are defined to depend on field variables and are passed into other user subroutines that are called at the material points.
+
+# Variable that can be updated
+
+# stateNew(nblock,nstatev)
+
+An array containing the solution-dependent state variables at the material points. In all cases stateNew can be updated in this subroutine, and the updated values are passed into other user subroutines that are called at the material points. The number of state variables associated with this material point is defined as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide.
+
+# Variables passed in for information
+
+# nblock
+
+Number of material points to be processed in this call to VUSDFLD.
+
+# nstatev
+
+Number of user-defined state variables that are associated with this material type (you define this as described in “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# nfieldv
+
+Number of user-defined external field variables.
+
+# nprops
+
+User-specified number of user-defined material properties.
+
+# ndir
+
+Number of direct components in a symmetric tensor.
+
+# nshr
+
+Number of indirect components in a symmetric tensor.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_057.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_057.md
new file mode 100644
index 00000000..7fc35a6e
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_057.md
@@ -0,0 +1,352 @@
+<!-- source-page: 561 -->
+
+# jElem
+
+Array of element numbers.
+
+# kIntPt
+
+Integration point number.
+
+# kLayer
+
+Layer number (for composite shells).
+
+# kSecPt
+
+Section point number within the current layer.
+
+# stepTime
+
+Value of time since the step began.
+
+# totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime-stepTime.
+
+# dt
+
+Time increment size.
+
+# cmname
+
+User-specified material name, left justified. It is passed in as an uppercase character string. Some internal material models are given names starting with the “ABQ\_” character string. To avoid conflict, you should not use “ABQ\_” as the leading string for cmname.
+
+# coordMp(nblock,\*)
+
+Material point coordinates. It is the midplane material point for shell elements and the centroid for beam elements.
+
+# direct(nblock,3,3)
+
+An array containing the direction cosines of the material directions in terms of the global basis directions. For material point k, direct(k,1,1), direct(k,2,1), direct(k,3,1) give the (1, 2, 3) components of the first material direction; direct(k,1,2), direct(k,2,2), direct(k,3,2) give the second material direction, etc. For shell and membrane elements, the first two directions are in the plane of the element and the third direction is the normal. This information is not available for beam elements.
+
+# T(nblock,3,3)
+
+An array containing the direction cosines of the material orientation components relative to the element basis directions. For material point k, this is the orientation that defines the material directions (direct) in terms of the element basis directions. For continuum elements T and direct are identical. For shell and membrane elements T(k,1,1) , T(k,1,2) , T(k,2,1) , T(k,2,2) , T(k,3,3) and all other components are zero, where is the
+
+<!-- source-page: 562 -->
+
+counterclockwise rotation around the normal vector that defines the orientation. If no orientation is used, T is an identity matrix. Orientation is not available for beam elements.
+
+charLength(nblock)
+
+Characteristic element length, which is either the default value based on the geometric mean or the user-defined characteristic element length defined in user subroutine VUCHARLENGTH. The default value is a typical length of a line across an element for a first-order element; it is half of the same typical length for a second-order element. For beams and trusses the default value is a characteristic length along the element axis. For membranes and shells it is a characteristic length in the reference surface. For axisymmetric elements it is a characteristic length in the r–z plane only. For cohesive elements it is equal to the constitutive thickness.
+
+props(nprops)
+
+User-supplied material properties.
+
+stateOld (nblock, nstatev)
+
+State variables at each material point at the beginning of the increment.
+
+Example: Damaged elasticity model
+
+Included below is an example of user subroutine VUSDFLD. In this example a truss element is loaded in tension. A damaged elasticity model is introduced: the modulus decreases as a function of the maximum tensile strain that occurred during the loading history. The maximum tensile strain is stored as a solution-dependent state variable (see “Defining solution-dependent field variables” in “Predefined fields,” Section 34.6.1 of the Abaqus Analysis User’s Guide).
+
+Input file   
+```csv
+*HEADING
+Damaged elasticity model with user subroutine vusdfld
+*ELEMENT, TYPE=T2D2, ELSET=ONE
+1, 1, 2
+*NODE, NSET=NALL
+1, 0., 0.
+2, 10., 0.
+*SOLID SECTION, ELSET=ONE, MATERIAL=ELASTIC
+1.
+*MATERIAL, NAME=ELASTIC
+*ELASTIC, DEPENDENCIES=1
+** Table of modulus values decreasing as a function
+** of field variable 1.
+2000., 0.3, 0., 0.00
+1500., 0.3, 0., 0.01
+1200., 0.3, 0., 0.02
+1000., 0.3, 0., 0.04 
+```
+
+<!-- source-page: 563 -->
+
+```csv
+* DENSITY
+1.0e-6
+* USER DEFINED FIELD
+* DEPVAR
+1
+1, EPSMAX, "Maximum strain value"
+* BOUNDARY
+1, 1, 2
+2, 2
+* AMPLITUDE, NAME=LOAD1
+0.0, 0.0, 1.0, 1.0
+* AMPLITUDE, NAME=LOAD2
+0.0, 0.0, 2.0, 1.0
+* AMPLITUDE, NAME=UNLOAD
+0.0, 1.0, 1.0, 0.0
+* STEP, NLGEOM=NO
+* DYNAMIC, EXPLICIT
+, 1.0
+* CLOAD, AMPLITUDE=LOAD1
+2, 1, 20.
+* OUTPUT, FIELD, VARIABLE=PRESELECT
+* OUTPUT, HISTORY, VARIABLE=PRESELECT
+* ELEMENT OUTPUT, ELSET=ONE
+S, E, SDV
+* NODE OUTPUT, NSET=NALL
+RF, CF, U
+* END STEP
+* STEP, NLGEOM=NO
+* DYNAMIC, EXPLICIT
+, 1.0
+* CLOAD, AMPLITUDE=UNLOAD
+2, 1, 20.
+* END STEP
+* STEP, NLGEOM=NO
+* DYNAMIC, EXPLICIT
+, 2.0
+* CLOAD, AMPLITUDE=LOAD2
+2, 1, 40.
+* END STEP 
+```
+
+<!-- source-page: 564 -->
+
+User subroutine   
+```fortran
+subroutine vusdfld(
+c Read only -
+* nblock, nstatev, nfieldv, nprops, ndir, nshr,
+* jElem, kIntPt, kLayer, kSecPt,
+* steTime, totalTime, dt, cmname,
+* coordMp, direct, T, charLength, props,
+* stateOld,
+c Write only -
+* stateNew, field )
+c
+    include 'vaba_param.inc'
+c
+    dimension jElem(nblock), coordMp(nblock,*),
+*    direct(nblock,3,3), T(nblock,3,3),
+*    charLength(nblock), props(nprops),
+*    stateOld(nblock,nstatev),
+*    stateNew(nblock,nstatev),
+*    field(nblock,nfieldv)
+    character*80 cmname
+c
+c Local arrays from vgetvrm are dimensioned to
+c maximum block size (maxblk)
+c
+    parameter( nrData=6 )
+    character*3 cData(maxblk*nrData)
+    dimension rData(maxblk*nrData), jData(maxblk*nrData)
+c
+    jStatus = 1
+    call vgetvrm('LE', rData, jData, cData, jStatus )
+c
+    if( jStatus .ne. 0 ) then
+    call xplb_abqerr(-2,'Utility routine VGETVRM '//
+*    'failed to get variable.',0,zero,' ') 
+    call xplb_exit
+    end if
+c
+    call setField( nblock, nstatev, nfieldv, nrData,
+*    rData, stateOld, stateNew, field)
+c
+    return 
+```
+
+<!-- source-page: 565 -->
+
+```prolog
+end
+subroutine setField( nblock, nstatev, nfieldv, nrData,
+*    strain, stateOld, stateNew, field )
+include 'vaba_param.inc'
+
+dimension stateOld(nblock, nstatev),
+*    stateNew(nblock, nstatev),
+*    field(nblock, nfieldv), strain(nblock, nrData)
+
+do k = 1, nblock
+
+Absolute value of current strain:
+    eps = abs( strain(k, 1) )
+
+Maximum value of strain up to this point in time:
+    epsmax = stateOld(k, 1)
+
+Use the maximum strain as a field variable
+    field(k, 1) = max( eps, epsmax )
+
+Store the maximum strain as a solution dependent state
+    stateNew(k, 1) = field(k, 1)
+
+end do
+
+return
+end 
+```
+
+<!-- source-page: 566 -->
+
+<!-- source-page: 567 -->
+
+# 1.2.25 VUTRS: User subroutine to define a reduced time shift function for a viscoelastic material.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Time domain viscoelasticity,” Section 22.7.1 of the Abaqus Analysis User’s Guide   
+• \*TRS   
+• \*VISCOELASTIC   
+• “Transient thermal loading of a viscoelastic slab,” Section 3.1.2 of the Abaqus Benchmarks Guide
+
+# Overview
+
+User subroutine VUTRS:
+
+• can be used to define a temperature-time shift for a time domain viscoelastic analysis;   
+• will be called for all material points of elements for which a user-defined shift function is specified to define the time-temperature correspondence as part of the viscoelastic material definition;   
+• can use and update solution-dependent state variables; and   
+• can have incoming field variables redefined by user subroutine VUSDFLD.
+
+# User subroutine interface
+
+```txt
+subroutine vuts(
+c Read only variables -
+    1 nblock, nstatev, nfieldv, nprops,
+    2 timeTime, totalTime, dt,
+    3 cmname, props, density, coordMp,
+    4 tempOld, fieldOld, stateOld,
+    5 tempNew, fieldNew,
+c Write only variables -
+    6 shift, stateNew )
+c
+    include 'vaba_param.inc'
+c
+    dimension props(nprops), density(nblock), coordMp(nblock,*),
+    1 tempOld(nblock), tempNew(nblock),
+    2 fieldOld(nblock, nfieldv), fieldNew(nblock, nfieldv),
+    3 stateOld(nblock, nstatev), stateNew(nblock, nstatev),
+    4 shift(nblock, 2) 
+```
+
+<!-- source-page: 568 -->
+
+```lua
+c
+character*80 cmname
+c
+do 100 k=1, nblock
+user coding to define shift(k,1) and shift(k,2)
+100 continue
+c
+return
+end 
+```
+
+# Variable to be defined
+
+shift(nblock,2)
+
+Array that defines the shift function, A ( ), at the material points. For material point k, shift(k,1) defines the shift function at the beginning of the increment, and shift(k,2) defines the shift function at the end of the increment. Abaqus/Explicit will apply an averaging scheme to these values that assumes that the natural logarithm of the shift function can be approximated by a linear function over the increment.
+
+If either shift(k,1) or shift(k,2) is less than or equal to zero, no time shift will be applied.
+
+# Variable that can be updated
+
+stateNew(nblock,nstatev)
+
+Array containing the solution-dependent state variables at the material points. This array will be passed in containing the values of these variables at the start of the increment unless they are updated in user subroutine VUSDFLD, in which case the updated values are passed in. If any of the solution-dependent state variables are being used in conjunction with the viscoelastic behavior, they must be updated in this subroutine to their values at the end of the increment.
+
+# Variables passed in for information
+
+nblock
+
+Number of material points to be processed in this call to VUTRS.
+
+nstatev
+
+Number of user-defined state variables that are associated with this material type (see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+nfieldv
+
+Number of user-defined external field variables.
+
+nprops
+
+User-specified number of user-defined material properties.
+
+<!-- source-page: 569 -->
+
+# stepTime
+
+Value of time since the step began.
+
+# totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime-stepTime.
+
+# dt
+
+Time increment size.
+
+# cmname
+
+Material name, left justified. It is passed in as an uppercase character string. Some internal material models are given names starting with the “ABQ\_” character string. To avoid conflict, “ABQ\_” should not be used as the leading string for cmname.
+
+# props(nprops)
+
+User-supplied material properties.
+
+# density(nblock)
+
+Current density at the material points in the midstep configuration. This value may be inaccurate in problems where the volumetric strain increment is very small. If an accurate value of the density is required in such cases, the analysis should be run in double precision. This value of the density is not affected by mass scaling.
+
+# coordMp(nblock,\*)
+
+Material point coordinates. It is the midplane material point for shell elements and the centroid for beam elements.
+
+# tempOld(nblock)
+
+Temperatures at each material point at the beginning of the increment.
+
+# fieldOld(nblock,nfieldv)
+
+Values of the user-defined field variables at each material point at the beginning of the increment.
+
+# stateOld(nblock,nstatev)
+
+State variables at each material point at the beginning of the increment.
+
+# tempNew(nblock)
+
+Temperatures at each material point at the end of the increment.
+
+# fieldNew(nblock,nfieldv)
+
+Values of the user-defined field variables at each material point at the end of the increment.
+
+<!-- source-page: 570 -->
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_058.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_058.md
new file mode 100644
index 00000000..0f315d6b
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_058.md
@@ -0,0 +1,384 @@
+<!-- source-page: 571 -->
+
+# 1.2.26 VUVISCOSITY: User subroutine to define the shear viscosity for equation of state models.
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “Equation of state,” Section 25.2.1 of the Abaqus Analysis User’s Guide   
+• \*EOS   
+• \*VISCOSITY   
+• “VUVISCOSITY,” Section 4.1.40 of the Abaqus Verification Guide
+
+# Overview
+
+User subroutine VUVISCOSITY:
+
+• is called at all material points of elements with an equation of state for which the material definition includes user-defined viscous shear behavior;   
+• can be used to define a material’s isotropic viscous behavior;   
+• can use and update solution-dependent state variables; and   
+• can be used in conjunction with user subroutine VUSDFLD to redefine any field variables before they are passed in.
+
+# User subroutine interface
+
+```prolog
+subroutine vuviscosity(
+C Read only -
+* nblock,
+* jElem, kIntPt, kLayer, kSecPt,
+* steppTime, totalTime, dt, cmname,
+* nstatev, nfieldv, nprops,
+* props, tempOld, tempNew, fieldOld, fieldNew,
+* stateOld,
+* shrRate,
+C Write only -
+* viscosity,
+* stateNew )
+C
+include 'vaba_param.inc'
+C
+dimension props(nprops), tempOld(nblock), tempNew(nblock), 
+```
+
+<!-- source-page: 572 -->
+
+```txt
+1 fieldOld(nblock, nfieldv), fieldNew(nblock, nfieldv),
+2 stateOld(nblock, nstatev), eqps(nblock), eqpsRate(nblock),
+3 viscosity(nblock),
+4 stateNew(nblock, nstatev), jElem(nblock)
+C
+    character*80 cmname
+C
+    do 100 km = 1, nblock
+    user coding
+100 continue
+C
+    return
+    end 
+```
+
+# Variables to be defined
+
+viscosity(nblock)
+
+Array containing the viscosity at the material points. (Units of FL−2 T.)
+
+stateNew(nblock,nstatev)
+
+Array containing the state variables at the material points at the end of the increment. The allocation of this array is described in “Solution-dependent state variables” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide.
+
+# Variables passed in for information
+
+nblock
+
+Number of material points to be processed in this call to VUVISCOSITY.
+
+jElem(nblock)
+
+Array of element numbers.
+
+kIntPt
+
+Integration point number.
+
+kLayer
+
+Layer number (for composite shells).
+
+kSecPt
+
+Section point number within the current layer.
+
+stepTime
+
+Value of time since the step began.
+
+totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime-stepTime.
+
+<!-- source-page: 573 -->
+
+# dt
+
+Time increment size.
+
+# cmname
+
+Material name, left justified. It is passed in as an uppercase character string. Some internal material models are given names starting with the “ABQ\_” character string. To avoid conflict, “ABQ\_” should not be used as the leading string for cmname.
+
+# nstatev
+
+Number of user-defined state variables that are associated with this material type (see “Allocating space” in “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide).
+
+# nfieldv
+
+Number of user-defined external field variables.
+
+# nprops
+
+User-specified number of user-defined material properties.
+
+# tempOld(nblock)
+
+Temperatures at the material points at the beginning of the increment.
+
+# tempNew(nblock)
+
+Temperatures at the material points at the end of the increment.
+
+# fieldOld(nblock,nfieldv)
+
+Values of the user-defined field variables at the material points at the beginning of the increment.
+
+# fieldNew(nblock,nfieldv)
+
+Values of the user-defined field variables at the material points at the end of the increment.
+
+# stateOld(nblock,nstatev)
+
+State variables at the material points at the beginning of the increment.
+
+# shrRate(nblock)
+
+Equivalent shear strain rate, , at the material points.
+
+# Example: Cross viscosity model
+
+As a simple example of the coding of subroutine VUVISCOSITY, consider the Cross viscosity model. The Cross model is commonly used when it is necessary to describe the low shear rate behavior of the viscosity. The viscosity is expressed as
+
+$$
+\eta = \frac {\eta_ {0}}{1 + (\lambda \dot {\gamma}) ^ {1 - n}},
+$$
+
+<!-- source-page: 574 -->
+
+where $\eta _ { 0 }$ is the Newtonian viscosity, is the flow index in the power law regime, and is a constant with units of time, such that $1 / \lambda$ corresponds to the critical shear rate at which the fluid changes from Newtonian to power law behavior.
+
+The subroutine would be coded as follows:
+
+```txt
+subroutine vuviscosity (
+C Read only -
+    * nblock,
+    * jElem, kIntPt, kLayer, kSecPt,
+    * steTime, totalTime, dt, cmname,
+    * nstatev, nfieldv, nprops,
+    * props, tempOld, tempNew, fieldOld, fieldNew,
+    * stateOld,
+    * shrRate,
+C Write only -
+    * viscosity,
+    * stateNew )
+C
+    include 'vaba_param.inc'
+C
+    dimension props(nprops),
+    * tempOld(nblock),
+    * fieldOld(nblock, nfieldv),
+    * stateOld(nblock, nstatev),
+    * shrRate(nblock),
+    * tempNew(nblock),
+    * fieldNew(nblock, nfieldv),
+    * viscosity(nblock),
+    * stateNew(nblock, nstatev)
+C
+    character*80 cmname
+C
+    parameter (one = 1.d0 )
+C
+C Cross viscosity
+C
+    eta0 = props(1)
+    rlambda = props(2)
+    rn = props(3)
+C
+    do k = 1, nblock
+    viscosity(k) = eta0 / (one + (rlambda * shrRate(k)) ** (one - rn))
+    end do 
+```
+
+<!-- source-page: 575 -->
+
+C
+
+return
+end
+
+<!-- source-page: 576 -->
+
+<!-- source-page: 577 -->
+
+# 1.2.27 VWAVE: User subroutine to define wave kinematics for an Abaqus/Aqua analysis.
+
+Products: Abaqus/Explicit Abaqus/Aqua
+
+# References
+
+• “Abaqus/Aqua analysis,” Section 6.11.1 of the Abaqus Analysis User’s Guide   
+• \*AQUA   
+• \*WAVE   
+• \*WIND
+
+# Overview
+
+User subroutine VWAVE:
+
+• will be called for a collection of points (typically load integration points) for which an Abaqus/Aqua load and a user-defined gravity wave are specified;   
+• can be used to define wave kinematics to provide unsteady contributions to fluid variables—such as velocity, acceleration, pressure, gradient of pressure along elevation, and the instantaneous freesurface elevation—as a function of time and space; and   
+• will be called twice within an increment for each Abaqus/Aqua load. The first call is used to obtain the instantaneous wave-surface elevation at the nodes of the elements on which the loads are applied. The second call is used to obtain desired fluid variables at the integration points for the load calculations.
+
+# User subroutine interface
+
+```python
+subroutine vwave(
+C Write only -
+    1 fWaveSurf, fUnsteadyVel, fFluidAcc,
+    2 fUnsteadyPress, fUnsteadyDPressDZ,
+C Read/Write -
+    1 ScaleSteady, ScaleUnsteady,
+C Read only -
+    1 kStep, kInc,
+    2 nblock, ndim, nprops, naquaconst, nwindconst,
+    3 nstatevar, nfieldvar, iElemType, iLoadType, sname,
+    4 lUpdFluidVar, Coord, Velocity, StateVar, FieldVar,
+    5 DirVec, AquaSteadyConstants, WindConstants,
+    6 fSteadyVel, Props, dt, timeTotal, timeStep)
+C
+    include 'vaba_param.inc' 
+```
+
+<!-- source-page: 578 -->
+
+```txt
+parameter ( j_upd_FreeSurf = 0,
+1 j_upd_FluidVarBuoyancy = 1,
+2 j_upd_FluidVarDrag = 2,
+3 j_upd_FluidVarInertia = 3, 
+```  
+The types of distributed loads:
+
+```javascript
+1 j_lcr_PB = 51,
+2 j_lcr_DragFDD = 53, j_lcr_DragWDD = 54,
+3 j_lcr_DragFDT = 55, j_lcr_DragFI = 56,
+4 j_lcr_DragFD1 = 57, j_lcr_DragFD2 = 58,
+5 j_lcr_DragWD1 = 59, j_lcr_DragWD2 = 60,
+6 j_lcr_DragFI1 = 61, j_lcr_DragFI2 = 62, 
+```  
+The types of concentrated loads:
+
+```python
+1 j_ccr_TSB = 1002, j_ccr_DragTFD = 1004,  
+2 j_ccr_DragTWD = 1005, j_ccr_DragTSI = 1006) 
+```
+
+dimension Props(nProps), Coord(nblock,ndim),   
+```txt
+1 Velocity(nblock,ndim), StateVar(nblock,nstatevar),
+2 FieldVar(nblock,nfieldvar), DirVec(nblock,ndim),
+3 fWaveSurf(nblock), fFluidAcc(nblock,ndim),
+4 fUnsteadyVel(nblock,ndim), fUnsteadyPress(nblock),
+5 fUnsteadyDPressDZ(nblock), fSteadyVel(nblock,ndim),
+6 AquaSteadyConstants(naquaconst), WindConstants(nwindconst) 
+```  
+character\*80 sname   
+The following if test and do loop structure illustrates proper usage of this user subroutine.
+
+if (lUpdFluidVar .eq. j\_upd\_FreeSurf) then This part is executed at the first call.   
+```txt
+do kn = 1, nblock
+    User coding to update fWaveSurf
+    optionally update StateVar
+end do 
+```  
+else   
+This part is executed at the second call.   
+if (lUpdFluidVar .eq. j\_upd\_FluidVarBuoyancy) then Update variables for buoyancy loads (PB, TSB):
+
+<!-- source-page: 579 -->
+
+```lua
+do kn = 1, nblock
+    user coding to update fUnsteadyPress, fUnsteadyDPressDZ
+    end do
+    optionally update multipliers ScaleSteady, ScaleUnsteady
+    else if (lUpdFluidVar .eq. j_upd_FluidVarDrag) then
+    Update variables for drag loads (FDD, FDT, FD1, FD2, TFD):
+    do kn = 1, nblock
+    User coding to update fUnsteadyVel
+    end do
+    optionally update multipliers ScaleSteady, ScaleUnsteady
+    else if (lUpdFluidVar .eq. j_upd_FluidVarInertia) then
+    Update variables for inertia loads (FI, FI1, FI2, TSI):
+    do kn = 1, nblock
+    User coding to update fFluidAcc
+    end do
+    optionally update multipliers ScaleSteady, ScaleUnsteady
+    end if
+    end if
+    return
+    end 
+```
+
+# Variables to be defined
+
+# fWaveSurf(nblock)
+
+This array contains the instantaneous fluid free surface elevation at the elemental nodes and is calculated when the flag lUpdFluidVar has the value j\_upd\_FreeSurf. The incoming array contains the still free-surface elevation value for each node. The nodal values, as seen in the first call, are used to identify the wet portion of an element.
+
+# fUnsteadyVel(nblock, ndim)
+
+This array contains the unsteady part of the fluid velocity at load integration points and is calculated when the flag lUpdFluidVar has the value j\_upd\_FluidVarDrag. The incoming array contains zeros.
+
+# fFluidAcc(nblock, ndim)
+
+This array contains the fluid acceleration at load integration points and is calculated when the flag lUpdFluidVar has the value j\_upd\_FluidVarInertia. The incoming array contains zeros.
+
+# fUnsteadyPress(nblock)
+
+This array contains the unsteady part of the fluid pressure at load integration points and is calculated when the flag lUpdFluidVar has the value j\_upd\_FluidVarBuoyancy. The incoming array contains zeros. The steady part of the pressure is not stored but is calculated by Abaqus/Explicit at
+
+<!-- source-page: 580 -->
+
+each load integration point based on the data provided under the \*AQUA option and is scaled by the ScaleSteady parameter.
+
+# fUnsteadyDPressDZ(nblock)
+
+This array contains the unsteady part of the gradient of fluid pressure along elevation at load integration points and is calculated when the flag lUpdFluidVar has the value j\_upd\_FluidVarBuoyancy. The incoming array contains zeros. Similar to steady pressures, the gradients are not stored but are calculated by Abaqus/Explicit at each load integration point based on the data provided in the fluid variable definition and scaled by the ScaleSteady parameter.
+
+# Variables that can be updated
+
+The fluid variables—such as velocity, acceleration, pressure, and pressure gradient along elevation—used in load calculations are split in steady and unsteady parts. The following real variables scale each of those parts:
+
+# ScaleSteady
+
+This variable is used by Abaqus/Explicit to scale the steady part of the fluid variables. For drag loads this variable is the amplitude value provided for this purpose on the load data line as explained in “Abaqus/Aqua analysis,” Section 6.11.1 of the Abaqus Analysis User’s Guide. For all other loads its incoming value is one. The user can optionally update this variable.
+
+# ScaleUnsteady
+
+This variable is used by Abaqus/Explicit to scale the unsteady part of the fluid variables. The user is expected to define unscaled values for unsteady fluid variables. For drag loads this variable is the amplitude value provided for this purpose on the load data line, as explained in “Abaqus/Aqua analysis,” Section 6.11.1 of the Abaqus Analysis User’s Guide. For all other loads its incoming value is one. The user can optionally update this variable.
+
+# Variables passed in for information
+
+kStep
+
+Step number.
+
+kInc
+
+Increment number.
+
+nblock
+
+Number of nodes or load integration points, where the wave effects are to be computed.
+
+ndim
+
+Dimension of the problem (two- or three-dimensional problem).
+
+nprops
+
+The number of properties (real numbers) for the user-defined wave.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_059.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_059.md
new file mode 100644
index 00000000..9c1298e7
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_059.md
@@ -0,0 +1,374 @@
+<!-- source-page: 581 -->
+
+# naquaconst
+
+The number of fluid property constants in the fluid variable definition.
+
+# nwindconst
+
+The number of constants included in the wind velocity profile.
+
+# nstatevar
+
+The number of state variables for the user-defined wave, which is specified using the DEPVAR parameter.
+
+# nfieldvar
+
+The number of field variables available on the element set on which the load is applied. At the first call the field variables will be available at nodes, and at the second call they will be available at the load integration points.
+
+# iElemType
+
+An integer indicating the type of element these points belong to: 1 for line elements, 2 for surface elements, and 3 for solid elements. This integer is used to interpret the value of the direction vector (DirVec) provided at each point.
+
+# iLoadType
+
+An integer indicating the type of Abaqus/Aqua load for which the wave kinematics are calculated. The parameters j\_lcr\_PB to j\_ccr\_TSI indicate the values by which the load types can be identified.
+
+# sname
+
+The name of the element set on which the load is applied.
+
+# lUpdFluidVar
+
+This integer flag is used to update different fluid variables. The flag acquires known values as declared in the list of parameters. This flag provides information regarding which variables are to be updated on each call. When the subroutine is called with lUpdFluidVar having a value of j\_upd\_FreeSurf, only the free surface elevation at each point is updated; otherwise, different sets of fluid variables are updated depending on different values of lUpdFluidVar, as explained in the subroutine structure above.
+
+# Coord(nblock,ndim)
+
+This array contains the global coordinates of all points in their current configuration at the start of the time increment.
+
+# Velocity(nblock,ndim)
+
+This array contains the structural velocities of all points at the mid-increment time level of the previous increment.
+
+<!-- source-page: 582 -->
+
+# StateVar(nblock,nstatevar)
+
+This array contains the user-defined solution-dependent state variables at all points. The state variables can optionally be updated during the first call, when the flag lUpdFluidVar has a value of j\_upd\_FreeSurf. At the second call the incoming values are interpolated values at the integration points and are read-only.
+
+# FieldVar(nblock,nfieldvar)
+
+This array contains the field variables at the load integration points at the start of the time increment.
+
+# DirVec(nblock,ndim)
+
+This array contains the direction vectors at the load integration points at the start of the time increment. For points lying on line elements, it is the tangent vector in the current configuration. For points lying on surface elements or solid faces, it is the surface-normal vector, pointing outward in the current configuration. The type of elements with which the points are associated can be found using the integer variable iElemType. The magnitude of this vector is the outer diameter for line elements under distributed loads (PB, FDD, WDD, FDT, FI), transitional section area for line elements under transitional distributed loads (FD1, FD2, WD1, WD2, FI1, FI2), and the nodal surface area for points under concentrated loads (TSB, TFD, TWD, TSI) or points lying on surface elements or solid faces.
+
+# fSteadyVel(nblock,ndim)
+
+The incoming array contains the steady fluid velocity.
+
+# AquaSteadyConstants(naquaconst)
+
+The user-specified fluid property constants in the fluid variable definition.
+
+# WindConstants(nwindconst)
+
+The user-specified constants in the wind profile definition. In the absence of a wind profile, all values are zeros.
+
+# Props(nprops)
+
+This real array contains properties for the user-defined wave, as listed on the data lines.
+
+# dt
+
+The time increment.
+
+# timeTotal
+
+The total time at the beginning of the increment over all steps.
+
+# timeStep
+
+The step time at the beginning of the increment within the step.
+
+<!-- source-page: 583 -->
+
+# 1.3 Abaqus/CFD subroutines
+
+• “SMACfdUserPressureBC,” Section 1.3.1   
+• “SMACfdUserVelocityBC,” Section 1.3.2
+
+<!-- source-page: 584 -->
+
+<!-- source-page: 585 -->
+
+# 1.3.1 SMACfdUserPressureBC: User subroutine to specify prescribed pressure boundary conditions.
+
+Product: Abaqus/CFD
+
+# Reference
+
+• \*FLUID BOUNDARY
+
+# Overview
+
+User subroutine SMACfdUserPressureBC can be used to define element face pressures.
+
+# Time incrementation
+
+During the analysis user subroutine SMACfdUserPressureBC is called a number of times to update the pressure and change in pressure. The returned variable should be set equal to the pressure at stepTime, where stepTime is the current step time.
+
+# User subroutine interface
+
+```c
+void SMACfdUserPressureBC(int nfacets, const int* labels, const int* sides, const int* instances, char** instanceNames, const double* xc, const double* yc, const double* zc, double amp, double totalTime, double stepTime, const char* surfaceName, double* bcvals); 
+```
+
+# Variable to be defined
+
+bcvals
+
+Values of the prescribed pressure at the element faces.
+
+# Variables passed in for information
+
+nfacets
+
+Number of element facets to be processed in this call to SMACfdUserPressureBC.
+
+labels
+
+User labels for the elements attached to the facets in the boundary condition.
+
+sides
+
+Side numbers for the facets in the boundary condition.
+
+<!-- source-page: 586 -->
+
+# instances
+
+Instance numbers of the elements in the boundary condition.
+
+# instanceNames
+
+Array of instance names in the model. Instance numbers provided for the elements are used to look up instance names in this array.
+
+xc
+
+Global X-coordinates for the centroid of the facets.
+
+yc
+
+Global Y-coordinates for the centroid of the facets.
+
+zc
+
+Global Z-coordinates for the centroid of the facets.
+
+amp
+
+Amplitude value corresponding to the associated amplitude function. This value is passed in for information only and will not contribute to the value of the prescribed variable automatically.
+
+# totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime-stepTime.
+
+# stepTime
+
+Current step time.
+
+# surfaceName
+
+Name of the surface used to define the boundary condition.
+
+<!-- source-page: 587 -->
+
+# 1.3.2 SMACfdUserVelocityBC: User subroutine to specify prescribed velocity boundary conditions.
+
+Product: Abaqus/CFD
+
+# Reference
+
+• \*FLUID BOUNDARY
+
+# Overview
+
+User subroutine SMACfdUserVelocityBC:
+
+• can be used to define element face velocities; and   
+• defines the magnitude of the associated boundary condition in the global directions.
+
+# Time incrementation
+
+During the analysis user subroutine SMACfdUserVelocityBC is called a number of times to update the velocity and change in velocity. The returned variable should be set equal to the velocity at stepTime, where stepTime is the current step time.
+
+# User subroutine interface
+
+```txt
+void SMACfdUserVelocityBC(int nfacets, int direction, const int* labels, const int* sides, const int* instances, char** instanceNames, const double* xc, const double* yc, const double* zc, double amp, double totalTime, double stepTime, const char* surfaceName, double* bcvals); 
+```
+
+# Variable to be defined
+
+bcvals
+
+Values of the prescribed velocity at the element faces in the indicated direction.
+
+# Variables passed in for information
+
+nfacets
+
+Number of element facets to be processed in this call to SMACfdUserVelocityBC.
+
+direction
+
+Global direction in which the velocity component is being defined.
+
+<!-- source-page: 588 -->
+
+# labels
+
+User labels for the elements attached to the facets in the boundary condition.
+
+# sides
+
+Side numbers for the facets in the boundary condition.
+
+# instances
+
+Instance numbers of the elements in the boundary condition.
+
+# instanceNames
+
+Array of instance names in the model. Instance numbers provided for the elements are used to look up instance names in this array.
+
+# xc
+
+Global X-coordinates for the centroid of the facets.
+
+#
+
+Global Y-coordinates for the centroid of the facets.
+
+# zc
+
+Global Z-coordinates for the centroid of the facets.
+
+# amp
+
+Amplitude value corresponding to the associated amplitude function. This value is passed in for information only and will not contribute to the value of the prescribed variable automatically.
+
+# totalTime
+
+Value of total time. The time at the beginning of the step is given by totalTime-stepTime.
+
+# stepTime
+
+Current step time.
+
+# surfaceName
+
+Name of the surface used to define the boundary condition.
+
+# Example: Imposition of a parabolic inlet flow velocity
+
+In this example a parabolic inlet flow velocity is imposed on a channel. User subroutine SMACfdUserVelocityBC given below illustrates how the return value array is to be computed.
+
+# Input file
+
+```txt
+*HEADING
+Test Abaqus/CFD velocity boundary condition user subroutine
+*NODE 
+```
+
+1, -10.0, 0.0, 0.0
+
+21, 10.0, 0.0, 0.0
+
+<!-- source-page: 589 -->
+
+```csv
+211, -10.0, 10.0, 0.0
+231, 10.0, 10.0, 0.0
+*NGEN, NSET=INLET
+1, 21, 1
+*NGEN, NSET=OUTLET
+211, 231, 1
+*NFILL, NSET=LOW
+INLET, OUTLET, 10, 21
+*NCOPY, CHANGE NUMBER=231, OLD SET=LOW, NEW SET=HIGH, SHIFT 0.0, 0.0, -1.0
+*NSET, NSET=NALL
+LOW, HIGH
+*ELEMENT, TYPE=FC3D8
+1, 1, 2, 233, 232, 22, 23, 254, 253
+*ELGEN, ELSET=EALL
+1, 20, 1, 1, 10, 21, 20
+*ELSET, GENERATE, ELSET=INLET
+1, 20, 1
+*ELSET, GENERATE, ELSET=OUTLET
+181, 200, 1
+*ELSET, GENERATE, ELSET=LEFT
+1, 181, 20
+*ELSET, GENERATE, ELSET=RIGHT
+20, 200, 20
+*SURFACE, TYPE=ELEMENT, NAME=INLET
+INLET, S1
+*SURFACE, TYPE=ELEMENT, NAME=OUTLET
+OUTLET, S2
+*SURFACE, TYPE=ELEMENT, NAME=TOPFACE
+EALL, S3
+*SURFACE, TYPE=ELEMENT, NAME=BOTFACE
+EALL, S5
+*SURFACE, TYPE=ELEMENT, NAME=LEFT
+LEFT, S6
+*SURFACE, TYPE=ELEMENT, NAME=RIGHT
+RIGHT, S4
+*MATERIAL, NAME=FLUID
+*DENSITY
+1.0,
+*VISCOSITY
+1.0E-3
+*CONDUCTIVITY 
+```
+
+<!-- source-page: 590 -->
+
+```csv
+1.0E-3
+*SPECIFIC HEAT, TYPE=CONSTANT PRESSURE
+1.0,
+*EXPANSION, ZERO=0.0
+1.0
+*FLUID SECTION, TYPE=SINGLE FLUID, ELSET=EALL
+FLUID
+*INITIAL CONDITIONS, TYPE=VELOCITY, ELEMENT AVERAGE
+EALL, 1, 0.0
+EALL, 2, 0.0
+EALL, 3, 0.0
+*STEP, NAME=PARABOLIC
+*CFD, INCOMPRESSIBLE NAVIER STOKES, INCREMENTATION=FIXED CFL
+0.01, 100.0, 0.025, 0.40, 1
+1.0E-10, 1.0, 0.0, 0.0, 1.0
+*FLUID BOUNDARY, TYPE=SURFACE
+INLET, VELY, 0.0
+INLET, VELZ, 0.0
+OUTLET, P, 0.0
+TOPFACE, VELZ, 0.0
+BOTFACE, VELZ, 0.0
+LEFT, VELX, 0.0
+LEFT, VELY, 0.0
+LEFT, VELZ, 0.0
+RIGHT, VELX, 0.0
+RIGHT, VELY, 0.0
+RIGHT, VELZ, 0.0
+*FLUID BOUNDARY, TYPE=SURFACE
+INLET, VELYNU
+*OUTPUT, FIELD, TIME INTERVAL=0.10
+*ELEMENT OUTPUT, ELSET=EALL
+V, PRESSURE
+*END STEP 
+```
+
+# User subroutine
+
+```c
+/* User defined velocity example */
+#include "SMACfdUserSubroutines.h"
+
+void SMACfdUserVelocityBC
+(int nfacets, int direction, const int* labels,
+const int* sides, const int* instances, char** instanceNames, 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_060.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_060.md
new file mode 100644
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+<!-- source-page: 591 -->
+
+```txt
+const double* xc, const double* yc, const double* zc, double amp, double totalTime, double stepTime, const char* surfaceName, double* bcvals)
+{
+    int i;
+    if (direction == 2) {
+    for (i = 0; i < nfacets; i++) {
+    bcvals[i] = 1.0 - xc[i] * xc[i] / 100.0;
+    }
+    }
+} 
+```
+
+<!-- source-page: 592 -->
+
+<!-- source-page: 593 -->
+
+# 2. Utility Routines
+
+• “Utility routines,” Section 2.1
+
+<!-- source-page: 594 -->
+
+<!-- source-page: 595 -->
+
+# 2.1 Utility routines
+
+• “Obtaining Abaqus environment variables,” Section 2.1.1   
+• “Obtaining the Abaqus job name,” Section 2.1.2   
+• “Obtaining the Abaqus output directory name,” Section 2.1.3   
+• “Obtaining parallel processes information,” Section 2.1.4   
+• “Obtaining part information,” Section 2.1.5   
+• “Obtaining material point information in an Abaqus/Standard analysis,” Section 2.1.6   
+• “Obtaining material point information in an Abaqus/Explicit analysis,” Section 2.1.7   
+• “Obtaining material point information averaged at a node,” Section 2.1.8   
+• “Obtaining node point information,” Section 2.1.9   
+• “Obtaining node to element connectivity,” Section 2.1.10   
+• “Obtaining stress invariants, principal stress/strain values and directions, and rotating tensors in an Abaqus/Standard analysis,” Section 2.1.11   
+• “Obtaining principal stress/strain values and directions in an Abaqus/Explicit analysis,” Section 2.1.12   
+• “Obtaining wave kinematic data in an Abaqus/Aqua analysis,” Section 2.1.13   
+• “Printing messages to the message or status file,” Section 2.1.14   
+• “Terminating an analysis,” Section 2.1.15   
+• “Obtaining sensor information,” Section 2.1.16   
+• “Accessing Abaqus materials,” Section 2.1.17   
+• “Accessing Abaqus thermal materials,” Section 2.1.18   
+• “Obtaining scalar state information in an Abaqus/CFD analysis,” Section 2.1.19   
+• “Obtaining vector state information in an Abaqus/CFD analysis,” Section 2.1.20   
+• “Obtaining the MPI communicator in an Abaqus/CFD analysis,” Section 2.1.21   
+• “Ensuring thread safety,” Section 2.1.22   
+• “Allocatable arrays,” Section 2.1.23
+
+<!-- source-page: 596 -->
+
+<!-- source-page: 597 -->
+
+# 2.1.1 OBTAINING Abaqus ENVIRONMENT VARIABLES
+
+Products: Abaqus/Standard Abaqus/Explicit
+
+# References
+
+• “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide   
+• “Using the Abaqus environment settings,” Section 3.3.1 of the Abaqus Analysis User’s Guide   
+• “UWAVE and UEXTERNALDB,” Section 4.1.27 of the Abaqus Verification Guide
+
+# Overview
+
+Utility routines GETENVVAR and VGETENVVAR can be called from any Abaqus/Standard or Abaqus/Explicit user subroutine, respectively, to obtain the value of an environment variable.
+
+# Interface
+
+```txt
+character*256 ENVVAR
+...
+CALL GETENVVAR('ENVVARNAME', ENVVAR, LENVVAR)
+CALL VGETENVVAR('ENVVARNAME', ENVVAR, LENVVAR)
+... 
+```
+
+# Variable to be provided to the utility routine
+
+# ENVVARNAME
+
+Environment variable name.
+
+# Variables returned from the utility routine
+
+# ENVVAR
+
+Character string to receive the value of the environment variable.
+
+# LENVVAR
+
+Length of the character string ENVVAR.
+
+<!-- source-page: 598 -->
+
+<!-- source-page: 599 -->
+
+# 2.1.2 OBTAINING THE Abaqus JOB NAME
+
+Products: Abaqus/Standard Abaqus/Explicit
+
+# References
+
+• “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide   
+• “UWAVE and UEXTERNALDB,” Section 4.1.27 of the Abaqus Verification Guide
+
+# Overview
+
+Utility routines GETJOBNAME and VGETJOBNAME can be called from any Abaqus/Standard or Abaqus/Explicit user subroutine, respectively, to obtain the name of the current job.
+
+# Interface
+
+```txt
+character*256 JOBNAME
+...
+CALL GETJOBNAME( JOBNAME, LENJOBNAME )
+CALL VGETJOBNAME( JOBNAME, LENJOBNAME )
+... 
+```
+
+# Variables returned from the utility routine
+
+# JOBNAME
+
+Character string to receive the value of the job name.
+
+# LENJOBNAME
+
+Length of the character string JOBNAME.
+
+<!-- source-page: 600 -->
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_061.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_061.md
new file mode 100644
index 00000000..ecc42f37
--- /dev/null
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@@ -0,0 +1,358 @@
+<!-- source-page: 601 -->
+
+# 2.1.3 OBTAINING THE Abaqus OUTPUT DIRECTORY NAME
+
+Products: Abaqus/Standard Abaqus/Explicit
+
+# References
+
+• “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide   
+• “UWAVE and UEXTERNALDB,” Section 4.1.27 of the Abaqus Verification Guide
+
+# Overview
+
+Utility routines GETOUTDIR and VGETOUTDIR can be called from any Abaqus/Standard or Abaqus/Explicit user subroutine, respectively, to obtain the output directory of the current job.
+
+# Interface
+
+```txt
+character*256 OUTDIR
+...
+CALL GETOUTDIR( OUTDIR, LENOUTDIR )
+CALL VGETOUTDIR( OUTDIR, LENOUTDIR )
+... 
+```
+
+# Variables returned from the utility routine
+
+# OUTDIR
+
+Character string to receive the value of the output directory name.
+
+# LENOUTDIR
+
+Length of the character string OUTDIR.
+
+<!-- source-page: 602 -->
+
+<!-- source-page: 603 -->
+
+# 2.1.4 OBTAINING PARALLEL PROCESSES INFORMATION
+
+Products: Abaqus/Standard Abaqus/Explicit
+
+# References
+
+• “Parallel execution: overview,” Section 3.5.1 of the Abaqus Analysis User’s Guide   
+• “VUSDFLD,” Section 4.1.39 of the Abaqus Verification Guide
+
+# Overview
+
+Several different utility routines are available to provide detailed information about your parallel processes.
+
+# GETNUMCPUS and VGETNUMCPUS (obtain the number of processes)
+
+Utility routine GETNUMCPUS can be called from any Abaqus/Standard user subroutine. GETNUMCPUS returns the number of MPI processes.
+
+Utility routine VGETNUMCPUS can be called from any Abaqus/Explicit user subroutine in a domainparallel run. VGETNUMCPUS provides the number of processes used for the parallel run.
+
+# Interface
+
+CALL GETNUMCPUS( NUMPROCESSES )CALL VGETNUMCPUS( NUMPROCESSES )：·
+
+# Variable returned from the utility routine
+
+# NUMPROCESSES
+
+Number of processes specified for the analysis.
+
+# GETRANK and VGETRANK (obtain the process number)
+
+Utility routine GETRANK can be called from any Abaqus/Standard user subroutine. GETRANK returns the rank of the MPI process from which the function is called. For example, in a hybrid MPI and thread parallel execution scheme, multiple threads may all return the rank of their parent MPI process (see “Parallel execution in Abaqus/Standard,” Section 3.5.2 of the Abaqus Analysis User’s Guide).
+
+Utility routine VGETRANK can be called from any Abaqus/Explicit user subroutine in a domain-parallel run. VGETRANK provides the individual process rank (see “Parallel execution in Abaqus/Explicit,” Section 3.5.3 of the Abaqus Analysis User’s Guide).
+
+<!-- source-page: 604 -->
+
+# Interface
+
+```txt
+CALL GETRANK( KPROCESSNUM )
+CALL VGETRANK( KPROCESSNUM )
+... 
+```
+
+# Variable returned from the utility routine
+
+# KPROCESSNUM
+
+Process number or rank. A process number is either zero or a positive integer.
+
+# GETNUMTHREADS (obtain the number of threads)
+
+Utility routine GETNUMTHREADS can be called from any Abaqus user subroutine. It returns the number of threads in a process. In a hybrid parallel execution mode, there will be several Abaqus MPI processes, each having several threads.
+
+# Interface
+
+```cpp
+Fortran:
+#include <SMAAspUserSubroutines.hdr>
+
+integer numThreads
+numThreads = GETNUMTHREADS()
+
+C++:
+#include <SMAAspUserSubroutines.h>
+
+int numThreads = GETNUMTHREADS(); 
+```
+
+<!-- source-page: 605 -->
+
+# get\_thread\_id
+
+You can determine the ID of the thread you are in by calling the utility function get\_thread\_id. The returned ID is an integer that Abaqus assigns to each of its threads. The main thread will have the ID=0, and each subsequent thread will have an ID of 1, 2, 3, 4, ..., N. This function can be called from any Abaqus user subroutine and from both the Fortran and C/C++ codes.
+
+# Interface
+
+```txt
+FOTRAN:
+#include <SMAAspUserSubroutines.hdr>
+
+INTEGER myThreadID
+
+myThreadID = get_thread_id()
+
+C++:
+#include <SMAAspUserSubroutines.h>
+
+int myThreadID = get_thread_id() 
+```
+
+# Variable returned from the utility routine
+
+thread\_id
+
+Current thread ID, an integer.
+
+# GETCOMMUNICATOR (Fortran)
+
+Utility function GETCOMMUNICATOR can be called from any Abaqus user subroutine. GETCOMMUNICATOR returns a communicator that Abaqus defines for its worker processes, similar to MPI\_COMM\_WORLD. In Fortran its type is an INTEGER. The communicator thus obtained can be used for subsequent MPI communication routines. In a nonparallel run, when the MPI subsystem is not initialized, communicators do not exist and GET\_COMMUNICATOR() will return 0. Another way of testing is to call the MPI\_Initialized(flag) function. This function will set the flag to 1 if MPI has been initialized.
+
+# Interface
+
+```cpp
+#include <SMAAspUserSubroutines.hdr>
+integer ABA_COMM_WORLD 
+```
+
+<!-- source-page: 606 -->
+
+```txt
+ABA_COMM_WORLD = GETCOMMUNICATOR()
+if (ABA_COMM_WORLD.ne.0) then
+    ...do some parallel work, using MPI ...
+else
+    ...do some work in a single process ...
+end if 
+```
+
+# Variable returned from the utility routine
+
+# INTEGER
+
+Communicator handle identifier of type INTEGER. In a non-MPI run, the value returned will be zero.
+
+# get\_communicator (C++)
+
+Utility function get\_communicator can be called from any Abaqus user subroutine. get\_communicator returns a communicator that Abaqus defines for its worker processes. In C++ this function is called get\_communicator() and returns a value of the type MPI\_Comm. The communicator thus obtained can be used for subsequent MPI communication routines. In a nonparallel run, when the MPI subsystem is not initialized, communicators do not exist and get\_communicator() will return 0. Another way of testing is to call the MPI\_Initialized(flag) function. This function will set the flag to 1 if MPI has been initialized.
+
+# Interface
+
+```c
+#include <SMAAspUserSubroutines.h>
+
+MPI_Comm ABA_COMM_WORLD = get_communicator()
+
+if (ABA_COMM_WORLD) {
+    ... do some parallel work, using MPI ...
+}
+else{
+    ...do some work in a single process ...
+} 
+```
+
+# Variable returned from the utility routine
+
+# MPI\_Comm
+
+Communicator handle of type MPI\_Comm. In a non-MPI run, the value returned will be zero.
+
+<!-- source-page: 607 -->
+
+# 2.1.5 OBTAINING PART INFORMATION
+
+Products: Abaqus/Standard Abaqus/Explicit
+
+# References
+
+• “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide   
+• “Defining an assembly,” Section 2.10.1 of the Abaqus Analysis User’s Guide   
+• “Pure bending of a cylinder: CAXA elements,” Section 1.3.33 of the Abaqus Verification Guide
+
+# Overview
+
+Several utility routines are available to allow you to obtain information about your part instances.
+
+Utility routines GETPARTINFO and VGETPARTINFO can be called from any Abaqus/Standard or Abaqus/Explicit user subroutine, respectively, to retrieve the part instance name and original node or element number corresponding to an internal node or element number. Utility routines GETINTERNAL and VGETINTERNAL can be called from any Abaqus/Standard or Abaqus/Explicit user subroutine, respectively, to retrieve the internal node or element number corresponding to a part instance name and original node or element number. The part file (jobname.prt) must be available. The expense of calling these routines is not trivial, so minimal use of them is recommended.
+
+GETPARTINFO and VGETPARTINFO (obtain part instance information given global node/element number)
+
+# Interface
+
+```txt
+CHARACTER*80 CPNAME
+...
+CALL GETPARTINFO(INTNUM, JTYP, CPNAME, LOCNUM, JRCD)
+or
+CALL VGETPARTINFO(INTNUM, JTYP, CPNAME, LOCNUM, JRCD) 
+```
+
+# Variables to be provided to the utility routine
+
+# INTNUM
+
+The internal (global) node or element number to be looked up.
+
+# JTYP
+
+An integer flag indicating whether INTNUM is a node or element number. Set JTYP=0 to look up a node number, and set JTYP=1 to look up an element number.
+
+<!-- source-page: 608 -->
+
+# Variables returned from the utility routine
+
+# CPNAME
+
+The name of the part instance that contains INTNUM. An empty part instance name indicates that the node or element is at the assembly level and is not included in any part instance.
+
+# LOCNUM
+
+The part-local node or element number corresponding to INTNUM.
+
+# JRCD
+
+Return code (0–no error, 1–error).
+
+# GETINTERNAL and VGETINTERNAL (obtain global node/element number given part instance information )
+
+# Interface
+
+```txt
+CHARACTER*80 CPNAME
+...
+CALL GETINTERNAL(CPNAME, LOCNUM, JTYP, INTNUM, JRCD)
+or
+CALL VGETINTERNAL(CPNAME, LOCNUM, JTYP, INTNUM, JRCD) 
+```
+
+# Variables to be provided to the utility routine
+
+# CPNAME
+
+The name of the part instance that contains LOCNUM.
+
+# LOCNUM
+
+The part-local node or element number to be looked up.
+
+# JTYP
+
+An integer flag indicating whether LOCNUM is a node or element number. Set JTYP=0 to look up a node number, and set JTYP=1 to look up an element number.
+
+# Variables returned from the utility routine
+
+# INTNUM
+
+The internal (global) node or element number corresponding to LOCNUM in part instance CPNAME.
+
+# JRCD
+
+Return code (0–no error, 1–error).
+
+<!-- source-page: 609 -->
+
+# 2.1.6 OBTAINING MATERIAL POINT INFORMATION IN AN Abaqus/Standard ANALYSIS
+
+# Product: Abaqus/Standard
+
+# References
+
+• “UVARM,” Section 1.1.58   
+• “USDFLD,” Section 1.1.53   
+• “UDMGINI,” Section 1.1.26   
+• “Damage and failure of a laminated composite plate,” Section 1.1.14 of the Abaqus Example Problems Guide   
+• “USDFLD,” Section 4.1.24 of the Abaqus Verification Guide   
+• “UVARM,” Section 4.1.26 of the Abaqus Verification Guide
+
+# Overview
+
+Utility routine GETVRM can be called from either user subroutine UVARM, UDMGINI, or USDFLD to access material integration point information.
+
+# Interface
+
+DIMENSION ARRAY(15), JARRAY(15) CHARACTER\*3 FLGRAY(15)
+
+CALL GETVRM('VAR',ARRAY,JARRAY,FLGRAY,JRCD,JMAC,JMATYP,MATLAYO, LACCFLA)
+
+# Variables to be provided to the utility routine
+
+# VAR
+
+Output variable key from the table in “Abaqus/Standard output variable identifiers,” Section 4.2.1 of the Abaqus Analysis User’s Guide. The applicable keys are listed in the output table as being available for results file output at the element integration points; e.g., S for stress.
+
+# JMAC
+
+Variable that must be passed into the GETVRM utility routine. The calling user subroutine provides this variable.
+
+# JMATYP
+
+Variable that must be passed into the GETVRM utility routine. The calling user subroutine provides this variable.
+
+<!-- source-page: 610 -->
+
+# MATLAYO
+
+Variable that must be passed into the GETVRM utility routine. The calling user subroutine provides this variable.
+
+# LACCFLA
+
+Variable that must be passed into the GETVRM utility routine. The calling user subroutine provides this variable.
+
+# Variables returned from the utility routine
+
+# ARRAY
+
+Real array containing individual components of the output variable.
+
+# JARRAY
+
+Integer array containing individual components of the output variable.
+
+# FLGRAY
+
+Character array containing flags corresponding to the individual components. Flags will contain either YES, NO, or N/A (not applicable).
+
+# JRCD
+
+Return code (0 – no error, 1 – output request error or all components of output request are zero).
+
+# Available output variable keys
+
+Only output variable keys that are valid for results file output are available for use with GETVRM. In general, if a key corresponds to a collective output variable, rather than an individual component, it can be used with GETVRM. For example, S for the stress tensor can be used, whereas any individual component of stress, say S11, cannot be used. The collective output variable keys are distinguished from their individual components by the fact that they have a bullet ( ) in the .fil column in the tables in “Abaqus/Standard output variable identifiers,” Section 4.2.1 of the Abaqus Analysis User’s Guide. Output variable keys that cannot be used with GETVRM are listed later in this section.
+
+You will be returned ARRAY, JARRAY, and FLGRAY, which correspond to the real-valued components, integer-valued components, and the flags associated with the request VAR, respectively. If any array component is not applicable for a given request, its value will be returned as the initialized value: 0.0 in ARRAY, 0 in JARRAY, and N/A in FLGRAY. The error flag JRCD=1 is returned from GETVRM any time a request key is not recognized, the request is not valid (such as requesting transverse shear stress for a shell element that uses thin shell theory), or all of the output components requested are zero; otherwise, JRCD=0.
+
+# Ordering of returned components
+
+The components for a request are written as follows. Single index components (and requests without components) are returned in positions 1, 2, 3, etc. Double index components are returned in the
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_062.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_062.md
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+<!-- source-page: 611 -->
+
+order 11, 22, 33, 12, 13, 23 for symmetric tensors, followed by 21, 31, 32 for unsymmetric tensors (deformation gradient). Thus, the stresses for a plane stress element are returned as ARRAY(1)=S11, ARRAY(2)=S22, ARRAY(3)=0.0, and ARRAY(4)=S12. Three values are always returned for principal value requests, the minimum value first and maximum value third, regardless of the dimensionality of the analysis.
+
+The description of the output variable (see “Abaqus/Standard output variable identifiers,” Section 4.2.1 of the Abaqus Analysis User’s Guide) determines which components are retrieved with GETVRM.
+
+# Analysis time for which values are returned
+
+When a material point quantity is requested with utility routine GETVRM, the time in the increment at which the values are returned will depend upon which user subroutine calls it. GETVRM returns values at the end of the current increment to user subroutine UVARM, whereas it returns values at the beginning of the current increment to user subroutine USDFLD.
+
+# Equilibrium state for values returned
+
+User subroutine UVARM may call GETVRM multiple times for each increment, as Abaqus/Standard iterates to a converged solution. Values returned from GETVRM calls preceding the final iteration for the increment will not represent the converged solution.
+
+# Example
+
+To illustrate the use of GETVRM, if the identifier PEQC is specified for use with a jointed material, ARRAY will be returned with the individual equivalent plastic strain components PEQC1, PEQC2, PEQC3, and PEQC4. Since there are no integer output variables associated with this identifier, JARRAY will be returned with default values of 0. The FLGRAY array will contain either YES or NO flags indicating whether each component is actively yielding. If the identifier PE is specified for a material with plasticity, ARRAY will be returned with plastic strain components PE11, PE22, PE33, PE12, PE13, PE23, the equivalent plastic strain PEEQ, and the plastic strain magnitude PEMAG. Since there are no integer values associated with this request, JARRAY will be 0. The FLGRAY array will have N/A for the first six components, either YES or NO in the seventh component (corresponding to PEEQ) indicating whether the material is currently yielding, and N/A in the eighth component. If the identifier HFL is specified, ARRAY will be returned with the magnitude HFLM and the components HFL1, HFL2, and HFL3 as described in “Abaqus/Standard output variable identifiers,” Section 4.2.1 of the Abaqus Analysis User’s Guide.
+
+# Accessing state-dependent variables
+
+If GETVRM is used to access state-dependent variables (output variable key SDV) and more than 15 state-dependent variables have been defined in the analysis, the dimension statement for ARRAY and JARRAY must be changed so that these arrays are dimensioned to the maximum number of state-dependent variables.
+
+<!-- source-page: 612 -->
+
+# Unsupported element types, procedures and output variable keys
+
+Since this capability pertains to material point quantities, it cannot be used for most of the element types that do not require a material definition. The following element types are, therefore, not supported:
+
+• DASHPOTx   
+• SPRINGx   
+• CONNxDx   
+• FRAMExD   
+• JOINTC   
+• JOINTxD   
+• DRAGxD   
+• PSIxx   
+• ITSxxx   
+• MASS   
+• ROTARYI   
+• all acoustic elements   
+• all contact elements   
+• all hydrostatic fluid elements
+
+If used with user subroutine UVARM, this capability is not available for linear perturbation procedures (“General and linear perturbation procedures,” Section 6.1.3 of the Abaqus Analysis User’s Guide):
+
+• static linear perturbation analysis (“Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide),   
+• “Eigenvalue buckling prediction,” Section 6.2.3 of the Abaqus Analysis User’s Guide,   
+• “Natural frequency extraction,” Section 6.3.5 of the Abaqus Analysis User’s Guide,   
+• “Transient modal dynamic analysis,” Section 6.3.7 of the Abaqus Analysis User’s Guide,   
+• “Mode-based steady-state dynamic analysis,” Section 6.3.8 of the Abaqus Analysis User’s Guide,   
+• “Direct-solution steady-state dynamic analysis,” Section 6.3.4 of the Abaqus Analysis User’s Guide,   
+• “Subspace-based steady-state dynamic analysis,” Section 6.3.9 of the Abaqus Analysis User’s Guide,   
+• “Response spectrum analysis,” Section 6.3.10 of the Abaqus Analysis User’s Guide, and   
+• “Random response analysis,” Section 6.3.11 of the Abaqus Analysis User’s Guide.
+
+The following output variable keys are not available for use with GETVRM:
+
+• SVOL   
+• TSHR   
+• CTSHR   
+• COORD
+
+<!-- source-page: 613 -->
+
+# 2.1.7 OBTAINING MATERIAL POINT INFORMATION IN AN Abaqus/Explicit ANALYSIS
+
+# Product: Abaqus/Explicit
+
+# References
+
+• “VUSDFLD,” Section 1.2.24   
+• “Damage and failure of a laminated composite plate,” Section 1.1.14 of the Abaqus Example Problems Guide
+
+# Overview
+
+Utility routine VGETVRM can be called from VUSDFLD to access selected output variables at the material points for the current block of elements being processed by VUSDFLD.
+
+# Interface
+
+```txt
+include 'vaba_param.inc'
+parameter( nrData=6 )
+character*3 cData(maxblk*nrData)
+dimension rData(maxblk*nrData), jData(maxblk*nrData)
+...
+call vgetvrm('VAR', rData, jData, cData, jStatus ) 
+```
+
+# Variable to be provided to the utility routine
+
+# VAR
+
+Output variable key. The available material point variables are given in “Available output variable keys,” below.
+
+# Variables returned from the utility routine
+
+# rData
+
+Real array containing individual components of the output variable.
+
+# jData
+
+Integer array containing individual components of the output variable.
+
+# cData
+
+Character array containing flags corresponding to the individual components. Flags will either be YES, NO, or N/A (not applicable).
+
+<!-- source-page: 614 -->
+
+# jStatus
+
+Return code (0: output request successful; 1: output request not available).
+
+# Available output variable keys
+
+The following output variable keys are currently supported:
+
+• S: All stress components.   
+• LE: All logarithmic strain components.   
+• THE: All thermal strain components.   
+• PE: All plastic strain components.   
+• PEEQ: Equivalent plastic strain.   
+• PEEQT: Equivalent plastic strain in uniaxial tension, defined as $\int \dot { \bar { \varepsilon } } _ { t } ^ { p l } d t$   
+• PEQC: All equivalent plastic strains for models that have more than one yield/failure surface.   
+• ALPHA: All total kinematic hardening shift tensor components.   
+• TEMP: Temperature.   
+• EVF: Volume fraction (Eulerian elements only).
+
+A requested output variable must be valid for the material model for the request to be successful.
+
+The returned arrays rData, jData, and cData correspond to the real-valued variable, integervalued variable, and string variable, respectively, that can be associated with the request output variable key VAR. If any output variable component is not applicable for a given request, its value will be returned as the initialized value: 0.0 in rData, 0 in jData, and N/A in cData. Currently the association of the integer-valued variable and the string variable with the request output variable is not supported. The error flag jStatus is returned with a value of 1 if a request key is not recognized, not valid, or not supported; otherwise, jStatus is returned with a value of 0.
+
+# Component ordering in symmetric tensors
+
+For symmetric tensors such as the stress and strain tensors, there are ndir+nshr components, where ndir and nshr are the number of direct and shear components, respectively, that are passed into user subroutine VUSDFLD. The component order is given as a natural permutation of the indices of the tensor. The direct components are first and then the indirect components, beginning with the 12-component. For example, a stress tensor contains ndir direct stress components and nshr shear stress components, which are returned as:
+
+<table><tr><td>Component</td><td>2D Case</td><td>3D Case</td></tr><tr><td>1</td><td> $\sigma_{11}$ </td><td> $\sigma_{11}$ </td></tr><tr><td>2</td><td> $\sigma_{22}$ </td><td> $\sigma_{22}$ </td></tr><tr><td>3</td><td> $\sigma_{33}$ </td><td> $\sigma_{33}$ </td></tr><tr><td>4</td><td> $\sigma_{12}$ </td><td> $\sigma_{12}$ </td></tr></table>
+
+<!-- source-page: 615 -->
+
+<table><tr><td>Component</td><td>2D Case</td><td>3D Case</td></tr><tr><td>5</td><td></td><td> $\sigma_{23}$ </td></tr><tr><td>6</td><td></td><td> $\sigma_{31}$ </td></tr></table>
+
+The shear strain components returned from utility subroutine VGETVRM are tensor components and not engineering components.
+
+# Analysis time for which values are returned
+
+Utility subroutine VGETVRM returns values of the requested variable that correspond to the beginning of the current increment.
+
+# Example
+
+To illustrate the use of VGETVRM, if the identifier PE is specified for a material with plasticity, rData will be returned with plastic strain components PE11, PE22, PE33, PE12, PE23, and PE13. jData will be 0 and cData array will have N/A for all components.
+
+# Unsupported element types, procedures, and output variable keys
+
+Since this capability pertains to material point quantities, it cannot be used for most of the element types that do not require a material definition. The following element types are, therefore, not supported:
+
+• DASHPOTx   
+• SPRINGx   
+• CONNxDx   
+• MASS   
+• ROTARYI   
+• all acoustic elements
+
+<!-- source-page: 616 -->
+
+<!-- source-page: 617 -->
+
+# 2.1.8 OBTAINING MATERIAL POINT INFORMATION AVERAGED AT A NODE
+
+# Product: Abaqus/Standard
+
+# References
+
+• “UMESHMOTION,” Section 1.1.46   
+• “Erosion of material (sand production) in an oil wellbore,” Section 1.1.22 of the Abaqus Example Problems Guide
+
+# Overview
+
+Utility routine GETVRMAVGATNODE can be called from user subroutine UMESHMOTION to access material integration point information averaged at a node.
+
+The results variables available from GETVRMAVGATNODE are nearly the same as those available from GETVRM; the exceptions follow from the restriction that, since it will average results, GETVRMAVGATNODE will operate only on real-valued results. Results values represented as integers or as flags are not available.
+
+# Interface
+
+```txt
+DIMENSION ARRAY(15), JELEMLIST(NELEMS)
+...
+CALL GETVRMAVGATNODE(NODE, JTYP, 'VAR', ARRAY, JRCD, JELEMLIST, NELEMS, JMATYP, JGVBLOCK) 
+```
+
+# Variables to be provided to the utility routine
+
+# NODE
+
+Node number.
+
+# JTYP
+
+An integer flag indicating how the material point information is averaged. Set JTYP=0 to extrapolate results, using element shape functions, and to average results at the node. Set JTYP=1 to perform a volume-weighted average of results.
+
+# VAR
+
+Output variable key from the table in “Abaqus/Standard output variable identifiers,” Section 4.2.1 of the Abaqus Analysis User’s Guide. The applicable keys are listed in the output table as being available for results file output at the element integration points; e.g., S for stress. One exception is the integration point coordinates variable COORD, which cannot be passed into the utility routine; you should use utility routine GETVRN instead to obtain nodal coordinates.
+
+<!-- source-page: 618 -->
+
+# JELEMLIST
+
+Array of element numbers for elements connected to NODE for which you want material point quantities considered in the average result. Results from each element in the list that contain the node will be extrapolated to that node and averaged. JELEMLIST can be obtained from utility routine GETNODETOELEMCONN.
+
+# NELEMS
+
+Length of JELEMLIST.
+
+# JGVBLOCK
+
+Variable that must be passed into the GETVRMAVGATNODE utility routine. This variable is available in user subroutine UMESHMOTION for this purpose.
+
+# JMATYP
+
+Variable that must be passed into the GETVRMAVGATNODE utility routine. This variable is available in user subroutine UMESHMOTION for this purpose.
+
+# Variables returned from the utility routine
+
+# ARRAY
+
+Real array containing individual components of the output variable.
+
+# JRCD
+
+Return code (0 – no error, 1 – output request error or all components of output request are zero).
+
+# Available output variable keys
+
+Only output variable keys that are valid for results file output are available for use with GETVRMAVGATNODE. In general, if a key corresponds to a collective output variable, rather than an individual component, it can be used with GETVRMAVGATNODE. For example, S for the stress tensor can be used, whereas any individual component of stress, say S11, cannot be used. The collective output variable keys are distinguished from their individual components by the fact that they have a bullet ( ) in the .fil column in the tables in “Abaqus/Standard output variable identifiers,” Section 4.2.1 of the Abaqus Analysis User’s Guide. Output variable keys that cannot be used with GETVRMAVGATNODE are listed later in this section.
+
+You will be returned ARRAY with components associated with the request VAR. If any array component is not applicable for a given request, its value will be returned as the initialized value: 0.0 in ARRAY. The error flag JRCD=1 is returned from GETVRMAVGATNODE any time a request key is not recognized, the request is not valid, or all of the output components requested are zero; otherwise, JRCD=0.
+
+<!-- source-page: 619 -->
+
+# Ordering of returned components
+
+The components for a request are written as follows. Single index components (and requests without components) are returned in positions 1, 2, 3, etc. Double index components are returned in the order 11, 22, 33, 12, 13, 23 for symmetric tensors, followed by 21, 31, 32 for unsymmetric tensors (deformation gradient). Thus, the stresses for a plane stress element are returned as ARRAY(1)=S11, ARRAY(2)=S22, ARRAY(3)=0.0, and ARRAY(4)=S12. Three values are always returned for principal value requests, the minimum value first and the maximum value third, regardless of the dimensionality of the analysis.
+
+The description of the output variable (see “Abaqus/Standard output variable identifiers,” Section 4.2.1 of the Abaqus Analysis User’s Guide) determines which components are retrieved with GETVRMAVGATNODE.
+
+# Analysis time for which values are returned
+
+GETVRMAVGATNODE returns values at the end of the current increment to user subroutine UMESHMOTION.
+
+# Accessing state-dependent variables
+
+If GETVRMAVGATNODE is used to access solution-dependent state variables (output variable key SDV) and more than 15 solution-dependent state variables have been defined in the analysis, the dimension statement for ARRAY must be changed so that these arrays are dimensioned to the maximum number of solution-dependent state variables.
+
+# Unsupported element types and output variable keys
+
+Since this capability pertains to material point quantities, it cannot be used for most of the element types that do not require a material definition. The following element types are, therefore, not supported:
+
+• DASHPOTx   
+• SPRINGx   
+• CONNxDx   
+• FRAMExD   
+• JOINTC   
+• JOINTxD   
+• DRAGxD   
+• PSIxx   
+• ITSxxx   
+• MASS   
+• ROTARYI
+
+<!-- source-page: 620 -->
+
+• all acoustic elements   
+• all hydrostatic fluid elements
+
+The following output variable keys are not available for use with GETVRMAVGATNODE:
+
+• SVOL   
+• TSHR   
+• CTSHR
+
+# Example: Obtaining plastic strain results
+
+To illustrate the use of GETVRMAVGATNODE, consider a case where the identifier PE is specified and JELEMLIST lists four three-dimensional elements, two of which have plastic yield behavior defined and two of which do not. ARRAY will be returned with the individual plastic strain components PE11, PE22, PE33, PE12, PE13, and PE23; the equivalent plastic strain PEEQ; and the plastic strain magnitude PEMAG. The result returned in ARRAY will be an average reflecting extrapolations of plastic strain results to NODE from only the two elements that have plastic yield behavior defined.
+
+# Example: Obtaining contact results
+
+A second illustration is relevant to the modeling of wear with UMESHMOTION. Consider a case where JELEMLIST is obtained from GETNODETOELEMCONN and where the identifier CSTRESS is specified. If NODE is associated with a contact pair slave surface, JELEMLIST will contain the internal element identifier for the contact element associated with the slave node pairing. ARRAY will be returned with the individual contact stress components CPRESS, CSHEAR1, and CSHEAR2. Similarly, if CDISP is specified, ARRAY will be returned with the individual contact stress components CDISP, CSLIP1, and CSLIP2.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_063.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_063.md
new file mode 100644
index 00000000..205df517
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+<!-- source-page: 621 -->
+
+# 2.1.9 OBTAINING NODE POINT INFORMATION
+
+Product: Abaqus/Standard
+
+# References
+
+• “UMESHMOTION,” Section 1.1.46   
+• “Erosion of material (sand production) in an oil wellbore,” Section 1.1.22 of the Abaqus Example Problems Guide
+
+# Overview
+
+Utility routine GETVRN can be called from user subroutine UMESHMOTION to access node point information.
+
+# Interface
+
+```txt
+DIMENSION ARRAY(15), JGVBLOCK(*)
+...
+CALL GETVRN(NODE, 'VAR', ARRAY, JRCD, JGVBLOCK, LTRN) 
+```
+
+# Variables to be provided to the utility routine
+
+# NODE
+
+Node number.
+
+# VAR
+
+Output variable key from the table in “Abaqus/Standard output variable identifiers,” Section 4.2.1 of the Abaqus Analysis User’s Guide. The applicable keys are listed in the output table as being available for results file output at nodes; e.g., U for displacement.
+
+# JGVBLOCK
+
+Variable that must be passed into the GETVRN utility routine. The variable is available in user subroutine UMESHMOTION for this purpose.
+
+# LTRN
+
+Variable indicating the coordinate system the nodal quantity is to be returned in. A value of 0 specifies that the results are to be returned in the global coordinate system, regardless of any transformation applied at the node. A value of 1 specifies that the results are to be returned in the local transformed system.
+
+<!-- source-page: 622 -->
+
+# Variables returned from the utility routine
+
+# ARRAY
+
+Real array containing individual components of the output variable.
+
+# JRCD
+
+Return code (0 – no error, 1 – output request error or all components of output request are zero).
+
+# Available output variable keys
+
+Only nodal output variable keys that are valid for results file output in the current step are available for use with GETVRN. In general, if a key corresponds to a collective output variable, rather than an individual component, it can be used with GETVRN. For example, U for displacement can be used, whereas any individual component of displacement, say U1, cannot be used. The collective output variable keys are distinguished from their individual components by the fact that they have a bullet ( ) in the .fil column in the tables in “Abaqus/Standard output variable identifiers,” Section 4.2.1 of the Abaqus Analysis User’s Guide.
+
+You will be returned ARRAY, which corresponds to the real-valued components associated with the request VAR. If any array component is not applicable for a given request, its value will be returned as the initialized value: 0.0. The error flag JRCD=1 is returned from GETVRN any time a request key is not recognized, the request is not valid (such as requesting pore pressure for a node not associated with a pore pressure or acoustic element, or requesting a variable not available for the current procedure), or all of the output components requested are zero; otherwise, JRCD=0.
+
+# Ordering of returned components
+
+The components for a vector request are returned in positions 1, 2, 3, etc.
+
+# Analysis time for which values are returned
+
+GETVRN returns values at the end of the current increment.
+
+<!-- source-page: 623 -->
+
+# 2.1.10 OBTAINING NODE TO ELEMENT CONNECTIVITY
+
+Product: Abaqus/Standard
+
+# References
+
+• “UMESHMOTION,” Section 1.1.46   
+• “Obtaining material point information averaged at a node,” Section 2.1.8   
+• “Erosion of material (sand production) in an oil wellbore,” Section 1.1.22 of the Abaqus Example Problems Guide
+
+# Overview
+
+Utility routine GETNODETOELEMCONN can be called from user subroutine UMESHMOTION to retrieve a list of elements connected to a specified node.
+
+# Interface
+
+```txt
+PARAMETER ( MAXNELEMS = 100 )
+DIMENSION JELEMLIST(MAXNELEMS), JELEMTYPE(MAXNELEMS), JGVBLOCK(*)
+...
+NELEMS = MAXNELEMS
+CALL GETNODETOELEMCORN(NODE, NELEMS, JELEMLIST, JELEMTYPE, JRCD, JGVBLOCK) 
+```
+
+# Variables to be provided to the utility routine
+
+# NODE
+
+User node number.
+
+# NELEMS
+
+You must set NELEMS to the maximum allowable length of the JELEMLIST and JELEMTYPE arrays. This value corresponds to the maximum expected number of elements attached to an adaptive mesh constraint node in your model. GETNODETOELEMCONN will assume that your JELEMLIST and JELEMTYPE arrays are NELEMS long. In the event that the actual element connectivity exceeds NELEMS, no result will be returned and the return code JRCD will indicate an error. An NELEMS value of 100 is typically more than adequate for common meshes. NELEMS is modified by GETNODETOELEMCONN and should not be a Fortran parameter-statement constant.
+
+# JGVBLOCK
+
+Variable that must be passed into the GETNODETOELEMCONN utility routine. This variable is available in user subroutine UMESHMOTION for this purpose.
+
+<!-- source-page: 624 -->
+
+# Variables returned from the utility routine
+
+# JELEMLIST
+
+Array of element numbers for elements connected to NODE. The list will contain elements only in adaptive mesh domains active in the step as well as any contact elements associated with the domain. The number of entries in this array corresponds to the returned value of NELEMS.
+
+# JELEMTYPE
+
+Array of element type designators describing the element types corresponding to each element entry in JELEMLIST. The number of entries in this array corresponds to the returned value of NELEMS.
+
+JELEMTYPE entries:
+
+1 indicates a solid element.
+
+2 indicates a contact element.
+
+# NELEMS
+
+Actual length of the JELEMLIST and JELEMTYPE arrays.
+
+# JRCD
+
+Return code (0 indicates no error, 1 indicates an output request error). An output request error indicates either that the requested variable is not available or that your NELEMS parameter setting is smaller than the element connectivity list at this node.
+
+<!-- source-page: 625 -->
+
+# 2.1.11 OBTAINING STRESS INVARIANTS, PRINCIPAL STRESS/STRAIN VALUES AND DIRECTIONS, AND ROTATING TENSORS IN AN Abaqus/Standard ANALYSIS
+
+Product: Abaqus/Standard
+
+# References
+
+• “UMAT,” Section 1.1.44   
+• “Calculation of principal stresses and strains and their directions: FPRIN,” Section 15.1.3 of the Abaqus Example Problems Guide
+
+# Overview
+
+Utility routines are available for calculating stress invariants, principal stress/strain values, and principal stress/strain directions from the relevant tensors, as well as for transforming tensors to a new basis.
+
+These utility routines are available for Abaqus/Standard user subroutines that store stress and strain components according to the convention presented in “Conventions,” Section 1.2.2 of the Abaqus Analysis User’s Guide. They are most commonly called from user subroutine UMAT.
+
+SINV (calculate stress invariants)
+
+# Interface
+
+CALL SINV(STRESS,SINV1,SINV2,NDI,NSHR)
+
+# Variables to be provided to the utility routine
+
+# STRESS
+
+A stress tensor.
+
+# NDI
+
+Number of direct components.
+
+# NSHR
+
+Number of shear components.
+
+# Variables returned from the utility routine
+
+# SINV1
+
+First invariant.
+
+$$
+\operatorname{SINV1} = \frac {1}{3} \operatorname{trace} \sigma ,
+$$
+
+<!-- source-page: 626 -->
+
+where is the stress tensor.
+
+# SINV2
+
+Second invariant.
+
+$$
+\mathrm{SINV2} = \sqrt {\frac {3}{2} \mathbf {S} : \mathbf {S}},
+$$
+
+where is the deviatoric stress tensor, defined as
+
+$$
+\mathbf {S} = \sigma - \frac {1}{3} \operatorname{trace} \sigma \mathbf {I}.
+$$
+
+# SPRINC (calculate principal values)
+
+# Interface
+
+CALL SPRINC(S,PS,LSTR,NDI,NSHR)
+
+# Variables to be provided to the utility routine
+
+S
+
+Stress or strain tensor.
+
+LSTR
+
+An identifier. LSTR=1 indicates that S contains stresses; LSTR=2 indicates that S contains strains.
+
+NDI
+
+Number of direct components.
+
+NSHR
+
+Number of shear components.
+
+# Variables returned from the utility routine
+
+PS(I), I=1,2,3
+
+The three principal values.
+
+# SPRIND (calculate principal values and directions)
+
+# Interface
+
+CALL SPRIND(S,PS,AN,LSTR,NDI,NSHR)
+
+<!-- source-page: 627 -->
+
+Variables to be provided to the utility routine   
+```txt
+S
+A stress or a strain tensor.
+LSTR
+An identifier. LSTR=1 indicates that S contains stresses; LSTR=2 indicates that S contains strains. 
+```
+
+```txt
+NDI Number of direct components. 
+```
+
+```txt
+NSHR Number of shear components. 
+```
+
+Variables returned from the utility routine   
+```txt
+PS(I), I=1,2,3
+The three principal values.
+AN(K1,I), I=1,2,3
+The direction cosines of the principal directions corresponding to PS(K1). 
+```  
+ROTSIG (rotate a tensor)
+
+Interface   
+```csv
+CALL ROTSIG(S,R,SPRIME,LSTR,NDI,NSHR)
+```
+
+Variables to be provided to the utility routine   
+```txt
+S
+A stress or strain tensor.
+NDI
+Number of direct components. 
+```
+
+```txt
+NSHR
+Number of shear components. 
+```
+
+```txt
+R Rotation matrix. 
+```
+
+```txt
+LSTR
+An identifier. LSTR = 1 indicates S contains stresses; LSTR = 2 indicates S contains strains. 
+```
+
+<!-- source-page: 628 -->
+
+# Variable returned from the utility routine
+
+# SPRIME
+
+The rotated stress or strain tensor.
+
+# Typical usage
+
+In user subroutine UMAT it is often necessary to rotate tensors during a finite-strain analysis. The matrix DROT that is passed into UMAT represents the incremental rotation of the material basis system in which the stress and strain are stored. For an elastic-plastic material that hardens isotropically, the elastic and plastic strain tensors must be rotated to account for the evolution of the material directions. In this case S is the elastic or plastic strain tensor and R is the incremental rotation DROT.
+
+<!-- source-page: 629 -->
+
+# 2.1.12 OBTAINING PRINCIPAL STRESS/STRAIN VALUES AND DIRECTIONS IN AN Abaqus/Explicit ANALYSIS
+
+Product: Abaqus/Explicit
+
+# Reference
+
+• “VUMAT,” Section 1.2.22
+
+# Overview
+
+Utility routines are available for calculating principal stress/strain values and principal stress/strain directions from the relevant tensors.
+
+These utility routines are available for Abaqus/Explicit user subroutines that store stress and strain components according to the convention presented in “Conventions,” Section 1.2.2 of the Abaqus Analysis User’s Guide. They are most commonly called from user subroutine VUMAT.
+
+VSPRINC (calculate principal values)
+
+# Interface
+
+call vsprinc( nblock, s, eigVal, ndir, nshr )
+
+Variables to be provided to the utility routine
+
+s(nblock,ndir+nshr)
+
+Stress or strain symmetric tensor.
+
+nblock
+
+Number of material points to be processed in this call to VSPRINC.
+
+ndir
+
+Number of direct components in the symmetric tensor.
+
+nshr
+
+Number of shear components in the symmetric tensor.
+
+Variable returned from the utility routine
+
+eigVal(nblock,I), I=1,2,3
+
+The three principal values.
+
+<!-- source-page: 630 -->
+
+# Interface
+
+call vsprind( nblock, s, eigVal, eigVec, ndir, nshr )
+
+# Variables to be provided to the utility routine
+
+s(nblock,ndir+nshr)
+
+Stress or strain symmetric tensor.
+
+nblock
+
+Number of material points to be processed in this call to VSPRIND.
+
+ndir
+
+Number of direct components in the symmetric tensor.
+
+nshr
+
+Number of shear components in the symmetric tensor.
+
+# Variables returned from the utility routine
+
+eigVal(nblock,I), I=1,2,3
+
+The three principal values.
+
+eigVec(nblock,I,K1), I=1,2,3
+
+The direction cosines of the principal directions corresponding to eigVal(K1).
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_064.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_064.md
new file mode 100644
index 00000000..b0df37df
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_064.md
@@ -0,0 +1,287 @@
+<!-- source-page: 631 -->
+
+# 2.1.13 OBTAINING WAVE KINEMATIC DATA IN AN Abaqus/Aqua ANALYSIS
+
+# Product: Abaqus/Aqua
+
+# References
+
+• “UEL,” Section 1.1.28   
+• “UWAVE,” Section 1.1.59   
+• “Abaqus/Aqua analysis,” Section 6.11.1 of the Abaqus Analysis User’s Guide   
+• \*AQUA   
+• “UEL,” Section 4.1.14 of the Abaqus Verification Guide
+
+# Overview
+
+Utility routines GETWAVE, GETWAVEVEL, GETWINDVEL, and GETCURRVEL are provided to access the fluid kinematic data for an Abaqus/Aqua analysis.
+
+These routines can be used only from within user subroutine UEL.
+
+# GETWAVE (get wave kinematics)
+
+# Interface
+
+```matlab
+PARAMETER (MWCOMP=number of wave components)
+DIMENSION WAMP (MWCOMP), WPERD (MWCOMP), WXLAMB (MWCOMP),
+1 WPHI (MWCOMP), WOFF (3), WANG (2, MWCOMP)
+...
+CALL GETWAVE (MWCOMP, NWCOMP, WAMP, WPERD, WXLAMB, WPHI, WOFF, WANG,
+1 ELEVB, ELEVS, JWTYPE, JRCD) 
+```
+
+# Variables returned from the utility routine
+
+# NWCOMP
+
+Number of wave components (always 1 for Stokes wave theory).
+
+# WAMP
+
+Array containing the amplitude of the wave components.
+
+# WPERD
+
+Array containing the period of the wave components.
+
+# WXLAMB
+
+Array containing the wavelength of the wave components.
+
+<!-- source-page: 632 -->
+
+# WPHI
+
+Array containing the phase angle of the wave components.
+
+# WOFF
+
+Used only for gridded wave data (JWTYPE=2), when WOFF gives the position of the origin of the gridded coordinate system with respect to the global system.
+
+# WANG(2,\*)
+
+For Stokes fifth-order wave theory WANG(1,1) and WANG(2,1) are the direction cosines of wave travel. For Airy wave theory WANG(1,K1) and WANG(2,K1) are the direction cosines of the direction of travel of the K1th wave. For gridded wave data WANG(1,1) and WANG(2,1) are the direction cosines of the wave data grid. In all cases these direction cosines are with respect to the global coordinate system.
+
+# ELEVB
+
+User-defined elevation of the seabed.
+
+# ELEVS
+
+User-defined elevation of the still water surface.
+
+# JWTYPE
+
+Integer flag indicating the wave type, as follows:
+
+```txt
+JWTYPE=0 Airy wave theory
+JWTYPE=1 Stokes fifth-order wave theory
+JWTYPE=2 Wave data obtained from gridded values 
+```
+
+# JRCD
+
+The error flag JRCD is returned from GETWAVE as 0 if all the wave kinematic data are read correctly and as −1 if an error occurred (for instance, NWCOMP is greater that MWCOMP).
+
+GETWAVEVEL, GETWINDVEL, and GETCURRVEL (get wave, wind, and current velocities)
+
+# Interface
+
+```txt
+CALL GETWAVEVEL (NDIM, X, V, A, LERROR, NOEL, XINTERMED)
+CALL GETWINDVEL (NDIM, X, V, NOEL, XINTERMED)
+CALL GETCURRVEL (NDIM, X, V, NOEL, XINTERMED) 
+```
+
+<!-- source-page: 633 -->
+
+# Variables to be provided to the utility routine
+
+# NDIM
+
+Dimensionality of the element. It should be set to 2 for two-dimensional cases (for example, beams in a plane) and 3 for three-dimensional cases (for example, beams in space).
+
+# X(1..NDIM)
+
+Global coordinates of the point.
+
+# Variables returned from the utility routine
+
+# V(1..NDIM)
+
+Velocity components in the global coordinate system.
+
+# A(1..NDIM)
+
+Wave acceleration components in the global coordinate system. This variable is returned by GETWAVEVEL only.
+
+# LERROR
+
+For gridded wave data LERROR is returned as 0 if the current point is within the grid or above the crest; it is returned as 1 if the point is outside the bounds of the grid. For Airy and Stokes waves LERROR is always returned as 0. If LERROR is returned as 1, the global coordinates of the nearest grid point are returned in X. LERROR is returned by GETWAVEVEL only.
+
+# NOEL
+
+Element number.
+
+# XINTERMED(NDIM)
+
+An array containing the intermediate configuration coordinates of the load integration point. For nonstochastic analysis this array is not used. In a stochastic analysis the wave field is based upon this configuration. Additional details are found in “UWAVE,” Section 1.1.59.
+
+<!-- source-page: 634 -->
+
+<!-- source-page: 635 -->
+
+# 2.1.14 PRINTING MESSAGES TO THE MESSAGE OR STATUS FILE
+
+Products: Abaqus/Standard Abaqus/Explicit
+
+# References
+
+• “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide   
+• “UWAVE and UEXTERNALDB,” Section 4.1.27 of the Abaqus Verification Guide   
+• “VUMAT: rotating cylinder,” Section 4.1.38 of the Abaqus Verification Guide
+
+# Overview
+
+Utility routines STDB\_ABQERR and XPLB\_ABQERR can be called from any Abaqus/Standard or Abaqus/Explicit user subroutine, respectively, to issue an informational, a warning, or an error message to the message (.msg) file in Abaqus/Standard or the status (.sta) file in Abaqus/Explicit.
+
+# Interface
+
+```txt
+DIMENSION INTV(*), REALV(*)
+CHARACTER*8 CHARV(*)
+...
+CALL STDB_ABQERR(LOP, STRING, INTV, REALV, CHARV)
+or
+CALL XPLB_ABQERR(LOP, STRING, INTV, REALV, CHARV)
+... 
+```
+
+# Variables to be provided to the utility routine
+
+# LOP
+
+Flag for the type of message to be issued.
+
+Set LOP = 1 if an informational message is to be issued.
+
+Set LOP = –1 if a warning message is to be issued.
+
+Set LOP = –2 if an error message is to be issued and the analysis is to be continued.
+
+Set LOP = –3 if an error message is to be issued and the analysis is to be stopped immediately.
+
+# STRING
+
+A string of at most 500 characters long between single quotes containing the message to be issued. If the string needs to be written on more than one line, several one line long strings (between single quotes) should be concatenated using the double forward slash (//) operator.
+
+Integer, real, and character variables can be referenced inside the message using the %I, %R, and %S inserts, respectively. The integer, real, or character variables are passed into the utility routine via
+
+<!-- source-page: 636 -->
+
+# PRINTING MESSAGES
+
+the INTV, REALV, and CHARV variables, respectively. The variables are then output in the order they are stored in these arrays.
+
+# INTV
+
+Array of integer variables to be output. The first %I in STRING will output INTV(1), the second INTV(2), and so on.
+
+# REALV
+
+Array of real variables to be output. The first %R in STRING will output REALV(1), the second REALV(2), and so on.
+
+# CHARV
+
+Array of at most 8 character long variables to be output. The first %S in STRING will output CHARV(1), the second CHARV(2), and so on.
+
+<!-- source-page: 637 -->
+
+# 2.1.15 TERMINATING AN ANALYSIS
+
+Products: Abaqus/Standard Abaqus/Explicit
+
+# References
+
+• “User subroutines: overview,” Section 18.1.1 of the Abaqus Analysis User’s Guide   
+• “UMAT and UHYPER,” Section 4.1.21 of the Abaqus Verification Guide   
+• “UWAVE and UEXTERNALDB,” Section 4.1.27 of the Abaqus Verification Guide   
+• “VUMAT: rotating cylinder,” Section 4.1.38 of the Abaqus Verification Guide
+
+# Overview
+
+Utility routines XIT and XPLB\_EXIT can be called from within any Abaqus/Standard or Abaqus/Explicit user subroutine, respectively, (except UEXTERNALDB) to terminate an analysis.
+
+XIT or XPLB\_EXIT should be used instead of STOP to ensure that all files associated with the analysis are closed properly.
+
+# Interface
+
+```csv
+CALL XIT
+or
+CALL XPLB_EXIT 
+```
+
+<!-- source-page: 638 -->
+
+<!-- source-page: 639 -->
+
+# 2.1.16 OBTAINING SENSOR INFORMATION
+
+Products: Abaqus/Standard Abaqus/Explicit
+
+# References
+
+• “UAMP,” Section 1.1.19   
+• “VUAMP,” Section 1.2.9   
+• “Crank mechanism,” Section 4.1.2 of the Abaqus Example Problems Guide
+
+# Overview
+
+Given the user-defined name for a sensor, utility routines can be used to obtain the sensor ID or the sensor value using a computationally efficient searching technique.
+
+Utility routines IGETSENSORID and GETSENSORVALUE can be called only from Abaqus/Standard user subroutine UAMP. Utility routines IVGETSENSORID and VGETSENSORVALUE can be called only from Abaqus/Explicit user subroutine VUAMP.
+
+# Interface
+
+```prolog
+character*80 mySensorName
+...
+iMySensorID = IGETSENSORID(mySensorName, jSensorLookUpTable)
+iMySensorID = IVGETSENSORID(mySensorName, jSensorLookUpTable)
+dMySensorValue = sensorValues(iMySensorID)
+...
+dMySensorValue = GETSENSORVALUE(mySensorName,
+C    jSensorLookUpTable, sensorValues)
+dMySensorValue = VGETSENSORVALUE(mySensorName,
+C    jSensorLookUpTable, sensorValues)
+... 
+```
+
+# Variables to be provided to the utility routine
+
+# mySensorName
+
+User-defined character string, uppercase, left justified.
+
+# jSensorLookUpTable
+
+Pointer to an object containing a binary tree look up table for sensors. The calling user subroutine provides this variable.
+
+<!-- source-page: 640 -->
+
+# sensorValues
+
+Array containing the latest sensor values for all sensors in the model.
+
+# Variables returned from the utility routine
+
+# iMySensorID
+
+Index in the sensorValues array for this sensor name.
+
+# dMySensorValue
+
+Sensor value for this sensor name.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_065.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_065.md
new file mode 100644
index 00000000..262a077d
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_065.md
@@ -0,0 +1,314 @@
+<!-- source-page: 641 -->
+
+# 2.1.17 ACCESSING Abaqus MATERIALS
+
+Product: Abaqus/Standard
+
+# References
+
+• “UELMAT,” Section 1.1.29   
+• “UELMAT,” Section 4.1.15 of the Abaqus Verification Guide
+
+# Overview
+
+Utility routine MATERIAL\_LIB\_MECH returns the stress and the material Jacobian at the element material point.
+
+The routine can be called only from Abaqus/Standard user subroutine UELMAT.
+
+# Interface
+
+```matlab
+dimension stress(*),ddsdde(ntens,*,stran(*),dstran(*), * defGrad(3,3),predef(npredf),dpredef(npredf),coords(3) ... call material_lib_mech(materialllib,stress,ddsdde,stran,dstran, * npt,dvdv0,dvmat,dfgrd,predef,dpredef,npredf,celent,coords) ... 
+```
+
+# Variables to be provided to the utility routine
+
+materiallib
+
+Variable containing information about the Abaqus material. This variable is passed into user subroutine UELMAT.
+
+stran
+
+Strain at the beginning of the increment.
+
+dstran
+
+Strain increment.
+
+npt
+
+Integration point number.
+
+dvdv0
+
+Ratio of the current volume to the reference volume at the integration point.
+
+<!-- source-page: 642 -->
+
+# dvmat
+
+Volume at the integration point.
+
+# dfgrd
+
+Array containing the deformation gradient at the end of the increment.
+
+# predef
+
+Array of interpolated values of predefined field variables at the integration point at the start of the increment.
+
+# dpredef
+
+Array of increments of predefined field variables.
+
+# npredf
+
+Number of predefined field variables, including temperature.
+
+# celent
+
+Characteristic element length.
+
+# coords
+
+An array containing the coordinates of this point. These are the current coordinates if geometric nonlinearities are accounted for during the step (see “Defining an analysis,” Section 6.1.2 of the Abaqus Analysis User’s Guide); otherwise, the array contains the original coordinates of the point.
+
+# Variables returned from the utility routine
+
+# stress
+
+Stress tensor at the end of the increment.
+
+# ddsdde
+
+Jacobian matrix of the constitutive model, $\partial \Delta \sigma / \partial \Delta \varepsilon$ , where $\Delta \sigma$ are the stress increments and $\Delta \varepsilon$ are the strain increments. ddsdde(i,j) defines the change in the ith stress component at the end of the time increment caused by an infinitesimal perturbation of the jth component of the strain increment array.
+
+<!-- source-page: 643 -->
+
+# 2.1.18 ACCESSING Abaqus THERMAL MATERIALS
+
+Product: Abaqus/Standard
+
+# References
+
+• “UELMAT,” Section 1.1.29   
+• “UELMAT,” Section 4.1.15 of the Abaqus Verification Guide
+
+# Overview
+
+Utility routine MATERIAL\_LIB\_HT returns heat fluxes, internal energy time derivative, volumetric heat generation rate, and their derivatives at the element material point.
+
+The routine can be called only from Abaqus/Standard user subroutine UELMAT.
+
+# Interface
+
+```txt
+dimension predef(npredef), dpredef(npredef), dtemdx(*),
+* rhodUdg(*), flux(*), dfdt(*), dfdg(ndim, *), drpldt(*),
+* coords(3)
+...
+call material_lib_ht(materialllib, rhoUdot, rhodUdt, rhodUdg,
+* flux, dfdt, dfdg, rpl, drpldt, npt, dvmat, predef,
+* dpredef, npredf, temp, dtemp, dtemdx, celent, coords)
+... 
+```
+
+# Variables to be provided to the utility routine
+
+materiallib
+
+Variable containing information about the Abaqus material. This variable is passed into user subroutine UELMAT.
+
+npt
+
+Integration point number.
+
+dvmat
+
+Volume at the integration point.
+
+predef
+
+Array of interpolated values of predefined field variables at the integration point at the start of the increment.
+
+<!-- source-page: 644 -->
+
+# dpredef
+
+Array of increments of predefined field variables.
+
+# npredf
+
+Number of predefined field variables, including temperature.
+
+# temp
+
+Temperature at the integration point at the start of the increment, .
+
+# dtemp
+
+Increment of temperature.
+
+# dtemdx
+
+Spatial gradients of temperature, , at the end of the increment.
+
+# celent
+
+Characteristic element length.
+
+# coords
+
+The array containing the original coordinates of this point.
+
+# Variables returned from the utility routine
+
+# rhoUdot
+
+Time derivative of the internal thermal energy per unit mass, U, multiplied by density at the end of increment.
+
+# rhodUdt
+
+Variation of internal thermal energy per unit mass with respect to temperature multiplied by density evaluated at the end of the increment.
+
+# rhodUdg
+
+Variation of internal thermal energy per unit mass with respect to the spatial gradients of temperature, , multiplied by density at the end of the increment.
+
+# flux
+
+Heat flux vector, , at the end of the increment.
+
+# dfdt
+
+Variation of the heat flux vector with respect to temperature, , evaluated at the end of the increment.
+
+# dfdg
+
+Variation of the heat flux vector with respect to the spatial gradients of temperature, , at the end of the increment
+
+<!-- source-page: 645 -->
+
+rpl
+
+Volumetric heat generation per unit time at the end of the increment.
+
+drpldt
+
+Variation of rpl with respect to temperature.
+
+<!-- source-page: 646 -->
+
+<!-- source-page: 647 -->
+
+# 2.1.19 OBTAINING SCALAR STATE INFORMATION IN AN Abaqus/CFD ANALYSIS
+
+# Product: Abaqus/CFD
+
+# References
+
+• “SMACfdUserPressureBC,” Section 1.3.1   
+• “SMACfdUserVelocityBC,” Section 1.3.2
+
+# Overview
+
+Utility routine SMACfdUserSubroutineGetScalar can be called from a user subroutine to access selected output variables for elements or surface facets that are part of a boundary condition definition.
+
+# Interface
+
+```txt
+#include <SMACfdUserSubroutines.h>
+const double* scalars = SMACfdUserSubroutineGetScalar("VAR"); 
+```
+
+# Variable to be provided to the utility routine
+
+# VAR
+
+Output variable key. The available variables are listed in “Available output variable keys.”
+
+# Variable returned from the utility routine
+
+# scalars
+
+Real array containing scalar values of the output variable.
+
+# Available output variable keys
+
+The following output variable keys are supported:
+
+• AREA: Area of the surface facet.   
+• DENSITY: Element density.   
+• DIV: Element divergence.   
+• EVOL: Element volume.   
+• TEMP: Element temperature.   
+• TURBEPS: Element energy dissipation rate.   
+• TURBKE: Element turbulent kinetic energy.   
+• TURBNU: Element turbulent eddy viscosity.
+
+<!-- source-page: 648 -->
+
+• TURBOMEGA: Element-specific energy dissipation rate.
+
+A requested output variable must be valid for the energy equation setting or turbulence model for the request to be successful.
+
+The returned array scalars corresponds to the real-valued variable that can be associated with the request output variable key VAR. If the surface associated with an output variable does not have any facets on the current processor, the pointer returned from the method will be 0. The method will throw an exception and terminate the analysis if an output variable is not available for the current model.
+
+# Analysis time for which values are returned
+
+Utility subroutine SMACfdUserSubroutineGetScalar returns values of the requested variable that correspond to the beginning of the current increment.
+
+<!-- source-page: 649 -->
+
+# 2.1.20 OBTAINING VECTOR STATE INFORMATION IN AN Abaqus/CFD ANALYSIS
+
+# Product: Abaqus/CFD
+
+# References
+
+• “SMACfdUserPressureBC,” Section 1.3.1   
+• “SMACfdUserVelocityBC,” Section 1.3.2
+
+# Overview
+
+Utility routine SMACfdUserSubroutineGetVector can be called from a user subroutine to access selected output variables for elements and surface facets that are part of a boundary condition definition.
+
+# Interface
+
+```c
+#include <SMACfdUserSubroutines.h>
+const double* vcomp = SMACfdUserSubroutineGetVector("VAR", comp); 
+```
+
+# Variables to be provided to the utility routine
+
+# VAR
+
+Output variable key. The available variables are listed in “Available output variable keys.”
+
+# comp
+
+Output variable vector component number; i.e., 1, 2, or 3.
+
+# Variable returned from the utility routine
+
+# vcomp
+
+Real array containing the values of the vector component for the output variable.
+
+# Available output variable keys
+
+The following output variable keys are supported:
+
+• NORMAL: Surface facet normal direction cosines.   
+• V: Surface facet normal velocity.
+
+The returned array component corresponds to the real-valued variable that can be associated with the request output variable key VAR’s component. If the surface associated with an output variable does not have any facets on the current processor, the pointer returned from the method will be 0. The method
+
+<!-- source-page: 650 -->
+
+will throw an exception and terminate the analysis if an output request is not available for the current model.
+
+# Analysis time for which values are returned
+
+Utility subroutine SMACfdUserSubroutineGetVector returns values of the requested variable that correspond to the beginning of the current increment.
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_066.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_066.md
new file mode 100644
index 00000000..dd2ee721
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_066.md
@@ -0,0 +1,245 @@
+<!-- source-page: 651 -->
+
+# 2.1.21 OBTAINING THE MPI COMMUNICATOR IN AN Abaqus/CFD ANALYSIS
+
+Product: Abaqus/CFD
+
+# References
+
+• “SMACfdUserPressureBC,” Section 1.3.1   
+• “SMACfdUserVelocityBC,” Section 1.3.2   
+• “System customization parameters,” Section 4.1.5 of the Abaqus Installation and Licensing Guide
+
+# Overview
+
+Utility routine SMACfdUserSubroutineGetMpiComm can be called from within a user subroutine to obtain the MPI communicator used in a parallel analysis job.
+
+# Interface
+
+```c
+#include <mpi.h>
+#include <SMACfdUserSubroutines.h>
+MPI_Comm comm = SMACfdUserSubroutineGetMpiComm(); 
+```
+
+# Variable returned from the utility routine
+
+comm
+
+MPI communicator.
+
+# Compile and link commands for utility usage
+
+Utility subroutine SMACfdUserSubroutineGetMpiComm requires the user to modify the compile and link commands for user subroutines to point to the MPI include files and libraries. These MPI files are not included in the release, but they are typically installed on the computer where MPI development is undertaken. The modification to the compile and link commands is done using the compile\_cpp and link\_sl options. The include directory for the mpi.h file must be added to the compile\_cpp variable, and the location of the MPI libraries must be added to the link\_sl variable. The syntax for the changes to the commands is compiler and linker dependent.
+
+<!-- source-page: 652 -->
+
+<!-- source-page: 653 -->
+
+# 2.1.22 ENSURING THREAD SAFETY
+
+Products: Abaqus/Standard Abaqus/Explicit
+
+# References
+
+• “Parallel execution: overview,” Section 3.5.1 of the Abaqus Analysis User’s Guide   
+• “Obtaining parallel processes information,” Section 2.1.4
+
+# Overview
+
+A number of primitives are provided to help code user subroutines for thread-parallel execution.
+
+<!-- source-page: 654 -->
+
+In thread-parallel execution mutexes can be used to protect a common block or a common file from being updated by multiple threads at the same time. Abaqus provides 100 predefined mutexes for use in user subroutines. They are referenced simply by number (1–100).
+
+Mutexes need to be initialized before they can be used. For example, MutexInit(1) initializes mutex 1. It is best to initialize mutexes at the very beginning of the analysis in user subroutines, such as user subroutines UEXTERNALDB and VEXTERNALDB.
+
+Once initialized, mutexes can safeguard sensitive sections of the code against concurrent access. For example, MutexLock(1) and MutexUnlock(1) will respectively lock and unlock mutex 1.
+
+Each mutex can protect a shared resource or a logical group of resources that are always accessed together. Different mutexes are provided so that users can protect a variety of shared resources or objects. For example, one mutex can protect a file and another mutex can protect common block variables. Thus, accessing a file can happen simultaneously with updating common block variables; but no two threads can write to the file at the same time, and no two threads can update the variables at the same time.
+
+To make data transfer and accumulation easier and safer between user subroutines in a multi-threaded environment, Abaqus provides utility functions to create dynamic storage in the form of thread-local arrays, which are private to each thread, and global arrays, which are shared. Any number of arrays of any size can be created at run time. Global arrays are accessible from all user subroutines and all threads. Thread-local arrays are private and exist only within the scope of each thread. They are accessible to all user subroutines running in that thread but not across threads. Since they are not visible to neighboring threads, they do not need to be protected from concurrent access. They are designed as a thread-agnostic replacement for the COMMON BLOCKS and SAVE variables (see “Allocatable arrays,” Section 2.1.23, for more information).
+
+All other techniques commonly used in parallel programming can also be employed; for example, restricting all file operations only to thread 0. Oftentimes, these alternatives are preferable to using mutexes because they may provide better performance.
+
+# Interface
+
+```txt
+Fortran:
+#include <SMAAspUserSubroutines.hdr>
+
+! Initialization in UEXTERNALDB/VEXTERNALDB
+call MutexInit(1) ! initialize Mutex #1
+! Use in all other user subs after being initialized
+call MutexLock(1) ! lock Mutex #1
+< critical section : update shared variables > 
+```
+
+<!-- source-page: 655 -->
+
+```txt
+call MutexUnlock(1) ! unlock Mutex #1
+C++:
+#include <SMAAspUserSubroutines.h>
+// Initialization in UEXTERNALDB/VEXTERNALD
+MutexInit(1); // initialize Mutex #1
+// Use in all other user subs after being initialized
+MutexLock(1); // lock Mutex #1
+< critical section : update shared variables >
+MutexUnlock(1); // unlock Mutex #1 
+```
+
+NOTE: IDs are arbitrary chosen by the user, from the pool of 1-100. Other threads, when encountering a locked mutex, will sleep. Once the first entering thread unlocks the mutex and leaves, other threads will be able to come in and execute the critical section (one at a time).
+
+<!-- source-page: 656 -->
+
+<!-- source-page: 657 -->
+
+# 2.1.23 ALLOCATABLE ARRAYS
+
+Products: Abaqus/Standard Abaqus/Explicit Abaqus/CFD
+
+# Reference
+
+• “Ensuring thread safety,” Section 2.1.22
+
+# Overview
+
+To facilitate data accumulation and transfer between user subroutines, you can use utility functions to create your own dynamic storage in the form of allocatable arrays. Thread-local and global arrays are supported. In addition to basic types, you can also vary the precision of real arrays according to the precision of Abaqus/Explicit and define arrays of user-defined data types.
+
+# SMALocalIntArrayCreate, SMALocalFloatArrayCreate
+
+You can create any number of thread-local or global arrays. You give each array an identifier (an arbitrary positive integer) at the time of its creation. You create an array in one user subroutine and reference it in another simply by its identifier. The arrays persist in memory until you explicitly delete them or until the analysis terminates.
+
+# Thread-local arrays
+
+A thread-local array is a mechanism to allocate storage that is local to a thread and does not need any locking for access. In a multi-threaded environment the thread safety of these arrays stems from their design and usage: they are deliberately not shared and, thus, do not need to be protected from competing threads. In fact, one thread cannot reference a local array of another thread. They can be accessed concurrently without any locking and, thus, are faster than global arrays.
+
+Thread-local arrays are unique in each thread. They are nonintersecting and nonoverlapping in memory, with each thread starting out with its own private copy of an array. For example, Thread 0 can have a local array with ID 1 and Thread 4 can have a local array with ID 1. Those two arrays are different and separate from each other. Similarly, it is possible to have an integer array with ID 1 and a float array with ID 1. Again, they are two different arrays. It is not possible to cross-reference these arrays across different threads. However, all user subroutines running in one thread can access all arrays of that thread. In a thread-agnostic way, these arrays are shared between user subroutines but not among threads. These routines are meant as a thread-safe replacement for COMMON BLOCKs and SAVE variables.
+
+The following utility subroutines are available to operate on thread-local arrays:
+
+• SMALocalIntArrayCreate, SMALocalFloatArrayCreate: to create or resize a local array.   
+• SMALocalIntArrayAccess, SMALocalFloatArrayAccess: to locate an existing local array.
+
+<!-- source-page: 658 -->
+
+• SMALocalIntArrayDelete, SMALocalFloatArrayDelete: to delete a local array.   
+• SMALocalIntArraySize, SMALocalFloatArraySize: to get the size of the array.
+
+These utility routines are accessible from both Fortran and C/C++. The details of their interfaces are described below.
+
+# Global arrays
+
+Global arrays are visible and accessible from all threads in an executable. To prevent race conditions, protect the creation and the write access to these arrays with mutexes (mutual exclusion locks). You can have each thread execute global array creation under a mutex protection. However, only the first thread to arrive will create the global array; the later threads will simply connect to the array already created. In addition, using mutexes on every write access will incur a performance penalty. In some situations it is possible to avoid unnecessary locking by restricting all threads to operate on nonintersecting ranges of a global array. Another alternative is to use thread-local arrays.
+
+The following utility routines are available to operate on global arrays:
+
+• SMAIntArrayCreate, SMAFloatArrayCreate: to create or resize a global array.   
+• SMAIntArrayAccess, SMAFloatArrayAccess: to locate an existing global array.   
+• SMAIntArrayDelete, SMAFloatArrayDelete: to delete a global array.   
+• SMAIntArraySize, SMAFloatArraySize: to get the size of the global array.
+
+These arrays are global and accessible from all threads within a process but not across different MPI processes. To share data between separate MPI processes, MPI facilities must be used. Abaqus supports the full use of MPI within user subroutines.
+
+# Interface
+
+Fortran:
+
+```txt
+INTEGER*8 SMALocalIntArrayCreate(ID, SIZE, INITVAL)
+INTEGER*8 SMALocalFloatArrayCreate(ID, SIZE, INITVAL) 
+```
+
+Example:
+
+```txt
+#include <SMAAspUserSubroutines.hdr> 
+```
+
+```txt
+integer a(100)
+pointer(ptra, a) 
+```
+
+```lisp
+real*8 b(*)
+pointer(ptrb, b) 
+```
+
+```txt
+! create a local array with ID=1 and SIZE=100
+ptra = SMALocalIntArrayCreate(1,100) 
+```
+
+<!-- source-page: 659 -->
+
+```txt
+a(1) = 11 ! use as a native Fortran array
+a(2) = 22 ! use as a native Fortran array
+! create a local float array with ID=1 and SIZE=100, and
+! initial value = -1.0
+ptrb = SMALocalFloatArrayCreate(1,100,-1.0) 
+```  
+C++:
+
+```c
+#include <SMAAspUserSubroutines.h>
+
+// Create a local integer array of with ID=1 and size=100
+int* a = SMALocalIntArrayCreate(1,100);
+
+// Create a local float array of with ID=1, size=20, and
+// initial value = -1.0
+real* b = SMALocalFloatArrayCreate(1,100,-1.0); 
+```  
+NOTE: Float Arrays can store both SINGLE PRECISION and DOUBLE PRECISION numbers. Internally, memory is allocated in units of 64 bits (double/real\*8).
+
+NOTE: To resize an array, simply call Create() with the same ID, but give it a new SIZE parameter. If the new size is larger, the old data are copied over to the new array. No data are lost during resizing.   
+For example:   
+```fortran
+! resize array with ID=1 to 300 integers
+ptra = SMALocalIntArrayCreate(1,300) 
+```  
+NOTE: In Create() functions, there is an optional third argument -- initial value. If not supplied, all Int arrays are initialized with INT\_MAX ( 2,147,483,647 ). All Float Arrays are initialized with Signaling NANs. The values of INT\_MAX and signaling NANs are accessible via the 'SMAAspNumericLimits.h' and 'SMAAspNumericLimit.hdr' header files.
+
+<!-- source-page: 660 -->
+
+# Variables to be provided to the utility routine
+
+# ID
+
+ID of the array (an integer), chosen by the user at the time of creation. Using this ID, an array can be opened in any other user subroutine.
+
+# SIZE
+
+Size of the array as the number of ints or doubles. The maximum size for thread-local arrays is INT\_MAX (2,147,483,647).
+
+# INITVAL
+
+Initial value for each item in the array. If this argument is not supplied, in the case of an integer a large value is used; in the case of a float NAN is used.
+
+# Variable returned from the utility routine
+
+# INTEGER\*8 ( address )
+
+Returns a pointer to the array created. This pointer can be associated with a native Fortran array or native C/C++ array. Each thread will receive a different pointer. Each thread will create and hold its own array. For example, Array(1) in Thread 0 is separate from Array(1) in Thread 4. These arrays are nonoverlapping and nonintersecting in any way.
+
+SMALocalIntArrayAccess, SMALocalFloatArrayAccess
+
+# Interface
+
+```txt
+Fortran interface:
+    INTEGER*8 SMALocalIntArrayAccess(ID)
+    INTEGER*8 SMALocalFloatArrayAccess(ID)
+
+Example:
+#include <SMAAspUserSubroutines.hdr>
+    integer a(100)
+    pointer(ptra, a)
+
+C Locate local Array(1) and associate a native array pointer with it
+    ptra = SMALocalIntArrayAccess(1) 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_067.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_067.md
new file mode 100644
index 00000000..c7420318
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_067.md
@@ -0,0 +1,383 @@
+<!-- source-page: 661 -->
+
+```txt
+a(1) = 11 ! use as a native Fortran array
+a(2) = 22 ! use as a native Fortran array 
+```
+
+C++ interface:   
+```c
+#include <SMAAspUserSubroutines.h>
+// Locate and open array with ID=1
+int* a = SMALocalArrayIntAccess(1);
+a[1] = 11; // use as a native array
+a[2] = 22; // use as a native array 
+```  
+NOTE: If a request is made to access an array that has not been created, the function will return 0.
+
+# Variable to be provided to the utility routine
+
+ID
+
+ID of the array (an integer), chosen by the user at the time of creation. Using this ID, an array can be opened in any other user subroutine.
+
+# Variable returned from the utility routine
+
+INTEGER\*8 ( address )
+
+Returns a pointer to the array created. This pointer can be associated with a native Fortran array or native C/C++ array. Each thread will receive a different pointer. Each thread, as it passes through this code, will create and hold its own array. For example, Array(1) in Thread 0 is a separate array from Array(1) in Thread 4. These arrays are nonoverlapping and nonintersecting in any way.
+
+SMALocalIntArraySize, SMALocalFloatArraySize
+
+Interface
+
+Fortran interface:
+
+```txt
+INTEGER*4 SMALocalIntArraySize(ID)
+INTEGER*4 SMALocalFloatArraySize(ID) 
+```
+
+Example:
+
+<!-- source-page: 662 -->
+
+```c
+#include <SMAAspUserSubroutines.hdr>
+    integer a_size, d_size
+
+C Get the size of Array(1) as the number of INTEGERs
+    a_size = SMALocalIntArraySize(1)
+
+! Get the size of Array(1) as the number of REALs
+    d_size = SMALocalFloatArraySize(1)
+    do k=1,a_size
+    ...
+    end do
+
+C++:
+    #include <SMAAspUserSubroutines.h>
+    // Lookup the size of Array(1) as the number of ints
+    int a_size = SMALocalIntArraySize(1);
+    // Lookup the size of Array(1) as the number of doubles
+    int d_size = SMALocalFloatArraySize(1);
+    for(int i=1; i<=size; i++) {
+    ...
+    } 
+```
+
+# Variable to be provided to the utility routine
+
+ID
+
+ID of the array (an integer), chosen by the user at the time of creation. Using this ID, an array can be opened in any other user subroutine.
+
+# Variable returned from the utility routine
+
+INTEGER\*4
+
+Size of the array.
+
+<!-- source-page: 663 -->
+
+Interface   
+```asm
+Fortran interface:
+    subroutine SMALocalIntArrayDelete(ID)
+    subroutine SMALocalFloatArrayDelete(ID)
+
+Example:
+#include <SMAAspUserSubroutines.hdr>
+    call SMALocalIntArrayDelete(1) ! Delete Array(1)
+
+C++ interface:
+#include <SMAAspUserSubroutines.h>
+    SMALocalIntArrayDelete(1); // Delete Array(1)
+
+NOTE: Deletion of arrays is optional. All storage allocated for these arrays will be freed when Abaqus threads terminate (at the very end of the analysis). It is, however, a good programming practice to delete all allocations explicitly, especially when they are no longer needed, as this will free up memory for something else. 
+```
+
+# Variable to be provided to the utility routine
+
+ID of the array (an integer), chosen by the user at the time of creation. Using this ID, an array can be opened in any other user subroutine.
+
+SMAIntArrayCreate, SMAFloatArrayCreate
+
+# Interface
+
+Fortran interface:
+
+INTEGER\*8 SMAIntArrayCreate(ID,SIZE,INITVAL)
+
+<!-- source-page: 664 -->
+
+INTEGER\*8 SMAFloatArrayCreate(ID,SIZE,INITVAL)   
+Example:   
+```fortran
+#include <SMAAspUserSubroutines.hdr>
+
+integer a(100)
+pointer(ptra, a)
+double b(100)
+pointer(ptrb, b)
+
+! create a global array with ID=1, SIZE=100, and
+! INITVAL=-1.0
+ptra = SMAIntArrayCreate(1, 100, -1.0)
+
+a(1) = 11 ! use as a native Fortran array
+a(2) = 22 ! use as a native Fortran array
+
+! create a global array with ID=2, SIZE=100, and
+! INITVAL=-1.0
+ptrb = SMAFloatArrayCreate(2, 100, -1.0) 
+```  
+C++ interface:
+
+```c
+#include <SMAAspUserSubroutines.h>
+
+// Create an integer array of with ID=1, size=100,
+// and initial value=-1.0
+int* a = SMAIntArrayCreate(1,100,-1.0);
+
+// Create a float array of with ID=2, size=20,
+// and initial value=-1.0
+Real* b = SMAFloatArrayCreate(2,20,-1.0); 
+```  
+NOTE: Float Arrays can store both SINGLE PRECISION and DOUBLE PRECISION numbers. Internally, they allocate storage in 64-bit units (double/real\*8).   
+NOTE: To resize an array, simply call Create() with the same ID, but give it a new SIZE parameter. If the size has increased, the old data will be copied over to the new array. No data is lost during resizing.
+
+<!-- source-page: 665 -->
+
+For example:
+
+```fortran
+! resize array with ID=1 to 300 integers
+ptra = SMAIntArrayCreate(1,300,-1) 
+```
+
+# Variables to be provided to the utility routine
+
+ID
+
+ID of the array (an integer), chosen by the user at the time of creation. Using this ID, an array can be opened in any other user subroutine.
+
+SIZE
+
+Size of the array as the number of ints or doubles. The maximum size is INT\_MAX.
+
+INITVAL
+
+Initial value for each item of the array. This argument is required.
+
+# Variables returned from the utility routine
+
+INTEGER\*8 ( address )
+
+Returns a pointer to the array created. This pointer can be associated with a native Fortran array or native C/C++ array. All threads with see the same address when they try to access this array through its ID.
+
+SMAIntArrayAccess, SMAFloatArrayAccess
+
+Interface
+
+Fortran interface:
+
+```cmake
+INTEGER*8 SMAIntArrayAccess(ID)
+INTEGER*8 SMAFloatArrayAccess(ID) 
+```
+
+Example:
+
+#include <SMAAspUserSubroutines.hdr>
+
+```txt
+integer a(100)
+pointer(ptra, a) 
+```
+
+C Locate Array(1) and associate a native array pointer with it
+
+<!-- source-page: 666 -->
+
+```txt
+ptra = SMAIntArrayAccess(1)
+
+a(1) = 11 ! use as a native Fortran array
+a(2) = 22 ! use as a native Fortran array
+
+C++ interface:
+#include <SMAAspUserSubroutines.h>
+
+// Locate and open array with ID=1
+int* a = SMAIntArrayAccess(1);
+
+a[1] = 11; // use as a native array
+a[2] = 22; // use as a native array 
+```
+
+NOTE: If a request is made to access an array which has not been created, the function will return 0.
+
+# Variable to be provided to the utility routine
+
+ID
+
+ID of the array (an integer), chosen by the user at the time of creation. Using this ID, an array can be opened in any other user subroutine.
+
+# Variable returned from the utility routine
+
+INTEGER\*8 ( address )
+
+Returns a pointer to the array, or 0 if an array with the requested ID does not exist. This pointer can be associated with a native Fortran or C/C++ array.
+
+SMAIntArraySize, SMAFloatArraySize
+
+# Interface
+
+```txt
+Fortran interface:
+INTEGER SMAIntArraySize(ID)
+INTEGER SMAFloatArraySize(ID) 
+```
+
+Example:
+
+<!-- source-page: 667 -->
+
+```cpp
+#include <SMAAspUserSubroutines.hdr>
+    integer a_size, d_size
+
+C Get the size of Array(1) as the number of INTEGERs
+    a_size = SMAIntArraySize(1)
+
+! Get the size of Array(1) as the number of REALs
+    d_size = SMAFloatArraySize(1)
+
+    do k=1,a_size
+    ...
+    end do
+
+C++ interface:
+    #include <SMAAspUserSubroutines.h>
+
+    // Lookup the size of Array(1) as the number of INTS
+    int a_size = SMAIntArraySize(1);
+
+    // Lookup the size of Array(1) as the number of doubles
+    int d_size = SMAFloatArraySize(1);
+
+    for(int i=1; i<=d_size; i++) {
+    ...
+    } 
+```
+
+# Variable to be provided to the utility routine
+
+# ID
+
+ID of the array (an integer), chosen by the user at the time of creation. Using this ID, an array can be opened in any other user subroutine.
+
+# Variable returned from the utility routine
+
+# INTEGER\*4
+
+Size of the array.
+
+<!-- source-page: 668 -->
+
+Interface   
+```txt
+Fortran:
+#include <SMAAspUserSubroutines.hdr>
+    call SMAIntArrayDelete(1) ! Delete global Array(1)
+C++:
+    #include <SMAAspUserSubroutines.h>
+    SMAIntArrayDelete(1); // Delete global Array(1)
+NOTE: Deletion of arrays is optional. All storage allocated for these arrays will be freed when Abaqus terminates (at the very end of the analysis). It is, however, a good programming practice to delete all allocations explicitly, especially when they are no longer needed, as this will free up memory for use somewhere else. 
+```
+
+# Variable to be provided to the utility routine
+
+# ID
+
+ID of the array (an integer), chosen by the user at the time of creation. Using this ID, an array can be opened in any other user subroutine.
+
+# Allocatable global arrays of variable precision
+
+The usage of real arrays is exactly the same as that of integer and floating point arrays except for the handling of precision. The precision of real arrays varies, changing along with the precision of Abaqus/Explicit. In single precision the values of real arrays are 32-bits long, and in double precision their values are 64-bits. For this automatic switching to work in Fortran, the type of such an array should not be declared explicitly. Abaqus relies on the implicit naming to alternate between single precision and double precision. In C/C++ the type of the native array should be Real\*. The typedef declaration changes between float and double depending on the precision of Abaqus/Explicit. The precision does not change during a run; it is determined at the beginning of the analysis and remains the same until the end.
+
+When you create real arrays, you give each array an identifier. Arrays can be created in one user subroutine and operated on in another simply by referencing this identifier. You need not capture the
+
+<!-- source-page: 669 -->
+
+pointer to the array and pass it between routines. The arrays persist in memory from the moment they are created until you delete them explicitly or until the analysis ends. The arrays do not disappear when any particular user subroutine terminates. They are accessible from all user subroutines and all threads. Each MPI process is separate in memory from other MPI processes and has its own arrays. There is no cross-referencing of these arrays across MPI processes.
+
+These arrays can be resized dynamically as needed. A call to Create() on an existing array but with a different size resizes the array. If the new size is larger than the previous size, there is no loss of data and the previous contents are carried over.
+
+# Interface
+
+Fortran:   
+```fortran
+#include <vaba_param.inc>
+#include <SMAAspUserSubroutines.hdr>
+
+! Note: we do not explicitly declare the type of 'ra', we
+! rely on rules of implicit typing: it will become real*4
+! or real*8 depending on the precision of Abaqus
+
+dimension ra(*)
+pointer(ptrra,ra)
+
+integer sz
+
+rinitval = -1.0e36 ! again, implicit typing
+
+! Creating an array
+
+! ID=1, SIZE=10, no initializer
+ptrra = SMARrealArrayCreate(1, 10)
+! ID=2, SIZE=10, rinitval used to initialize
+ptrra = SMARrealArrayCreate(2, 10, rinitval)
+! ID=3, SIZE=10, initial value is -3.3d0
+ptrra = SMARrealArrayCreate(3, 10, -3.3d0)
+! ID=4, SIZE=10, initial value is -3.3
+ptrra = SMARrealArrayCreate(4, 10, -3.3)
+
+! Use ( from another subroutine )
+
+ptrra = SMARrealArrayAccess(1) 
+```
+
+<!-- source-page: 670 -->
+
+```txt
+if (ptrra.eq.0) then
+    write(*,*) '### Array',i, 'does not exist'
+end if
+! Use as a native array in Fortran
+ra(1) = 11.11
+ra(2) = 22.22
+! Looping
+! Find out the current size of the array #1
+sz = SMARealArraySize(1)
+do k=1,sz
+    write(*,*) k, '=', ra(k)
+end do
+! Resizing
+ptrra = SMARealArrayCreate(1, 1000, -1.0)
+! Array #1 is resized; the original 10 entries
+! are intact and carried over to the new array;
+! all new entries are set to -1.0
+! Deletion
+call SMARealArrayDelete(1)
+call SMARealArrayDelete(2)
+call SMARealArrayDelete(3)
+call SMARealArrayDelete(4)
+C/C++:
+#include <omi_for_types.h>
+#include <SMAAspUserSubroutines.h>
+Real* ra = 0;    // Type 'Real' switches precision with Explicit
+int    sz = 0; 
+```
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_068.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_068.md
new file mode 100644
index 00000000..d38c4ad3
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_068.md
@@ -0,0 +1,253 @@
+<!-- source-page: 671 -->
+
+```c
+// Examples of Array Creation
+// ID=1, SIZE=10, no initializer used
+ra = SMARealArrayCreate(1, 10);
+// ID=2, SIZE=10, initial value = -1.0
+ra = SMARealArrayCreate(2, 10, -1.0);
+// Access from another User Subroutine
+ra = SMARealArrayAccess(1);
+if (ra == 0) {
+    fprintf(stderr,
+    "*** Error: array %d does not exist ***\n", 1);
+}
+// Looping over the entries
+// obtain the current size of array #1
+sz = SMARealArraySize(1);
+for (int i=0; i<sz; i++) {
+    sum = sum + ra[i];
+}
+// Deletion
+SMARealArrayDelete(1);
+SMARealArrayDelete(2); 
+```
+
+# Variables to be provided to the utility routine
+
+# ID
+
+ID of the array (an integer), chosen by the user at the time of creation. Using this ID, an array can be opened in any other user subroutine.
+
+# SIZE
+
+Size of the array as the number of items. The maximum size is INT\_MAX (2,147,483,647).
+
+# INITVAL
+
+Initial value for each item of the array. If the argument is not supplied, zero is used as the initial value.
+
+<!-- source-page: 672 -->
+
+# Variable returned from the utility routine
+
+# INTEGER\*8 (address)
+
+Returns a pointer to the array created. This pointer can be associated with a native Fortran array or a native C/C++ array. These arrays are global. All threads will see and access exactly the same global array with a given ID.
+
+# Allocatable global arrays of user-defined types
+
+The usage and syntax of arrays of structures are exactly the same as those of integer, floating point, and real arrays. These arrays are designed to store any user-defined types or classes, defined either in Fortran or in C/C++. The only information an array needs to know about these structures is their memory size. Most compilers provide the sizeof() operator, which returns the size of any object in memory in bytes. This size is one additional argument to the routines that operate on arrays of structures.
+
+When you create arrays of structures, you give each array an identifier. Arrays can be created in one user subroutine and operated on in another simply by referencing this identifier. You need not capture the pointer to the array and pass it between routines. The arrays persist in memory from the moment they are created until you delete them explicitly or until the analysis ends. The arrays do not disappear when any particular user subroutine terminates. They are accessible from all user subroutines and from all threads. Each MPI process is separate in memory from other MPI processes and has its own arrays. There is no cross-referencing of these arrays across MPI processes.
+
+These arrays can be resized dynamically as needed. A call to Create() on an existing array but with a different size resizes the array. If the new size is larger than the previous size, there is no data loss and the previous contents are carried over.
+
+# Interface
+
+# Fortran:
+
+```gradle
+! Include a user module called, for example, 'mod', ! which defines some user structure 'UserStruct' 
+```
+
+use mod
+
+```cpp
+#include <aba_param.inc> ! include this for Abaqus/Standard
+#include <vaba_param.inc> ! include this for Abaqus/Explicit 
+```
+
+```txt
+#include <SMAAspUserSubroutines.hdr> 
+```
+
+```cpp
+type(UserStruct):: us(10)
+type(UserStruct):: structs(10)
+type(UserStruct):: initval,s 
+```
+
+<!-- source-page: 673 -->
+
+```fortran
+pointer(ptrstructs, structs)
+
+integer:: size1, size2, size3, size4
+integer(kind=8) :: arraySize
+
+! Create an initializer for the values of the array
+!(optional)
+
+initval%a = 100
+initval%b = 200
+initval%c = 300
+
+! Different ways of obtaining the size of a structure
+size1 = storage_size( us(1) ) / 8    ! returns the size
+    ! in bits
+size2 = sizeof( us(1) )
+size3 = storage_size( initval ) / 8    ! returns the size
+    ! in bits
+size4 = sizeof( initval )
+
+! Creating an array
+write(*,*) 'Array without initializers:'
+ptrstructs = SMAStructArrayCreate(1, 10, sizeof(initval))
+write(*,*) 'Array with initializers:'
+ptrstructs = SMAStructArrayCreate(2, 10, sizeof(initval), initval)
+
+! Use ( from another subroutine )
+ptrstructs = SMAStructArrayAccess(2)
+if (ptrstructs.eq.0) then 
+```
+
+<!-- source-page: 674 -->
+
+```txt
+write(*,*) '### Array 2 does not exist'
+end if 
+```
+
+! Use as a native array in Fortran   
+```txt
+structs(5).a = -51
+structs(5).b = -52
+structs(5).c = -53 
+```
+
+```txt
+structs(10).a = 111
+structs(10).b = 222
+structs(10).c = 333 
+```  
+! Looping over the entries arraySize = SMAStructArraySize(2) do k=1,arraySize s = structs(k); call PrintStruct(s) end do
+
+! Resize an array without using initializer ptrstructs = SMAStructArrayCreate(2, 100, sizeof(initval)) arraySize = SMAStructArraySize(2)   
+! Resize array 2 with initializer ptrstructs = SMAStructArrayCreate(2, 200, sizeof(initval), & initval)   
+```objectivec
+arraySize = SMAStructArraySize(2) 
+```  
+! Deletion call SMAStructArrayDelete(1)
+
+<!-- source-page: 675 -->
+
+```c
+call SMAStructArrayDelete(2)
+
+C/C++:
+
+#include <omi_for_types.h>
+#include <SMAAspUserSubroutines.h>
+
+// Include the definition of a user-defined type,
+// for example, A
+
+#include <A.h>
+
+// Create an (optional) initializer for user structs
+A init = { -1, -2, -3 };
+
+// Creating arrays
+
+// no initializer
+SMAStructArrayCreate(1, 10, sizeof(A));
+// with initializer
+SMAStructArrayCreate(2, 10, sizeof(A), &init);
+
+// Accessing arrays (from another subroutine)
+
+A* array = (A*) SMAStructArrayAccess(1);
+
+// Modifying values in the array
+
+A* s1 = &array[5]; // We use a pointer to modify the value in
+// the array itself. Without a pointer, s1
+// will contain a copy of the entry in
+// the array, and any modifications to
+// this copy will not affect the value in
+// the original array.
+
+s1->a = -111;
+s1->b = -222;
+s1->c = -333; 
+```
+
+<!-- source-page: 676 -->
+
+```c
+// Looping over the entries
+size_t sz = SMAStructArraySize(1);
+printf("Array 1: \n");
+for (size_t i=0; i < sz; i++) {
+    PrintStruct(i, &array[i]);
+}
+// Deletion
+SMAStructArrayDelete(1);
+SMAStructArrayDelete(2); 
+```
+
+# Variables to be provided to the utility routine
+
+# ID
+
+ID of the array (an integer), chosen by the user at the time of creation. Using this ID, an array can be opened in any other user subroutine.
+
+# NUM\_ITEMS
+
+Size of the array as the number of items. The maximum size is INT\_MAX (2,147,483,647).
+
+# ITEM\_SIZE
+
+Size of one item (struct) in bytes.
+
+# INITVAL
+
+Initial value for each item (struct) in the array. If this value is not supplied, the memory is simply zeroed out.
+
+# Variable returned from the utility routine
+
+# INTEGER\*8 (address)
+
+Returns a pointer to the array created. This pointer can be associated with a native Fortran array or a native C/C++ array. These arrays are global. All threads will see and access exactly the same global array with a given ID.
+
+<!-- source-page: 677 -->
+
+# Appendix A: Index
+
+• “User subroutines index,” Section A.1   
+• “User subroutine functions listing,” Section A.2
+
+<!-- source-page: 678 -->
+
+<!-- source-page: 679 -->
+
+# A.1 User subroutines index
+
+The following tables categorize each user subroutine according to its primary function. The topics are listed alphabetically.
+
+Table A–1 Abaqus/Standard user subroutines. 
+
+<table><tr><td>Function</td><td>Related user subroutines</td></tr><tr><td>Amplitudes, User-defined</td><td>UAMP</td></tr><tr><td>Boundary Conditions</td><td>DISP, UDEMPOTENTIAL</td></tr><tr><td>Constraints</td><td>MPC</td></tr><tr><td>Contact Behavior</td><td>FRIC, FRIC_COEF, GAPCON, GAPELECTR, UINTER</td></tr><tr><td>Contact Surfaces</td><td>RSURFU</td></tr><tr><td>Element Output</td><td>UVARM</td></tr><tr><td>Elements, User-defined</td><td>UEL, UELMAT</td></tr><tr><td>Fields, Predefined</td><td>UFIELD, UMASFL, UPRESS, USDFLD, UTEMP</td></tr><tr><td>Fluid Pipe Section Behavior</td><td>UFLUIDCONNECTORLOSS, UFLUIDCONNECTORVALVE, UFLUIDPIPEFRICTION</td></tr><tr><td>Initial Conditions</td><td>HARDINI, SDVINI, SIGINI, UPOREP, VOIDRI</td></tr><tr><td>Interfacing with External Resources</td><td>UETERNALDB, URDFIL</td></tr><tr><td>Loads, Distributed</td><td>DLOAD, UTRACLOAD</td></tr><tr><td>Loads, Thermal</td><td>FILM, HETVAL</td></tr><tr><td>Loads, Electromagnetic</td><td>UDECURRENT, UDSECURRENT</td></tr><tr><td>Material Properties</td><td>CREEP, UANISOHYPER_INV, UANISOHYPER_STRAIN, UCREEPNETWORK, UDMGINI, UEXPAN, UFLUID, UFLUIDLEAKOFF, UHARD, UHYPEL, UHYPER, UMULLINS, UTRS, UTRSNETWORK, UXFEMNONLOCALWEIGHT</td></tr><tr><td>Materials, User-defined</td><td>UMAT, UMATHT</td></tr><tr><td>Motion, Prescribed</td><td>UMESHMOTION, UMOTION</td></tr><tr><td>Orientation</td><td>ORIENT</td></tr><tr><td>Pore Fluid Flow</td><td>DFLOW, DFLUX, FLOW</td></tr><tr><td>Random Response</td><td>UCORR, UPSD</td></tr></table>
+
+<!-- source-page: 680 -->
+
+<table><tr><td>Function</td><td>Related user subroutines</td></tr><tr><td>Shell Section Behavior</td><td>UGENS</td></tr><tr><td>Wave Kinematics</td><td>UWAVE</td></tr></table>
+
+Table A–2 Abaqus/Explicit user subroutines. 
+
+<table><tr><td>Function</td><td>Related user subroutines</td></tr><tr><td>Amplitudes, User-defined</td><td>VUAMP</td></tr><tr><td>Boundary Conditions</td><td>VDISP</td></tr><tr><td>Contact Behavior</td><td>VFRIC, VFRIC_COEF, VFRICTION, VUINTER, VUINTERACTION</td></tr><tr><td>Elements, User-defined</td><td>VUEL</td></tr><tr><td>Fields, Predefined</td><td>VUFIELD, VUSDFLD</td></tr><tr><td>Fluid Exchange, User-defined</td><td>VUFLUIDEXCH, VUFLUIDEXCHEFFAREA</td></tr><tr><td>Interfacing with External Resources</td><td>VEXTERNALDB</td></tr><tr><td>Loads, Distributed</td><td>VDLOAD</td></tr><tr><td>Loads, Thermal</td><td>VDFLUX</td></tr><tr><td>Material Properties</td><td>VFABRIC, VUANISOHYPER_INV, VUANISOHYPER_STRAIN, VUCHARLENGTH, VUCREEPNETWORK, VUEOS, VUHARD, VUMULLINS, VUTRS, VUVISCOSITY</td></tr><tr><td>Materials, User-defined</td><td>VUMAT</td></tr><tr><td>Wave Kinematics</td><td>VWAVE</td></tr></table>
+
+Table A–3 Abaqus/CFD user subroutines. 
+
+<table><tr><td>Function</td><td>Related user subroutines</td></tr><tr><td>Boundary Conditions</td><td>SMACfdUserPressureBC, SMACfdUserVelocityBC</td></tr></table>
diff --git a/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_069.md b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_069.md
new file mode 100644
index 00000000..6b2e2f3b
--- /dev/null
+++ b/.raw/AbaqusUserSubroutineManual/AbaqusUserSubroutineManual_069.md
@@ -0,0 +1,47 @@
+<!-- source-page: 681 -->
+
+# A.2 User subroutine functions listing
+
+The following tables describe the function of each available user subroutine.
+
+Abaqus/Standard User Subroutines 
+
+<table><tr><td>Name</td><td>Function</td></tr><tr><td>CREEP</td><td>User subroutine to define time-dependent, viscoplastic behavior (creep and swelling).</td></tr><tr><td>DFLOW</td><td>User subroutine to define nonuniform pore fluid velocity in a consolidation analysis.</td></tr><tr><td>DFLUX</td><td>User subroutine to define nonuniform distributed flux in a heat transfer or mass diffusion analysis.</td></tr><tr><td>DISP</td><td>User subroutine to specify prescribed boundary conditions.</td></tr><tr><td>DLOAD</td><td>User subroutine to specify nonuniform distributed loads.</td></tr><tr><td>FILM</td><td>User subroutine to define nonuniform film coefficient and associated sink temperatures for heat transfer analysis.</td></tr><tr><td>FLOW</td><td>User subroutine to define nonuniform seepage coefficient and associated sink pore pressure for consolidation analysis.</td></tr><tr><td>FRIC</td><td>User subroutine to define frictional behavior for contact surfaces.</td></tr><tr><td>FRIC_COEF</td><td>User subroutine to define the frictional coefficient for contact surfaces.</td></tr><tr><td>GAPCON</td><td>User subroutine to define conductance between contact surfaces or nodes in a fully coupled temperature-displacement analysis, coupled thermal-electrical-structural analysis, or pure heat transfer analysis.</td></tr><tr><td>GAPELECTR</td><td>User subroutine to define electrical conductance between surfaces in a coupled thermal-electrical or a coupled thermal-electrical-structural analysis.</td></tr><tr><td>HARDINI</td><td>User subroutine to define initial equivalent plastic strain and initial backstress tensor.</td></tr><tr><td>HETVAL</td><td>User subroutine to provide internal heat generation in heat transfer analysis.</td></tr><tr><td>MPC</td><td>User subroutine to define multi-point constraints.</td></tr><tr><td>ORIENT</td><td>User subroutine to provide an orientation for defining local material directions or local directions for kinematic coupling constraints or local rigid body directions for inertia relief.</td></tr><tr><td>RSURFU</td><td>User subroutine to define a rigid surface.</td></tr><tr><td>SDVINI</td><td>User subroutine to define initial solution-dependent state variable fields.</td></tr><tr><td>SIGINI</td><td>User subroutine to define an initial stress field.</td></tr><tr><td>UAMP</td><td>User subroutine to specify amplitudes.</td></tr><tr><td>UANISOHYPER_INV</td><td>User subroutine to define anisotropic hyperelastic material behavior using the invariant formulation.</td></tr></table>
+
+<!-- source-page: 682 -->
+
+<table><tr><td>UANISOHYPER_STRAIN</td><td>User subroutine to define anisotropic hyperelastic material behavior based on Green strain.</td></tr><tr><td>UCORR</td><td>User subroutine to define cross-correlation properties for random response loading.</td></tr><tr><td>UCREEPNETWORK</td><td>User subroutine to define time-dependent behavior (creep) for models defined within the parallel rheological framework.</td></tr><tr><td>UDECURRENT</td><td>User subroutine to define nonuniform volume current density in an eddy current or magnetostatic analysis.</td></tr><tr><td>UDEMPOTENTIAL</td><td>User subroutine to define nonuniform magnetic vector potential on a surface in an eddy current or magnetostatic analysis.</td></tr><tr><td>UDMGINI</td><td>User subroutine to define the damage initiation criterion.</td></tr><tr><td>UDSECURRENT</td><td>User subroutine to define nonuniform surface current density in an eddy current or magnetostatic analysis.</td></tr><tr><td>UEL</td><td>User subroutine to define an element.</td></tr><tr><td>UELMAT</td><td>User subroutine to define an element with access to Abaqus materials.</td></tr><tr><td>UEXPAN</td><td>User subroutine to define incremental thermal strains.</td></tr><tr><td>UEXTERNALDB</td><td>User subroutine to manage user-defined external databases and calculate model-independent history information.</td></tr><tr><td>UFIELD</td><td>User subroutine to specify predefined field variables.</td></tr><tr><td>UFLUID</td><td>User subroutine to define fluid density and fluid compliance for hydrostatic fluid elements.</td></tr><tr><td>UFLUIDCONNECTORLOSS</td><td>User subroutine to define the loss coefficient for fluid flow in fluid pipe connector elements.</td></tr><tr><td>UFLUIDCONNECTORVALVE</td><td>User subroutine to define the valve opening to control flow in fluid pipe connector elements.</td></tr><tr><td>UFLUIDLEAKOFF</td><td>User subroutine to define the fluid leak-off coefficients for pore pressure cohesive elements.</td></tr><tr><td>UFLUIDPIPEFRICTION</td><td>User subroutine to define the frictional coefficient for fluid flow in fluid pipe elements.</td></tr><tr><td>UGENS</td><td>User subroutine to define the mechanical behavior of a shell section.</td></tr><tr><td>UHARD</td><td>User subroutine to define the yield surface size and hardening parameters for isotropic plasticity or combined hardening models.</td></tr><tr><td>UHYPEL</td><td>User subroutine to define a hypoelastic stress-strain relation.</td></tr><tr><td>UHYPER</td><td>User subroutine to define a hyperelastic material.</td></tr><tr><td>UINTER</td><td>User subroutine to define surface interaction behavior for contact surfaces.</td></tr><tr><td>UMASFL</td><td>User subroutine to specify prescribed mass flow rate conditions for a convection/diffusion heat transfer analysis.</td></tr><tr><td>UMAT</td><td>User subroutine to define a material&#x27;s mechanical behavior.</td></tr><tr><td>UMATHT</td><td>User subroutine to define a material&#x27;s thermal behavior.</td></tr></table>
+
+<!-- source-page: 683 -->
+
+<table><tr><td>UMESHMOTION</td><td>User subroutine to specify mesh motion constraints during adaptive meshing.</td></tr><tr><td>UMOTION</td><td>User subroutine to specify motions during cavity radiation heat transfer analysis or steady-state transport analysis.</td></tr><tr><td>UMULLINS</td><td>User subroutine to define damage variable for the Mullins effect material model.</td></tr><tr><td>UPOREP</td><td>User subroutine to define initial fluid pore pressure.</td></tr><tr><td>UPRESS</td><td>User subroutine to specify prescribed equivalent pressure stress conditions.</td></tr><tr><td>UPSD</td><td>User subroutine to define the frequency dependence for random response loading.</td></tr><tr><td>URDFIL</td><td>User subroutine to read the results file.</td></tr><tr><td>USDFLD</td><td>User subroutine to redefine field variables at a material point.</td></tr><tr><td>UTEMP</td><td>User subroutine to specify prescribed temperatures.</td></tr><tr><td>UTRACLOAD</td><td>User subroutine to specify nonuniform traction loads.</td></tr><tr><td>UTRS</td><td>User subroutine to define a reduced time shift function for a viscoelastic material.</td></tr><tr><td>UTRSNETWORK</td><td>User subroutine to define a reduced time shift function for models defined within the parallel rheological framework.</td></tr><tr><td>UVARM</td><td>User subroutine to generate element output.</td></tr><tr><td>UWAVE</td><td>User subroutine to define wave kinematics for an Abaqus/Aqua analysis.</td></tr><tr><td>UXFEMNONLOCALWEIGHT</td><td>User subroutine to define the weight function used to compute the average stress/strain to determine the crack propagation direction.</td></tr><tr><td>VOIDRI</td><td>User subroutine to define initial void ratios.</td></tr></table>
+
+Abaqus/Explicit User Subroutines 
+
+<table><tr><td>Name</td><td>Function</td></tr><tr><td>VDFLUX</td><td>User subroutine to specify nonuniform distributed fluxes in an explicit dynamic coupled temperature-displacement analysis.</td></tr><tr><td>VDISP</td><td>User subroutine to specify prescribed boundary conditions.</td></tr><tr><td>VDLOAD</td><td>User subroutine to specify nonuniform distributed loads.</td></tr><tr><td>VEXTERNALDB</td><td>User subroutine that gives control to the user at key moments of the analysis so that data can be exchanged dynamically among Abaqus user subroutines and with external programs or files.</td></tr><tr><td>VFABRIC</td><td>User subroutine to define fabric material behavior.</td></tr><tr><td>VFRIC</td><td>User subroutine to define frictional behavior for contact surfaces.</td></tr><tr><td>VFRIC_COEF</td><td>User subroutine to define the frictional coefficient for contact surfaces.</td></tr><tr><td>VFRICTION</td><td>User subroutine to define frictional behavior for contact surfaces.</td></tr><tr><td>VUAMP</td><td>User subroutine to specify amplitudes.</td></tr></table>
+
+<!-- source-page: 684 -->
+
+<table><tr><td>VUANISOHYPER_INV</td><td>User subroutine to define anisotropic hyperelastic material behavior using the invariant formulation.</td></tr><tr><td>VUANISOHYPER_STRAIN</td><td>User subroutine to define anisotropic hyperelastic material behavior based on Green strain.</td></tr><tr><td>VUCHARLENGTH</td><td>User subroutine to define characteristic element length at a material point.</td></tr><tr><td>VUCREEPNETWORK</td><td>User subroutine to define time-dependent behavior (creep) for models defined within the parallel rheological framework.</td></tr><tr><td>VUEL</td><td>User subroutine to define an element.</td></tr><tr><td>VUEOS</td><td>User subroutine to define equation of state material model.</td></tr><tr><td>VUFIELD</td><td>User subroutine to specify predefined field variables.</td></tr><tr><td>VUFLUIDEXCH</td><td>User subroutine to define the mass flow rate/heat energy flow rate for fluid exchange.</td></tr><tr><td>VUFLUIDEXCHEFFAREA</td><td>User subroutine to define the effective area for fluid exchange.</td></tr><tr><td>VUHARD</td><td>User subroutine to define the yield surface size and hardening parameters for isotropic plasticity or combined hardening models.</td></tr><tr><td>VUINTER</td><td>User subroutine to define the interaction between contact surfaces.</td></tr><tr><td>VUINTERACTION</td><td>User subroutine to define the contact interaction between surfaces with the general contact algorithm.</td></tr><tr><td>VUMAT</td><td>User subroutine to define material behavior.</td></tr><tr><td>VUMULLINS</td><td>User subroutine to define damage variable for the Mullins effect material model.</td></tr><tr><td>VUSDFLD</td><td>User subroutine to redefine field variables at a material point.</td></tr><tr><td>VUTRS</td><td>User subroutine to define a reduced time shift function for a viscoelastic material.</td></tr><tr><td>VUVISCOSITY</td><td>User subroutine to define the shear viscosity for equation of state models.</td></tr><tr><td>VWAVE</td><td>User subroutine to define wave kinematics for an Abaqus/Aqua analysis.</td></tr></table>
+
+Abaqus/CFD User Subroutines
+
+<table><tr><td>Name</td><td>Function</td></tr><tr><td>SMACfdUserPressureBC</td><td>User subroutine to specify prescribed pressure boundary conditions.</td></tr><tr><td>SMACfdUserVelocityBC</td><td>User subroutine to specify prescribed velocity boundary conditions.</td></tr></table>
+
+<!-- source-page: 685 -->
+
+# About SIMULIA
+
+Dassault Systèmes SIMULIA applications, including Abaqus, Isight, Tosca, and Simulation Lifecycle Management, enable users to leverage physics-based simulation and high-performance computing to explore real-world behavior of products, nature, and life. As an integral part of Dassault Systèmes’ 3DEXPERIENCE platform, SIMULIA applications accelerate the process of making highly informed, mission-critical design and engineering decisions before committing to costly and time-consuming physical prototypes. www.3ds.com/simulia
+
+# Our 3DEXPERIENCE Platform powers our brand applications, serving 12 industries, and provides a rich portfolio of industry solution experiences.
+
+Dassault Systèmes, the 3DEXPERIENCE Company, provides business and people with virtual universes to imagine sustainable innovations. Its world-leading solutions transform the way products are designed, produced, and supported. Dassault Systèmes’ collaborative solutions foster social innovation, expanding possibilities for the virtual world to improve the real world. The group brings value to over 170,000 customers of all sizes in all industries in more than 140 countries. For more information, visit www.3ds.com.
+
+![](images/page-685_f1615bc434e0713fdb26a727f7bc9fde03a0ea90803fde773501f27db0401bba.jpg)
+
+<details>
+<summary>flowchart</summary>
+
+Circular diagram illustrating the integration of 3D modeling apps, information intelligence apps, and simulation apps, with associated platform icons and labels.
+</details>
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+<!-- source-page: 1 -->
+
+# PREFACE
+
+The finite element method is firmly established as a powerful and popular analysis tool. It is applied to many different problems of continua but is most widely used for structural mechanics. Accordingly, structural mechanics is emphasized in this book, with lesser excursions into other areas such as heat conduction.
+
+The finite element literature is very large. In a book this size it would scarcely be possible even to list all publications, let alone discuss all useful procedures. This text is introductory and is oriented more toward the eventual practitioner than toward the theoretician. The book contains enough material for a two-semester course.
+
+We assume that the reader has the following background. Undergraduate courses in calculus, statics, dynamics, and mechanics of materials must be mastered. Matrix operations (summarized in Appendix A) must be understood. More advanced studies—theory of elasticity, energy methods, numerical analysis, and so on—are not essential. Occasionally these studies must be called upon, but only for their elementary concepts.
+
+The specific elements discussed are often quite good, but we do not claim that they are the best available. Rather, these elements illustrate useful concepts and procedures. Similarly, blocks of Fortran code in the book illustrate the steps of an element formulation, of an algorithm for equation solving, or of finite element bookkeeping, but they may not be the most efficient coding available. These blocks of code can form the basis of various semester projects if so desired. However, the principal purpose of most of these blocks of code is to state precisely the content of certain procedures, and they thereby serve as aids to understanding.
+
+Software entitled FEMCOD is intended for use with the book. FEMCOD is a “framework” program for time-independent finite element analysis: it provides the machinery for input of data, assembly of elements, assignment of loads and boundary conditions, and solution of equations. The user may supply coding for a particular element and for postprocessing (such as stress calculation). To institutions that adopt this textbook, FEMCOD, with instructions for use and examples, is available on diskette from the publisher (John Wiley & Sons, Inc., 605 Third Avenue, New York, N.Y., 10158).
+
+Our presentation of structural dynamics is based partially on the finite element course notes of Ted Belytschko. We gratefully acknowledge his advice and assistance. The inspiration for the discussion of optimal lumping came originally from Isaac Fried. We are also grateful to T. J. R. Hughes, W. K. Liu, and V. Snyder for their insights. Not the least of our thanks is to Beth Brown, who typed and retyped with her usual intelligence and dependability, despite substantial other commitments, and without ever suggesting that the task might be tiresome.
+
+Madison, Wisconsin
+
+October 1988
+
+R. D. Cook
+
+D. S. MALKUS
+
+M. E. PLESHA
+
+<!-- source-page: 2 -->
+
+1. 2017年1月1日，公司与上海浦东发展银行股份有限公司签订了《最高额保证合同》，约定在2017年1月1日（含）内履行了保证责任。
+
+<!-- source-page: 3 -->
+
+CONCEPTS
+
+AND
+
+APPLICATIONS
+
+OF
+
+FINITE ELEMENT ANALYSIS
+
+DSY
+
+<!-- source-page: 4 -->
+
+$$
+\begin{array}{r}\cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ 1 \end{array}
+$$
+
+<!-- source-page: 5 -->
+
+# CONCEPTS
+
+# AND
+
+# APPLICATIONS
+
+# OF
+
+# FINITE ELEMENT ANALYSIS
+
+THIRD EDITION
+
+ROBERT D. COOK
+
+DAVID S. MALKUS
+
+HAEL E. PLESHA
+
+and :
+
+Wisconsin--Madison, His res
+
+el-rem method to probl
+
+flow covercity of Wisconsin—Madison
+
+<!-- source-page: 6 -->
+
+![](images/page-006_a45620d97abb66febdab30a3152ac19692725a48a076e2bb885881164705379c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+THIRD TO
+</details>
+
+HOOH
+DIVAO
+BAF
+M
+M
+
+247076
+
+TA646 C66 1989 METU LIBRAJ
+
+Copyright © 1974, 1981, 1989, by John Wiley & Sons, Inc.
+
+All rights reserved. Published simultaneously in Canada.
+
+![](images/page-006_db5464dda7dcbf094a0c78b63809388c0a34976ebd052fbc61da41aa5228eb01.jpg)
+
+Reproduction or translation of any part of this work beyond that permitted by Sections 107 and 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons.
+
+Library of Congress Cataloging in Publication Data:
+
+Cook, Robert Davis.
+
+Concepts and applications of finite element analysis.
+
+Bibliography: p.
+
+Includes index.
+
+1. Structural analysis (Engineering) 2. Finite element method. I. Malkus, David S. II. Plesha, Michael E. III. Title.
+
+TA646.C66 1989 624.1'71 88-27929
+
+<!-- source-page: 7 -->
+
+# About the Authors
+
+Robert D. Cook received his Ph.D. degree from the University of Illinois in 1963. He then went to the University of Wisconsin—Madison, where he is Professor of Engineering Mechanics. His research interests include stress analysis and finite element methods. He is a member of the American Society of Mechanical Engineers. With Warren C. Young, he is coauthor of Advanced Mechanics of Materials (Macmillan, 1985).
+
+The first edition of Concepts and Applications of Finite Element Analysis was published in 1974 and the second in 1981, both with Dr. Cook as sole author.
+
+David S. Malkus received his Ph.D. from Boston University in 1976. He spent two years at the National Bureau of Standards and seven years in the Mathematics Department of Illinois Institute of Technology. He is now Professor of Engineering Mechanics and a professor in the Center for Mathematical Sciences at the University of Wisconsin—Madison. His research interests concern the application of the finite element method to problems of structural and continuum mechanics, in particular the flow of non-Newtonian fluids. He is a member of the Rheology Research Center (University of Wisconsin—Madison), the American Academy of Mechanics, the Society for Industrial and Applied Mathematics, and the Society of Rheology.
+
+Michael E. Plesha received his B.S. degree from the University of Illinois at Chicago, and his M.S. and Ph.D. degrees from Northwestern University, the Ph.D. degree in 1983. After a short stay at Michigan Technological University, he joined the Engineering Mechanics Department at the University of Wisconsin—Madison, where he is an associate professor. His research interests include constitutive modeling and finite element analysis of contact-friction problems, transient finite element analysis, and geomechanics.
+
+<!-- source-page: 8 -->
+
+<!-- source-page: 9 -->
+
+# PREFACE
+
+The finite element method is firmly established as a powerful and popular analysis tool. It is applied to many different problems of continua but is most widely used for structural mechanics. Accordingly, structural mechanics is emphasized in this book, with lesser excursions into other areas such as heat conduction.
+
+The finite element literature is very large. In a book this size it would scarcely be possible even to list all publications, let alone discuss all useful procedures. This text is introductory and is oriented more toward the eventual practitioner than toward the theoretician. The book contains enough material for a two-semester course.
+
+We assume that the reader has the following background. Undergraduate courses in calculus, statics, dynamics, and mechanics of materials must be mastered. Matrix operations (summarized in Appendix A) must be understood. More advanced studies—theory of elasticity, energy methods, numerical analysis, and so on—are not essential. Occasionally these studies must be called upon, but only for their elementary concepts.
+
+The specific elements discussed are often quite good, but we do not claim that they are the best available. Rather, these elements illustrate useful concepts and procedures. Similarly, blocks of Fortran code in the book illustrate the steps of an element formulation, of an algorithm for equation solving, or of finite element bookkeeping, but they may not be the most efficient coding available. These blocks of code can form the basis of various semester projects if so desired. However, the principal purpose of most of these blocks of code is to state precisely the content of certain procedures, and they thereby serve as aids to understanding.
+
+Software entitled FEMCOD is intended for use with the book. FEMCOD is a “framework” program for time-independent finite element analysis: it provides the machinery for input of data, assembly of elements, assignment of loads and boundary conditions, and solution of equations. The user may supply coding for a particular element and for postprocessing (such as stress calculation). To institutions that adopt this textbook, FEMCOD, with instructions for use and examples, is available on diskette from the publisher (John Wiley & Sons, Inc., 605 Third Avenue, New York, N.Y., 10158).
+
+Our presentation of structural dynamics is based partially on the finite element course notes of Ted Belytschko. We gratefully acknowledge his advice and assistance. The inspiration for the discussion of optimal lumping came originally from Isaac Fried. We are also grateful to T. J. R. Hughes, W. K. Liu, and V. Snyder for their insights. Not the least of our thanks is to Beth Brown, who typed and retyped with her usual intelligence and dependability, despite substantial other commitments, and without ever suggesting that the task might be tiresome.
+
+Madison, Wisconsin
+
+October 1988
+
+R. D. Cook
+
+D. S. MALKUS
+
+M. E. PLESHA
+
+<!-- source-page: 10 -->
+
+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.
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_002.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_002.md
new file mode 100644
index 00000000..86b57e1e
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_002.md
@@ -0,0 +1,382 @@
+<!-- source-page: 11 -->
+
+# CONTENTS
+
+# NOTATION
+
+# Chapter 1 INTRODUCTION 1
+
+1.1 The Finite Element Method 1   
+1.2 The Element Characteristic Matrix 7   
+1.3 Element Assembly and Solution for Unknowns 11   
+1.4 Summary of Finite Element History 14   
+1.5 Strain-Displacement Relations 15   
+1.6 Theory of Stress and Deformation 17   
+1.7 Stress-Strain-Temperature Relations 20   
+1.8 Warning: The Computed Answer May Be Wrong 24
+Problems 25
+
+# Chapter 2 THE STIFFNESS METHOD AND THE PLANE TRUSS 31
+
+2.1 Introduction 31   
+2.2 Structure Stiffness Equations 32   
+2.3 Properties of [K]. Solution for Unknowns 34   
+2.4 Element Stiffness Equations 36   
+2.5 Assembly of Elements. Plane Truss Example 38   
+2.6 Assembly Regarded as Satisfying Equilibrium 40   
+2.7 Assembly as Dictated by Node Numbers 41   
+2.8 Node Numbering That Exploits Matrix Sparsity 44   
+2.9 Automatic Assignment of Node Numbers 47   
+2.10 Displacement Boundary Conditions 48   
+2.11 Gauss Elimination Solution of Equations 53   
+2.12 Stress Computation. Support Reactions 55   
+2.13 Summary of Procedure 57
+Problems 59
+
+# Chapter 3 STATIONARY PRINCIPLES, THE RAYLEIGH-RITZ METHOD, AND INTERPOLATION 69
+
+3.1 Introduction 69   
+3.2 Principle of Stationary Potential Energy 70   
+3.3 Problems Having Many D.O.F. 73   
+3.4 Potential Energy of an Elastic Body 75
+
+<!-- source-page: 12 -->
+
+3.5 The Rayleigh-Ritz Method 78   
+3.6 Comments on the Rayleigh-Ritz Method Based on Assumed Displacement Fields 81   
+3.7 Stationary Principles and Governing Equations 83   
+3.8 A Piecewise Polynomial Field 88   
+3.9 - Finite Element Form of the Rayleigh-Ritz Method 90   
+3.10 Finite Element Formulations Derived from a Functional 93   
+3.11 Interpolation 95   
+3.12 Shape Functions for $C^0$ Elements 96   
+3.13 Shape Functions for $C^1$ Elements 99 Problems 101
+
+# Chapter 4 DISPLACEMENT-BASED ELEMENTS FOR STRUCTURAL MECHANICS
+
+109
+
+4.1 Formulas for Element Matrices [k] and $\{\mathbf{r}_e\}$ 109   
+4.2 Overview of Element Stiffness Matrices 113   
+4.3 Consistent Element Nodal Loads $\{\mathbf{r}_e\}$ 118   
+4.4 Equilibrium and Compatibility in the Solution 124   
+4.5 Convergence Requirements 126   
+4.6 The Patch Test 129   
+4.7 Stress Calculation 132   
+4.8 Other Formulation Methods 136 Problems 137
+
+# Chapter 5 STRAIGHT-SIDED TRIANGLES AND TETRAHEDRA
+
+147
+
+5.1 Natural Coordinates (Linear) 147   
+5.2 Natural Coordinates (Area and Volume) 149   
+5.3 Interpolation Fields for Plane Triangles 153   
+5.4 The Linear Triangle 154   
+5.5 The Quadratic Triangle 157   
+5.6 The Quadratic Tetrahedron 159 Problems 159
+
+# Chapter 6 THE ISOPARAMETRIC FORMULATION
+
+163
+
+6.1 Introduction 163   
+6.2 An Isoparametric Bar Element 164   
+6.3 Plane Bilinear Isoparametric Element 166   
+6.4 Summary of Gauss Quadrature 170   
+6.5 Computer Subroutines for the Bilinear Isoparametric Element 173   
+6.6 Quadratic Plane Elements 176
+
+<!-- source-page: 13 -->
+
+# CONTENTS
+
+xi
+
+6.7 Hexahedral (Solid) Isoparametric Elements 180
+
+6.8 Triangular Isoparametric Elements 182
+
+6.9 Consistent Element Nodal Loads $\{\mathbf{r}_e\}$ 185
+
+6.10 The Validity of Isoparametric Elements 186
+
+6.11 Appropriate Order of Quadrature 188
+
+6.12 Element and Mesh Instabilities 190
+
+6.13 Remarks on Stress Computation 194
+
+6.14 Examples. Effect of Element Geometry 196
+
+Problems 199
+
+# Chapter 7 COORDINATE TRANSFORMATION 209
+
+7.1 Introduction 209
+
+7.2 Transformation of Vectors 209
+
+7.3 Transformation of Stress, Strain, and Material Properties 211
+
+7.4 Transformation of Stiffness Matrices 213
+
+7.5 Examples: Transformation of Stiffness Matrices 214
+
+7.6 Inclined Support 216
+
+7.7 Joining Dissimilar Elements to One Another 218
+
+7.8 Rigid Links. Rigid Elements 220
+
+Problems 222
+
+# Chapter 8 TOPICS IN STRUCTURAL MECHANICS 228
+
+8.1 D.O.F. Within Elements. Condensation 228
+
+8.2 Condensation and Recovery Algorithms 231
+
+8.3 Parasitic Shear. Incompatible Elements 232
+
+8.4 Rotational D.O.F. in Plane Elements 236
+
+8.5 Assumed-Stress Hybrid Formulation 239
+
+8.6 A Plane Hybrid Triangle with Rotational D.O.F. 242
+
+8.7 User-Defined Elements. Elastic Kernel 244
+
+8.8 Higher Derivatives as Nodal D.O.F. 246
+
+8.9 Fracture Mechanics. Singularity Elements 247
+
+8.10 Elastic Foundations 250
+
+8.11 Media of Infinite Extent 252
+
+8.12 Finite Elements and Finite Differences 256
+
+8.13 Reanalysis Methods 256
+
+8.14 Substructuring 257
+
+8.15 Structural Symmetry 260
+
+8.16 Cyclic Symmetry 262
+
+Problems 263
+
+<!-- source-page: 14 -->
+
+# Chapter 9 CONSTRAINTS
+
+9.1 Constraints. Transformations 272   
+9.2 Lagrange Multipliers 275   
+9.3 Penalty Functions 276   
+9.4 Naturally Arising Penalty Formulations. Numerical Integration and Constraints 278   
+9.5 Constraint Counting 283   
+9.6 Additional Techniques for Incompressible Media 285
+Problems 288
+
+# Chapter 10 SOLIDS OF REVOLUTION
+
+10.1 Introduction 293
+
+10.2 Elasticity Relations for Axial Symmetry 294
+
+10.3 Finite Elements for Axial Symmetry 295
+
+10.4 Fourier Series 298
+
+10.5 Loads Without Axial Symmetry: Introduction 301
+
+10.6 Loads Without Axial Symmetry: Element Matrices 304
+
+10.7 Related Problems 307
+Problems 308
+
+# Chapter 11 BENDING OF FLAT PLATES
+
+11.1 Plate-Bending Theory 314   
+11.2 Finite Elements for Plates 319   
+11.3 Mindlin Plate Elements 323   
+11.4 A Triangular Discrete Kirchhoff Element 328   
+11.5 Boundary Conditions and Test Cases 332
+,Problems 335
+
+# Chapter 12 SHELLS
+
+12.1 Shell Geometry and Behavior. Shell Elements 340   
+12.2 Circular Arches and Arch Elements 343   
+12.3 Flat Elements for Shells 351   
+12.4 Shells of Revolution 352   
+12.5 Isoparametric General Shell Elements 358
+
+Problems 362
+
+# Chapter 13 FINITE ELEMENTS IN DYNAMICS AND VIBRATIONS
+
+13.1 Introduction 367   
+13.2 Dynamic Equations. Mass and Damping Matrices 368   
+13.3 Mass Matrices, Consistent and Diagonal 370
+
+<!-- source-page: 15 -->
+
+13.4 Damping 376   
+13.5 Natural Frequencies and Mode Shapes 378   
+13.6 Time-History Analysis. Modal Methods 381   
+13.7 Mass Condensation. Guyan Reduction 387   
+13.8 Component Mode Synthesis 391   
+13.9 Time-History Analysis. Direct Integration Methods 395   
+13.10 Explicit Direct Integration Methods 397   
+13.11 Implicit Direct Integration Methods 405   
+13.12 Other Implicit and Explicit Methods. Mixed Methods 407   
+13.13 Stability Analysis. Accuracy of Direct Integration Methods 410   
+13.14 Concluding Remarks on Time-History Analysis 417
+Problems 418
+
+# Chapter 14 STRESS STIFFENING AND BUCKLING 429
+
+14.1 Introduction 429   
+14.2 Stress Stiffness Matrices for Beams and Bars 432   
+14.3 Stress Stiffness Matrix of a Plate Element 435   
+14.4 A General Formulation for $[k_{\sigma}]$ 437   
+14.5 Bifurcation Buckling 441   
+14.6 Remarks on $[\mathbf{K}_{\sigma}]$ and Its Uses 444   
+14.7 Remarks on Buckling and Buckling Analysis 446
+Problems 448
+
+# Chapter 15 WEIGHTED RESIDUAL METHODS 455
+
+15.1 Introduction 455   
+15.2 Some Weighted Residual Methods 455   
+15.3 Example Solutions 458   
+15.4 Galerkin Finite Element Method 461   
+15.5 Integration by Parts 466   
+15.6 Two-Dimensional Problems 468   
+Problems 470
+
+# Chapter 16 HEAT CONDUCTION AND SELECTED FLUID PROBLEMS 474
+
+16.1 Introduction to Heat Conduction Problems 474   
+16.2 A One-Dimensional Example 475   
+16.3 Heat Conduction in a Plane 477   
+16.4 General Solids and Solids of Revolution 479   
+16.5 Finite Element Formulation 480   
+16.6 Thermal Transients 484   
+16.7 Related Problems. Fluid Flow 486
+
+<!-- source-page: 16 -->
+
+16.8 Fluid Vibration and Waves, Pressure Formulation 488
+
+16.9 Fluid-Structure Interaction 491
+
+Problems 495
+
+# Chapter 17 AN INTRODUCTION TO SOME NONLINEAR PROBLEMS
+
+501
+
+17.1 Introduction 501
+
+17.2 Some Solution Methods 502
+
+17.3 One-Dimensional Elastic-Plastic Analysis 510
+
+17.4 Small-Strain Plasticity Relations 515
+
+17.5 Elastic-Plastic Analysis Procedures 519
+
+17.6 Nonlinear Dynamic Problems 522
+
+17.7 A Problem Having Geometric Nonlinearity 529
+
+17.8 Other Nonlinear Problems 532
+
+Problems 533
+
+# Chapter 18 NUMERICAL ERRORS AND CONVERGENCE 542
+
+18.1 Introduction. Error Classification 542
+
+18.2 Ill-Conditioning 543
+
+18.3 The Condition Number 546
+
+18.4 Diagonal Decay Error Tests 550
+
+18.5 Residuals 552
+
+18.6 Discretization Error: Analysis 553
+
+18.7 Discretization Error: Estimation and Extrapolation 558
+
+18.8 Tests of Element Quality 563
+
+18.9 Concluding Remarks 566
+
+Problems 566
+
+# Chapter 19 MODELING, PROGRAMS, AND PROGRAMMING 573
+
+19.1 Modeling 573
+
+19.2 Programming and Programs 584
+
+# Appendix A MATRICES: SELECTED DEFINITIONS AND MANIPULATIONS
+
+589
+
+# Appendix B SIMULTANEOUS ALGEBRAIC EQUATIONS 592
+
+B.1 Introduction 592
+
+B.2 Solution of Simultaneous Linear Algebraic Equations by Gauss Elimination 593
+
+<!-- source-page: 17 -->
+
+CONTENTS
+
+XV
+
+# Appendix C EIGENVALUES AND EIGENVECTORS 598
+
+C.1 The Eigenproblem 598
+
+C.2 The Standard Eigenproblem 598
+
+C.3 The General Eigenproblem 599
+
+C.4 Remarks on Special Forms 602
+
+C.5 Solution Algorithms 603
+
+# REFERENCES 605
+
+# INDEX 623
+
+<!-- source-page: 18 -->
+
+<!-- source-page: 19 -->
+
+# NOTATION
+
+What follows is a list of principal symbols. Less frequently used symbols, and symbols that have different meanings in different contexts, are defined where they are used. Matrices and vectors are denoted by boldface type.
+
+MATHEMATICAL SYMBOLS 
+
+<table><tr><td>[ ]</td><td>Rectangular or square matrix.</td></tr><tr><td>{ }, [ ], [ ]</td><td>Column, row, and diagonal matrices.</td></tr><tr><td>[ ]T</td><td>Matrix, transpose.</td></tr><tr><td>[ ]-1, [ ]-T</td><td>Matrix inverse and inverse transpose; that is, ([ ]-1)T ≡ ([ ]T)-1.</td></tr><tr><td>|| ||</td><td>Norm of a matrix or a vector.</td></tr><tr><td>.</td><td>Time differentiation; for example, i = du/dt, ii = d2u/dt2.</td></tr><tr><td>,</td><td>Partial differentiation if the following subscript(s) is literal; for example, w,x = ∂w/∂x, w,xy = ∂2w/∂x ∂y.</td></tr><tr><td>{ ∂Π/∂a}</td><td>Represents [ ∂Π/∂a1 ∂Π/∂a2 · · · ∂Π/∂an]T, where Π is a scalar function of a1, a2, . . . , an.</td></tr></table>
+
+LATIN SYMBOLS
+
+<table><tr><td>A</td><td>Area or cross-sectional area.</td></tr><tr><td>[A]</td><td>Relates {d} to {a}; {d} = [A]{a}.</td></tr><tr><td>{a}</td><td>Generalized coordinates.</td></tr><tr><td>B</td><td>Bulk modulus,  $B = E/(3 - 6\nu)$ .</td></tr><tr><td>[B]</td><td>Spatial derivative(s) of the field variable(s) are [B]{d}.</td></tr><tr><td> $C^{m}$ </td><td>Field continuity of degree m (Section 3.11).</td></tr><tr><td>[C]</td><td>Damping matrix. Constraint matrix.</td></tr><tr><td>d.o.f.</td><td>Degree(s) of freedom.</td></tr><tr><td>D</td><td>Displacement. Flexural rigidity of a plate or shell.</td></tr><tr><td>{D}, {d}</td><td>Nodal d.o.f. of structure and element, respectively.</td></tr><tr><td>E</td><td>Modulus of elasticity.</td></tr><tr><td>[E]</td><td>Matrix of elastic stiffnesses (Section 1.7).</td></tr><tr><td>{F}</td><td>Body forces per unit volume.</td></tr><tr><td>G</td><td>Shear modulus.</td></tr><tr><td>I</td><td>Moment of inertia of cross-sectional area.</td></tr><tr><td>[I]</td><td>Unit matrix (also called identity matrix).</td></tr><tr><td>J</td><td>Determinant of [J] (called the Jacobian).</td></tr><tr><td>[J]</td><td>The Jacobian matrix.</td></tr><tr><td>k</td><td>Spring stiffness. Thermal conductivity.</td></tr><tr><td>[K], [k]</td><td>Structure and element conventional stiffness matrices.</td></tr><tr><td> $[K_{\sigma}]$ ,  $[k_{\sigma}]$ </td><td>Structure and element stress stiffness matrices.</td></tr><tr><td>L,  $L_{T}$ </td><td>Length of element, length of structure.</td></tr><tr><td> $\ell$ , m, n</td><td>Direction cosines.</td></tr><tr><td> $n_{\text{eq}}$ </td><td>Number of equations.</td></tr><tr><td>[M], [m]</td><td>Structure and element mass matrices.</td></tr></table>
+
+<!-- source-page: 20 -->
+
+<table><tr><td>[N], [N]</td><td>Shape (or basis, or interpolation) functions.</td></tr><tr><td>O</td><td>Order; for example,  $O(h^{2}) =$  a term of order  $h^{2}$ .</td></tr><tr><td>[0], {0}</td><td>Null matrix, null vector.</td></tr><tr><td>{P}</td><td>Externally applied concentrated loads on structure nodes.</td></tr><tr><td>q</td><td>Distributed load (surface or line).</td></tr><tr><td>{R}</td><td>Total load on structure nodes; {R} = {P} +  $\Sigma$  {re}.</td></tr><tr><td>{re}</td><td>Loads applied to nodes by element, for example, by temperature change or distributed load (Eq. 4.1-6).</td></tr><tr><td>S, Se</td><td>Surface, element surface.</td></tr><tr><td>T</td><td>Temperature.</td></tr><tr><td>t</td><td>Thickness. Time.</td></tr><tr><td>[T]</td><td>Transformation matrix.</td></tr><tr><td>U, U0</td><td>Strain energy, strain energy per unit volume.</td></tr><tr><td>u, v, w</td><td>Displacements, for example, in directions x, y, z.</td></tr><tr><td>{u}</td><td>Vector of displacements; {u} = [u v w]T.</td></tr><tr><td>V, Ve</td><td>Volume, element volume.</td></tr><tr><td>x, y, z</td><td>Cartesian coordinates.</td></tr></table>
+
+GREEK SYMBOLS 
+
+<table><tr><td> $\alpha$ </td><td>Coefficient of thermal expansion, penalty number.</td></tr><tr><td> $[\Gamma]$ </td><td>Jacobian inverse;  $[\Gamma] = [J]^{-1}$ .</td></tr><tr><td> $\{ \epsilon \}, \{ \epsilon_0 \}$ </td><td>Strains, initial strains.</td></tr><tr><td> $[\kappa], \{ \kappa \}$ </td><td>Matrix of thermal conductivities, vector of curvatures.</td></tr><tr><td> $\lambda$ </td><td>Eigenvalue. Lagrange multiplier.</td></tr><tr><td> $\nu$ </td><td>Poisson&#x27;s ratio of an isotropic material.</td></tr><tr><td> $\xi, \eta, \zeta$ </td><td>Isoparametric coordinates.</td></tr><tr><td> $\xi_1, \xi_2, \xi_3$ </td><td>Area coordinates.</td></tr><tr><td> $\Pi$ </td><td>A functional; for example,  $\Pi_p =$  potential energy.</td></tr><tr><td> $\rho$ </td><td>Mass density.</td></tr><tr><td> $\{ \sigma \}, \{ \sigma_0 \}$ </td><td>Stresses, initial stresses.</td></tr><tr><td> $\phi$ </td><td>A dependent variable. Meridian angle of a shell.</td></tr><tr><td> $\{ \Phi \}$ </td><td>Surface tractions.</td></tr><tr><td> $\omega$ </td><td>Circular frequency in radians per second.</td></tr></table>
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_003.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_003.md
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_003.md
@@ -0,0 +1,524 @@
+<!-- source-page: 21 -->
+
+# INTRODUCTION
+
+A brief overview of the finite element method and its concepts is presented. Background information used for finite element applications in structural mechanics is discussed.
+
+# 1.1 THE FINITE ELEMENT METHOD
+
+The finite element method is a numerical procedure for analyzing structures and continua. Usually the problem addressed is too complicated to be solved satisfactorily by classical analytical methods. The problem may concern stress analysis, heat conduction, or any of several other areas. The finite element procedure produces many simultaneous algebraic equations, which are generated and solved on a digital computer. Finite element calculations are performed on personal computers, mainframes, and all sizes in between. Results are rarely exact. However, errors are decreased by processing more equations, and results accurate enough for engineering purposes are obtainable at reasonable cost.
+
+The finite element method originated as a method of stress analysis. Today finite elements are also used to analyze problems of heat transfer, fluid flow, lubrication, electric and magnetic fields, and many others. Problems that previously were utterly intractable are now solved routinely. Finite element procedures are used in the design of buildings, electric motors, heat engines, ships, airframes, and spacecraft. Manufacturing companies and large design offices typically have one or more large finite element programs in-house. Smaller companies usually have access to a large program through a commercial computing center or use a smaller program on a personal computer.
+
+Figure 1.1-1 shows a very simple problem that illustrates discretization, a basic finite element concept. Imagine that the displacement of the right end of the bar is required. The classical approach is to write the differential equation of the continuously tapered bar, solve this equation for axial displacement u as a function of x, and finally substitute $x = L_{T}$ to find the required end displacement. The finite element approach to this problem does not begin with a differential equation. Instead, the bar is discretized by modeling it as a series of finite elements, each uniform but of a different cross-sectional area A (Fig. 1.1-1b). In each element, u varies linearly with x; therefore, for $0 < x < L_{T}$ , u is a piecewise-smooth function of x. The elongation of each element can be determined from the elementary formula PL/AE. The end displacement, at $x = L_{T}$ , is the sum of the element elongations. Accuracy improves as more elements are used.
+
+In the foregoing example, and in general, the finite element method models a structure as an assemblage of small parts (elements). Each element is of simple geometry and therefore is much easier to analyze than the actual structure. In
+
+<!-- source-page: 22 -->
+
+![](images/page-022_1ce2b51979ade83816cb362d6aab0d94ec07f564a6537687da74efe31101b319.jpg)
+
+<details>
+<summary>text_image</summary>
+
+x,u
+L_T
+P
+</details>
+
+(a)
+
+![](images/page-022_c8454a4ff6f60486b72a2eb91b9609dc4f7307670d776c8206a927410b1cd248.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L
+x
+P
+4@L = L_T
+</details>
+
+(b)   
+Figure 1.1-1. (a) A tapered bar under end load P. (b) A model built of four uniform (nontapered) elements of equal length.
+
+essence, we approximate a complicated solution by a model that consists of piecewise-continuous simple solutions. Elements are called “finite” to distinguish them from differential elements used in calculus.
+
+In a heat transfer context, Fig. 1.1-1 might represent a bar with insulated sides, prescribed temperature at the left end, and prescribed heat flow at the right end. One might ask for the temperature in the bar as a function of x and time.
+
+Figure 1.1-2a shows a plane structure. Displacements and stresses caused by pressure p are required. The finite element model, Fig. 1.1-2b, consists of plane areas, some triangular and some quadrilateral (if done properly, there is no difficulty in combining the different element types). Black dots, called nodes or node points, indicate where elements are connected to one another. In this model each node has two degrees of freedom (d.o.f.): that is, each node can displace in both the x direction and the y direction. Thus, if there are n nodes in Fig. 1.1-2b, there are 2n d.o.f. in the model. (In the real structure there are infinitely many d.o.f. because the structure has infinitely many particles.) Algebraic equations that describe the finite element model are solved to determine the d.o.f. Use of only 2n d.o.f. in analysis is similar to use of the first 2n terms of a convergent infinite series. (In heat transfer, each node has only one d.o.f.—namely, the temperature of the node. Thus a finite element model of n nodes has n d.o.f.)
+
+We see that in going from Fig. 1.1-2a to 1.1-2b the distributed pressure p has been converted to concentrated forces at nodes. The analysis procedure gives a prescription for making conversion, as will be shown subsequently.
+
+From Fig. 1.1-2 it may appear that discretization is accomplished simply by
+
+![](images/page-022_b79b51c5e1224dd4d49d75deec05281d6377c9526360fb9e0c6791dccb06679d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y, v
+Pressure p
+Hole
+x, u
+</details>
+
+(a)
+
+![](images/page-022_b27e91a885a5e6e47a4eb43f52c9980841dce8668d5084f143ceb3366fac105a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y, v
+x, u
+</details>
+
+{b}
+
+![](images/page-022_5411f278079c39a6453daa6dddb588964b21b8cb4d70fe3d7a4ba31af4e20c9e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y, v
+p4 → 4
+q4
+q3
+3
+p3
+p1 → 1
+1
+2
+p2
+q2
+x, u
+q1
+</details>
+
+{c}   
+Figure 1.1-2. (a) A plane structure of arbitrary shape. (b) A possible finite element model of the structure. (c) A plane rectangular element showing nodal forces $p_{i}$ and $q_{i}$ . The dashed line shows the deformation mode associated with x-direction displacement of node 3.
+
+<!-- source-page: 23 -->
+
+sawing the continuum into pieces and then pinning the pieces together again at node points. But such a model would not deform like the continuum. Under load, strain concentrations would appear at the nodes, and the elements would tend to overlap or separate along the saw cuts. Clearly, the actual structure does not behave in this way, so the elements must be restricted in their deformation patterns. For example, if elements are allowed to have only such deformation modes as will keep edges straight (Fig. 1.1-2c), then adjacent elements will neither overlap nor separate. In this way we satisfy the basic requirement that deformations of a continuous medium must be compatible.
+
+An important ingredient in a finite element analysis is the behavior of the individual elements. A few good elements may produce better results than many poorer elements. We can see that several element types are possible by considering Fig. 1.1-3. Function $\phi$ , which might represent any of several physical quantities, varies smoothly in the actual structure. A finite element model typically yields a piecewise-smooth representation of $\phi$ . Between elements there may be jumps in the $x$ and $y$ derivatives of $\phi$ . Within each element $\phi$ is a smooth function that is usually represented by a simple polynomial. What shall the polynomial be? For the triangular element, the linear polynomial
+
+$$
+\phi = a _ {1} + a _ {2} x + a _ {3} y \tag {1.1-1}
+$$
+
+is appropriate, where the $a_{i}$ are constants. These constants can be expressed in terms of $\phi_{1}$ , $\phi_{2}$ , and $\phi_{3}$ , which are the values of $\phi$ at the three nodes. Triangles model the actual $\phi$ by a surface of flat triangular facets. For the four-node quadrilateral, the “bilinear” function
+
+$$
+\phi = a _ {1} + a _ {2} x + a _ {3} y + a _ {4} x y \tag {1.1-2}
+$$
+
+is appropriate. The eight-node quadrilateral in Fig. 1.1-3 has eight $a_{i}$ in its polynomial expansion and can represent a parabolic surface.
+
+Equations 1.1-1 and 1.1-2 are interpolations of function $\phi$ in terms of the position $(x,y)$ within an element. That is, when the $a_{i}$ have been determined in terms of nodal values $\phi_{i}$ , Eqs. 1.1-1 and 1.1-2 define $\phi$ within an element in terms of the $\phi_{i}$ and the coordinates. Clearly, if the mesh of elements is not too coarse and if the $\phi_{i}$ happened to be exact, then $\phi$ away from nodes would be a good approx-
+
+![](images/page-023_ca61767818fb6645ebe73e5fc3d1ed8aa999a92af75276d92d882740218ebb02.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Meaning
+Torsion: w
+Fluid flow
+Seepage f
+Magnetos
+Electric fi
+Heat cono
+</details>
+
+Meaning of $\phi$ in various problems:   
+Torsion: warping function or stress function   
+Fluid flow: stream function or velocity potential   
+Seepage flow: hydraulic head   
+Magnetostatic: magnetic potential   
+Electric field: field potential (voltage)   
+Heat conduction: temperature
+
+Figure 1.1-3. A function $\phi = \phi(x, y)$ that varies smoothly over a rectangular region in the xy plane, and typical elements that might be used to approximate it.
+
+<!-- source-page: 24 -->
+
+imation. Nodal values $\phi_i$ are close to exact if the mesh is not too coarse and if element properties are properly formulated.
+
+How can the user decide which element to use? Unfortunately, the answer is not simple. An element that is good in one problem area (such as magnetic fields) may be poor in another (such as stress analysis). Even in a specific problem area an element may behave well or badly, depending on the particular geometry, loading, and boundary conditions. A competent user of finite elements must be familiar with how various elements behave under various conditions.
+
+We may now venture some definitions. The finite element method is a method of piecewise approximation in which the approximating function $\phi$ is formed by connecting simple functions, each defined over a small region (element). A finite element is a region in space in which a function $\phi$ is interpolated from nodal values of $\phi$ on the boundary of the region in such a way that interelement continuity of $\phi$ tends to be maintained in the assemblage.
+
+A finite element analysis typically involves the following steps. Again we will cite stress analysis and heat transfer as typical applications. Steps 1, 4, and 5 require decisions by the analyst and provide input data for the computer program. Steps 2, 3, 6, and 7 are carried out automatically by the computer program.
+
+√1. Divide the structure or continuum into finite elements. Mesh generation programs, called preprocessors, help the user in doing this work.   
+√2. Formulate the properties of each element. In stress analysis, this means determining nodal loads associated with all element deformation states that are allowed. In heat transfer, it means determining nodal heat fluxes associated with all element temperature fields that are allowed.   
+3. Assemble elements to obtain the finite element model of the structure.   
+√4. Apply the known loads: nodal forces and/or moments in stress analysis, nodal heat fluxes in heat transfer.   
+5. In stress analysis, specify how the structure is supported. This step involves setting several nodal displacements to known values (which often are zero). In heat transfer, where typically certain temperatures are known, impose all known values of nodal temperature.   
+6. Solve simultaneous linear-algebraic equations to determine nodal d.o.f. (nodal displacements in stress analysis, nodal temperatures in heat transfer).   
+7. In stress analysis, calculate element strains from the nodal d.o.f. and the element displacement field interpolation, and finally calculate stresses from strains. In heat transfer, calculate element heat fluxes from the nodal temperatures and the element temperature field interpolation. Output interpretation programs, called postprocessors, help the user sort the output and display it in graphical form.
+
+The power of the finite element method resides principally in its versatility. The method can be applied to various physical problems. The body analyzed can have arbitrary shape, loads, and support conditions. The mesh can mix elements of different types, shapes, and physical properties. This great versatility is contained within a single computer program. User-prepared input data controls the selection of problem type, geometry, boundary conditions, element selection, and so on.
+
+Another attractive feature of finite elements is the close physical resemblance
+
+<!-- source-page: 25 -->
+
+between the actual structure and its finite element model. The model is not simply an abstraction. This seems especially true in structural mechanics, and may account for the finite element method having its origins there.
+
+The finite element method also has disadvantages. A specific numerical result is found for a specific problem: a finite element analysis provides no closed-form solution that permits analytical study of the effects of changing various parameters. A computer, a reliable program, and intelligent use are essential. A general-purpose program has extensive documentation, which cannot be ignored. Experience and good engineering judgment are needed in order to define a good model. Many input data are required and voluminous output must be sorted and understood.
+
+Example Applications. Figure 1.1-4 shows a finite element model of an axially symmetric rocket nozzle [10.1].¹ The axis, not shown, is horizontal and lies above the cross section in Fig. 1.1-4. Each element is a toroidal ring of triangular cross section. Each element has a node (actually a nodal circle) at each vertex. Each nodal circle has axial and radial displacements as d.o.f. Stresses caused by temperature gradient and internal pressure are desired.
+
+Figure 1.1-5 shows three ways of modeling an arch dam using “isoparametric” solid elements (discussed in Chapter 6). One might ask for the stresses produced by hydrostatic and gravity loads. Or, the response to earthquake motion might be required, in which fluid-structure interaction is taken into account.
+
+Figures 1.1-6 and 1.1-7 show typical problems in structural mechanics. The structure in Fig. 1.1-6 consists primarily of plate-bending elements. The structure in Fig. 1.1-7 consists of three-dimensional solid elements. The postprocessor has removed hidden lines. The deformation and stress plots display the results of analysis.
+
+Figure 1.1-8 shows a nonstructural problem. Lines of magnetic flux are crowded
+
+![](images/page-025_5287c1b738da646d9af47a9ea4913fd25fc112d0d4457c4fe03a76839cb7914b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+21.37 Diameter
+Graphite
+Insulator
+Glass filament
+Glass fabric
+Steel shell
+Asbestos
+</details>
+
+Figure 1.1-4. Cross section of a multimaterial rocket nozzle, showing construction (left portion) and possible finite element mesh (right portion). This problem was solved in the early days of finite element technology [10.1].
+
+$^{1}$ Numbers within brackets indicate references listed at the back of the book.
+
+<!-- source-page: 26 -->
+
+![](images/page-026_5aa5ea983ff6bc0cc4c52e25c5583b2a1a3da7dc75a4c16d65e5513ebd7a7514.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Pure geometric line drawing of a 3D curved structure with internal grid lines (no text or symbols)
+</details>
+
+{a}
+
+![](images/page-026_544a8b05e0e8046c7f2235c8fef97ac2bd1cc9a4fab25eb649ee9e9716476c2a.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Pure geometric line drawing of a 3D geometric shape with grid lines, no text or symbols present
+</details>
+
+(b)
+
+![](images/page-026_7a0347554e026a78faf1e8f28bd302ef1e226e42ed5b136068ff73ba79a2f391.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Simple line drawing of a curved, segmented shape with internal lines and dots, no text or symbols present.
+</details>
+
+{c}   
+Figure 1.1-5. Half of an arch dam, modeled by $(a,b)$ quadrilateral and triangular “quadratic” elements, and $(c)$ a single “cubic” element [1.1]. Nodes of a typical element are shown by dots.
+
+near the gap between rotor and stator, which means that gradients are large in this region. Areas of large gradient are areas of particular interest. The analyst places more elements there in order to calculate the magnetic field in greater detail. The mesh shown is adequate for the analysis of magnetic flux, but is probably too crude to be used for stress analysis.
+
+Clearly some problems use a great many d.o.f. How many d.o.f. must a problem have to be considered “large”? In 1960, perhaps 1000; in 1980, over 10,000. Improvements in hardware and software have made this increase possible.
+
+Why Study the Theory of Finite Elements? Many satisfactory elements have already been formulated and reside in popular computer programs. The practitioner desires to understand how various elements behave. Clearly, engineers who understand analysis tools will be able to use them to better advantage and will be less likely to misuse them. Such an understanding cannot be achieved if theory is ignored. In this book we intend to aid the eventual practitioner and to present theory to an adequate but not excessive degree. We recognize that for engineers the study of finite elements is more than a theoretical study of mathematical foundations and formulation procedures for various types of finite elements.
+
+Complete computer codes need not be studied in detail, but concepts and assumptions' behind the coding should be mastered. Otherwise, the treatment of loads and boundary conditions may be confusing, the variety of program options and element types may be baffling, and error messages may provide no clue as to the source of difficulty or how to correct it.
+
+![](images/page-026_a594e1a0e682807d28cf8c72afdc0213a8926c3499ca22ab07edd86466fcf913.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+3D wireframe diagram of a twisted rope or cable structure (no text or symbols)
+</details>
+
+Figure 1.1-6. A detailed model of half of an automobile frame, used to find deformations, stresses, natural frequencies, and mode shapes. (Courtesy of A. O. Smith Corp., Data Systems Division, Milwaukee, Wisconsin.)
+
+<!-- source-page: 27 -->
+
+![](images/page-027_903fa76d582a9e9b1c5b5f1de57074c2d8682a5e39c8c373b7f4a267d5f7a837.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+3D wireframe model of a curved surface with grid lines, no text or symbols present
+</details>
+
+BEARING HOUSING MODEL
+
+![](images/page-027_bd4a3aa85d9e7fba11bfb42e4f880614429a3503820ca2ce8de3f3b23260a685.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+3D wireframe model of a curved, folded structure (no text or symbols)
+</details>
+
+DEFORMED MODEL
+
+![](images/page-027_a80d11d707b0f029a5a3dde521eb708f722d1a18247c7df2b1cfd0b472d01013.jpg)
+
+<details>
+<summary>text_image</summary>
+
+100
+300
+100
+300
+500
+700
+900
+</details>
+
+STRESS CONTOUR   
+Figure 1.1-7. Finite element mesh and computed deformations and stresses in a portion of a bearing housing. (Courtesy of Algor Interactive Systems Inc., Pittsburgh, Pennsylvania.)
+
+# 1.2 THE ELEMENT CHARACTERISTIC MATRIX
+
+![](images/page-027_b65af15a8bda765d4095906e01801e725f5e9c1f48d3f31c9221aa5c72fb76f2.jpg)
+
+The element characteristic matrix has different names in different problem areas. In structural mechanics it is called a stiffness matrix: it relates nodal displacements to nodal forces. In heat conduction it is called a conductivity matrix: it relates nodal temperatures to nodal fluxes. There are three important ways to derive an element characteristic matrix.
+
+1. The direct method is based on physical reasoning. It is limited to very simple elements, but is worth studying because it enhances our physical understanding of the finite element method.
+
+![](images/page-027_5142c40a9b9d17abc25b35e6c360ec65ed85bce4b0cba67300b2a158ca802c6b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+stator
+rotor
+</details>
+
+![](images/page-027_aef1db9f858fee8e9bfc1fcc5b8aaa1bf6103719d722c9f070b877c5b83ecb55.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Topographic contour lines diagram of a building structure (no text or labels)
+</details>
+
+Figure 1.1-8. Part of an induction motor. Elements model the solid parts as well as the spaces between them. Symmetry is exploited by modeling only a half-pole. The computed magnetic flux contours for zero rotor speed are shown by the right-hand figure. (Courtesy of A. O. Smith Corp., Data Systems Division, Milwaukee, Wisconsin.)
+
+<!-- source-page: 28 -->
+
+2. The variational method is applicable to problems that can be stated by certain integral expressions such as the expression for potential energy. This method is discussed in Chapters 3 and 4.   
+③. Weighted residual methods are particularly suited to problems for which differential equations are known but no variational statement is available. For stress analysis and some other problem areas, the variational method and the most popular weighted residual method (the Galerkin method) yield identical finite element formulations. Weighted residual methods are discussed in Chapter 15.
+
+In the present section we consider applications of the direct method.
+
+The Elastic Bar: Direct Method. Consider a weightless straight bar of length L, elastic modulus E, and cross-sectional area A. We regard the bar as a finite element and place a node at each end. If only axial loads and axial displacements are allowed, nodal d.o.f. are displacements $u_{i}$ and $u_{j}$ (Fig. 1.2-1). The element stiffness matrix is formulated by determining the nodal forces that must be applied in order to produce nodal displacements $u_{i}$ and $u_{j}$ . Our sign convention is that both force and displacement are positive when directed toward the right. Accordingly, when $u_{i} > 0$ but $u_{j} = 0$ (Fig. 1.2-1a), nodal forces consistent with static equilibrium and a linearly elastic material are
+
+$$
+F _ {i} = \frac {A E}{L} u _ {i} \quad \text { and } \quad F _ {j} = - \frac {A E}{L} u _ {i} \tag {1.2-1}
+$$
+
+Similarly, when $u_{i} = 0$ but $u_{j} > 0$ (Fig. 1.2-1b),
+
+$$
+F _ {i} = - \frac {A E}{L} u _ {j} \quad \text { and } \quad F _ {j} = \frac {A E}{L} u _ {j} \tag {1.2-2}
+$$
+
+If both $u_{i}$ and $u_{j}$ can be simultaneously nonzero, then nodal forces are $F_{i} = (AE/L)(u_{i} - u_{j})$ and $F_{j} = (AE/L)(-u_{i} + u_{j})$ . In matrix format these two equations are
+
+$$
+\left[ \begin{array}{c c} A E / L & - A E / L \\ - A E / L & A E / L \end{array} \right] \left\{ \begin{array}{l} u _ {i} \\ u _ {j} \end{array} \right\} = \left\{ \begin{array}{l} F _ {i} \\ F _ {j} \end{array} \right\} \tag {1.2-3}
+$$
+
+or
+
+$$
+[ \mathbf {k} ] \{\mathbf {d} \} = \{\overline {{\mathbf {r}}} \} \tag {1.2-4}
+$$
+
+where [k] is the element stiffness matrix. By considering first $u_{i}=1$ and $u_{j}=0$ , then $u_{i}=0$ and $u_{j}=1$ , each time computing nodal forces $F_{i}$ and $F_{j}$ by the matrix-
+
+![](images/page-028_4172349dde6c39e098ca1ceea939e18cffe4d956c0aa2ae1536f5657174cb69f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+F_i
+u_i
+F_j (= -F_i)
+i
+L
+j
+(a)
+</details>
+
+![](images/page-028_fbe3792250c8fd26b1f137762f62955461644b11f1b389e8130768075376f14a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+F_i (= -F_j)
+i
+L
+u_j
+j
+F_j
+(b)
+</details>
+
+Figure 1.2-1. A uniform bar, showing nodal forces associated with nodal displacements $u_{i}$ and $u_{j}$ . Displacements $u_{i}$ and $u_{j}$ are greatly exaggerated in the drawing.
+
+<!-- source-page: 29 -->
+
+times-vector multiplication indicated in Eq. 1.2-3, we reach the following conclusion: a column of [k] lists nodal loads that must be applied to nodal d.o.f. in order to create the deformation state associated with unit value of the corresponding element d.o.f. while all other element d.o.f. are zero. Later in this section we will use this procedure in calculating [k] for a beam element, Fig. 1.2-2.
+
+Remarks. No approximation was used in deriving the foregoing stiffness matrix. It is exact. Of course, if used to construct a stepwise model of a continuously tapered bar, as in Fig. 1.1-1, the stepwise model is inexact and computed displacements will differ from those of the actual structure.
+
+The stiffness matrix is symmetric. This is always true when displacements are directly proportional to applied loads.
+
+By trying to apply the direct method of element derivation to Fig. 1.1-2c, we can see that the direct method is limited to simple elements: physical reasoning, based on elementary mechanics of materials, does not quantify the eight nodal forces associated with a nodal displacement such as $u_{3}$ .
+
+Heat or Current Conduction. A uniform bar element for heat conduction analysis has an element characteristic matrix that resembles stiffness matrix [k] of the elastic bar. With s a coordinate along the bar, the Fourier heat conduction equation becomes
+
+$$
+q = - k \frac {d T}{d s} = - k \frac {T _ {j} - T _ {i}}{L} \tag {1.2-5}
+$$
+
+where $q =$ heat flux per unit area, $k_{\perp} =$ thermal conductivity, and $T =$ temperature. If $A$ is the cross-sectional area and nodal heat flux is considered to be positive when directed into the bar at either end, then the equation analogous to Eq. 1.2-3 is
+
+$$
+\frac {A k}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right] \left\{ \begin{array}{l} T _ {i} \\ T _ {j} \end{array} \right\} = \left\{ \begin{array}{l} A q _ {i} \\ A q _ {j} \end{array} \right\} \tag {1.2-6}
+$$
+
+Similarly, if the bar is regarded as an electrical resistor, Ohm's law becomes $I = (V_{i} - V_{j}) / r$ , where $r$ is the resistance of the bar. In matrix form,
+
+$$
+\left[ \begin{array}{c c} 1 / r & - 1 / r \\ - 1 / r & 1 / r \end{array} \right] \left\{ \begin{array}{l} V _ {i} \\ V _ {j} \end{array} \right\} = \left\{ \begin{array}{l} I _ {i} \\ I _ {j} \end{array} \right\} \tag {1.2-7}
+$$
+
+where $V_{i}$ and $V_{j}$ are nodal voltages. Nodal currents $I_{i}$ and $I_{j}$ are considered positive when flowing into the element.
+
+One can envision networks of the foregoing heat conduction or current flow elements, in which a single node may be shared by several elements. If no external source supplies heat or current to such a node, the net flow into the node is zero.
+
+The Elastic Beam: Direct Method. Consider a uniform beam that deforms in the plane of the paper and undergoes no axial deformation. This element has four d.o.f.: a lateral displacement w and a rotation $\theta$ at each end (Fig. 1.2-2). Nodal forces $F_{i}$ and $F_{j}$ correspond to nodal displacements $w_{i}$ and $w_{j}$ . Nodal moments $M_{i}$
+
+<!-- source-page: 30 -->
+
+$$
+\vert C C (W) [ H ] / U _ {P}
+$$
+
+INTRODUCTION   
+![](images/page-030_ff51f3f6944e93e860290c9d9d356e84e547919ba46a201ae718ac64ac9917b8.jpg)
+
+<details>
+<summary>text_image</summary>
+
+i
+θi
+j
+θj
+wi
+L
+wj
+</details>
+
+(a)
+
+![](images/page-030_f3fa0cf38b01b7c015cae1c5ccd2184bc2637b61887f3ed8e7a8eb07d14f1434.jpg)
+
+<details>
+<summary>text_image</summary>
+
+i
+M_i
+j
+M_j
+F_j
+F_i
+L
+</details>
+
+(b)
+
+![](images/page-030_193ac308078348a22b584db8e4515889457b85d1b4a9841c2dc2764a23c3a13a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+w_i = 1
+k_{11}
+k_{21}
+k_{31}
+k_{41}
+</details>
+
+(r)
+
+![](images/page-030_38c332a051d02ccf0478a01380bc6988b41bd09add58dc5a40204b12d9f72b6b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k₂₂
+θᵢ=1
+k₁₂
+k₃₂
+k₄₂
+</details>
+
+(d)
+
+![](images/page-030_3bf824f61ffc411d0a34786bf5b93586441b10c20bd3874e3ee980939f03f309.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k₂₃
+k₁₃
+k₃₃
+k₄₃
+wⱼ=1
+</details>
+
+(e)
+
+![](images/page-030_0b3ef3c8e368e4c728a4c05288c5e55d3d13db620d5e7cab285c805fdec93785.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k24
+k14
+θj=1
+k34
+k44
+</details>
+
+(1)   
+Figure 1.2-2. (a) A uniform beam element and its four nodal d.o.f. (b) Associated nodal forces and moments. (c-e) Deformation states associated with activation of each d.o.f. in turn, showing the required nodal forces and moments labeled according to their position in [k].
+
+and $M_{j}$ correspond to nodal rotations $\theta_{i}$ and $\theta_{j}$ . Positive directions of these quantities are shown in Figs. 1.2-2a and 1.2-2b.
+
+The element equation is $[\mathbf{k}]\{\mathbf{d}\} = \{\overline{\mathbf{r}}\}$ , where $[\mathbf{k}]$ is a 4 by 4 stiffness matrix for the present beam element. The element nodal displacement vector is
+
+$$
+\{\mathbf {d} \} = \left[ \begin{array}{l l l l} w _ {i} & \theta_ {i} & w _ {j} & \theta_ {j} \end{array} \right] ^ {T} \tag {1.2-8}
+$$
+
+(A different ordering of d.o.f. in $\{d\}$ would change the ordering of coefficients in $[k]$ but would not change their numerical values.) Vector $\{\bar{r}\}$ contains nodal forces and moments applied to the element to maintain the deformation state $\{d\}$ . Written out, the element stiffness equation is
+
+$$
+\left. \begin{array}{l} \text {Apo} \left[ \begin{array}{l l l l} k _ {1 1} & k _ {1 2} & k _ {1 3} & k _ {1 4} \\ k _ {2 1} & k _ {2 2} & k _ {2 3} & k _ {2 4} \\ k _ {3 1} & k _ {3 2} & k _ {3 3} & k _ {3 4} \\ k _ {4 1} & k _ {4 2} & k _ {4 3} & k _ {4 4} \end{array} \right] \left\{ \begin{array}{l} w _ {i} \\ \theta_ {i} \\ w _ {j} \\ \theta_ {j} \end{array} \right\} = \left\{ \begin{array}{l} F _ {i} \\ M _ {i} \\ F _ {j} \\ M _ {j} \end{array} \right\} \end{array} \right\} \tag {1.2-9}
+$$
+
+We must express each stiffness coefficient in [k] in terms of element geometry and elastic modulus. To thus determine the stiffness coefficients in a single column of [k], we set the corresponding d.o.f. to unity while keeping all other d.o.f. zero, and calculate the values of $F_{i}$ , $M_{i}$ , $F_{j}$ , and $M_{j}$ needed in order to produce this deformation state. Thus, the first column of [k] is determined by activating only the first d.o.f. Specifically, for the case $w_{i} = 1$ and $\theta_{i} = w_{j} = \theta_{j} = 0$ , we see
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--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_004.md
@@ -0,0 +1,481 @@
+<!-- source-page: 31 -->
+
+from Eq. 1.2-9 that $k_{11} = F_{i}$ , $k_{21} = M_{i}$ , $k_{31} = F_{j}$ , and $k_{41} = M_{j}$ . These quantities are shown in Fig. 1.2-2c, all directed in the assumed positive sense. Let E = elastic modulus and I = moment of inertia of the cross-sectional area. We regard Fig. 1.2-2c as a beam cantilevered from its right end and loaded at its left end, and apply equations of beam theory and statics. Thus
+
+$$
+w = 1 \text {   at   node   } i \quad 1 = \frac {k _ {1 1} L ^ {3}}{3 E I} - \frac {k _ {2 1} L ^ {2}}{2 E I} \tag {1.2-10}
+$$
+
+$$
+\theta = 0 \text {   at   node   } i \quad 0 = \frac {k _ {1 1} L ^ {2}}{2 E I} - \frac {k _ {2 1} L}{E I} \tag {1.2-11}
+$$
+
+$$
+\Sigma (\text { forces }) = 0 \quad 0 = k _ {1 1} + k _ {3 1} \tag {1.2-12}
+$$
+
+$$
+\Sigma (\text { moments }) = 0 \quad 0 = k _ {2 1} + k _ {4 1} - k _ {1 1} L \tag {1.2-13}
+$$
+
+Solution of these equations yields
+
+$$
+k _ {1 1} = - k _ {3 1} = \frac {1 2 E I}{L ^ {3}} / \quad \text { and } \quad k _ {2 1} = k _ {4 1} = \frac {6 E I}{L ^ {2}} \tag {1.2-14}
+$$
+
+Stiffness coefficients in columns 2, 3, and 4 of [k] are determined by applying similar arguments to Figs. 1.2-2d, 1.2-2e, and 1.2-2f in turn. The resulting stiffness matrix is exact, not approximate (provided that transverse shear deformation is ignored and deflections are small, as is commonly the case).
+
+Having defined [k] in terms of E, I, and L, one is prepared to solve many problems of plane beams, such as that in Fig. 1.2-3. Generating the finite element model involves converting the distributed load into concentrated nodal loads. The conversion recipe for the loading shown in Fig. 1.2-3 is discussed in Section 4.3.
+
+One can combine the stiffness matrices of Eqs. 1.2-3 and 1.2-9 to produce a 6 by 6 matrix [k] for an element that has two translational and one rotational d.o.f. at each end. Such elements can be used to analyze a plane frame. These elements are discussed in Sections 4.2 and 7.5.
+
+# 1.3 ELEMENT ASSEMBLY AND SOLUTION FOR UNKNOWNS
+
+$$
+c P _ {- 1} = 1 + 1 \frac {1}{2}
+$$
+
+We consider a very simple example, which illustrates briefly how elements are put together to form a finite element structure and how a solution for displacements and stresses is obtained. These matters are discussed in detail in Chapter 2. In
+
+![](images/page-031_eedac9f0820649cd3263edc401c920ab901078f0977c1d437685013499cdd0f4.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L1
+L2
+</details>
+
+(a)
+
+![](images/page-031_10b61df52ac4c1946b7cdb93d1f003d29a00341bd943748903dd40be99714f0b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+qL₂/2
+qL₂²/12
+qL₂²/12
+qL₂/2
+①
+②
+L₁
+L₂
+</details>
+
+(b)   
+Figure 1.2-3. (a) Cantilever beam carrying a uniformly distributed load q (force per unit length). (b) A two-element model, showing nodal loads produced by q.
+
+<!-- source-page: 32 -->
+
+nonstructural problems the matrices have other names, but manipulations are the same.
+
+Assembly. An axially loaded bar structure is shown in Fig. 1.3-1a; a two-element model of it is shown in Fig. 1.3-1b. The stiffness matrix of a typical element is given by Eq. 1.2-3. Stiffness coefficients associated with elements 1 and 2 in Fig. 1.3-1 are abbreviated as
+
+$$
+k _ {1} = \frac {A _ {1} E _ {1}}{L _ {1}} \quad \text { and } \quad k _ {2} = \frac {A _ {2} E _ {2}}{L _ {2}} \tag {1.3-1}
+$$
+
+The d.o.f. are axial displacements $u_{1}$ , $u_{2}$ , and $u_{3}$ , where 1, 2, and 3 are arbitrary labels assigned to identify the structure nodes. $^{2}$ Now imagine two hypothetical states: in the first, only element 1 is present; in the second, only element 2 is present. Thus the respective structure stiffness matrices would be
+
+$$
+\left[ \begin{array}{c c c} u _ {1} & u _ {2} & u _ {3} \\ k _ {1} & - k _ {1} & 0 \\ - k _ {1} & k _ {1} & 0 \\ 0 & 0 & 0 \end{array} \right] \quad \text { and } \quad \left[ \begin{array}{c c c} u _ {1} & u _ {2} & u _ {3} \\ 0 & 0 & 0 \\ 0 & k _ {2} & - k _ {2} \\ 0 & - k _ {2} & k _ {2} \end{array} \right] \tag {1.3-2}
+$$
+
+element 1 only
+
+element 2 only
+
+where column headings indicate the d.o.f. associated with the matrix coefficients. As elements are put together to form a finite element model, element matrices are put together to form the structure matrix. By direct addition of the preceding matrices, the structure stiffness matrix is
+
+$$
+\left[ \begin{array}{l} F _ {0 1} \quad \text { Set } + i \cdot c \\ S + t u c t + v c \end{array} \right] \quad \left[ \begin{array}{l} \left[ \mathbf {K} \right] = \left[ \begin{array}{c c c} u _ {1} & u _ {2} & u _ {3} \\ k _ {1} & - k _ {1} & 0 \\ - k _ {1} & k _ {1} + k _ {2} & - k _ {2} \\ 0 & - k _ {2} & k _ {2} \\ \frac {0}{o} & \frac {1}{o} & \frac {1}{o} \end{array} \right] \end{array} \right. \tag {1.3-3}
+$$
+
+One can easily check that each column of [K] represents an equilibrium set of nodal forces associated with activation of the corresponding d.o.f. This should be no surprise, as the method of activating each d.o.f. in turn can be used to generate either an element matrix [k] or a structure matrix [K].
+
+![](images/page-032_51ba29e95584e33b3a0f36b05a88a56bf0fa81142a709e1918f880d212c45c22.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P ← A₁, E₁
+L₁ ← L₂
+2
+A₂, E₂
+3
+</details>
+
+(a)
+
+![](images/page-032_4a3927ae92371a744cfd6007d3723aafcc879ba94bf57b6152442ec89b7c5043.jpg)
+
+<details>
+<summary>text_image</summary>
+
+u₁
+u₂
+u₃
+1
+2
+3
+①
+②
+L₁
+L₂
+</details>
+
+(b)   
+Figure 1.3-1. (a) An axially loaded bar. (b) Finite elements used to model the bar.
+
+$^{2}$ Node labels i and j in Figs. 1.2-1 and 1.2-2 are called element or local node labels. Numbers would serve as well. They are used only temporarily, when element properties are defined. When elements are assembled, local labels are replaced by structure or global node labels (such as 1, 2, and 3 in Fig. 1.3-1). These matters are discussed in Section 2.7.
+
+<!-- source-page: 33 -->
+
+Matrix [K] of Eq. 1.3-3 is singular. Physically, this means that the structure in Fig. 1.3-1b is unsupported and can undergo a rigid-body translation. For solving a particular problem, at least one of the d.o.f. must be prescribed.
+
+Solution for Unknowns. To impose the displacement boundary condition appropriate to Fig. 1.3-1, we must enforce the constraint $u_{3} = 0$ . This can be done by discarding row 3 and column 3 from [K] of Eq. 1.3-3. What remains is a 2 by 2 system to be solved for $u_{1}$ and $u_{2}$ . (This system can also be obtained by activating $u_{1}$ and $u_{2}$ . in turn, while $u_{3}$ is kept at zero, and calculating the loads that must be applied to d.o.f. $u_{1}$ and $u_{2}$ .) Thus the problem of Fig. 1.3-1a is described by the matrix equation
+
+$$
+\left[ \begin{array}{c c} k _ {1} & - k _ {1} \\ - k _ {1} & k _ {1} + k _ {2} \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \end{array} \right\} = \left\{ \begin{array}{c} - P \\ 0 \end{array} \right\} \tag {1.3-4}
+$$
+
+The right-hand side indicates that node 1 carries a load P in the negative direction and that node 2 carries no externally applied load. The stiffness matrix in Eq. 1.3-4 is nonsingular. Therefore, Eqs. 1.3-4 can be solved for $u_{1}$ and $u_{2}$ . These results are
+
+$$
+u _ {1} = - \frac {P}{k _ {1}} - \frac {P}{k _ {2}} \quad \text { and } \quad u _ {2} = - \frac {P}{k _ {2}} \tag {1.3-5}
+$$
+
+Finally we convert displacements to strains and strains to stresses as follows. In elements 1 and 2, respectively, with $u_{3} = 0$ ,
+
+$$
+\begin{array}{l} \text {Up to Now} \\ \text {App. of Direct,} \\ \text {Sail. Mesh,} \\ \text {(No diff.)} \end{array} \left\{ \begin{array}{l} \checkmark \sigma_ {1} = E _ {1} \epsilon_ {1} = E _ {1} \frac {u _ {2} - u _ {1}}{L _ {1}} = \frac {E _ {1}}{L _ {1}} \frac {P}{A _ {1} E _ {1} / L _ {1}} = \frac {P}{A _ {1}} \\ \checkmark \sigma_ {2} = E _ {2} \epsilon_ {2} = E _ {2} \frac {u _ {3} - u _ {2}}{L _ {2}} = \frac {E _ {2}}{L _ {2}} \frac {P}{A _ {2} E _ {2} / L _ {2}} = \frac {P}{A _ {2}} \end{array} \right. \tag {1.3-6a}
+$$
+
+The preceding displacements and stresses are exact for this particular problem.
+
+Notation. Our symbols for element and structure stiffness equations are, respectively,
+
+$$
+[ \mathbf {k} ] \{\mathbf {d} \} = \{\overline {{{\mathbf {r}}}} \} \quad \text { and } \quad [ \mathbf {K} ] \{\mathbf {D} \} = \{\mathbf {R} \} \tag {1.3-7}
+$$
+
+where [k] and [K] are stiffness matrices, {d} and {D} are vectors of nodal d.o.f., and {r} and {R} are vectors of nodal loads. Subsequently (in Section 2.6) it will be worthwhile to distinguish between loads applied to an element and loads applied by an element. Displacements {d} produce loads {r} consistent with static equilibrium and applied to an element. If we define {r} as loads equal and opposite to {r}, that is,
+
+$$
+\{\mathbf {r} \} = - \{\vec {\mathbf {r}} \} \tag {1.3-8}
+$$
+
+then $\{r\}$ represents loads applied to nodes by deformed elements. Similarly, in the assembled structure, $\{R\}$ represents loads applied to nodes, now by external sources rather than by deformed elements.
+
+<!-- source-page: 34 -->
+
+# 1.4 SUMMARY OF FINITE ELEMENT HISTORY
+
+①Beginning in 1906, researchers suggested a “lattice analogy” for stress analysis [1.2–1.4]. The continuum was replaced by a regular pattern of elastic bars. Properties of the bars were chosen in a way that caused displacements of the joints to approximate displacements of points in the continuum. The method sought to capitalize on well-known methods of structural analysis.
+
+Courant appears to have been the first to propose the finite element method as we know it today. In a 1941 mathematics lecture, published in 1943, he used the principle of stationary potential energy and piecewise polynomial interpolation over triangular subregions to study the Saint-Venant torsion problem [1.5]. Courant's work was ignored until engineers had independently developed it.
+
+None of the foregoing work was of much practical value at the time because there were no computers available to generate and solve large sets of simultaneous algebraic equations. It is no accident that the development of finite elements coincided with major advances in digital computers and programming languages.
+
+By 1953 engineers had written stiffness equations in matrix format and solved the equations with digital computers [1.6]. Most of this work took place in the aerospace industry. At the time, a large problem was one with 100 d.o.f. In 1953, at the Boeing Airplane Company, Turner suggested that triangular plane stress elements be used to model the skin of a delta wing [1.7]. This work, published almost simultaneously with similar work done in England [1.8,1.9], marks the beginning of widespread use of finite elements. Much of this early work went unrecognized because of company policies against publication [1.10].
+
+The name “finite element method” was coined by Clough in 1960. The practical value of the method was soon obvious. New elements for stress analysis applications were developed, largely by intuition and physical argument. In 1963 the finite element method gained respectability when it was recognized as having a sound mathematical foundation: it can be regarded as the solution of a variational problem by minimization of a functional. Thus the method was seen as applicable to all field problems that can be cast in a variational form. Papers about the application of finite elements to problems of heat conduction and seepage flow appeared in 1965.
+
+Large general-purpose finite element computer programs emerged during the late 1960s and early 1970s. Examples include ANSYS, ASKA, and NASTRAN. Each of these programs included several kinds of elements and could perform static, dynamic, and heat transfer analysis. Additional capabilities were soon added. Also added were preprocessors (for data input) and postprocessors (for results evaluation). These processors rely on graphics and make it easier, faster, and cheaper to do finite element analysis. Graphics development became intensive in the early 1980s as hardware and software for interactive graphics became available and affordable.
+
+A general-purpose finite element program typically contains over 100,000 lines of code and usually resides on a mainframe or a superminicomputer. However, in the mid-1980s, adaptations of general-purpose programs began to appear on personal computers. Hundreds of analysis and analysis-related programs are now available, large and small, general and narrow, cheap and expensive, for lease or for purchase.
+
+Ten papers about finite elements were published in 1961, 134 in 1966, and 844 in 1971. By 1976, two decades after engineering applications began, the cumulative
+
+<!-- source-page: 35 -->
+
+total of publications about finite elements exceeded 7000. By 1986, the total was about 20,000.
+
+# 1.5 STRAIN-DISPLACEMENT RELATIONS
+
+The relation between strain and displacement is a key ingredient in the formulation of finite elements for stress analysis problems. In the present section we consider general strain-displacement relations in Cartesian coordinates. Alternative special forms of the relations, such as forms used for solids of revolution and for plate bending, are stated where they are used.
+
+In Fig. 1.5-1, perpendicular lines 01 and 02 are drawn on a plane sheet of material before the sheet is loaded. As a result of loading the lines become $0^{\prime}1^{\prime}$ and $0^{\prime}2^{\prime}$ . Displacements u and v are functions of the coordinates: $u = u(x, y)$ and $v = v(x, y)$ . We assume that displacement increments $^{3}$ such as $\overline{u} \cup_{a} dx$ are small in comparison with u and v. By definition, normal strain is the ratio of change in length to original length. Therefore,
+
+$$
+\epsilon_ {x} = \frac {L _ {0 ^ {\prime} 2 ^ {\prime}} - L _ {0 2}}{L _ {0 2}} = \frac {[ d x + (u + \frac {d y}{d x} d x) - u ] - d x}{d x} = u _ {x} = \frac {d y}{d x} \tag {1.5-1}
+$$
+
+A similar analysis yields the y-direction normal strain as
+
+$$
+\epsilon_ {y} = v _ {, y} \text {   与   } \frac {2 0}{2 0} \tag {1.5-2}
+$$
+
+Shear strain in the “engineering definition” is defined as the amount of change in a right angle. Because displacement increments are small, $\beta_{1} \approx \tan \beta_{1}$ and $\beta_{2} \approx \tan \beta_{2}$ . Therefore, the engineering shear strain is
+
+$$
+\gamma_ {x y} = \underline {{{\beta_ {1}}}} + \underline {{{\beta_ {2}}}} = \frac {(u + u , y d y) - u}{d y} + \frac {(v + v , x d x) - v}{d x} = u, y + v, x \tag {1.5-3}
+$$
+
+![](images/page-035_7087f674e3d72b86029ee1da62dded007a17c6a06cda524124589cb23ec485c2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+u + u,y dy
+v + v,y dy
+1'
+1
+β₁
+β₂
+2'
+0'
+v
+0
+2
+v + v₁,x dx
+x,u
+u
+dx
+u + u₁,x dx
+</details>
+
+Figure 1.5-1. Displacement and distortion of differential lengths dx and dy.
+
+$^{3}$ The notation $u_{,x}$ means $\partial u/\partial x$ . In general, a comma followed by a literal subscript indicates partial differentiation with respect to that subscript. Thus, for example, $\phi_{,xy} = \partial^{2}\phi/\partial x \partial y$ .
+
+<!-- source-page: 36 -->
+
+Collecting results, we have the two-dimensional strain-displacement relations:
+
+$$
+\epsilon_ {x} = u _ {, x} \checkmark \epsilon_ {y} = v _ {, y} \checkmark \gamma_ {x y} = u _ {, y} + v _ {, x} \checkmark \tag {1.5-4}
+$$
+
+In three dimensions, with w the z-direction displacement, a similar analysis yields the strains of Eqs. 1.5-4 and also
+
+$$
+\epsilon_ {z} = w _ {, z} \checkmark \quad \gamma_ {y z} = v _ {, z} + w _ {, y} \checkmark \quad \gamma_ {z x} = u _ {, z} + w _ {, x} \checkmark \tag {1.5-5}
+$$
+
+where now u, v, and w are all functions of x, y, and z. The foregoing relations can be stated in matrix operator form as follows. In two and three dimensions, respectively,
+
+$$
+\left\{ \begin{array}{l} \dot {\epsilon_ {x}} \\ \epsilon_ {y} \\ \gamma_ {x y} \end{array} \right\} = \left[ \begin{array}{c c} \frac {\partial}{\partial x} & 0 \\ 0 & \frac {\partial}{\partial y} \\ \frac {\partial}{\partial y} & \frac {\partial}{\partial x} \end{array} \right] \left\{ \begin{array}{l} u \\ v \end{array} \right\} \quad \text { and } \quad \left\{ \begin{array}{l} \epsilon_ {x} \\ \epsilon_ {y} \\ \epsilon_ {z} \\ \gamma_ {x y} \\ \gamma_ {y z} \\ \gamma_ {z x} \end{array} \right\} = \left[ \begin{array}{c c c} \frac {\partial}{\partial x} & 0 & 0 \\ 0 & \frac {\partial}{\partial y} & 0 \\ 0 & 0 & \frac {\partial}{\partial z} \\ \frac {\partial}{\partial y} & \frac {\partial}{\partial x} & 0 \\ 0 & \frac {\partial}{\partial z} & \frac {\partial}{\partial y} \\ \frac {\partial}{\partial z} & 0 & \frac {\partial}{\partial x} \end{array} \right] \left\{ \begin{array}{l} u \\ v \\ w \end{array} \right\} \tag {1.5-6}
+$$
+
+Plane Beams. We adopt standard beam theory in which transverse shear deformation is ignored and all deformations and strains are expressed in terms of the lateral displacement w. In Fig. 1.5-2, rotation $w_{,x}$ is assumed to be small, and plane cross sections are assumed to remain plane and normal to the deformed
+
+![](images/page-036_d3fb52536e48d269395df48e5f6679a77edb626ccfc38ea480565b8a6eadba6d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+dx
+P
+z
+C
+t/2
+x,u
+t/2
+</details>
+
+(a)
+
+![](images/page-036_5fa15b7b060502f005852b25aac7ed2a46356ea4d907020e50ad3c1e4d566e2d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+u = -zw_x
+w_x
+P
+z
+C
+w
+w_x
+dx
+</details>
+
+(b)   
+Figure 1.5-2. (a) Differential slice of a beam that lies along the x axis, before loading. (b) The same slice after loading. Transverse shear deformation is assumed to be negligible.
+
+<!-- source-page: 37 -->
+
+axis of the beam after it is bent. Therefore, axial displacement u is seen to be $u = -zw_{,x}$ . The negative sign means that with z and $w_{,x}$ both positive, displacement u is in the negative x direction. Accordingly, since $\epsilon_{x} = u_{,x}$ ,
+
+$$
+\epsilon_ {x} = - z w _ {, x x} \tag {1.5-7}
+$$
+
+where $w_{xx}$ is called the curvature of the beam.
+
+Notation. A compact notation serves to symbolize any strain-displacement relation. Let $\{\epsilon\}$ represent the strains and $\{\mathbf{u}\}$ represent the displacements. Then the equation
+
+$$
+\{\epsilon \} = [ \partial ] \{\mathbf {u} \} \tag {1.5-8}
+$$
+
+represents either of Eqs. 1.5-6. It is only necessary to infer the proper size and proper contents of the three matrices $\{\epsilon\}$ , $[\partial]$ , and $\{\mathbf{u}\}$ . For beam bending, the matrices are all 1 by 1 (scalars):
+
+$$
+\text { In   beams: } \quad \epsilon_ {x} = [ \partial ] w, \quad \text { where } \quad [ \partial ] = - z \frac {d ^ {2}}{d x ^ {2}} \tag {1.5-9}
+$$
+
+# 1.6 THEORY OF STRESS AND DEFORMATION
+
+Fundamental concepts, definitions, and equations used in the analysis of stress and deformation are discussed in the discipline called theory of elasticity. These fundamentals are used in solving problems by both classical and finite element methods. Elasticity theory states conditions that must be met by an exact solution, and therefore helps us in judging the shortcomings or range of applicability of approximate solutions. For simplicity, the following summary is presented in Cartesian coordinates only, and the two-dimensional case is emphasized. More general arguments may be found in texts on theory of elasticity.
+
+Equilibrium. Figure 1.6-1a shows a plane differential element (not a finite element!). We will develop equations stating that the differential element is in equilibrium under forces applied to it. Forces come from stresses on the edges and from body forces.
+
+Body forces $F_{x}$ and $F_{y}$ are applied to all material points and have dimensions of force per unit volume. They can arise from gravity, acceleration, a magnetic field, and so on. They are considered positive when acting in positive coordinate directions. On each differential element of volume (dV = t dx dy, where t = thickness), $F_{x}$ and $F_{y}$ produce differential forces $F_{x}$ dV and $F_{y}$ dV.
+
+In general, stresses and body forces are functions of the coordinates. Thus, for example, $\sigma_{x,x}$ is the rate of change of $\sigma_x$ with respect to $x$ , and $\sigma_{x,x} dx$ is the amount of change of $\sigma_x$ over distance $dx$ . For constant thickness $t$ , static equilibrium of forces in the $x$ direction, $\Sigma f_x = 0$ , requires that
+
+$$
+\begin{array}{l} - \sigma_ {x} t d y - \tau_ {x y} t d x + (\sigma_ {x} + \sigma_ {x, x} d x) t d y \\ + \left(\tau_ {x y} + \tau_ {x y, y} d y\right) t d x + F _ {x} t d x d y = 0 \tag {1.6-1} \\ \end{array}
+$$
+
+<!-- source-page: 38 -->
+
+![](images/page-038_2005f604845d19f2b4622c913c6153908cf1c6509004368d24c797ad32847e3c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+σy + σy,y dy
+τxy + τxy,y dy
+Fy
+dy
+Fx
+τxy + τxy,x dx
+σx + σx,x dx
+dx
+τxy
+σy
+</details>
+
+(a)
+
+![](images/page-038_3d713c82eae26e424644d81036d3789bec1c9112ee735378e6ff29d185640afe.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+Normal
+Φy
+dy
+ds
+Φx
+σx
+τxy
+dx
+τxy
+σy
+x
+Structure boundary
+</details>
+
+(b)   
+Figure 1.6-1. (a) Stresses and body forces that act on a plane differential element of constant thickness. (b) Surface tractions $\Phi_{x}$ and $\Phi_{y}$ on an arbitrarily oriented edge in the xy plane.
+
+There is a corresponding $y$ -direction equilibrium equation, $\Sigma f_y = 0$ . After simplification, the two equilibrium equations are
+
+$$
+\sqrt {x} \text { direction: } \sigma_ {x, x} + \tau_ {x y, y} + F _ {x} = 0 \tag {1.6-2a}
+$$
+
+$$
+\text { y   direction: } \quad \tau_ {x y, x} + \sigma_ {y, y} + F _ {y} = 0 \tag {1.6-2b}
+$$
+
+In many problems the effects of body forces are far less important than the effects of loads applied to the surface of the structure. Then $F_{x}$ and $F_{y}$ are set to zero. Note that Eqs. 1.6-2 are derived from equilibrium considerations alone: as no material properties are invoked, Eqs. 1.6-2 are applicable whether or not the body is linearly elastic.
+
+Although derived for the case of static equilibrium, Eqs. 1.6-2 are also valid if acceleration is present. The right-hand sides of the Newton's law equations $\Sigma f_{x} = ma_{x}$ and $\Sigma f_{y} = ma_{y}$ become negligible in comparison with $\Sigma f_{x}$ and $\Sigma f_{y}$ as the element size shrinks to zero.
+
+Compatibility. When a body is deformed without breaking, no cracks appear in stretching, no kinks appear in bending, and no part overlaps another. Stated more elegantly, this is the compatibility condition: the displacement field is continuous and single valued.
+
+The compatibility equation $\epsilon_{x,yy} + \epsilon_{y,xx} = \gamma_{xy,xy}$ states the relation that exists among the strains if a plane displacement field is compatible. Most finite element methods are based on displacements rather than stresses. Thus, each element invokes a displacement field that is continuous and single valued. Therefore the compatibility equation is automatically satisfied, as one may see by substituting Eqs. 1.5-4 into it. (However, the displacement field may not satisfy the equilibrium equations. This point is discussed further in the following.)
+
+Boundary Conditions. Boundary conditions consist of prescriptions of displacement and of stress. For example, in Fig. 1.1-2a the left edge does not move, so the displacement boundary condition along x = 0 is u = v = 0. Along the top edge, the stress boundary condition is $\sigma_{y} = \tau_{xy} = 0$ to the left of the zone that carries pressure p, and $\sigma_{y} = -p$ and $\tau_{xy} = 0$ beneath pressure p. On the right
+
+<!-- source-page: 39 -->
+
+edge, $\sigma_{x} = \tau_{xy} = 0$ . Along the curved lower edge, surface tractions $\Phi_x$ and $\Phi_y$ are zero. Surface tractions are related to stresses in the manner now described.
+
+Surface tractions $\Phi_x$ and $\Phi_y$ in Fig. 1.6-1b have units of stress. They are applied on a boundary (in contrast to body forces, which act throughout a volume). Tractions are $x$ - and $y$ -direction force increments divided by the boundary area increment $dA$ on which they act. In Fig. 1.6-1b, $dA = t ds$ , where $t =$ thickness. Note that $dA$ is not either of the projected areas $t dx$ or $t dy$ . Equilibrium of $x$ - and $y$ -direction forces in Fig. 1.6-1b requires that $\Phi_x t ds = \sigma_x t dy + \tau_{xy} t dx$ and $\Phi_y t ds = \tau_{xy} t dy + \sigma_y t dx$ . But $dy / ds = \ell$ and $dx / ds = m$ , where $\ell$ and $m$ are direction cosines of the outward normal to the boundary. Thus
+
+$$
+x \text {   direction:   } \Phi_ {x} = \ell \sigma_ {x} + m \tau_ {x y} \tag {1.6-3a}
+$$
+
+$$
+y \text {   direction:   } \Phi_ {y} = \ell \tau_ {x y} + m \sigma_ {y} \tag {1.6-3b}
+$$
+
+Equations 1.6-3 define a relation among stresses at an arbitrarily curved edge when tractions $\Phi_{x}$ and $\Phi_{y}$ are prescribed along that edge. Like Eqs. 1.6-2, Eqs. 1.6-3 do not require that the body be linearly elastic. One can check that Eqs. 1.6-3 yield the correct stress boundary conditions along the top and right edges of Fig. 1.1-2a.
+
+Three Dimensions. In solids, arguments analogous to those preceding lead to analogous results. The equilibrium and surface traction equations are, respectively,
+
+$$
+\sigma_ {x, x} + \tau_ {x y, y} + \tau_ {z x, z} + F _ {x} = 0 \quad \Phi_ {x} = \ell \sigma_ {x} + m \tau_ {x y} + n \tau_ {z x}
+$$
+
+$$
+\tau_ {x y, x} + \sigma_ {y, y} + \tau_ {y z, z} + F _ {y} = 0 \quad \text { and } \quad \Phi_ {y} = \ell \tau_ {x y} + m \sigma_ {y} + n \tau_ {y z} \tag {1.6-4}
+$$
+
+$$
+\tau_ {z x, x} + \tau_ {y z, y} + \sigma_ {z, z} + F _ {z} = 0 \quad \Phi_ {z} = \ell \tau_ {z x} + m \tau_ {y z} + n \sigma_ {z}
+$$
+
+where $\ell$ , m, and n are direction cosines of an outward normal to the surface. There are six compatibility equations that relate the six strains used in solids. They are not presented here.
+
+We observe that if stresses $\{\sigma\}$ are arrayed in the order used in Section 1.7, then the notation of Eq. 1.5-8 permits us to write the equilibrium equations in either two or three dimensions as
+
+$$
+[ \partial ] ^ {T} \{\boldsymbol {\sigma} \} + \{\mathbf {F} \} = \{\mathbf {0} \} \tag {1.6-5}
+$$
+
+where $[\partial]$ is given by Eqs. 1.5-6 and $\{\mathbf{F}\}$ is $\lfloor F_x\quad F_y\rfloor^T$ or $\lfloor F_x\quad F_y\quad F_z\rfloor^T$ .
+
+Remarks. Consider a linearly elastic body. If one finds a stress field or a displacement field that simultaneously satisfies equilibrium, compatibility, and boundary conditions, then one has found a solution to the problem posed. The solution is both unique and exact within the assumptions made (such as linearity and homogeneity).
+
+How do these observations relate to finite element analysis? If elements are based on polynomial displacement fields, then compatibility prevails within elements. Suitably chosen displacement fields also provide compatibility between elements and satisfy displacement boundary conditions. The differential equations
+
+<!-- source-page: 40 -->
+
+of equilibrium and boundary conditions on stress are satisfied only approximately. The approximation improves as more elements are used and, barring computational difficulties, the exact solution is achieved in the limit of an infinitely refined mesh.
+
+Some finite elements are incompatible: adjacent elements can overlap or gap apart between nodes. Compatibility is then satisfied only within elements and at node points. Thus there is only approximate satisfaction of equilibrium, stress boundary conditions, and compatibility. If interelement compatibility tends to be restored as more elements are used, it is still possible to converge toward the exact solution as the mesh is refined. (Incompatible elements are subsequently treated in more detail.)
+
+Other finite element methods are based on fields other than displacement. A stress field model may satisfy equilibrium equations a priori. Mesh refinement would then yield a better approximation of compatibility conditions.
+
+# 1.7 STRESS-STRAIN-TEMPERATURE RELATIONS
+
+The finite element method deals easily with rather general material properties and with both thermal and mechanical loading. We therefore devote this entire section to elastic properties and thermal relations.
+
+General. The stress vector $\{\sigma\}$ and the strain vector $\{\epsilon\}$ are, respectively,
+
+$$
+\{\boldsymbol {\sigma} \} = \left\lfloor \sigma_ {x} \quad \sigma_ {y} \quad \sigma_ {z} \quad \tau_ {x y} \quad \tau_ {y z} \quad \tau_ {z x} \right] ^ {T} \tag {1.7-1a}
+$$
+
+and
+
+$$
+\{\epsilon \} = \left\lfloor \epsilon_ {x} \quad \epsilon_ {y} \quad \epsilon_ {z} \quad \gamma_ {x y} \quad \gamma_ {y z} \quad \gamma_ {z x} \right\rfloor^ {T} \tag {1.7-1b}
+$$
+
+Ignoring the effect of temperature change, which will be treated subsequently, we symbolize the isothermal stress–strain relation as
+
+$$
+\{\epsilon \} = [ \mathrm{C} ] \{\sigma \} \quad \text { or   as } \quad \{\sigma \} = [ \mathrm{E} ] \{\epsilon \} \tag {1.7-2}
+$$
+
+where [C] is a symmetric matrix of material compliances, [E] is a symmetric matrix of material stiffnesses, and $[\mathbf{E}] = [\mathbf{C}]^{-1}$ (examples follow). In the most general case of anisotropy, [C] and [E] each contain 21 independent coefficients.
+
+An orthotropic material is an anisotropic material that displays extreme values of stiffness in mutually perpendicular directions. These directions are called principal directions of the material. An example is wood cut from a log, which is stiffest in the axial direction, least stiff in the circumferential direction, and of intermediate stiffness in the radial direction. If x, y, and z are principal directions, then normal stresses $\sigma_{x}$ , $\sigma_{y}$ , and $\sigma_{z}$ are independent of shear strains $\gamma_{xy}$ , $\gamma_{yz}$ , and $\gamma_{zy}$ . [E] for an orthotropic material contains only nine independent coefficients.
+
+Equations 1.7-2 express Hooke's law: stress is directly proportional to strain. This rule is an approximation limited to small strains and certain materials.
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index 00000000..87f4037c
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_005.md
@@ -0,0 +1,513 @@
+<!-- source-page: 41 -->
+
+Isotropy. An isotropic material has no preferred directions. Material properties are commonly expressed as a combination of two of the following: elastic modulus E, Poisson's ratio $\nu$ , and shear modulus G. In its upper triangle, [E] contains 12 zero coefficients and the 9 nonzero coefficients
+
+$$
+E _ {1 1} = E _ {2 2} = E _ {3 3} = (1 - \nu) c
+$$
+
+$$
+E _ {1 2} = E _ {1 3} = E _ {2 3} = \nu c \tag {1.7-3}
+$$
+
+$$
+E _ {4 4} = E _ {5 5} = E _ {6 6} = G
+$$
+
+where
+
+$$
+c = \frac {E}{(1 + \nu) (1 - 2 \nu)} \quad \text { and } \quad G = \frac {E}{2 (1 + \nu)}
+$$
+
+Because of the relation $E = 2(1 + \nu)G$ and the symmetry of [E], we see that [E] for an isotropic material contains only two independent coefficients.
+
+Plane strain is defined as a deformation state in which $w = 0$ everywhere and $u$ and $v$ are functions of $x$ and $y$ but not of $z$ . Thus, $\epsilon_z = \gamma_{yz} = \gamma_{zx} = 0$ . A typical slice of an underground tunnel that lies along the $z$ axis might deform in essentially plane strain conditions. The stress-strain relation $\{\sigma\} = [\mathrm{E}]\{\epsilon\}$ for isotropic and isothermal plane strain is
+
+$$
+\left\{ \begin{array}{l} \sigma_ {x} \\ \sigma_ {y} \\ \tau_ {x y} \end{array} \right\} = \frac {E}{(1 + \nu) (1 - 2 \nu)} \left[ \begin{array}{c c c c} 1 & - \nu & \nu & 0 \\ & \nu & 1 - \nu & 0 \\ & 0 & 0 & \frac {1 - 2 \nu}{2} \end{array} \right] \left\{ \begin{array}{l} \epsilon_ {x} \\ \epsilon_ {y} \\ \gamma_ {x y} \end{array} \right\} \tag {1.7-4}
+$$
+
+In Eq. 1.7-4, [E] is obtained by discarding rows and columns-3, -5, and 6 from the 6 by 6 matrix [E] of an isotropic solid.
+
+$\text{Stress } \sigma_z \text{ does not appear in Eq. 1.7-4, even though it is usually not zero. If needed, } \sigma_z \text{ can be obtained from the relation } \epsilon_z = 0 = (\sigma_z - \nu \sigma_y - \nu \sigma_x)/E \text{ after } \sigma_y \text{ and } \sigma_x \text{ are known.}$
+
+Plane stress is a condition that prevails in a flat plate in the xy plane, loaded only in its own plane and without z-direction restraint, so that $\sigma_{z} = \tau_{yz} = \tau_{zx} = 0$ . Then, for isotropic and isothermal conditions,
+
+$$
+\checkmark \left\{ \begin{array}{l} \epsilon_ {x} \\ \epsilon_ {y} \\ \gamma_ {x y} \end{array} \right\} = \frac {1}{E} \left[ \begin{array}{c c c} 1 & - \nu & 0 \\ - \nu & 1 & 0 \\ 0 & 0 & E / G \end{array} \right] \left\{ \begin{array}{l} \sigma_ {x} \\ \sigma_ {y} \\ \tau_ {x y} \end{array} \right\}
+$$
+
+or
+
+$$
+\left\{ \begin{array}{l} \sigma_ {x} \\ \sigma_ {y} \\ \tau_ {x y} \end{array} \right\} = \frac {E}{1 - \nu^ {2}} \left[ \begin{array}{c c c} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac {1 - \nu}{2} \end{array} \right] \left\{ \begin{array}{l} \epsilon_ {x} \\ \epsilon_ {y} \\ \gamma_ {x y} \end{array} \right\} \tag {1.7-5}
+$$
+
+where $E / G = 2(1 + \nu)$ . The square matrices, including their scalar multipliers $1 / E$ and $E / (1 - \nu^2)$ , are, respectively, [C] and [E].
+
+Axially symmetric solids require a 4 by 4 matrix [E]. This problem is discussed in Chapter 10.
+
+<!-- source-page: 42 -->
+
+Beam Bending. Consider again the beam of Fig. 1.5-2. Uniaxial stress prevails, so $\sigma_{x}=E\epsilon_{x}$ . Combining this equation with Eq. 1.5-7, we obtain $\sigma_{x}=-Ezw_{,xx}$ . Let the beam have cross-sectional area A and a constant modulus E. Multiplying by z dA and integrating over a cross section of depth t, we obtain
+
+$$
+\int_ {- t / 2} ^ {t / 2} \sigma_ {x} z d A = - E w _ {, x x} \int_ {- t / 2} ^ {t / 2} z ^ {2} d A \tag {1.7-6}
+$$
+
+The former integral is identified as bending moment M, positive when it bends the beam of Fig. 1.5-2 concave down. The latter integral is identified as the moment of inertia I of the cross-sectional area. Thus Eq. 1.7-6 becomes
+
+$$
+M = - E I w _ {, x x} \tag {1.7-7}
+$$
+
+This is a familiar expression from elementary mechanics of materials. It is the form of $\{\sigma\} = [E]\{\epsilon\}$ that applies to beam bending and is called a load–displacement relation. A similar expression, expanded to two dimensions, is used for plate bending (Chapter 11).
+
+Initial Stress and Strain. Thermal Effects. The term “initial stress” signifies a stress present before deformations are allowed. Effectively, it is a residual stress to be superposed on stress caused by deformation. With the addition of initial stresses $\{\sigma_{0}\}$ and initial strains $\{\epsilon_{0}\}$ , Eq. 1.7-2 becomes
+
+$$
+\{\boldsymbol {\sigma} \} = [ \mathrm{E} ] (\{\boldsymbol {\epsilon} \} - \{\boldsymbol {\epsilon} _ {0} \}) + \{\boldsymbol {\sigma} _ {0} \} \tag {1.7-8}
+$$
+
+As examples, $\{\epsilon_{0}\}$ might describe moisture-induced swelling and $\{\sigma_{0}\}$ might describe stresses produced by heating. Alternatively, both effects can be placed in $\{\epsilon_{0}\}$ , or $\{\epsilon_{0}\}$ and $\{\sigma_{0}\}$ can be viewed as alternative ways to express the same thing. For example, free expansion of an orthotropic material with principal axes xyz produces the initial strains
+
+$$
+\left\{\epsilon_ {0} \right\} = \left\lfloor \alpha_ {x} T \quad \alpha_ {y} T \quad \alpha_ {z} T \quad 0 \quad 0 \quad 0 \right] ^ {T} \tag {1.7-9}
+$$
+
+where $T$ is the temperature relative to an arbitrary reference temperature at which the body is free of stress, and the $\alpha$ 's are coefficients of thermal expansion in the principal material directions. Thus, to account for the effects of temperature change, we can substitute Eq. 1.7-9 into Eq. 1.7-8 and set $\{\sigma_0\} = \{0\}$ . Alternatively, we can substitute $\{\sigma_0\} = -[\mathbf{E}] \left[ \alpha_x T \quad \alpha_y T \quad \alpha_z T \quad 0 \quad 0 \quad 0 \right]^T$ into Eq. 1.7-8 and set $\{\epsilon_0\} = \{0\}$ . Stresses $\{\sigma\} = \{\sigma_0\}$ prevail when mechanical strains $\{\epsilon\}$ are prohibited. In the special case of isotropy we have, in three dimensions,
+
+$$
+\left\{\sigma_ {0} \right\} = - \frac {E \alpha T}{1 - 2 \nu} \left[ \begin{array}{l l l l l l} 1 & 1 & 1 & 0 & 0 & 0 \end{array} \right] ^ {T} \tag {1.7-10}
+$$
+
+<!-- source-page: 43 -->
+
+and, in plane stress,
+
+$$
+\left\{\epsilon_ {0} \right\} = \left\lfloor \alpha T \quad \alpha T \quad 0 \right] ^ {T} \quad \text {and} \quad \left\{\sigma_ {0} \right\} = - \frac {E \alpha T}{1 - \nu} \left\lfloor 1 \quad 1 \quad 0 \right] ^ {T} \tag {1.7-11}
+$$
+
+and, finally, in plane strain,
+
+$$
+\left\{\boldsymbol {\epsilon} _ {0} \right\} = (1 + \nu) \left\lfloor \alpha T \quad \dot {\alpha} T \quad 0 \right] ^ {T} \quad \text { and } \quad \left\{\boldsymbol {\sigma} _ {0} \right\} = - \frac {E \alpha T}{1 - 2 \nu} \left\lfloor 1 \quad 1 \quad 0 \right] ^ {T} \tag {1.7-12}
+$$
+
+Temperature-dependent moduli can be accommodated by using the [E] appropriate to the temperature that prevails. A temperature-dependent expansion coefficient is more troublesome. By definition, $\alpha = \partial\epsilon/\partial T$ when deformation is unrestrained, so $\epsilon = \alpha T$ only if $\alpha$ is independent of T. Otherwise,
+
+$$
+\epsilon = \int_ {0} ^ {T} \alpha d T = \overline {{{\alpha}}} T, \quad \text { where } \quad \overline {{{\alpha}}} = \frac {1}{T} \int_ {0} ^ {T} \alpha d T \tag {1.7-13}
+$$
+
+Here $\bar{\alpha}$ is an average expansion coefficient, valid only over the temperature range from 0 to T.
+
+Remarks. For plane stress or plane strain conditions to prevail in the xy plane, the xy plane must be a plane of elastic symmetry. Thus, if the material is orthotropic, the z axis must be a principal material direction. If, in addition, the x and y axes are principal material directions, then $E_{13} = E_{31} = E_{23} = E_{32} = 0$ in the 3 by 3 matrix [E].
+
+Poisson's ratio $\nu$ is little affected by temperature. Modulus $E$ is affected more: for stainless steel $E$ decreases about $20\%$ when the temperature rises from $0^{\circ}$ to $450^{\circ}\mathrm{C}$ . Barring plastic flow, elastic properties are almost independent of stress. For example, an increase in hydrostatic pressure from 0 to 350 MPA increases the moduli of steel and aluminum $0.8\%$ and $2.6\%$ , respectively. Like $E$ , thermal expansion coefficient $\alpha$ is relatively insensitive to stress but may vary appreciably with temperature.
+
+When strain rates are high, as in wave propagation, modulus E is higher than its static value. The difference is appreciable for rubber-like materials but negligible for common metals unless strain rates are extreme, as in explosive forming processes.
+
+If a body is isotropic and linearly elastic, and its supports do not inhibit thermal expansion or contraction, then the body deforms but remains free of stress when T is a harmonic function; that is, when $\nabla^{2}T = 0$ . A special case of this is when T is a linear function of the coordinates, for which both isotropic and rectilinearly orthotropic bodies remain free of stress (but not curvilinearly orthotropic bodies, such as tree trunks).
+
+Capabilities of the finite element method far exceed the knowledge of material behavior on which an analysis must be based. If test data are lacking, as is often the case with anisotropic materials, we can only estimate the elastic constants. Even when the constants are known, anisotropy has an adverse effect on the accuracy of finite element solutions $[8.34]$ .
+
+<!-- source-page: 44 -->
+
+# 1.8 WARNING: THE COMPUTED ANSWER MAY·BE WRONG
+
+Users of finite element programs may be so impressed by the power of the method that its limitations are ignored. Whether computer-based or not, analytical methods rely on assumptions and on theory that is not universally applicable. The analyst may overlook or misjudge important aspects of physical behavior. There may be an error in the computer program. A large program has many options and many computational paths. Perhaps some paths have never before been exercised, were not anticipated by the program designers, and have never been checked. Far more likely causes of incorrect results are user errors, such as using an inappropriate program or supplying an appropriate program with the wrong data. A poor mesh may be used, an inappropriate element type may be chosen, yielding or buckling may be overlooked, support conditions may be misrepresented, and so on. Users must remember that a structure is not obliged to behave as a computer says it should, regardless of how expensive the program, how many digits are printed in the results, or how elegant the graphic display. Computer graphics has achieved such a level of polish and versatility as to inspire great trust in the underlying analysis, a trust that may be unwarranted. (One can now make mistakes with more confidence than ever before.)
+
+The finite element method is a most versatile tool, but not the best analytical tool for every problem. It is foolish, but not unheard of, to use three-dimensional finite elements to compute stresses obtainable by the flexure formula $\sigma = Mc/I$ . In other cases experiment may be the most appropriate method, especially if experiment is needed anyway to obtain data needed for analysis (data such as material properties, the effective stiffness of joints, damping properties, the time history of loads, etc.). If an analysis is to be done by numerical methods, finite elements are not the only choice. For example, finite difference methods are effective for shells of revolution, and boundary elements are effective for some problems with boundaries at infinity.
+
+Powerful computer programs cannot be used without training. Their results cannot be trusted if users have no knowledge of their internal workings and little understanding of the physical theories on which they are based. An error caused by misunderstanding or oversight is not correctible by mesh refinement or by use of a more powerful computer. Some authorities have suggested that users be “qualified,” somewhat in the manner of practitioners having to be licensed before engaging in a profession in which the potential for damage to the public is substantial. Although the finite element method can make a good engineer better, it can make a poor engineer more dangerous.
+
+In years past, when analysis was done by hand, the analyst was required to invent a mathematical model before undertaking its analysis. Invention of a good model required sound physical understanding of the problem. Understanding can now be replaced by activation of a computer program. Having had little need to sharpen intuitions by devising simple models, the computer user may lack the physical understanding needed to prepare a good model and to check computed results. Or, what the user perceives as understanding may instead be familiarity with previous computer output.
+
+Computed results must in some way be judged or compared with expectations. Alternative results, useful for comparison, might be obtained from a different computer program that relies on a different analytical basis, from a simplified
+
+<!-- source-page: 45 -->
+
+model amenable to hand calculation, from the behavior of similar structures already built, and from experiment. Experiment may be expensive and has its own pitfalls, but is desirable if the analytical process is pushed beyond previous experience and established practice.
+
+The overall message of this discussion is that a competent analyst must have sound engineering judgment and experience, and that doubts raised in the course of an analysis should be taken seriously.
+
+# PROBLEMS
+
+# Section 1.1
+
+1.1 In Fig. 1.1-1a, let cross-sectional area A vary linearly from $3A_{0}$ at x = 0 to $A_{0}$ at $x = L_{T}$ . Model the bar by uniform elements. Let the cross-sectional area of each element be that of the actual bar at the x coordinate of the element midpoint. Assume that elastic modulus E is constant. Solve for the displacement of load P in the following ways, and compute the percentage error of each result. The exact answer is $PL_{T} \ln 3/2EA_{0}$ .
+
+(a) Use a single uniform element. Let $L_{1} = L_{T}$ (and $A = 2A_{0}$ ).   
+(b) Use two uniform elements. Let $L_{1} = L_{2} = L_{T} / 2$ .   
+(c) Use three uniform elements. Let $L_{1} = L_{2} = L_{3} = L_{T} / 3$ .   
+(d) Use four uniform elements, each of length $L_{T} / 4$ .
+
+1.2 Derive the “exact answer” stated in Problem 1.1.
+
+1.3 The bar shown has a uniform circular cross section of radius $a$ . It carries only torsional loads. Develop the 2 by 2 element stiffness matrix in terms of $a$ , $L$ , and shear modulus $G$ .
+
+![](images/page-045_f33eaf59257da67da26b97ce6e0e440e181bd1761c6495c43963651267b035f1.jpg)
+
+<details>
+<summary>text_image</summary>
+
+θ₁,T₁
+L
+S₂
+θ₂,T₂
+</details>
+
+Problem 1.3
+
+1.4 Let $\phi$ be a function that is interpolated over an element, where $\phi$ is a function of $x, y$ , element dimensions, and nodal values of $\phi$ . For each element shown, write this expression for $\phi$ .
+
+(a) Consider an element of length $L$ . Let $\phi$ vary linearly with axial coordinate $x$ .   
+(b) Consider the triangular element (see Eq. 1.1-1).   
+(c) Consider the rectangular element (see Eq. 1.1-2).
+
+![](images/page-045_16b457afeab78613290f5cb1b8adc483a5eeba05537525d37b3096b3f996924e.jpg)  
+(a)
+
+![](images/page-045_ee95b7c65d067df4cc9ecfca17cdede00340c2ca69359403abcc434f3ba63676.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+3 ← a →
+b
+1 2 x
+</details>
+
+(b)
+
+![](images/page-045_fbfcc81e8c821f51777ec12c191520d25f3ec761894535a338553795d665a9b6.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+a
+4
+3
+b
+1
+2
+x
+</details>
+
+(c)   
+Problem 1.4
+
+<!-- source-page: 46 -->
+
+1.5 The quadrilateral element shown might be used in the finite element model of Fig. 1.1-2b. Imagine that its $x$ -direction displacement field is $u = a_1 + a_2x + a_3y + a_4xy$ , where the $a_l$ are constants. How does $u$ vary with $x$ or $y$ along each side? Do you think this element will be compatible with its neighbors?
+
+![](images/page-046_3dc2297b85c9be366af568bb5228631d5536205b0a5e81e351e1f3887fad7d16.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+4 3
+45°
+1 2 x
+</details>
+
+Problem 1.5
+
+# Section 1.2
+
+1.6 Following the procedure used in Eqs. 1.2-10 to Eq. 1.2-14, derive the stiffness coefficients in columns 2, 3, and 4 of the stiffness matrix of a uniform beam element.   
+1.7 The bar shown can have both axial and bending deformations. Regard the bar as a single element with the four translational and two rotational d.o.f. shown. Without calculation, determine the algebraic sign of each coefficient in the 6 by 6 element stiffness matrix [k] (or enter zero for a null coefficient). Assume that displacements and rotations are small. Suggestion: Sketch each of six separate deformation states. Show the forces and moments needed to produce these states while satisfying static equilibrium. Finally, compare the directions of these loads with the assumed positive directions (which are those of the six nodal d.o.f.).
+
+![](images/page-046_34e167ffc278a0f949b4610ccaa2c8afad2c185f36e679e2a44b41bcc53f18d5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+d₁
+d₂
+d₃
+d₆
+d₄
+d₅
+</details>
+
+Problem 1.7
+
+![](images/page-046_5ebfb77f33bceb4e1709f72793a4e58566eadb62810312d148c276a8ae8f377c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+d₃
+d₁
+d₂
+d₆
+d₅
+d₄
+45°
+</details>
+
+Problem 1.8
+
+1.8 Repeat the instructions of Problem 1.7, but with reference to the inclined bar shown. Assume that the bar has an axial stiffness much greater than its bending stiffness. Also assume that displacements and rotations are small.   
+1.9 Repeat the instructions of Problem 1.7, but with reference to the bent bar shown. Suggestion: Assume that $[k]$ is symmetric. Also, start with column 2 of $[k]$ , then proceed to columns 1, 3, 4, 5, and 6.
+
+<!-- source-page: 47 -->
+
+![](images/page-047_2a85ba1ca0c2c486d384ba260b73399ba0bae39843090ee09a53eae21503db80.jpg)
+
+<details>
+<summary>text_image</summary>
+
+d₁
+d₂
+d₃
+d₄
+d₅
+d₆
+</details>
+
+Problem 1.9
+
+# Section 1.3
+
+1.10 The structure shown consists of a rigid, weightless bar and two linear springs of stiffnesses $k_{1}$ and $k_{2}$ . Only small vertical displacements are permitted. The stiffness matrix [K] of this structure is 2 by 2 but can have various forms depending on the choice of d.o.f. Write [K] for each of the following choices of d.o.f.
+
+(a) Displacements $v_{1}$ at $x = 0$ and $v_{2}$ at $x = L$ (shown in the second sketch).   
+(b) Displacements $v_{1}$ at $x = 0$ and $v_{A}$ at $x = L / 2$ .   
+(c) Displacements $v_{2}$ at $x = L$ and $v_{B}$ at $x = 2L$ .   
+(d) Displacement $v_{1}$ at $x = 0$ and a small rotation $\theta$ about $x = 0$ .   
+(e) Displacement $v_{B}$ at $x = 2L$ and a small rotation $\theta$ about $x = 2L$ .
+
+![](images/page-047_7f334236f13089b44c01194f6f38376de8bb965c83393a682215f4fd92ff7717.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+L
+L
+A
+B
+x
+k₁
+k₂
+</details>
+
+![](images/page-047_fb42f52e47f59b8cd80e6148a4f971b5e5472115346cf554f230fa96bbcec900.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Sketch for part (a)
+v₁
+v₂
+k₁
+k₂
+</details>
+
+Problem 1.10
+
+1.11 The angled bar is rigid and weightless. With its two linear springs it forms a structure whose stiffness matrix [K] is 2 by 2. Various forms of [K] are
+
+![](images/page-047_e324adeadee1396280ebec7bac6cc116167c33a80650ce8bd69e71e0bbbbfcbb.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k₂
+k₁
+a
+b
+</details>
+
+![](images/page-047_7895e7e95859a1b5f8b7e9a33a244ff61356a961069fa26ba8c76fa672a2cc93.jpg)
+
+<details>
+<summary>text_image</summary>
+
+v₁
+v₂
+</details>
+
+(a)
+
+![](images/page-047_cd64884958711b5f63c9d2816e7fb0e4162f1c32619b039694a2b8436509e857.jpg)
+
+<details>
+<summary>text_image</summary>
+
+θ₁
+v₁
+</details>
+
+(b)
+
+![](images/page-047_f09207c5cdee2a10e2af6a595d7a91e295232f676f35d4fc0835af2939812246.jpg)
+
+<details>
+<summary>text_image</summary>
+
+a/2
+v₁ v₃
+</details>
+
+(c)   
+Problem 1.11
+
+<!-- source-page: 48 -->
+
+possible for various choices of d.o.f. Write [K] for each of the choices (a), (b), and (c) shown in the sketch. Displacements are small in each case.
+
+1.12 The structure shown consists of linear springs whose stiffnesses are $k_{1}, k_{2}, k_{3}$ , and $k_{4}$ . Only horizontal displacements are allowed.
+
+(a) In matrix form, write the three equilibrium equations of the structure. The d.o.f. are $u_{1}, u_{2}$ , and $u_{3}$ .
+
+(b) Let $k_{1} = k_{2} = k_{3} = k_{4} = k$ and $F_{1} = F_{2} = 0$ . Determine $u_{1}, u_{2}$ , and $u_{3}$ in terms of $k$ and $F_{3}$ .
+
+![](images/page-048_59e6973f4695f1f3fc6f2d0ae7b15d26cfaac00d0fbafff0ec3e17fd2e218518.jpg)
+
+<details>
+<summary>text_image</summary>
+
+u₁,F₁
+u₂,F₂
+u₃,F₃
+k₁
+k₂
+k₃
+k₄
+</details>
+
+Problem 1.12
+
+1.13 The structure shown consists of rigid bars $AB$ and $CD$ and linear springs that connect $A$ to $C$ and $B$ to $D$ . Only horizontal motions and small rotations of the bars are permitted. In terms of $k_1, k_2$ , and $a$ , determine the 4 by 4 stiffness matrix that operates on $[u_1 \theta_1 u_2 \theta_2]^T$ to yield $[F_1 M_1 F_2 M_2]^T$ . Suggestion: Since rotations are small, $u_A = u_1 - a\theta_1$ , with similar relations for $u_B, u_C$ , and $u_D$ .
+
+![](images/page-048_85f9e0a42782158a59607ceb68e366c51c668ebde7e64fc33ddce84b6498b38e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+A
+k₁
+C
+1
+θ₁,M₁
+a
+θ₂,M₂
+u₁,F₁
+2
+u₂,F₂
+B
+k₂
+D
+</details>
+
+Problem 1.13
+
+1.14 The uniform, linearly elastic bar shown is fixed at both ends. Force $P$ is applied at node 2. Use the finite element method to compute nodal displacements and stresses in each element in terms of $P, L, A$ , and $E$ . Compare these results with exact values.
+
+1.15 The uniform, linearly elastic bar shown carries a uniformly distributed load $q_0$ . Assume that $q_0$ produces the respective nodal loads $q_0L / 2$ , $q_0L$ , and $q_0L / 2$ .
+
+(a) Compute nodal displacements by the finite element method and compare them with the exact values, which are $u_{2} = 3q_{0}L^{2} / 2AE$ and $u_{3} = 2q_{0}L^{2} / AE$ .
+
+(b) Using the procedure of Eqs. 1.3-6, compute the axial stress in each element. On a single set of axes, plot these stresses as well as the actual stress distribution along the bar. Do the results suggest a rule regarding where stresses should be calculated in a finite element?
+
+<!-- source-page: 49 -->
+
+![](images/page-049_0da1262f41b2075cc269936eee0186384ac218be15e8c5792e2ee3b5240c5f9c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P 2 3
+1
+L L L
+4
+</details>
+
+Problem 1.14
+
+![](images/page-049_e9a9dc38aa9b8051a903bc5609f26940f15d89daade391a9fd40b71d2dee792c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+q₀
+2
+3
+L
+L
+</details>
+
+Problem 1.15
+
+# Section 1.6
+
+1.16 (a) The sketch shows a plane differential element similar to that in Fig. 1.6-1a but in polar coordinates. The stresses are $\sigma_r$ (radial), $\sigma_\theta$ (circumferential), and $\tau_{r\theta}$ (shear). Derive the differential equations of equilibrium in polar coordinates.
+
+(b) Similarly, use cylindrical coordinates to derive equilibrium equations analogous to the first set of Eqs. 1.6-4.
+
+![](images/page-049_a8e5e294359ccea51032bc53731007242209f78f974491f6f64e281fe177974f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Fθ
+Fτ
+τrθ
+σr
+dθ
+τrθ
+σθ
+dr
+</details>
+
+Problem 1.16
+
+1.17 Imagine that stresses in the $xy$ plane are given as $\sigma_x = -6a_1x^2$ , $\sigma_y = 12a_1x^2$ , and $\tau_{xy} = 12a_1y^2$ , where $a_1$ is a constant.
+
+(a) Consider the square region $0 \leqslant x \leqslant b$ , $0 \leqslant y \leqslant b$ . Write expressions for tractions $\Phi_x$ and $\Phi_y$ on each side of this square, in terms of $x, y, b$ , and $a_1$ .   
+(b) If body forces are zero, is the given stress field in fact possible? Explain.
+
+1.18 In a certain approximation method, stresses in a plane region are assumed to have the forms
+
+$$
+\sigma_ {x} = a _ {1} + a _ {2} x + a _ {3} y, \quad \sigma_ {y} = a _ {4} + a _ {5} x + a _ {6} y, \quad \tau_ {x y} = a _ {7} + a _ {8} x + a _ {9} y
+$$
+
+where each $a_{i}$ is a constant. Let all body forces vanish. What must be the relation among the $a_{i}$ if equilibrium is to be satisfied?
+
+1.19 Determine whether or not the following stress field is a valid solution of a plane elasticity problem: $\sigma_x = 3a_1x^2y$ , $\sigma_y = a_1y^3$ , and $\tau_{xy} = -3a_1xy^2$ , where $a_1$ is a constant. The body is isotropic and linearly elastic and body forces are zero.
+
+# Section 1.7
+
+1.20 Judging by Eq. 1.7-4, what property does a material display if $\nu = 0.5$ ?   
+1.21 Combine the latter form of Eqs. 1.7-5, the strain-displacement relations, and the equilibrium equations with $F_{x} = F_{y} = 0$ , and show that
+
+<!-- source-page: 50 -->
+
+$$
+u _ {, x x} + u _ {, y y} = (1 + \nu) (u _ {, y y} - v _ {, x y}) / 2
+$$
+
+This equation, and its companion (obtained by interchange of u with v and x with y), are known as the equilibrium equations expressed in terms of displacements.
+
+1.22 Let displacements in a plane stress problem be given by
+
+$$
+\begin{array}{l} u = a _ {1} + a _ {2} x + a _ {3} y + a _ {4} x ^ {2} + a _ {5} x y + a _ {6} y ^ {2} \\ v = a _ {7} + a _ {8} x + a _ {9} y + a _ {1 0} x ^ {2} + a _ {1 1} x y + a _ {1 2} y ^ {2} \\ \end{array}
+$$
+
+where each $a_{i}$ is a constant. Let all body forces vanish. What relation among the $a_{i}$ is needed to satisfy equilibrium? The medium is isotropic.
+
+1.23 Start with Eqs. 1.7-8 and 1.7-9, and derive the plane strain form of $\{\epsilon_0\}$ (Eq. 1.7-12).
+
+[Unreadable]
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@@ -0,0 +1,456 @@
+<!-- source-page: 51 -->
+
+# THE STIFFNESS METHOD AND THE
+
+# PLANE TRUSS
+
+Matrix procedures used in the analysis of framed structures and other finite element structures are described. The plane truss is used as the principal vehicle for the discussion.
+
+# 2.1 INTRODUCTION
+
+Certain matrix procedures of structural mechanics are described in this chapter. These methods are also used in finite element analysis of many other physical problems. These procedures include assembly of elements to form a structure, imposition of boundary or support conditions, solution of simultaneous equations to obtain nodal quantities, and processing of elements to obtain quantities such as stresses or flows. The plane truss is a very simple structure that serves to explain these concepts and procedures.
+
+The truss can be called a “discrete element” structure. Its elements are the individual bars, already present as separate pieces. Thus we bypass the important finite element processes of dividing a continuum into appropriate elements and idealizing the behavior of each element. In the present chapter we are primarily concerned with manipulation procedures that apply to a previously discretized structure.
+
+Each bar of a truss is assumed to be uniform, linearly elastic, pin-connected to nodes at its ends, and axially loaded. Displacements shown in sketches are greatly exaggerated. Actual displacements are assumed to be small, so that if $\theta$ is the angle of rotation under load of any bar, then $\sin \theta \approx \theta$ and $\cos \theta \approx 1$ . We consider only statically loaded structures. Within these restrictions, the analysis is exact, not approximate.
+
+Degrees of Freedom (d.o.f.). A structure has n d.o.f. if n independent quantities are needed to uniquely define the deformed configuration of the structure. The structure stiffness matrix will have n rows and n columns. In a plane truss, n is equal to two times the number of nodes allowed to displace. The individual d.o.f. are the x- and y-direction displacement components of each structure node. In nonstructural problems, d.o.f. are analogously defined: they are the independent quantities needed to define a field, such as the temperature field in a heat conduction problem. In heat conduction analysis, there is a single d.o.f. per node—namely, the nodal temperature.
+
+<!-- source-page: 52 -->
+
+# 2.2 STRUCTURE STIFFNESS EQUATIONS
+
+We begin by generating the structure stiffness matrix [K] of a plane truss by a direct attack on the structure as a whole. The result will be used to illustrate certain concepts. Later we will show how [K] can be built by assembly of element matrices, which is the process actually used in computer programs.
+
+Consider, for example, the three-bar truss of Fig. 2.2-1. Nodes and elements (bars) are numbered arbitrarily. For element i, where i = 1, 2, 3 in this example, let $A_{i} = \text{cross-sectional area}$ , $E_{i} = \text{elastic modulus}$ , and $L_{i} = \text{length}$ . From elementary mechanics of materials, axial force $F_{i}$ and change in length $e_{i}$ have the relation
+
+$$
+\sqrt {e _ {i}} = \frac {F _ {i} L _ {i}}{A _ {i} E _ {i}} \tag {2.2-1}
+$$
+
+Stiffness is defined as the ratio of force to displacement and is by custom given the symbol k. Thus, the axial stiffness of any uniform bar of a truss is
+
+$$
+\sqrt {k _ {i}} = \frac {F _ {i}}{e _ {i}} = \frac {A _ {i} E _ {i}}{L _ {i}} \tag {2.2-2}
+$$
+
+Supports at nodes 2 and 3 in Fig. 2.2-1 are temporarily removed, so that nonzero values can be assigned to all nodal displacements. Now let a node be displaced a small amount, first in the x direction and then in the y direction, while all other nodes are held at zero displacement. Thus there are six possible deformation states for a three-node plane truss. In each of these six states we calculate forces that must be applied to the nodes to maintain the deformation state. These forces, acting on the truss cut free of its supports, place the truss in static equilibrium. The first two free-body diagrams are shown in Fig. 2.2-2.
+
+As an example of the force computation, consider the forces in Fig. 2.2-2a. Because displacement $u_{1}$ is small, its component along bar 2, which is the change in length of bar 2, is $e_{2} = 0.6u_{1}$ . This deformation produces the axial force $F_{2} = k_{2}e_{2}$ , whose horizontal and vertical components have the respective magnitudes $0.6F_{2} = 0.36k_{2}u_{1}$ and $0.8F_{2} = 0.48k_{2}u_{1}$ . Bar 3 has elongation $e_{3} = u_{1}$ and contributes forces $F_{3} = k_{3}e_{3} = k_{3}u_{1}$ .
+
+![](images/page-052_6ad2052aa3bb1301c1e28eda5e0619245f4c01969be86a23c839cd3b2f5d17f2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+3
+36.9°
+3
+4
+①
+②
+P
+2
+③
+1
+x,u
+</details>
+
+Figure 2.2-1. A three-bar plane truss. D.o.f. $u_{2}$ , $v_{2}$ , and $u_{3}$ are restrained. D.o.f. $u_{1}$ , $v_{1}$ , and $v_{3}$ are active (allowed to displace). Externally applied loading consists of force P.
+
+<!-- source-page: 53 -->
+
+![](images/page-053_ce94162f2a8de3097e666b5fa03cc5cc4d009872f511dde39c174ede4f8cc93e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+0.48k₂u₁
+0.36k₂u₁
+3
+①
+②
+u₁
+(0.36k₂ + k₃)u₁
+k₃u₁
+2
+③
+u₁
+0.48k₂u₁
+</details>
+
+(a)
+
+![](images/page-053_f4e3f506a8bd71f1fc36f9ebcdb2a268c576c577d5150146d35203bfd4039f1d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+0.64k₂v₁
+0.48k₂v₁
+3
+①
+②
+0.48k₂v₁
+1
+2
+③
+0.64k₂v₁
+v₁
+</details>
+
+(b)   
+Figure 2.2-2. Nodal loads consistent with the respective displacement states $\{\mathbf{D}\} = [u_1 0 0 0 0 0]^T$ and $\{\mathbf{D}\} = [0 v_1 0 0 0 0]^T$ .
+
+Let $\{\mathbf{Q}_1\}$ represent the vector of forces in Fig. 2.2-2a associated with unit displacement, $u_1 = 1$ . Thus forces that appear in Fig. 2.2-2a are $\{\mathbf{Q}_1\} u_1$ :
+
+$$
+\left\{\mathbf {Q} _ {1} \right\} u _ {1} = \left\lfloor k _ {3} + 0. 3 6 k _ {2} - 0. 4 8 k _ {2} - k _ {3} 0 - 0. 3 6 k _ {2} 0. 4 8 k _ {2} \right] ^ {T} u _ {1} \tag {2.2-3}
+$$
+
+Similarly, forces that appear in Fig. 2.2-2b are $\{Q_{2}\}v_{1}$ , where $\{Q_{2}\}$ is the force vector associated with the unit displacement $v_{1}=1$ :
+
+$$
+\{\mathbf {Q} _ {2} \} v _ {1} = \left[ - 0. 4 8 k _ {2} \quad 0. 6 4 k _ {2} \quad 0 \quad 0 \quad 0. 4 8 k _ {2} \quad - 0. 6 4 k _ {2} \right] ^ {T} v _ {1} \tag {2.2-4}
+$$
+
+Let $\{\mathbf{Q}_3\}, \{\mathbf{Q}_4\}, \{\mathbf{Q}_5\}$ , and $\{\mathbf{Q}_6\}$ represent the equilibrium nodal force vectors associated with the remaining four unit displacement states $u_2 = 1$ , $v_2 = 1$ , $u_3 = 1$ , and $v_3 = 1$ . Then, if all six nodal d.o.f. may be nonzero simultaneously, the associated nodal loads are obtained by adding the six separate cases,
+
+$$
+\left[ \begin{array}{l l l l l l} \mathbf {Q} _ {1} & \mathbf {Q} _ {2} & \mathbf {Q} _ {3} & \mathbf {Q} _ {4} & \mathbf {Q} _ {5} & \mathbf {Q} _ {6} \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \\ u _ {2} \\ v _ {2} \\ u _ {3} \\ v _ {3} \end{array} \right\} = \left\{ \begin{array}{l} p _ {1} \\ q _ {1} \\ p _ {2} \\ q _ {2} \\ p _ {3} \\ q _ {3} \end{array} \right\} \tag {2.2-5}
+$$
+
+where forces applied to node i are called $p_{i}$ and $q_{i}$ , positive in +x and +y directions, respectively. Written out, Eq. 2.2-5 is
+
+$$
+\left[ \begin{array}{c c c c c c} k _ {3} + 0. 3 6 k _ {2} & - 0. 4 8 k _ {2} & - k _ {3} & 0 & - 0. 3 6 k _ {2} & 0. 4 8 k _ {2} \\ - 0. 4 8 k _ {2} & 0. 6 4 k _ {2} & 0 & 0 & 0. 4 8 k _ {2} & - 0. 6 4 k _ {2} \\ - k _ {3} & 0 & k _ {3} & 0 & 0 & 0 \\ 0 & 0 & 0 & k _ {1} & 0 & - k _ {1} \\ - 0. 3 6 k _ {2} & 0. 4 8 k _ {2} & 0 & 0 & 0. 3 6 k _ {2} & - 0. 4 8 k _ {2} \\ 0. 4 8 k _ {2} & - 0. 6 4 k _ {2} & 0 & - k _ {1} & - 0. 4 8 k _ {2} & k _ {1} + 0. 6 4 k _ {2} \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ \widehat {v _ {1}} \\ u _ {2} \\ v _ {2} \\ u _ {3} \\ v _ {3} \end{array} \right\} = \left\{ \begin{array}{l} p _ {1} \\ q _ {1} \\ p _ {2} \\ q _ {2} \\ p _ {3} \\ q _ {3} \end{array} \right\} \tag {2.2-6}
+$$
+
+<!-- source-page: 54 -->
+
+In our standard abbreviation, these structure stiffness equations are
+
+$$
+[ \mathbf {K} ] \{\mathbf {D} \} = \{\mathbf {R} \} \tag {2.2-7}
+$$
+
+where [K] is the structure stiffness matrix. As shown by the development, Eqs. 2.2-7 are equilibrium equations.
+
+The physical meaning of [K], as well as a procedure for formulating [K], are contained in the following statement. The jth column of [K] is the vector of loads that must be applied to nodal d.o.f. in order to maintain the deformation state associated with unit value of d.o.f. j while all other nodal d.o.f. are zero. For a frame, “loads” include displacements and rotations. By this procedure—activating one d.o.f. at a time—we can generate the stiffness matrix of any truss or frame, regardless of the number of bars or the degree of static indeterminacy. The stiffness matrix is square; that is, there are as many equations as there are d.o.f.
+
+# 2.3 PROPERTIES OF [K]. SOLUTION FOR UNKNOWNS
+
+Properties of [K]. Each diagonal stiffness coefficient $K_{ii}$ in Eq. 2.2-6 is positive. This is physically reasonable, as it means that a force $R_{i}$ directed toward (say) the right will not produce a displacement directed toward the left (here it helps to imagine that all d.o.f. in {D} except $D_{i}$ are constrained to be zero). In general, for any structure, no diagonal coefficient $K_{ii}$ is negative or zero unless the structure is unstable.
+
+[ K ] is symmetric. This is true of any structure that displays a linear relationship between applied loads and the resulting displacements. The symmetry of [K] may be proved by use of the procedure suggested in Problem 2.7.
+
+If the structure is not attached to supports, its [K] does not resist rigid-body motion of the structure. An infinite number of vectors $\{\mathbf{D}\}$ that represent rigid-body motion can be written. For the plane truss of Fig. 2.2-1, with all supports removed, four of them are
+
+$$
+\{\mathbf {D} \} _ {1} = \left[ \begin{array}{l l l l l l} \delta & 0 & \delta & 0 & \delta & 0 \end{array} \right] ^ {T} \quad \{\mathbf {D} \} _ {2} = \left[ \begin{array}{l l l l l l} 0 & \delta & 0 & \delta & 0 & \delta \end{array} \right] ^ {T} \tag {2.3-1}
+$$
+
+$$
+\{\mathbf {D} \} _ {3} = \left[ \begin{array}{c c c c c c} \delta & \delta & \delta & \delta & \delta & \delta \end{array} \right] ^ {T} \quad \{\mathbf {D} \} _ {4} = \left[ \begin{array}{c c c c c c} 4 \theta & 3 \theta & 4 \theta & 0 & 0 & 0 \end{array} \right] ^ {T}
+$$
+
+where $\delta$ is a small displacement and $\theta$ is a small angle of rotation. Respectively, the foregoing vectors represent translation along the $x$ axis, translation along the $y$ axis, translation along the line $x = y$ , and rotation about node 3. For any plane structure only three of the infinitely many rigid-body $\{\mathbf{D}\}_{i}$ are linearly independent. The choice of three is not unique. For example, from Eqs. 2.3-1 we could choose $\{\mathbf{D}\}_{4}$ and any two of $\{\mathbf{D}\}_{1}, \{\mathbf{D}\}_{2}$ , and $\{\mathbf{D}\}_{3}$ . The first three $\{\mathbf{D}\}_{i}$ in Eqs. 2.3-1 are linearly dependent because $\{\mathbf{D}\}_{3} = \{\mathbf{D}\}_{i} + \{\mathbf{D}\}_{2}$ .
+
+A rigid-body motion does not deform a structure. Therefore, $[\mathbf{K}]\{\mathbf{D}\}_{i} = \{\mathbf{0}\}$ for any rigid-body motion $\{\mathbf{D}\}_{i}$ . With reference to our three-bar truss example, and for $\delta = 1$ in Eqs. 2.3-1, the equations $[\mathbf{K}]\{\mathbf{D}\}_{3} = \{\mathbf{0}\}$ state that coefficients in each row of $[\mathbf{K}]$ sum to zero. Note, however, that row sums of $[\mathbf{K}]$ will not vanish for any and all structures, since setting each entry in $\{\mathbf{D}\}$ to unity does not in general
+
+<!-- source-page: 55 -->
+
+constitute rigid-body motion. Cases in point include structures that contain beam or plate elements, for which rotational d.o.f. are present. To summarize: for an unsupported structure, (a) $[K]\{D\} = \{0\}$ when $\{D\}$ represents rigid-body motion, and (b) each column of $[K]$ represents a set of nodal forces and/or moments in static equilibrium.
+
+Solution for Unknowns. The stiffness matrix of Eq. 2.2-6 is singular. Its order is 6 but its rank is 3. [K] cannot be inverted, nor can a unique {D} be obtained by solving equations. The physical reason for this is that rigid-body motion is still possible. Without supports, the structure will float away if the slightest external load is applied. Before continuing with the truss example we state a more general argument about solution for unknowns, as follows.
+
+One must remove the singularity of [K] in order to solve for the unknown d.o.f. in {D}. We now show a formal procedure by which this may be done. Let $\{\mathbf{D}_c\}$ and $\{\mathbf{R}_c\}$ be known d.o.f. and known loads, and $\{\mathbf{D}_x\}$ and $\{\mathbf{R}_x\}$ be as yet unknown d.o.f. and loads. By partitioning, accompanied by such rearrangement of matrix coefficients as may be necessary, the structural equations [K]{D} = {R} can be written in the form
+
+$$
+\left[ \begin{array}{l l} \mathbf {K} _ {1 1} & \mathbf {K} _ {1 2} \\ \mathbf {K} _ {2 1} & \mathbf {K} _ {2 2} \end{array} \right] \left\{ \begin{array}{l} \mathbf {D} _ {x} \\ \mathbf {D} _ {c} \end{array} \right\} = \left\{ \begin{array}{l} \mathbf {R} _ {c} \\ \mathbf {R} _ {x} \end{array} \right\} \tag {2.3-2}
+$$
+
+or, in a more expanded form,
+
+$$
+[ \mathbf {K} _ {1 1} ] \{\mathbf {D} _ {x} \} + [ \mathbf {K} _ {1 2} ] \{\mathbf {D} _ {c} \} = \{\mathbf {R} _ {c} \} \tag {2.3-3}
+$$
+
+$$
+[ \mathbf {K} _ {2 1} ] \{\mathbf {D} _ {x} \} + [ \mathbf {K} _ {2 2} ] \{\mathbf {D} _ {c} \} = \{\mathbf {R} _ {x} \} \tag {2.3-4}
+$$
+
+(Note that at this stage we know either a d.o.f. or its corresponding load, but not both.) $[K_{11}]$ is nonsingular if the prescribed d.o.f. $\{D_{c}\}$ are sufficient in arrangement and number to prevent rigid-body motion. Therefore, the unknown d.o.f. $\{D_{x}\}$ can be found from Eq. 2.3-3:
+
+$$
+\{\mathbf {D} _ {x} \} = [ \mathbf {K} _ {1 1} ] ^ {- 1} \left(\{\mathbf {R} _ {c} \} - [ \mathbf {K} _ {1 2} ] \{\mathbf {D} _ {c} \}\right) \tag {2.3-5}
+$$
+
+Finally, unknown loads $\{R_{x}\}$ can be found from Eq. 2.3-4 after substitution of d.o.f. $\{D_{x}\}$ , which are now known. In structural mechanics, $\{R_{x}\}$ usually represents support reactions.
+
+In practice, the foregoing rearrangement of coefficients, partitioning, and matrix inversion are avoided by use of other operations that implicitly accomplish the same ends. These operations are discussed in Section 2.10.
+
+We now apply the foregoing solution procedure to the truss of Fig. 2.2-1. Support conditions $\{\mathbf{D}_c\}$ are
+
+$$
+u _ {2} = v _ {2} = u _ {3} = 0 \quad \checkmark \tag {2.3-6}
+$$
+
+which means that $\{\mathbf{D}_c\} = \{\mathbf{0}\}$ . Known loads $\{\mathbf{R}_c\}$ , which correspond to as yet unknown d.o.f., are
+
+$$
+p _ {1} = 0 \quad q _ {1} = - P \quad q _ {3} = 0 \tag {2.3-7}
+$$
+
+<!-- source-page: 56 -->
+
+Equation 2.3-3 becomes $[\mathbf{K}_{11}]\{\mathbf{D}_x\} = \{\mathbf{R}_c\}$ , or
+
+$$
+\left[ \begin{array}{c c c} k _ {3} + 0. 3 6 k _ {2} & - 0. 4 8 k _ {2} & 0. 4 8 k _ {2} \\ - 0. 4 8 k _ {2} & 0. 6 4 k _ {2} & - 0. 6 4 k _ {2} \\ 0. 4 8 k _ {2} & - 0. 6 4 k _ {2} & k _ {1} + 0. 6 4 k _ {2} \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \\ v _ {3} \end{array} \right\} = \left\{ \begin{array}{c} 0 \\ - P \\ 0 \end{array} \right\} \tag {2.3-8}
+$$
+
+Since $[K_{11}]$ is nonsingular, Eq. 2.3-8 has a unique solution and can be solved for $u_{1}$ , $v_{1}$ , and $v_{3}$ . Then, since $\{D_{c}\} = \{0\}$ , Eq. 2.3-4 becomes $\{R_{x}\} = [K_{21}]\{D_{x}\}$ , or
+
+$$
+\left\{ \begin{array}{l} p _ {2} \\ q _ {2} \\ p _ {3} \end{array} \right\} = \left[ \begin{array}{c c c} - k _ {3} & 0 & 0 \\ 0 & 0 & - k _ {1} \\ - 0. 3 6 k _ {2} & 0. 4 8 k _ {2} & - 0. 4 8 k _ {2} \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \\ v _ {3} \end{array} \right\} \tag {2.3-9}
+$$
+
+which can be solved for the support reactions $p_2, q_2$ , and $p_3$ by substituting the known values of $u_1, v_1$ , and $v_3$ .
+
+The case $\{\mathbf{D}_c\} = \{\mathbf{0}\}$ , which says that all prescribed d.o.f. are zero, is common. When $\{\mathbf{D}_c\} = \{\mathbf{0}\}$ one can obtain $[\mathbf{K}_{11}]$ from $[\mathbf{K}]$ by discarding each row $i$ and column $i$ for which $D_i = 0$ . In the present example problem, we obtain Eq. 2.3-8 by discarding rows and columns 3, 4, and 5 from Eq. 2.2-6.
+
+Note also that $[K_{11}]$ can be obtained by applying the procedure described in Section 2.2, using only active d.o.f. and calculating loads associated with these d.o.f. Thus, for the truss of Fig. 2.2-1, one obtains the first column of $[K_{11}]$ by setting $u_{1}=1$ and $u_{2}=v_{2}=u_{3}=0$ , then computing the nodal loads $p_{1}, q_{1}$ , and $q_{3}$ .
+
+$$
+\begin{array}{l} p u ^ {k} \\ 3 8 3 \end{array}
+$$
+
+# 2.4 ELEMENT STIFFNESS EQUATIONS
+
+In Section 2.2, [K] is generated directly by considering the structure as a whole. This approach clarifies the physical meaning of [K] but does not lend itself to computer implementation. In practice, [K] is built by summation of coefficients from element stiffness matrices [k]. The summation process is easily computerized. In the present section we formulate the necessary element [k] matrix for a uniform plane truss member.
+
+Let the element of Fig. 2.4-1 have constant cross-sectional area $A$ and elastic modulus $E$ . Everything needed to generate [k] can be found from $A, E$ , and the four nodal coordinates $x_{i}, x_{j}, y_{i}$ , and $y_{j}$ . First, we compute
+
+$$
+L = [ (x _ {j} - x _ {i}) ^ {2} + (y _ {j} - y _ {i}) ^ {2} ] ^ {1 / 2} \tag {2.4-1a}
+$$
+
+$$
+s = \sin \beta = \frac {y _ {j} - y _ {i}}{L}. \tag {2.4-1b}
+$$
+
+$$
+c = \cos \beta = \frac {x _ {j} - x _ {i}}{L} \tag {2.4-1c}
+$$
+
+Next, as in Section 2.2, we generate columns of [k] by activating each d.o.f. in turn while keeping the others zero. The first of these four cases is shown in Fig. 2.4-2. Axial shortening $cu_{i}$ produces an axial compressive force $F = (AE/L)cu_{i}$ ,
+
+<!-- source-page: 57 -->
+
+![](images/page-057_a06a1d1eb504e6d66b8a3db8a0b95386fa262ee45e04a96a5eff94a0bbd9d79d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+L
+j
+i
+β
+x,u
+</details>
+
+Figure 2.4-1. A uniform truss element, arbitrarily oriented in the xy plane.
+
+![](images/page-057_48ef3968979592ea0aeaead02839f773d73e19769a2c4b6d5fdb879a3d351131.jpg)
+
+<details>
+<summary>text_image</summary>
+
+c = cos β
+j
+p_j
+q_j
+β + dβ
+p_i
+u_i
+q_i
+i
+c u_i
+</details>
+
+Figure 2.4-2. The truss element after nodal displacements $u_{i} > 0$ , $v_{i} = u_{j} = v_{j} = 0$ have been imposed.
+
+whose $x$ and $y$ components are $p_i = -p_j = Fc$ and $q_i = -q_j = Fs$ . These components provide static equilibrium. Thus
+
+$$
+\frac {A E}{L} \left\{ \begin{array}{l} c ^ {2} \\ c s \\ - c ^ {2} \\ - c s \end{array} \right\} u _ {i} = \left\{ \begin{array}{l} p _ {i} \\ q _ {i} \\ p _ {j} \\ q _ {j} \end{array} \right\} \tag {2.4-2}
+$$
+
+Similar results are given by the remaining displacements, $v_{i}$ , $u_{j}$ , and $v_{j}$ , when each acts alone. If all four d.o.f. may be nonzero simultaneously, we superpose results, just as in Eq. 2.2-5, and obtain
+
+$$
+\frac {A E}{L} \left[ \begin{array}{c c c c} c ^ {2} & c s & - c ^ {2} & - c s \\ c s & s ^ {2} & - c s & - s ^ {2} \\ - c ^ {2} & - c s & c ^ {2} & c s \\ - c s & - s ^ {2} & c s & s ^ {2} \end{array} \right] \left\{ \begin{array}{l} u _ {i} \\ v _ {i} \\ u _ {j} \\ v _ {j} \end{array} \right\} = \left\{ \begin{array}{l} p _ {i} \\ q _ {i} \\ p _ {j} \\ q _ {j} \end{array} \right\} \tag {2.4-3}
+$$
+
+where $c = \cos \beta$ and $s = \sin \beta$ . The square matrix, including the factor $AE / L$ , is the element stiffness matrix [k]. We abbreviate Eq. 2.4-3 as
+
+$$
+[ \mathbf {k} ] \{\mathbf {d} \} = \{\widetilde {\mathbf {r}} \} \tag {2.4-4}
+$$
+
+The jth column of [k] is the vector of loads that must be applied to element nodes to maintain the deformation state when $d_{j} = 1$ and all other element nodal d.o.f. are zero.
+
+Subsequently it will be desirable to distinguish between loads applied by the element and loads applied to the element. Loads $\{\bar{\mathbf{r}}\} = [\mathbf{k}]\{\mathbf{d}\}$ are applied to the element in order to sustain nodal d.o.f. $\{\mathbf{d}\}$ .
+
+Why are rotational d.o.f. not present in $\{d\}$ ? Such d.o.f. would be present if bending stiffness of the bar were taken into account. Then the structure would be a beam or a plane frame. By definition, truss bars do not resist bending. In a truss, rotational d.o.f. are not introduced because there is no resistance to them. Their presence would make $[K]$ a singular matrix.
+
+Special Cases. If $\beta = 0$ , as for bar 3 in Fig. 2.2-1, [k] remains 4 by 4 but contains only four nonzero coefficients, $k_{11} = k_{33} = -k_{13} = -k_{31} = AE / L$ . Therefore, displacements $v_{i}$ and $v_{j}$ produce no nodal loads $\{\overline{\mathbf{r}}\}$ . This is correct: in accordance
+
+<!-- source-page: 58 -->
+
+with our assumptions, small lateral displacements $v_{i}$ and $v_{j}$ do not strain the bar and therefore generate no force. Similar remarks apply if $\beta = \pi / 2$ , $\pi$ , and so on. That some diagonal coefficients in [k] are null does not necessarily mean that any diagonal coefficient in the assembled [K] will be null.
+
+Imagine that $\beta = 0$ and that $v_{i}$ and $v_{j}$ are suppressed by striking out rows and columns 2 and 4 from Eq. 2.4-3. What remains is
+
+$$
+\frac {A E}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right] \left\{ \begin{array}{l} u _ {i} \\ u _ {j} \end{array} \right\} = \left\{ \begin{array}{l} p _ {i} \\ p _ {j} \end{array} \right\} \tag {2.4-5}
+$$
+
+This is the stiffness equation of a bar allowed only the axial d.o.f. $u_{i}$ and $u_{j}$ . It was previously derived (Eq. 1.2-3). The orientation of the bar in space does not matter if $u_{i}, u_{j}, p_{i}$ , and $p_{j}$ remain axially directed. In Section 7.5 we will see that Eq. 2.4-3 can be obtained from Eq. 2.4-5 by a simple coordinate transformation procedure.
+
+The 2 by 2 stiffness matrix in Eq. 2.4-5 is used in various examples throughout this book.
+
+$$
+\begin{array}{l} p _ {0} ^ {\prime} C \\ 3 8 3 \end{array}
+$$
+
+# 2.5 ASSEMBLY OF ELEMENTS. PLANE TRUSS EXAMPLE
+
+The process of assembling elements to form a structure can be symbolized as $[\mathbf{K}] = \Sigma[\mathbf{k}]$ . In this section we consider arguments that apply to the three-bar truss of Fig. 2.2-1. In subsequent sections we present more general arguments and provide computer algorithms. Fortunately, the concepts and procedures of assembly depend very little on whether the structure is a truss, a frame, or a discretized continuum.
+
+Physically, construction of the truss of Fig. 2.2-1 can be visualized as follows. Structure nodes are positioned in space and are assigned labels, such as 1, 2, and 3 in Fig. 2.2-1. Bars are at first unassembled, but each bar is tagged with a node label at each end to show where it is to be placed. One by one, the bars are attached to the appropriate structure nodes. The structure gains stiffness as each bar is added.
+
+Symbolically, the foregoing process is that of starting with a null structure stiffness matrix [K], then adding to it the [k] of each element. When the last element has been added, the structure is complete and [K] is complete.
+
+The summation $[K] = \Sigma [k]$ can be performed if each $[k]$ is made to operate on $\{D\}$ , the vector of structure d.o.f. If the structure has n d.o.f., this means that each $[k]$ must be expanded to become an n by n matrix. Such expansion to “structure size” is a helpful conceptual device. Computationally it would be cumbersome. In Section 2.7 we will show how to perform the summation without expansion.
+
+To begin, we write [k] for each bar in Fig. 2.2-1 as a 4 by 4 matrix. Element node labels $i$ and $j$ can be interchanged: in other words, adding $\pi$ to angle $\beta$ does not change Eq. 2.4-3. To this extent, node labels on the bars are arbitrary. Let $k_{1}, k_{2}$ , and $k_{3}$ represent the $AE/L$ factors of the respective bars, and apply Eq. 2.4-3.
+
+<!-- source-page: 59 -->
+
+Bar 1: Let $i = 2$ and $j = 3$ . Hence $\beta = 90^{\circ}$ , $c = 0$ , and $s = 1$ .
+
+$$
+[ \mathbf {k} ] _ {1} \{\mathbf {d} \} _ {1} = k _ {1} \left[ \begin{array}{c c c c} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & - 1 \\ 0 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 1 \end{array} \right] \left\{ \begin{array}{l} u _ {2} \\ v _ {2} \\ u _ {3} \\ v _ {3} \end{array} \right\} \tag {2.5-1}
+$$
+
+Bar 2: Let $i = 1$ and $j = 3$ . Hence $\beta = 126.9^{\circ}$ , $c = -0.6$ , and $s = 0.8$ .
+
+$$
+[ \mathbf {k} ] _ {2} \{\mathbf {d} \} _ {2} = k _ {2} \left[ \begin{array}{c c c c} 0. 3 6 & - 0. 4 8 & 0. 3 6 & 0. 4 8 \\ - 0. 4 8 & 0. 6 4 & 0. 4 8 & - 0. 6 4 \\ - 0. 3 6 & 0. 4 8 & 0. 3 6 & - 0. 4 8 \\ 0. 4 8 & - 0. 6 4 & 0. 4 8 & 0. 6 4 \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \\ u _ {3} \\ v _ {3} \end{array} \right\} \tag {2.5-2}
+$$
+
+Bar 3: Let $i = 1$ and $j = 2$ . Hence $\beta = 180^{\circ}$ , $c = -1$ , and $s = 0$ .
+
+$$
+[ \mathbf {k} ] _ {3} \{\mathbf {d} \} _ {3} = k _ {3} \left[ \begin{array}{c c c c} 1 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 \\ - 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \\ u _ {2} \\ v _ {2} \end{array} \right\} \tag {2.5-3}
+$$
+
+Each of the foregoing three [k] matrices must be expanded from 4 by 4 to 6 by 6. This is done by adding two rows of zeros and two columns of zeros—at the start in Eq. 2.5-1, in the middle in Eq. 2.5-2, and at the end in Eq. 2.5-3. Thus each element displacement vector becomes identical to the structure displacement
+
+vector {D}:
+
+$$
+\left| \begin{array}{c c} \left. \begin{array}{c c c c c c} \mathbf {g} _ {1} & \varepsilon^ {\prime} \rho_ {m a x} \dot {\varepsilon} ^ {d} & \dot {\varepsilon} \varepsilon^ {l} & \rho_ {m a x} ^ {2} & \mathcal {L} _ {f} ^ {j} & \nu_ {m a x} ^ {i} \\ \hline \{\mathbf {d} \} _ {1} = \{\mathbf {d} \} _ {2} = \{\mathbf {d} \} _ {3} = \{\mathbf {D} \} = \left\lfloor u _ {1} v _ {1} u _ {2} v _ {2} u _ {3} v _ {3} \right\rfloor^ {T} & \end{array} \right. & \sqrt {\frac {1}{2} \frac {1}{2} \frac {1}{2}} \\ \hline \end{array} \right.
+$$
+
+Equation 2.5-4 enforces compatibility; that is, it makes end points of the pin-connected bars coincident under any displacement $\{D\}$ .
+
+The addition of rows and columns of zeros can be physically justified as follows. Consider, for example, the expanded [k] of element 2, in which rows 3 and 4 and columns 3 and 4 contain only zeros. Element 2 and node 2 are not connected. Therefore, no displacement of node 2 can strain element 2. Thus displacements $u_{2}$ and $v_{2}$ are associated with zero force, which accounts for the two columns of zeros. In addition, none of the six d.o.f. can produce forces at node 2 because there is no connecting material to resist strain or transmit load. This accounts for the two rows of zeros.
+
+The reader can easily check that the expanded [k]'s from Eqs. 2.5-1, 2.5-2, and 2.5-3 do indeed add up to the structure [K] in Eq. 2.2-6. When one regards a column of [K] as a set of resisting forces, with each contributory force coming from elements connected to a common node, it becomes clear that one obtains [K] by adding element stiffness matrices.
+
+Stiffness Coefficients That Remain Zero. When is $K_{ij} = 0$ in the assembled structure? A column of [K] represents nodal loads associated with activation of one and only one d.o.f. Activation of a d.o.f. creates nodal loads in only the element or elements that contain the d.o.f. in question. Other elements are not strained and produce no nodal loads. Therefore, in a column $j$ of [K], coefficient $K_{ij}$ is
+
+<!-- source-page: 60 -->
+
+zero unless structure d.o.f. i and j are both present in at least one element. ( $K_{ij}$ may be zero even if d.o.f. i and j are shared by an element. For example, in Eq.
+
+2.2-6, $K_{41} = K_{14} = 0$ because/bar 3 of the truss happens to be horizontal.)
+
+# 2.6 ASSEMBLY REGARDED AS SATISFYING EQUILIBRIUM
+
+$$
+\begin{array}{r l} {\mathrm {d} \theta} & {= \mathrm {d} \theta^ {\prime} / \mathrm {d} \theta^ {\prime} / \mathrm {d} \theta^ {\prime} / \mathrm {d} \theta^ {\prime} / \mathrm {d} \theta^ {\prime}} \\ {\mathrm {d} \theta} & {= \mathrm {d} \theta^ {\prime} / \mathrm {d} \theta^ {\prime} / \mathrm {d} \theta^ {\prime} / \mathrm {d} \theta^ {\prime} / \mathrm {d} \theta^ {\prime}} \end{array}
+$$
+
+In stress analysis, assembly of elements can be regarded as a process of writing equations stating that each node of the structure is in static equilibrium under all loads applied to it. Nodal loads come from elements because of self-weight, temperature change, and lack of fit, from deformations associated with nodal displacements, and from external sources. The equilibrium argument is now explained with particular reference to the plane truss.
+
+Loads applied to nodes because of gravity in the negative y direction are shown in Fig. 2.6-1a. Written formally for a four d.o.f. element, these loads are
+
+$$
+\{\mathbf {r} _ {w} \} = \frac {W}{2} \left[ \begin{array}{l l l l} 0 & - 1 & 0 & - 1 \end{array} \right] ^ {T} \tag {2.6-1}
+$$
+
+where the total element weight W is equally apportioned to the two nodes. (We will assume that bending of an individual bar under its own weight can be neglected.) If a fully restrained bar is initially stress-free and then is uniformly heated T degrees, it sustains an axial compressive force $F = \alpha EAT$ , where $\alpha$ is the coefficient of thermal expansion (Fig. 2.6-1b). The resulting nodal load vector is
+
+$$
+\{\mathbf {r} _ {T} \} = \alpha E A T \left[ - c - s c s \right] ^ {T} \tag {2.6-2}
+$$
+
+where $c = \cos \beta$ and $s = \sin \beta$ . The same forces $\{r_{T}\}$ would arise from the force-fitting of a bar that is initially $\alpha LT$ units too long. We will use $\{r_{e}\}$ to symbolize element loads. For the loads mentioned here,
+
+$$
+\{\mathbf {r} _ {e} \} = \{\mathbf {r} _ {W} \} + \{\mathbf {r} _ {T} \} \tag {2.6-3}
+$$
+
+Loads $\{\tilde{r}\} = [k]\{d\}$ are loads applied to an element to sustain its deformation state $\{d\}$ . Therefore, equal and opposite loads $\{r\} = -\{\tilde{r}\}$ are applied by the element
+
+![](images/page-060_8df09e8bcc260435936b172be1f2e943e89857eb90b99b0ed17d2d2d079f36a0.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+W
+j
+=
+W/2
+j
+i
+x
+i
+</details>
+
+(a)
+
+![](images/page-060_d2ddf0fe28dec72a4cb6bc62c91b8e8b9649080dc54fabab08281b176cd8006f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+F = αEAT
+y
+β
+x
+i
+F
+F
+j
+F
+Fc
+Fc
+i
+Fs
+Fs
+j
+Fc
+c = cos β
+s = sin β
+</details>
+
+(b)   
+Figure 2.6-1. (a) Allocation of the weight W of a truss bar to its nodes. (b) Nodal loads associated with uniform heating of T degrees above the unstressed temperature.
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_007.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_007.md
new file mode 100644
index 00000000..d2481854
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_007.md
@@ -0,0 +1,617 @@
+<!-- source-page: 61 -->
+
+to structure nodes, in accord with Newton's third law. That is, nodal loads associated with element deformation are
+
+$$
+\{\mathbf {r} \} = - [ \mathbf {k} ] \{\mathbf {d} \} \tag {2.6-4}
+$$
+
+Finally, loads applied to structure nodes by external sources are called $\{P\}$ . For example, $\{P\} = \left[0 - P \quad 0 \quad 0 \quad 0 \quad 0\right]^{T}$ for the truss of Fig. 2.2-1. Loads applied by fixed supports are usually not included in $\{P\}$ because they would immediately be discarded by standard methods of imposing support conditions (see Section 2.10).
+
+The set of equations that places each node in static equilibrium is
+
+$$
+\{\mathbf {P} \} + \sum_ {n = 1} ^ {\text {numel}} \{\mathbf {r} \} _ {n} + \sum_ {n = 1} ^ {\text {numel}} \{\mathbf {r} _ {e} \} _ {n} = \{\mathbf {0} \} \tag {2.6-5}
+$$
+
+where $\text{numel}$ is the number of elements in the structure. Summations are written because a typical node is connected to more than one element. However, a node receives $\{\mathbf{r}\}$ and $\{\mathbf{r}_e\}$ contributions only from the elements to which it is connected; thus, Eq. 2.6-5 implies the expansion of element vectors to “structure size” by addition of many zeros.
+
+Substitution of Eq. 2.6-4 into 2.6-5 yields
+
+$$
+[ \mathbf {K} ] \{\mathbf {D} \} = \{\mathbf {R} \} \tag {2.6-6}
+$$
+
+where
+
+$$
+[ \mathbf {K} ] = \sum_ {n = 1} ^ {\text {numel}} [ \mathbf {k} ] _ {n} \quad \text {and} \quad \{\mathbf {R} \} = \{\mathbf {P} \} + \sum_ {n = 1} ^ {\text {numel}} \{\mathbf {r} _ {e} \} _ {n} \tag {2.6-7}
+$$
+
+Summations imply the expansion of element arrays [k] and $\{\mathbf{r}_e\}$ to "structure size" so that $\{\mathbf{d}\}_n$ of each element $n$ becomes identical to the structure displacement vector $\{\mathbf{D}\}$ .
+
+For a plane truss, Eq. 2.6-6 contains two equations per node. For a space truss there would be three equations per node.
+
+# 2.7 ASSEMBLY AS DICTATED BY NODE NUMBERS
+
+Element node labels, such as $i$ and $j$ in Fig. 2.4-1, serve only as convenient tags during the generation of element matrices. In the assembly process it is the structure node labels, such as 1, 2, and 3 in Fig. 2.2-1, that determine the locations in [K] and {R} to which coefficients in element arrays [k] and {r\_e} are assigned. This is true of any finite element, regardless of its type, size, shape, or number of nodes.
+
+As a simple example, consider a hypothetical structure that has two triangular elements and one d.o.f. per node (Fig. 2.7-1). This structure is not a truss. We need not know what physical problem is being modeled. We need say only that the characteristic matrix [k] of each element is 3 by 3, that structure nodes of
+
+<!-- source-page: 62 -->
+
+![](images/page-062_0d74db5330a555c4b8c022b46cb0b0a9623abc003ed48759099e38cd378e2627.jpg)
+
+<details>
+<summary>text_image</summary>
+
+3.00 / node
+2
+k
+i
+①
+j
+②
+k
+i
+①
+j
+②
+3
+1
+4
+=
+</details>
+
+Figure 2.7-1. A hypothetical four-node structure built of two triangular elements. This structure is not a truss. Each node has a single d.o.f. Node labels are assigned arbitrarily.
+
+element 1 are numbered 1, 4, and 2, and that structure nodes of element 2 are numbered 4, 3, and 2. For elements 1 and 2 in Fig. 2.7-1, we write
+
+$$
+\Gamma_ {i} \leq [ \mathbf {k} ] _ {1} \{\mathbf {d} \} _ {1} = \left[ \begin{array}{l l l} a _ {1} & a _ {2} & a _ {3} \\ a _ {4} & a _ {5} & a _ {6} \\ a _ {7} & a _ {8} & a _ {9} \end{array} \right] \left\{ \begin{array}{l} d _ {i} \\ d _ {j} \\ d _ {k} \end{array} \right\} \quad \text { and } \quad [ \mathbf {k} ] _ {2} \{\mathbf {d} \} _ {2} = \left[ \begin{array}{l l l} b _ {1} & b _ {2} & b _ {3} \\ b _ {4} & b _ {5} & b _ {6} \\ b _ {7} & b _ {8} & b _ {9} \end{array} \right] \left\{ \begin{array}{l} d _ {i} \\ d _ {j} \\ d _ {k} \end{array} \right\} = \left[ \begin{array}{l} \vdots \\ (2. 7 - 1) \end{array} \right.
+$$
+
+where $d$ 's are nodal d.o.f. Letter subscripts indicate element node labels. It does not matter how the $a$ 's and $b$ 's are calculated; it matters only that they exist. In the following explanation we ignore symmetry of the [k]'s to show more clearly what happens to the $a$ 's and $b$ 's. Let nodal "loads" be called $\{\overline{\mathbf{r}}\}$ , where $\{\overline{\mathbf{r}}\} = [\mathbf{k}]\{\mathbf{d}\}$ . Now consider element 1. Its nodal loads, first in element labeling and then in structure labeling, are
+
+$$
+\begin{array}{l} \bar {r} _ {i} = a _ {1} d _ {i} + a _ {2} d _ {j} + a _ {3} d _ {k} \quad \checkmark \quad \bar {r} _ {1} = a _ {1} D _ {1} + a _ {2} D _ {4} + a _ {3} D _ {2} \\ \bar {r} _ {j} = a _ {4} d _ {i} + a _ {5} d _ {j} + a _ {6} d _ {k} \quad \text { and } \quad \bar {r} _ {4} = a _ {4} D _ {1} + a _ {5} D _ {4} + a _ {6} D _ {2} \tag {2.7-2} \\ \vec {r} _ {k} = a _ {7} d _ {i} + a _ {8} d _ {j} + a _ {9} d _ {k} \quad \vec {r} _ {2} = a _ {7} D _ {1} + a _ {8} D _ {4} + a _ {9} D _ {2} \\ \end{array}
+$$
+
+To the latter group of equations we can add the equation $\bar{r}_{3}=0$ because node 3 is not attached to element 1. After this addition, and after rearrangement to place the D's in numerical order, we have for element 1
+
+$$
+\left\{ \begin{array}{l} \bar {r} _ {1} \\ \bar {r} _ {2} \\ \bar {r} _ {3} \\ \bar {r} _ {4} \end{array} \right\} = \left[ \begin{array}{c c c c} a _ {1} & a _ {3} & 0 & a _ {2} \\ a _ {7} & a _ {9} & 0 & a _ {8} \\ 0 & 0 & 0 & 0 \\ a _ {4} & a _ {6} & 0 & a _ {5} \end{array} \right] \left\{ \begin{array}{l} D _ {1} \\ D _ {2} \\ D _ {3} \\ D _ {4} \end{array} \right\} \tag {2.7-3}
+$$
+
+in which the square matrix is $[k]_{1}$ . Element 2 can be treated similarly. Then, because the two matrices $[k]_{1}$ and $[k]_{2}$ have the same size and operate on the same vector of d.o.f. $\{D\}$ , we can write $[K]\{D\} = (\Sigma [k])(\{D\})$ , where
+
+$$
+[ \mathbf {K} ] = [ \mathbf {k} ] _ {1} + [ \mathbf {k} ] _ {2} = \left[ \begin{array}{l l l l} a _ {1} & a _ {3} & 0 & a _ {2} \\ a _ {7} & a _ {9} & 0 & a _ {8} \\ 0 & 0 & 0 & 0 \\ a _ {4} & a _ {6} & 0 & a _ {5} \end{array} \right] + \left[ \begin{array}{l l l l} 0 & 0 & 0 & 0 \\ 0 & b _ {9} & b _ {8} & b _ {7} \\ 0 & b _ {6} & b _ {5} & b _ {4} \\ 0 & b _ {3} & b _ {2} & b _ {1} \end{array} \right] \tag {2.7-4}
+$$
+
+We see that coefficients below the diagonal of an element [k] matrix (before reordering) may appear above the diagonal in [K]. This happens in the present example (but not in the truss example of Section 2.5) because expansion and
+
+<!-- source-page: 63 -->
+
+![](images/page-063_0d3462f689cc60572efe1b7897d45a849e68898814c0be54dd5655623e98fa3c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+3
+①
+2
+5
+②
+1
+4
+</details>
+
+$$
+\left\{\mathbf {R} \right\} = \left\{ \begin{array}{l} \triangle \\ \triangle \\ \triangle \end{array} \right\} + \left\{ \begin{array}{l} \square \\ \square \\ \square \\ \square \end{array} \right\} = \left\{ \begin{array}{l} \square \\ \square \\ \triangle \\ \square \\ \square \end{array} \right\} \text {Structure}
+$$
+
+$$
+\begin{array}{l} [ \mathbf {K} ] = \left[ \begin{array}{c c} & \triangle \triangle \quad \triangle \\ & \triangle \triangle \quad \triangle \\ & \triangle \triangle \quad \triangle \end{array} \right] + \left[ \begin{array}{c c} \square \square & \square \square \\ \square \square & \square \square \\ & \square \square & \square \square \\ \square \square & \square \square \end{array} \right] = \left[ \begin{array}{c c} \square \square & \square \square \\ \square \triangle \triangle \square \triangle \\ \triangle \triangle & \triangle \\ \square \square & \square \square \\ \square \triangle \triangle \square \triangle \end{array} \right] \\ ⓘ \\ ② \\ \end{array}
+$$
+
+Figure 2.7-2. A hypothetical structure having one d.o.f. per node and built from a three-node element and a four-node element.
+
+rearrangement of coefficients are needed to make element d.o.f. vectors {d} identical to the structure d.o.f. vector {D}.
+
+If element d.o.f. labels are interchanged, then coefficients $a_i$ and $b_i$ in Eqs. 2.7-1 will be rearranged in the element [k] matrices. However, if structure node labels are preserved, the $a_i$ and $b_i$ are assigned to the same locations in the structure matrix [K] as before. For example, if in Fig. 2.7-1 the labels $ijk$ are permuted to $jki$ , then the first of Eqs. 2.7-2 becomes $\bar{r}_j = a_1d_j + a_2d_k + a_3d_i$ , but $i = 2, j = 1$ , and $k = 4$ , so that $\bar{r}_1 = a_1D_1 + a_2D_4 + a_3D_2$ as before.
+
+Another example of assembly appears in Fig. 2.7-2. Again the structure is hypothetical and is not a truss. Element matrices are shown already expanded to “structure size.” Because element node labels are not shown and specific $k_{ij}$ are not identified, Fig. 2.7-2 shows only the matrix topology of assembly. Note that $\{R\}$ and $[K]$ have the same row topology.
+<table><tr><td>DO 500 N=1, NUMELCALL ELEMENTKK(1) = NOD(1,N)KK(2) = NOD(2,N)KK(3) = NOD(3,N)DO 400 I=1,3K = KK(I) →R(K) = R(K)+RE(I)DO 300 J=1,3L = KK(J)S(K,L) = S(K,L)+SE(I,J)300 CONTINUE400 CONTINUE500 CONTINUE</td><td>Terms:S = [K]SE = [k]R = {R}RE = {r_e}</td><td>Example (Fig. 2.7-1):NOD(1,1) = 1NOD(2,1) = 4NOD(3,1) = 2NOD(1,2) = 4NOD(2,2) = 3NOD(3,2) = 2</td></tr><tr><td>(a)</td><td>(b)</td><td>(c)</td></tr></table>
+
+Figure 2.7-3. (a) Fortran coding for assembly of element matrices. Each element has three nodes and one d.o.f. per node. NUMEL = number of elements in the structure. (b) Typical structural notation (as in Eqs. 2.6-7). (c) Example of array NOD.
+
+<!-- source-page: 64 -->
+
+DO 500 N=1, NUMEL
+CALL ELEMENT
+KK(2) = 2*NOD(1,N)
+KK(1) = KK(2) - 1
+KK(4) = 2*NOD(2,N)
+KK(3) = KK(4) - 1
+DO 400 I=1,4
+K = KK(I)
+R(K) = R(K) + RE(I)
+DO 300 J=1,4
+L = KK(J)
+S(K,L)=S(K,L)+SE(I,J)
+300 CONTINUE
+400 CONTINUE
+500 CONTINUE   
+(a)
+
+![](images/page-064_16099ed9a0683b149c1ffc142620a3b0cbc6a1031abd87220ef7b0c4a4716887.jpg)
+
+<details>
+<summary>text_image</summary>
+
+D6 = v3
+j=3
+D5 = u3
+②
+D2 = v1
+D1 = u1
+i=1
+</details>
+
+(b)
+
+For bar 2 (N = 2):
+
+$$
+\mathrm{NOD} (1, 2) = 1
+$$
+
+$$
+\mathrm{NOD} (2, 2) = 3
+$$
+
+$$
+\mathrm{KK} (2) = 2 * 1 = 2
+$$
+
+$$
+\mathrm{KK} (1) = 2 - 1 = 1
+$$
+
+$$
+\mathrm{KK} (4) = 2 * 3 = 6
+$$
+
+$$
+\mathrm{KK} (3) = 6 - 1 = 5
+$$
+
+(c)
+
+Figure 2.7-4. (a) Fortran coding for assembly of element matrices. Each element has two nodes and two d.o.f. per node. NUMEL = number of elements in the structure. (b) Bar 2 of the truss of Fig. 2.2-1, showing the numbering of nodes and d.o.f. (c) Contents of arrays for bar 2.
+
+Expansion to “structure size” is a conceptual device that need not actually be carried out. Assembly can be stated as an addition algorithm that assigns element coefficients to positions dictated by structure node numbers associated with the element. Such an algorithm appears in Fig. 2.7-3. Here it is assumed that each element has three nodes, each with a single d.o.f., as in Fig. 2.7-1. Structure node numbers that correspond to element node labels i, j, and k are assumed to have been previously stored in rows 1, 2, and 3 of array NOD, which has as many columns as there are elements in the structure. Subroutine ELEMENT (not shown) is assumed to return element matrices $[k]$ and $\{r_{e}\}$ in arrays SE and RE, respectively, via a COMMON block (not shown). Information in SE and RE is repeatedly created and destroyed as subroutine ELEMENT is called for each element in turn. Information in SE and RE is added into structural arrays S and R as part of the assembly process $[K] = \Sigma [k]$ and $\{R\} = \Sigma \{r_{e}\} + \{P\}$ . Externally applied loads $\{P\}$ must be separately added to $\{R\}$ after completing the algorithm in Fig. 2.7-3. It is assumed that arrays S and R are null before executing this assembly algorithm.
+
+A similar assembly algorithm, applicable to a plane truss, is shown in Fig. 2.7-4. Each element has two nodes and each node has two d.o.f. Otherwise, this algorithm is like that of Fig. 2.7-3.
+
+The algorithm of Fig. 2.7-4 can easily be altered to deal with a space truss, where each node has three d.o.f. and each [k] is 6 by 6. Array KK must contain six entries. The first three are KK(3) = 3\*NOD(1,N), KK(2) = KK(3)-1, and KK(1) = KK(3)-2. Loop indices 1 and J must run from 1 to 6.
+
+# 2.8 NODE NUMBERING THAT EXPLOITS MATRIX SPARSITY
+
+A finite element structure with many d.o.f. has a sparse coefficient matrix [K]. That is, most of the individual coefficients $K_{ij}$ are zero. Sparsity should be exploited in order to economize on computer storage space and running time. Spars-
+
+<!-- source-page: 65 -->
+
+ity may be exploited by various schemes. In the present section we emphasize bandedness, which is among the simpler schemes.
+
+The number of nonzero coefficients in [K], and their numerical values, are independent of how structure nodes are numbered. A change in structure node numbers changes only the arrangement of nonzero $K_{ij}$ . Figure 2.8-1 is a case in point. The topology of nonzero coefficients in Fig. 2.8-1 can be understood by recalling that for any structure, a structure stiffness coefficient $K_{ij}$ can be nonzero only if d.o.f. $i$ and $j$ are both present in at least one element.
+
+Consider next the plane truss of Fig. 2.8-2. [K] is 12 by 12. For the first numbering, the topology of [K] is shown in Fig. 2.8-3a. The semibandwidth (also called the half-bandwidth) is given the symbol $b$ . Here $b = 6$ . Matrix [K] is symmetric and has a total bandwidth $2b - 1$ . Bandwidth $2b - 1$ indicates the horizontal span of the zone in which all nonzero $K_{ij}$ reside. This zone lies along the principal diagonal of [K]. Some zeros may appear within the band, but only zeros appear outside it.
+
+![](images/page-065_78e20c699d896e7327421a74ae9464bf0577580c5d059de34294930a77d9790c.jpg)
+
+A small semibandwidth is usually achieved by placing consecutive node numbers along the shorter dimension of a structure. The reader may check that the alternative numbering, in Fig. 2.8-2b, achieves the maximum possible semibandwidth for this problem (b = 12). A small value of b is desired.
+
+The entire information content of a symmetric banded matrix resides in coefficients within the semiband. In practice, matrix order $n_{eq}$ may greatly exceed semibandwidth b. If we store and process only the semiband rather than all coefficients in the upper triangle of [K], we decrease storage requirements by a factor of about $n_{\mathrm{eq}}b/(n_{\mathrm{eq}}^{2}/2)=2b/n_{\mathrm{eq}}$ . In addition, as compared with processing a full but symmetric matrix, we reduce equation-solving expense by a factor of about $3b^{2}/n_{eq}^{2}$ . For example, if $n_{eq}=10b$ , then the time needed to solve for d.o.f. {D} is reduced by a factor of about 30.
+
+A simple storage format for the semiband is shown in Fig. 2.8-3b. Each row is shifted left: 1 space for row 2, 2 spaces for row 3, and in general i - 1 spaces for row i. Thus all diagonal coefficients $K_{ii}$ of the matrix are stored in column 1 of the semiband array. To program the assembly of [K] in this form we need change only the innermost loop of the algorithm in Fig. 2.7-4. The required form of this innermost loop is shown in Fig. 2.8-4. The IF statement avoids coefficients $K_{ij}$ below the main diagonal of [K], which would fall outside the stored semiband.
+
+![](images/page-065_2d4384dd3d12336b933585603f15b2ceb7c415a5e79817c88ddda272a8fc49d8.jpg)  
+Figure 2.8-1. A hypothetical four-element structure. Each element has two nodes. Each node has one d.o.f. Two different numberings (one the reverse of the other) and their associated stiffness matrices are shown. Capital letters indicate nonzero stiffness coefficients.
+
+<!-- source-page: 66 -->
+
+![](images/page-066_e669aa439722ca0cc04dbfcee0c19df31f17251cda12a66569238e7502fddacc.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+1
+2
+3
+4
+5
+6
+x,u
+(good)
+</details>
+
+(a)
+
+![](images/page-066_c9e2dd110352350535a942d2cc3e0cf6c679ed0353f5ce35e837a7c3f6009a44.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+6
+5
+4
+1
+2
+3
+(poor)
+x,u
+</details>
+
+(b)   
+Figure 2.8-2. A plane truss, showing node numberings that are (a) favorable and (b) unfavorable for achieving a banded stiffness matrix [K].
+
+Some Computational Details. From Fig. 2.8-1 we see that the rightmost nonzero coefficient $K_{ij}$ in any row may be close to the diagonal or far away. To compute $b$ we must look for the $K_{ij}$ farthest away. Specifically, $b$ is the maximum of all $n_{\text{eq}}$ values of $b_i$ , where $b_i$ is the number of columns from and including the diagonal to the rightmost nonzero $K_{ij}$ in row $i$ . Or, we can compute $b$ by adding one to the magnitude of the maximum difference in active global d.o.f. in an element, using the element that displays the largest difference. For example, in Fig. 2.8-1a we find from element 2-5 that $b = (5 - 2) + 1 = 4$ . In Fig. 2.8-2a we obtain $b = 6$ from elements 1-3, 2-4, 3-5, and 4-6; respectively, they give the d.o.f. differences $6 - 1 = 5, 8 - 3 = 5, 10 - 5 = 5$ , and $12 - 7 = 5$ . (D.o.f. that are suppressed, as by a fixed support, may not be listed in $\{\mathbf{D}\}$ . Then the numbering of active d.o.f. will not correspond to node numbers in such a convenient way. Information needed to determine $b$ can still be found in columns of array ID. See Section 2.10.)
+
+Both parts of Fig. 2.8-1 display a solid line that bounds the uppermost nonzero coefficient in each column of the matrix. This line is called the skyline (or envelope, or profile). We see that the matrices in Fig. 2.8-1 have the same semibandwidth but different skylines. As an alternative to the band storage scheme of Fig. 2.8-3, one could elect to store only the portion of each matrix column between
+
+![](images/page-066_7cbd8eeed85feee91ef2de62b97119f0cbb2deb0109f1eac356326bb4058fd67.jpg)
+
+<details>
+<summary>text_image</summary>
+
+b = 6
+X X X X X X
+X X X X X O
+X X X X X X
+X X X X X O
+X X X X X X
+X X X X X X
+X X X X X
+X X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+X X X X
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+x
+12
+Symmetric
+Zero
+Zero
+</details>
+
+(a)
+
+![](images/page-066_3994d2cd65714c45ac42354fd88be1736bc0348f76db85063c0b568a75462296.jpg)
+
+<details>
+<summary>text_image</summary>
+
+b=6
+X X X X X X
+X X X X X O
+X X X X X X
+X X X X X O
+X X X X X X
+X X X X X
+X X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+X X X
+Zero
+</details>
+
+(b)   
+Figure 2.8-3. (a) [K] for the plane truss of Fig. 2.8-2a (the “good” numbering). X = nonzero coefficient. (b) Band form storage of the same matrix.
+
+<!-- source-page: 67 -->
+
+![](images/page-067_9eaa4088c8d884611e4455335fc4a3e46b2493d3fa4c81a6db2220154b5bc99d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+DO 300 J=1,4
+IF (KK(J) ,LT. K) GO TO 300
+L = KK(J) - K + 1
+S(K,L) = S(K,L) + SE(I,J)
+300 CONTINUE
+</details>
+
+Figure 2.8-4. Altered form of the innermost loop of Fig. 2.7-4 to achieve the semiband storage format of Fig. 2.8-3b.
+
+the skyline and the diagonal. Then Fig. 2.8-1b would be preferable to Fig. 2.8-1a. Another reason to prefer Fig. 2.8-1b is that equation solving creates fills; that is, a zero beneath the skyline is often changed to nonzero by the equation-solving process. There are three such zeros in Fig. 2.8-1a but none in Fig. 2.8-1b. Storage space must be reserved for fills, and fills must be processed after they are created.
+
+The attention given to exploiting sparsity is worthwhile in reducing computation cost. This is clear from Fig. 2.8-5: for every nonzero coefficient in [K] there are over 100 zero coefficients. Many different storage schemes and equation-solving algorithms are available. Extensive discussion appears in Ref. 2.1.
+
+# 2.9 AUTOMATIC ASSIGNMENT OF NODE NUMBERS
+
+Imagine that a node must be added to the left of the truss of Fig. 2.8-2a. Two new bars will connect the new node to existing nodes 1 and 2. The entire structure must be renumbered if low bandwidth is to be preserved. It would be far more convenient if the new node could be given the next available number, 7, and the computer program could do the rest. That is, the computer program should accept arbitrary node numbers, adopt new numbers for efficient internal operations, and produce results in the user's original numbering system.
+
+![](images/page-067_b0ec3100c7195f409e897149a58f7f165690c6c239f7e295baf57c7e553134cf.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+3D wireframe model of a mechanical component with internal mesh structure (no text or symbols)
+</details>
+
+Figure 2.8-5. Finite element model of a baseplate, built of 2633 bar, plate, shell, and solid elements. There are 1005 nodes and from 3 to 6 d.o.f. per node. The semibandwidth is 852 and the density of [K] is 0.85% [2.2] (Courtesy of G.C. Everstine, David W. Taylor Naval Ship R&D Center, Bethesda, Maryland.)
+
+<!-- source-page: 68 -->
+
+![](images/page-068_61782e59f6f92fa8f21a43f769870f441b291b16e5aa6886a87e18a29a148866.jpg)  
+Figure 2.9-1. A graph that might represent a piping system. Node numbers have been chosen by a formal algorithm rather than by inspection.
+
+Figure 2.9-1 shows the numbering achieved by a node-renumbering algorithm. For a single d.o.f. per node the semibandwidth is $b = 8$ . Numbering by inspection, one would probably not achieve so low a value of $b$ without considerable thought and several trials.
+
+Renumbering algorithms exploit graph theory and its terminology [2.2,2.3]. Details are not presented here. Broadly speaking, a typical reordering algorithm examines a few promising numbering patterns (of the $N!$ possibilities for a $N$ by $N$ matrix) and selects the best one. The algorithm does not guarantee an optimum numbering, or even guarantee improvement over the numbering supplied to it. But the goal of reducing bandwidth—or skyline, or fills—is usually achieved. No single strategy is best for all goals or for all finite element meshes. Interestingly, simple reversal of node numbers (as in Fig. 2.8-1) may reduce the skyline while leaving bandwidth unchanged.
+
+A “frontal” or “wave front” equation solver processes equations in element order rather than in node order. One then uses an algorithm that produces good element numbering.
+
+A table of nodal connectivity is needed for node numbering. There is some computational expense in generating this table and doing the renumbering. But automatic renumbering is cost-effective, especially if the same numbering is used in repeated solutions, as is the case in nonlinear problems. Automatic renumbering is always worthwhile from the viewpoint of user convenience.
+
+# 2.10 DISPLACEMENT BOUNDARY CONDITIONS
+
+In the structure stiffness equations $[K]\{D\} = \{R\}$ , matrix $[K]$ is singular and no unique solution for d.o.f. $\{D\}$ is possible if the structure is unsupported. Some d.o.f. in $\{D\}$ must be prescribed to enable a solution. Similarly, in a nonstructural problem where the matrices have other physical meanings, one or more d.o.f. $D_{i}$ must be prescribed. The method described by Eqs. 2.3-2 to 2.3-4 usually requires row and column interchanges. Therefore, it is not well suited to computer programming. In this section we consider alternative procedures for imposing prescribed values of one or more d.o.f. $D_{i}$ . Initially we assume that all prescribed $D_{i}$ are prescribed as zero. Prescribed nonzero $D_{i}$ are considered subsequently. ( $\frac{1}{2}$ )
+
+A general-purpose program for structural analysis typically allows six d.o.f. per node (displacement in each coordinate direction and rotation about each coordinate axis). Often, not all of these d.o.f. are needed in the analysis of a particular structure. Indeed, for a plane structure, some must be eliminated: if no element
+
+<!-- source-page: 69 -->
+
+resists a displacement $D_{i}$ , then $K_{ii} = 0$ and [K] is singular. A plane structure, by definition, resists only in-plane distortions. Therefore, nodal d.o.f. that represent $z$ -direction motion and rotations about $x$ and $y$ axes must be eliminated (i.e., prohibited) at all nodes. For a plane truss (but not a plane frame) we must also prohibit rotation $\theta_{z}$ about the $z$ -axis at all nodes. If $\theta_{z}$ is prohibited at all nodes, then truss elements, and the truss itself, can still have rigid body rotation in the $xy$ plane because the prohibited rotations are not among their nodal d.o.f. (One can imagine that nodes of a plane truss are frictionless pins that connect bars together. A rotation $\theta_{z}$ of a pin does not deform the truss. Accordingly, $\theta_{z}$ is not resisted, and the associated rotational stiffness is zero.)
+
+ID Array. We introduce a “destination array” ID, which is to be filled with numbers that indicate the locations in [K] to which element coefficients $k_{ij}$ are to be assigned. Array ID has as many columns as there are nodes in the structure and as many rows as the maximum number d.o.f. allowed per node (typically six rows, for three displacement d.o.f. and three rotation d.o.f.). By use of array ID we will directly assemble matrix $[K_{11}]$ of Eq. 2.3-3, although we will call it simply [K] in what follows. Coefficients in $[K_{12}]$ of Eq. 2.3-3 will be discarded, which is acceptable if $\{D_{c}\} = \{0\}$ as is currently assumed.
+
+Consider, for example, Fig. 2.10-1. If this structure is to be analyzed by use of a program that allows six d.o.f. per node, array ID has 6 rows and 8 columns. We start with ID null and, by means of input data, insert a 1 for each d.o.f. to be eliminated because it has a prescribed zero displacement. Support conditions in Fig. 2.10-1 dictate that $v_{1} = u_{5} = v_{5} = u_{7} = 0$ . In addition, at each node i we must suppress z-direction displacement $w_{i}$ (normal to the xy plane) and rotations $\theta_{xi}, \theta_{yi}$ , and $\theta_{zi}$ about x, y, and z axes, respectively. The resulting ID array is shown in Fig. 2.10-2.
+
+The next step is to convert array ID to a list of equation numbers by counting zeros in successive columns and converting each 1 to a zero. This counting, accomplished by the algorithm of Fig. 2.10-4, produces the result shown in Fig. 2.10-3. Zeros now indicate d.o.f. that are not to appear in vector $\{D\}$ of active d.o.f. Nonzeros indicate equation numbers associated with active d.o.f. For example, to locate the $D_{i}$ associated with node 7, we go to column 7 in Fig. 2.10-3, and find that all d.o.f. are suppressed except $v_{7}$ , which appears as $D_{10}$ in $\{D\}$ . Matrix $[K]$ for the supported structure is 12 by 12, where NEQ = 12 is computed in Fig. 2.10-4.
+
+To assemble structural equations in the band format of Fig. 2.8-3b, while allowing only active d.o.f. to be present in $\{D\}$ , we can make use of array ID. An assembly algorithm, obtained by combining and modifying Figs. 2.7-4 and 2.8-4, is shown in Fig. 2.10-5. Array KK is filled with structural equation numbers for
+
+![](images/page-069_fc5653c991ec022af8b3d3e4fa0f2b87877688d154749872b1d378e6712b3645.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+x,u
+2
+4
+6
+8
+1
+3
+5
+7
+</details>
+
+Figure 2.10-1. A plane truss, showing support conditions.
+
+<!-- source-page: 70 -->
+
+$$
+\begin{array}{l} \text {[ID]} = \left[ \begin{array}{c c c c c c c c} 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{array} \right] \begin{array}{l} u \\ v \\ w \\ \theta_ {x} \\ \theta_ {y} \\ \theta_ {z} \end{array} \end{array}
+$$
+
+Figure 2.10-2. Array ID for the truss of Fig. 2.10-1, after input data has supplied 1's for d.o.f. to be suppressed. Types of nodal d.o.f. associated with each row are shown at the right.
+
+$$
+\begin{array}{l} \text {[ID]} \\ \text {(converted)} \end{array} = \left[ \begin{array}{c c c c c c c c} 1 & 2 & 4 & 6 & 0 & 8 & 0 & 1 1 \\ 0 & 3 & 5 & 7 & 0 & 9 & 1 0 & 1 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right]
+$$
+
+Figure 2.10-3. Array ID for the truss of Fig. 2.10-1, after conversion to a table of equation numbers (by means of Fig. 2.10-4 with NUMNP=8 and NDOF=6). It is now a “destination array.”
+
+each bar in turn. For example, bar 1-3 of Fig. 2.10-1 yields KK entries of 1, 0, 4, and 5, from columns 1 and 3 of array ID. The first IF statement in Fig. 2.10-5 discards $K_{ij}$ and $R_i$ in rows associated with suppressed d.o.f., such as the preceding KK(2) = 0, which indicates that $v_1 = 0$ . The second IF statement discards $K_{ij}$ in columns associated with suppressed d.o.f. and also discards $k_{ij}$ that would fall below the diagonal of [K]. Any externally applied loads {P} must be separately added to {R} after completing the assembly algorithm.
+
+Figure 2.10-5 requires element [k]'s to be 4 by 4. For generality, an arbitrary size should be allowed. Such generalizations are discussed in Ref. 2.1. As an exercise, one may imagine that the structure in Fig. 2.10-1 is a plane frame, whose nodal d.o.f. are u, v, and $\theta_{z}$ . How is the preceding discussion altered? If $\theta_{z}$ is not suppressed at any node, Fig. 2.10-2 is altered only by setting all entries in row 6 to zero. Figure 2.10-5 must be altered to allow for larger element arrays and one more d.o.f. per node.
+
+Penalty Method. We now describe a method of imposing boundary conditions that allows prescribed d.o.f. to be either zero or nonzero. Consider again the truss of Fig. 2.2-1. Let nodal loads be $R_{1}$ , $R_{2}$ , and $R_{3}$ , as shown in Fig. 2.10-6. Imagine that vertical displacement $v_{1}$ is to be forced to have a value $\overline{v}_{1}$ . This condition can be treated as follows. Let $k_{s}$ be a large positive stiffness, say $10^{6}$ times $K_{22}$ . Add a spring of stiffness $k_{s}$ as shown in Fig. 2.10-6, and apply the large force $k_{s}\overline{v}_{1}$ in the direction of $\overline{v}_{1}$ . Load $R_{2}$ is discarded. Now solve for all three d.o.f. ( $u_{1}$ , $v_{1}$ , and $v_{3}$ ) in the usual way. If force $k_{s}\overline{v}_{1}$ were applied to only the added spring, its displacement would be precisely $\overline{v}_{1}$ . In our model, the added spring is only slightly
+```txt
+NEQ = 0
+DO 62 N=1, NUMNP
+DO 60 J=1, NDOF
+C --- Transfer if D.O.F. is fixed. Otherwise increment NEQ.
+IF (ID(J,N) .GT. 0) GO TO 58
+NEQ = NEQ + 1
+ID(J,N) = NEQ
+GO TO 60
+58 ID(J,N) = 0
+60 CONTINUE
+62 CONTINUE 
+```  
+Figure 2.10-4. Fortran statements that generate a table of equation numbers. This algorithm converts Fig. 2.10-2 to Fig. 2.10-3. Here NUMNP = total number of structure nodes, NDOF = number of d.o.f. per structure node allowed by the program, and NEQ = number of active d.o.f. = order of [K] for the constrained structure.
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_008.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_008.md
new file mode 100644
index 00000000..caaa0513
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_008.md
@@ -0,0 +1,452 @@
+<!-- source-page: 71 -->
+
+DO 500 N=1, NUMEL
+CALL ELEMENT
+I = NOD(1,N)
+J = NOD(2,N)
+KK(1) = ID(1,I)
+KK(2) = ID(2,I)
+KK(3) = ID(1,J)
+KK(4) = ID(2,J)
+DO 400 I=1,4
+IF (KK(I).LE.0) GO TO 400
+K = KK(I)
+R(K) = R(K) + RE(I)
+DO 300 J=1,4
+IF (KK(J).LT.K) GO TO 300
+L = KK(J) - K + 1
+S(K,L) = S(K,L) + SE(I,J)
+300 CONTINUE
+400 CONTINUE
+500 CONTINUE   
+Figure 2.10-5. Assembly of active stiffness equations in the banded format described by Fig. 2.8-3b. Arrays S and R must initially be null.
+
+restrained by the comparatively flimsy truss, and we compute $v_{1}$ to be only slightly less than $\overline{v}_{1}$ . Computed values of $u_{1}$ and $v_{3}$ are those appropriate to the remaining loads $R_{1}$ and $R_{3}$ and the computed value of $v_{1}$ . The value $\overline{v}_{1} = 0$ is permissible, in which case the force $k_{s}\overline{v}_{1}$ is zero.
+
+Any or all structural d.o.f. can be prescribed in the foregoing way. Each prescription adds a large diagonal stiffness to [K], and also adds a large load to {R} if the prescribed d.o.f. is nonzero. Mathematically, this procedure is called a penalty method and $k_{s}$ is called a penalty number. As $k_{s}$ approaches infinity, the constraint $v_{1} = \overline{v}_{1}$ is exactly enforced. Of course, for computational purposes, $k_{s}$ is given a finite value. A large $k_{s}$ greatly increases the maximum eigenvalues of [K] and may therefore cause trouble in a dynamic analysis.
+
+Each prescription of a d.o.f. decreases the number of unknowns in $\{D\}$ by one. Accordingly, the size of the system $[K]\{D\} = \{R\}$ should contract. This is true if formal procedures such as Eqs. 2.3-2 through 2.3-5 are used, but rearrangement of coefficients in $[K]$ is then required, which we wish to avoid in computation. Instead, by using the penalty method, we have elected to keep $\{D\}$ the same size, and populated only with unknowns, by changing the structure: each prescription of a d.o.f. adds a stiff element, creating a new structure but leaving the number of active d.o.f. unchanged.
+
+![](images/page-071_10972378fb87b8219740e198225383bba7d5e4eab2acb89ba369f729849a5adb.jpg)
+
+<details>
+<summary>text_image</summary>
+
+R₃
+3
+R₂
+kₓv̄₁
+2
+1
+R₁
+kₓ
+</details>
+
+[K] {D} = {R} for active d.o.f.:
+
+$$
+\left[ \begin{array}{c c c} K _ {1 1} & K _ {1 2} & K _ {1 3} \\ K _ {2 1} & K _ {2 2} + k _ {s} & K _ {2 3} \\ K _ {3 1} & K _ {3 2} & K _ {3 3} \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \\ v _ {3} \end{array} \right\} = \left\{ \begin{array}{l} R _ {1} \\ k _ {s} \bar {v} _ {1} \\ R _ {3} \end{array} \right\}
+$$
+
+Figure 2.10-6. A method of imposing the prescribed displacement $v_{1} = \overline{v}_{1}$ by adding a large stiffness $k_{s}$ . External loads $R_{1}$ and $R_{3}$ may continue to act.
+
+<!-- source-page: 72 -->
+
+The procedure illustrated in Fig. 2.10-6, while couched in structural terminology, is generally applicable. For example, in heat conduction analysis [K] would be a conductance matrix, {D} a vector of temperatures, and {R} a vector of heat fluxes. The matrix operations in Fig. 2.10-6 then represent the prescription of a nodal temperature by adding a large conductance to [K] and placing a large flux in {R}.
+
+Caution. A very stiff element should be parallel to a d.o.f. as is the case for $k_{s}$ in Fig. 2.10-6. If $k_{s}$ were inclined, or were placed within a structure, it would contribute to both diagonal and off-diagonal coefficients in [K]. This circumstance can lead to numerical difficulties (see Section 18.2).
+
+More About Prescribed Nonzero D.O.F. Another method of imposing a prescribed displacement, either zero or nonzero, is illustrated with reference to the truss of Fig. 2.10-6. Again imagine that [K] is 3 by 3 for the supported structure and that displacement $v_{1} = \bar{v}_{1}$ is to be imposed. As a first step we take known forces to the right side (Fig. 2.10-7a). But now the square matrix is unsymmetric and singular. We can restore symmetry and nonsingularity by replacing the second equation by the trivial equation $v_{1} = \bar{v}_{1}$ (Fig. 2.10-7b). Solution of the latter set of equations gives $v_{1} = \bar{v}_{1}$ and values of $u_{1}$ and $v_{3}$ appropriate to the system
+
+$$
+\left[ \begin{array}{l l} K _ {1 1} & K _ {1 3} \\ K _ {3 1} & K _ {3 3} \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ v _ {3} \end{array} \right\} = \left\{ \begin{array}{l} R _ {1} - K _ {1 2} \bar {v} _ {1} \\ R _ {3} - K _ {3 2} \bar {v} _ {1} \end{array} \right\} \tag {2.10-1}
+$$
+
+The effect of the treatment in Fig. 2.10-7 is to obtain Eqs. 2.10-1 but without changing the size of [K].
+
+When several d.o.f. are prescribed, one merely applies the foregoing treatment to each d.o.f. in turn. The result is a [K] with several rows and columns that are null except for 1's on the diagonal. [K] remains symmetric and banded. Results are exact, not approximate. Prescribed d.o.f. may be zero or nonzero. The method is not limited to structural problems.
+
+Except for the load terms $K_{12}\overline{v}_{1}$ and $K_{32}\overline{v}_{1}$ , Eq. 2.10-1 could be obtained by use of array ID. This observation suggests that we use array ID as before, so as to retain only d.o.f. not prescribed, but augment the procedure so as to obtain the extra load terms. More specifically, as each element is assembled, calculate loads $\{\overline{r}\} = [k]\{d\}$ produced by prescribed d.o.f. in $\{d\}$ , subtract $\{\overline{r}\}$ from element loads $\{r_{e}\}$ , then assemble the net loads as before. If no d.o.f. are prescribed for the element at hand, or if the prescribed d.o.f. are zero, then $\{\overline{r}\} = \{0\}$ , and $\{R\}$ is not changed.
+
+$$
+\left[ \begin{array}{l l l} K _ {1 1} & 0 & K _ {1 3} \\ K _ {2 1} & 0 & K _ {2 3} \\ K _ {3 1} & 0 & K _ {3 3} \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \\ v _ {3} \end{array} \right\} = \left\{ \begin{array}{l l l} R _ {1} - K _ {1 2} \bar {v} _ {1} \\ R _ {2} - K _ {2 2} \bar {v} _ {1} \\ R _ {3} - K _ {3 2} \bar {v} _ {1} \end{array} \right\}, \quad \left[ \begin{array}{l l l} K _ {1 1} & 0 & K _ {1 3} \\ 0 & 1 & 0 \\ K _ {3 1} & 0 & K _ {3 3} \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \\ v _ {3} \end{array} \right\} = \left\{ \begin{array}{l} R _ {1} - K _ {1 2} \bar {v} _ {1} \\ \bar {v} _ {1} \\ R _ {3} - K _ {3 2} \bar {v} _ {1} \end{array} \right\}
+$$
+
+(a)
+
+(b)
+
+Figure 2.10-7. Use of the “zero-one” treatment to impose displacement $v_{1} = \overline{v}_{1}$ on the truss of Fig. 2.10-6. (a) Intermediate form. (b) Final form.
+
+<!-- source-page: 73 -->
+
+Bandwidth Calculation. A disadvantage of the treatment in Fig. 2.10-7 is that if many d.o.f. are prescribed, many useless zeros are stored and processed. $^{1}$ Moreover, calculation of semibandwidth b should recognize that whole groups of d.o.f. may be suppressed (e.g., rows 3, 4, 5, and 6 in the ID arrays of Figs. 2.10-2 and 2.10-3). As explained in Section 2.8, semibandwidth b can be calculated from knowledge of which structural d.o.f. are associated with nodes of each element. This information resides in columns of ID. Consider, for example, bar 2-4 in Fig. 2.10-1. We consult columns 2 and 4 in Fig. 2.10-3, ignore the zeros, and find that the largest difference among the d.o.f. numbers 2, 3, 6, and 7 is $7 - 2 = 5$ . No other element yields a larger difference. Therefore, adding 1 to include the diagonal of [K], we conclude that $b = 5 + 1 = 6$ for this problem.
+
+# 2.11 GAUSS ELIMINATION SOLUTION OF EQUATIONS
+
+Structural equations $[K]\{D\} = \{R\}$ can be solved by a direct method or an indirect (iterative) method. In either case there are many algorithms to choose from. Direct algorithms are favored in practice. Computational aspects of equation solving are discussed in Appendix B. In the present section we summarize Gauss elimination, which is a direct method, and illustrate its physical meaning.
+
+Consider the application of Gauss elimination to the $n_{eq}$ by $n_{eq}$ system of stiffness equations $[K]\{D\} = \{R\}$ . The first equation is solved for $D_{1}$ , then substituted into the subsequent equations. Thus, $D_{1}$ is said to be “eliminated.” Then the second equation is solved for $D_{2}$ and substituted into subsequent equations, and so on. This forward-reduction process alters $\{R\}$ and changes $[K]$ to upper triangular form with 1’s on the diagonal. Finally, numerical values of unknowns are computed by back-substitution, so that $D_{n_{eq}}$ is found first and $D_{1}$ is found last.
+
+An example appears in Fig. 2.11-1. All d.o.f. are restrained except $u_2$ , $u_3$ , and $u_4$ . Starting with the original matrix equation, Fig. 2.11-1b, we divide the first row by 2 and add it to the second row. This completes the substitution of $u_2$ into the remaining equations (Fig. 2.11-1c). Since $u_2$ does not appear in the third equation, the third row is unaffected. A similar substitution, now of row 2 into row 3 by multiplication of row 2 by 2/3 and addition, is shown in Fig. 2.11-1d. Figure 2.11-1e shows the result of dividing each equation by its diagonal coefficient (in an actual algorithm, this step may not be postponed until last). Solution for the d.o.f. by back-substitution is shown in Fig. 2.11-1f.
+
+The foregoing process admits a physical interpretation: that each elimination releases the corresponding d.o.f., freeing it to move as dictated by applied loads and elastic properties of the structure. Consider the result of eliminating $u_{2}$ , Fig. 2.11-1c. In the original structure, the diagonal coefficient is $K_{22} = 12$ . That is, a force of 12 is needed to produce $u_{3} = 1$ while $u_{2} = u_{4} = 0$ ; or, $K_{22} = 6 + 6$ is the sum of the adjacent bar stiffnesses seen by d.o.f. $u_{3}$ . Elimination of $u_{2}$ effectively eliminates the constraint $u_{2} = 0$ and places bars 1 and 2 in series, forming
+
+<!-- source-page: 74 -->
+
+![](images/page-074_b883c279ee4626f7776881329f9db61763571ffec281389792c038f014ddca26.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+1
+P=24
+① 2 ② 3 ③ 4
+x,u
+(a)
+</details>
+
+$$
+\left[ \begin{array}{r r r} 1 2 & - 6 & 0 \\ - 6 & 1 2 & - 6 \\ 0 & - 6 & 6 \end{array} \right] \left\{ \begin{array}{l} u _ {2} \\ u _ {3} \\ u _ {4} \end{array} \right\} = \left\{ \begin{array}{l} 2 4 \\ 0 \\ 0 \end{array} \right\} \tag {b}
+$$
+
+$$
+\left[ \begin{array}{c c c} 1 2 & - 6 & 0 \\ 0 & 9 & - 6 \\ 0 & - 6 & 6 \end{array} \right] \left\{ \begin{array}{l} u _ {2} \\ u _ {3} \\ u _ {4} \end{array} \right\} = \left\{ \begin{array}{l} 2 4 \\ 1 2 \\ 0 \end{array} \right\} \tag {c}
+$$
+
+$$
+\left[ \begin{array}{c c c} 1 2 & - 6 & 0 \\ 0 & 9 & - 6 \\ 0 & 0 & 2 \end{array} \right] \left\{ \begin{array}{l} u _ {2} \\ u _ {3} \\ u _ {4} \end{array} \right\} = \left\{ \begin{array}{l} 2 4 \\ 1 2 \\ 8 \end{array} \right\} \tag {d}
+$$
+
+$$
+\left[ \begin{array}{c c c} 1 & - \frac {1}{2} & 0 \\ 0 & 1 & - \frac {2}{3} \\ 0 & 0 & 1 \end{array} \right] \left\{ \begin{array}{l} u _ {2} \\ u _ {3} \\ u _ {4} \end{array} \right\} = \left\{ \begin{array}{l} 2 \\ \frac {4}{3} \\ 4 \end{array} \right\} \tag {e}
+$$
+
+$$
+\left\{ \begin{array}{l} u _ {2} \\ u _ {3} \\ u _ {4} \end{array} \right\} = \left\{ \begin{array}{c} 2 + \frac {1}{2} u _ {3} \\ \frac {4}{3} + \frac {2}{3} u _ {4} \\ 4 \end{array} \right\} = \left\{ \begin{array}{l} 4 \\ 4 \\ 4 \end{array} \right\} \tag {f}
+$$
+
+Figure 2.11-1. (a) Three-bar truss, with load $P = 24$ at node 2. (b) The structure equations $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{R}\}$ , if $AE / L = 6$ for each bar and only active d.o.f. are retained. (c-f) Stages in a Gauss elimination solution for nodal d.o.f. $u_{2}, u_{3}$ , and $u_{4}$ (which are respectively $D_{1}, D_{2}$ , and $D_{3}$ ).
+
+a bar with nodes 1 and 3 whose axial stiffness is 3 rather than 6. Adjacent bar stiffnesses seen by d.o.f. $u_{3}$ are now 3 (from bars 1 and 2 in series) plus 6 (from bar 3), for a total of 9. Similarly, after elimination of $u_{3}$ and $u_{4}$ (Fig. 2.11-1d), the stiffness coefficient $K_{33} = 2$ represents the stiffness seen by d.o.f. $u_{4}$ when all three elements are connected in series with $u_{2}$ and $u_{3}$ free to move.
+
+From the foregoing physical argument we conclude that each elimination reduces the stiffnesses seen by d.o.f. not yet eliminated, but does not reduce these stiffnesses to zero unless the structure is badly modeled or is without adequate support (Fig. 2.11-2). Accordingly, if a structure is properly modeled and adequately supported, we can proceed as in Fig. 2.11-1: use the ith equation to eliminate the ith d.o.f., without rearranging coefficients and without special coding to avoid zeros on the diagonal. If a zero diagonal coefficient is encountered, the user should check for an error in modeling or support conditions. (However, if [K] is not a true stiffness matrix, as for a “mixed” structural model or a non-structural problem, zero and/or negative diagonal coefficients do not necessarily signal an error.)
+
+After elimination of d.o.f. $D_{1}$ through $D_{i}$ , where $1 \leqslant i < n_{\text{eq}}$ , the lower right portion of [K] below row $i$ remains symmetric if [K] was originally symmetric. Semibandwidth $b$ is also preserved. Each elimination affects only $b$ rows and up to $b$ coefficients per row. These attributes are exploited in programming (see Appendix B and Ref. 2.1).
+
+Often it is necessary to compute the response of a structure to several different sets of loads. Then one must process a single [K] but several vectors {R}. The processing of [K] need not be repeated in order to treat another {R}. This is fortunate, as the processing of [K] is by far the more expensive operation.
+
+The notation $\{\mathbf{D}\} = [\mathbf{K}]^{-1}\{\mathbf{R}\}$ does not necessarily mean that matrix inversion is used. Often it means only that the equations $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{R}\}$ are to be solved for
+
+<!-- source-page: 75 -->
+
+![](images/page-075_2f81917d5914ab86096efbeef9520c0e5bf873522601922fcb43811b1083faa6.jpg)
+
+<details>
+<summary>text_image</summary>
+
+w₁
+w₂
+P₁ ← L → P₂
+</details>
+
+$$
+\frac {1 2 E I}{L ^ {3}} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right] \left\{ \begin{array}{l} w _ {1} \\ w _ {2} \end{array} \right\} = \left\{ \begin{array}{l} P _ {1} \\ P _ {2} \end{array} \right\}
+$$
+
+(a)   
+![](images/page-075_b20ab59af7a65b0dfc5189e7c7b0733574379451e03a0a385d5ab57c6110c2eb.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+1
+2
+3
+x,u
+L
+L
+P
+</details>
+
+$$
+\left[ \begin{array}{c c} 2 A E / L & 0 \\ 0 & 0 \end{array} \right] \left\{ \begin{array}{l} u _ {2} \\ v _ {2} \end{array} \right\} = \left\{ \begin{array}{l} 0 \\ P \end{array} \right\}
+$$
+
+(b)
+
+Figure 2.11-2. Structures for which Gauss elimination fails because [K] is singular. (a) One-element beam, with $\theta_{1} = \theta_{2} = 0$ the only d.o.f. prescribed. (b) Two-element truss with bars collinear at node 2.
+
+{D} by any convenient or efficient method. Solving equations is faster than inverting a matrix. In addition, the inverse of a banded matrix is a full matrix. For these reasons matrix inversion is usually avoided.
+
+# 2.12 STRESS COMPUTATION. SUPPORT REACTIONS
+
+Stress Computation. After solving the global equations $[K]\{D\} = \{R\}$ for $\{D\}$ , all nodal d.o.f. of the structure are known. To compute stress in a given element we extract nodal d.o.f. $\{d\}$ of that element from $\{D\}$ , compute element strains from $\{d\}$ , and finally compute stresses from strains.
+
+Extraction of $\{d\}$ from $\{D\}$ is straightforward if all d.o.f. of the structure reside in $\{D\}$ in a regular pattern. For a plane truss of n nodes, we may have $\{D\} = [u_{1} v_{1} u_{2} v_{2} \cdots u_{n} v_{n}]^{T}$ . This is the case if the boundary condition treatment of Fig. 2.10-7 is applied to all prescribed d.o.f. of the structure. Thus, if i and j represent structure node numbers of a particular bar, then d.o.f. in $\{d\}$ are $u_{i} = D_{2i-1}$ , $v_{i} = D_{2i}$ , $u_{j} = D_{2j-1}$ , and $v_{j} = D_{2j}$ . It does not matter that some of these d.o.f. were initially prescribed (usually as zero) rather than calculated by solving equations.
+
+Things are not as simple if assembly makes use of array ID and the algorithm of Fig. 2.10-5. Now d.o.f. that are initially prescribed do not appear in $\{D\}$ , thus destroying the regular pattern. However, columns of array ID associated with element nodes still contain the location in $\{D\}$ of each nodal d.o.f. calculated by solving equations. An algorithm for extracting $\{d\}$ from $\{D\}$ appears in Fig. 2.12-1, in which it is assumed that all prescribed d.o.f. are zero. Prescribed nonzero d.o.f. require that the statement $\mathrm{DE}(M) = 0$ be altered.
+
+Elongation e of a plane truss bar is computed from components of nodal d.o.f. parallel to the bar (see Fig. 2.4-2):
+
+$$
+e = (u _ {j} - u _ {i}) \cos \beta + (v _ {j} - v _ {i}) \sin \beta \tag {2.12-1}
+$$
+
+Axial strain is $\epsilon = e/L$ . The bar is in uniaxial stress. Therefore, the axial stress caused by strain is
+
+<!-- source-page: 76 -->
+
+M = 0
+DO 220 K=1,NNEL
+C --- N is the structure number of node K of the Nth element.
+N = NOD(K,NTH)
+DO 200 L=1,NDOF
+M = M + 1
+DE(M) = 0.
+J = ID(L,N)
+C --- J is zero only if the D.O.F. is fixed.
+IF (J .GT. 0) DE(M) = D(J)
+200 CONTINUE
+220 CONTINUE   
+Figure 2.12-1. Fortran code to extract the displacement vector DE of the Nth element from the structure solution vector D when prescribed d.o.f. are zero and D lists only nonzero d.o.f. Here NNEL = number of nodes per element and NDOF = number of degrees of freedom per node allowed by the program (e.g., NDOF = 6 in Fig. 2.10-3).
+
+$$
+\sigma = E \epsilon = E \frac {e}{L} \tag {2.12-2}
+$$
+
+If the bar has thermal expansion coefficient $\alpha$ and is uniformly heated T degrees from its stress free state, then initial stress must be superposed on stress owing to mechanical strain. Thus, instead of Eq. 2.12-2, we have
+
+$$
+\sigma = E \epsilon - E \alpha T = E \left(\frac {e}{L} - \alpha T\right) \tag {2.12-3}
+$$
+
+The physical argument associated with thermal stress analysis is as follows. The argument applies to finite element structures in general, not only to a truss. With all d.o.f. fixed, compute loads that each element applies to its nodes because of heating or cooling (e.g., as in Fig. 2.6-1b). Add mechanical loads (if any). Release the d.o.f.; that is, find nodal displacements by solving $[K]\{D\} = \{R\}$ for $\{D\}$ . Compute stress caused by nodal displacements (e.g., Ee/L in Eq. 2.12-3). Algebraically add stress associated with heating or cooling of the restrained element (e.g., $-E\alpha T$ in Eq. 2.12-3). Note that stress will be zero if thermal strain is uninhibited (e.g., if $e = \alpha TL$ in Eq. 2.12-3).
+
+In stress analysis, displacements $\{d\}$ yield strains $\{\epsilon\}$ , and strains yield stresses $\{\sigma\}$ when multiplied by elastic constants. In more general terms, $\{\epsilon\}$ is the gradient of the element displacement field produced by element d.o.f. $\{d\}$ . A similar remark applies to nonstructural problems, where typically a flow quantity is analogous to $\{\sigma\}$ . As examples, if nodal d.o.f. $\{d\}$ are voltages or temperatures, the gradient of voltage or temperature produced by $\{d\}$ yields a flow of current when multiplied by electrical conductivity, or of heat when multiplied by thermal conductivity.
+
+Support Reactions. When all d.o.f. are known, support reactions $\{\mathbf{R}_x\}$ can be obtained from Eq. 2.3-4. Unfortunately, while imposing displacement boundary conditions by methods discussed in Section 2.10, the $K_{ij}$ belonging to arrays $[\mathbf{K}_{21}]$ and $[\mathbf{K}_{22}]$ of Eq. 2.3-4 have been discarded. One can either save the necessary coefficients in a file before imposing displacements, or regenerate the coefficients later.
+
+A particular reaction $R_{i}$ in the list $\{R_{x}\}$ can be computed as
+
+$$
+\sum_ {j} K _ {i j} D _ {j} = R _ {i} \quad \text { or } \quad \sum_ {m} \left(\sum_ {j} k _ {i j} d _ {j}\right) = R _ {i} \tag {2.12-4}
+$$
+
+<!-- source-page: 77 -->
+
+The first equation uses the j nonzero entries in row i of [K]. The second equation is a similar sum, taken over the m elements joined to the node that $R_{i}$ acts upon.
+
+What is the meaning of $\{R\}$ if large stiffnesses have been added in the process of imposing nonzero values of certain d.o.f. (as in Fig. 2.10-6)? If $[K]$ pertains to the original structure, before the addition of large stiffnesses, then loads $\{R\} = [K]\{D\}$ include loads originally applied to d.o.f. that are free to move and loads that must be applied to the original structure in order to produce the prescribed d.o.f.
+
+# 2.13 SUMMARY OF PROCEDURE
+
+The principal computational steps of linear static stress analysis by the finite element-method are now listed. Analogous steps are used for linear time-independent analysis of a nonstructural problem.
+
+1. Input and initialization. Input the number of nodes and elements, nodal coordinates, structure node numbers of each element, material properties, temperature changes, mechanical loads, and boundary conditions. Reserve storage space for structure arrays [K] and {R}. Initialize [K] and {R} to null arrays. If array ID is used to manage boundary conditions, initialize ID and then convert it to a table of equation numbers.   
+2. Compute element properties. For each element: compute element property matrix [k] and element load vector $\{r_{e}\}$ .   
+3. Assemble the structure. Add [k] into [K] and $\{r_{e}\}$ into $\{R\}$ . Go back to step 2. Repeat steps 2 and 3 until all elements are assembled. Add external loads $\{P\}$ to $\{R\}$ . Impose displacement boundary conditions (if not imposed implicitly during assembly by use of array ID).   
+4. Solve the equations $[K]\{D\} = \{R\}$ for $\{D\}$ .   
+5. Stress calculation. For each element, extract $\{d\}$ from $\{D\}$ . Compute mechanical strains produced by $\{d\}$ . Include initial strains, if any, and convert resultant strains to stresses.
+
+The foregoing steps outline an austere computer program, without preprocessors or postprocessors, automatic node renumbering, and other conveniences for the user. Modifications of the procedure are possible, such as computing properties of all elements before assembling any, and alternating steps of assembly with steps of equation solving.
+
+Example. We illustrate the foregoing steps by applying them to the three-bar truss of Fig. 2.13-1. Only axial displacements and axial loads are present. For clarity we will use symbols as well as numbers. In actual computation only numbers would be present.
+
+1. Input and initialization. Read the number of nodes and the number of elements: NUMNP=4 and NUMEL=3. Nodal coordinates are
+
+$$
+\begin{array}{l} x _ {1} = 0 \quad x _ {2} = L \quad x _ {3} = 2 L \quad x _ {4} = 3 L \\ y _ {1} = z _ {1} = 0 \quad y _ {2} = z _ {2} = 0 \quad y _ {3} = z _ {3} = 0 \quad y _ {4} = z _ {4} = 0 \\ \end{array}
+$$
+
+<!-- source-page: 78 -->
+
+![](images/page-078_1390c1da18222542a61d46d4160036cb6eb791eb473a4355f21fb509a862a80f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y, v
+3 @ L = 3L
+1 2 3 4 x, u
+F
+</details>
+
+[a]
+
+![](images/page-078_5af2ec77da9f3e0e384ec0fdd1ba9bfb126932344b9ba9cecc18003cf93c5fc5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+2F/3 + 2AEαT/3
+F/3 - 2AEαT/3
+1 4
+</details>
+
+(b)   
+Figure 2.13-1. (a) Example problem. The bar is divided into three identical elements and has four nodes. Elements 1 and 2 only are uniformly heated T degrees. (b) Support reactions predicted by elementary mechanics of materials theory.
+
+Node numbers associated with the three elements are
+
+$$
+\mathrm{NOD} (1, 1) = 1 \quad \mathrm{NOD} (1, 2) = 2 \quad \mathrm{NOD} (1, 3) = 3
+$$
+
+$$
+\mathrm{NOD} (2, 1) = 2 \quad \mathrm{NOD} (2, 2) = 3 \quad \mathrm{NOD} (2, 3) = 4
+$$
+
+Read cross-sectional area A, elastic modulus E, and coefficient of thermal expansion $\alpha$ (the same for each element in this example). The left two elements only are uniformly heated T degrees above the stress-free temperature of the structure. External force F is applied in the negative direction at node 2. Boundary conditions prohibit all nodal motions except $u_{2}$ and $u_{3}$ . For the sake of explanation we presume that the computer program allows only three d.o.f. per node (u, v, and w). Thus for array ID we have
+
+$$
+\begin{array}{l} \text {[ID]} = \left[ \begin{array}{l l l l} 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{array} \right] \quad \text { and } \quad \text {[ID]} = \left[ \begin{array}{l l l l} 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \\ \text {(input)} \end{array}
+$$
+
+2. Compute element properties. The stiffness matrix of each element is
+
+$$
+[ \mathbf {k} ] = \frac {A E}{L} \left[ \begin{array}{c c c c c c} 1 & 0 & 0 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ - 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right]
+$$
+
+where, in element 1 for example, the associated structural nodal d.o.f. are $\{\mathbf{d}\} = [u_1 v_1 w_1 u_2 v_2 w_2]^T$ . From Eq. 2.6-2, nodal loads of the three elements are:
+
+$$
+\{\mathbf {r} _ {e} \} _ {1} = \{\mathbf {r} _ {e} \} _ {2} = \alpha E A T \left[ - 1 \quad 0 \quad 0 \quad 1 \quad 0 \quad 0 \right] ^ {T}
+$$
+
+$$
+\{\mathbf {r} _ {e} \} _ {3} = \alpha E A T \left[ \begin{array}{l l l l l l} 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right] ^ {T}
+$$
+
+3. Assemble the structure. In the assembly algorithm of Fig. 2.10-5, the contents of array KK for the successive elements are $[0\ 0\ 0\ 1\ 0\ 0]$ , $[1\ 0\ 0\ 2\ 0\ 0]$ , and $[2\ 0\ 0\ 0\ 0\ 0]$ . Information associated with nodes 1 and 4 is discarded, and the “active” structure stiffness matrix [K] is 2 by 2. For illustration, consider where this 2 by 2 matrix would appear in a 4 by 4 stiffness matrix that operates on d.o.f. $u_{1}$ through $u_{4}$ . (The entire 4 by 4 matrix need not actually be formed.) The contribution of the leftmost element to structure arrays [K] and {R} is
+
+<!-- source-page: 79 -->
+
+$$
+\frac {A E}{L} \left[ \begin{array}{c c c c} u _ {1} & u _ {2} & u _ {3} & u _ {4} \\ 1 & - 1 & \cdot & \cdot \\ - 1 & \boxed {1} & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \end{array} \right] \quad \text {and} \quad \alpha E A T \left\{ \begin{array}{c} - 1 \\ \boxed {1} \\ \cdot \\ \cdot \end{array} \right\}
+$$
+
+where dashed lines enclose [K] and {R}. Dots indicate locations in the structure arrays that receive no contribution from the element. In similar notation, the assembly of all three elements is written
+
+$$
+[ \mathbf {K} ] = \frac {A E}{L} \left[ \begin{array}{l l} 1 & \cdot \\ \cdot & \cdot \end{array} \right] + \frac {A E}{L} \left[ \begin{array}{l l} 1 & - 1 \\ - 1 & 1 \end{array} \right] + \frac {A E}{L} \left[ \begin{array}{l l} \cdot & \cdot \\ \cdot & 1 \end{array} \right] = \frac {A E}{L} \left[ \begin{array}{l l} 2 & - 1 \\ - 1 & 2 \end{array} \right]
+$$
+
+$$
+\{\mathbf {R} \} = \alpha E A T \left\{ \begin{array}{l} 1 \\ \cdot \end{array} \right\} _ {0} ^ {0} + \alpha E A T \left\{ \begin{array}{l} - 1 \\ 1 \end{array} \right\} _ {0} ^ {0} + \alpha E A T \left\{ \begin{array}{l} \cdot \\ 0 \end{array} \right\} _ {0} ^ {0} + \left\{ \begin{array}{l} - F \\ 0 \end{array} \right\} _ {0} ^ {0} = \left\{ \begin{array}{l} - F \\ \alpha E A T \end{array} \right\}
+$$
+
+4. Solve the equations $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{R}\}$ for $\{\mathbf{D}\}$ .
+
+$$
+D _ {1} = u _ {2} = - \frac {2 F L}{3 A E} + \frac {\alpha L T}{3} \quad \text { and } \quad D _ {2} = u _ {3} = - \frac {F L}{3 A E} + \frac {2 \alpha L T}{3}
+$$
+
+5. Stress calculation. With $u_{1} = u_{4} = 0$ and the rightmost element not heated,
+
+$$
+\{\mathbf {d} \} _ {1} = \left\{ \begin{array}{l} 0 \\ u _ {2} \end{array} \right\} \quad \text { and } \quad \sigma_ {1} = E \left(\frac {u _ {2} - 0}{L} - \alpha T\right) = - \frac {2 F}{3 A} - \frac {2 \alpha E T}{3}
+$$
+
+$$
+\{\mathbf {d} \} _ {2} = \left\{ \begin{array}{l} u _ {2} \\ u _ {3} \end{array} \right\} \quad \text { and } \quad \sigma_ {2} = E \left(\frac {u _ {3} - u _ {2}}{L} - \alpha T\right) = \frac {F}{3 A} - \frac {2 \alpha E T}{3}
+$$
+
+$$
+\{\mathbf {d} \} _ {3} = \left\{ \begin{array}{l} u _ {3} \\ 0 \end{array} \right\} \quad \text { and } \quad \sigma_ {3} = E \left(\frac {0 - u _ {3}}{L} - 0\right) = \frac {F}{3 A} - \frac {2 \alpha E T}{3}
+$$
+
+These results agree with results given by elementary mechanics of materials. Thermal stress is constant over the entire length of the structure, as should be expected.
+
+# PROBLEMS
+
+# Section 2.2
+
+2.1 For the plane truss of Fig. 2.2-1, sketch the remaining four free-body diagrams not shown in Fig. 2.2-2 and write equations analogous to Eqs. 2.2-3 and 2.2-4.   
+2.2 Four springs, each of stiffness k, are constrained to slide in a circular frictionless track as shown. Nodes 1, 2, 3, and 4 are allowed only small displacements u, tangent to the circular track and positive counterclockwise. Write the structure stiffness matrix [K]. What is its rank?   
+2.3 A plane truss is shown in the sketch. Set up an initially null matrix [K] having 12 rows and 12 columns. Indentify locations of nonzero coefficients $K_{ij}$ by inserting at proper positions in [K] a “+” sign if $K_{ij} > 0$ or a “-” sign if $K_{ij} < 0$ .
+
+<!-- source-page: 80 -->
+
+![](images/page-080_d2727b15c74c67fe6b7ff0b0dcba4e917f457086f030abba2555a373243a0716.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+3
+4
+</details>
+
+Problem 2.2
+
+![](images/page-080_477f389f4686d6ce448ac0cee6f8123fbf5375af2a8672710ec93d0209e219b0.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+x,u
+2 4 6
+1 3 5
+</details>
+
+Problem 2.3
+
+2.4 (a) Follow the instructions of Problem 2.3, but with reference to the six-d.o.f. beam shown in Problem 2.12. Let the beam be uniform and let $L_{1} = L_{2}$ .
+
+(b) Repeat part (a), but let $L_{1} > L_{2}$ .   
+(c) Repeat part (a), but let $L_{1} < L_{2}$ .
+
+2.5 Follow the instructions of Problem 2.3 but with reference to the eight-d.o.f. truss shown. All bars have the same A and the same E. Each of the nodal d.o.f. ( $D_{1}$ through $D_{8}$ ) is parallel or perpendicular to one or more bars.
+
+![](images/page-080_58d6c130e9f419c631882c9a7cb002098f32ea620ea7c936509609874040f3d1.jpg)
+
+<details>
+<summary>text_image</summary>
+
+D₄
+D₃
+45°
+D₈
+D₇
+L
+D₂
+45°
+D₅
+D₁
+D₆
+</details>
+
+Problem 2.5
+
+![](images/page-080_83227a24b7f09eb570bafa59caf021e22088afcacb24bf01f354bfa03fd050f5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+4
+2
+F
+3
+4
+1
+2
+3
+4
+3
+</details>
+
+Problem 2.6
+
+2.6 The plane truss shown has bars of stiffness $k_{1}, k_{2}, k_{3}$ , and $k_{4}$ , where $k_{i} = A_{i}E_{i} / L_{i}$ . Free all d.o.f., and write the structure stiffness matrix in terms of the $k_{i}$ . Let nodal d.o.f. have the order $\{\mathbf{D}\} = [u_{1} \quad v_{1} \quad u_{2} \quad v_{2} \quad u_{3} \quad v_{3} \quad u_{4} \quad v_{4}]^{T}$ .
+
+# Section 2.3
+
+2.7 The Betti–Maxwell reciprocal theorem states that if two sets of loads $\{R\}_{1}$ and $\{R\}_{2}$ act on a structure, work done by the first set in acting through displacements caused by the second set is equal to work done by the second set in acting through displacements caused by the first set. Symbolically, $\{D\}_{1}^{T}\{R\}_{2} = \{D\}_{2}^{T}\{R\}_{1}$ . Substitute $\{D\}_{1} = [K]^{-1}\{R\}_{1}$ and $\{D\}_{2} = [K]^{-1}\{R\}_{2}$ and show that [K] is symmetric.   
+2.8 Consider the truss of Fig. 2.2-1. Write rigid-body motion vectors for the following cases and show that each produces zero forces $\{R\}$ . Are the three cases linearly independent?
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+<!-- source-page: 81 -->
+
+(a) Translation in the direction of bar 2.   
+(b) Rotation through a small angle about node 1 (the point $x = L_{1}, y = 0$ ).   
+(c) Rotation through a small angle about the point $x = L_{3}, y = L_{1}$ .
+
+2.9 For the truss of Fig. 2.2-1, let bar lengths be $L_{1} = 4$ , $L_{2} = 5$ , and $L_{3} = 3$ . Now consider the rigid-body motion $\{\mathbf{D}\} = [1\quad 3\quad 4\quad 0\quad 0\quad -4]^{T}$ . Sketch the displaced truss. Explain why the product $[\mathbf{K}]\{\mathbf{D}\}$ is not zero.
+
+2.10 Consider the circular four-spring structure of Problem 2.2. Write the displacement vector $\{\mathbf{D}\}$ for each possible rigid-body motion, and show that $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{0}\}$ .
+
+2.11 Imagine that a $90^{\circ}$ curved-beam element is formulated using the six d.o.f. shown.
+
+(a) Does each row (or each column) of [k] sum to zero? Why or why not?   
+(b) Sketch the approximate displaced shape if the d.o.f. are $\{\mathbf{d}\} = c[1\quad 1\quad 1\quad 1\quad 1\quad 1]^T$ , where $c$ is a small number.   
+(c) Write $\{\mathbf{d}\}$ (all six terms) such that $[\mathbf{k}]\{\mathbf{d}\} = \{\mathbf{0}\}$ . There are infinitely many possibilities. Can you write three $\{\mathbf{d}\}$ 's that are linearly independent?
+
+![](images/page-081_5c1c1257b8968ac8d01bfc828687650896588151682ce017a79797dc722bf7ca.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+d5
+d4
+d6
+d1
+d3
+x,u
+d2
+R
+</details>
+
+Problem 2.11
+
+![](images/page-081_93299dc3b1bda7b57761dca332190a356d45620e28d9c4f5e2c3553bff39b44e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+D₁
+D₃
+D₅
+D₂
+1
+2
+3
+D₆
+L₁
+L₂
+</details>
+
+Problem 2.12
+
+2.12 The beam shown contains two elements. Each node has two d.o.f., one in translation and one in rotation.
+
+(a) How many rigid-body motions are possible? Write a suitable d.o.f. vector $\{\mathbf{D}\}$ for each.
+
+(b) Let $\{\mathbf{D}\} = c[1\quad 1\quad 1\quad 1\quad 1\quad 1]^T$ , where $c$ is a small number. Sketch the deformed structure. Is $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{0}\}$ ? (Do not derive $[\mathbf{K}]$ .)
+
+(c) Let $\{\mathbf{D}\} = c[1 0 0 0 0 0]^T$ . Sketch the deformed structure and show qualitatively by properly directed arrows the nodal loads required.
+
+2.13 Consider the plane truss of Problem 2.6.
+
+(a) Impose support conditions implied by the sketch. That is, by discarding the appropriate rows and columns, obtain a smaller [K] that operates on only the active d.o.f.   
+(b) For the loading by force $F$ shown, write the 4 by 1 vector $\{\mathbf{D}\}$ by inspection (not by solving simultaneous equations). Hence, find the nodal loads $\{\mathbf{R}\} = [\mathbf{K}]\{\mathbf{D}\}$ . Are these loads physically reasonable?
+
+2.14 The two-element structure shown is built of standard beam elements with two d.o.f. per node (see Fig. 1.2-2). By an error, the boundary conditions
+
+<!-- source-page: 82 -->
+
+![](images/page-082_33d270ff1d78103b020acd98ae6ba8c7c5b0873d3837c5491025ff2733c87225.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+1
+2
+3
+P
+x
+L/2
+L/2
+</details>
+
+Problem 2.14
+
+given to a computer program are $w_{1} = \theta_{1} = \theta_{2} = \theta_{3} = 0$ . The expected result, $w_{3} = PL^{3}/3EI$ , is not computed. What value of $w_{3}$ is in fact computed by the program? (The question can be answered by sketching the deformed structure and applying elementary beam theory.)
+
+2.15 (a) Let $k_{1} = k_{2} = k_{3} = k$ in Eq. 2.3-8. Solve for $u_{1}, v_{1}$ , and $v_{3}$ in terms of $P$ and $k$ .   
+(b) Using the results of part (a), compute $p_2, q_2$ , and $p_3$ (Eq. 2.3-9). Show these forces applied to a free-body diagram of the truss, and check that static equilibrium conditions are satisfied.
+
+# Section 2.4
+
+2.16 (a) Derive a 4 by 4 element stiffness matrix for a uniform plane truss member, using d.o.f. shown in the sketch.
+
+(b) Check that nodal loads $[k]\{d\}$ are zero for the following rigid-body motions: x-direction translation, y-direction translation, and a small counterclockwise rotation about node i.
+
+![](images/page-082_f3f5d58ddaee4b0bcde5c6a209d29478b5f06c42a402a2474c408d3eb110c52a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L
+u_j
+j
+v_j
+u_i
+β
+i
+v_i
+</details>
+
+Problem 2.16
+
+2.17 Consider a straight, uniform shaft of solid circular cross section, with a node at each end.
+
+(a) Let nodal d.o.f. be angular rotation vectors parallel to the bar, one at each end. Nodal loads are axially directed torque vectors. What is [k], in terms of the length, shear modulus, and radius of the cross section?   
+(b) Let the bar be inclined at angle $\beta$ to the $x$ axis, with $\theta_{x}$ and $\theta_{y}$ as d.o.f. at each node (rotations about the $x$ and $y$ coordinate axes). What is [k]? As in part (a), consider torsional stiffness only.
+
+2.18 A uniform bar of axial stiffness $k = AE / L$ is arbitrarily oriented in space. Cosines of angles between the bar and the $x, y$ , and $z$ coordinate axes are $\ell, m$ , and $n$ . Nodal d.o.f. are translations $u, v$ , and $w$ at each end. Derive the 6 by 6 element stiffness matrix.
+
+<!-- source-page: 83 -->
+
+# Section 2.5
+
+2.19 For each of the following structures, generate the structure stiffness matrix by writing element matrices “structure size” and assembling them.
+
+(a) The four-spring circular structure of Problem 2.2.   
+(b) The four-bar truss of Problem 2.6 (without supports).
+
+# Section 2.6
+
+2.20 The uniform bar shown hangs under its own weight $W$ . Compute the deflection of the lower end in terms of $W$ , $L$ , $A$ , and $E$ . (Obtain $[\mathbf{K}_{11}]$ of Eq. 2.3-3 by retaining only active d.o.f. The uppermost node is fixed.)
+
+(a) Use one element of length $L$ .   
+(b) Use two elements, each of length $L / 2$ .
+
+![](images/page-083_00660aff4be7eb7a06eb6fa2270ca5f2f8f087e23d1cdd2557dddc3aece21bf7.jpg)  
+Problem 2.20
+
+![](images/page-083_b93ce40f077a10321cf3b15f34b51b7f935fbccfd39dad9c902503651039ac55.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+1
+2
+3
+x,u
+2L
+L
+</details>
+
+Problem 2.21
+
+2.21 The uniform bar shown is built of two elements. Both elements are uniformly heated T degrees. Obtain $[K_{11}]$ of Eq. 2.3-3 by retaining only the active d.o.f. $(u_{2}$ and $u_{3})$ . Solve for $u_{2}$ and $u_{3}$ in terms of $\alpha$ , L, and T.
+
+# Section 2.7
+
+2.22 As suggested in Section 2.7, permute node labels of both elements in Fig. 2.7-1 so that $j$ replaces $i, k$ replaces $j$ , and $i$ replaces $k$ . Maintain structure node labels 1, 2, 3, and 4 where they are shown in Fig. 2.7-1. Show that [K] of Eq. 2.7-4 is again produced.   
+2.23 Change structure node labels in Fig. 2.7-1 from 1, 2, 3, and 4 to 1, 3, 4, and 7, respectively. Thus, the two elements shown are regarded as a fragment of a larger structure. To what row and column location in a 7 by 7 array [K] is each of the $a$ 's and $b$ 's in Eqs. 2.7-1 assigned?   
+2.24 Add the following elements to Fig. 2.7-2. Show the locations of nonzero element coefficients in $\{R\}$ and in $[K]$ , as in Fig. 2.7-2.
+
+(a) Attach a triangular element 1-4-6 to nodes 1 and 4 of existing element 2.   
+(b) Attach a rectangular element 3-5-7-8 to nodes 3 and 5 of existing element 1.
+
+2.25 Manually apply the assembly algorithm of Fig. 2.7-3 to matrix $[\mathbf{k}]_{\mathrm{I}}$ of Eq. 2.7-1. Specifically, by supplying numerical indexes for arrays, discover where the $a$ 's are placed in array S for
+
+(a) I = 1 in the DO 400 loop.   
+(b) I = 2 in the DO 400 loop.   
+(c) $l = 3$ in the DO 400 loop.
+
+<!-- source-page: 84 -->
+
+2.26 Imagine that coefficients in the plane truss element stiffness matrix are arranged to suit the order of d.o.f. $\{\mathbf{d}\} = \left[u_i \quad u_j \quad v_i \quad v_j\right]^T$ . If structure d.o.f. still have the order $\{\mathbf{D}\} = \left[u_1 \quad v_1 \quad u_2 \ldots u_N \quad v_N\right]^T$ , revise the assembly algorithm of Fig. 2.7-4 as required.   
+2.27 Revise the assembly algorithm of Fig. 2.7-4 to deal with the following elements:
+
+(a) a plane frame element (three d.o.f. per node).
+
+(b) a space frame element (six d.o.f. per node).
+
+# Section 2.8
+
+2.28 (a) For each of the plane trusses shown, show the topology of the structure stiffness matrix, in the manner of Fig. 2.8-3a. However, let each X represent a 2 by 2 submatrix: thus, the sketch of [K] will have eight rows and eight columns of submatrices X.   
+(b) How would your answer to part (a) change if the structure were a plane frame? Or a network of electrical resistors?
+
+![](images/page-084_15a44b4e6b11cc674f2374736d4e04eb3977a20371be60f658519829d3b8afca.jpg)
+
+<details>
+<summary>text_image</summary>
+
+2 4 6 8
+1 3 5 7
+</details>
+
+![](images/page-084_6e6eaf218ec3a3669b2ad6851a9651178ead122997fb59a4c2ce4e160b06a8fd.jpg)
+
+<details>
+<summary>flowchart</summary>
+
+```mermaid
+graph TD
+    1 --> 2
+    2 --> 3
+    3 --> 4
+    4 --> 5
+    5 --> 6
+    6 --> 7
+    7 --> 8
+    1 --> 2
+    2 --> 3
+    3 --> 4
+    4 --> 5
+    5 --> 6
+    6 --> 7
+    7 --> 8
+    8 --> 1
+```
+</details>
+
+Problem 2.28
+
+![](images/page-084_3ab54661ea0767d34a8d30ec08b76a93d3c9ba4539d12c8ebd84699745f7055b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+A
+B
+C D E F G H
+J
+I
+</details>
+
+(a)
+
+![](images/page-084_15622794909c53225025e39338d75b344b071e5a9b8ed89fa179d49c5ae87f8f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+A
+H
+B
+G
+I
+C
+F
+D
+E
+</details>
+
+(b)
+
+![](images/page-084_9d5eccb367720c3513f8627fe3ffcc6d7d5ba5850d6c0f800d52c1dfca4e5bc9.jpg)
+
+<details>
+<summary>text_image</summary>
+
+A
+B
+C
+D
+E
+I
+H
+J
+K
+L
+F
+G
+N
+M
+</details>
+
+(c)
+
+![](images/page-084_26b5543368c1f75c6a58c9a3317c7dee11ee6e20e5037500002053fabef9f393.jpg)
+
+<details>
+<summary>text_image</summary>
+
+H
+B
+C
+I
+G
+E
+D
+A
+F
+J
+</details>
+
+(d)   
+Problem 2.29
+
+<!-- source-page: 85 -->
+
+2.29 Each of the structures shown is hypothetical and has one d.o.f. per node. Solid lines indicate connectivity between nodes. (Letters are for use in another problem.) Number the nodes so as to achieve minimum bandwidth of the coefficient matrix [K]. Sketch the topology of the assembled matrix [K] (as in Fig. 2.8-3a).   
+2.30 In an element having many nodes, let $i$ and $j$ represent respectively the highest and lowest structure node numbers connected to that element. If $i - j$ happens to be the largest difference for any element of the structure, and if $n$ is the number of d.o.f. per node, what is semibandwidth $b$ in terms of $i, j$ , and $n$ ?   
+2.31 Apply the formula for $b$ devised in Problem 2.30 to the following structures: (a) the plane truss in Fig. 2.2-1. (b) the first structure shown in Problem 2.28, regarded as a plane frame (three d.o.f. per node).   
+2.32 Consider the structures of Problem 2.29. Assign node numbers by the following system. Pick a starting node (say A) and call it 1. Number as 2, 3, and so on, nodes that share an element with node 1 (thus, in (b), node numbers become H = 2 and B = 3). Next, number nodes that share an element with nodes 2, 3, and so on. (Figure 2.9-1 shows the results of such a scheme, but starting with the highest number and counting down.) For one d.o.f. per node, what semibandwidth b do you obtain?   
+2.33 Reverse the node numberings found in Problem 2.32. For each structure, how many fills are there during equation solving, both in the original numbering of Problem 2.32 and in the reversed numbering?
+
+# Section 2.10
+
+2.34 For each of the plane trusses shown in Problem 2.28, write the "input" and "converted" forms of array ID (see Figs. 2.10-2 and 2.10-3). As support conditions, assume that all d.o.f. are set to zero at the upper left node and at the lower right node.   
+2.35 Repeat Problem 2.34, but regard each structure as a plane frame (nodal d.o.f. u, v, and $\theta_{z}$ ).   
+2.36 (a) The unsupported plane truss shown has eight d.o.f. Set up an 8 by 8 stiffness array [K]. Write a bar number in those positions of [K] that
+
+![](images/page-085_bc44cc8ef449958287fcf40855b57012b550ceb4743c986405a8f8a6df1af590.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+3
+⑤
+4
+②
+③
+④
+1
+2
+x,u
+</details>
+
+(a)
+
+$$
+\left[ \begin{array}{c c c c c} 1 & & & & \\ & 5 & & 5 & \\ & & 2 & & \\ & 5 & & 3, 5 & 3 \\ & & & 3 & 3, 4 \end{array} \right]
+$$
+
+(b)
+
+Problem 2.36
+
+<!-- source-page: 86 -->
+
+receive nonzero stiffness contributions from that bar (e.g., write $K_{22} = K_{26} = K_{62} = K_{66} = 2$ , from bar 2).
+
+(b) After support conditions are imposed by use of array ID, a similarly diagrammed matrix [K] appears as shown. Sketch the truss, showing the supports implied.
+
+2.37 Imagine that the structure in Fig. 2.10-1 is a plane frame, for which nodal d.o.f. are $u, v,$ and $\theta_{z}$ . Supports apply no nodal moments.
+
+(a) Modify the ID arrays in Figs. 2.10-2 and 2.10-3 as required.
+
+(b) Modify the assembly algorithm of Fig. 2.10-5 as required.
+
+2.38 Consider the 2 by 2 system of equations
+
+$$
+\left[ \begin{array}{l l} K _ {1} & K _ {2} \\ K _ {3} & K _ {4} \end{array} \right] \left\{ \begin{array}{l} x \\ y \end{array} \right\} = \left\{ \begin{array}{l} 0 \\ b \end{array} \right\}
+$$
+
+Use the penalty method to impose the result $x = c$ . Show that exact values of $x$ and $y$ are approached as the added stiffness approaches infinity.
+
+2.39 How would you impose a prescribed relative displacement by the penalty method? Consider, for example, imposing $u_4 - u_2 = c$ in Fig. 2.11-1a, where $c$ is a constant. Give a physical explanation, then state exactly which coefficients in the equations $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{R}\}$ must be changed and how you would change them. (This procedure is not recommended; see the Caution in Section 2.10.)
+
+2.40 Consider the axially loaded structure in Fig. 2.11-1a and the $[K]\{D\} = \{R\}$ equation in Fig. 2.11-1b. Impose the displacement $u_{3} = 6$ and solve for $u_{2}$ and $u_{4}$ .
+
+(a) Use the penalty method of Fig. 2.10-6.
+
+(b) Use the “zero-one” procedure of Fig. 2.10-7.
+
+(c) Use Eq. 2.10-1, and determine the supplementary terms on the right-hand side by summing element contributions, as suggested below Eq. 2.10-1.
+
+2.41 Consider again the four-spring circular structure of Problem 2.2. No forces are applied, but displacements $u_{2} = u_{4} = c$ are prescribed, where c is a constant. Impose these displacements and solve for $u_{1}$ and $u_{3}$ . Use the “zero-one” procedure of Fig. 2.10-7.
+
+2.42 From the “converted” ID arrays found in Problem 2.34, compute semi-bandwidth b by applying the method described at the end of Section 2.10.
+
+2.43 Write a Fortran algorithm that calculates semibandwidth $b$ according to the procedure outlined at the end of Section 2.10.
+
+# Section 2.11
+
+2.44 Let $k_{1} = k_{2} = k_{3} = k$ in Eq. 2.3-8. Calculate $u_{1}, v_{1}$ , and $v_{3}$ in terms of $P$ and $k$ by applying the Gauss elimination method.
+
+2.45 Consider the circular four-spring structure of Problem 2.2. Without imposing any support condition, show the four modified [K]'s produced by successive steps of Gauss elimination (as in Fig. 2.11-1).
+
+2.46 A one-element cantilever beam is shown. Also shown is the stiffness matrix that operates on the unrestrained d.o.f. $w_{2}$ and $\theta_{2}$ . In parts (a) and (b) carry
+
+<!-- source-page: 87 -->
+
+![](images/page-087_041780e3fb3f92e6f6d479801fbc622f00f63467f6a476c676e0f12d2c222d84.jpg)
+
+$$
+[ K ] = \left[ \begin{array}{c c} 1 2 E I / L ^ {3} & - 6 E I / L ^ {2} \\ - 6 E I / L ^ {2} & 4 E I / L \end{array} \right] \begin{array}{l} w _ {2} \\ \theta_ {2} \end{array}
+$$
+
+Problem 2.46
+
+out one step of Gauss elimination, and explain the physical meaning of the diagonal coefficient that remains.
+
+(a) Eliminate $w_{2}$ (reduction of [K] to upper triangular form).   
+(b) Eliminate $\theta_{2}$ (reduction of [K] to lower triangular form).
+
+2.47 Consider reduction of [K] to upper triangular form by Gauss elimination. Coefficients that are initially zero may become nonzero in this process. In the following matrices, which zeros above the diagonal become nonzero?
+
+(a) [K] of Eq. 2.2-6.   
+(b) [K] of Fig. 2.7-2.
+
+2.48 Imagine that no boundary conditions are imposed, so that too many structure d.o.f. remain active. A solution for the d.o.f. by Gauss elimination is started, but fails during the attempt to eliminate the $n$ th d.o.f. For the following structures, what is $n$ , and why?
+
+(a) The plane truss of Fig. 2.8-2a (allow two d.o.f. per node).   
+(b) Imagine that Fig. 2.8-2a represents a plane frame (allow three d.o.f. per node).   
+(c) The network of Fig. 2.9-1 (allow one d.o.f. per node).   
+(d) The beam of Problem 2.12 (allow the six d.o.f. shown).
+
+# Section 2.12
+
+2.49 Consider the uniform hanging bar of Problem 2.20. Assume that finite element analysis yields nodal displacements that are exact. Plot the correct distribution of axial stress (from $W / A$ at the top to zero at the bottom). On the same plot show the stress distribution predicted by finite elements using
+
+(a) one element.   
+(b) two identical elements.   
+(c) four identical elements.
+
+2.50 The uniform bar shown is built of two identical bar elements and is loaded by axially directed forces $P_{2}$ and $P_{3}$ at nodes 2 and 3, respectively. Impose
+
+![](images/page-087_1deb5b8c2a24737c006cb7c1be4e808dbbb9aea12999060692ae099b4f04fc43.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+3
+P₂
+P₃
+L
+L
+</details>
+
+Problem 2.50
+
+![](images/page-087_fcac21768e9a757a9ca9460479592515dd9e29d9539b5c40382e8e7f2641a625.jpg)
+
+<details>
+<summary>text_image</summary>
+
+3
+3
+4
+y,v
+2
+1
+x,u
+</details>
+
+Problem 2.51
+
+<!-- source-page: 88 -->
+
+the displacement $u_{2} = \overline{u}_{2}$ by the penalty method described in Section 2.10, using $k_{s} = 1000AE/L$ . Solve for $u_{2}$ and $u_{3}$ , then compute loads $\{R\} = [K]\{D\}$ , where [K] pertains to the original structure. Interpret the result for the special cases $\overline{u}_{2} = 0$ and $P_{3} = 0$ .
+
+2.51 Let $AE / L = 2(10)^6$ N/m for each bar of the two-bar truss shown.
+
+(a) Set up the 2 by 2 structure stiffness matrix that operates on $\{\mathbf{D}\} = |u_1, v_1|^T$ .   
+(b) Let there be a prescribed downward displacement of 0.0001 m at node 1. No horizontal load is applied and no horizontal displacement is prescribed at node 1. Modify the structure equations using the penalty method procedure described in Section 2.10, using $k_{s} = 1000 \, AE/L$ .   
+(c) Solve for $u_{1}$ by Gauss elimination.   
+(d) Solve for the vertical force applied to node 1.
+
+# Section 2.13
+
+2.52 Repeat the example given in Section 2.13, but allow node 1 to move axially, so that the active d.o.f. are $u_{1}, u_{2}$ , and $u_{3}$ .   
+2.53 Analyze the truss shown for nodal displacements and element stresses. Follow the steps used in the example problem of Section 2.13. Let $E = 200$ GPa for each bar.
+
+![](images/page-088_99bbcb43eac3309d60ca6b9f86b7be3c64240eeb5bf3773825999f24993a8bfb.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P=10,000 N
+400 mm
+y,v
+1
+A=200 mm²
+2
+500 mm
+A=140 mm²
+x,u
+P
+4
+3
+</details>
+
+Problem 2.53
+
+2.54 Using the steps listed in Section 2.13 as a guide, write a computer program for the analysis of plane trusses. An algorithm for equation solving is given in Appendix B.
+
+<!-- source-page: 89 -->
+
+# STATIONARY PRINCIPLES, THE RAYLEIGH-RITZ METHOD, AND INTERPOLATION
+
+The equilibrium configuration of a system is found by analysis of its potential energy. Expressions for potential energy are presented. These and other integral expressions, called functionals, are introduced as a starting point for an approximation technique—namely, the Rayleigh–Ritz method—whose modern form is the finite element method. Interpolation, necessary to the method, is described.
+
+# 3.1 INTRODUCTION
+
+In preceding chapters, element stiffness matrices [k] have been formulated by direct physical argument. This is easily done for truss and beam elements by activating d.o.f. in turn and computing the nodal loads required to maintain the deformation state. Finite elements obtained by discretization of a continuum are not as easily formulated. (For example, is there an easy way to find nodal forces that appear in response to displacement $u_{3}$ in Fig. 1.1-2c?) A systematic and general way of obtaining [k] is needed. One of the best ways is the Rayleigh–Ritz method. An alternative, the method of weighted residuals, is discussed in Chapter 15.
+
+The Rayleigh–Ritz method has a classical form and a finite element form. In the classical form, an approximating field is defined over the entire region of interest. In the finite element form, the approximating field is defined in piecewise fashion. As degrees of freedom, finite elements use nodal values of the field (and perhaps nodal values of one or more spatial derivatives of the field as well). By degrees of freedom (d.o.f.) we mean independent quantities used to define a configuration of a system that violates neither compatibility conditions nor constraints such as support conditions. Using more general terms, one can say that d.o.f. are quantities used to define the spatial variation of an approximating field.
+
+In order to analyze a continuum by use of the Rayleigh–Ritz method, one must have a functional. A functional is an integral expression that implicitly contains differential equations that describe the problem. In structural mechanics the most widely used functional is the expression for potential energy. Functionals are also available for problems of heat conduction, acoustic modes in cavities, certain types of fluid flow, and other problems. We will present some of these functionals and will show how they are used to produce finite element formulations.
+
+Differential equations are said to state a problem in the strong form. An integral expression such as a functional that implicitly contains the differential equations is called the weak form. The strong form states conditions that must be met at every material point, whereas the weak form states conditions that must be met
+
+only in an average sense.
+
+diff eq—evolypic
+inv. eq—imavex
+
+<!-- source-page: 90 -->
+
+A functional, such as that for potential energy $\Pi_p$ , contains integrals that span the line, area, or volume of interest. After applying the Rayleigh-Ritz method, the $\Pi_p$ expression contains no integrals and is no longer called a functional. Rather, $\Pi_p$ is then a function of a finite number of d.o.f. Indeed, for an initially discrete structure such as a truss, no integrals need be invoked in writing the $\Pi_p$ expression. We will consider these “initially discrete” forms first, then return to integral forms later in this chapter.
+
+Physical insight was responsible for the early rapid development of the finite element method and for its ready appeal to stress analysts. A more mathematical approach augments physical understanding by placing the finite element method on a sound foundation, thus allowing statements to be made regarding bounds and convergence, and suggesting solution tactics that are not apparent from physical reasoning alone.
+
+# 3.2 PRINCIPLE OF STATIONARY POTENTIAL ENERGY
+
+In the present section we consider time-independent problems of structural mechanics. We define a system as the physical structure and the loads applied to it. The configuration of a system is the set of positions of all particles of the structure. Let the system have a reference configuration $C_R$ and a displaced configuration $C_D$ . A system is called conservative if work done by internal forces and work done by external loads are each independent of the path taken between $C_R$ and $C_D$ . In an elastic structure, work done by internal forces is equal in magnitude to the change in strain energy.
+
+The loaded spring of Fig. 3.2-1 is a case in point. Let $C_R$ and $C_D$ refer to unstretched and stretched configurations, respectively. If the spring dissipates no energy, then the work of internal forces (i.e., strain energy in the spring) depends only on stretch $D$ , not on whether the passage from $C_R$ to $C_D$ is via path $A$ or path $B$ . Similarly, if external load $P$ has constant magnitude and constant direction, it does negative work of magnitude $PD$ regardless of the path taken from $C_R$ to $C_D$ . We conclude that because internal forces and external loads are both conservative, so is the system.
+
+Boundary conditions are of two types: essential (or principal) and nonessential (often called natural). In the finite element method, essential boundary conditions are prescribed values of nodal d.o.f., and nonessential boundary conditions are prescribed values of higher derivatives of the field quantity than are usually used as nodal d.o.f. For example, if standard beam elements are used, nodal d.o.f. are lateral deflection w and its first derivative, $w_{xx}$ . When these elements are used to analyze the beam of Fig. 3.2-2a, essential boundary conditions (which can also be called geometric or kinematic in this problem) are that w = 0 and $w_{xx} = 0$ .
+
+![](images/page-090_3301d28400a21510117d85c08650720b4fce6e9b4f550dfbab62735c2975c56a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k
+C_R
+P
+C_D
+B
+A
+D
+</details>
+
+Figure 3.2-1. A linear spring of stiffness k loaded by a constant force P that acts parallel to the x axis. Hypothetical displacement paths A and B of the loaded point are shown by dashed lines.
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_010.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_010.md
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--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_010.md
@@ -0,0 +1,494 @@
+<!-- source-page: 91 -->
+
+![](images/page-091_aaf494c08e2b5972502d8c0445f23bef576a61f996e101e2aefd246e02a3739c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+P
+x
+L
+</details>
+
+(a)
+
+![](images/page-091_102721bc52b224ec333477e3a895e28cd1d9278abbc9a1449905245c8e5972ee.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+A
+B
+P
+x
+</details>
+
+(b)   
+Figure 3.2-2. (a) A cantilever beam. (b) An inadmissible configuration (upper dashed line) and two admissible configurations (lower dashed lines).
+
+at $x = 0$ . Nonessential boundary conditions are that $w_{,xx} = 0$ and $w_{,xxx} = 0$ at $x = L$ , since bending moment $M = EIw_{,xx}$ and transverse shear force $V = EIw_{,xxx}$ are both zero at $x = L$ .
+
+An admissible configuration is any configuration that satisfies internal compatibility and essential boundary conditions. Examples appear in Fig. 3.2-2b. The uppermost curve, which is inadmissible, has four faults: it violates the two essential boundary conditions w = 0 and $w_{,x} = 0$ at x = 0, and it violates compatibility because of the jump at A and the cusp at B. The lower two curves are both admissible, even though only the lower one seems physically reasonable. An admissible configuration need not satisfy nonessential boundary conditions. Thus, at x = L, neither of the two lower curves need display $w_{,xx} = 0$ or $w_{,xxx} = 0$ .
+
+A conservative mechanical system has a potential energy. That is, one can express the energy content of the system in terms of its configuration, without reference to whatever deformation history or path may have led to that configuration [3.1]. Potential energy, also called total potential energy, includes (a) the strain energy of elastic distortion, and (b) the potential possessed by applied loads, by virtue of their having the capacity to do work if displaced through a distance. The principle of stationary potential energy states that
+
+Among all admissible configurations of a conservative system, those that satisfy the equations of equilibrium make the potential energy stationary with respect to small admissible variations of displacement. $\mathrm{d}T = 0$ (dy) This principle is applicable whether or not the load versus deformation relation is linear. If the stationary condition is a relative minimum, the equilibrium state is stable. Note that loads are kept constant while displacements are varied.
+
+Example. Linear Spring with Axial Load. A very simple system is shown in Fig. 3.2-3. Its (total) potential energy $\Pi_{p}$ has two parts;
+
+![](images/page-091_2bf6e4f9e334ff08c86071a63f7c921e36173c0234eeac29db1e6dbe65eea9f2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k
+P
+x
+L
+</details>
+
+(a)
+
+![](images/page-091_636096cd6b67046753d380c67b716e03c9289a005b14fedd2206fa50a2547cfe.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k
+D
+P
+L + D
+x
+</details>
+
+(b)   
+Figure 3.2-3. (a) Unstretched (reference) configuration of a linear spring of stiffness k. (b) Configuration after force P is applied, stretching the spring D units.
+
+<!-- source-page: 92 -->
+
+$$
+\Pi_ {p} = U + \Omega \tag {3.2-1}
+$$
+
+where U is the strain energy of the system, which in the present example is given by
+
+$$
+U = \frac {1}{2} k D ^ {2} \tag {3.2-2}
+$$
+
+The potential of loads, called $\Omega$ , is here given by
+
+$$
+\Omega = - P D \tag {3.2-3}
+$$
+
+The load is regarded as always acting at its full value P. In moving through displacement D it does work in the amount PD, thereby losing potential of equal amount; hence the negative sign in the expression $\Omega = -PD$ . The potential energy
+
+$$
+\left[ \Pi_ {p} = \frac {1}{2} k D ^ {2} - P D\right) \tag {3.2-4}
+$$
+
+can be regarded as the total internal and external work done in changing the configuration from the reference state $D = 0$ to the displaced state $D \neq 0$ . Note that if $P$ were directed toward the left, while $D$ remains positive toward the right, then $\Omega$ would become $+PD$ . This is in essence the same as increasing potential energy by increasing the elevation of a weight.
+
+If only displacements along the $x$ axis are allowed, then the single d.o.f. $D$ defines all admissible configurations. The equilibrium configuration $D_{\mathrm{eq}}$ is found from the stationary value of $\Pi_p$ :
+
+$$
+\left\{ \begin{array}{l} d \Pi_ {p} = (k D _ {\mathrm{eq}} - P) d D = 0, \quad \text { hence } \quad D _ {\mathrm{eq}} = \frac {P}{k} \end{array} \right. \tag {3.2-5}
+$$
+
+The equation $(kD_{\mathrm{eq}} - P)dD = 0$ is an instance of the virtual work principle: zero network is done by all forces during a small admissible displacement $dD$ from the equilibrium configuration. This is graphically apparent in Fig. 3.2-4. We see also that $\Pi_p$ is a relative minimum, which means that the equilibrium state is stable.
+
+The reference datum for $\Omega$ can be arbitrarily changed by a constant. For example, if we say that $\Omega$ is zero at the equilibrium configuration, then $\Omega = P(D_{\mathrm{eq}} - D)$ . The added constant $PD_{\mathrm{eq}}$ disappears in the process of writing $d\Pi_p = 0$ , and the same value of $D_{\mathrm{eq}}$ is again obtained.
+
+![](images/page-092_995b7af086a9d68d170fd0dbade610148896f302ff30bb84ab9ce0c3b6f2691c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Potential
+U=½kD²
+Πp=U+Ω
+D
+D_eq
+Ω=-PD
+</details>
+
+Figure 3.2-4. Graphical interpretation of the potential energy relations for the problem of Fig. 3.2-3.
+
+<!-- source-page: 93 -->
+
+In Eq. 3.2-2 we write $\Omega = -PD$ rather than $\Omega = -PD/2$ . This is because P is regarded as always acting at full intensity. True, we could bypass the stationary potential energy principle and imagine that D is produced by a gradually increasing load whose final value is P. Thus, equating work done against a linear spring to strain energy stored, we have
+
+$$
+\frac {1}{2} P D _ {\mathrm{eq}} = \frac {1}{2} k D _ {\mathrm{eq}} ^ {2} \quad \text { from   which } \quad D _ {\mathrm{eq}} = \frac {P}{k} \tag {3.2-6}
+$$
+
+This energy balance argument is valid but rarely helpful. It yields but one equation, even if there are a great many d.o.f. that must be determined.
+
+# 3.3 PROBLEMS HAVING MANY D.O.F.
+
+A finite element analysis typically uses hundreds of d.o.f. They may be the x and y displacements of nodes (as in a plane stress problem), or lateral displacement w and its first derivative $w_{xx}$ at nodes (as in a beam problem), and so on. Let n be the number of d.o.f. that must be calculated, and let them be collected in the structure displacement vector $\{D\} = \left[D_{1}, D_{2}, \ldots, D_{n}\right]^{T}$ . We assume here that support conditions are already imposed, so that arbitrary values of the $D_{i}$ always create admissible configurations.
+
+Potential $\Pi_p$ is a function of the $D_i$ . Symbolically, $\Pi_p = \Pi_p (D_1, D_2, \ldots, D_n)$ . Applying the principle of stationary potential energy, we write
+
+$$
+d \Pi_ {p} = \frac {\partial \Pi_ {p}}{\partial D _ {1}} d D _ {1} + \frac {\partial \Pi_ {p}}{\partial D _ {2}} d D _ {2} + \dots + \frac {\partial \Pi_ {p}}{\partial D _ {n}} d D _ {n} = 0 \tag {3.3-1}
+$$
+
+$$
+\text {   S   I   A   (  } \quad \text {   d   } \text {   1   } \quad \text {   d   } \text {   1   } \quad \text {   d   } \text {   1   } \quad \text {   d   } \text {   1   } \quad \text {   d   } \text {   1   } \quad \text {   d   } \text {   1   } \quad \text {   d   } \text {   p   } \text {   2   } \quad \text {   2   } \quad \text {   2   } \quad \text {   2   } \quad \text {   2   } \quad \text {   2   } \quad \text {   2   } \quad \text {   2   } \quad \text {   2   } \quad \text {   2   } \quad \text {   2   } \text {   2   } \quad \text {   2   } \quad \text {   2   } \quad \text {   2   } \quad \text {   2   } \quad \text {   2   } \quad \text {   2   } \quad \text {   2   } \quad \text {   2   } \quad \text {   2   }
+$$
+
+The stationary principle states that equilibrium prevails when $d\Pi_{p} = 0$ for any small admissible variation of the configuration. We can imagine that only $dD_{1}$ is nonzero, or that only $dD_{2}$ and $dD_{3}$ are nonzero, and so on. For any and all such choices, $d\Pi_{p}$ must vanish. This is possible only if coefficients of the $dD_{i}$ vanish separately. Thus, for $i = 1, 2, 3, \ldots, n$ ,
+
+$$
+\boxed {\frac {\partial \Pi_ {p}}{\partial D _ {i}} = 0} \text {   or,   in   alternative   notation,   } \left\{\frac {\partial \Pi_ {p}}{\partial \mathbf {D}} \right\} = \{\mathbf {0} \} \tag {3.3-2}
+$$
+
+These are n equations to be solved for the n values of d.o.f. $D_{i}$ that define the static equilibrium configuration.
+
+Example. Springs in Series. The structure shown in Fig. 3.3-1 has the potential
+
+$$
+\Pi_ {p} = \frac {1}{2} k _ {1} D _ {1} ^ {2} + \frac {1}{2} k _ {2} \left(D _ {2} - D _ {1}\right) ^ {2} + \frac {1}{2} k _ {3} \left(D _ {3} - D _ {2}\right) ^ {2} - P _ {1} D _ {1} - P _ {2} D _ {2} - P _ {3} D _ {3} \tag {3.3-3}
+$$
+
+Equations 3.3-2 and 3.3-3 yield, for $i = 1, 2, 3$ ,
+
+$$
+k _ {1} D _ {1} - k _ {2} (D _ {2} - D _ {1}) - P _ {1} = 0
+$$
+
+$$
+k _ {2} (D _ {2} - D _ {1}) - k _ {3} (D _ {3} - D _ {2}) - P _ {2} = 0 \tag {3.3-4}
+$$
+
+$$
+k _ {3} (D _ {3} - D _ {2}) - P _ {3} = 0
+$$
+
+<!-- source-page: 94 -->
+
+![](images/page-094_57ce754ebb480510eb82bd9ee799ad9aeb037ced9c4f21c3c6a0d22ac48af130.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k₁
+D₁
+k₂
+D₂
+k₃
+D₃
+P₁
+P₂
+P₃
+</details>
+
+Figure 3.3-1. A three d.o.f. system of three linear springs and three axial loads $P_{1}, P_{2}$ , and $P_{3}$ . D.o.f. $D_{i}$ are axial displacements relative to a fixed point, such as the left support. The springs are unstretched when $D_{1} = D_{2} = D_{3} = 0$ .
+
+In the matrix form $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{R}\}$ , Eqs. 3.3-4 are
+
+$$
+\left[ \begin{array}{c c c} k _ {1} + k _ {2} & - k _ {2} & 0 \\ - k _ {2} & k _ {2} + k _ {3} & - k _ {3} \\ 0 & - k _ {3} & k _ {3} \end{array} \right] \left\{ \begin{array}{l} D _ {1} \\ D _ {2} \\ D _ {3} \end{array} \right\} = \left\{ \begin{array}{l} P _ {1} \\ P _ {2} \\ P _ {3} \end{array} \right\} \tag {3.3-5}
+$$
+
+The correctness of stiffness matrix [K] in Eq. 3.3-5 can be checked by the procedure of activating one d.o.f. at a time, as described in Section 2.2.
+
+![](images/page-094_5dac28560dd8405cdf3f69794c3ba4ce7fcae92dd789f0f50cba1da2d7abd0d9.jpg)
+
+Example. Plane Truss Problems. Consider the plane truss element of Figs. 2.4-1 and 2.4-2. Its potential expression is
+
+$$
+\Pi_ {p} = \frac {1}{2} k e ^ {2} - p _ {i} u _ {i} - q _ {i} v _ {i} - p _ {j} u _ {j} - q _ {j} v _ {j} \tag {3.3-6}
+$$
+
+where $k = AE / L$ and elongation $e$ is given by
+
+$$
+e = (u _ {j} - u _ {i}) \cos \beta + (v _ {j} - v _ {i}) \sin \beta \tag {3.3-7}
+$$
+
+Setting to zero the four derivatives of $\Pi_p$ with respect to d.o.f. $u_i, v_i, u_j$ , and $v_j$ , we obtain the four rows of Eq. 2.4-3.
+
+The same procedure can be applied to an entire structure. Consider the three-bar truss of Fig. 2.2-1, with support conditions $u_{2} = v_{2} = u_{3} = 0$ already imposed. Its potential energy is
+
+$$
+\Pi_ {p} = \frac {1}{2} k _ {3} u _ {1} ^ {2} + \frac {1}{2} k _ {1} v _ {3} ^ {2} + \frac {1}{2} k _ {2} \left[ (0 - u _ {1}) (- 0. 6) + (v _ {3} - v _ {1}) (0. 8) \right] ^ {2} + P v _ {1} \tag {3.3-8}
+$$
+
+The $Pv_{1}$ term bears a positive sign because the potential of the (downward) load P is increased by a positive (upward) displacement $v_{1}$ . The derivatives of $\Pi_{p}$ with respect to $u_{1}$ , $v_{1}$ and $v_{3}$ , when equated to zero, are found to yield Eqs. 2.3-8.
+
+From the foregoing examples we draw the following conclusions, which are true in general.
+
+A system that has linear load versus displacement characteristics has a symmetric stiffness matrix; that is, $K_{ij} = K_{ji}$ . This happens because each symmetrically located pair of off-diagonal coefficients comes from a single term in $\Pi_p$ whose form is a constant times $D_i D_j$ . Thus $K_{ij} = \partial^2 \Pi_p / \partial D_i \partial D_j = \overline{\partial^2 \Pi_p / \partial D_j} \partial D_i = K_{ji}$ .
+
+2. If $D_{i}$ is a nodal displacement (or rotation), the equation $\sqrt{\partial\Pi_{p}/\partial D_{i}} = 0$ is a nodal equilibrium equation stating that forces (or moments) applied to the
+
+<!-- source-page: 95 -->
+
+node sum to zero in the direction of $D_{i}$ . Included in the sum are (a) loads applied externally, and (b) loads applied internally, owing to deformation of structural components (and perhaps also to thermal load, body force, etc.).
+
+3. Static indeterminacy does not alter the procedure or make a problem more difficult. For example, in Fig. 3.3-1 we could connect a fourth spring between the fixed support and node 3. Then $\Pi_p$ is augmented by $k_4D_3^2/2$ , and the last stiffness coefficient in Eq. 3.3-5 is changed from $k_3$ to $k_3 + k_4$ , but the same three d.o.f. still suffice.
+
+√ 4. The potential energy of a structure can be written in the form
+
+$$
+\left| \Pi_ {p} = U + \Omega , \right. \text {   where   } \left. U = \frac {1}{2} \{\mathbf {D} \} ^ {T} [ \mathbf {K} ] \{\mathbf {D} \} \right| \text {   and   } \left. \Omega = - \{\mathbf {D} \} ^ {T} \{\mathbf {R} \} \right. \tag {3.3-9}
+$$
+
+If $U = 0$ , then either $\{\mathbf{D}\} = \{\mathbf{0}\}$ or $\{\mathbf{D}\}$ expresses a rigid-body motion. If the structure is stable and is supported so that rigid-body motion is not possible (as in Eqs. 3.3-3 and 3.3-5), then $\frac{1}{2}\{\mathbf{D}\}^T [\mathbf{K}]\{\mathbf{D}\} > 0$ for any nonzero $\{\mathbf{D}\}$ ; that is, $[\mathbf{K}]$ is said to be positive definite.
+
+# 3.4 POTENTIAL ENERGY OF AN ELASTIC BODY
+
+The potential energy of an elastic body consists of the strain energy contained in elastic distortions and the potential of loads that act within the body or on its surface. The potential energy expression can be used to formulate element stiffness matrices and element load vectors. Simple finite element formulations appear later in this chapter. Additional formulations appear in subsequent chapters.
+
+In this section we present formulas, argue their validity, show that special cases yield correct results, and consider examples. Derivations and detailed arguments may be found elsewhere [3.1,3.2].
+
+Consider a linearly elastic body that carries conservative loads. Let its volume be V and its surface area be S. The expression for its potential energy is
+
+$$
+\begin{array}{l} \Pi_ {p} = \int_ {V} \left(\frac {1}{2} \{\epsilon \} ^ {T} [ \mathbf {E} ] \{\epsilon \} - \{\epsilon \} ^ {T} [ \mathbf {E} ] \{\epsilon_ {0} \} + \{\epsilon \} ^ {T} \{\sigma_ {0} \}\right) d V \\ - \int_ {V} \{\mathbf {u} \} ^ {T} \{\mathbf {F} \} d V - \int_ {S} \{\mathbf {u} \} ^ {T} \{\boldsymbol {\Phi} \} d S - \{\mathbf {D} \} ^ {T} \{\mathbf {P} \} \tag {3.4-1} \\ \end{array}
+$$
+
+The notation is explained in Section 1.6 and in the following.
+
+Explanation and Justification. In the first integral of Eq. 3.4-1, the expression within parentheses represents $U_{0}$ , the strain energy per unit volume. The expression is derived as follows. Consider a unit cube (i.e., a cube of unit length along each edge). Stresses that act on faces of the cube do work during infinitesimal straining of the cube. This work is stored as an increment of strain energy $dU_{0}$ . On a unit cube, stress and force have the same magnitude on each face, and strain and elongation have the same magnitude along each edge. Therefore, since work is equal to force times displacement,
+
+$$
+\left| d U _ {0} = \sigma_ {x} d \epsilon_ {x} + \sigma_ {y} d \epsilon_ {y} + \sigma_ {z} d \epsilon_ {z} + \tau_ {x y} d \gamma_ {x y} + \tau_ {y z} d \gamma_ {y z} + \tau_ {z x} d \gamma_ {z x} \right. \tag {3.4-2}
+$$
+
+<!-- source-page: 96 -->
+
+Changes in stress produced by infinitesimal strain increments have been discarded from $dU_0$ because they produce terms of higher order. For example, $(\sigma_x + d\sigma_x)$ $d\epsilon_x \approx \sigma_x d\epsilon_x$ . From Eq. 3.4-2 we conclude that $\partial U_0 / \partial \epsilon_x = \sigma_x$ , $\overline{\partial U_0 / \partial \epsilon_y} = \overline{\sigma_y, \ldots, \partial U_0 / \partial \gamma_{zx}} = \tau_{zx}$ . Expressing these six derivatives in matrix format and using the stress-strain relation (Eq. 1.7-8), we obtain
+
+$$
+\left\{\frac {\partial U _ {0}}{\partial \epsilon} \right\} = \{\boldsymbol {\sigma} \} \quad \text { or } \quad \left\{\frac {\partial U _ {0}}{\partial \epsilon} \right\} = [ \mathrm{E} ] \{\boldsymbol {\epsilon} \} - [ \mathrm{E} ] \{\boldsymbol {\epsilon} _ {0} \} + \{\boldsymbol {\sigma} _ {0} \} \tag {3.4-3}
+$$
+
+Integration of the latter equation with respect to the strains yields the parenthetic expression in Eq. 3.4-1. That the integration is correct may be shown by applying differentiation rules given in Appendix A. A constant of integration has been discarded. It is superfluous because it disappears during the differentiation process that makes $\Pi_{n}$ stationary.
+
+Integrals in Eq. 3.4-1 that contain body forces $\{F\}$ and surface tractions $\{\Phi\}$ represent work done (hence potential lost) by $\{F\}$ and $\{\Phi\}$ as the body is deformed. Displacements in the x, y, and z coordinate directions are
+
+$$
+\{\mathbf {u} \} ^ {T} = \left[ \begin{array}{l l l} u & v & w \end{array} \right] \tag {3.4-4}
+$$
+
+Thus potential changes associated with $\{F\}$ and $\{\Phi\}$ , per unit volume and per unit area, respectively, are
+
+$$
+- F _ {x} u - F _ {y} v - F _ {z} w \quad \text { and } \quad - \Phi_ {x} u - \Phi_ {y} v - \Phi_ {z} w \tag {3.4-5}
+$$
+
+These are the products $-\{\mathbf{u}\}^T\{\mathbf{F}\}$ and $-\{\mathbf{u}\}^T\{\Phi\}$ . It is assumed that positive senses correspond—for example, that $F_x$ , $\Phi_x$ , and $u$ are all considered positive when acting in the $+x$ direction. Integrals that contain $\{\mathbf{F}\}$ and $\{\Phi\}$ are evaluated only over the portions of $V$ and $S$ where $\{\mathbf{F}\}$ and $\{\Phi\}$ are prescribed.
+
+The final term in Eq. 3.4-1, $-\{\mathbf{D}\}^{T}\{\mathbf{P}\} = -D_{1}P_{1} - D_{2}P_{2} - \cdots - D_{n}P_{n}$ , accounts for work done (hence potential lost) by concentrated forces and/or moments applied to the body. The potential of these loads could be included in the surface integral by imagining large tractions to be applied over small areas, but is easier to regard the potential of such loads as $-\{\mathbf{D}\}^{T}\{\mathbf{P}\}$ . As usual, the same sense is considered positive for a displacement or rotation $D_{i}$ and for its corresponding force or moment $P_{i}$ .
+
+In a typical problem, many of the load terms $\{\epsilon_0\}$ , $\{\sigma_0\}$ , $\{\mathbf{F}\}$ , $\{\Phi\}$ , and $\{\mathbf{P}\}$ are zero. For example, if heating is localized $\{\epsilon_0\}$ is zero over most of the body. If gravity or acceleration loads are considered unimportant, $\{\mathbf{F}\} = \{\mathbf{0}\}$ . Surface tractions $\{\Phi\}$ usually act on only a portion of surface $S$ and may be absent altogether. Indeed, all these load terms might be zero if nonzero values of one or more d.o.f. are prescribed instead.
+
+Equation 3.4-1 is not restricted to rectangular coordinates. It requires only that x, y, and z refer to three mutually perpendicular directions at each material point.
+
+Particular Cases. In its most general form, Eq. 3.4-1 includes all six strains in $\{\epsilon\}$ . Material property matrix [E] is 6 by 6. Its coefficients are stated in Eq. 1.7-3 for the case of isotropy.
+
+If the problem is one of plane stress or plane strain, then $\{\epsilon\}^T = \left[\epsilon_x \quad \epsilon_y \quad \gamma_{xy}\right]$ and [E] is a 3 by 3 array whose coefficients are given by Eq. 1.7-4 or Eq. 1.7-5 for the case of isotropy.
+
+<!-- source-page: 97 -->
+
+The simplest special case is that of uniaxial stress. For this particular case, and perhaps for others as well, it may be easiest to derive the strain energy expression afresh, as follows, rather than specialize Eq. 3.4-1. Equations 3.4-2 and 3.4-3 become $dU_{0} = \sigma_{x} d\epsilon_{x}$ and $dU_{0}/d\epsilon_{x} = \sigma_{x} = E\epsilon_{x} - E\epsilon_{x0} + \sigma_{x0}$ . After integration and inclusion of the load terms, we obtain in place of Eq. 3.4-1
+
+$$
+\Pi_ {p} = \int_ {0} ^ {L} \left(\frac {1}{2} E \epsilon_ {x} ^ {2} - \epsilon_ {x} E \epsilon_ {0} + \epsilon_ {x} \sigma_ {0}\right) A d x - \int_ {0} ^ {L} u F _ {x} A d x - \{\mathbf {D} \} ^ {T} \{\mathbf {P} \} \tag {3.4-6}
+$$
+
+where $E =$ elastic modulus, $dV = A dx$ , $A =$ cross-sectional area, and $L =$ length. In the second integral, the integrand could be regarded as $uF_x dV$ or as $uq dx$ , where $q = F_x A$ : axial body force $F_x$ and axial line load $q$ have the same effect when the body is mathematically one-dimensional.
+
+In beam bending, Fig. 3.4-1, each z = constant layer of the beam is regarded as being in a state of uniaxial stress if, as is common, transverse shear deformation is neglected. Accordingly, the expression for strain energy in a beam can be derived from Eq. 3.4-6. Let b represent the width of the beam. Since $\epsilon_{x} = u_{,x}$ and $u = -zw_{,x}$ , the first term in Eq. 3.4-6 yields
+
+$$
+\int \frac {1}{2} E \epsilon_ {x} ^ {2} d V = \iint \frac {1}{2} E (- z w _ {, x x}) ^ {2} b d z d x = \int \frac {1}{2} E I w _ {, x x} ^ {2} d x \tag {3.4-7}
+$$
+
+where, if the cross section is rectangular, $I = bt^{3}/12$ is the moment of inertia of cross-sectional area A. If terms analogous to $\epsilon_{0}$ and $\sigma_{0}$ are omitted (but which the reader may add as an exercise), the expression for potential of a straight beam without transverse shear deformation is
+
+$$
+\Pi_ {p} = \int_ {0} ^ {L} \frac {1}{2} E I w _ {, x x} ^ {2} d x - \int_ {0} ^ {L} w q d x - \{\mathbf {w} \} ^ {T} \{\mathbf {F} \} - \{\boldsymbol {\theta} \} ^ {T} \{\mathbf {M} \} \tag {3.4-8}
+$$
+
+where $\{\mathbf{w}\}^T = \lfloor w_1 \quad w_2 \quad \ldots \rfloor$ and $\{\theta\}^T = \lfloor \theta_1 \quad \theta_2 \quad \ldots \rfloor$ are lateral deflections and rotations $(\theta = w_{,x})$ at locations where lateral forces $\{\mathbf{F}\}$ and moments $\{\mathbf{M}\}$ are applied as loads. The second integral in Eq. 3.4-8 accounts for work done by lateral force increments $q dx$ during lateral displacement $w$ . Axial stress in the beam is $\sigma_x = E\epsilon_x = Eu_x = -Ezw_{,xx}$ .
+
+![](images/page-097_27ab4bf65023a87900c7bf739698fe20c9f95ca9b05c2c58e7ca78f4bc5a481e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+F₁ F₂ q
+M₁ M₂
+x,u
+L
+(a)
+</details>
+
+![](images/page-097_f493192853eec5367673cd16d9b04245128c8c0a5960917d9acffad24a6f0a42.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+w_x
+w_x
+w
+x,u
+u = -zw_x
+t/2
+z
+dx
+t/2
+(b)
+</details>
+
+Figure 3.4-1. (a) A beam loaded by lateral forces $F_{i}$ , moments $M_{i}$ , and distributed lateral load q (force per unit length). (b) A slice cut from the beam, shown after it has undergone lateral deflection w and small rotation $w_{xx}$ . Transverse shear deformation is ignored.
+
+<!-- source-page: 98 -->
+
+![](images/page-098_d235243ca2e143640a900533dfed990106a76d3c95e51514e8d18dfa3a96e55c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+D
+P
+x,u
+L
+</details>
+
+Figure 3.4-2. Uniform bar under axial load P.
+
+Plates in bending are analogous to beams in that strain energy is conveniently expressed in terms of curvatures rather than strains. Specifically, in a thin plate having lateral deflection w, strain energy can be expressed in terms of $w_{,xx}, w_{,xy}$ , and $w_{,yy}$ instead of $\epsilon_{x}, \epsilon_{y}$ , and $\gamma_{xy}$ . Potential expressions for problems of plates, shells, and other special cases are introduced in subsequent chapters where they are used.
+
+Example. Bar Under Axial Load. The correctness of Eq. 3.4-6 can be established by solving simple test problems. For example, let the uniform bar of Fig. 3.4-2 carry an end load P and be uniformly heated T degrees. D designates the end displacement produced by P and T. With $\epsilon_{0} = \alpha T$ , $\sigma_{0} = 0$ , and $\epsilon_{x} = D/L$ , Eq. 3.4-6 becomes
+
+$$
+\Pi_ {p} = \int_ {0} ^ {L} \left(\frac {1}{2} E \frac {D ^ {2}}{L ^ {2}} - \frac {D}{L} E \alpha T\right) A d x - D P = \frac {E A D ^ {2}}{2 L} - D E A \alpha T - D P \tag {3.4-9}
+$$
+
+End displacement $D$ is found from the equation $d\Pi_{\rho} / dD = 0$ :
+
+$$
+D = \frac {P L}{A E} + \alpha T L \tag {3.4-10}
+$$
+
+Finally, the axial stress $\sigma_{x}$ is, with $\epsilon_{x} = D / L$ ,
+
+$$
+\sigma_ {x} = E \epsilon_ {x} - E \epsilon_ {0} = E \left(\frac {P}{A E} + \alpha T\right) - E \alpha T = \frac {P}{A} \tag {3.4-11}
+$$
+
+which is the result expected.
+
+# 3.5 THE RAYLEIGH-RITZ METHOD
+
+A structure composed of discrete members, such as a truss or a frame, can be represented exactly by a finite number of d.o.f.; these d.o.f. are motions of the joints. A continuum, such as an elastic solid, has infinitely many d.o.f.; these d.o.f. are the displacements of every material point. The behavior of a continuum is described by partial differential equations. For all but the simplest problems there is little hope of discovering a stress field or a displacement field that solves the differential equations and satisfies boundary conditions. The need to solve differential equations can be avoided by applying the Rayleigh–Ritz method to a functional such as $\Pi_{p}$ that describes the problem. The result is a substitute problem that has a finite number of d.o.f. and is described by algebraic equations rather than by differential equations. A Rayleigh–Ritz solution is rarely exact but becomes more accurate as more d.o.f. are used.
+
+<!-- source-page: 99 -->
+
+The Rayleigh–Ritz method began in 1870 with studies of vibration problems by Lord Rayleigh. He used an approximating field that contained a single d.o.f. In 1909, Ritz generalized the method by building an approximating field from several functions, each satisfying essential (i.e., kinematic) boundary conditions, and each associated with a separate d.o.f. Ritz applied the method to equilibrium problems and to eigenvalue problems. The procedure for an equilibrium (static) problem is as follows.
+
+Consider an elastic solid. Displacements and stresses produced by applied loads are required. The displacement of a point is described by the displacement components $u, v,$ and $w$ . A Rayleigh-Ritz solution begins with approximating fields for $u, v,$ and $w$ . Each field is a series, whose typical term is a function of the coordinates, $f_{i} = f_{i}(x,y,z)$ , times an amplitude $a_{i}$ whose value is yet to be determined. The $a_{i}$ may be called generalized coordinates. We write
+
+$$
+u = \sum_ {i = 1} ^ {\ell} a _ {i} f _ {i} \quad v = \sum_ {i = \ell + 1} ^ {m} a _ {i} f _ {i} \quad w = \sum_ {i = m + 1} ^ {n} a _ {i} f _ {i} \tag {3.5-1}
+$$
+
+Each of the functions $f_{i} = f_{i}(x,y,z)$ must be admissible; that is, each must satisfy compatibility conditions and essential boundary conditions. It is not required that any of the $f_{i}$ satisfy nonessential boundary conditions (but doing so yields a more accurate approximation for a given number of d.o.f.). Usually, but not necessarily, the $f_{i}$ are polynomials. The analyst must estimate how many terms are needed in each series in order to achieve the accuracy required. Thus the series are truncated rather than infinite, having, respectively, $\ell, m - \ell$ , and $n - m$ terms, for a total of $n$ terms.
+
+The d.o.f. of the problem are the $n$ amplitudes $a_{i}$ . They are determined as follows. Substitute Eqs. 3.5-1 into the strain-displacement relations (Eqs. 1.5-6) to find strains $\{\epsilon\}$ , then use Eq. 3.4-1 to evaluate $\Pi_{p}$ . Thus $\Pi_{p}$ becomes a function of d.o.f. $a_{i}$ , just as $\Pi_{p}$ is a function of d.o.f. $D_{i}$ in Eq. 3.3-1. According to the principle of stationary potential energy, the equilibrium configuration is defined by the $n$ algebraic equations
+
+$$
+\frac {\partial \Pi_ {p}}{\partial a _ {i}} = 0 \quad \text { for } \quad i = 1, 2, \dots , n \tag {3.5-2}
+$$
+
+After Eqs. 3.5-2 are solved for numerical values of the $a_{i}$ , the displacement fields of Eqs. 3.5-1 are completely defined. Differentiation of the displacement fields yields strains, which enter the stress-strain relations to produce stresses.
+
+The foregoing procedure has two principal steps. First, establish a trial family of admissible solutions. Second, apply a criterion to select the best form of the family. Here the criterion is that $\Pi_{p}$ be stationary. Alternative criteria are available, such as methods of weighted residuals (Chapter 15).
+
+Equations 3.5-1 create a substitute problem because the infinitely many d.o.f. of the real structure are replaced by the finite number of d.o.f. in the mathematical model. A Rayleigh–Ritz solution is usually approximate because the functions $f_{i}$ are usually incapable of exactly representing the actual displacements. The solution process selects amplitudes $a_{i}$ so as to combine the functions $f_{i}$ to best advantage. When $\Pi_{p}$ is the functional, “best” means tending to satisfy differential equations of equilibrium and stress boundary conditions more and more closely as more and more terms $a_{i}f_{i}$ are added to the series.
+
+<!-- source-page: 100 -->
+
+Equations 3.5-2 are found to be stiffness equations. They can be written in the usual form $[K]\{D\} = \{R\}$ , where $\{D\} = \left[a_{1} - a_{2} - \ldots - a_{n}\right]^{T}$ . Not all $D_{i}$ have units of displacement and not all $R_{i}$ have units of force, but each product $R_{i}D_{i}$ has units of work or energy.
+
+Example. Bar Under Axial Load. Consider the uniform bar of Fig. 3.5-1a. The load is distributed along the length of the bar in linearly varying fashion: $q = cx$ , where $c$ is a constant that has units of force divided by the square of length. Axial displacement $u$ and axial stress $\sigma_x$ are to be computed by the Rayleigh-Ritz method.
+
+Axial strain is $\epsilon_{x} = u_{,x}$ . Thus eq. 3.4-6 becomes
+
+$$
+\Pi_ {\rho} = \int_ {0} ^ {L _ {T}} \frac {1}{2} E u _ {, x} ^ {2} A d x - \int_ {0} ^ {L _ {T}} u (c x) d x \tag {3.5-3}
+$$
+
+Equation 3.5-1 becomes, with $f_{i} = f_{i}(x)$ and only polynomial functions considered,
+
+$$
+u = \sum_ {i = 1} ^ {n} a _ {i} f _ {i} = a _ {1} x + a _ {2} x ^ {2} + a _ {3} x ^ {3} + \dots + a _ {n} x ^ {n} \tag {3.5-4}
+$$
+
+Note that there is no initial term $a_0$ : the displacement mode $u = a_0$ is inadmissible because it violates the essential boundary condition—namely $u = 0$ at $x = 0$ .
+
+The simplest approximation results from using only the first term of the series, $u = a_{1}x$ . From Eqs. 3.5-2, 3.5-3, and 3.5-4,
+
+$$
+\Pi_ {p} = \frac {A E L _ {T}}{2} a _ {1} ^ {2} - \frac {c L _ {T} ^ {3}}{3} a _ {1} \tag {3.5-5a}
+$$
+
+$$
+\frac {d \Pi_ {p}}{d a _ {1}} = 0 \quad \text { yields } \quad a _ {1} = \frac {c L _ {T} ^ {2}}{3 A E} \tag {3.5-5b}
+$$
+
+$$
+\text { hence } \quad u = \frac {c L _ {T} ^ {2}}{3 A E} x ^ {\nu} \quad \text { and } \quad \sigma_ {x} = E u _ {, x} = \frac {c L _ {T} ^ {2}}{3 A} \tag {3.5-5c}
+$$
+
+Before commenting on these results we consider a two-term solution, using the field
+
+![](images/page-100_063652fca2065b97770a4e3a6f81afc4c35a0bdb0e3ee7770d74f43790ac676e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+q = cx
+x,u
+L_T
+(a)
+</details>
+
+![](images/page-100_144ba22e5694a4889b58372362397f2e93919bbe8a274d5111863238b93fdb4e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Exact and two
+terms (almost
+coincident)
+One term
+0
+x
+L_T
+(b)
+</details>
+
+![](images/page-100_3f7a88c1a8adeb95dd543abd7a7d6197127a89a78785ce2a791c8b4f141402c6.jpg)
+
+<details>
+<summary>line</summary>
+| x    | One Term | Exact | Two Terms |
+| ---- | -------- | ----- | --------- |
+| 0    | 0        | 0     | 0         |
+| L_T  | 0        | 0     | 0         |
+</details>
+
+Figure 3.5-1. (a) Uniform bar under linearly varying distributed axial load of intensity q = cx, where c is a constant. (b) Exact and approximate axial displacements. (c) Exact and approximate axial stresses.
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_011.md
@@ -0,0 +1,396 @@
+<!-- source-page: 101 -->
+
+$u = a_{1}x + a_{2}x^{2}$ . After substituting into Eq. 3.5-3 and writing $\frac{\partial\Pi_{p}}{\partial a_{1}} = 0$ and $\frac{\partial\Pi_{p}}{\partial a_{2}} = 0$ , we obtain
+
+$$
+A E L _ {T} \left[ \begin{array}{l l} 1 & L _ {T} \\ L _ {T} & 4 L _ {T} ^ {2} / 3 \end{array} \right] \left\{ \begin{array}{l} a _ {1} \\ a _ {2} \end{array} \right\} = \frac {c L _ {T} ^ {3}}{1 2} \left\{ \begin{array}{l} 4 \\ 3 L _ {T} \end{array} \right\} \quad \text { or } \quad \left\{ \begin{array}{l} a _ {1} \\ a _ {2} \end{array} \right\} = \frac {c L _ {T}}{1 2 A E} \left\{ \begin{array}{l} 7 L _ {T} \\ - 3 \end{array} \right\} \tag {3.5-6a}
+$$
+
+hence $u = \frac{cL_T}{12AE} (7L_Tx - 3x^2)$ and $\sigma_x = Eu_x = \frac{cL_T}{12A} (7L_T - 6x)$ (3.5-6b)
+
+Figures 3.5-1b and 3.5-1c show the comparison between exact and approximate results. As might be expected, the two-term results (Eqs. 3.5-6) are better than the one-term results (Eqs. 3.5-5). With the body force term $F_{x} = cx/A$ , the differential equation of equilibrium (Eq. 1.6-2a) becomes
+
+$$
+\sigma_ {x, x} + \frac {c x}{A} = 0 \quad \text { or } \quad A E u _ {, x x} + c x = 0 \tag {3.5-7}
+$$
+
+The latter equation results from the substitution $\sigma_{x}=Eu_{,x}$ . Neither Eq. 3.5-7, nor the natural boundary condition $\sigma_{x}=0$ at $x=L_{T}$ , is satisfied by the foregoing two approximate solutions.
+
+Notice that approximate displacements are more accurate than approximate stresses. This is to be expected, because stresses are calculated from derivatives of the approximating field. (To see that differentiation emphasizes discrepancies, consider the functions $f_{1}=4x(1-x)$ and $f_{2}=\sin\pi x$ . In the range $0<x<1$ , functions $f_{1}$ and $f_{2}$ look much alike, but successive derivatives of $f_{1}$ and $f_{2}$ have less and less resemblance.)
+
+The exact solution of the problem of Fig. 3.5-1a is
+
+$$
+u = \frac {c}{6 A E} (3 L _ {T} ^ {2} x - x ^ {3}) \quad \begin{array}{l} \text {   解   } \text {   1,   } \text {   2,   } \text {   and   } \text {   derivative   } \text {   3,   } \text {   4,   } \\ \text {   Shall   } \text {   non   } \text {   is   } \text {   denote   } \text {   each   } \\ \text {   (3.5 - 8)   } \text {   2   } \text {   3,   } \text {   4,   } \end{array}
+$$
+
+Use of the series $u = a_{1}x + a_{2}x^{2} + a_{3}x^{3}$ in the Rayleigh-Ritz method produces
+
+$$
+a _ {1} = \frac {c L _ {T} ^ {2}}{2 A E}, \quad a _ {2} = 0 \quad a _ {3} = - \frac {c}{6 A E} \tag {3.5-9}
+$$
+
+which is the exact solution. Use of still more terms leads again to the exact solution: one finds $a_{1}, a_{2}$ , and $a_{3}$ to be the values given in Eq. 3.5-9 and $a_{4} = a_{5} = a_{6} = \cdots = a_{n} = 0$ . In general, the Rayleigh-Ritz method yields the exact solution if the approximating field is capable of representing the exact field by appropriate choice of d.o.f. $a_{i}$ . In practice this circumstance is rare.
+
+# 3.6 COMMENTS ON THE
+
+# RAYLEIGH-RITZ METHOD BASED
+
+# ON ASSUMED DISPLACEMENT FIELDS
+
+Approximating fields must be admissible and should be easy to use. Only polynomials, and occasionally sine and cosine functions, are simple enough to be practicable. Beyond this there are no easy answers to important questions: What sort of assumption for the field is best? What particular terms and how many of them? How good are the computed results? These difficulties and uncertainties are increased for multidimensional problems.
+
+<!-- source-page: 102 -->
+
+Let a problem be solved repeatedly, each time with another term added to the assumed field (as in Eqs. 3.5-5 and 3.5-6, for example). Thus we generate a sequence of trial solutions. We expect the sequence to converge: to the exact $\Pi_p$ , to the exact displacements, and to the exact stresses. A necessary condition for convergence is that trial field be complete.
+
+Completeness is achieved if the exact displacements, and their derivatives that appear in $\Pi_p$ , can be matched arbitrarily closely if enough terms appear in the trial field. A polynomial series is complete if it is of high enough degree and if no terms are omitted. Fourier series are also complete.
+
+Completeness demands that the lowest-order admissible terms be included. For example, consider the end-loaded bar of Fig. 3.4-2. If we omit the term $a_1x$ from Eq. 3.5-4, we omit the very term that contains the exact answer—namely, $u = (P/AE)x$ . Thus completeness is destroyed, and the sequence of approximate solution does not produce the exact answer even if the number of terms approaches infinity. This is easy to see: if Eq. 3.5-4 begins with $a_2x^2$ , then $\sigma_x = Eu_{,x} = 0$ at $x = 0$ , which is incorrect. The term $a_1x$ represents the essential constant-strain capability. (In a finite element context, this requirement means that each element must be capable of representing a state of constant strain.)
+
+Completeness also requires that no series terms be omitted. Referring again to Eq. 3.5-4, a two-term approximation should be $a_1x + a_2x^2$ but not $a_1x + a_3x^3$ , and not $a_1x + a_4x^4$ , and so on.
+
+In two dimensions, a polynomial is of degree $n$ if it contains a term of the form $x^{\ell}y^{m}$ , where $\ell$ and $m$ are nonnegative integers and $\ell + m = n$ . The polynomial is complete if it contains all combinations of $\ell$ and $m$ for which $\ell + m = n$ and if no lower-order terms are omitted. For example, a complete quadratic, $n = 2$ , has the form $u = a_{1} + a_{2}x + a_{3}y + a_{4}x^{2} + a_{5}xy + a_{6}y^{2}$ . A complete polynomial of degree $n$ in two dimensions contains $(n + 1)(n + 2)/2$ terms (see Fig. 3.6-1).
+
+In three dimensions, similar remarks apply, so that a complete quadratic contains 10 terms, which include a constant term, the linear terms $x, y$ , and $z$ , and the quadratic terms $x^2, xy, y^2, yz, z^2$ , and $zx$ .
+
+A Rayleigh-Ritz solution is either exact or it is too stiff. This happens because the mathematical structure is permitted to displace only into shapes that can be described by superposing the finite number of functions $f_{i}$ present in the assumed displacement field that the analyst selects. Therefore, the correct shape is excluded, unless the assumed field happens to contain it. Effectively, the assumed field imposes constraints that prevent the structure from deforming the way it wants to. Constraints stiffen a structure. In effect, the solution method creates a substitute structure that is stiffer than the real one.
+
+The work $W$ done by loads that gradually increase from zero to $\{\mathbf{R}\}$ is $W = \{\mathbf{D}\}^T \{\mathbf{R}\} / 2$ if the structure is linearly elastic. An approximate solution yields d.o.f.
+
+$$
+
+$$
+
+Figure 3.6-1. Pascal triangle, showing the number of terms in complete polynomials in two independent variables x and y.
+
+<!-- source-page: 103 -->
+
+$\{D\}$ such that work W is less than the exact value. This does not necessarily mean that every d.o.f. in $\{D\}$ is underestimated. But if the structure carries a single load P, we can say that its computed displacement D is a lower bound to the correct magnitude.
+
+Strain energy U is numerically equal to W, so the approximate solution underestimates U when loads are prescribed. If displacements are prescribed instead, U is overestimated because extra force is needed to deform an overly stiff structure. When loads and displacements are prescribed, U may be high or low.
+
+Stresses are calculated from displacements, so we expect that a too-stiff structure will underestimate stress magnitudes. However, as seen in Fig. 3.5-1c, approximate stresses may be too low in one place but too high in another, even when the stress is derived from an approximate displacement field that is everywhere too low. Accordingly, a rule about stress magnitudes would be either so crude or so equivocal as to be of little value.
+
+# 3.7 STATIONARY PRINCIPLES AND GOVERNING EQUATIONS
+
+The principle of stationary potential energy is but one of many stationary principles of mathematical physics. Central to each is a functional, of which $\Pi_p$ is but one. Rayleigh-Ritz approximations and finite element formulations can be derived from functionals. For this reason the following brief remarks are offered.
+
+Consider the functional for $\Pi_p$ , Eq. 3.4-1. It depends on displacements $\{\mathbf{u}\}$ and on strains $\{\epsilon\}$ , which are derivates of $\{\mathbf{u}\}$ . The term “functional” indicates that $\Pi_p$ depends not on $\{\mathbf{u}\}$ and its derivatives at a point but upon their integrated effect over a region of interest. The stationary condition $d\Pi_p = 0$ may be applied directly to Eq. 3.4-1, without first expressing $\Pi_p$ in terms of a finite number of d.o.f. This is accomplished by using the calculus of variations, the procedures of which are beyond the scope of this book [3.1]. However, the end results of setting $d\Pi_p$ to zero are found to be the differential equations of equilibrium (Eqs. 1.6-2) and the nonessential boundary conditions (the stress boundary conditions, Eqs. 1.6-4). Thus, if the field $\{\mathbf{u}\}$ is admissible, the statement $d\Pi_p = 0$ implies all components of a valid solution: satisfaction of equilibrium, compatibility, and boundary conditions. In an approximate solution, equilibrium conditions and stress boundary conditions are satisfied only in an average or integral sense, not at every point.
+
+In other physical problems there exist other functionals $\Pi$ . Instead of displacements $\{u\}$ , the primary field may be temperature, or pressure, or voltage, and so on. In each case the functional $\Pi$ can be tested for correctness by applying the calculus of variations to see if the condition $d\Pi = 0$ yields the appropriate governing differential equation and nonessential boundary conditions.
+
+Boundary Conditions. In order to use variational methods, one must be able to distinguish between essential and nonessential boundary conditions. For a problem having one dependent field variable, the rule is as follows. Let 2m be the highest-order derivative of the dependent field variable in the governing differential equation. (Derivatives of order m then appear in the functional.) Essential boundary conditions involve derivatives of order zero through m - 1, the zeroth derivative being the dependent variable itself. Nonessential boundary conditions
+
+<!-- source-page: 104 -->
+
+involve derivatives of order $m$ and higher, up to and including $2m - 1$ . The following are examples.
+
+<table><tr><td>Problem</td><td>Bar (Fig. 3.5-1a)</td><td>Beam bending</td><td>Two-dimensional heat conduction</td></tr><tr><td>Differential equation</td><td> $AEu_{,xx} + q = 0$ </td><td> $EIw_{,xxxx} - q = 0$ </td><td> $k\nabla^{2}T + Q = cp\dot{T}$ </td></tr><tr><td>2m, m - 1, 2m - 1</td><td>2, 0, 1</td><td>4, 1, 3</td><td>2, 0, 1</td></tr><tr><td>Essential B.C.</td><td>On u only</td><td>On w and  $w_{,x}$ </td><td>On T only</td></tr><tr><td>Nonessential B.C.</td><td>On  $\sigma_{x} = Eu_{,x}$ </td><td>On  $M = EIw_{,xx}$ and  $V = EIw_{,xxx}$ </td><td>On  $q = -k(T_{,x} \ell_{B} + T_{,y}m_{B})$ </td></tr></table>
+
+In these examples the dependent field variables are axial displacement $u$ , lateral displacement $w$ , and temperature $T$ . Nonessential boundary conditions concern axial stress $\sigma_x$ , bending moment $M$ , transverse shear force $V$ , and heat flow $q$ . In the heat conduction example, $k$ is thermal conductivity, $\dot{T}$ means the time derivative of $T$ , and $\ell_B$ and $m_B$ are direction cosines of a normal to the boundary (see Chapter 16).
+
+The foregoing remarks are little changed if there is more than one field. Imagine, for example, that there are dependent field variables $u$ and $v$ , with second derivatives $u,_{xx}, u,_{xy}, u,_{yy}, v,_{xx}, v,_{xy}$ , and $v,_{yy}$ in the governing differential equations and first derivatives $u,_{x}, u,_{y}, v,_{x}$ and $v,_{y}$ in the functional. Then $2m = 2$ and $m = 1$ for both $u$ and $v$ . Essential boundary conditions are prescriptions of $u$ and $v$ at particular locations. Nonessential boundary conditions involve first derivatives of $u$ and $v$ , either singly or in combination.
+
+Functionals and Governing Differential Equations. Imagine that a functional $\Pi$ depends on two dependent field variables, $u = u(x,y)$ and $v = v(x,y)$ , in which independent variables x and y are Cartesian coordinates:
+
+$$
+\Pi = \iint F (x, y, u, v, u, u, x, u, y, v, x, v, y, \dots , v, y y) d x d y \tag {3.7-1}
+$$
+
+Here we will assume that F contains no derivatives of order higher than second. There are as many “Euler equations” as there are dependent field variables. An Euler equation is a governing differential equation of the physical problem. Methods of calculus of variations extract from Eq. 3.7-1 the Euler equations
+
+$$
+\frac {\partial F}{\partial u} - \frac {\partial}{\partial x} \frac {\partial F}{\partial u _ {, x}} - \frac {\partial}{\partial y} \frac {\partial F}{\partial u _ {, y}} + \frac {\partial^ {2}}{\partial x ^ {2}} \frac {\partial F}{\partial u _ {, x x}} + \frac {\partial^ {2}}{\partial x \partial y} \frac {\partial F}{\partial u _ {, x y}} + \frac {\partial^ {2}}{\partial y ^ {2}} \frac {\partial F}{\partial u _ {, y y}} = 0 \tag {3.7-2a}
+$$
+
+$$
+\frac {\partial F}{\partial v} - \frac {\partial}{\partial x} \frac {\partial F}{\partial v _ {, x}} - \frac {\partial}{\partial y} \frac {\partial F}{\partial v _ {, y}} + \frac {\partial^ {2}}{\partial x ^ {2}} \frac {\partial F}{\partial v _ {, x x}} + \frac {\partial^ {2}}{\partial x \partial y} \frac {\partial F}{\partial v _ {, x y}} + \frac {\partial^ {2}}{\partial y ^ {2}} \frac {\partial F}{\partial v _ {, y y}} = 0 \tag {3.7-2b}
+$$
+
+Equations 3.7-1 and 3.7-2 both describe the same problem, Eq. 3.7-1 being called the “weak form” and Eqs. 3.7-2 the “strong form.”
+
+As a specific example of Eq. 3.7-2, consider the axially loaded uniform bar described by Fig. 3.5-1a. Here there is one independent variable, one dependent variable, and no second derivative. Equation 3.7-1 reduces to Eq. 3.5-3. There is but one Euler equation, Eq. 3.7-2a, which reduces to
+
+<!-- source-page: 105 -->
+
+$$
+\frac {\partial F}{\partial u} - \frac {d}{d x} \frac {\partial F}{\partial u _ {, x}} = 0 \quad \text { in   which } \quad F = \frac {1}{2} A E u _ {, x} ^ {2} - u (c x) \tag {3.7-3}
+$$
+
+Derivatives in the Euler equation are
+
+$$
+\frac {\partial F}{\partial u} = - c x \quad \text { and } \quad \frac {d}{d x} \frac {\partial F}{\partial u _ {, x}} = \frac {d}{d x} (A E u _ {, x}) = A E u _ {, x x} \tag {3.7-4}
+$$
+
+from which Eq. 3.5-7 is obtained, as expected.
+
+As another example, consider plane heat conduction in an isotropic material. A suitable functional is
+
+$$
+\Pi = \iint \left(\frac {1}{2} k T _ {, x} ^ {2} + \frac {1}{2} k T _ {, y} ^ {2} - Q T + \rho c T \dot {T}\right) d x d y \quad \text { or } \quad \Pi = \iint F d x d y \tag {3.7-5}
+$$
+
+in which T = temperature, k = thermal conductivity, Q = internally generated heat flow, $\rho = mass density$ , c = specific heat, and $\dot{T}$ is the time derivative $\partial T/\partial t$ . Unit thickness is assumed. (Equation 3.7-5 omits certain boundary terms of practical interest. See Section 16.3 for a more detailed treatment.) Again there is one dependent field variable, T, and one Euler equation—namely,
+
+$$
+\frac {\partial F}{\partial T} - \frac {\partial}{\partial x} \frac {\partial F}{\partial T _ {, x}} - \frac {\partial}{\partial y} \frac {\partial F}{\partial T _ {, y}} = 0 \tag {3.7-6}
+$$
+
+where $F$ is the integrand of Eq. 3.7-5. Equations 3.7-5 and 3.7-6 yield
+
+$$
+k \left(T, _ {x x} + T, _ {y y}\right) + Q - \rho c T = 0 \tag {3.7-7}
+$$
+
+as the differential equation that describes the temperature distribution in the region of interest.
+
+Potential energy functionals $\Pi_{p}$ for problems of beam bending and plate bending contain second derivatives of lateral displacement w. These and other examples are left as exercises.
+
+The calculus of variations also produces natural boundary conditions. An explanation of their derivation and interpretation takes more space than we can allot to it. The reader is referred to other texts $[3.1,3.2]$ .
+
+Finally, we remark that although there is always a differential equation associated with a functional, the reverse is not necessarily true. For example, a differential equation that contains an odd-numbered derivative does not have an associated functional of the form of Eq. 3.7-1.
+
+Variational Methods: A Brief Example. Figure 3.7-1 shows a uniform bar loaded by distributed axial load $q = q(x)$ and prescribed stress $\sigma_{L}$ at x = L. We will use this problem to illustrate that the calculus of variations produces the governing differential equation and the nonessential boundary condition. In this way we will discover the origin of terms seen in Eqs. 3.7-3 and 3.7-4. The development will
+
+<!-- source-page: 106 -->
+
+![](images/page-106_912cef4a7a4975b2f27e4f7c43a960b9504365c86cfb14d2566aaddd94fd50b9.jpg)
+
+<details>
+<summary>text_image</summary>
+
+x
+q
+σL
+L
+</details>
+
+(a)
+
+![](images/page-106_601441cc18228f327e5c15f2176871570d9531d92d20e9687a03515ba2ab4e15.jpg)
+
+<details>
+<summary>text_image</summary>
+
+q dx
+σₓ A ← → (σₓ + dσₓ)A
+← dx ←
+</details>
+
+(b)   
+Figure 3.7-1. (a) Uniform elastic bar loaded by distributed axial load q and end stress $\sigma_{L}$ . (b) Forces that act on a differential element of the bar.
+
+also be seen to produce the virtual work equation and to suggest an alternative formulation method (the method of weighted residuals).
+
+In Eq. 3.4-6, let $\epsilon_{x}=u_{,x}$ , $F_{x}=q/A$ , $D=u_{L}$ , and $P=A\sigma_{L}$ . Thus the potential energy functional for the bar in Fig. 3.7-1a is
+
+$$
+\Pi_ {p} = \frac {A E}{2} \int_ {0} ^ {L} u _ {, x} ^ {2} d x - \int_ {0} ^ {L} q u d x - (A \sigma_ {L}) u _ {L} \tag {3.7-8}
+$$
+
+We presume that $u = u(x)$ is an admissible displacement field for this problem-- that is, a field that is continuous and satisfies the essential boundary condition $u = 0$ at $x = 0$ . Let $u$ be perturbed by an amount $\delta u$ , which we elect to write as $\delta u = e\eta$ , where $e$ is a small number and $\eta = \eta(x)$ is an admissible field. Thus the perturbed field $u + e\eta$ is also admissible and satisfies the same essential boundary condition as $u$ . Hence, $u_{,x}$ becomes $u_{,x} + e\eta_{,x}$ , $u_L$ becomes $u_L + e\eta_L$ , and $\Pi_p$ becomes $\Pi_p + \delta\Pi_p$ . The change in energy $(\Pi_p + \delta\Pi_p) - \Pi_p$ is
+
+$$
+\delta \Pi_ {p} = e \left[ A E \int_ {0} ^ {L} u _ {, x} \eta_ {, x} d x - \int_ {0} ^ {L} q \eta d x - (A \sigma_ {L}) \eta_ {L} \right] + e ^ {2} \frac {A E}{2} \int_ {0} ^ {L} \eta_ {, x} ^ {2} d x \tag {3.7-9}
+$$
+
+According to the potential energy principle, stable equilibrium occurs when $\Pi_p$ is a relative minimum. This implies that $\delta\Pi_p > 0$ for any admissible $\eta$ . Now $e^2\eta_{,x}^2$ is never negative, and the remaining term $e[-\cdot]$ changes sign when $e$ changes sign. We conclude that if $\delta\Pi_p$ is to be positive for all small values of $e$ , the bracketed expression in Eq. 3.7-9 must vanish. Setting this expression to zero, and integrating its first term by parts according to the standard formula $\int u dv = -\int v du + uv$ , we obtain
+
+$$
+0 = - A E \int_ {0} ^ {L} u _ {, x x} \eta d x + \left[ A E u _ {, x} \eta \right] _ {0} ^ {L} - \int_ {0} ^ {L} q \eta d x - (A \sigma_ {L}) \eta_ {L} \tag {3.7-10}
+$$
+
+But $\eta = 0$ at $x = 0$ , so Eq. 3.7-10 becomes
+
+$$
+0 = - \int_ {0} ^ {L} (A E u _ {, x x} + q) \eta d x + A (E u _ {, x} - \sigma_ {L}) \eta_ {L} \tag {3.7-11}
+$$
+
+Since $\eta = \eta(x)$ is admissible but otherwise arbitrary, an arbitrary value of $\eta_{L}$ can be assigned while infinitely many functions $\eta$ are yet possible in the range 0 <
+
+<!-- source-page: 107 -->
+
+x < L. Accordingly, Eq. 3.7-11 can be satisfied only if the coefficients of $\eta$ and $\eta_{L}$ vanish separately. Thus we obtain
+
+$$
+A E u _ {, x x} + q = 0 \quad \text { for } \quad 0 <   x <   L \tag {3.7-12a}
+$$
+
+$$
+E u _ {, x} - \sigma_ {L} = 0 \quad \text { at } x = L \tag {3.7-12b}
+$$
+
+Equation 3.7-12a is the governing differential equation. It can be written in the alternative form $A\sigma_{x,x'} + q = 0$ , and can also be derived by considering the equilibrium of axial forces in Fig. 3.7-1b. Equation 3.7-12b is a nonessential (or natural) boundary condition, which says that $\epsilon_x = \sigma_L / E$ at $x = L$ .
+
+The vanishing of the bracketed expression in Eq. 3.7-9 can be regarded as an expression of the virtual work principle, which states that the total work of internal and external forces must vanish for any admissible infinitesimal displacement from an equilibrium configuration. In Eq. 3.7-9 internal forces $AEu_{,x} \, dx = A\sigma_{x} \, dx$ do work (and store strain energy) when strains $\eta_{,x}$ occur, and external forces q dx and $A\sigma_{L}$ do negative work (and lose potential energy) when positive displacements $\eta$ and $\eta_{L}$ occur.
+
+Matrices used in finite element analysis can be generated from either Eq. 3.7-8 or the bracketed expression in Eq. 3.7-9. (In Eq. 3.7-9, if we identify $u_{,x}$ as $\epsilon_{x}$ and $e\eta_{,x}$ as $\delta\epsilon_{x}$ , the first integrand becomes $\delta\epsilon_{x}AE\epsilon_{x}$ , which will subsequently be recognized as a familiar form.)
+
+The vanishing of the bracketed expression in Eq. 3.7-9 can also be obtained by “working backward,” as follows. Imagine that we seek an approximate solution $\bar{u} = \bar{u}(x)$ —for example, the admissible polynomial $\bar{u} = a_{1}x + a_{2}x^{2} + a_{3}x^{3} + \cdots$ , where the $a_{i}$ are constants that must be determined. Now $\bar{u}$ does not satisfy Eq. 3.7-12a for all x: a “residual,” $R = R(x) = AE\bar{u}_{xx} + q \neq 0$ , is left over. Nevertheless, we can select the $a_{i}$ so as to satisfy Eq. 3.7-12a in an average or integral sense by writing
+
+$$
+\int_ {0} ^ {L} (A E \tilde {u}, _ {x x} + q) \eta d x = 0 \tag {3.7-13}
+$$
+
+where $\eta = \eta(x)$ may now be called a “weight function.” Applying integration by parts to Eq. 3.7-13, we obtain
+
+$$
+- A E \int_ {0} ^ {L} \tilde {u} _ {, x} \eta_ {, x} d x + \left[ A E \tilde {u} _ {, x} \eta \right] _ {0} ^ {L} + \int_ {0} ^ {L} q \eta d x = 0 \tag {3.7-14}
+$$
+
+But $\eta = 0$ at $x = 0$ . In addition, at $x = L$ , we may replace $E\hat{u}_{,x}$ by $\sigma_L$ , thus introducing the nonessential boundary condition. Equation 3.7-14 becomes
+
+$$
+- A E \int_ {0} ^ {L} \tilde {u} _ {, x} \eta_ {, x} d x + \int_ {0} ^ {L} \dot {q} \eta d x + (A \sigma_ {L}) \eta_ {L} = 0 \tag {3.7-15}
+$$
+
+which agrees with the vanishing of the bracketed expression in Eq. 3.7-9. This method of formulating a problem for approximate solution is called a weighted residual method. It can be applied to problems for which one knows the differential equation but not the functional or the variational principle. If $\eta_{i} = \partial\bar{u}/\partial a_{i}$ the method is known as the Galerkin method (for which Eq. 3.7-15 yields as many
+
+<!-- source-page: 108 -->
+
+equations as there are $a_i$ to be determined). A more detailed discussion appears in Chapter 15.
+
+# 3.8 A PIECEWISE POLYNOMIAL FIELD
+
+In this section we use a one-dimensional example to illustrate how a displacement field can be written in terms of physical displacements $\{\mathbf{d}\}$ rather than parameters $a_{i}$ (as in Eq. 3.5-4). The use of $\{\mathbf{d}\}$ , in combination with a piecewise polynomial field, leads to the finite element method in a form that is easy to program for computer solution. We begin by using the $a_{i}$ , then show how to replace them by functions of nodal d.o.f. $d_{i}$ .
+
+Consider the bar of Fig. 3.8-1a. It is to be loaded axially. Axial displacement $u$ over the length from $x = 0$ to $x = L_T$ is to be approximated as three separate linear fields,
+
+$$
+u = a _ {1} + a _ {2} x \quad \text { for } \quad 0 \leqslant x \leqslant x _ {2} \tag {3.8-1a}
+$$
+
+$$
+u = a _ {3} + a _ {4} x \quad \text { for } \quad x _ {2} \leqslant x \leqslant x _ {3} \tag {3.8-1b}
+$$
+
+$$
+u = a _ {5} + a _ {6} x \quad \text { for } \quad x _ {3} \leqslant x \leqslant x _ {4} \tag {3.8-1c}
+$$
+
+where the $a_{i}$ are d.o.f. to be determined in a subsequent Rayleigh–Ritz solution. For the field of Eqs. 3.8-1 to be admissible we must have u = 0 at x = 0. In addition, the first and second expressions must yield the same u at $x = x_{2}$ , and the second and third expressions must yield the same u at $x = x_{3}$ . These three conditions yield $a_{1} = 0$ , $a_{3} = (a_{2} - a_{4})x_{2}$ , and $a_{5} = (a_{2} - a_{4})x_{2} + (a_{4} - a_{6})x_{3}$ . Thus Eqs. 3.8-1 assume the form
+
+$$
+u = a _ {2} x \quad \text { for } \quad 0 \leqslant x \leqslant x _ {2} \tag {3.8-2a}
+$$
+
+$$
+u = a _ {2} x _ {2} + a _ {4} (x - x _ {2}) \quad \text { for } \quad x _ {2} \leqslant x \leqslant x _ {3} \tag {3.8-2b}
+$$
+
+$$
+u = a _ {2} x _ {2} + a _ {4} \left(x _ {3} - x _ {2}\right) + a _ {6} \left(x - x _ {3}\right) \quad \text { for } x _ {3} \leqslant x \leqslant x _ {4} \tag {3.8-2c}
+$$
+
+![](images/page-108_69de6dbfab8602cf86c280566d22c5fa222e47d5ea840d59b61877c0f04e1630.jpg)
+
+<details>
+<summary>text_image</summary>
+
+u
+x
+①
+②
+③
+1
+2
+3
+4
+x,u
+x₂
+x₃
+x₄
+Lₜ
+</details>
+
+(a)
+
+![](images/page-108_19331717b8df3f043e6b99e3773ee4a7015081c523398a370fa56ac736f7039d.jpg)
+
+![](images/page-108_05b4938086448af2c01ffa52cdaf6277934d7db4220a392a43c2582dad66f88b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+generic element
+i
+j
+u
+s
+L
+</details>
+
+(b)   
+Figure 3.8-1. (a) Bar whose axial displacement field $u = u(x)$ is approximated by a piecewise linear fit. (b) Separate regions (elements) of the bar.
+
+<!-- source-page: 109 -->
+
+The foregoing procedure has drawbacks. First, the $a_{i}$ do not have an obvious physical meaning. Second, neither the passage from Eqs. 3.8-1 to 3.8-2 nor the subsequent determination of d.o.f. $a_{i}$ from the equations $\partial\Pi_{p}/\partial a_{i}=0$ is readily coded, especially if the user of the program is to be allowed to use various loadings, other boundary conditions, and more $a_{i}$ than six. These drawbacks are neatly avoided by expressing the displacement field in terms of nodal d.o.f. rather than the $a_{i}$ . The procedure is as follows.
+
+Shape Function Matrix. Consider the generic element, Fig. 3.8-1b. Its axial displacement u is to be linear in axial coordinate s. The displacement field must yield $u = u_{i}$ at one end and $u = u_{j}$ at the other. By inspection, we write
+
+$$
+\begin{array}{l l l l}&\rightarrow u = \frac {L - s}{L} u _ {i} + \frac {s}{L} u _ {j}&\text { or }&u = \lfloor \mathbf {N} \rfloor \{\mathbf {d} \}\\\text { where }&u = a _ {1} + a _ {2} s \Rightarrow u = [ 1 \leq ] \left[\begin{array}{l}a _ {1}\\a _ {2}\end{array}\right]&\Rightarrow&u _ {\xi} = u _ {i} \oplus s = 0 \Rightarrow u _ {i} = a _ {1} + a _ {2} =\\&&&u = u _ {j} \oplus s = L \Rightarrow u _ {j} = a _ {1} + a _ {2} =\\&&&\left[\begin{array}{l}a _ {1} ^ {\prime}\\a _ {j} ^ {\prime}\end{array}\right] = \left[\begin{array}{l}1 0\\1 L\end{array}\right] \left[\begin{array}{l}a _ {1}\\a _ {2}\end{array}\right]\\&\lfloor \mathbf {N} \rfloor = \left\lfloor \frac {L - s}{L} \frac {s}{L} \right\rfloor&\text { and }&\{\mathbf {d} \} = \left\{\begin{array}{l}u _ {i}\\u _ {j}\end{array}\right\}\end{array}\tag {3.8-4}
+$$
+
+Checking, we see the $u$ is indeed linear in $s$ and assumes the values $u = u_{i}$ at $s = 0$ and $u = u_{j}$ at $s = L$ . Therefore, the expression written is the one desired.
+
+Matrix [N] is usually called a shape function matrix. Its terms $N_i$ may be called shape, basis, or interpolation functions. Each of the two terms $N_1$ and $N_2$ defines how displacement $u$ varies with $x$ when the corresponding d.o.f. has unit value while the other d.o.f. is zero. Matrix [N] describes how $u$ is to be interpolated from nodal values $u_i$ and $u_j$ over the elements—that is, over the range $0 \leqslant s \leqslant L$ . Here the shape function matrix happens to be a row matrix. For many other elements, treated in subsequent chapters, it is a rectangular matrix. Often, as in Eq. 3.8-3, it is possible to write [N] by inspection and a bit of trial. Alternatively, [N] can be formally derived, as we now illustrate.
+
+Consider again the generic element in Fig. 3.8-1b. We begin with the linear displacement field
+
+$$
+u = a _ {1} + a _ {2} s \quad \text { or } \quad u = \left\lfloor 1 - s \right\rfloor \left\{ \begin{array}{l} a _ {1} \\ a _ {2} \end{array} \right\} \tag {3.8-5}
+$$
+
+This field must take on the values $u = u_{i}$ at $s = 0$ and $u = u_{j}$ at $s = L$ :
+
+$$
+\left\{ \begin{array}{l} u _ {i} \\ u _ {j} \end{array} \right\} = \left[ \begin{array}{l l} 1 & 0 \\ 1 & L \end{array} \right] \left\{ \begin{array}{l} a _ {1} \\ a _ {2} \end{array} \right\} \quad \text { or } \quad \{\mathbf {d} \} = [ \mathbf {A} ] \{\mathbf {a} \} \tag {3.8-6}
+$$
+
+Solving for $\{\mathbf{a}\}$ and substituting into Eq. 3.8-5, we obtain
+
+$$
+\{\mathbf {a} \} = [ \mathbf {A} ] ^ {- 1} \{\mathbf {d} \} \quad \text { and } \quad u = \left[ \begin{array}{l l} 1 & s \end{array} \right] [ \mathbf {A} ] ^ {- 1} \left\{ \begin{array}{l} u _ {i} \\ u _ {j} \end{array} \right\} \tag {3.8-7}
+$$
+
+<!-- source-page: 110 -->
+
+The shape function matrix is
+
+$$
+\begin{array}{l} \text { e   function   matrix   is } \\ \left\lfloor \mathrm{N} \right\rfloor = \left\lfloor 1 \quad s \right\rfloor [ \mathrm{A} ] ^ {- 1} = \left\lfloor 1 \quad s \right\rfloor \left[ \begin{array}{c c} 1 & 0 \\ - 1 / L & 1 / L \end{array} \right] = \left\lfloor \frac {L - s}{L} \cdot \frac {s}{L} \right\rfloor \end{array} \tag {3.8-8}
+$$
+
+which agrees with Eq. 3.8-4.
+
+In all elements of Fig. 3.8-1, displacement u has the same form—a linear variation—but not the same value because lengths L and nodal d.o.f. {d} are in general different for different elements. Compatibility between elements is assured because elements share a common d.o.f. where they meet; for example, at $x = x_{2}$ , $u = u_{2}$ in element 1 and in element 2.
+
+Additional examples of displacement fields and shape function matrices appear in Sections 3.12 and 3.13 and in subsequent chapters.
+
+# 3.9 FINITE ELEMENT FORM OF THE RAYLEIGH-RITZ METHOD
+
+The finite element method can be defined as a Rayleigh–Ritz method in which the approximating field is interpolated in piecewise fashion from d.o.f. that are nodal values of the field. This viewpoint is illustrated by the following treatment of the axially loaded bar depicted in Fig. 3.8-1. As in preceding sections of the present chapter, we will form potential $\Pi_{p} = U + \Omega$ , make $\Pi_{p}$ stationary with respect to the d.o.f., then solve the resulting equations $[K]\{D\} = \{R\}$ for d.o.f. $\{D\}$ . In an actual computer program equations $[K]\{D\} = \{R\}$ would be written directly, without using $\Pi_{p}$ at all.
+
+Element Stiffness Matrix. Consider a typical bar element of length L that lies along the x axis (Fig. 3.8-1b). Axial strain is $\epsilon_{x} = du/dx = du/ds$ . Hence, from Eq. 3.8-3,
+
+$$
+\epsilon_ {x} = \lfloor \mathbf {B} \rfloor \{\mathbf {d} \}, \quad \text { where } \quad \lfloor \mathbf {B} \rfloor = \frac {d}{d s} \lfloor \mathbf {N} \rfloor = \left\lfloor - \frac {1}{L} \quad \frac {1}{L} \right\rfloor \tag {3.9-1}
+$$
+
+Matrix $[B]$ is called the strain-displacement matrix. From the first integral in Eq. 3.4-6, with $L_{T} = L$ and dx = ds, strain energy in an element is
+
+$$
+U = \int_ {0} ^ {L} \frac {1}{2} E \epsilon_ {x} ^ {2} A d s = \frac {1}{2} \int_ {0} ^ {L} \epsilon_ {x} ^ {T} A E \epsilon_ {x} d s \tag {3.9-2}
+$$
+
+The purpose of writing $\epsilon_{x}^{T}\epsilon_{x}$ instead of $\epsilon_{x}^{2}$ is to simplify subsequent differentiation of matrix forms. From Eqs. 3.9-1 and 3.9-2,
+
+$$
+U = \frac {1}{2} \{\mathbf {d} \} ^ {T} [ \mathbf {k} ] \{\mathbf {d} \}, \quad \text { where } \quad [ \mathbf {k} ] = \int_ {0} ^ {L} [ \mathbf {B} ] ^ {T} A E [ \mathbf {B} ] d s \tag {3.9-3}
+$$
+
+If AE is constant, the 2 by 2 element stiffness matrix [k] is found to be the familiar result for a bar element, seen previously in Eqs. 1.2-3 and 2.4-5.
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_012.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_012.md
new file mode 100644
index 00000000..f0d8958f
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_012.md
@@ -0,0 +1,469 @@
+<!-- source-page: 111 -->
+
+Loads. From the second integral in Eq. 3.4-6, with the substitution of $u = u^{T}$ for axial displacement and $F_{x} = q/A$ for axial body force, the potential of the load is
+
+$$
+\begin{array}{r l} \Omega & = - \int_ {0} ^ {L} \left(f _ {x} A\right) d s = - \int_ {0} ^ {L} u ^ {T} q d s \\ & \quad f _ {x} = \frac {f - c e}{\Delta t} = \frac {q}{A} = \frac {c x}{A} \end{array} \tag {3.9-4}
+$$
+
+From Eqs. 3.8-3 and 3.9-4,
+
+$$
+\Omega = - \{\mathbf {d} \} ^ {T} \left\{\mathbf {r} _ {e} \right\}, \quad \text { where } \quad \left\{\mathbf {r} _ {e} \right\} = \int_ {0} ^ {L} \left\lfloor \mathbf {N} \right] ^ {T} q d s \tag {3.9-5}
+$$
+
+Vector $\{\mathbf{r}_e\}$ is called a consistent load vector. It tells how a distributed load should be allocated to nodes in a way that is consistent with the displacement field assumed. Further explanation appears in Section 4.3.
+
+For the present illustration we will take q = cx, which is the linearly varying distributed axial load used in Fig. 3.5-1. In elements 1, 2, and 3, respectively, q = cs, $q = c(x_{2} + s)$ , and $q = c(x_{3} + s)$ . For convenience we now give all elements the same length, $L = L_{T}/3$ . Thus, for elements 1, 2, and 3,
+
+$$
+\left\{\mathbf {r} _ {e} \right\} _ {1} = \frac {c L ^ {2}}{6} \left\{ \begin{array}{l} 1 \\ 2 \end{array} \right\} \quad \left\{\mathbf {r} _ {e} \right\} _ {2} = \frac {c L ^ {2}}{6} \left\{ \begin{array}{l} 4 \\ 5 \end{array} \right\} \quad \left\{\mathbf {r} _ {e} \right\} _ {3} = \frac {c L ^ {2}}{6} \left\{ \begin{array}{l} 7 \\ 8 \end{array} \right\} \tag {3.9-6}
+$$
+
+Global (Structural) Equations. The total potential of the three-element structure is the sum of the three element contributions:
+
+$$
+\Pi_ {p} = U + \Omega = U _ {1} + U _ {2} + U _ {3} + \Omega_ {1} + \Omega_ {2} + \Omega_ {3} \tag {3.9-7}
+$$
+
+Let element matrices be expanded to “structure size” as explained in Section 2.5, so that nodal d.o.f. vector $\{d\}$ of each element is replaced by the “global” vector $\{D\} = \left[u_{1} \quad u_{2} \quad u_{3} \quad u_{4}\right]^{T}$ , which contains all d.o.f. of the structure. Thus, from Eqs. 3.9-3 and 3.9-5, with AE taken as constant over the entire length $L_{T} = 3L$ ,
+
+$$
+\begin{array}{l} \Pi_ {p} = \frac {1}{2} \{\mathbf {D} \} ^ {T} \left(\frac {A E}{L} \left[ \begin{array}{r r r r} 1 & - 1 & 0 & 0 \\ - 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] + \frac {A E}{L} \left[ \begin{array}{r r r r} 0 & 0 & 0 & 0 \\ 0 & 1 & - 1 & 0 \\ 0 & - 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \right. \\ \left. + \frac {A E}{L} \left[ \begin{array}{c c c c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & - 1 \\ 0 & 0 & - 1 & 1 \end{array} \right]\right) \{\mathbf {D} \} = \{\mathbf {D} \} ^ {T} \left(\frac {c L ^ {2}}{6} \left\{ \begin{array}{l} 1 \\ 2 \\ 0 \\ 0 \end{array} \right\} + \frac {c L ^ {2}}{6} \left\{ \begin{array}{l} 0 \\ 4 \\ 5 \\ 0 \end{array} \right\} + \frac {c L ^ {2}}{6} \left\{ \begin{array}{l} 0 \\ 0 \\ 7 \\ 8 \end{array} \right\} + \left\{ \begin{array}{l} R _ {1} \\ 0 \\ 0 \\ 0 \end{array} \right\}\right) \tag {3.9-8} \\ \end{array}
+$$
+
+in which $R_{1}$ represents the support reaction at x = 0. Four equilibrium equations $[K]\{D\} = \{R\}$ are provided by the stationary condition $\{\partial\Pi_{p}/\partial D\} = \{0\}$ . Differentiation rules given in Appendix A make this process easy. The resulting global equations $[K]\{D\} = \{R\}$ are
+
+<!-- source-page: 112 -->
+
+$$
+\frac {A E}{L} \left[ \begin{array}{r r r r} 1 & - 1 & 0 & 0 \\ - 1 & 2 & - 1 & 0 \\ 0 & - 1 & 2 & - 1 \\ 0 & 0 & - 1 & 1 \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \\ u _ {3} \\ u _ {4} \end{array} \right\} = \frac {c L ^ {2}}{6} \left\{ \begin{array}{l} 1 \\ 6 \\ 1 2 \\ 8 \end{array} \right\} + \left\{ \begin{array}{l} R _ {1} \\ 0 \\ 0 \\ 0 \end{array} \right\} \tag {3.9-9}
+$$
+
+The essential boundary condition $u_{1} = 0$ is imposed by striking out the first equation and the first column of the matrix, as explained in Section 2.3. Solution of the remaining three equations yields nodal d.o.f. $u_{2}, u_{3}$ , and $u_{4}$ . The complete solution vector is
+
+$$
+\{\mathbf {D} \} = \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \\ u _ {3} \\ u _ {4} \end{array} \right\} = \frac {c L ^ {3}}{3 A E} \left\{ \begin{array}{l} 0 \\ 1 3 \\ 2 3 \\ 2 7 \end{array} \right\} \tag {3.9-10}
+$$
+
+These are the axial displacements at $x = 0$ , $x = L$ , $x = 2L$ , and $x = 3L$ .
+
+Inspection of the Solution. Axial displacements are plotted in Fig. 3.9-1a. Upon comparing Eq. 3.9-10 with the exact solution, Eq. 3.5-8, we find that nodal d.o.f. $u_{1}$ , $u_{2}$ , and $u_{3}$ are exact. This happens only because the present problem is of a special mathematical type [3.3]. In most problems, nodal d.o.f. $\{D\}$ are not exact. Between nodes, the linear finite element field cannot match the exact cubic field. For example, at the midpoint $x = 3L_{T}/2$ , for which s = L/2 in element 2, Eq. 3.8-3 yields
+
+$$
+u = \left\lfloor \frac {L - (L / 2)}{L} \quad \frac {L / 2}{L} \right\rfloor \left\{ \begin{array}{l} u _ {2} \\ u _ {3} \end{array} \right\} = \frac {u _ {2} + u _ {3}}{2} = \frac {6 c L ^ {3}}{A E} \tag {3.9-11}
+$$
+
+The exact result at $x = 3L_{T} / 2$ , from Eq. 3.5-8, is $u = 6.1875cL^{3} / AE$ .
+
+The state of stress is uniaxial. Therefore, in a typical element, using Eq. 3.9-1,
+
+![](images/page-112_6961e3af88652f2f32690c737aab8f0858557367de1b5f8f9a3e75528f1c00b4.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+u₁=0
+u₂
+q=cx
+u₃
+u₄
+1
+2
+3
+4
+x
+3@L=Lₜ
+</details>
+
+(a)
+
+![](images/page-112_1d5ceeb2d19fa1894b147fa2ba30bb9752cc0415a4dcbe6812f7fb3b61e5b759.jpg)
+
+<details>
+<summary>line</summary>
+| x    | σ_x (dashed line) | σ_x = c/(2A) (9L² - x²) |
+| ---- | ----------------- | ------------------------ |
+| 0    | ~1.0              | ~1.0                     |
+| L    | ~0.8              | ~0.8                     |
+| 2L   | ~0.6              | ~0.6                     |
+| 3L   | ~0.0              | ~0.0                     |
+</details>
+
+(b)   
+Figure 3.9-1. Axial displacements and axial stresses in a bar under linearly varying axial load q = cx, modeled by three elements of equal length. The dashed line represents the exact stress field.
+
+<!-- source-page: 113 -->
+
+$$
+\sigma_ {x} = E \epsilon_ {x} = E [ \mathbf {B} ] \{\mathbf {d} \} = \frac {E}{L} [ - 1 1 ] \{\mathbf {d} \} \tag {3.9-12}
+$$
+
+Applying Eq. 3.9-12 to the three elements in turn, we obtain the stresses plotted in Fig. 3.9-1b. As is typical in finite element problems, stresses change abruptly at nodes and are less accurate than displacements. A bar element based on a linear axial displacement field will always approximate the exact stress curve in stairstep fashion. Nevertheless, the exact curve can be approached arbitrarily closely by using more and more elements.
+
+That stresses are most accurate near element centers is a consequence of the mean value theorem for derivatives. As applied to Fig. 3.5-1b, the theorem says that slopes of the two curves agree near the middle of the interval even though slopes disagree at x = 0 and at $x = L_{T}$ . The same argument applies to Fig. 3.9-1a: slopes of the piecewise linear curve match the exact $u_{,x}$ from Eq. 3.5-8 somewhere near element centers but not at nodes.
+
+# 3.10 FINITE ELEMENT FORMULATIONS DERIVED FROM A FUNCTIONAL
+
+In Section 3.9, a finite element solution based on a two-d.o.f. bar element is derived from the potential energy functional $\Pi_{p}$ . In the present section we consider another example, this time in two-dimensional heat conduction, but without requiring that the element have a particular shape or a particular number of nodes. Similar formulations for stress analysis are considered in Chapter 4.
+
+Let temperature T within a plane element be interpolated from n nodal temperatures $\{T_{e}\}$ ,
+
+$$
+T = \left\lfloor \mathrm{N} \right] \left\{\mathrm{T} _ {e} \right\} \tag {3.10-1}
+$$
+
+where each of the n shape functions in $[N]$ is a function of x and y. If elements have corner nodes only, then n = 3 for a triangle, n = 4 for a rectangle, and so on.
+
+A functional for plane heat conduction is given in Eq. 3.7-5. With $T = T^T$ , $T_{,x}^2 = T_{,x}^T T_{,x}$ , and $T_{,y}^2 = T_{,y}^T T_{,y}$ , Eq. 3.7-5 has the form
+
+$$
+\Pi = \iint \frac {1}{2} \left(T _ {, x} ^ {T} T _ {, x} + T _ {, y} ^ {T} T _ {, y}\right) k d x d y - \iint T ^ {T} Q d x d y + \iint T ^ {T} \dot {T} \rho c d x d y \tag {3.10-2}
+$$
+
+As in Eq. 3.9-2, the transposition symbol is placed on the scalars to make it easier to differentiate subsequent matrix expressions. Differentiation of Eq. 3.10-1 yields
+
+$$
+T _ {, x} = \left\lfloor \mathrm{N} _ {, x} \right\rfloor \left\{\mathrm{T} _ {e} \right\} \quad T _ {, y} = \left\lfloor \mathrm{N} _ {, y} \right\rfloor \left\{\mathrm{T} _ {e} \right\} \quad \dot {T} = \left\lfloor \mathrm{N} \right\rfloor \left\{\dot {\mathrm{T}} _ {e} \right\} \tag {3.10-3}
+$$
+
+in which, for example,
+
+$$
+\left[ \mathrm{N} _ {, x} \right] = \left[ N _ {1, x} \quad N _ {2, x} \quad \dots \quad N _ {n, x} \right] \tag {3.10-4}
+$$
+
+<!-- source-page: 114 -->
+
+The expression for $\dot{T}$ in Eqs. 3.10-3 indicates that nodal temperatures are regarded as functions of time, but the $N_{i}$ are independent of time. Thus $T$ and $\dot{T}$ are interpolated from nodal values $\{\mathbf{T}_{e}\}$ and $\{\dot{\mathbf{T}}_{e}\}$ by means of the same shape function matrix $[\mathbf{N}]$ .
+
+Substitution of Eqs. 3.10-3 into 3.10-2 yields, for a single element,
+
+$$
+\Pi = \frac {1}{2} \{\mathbf {T} _ {e} \} ^ {T} [ \mathbf {k} ] \{\mathbf {T} _ {e} \} + \{\mathbf {T} _ {e} \} ^ {T} [ \mathbf {c} ] \{\dot {\mathbf {T}} _ {e} \} - \{\mathbf {T} _ {e} \} ^ {T} \{\mathbf {r} _ {Q} \} \tag {3.10-5}
+$$
+
+in which we have defined terms as follows:
+
+$$
+[ \mathbf {k} ] = \int \int (\lfloor \mathbf {N}, _ {x} \rfloor^ {T} \lfloor \mathbf {N}, _ {x} \rfloor + \lfloor \mathbf {N}, _ {y} \rfloor^ {T} \lfloor \mathbf {N}, _ {y} \rfloor) k d x d y \tag {3.10-6a}
+$$
+
+$$
+[ \mathbf {c} ] = \int \int [ \mathbf {N} ] ^ {T} [ \mathbf {N} ] \rho c d x d y \tag {3.10-6b}
+$$
+
+$$
+\left\{\mathbf {r} _ {Q} \right\} = \iint [ \mathbf {N} ] ^ {T} Q d x d y \tag {3.10-6c}
+$$
+
+Finite element equations are obtained by making $\Pi$ stationary with respect to variations of nodal temperature:
+
+$$
+\left\{\frac {\partial \Pi}{\partial \mathbf {T} _ {e}} \right\} = \{\mathbf {0} \} \quad \text { yields } \quad [ \mathbf {k} ] \{\mathbf {T} _ {e} \} + [ \mathbf {c} ] \{\dot {\mathbf {T}} _ {e} \} = \{\mathbf {r} _ {Q} \} \tag {3.10-7}
+$$
+
+Upon assembly of elements, $\{T_{e}\}$ is replaced by the global vector $\{T\}$ , which contains all nodal temperatures of the structure. The global equations are therefore
+
+$$
+(\sum [ \mathbf {k} ]) \{\mathbf {T} \} + (\sum [ \mathbf {c} ]) \{\dot {\mathbf {T}} \} = \sum \{\mathbf {r} _ {Q} \} \tag {3.10-8}
+$$
+
+in which summation signs imply the usual assembly process of summing overlapping terms of element matrices. (Equation 3.10-8 is again obtained if assembly is indicated earlier, i.e., by summing element contributions from Eq. 3.10-5 to a global II.)
+
+global II.) For the sake of having notation like that used in structural mechanics, we can write [k] of Eq. 3.10-6a in the form
+
+$$
+[ \mathbf {k} ] = \int \int [ \mathbf {B} ] ^ {T} k [ \mathbf {B} ] t d x d y, \quad \text { where } \quad [ \mathbf {B} ] = \left[ \begin{array}{l} \mathbf {N} _ {, x} \\ \mathbf {N} _ {, y} \end{array} \right] \tag {3.10-9}
+$$
+
+Thickness $t$ -is taken as unity in the preceding development.
+
+Remarks. The foregoing derivation has significance that goes beyond the problem of plane heat conduction. The derivation shows that a finite element formulation of a physical problem is available from only two basic ingredients—namely, a functional that describes the physical problem and a shape function matrix [N] that describes the element. From these we obtain definitions of element properties (e.g., Eqs. 3.10-6) and algebraic equations of the structure (e.g., Eqs. 3.10-8).
+
+To obtain numerical results one must next attend to specifics by choosing the element shape, number of d.o.f., distribution of d.o.f. over the element, and the shape function matrix [N]. These choices have great influence on the efficiency of calculation and the accuracy of results.
+
+<!-- source-page: 115 -->
+
+# 3.11 INTERPOLATION
+
+To interpolate is to approximate the value of a function between known values by operating on the known values with a formula different from the function itself. This is done in each of the three spans of length L in Fig. 3.9-1a, where a linear operator $[N]$ is applied to known values $u_{1}, u_{2}, u_{3}$ , and $u_{4}$ that happen to lie on a cubic curve. In a finite element context, the “known values” are d.o.f. to be found by solving algebraic equations, and they are usually approximate rather than exact. Operator $[N]$ , the shape function matrix, serves as a basis from which a finite element can be formulated.
+
+One can regard interpolation as the basic motivation of the finite element method, in that a sufficiently small portion of even a complicated field can be modeled well enough by a simpler interpolating field. A linear interpolating field, as used in each element of Fig. 3.9-1a, may be adequate if many elements are used. Elements based on a quadratic field or a cubic field would provide a better fit of the actual field: fewer elements would be needed, but each element would be more complicated. The limiting case of a single element with many d.o.f. yields the classical Rayleigh–Ritz method.
+
+Should one use many simple elements or a few complicated elements? There is no easy answer. A good analyst is familiar with how various elements behave in various circumstances. In solving a transient or nonlinear problem, for example, many analysts prefer simpler (and therefore cheaper) elements because of the need to seek low cost in every computational step.
+
+Degree of Continuity. For future use, we introduce the following symbolism to define the degree of continuity of a function or a field. A field is said to have $C^m$ continuity if derivatives of the field through order $m$ are continuous. Thus $\phi = \phi(x)$ is $C^0$ continuous if $\phi$ is continuous but $\phi_{,x}$ is not. An example of $C^0$ continuity appears in Fig. 3.11-1a. Another example of $C^0$ continuity is axial displacement $u$ in Fig. 3.9-1. Figure 3.11-1b shows an example of $C^1$ continuity: both $\phi$ and $\phi_{,x}$ are continuous but $\phi_{,xx}$ is discontinuous at $x = x_c$ . In general, it is necessary that derivatives of $\phi$ of order $m$ be used as nodal d.o.f. if the field $\phi$ produced by a mesh of finite elements is to be $C^m$ continuous.
+
+Element Node Identification. Heretofore we have labeled element nodes with letters and structure nodes with numbers. We will subsequently encounter ele-
+
+![](images/page-115_8859507308034f128b6d3be136b38860f8dca6e02b1e40158b562460d72080a7.jpg)
+
+<details>
+<summary>line</summary>
+
+| x     | φ     |
+|-------|-------|
+| x_c   | φ_x   |
+</details>
+
+(a)
+
+![](images/page-115_8e1ec4cde7dff6d38f69dc3a9f2ddc7d4e47d0995b5b14b0d318f01e184259a3.jpg)
+
+<details>
+<summary>text_image</summary>
+
+φ or φₓ
+φ
+φₓ
+x
+xₑ
+</details>
+
+(b)   
+Figure 3.11-1. Function $\phi = \phi(x)$ is (a) $C^0$ continuous and (b) $C^1$ continuous.
+
+<!-- source-page: 116 -->
+
+ments that have many nodes, for which letters would be awkward as node labels. Therefore, we will henceforth use numbers as element node labels. At this stage in our study, the reader should be able to distinguish between an element and a structure without the artifice of separate labeling systems.
+
+# 3.12 SHAPE FUNCTIONS FOR $C^0$
+
+# ELEMENTS
+
+A $C^0$ element provides interelement continuity of the field quantity $\phi$ but not interelement continuity of all first derivatives of $\phi$ . Thus, in a mesh of $C^0$ elements, $\phi_{xx}, \phi_{yy}$ , and/or $\phi_{,z}$ exhibit a jump as one passes from one element into another.
+
+Interpolation formulas lead to shape functions [N], from which finite elements can be formulated. In this section and the next we consider shape functions for some simple elements. These and other elements are considered in more detail in subsequent chapters.
+
+A field $\phi$ is interpolated over an element from $n$ element nodal values $\{\phi_e\} = [\phi_1 \quad \phi_2 \ldots \phi_n]^T$ according to the formula
+
+$$
+\phi = \lfloor \mathrm{N} \rfloor \{\phi_ {e} \} \quad \text { that   is, } \quad \phi = \sum_ {i = 1} ^ {n} N _ {i} \phi_ {i} \tag {3.12-1}
+$$
+
+where the $N_{i}$ are functions of the coordinates. A shape function $N_{i}$ defines the distribution of $\phi$ within the element when the $i$ th nodal d.o.f. $\phi_{i}$ nas unit value and all other nodal $\phi$ 's are zero.
+
+One Dimension. Linear interpolation in one dimension is depicted in Fig. 3.12-1a. The interpolated function $\phi = \phi(x)$ is to have value $\phi_{1}$ at $x = x_{1}$ and
+
+![](images/page-116_6825f859bf574f5f3a09d501e9fe7a84e114efb113e4c415bb4a8192c28a0ad0.jpg)
+
+$$
+N _ {1} = \frac {L - x}{L}
+$$
+
+$$
+N _ {2} = \frac {x}{L}
+$$
+
+$$
+\phi = \left\lfloor N \right\rfloor \left\{ \begin{array}{l} \phi_ {1} \\ \phi_ {2} \end{array} \right\}
+$$
+
+$$
+N _ {1} = \frac {(x _ {2} - x) (x _ {3} - x)}{(x _ {2} - x _ {1}) (x _ {3} - x _ {1})}
+$$
+
+$$
+N _ {2} = \frac {(x _ {1} - x) (x _ {3} - x)}{(x _ {1} - x _ {2}) (x _ {3} - x _ {2})}
+$$
+
+$$
+N _ {3} = \frac {(x _ {1} - x) (x _ {2} - x)}{(x _ {1} - x _ {3}) (x _ {2} - x _ {3})}
+$$
+
+$$
+\varphi = [ N ] \left\{ \begin{array}{l} \phi_ {1} \\ \phi_ {2} \\ \phi_ {3} \end{array} \right\}
+$$
+
+Figure 3.12-1. (a) Linear interpolation and shape functions. (b) Quadratic interpolation and shape functions.
+
+<!-- source-page: 117 -->
+
+value $\phi_{2}$ at $x = x_{2}$ . Two data points define a linear polynomial $\phi = a_{1} + a_{2}x$ . The procedure for obtaining shape functions $N_{i}$ from this polynomial is discussed in Section 3.8. In the notation of Fig. 3.12-1a, with $x_{1} = 0$ and $x_{2} = L$ , the result is
+
+$$
+[ \mathrm{N} ] = \left\lfloor \frac {L - x}{L} \quad \frac {x}{L} \right\rfloor \tag {3.12-2}
+$$
+
+Quadratic interpolation in one dimension is depicted in Fig. 3.12-1b. Here there are three data points. They define a parabola $\phi = a_{1} + a_{2}x + a_{3}x^{2}$ , which must display the values $\phi = \phi_{1}$ , $\phi = \phi_{2}$ , and $\phi = \phi_{3}$ at $x = x_{1}$ , $x = x_{2}$ , and $x = x_{3}$ , respectively. The $x_{i}$ values need not be uniformly spaced. Quadratic shape functions can be determined by the procedure used for linear shape functions in Eqs. 3.8-6 to 3.8-8, but now there are three $a_{i}$ and the algebra is more tedious. The three shape functions are most easily obtained from Lagrange's interpolation formula, which is discussed subsequently. The result is
+
+$$
+[ \mathrm{N} ] = \left\lfloor \frac {(x _ {2} - x) (x _ {3} - x)}{(x _ {2} - x _ {1}) (x _ {3} - x _ {1})} \frac {(x _ {1} - x) (x _ {3} - x)}{(x _ {1} - x _ {2}) (x _ {3} - x _ {2})} \frac {(x _ {1} - x) (x _ {2} - x)}{(x _ {1} - x _ {3}) (x _ {2} - x _ {3})} \right\rfloor \tag {3.12-3}
+$$
+
+In Eqs. 3.12-2 and 3.12-3 we note the following characteristics, which are true of all $C^0$ polynomial shape functions (in one dimension).
+
+1. All shape functions $N_{i}$ , and function $\phi$ itself, are polynomials of the same degree.
+
+2. For any shape function $N_{i}, N_{i} = 1$ when $x = x_{i}$ and $N_{i} = 0$ when $x = x_{j}$ where $i \neq j$ .
+
+3. $C^0$ shape functions sum to unity. This is not obvious in Eq. 3.12-3 but can be shown as follows. If $\phi_i = 1$ at all $n$ data points, then $\phi = 1$ everywhere in the interpolated function $\phi$ . Equation 3.12-1 becomes
+
+$$
+1 = \sum_ {i = 1} ^ {n} N _ {i} \tag {3.12-4}
+$$
+
+For $C^{1}$ elements, in which derivatives of $\phi$ are also used as nodal d.o.f., Eq. 3.12-4 is valid if the $N_{i}$ are those associated with translational d.o.f. only.
+
+Lagrange's Interpolation Formula. A function $\phi = \phi(x)$ , of degree $n - 1$ and defined by $n$ values $\phi_i$ at corresponding abscissae $x_i$ , has the form
+
+$$
+\phi = \sum_ {i = 1} ^ {n} N _ {i} \phi_ {i} \quad \text { or } \quad \phi = N _ {1} \phi_ {1} + N _ {2} \phi_ {2} + \dots + N _ {n} \phi_ {n} \tag {3.12-5}
+$$
+
+<!-- source-page: 118 -->
+
+in which shape functions $N_{i}$ have been devised by Lagrange as follows:
+
+$$
+N _ {1} = \frac {(x _ {2} - x) (x _ {3} - x) (x _ {4} - x) \cdot \cdot \cdot (x _ {n} - x)}{(x _ {2} - x _ {1}) (x _ {3} - x _ {1}) (x _ {4} - x _ {1}) \cdot \cdot \cdot (x _ {n} - x _ {1})}
+$$
+
+$$
+N _ {2} = \frac {(x _ {1} - x) (x _ {3} - x) (x _ {4} - x) \cdot \cdot \cdot (x _ {n} - x)}{(x _ {1} - x _ {2}) (x _ {3} - x _ {2}) (x _ {4} - x _ {2}) \cdot \cdot \cdot (x _ {n} - x _ {2})} \tag {3.12-6}
+$$
+
+$$
+N _ {n} = \frac {(x _ {1} - x) (x _ {2} - x) (x _ {3} - x) \cdot \cdot \cdot (x _ {n - 1} - x)}{(x _ {1} - x _ {n}) (x _ {2} - x _ {n}) (x _ {3} - x _ {n}) \cdot \cdot \cdot (x _ {n - 1} - x _ {n})}
+$$
+
+Note that the $N_{i}$ have characteristics 1 and 2 just cited. Characteristic 3 is present but is not obvious. Note also that the $N_{i}$ in Eqs. 3.12-2 and 3.12-3 are special cases of Eq. 3.12-6, for which $n = 2$ and $n = 3$ , respectively.
+
+If there is a “true curve” for which the interpolated curve $\phi = \Sigma N_{i}\phi_{i}$ is but an approximation, the two curves are coincident only at the n values of $x_{i}$ that provide the $\phi_{i}$ used for interpolation. Moreover, the interpolated curve yields only exact ordinates $\phi_{i}$ , not exact slopes $\phi_{,xi}$ as well. An example appears in Fig. 3.12-2.
+
+Two Dimensions. Imagine that a dependent variable $\phi = \phi(x, y)$ is to be interpolated from four nodal $\phi_{l}$ at corners of a rectangle (Fig. 3.12-3). Here $\phi$ has the form
+
+$$
+\phi = a _ {1} + a _ {2} x + a _ {3} y + a _ {4} x y \tag {3.12-7}
+$$
+
+Shape functions are products of the $N_{i}$ of Lagrange's formula. We argue as follows.
+
+In Fig. 3.12-3, one can linearly interpolate $\phi$ along the left edge between nodal values $\phi_1$ and $\phi_4$ , and along the right edge between nodal values $\phi_2$ and $\phi_3$ . Thus, in Eqs. 3.12-6, $y$ replaces $x$ and $n = 2$ . Calling the edge values $\phi_{14}$ and $\phi_{23}$ , we have
+
+![](images/page-118_a6d71ebc485b2767aaaf4399dd977c5e27353ccb3e68743b083a98b241197952.jpg)
+
+<details>
+<summary>line</summary>
+| x    | φ    |
+| ---- | ---- |
+| x₁   | 1    |
+| x₂   | 2    |
+| x₃   | 3    |
+| x₄   | 4    |
+</details>
+
+Figure 3.12-2. Possible discrepancies between a “true curve” (solid line) and the fit produced by Lagrange’s formula (dashed line).
+
+![](images/page-118_1340cf470e0ec291b3d82e0ae78be1108dcb6b611f743cfc3f0a45156e6db45a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+a a
+4 3
+b
+x
+1 2
+</details>
+
+Figure 3.12-3. Four-node "bilinear" element.
+
+<!-- source-page: 119 -->
+
+$$
+\phi_ {1 4} = \frac {b - y}{2 b} \phi_ {1} + \frac {b + y}{2 b} \phi_ {4} \quad \text { and } \quad \phi_ {2 3} = \frac {b - y}{2 b} \phi_ {2} + \frac {b + y}{2 b} \phi_ {3} \tag {3.12-8}
+$$
+
+Next we linearly interpolate in the $x$ direction between $\phi_{14}$ and $\phi_{23}$ :
+
+$$
+\phi = \frac {a - x}{2 a} \phi_ {1 4} + \frac {a + x}{2 a} \phi_ {2 3} \tag {3.12-9}
+$$
+
+Substitution of Eq. 3.12-8 into Eq. 3.12-9 yields $\phi = \Sigma N_{i}\phi_{i}$ , where
+
+$$
+N _ {1} = \frac {(a - x) (b - y)}{4 a b} \quad N _ {2} = \frac {(a + x) (b - y)}{4 a b} \tag {3.12-10}
+$$
+
+$$
+N _ {3} = \frac {(a + x) (b + y)}{4 a b} \quad N _ {4} = \frac {(a - x) (b + y)}{4 a b}
+$$
+
+One can easily check that each $N_{i} = 1$ at the coordinates of node $i$ , is zero at other nodes, and that $N_{1} + N_{2} + N_{3} + N_{4} = 1$ .
+
+The element associated with Eqs. 3.12-10 is called “bilinear,” as each of its shape functions is a product of two linear polynomials. Similarly, a nine-node element (nodes at corners, midsides, and the center) is called “biquadratic,” a 16-node element in which four of the nodes are internal is called “bicubic,” and so on. Shape functions for all these elements are products of one-dimensional Lagrange interpolation shape functions [3.4]. These elements, and analogous elements in three dimensions, are called Lagrange elements.
+
+Additional Dependent Variables. The element of Fig. 3.12-3 can be used to solve problems of plane stress and plane strain. For such problems there are two dependent field variables—namely, $u = u(x, y)$ and $v = v(x, y)$ . The four-node element then has eight d.o.f. Displacements u and v are each interpolated from four nodal values, that is,
+
+$$
+u = \sum_ {i = 1} ^ {4} N _ {i} u _ {i} \quad \text { and } \quad v = \sum_ {i = 1} ^ {4} N _ {i} v _ {i} \tag {3.12-11}
+$$
+
+in which the $N_{i}$ are defined by Eqs. 3.12-10.
+
+# 3.13 SHAPE FUNCTIONS FOR $C^1$ ELEMENTS
+
+A $C^{1}$ element provides interelement continuity of the field quantity $\phi$ and its first derivatives at nodes, but not interelement continuity of all second derivatives of $\phi$ . An example appears in the analysis of a thin plate in bending, where the field quantity is displacement w in the z direction, where $w = w(x, y)$ , and x and y are coordinates in the plane of the plate. Typical thin-plate elements use w, $w_{,x}$ , and $w_{,y}$ as nodal d.o.f. Second derivatives $w_{,xx}$ , $w_{,yy}$ , and $w_{,xy}$ are not all continuous across interelement boundaries. (Even the first derivative $w_{,n}$ , where n is a direction normal to an edge, is typically discontinuous except at nodes.)
+
+<!-- source-page: 120 -->
+
+![](images/page-120_1d5eda54685c0b4121a418d8e3830b9a5634b63b441c27cd93a72ba7aacaf3d5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+φ
+1
+θ₁
+2
+θ₂
+φ₁
+0
+L
+x
+</details>
+
+Figure 3.13-1. A curve interpolated between two points at which ordinates $\phi_{1}$ and $\phi_{2}$ and inclinations $\theta_{1}$ and $\theta_{2}$ are known.
+
+One Dimension. Fitting a curve to both ordinate and slope information at data points is known as Hermitian interpolation. The simplest and most common Hermitian interpolation is between two points at which both ordinate and slope are known (Fig. 3.13-1). We will assume that slope $\phi_{,x}$ is small, so that rotation $\theta$ is practically the same as $\phi_{,x}$ . Four data items define a cubic curve,
+
+$$
+\phi = a _ {1} + a _ {2} x + a _ {3} x ^ {2} + a _ {4} x ^ {3} \quad \text { or } \quad \phi = \lfloor \mathbf {X} \rfloor \{\mathbf {a} \} \tag {3.13-1a}
+$$
+
+where
+
+$$
+[ \mathbf {X} ] = \left[ \begin{array}{l l l l} 1 & x & x ^ {2} & x ^ {3} \end{array} \right] \quad \text { and } \quad \{\mathbf {a} \} = \left[ \begin{array}{l l l l} a _ {1} & a _ {2} & a _ {3} & a _ {4} \end{array} \right] ^ {T} \tag {3.13-1b}
+$$
+
+To express the $a_{i}$ in terms of ordinates and slopes at $x = 0$ and at $x = L$ , we make the substitutions
+
+$$
+\phi = \phi_ {1} \quad \text { and } \quad \phi_ {, x} = \theta_ {1} \quad \text { at } x = 0 \tag {3.13-2}
+$$
+
+$$
+\phi = \phi_ {2} \quad \text { and } \quad \phi_ {, x} = \theta_ {2} \quad \text { at } x = L
+$$
+
+Thus Eq. 3.13-1 yields
+
+$$
+\left\{ \begin{array}{l} \phi_ {1} \\ \theta_ {1} \\ \phi_ {2} \\ \theta_ {2} \end{array} \right\} = \left[ \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & L & L ^ {2} & L ^ {3} \\ 0 & 1 & 2 L & 3 L ^ {2} \end{array} \right] \left\{ \begin{array}{l} a _ {1} \\ a _ {2} \\ a _ {3} \\ a _ {4} \end{array} \right\} \quad \text { or } \quad \{\mathbf {d} \} = [ \mathbf {A} ] \{\mathbf {a} \} \tag {3.13-3}
+$$
+
+Therefore $\{\mathbf{a}\} = [\mathbf{A}]^{-1}\{\mathbf{d}\}$ , and Eq. 3.13-1 becomes
+
+$$
+\phi = \lfloor \mathbf {N} \rfloor \{\mathbf {d} \}, \quad \text { where } \quad \lfloor \mathbf {N} \rfloor = \lfloor \mathbf {X} \rfloor [ \mathbf {A} ] ^ {- 1} \tag {3.13-4}
+$$
+
+Shape function matrix [N] is 1 by 4. The four $N_{i}$ are shown in Fig. 3.13-2. As expected, three of the $N_{i}$ and three of the $dN_{i}/dx$ are zero at end x = 0, whereas the remaining $N_{i}$ and the remaining $dN_{i}/dx$ have unit value. The same is true at end x = L. This behavior is required if Eqs. 3.13-2 are to be satisfied by the interpolation $\phi = \Sigma N_{i}d_{i}$ . These shape functions may be used to generate the stiffness matrix of a beam element (Section 4.2).
+
+Two Dimensions. Hermitian interpolation of a function $w = w(x, y)$ can be used to generate elements for the analysis of thin plates in bending. A great many
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_013.md
@@ -0,0 +1,504 @@
+<!-- source-page: 121 -->
+
+![](images/page-121_481ecc498735620058b0ccd9335e8f19e529c079bc83b993631ca17862b1951b.jpg)  
+Figure 3.13-2. Shape functions of a cubic curve fitted to ordinates and slopes at x = 0 and at x = L.
+
+different interpolation schemes are possible. All are lengthy to write out, and their comparative merits are intimately connected with plate theory (Chapter 11).
+
+# PROBLEMS
+
+# Section 3.2
+
+3.1 The system shown consists of a rigid half-cylinder that can roll without friction on a horizontal surface, a linear nondissipative spring, and a force P of constant magnitude that can move around the cylinder, but is always directed toward point C on the cylinder. Show that this system is not conservative.
+
+![](images/page-121_8eb3ea8b2da2c7b41e4a890c0c4ac34d9362b82b9bc247e8ed6cafb7e270dded.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+C
+k
+</details>
+
+Problem 3.1
+
+3.2 Redefine $\Omega$ to be $\Omega = P(D_{\mathrm{eq}} - D)$ , as suggested near the end of Section 3.2. Redraw Fig. 3.2-4 as required.   
+3.3 Reverse the direction of load P in Fig. 3.2-3. Solve for the equilibrium value of D by use of $\Pi_{p}$ . Revise Fig. 3.2-4 as required.   
+3.4 Imagine that the spring in Fig. 3.2-3 is not linear but exerts a force proportional to the square of its stretch. Write an expression for $\Pi_p$ , and from it determine the equilibrium value of $D$ .
+
+# Section 3.3
+
+3.5 Redefine $D_{2}$ and $D_{3}$ in Fig. 3.3-1 so that $D_{2}$ is an axial displacement relative to $D_{1}$ and $D_{3}$ is an axial displacement relative to $D_{2}$ . Write an expression for $\Pi_{p}$ , analogous to Eq. 3.3-3. For the special case $k_{1} = k_{2} = k_{3} = k$ and $P_{1} = P_{2} = P_{3} = P$ , solve for the $D_{i}$ and show that they give the same absolute axial displacements as Eq. 3.3-3.
+
+<!-- source-page: 122 -->
+
+![](images/page-122_e38d5061b72ba39656624174fd5b4cb6e75ac48f850052cc09a7a2e81568f0f4.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+L
+d4
+d3
+d2
+A,E
+β
+d1
+x
+</details>
+
+Problem 3.7
+
+3.6 Show that Eqs. 3.3-6 and 3.3-7 yield the rows of Eq. 2.4-3 from the stationary condition $d\Pi_{p} = 0$ .
+
+3.7 Displacement d.o.f. $d_{i}$ at ends of a uniform bar element have the directions shown. Write an expression for $\Pi_p$ in terms of these d.o.f. From the stationary condition $d\Pi_p = 0$ , obtain the element stiffness matrix that operates on these four d.o.f.
+
+3.8 Verify that Eqs. 2.3-8 are produced when the stationary condition $d\Pi_{p} = 0$ is applied to Eq. 3.3-8.
+
+3.9 Use the method of stationary potential energy to derive the 4 by 4 stiffness matrix for the structure described in (a) Problem 1.13, and (b) Problem 2.2.
+
+3.10 Verify that Eqs. 3.3-5 and 3.3-9 yield Eq. 3.3-3.
+
+# Section 3.4
+
+Section 3.4
+3.11 Write the term $\frac{1}{2}\{\epsilon\}^{T}[E]\{\epsilon\}$ in Eq. 3.4-1 for an isotropic material in a condition of plane stress in the xy plane (for which [E] is 3 by 3). Specialize this expression for the case of uniaxial stress $\sigma_{x}$ . Check your result against Eq. 3.4-6.
+
+3.12 In the beam of Fig. 3.4-1b, let initial strain and initial stress be given by $\epsilon_0 = -z\kappa_0$ and $\sigma_0 = -m_0z/I$ , respectively. Here $\kappa_0$ and $m_0$ are regarded as initial curvature and initial moment, both considered positive when associated with a concave-up condition of the beam. Determine the contributions of $\kappa_0$ and $m_0$ to Eq. 3.4-8.
+
+3.13 Repeat the example of Eqs. 3.4-9 to 3.4-11, but account for heating by use of $\sigma_0$ rather than $\epsilon_0$ .
+
+# Section 3.5
+
+Section 3.5
+3.14 Verify that the first of Eqs. 3.5-6a is indeed given by the conditions $\partial\Pi_{p}/\partial a_{1}=\partial\Pi_{p}/\partial a_{2}=0.$
+
+3.15 Verify that the $a_{i}$ of Eq. 3.5-9 result from the use of the three-term polynomial $u = a_{1}x + a_{2}x^{2} + a_{3}x^{3}$ in a Rayleigh-Ritz solution.
+
+3.16 Consider the two approximate solutions and the one exact solution in Section 3.5 (Eqs. 3.5-5, 3.5-6, and 3.5-8). At what point or points is the differential equation of equilibrium satisfied by each solution?
+
+3.17 Obtain one-term and two-term solutions for a uniform axially loaded bar, analogous to Eqs. 3.5-5 and 3.5-6, if load $q$ is replaced by concentrated forces at $x = L_T / 3$ , $x = 2L_T / 3$ , and $x = L_T$ . Each of the three forces is directed to the right and is of magnitude $P$ .
+
+<!-- source-page: 123 -->
+
+3.18 Consider a cantilever beam of length L, fixed at end x = 0 and carrying a moment load $M_{L}$ at x = L. Write an admissible series for lateral displacement w based on either sine or cosine functions.
+
+3.19 Consider a uniform cantilever beam of length $L$ , fixed at end $x = 0$ and carrying a transverse force $F$ at $x = L$ .
+
+(a) Let the lateral displacement field be $w = a_{1}x^{3}$ , where $a_{1}$ is a constant. Is this field admissible? Explain.   
+(b) Write a polynomial field for $w$ that is better than that of part (a). Let the field contain three terms, each of the form $a_{i}x^{j}$ , where $i = 1,2,3$ and $j$ is an integer such that the term is admissible.   
+(c) Without calculation, can you predict the quality of the answers obtainable from the field of part (b) and the numerical value of any of the $a_{i}$ ?   
+(d) Use the field of part (a) to find the deflection of force $F$ .
+
+3.20 A uniformly loaded beam of constant flexural stiffness $EI$ is simply supported at its ends $x = 0$ and $x = L$ . In parts (a) and (b), determine the deflection and bending moment predicted at $x = L / 2$ by a Rayleigh-Ritz solution that has a single d.o.f. Compare exact and approximate results.
+
+(a) Use a single-d.o.f. algebraic expression—that is, a d.o.f. $a_1$ times a function that contains $x$ and $x^2$ .   
+(b) Use one term of a sine series.   
+(c) Why should you anticipate that part (b) will be better than part (a) if part (a) is the simplest admissible function?
+
+3.21 The uniform cantilever beam shown carries uniformly distributed load of intensity q, tip force $P_{L}$ , and tip moment $M_{L}$ . In parts (a) and (b), compute Rayleigh–Ritz approximations for displacement and rotation at the tip. Compare these results with formulas from beam theory, and explain why the Rayleigh–Ritz result is or is not exact.
+
+(a) Use one term of a polynomial series.
+
+(b) Use two terms of a polynomial series.
+
+(c) In part (a), for which of the given loadings is $w = w(x)$ exact away from the tip—that is, for $0 < x < L$ ? Why?
+
+![](images/page-123_a9d9518996ec30aadd58b4c1c765b113ff7b51748592d674e721f332bc660f59.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+q
+P_L
+x
+L
+M_L
+</details>
+
+Problem 3.21
+
+![](images/page-123_4ff90313374d222de3666501a24e65402ed81ad3f09720c7edadaf1ad0325c98.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z₁w
+P
+x
+L/2
+L/2
+</details>
+
+Problem 3.23
+
+3.22 If loads $P_{L}$ and $M_{L}$ in Problem 3.21 were applied at x = L/2 rather than at the tip, how many series terms would be needed in order to obtain the exact displacement at x = L? Explain.
+
+3.23 The uniform beam shown is simply supported and carries a force P at its center. Use the infinite series
+
+$$
+w = \sum_ {i = 1} ^ {n} a _ {i} \sin \frac {i \pi x}{L}
+$$
+
+<!-- source-page: 124 -->
+
+in a Rayleigh–Ritz solution for deflection and bending moment at the center. Compare exact and approximate results for n = 1, 2, 3, and 4.
+
+3.24 Repeat Problem 3.23 with load P replaced by a uniformly distributed downward load of intensity q.
+
+# Section 3.6
+
+3.25 A complete polynomial of degree n in x, y, and z contains $(n + 3)!/6n!$ terms. With this in mind, construct a “Pascal tetrahedron” analogous to the Pascal triangle in Fig. 3.6-1.
+
+3.26 (a) Compute the work done by applied load $q = cx$ in going through the exact displacement $u$ of Eq. 3.5-8.
+
+(b) Similarly, compute the work done by load $q$ in each of the two approximate solutions (Eqs. 3.5-5 and 3.5-6). What conclusion can you draw?
+
+3.27 (a) Compute the strain energy associated with the exact solution given in Eq. 3.5-8. How is this energy related to the work computed in Problem 3.26a, and why?
+
+(b) Similarly, compute the strain energy associated with the two approximate solutions (Eqs. 3.5-5 and 3.5-6), and compare answers with work values computed in Problem 3.26b.
+
+# Section 3.7
+
+3.28 A certain functional is $\Pi = \int F dx$ , in which $F = c_{1}\phi_{,xx}^{2} + c_{2}\phi_{,x}^{2} + c_{3}\phi^{2} + c_{4}\phi + c_{5}$ and the five $c_{i}$ are constants. What is the Euler equation?
+
+3.29 A certain physical problem has the functional
+
+$$
+\Pi = \int_ {0} ^ {L} \left(\frac {1}{2} \phi_ {, x} ^ {2} - 5 0 \phi\right) d x
+$$
+
+Essential boundary conditions are $\phi = 0$ at $x = 0$ and $\phi = 20$ at $x = L$ . What is $\phi$ as a function of $x$ and $L$ ?
+
+3.30 Discard terms that contain $\{\mathbf{F}\}$ and $\{\mathbf{M}\}$ from Eq. 3.4-8, and find the Euler equation of a uniform beam under lateral load $q$ . What is the Euler equation if $EI$ is not constant?
+
+3.31 The potential energy of an isotropic plate that carries lateral pressure q is
+
+$$
+\Pi_ {p} = \frac {D}{2} \int \int \left\{(w _ {, x x} + w _ {, y y}) ^ {2} - 2 (1 - \nu) [ w _ {, x x} w _ {, y y} - w _ {, x y} ^ {2} ] - \frac {2 q}{D} \right\} d x d y
+$$
+
+where $D$ is a constant called flexural rigidity. Show that the Euler equation is $\nabla^{4}w = q / D$ , where $\nabla^{4}$ is the biharmonic operator.
+
+3.32 Consider the functional
+
+$$
+\Pi = \int \left(\frac {M ^ {2}}{2 E I} + M _ {, x} w _ {, x} + q w\right) d x
+$$
+
+for a uniform beam that carries distributed lateral load q. Bending moment M and lateral deflection w are each regarded as dependent variables. Derive the two Euler equations. Do they have the form expected from beam theory?
+
+<!-- source-page: 125 -->
+
+![](images/page-125_c861d870cdb66be73fbe84c163a1e85738c04e5bda0a3423dbd85678d9ae6760.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+i j
+x,u
+x_i
+x_j
+</details>
+
+Problem 3.33
+
+![](images/page-125_11c4a2ec04860b11fbf31b56cd044b84e33ba6f8a3edbae6867b85bc2a0974c8.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+l
+l
+i
+k
+j
+x,u
+</details>
+
+Problem 3.34
+
+# Section 3.8
+
+3.33 The bar element shown is to have an axial displacement field $u$ that is linear in $x$ and depends on nodal d.o.f. $u_{i}$ and $u_{j}$ . The shape function matrix $[\mathbf{N}]$ is a function of $x_{i}, x_{j}$ , and $x$ . Write $[\mathbf{N}]$ by inspection if you can. Then derive $[\mathbf{N}]$ by use of the [A]-matrix method described in Section 3.8.   
+3.34 The three-node bar element shown is to have an axial displacement field $u$ that is quadratic in $x$ and depends on nodal d.o.f. $u_i, u_j$ , and $u_k$ . Determine the shape function matrix $\lfloor \mathbf{N} \rfloor$ in terms of $x$ and $\ell$ .
+
+# Section 3.9
+
+3.35 Use $\{\mathbf{D}\}$ as given by Eq. 3.9-10, and $[\mathbf{K}]$ and $\{\mathbf{R}\}$ as given in Eq. 3.9-9, to compute $U = \frac{1}{2}\{\mathbf{D}\}^T [\mathbf{K}]\{\mathbf{D}\}$ and $\Omega = -\{\mathbf{D}\}^T \{\mathbf{R}\}$ . How is $U$ related to $\Omega$ ? What are the precentage errors of $U$ and $\Omega$ ? (Exact values of $U$ and $\Omega$ are computed in Problems 3.26a and 3.27a.)   
+3.36 Consider a uniform bar under axial load. Displacements and stresses may be obtained by either the classical or the finite element form of the Rayleigh-Ritz method. Which form (if either) gives exact results, and how many d.o.f. are required for exactness, if the axial loading is (a) distributed in the form $q = c(L_T - x)$ , and (b) concentrated, with equal forces $P$ at $x = L$ , $x = 2L$ , and $x = 3L$ ? The total length of the bar is $L_T = 3L$ .   
+3.37 Consider a bar of length $L_{T}$ that is fixed at $x = 0$ and free at $x = L_{T}$ . The bar carries a distributed axial loading of intensity $q = c(L_{T} - x)$ , where $c$ is a constant. Generate a finite element solution like that in Section 3.9, using
+
+(a) a single element of length $L_{T}$ .   
+(b) two elements, each of length $L = L_{T} / 2$ .   
+(c) three elements, each of length $L = L_{T} / 3$ .
+
+3.38 Repeat the finite element solution of Section 3.9 but let the respective elements have lengths $L_{T} / 6$ , $L_{T} / 3$ , and $L_{T} / 2$ (reading left to right).
+
+3.39 (a) Apply Eq. 3.9-3, with $\lfloor \mathbf{B} \rfloor$ taken from Eq. 3.9-1, to determine the stiffness matrix of a bar element that is tapered: its cross-sectional area is $A = A_0(3L - 2x)/L$ , where $A_0$ is the cross-sectional area at the right end ( $x = L$ ).
+
+(b) What is the exact stiffness matrix of this tapered bar? Find out by displacing d.o.f. one at a time and using an elementary mechanics of materials analysis to compute the axial force required.
+
+3.40 The rigid bar shown rests on an elastic foundation. When displaced laterally an amount w, the foundation applies a force kw dx to a length dx of the bar, where k is a constant. Determine the 2 by 2 stiffness matrix that operates
+
+<!-- source-page: 126 -->
+
+![](images/page-126_306a1176df40e7e47153cbb55e523848aaa364313c9b4e7908af09d04f173aa8.jpg)
+
+<details>
+<summary>text_image</summary>
+
+w_i
+L
+w_j
+i
+j
+</details>
+
+Problem 3.40
+
+on $w_{i}$ and $w_{j}$ . Suggestion: Express strain energy U in terms of k, L, $w_{i}$ , and $w_{j}$ , then write U in the form $\{d\}^{T}[k]\{d\}/2$ , and identify [k].
+
+# Section 3.10
+
+3.41 As suggested below Eq. 3.10-8, indicate assembly of elements when $\Pi$ is written, and see whether Eq. 3.10-8 again results.   
+3.42 For each of the following problems, obtain finite element formulations in a form analogous to Eqs. 3.10-6 and 3.10-7. Details such as specific element shape functions are not required.
+
+(a) Plane beam (use the integrals in Eq. 3.4-8).   
+(a) Plane beam (see Problem 3.32). Use two interpolating fields, one for $M$ that depends on nodal moments $\{\mathbf{M}_e\}$ and one for $w$ that depends on nodal lateral deflections $\{\mathbf{w}_e\}$ .   
+(c) Acoustic modes in a cavity with rigid walls. The functional is
+
+$$
+\Pi = \int \left(p _ {, x} ^ {2} + p _ {, y} ^ {2} + p _ {, z} ^ {2} - \frac {\omega^ {2}}{c ^ {2}} p ^ {2}\right) d V
+$$
+
+where $p = p(x,y,z)$ is the amplitude of gas pressure that varies with time, $\omega$ is the circular frequency, and $c$ is the speed of sound. Let $p = \lfloor \mathbf{N} \rfloor \{\mathbf{p}_e\}$ , where $\{\mathbf{p}_e\}$ represents nodal pressures.
+
+3.43 The uniform bar shown is to act as a heat conduction element, with $T =$ temperature, $k =$ thermal conductivity, $q =$ axial heat flux in the bar per unit of cross-sectional area $A$ , and $H =$ lateral heat flux per unit length. From the functional
+
+$$
+\Pi = A q _ {j} T _ {j} - A q _ {i} T _ {i} + \frac {1}{2} \int_ {0} ^ {L} k T _ {, x} ^ {2} A d x - \int_ {0} ^ {L} H T d x
+$$
+
+determine expressions for [k] and $\{\mathbf{r}_Q\}$ in the element equations $[\mathbf{k}]\{\mathbf{T}_e\} = \{\mathbf{r}_O\}$ , where $\{\mathbf{T}_e\} = \lfloor T_l - T_j \rfloor^T$ . Let $T = \lfloor \mathbf{N} \rfloor \{\mathbf{T}_e\}$ and $H = \lfloor \mathbf{N} \rfloor \left\lfloor H_i - H_j \right\rfloor^T$ , where
+
+$$
+[ \mathbf {N} ] = \left\lfloor \frac {L - x}{L} \frac {x}{L} \right\rfloor .
+$$
+
+![](images/page-126_03f2a9dff833f1a74cf4ff6416e0a44d872728a8c80bdfbede75b659400247da.jpg)
+
+<details>
+<summary>text_image</summary>
+
+H = H(x)
+y
+q_i A i j q_j A x
+L
+</details>
+
+Problem 3.43
+
+<!-- source-page: 127 -->
+
+# Section 3.12
+
+3.44 In Fig. 3.12-1b, let $x_{1} = 0$ , $x_{2} = 2$ , and $x_{3} = 3$ . Then do as follows.
+
+(a) Verify numerically that $\Sigma N_{i} = 1$ .   
+(b) According to Eq. 3.12-4, $\Sigma N_{i,x} = 0$ . Verify this property numerically.
+
+3.45 For the four points in Fig. 3.12-2, let the respective $x_{i}$ be 1, 3, 5, and 8, and let the respective $\phi_{i}$ be 2, 2, 2, and 5.
+
+(a) Use Lagrange's formula to obtain the interpolating curve.   
+(b) What values of $\phi$ does Lagrange's formula predict at $x = 2$ , $x = 4$ , and $x = 7$ ?
+
+3.46 Sketch the four $N_{i}$ of Eqs. 3.12-10 in a fashion analogous to Fig. 3.12-1. That is, sketch the element in isometric view, with $\phi = N_{i}\phi_{i}$ shown normal to the $xy$ plane, as in Fig. 1.1-3.
+
+3.47 (a) Determine shape functions $N_{i}$ for the nine-node Lagrange element shown.
+
+(b) What is $\phi = \phi(x)$ for this element when written in the form of Eq. 3.12-7?   
+(c) Show how $N_{1}$ varies over the element by making an isometric sketch and showing $\phi = N_{1}\phi_{1}$ normal to the $xy$ plane (similar to Fig. 1.1-3). Do the same for $N_{8}$ and $N_{9}$ .
+
+![](images/page-127_f5ec44ea161e5cd664011eb092571294b20a5942d141d73f6dad94c84c27edd9.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+a a
+4 7 3
+8 9 6
+1 5 2
+x
+b
+b
+</details>
+
+Problem 3.47
+
+3.48 The element shown has three linear edges and one quadratic edge. Determine the five shape functions $N_{i}$ . Suggestion: after interpolating along edges $y = -b$ and $y = +b$ , interpolate linearly in the $y$ direction.   
+3.49 Determine shape functions $N_{i}$ for the eight-node rectangular parallelepiped shown. Overall side lengths are 2a, 2b, and 2c. Suggestion: Infer answers from the pattern seen in Eqs. 3.12-10, then test the answers.
+
+![](images/page-127_0e20b10db202fb8e3d5d7b90d4319615ee44f72743336a3ef184036809f203cc.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+a a
+4 3
+b
+x
+1 5 2
+b
+</details>
+
+Problem 3.48
+
+![](images/page-127_5531cc8de9126880771aba30047198e42ed5e6503868cd2e76ca049962094ef4.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+3
+4
+8
+7
+2c
+2b
+x
+z
+2
+1
+2a
+5
+6
+</details>
+
+Problem 3.49
+
+<!-- source-page: 128 -->
+
+# Section 3.13
+
+3.50 Determine matrix $[A]^{-1}$ of Eq. 3.13-4 and verify the shape functions given in Fig. 3.13-2. Suggestion: Regard Eqs. 3.13-3 as four equations to be solved for the four $a_{i}$ . Arrange the results in matrix format and identify a 4 by 4 matrix as $[A]^{-1}$ .   
+3.51 Imagine that a curve $\phi = \phi(x)$ is to be fitted to three data values: $\phi_1$ and $\theta_1$ at $x = 0$ and $\phi_2$ at $x = L$ (analogous to Fig. 3.13-1, but with $\theta_2$ unspecified). Determine the shape functions. Also sketch them and check their behavior at $x = 0$ and at $x = L$ (in the fashion of Fig. 3.13-2).   
+3.52 Show that cubic shape functions (Fig. 3.13-2) do not provide $C^2$ continuity. Suggestion: Examine the node shared by two adjacent elements, only one of which has nonzero d.o.f.   
+3.53 Imagine that at points $A, B$ , and $C$ in the sketch one knows both ordinate and slope data. Slope is indicated by a short line through a data point. Without calculation, sketch
+
+(a) a Lagrange interpolation curve through all three points.
+
+(b) piecewise interpolation of $C^0$ continuity.
+
+(c) piecewise interpolation of $C^1$ continuity.
+
+![](images/page-128_16546ca623e4ed6997bbd9643ae06344a1916624ad9139838bfbae7c56b69efc.jpg)
+
+<details>
+<summary>text_image</summary>
+
+φ
+A
+B
+C
+Problem 3.53
+x
+</details>
+
+3.54 Shape functions $N_{i}$ of $C^0$ elements satisfy the relation $\Sigma N_{i} = 1$ . Such is not the case for the $N_{i}$ of Fig. 3.13-2. Why?
+
+<!-- source-page: 129 -->
+
+# DISPLACEMENT-BASED ELEMENTS FOR STRUCTURAL MECHANICS
+
+General expressions for the element stiffness matrix [k] and the element load vector $\{r_{e}\}$ are derived, then used to formulate simple elements. Also discussed are how equilibrium and compatibility are approximated by the solution, requirements for convergence with mesh refinement, and procedures for stress computation.
+
+# 4.1 FORMULAS FOR ELEMENT
+
+MATRICES [k] AND $\{\mathbf{r}_e\}$
+
+Our discussion is restricted to elements based on displacement fields. Other elements, often equally good but less popular, are based on stress fields. Stress field elements are not discussed in this chapter.
+
+The derivation of finite element formulas is a straightforward procedure, which can be verbally summarized as follows [4.1]. Displacements are taken as the dependent variables. Therefore, the appropriate functional for a Rayleigh-Ritz solution is $\Pi_p$ , the expression for potential energy. We select an admissible displacement field, defined in piecewise fashion so that displacements within any element are interpolated from nodal d.o.f. of that element, then evaluate $\Pi_p$ in terms of nodal d.o.f. Using the principle of stationary potential energy, we write $d\Pi_p = 0$ , from which we obtain algebraic equations to be solved for the nodal d.o.f. In the course of this argument we identify certain expressions as the element stiffness matrix [k] and the element load vector $\{\mathbf{r}_e\}$ . Details of this derivation are now described.
+
+The starting point is the expression for potential energy in a linearly elastic body, Eq. 3.4-1, which is repeated here as Eq. 4.1-1,
+
+$$
+\Pi_ {p} = \int_ {V} \left(\frac {1}{2} \{\boldsymbol {\epsilon} \} ^ {T} [ \mathbf {E} ] \{\boldsymbol {\epsilon} \} - \{\boldsymbol {\epsilon} \} ^ {T} [ \mathbf {E} ] \{\boldsymbol {\epsilon} _ {0} \} + \{\boldsymbol {\epsilon} \} ^ {T} \{\boldsymbol {\sigma} _ {0} \}\right) d V \tag {4.1-1}
+$$
+
+$$
+- \int_ {V} \{\mathbf {u} \} ^ {T} \{\mathbf {F} \} d V - \int_ {S} \{\mathbf {u} \} ^ {T} \{\boldsymbol {\Phi} \} d S - \{\mathbf {D} \} ^ {T} \{\dot {\mathbf {P}} \}
+$$
+
+in which $\{\mathbf{u}\} = \lfloor u \quad v \quad w \rfloor^T$ , the displacement field
+
+$\{\epsilon\} = \left\lfloor \epsilon_x \quad \epsilon_y \quad \epsilon_z \quad \gamma_{xy} \quad \gamma_{yz} \quad \gamma_{zx} \right\rfloor^T$ , the strain field
+
+[E] = the material property matrix (e.g., Eq. 1.7-3)
+
+$\{\pmb{\epsilon}_0\} ,\{\pmb{\sigma}_0\} =$ initial strains and initial stresses (Eq. 1.7-8)
+
+$\{\mathbf{F}\} = \left\lfloor F_x\quad F_y\quad F_z\right\rfloor^T$ , body forces (Eq. 1.6-4)
+
+$\{\Phi\} = \left\lfloor \Phi_x \quad \Phi_y \quad \Phi_z \right\rfloor^T$ , surface tractions (Eq. 1.6-4)
+
+<!-- source-page: 130 -->
+
+$\{D\} = \text{nodal d.o.f. of the structure}$
+
+$\{\mathbf{P}\} =$ loads applied to d.o.f. by external agencies
+
+S, V = surface area and volume of the structure
+
+Shorter expressions may be used for problems that are two- or one-dimensional, as will be seen subsequently.
+
+s will be seen subsequently.
+Displacements within an element are interpolated from element nodal d.o.f. {d},
+
+$$
+\{\mathbf {u} \} = [ \mathbf {N} ] \{\mathbf {d} \} \tag {4.1-2}
+$$
+
+where [N] is the shape function matrix. The particular form of [N] need not be specified yet, but a form must eventually be written. The form selected has much to do with the quality of the approximate solution.
+
+From here onward, the derivation requires only straightforward manipulation. Strains are obtained from displacements by differentiation. Thus
+
+$$
+\{\epsilon \} = [ \partial ] \{\mathbf {u} \} \quad \text { yields } \quad \{\epsilon \} = [ \mathbf {B} ] \{\mathbf {d} \}, \quad \text { where } \quad [ \mathbf {B} ] = [ \partial ] [ \mathbf {N} ] \tag {4.1-3}
+$$
+
+The differential operator matrix $[\partial]$ is given by Eq. 1.5-6; its size is 6 by 3 for three-dimensional problems and 3 by 2 for two-dimensional problems. Substitution of the expressions for $\{\mathbf{u}\}$ and $\{\epsilon\}$ into Eq. 4.1-1 yields
+
+$$
+\Pi_ {p} = \frac {1}{2} \sum_ {n = 1} ^ {\text { numel }} \{\mathbf {d} \} _ {n} ^ {T} [ \mathbf {k} ] _ {n} \{\mathbf {d} \} _ {n} - \sum_ {n = 1} ^ {\text { numel }} \{\mathbf {d} \} _ {n} ^ {T} \{\mathbf {r} _ {e} \} _ {n} - \{\mathbf {D} \} ^ {T} \{\mathbf {P} \} \tag {4.1-4}
+$$
+
+where summation symbols indicate that we include contributions from all numel elements of the structure, and we have defined
+
+$$
+\text { element   stiffness   matrix } \quad \boxed {[ \mathbf {k} ] = \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] [ \mathbf {B} ] d V} \tag {4.1-5}
+$$
+
+element load vector
+
+$$
+\begin{array}{r l} \{\mathbf {r} _ {e} \} & = \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] \{\boldsymbol {\epsilon} _ {0} \} d V - \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} \{\boldsymbol {\sigma} _ {0} \} d V \\ & + \int_ {V _ {e}} [ \mathbf {N} ] ^ {T} \{\mathbf {F} \} d V + \int_ {S _ {e}} [ \mathbf {N} ] ^ {T} \{\boldsymbol {\Phi} \} d S \end{array} \tag {4.1-6}
+$$
+
+where $V_{e}$ denotes the volume of an element and $S_{e}$ its surface. In the surface integral, [N] is evaluated on $S_{e}$ .
+
+Despite its formidable appearance, the expression for $\{\mathbf{r}_e\}$ is less important than the expression for [k]. In essence, the $\{\mathbf{r}_e\}$ expression states how certain loads can be dealt with to best advantage. Use of an ad hoc method instead may provide acceptable accuracy. Examples follow (see below Eq. 4.3-14).
+
+To complete the derivation we must determine the algebraic equations to be solved for nodal d.o.f., as follows. Every d.o.f. in an element vector $\{\mathbf{d}\}$ also appears in the vector of global (i.e., structural) d.o.f. $\{\mathbf{D}\}$ . Therefore, as argued in Sections 2.6 and 2.7, $\{\mathbf{D}\}$ can replace $\{\mathbf{d}\}$ in Eq. 4.1-4 if $[\mathbf{k}]$ and $\{\mathbf{r}_e\}$ of every element are conceptually expanded to structure size. Thus Eq. 4.1-4 becomes
+
+$$
+\Pi_ {p} = \frac {1}{2} \{\mathbf {D} \} ^ {T} [ \mathbf {K} ] \{\mathbf {D} \} - \{\mathbf {D} \} ^ {T} \{\mathbf {R} \} \tag {4.1-7}
+$$
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_014.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_014.md
new file mode 100644
index 00000000..e8a77f1e
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_014.md
@@ -0,0 +1,902 @@
+<!-- source-page: 131 -->
+
+where
+
+$$
+[ \mathbf {K} ] = \sum_ {n = 1} ^ {\text { numel }} [ \mathbf {k} ] _ {n} \quad \text { and } \quad \{\mathbf {R} \} = \{\mathbf {P} \} + \sum_ {n = 1} ^ {\text { numel }} \{\mathbf {r} _ {e} \} _ {n} \tag {4.1-8}
+$$
+
+Summations indicate assembly of element matrices by addition of overlapping terms, as in Section 2.7. Now $\Pi_{p}$ is a function of d.o.f. $\{D\}$ . Making $\Pi_{p}$ stationary with respect to small changes in the $D_{i}$ by use of convenient differentiation rules given in Appendix A, we write
+
+$$
+\left\{\frac {\partial \Pi_ {p}}{\partial \mathbf {D}} \right\} = \{\mathbf {0} \} \quad \text { yields } \quad [ \mathbf {K} ] \{\mathbf {D} \} = \{\mathbf {R} \} \tag {4.1-9}
+$$
+
+The latter matrix equation is a set of simultaneous algebraic equations to be solved for d.o.f. {D}. As noted in the latter part of Section 3.3, $\partial \Pi_{p} / \partial D_{i} = 0$ is a nodal equilibrium equation, [K] is a symmetric matrix, and $K_{ij} = \partial^2\Pi_p / \partial D_i\partial D_j$ .
+
+Some Particular Cases. The preceding derivation may have suggested that $\{\epsilon\}$ must always be a 6 by 1 vector, as for a three-dimensional problem. However, no change in Eqs. 4.1-5 and 4.1-6 need be made if the problem is two-dimensional. For plane stress or plane strain, $\{\epsilon\}$ is 3 by 1 and contains only $\epsilon_x$ , $\epsilon_y$ , and $\gamma_{xy}$ , and matrix [E] is 3 by 3 (see Eq. 1.7-5, for example). For a symmetrically loaded solid of revolution, $\{\epsilon\}$ is 4 by 1 and contains $\epsilon_r$ , $\epsilon_z$ , $\epsilon_\theta$ , and $\gamma_{zr}$ , and [E] is 4 by 4.
+
+Formulas for [k] and $\{\mathbf{r}_e\}$ applicable to a bar under axial load can be obtained by specialization of Eqs. 4.1-5 and 4.1-6. However, it is easier to rederive the formulas, using at the outset special forms applicable to a bar. Such a derivation has already been given in Section 3.9. In that development, note Eqs. 3.9-2 and 3.9-3 in particular. They state that the strain energy of a one-element structure is
+
+$$
+U _ {e} = \int_ {V _ {e}} (\text { strain   energy   density }) d V = \frac {1}{2} \{\mathbf {d} \} ^ {T} [ \mathbf {k} ] \{\mathbf {d} \} \tag {4.1-10}
+$$
+
+Equation 4.1-10 summarizes the argument that leads to Eq. 4.1-5: namely, that an element stiffness matrix is obtained by substitution of an interpolation scheme into a strain energy expression. Equation 4.1-6 can be explained similarly. Element nodal loads $\{r_{e}\}$ are obtained by substitution of an interpolation scheme into an expression for work done by distributed loads that act on an element:
+
+$$
+\Omega_ {e} = - \int_ {V _ {e}} (\text { work   per   unit   volume }) d V = - \{\mathbf {d} \} ^ {T} \{\mathbf {r} _ {e} \} \tag {4.1-11}
+$$
+
+(Work done by externally applied nodal loads $\{\mathbf{P}\}$ is excluded from $\{\mathbf{r}_e\}$ .)
+
+In flexural problems, such as beam and plate bending, it is convenient to define [B] in such a way that the product [B]{d} represents curvatures rather than strains. Consider beam bending. Expressions for $U_{e}$ and $\Omega_{e}$ come from the first two integrals in Eq. 3.4-8. Noting that a scalar is its own transpose, we write $w_{,xx}^{2} = w_{,xx}^{T}w_{,xx}$ and $w = w^{T}$ . Hence
+
+<!-- source-page: 132 -->
+
+$$
+U _ {e} = \frac {1}{2} \int_ {0} ^ {L} w _ {, x x} ^ {T} E I w _ {, x x} d x \quad \text { and } \quad \Omega_ {e} = - \int_ {0} ^ {L} w ^ {T} q d x \tag {4.1-12}
+$$
+
+where $w$ is lateral displacement and $L$ is the element length. Interpolating $w$ from element nodal d.o.f. $\{\mathbf{d}\}$ , we have
+
+$$
+w = \lfloor \mathbf {N} \rfloor \{\mathbf {d} \} \quad \text { and } \quad w _ {, x x} = \lfloor \mathbf {B} \rfloor \{\mathbf {d} \}, \quad \text { where } \quad \lfloor \mathbf {B} \rfloor = \frac {d ^ {2}}{d x ^ {2}} \lfloor \mathbf {N} \rfloor \tag {4.1-13}
+$$
+
+Therefore, in view of Eqs. 4.1-10 and 4.1-11, [k] and $\{\mathbf{r}_e\}$ for a straight beam element are
+
+$$
+\left\lfloor [ \mathbf {k} ] = \int_ {0} ^ {L} \left\lfloor \mathbf {B} \right\rfloor^ {T} E I \left\lfloor \mathbf {B} \right\rfloor d x\right) \quad \text { and } \quad \left\{\mathbf {r} _ {e} \right\} = \int_ {0} ^ {L} \left\lfloor \mathbf {N} \right\rfloor^ {T} q d x \tag {4.1-14}
+$$
+
+Analogous expressions are written for flat plates in bending, where EI becomes a matrix of flexural rigidities and integration is over the area of the plate midsurface.
+
+a-Basis Formulation. If displacements are initially expressed in terms of d.o.f. {a}, one usually replaces {a} by element nodal d.o.f. {d} before generating a stiffness matrix. The process of replacing {a} by {d} is explained in Sections 3.8 and 3.13. Sometimes it is convenient to delay the replacement of {a} by {d}. The procedure is as follows.
+
+Displacements and strains, expressed in terms of $\{a\}$ , are
+
+$$
+\{\mathbf {u} \} = [ \mathbf {N} _ {a} ] \{\mathbf {a} \} \quad \text { and } \quad \{\boldsymbol {\epsilon} \} = [ \mathbf {B} _ {a} ] \{\mathbf {a} \}, \quad \text { where } \quad [ \mathbf {B} _ {a} ] = [ \partial ] [ \mathbf {N} _ {a} ] \tag {4.1-15}
+$$
+
+(For example, $[\mathbf{N}_a] = \lfloor 1 \quad \mathrm{s} \rfloor$ in Eq. 3.8-5.) But $\{\mathbf{a}\} = [\mathbf{A}]^{-1}\{\mathbf{d}\}$ , so
+
+$$
+[ \mathbf {N} ] = [ \mathbf {N} _ {a} ] [ \mathbf {A} ] ^ {- 1} \quad \text { and } \quad [ \mathbf {B} ] = [ \mathbf {B} _ {a} ] [ \mathbf {A} ] ^ {- 1} \tag {4.1-16}
+$$
+
+Substitution of Eqs. 4.1-16 into Eqs. 4.1-5 and 4.1-6 yields the “d-basis” matrices [k] and $\{r_{e}\}$ ,
+
+$$
+[ \mathbf {k} ] = [ \mathbf {A} ] ^ {- T} [ \mathbf {k} _ {a} ] [ \mathbf {A} ] ^ {- 1} \quad \text { and } \quad \{\mathbf {r} _ {e} \} = [ \mathbf {A} ] ^ {- T} \{\mathbf {r} _ {e a} \} \tag {4.1-17}
+$$
+
+where the “a-basis” matrices are defined as
+
+$$
+[ \mathbf {k} _ {a} ] = \int_ {V _ {e}} [ \mathbf {B} _ {a} ] ^ {T} [ \mathbf {E} ] [ \mathbf {B} _ {a} ] d V \quad \text { and } \quad \{\mathbf {r} _ {e a} \} = \int_ {V _ {e}} [ \mathbf {B} _ {a} ] ^ {T} [ \mathbf {E} ] \{\boldsymbol {\epsilon} _ {0} \} d V - \dots \tag {4.1-18}
+$$
+
+The $a$ -basis matrices $[\mathbf{k}_a]$ and $\{\mathbf{r}_{ea}\}$ are converted to $d$ -basis matrices $[\mathbf{k}]$ and $\{\mathbf{r}_e\}$ before global equations are assembled, as the relationship between d.o.f. $\{\mathbf{a}\}$ in neighboring elements is complicated and unwieldy.
+
+An example application appears in Eqs. 4.2-6 and 4.2-7.
+
+<!-- source-page: 133 -->
+
+# 4.2 OVERVIEW OF ELEMENT STIFFNESS MATRICES
+
+In this section we outline the formulation of selected element stiffness matrices, with the intent of showing the conceptual simplicity of the process. Details of manipulations may be found in the text sections cited.
+
+Bar. Figure 4.2-1 shows a straight bar whose nodal d.o.f. are axial displacements $u_{1}$ and $u_{2}$ . A linear axial displacement field, as used in Section 3.9, is appropriate,
+
+$$
+u = \lfloor \mathbf {N} \rfloor \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \end{array} \right\}, \quad \text { where } \quad \lfloor \mathbf {N} \rfloor = \left\lfloor \frac {L - x}{L} \frac {x}{L} \right\rfloor \tag {4.2-1}
+$$
+
+Using Eq. 3.9-3, with $\lfloor \mathbf{B} \rfloor = d\lfloor \mathbf{N} \rfloor / dx = \lfloor -1 - 1 \rfloor / L$ , we obtain
+
+$$
+[ \mathbf {k} ] = \int_ {0} ^ {L} \left\lfloor \mathbf {B} \right] ^ {T} A E \left\lfloor \mathbf {B} \right\rfloor d x = \frac {A E}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right] \tag {4.2-2}
+$$
+
+where the latter expression is for a uniform bar, AE = constant. The integral expression for [k] does not demand that AE be independent of x.
+
+Beam. Figure 4.2-2 shows a four-d.o.f. straight beam element. Rotation $\theta$ is assumed to be small, so that $\theta \approx dw/dx$ . Four d.o.f. define a cubic lateral displacement field,
+
+$$
+w = \lfloor \mathbf {N} \rfloor \left\lfloor w _ {1} \quad \theta_ {1} \quad w _ {2} \quad \theta_ {2} \right\rfloor^ {T} \tag {4.2-3}
+$$
+
+where the four $N_{i}$ are given in Fig. 3.13-2. The curvature field is $w_{xx} = [\mathbf{B}]\{\mathbf{d}\}$ , where
+
+$$
+\lfloor \mathbf {B} \rfloor = \frac {d ^ {2}}{d x ^ {2}} \lfloor \mathbf {N} \rfloor = \left\lfloor - \frac {6}{L ^ {2}} + \frac {1 2 x}{L ^ {3}} \quad - \frac {4}{L} + \frac {6 x}{L ^ {2}} \quad \frac {6}{L ^ {2}} - \frac {1 2 x}{L ^ {3}} \quad - \frac {2}{L} + \frac {6 x}{L ^ {2}} \right\rfloor \tag {4.2-4}
+$$
+
+For constant $EI$ , the element stiffness matrix given by Eq. 4.1-14 is
+
+![](images/page-133_3f8ea6bbab99fafe53e75f269316ef173f9026012114ba792340bea4c96c55ee.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+L
+u₁
+1
+A,E
+2
+u₂
+x,u
+</details>
+
+Figure 4.2-1. Bar element with two d.o.f. ( $u_{1}$ and $u_{2}$ ).
+
+![](images/page-133_f988a286e8d6d2bd9a9fb2a647c8b2fd1bc981a7f3d4f955b0186346472f94d9.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+L
+w₁ θ₁
+1 E,I
+w₂ θ₂
+2 x
+</details>
+
+Figure 4.2-2. Standard four-d.o.f. beam element.
+
+<!-- source-page: 134 -->
+
+$$
+[ \mathbf {k} ] = \int_ {0} ^ {L} [ \mathbf {B} ] ^ {T} E I [ \mathbf {B} ] d x = \frac {E I}{L ^ {3}} \left[ \begin{array}{c c c c} 1 2 & 6 L & - 1 2 & 6 L \\ 6 L & 4 L ^ {2} & - 6 L & 2 L ^ {2} \\ - 1 2 & - 6 L & 1 2 & - 6 L \\ 6 L & 2 L ^ {2} & - 6 L & 4 L ^ {2} \end{array} \right] \tag {4.2-5}
+$$
+
+This [k] operates on d.o.f. $\{\mathbf{d}\}$ having the order shown in Eq. 4.2-3. A slightly modified form of Eq. 4.2-5 is able to account for transverse shear deformation [4.2,4.11].
+
+The manipulations needed to obtain [k] are shortened by using the “a-basis” of Eqs. 4.1-17 and 4.1-18. With [X] and [A] given by Eqs. 3.13-1 and 3.13-3, we write
+
+$$
+\left\lfloor \mathbf {B} \right\rfloor = \frac {d ^ {2}}{d x ^ {2}} \left\lfloor \mathbf {X} \right\rfloor [ \mathbf {A} ] ^ {- 1} = \left\lfloor 0 0 2 6 x \right\rfloor [ \mathbf {A} ] ^ {- 1} \tag {4.2-6}
+$$
+
+$$
+[ \mathbf {k} ] = [ \mathbf {A} ] ^ {- T} \int_ {0} ^ {L} \left\lfloor 0 \quad 0 \quad 2 \quad 6 x \right] ^ {T} E I \left\lfloor 0 \quad 0 \quad 2 \quad 6 x \right\rfloor d x [ \mathbf {A} ] ^ {- 1} \tag {4.2-7}
+$$
+
+and obtain the same [k] as given in Eq. 4.2-5.
+
+The reader should understand the sign conventions for nodal moments and bending moment. Nodal moments $M_1$ and $M_2$ are positive when acting in the directions of $\theta_1$ and $\theta_2$ in Fig. 4.2-2. Bending moment $M = EIw_{,xx}$ is positive when it creates tension on the bottom of the beam. Therefore, $M = -M_1$ at the left end and $M = +M_2$ at the right end.
+
+Plane Frame. A plane frame member can deform both axially and in bending. Effectively, to obtain a plane frame element we superpose the bar and beam elements of Figs. 4.2-1 and 4.2-2 (and of Eqs. 4.2-2 and 4.2-5). Nodal d.o.f. are $\{\mathbf{d}\} = \left[u_1 w_1 \theta_1 u_2 w_2 \theta_2\right]^T$ . If the element is uniform and lies along the $x$ axis, its stiffness matrix is
+
+$$
+[ \mathbf {k} ] = \frac {A E}{L} \left[ \begin{array}{c c c c c c} 1 & 0 & 0 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ - 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right] + \frac {E I}{L ^ {3}} \left[ \begin{array}{c c c c c c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 2 & 6 L & 0 & - 1 2 & 6 L \\ 0 & 6 L & 4 L ^ {2} & 0 & - 6 L & 2 L ^ {2} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & - 1 2 & - 6 L & 0 & 1 2 & - 6 L \\ 0 & 6 L & 2 L ^ {2} & 0 & - 6 L & 4 L ^ {2} \end{array} \right] \tag {4.2-8}
+$$
+
+If a frame element is arbitrarily oriented in the plane, its stiffness matrix is easily determined from [k] of Eq. 4.2-8 by a coordinate transformation (see Section 7.5). Similarly, a plane truss element of arbitrary orientation can be obtained by coordinate transformation of Eq. 4.2-2 (again, see Section 7.5).
+
+Constant-Strain Triangle. This element, shown in Fig. 4.2-3, is one of the earliest finite elements [1.8]. It can be used to solve problems of plane stress and plane strain. However, it is not a very good element for this purpose. We introduce it here primarily because it serves as a good example in subsequent discussions of why elements behave as they do.
+
+<!-- source-page: 135 -->
+
+![](images/page-135_eaf2138557259634a44cbee8283bdff542946be0a66ef5c5ff7e88131ec94427.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+v₃
+u₃
+3
+u₂
+2
+v₂
+u₁
+1
+v₁
+x,u
+</details>
+
+Figure 4.2-3. Constant-strain triangle (six d.o.f.).
+
+![](images/page-135_dc6a28a072f0b6615f30bb51a894e1f0a1389142da9928636c01a0726668ba5b.jpg)
+
+<details>
+<summary>flowchart</summary>
+
+```mermaid
+graph TD
+    A["1"] --> B["2"]
+    B --> C["3"]
+    C --> D["4"]
+    D --> E["5"]
+    E --> F["6"]
+    F --> G["7"]
+    G --> H["8"]
+    H --> I["9"]
+    I --> J["10"]
+    J --> K["11"]
+    K --> L["12"]
+    L --> M["13"]
+    M --> N["14"]
+    N --> O["15"]
+    O --> P["16"]
+    P --> Q["17"]
+    Q --> R["18"]
+    R --> S["19"]
+    S --> T["20"]
+    T --> U["21"]
+    U --> V["22"]
+    V --> W["23"]
+    W --> X["24"]
+    X --> Y["25"]
+    Y --> Z["26"]
+    Z --> A["4"]
+    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
+    style AA fill:#f9f,stroke:#333
+    style AB fill:#f9f,stroke:#333
+    style AC fill:#f9f,stroke:#333
+    style AD fill:#f9f,stroke:#333
+    style AE fill:#f9f,stroke:#333
+    style AF fill:#f9f,stroke:#333
+    style AG fill:#f9f,stroke:#333
+    style AH fill:#f9f,stroke:#333
+    style AI fill:#f9f,stroke:#333
+    style AJ fill:#f9f,stroke:#333
+    style AK fill:#f9f,stroke:#333
+    style AL fill:#f9f,stroke:#333
+    style AM fill:#f9f,stroke:#333
+    style AN fill:#f9f,stroke:#333
+    style AO fill:#f9f,stroke:#333
+    style AP fill:#f9f,stroke:#333
+    style AQ fill:#f9f,stroke:#333
+    style AR fill:#f9f,stroke:#333
+    style AS fill:#f9f,stroke:#333
+    style AT fill:#f9f,stroke:#333
+    style AU fill:#f9f,stroke:#333
+    style AV fill:#f9f,stroke:#333
+    style AW fill:#f9f,stroke:#333
+    style AX fill:#f9f,stroke:#333
+    style AY fill:#f9f,stroke:#333
+    style AZ fill:#f9f,stroke:#333
+    style BA fill:#f9f,stroke:#333
+    style BB fill:#f9f,stroke:#333
+    style BC fill:#f9f,stroke:#333
+    style BD fill:#f9f,stroke:#333
+    style BE fill:#f9f,stroke:#333
+    style BF fill:#f9f,stroke:#333
+    style BG fill:#f9f,stroke:#333
+    style BH fill:#f9f,stroke:#333
+    style BI fill:#f9f,stroke:#333
+    style BJ fill:#f9f,stroke:#333
+    style BK fill:#f9f,stroke:#333
+    style BL fill:#f9f,stroke:#333
+    style BM fill:#f9f,stroke:#333
+    style BN fill:#f9f,stroke:#333
+    style BO fill:#f9f,stroke:#333
+    style BP fill:#f9f,stroke:#333
+    style B1 fill:#f9f,stroke:#333
+    style CA fill:#f9f,stroke:#333
+    style CB fill:#f9f,stroke:#333
+    style CC fill:#f9f,stroke:#333
+    style CD fill:#f9f,stroke:#333
+    style DE fill:#f9f,stroke:#333
+    style DF fill:#f9f,stroke:#333
+    style DG fill:#f9f,stroke:#333
+    style BE1 fill:#f9f,stroke:#333
+    style BF1 fill:#f9f,stroke:#333
+    style BG1 fill:#f9f,stroke:#333
+    style CA1 fill:#f9f,stroke:#333
+    style CB1 fill:#f9f,stroke:#333
+    style AD1 fill:#f9f,stroke:#333
+    style AE1 fill:#f9f,stroke:#333
+    style AF1 fill:#f9f,stroke:#333
+    style AG1 fill:#f9f,stroke:#333
+    style AH1 fill:#f9f,stroke:#333
+    style AI1 fill:#f9f,stroke:#333
+    style AJ1 fill:#f9f,stroke:#333
+    style AK1 fill:#f9f,stroke:#333
+    style AL1 fill:#f9f,stroke:#333
+    style AM1 fill:#f9f,stroke:#333
+    style AN1 fill:#f9f,stroke:#333
+    style AO1 fill:#f9f,stroke:#333
+    style AP1 fill:#f9f,stroke:#333
+    style AQ1 fill:#f9f,stroke:#333
+    style AR1 fill:#f9f,stroke:#333
+    style AS1 fill:#f9f,stroke:#333
+    style AT1 fill:#f9f,stroke:#333
+    style AU1 fill:#f9f,stroke:#333
+    style AV1 fill:#f9f,stroke:#333
+    style AW1 fill:#f9f,stroke:#333
+    style AX1 fill:#f9f,stroke:#333
+    style AY1 fill:#f9f,stroke:#333
+    style AZ1 fill:#f9f,stroke:#333
+    style BA1 fill:#f9f,stroke:#333
+    style BB1 fill:#f9f,stroke:#333
+    style BC1 fill:#f9f,stroke:#333
+    style BD1 fill:#f9f,stroke:#333
+    style BE1 fill:#f9f,stroke:#333
+    style BF1 fill:#f9f,stroke:#333
+    style BG1 fill:#f9f,stroke:#333
+    style BH1 fill:#f9f,stroke:#333
+    style BI1 fill:#f9f,stroke:#333
+    style BJ1 fill:#f9f,stroke:#333
+    style BK1 fill:#f9f,stroke:#333
+    style BL1 fill:#f9f,stroke:#333
+    style BM1 fill:#f9f,stroke:#333
+    style BN1 fill:#f9f,stroke:#333
+    style BO1 fill:#f9f,stroke:#333
+    style BP1 fill:#f9f,stroke:#333
+    style CA1 fill:#f9f,stroke:#333
+    style DE1 fill:#f9f,stroke:#333
+    style BF1 fill:#f9f,stroke:#333
+    style BG1 fill:#f9f,stroke:#333
+    style BH1 fill:#f9f,stroke:#333
+    style BI1 fill:#f9f,stroke:#333
+    style AD1 fill:#f9f,stroke:#333
+    style AS1 fill:#f9f,stroke:#333
+    style AT1 fill:#f9f,stroke:#333
+    style AU1 fill:#f9f,stroke:#333
+    style AV1 fill:#f9f,stroke:#333
+    style AW1 fill:#f9f,stroke:#333
+    style AX1 fill:#d9f9f,stroke:#333
+    style AZ1 fill:#f9f,stroke:#333
+    style BA1 fill:#f9f,stroke:#333
+    style BB1 fill:#f9f,stroke:#333
+    style BC1 fill:#f9f,stroke:#333
+    style BD1 fill:#f9f,stroke:#333
+    style BE1 fill:#f9f.stroke:#333
+    style BF1 fill:#f9f,stroke:#333
+    style BG1 fill:#f9f,stroke:#333
+    style BH1 fill:#f9f,stroke:#333
+    style BI1 fill:#f9f,stroke:#333
+    style AD1 fill:#f9f,stroke:#333
+    style AS1 fill:#f9f,stroke:#380
+    style AT1 fill:#f9f,stroke:#380
+    style AU1 fill:#f9f,stroke:#380
+    style AV1 fill:#f9f,stroke:#380
+    style AZ1 fill:#f9f,stroke:#380
+    style AS1 fill:#f9f,stroke:#380
+    style BW1 fill:#f9f,stroke:#380
+    style AX1 fill:#f9f,stroke:#380
+    style AZ1 fill:#f9f,stroke:#380
+    style BA1 fill:#f9f,stroke:#380
+    style BB1 fill:#f9f,stroke:#380
+    style BC1 fill:#f9f,stroke:#380
+    style BD1 fill:#f9f,stroke:#380
+    style BE1 fill:#f9f,stroke:#380
+    style BF1 fill:#f9f,stroke:#380
+    style BG1 fill:#f9f,stroke:#380
+    style BH1 fill:#f9f,stroke:#380
+    style BI1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style AS1 fill:#f9f,stroke:#380
+    style AT1 fill:#f9f,stroke:#380
+    style AU1 fill:#f9f,stroke:#380
+    style AV1 fill:#f9f,stroke:#380
+    style AZ1 fill:#f9f,stroke:#380
+    style AS1 fill:#f9f,stroke:#380
+    style BB1 fill:#f9f,stroke:#380
+    style BC1 fill:#f9f,stroke:#380
+    style BD1 fill:#f9f,stroke:#380
+    style BE1 fill:#f9f,stroke:#380
+    style BF1 fill:#f9f,stroke:#380
+    style BG1 fill:#d9f9f,stroke:#380
+    style BH1 fill:#f9f,stroke:#380
+    style BI1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style AS1 fill:#f9f,stroke:#380
+    style AT1 fill:#f9f,stroke:#380
+    style AU1 fill:#f9f.stroke:#380
+    style AV1 fill:#f9f,stroke:#380
+    style AZ1 fill:#f9f,stroke:#380
+    style AS1 fill:#f9f,stroke:#380
+    style BB1 fill:#f9f,stroke:#380
+    style BC1 fill:#f9f,stroke:#380
+    style BD1 fill:#f9f,stroke:#333
+    style BE1 fill:#f9f,stroke:#333
+    style BF1 fill:#f9f,stroke:#333
+    style BG1 fill:#f9f,stroke:#333
+    style BH1 fill:#f9f,stroke:#333
+    style BI1 fill:#f9f,stroke:#333
+    style AD1 fill:#f9f,stroke:#333
+    style AS1 fill:#d9f9f,stroke:#333
+    style AT1 fill:#f9f,stroke:#333
+    style AU1 fill:#f9f,stroke:#333
+    style AV1 fill:#f9f,stroke:#333
+    style AZ1 fill:#f9f,stroke:#333
+    style AS1 fill:#f9f,stroke:#333
+    style BB1 fill:#f9f,stroke:#333
+    style BC1 fill:#f9f,stroke:#333
+    style BD1 fill:#f9f,stroke:#333
+    style BE1 fill:#f9f,stroke:#333
+    style BF1 fill:#f9f,stroke:#333
+    style BG1 fill:#d9f9f,stroke:#333
+    style BH1 fill:#f9f,stroke:#333
+    style BI1 fill:#f9f,stroke:#333
+    style AD1 fill:#f9f,stroke:#333
+    style AS1 fill:#f9f,stroke:#333
+    style AT1 fill:#f9f,stroke:#333
+    style AU1 fill:#f9f.stroke:#333
+    style AV1 fill:#f9f,stroke:#333
+    style AZ1 fill:#f9f,stroke:#333
+    style AS1 fill:#f9f,stroke:#333
+    style BB1 fill:#f9f,stroke:#333
+    style BC1 fill:#f9f,stroke:#333
+    style BD1 fill:#f9f,stroke:#380
+    style BE1 fill:#f9f,stroke:#380
+    style BF1 fill:#f9f,stroke:#380
+    style BG1 fill:#f9f,stroke:#380
+    style BH1 fill:#f9f,stroke:#380
+    style BI1 fill:#f9f,stroke:#380
+    style AC1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style AS1 fill:#f9f,stroke:#380
+    style AT1 fill:#f9f,stroke:#380
+    style AU1 fill:#f9f,stroke:#380
+    style AV1 fill:#f9f,stroke:#380
+    style AZ1 fill:#d9f9f,stroke:#333
+    style AS1 fill:#f9f,stroke:#380
+    style BB1 fill:#f9f,stroke:#380
+    style BC1 fill:#f9f,stroke:#380
+    style BD1 fill:#f9f,stroke:#380
+    style BE1 fill:#f9f,stroke:#380
+    style BF1 fill:#f9f,stroke:#380
+    style BN1 fill:#f9f,stroke:#380
+    style CA1 fill:#f9f,stroke:#380
+    style D1 fill:#f9f,stroke:#380
+    style BE1 fill:#f9f,stroke:#380
+    style BF1 fill:#f9f,stroke:#380
+    style BG1 fill:#f9f,stroke:#380
+    style BH1 fill:#f9f,stroke:#380
+    style BI1 fill:#f9f,stroke:#380
+    style AC1 fill:#d9f9f,stroke:#333
+    style AD1 fill:#f9f,stroke:#333
+    style AS1 fill:#f9f,stroke:#380
+    style AT1 fill:#f9f,stroke:#380
+    style AU1 fill:#f9f,stroke:#380
+    style AV1 fill:#f9f,stroke:#380
+    style AZ1 fill:#f9f.stroke:#380
+    style AS1 fill:#f9f,stroke:#380
+    style BB1 fill:#f9f,stroke:#380
+    style BC1 fill:#f9f,stroke:#380
+    style BD1 fill:#f9f,stroke:#380
+    style BE1 fill:#f9f,stroke:#380
+    style BF1 fill:#f9f,stroke:#333
+    style BN1 fill:#f9f,stroke:#380
+    style BH1 fill:#f9f,stroke:#380
+    style BI1 fill:#f9f,stroke:#380
+    style AC1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style AS1 fill:#f9f,stroke:#380
+    style AT1 fill:#d9f9f,stroke:#333
+    style AU1 fill:#f9f,stroke:#380
+    style AV1 fill:#f9f,stroke:#380
+    style AZ1 fill:#f9f,stroke:#380
+    style AS1 fill:#f9f,stroke:#380
+    style BB1 fill:#f9f,stroke:#380
+    style BC1 fill:#f9f,stroke:#380
+    style BN1 fill:#f9f,stroke:#380
+    style BH1 fill:#f9f,stroke:#380
+    style BI1 fill:#f9f,stroke:#380
+    style AC1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style AS1 fill:#f9f,stroke:#380
+    style AN1 fill:#f9f,stroke:#380
+    style AT1 fill:#f9f,stroke:#380
+    style AU1 fill:#f9f,stroke:#380
+    style AV1 fill:#f9f,stroke:#380
+    style AZ1 fill:#f9f,stroke:#380
+    style AS1 fill:#f9f,stroke:#380
+    style BB1 fill:#d9f9f,stroke:#333
+    style BC1 fill:#f9f,stroke:#380
+    style BN1 fill:#f9f,stroke:#380
+    style BH1 fill:#f9f,stroke:#380
+    style BI1 fill:#f9f,stroke:#380
+    style AC1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style AB1 fill:#f9f,stroke:#380
+    style AC1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style BN1 fill:#f9f,stroke:#380
+    style BH1 fill:#f9f,stroke:#380
+    style BI1 fill:#f9f,stroke:#380
+    style AC1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style AB1 fill:#d9f9f,stroke:#333
+    style AC1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style BN1 fill:#f9f,stroke:#380
+    style BH1 fill:#f9f,stroke:#380
+    style BI1 fill:#f9f,stroke:#380
+    style AC1 fill:#f9f,stroke:#380
+    style AB1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style BN1 fill:#f9f,stroke:#380
+    style BH1 fill:#f9f,stroke:#380
+    style BI1 fill:#f9f,stroke:#380
+    style AC1 fill:#f9f,stroke:#380
+    style AD1 fill:#d9f9f,stroke:#333
+    style AB1 fill:#f9f,stroke:#380
+    style AC1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style BN1 fill:#f9f,stroke:#380
+    style BH1 fill:#f9f,stroke:#380
+    style BI1 fill:#f9f,stroke:#380
+    style Ac1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style AB1 fill:#f9f,stroke:#380
+    style AC1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style BN1 fill:#f9f,stroke:#380
+    style BH1 fill:#d9f9f,stroke:#333
+    style BI1 fill:#f9f,stroke:#380
+    style AC1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style BN1 fill:#f9f,stroke:#380
+    style BH1 fill:#f9f,stroke:#380
+    style BI1 fill:#f9f,stroke:#380
+    style AC1 fill:#d9f9f,stroke:#333
+    style AD1 fill:#f9f,stroke:#380
+    style AB1 fill:#f9f,stroke:#380
+    style AC1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style BN1 fill:#f9f,stroke:#380
+    style BH1 fill:#f9f,stroke:#380
+    style BE1 fill:#f9f,stroke:#380
+    style AC1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style BN1 fill:#f9f,stroke:#380
+    style BH1 fill:#f9f,stroke:#380
+    style BI1 fill:#f9f,stroke:#380
+    style AC1 fill:#e9f9f,stroke:#333
+    style AD1 fill:#f9f,stroke:#380
+    style AB1 fill:#f9f,stroke:#380
+    style AC1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style BN1 fill:#f9f,stroke:#380
+    style BH1 fill:#f9i9f,stroke:#333
+    style BI1 fill:#f9i9f,stroke:#380
+    style AC1 fill:#f9i9f,stroke:#380
+    style AD1 fill:#f9i9f,stroke:#380
+    style BN1 fill:#f9i9f,stroke:#380
+    style BH1 fill:#f9i9f,stroke:#380
+    style BI1 fill:#f9i9f,stroke:#380
+    style AC1 fill:#f9i9f,stroke:#380
+    style AD1 fill:#f9i9f,stroke:#380
+    style BN1 fill:#f9i9f,stroke:#380
+    style BH1 fill:#f9i9f,stroke:#380
+    style BI1 fill:#f9f,stroke:#380
+    style AC1 fill:#f9f,stroke:#380
+    style AD1 fill:#f9f,stroke:#380
+    style BN1 fill:#f9f,stroke:#380
+    style BH1 fill:#f9i9f,stroke:#380
+    style BI1 fill:#f9i9f,stroke:#380
+    style AC1 fill:#f9i9f,stroke:#380
+    style AD1 fill:#f9i9f,stroke:#380
+    style BN1 fill:#f9i9f,stroke:#380
+```
+```
+</details>
+
+Figure 4.2-4. Plane rectangular bilinear element (eight d.o.f.).
+
+The constant-strain triangle is based on a complete linear polynomial for both x- and y-direction displacements:
+
+$$
+u = a _ {1} + a _ {2} x + a _ {3} y \tag {4.2-9a}
+$$
+
+$$
+v = a _ {4} + a _ {5} x + a _ {6} y \tag {4.2-9b}
+$$
+
+Strains are $\epsilon_x = u_{,x}, \epsilon_y = v_{,y}$ , and $\gamma_{xy} = u_{,y} + v_{,x}$ . Thus
+
+$$
+\epsilon_ {x} = a _ {2} \quad \epsilon_ {y} = a _ {6} \quad \gamma_ {x y} = a _ {3} + a _ {5} \tag {4.2-10}
+$$
+
+We see that strains are independent of $x$ and $y$ within the element; hence the name "constant-strain triangle."
+
+The algebra of generating the element stiffness matrix is most easily carried out in area coordinates, as described in Chapter 5. Strain-displacement matrix [B] is 3 by 6 and contains only constants, which depend on the $x$ and $y$ coordinates of the three nodes. Hence, if material property matrix [E] and element thickness $t$ are constant over the element,
+
+$$
+\underset {6 \times 6} {[ \mathbf {k} ]} = \iint_ {6 \times 3} [ \mathbf {B} ] ^ {T} \underset {3 \times 3} {[ \mathbf {E} ]} \underset {3 \times 6} {[ \mathbf {B} ]} t d x d y = A t [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] [ \mathbf {B} ] \tag {4.2-11}
+$$
+
+where $A$ is the area of the triangle.
+
+The behavior of the constant-strain triangle is illustrated by numerical examples in Fig. 5.5-2.
+
+Plane Rectangular Bilinear Element. This element, shown in Fig. 4.2-4, is based on the bilinear displacement field
+
+$$
+u = a _ {1} + a _ {2} x + a _ {3} y + a _ {4} x y \tag {4.2-12a}
+$$
+
+$$
+v = a _ {5} + a _ {6} x + a _ {7} y + a _ {8} x y \tag {4.2-12b}
+$$
+
+In terms of nodal d.o.f., the displacement field $\{\mathbf{u}\} = [\mathbf{N}]\{\mathbf{d}\}$ is
+
+<!-- source-page: 136 -->
+
+$$
+\left\{ \begin{array}{l} u \\ v \end{array} \right\} = \left[ \begin{array}{c c c c c c c c} N _ {1} & 0 & N _ {2} & 0 & N _ {3} & 0 & N _ {4} & 0 \\ 0 & N _ {1} & 0 & N _ {2} & 0 & N _ {3} & 0 & N _ {4} \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \\ u _ {2} \\ \vdots \\ v _ {4} \end{array} \right\} \tag {4.2-13}
+$$
+
+Shape functions $N_{i}$ of this element are presented as Eqs. 3.12-10. Each $N_{i}$ has the form $(a \pm x)(b \pm y)/4ab$ . (With an adequate understanding of shape functions, the reader should have no difficulty in choosing the proper algebraic signs for each of the $N_{i}$ by inspection.) The strain-displacement matrix is
+
+$$
+[ \mathbf {B} ] _ {3 \times 8} = \left[ \begin{array}{c c} \partial / \partial x & 0 \\ 0 & \partial / \partial y \\ \partial / \partial y & \partial / \partial x \end{array} \right] [ \mathbf {N} ] = \frac {1}{4 a b} \left[ \begin{array}{c c c c} - (b - y) & 0 & (b - y) & \dots \\ 0 & - (a - x) & 0 & \dots \\ - (a - x) & - (b - y) & - (a + x) & \dots \end{array} \right] \tag {4.2-14}
+$$
+
+We see that $\epsilon_x$ depends on $y$ , $\epsilon_y$ depends on $x$ , and $\gamma_{xy}$ depends on both $x$ and $y$ . The element stiffness matrix is
+
+$$
+\underset {8 \times 8} {[ \mathrm{k} ]} = \int_ {- b} ^ {b} \int_ {- a} ^ {a} \underset {8 \times 3} {[ \mathrm{B} ] ^ {T}} \underset {3 \times 3} {[ \mathrm{E} ]} \underset {3 \times 8} {[ \mathrm{B} ]} t d x d y \tag {4.2-15}
+$$
+
+where $t$ is the element thickness. The integrand involves polynomials in $x$ and $y$ and is easily evaluated.
+
+If elements are rectangular, the bilinear element and the four-node plane isoparametric element of Section 6.3 are identical. Numerical examples that use isoparametric elements appear in Table 6.14-1.
+
+Solid Rectangular Trilinear Element. This element, shown in Fig. 4.2-5, is a simple generalization of the plane bilinear element. Its x-direction displacement is
+
+$$
+u = a _ {1} + a _ {2} x + a _ {3} y + a _ {4} z + a _ {5} x y + a _ {6} y z + a _ {7} z x + a _ {8} x y z \tag {4.2-16}
+$$
+
+![](images/page-136_ecfa1975fde61ed6dea917f496bfde86d6368b46284110036e22798fe60d9a79.jpg)
+
+<details>
+<summary>text_image</summary>
+
+2c
+y,v
+3
+4
+8
+7
+2b
+2
+z,w
+1
+6
+x,u
+5
+2a
+</details>
+
+(a)
+
+![](images/page-136_1138cc397c3f1d71ef38d3d3ef17c14786c61b53e8e1730945dcc58dfd50847f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+v_i
+i
+u_i
+w_i
+</details>
+
+(b)   
+Figure 4.2-5. (a) Solid rectangular trilinear element (24 d.o.f.). (b) D.o.f. at a typical node i.
+
+<!-- source-page: 137 -->
+
+and similarly for $v$ and $w$ , for a total of 24 d.o.f. In terms of nodal d.o.f., the displacement field $\{\mathbf{u}\} = [\mathbf{N}]\{\mathbf{d}\}$ is
+
+$$
+\left\{ \begin{array}{l} u \\ v \\ w \end{array} \right\} = \left[ \begin{array}{c c c c c c c} N _ {1} & 0 & 0 & N _ {2} & 0 & 0 & \dots \\ 0 & N _ {1} & 0 & 0 & N _ {2} & 0 & \dots \\ 0 & 0 & N _ {1} & 0 & 0 & N _ {2} & \dots \end{array} \right] \left\{ \begin{array}{c} u _ {1} \\ v _ {1} \\ w _ {1} \\ u _ {2} \\ \vdots \\ w _ {8} \end{array} \right\}. \tag {4.2-17}
+$$
+
+where each $N_{i}$ has the form
+
+$$
+\frac {(a \pm x) (b \pm y) (c \pm z)}{8 a b c} \tag {4.2-18}
+$$
+
+The signs are all negative for $N_2$ , all positive for $N_8$ , and so on. The strain-displacement matrix is $[\mathbf{B}] = [\partial][\mathbf{N}]$ , where $[\partial]$ is given in Eqs. 1.5-6. The element stiffness matrix is
+
+$$
+\underset {2 4 \times 2 4} {[ \mathbf {k} ]} = \int_ {- c} ^ {c} \int_ {- b} ^ {b} \int_ {- a} ^ {a} \underset {2 4 \times 6} {[ \mathbf {B} ] ^ {T}} \underset {6 \times 6} {[ \mathbf {E} ]} \underset {6 \times 2 4} {[ \mathbf {B} ]} d x d y d z \tag {4.2-19}
+$$
+
+Again the integrations are straightforward.
+
+Note that on any face (e.g., z = c), Eqs. 4.2-16 and 4.2-17 yield forms used for the bilinear element, Eqs. 4.2-12 and 4.2-13, respectively.
+
+A possible application of solid elements is shown in Fig. 4.2-6.
+
+Remark. According to terminology discussed in Section 3.11, the bar, triangular, bilinear, and trilinear elements are all $C^{0}$ elements, and the beam is a $C^{1}$ element. The frame element is $C^{0}$ in axial deformation and $C^{1}$ in bending.
+
+![](images/page-137_94947842a09abd5d38e8fe3ae4dd024270b5737953394672f1f09813adc93501.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+3D wireframe model of a curved surface with grid lines, no text or symbols present
+</details>
+
+Figure 4.2-6. Quarter of a railway wheel, modeled by solid elements of a type not restricted to rectangular shape. Hidden lines are removed. (Courtesy of Algor Interactive Systems, Inc., Pittsburgh, Pennsylvania.)
+
+<!-- source-page: 138 -->
+
+# 4.3 CONSISTENT ELEMENT NODAL
+
+# LOADS $\{\mathbf{r}_e\}$
+
+In this section we consider the element nodal load vector $\{\mathbf{r}_e\}$ , which is stated in Eq. 4.1-6. Equation 4.1-6 converts loads distributed throughout an element or write surface to discrete loads at element nodes.
+
+Any of the four integrals in Eq. 4.1-6 may vanish. For example, the surface integral is zero unless the element has an edge on the structure boundary and traction is applied to that edge. Then we integrate over only that edge of the element. All four integrals vanish if externally applied nodal loads $\{P\}$ make up the entire load vector $\{R\}$ .
+
+Initial Strain and Initial Stress. Terms in Eq. 4.1-6 that contain $\{\epsilon_{0}\}$ and $\{\sigma_{0}\}$ produce nodal loads that account for heating or cooling of the element, swelling (due perhaps to irradiation), and initial lack of fit. These nodal loads are self-equilibrating; that is, $\{r_{e}\}$ produces zero resultant force and zero resultant moment. The familiar bar element provides a convenient example (Fig. 4.3-1). Here
+
+$$
+\lfloor \mathbf {N} \rfloor = \left\lfloor \frac {L - x}{L} \quad \frac {x}{L} \right\rfloor \quad \text { and } \quad \lfloor \mathbf {B} \rfloor = \frac {1}{L} \lfloor - 1 \quad 1 \rfloor \tag {4.3-1}
+$$
+
+Imagine that the bar is $\Delta L$ units too long, so that the initial strain is $\epsilon_{0} = \Delta L/L$ . Also, let the bar be heated $T^{\circ}$ so that (with expansion prohibited) the initial stress caused by heating is $\sigma_{0} = -E\alpha T$ . The first two integrals in Eq. 4.1-6 yield
+
+$$
+\left\{\mathbf {r} _ {e} \right\} _ {2 \times 1} = \int_ {0} ^ {L} \frac {1}{L} \left\{ \begin{array}{c} - 1 \\ 1 \end{array} \right\} E \frac {\Delta L}{L} A d x - \int_ {0} ^ {L} \frac {1}{L} \left\{ \begin{array}{c} - 1 \\ 1 \end{array} \right\} (- E \alpha T) A d x \tag {4.3-2a}
+$$
+
+hence
+
+$$
+\left\{\mathbf {r} _ {e} \right\} _ {2 \times 1} = \left\{ \begin{array}{c} - F \\ F \end{array} \right\}, \quad \text { where } \quad F = E A \left(\frac {\Delta L}{L} + \alpha T\right) \tag {4.3-2b}
+$$
+
+Forces $F$ are shown in Fig. 4.3-1b. It is a worthwhile exercise for the reader to show that the same forces $F$ are produced by regarding the lack of fit as an initial stress and the temperature effect as an initial strain.
+
+Although forces $F$ sum to zero, they act to deform the bar an amount $FL/AE$ if axial deformation of the bar is uninhibited. This deformation creates a stress $\sigma = F/A$ , which is exactly canceled by the initial stress $\sigma = -E\epsilon_0 + \sigma_0$ . Thus we obtain axial strain without axial stress, which is entirely correct for an unrestrained bar.
+
+![](images/page-138_58369fb9cdad017881891a2d4795d0ee30f07943aeef18c0a18ea7d02e7af1fb.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+L
+1
+2
+x,u
+</details>
+
+(a)
+
+![](images/page-138_1aa2dccd5f9cc5cd1ec8a81df3d31a95d67b2fa8a63fdc27718dc03aac8e7664.jpg)  
+(b)   
+Figure 4.3-1. (a) Bar element. (b) Nodal forces caused by heating and initial lack of fit (from Eq. 4.3-2).
+
+<!-- source-page: 139 -->
+
+In this example, nodal forces could easily be deduced by direct physical argument. Elements more complicated than bars and beams often require formal use of Eq. 4.1-6.
+
+Mechanical Loads. Loads $\{r_{e}\}$ produced by body forces $\{F\}$ and surface tractions $\{\Phi\}$ are given by the latter two integrals in Eq. 4.1-6. These loads are called work-equivalent loads for the following reason: work done by nodal loads $\{r_{e}\}$ in going through nodal displacements $\{d\}$ is equal to work done by distributed loads $\{F\}$ and $\{\Phi\}$ in going through the displacement field associated with the element shape function. To show this we argue as follows. Work W done by loads $\{r_{e}\}$ during small nodal displacements $\{d\}$ is $W = \{d\}^{T}\{r_{e}\}$ . Taking for example the surface integral in Eq. 4.1-6, and substituting the displacement field $\{u\}^{T} = \{d\}^{T}[N]^{T}$ , we have
+
+$$
+W = \{\mathbf {d} \} ^ {T} \left\{\mathbf {r} _ {e} \right\} = \int_ {S _ {e}} \left\{\mathbf {d} \right\} ^ {T} [ \mathbf {N} ] ^ {T} \left\{\boldsymbol {\Phi} \right\} d S = \int_ {S _ {e}} \left\{\mathbf {u} \right\} ^ {T} \left\{\boldsymbol {\Phi} \right\} d S \tag {4.3-3}
+$$
+
+The latter integral sums the work of force increments $\{\Phi\}$ dS in going through displacements $\{u\}$ , where $\{u\}$ are field displacements created by $\{d\}$ via shape functions [N]. (See Eqs. 4.3-8 and 4.3-9 for an illustrative example.)
+
+Loads $\{r_{e}\}$ calculated by Eq. 4.1-6 are also called consistent because they are based on the same shape functions as used to calculate the element stiffness matrix. Finally, loads $\{r_{e}\}$ calculated by Eq. 4.1-6 are statically equivalent to the original distributed loading; that is, both $\{r_{e}\}$ and the original loading have the same resultant force and the same moment about an arbitrarily chosen point. That this is true may be seen by considering the work equivalence of the two loadings during a rigid-body translation and a small rigid-body rotation about an arbitrarily chosen point.
+
+We define inconsistent loading or lumping as the conversion of a distributed load to nodal loads that are inconsistent with Eq. 4.1-6, but are statically equivalent to the distributed load in that they provide the same resultant force. Typically, lumping is achieved by (a) computing the total force on an element caused by distributed loading, then assigning the same fraction of the total force to each element node, and (b) ignoring any nodal moments that would be present in the consistent vector $\{r_{e}\}$ . As will be seen in subsequent examples, the consistent method generally does not allot the total force on an element equally to element nodes. Lumping can yield poor answers in a coarse mesh, produce locally poor answers in a fine mesh, and lead to failure of the patch test (the patch test is discussed in Section 4.6).
+
+Concentrated Loads Not at Nodes. We denote by $\{p\}$ a concentrated force that has components $p_{x}$ , $p_{y}$ , and $p_{z}$ . The contribution of $\{p\}$ to $\{r_{e}\}$ can be evaluated from the surface integral in Eq. 4.1-6 by regarding a concentrated force as a large traction $\{\Phi\}$ acting on a small area dS. Thus $\{p\} = \{\Phi\}$ dS. The integral of $[N]^{T}\{\Phi\}$ dS is simply $[N]^{T}\{p\}$ where the concentrated force acts and is zero elsewhere. Thus, if there are n concentrated forces applied to an element, Eq. 4.1-6 yields
+
+$$
+\text { Owing   to   concentrated   forces } \quad \{\mathbf {p} \} _ {i}, \quad \{\mathbf {r} _ {e} \} = \sum_ {i = 1} ^ {n} [ \mathbf {N} ] _ {i} ^ {T} \{\mathbf {p} \} _ {i} \tag {4.3-4}
+$$
+
+where $[N]_{i}$ is the value of [N] at the location of $\{p\}_{i}$ .
+
+<!-- source-page: 140 -->
+
+![](images/page-140_391b5d957d24590db65e44de966588561861380653dce6a0352771f54cdd77bd.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+q = \frac{L - x}{L} q_1 + \frac{x}{L} q_2
+F
+1
+2
+x_1 u
+\frac{2L}{3}
+L
+</details>
+
+(a)
+
+![](images/page-140_5d639af5b32f1ae63186ba34b16b2614e324a6a8a02b83b434beba12a76283f9.jpg)
+
+<details>
+<summary>text_image</summary>
+
+N₁ = \frac{L - x}{L}
+1
+\frac{1}{3}
+\frac{2L}{3}
+L
+</details>
+
+(b)
+
+![](images/page-140_3bb27e7822857568f3236966d2458ea08d413f74b5f5250530b81ce44507a6df.jpg)
+
+<details>
+<summary>text_image</summary>
+
+N₂ = x/L
+2/3
+2L/3
+L
+</details>
+
+(c)   
+Figure 4.3-2. (a) Linearly varying distributed load $q$ and concentrated force $F$ on a bar. (b,c) Shape functions for axial displacement $u$ .
+
+Similarly, if $n$ concentrated moments $\{\mathbf{m}\} = \left\lfloor m_x - m_y - m_z \right]^T$ act at $n$ nonnodal locations on an element,
+
+Owing to concentrated moments $\{\mathbf{m}\}_{i}, \{\mathbf{r}_e\} = \sum_{i=1}^{n} [\mathbf{N}']_i^T \{\mathbf{m}\}_{i}$ (4.3-5)
+
+where $[N']$ contains derivatives of [N] and is evaluated at the location of $\{m\}_{i}$ . Derivatives are needed to calculate rotations $[N']_{i}\{d\}$ , through which moments $\{m\}_{i}$ act in doing work equal to $\{d\}^{T}\{r_{e}\}$ . An example of loads that result from Eq. 4.3-5 appears in Fig. 4.3-6c.
+
+Example. Bar Element. The bar element in Fig. 4.3-2 carries a distributed axial load $q$ , having dimensions of force per unit length, which varies linearly from intensity $q_{1}$ at $x = 0$ to intensity $q_{2}$ at $x = L$ . In addition, a concentrated axial force $F$ acts at $x = 2L / 3$ . Consistent nodal loads $\{\mathbf{r}_e\}$ are required.
+
+2L/3. Consistent nodal loads $[t_e]$ are required. To account for force $F$ we use Eq. 4.3-4. To account for load $q$ we can use the last integral in Eq. 4.1-6, writing the force increment as $q dx$ instead of $\{\Phi\} dS$ . Thus
+
+$$
+\left\{\mathbf {r} _ {e} \right\} _ {2 \times 1} = \int_ {0} ^ {L} \left\lfloor \mathbf {N} \right] ^ {T} q d x + \left\lfloor \mathbf {N} \right\rfloor_ {2 L / 3} F \tag {4.3-6}
+$$
+
+from which, with $\lfloor \mathbf{N}\rfloor = \left\lfloor \frac{L - x}{L} \frac{x}{L} \right\rfloor$ and $q$ as given in Fig. 4.3-2a,
+
+$$
+\left\{\mathbf {r} _ {e} \right\} = \frac {L}{6} \left\{ \begin{array}{l} 2 q _ {1} + q _ {2} \\ q _ {1} + 2 q _ {2} \end{array} \right\} + \left\{ \begin{array}{l} F / 3 \\ 2 F / 3 \end{array} \right\} \tag {4.3-7}
+$$
+
+We see that if $q_{1} = q_{2} = q$ , a constant, then the total load $qL$ is equally distributed to element nodes. The fraction of force $F$ allocated to each node is dictated by Eq. 4.3-4 and is shown graphically as an ordinate of the appropriate shape function in Fig. 4.3-2. (In civil engineering parlance, $N_{1}$ and $N_{2}$ are influence lines for reactions at the nodes.)
+
+The concept of work-equivalent loads is discussed in connection with Eq. 4.5.9.16 show that $\{\mathbf{r}_e\}$ of eq 4.3-7 is indeed a set of work-equivalent loads, we must show that
+
+$$
+\left\lfloor u _ {1} \quad u _ {2} \right\rfloor \left\{\mathrm{r} _ {e} \right\} = \int_ {0} ^ {L} u q d x + F u _ {2 L / 3} \tag {4.3-8}
+$$
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_015.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_015.md
new file mode 100644
index 00000000..a9011657
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_015.md
@@ -0,0 +1,383 @@
+<!-- source-page: 141 -->
+
+![](images/page-141_b19c9426022474429c0c35a4ed5fd79394664085a2968d042f76536096a9ce8b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+q1
+q2
+L
+s
+y
+x
+=
+L/6 (2q1 + q2)
+L/6 (q1 + 2q2)
+</details>
+
+Figure 4.3-3. Linearly varying load on a linear edge, and the consistent nodal loads produced.
+
+where
+
+$$
+u = \frac {L - x}{L} u _ {1} + \frac {x}{L} u _ {2} \quad \text { and } \quad q = \frac {L - x}{L} q _ {1} + \frac {x}{L} q _ {2} \tag {4.3-9}
+$$
+
+Substitution of Eqs. 4.3-7 and 4.3-9 into Eq. 4.3-8 shows that Eq. 4.3-8 is indeed satisfied.
+
+Plane Elements. Figure 4.3-3 shows a distributed load of intensity q that acts normal to a linear edge of a plane element. The inclination of the edge with respect to global coordinates xy does not matter. So long as edge-normal displacement varies linearly with edge-tangent coordinate s, loads are allocated to nodes as shown. This is of course the same allocation as seen in Eq. 4.3-7. An edge-parallel traction that varies linearly with s would be allocated to nodes in the same proportions.
+
+Figure 4.3-4 shows a uniformly distributed load of intensity q that acts normal to a quadratic edge. That is, the edge-normal displacement v is $v = \left[N\right]\left[v_{4} \quad v_{7} \quad v_{3}\right]^{T}$ , where
+
+$$
+\lfloor \mathrm{N} \rfloor = \left\lfloor \frac {2 x ^ {2}}{L ^ {2}} - \frac {x}{L} \quad 1 - \frac {4 x ^ {2}}{L ^ {2}} \quad \frac {2 x ^ {2}}{L ^ {2}} + \frac {x}{L} \right\rfloor \tag {4.3-10}
+$$
+
+![](images/page-141_32612abad4a2218c30c3e903cae76f79cb57bf5001ef53d25b7ca678aad734cf.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L
+2
+L
+2
+q
+b
+4
+7
+y
+x
+3
+8
+6
+a
+a
+1
+5
+2
+=
+qL
+6
+2qL
+3
+qL
+6
+4
+7
+3
+8
+6
+1
+5
+2
+</details>
+
+Figure 4.3-4. Uniform traction of intensity q on a quadratic edge, and the consistent nodal loads produced.
+
+<!-- source-page: 142 -->
+
+The edge is straight and node 7 is at the midpoint. Work-equivalent loads at nodes 4, 7, and 3 are
+
+$$
+\left\{\mathbf {r} _ {e} \right\} = \int_ {- L / 2} ^ {L / 2} \left\lfloor \mathbf {N} \right] ^ {T} q d x = \left\{ \begin{array}{l} q L / 6 \\ 2 q L / 3 \\ q L / 6 \end{array} \right\} \tag {4.3-11}
+$$
+
+We see that a uniform traction does not produce the same force at each node on a quadratic edge.
+
+Nodal loads produced by a concentrated force within a plane element are evaluated by means of Eq. 4.3-4. Shape functions for the rectangular element of Fig. 4.3-4 are given by Eqs. 6.6-1 if one substitutes $\xi = x / a$ and $\eta = y / b$ . If a concentrated force $F$ acts toward (say) the right at the center of this element (where $\xi = \eta = 0$ in Eqs. 6.6-1), Eq. 4.3-4 dictates that rightward forces $F / 2$ appear at nodes 5, 6, 7, and 8 and that leftward forces $F / 4$ appear at nodes 1, 2, 3, and 4. The nodal forces sum to $F$ , as they must, but the appearance of nodal forces whose sense is opposite to that of $F$ is unexpected from the standpoint of “common sense.” Note that in a nonstructural problem, the analogue of a concentrated force $F$ is a point source or a sink.
+
+If a uniform body force acts on a plane four-node rectangular element, one-quarter of the total force appears at each node. If the body force is again uniform and the element again plane and rectangular, but now with eight nodes as in Fig. 4.3-4, the total force is allocated to nodes in the proportions shown on the upper face of the element in Fig. 4.3-5b.
+
+Solid Elements. Again we use Eq. 4.1-6. If there is a body force $\{\mathbf{F}\}$ , integration spans the entire element volume and all shape functions of the element are involved. If there is a traction $\{\Phi\}$ on one face, integration spans only the loaded face, and only the shape functions associated with nodes on that face are involved.
+
+Imagine, for example, that a uniform normal stress $\sigma_{c}$ acts on a rectangular face that has corner and midside nodes (Fig. 4.3-5). Shape functions for this face may be taken from Eqs. 6.6-1 with $\xi = x / a$ and $\eta = y / b$ . The last integral in Eq. 4.1-6 yields the nodal loads shown in Fig. 4.3-5. The total force is $4P + 4Q = \sigma_{c}A$ , as required, but loads at corner nodes have a direction opposite to that of $\sigma_{c}$ .
+
+![](images/page-142_732ed3a23f4ddd39cb7457512605175f70a9993f6541c383770e4b640b458d0d.jpg)
+
+$$
+P = \frac {\sigma_ {c} A}{3}
+$$
+
+$$
+Q = \frac {\sigma_ {c} A}{1 2}
+$$
+
+Figure 4.3-5. (a) Uniform stress $\sigma_{c}$ on a rectangular face area of $A$ . Side nodes are at midsides. (b) Consistent nodal loads.
+
+<!-- source-page: 143 -->
+
+![](images/page-143_b7cddd6812bc8f7bd5bce2fe9e7372a3b9124e1225e33efe7941922090611f21.jpg)  
+Figure 4.3-6. Consistent nodal loads associated with loads q (constant), P, and M on a standard four-d.o.f. beam element (Eq. 4.2-3).
+
+Beam Elements. Figure 4.3-6 shows nodal loads produced by typical loading patterns on a beam element. Nodal loads are calculated by use of Eqs. 4.1-6, 4.3-4, and 4.3-5. Specifically,
+
+$$
+\left\{\mathbf {r} _ {e} \right\} _ {4 \times 1} = \int_ {0} ^ {L} \left\lfloor \mathbf {N} \right] ^ {T} q d x + \left\lfloor \mathbf {N} \right\rfloor_ {L / 2} ^ {T} P + \left\lfloor \frac {d \mathbf {N}}{d x} \right\rfloor_ {L / 2} ^ {T} M \tag {4.3-12}
+$$
+
+Loads $\{r_{e}\}$ are of course work-equivalent to the original loads q, P, or M, in the sense defined in connection with Eq. 4.3-3. They are also statically equivalent; that is, loads $\{r_{e}\}$ produce the same resultant force and the same resultant moment about an arbitrary point as do the original loads, as the reader can easily show.
+
+Error Produced by Lumping. The following example shows the merit of using consistent nodal loads rather than a lumping. Consider the uniformly loaded cantilever beam of Fig. 4.3-7a. Consistent nodal loads for a one-element model are shown in Fig. 4.3-7b. Taking [k] from Eq. 4.2-5 and fixing the left end of the beam, we arrive at the following set of equations to be solved for $w_{2}$ and $\theta_{2}$ ,
+
+$$
+\frac {E I}{L _ {T} ^ {3}} \left[ \begin{array}{c c} 1 2 & - 6 L _ {T} \\ - 6 L _ {T} & 4 L _ {T} ^ {2} \end{array} \right] \left\{ \begin{array}{l} w _ {2} \\ \theta_ {2} \end{array} \right\} = \left\{ \begin{array}{l} q L _ {T} / 2 \\ - q L _ {T} ^ {2} / 1 2 \end{array} \right\} \tag {4.3-13}
+$$
+
+![](images/page-143_df526b8ea12abaf49e30f001000885cc302c4bd034bf02e551f01f3514feaee6.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+q
+x
+LT
+</details>
+
+(a)
+
+![](images/page-143_00f41236403f7d4e3f174a9d0e2d7a593b713753ea9530ef6ba0f8b84c547a11.jpg)
+
+<details>
+<summary>text_image</summary>
+
+qL_T
+2
+qL_T^2
+12
+2
+L_T
+</details>
+
+(b)
+
+![](images/page-143_3b870de66591b60646f5a7557c56e36bb9da7297a3b0cf09591a687835423cd5.jpg)  
+(c)   
+Figure 4.3-7. (a) Uniformly loaded cantilever beam. (b) Consistent loads at node 2 of a one-element model. (c) Lumped (inconsistent) loads at node 2.
+
+<!-- source-page: 144 -->
+
+from which
+
+$$
+w _ {2} = \frac {q L _ {T} ^ {4}}{8 E I} \quad \text { and } \quad \theta_ {2} = \frac {q L _ {T} ^ {3}}{6 E I} \tag {4.3-14}
+$$
+
+These are the exact values of $w_{2}$ and $\theta_{2}$ . (Values of $w$ for $0 < x < L_{T}$ are not exact. The approximating field $w = \sum N_{i}d_{i}$ is cubic in $x$ , but the exact field for a uniformly distributed load is quartic in $x$ .)
+
+If the beam is divided into two or more elements of equal length L, moment loads cancel at all interior nodes. Accordingly, lumped loading differs from consistent loading only in that lumped loading omits the clockwise moment $qL^{2}/12$ at the beam tip, thus causing tip deflection and tip rotation to be overestimated. With n the number of equal-length elements, lumped loading yields the following percentage errors in deflection and rotation at the right end.
+<table><tr><td></td><td>n = 1</td><td>n = 2</td><td>n = 3</td><td>n = 4</td></tr><tr><td>Deflection error</td><td>33.3%</td><td>8.3%</td><td>3.7%</td><td>2.1%</td></tr><tr><td>Rotation error</td><td>50.0%</td><td>12.5%</td><td>5.6%</td><td>3.1%</td></tr></table>
+
+Consistent loading produces exact values of end deflection and end rotation for all values of n.
+
+# 4.4 EQUILIBRIUM AND COMPATIBILITY IN THE SOLUTION
+
+In an exact solution, according to the theory of elasticity, every differential element of a continuum is in static equilibrium, and compatibility prevails everywhere. An approximate finite element solution does not fulfill these requirements in every sense. In the present section we note the extent to which equilibrium and compatibility conditions may be satisfied at nodes, across interelement boundaries, and within individual elements.
+
+1. Equilibrium of nodal forces and moments is satisfied. The structural equations $\{R\} - [K]\{D\} = \{0\}$ are nodal equilibrium equations. Therefore, the solution vector $\{D\}$ is such that nodal forces and moments have a zero resultant at every node.   
+2. Compatibility prevails at nodes. Loosely speaking, elements connected to one another have the same displacements at the connection point. More precisely, elements are compatible at nodes to the extent of nodal d.o.f. they share. The latter statement allows the modeling of a physical hinge or roller between adjacent nodes that would otherwise be fully connected; one then connects some but not all nodal d.o.f. For example, if adjacent beam elements meet at a node where they share only translational d.o.f., a hinge connection is created.
+
+<!-- source-page: 145 -->
+
+![](images/page-145_67d1683ebb6fbf1db5c7166743e18aba83615617d203d9cf97127da2d480dd31.jpg)
+
+<details>
+<summary>flowchart</summary>
+
+```mermaid
+graph TD
+    1 --> 2
+    2 --> 3
+    3 --> 4
+    1 -->|1| 2
+    2 -->|2| 3
+    3 -->|σx2| 4
+    4 -->|u4| 3
+```
+</details>
+
+Figure 4.4-1. Differential element (shaded) that spans an interelement boundary.
+
+![](images/page-145_3142e514b8a1cc9791e51176488820ff5664a3c57eb88df9b3b4ea5944006039.jpg)
+
+<details>
+<summary>text_image</summary>
+
+2
+4
+6
+y,v
+x,u
+1
+3
+5
+v̄
+v̄
+</details>
+
+Figure 4.4-2. Adjacent incompatible plane elements. All nodes but 5 and 6 have zero displacement.
+
+3. Equilibrium is usually not satisfied across interelement boundaries. Figure 4.4-1 provides a simple example. Imagine that the elements are constant-strain triangles (Eq. 4.2-9) and that node 4 is the only node displaced, as shown. Then $\sigma_{x2}$ is the only nonzero stress, and the shaded differential element is not in equilibrium. Other types of finite elements behave similarly. (Interelement continuity of stresses may be displayed by elements that include strains among their nodal d.o.f., but such elements are not commonly used.)
+
+When examining finite element stress output, one should not expect that stresses in adjacent elements will be the same along the common edge, that stresses at nodes will be the same in all elements that share the node, or that stress boundary conditions will be exactly satisfied (e.g., in Fig. 1.1-2b we will not compute precisely $\sigma_{x} = \tau_{xy} = 0$ along the right edge). For a properly constructed mesh these discrepancies will be small and will become smaller with mesh refinement.
+
+4. Compatibility may or may not be satisfied across interlement boundaries. Interelement compatibility is satisfied by all elements we have discussed thus far. For example, with both the constant-strain triangle and the plane bilinear element (both discussed in Section 4.2), compatibility is guaranteed because element sides remain straight even after the element is deformed. (That sides remain straight can be shown by noting that edges are initially straight and displacements along a side are linear functions of the coordinates.)
+
+Other elements, not yet discussed, may be incompatible. Several successful plate elements are incompatible in rotation about an interlement boundary. Figure 4.4-2 is another case in point. Stretching of the right edge causes vertical edges of the right element to bend, and the interelement gap (shaded) appears. (This element, called either incompatible or nonconforming, is discussed in Section 8.3.)
+
+Incompatibilities between elements should tend toward zero as more and more elements are used to model a structure. Indeed, this must happen if an incompatible element is to be considered reliable enough for general use.
+
+5. Equilibrium is usually not satisfied within elements. In the absence of body forces, the differential equations of equilibrium (Eqs. 1.6-2) are exactly satisfied by the constant-strain triangle (Eq. 4.2-9) but not by the bilinear rec-
+
+<!-- source-page: 146 -->
+
+tangle (Eq. 4.2-12) unless $a_4 = a_8 = 0$ . In general, satisfaction of the differential equations of equilibrium at every point in an element demands a relation among element d.o.f. that usually does not result from solution of the global finite element equations $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{R}\}$ .
+
+If exact results are to be approached as a mesh is refined, then satisfaction of the equilibrium equations must be approached throughout each element. For elements having few d.o.f., this means that a constant-strain condition must be approached within each element. Higher-order elements, such as the quadratic element of Fig. 4.3-4, can satisfy the differential equations of equilibrium while displaying either constant or linear-strain fields. However, an important property of any reliable element is that its displacement field be capable of representing all possible states of constant strain. $^{1}$
+
+6. Compatibility is satisfied within elements. We require only that the element displacement field be continuous and single-valued. These properties are automatically provided by polynomial fields.
+
+# 4.5 CONVERGENCE REQUIREMENTS
+
+If a particular problem is repeatedly analyzed, each time using a finer mesh of elements, we generate a sequence of approximate solutions. How can we be assured that the sequence converges to the theoretically exact result? In the following, requirements for convergence are stated in general terms, then interpreted in terms appropriate to structural mechanics and elements based on displacement fields.
+
+General. Let the field variable be $\phi = \phi(x,y,z)$ , and let there be a functional $\Pi = \Pi(\phi)$ that yields the governing differential equation of the physical problem from the stationary condition $d\Pi = 0$ . Assume that $\Pi$ contains derivatives of $\phi$ through order $m$ . If the exact $\phi$ is to be approached as the mesh is refined, then:
+
+1. Within each element, the assumed field for $\phi$ must contain a complete polynomial of degree $m$ . (Completeness is discussed in Section 3.6.)   
+2. Across boundaries between elements, there must be continuity of $\phi$ and its derivatives through order $m - 1$ .   
+3. Let elements be used in a mesh (rather than tested individually), and let boundary conditions on the mesh be appropriate to a constant value of any of the $m$ th derivatives of $\phi$ . Then, as the mesh is refined, each element must come to display that constant value.
+
+For example, if $\phi = \phi(x, y)$ and $\Pi$ contains first derivatives of $\phi$ , then the lowest-order acceptable field has the form $\phi = a_1 + a_2x + a_3y$ in each element, only $\phi$ itself need be continuous across interelement boundaries, and each element of an appropriately loaded mesh must display a constant value of $\phi_x$ (or of $\phi_y$ for other appropriate loading), at least as the mesh is refined.
+
+<!-- source-page: 147 -->
+
+Requirement 1 ensures that $\phi$ will be continuous within elements and is necessary (but not always sufficient) in order for Requirement 3 to be satisfied.
+
+Requirement 2 is met at all stages of mesh refinement by compatible elements. Incompatible elements must become compatible as the mesh is refined ad infinitum.
+
+Requirement 3, which must be met in the limit of mesh refinement, is also met even in a coarse mesh by most commonly used elements.
+
+The order of differentiation $m$ can be determined by examination of either the governing differential equation or its associated functional $\Pi$ . The correspondence between differential equation and $\Pi$ is as follows: if derivatives of $\phi$ of order $2m$ appear in the differential equation, then derivatives of $\phi$ of order $m$ appear in $\Pi$ . If $2m$ is odd, so that no variational principle exists, one can yet obtain a finite element formulation by weighted residual methods such as the Galerkin method. Then one regards $m$ as the highest-order derivative of $\phi$ to be found in integral expressions used to generate the finite element matrices. Usually these expressions result from integrations by parts that reduce $m$ as much as possible.
+
+Satisfaction of Requirements 1, 2, and 3 guarantees convergence to correct results, but says nothing about accuracy in a coarse mesh or the rate of convergence with mesh refinement. However, if the requirements are met at all stages of mesh refinement, and if each refinement is achieved by dividing the elements of the previous mesh into two or more elements, then convergence is monotonic [4.3]. This manner of subdivision means that each new mesh contains the old mesh, especially in the mathematical sense of having the old trial space embedded in the new.
+
+Structural Mechanics. When elements are based on displacement fields, there is often more than one field required (e.g., fields for both u and v are needed in a plane problem). The order of differentiation, m, is determined from the strain energy term in $\Pi_{p}$ . Sometimes m has two values for one element. A case in point is a thin flat element that must both stretch and bend. Here m = 1 for displacements u and v tangent to the element midsurface and m = 2 for displacement w normal to the element midsurface. Interelement compatibility of u, v, w, $w_{,x}$ and $w_{,y}$ is required, at least as the mesh is refined ad infinitum.
+
+Together, Requirements 1 and 3 say that a mesh of elements must, when given appropriate boundary conditions, display rigid-body motion or a state of constant strain. For the flat element mentioned in the preceding paragraph, we must find that strains $\{\epsilon\} = [B]\{d\}$ are zero when $\{d\}$ represents translation along (or small rotation about) any coordinate axis, and that the mesh gives constant values of strains $\epsilon_{x} = u_{,x}, \epsilon_{y} = v_{,y}$ , and $\gamma_{xy} = u_{,y} + v_{,x}$ when appropriate stretching loads are applied to boundaries of the mesh. When bending or twisting loads are applied, a constant-strain state must be observed in a layer parallel to the element mid-surface, which means that the mesh must be able to display constant curvatures $w_{,xx}$ and $w_{,yy}$ and the constant twist $w_{,xy}$ .
+
+As an example of rigid-body motion, consider the constant-strain triangle, Eq. 4.2-9. The motion $u = a_{1}$ is a translation in the x direction, in which all points have displacement $a_{1}$ . Similarly, $v = a_{4}$ is a translation in the y direction. With $a_{6} = -a_{3}$ , the motion $u = a_{3}y$ and $v = -a_{3}x$ is a small clockwise rigid-body rotation through an angle $a_{3}$ about the point x = y = 0 (Fig. 4.5-1). One easily checks that $\epsilon_{x} = \epsilon_{y} = \gamma_{xy} = 0$ for this rotation.
+
+All elements discussed in Section 4.2 can display rigid-body translation and
+
+<!-- source-page: 148 -->
+
+![](images/page-148_49e84079954531e70a95710a41dd9914486828a43048b7c912cb99b09ead6d0e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+u = a₃y
+y
+a₃
+a₃
+v = -a₃x
+x
+</details>
+
+Figure 4.5-1. Rigid-body rotation through a small angle $a_{3}$ .
+
+![](images/page-148_c23958150da503293a298c550c41926fb746a6b323bc64d28ab6c178939ae26c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+εₓ
+Lₜ
+a₂
+L
+x,u
+s
+q = q(x)
+</details>
+
+Figure 4.5-2. Axial strain $\epsilon_{x}$ in a bar under axial load q. A typical element has length L.
+
+small rigid-body rotation without strain. An example of an element that cannot, at least in a coarse mesh, is an element used to model a shell of revolution—for example, a spherical shell [4.4]. The element displacement field is written in curvilinear coordinates rather than in Cartesian coordinates. When element nodal d.o.f. {d} represent a rigid-body translation along the axis of revolution, this shell element yields zero strains only if the arc subtended by the element meridian approaches zero. As the mesh is refined, the arc subtended by each element is defined by the convergence to correct results is obtained.
+
+A physical explanation of Requirement 3 can be given with reference to Fig. 4.5-2. The structure is modeled by standard bar elements whose displacement field is of the form $u = a_1 + a_2s$ and whose strain is therefore $\epsilon_x = a_2$ . If $L_T >> L$ , the actual strain distribution over length $L$ departs very little from the constant value $\epsilon_x = a_2$ . Clearly, as $L$ shrinks, the actual curve can be matched arbitrarily closely in stairstep fashion. One could not converge to an arbitrarily close match by using an element based on the field $u = a_1 + a_2s^2$ . Here $\epsilon_x$ is not constant; it is $\epsilon_x = 2a_2s$ , for which $\epsilon_x = 0$ at the left end of every element. Such a defect is not correctible by mesh refinement.
+
+Requirement 3 is stated in terms of a mesh of elements rather than in terms of a single element for the following reason. It is possible for an element to be based on a polynomial field that contains constant-strain terms, yet fail to display constant strain when a mesh of arbitrarily shaped elements is appropriately supported and loaded. None of the elements discussed in Section 4.2 exhibits this difficulty.
+
+Most elements meet Requirement 3 when used as a coarse mesh, but some elements require mesh refinement. This matter is discussed in Section 4.6, in connection with the “weak” patch test.
+
+Geometric Isotropy. Computed results for a given structure should not depend on how the mesh is oriented in global coordinates. For example, the computed displacement of load P in Fig. 4.5-3 should be the same whether the element has orientation (a) or orientation (b). Elements that are well behaved in this regard are called geometrically isotropic (also called geometrically invariant and spatially isotropic). Although geometric invariance is not required for convergence with mesh refinement, it is very desirable that elements have no “preferred directions” so that the user will not encounter results that seem puzzling or even alarming.
+
+If a plane element is to be geometrically isotropic, it is not a displacement expansions for $u$ and $v$ have the same form and include terms sym-
+
+<!-- source-page: 149 -->
+
+![](images/page-149_5313d5466b3f688e21f311388a0be7d8526c02809b07dee2fbe20992ecc78588.jpg)  
+Figure 4.5-3. A rectangular plane element, loaded by force P at one corner.
+
+metric about the axis of a Pascal triangle (Fig. 3.6-1). The constant-strain triangle (Eq. 4.2-9) and the bilinear element (Eq. 4.2-12) both satisfy these requirements. In particular, to produce a bilinear element one should not supplement the complete linear fields of Eq. 4.2-9 with (say) $u = a_4x^2$ and $v = a_8x^2$ , as this would destroy geometric isotropy. One selects instead the modes $u = a_4xy$ and $v = a_8xy$ , which favor neither $x$ nor $y$ . If used in the problem of Fig. 4.5-3, the bilinear element yields the same displacement of load $P$ for orientation (a) and for orientation (b). (We have not yet explained how to deal with arbitrary orientations of this element; see Chapter 6.)
+
+Similar remarks apply to solid elements. The trilinear element, Eq. 4.2-16, uses all linear terms, a balanced selection of quadratic terms, and a single cubic term that favors none of the coordinate directions over another.
+
+# 4.6 THE PATCH TEST
+
+The patch test was originated by Irons [4.5,4.6]. It is a simple test that can be performed numerically, so as to check the validity of an element formulation and its programmed implementation. We assume that the element is stable in the sense described below. Then, if the element passes the patch test; we have assurance that all convergence criteria noted in Section 4.5 are met. Therefore, when this type of element is used to model any other structure, mesh refinement will produce a sequence of approximate solutions that converges to the exact solution. In other words, the patch test serves as a necessary and sufficient condition for correct convergence of a finite element formulation. The test can be described in a general way, but we will describe it in terms appropriate to structural mechanics.
+
+Procedure. One assembles a small number of elements into a “patch,” taking care to place at least one node within the patch, so that the node is shared by two or more elements, and so that one or more interelement boundaries exist. Figure 4.6-1 shows an acceptable two-dimensional patch, built of four-node elements of a type we have not yet discussed. Boundary nodes of the patch are loaded by consistently derived nodal loads appropriate to a state of constant stress. Internal nodes are neither loaded nor restrained. The patch is provided with just
+
+<!-- source-page: 150 -->
+
+![](images/page-150_a1be6868bd2cbc59777b80445ee4eb99dc6855c3cc08d9146788c2a7da9867ac.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+1.5F
+3
+2
+4
+2
+F
+6
+9
+2
+3
+2F
+5
+8
+1
+2
+1
+1
+4
+7
+F
+x,u
+Thickness = 1
+F = 1/4 (4)(1)σc
+F = σc
+σc
+</details>
+
+Figure 4.6-1. Patch of four-node elements, loaded by forces $F$ consistent with the uniform stress state $\sigma_x = \sigma_c$ , $\sigma_y = \tau_{xy} = 0$ .
+
+enough supports to prevent rigid-body motion. One next executes a standard solution and examines the computed stresses. If, throughout the element, computed stresses agree with exact stresses for the physical problem modeled, then the patch test is passed. By “agree” we mean exact agreement, allowing only for physical noise associated with the finite length of computer words.
+
+The patch test must be repeated for all other constant-stress states demanded of the element being tested. In Fig. 4.6-1, a test for constant $\sigma_{x}$ is depicted; we must test the patch again for constant $\sigma_{y}$ , and again for constant $\tau_{xy}$ (each time using appropriate nodal loads). Solid elements must be patch-tested for constant states of $\sigma_{x}$ , $\sigma_{y}$ , $\sigma_{z}$ , $\tau_{xy}$ , $\tau_{yz}$ , and $\tau_{zx}$ . A patch of plate-bending elements must display constant bending moments $M_{x}$ and $M_{y}$ and constant twisting moment $M_{xy}$ .
+
+It is only for convenience that we examine stress states rather than strains. It is typically, a computer program outputs stresses rather than strains. Computed nodal d.o.f. can also be examined; if they are incorrect, strains and stresses will also be incorrect. If displacements are correct but stresses are incorrect, one strain is that the stress calculation subroutine may be in error.
+
+Support conditions must not prevent the constant state from occurring. In Fig. 4.6-1, complete fixity of nodes 1, 2, and 3 would be fatal to the patch test, as Poisson contraction in the y direction would be prevented along the left edge. It would be acceptable to impose the boundary condition $u_{3} = 0$ rather than a load at node 3—that is, to replace the force 1.5F by a roller support.
+
+Stability. At the outset we assumed that the element to be patch-tested is stable. A stable element is one that admits no zero-energy deformation states when adequately supported against rigid-body motion. This matter is discussed in detail in Section 6.12. Unstable elements should be used with caution. They can produce an unstable mesh, whose displacements are excessive and quite unrepresentative of the actual structure.
+
+of the actual structure.
+
+Instabilities can be detected by an eigenvalue test (see Section 18.8). They can also be detected by a “perturbed” patch test, as follows [4.6]. Let the patch be just adequately supported, and add a small amount to one of the consistently derived nodal loads (e.g., apply an additional load equal to an existing load times
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_016.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_016.md
new file mode 100644
index 00000000..2c891d53
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_016.md
@@ -0,0 +1,474 @@
+<!-- source-page: 151 -->
+
+the square root of the round-off limit, $10^{-6}$ for round-offs of order $10^{-12}$ . If computed displacements change a large amount as a result, the patch is unstable. If this test is applied to a single element, one can detect if the element is unstable. (An unstable element does not necessarily produce an unstable mesh.)
+
+"Weak" Patch Test. An element that fails to display constant stress in a patch of large elements has not necessarily failed the patch test. If, as the mesh is repeatedly subdivided, elements come to display the expected state of constant stress, then the element is said to have passed a "weak" patch test, and convergence to correct results is assured.
+
+For example, a certain incompatible plane element fails the patch test if of arbitrary quadrilateral shape, but passes if each element of the patch is a parallelogram. If a mesh of arbitrary quadrilaterals is subdivided again and again, the newly created elements approach parallelograms in shape. Thus the element is found to pass a weak patch test. This result might be anticipated by applying the standard patch test to a mesh of parallelograms, Fig. 4.6-2, using nodal loads computed by means of Eqs. 1.6-3 and 4.1-6.
+
+As another example, consider an axially symmetric geometry. One uses elements whose strain fields include the circumferential strain $\epsilon_{\theta}=u/r$ (Fig. 4.6-3). As a mesh is refined, h/a and $\theta_{e}$ become small for each element. Accordingly, one applies the weak patch test to see whether $\epsilon_{\theta}$ becomes constant over each element as h/a and $\theta_{e}$ approach zero.
+
+Remarks. It is possible for an element that fails the patch test to give better answers in a coarse mesh than an equal number of elements that pass. Nevertheless, an element that fails cannot be trusted. An element that fails may provide convergence, but may converge to an incorrect result. This behavior is observed in certain plate-bending problems when the elements used do not permit a state of constant twist.
+
+“Higher-order” patch tests are possible. For example, a plane element whose displacement field includes a complete quadratic expansion for u and v (Fig. 3.6-1) should be able to represent exactly a field of pure bending. “Robustness” can also be checked [4.6]. For example, Poisson’s ratio v should have no effect on pure bending of a plane mesh; by computation, one discovers to what extent an element is insensitive to v.
+
+Apart from their use in testing elements, patch tests provide good example problems for learning to use an unfamiliar computer program.
+
+![](images/page-151_6a2bd2705f8edab30d4c97b1fdf5bf789450e2452128b069c18268df4e926144.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Pure geometric line drawing of a 3D geometric structure with no text, numbers, or symbols
+</details>
+
+Figure 4.6-2. Patch of parallelogram elements, ready for application of patch-test loadings.
+
+![](images/page-151_ad80be4d7891cde0a0c051500e96624abb19fe9df252c825ff5bf418e7a3810b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+θ,v
+θe
+r,u
+4 3
+1 2
+s
+a h
+</details>
+
+Figure 4.6-3. A plane element in polar coordinates. Displacements u and v are respectively radial and circumferential.
+
+<!-- source-page: 152 -->
+
+# 4.7 STRESS CALCULATION
+
+Stress $\{\sigma\}$ in an element can be calculated when its nodal d.o.f. $\{\mathbf{d}\}$ are known. These d.o.f. are available after the structural equations $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{R}\}$ have been solved. Equation 1.7-8, repeated here, is
+
+$$
+\{\boldsymbol {\sigma} \} = [ \mathrm{E} ] (\{\boldsymbol {\epsilon} \} - \{\boldsymbol {\epsilon} _ {0} \}) + \{\boldsymbol {\sigma} _ {0} \} \tag {4.7-1}
+$$
+
+in which mechanical strains $\{\epsilon\} = [B]\{d\}$ are produced by displacements of the nodes. Typically, $\{\sigma_{0}\}$ is omitted and $\{\epsilon_{0}\}$ is used to account for thermal strains. Thus, in a plane problem with isotropic material, with T the temperature relative to a stress-free temperature and $\alpha$ the coefficient of thermal expansion,
+
+$$
+\left\{ \begin{array}{l} \sigma_ {x} \\ \sigma_ {y} \\ \tau_ {x y} \end{array} \right\} = \frac {E}{1 - \nu^ {2}} \left[ \begin{array}{c c c} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac {1 - \nu}{2} \end{array} \right] \left([ \mathbf {B} ] \{\mathbf {d} \} - \left\{ \begin{array}{l} \alpha T \\ \alpha T \\ 0 \end{array} \right\}\right) \tag {4.7-2}
+$$
+
+Matrix [B] is a function of the coordinates and must be evaluated at the location in the element where stresses are desired.
+
+The calculation $\{\epsilon\} = [\mathbf{B}]\{\mathbf{d}\}$ involves differentiation of the displacement. Accordingly, one expects that stresses will be less accurate than displacements. In low-order elements stresses are often most accurate at the element centroid, less accurate at midsides, and least accurate at corners. Elements of higher order usually display multiple points of optimal accuracy for stresses. The locations of these points depend on the element geometry and the displacement field, and can often be predicted before doing numerical calculations. Stresses at other locations are usually best found by extrapolation from the optimal points (Section 6.13).
+
+The most commonly used elements display only the minimum of interelement continuity. $C^0$ elements are popular for determining displacements in plane and solid elasticity, and $C^1$ elements are popular for determining displacements and slopes in plate bending. Stresses depend on displacement derivatives in $C^0$ elements (or on curvatures in commonly used $C^1$ elements). Therefore, unless a state of constant stress prevails, stresses are discontinuous across boundaries between $C^0$ elements, and $C^0$ elements that share a node do not all display the same state of stress at that node. A substantial difference in stress between elements suggests a need for mesh refinement.
+
+The average stress at a node is more to be trusted than the nodal stress in any one element attached to the node. A weighted nodal average—for example, with each element contribution weighted in proportion to its interior corner angle at the shared node—may be more reliable than a simple nodal average. However, averaging of stresses on either side of a physical discontinuity such as a step change in thickness should be avoided. Nodal average stresses may be used by a postprocessor in graphic display of results. A caution: a smooth stress field produced by a postprocessor may hide large stress differences in adjacent elements that indicate a need for mesh refinement.
+
+Imagine that the average strain along a line joining any two nodes is useful. The following trick is useful. Connect a straight bar element to the two nodes, but make its cross-sectional area very small, so that its stiffness is negligible in
+
+<!-- source-page: 153 -->
+
+comparison with the stiffness of the existing structure. The computer program will calculate the bar stress, which the user can divide by E to obtain the strain desired. Effectively, we have attached a strain gage to the structure.
+
+Body Forces and Surface Tractions. In the absence of $\{\epsilon_{0}\}$ and $\{\sigma_{0}\}$ , stress is due entirely to mechanical loads. Imagine that the mechanical load is entirely body force, as in the axially loaded bar of Fig. 4.7-1. If a single bar element is used, both of its nodes are fixed, and we compute $\{d\}=\{0\}$ and $\{\sigma\}=\{0\}$ . Two-element and four-element models yield increasingly better results, as expected. The exact stress variation can be obtained from any of these models if the effect of body force within each separate element is taken into account. For example, with a one-element model of weight W, we add the stress
+
+$$
+\sigma_ {y} = \frac {W}{A} \left(\frac {y}{L _ {T}} - \frac {1}{2}\right) \tag {4.7-3}
+$$
+
+to the stresses $\{\sigma\} = [\mathbf{E}][\mathbf{B}]\{\mathbf{d}\}$ , which are zero for a one-element model. A similar formula can be written for $\sigma_y$ in an arbitrary element of a multi-element model.
+
+Analogous refinement can be made when calculating bending moments in a beam element. The bending moment $M = EI[B]\{d\}$ can be augmented by the bending moment produced by lateral loads on an element whose ends are completely fixed. This refinement may be worthwhile because beams are frequently analyzed and their loadings are comparatively easy to categorize and incorporate in a computer program.
+
+For other elements, such adjustments are ignored in practice. The adjustments are less important for plane and solid elements than for beam elements. Moreover, they are not easily formulated for an element of general shape.
+
+Thermal Stress. As a usual rule, for analysis of thermal stress one should construct a finite element model that permits the strain field to have about the same level of complexity as seen in the temperature field. Thus, one avoids a mismatch between $\{\epsilon\}$ and $\{\epsilon_{0}\}$ in Eq. 4.7-1. A mismatch can produce an unreliable stress prediction, especially among simpler elements, as the following example shows.
+
+![](images/page-153_84fa856d476e38414190868b1ea3c0e7c103de3f5ef542f7d405e847bf9ecd21.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+Gravity
+L_T
+x
+</details>
+
+(a)
+
+![](images/page-153_58599379639062496e9ec9b67eb19a3df907185602d8353f0c18b092a7f7ea15.jpg)
+
+<details>
+<summary>text_image</summary>
+
+-σe
++σ
+</details>
+
+(b)
+
+![](images/page-153_35a149ce7f64c01b562f83556809d00433237eff7897cde2b41af906c0ed4a5c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+r_c + \frac{\sigma_c}{2}
+-\frac{\sigma_c}{2} - \frac{3\sigma_c}{4}
+</details>
+
+(c)
+
+![](images/page-153_1ae618be65a01eb241ef11b665aaeea8f0bd3677c8e2287984358ef4710c3fd5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+- \frac{\sigma_c}{4} + \frac{3\sigma_c}{4}
++ \frac{\sigma_c}{4}
+</details>
+
+(d)   
+Figure 4.7-1. (a) Uniform bar of weight W and cross-sectional area A, fixed at both ends. (b) Axial stress variation, where $\sigma_{c} = W/2A$ . (c,d) Stresses predicted by two-element and four-element models.
+
+<!-- source-page: 154 -->
+
+![](images/page-154_71b4179ce9051244dbc84d5399ff09de2d2f91fb74cf438dab13523c9c27d549.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+3
+x,u
+1
+2
+</details>
+
+Figure 4.7-2. A triangular element.
+
+Let the triangular element of Fig. 4.7-2 have the linear temperature field
+
+$$
+T = c _ {1} x + c _ {2} y \tag {4.7-4}
+$$
+
+where $c_{1}$ and $c_{2}$ are constants. If the centroid of the triangle is at $x = y = 0$ , and if the element is an isotropic constant-strain triangle, then Eq. 4.1-6 shows that nodal loads $\{\mathbf{r}_e\}$ produced by the temperature field are all zero. Therefore, $\{\mathbf{d}\} = \{\mathbf{0}\}$ , and, by Eq. 4.7-2,
+
+$$
+\left\{ \begin{array}{l} \sigma_ {x} \\ \sigma_ {y} \\ \tau_ {x y} \end{array} \right\} = - \frac {E \alpha}{1 - \nu} \left(c _ {1} x + c _ {2} y\right) \left\{ \begin{array}{l} 1 \\ 1 \\ 0 \end{array} \right\} \tag {4.7-5}
+$$
+
+These stresses can be large. However, they are spurious: theory of elasticity shows that a linear temperature field produces deformation of the triangle but zero stresses. Equation 4.7-5 would indeed yield zero stresses if evaluated at the centroid of the triangle, $x = y = 0$ . This amounts to using a constant temperature field in the constant-strain triangle when calculating stresses. Generalizing, we infer that, for the purpose of stress analysis, one should reduce (if necessary) the order of the element temperature field to the same order as the element strain field. Thus, if Fig. 4.7-2 is regarded as a quadratic triangle (three corner nodes and three midside nodes), the element has a linear strain field. The temperature field of Eq. 4.7-4 is also linear, so the element correctly yields $\sigma_x = \sigma_y = \tau_{xy} = 0$ .
+
+$\tau_{xy} = 0$ . As a counterexample [4.7, 4.8], imagine that a uniform beam is fixed between rigid walls (e.g., for a one-element model, set $u_{1} = u_{2} = u_{3} = u_{4} = 0$ in Fig. 4.2-4). Let $\nu = 0$ and let the temperature vary linearly with distance $y$ from the midsurface of the beam, $T = T_{0}y$ . Use of the actual temperature field $T = T_{0}y$ yields $\sigma_{x} = \sigma_{y} = -E\alpha T$ , which is correct for $\sigma_{x}$ but wrong for $\sigma_{y}$ . Use of the reduced temperature field ( $T = 0$ in this case) yields $\sigma_{x} = \sigma_{y} = 0$ , which is wrong for $\sigma_{x}$ but correct for $\sigma_{y}$ .
+
+for $\sigma_{x}$ but correct for $\sigma_{y}$ . Accordingly, we cannot say whether it is always better to accept a complicated temperature field or to smooth it in element by element fashion, as neither strategy will be best in all cases [4.7, 4.8].
+
+Iterative Improvement. Computed stresses can be iteratively improved as follows [4.9, 4.10]. First, solve the problem in standard fashion, and apply Eq. 4.7-1 to obtain stresses in each element. Next, at each node, use stresses from the sur-
+
+<!-- source-page: 155 -->
+
+rounding elements to produce nodal average stresses. In a typical element, interpolate to find stresses $\{\sigma\}$ within the element,
+
+$$
+\{\sigma \} = [ N ] \{\overline {{\sigma}} \} \tag {4.7-6}
+$$
+
+where [N] is a shape function matrix and $\{\overline{\sigma}\}$ contains nodal average stresses for the element at hand. Terms may be added to the right-hand side of Eq. 4.7-6 in order to involve stresses within the element as well [4.10]. Known stress boundary conditions can be accounted for in Eq. 4.7-6. We recall from Eq. 4.1-6 that the integral of $[B]^{T}\{\sigma_{0}\}$ is a vector of nodal loads $\{r_{e}\}$ that is statically equivalent to the stress distribution $\{\sigma_{0}\}$ . If, instead of $\{\sigma_{0}\}$ , we use the total stress $\{\sigma\}$ from Eq. 4.7-6, the sum of nodal loads $\{r_{e}\}$ over all numel elements of the structure should be statically equivalent to the entire structure load vector $\{R\}$ . So we write
+
+$$
+[ \mathbf {K} ] \{\Delta \mathbf {D} \} = \{\mathbf {R} \} - \sum_ {n = 1} ^ {\text {numel}} \left(\int_ {V _ {e}} [ \mathbf {B} ] ^ {T} \{\boldsymbol {\sigma} \} d V\right) _ {n} \tag {4.7-7}
+$$
+
+where $\{\sigma\}$ comes from Eq. 4.7-6. If $\{\sigma\}$ is exact, the right-hand side of Eq. 4.7-7 is zero. Otherwise, it is a load imbalance that drives the solution toward a configuration that reduces the imbalance. We compute increments $\{\Delta D\}$ from Eq. 4.7-7. The new configuration is $\{D\}_{new} = \{D\}_{old} + \{\Delta D\}$ . New stresses are computed based on $\{D\}_{new}$ , Eqs. 4.7-6 and 4.7-7 are applied to the new stresses, and another $\{\Delta D\}$ is determined. The process repeats until convergence. Note that Eq. 4.7-7 does not require repeated construction and reduction of [K].
+
+Example applications appear in Fig. 4.7-3 and Table 4.7-1. The four-node elements used are described in Section 6.3. Symmetry is exploited by analyzing only one quadrant of the object. Stress at point A is obtained by extrapolation of stresses at four integration stations in the adjacent element. The “standard method” in Table 4.7-1 is the usual single-pass analysis.
+
+Stress Concentrations. A stress raiser, such as a hole or a notch, might be analyzed by a brute-force approach—that is, by using a profusion of elements to surround
+
+![](images/page-155_b51dee29973ee2a38882ba452a6af36ccfd8e43e650410e470d0e214e2bbd219.jpg)
+
+<details>
+<summary>text_image</summary>
+
+σ
+A
+(a)
+σ
+</details>
+
+![](images/page-155_dc3b81139e2f407cbcbd75837572fb7891d726bd6373b23f5593b57333321fe6.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P ← A → P
+(b)
+</details>
+
+![](images/page-155_944315bec1d313899f0b420660ddfe89013dee9656a8a3064a28a1ebcb3ce88a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+2 by 2
+mesh
+(c)
+</details>
+
+Figure 4.7-3. Flat plate with a central hole of diameter equal to half the plate width. (a) Uniform tensile load on portions distant from the hole. (b) Point forces on a square plate. (c) Typical mesh used on quadrant shaded in parts a and b.
+
+<!-- source-page: 156 -->
+
+TABLE 4.7-1. HORIZONTAL STRESS AT POINT A IN FIG. 4.7-3 USING THE STANDARD METHOD AND THE METHOD OF EQ. 4.7-7 (SIX ITERATIONS) [4.12]. 
+
+<table><tr><td rowspan="2">Loading Case</td><td colspan="3">2 by 2 Mesh in Quadrant</td><td colspan="3">4 by 4 Mesh in Quadrant</td></tr><tr><td>Standard</td><td>Six Iterations</td><td>Exact</td><td>Standard</td><td>Six Iterations</td><td>Exact</td></tr><tr><td>Fig. 4.7-3a</td><td>30.9</td><td>38.1</td><td>43.2</td><td>36.6</td><td>39.5</td><td>43.2</td></tr><tr><td>Fig. 4.7-3b</td><td>73.6</td><td>85.4</td><td> $112.3^a$ </td><td>95.3</td><td>96.9</td><td> $112.3^a$ </td></tr></table>
+
+$^{a}$ From an 8 by 8 mesh of nine-node elements.
+
+the discontinuity. This is both tedious and expensive. An alternative that uses a coarse mesh may be available, as follows [4.13].
+
+Analyze the problem of interest using the coarsest mesh that will model the geometry. Let the computed stress at the point of interest be called $\sigma_{a}$ . Next, use a locally identical mesh to analyze the most closely related condition for which results are already available—for example, in a table of stress concentration factors. For this “secondary” case, call the computed stress $\sigma_{r}$ and the tabulated stress $\sigma_{t}$ . The ratio $\sigma_{t}/\sigma_{r}$ is regarded as a correction factor that can be applied to closely related geometries. Thus the stress in the case of interest is now estimated to be $(\sigma_{t}/\sigma_{r})\sigma_{a}$ rather than $\sigma_{a}$ .
+
+to be $(\sigma_t / \sigma_r)\sigma_a$ rather than $\sigma_a$ . The success of this method depends on the availability of tabulated cases and on the skill of the analyst in selecting an appropriate secondary case.
+
+As an example, consider the determination of the peak stress at A in the point-loaded plate of Fig. 4.7-3b. As the secondary case we take the tensile strip of Fig. 4.7-3a. The tabulated stress concentration factor for the tensile strip is 2.16, which yields the stress at point A as 43.2 for the load applied. Thus, from the 2 by 2 and 4 by 4 mesh results in Table 4.7-1, respectively, we estimate that the stress at point A in the point-loaded plate is
+
+$$
+\frac {4 3 . 2}{3 0 . 9} 7 3. 6 = 1 0 3 \quad \text { and } \quad \frac {4 3 . 2}{3 6 . 6} 9 5. 3 = 1 1 2 \tag {4.7-8}
+$$
+
+Both of these results are better than the results given by six iterations in Table 4.7-1.
+
+4.7-1.
+If the case of interest and the secondary case are identical, one of course obtains the tabulated stress. This means only that a finite element analysis is unnecessary, except perhaps as used for the entire structure in order to obtain the load applied to the portion of interest or the average stress field in a certain region (e.g., to obtain stress $\sigma$ in Fig. 4.7-3a).
+
+# 4.8 OTHER FORMULATION METHODS
+
+The present chapter is devoted to elements whose properties are based on assumed displacement fields. This type of element is the most popular. The reader should be aware that several other element types are possible and are in use. If an alternatively derived element has displacement d.o.f., it can be used in combination with displacement-based elements. Indeed, the user of an analysis program may be unaware that some elements in the program are not displacement-based.
+
+<!-- source-page: 157 -->
+
+Displacement fields and strain energy expressions provide only one of many approaches to formulation of finite elements.
+
+A type of “hybrid” element makes use of an assumed stress distribution within the element and assumed displacements along its edges. The formulation procedure yields an element with displacement d.o.f. A “mixed” element has both displacement d.o.f. and force d.o.f. Its characteristic matrix is not a stiffness matrix; rather, it contains a flexibility submatrix and a submatrix that couples the displacement and force d.o.f. Both hybrid elements and mixed elements can be formulated from appropriate functionals.
+
+For some physical problems, such as certain problems in fluid mechanics, no variational principle exits. Finite element formulations can yet be developed by weighted residual methods, chief among which is the Galerkin method.
+
+# PROBLEMS
+
+# Section 4.1
+
+4.1 If element d.o.f. are given virtual (i.e., small imaginary) displacements $\{\delta \mathbf{d}\}$ , strains are changed in the amount $\{\delta \boldsymbol{\epsilon}\} = [\mathbf{B}]\{\delta \mathbf{d}\}$ . Loads acting on the structure do work that is stored as the strain energy $\delta U = \int \{\delta \boldsymbol{\epsilon}\}^T \{\boldsymbol{\sigma}\} dV$ . Complete this virtual work argument to obtain the structure equations $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{R}\}$ . In the process, identify formulas for $[\mathbf{k}]$ and $\{\mathbf{r}_e\}$ .   
+4.2 Derive the stiffness matrix of a bar, Fig. 4.2-1, by use of the $a$ -basis method, Eqs. 4.1-17 and 4.1-18.
+
+# Section 4.2
+
+4.3 The three-node bar element shown is uniform and is allowed only axial displacement u. Its nodal d.o.f. are $u_{1}$ , $u_{2}$ , and $u_{3}$ . Its shape functions are given by Eq. 4.3-10. Determine the element stiffness matrix.
+
+![](images/page-157_3c8e734de4aa5e9911a100917cdaeb31b87ad0a0f81e67cd117add408864b832.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+L/2 L/2
+1 2 3 x,u
+→ u₁ → u₂ → u₃
+</details>
+
+Problem 4.3
+
+![](images/page-157_774aeca5047f5f92ff34aa19cb8d023c1611b0acc9e3de6ee1fa66aaf664922d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+L
+v₂
+u₂
+2
+u₁
+1
+β
+x,u
+v₁
+</details>
+
+Problem 4.4
+
+4.4. For the plane truss element shown, write an expression for axial displacement as a function of the four nodal d.o.f., angle $\beta$ , and an axial coordinate along the bar. Use this expression to determine the 4 by 4 element stiffness matrix. Check your result against Eq. 2.4-3.   
+4.5 Imagine that a beam element has positive directions for nodal d.o.f. and nodal load as shown in the sketch, rather than the directions shown in Fig. 4.2-2. How does this sign convention change [k] of Eq. 4.2-5? What is awkward about this sign convention?
+
+<!-- source-page: 158 -->
+
+![](images/page-158_62541c40ebec0c067c156ead45aa735d9a9cbc1c0288c43a78648f398e951943.jpg)
+
+<details>
+<summary>text_image</summary>
+
+w₁,F₁
+θ₁,M₁
+w₂,F₂
+θ₂,M₂
+L
+</details>
+
+Problem 4.5
+
+4.6 (a) Verify the correctness of the beam element stiffness matrix, Eq. 4.2-5. Use either Eq. 4.2-5 or Eq. 4.2-7.
+
+(b) Verify that $[\mathbf{B}]\{\mathbf{d}\}$ and $[\mathbf{k}]\{\mathbf{d}\}$ are both zero for a small rigid-body rotation of the beam element about its left end.
+
+4.7 Imagine that a pin-jointed plane truss is modeled by plane frame elements. If rotational d.o.f. $\theta$ are suppressed at all nodes, is the truss correctly modeled? Explain.
+
+4.8 (a) Imagine that the uniform bar element of Fig. 4.2-1 is to have two d.o.f. at each node—namely, axial displacement $u$ and axial strain $\epsilon_x$ . Thus $\{\mathbf{d}\} = \left[u_1 \quad \epsilon_{x1} \quad u_2 \quad \epsilon_{x2}\right]^T$ . Derive the 4 by 4 element stiffness matrix. (b) Should this element be used to model a bar that has abrupt changes in cross section, as in Fig. 1.1-1b? Explain.
+
+4.9 Consider a plane grillage, which consists of bars that all occupy the $xy$ plane. Let each bar be uniform and parallel to either the $x$ axis or the $y$ axis. Each bar can resist bending, in either the $yz$ or the $zx$ plane, and torsion about its axis. Write the 6 by 6 stiffness matrix of an element that lies parallel to the $y$ axis. Express your answer in terms of $E$ , $I_x$ , $G$ , $J$ , and element length $L$ . Indicate the d.o.f. on which [k] operates, arrayed in order consistent with [k].
+
+4.10 The cantilever beam shown is modeled by an extremely coarse mesh of constant-strain triangles. The beam is loaded by an end moment. Qualitatively plot the variation of $\sigma_{x}$ versus $x$ along the $x$ axis.
+
+4.11 (a) For the particular triangular element shown, evaluate the strain-displacement matrix [B] in terms of dimensions $a$ and $b$ . Let d.o.f. in $\{\mathbf{d}\}$ have the ordering $\lfloor u_1, v_1, u_2, v_2, u_3, v_3 \rfloor^T$ .
+
+(b) Let $\{\mathbf{d}\}$ contain displacements consistent with Eq. 4.2-9—that is, $u_{1} = a_{1}, u_{2} = a_{1} + a_{2}a$ , and so on. Show that $[\mathbf{B}]\{\mathbf{d}\}$ yields the strains of Eq. 4.2-10.
+
+4.2-10. (c) Determine [k] in terms of $a, b, E$ , and $t$ if thickness $t$ is constant and Poisson's ratio is zero.
+
+(d) Fix nodes 1 and 2, apply a $y$ -direction force $P$ to node 3, and solve for $u_{3}$ and $v_{3}$ . Also compute stresses in the element.
+
+![](images/page-158_2d8d9bd0988519ea7979021a3c5cc673156c88baaa3836b5f9d566246300e47e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+c
+x
+F
+M = 2cF
+c
+F
+</details>
+
+Problem 4.10
+
+![](images/page-158_9a247e78b9eb3da32aea79dc053d6808b3b27db56c174ef17931e6e1c8df485d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+a
+3
+b
+1
+2
+x,u
+</details>
+
+Problem 4.11
+
+<!-- source-page: 159 -->
+
+![](images/page-159_f3d36d4ff7d92232c032cfe654ad2373584e6281d75ff536c0567125fd245481.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y, v
+E = 10^7
+v = 0
+t = 1
+4
+3
+j
+x, u
+1
+2
+2
+</details>
+
+![](images/page-159_0be2b4a9749584c0ea7d5ad818930d4f8ebe19c8b99d9a8624c65356a80fc551.jpg)
+
+<details>
+<summary>text_image</summary>
+
+20
+31
+j
+19
+30
+</details>
+
+Problem 4.12
+
+4.12 Element $j$ is of the type shown in Fig. 4.2-4. It has stiffness matrix $[\mathbf{k}]$ and nodal displacement vector $\{\mathbf{d}\} = \lfloor u_1, v_1, u_2, v_2, \ldots, v_4 \rfloor^T$ . Element $j$ is to be attached to nodes 19, 20, 30, and 31 of the structure, a fragment of which is shown. The structure stiffness matrix $[\mathbf{K}]$ is to be stored in full (not banded) format, and no d.o.f. are as yet removed by imposing support conditions. What is the numerical contribution of element $j$ to the single coefficient in $[\mathbf{K}]$ at the intersection of
+
+(a) row 48 and column 39?   
+(b) row 37 and column 37?   
+(c) row 59 and column 61?
+
+4.13 Consider the plane bilinear element of Fig. 4.2-4 and the linear displacement field $u = a_{1} + a_{2}x + a_{3}y$ , $v = a_{4} + a_{5}x + a_{6}y$ . Let nodal d.o.f. $\{\mathbf{d}\}$ be consistent with this field; that is, let $u_{1} = a_{1} - a_{2}a - a_{3}b$ , and so on.
+
+(a) Show that Eq. 4.2-13 yields the given $u$ and $v$ fields.   
+(b) Show that Eq. 4.2-14 yields the constant-strain state $\epsilon_{x} = a_{2}, \epsilon_{y} = a_{6}$ , and $\gamma_{xy} = a_{3} + a_{5}$ .
+
+4.14 The best approximation of a state of pure bending that a plane bilinear element can display is $u = \overline{u}xy / ab$ and $v = 0$ , where $\overline{u}$ is the magnitude of a corner displacement.
+
+(a) Sketch the deformed element. Show nodal forces $F$ that produce the deformation state.   
+(b) If Poisson's ratio is taken as zero, strain energy per unit volume is $(E\epsilon_x^2 + E\epsilon_y^2 + G\gamma_{xy}^2)/2$ . Use this information to determine $F$ as a function of $a, b, E, G, \overline{u}$ , and element thickness $t$ .   
+(c) What is the correct value of $F$ , according to elementary beam theory?   
+(d) In parts (b) and (c), one can define a stiffness measure $S$ as $S = F / \overline{u}$ . Give a physical explanation as to why $S_{(b)} > S_{(c)}$ .   
+(e) Show that the ratio $S_{(b)} / S_{(c)}$ approaches unity only as $a / b$ approaches zero. What happens as $a / b$ becomes large?
+
+4.15 For the rectangular solid element of Fig. 4.2-5, write out the first three columns of [N] and the first three columns of [B].
+
+4.16 The block of material shown is loaded by axial force $P = \sigma_c bt$ , which produces axial deflection $D$ . Axial stiffness is $k = P / D$ .
+
+(a) Show that $k$ is inversely proportional to $L$ if cross-sectional area $A = bt$ remains constant.   
+(b) Show that $k$ is independent of $b$ and $L$ if $t$ remains constant and the aspect ratio $b / L$ is not changed.
+
+<!-- source-page: 160 -->
+
+![](images/page-160_2e25f14175c3fdcfd075ff6c33bb9d26fcc524a73330dc565ad42c31874a9f04.jpg)
+
+<details>
+<summary>text_image</summary>
+
+b
+L
+t
+σc
+</details>
+
+Problem 4.16
+
+(c) Show that $k$ is directly proportional to a linear dimension if the shape of the element is not changed. (These behaviors are in fact observed in axial, plane, and solid elements, respectively.)
+
+# Section 4.3
+
+4.17 Show that integrals containing $\{\epsilon_0\}$ and $\{\sigma_0\}$ in Eq. 4.1-6 yield nodal loads $\{\mathbf{r}_e\}$ that are self-equilibrating—that is, they sum to zero net force on an element. For simplicity, restrict your argument to a $C^0$ element, such as a plane bilinear element.   
+4.18 Write a work equation analogous to Eq. 4.3-3, but include $\{\epsilon_0\}$ and $\{\sigma_0\}$ rather than $\{\Phi\}$ . Interpret the result: that is, can loads $\{\mathbf{r}_e\}$ still be called work-equivalent?   
+4.19 (a) A two-node bar element is uniformly heated an amount $T$ . The cross-sectional area of the bar varies linearly from $A_{1}$ at $x = 0$ to $A_{2}$ at $x = L$ . What loads $\{\mathbf{r}_e\}$ are predicted by Eq. 4.1-6? Continue to use the linear shape functions given in Fig. 4.3-2. (b) Use the methods of elementary mechanics of materials to compute the forces this bar would apply to rigid walls if the bar were placed between the walls and uniformly heated an amount $T$ . Why do these forces differ from loads $\{\mathbf{r}_e\}$ of part (a)?   
+4.20 A three-node bar element is subjected to a temperature change that varies linearly with $x$ , as shown. What axially directed nodal loads appear? Shape functions are given by Eq. 4.3-10.
+
+![](images/page-160_dcf70afc7cc3d8debb07366e32cc89b61eb89478b837aeda570ef211d9109f50.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+T₃
+L/2
+L/2
+3
+x
+</details>
+
+Problem 4.20
+
+4.21 Let temperature $T$ in the plane bilinear element of Fig. 4.2-4 vary as $T = T_0x$ , where $T_0$ is a constant. Material properties $E, \nu,$ and $\alpha$ are constant over the element, as is thickness $t$ . What nodal loads $\{\mathbf{r}_e\}$ are produced by $T$ ?   
+4.22 Imagine that a uniform prestress $\sigma_x = \overline{\sigma}$ exists throughout the trilinear element of Fig. 4.2-5. All other stresses are zero. What nodal loads $\{\mathbf{r}_e\}$ are produced?
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_017.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_017.md
new file mode 100644
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--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_017.md
@@ -0,0 +1,569 @@
+<!-- source-page: 161 -->
+
+4.23 Consider the uniform, three-node bar element of Problem 4.3. Let the element be fixed at the left end and loaded by a uniformly distributed axial load of intensity $q$ . The respective rows of [k] are $[7, -8, 1]$ , $[-8, 16, -8]$ , and $[1, -8, 7]$ , each times $AE / 3L$ . Calculate $u_2$ and $u_3$ if nodal loads are (a) calculated in consistent fashion. (b) of magnitude $qL / 3$ at each of the three nodes.
+
+4.24 Verify the correctness of the three nodal loads shown in Fig. 4.3-4.
+
+4.25 (a) Compute loads allocated to nodes 4, 7, and 3 in Fig. 4.3-4 by an edge-normal traction that varies linearly with $x$ , from intensity $-\overline{q}$ (compression) at node 4 to intensity $+\overline{q}$ (tension) at node 3.
+
+(b) Combine the result of part (a) with the loads of Eq. 4.3-11 to determine $\{\mathbf{r}_e\}$ for a traction that varies linearly from intensity $q_{4}$ at node 4 to intensity $q_{3}$ at node 3.
+
+4.26 (a) Force $F$ acts on one edge of the plane bilinear element at $y = b / 2$ , as shown. What 8 by 1 load vector $\{\mathbf{r}_e\}$ results?
+
+(b) What would be the three nonzero nodal loads if the left edge had three uniformly spaced nodes? (See Eq. 4.3-10.)
+
+![](images/page-161_2cedf2a31b83361f47b32121c9d302daa710b6eafe63c10c0610d67feed2f339.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+a a
+F 4 3
+b
+x,u
+b
+1 2
+</details>
+
+Problem 4.26
+
+![](images/page-161_3a3e49c78580b167d5cdbaa4a29836531abbb8c9ca713e629903686e4cac117a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+a
+a
+4
+3
+a/2
+b
+x,u
+4
+Q
+3
+10
+1
+2
+b
+</details>
+
+Problem 4.27
+
+4.27 A 10-unit force acts at point $Q$ in the plane bilinear element shown. What 8 by 1 load vector $\{\mathbf{r}_e\}$ results?
+
+4.28 Let a concentrated force $F$ act in the $y$ direction on the top edge of the element in Fig. 4.3-4. Where on this edge—that is, at what value of $x / L$ —must $F$ act if nodal loads at nodes 4 and 7 are to be equal? What then is the load at node 3?
+
+4.29 A uniform body force $F_{x}$ acts in the positive x direction on a plane rectangular bilinear element (Fig. 4.2-4). Use Eq. 4.1-6 to compute values of the nodal loads in terms of $F_{x}$ , a, b, and uniform element thickness t.
+
+4.30 Verify the nodal loads shown in Fig. 4.3-5b by integration of $[N]^{T}\{\Phi\}$ over a rectangular face. Shape functions are given in Eqs. 6.6-1. For simplicity you may use $\xi = x$ and $\eta = y$ as coordinates whose origin is at the center of a square face two units on a side.
+
+4.31 Imagine that the temperature of a uniform beam element varies linearly through its depth, from -T along the lower surface to +T along the upper surface. Nodal moments, but not nodal forces, appear in $\{r_{e}\}$ . Compute these moments by use of mechanics of materials arguments rather than by use of Eq. 4.1-6.
+
+<!-- source-page: 162 -->
+
+4.32 Use the virtual work concept to compute the moment $M_{1} = qL^{2} / 12$ at node 1 in Fig. 4.3-6a. That is, compute work done by load $q$ as it moves through displacements created by virtual rotation $\delta \theta_{1}$ (with other nodal d.o.f. kept at zero), and equate it to work done by $M_{1}$ in moving through displacement $\delta \theta_{1}$ .
+
+4.33 A concentrated lateral force $F$ is applied to a standard beam element at a distance $x$ from the left end. For what value of $x$ between 0 and $L$ will $\{\mathbf{r}_e\}$ contain the largest magnitude of moment at node 1? Suggestion: Note the interpretation made in Fig. 4.3-2b.
+
+4.34 Let a uniformly distributed load of intensity $q$ act over only the left half of a beam element. Compute the 4 by 1 load vector $\{\mathbf{r}_e\}$ . Check that loads in $\{\mathbf{r}_e\}$ are statically equivalent to the original load $q$ .
+
+4.35 The uniform cantilever beam shown carries Model the beam by a single beam element.
+
+(a) Calculate $w_{2}$ (the lateral deflection at the right end). Use the consistent load vector at node 2. Express your answer in terms of $P, L; E, I,$ and $x$ .
+
+(b) Again calculate $w_{2}$ , but use load lumping: let the lateral force be $Px / L$ and ignore the moment load at node 2.
+
+(c) Compute the exact $w_{2}$ according to elementary beam analysis.  
+(d) Compute the ratio of the finite element $w_{2}$ to the exact $w_{2}$ . Do this for part (a) and for part (b). Plot these ratios versus $x / L$ .
+
+(e) Compute the bending moment at the left end, first as given by a consistent loads, then as given by lumped loads. Compute the ratio of each moment to the exact value, and plot the two ratios versus $x$ for $0 < x < L$ .
+
+![](images/page-162_cb8e14c4fd53171da922aedae9913c63487a0636dc4bdaf340c9d5189de4cd47.jpg)
+
+<details>
+<summary>text_image</summary>
+
+x
+P
+L
+</details>
+
+Problem 4.35
+
+![](images/page-162_89304b276ca720f486268be0d6929cc0590638354a1323dd9a3f55a6be2eae50.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z
+M_c
+x
+L/2
+L/2
+</details>
+
+Problem 4.36
+
+4.36 A uniform simply supported beam is loaded by moment $M_c$ at midspan, as shown. If the entire beam is modeled by one element, what rotation at midspan is computed? (The exact answer is $\theta = -ML / 12EI$ .)
+
+4.37 Verify the percentage errors listed at the end of section 16.5. Use beam theory rather than equations $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{R}\}$ .
+
+4.38 Consider the bending moment $M$ at the left end of the distance of the cantilever beam element in Fig. 4.3-7a. Compute $M$ from the equation $M = EI[\mathbf{B}]\{\mathbf{d}\}$ , where $[\mathbf{B}]$ is given by Eq. 4.2-4 and nonzero d.o.f. are those used in Eq. 4.3-13. What percentage errors in $M$ are given by the consistent loading and by the lumped loading in Fig. 4.3-7?
+
+<!-- source-page: 163 -->
+
+# Section 4.4
+
+4.39 Imagine that the right-hand element in Fig. 4.4-2 is not an incompatible element, but rather the bilinear element of Fig. 4.2-4. Let vertical displacements $\bar{v}$ be imposed as shown at nodes 5 and 6. The remaining d.o.f. are zero.
+
+(a) Show that edges 3–5 and 4–6 remain straight.   
+(b) Compute the stresses on both sides of interelement boundary 3-4. Express your answers in terms of $E$ , $\nu$ , $\overline{\nu}$ , and the coordinates.
+
+4.40 A uniform beam is modeled by bilinear plane elements, as shown. For each of the loadings (1), (2), and (3), answer the following questions about stresses displayed by the finite element solution. In parts (c) and (d), plot stresses qualitatively (without numerical calculation).
+
+(a) Is $\sigma_{x}$ continuous across the interelement boundary at $A$ ? Explain.   
+(b) Is $\sigma_y$ zero at $B$ ? Explain.   
+(c) Plot $\tau_{xy}$ along the line $y = 0$ .   
+(d) Plot $\sigma_{x}$ along the line $y = 0$ .
+
+![](images/page-163_db14014ecdfdc5a87dfa8584f32af58f3a12d1786ed8dcf3416e7a10c463d9b9.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+B
+A
+C
+D
+x,u
+</details>
+
+![](images/page-163_691697e29ab43071e43d143c05c5102fc387e186552c62e2a7030616fc9f25ee.jpg)  
+{1}
+
+![](images/page-163_ddd264ba65663201fcd8eafc2a52654f7c6e34855df281fcf1fb79f713e00451.jpg)  
+(2)
+
+![](images/page-163_c4e2059db571aa27912310755d0c26232f5cdf779ceb2587e7768e7f3011b194.jpg)  
+{3}   
+Problem 4.40
+
+4.41 Apply the displacement field of the bilinear element (Eq. 4.2-12) to the trapezoidal element shown. Replace the $a_{i}$ by nodal d.o.f. $u_{i}$ , then evaluate $u$ along the line $x = y$ . Hence, demonstrate that this element is incompatible. (The isoparametric formulation, Chapter 6, can produce a compatible element of this shape.)
+
+![](images/page-163_d9b74013bc7a33868171f83324696fc63de0dc114e3ebb13eac7c1fac0a7ddca.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+1 1
+4 3
+1
+2
+x,u
+</details>
+
+Problem 4.41
+
+4.42 Assume that a plane body is isotropic and that body forces $F_{x}$ and $F_{y}$ are constant. In terms of $F_{x}, F_{y}, E$ , and $\nu$ , what must be the values of $a_{4}$ and $a_{8}$ in Eq. 4.2-12 if the differential equations of equilibrium are to be satisfied?
+
+# Section 4.5
+
+4.43 The bar element shown is uniform and has nodes 1 and 2. Let the assumed axial displacement field have the form $u = \lfloor 1 - x^2 \rfloor \lfloor a_1 - a_2 \rfloor^T$ . First, replace
+
+<!-- source-page: 164 -->
+
+![](images/page-164_4bdbd8269c20117679cfd1de054f88cf3c54087139c8e91b389b8380e58cafbe.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L
+x,u
+1
+2
+</details>
+
+Problem 4.43
+
+![](images/page-164_a9dce7d5f475205379cef8ad434010f8e474b14dc0febdeb8905c84bebdefc52.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L/2 → L/2 →
+→ x,u
+1 2
+</details>
+
+Problem 4.44
+
+d.o.f. $a_1$ and $a_2$ by nodal d.o.f. $u_1$ and $u_2$ . Then determine the strain-displacement matrix [B] and the element stiffness matrix [k], in terms of $A, E,$ and $L$ . What defects do you see in these results, and what is their source?
+
+4.44 Repeat Problem 4.43 with reference to the element shown (nodes at $x = \pm L / 2$ ) and the assumed axial displacement field $u = \lfloor x - x^2 \rfloor \lfloor a_1 - a_2 \rfloor^T$ .
+
+4.45 (a) Compute the axial deflection at $x = 2L$ and the axial stress at $x = 0$ in the uniform two-element model shown. Use the element stiffness matrix derived in Problem 4.43 and evaluate stress by the calculation $\sigma_x = E[\mathbf{B}]\{\mathbf{d}\}$ . Are the results correct?
+
+$E[\mathbf{B}]\{\mathbf{d}\}$ . Are the results correct. (b) Repeat part (a), but use the element stiffness matrix derived in Problem 4.44.
+
+![](images/page-164_d46cf5f99be8aacfc705ce37e5640e0b8a633ff94e508f653e9521f5db8cadc6.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+L
+L
+P
+x,u
+1
+2
+3
+</details>
+
+Problem 4.45
+
+4.46 If each of the two coefficients $xy$ in Eq. 4.2-12 is replaced by $x^2 + y^2$ , the element becomes incompatible. Why? Suggestion: Let two adjacent elements have two corner nodes in common. Along the boundary between elements, d.o.f. of these corner nodes must produce the same boundary displacement in each element if the elements are to be compatible. But how many d.o.f. are needed to define a quadratic curve? And what does this imply?
+
+4.47 It is proposed that a beam element be based on a cubic polynomial but that d.o.f. are to be only lateral displacements $w_{i}$ , where $i = 1, 2, 3, 4$ . Nodes are to be at either end and at the third points. What convergence criterion is violated by this element?
+
+4.48 What quadratic terms are omitted from the displacement field of the solid trilinear element (Eq. 4.2-16)? What cubic terms are omitted?
+
+4.49 In three dimensions, which of the cubic terms would you add to a complete quadratic field (10 terms), if there are to be no preferred directions and the total number of terms in the expansion is to be (a) 11, (b) 13, (c) 14, (d) 16, (e) 17, (f) 19?
+
+# Section 4.6
+
+4.50 Imagine that the mesh of Fig. 4.6-1 passes the patch test for constant values of $\sigma_x$ , $\sigma_y$ , and $\tau_{xy}$ . If now all nine nodes are assigned displacements consistent with a field of constant strain (e.g., Eq. 4.2-9), what loads at node 5 should result from the calculation [K]{D}, and why?
+
+<!-- source-page: 165 -->
+
+4.51 For the four-element patch shown in Fig. 4.6-1, determine the nodal loads appropriate to the following patch tests. In each case show the loads on a sketch.
+
+(a) Uniaxial stress $\sigma_y = \sigma_c$ .   
+(b) Shear stress $\tau_{xy} = \tau_c$ .
+
+4.52 Sketch an assembly of hexahedral solid elements, suitable for use as a patch test mesh. Let the elements have corner nodes only and some corner angles other than $90^{\circ}$ . Show supports and nodal loads appropriate to a test for uniform tensile stress $\sigma_{z}$ .
+
+4.53 For the beam element of Fig. 4.2-2, consider the lateral displacement field
+
+$$
+w = \frac {L - x}{L} w _ {1} + \frac {x}{2} \left(\frac {L - x}{L}\right) \theta_ {1} + \frac {x}{L} w _ {2} - \frac {x}{2} \left(\frac {L - x}{L}\right) \theta_ {2}
+$$
+
+(a) Show that this field includes the required rigid-body motion capability.   
+(b) If nodal d.o.f. consistent with a state of constant curvature are prescribed, is the correct $w_{xxx}$ obtained?   
+(c) Determine the element stiffness matrix based on the given field. What defects does it have?
+
+4.54 Divide the element shown into four elements by connecting midpoints of opposite sides. Subdivide the new elements again in the same way. In the limit, what shape does each element approach?
+
+![](images/page-165_2716b26151628abbacc19da457c2d3327f9f96c54bc0f6fdba28ada96b2c1cdb.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Simple geometric diagram of a quadrilateral with four vertices marked by dots (no text or labels)
+</details>
+
+Problem 4.54
+
+4.55 Imagine that the four-element patch of Fig. 4.6-1 is to be tested to see how well it models the pure bending states given in parts (a) and (b). For each of these states, what should be the nodal loads? Rearrange support conditions as may be convenient and appropriate.
+
+(a) $\sigma_{x}$ varies linearly from $-6$ at $y = 0$ to $+6$ at $y = 4$ .   
+(b) $\sigma_{y}$ varies linearly from $-4$ at $x = 0$ to $+4$ at $x = 6$ .
+
+4.56 In Fig. 4.6-3, let u vary linearly with s along edge 1-2. Interpolate u from nodal d.o.f. $u_{1}$ and $u_{2}$ . Assume that all material points in the element move only radially, so that v = 0 and $u = u(r)$ . Show that circumferential strain $\epsilon_{\theta}$ becomes independent of s as a becomes much larger than h.
+
+# Section 4.7
+
+4.57 Nodal d.o.f. of (say) plane elements need not be restricted to displacements u and v. One might also use all four first derivatives of u and v, for a total of six d.o.f. per node. Such an approach has both advantages and disadvantages. What do you think they are?
+
+4.58 Model the bar of Fig. 4.7-1 by two elements, each of length $L_{T}/2$ . Solve for the nodal d.o.f. and the resulting stress in the upper element. Augment this stress by the appropriate form of Eq. 4.7-3. Is the exact stress field obtained?
+
+<!-- source-page: 166 -->
+
+4.59 Consider the bending moment at the center of the uniformly loaded cantilever beam of Fig. 4.3-7.
+
+(a) What is the exact value?   
+(a) What is the exact value.  
+(b) What value is given by $M = EI[\mathbf{B}]\{\mathbf{d}\}$ and the exact values of d.o.f. $w_{2}$ and $\theta_{2}$ ? Use a one-element model.   
+(c) What value is given by adding the result of part (b) to the bending moment at the center of a uniformly loaded clamped-clamped beam of length $L$ ?
+
+4.60 In Problem 4.35e, nodal d.o.f. calculated by use of consistent nodal loads produce a bending moment $M = (2Lx^2 - x^3)P / L^2$ at the fixed end of the cantilever beam. If this $M$ is augmented by the bending moment at the left end of a clamped-clamped beam under load $P$ at position $x$ , is the exact moment produced?   
+4.61 Calculate the bending moment at $x = L / 4$ in the one-element simply supported beam of Problem 4.36. Compute this moment by adding (a) $M$ caused by nodal d.o.f. $\theta_{1}$ and $\theta_{2}$ , and (b) $M$ in a fixed-fixed beam of length $L$ with $M_{c}$ applied at midspan.   
+4.62 The uniform bar element shown is given the temperature variation $T = cx / L$ , where $c$ is a constant.
+
+(a) Solve for $u_{2}$ , then for axial stress $\sigma_{x} = E(\epsilon_{x} - \epsilon_{x0})$ .
+
+(a) Solve for $u_{2}$ , then for axial stress $c_{x}$ and $c_{y}$ for the constant temperature. (b) Repeat part (a), but for stress calculation use an appropriate constant temperature $T_{0}$ rather than $T = cx / L$ .
+
+(c) Why do the results of parts (a) and (b) differ? Is either correct?
+
+![](images/page-166_53a5f84aa0b7d87dee5f71e60be599f2f0b9688fc2a9827b5b4327f36011f09a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+L
+1
+2
+x,u
+</details>
+
+Problem 4.62
+
+4.63 The sketch shows a uniform bar of length 2L, loaded by a uniformly distributed axial load of intensity q. Also shown is a two-element model and the consistently derived loads at nodes 2 and 3. Carry out two cycles of iterative improvement (see Eq. 4.7-7). To what result does the solution appear to be converging, and what then are the nodal average stresses?
+
+![](images/page-166_07978d12d68ad63e072fe7e42dd2bcffb169cb3c1a4aaf08662362a82f4e7a22.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+q
+x,u
+2L
+</details>
+
+![](images/page-166_509e551b7746836984665cc4cfb58a318746a363e9f721b348e0c8bad3ccb1b1.jpg)
+
+<details>
+<summary>text_image</summary>
+
+①
+qL
+②
+qL/2
+1
+2
+3
+L
+L
+</details>
+
+Problem 4.63
+
+<!-- source-page: 167 -->
+
+# STRAIGHT-SIDED TRIANGLES AND TETRAHEDRA
+
+Natural coordinates are introduced. Triangles and tetrahedra are discussed, with the emphasis on triangles. Shape functions are general, but integration formulas and element matrices presented are restricted to elements having straight sides and evenly spaced side nodes. These restrictions are removed in Chapter 6.
+
+# 5.1 NATURAL COORDINATES (LINEAR)
+
+Natural coordinates are dimensionless. They are defined with reference to the element rather than with reference to the global coordinate system in which the element resides. They are used in preference to Cartesian coordinates because they simplify the process of formulating element matrices. The simplest instance, that of natural coordinates for a straight line, is considered in the present section. Similarly defined natural coordinates for triangles and tetrahedra are considered in Section 5.2. Differently defined natural coordinates for quadrilaterals and hexahedra are used in Chapter 6.
+
+Natural coordinates $\xi_{1}$ and $\xi_{2}$ in Fig. 5.1-1 are defined as ratios of lengths:
+
+$$
+\xi_ {1} = \frac {L _ {1}}{L} \quad \text { and } \quad \xi_ {2} = \frac {L _ {2}}{L} \tag {5.1-1}
+$$
+
+Since $L_{1} + L_{2} = L$ , coordinates $\xi_{1}$ and $\xi_{2}$ are not independent. They satisfy the constraint relation
+
+$$
+\xi_ {1} + \xi_ {2} = 1 \tag {5.1-2}
+$$
+
+Note that $\xi_{1}$ and $\xi_{2}$ are each either zero or unity at end points 1 and 2. Definitions
+
+![](images/page-167_bc5f20070596fc02a7e1cf54b1bae2b9996270dcc511645cfa05581470007087.jpg)
+
+<details>
+<summary>text_image</summary>
+
+0
+•
+1
+P
+2
+x₁
+L₂
+L₁
+x = ξ₁x₁ + ξ₂x₂
+x₂ = x₁ + L
+</details>
+
+![](images/page-167_cbdcd1bc6acc8276a4dced6f762627d61427cbfdc239ee7f748197809bc801cf.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L
+1
+2
+ξ₁ = L₁/L
+1
+ξ₂ = L₂/L
+1
+</details>
+
+Figure 5.1-1. Natural coordinates $\xi_{1}$ and $\xi_{2}$ along a straight line. Point $P$ is arbitrarily located; that is, $x$ can have any value.
+
+<!-- source-page: 168 -->
+
+5.1-1 are independent of global coordinate $x$ . Even so, $\xi_1$ and $\xi_2$ can be used to state the position of the arbitrary point $P$ in terms of $x_1$ and $x_2$ :
+
+$$
+x = \xi_ {1} x _ {1} + \xi_ {2} x _ {2} \tag {5.1-3}
+$$
+
+For example, the centroid of line 1-2 is at $\xi_{1} = \xi_{2} = \frac{1}{2}$ , where $x = (x_{1} + x_{2}) / 2$ . Equations 5.1-2 and 5.1-3, and the inverse relations that state $\xi_{1}$ and $\xi_{2}$ in terms of $x$ , are
+
+$$
+\left\{ \begin{array}{l} 1 \\ x \end{array} \right\} = \left[ \begin{array}{l l} 1 & 1 \\ x _ {1} & x _ {2} \end{array} \right] \left\{ \begin{array}{l} \xi_ {1} \\ \xi_ {2} \end{array} \right\} \quad \text { and } \quad \left\{ \begin{array}{l} \xi_ {1} \\ \xi_ {2} \end{array} \right\} = \frac {1}{L} \left[ \begin{array}{l l} x _ {2} & - 1 \\ - x _ {1} & 1 \end{array} \right] \left\{ \begin{array}{l} 1 \\ x \end{array} \right\} \tag {5.1-4}
+$$
+
+Equations 5.1-4 provide a linear mapping between the $x$ and $\xi$ coordinate systems. Interpolation along line 1-2 can be done in natural coordinates. Linear interpolation of a function $\phi$ , from nodal values $\phi_1$ and $\phi_2$ , is
+
+$$
+\phi = \lfloor \mathbf {N} \rfloor \left\{ \begin{array}{l} \phi_ {1} \\ \phi_ {2} \end{array} \right\}, \quad \text { where } \quad \lfloor \mathbf {N} \rfloor = \left\lfloor \xi_ {1} \quad \xi_ {2} \right\rfloor \tag {5.1-5}
+$$
+
+In this instance the individual shape functions are $N_{1} = \xi_{1}$ and $N_{2} = \xi_{2}$ . Quadratic interpolation from nodal values $\phi_{1}$ at $\xi_{1} = 1$ , $\phi_{2}$ at $\xi_{2} = 1$ , and $\phi_{3}$ at $\xi_{1} = \xi_{2} = \frac{1}{2}$ , is
+
+$$
+\phi = \xi_ {1} (2 \xi_ {1} - 1) \phi_ {1} + \xi_ {2} (2 \xi_ {2} - 1) \phi_ {2} + 4 \xi_ {1} \xi_ {2} \phi_ {3} \tag {5.1-6}
+$$
+
+The three quadratic shape functions are shown in Fig. 5.1-2. Equation 5.1-6 can be derived by starting with $\phi = a_{1}\xi_{1}^{2} + a_{2}\xi_{2}^{2} + a_{3}\xi_{1}\xi_{2}$ (which is quadratic in $x$ ), evaluating the $a_{i}$ by substitutions such as $\phi = \phi_{1}$ at $\xi_{1} = 1$ , and using Eq. 5.1-2. Alternatively, one can use insight: by comparing curves in Figs. 5.1-1 and 5.1-2, we see that $N_{1} = \xi_{1} - N_{3}/2$ and $N_{2} = \xi_{2} - N_{3}/2$ .
+
+Stiffness matrices can be formulated by use of natural cross-sections to imagine that line 1–2 in Fig. 5.1-1 is a two-node bar element of cross-sectional area A and elastic modulus E. From Eq. 5.1-5 with $\phi = u$ , axial displacement is $u = \xi_{1}u_{1} + \xi_{2}u_{2}$ . Therefore, the axial strain is
+
+$$
+\epsilon_ {x} = \frac {d u}{d x} = \frac {\partial u}{\partial \xi_ {1}} \frac {\partial \xi_ {1}}{\partial x} + \frac {\partial u}{\partial \xi_ {2}} \frac {\partial \xi_ {2}}{\partial x} = u _ {1} \left(- \frac {1}{L}\right) + u _ {2} \left(\frac {1}{L}\right) \tag {5.1-7}
+$$
+
+in which the factors $\pm 1 / L$ are obtained from the second of Eqs. 5.1-4; that is,
+
+$$
+\frac {\partial \xi_ {1}}{\partial x} = \frac {\partial}{\partial x} \left(\frac {x _ {2} - x}{L}\right) = - \frac {1}{L} \quad \text { and } \quad \frac {\partial \xi_ {2}}{\partial x} = \frac {\partial}{\partial x} \left(\frac {x - x _ {1}}{L}\right) = \frac {1}{L} \tag {5.1-8}
+$$
+
+![](images/page-168_7fb063831ba696e91e90b59a6996efc505a7719ff30ede77e405e0a09c47211c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1 3 2
+← L/2 ← L/2 →
+1
+N₁ = ξ₁(2ξ₁ - 1)
+N₂ = ξ₂(2ξ₂ - 1)
+1
+N₃ = 4ξ₁ξ₂
+</details>
+
+Figure 5.1-2. Quadratic shape functions over a span $L$ in natural coordinates $\xi_1$ and $\xi_2$ , with 3 a central node.
+
+<!-- source-page: 169 -->
+
+The element stiffness matrix that operates on $u_{1}$ and $u_{2}$ is
+
+$$
+[ \mathbf {k} ] = \int_ {L} A E \left\lfloor \mathbf {B} \right\rfloor^ {T} \left\lfloor \mathbf {B} \right\rfloor d L, \quad \text { where } \quad \left\lfloor \mathbf {B} \right\rfloor = \left\lfloor - \frac {1}{L} \quad \frac {1}{L} \right\rfloor \tag {5.1-9}
+$$
+
+The familiar result, Eq. 2.4-5, follows immediately.
+
+For elements of quadratic or higher order, the integrand of [k] contains functions of $\xi_{1}$ and $\xi_{2}$ . Integration of polynomials in $\xi_{1}$ and $\xi_{2}$ can be done by the formula
+
+$$
+\int_ {L} \xi_ {1} ^ {k} \xi_ {2} ^ {\ell} d L = L \frac {k ! \ell !}{(1 + k + \ell) !} \tag {5.1-10}
+$$
+
+where k and $\ell$ are nonnegative integers and L is the distance between the end points $\xi_{1}=1$ and $\xi_{2}=1$ . When it appears, the factorial 0! is defined as unity. Integration in Eq. 5.1-9 involves integration of only dL. Therefore, $k=\ell=0$ in Eq. 5.1-10, and integration yields L. As additional examples of the application of Eq. 5.1-10,
+
+$$
+\int_ {L} x d L = \int_ {L} \left(\xi_ {1} x _ {1} + \xi_ {2} x _ {2}\right) d L = \frac {L}{2} \left(x _ {1} + x _ {2}\right) \tag {5.1-11}
+$$
+
+$$
+\int_ {L} \xi_ {1} \xi_ {2} ^ {2} d L = L \frac {2}{4 !} = \frac {L}{1 2} \tag {5.1-12}
+$$
+
+# 5.2 NATURAL COORDINATES (AREA AND VOLUME)
+
+As defined in Section 5.1, natural coordinates for a line are ratios of lengths. Analogous natural coordinates for triangles and tetrahedra are respectively defined as ratios of areas and ratios of volumes. In the present section we assume that sides of triangles and edges of tetrahedra are straight.
+
+Area Coordinates. In Fig. 5.2-1, an arbitrarily located point P divides a triangle 1–2–3 into three subareas $A_{1}$ , $A_{2}$ , and $A_{3}$ . Area coordinates are defined as ratios of areas $^{1}$ :
+
+$$
+\xi_ {1} = \frac {A _ {1}}{A} \quad \xi_ {2} = \frac {A _ {2}}{A} \quad \xi_ {3} = \frac {A _ {3}}{A} \tag {5.2-1}
+$$
+
+where A is the area of triangle 1–2–3. Since $A = A_{1} + A_{2} + A_{3}$ , the $\xi_{i}$ are not independent. They satisfy the constraint equation
+
+$$
+\xi_ {1} + \xi_ {2} + \xi_ {3} = 1 \tag {5.2-2}
+$$
+
+<!-- source-page: 170 -->
+
+![](images/page-170_c64cf1ac2edd2e892aa392fd22831663467741f620a25ee2618bd71dd5a0236b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+3
+Side 1
+Side 2
+A₂
+P
+A₁
+A₃
+1
+2
+Side 3
+x
+y
+</details>
+
+![](images/page-170_6f3132f364cad6ed7f76abfac858aeb531874e0f24ebb2356aaa62a2ae9a833d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ξ₃=1
+ξ₂ = 0
+ξ₃ = 1/2
+ξ₁ = 0
+ξ₁ = 1
+ξ₂ = 1/2
+ξ₃ = 0
+ξ₁ = 1/2
+</details>
+
+Figure 5.2-1. Natural (area) coordinates for a triangle.
+
+The centroid of a straight-sided triangle is at $\xi_1 = \xi_2 = \xi_3 = \frac{1}{3}$ .
+
+The constraint equation and the linear relation between Cartesian and area coordinates are expressed by equations analogous to Eqs. 5.1-4,
+
+$$
+\left\{ \begin{array}{l} 1 \\ x \\ y \end{array} \right\} = [ \mathbf {A} ] \left\{ \begin{array}{l} \xi_ {1} \\ \xi_ {2} \\ \xi_ {3} \end{array} \right\} \quad \text { and } \quad \left\{ \begin{array}{l} \xi_ {1} \\ \xi_ {2} \\ \xi_ {3} \end{array} \right\} = [ \mathbf {A} ] ^ {- 1} \left\{ \begin{array}{l} 1 \\ x \\ y \end{array} \right\} \tag {5.2-3}
+$$
+
+where, with $x_{ij} = x_i - x_j$ and $y_{ij} = y_i - y_j$ ,
+
+$$
+[ \mathbf {A} ] = \left[ \begin{array}{l l l} 1 & 1 & 1 \\ x _ {1} & x _ {2} & x _ {3} \\ y _ {1} & y _ {2} & y _ {3} \end{array} \right] \quad \text { and } \quad [ \mathbf {A} ] ^ {- 1} = \frac {1}{2 A} \left[ \begin{array}{l l l} x _ {2} \dot {y} _ {3} - x _ {3} y _ {2} & y _ {2 3} & x _ {3 2} \\ x _ {3} y _ {1} - x _ {1} y _ {3} & y _ {3 1} & x _ {1 3} \\ x _ {1} y _ {2} - x _ {2} y _ {1} & y _ {1 2} & x _ {2 1} \end{array} \right] \tag {5.2-4}
+$$
+
+That the first of Eqs. 5.2-3 is correct may be checked by noting that it yields the correct $x$ and $y$ values at the vertices (and at other points, such as midsides), and that $x$ and $y$ vary linearly elsewhere.
+
+Twice the area of triangle 1-2-3 is
+
+$$
+2 A = \det [ \mathbf {A} ] = x _ {2 1} y _ {3 1} - x _ {3 1} y _ {2 1} \tag {5.2-5}
+$$
+
+where $x_{21} = x_2 - x_1$ , and so on. The latter expression for $2A$ is neither unique nor obvious but can be obtained by manipulation of the determinant. If labels 1-2-3 are reversed, so that the order 1-2-3 is clockwise around the triangle, Eq. 5.2-5 gives a negative number for $2A$ .
+
+The first of Eqs. 5.2-3 allows one to find the Cartesian coordinates of a point when its area coordinates are given. A point in the triangle is uniquely located by specifying any two of its area coordinates. If all three are specified they must satisfy the constraint equation, Eq. 5.2-2. Further discussion of interpolation in area coordinates appears in subsequent sections, where properties of specific elements are formulated.
+
+Formulation of element matrices requires that a function $\phi$ , expressed in terms of area coordinates, be differentiated with respect to Cartesian coordinates. By the chain rule, with $\phi = \phi(\xi_1, \xi_2, \xi_3)$ ,
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_018.md
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+<!-- source-page: 171 -->
+
+$$
+\frac {\partial \phi}{\partial x} = \frac {\partial \phi}{\partial \xi_ {1}} \frac {\partial \xi_ {1}}{\partial x} + \frac {\partial \phi}{\partial \xi_ {2}} \frac {\partial \xi_ {2}}{\partial x} + \frac {\partial \phi}{\partial \xi_ {3}} \frac {\partial \xi_ {3}}{\partial x} \tag {5.2-6a}
+$$
+
+$$
+\frac {\partial \phi}{\partial y} = \frac {\partial \phi}{\partial \xi_ {1}} \frac {\partial \xi_ {1}}{\partial y} + \frac {\partial \phi}{\partial \xi_ {2}} \frac {\partial \xi_ {2}}{\partial y} + \frac {\partial \phi}{\partial \xi_ {3}} \frac {\partial \xi_ {3}}{\partial y} \tag {5.2-6b}
+$$
+
+From the second of Eqs. 5.2-3,
+
+$$
+\frac {\partial \xi_ {1}}{\partial x} = \frac {y _ {2 3}}{2 A} \quad \frac {\partial \xi_ {2}}{\partial x} = \frac {y _ {3 1}}{2 A} \quad \frac {\partial \xi_ {3}}{\partial x} = \frac {y _ {1 2}}{2 A} \tag {5.2-7a}
+$$
+
+$$
+\frac {\partial \xi_ {1}}{\partial y} = \frac {x _ {3 2}}{2 A} \quad \frac {\partial \xi_ {2}}{\partial y} = \frac {x _ {1 3}}{2 A} \quad \frac {\partial \xi_ {3}}{\partial y} = \frac {x _ {2 1}}{2 A} \tag {5.2-7b}
+$$
+
+where $y_{23} = y_2 - y_3$ , and so on. Equations 5.2-7 apply to straight-sided triangles whose side nodes (if any) are evenly spaced.
+
+Integration of a polynomial in area coordinates over the triangle area is accomplished by a formula analogous to Eq. 5.1-10. If $k, \ell,$ and $m$ are nonnegative integers, then
+
+$$
+\int_ {A} \xi_ {1} ^ {k} \xi_ {2} ^ {\ell} \xi_ {3} ^ {m} d A = 2 A \frac {k ! \ell ! m !}{(2 + k + \ell + m) !} \tag {5.2-8}
+$$
+
+where $A$ is the entire area of the triangle in Fig. 5.2-1. Integration along a side, such as side 3 of the triangle, where $\xi_{3} = 0$ , is accomplished by Eq. 5.1-10.
+
+The following form of the area integration formula is often useful [5.1]. It allows the integrand to be expressed in terms of Cartesian coordinates rather than area coordinates. Let $x$ and $y$ be centroidal axes, so that vertex coordinates satisfy the equations $x_{1} + x_{2} + x_{3} = 0$ and $y_{1} + y_{2} + y_{3} = 0$ . Then, if $r$ and $s$ are nonnegative integers,
+
+$$
+\int_ {A} x ^ {r} y ^ {s} d A = C _ {r + s} A \left(x _ {1} ^ {r} y _ {1} ^ {s} + x _ {2} ^ {r} y _ {2} ^ {s} + x _ {3} ^ {r} y _ {3} ^ {s}\right) \tag {5.2-9}
+$$
+
+where
+
+$$
+\begin{array}{c c c c c c} r + s & 1 & 2 & 3 & 4 & 5 \\ \hline C _ {r + s} & 0 & 1 / 1 2 & 1 / 3 0 & 1 / 3 0 & 2 / 1 0 5 \end{array}
+$$
+
+For example,
+
+$$
+\int_ {A} x ^ {2} d A = \frac {A}{1 2} \left(x _ {1} ^ {2} + x _ {2} ^ {2} + x _ {3} ^ {2}\right) \tag {5.2-10}
+$$
+
+where, from Eq. 5.2-5, $A = x_{21}y_{31} - x_{31}y_{21}$ . Equation 5.2-9 can be derived from Eqs. 5.2-3, 5.2-4, and 5.2-8, but the manipulations are tedious.
+
+Numerical integration formulas for triangles appear in Section 6.8.
+
+Volume Coordinates. Volume coordinates for a tetrahedron are a direct extension of area coordinates for a triangle, so we will be brief.
+
+<!-- source-page: 172 -->
+
+![](images/page-172_6f5e853c97e34677416d33b7377480254608c2da51c3187438f3f874533fe1be.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z
+4
+V₁
+3
+P
+1
+y
+2
+x
+</details>
+
+Figure 5.2-2. Tetrahedron 1–2–3–4. Volume $V_{1}$ refers to subtetrahedron P–2–3–4, which is identified by hatching.
+
+Point P is the common vertex of four subtetrahedra (Fig. 5.2-2). Volume coordinates are $\xi_{i}=V_{i}/V$ , where the volume of tetrahedron 1–2–3–4 is $V=\Sigma V_{i}$ and i runs from 1 to 4. Cartesian and volume coordinates have the relation
+
+$$
+\left\{ \begin{array}{l} 1 \\ x \\ y \\ z \end{array} \right\} = \left[ \begin{array}{c c c c} 1 & 1 & 1 & 1 \\ x _ {1} & x _ {2} & x _ {3} & x _ {4} \\ y _ {1} & y _ {2} & y _ {3} & y _ {4} \\ z _ {1} & z _ {2} & z _ {3} & z _ {4} \end{array} \right] \left\{ \begin{array}{l} \xi_ {1} \\ \xi_ {2} \\ \xi_ {3} \\ \xi_ {4} \end{array} \right\} \tag {5.2-11}
+$$
+
+The determinant of the square matrix is $6V$ , where $V$ is positive if nodes are numbered so that the sequence 1-2-3 runs counterclockwise when viewed from node 4. Relations analogous to Eqs. 5.2-7 contain terms from the inverse of the square matrix in Eq. 5.2-11.
+
+The integration formula in volume coordinates is
+
+$$
+\int_ {V} \xi_ {1} ^ {k} \xi_ {2} ^ {\ell} \xi_ {3} ^ {m} \xi_ {4} ^ {n} d V = 6 V \frac {k ! \ell ! m ! n !}{(3 + k + \ell + m + n) !} \tag {5.2-12}
+$$
+
+A formula analogous to Eq. 5.2-9, but for a special case, is
+
+$$
+\int_ {V} x ^ {r} y ^ {s} z ^ {t} d V = \frac {V}{2 0} \sum_ {i = 1} ^ {4} x _ {i} ^ {r} y _ {i} ^ {s} z _ {i} ^ {t} \quad (\text { if } r + s + t = 2) \tag {5.2-13}
+$$
+
+provided that the centroid of the tetrahedron is at $x = y = z = 0$ .
+
+Remarks. Triangles and tetrahedra are instances of a simplex, which is a figure in $n$ -dimensional space that has $n + 1$ vertices and is bounded by $n + 1$ surfaces of dimensionality $n - 1$ . Other names for area coordinates are areal, triangular, and trilinear coordinates. Volume coordinates may also be called tetrahedronal coordinates. Area and volume coordinates are known to mathematicians as simplex or barycentric coordinates. They are not new [5.2,5.3] but seem to have been independently devised when finite element theory found a need for them.
+
+<!-- source-page: 173 -->
+
+# 5.3 INTERPOLATION FIELDS FOR PLANE TRIANGLES
+
+Triangular elements allow a complete polynomial in Cartesian coordinates to be used for the field quantity (e.g., for temperature or for displacement). In other words, in Fig. 5.3-1, with internal nodes present for cubic and higher-order elements, all terms of a truncated Pascal triangle are used in the shape functions: through the second row for a linear element, through the third row for a quadratic element, and so on.
+
+In order to generate finite elements, we seek shape functions $N_{i} = N_{i}(\xi_{1}, \xi_{2}, \xi_{3})$ in the relation $\phi = \Sigma N_{i}\phi_{i}$ , where the $\phi_{i}$ are nodal d.o.f. One way to generate shape functions $N_{i}$ is the usual way of starting with a polynomial that contains constants $a_{i}$ that must be determined. Consider a function $\phi = \phi(\xi_{1}, \xi_{2}, \xi_{3}) = \phi(x, y)$ , where $\phi$ is given by the expansion
+
+$$
+\phi = \sum_ {i = 1} ^ {n} a _ {i} \xi_ {1} ^ {q} \xi_ {2} ^ {r} \xi_ {3} ^ {s} \tag {5.3-1}
+$$
+
+in which q, r, and s are nonnegative integers that range over the n possible combinations for which $q + r + s = p$ . Thus $\phi$ is a complete polynomial of degree p in Cartesian coordinates [5.4]. For example, for the quadratic triangle in Fig. 5.3-2b, n = 6, p = 2, and
+
+$$
+\phi = a _ {1} \xi_ {1} ^ {2} + a _ {2} \xi_ {2} ^ {2} + a _ {3} \xi_ {3} ^ {2} + a _ {4} \xi_ {1} \xi_ {2} + a _ {5} \xi_ {2} \xi_ {3} + a _ {6} \xi_ {3} \xi_ {1} \tag {5.3-2}
+$$
+
+which, for a straight-sided triangle, is equivalent to
+
+$$
+\phi = b _ {1} + b _ {2} x + b _ {3} y + b _ {4} x ^ {2} + b _ {5} x y + b _ {6} y ^ {2} \tag {5.3-3}
+$$
+
+where the $b_{i}$ are constants related to constants $a_{i}$ of Eq. 5.3-2.
+
+To obtain shape functions $N_{i}$ from Eq. 5.3-1, we express the $a_{i}$ in terms of nodal d.o.f. $\phi_{i}$ . Consider again the quadratic triangle. Side node 4 is at $\xi_{1} = \xi_{2} = \frac{1}{2}$ and $\xi_{3} = 0$ , side node 5 is at $\xi_{2} = \xi_{3} = \frac{1}{2}$ and $\xi_{1} = 0$ , and side node 6 is at $\xi_{3} = \xi_{1} = \frac{1}{2}$ and $\xi_{2} = 0$ . In Eq. 5.3-2 we set $\phi = \phi_{1}$ for $\xi_{1} = 1$ and $\xi_{2} = \xi_{3} = 0$ , $\phi = \phi_{2}$ for $\xi_{2} = 1$ and $\xi_{3} = \xi_{1} = 0$ , and so on through $\phi = \phi_{6}$ for $\xi_{3} = \xi_{1} = \frac{1}{2}$ and $\xi_{2} = 0$ . After we have solved for the $a_{i}$ , the coefficients of the $\phi_{i}$ are identified as shape functions.
+
+![](images/page-173_713fbc63b36d1dc619866c8c27bde9311ef17304a9c2edc4b1c24a7eba6f5374.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Pascal triangle
+Polynomial
+degree, p
+Number of
+terms, n
+Triangular element (number of
+nodes = number of terms)
+1
+x y
+x² xy y²
+x³ x²y xy² y³
+x⁴ x³y x²y² xy³ y⁴
+0 (constant)
+1 (linear)
+2 (quadratic)
+3 (cubic)
+4 (quartic)
+1
+3 ←
+6 ←
+10 ←
+15 ←
+n=(p+1)(p+2)/2
+</details>
+
+Figure 5.3-1. Relation between type of plane triangular element and number of polynomial coefficients used for interpolation.
+
+<!-- source-page: 174 -->
+
+![](images/page-174_fadf7625ae01a63f82c6da36d2df6b222f30e8cda05050b4c227764880f4e450.jpg)
+
+<details>
+<summary>text_image</summary>
+
+3
+1
+2
+</details>
+
+(a)
+
+![](images/page-174_6922c276d74810d690b5de7ffb5a0b99fb07c0fc74793df24322c840bcd7587e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+3
+4
+5
+6
+</details>
+
+(b)
+
+![](images/page-174_7b5471d1ff5cd18175ca1f689171f3c2c0ffa326f477bb15c1044127adf8e10f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+3
+7
+8
+9
+10
+1
+4
+5
+2
+</details>
+
+(c)   
+Figure 5.3-2. Triangular elements. (a) Linear. (b) Quadratic. (c) Cubic. Node 10 is at the centroid, $\xi_{1} = \xi_{2} = \xi_{3} = \frac{1}{3}$ .
+
+An alternative way of determining the shape functions is available. With nodes permitted in the element interior, but no derivative d.o.f. permitted at any node, the elements can be identified as Lagrangian. Hence, the Lagrange interpolation formula produces shape functions $N_{i}$ directly. The procedure is not difficult but cannot be explained briefly. Details appear in [5.4].
+
+Shape functions for the elements of Fig. 5.3-2 are as follows. Note that (as expected) each $N_{i}$ is unity at node $i$ but vanishes at all other nodes. For the linear triangle, Fig. 5.3-2a,
+
+$$
+N _ {1} = \xi_ {1} \quad N _ {2} = \xi_ {2} \quad N _ {3} = \xi_ {3} \tag {5.3-4}
+$$
+
+Thus, individual shape functions of a linear triangle are the area coordinates themselves. For the quadratic triangle, Fig. 5.3-2b,
+
+$$
+N _ {1} = \xi_ {1} \left(2 \xi_ {1} - 1\right) \quad N _ {2} = \xi_ {2} \left(2 \xi_ {2} - 1\right) \quad N _ {3} = \xi_ {3} \left(2 \xi_ {3} - 1\right) \tag {5.3-5}
+$$
+
+$$
+N _ {4} = 4 \xi_ {1} \xi_ {2} \quad N _ {5} = 4 \xi_ {2} \xi_ {3} \quad N _ {6} = 4 \xi_ {3} \xi_ {1}
+$$
+
+For the cubic triangle, Fig. 5.3-2c,
+
+$$
+N _ {i} = \frac {1}{2} \xi_ {i} (3 \xi_ {i} - 1) (3 \xi_ {i} - 2) \quad \text { for } i = 1, 2, 3
+$$
+
+$$
+N _ {4} = \frac {9}{2} \xi_ {2} \xi_ {1} (3 \xi_ {1} - 1) \quad N _ {6} = \frac {9}{2} \xi_ {3} \xi_ {2} (3 \xi_ {2} - 1) \tag {5.3-6}
+$$
+
+$$
+N _ {8} = \frac {9}{2} \xi_ {1} \xi_ {3} (3 \xi_ {3} - 1) \quad N _ {1 0} = 2 7 \xi_ {1} \xi_ {2} \xi_ {3}
+$$
+
+and $N_5, N_7$ , and $N_9$ are obtained from $N_4, N_6$ , and $N_8$ , respectively, by interchanging subscripts; for example, $N_5 = 9\xi_1\xi_2(3\xi_2 - 1)/2$ .
+
+Equations 5.3-5 and 5.3-6 can also be used for triangular elements having curved sides. The procedure is described in Section 6.8.
+
+# 5.4 THE LINEAR TRIANGLE
+
+Scalar Field Element. For simplicity, we begin with a scalar field problem. Each node has a single d.o.f. An example application is heat conduction, for which $\phi$ represents temperature (see Section 3.10 or Chapter 16). For the three-node triangle, from Eq. 5.3-4,
+
+<!-- source-page: 175 -->
+
+$$
+\phi = \lfloor \mathrm{N} \rfloor \left\{ \begin{array}{l} \phi_ {1} \\ \phi_ {2} \\ \phi_ {3} \end{array} \right\}, \quad \text { where } \quad \lfloor \mathrm{N} \rfloor = \left\lfloor \xi_ {1} \quad \xi_ {2} \quad \xi_ {3} \right\rfloor \tag {5.4-1}
+$$
+
+In order to generate the characteristic matrix [k], derivatives of $\phi$ with respect to $x$ and $y$ are needed. Using Eqs. 5.2-6 and 5.4-1, we have
+
+$$
+\frac {\partial \phi}{\partial x} = \sum_ {i = 1} ^ {3} \frac {\partial N _ {i}}{\partial x} \phi_ {i} = \sum_ {i = 1} ^ {3} \frac {\partial \xi_ {i}}{\partial x} \phi_ {i} \tag {5.4-2}
+$$
+
+and similarly for $\partial \phi / \partial y$ , where derivatives $\partial \xi_i / \partial x$ and $\partial \xi_i / \partial y$ are given by Eqs. 5.2-7. Hence,
+
+$$
+\left\{ \begin{array}{l} \phi_ {, x} \\ \phi_ {, y} \end{array} \right\} = \left[ \begin{array}{l} \mathrm{N} _ {, x} \\ \mathrm{N} _ {, y} \end{array} \right] \left\{ \begin{array}{l} \phi_ {1} \\ \phi_ {2} \\ \phi_ {3} \end{array} \right\} = [ \mathbf {B} ] \left\{ \begin{array}{l} \phi_ {1} \\ \phi_ {2} \\ \phi_ {3} \end{array} \right\}, \quad \text { where } \quad [ \mathbf {B} ] = \frac {1}{2 A} \left[ \begin{array}{l l l} y _ {2 3} & y _ {3 1} & y _ {1 2} \\ x _ {3 2} & x _ {1 3} & x _ {2 1} \end{array} \right] \tag {5.4-3}
+$$
+
+and $A$ is the area of the triangle. The element characteristic matrix is
+
+$$
+[ \mathbf {k} ] = \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} k [ \mathbf {B} ] d V \tag {5.4-4}
+$$
+
+where $k$ is a material property. If $k$ and element thickness $t$ are constant over the element, then $[\mathbf{k}] = ktA[\mathbf{B}]^T[\mathbf{B}]$ .
+
+Constant-Strain Triangle. In plane stress analysis the manipulations are quite similar to those presented in Eqs. 5.4-1 to 5.4-4. Displacement fields $u = u(x,y)$ and $v = v(x,y)$ are each interpolated from nodal d.o.f. $u_i$ and $v_i$ . The linear element has six displacements in the vector of nodal d.o.f. $\{\mathbf{d}\}$ . With $\{\mathbf{d}\} = \left[u_1 \quad v_1 \quad u_2 \quad v_2 \quad u_3 \quad v_3\right]^T$ ,
+
+$$
+\left\{ \begin{array}{l} u \\ v \end{array} \right\} = [ \mathrm{N} ] \{\mathbf {d} \}, \quad \text { where } \quad [ \mathrm{N} ] = \left[ \begin{array}{c c c c c c} \xi_ {1} & 0 & \xi_ {2} & 0 & \xi_ {3} & 0 \\ 0 & \xi_ {1} & 0 & \xi_ {2} & 0 & \xi_ {3} \end{array} \right] \tag {5.4-5}
+$$
+
+Strains $\{\pmb{\epsilon}\} = [\mathbf{B}]\{\mathbf{d}\}$ are given by Eqs. 1.5-6 and 1.5-8. Using these equations, and also Eqs. 5.2-6 and 5.2-7, we obtain
+
+$$
+[ \mathbf {B} ] = [ \partial ] [ \mathbf {N} ] = \frac {1}{2 A} \left[ \begin{array}{c c c c c c} y _ {2 3} & 0 & y _ {3 1} & 0 & y _ {1 2} & 0 \\ 0 & x _ {3 2} & 0 & x _ {1 3} & 0 & x _ {2 1} \\ x _ {3 2} & y _ {2 3} & x _ {1 3} & y _ {3 1} & x _ {2 1} & y _ {1 2} \end{array} \right] \tag {5.4-6}
+$$
+
+With [E] given by Eq. 1.7-4 or Eq. 1.7-5, the stiffness matrix of the linear triangle is
+
+$$
+[ \mathbf {k} ] = \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] [ \mathbf {B} ] d V \tag {5.4-7}
+$$
+
+<!-- source-page: 176 -->
+
+![](images/page-176_0cd723ff44f44d90d5424f1d2ade70c9b444a83c98f029e9ebe9cbe53af56adf.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+l
+l
+2
+x,u
+3
+1
+c
+c
+M
+M
+</details>
+
+Figure 5.4-1. A linear triangle in a beam subjected to pure bending.
+
+If [E] and element thickness $t$ are constant, then $[\mathbf{k}] = tA[\mathbf{B}]^T [\mathbf{E}][\mathbf{B}]$ .
+
+The element associated with Eq. 5.4-5 is called the constant-strain triangle (CST) because its strain field contains only constants. Because $x$ and $y$ terms are absent, the CST element behaves poorly in bending. Consider, for example, Fig. 5.4-1. Let the origin $x = y = 0$ be motionless, and let bending moment $M$ be of such magnitude that $u = \overline{u}$ at $x = \ell$ on top of the beam. Hence, the correct displacement field associated with pure bending, and the resulting strain field, are
+
+$$
+u = \frac {\bar {u}}{c \ell} x y \quad v = \frac {\bar {u}}{2 c \ell} (- x ^ {2} - \nu y ^ {2}) \tag {5.4-8}
+$$
+
+$$
+\epsilon_ {x} = \frac {\overline {{{u}}}}{c \ell} y \quad \epsilon_ {y} = - \nu \frac {\overline {{{u}}}}{c \ell} y \quad \gamma_ {x y} = 0
+$$
+
+To see how the CST represents these bending strains, we impose on the CST nodal displacements $\{d\}$ consistent with the bending field of Eq. 5.4-8 ( $u_{1} = -\overline{u}$ , $u_{2} = \overline{u}$ , $u_{3} = 0$ , and similar prescriptions of $v_{1}$ , $v_{2}$ , and $v_{3}$ from Eqs. 5.4-8). Hence,
+
+![](images/page-176_2fb253b6adc5d1889fb15c9c29b548f82496546559e0c271125afe1b0ecac28c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+← h →
+B
+v=0.25
+E
+x
+D
+h
+A
+C
+4h
+</details>
+
+![](images/page-176_00c71a62b7ff0eafc4dc305383585ffdeb89fb1df24901f90ddfa49059093286.jpg)
+
+<details>
+<summary>text_image</summary>
+
+32 linear-strain triangles
+160 d.o.f., v_A = 0.998, σ_XB = 0.986
+</details>
+
+![](images/page-176_55e173d943d17cd9f13401f2c0c65aff1e8c22b5fc8aa68464fb5a52b81fc2eb.jpg)
+
+<details>
+<summary>text_image</summary>
+
+128 constant-strain triangles
+160 d.o.f., v_A = 0.859, σ_XB = 0.854
+</details>
+
+![](images/page-176_ecd82791e9ca132e7518c63f06fb392bd4ae3074d3188fe85d93813234fbb326.jpg)
+
+<details>
+<summary>text_image</summary>
+
+512 constant-strain triangles
+576 d.o.f., v_A = 0.961, σ_XB = 0.956
+</details>
+
+Figure 5.4-2. Tip deflection $v_{A}$ and flexural stress $\sigma_{xB}$ in an isotropic cantilever beam of constant thickness [5.5]. The transverse load is parabolically distributed over the right end. Values reported are ratios of computed values to values predicted by theory of elasticity.
+
+<!-- source-page: 177 -->
+
+from $\{\pmb{\epsilon}\} = [\mathbf{B}]\{\mathbf{d}\}$ , with [B] given by Eq. 5.4-6, we obtain strains in the CST of Fig. 5.4-1. They are
+
+$$
+\epsilon_ {x} = 0 \quad \epsilon_ {y} = 0 \quad \gamma_ {x y} = \left(\frac {1}{c} - \nu \frac {c}{4 \ell^ {2}}\right) \bar {u} \tag {5.4-9}
+$$
+
+Clearly these strains are quite wrong. Despite their poor performance in bending, CST elements can adequately model a beam in bending if a great many elements are used through the depth of the beam.
+
+Figure 5.4-2 gives numerical evidence of how the CST behaves. Theoretical values allow for transverse shear deformation and assume that the end at $x = 0$ is free to warp while points $C, D$ , and $E$ remain on a vertical line.
+
+Elements better than the CST are available. Routine use of the CST in stress analysis is not recommended.
+
+# 5.5 THE QUADRATIC TRIANGLE
+
+The quadratic triangle with straight sides and midside nodes, Fig. 5.5-1, dates from 1964. It is an excellent element for stress analysis. Its strain field contains a complete linear polynomial for $\epsilon_{x}$ , $\epsilon_{y}$ , and $\gamma_{xy}$ . Accordingly, it is also known as the linear-strain triangle. Its sides can deform into quadratic curves, which means that a uniform edge traction is allocated to nodal loads $\{r_{e}\}$ in the 1–4–1 proportion seen in Fig. 4.3-4.
+
+In the present section we formulate the element stiffness matrix under the restrictions that sides are straight and side nodes are at midsides. In Section 6.8 these restrictions are removed.
+
+For convenience of notation in the following development, we arrange nodal d.o.f. in the order
+
+$$
+\{\mathbf {d} \} = \left\lfloor u _ {1} \quad u _ {2} \quad u _ {3} \quad u _ {4} \quad u _ {5} \quad u _ {6} \quad v _ {1} \quad v _ {2} \quad v _ {3} \quad v _ {4} \quad v _ {5} \quad v _ {6} \right] ^ {T} \tag {5.5-1}
+$$
+
+The $x$ - and $y$ -direction displacement fields are
+
+$$
+u = \sum_ {i = 1} ^ {6} N _ {i} u _ {i} \quad \text { and } \quad v = \sum_ {i = 1} ^ {6} N _ {i} v _ {i} \tag {5.5-2}
+$$
+
+![](images/page-177_d3e271bd9d7ed49514b2136fc516b27eec077ef8451c06717082c708bcfc0679.jpg)
+
+<details>
+<summary>text_image</summary>
+
+3
+6
+1
+4
+5
+2
+</details>
+
+(a)
+
+![](images/page-177_a626e2a037b5092a72197c6cc0b88cfc33d1eca29a9bc3150689f4b98e5e8be2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+φ₂
+</details>
+
+(b)
+
+![](images/page-177_82ea563ad6e5bda862c35006ff147f58be8f42dc24a1f7aeeef04bcf9a109b28.jpg)
+
+<details>
+<summary>text_image</summary>
+
+φ₅
+</details>
+
+(c)   
+Figure 5.5-1. Quadratic triangle. (a) Node numbering. (b) A vertex-node shape function. (c) A side-node shape function.
+
+<!-- source-page: 178 -->
+
+where the $N_{i}$ are given by Eqs. 5.3-5. Next
+
+$$
+\epsilon_ {x} = \frac {\partial u}{\partial x} = \sum_ {i = 1} ^ {6} \frac {\partial N _ {i}}{\partial x} u _ {i} \quad \text { and } \quad \frac {\partial N _ {i}}{\partial x} = \sum_ {j = 1} ^ {3} \frac {\partial N _ {i}}{\partial \xi_ {j}} \frac {\partial \xi_ {j}}{\partial x} \tag {5.5-3}
+$$
+
+where the latter equation is a restatement of Eq. 5.2-6a. Thus, from Eqs. 5.2-7a, 5.3-5, and 5.5-3,
+
+$$
+\begin{array}{l} \epsilon_ {x} = \frac {1}{2 A} \left[ (4 \xi_ {1} - 1) y _ {2 3} \quad (4 \xi_ {2} - 1) y _ {3 1} \quad (4 \xi_ {3} - 1) y _ {1 2} \right. \tag {5.5-4} \\ 4 \left(\xi_ {2} y _ {2 3} + \xi_ {1} \dot {y} _ {3 1}\right) \quad 4 \left(\xi_ {3} y _ {3 1} + \xi_ {2} y _ {1 2}\right) \quad 4 \left(\xi_ {1} y _ {1 2} + \xi_ {3} y _ {2 3}\right) \Bigg \} \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \\ u _ {3} \\ u _ {4} \\ u _ {5} \\ u _ {6} \end{array} \right\} \\ \end{array}
+$$
+
+Expressions for strains $\epsilon_{y}$ and $\gamma_{xy}$ are developed similarly. The complete result is
+
+$$
+\{\epsilon \} = \left\{ \begin{array}{l} \epsilon_ {x} \\ \epsilon_ {y} \\ \gamma_ {x y} \end{array} \right\} = [ \mathbf {B} ] \{\mathbf {d} \}, \quad \text { where } \quad \left[ \mathbf {B} \right] _ {3 \times 1 2} = \left[ \begin{array}{l l} \mathbf {B} _ {x} & \mathbf {0} \\ \mathbf {0} & \mathbf {B} _ {y} \\ \mathbf {B} _ {y} & \mathbf {B} _ {x} \end{array} \right] \tag {5.5-5}
+$$
+
+$\mathbf{B}_y$ is the row matrix in Eq. 5.5-4 and $\mathbf{B}_y$ is a similar row matrix.
+
+$B_{x}$ is the row matrix in Eq. 5.5.1 and by $\epsilon$ . Because each strain in $\{\epsilon\}$ is a complete linear field, the following convenient trick can be used [5.5]. We interpolate strains $\{\epsilon\}$ from strains $\{\epsilon_{c}\}$ at corner nodes 1, 2, and 3,
+
+$$
+\{\boldsymbol {\epsilon} \} = [ \mathbf {Q} ] \{\boldsymbol {\epsilon} _ {c} \} = [ \mathbf {Q} ] \left[ \begin{array}{l l l l l l l l l} \epsilon_ {x 1} & \epsilon_ {x 2} & \epsilon_ {x 3} & \epsilon_ {y 1} & \epsilon_ {y 2} & \epsilon_ {y 3} & \gamma_ {x y 1} & \gamma_ {x y 2} & \gamma_ {x y 3} \end{array} \right] ^ {T} \tag {5.5-6}
+$$
+
+where $\epsilon_{x1}$ is $\epsilon_x$ at node 1, and so on, and, making use of Eq. 5.3-4,
+
+$$
+[ \mathbf {Q} ] = \left[ \begin{array}{c c c c c c c c c} \xi_ {1} & \xi_ {2} & \xi_ {3} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \xi_ {1} & \xi_ {2} & \xi_ {3} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \xi_ {1} & \xi_ {2} & \xi_ {3} \end{array} \right] \tag {5.5-7}
+$$
+
+Corner strains $\{\epsilon_c\}$ are written in terms of nodal d.o.f. $\{\mathbf{d}\}$ by evaluating Eq. 5.5-5 at nodes 1, 2, and 3. Thus
+
+$$
+\{\epsilon_ {c} \} = [ \mathbf {H} ] \{\mathbf {d} \} \quad \text { and } \quad \{\epsilon \} = [ \mathbf {Q} ] [ \mathbf {H} ] \{\mathbf {d} \} \tag {5.5-8}
+$$
+
+where [H] is a 9 by 12 matrix of element dimensions: $H_{11} = 3y_{23} / 2A$ , $H_{12} = -y_{31} / 2A$ , and so on. Combining Eqs. 5.5-6 and 5.5-8, we obtain
+
+$$
+\{\epsilon \} = [ \mathbf {B} ] \{\mathbf {d} \}, \quad \text { where } \quad \underset {3 \times 1 2} {[ \mathbf {B} ]} = \underset {3 \times 9} {[ \mathbf {Q} ]} \underset {9 \times 1 2} {[ \mathbf {H} ]} \tag {5.5-9}
+$$
+
+The element stiffness matrix is
+
+<!-- source-page: 179 -->
+
+![](images/page-179_66b1c596746b7c4ac018c9df5132e0609bcbbcbd8eca877cbe28b1b33b252d9a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+4
+10
+8
+9
+3
+7
+6
+1
+5
+2
+</details>
+
+<table><tr><td>Node</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td></tr><tr><td> $\xi_1$ </td><td>1</td><td>0</td><td>0</td><td>0</td><td> $\frac{1}{2}$ </td><td>0</td><td> $\frac{1}{2}$ </td><td> $\frac{1}{2}$ </td><td>0</td><td>0</td></tr><tr><td> $\xi_2$ </td><td>0</td><td>1</td><td>0</td><td>0</td><td> $\frac{1}{2}$ </td><td> $\frac{1}{2}$ </td><td>0</td><td>0</td><td> $\frac{1}{2}$ </td><td>0</td></tr><tr><td> $\xi_3$ </td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td> $\frac{1}{2}$ </td><td> $\frac{1}{2}$ </td><td>0</td><td>0</td><td> $\frac{1}{2}$ </td></tr><tr><td> $\xi_4$ </td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td> $\frac{1}{2}$ </td><td> $\frac{1}{2}$ </td><td> $\frac{1}{2}$ </td></tr></table>
+
+Figure 5.6-1. The quadratic tetrahedron.
+
+$$
+[ \mathbf {k} ] = \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] [ \mathbf {B} ] d V = [ \mathbf {H} ] ^ {T} \int_ {V _ {e}} [ \mathbf {Q} ] ^ {T} [ \mathbf {E} ] [ \mathbf {Q} ] d V [ \mathbf {H} ] \tag {5.5-10}
+$$
+
+where $dV = t \, dA$ and $t$ is the element thickness. Again, note the ordering of nodal d.o.f. assumed in Eq. 5.5-1. The latter form of Eq. 5.5-10 can be used to advantage in programming. If $t$ is constant or a polynomial in area coordinates, integrations are easily done by means of Eq. 5.2-8.
+
+Behaviors of the constant- and linear-strain triangles are compared in Fig. 5.4-2. We see that the linear-strain triangle is much better in this problem despite using fewer d.o.f.
+
+Fortran coding for straight-sided quadratic triangles of constant thickness may be found in [5.5]. Alternative compact coding for linear, quadratic, and cubic triangles is discussed in [5.6].
+
+# 5.6 THE QUADRATIC TETRAHEDRON
+
+Tetrahedral elements are a straightforward extension of triangular elements. Accordingly, we cite only the quadratic element, Fig. 5.6-1, whose shape functions are
+
+$$
+N _ {i} = \xi_ {i} (2 \xi_ {i} - 1) \quad \text { for } i = 1, 2, 3, 4
+$$
+
+$$
+N _ {5} = 4 \xi_ {1} \xi_ {2} \quad N _ {6} = 4 \xi_ {2} \xi_ {3} \quad N _ {7} = 4 \xi_ {3} \xi_ {1} \tag {5.6-1}
+$$
+
+$$
+N _ {8} = 4 \xi_ {1} \xi_ {4} \quad N _ {9} = 4 \xi_ {2} \xi_ {4} \quad N _ {1 0} = 4 \xi_ {3} \xi_ {4}
+$$
+
+Equation 5.5-10 again describes the element stiffness matrix, where $[k]$ is now 30 by 30, $[E]$ is 6 by 6, and $[B]$ is 6 by 30.
+
+# PROBLEMS
+
+# Section 5.1
+
+5.1 (a) Derive Eq. 5.1-6 by using the substitution procedure suggested below that equation.
+
+(b) Similarly, derive shape functions for cubic interpolation along a line. Nodes are at $\xi_{1} = 1$ , $\xi_{1} = \frac{2}{3}$ , $\xi_{1} = \frac{1}{3}$ , and $\xi_{1} = 0$ . Start with $\phi = a_{1}\xi_{1}^{3} + a_{2}\xi_{1}^{2}\xi_{2} + a_{3}\xi_{1}\xi_{2}^{2} + a_{4}\xi_{2}^{3}$ .
+
+<!-- source-page: 180 -->
+
+5.2 Use Eq. 5.1-10 to integrate the following functions over span $L$ in Fig. 5.1-1.
+
+(a) $x^{2}$   
+(b) $\xi_{1}^{3}\xi_{2}^{2}$   
+(c) $\xi_1x$
+
+5.3 In Eq. 5.1-9, let $A$ vary linearly from $A_1$ at node 1 to $A_2$ at node 2. Determine the resulting stiffness matrix [k].
+
+5.4 In Eq. 5.1-6, let $\phi$ represent axial displacement $u$ of a uniform bar element. Determine the 3 by 3 element stiffness matrix. Let node 3 lie at the center of the bar (hence, $\xi_{1}$ and $\xi_{2}$ are linear functions of $x$ ).
+
+# Section 5.2
+
+5.5 Let $x_{1}$ and $y_{1}$ in [A] of Eq. 5.2-4 be replaced by $x$ and $y$ . Then, Eq. 5.2-5 yields $2A_{1} = \det[A]$ . Similar expressions for $2A_{2}$ and $2A_{3}$ can be written. Hence, $\left|\xi_{1} \xi_{2} \xi_{3}\right|^{T} = \left[2A_{1} 2A_{2} 2A_{3}\right]^{T} / 2A = [A]^{-1}\left[1 x y\right]^{T}$ . In this way verify the expression for $[A]^{-1}$ in Eq. 5.2-4.   
+5.6 (a) Verify that $[\mathbf{A}]^{-1}$ is correctly stated in Eq. 5.2-4 by forming the product $[\mathbf{A}][\mathbf{A}]^{-1}$ .   
+(b) Show that the equation of side 1 of the triangle is $x_{2}y_{3} - x_{3}y_{2} + y_{23}x + x_{32}y = 0$ .   
+(c) Derive the latter expression for 2A in Eq. 5.2-5 from det[A].   
+(d) In Fig. 5.2-1, drop lines from the triangle vertices to the $x$ axis. One can now identify three trapezoids, whose upper sides are 1-2, 2-3, and 3-1. Two trapezoidal areas minus a third equals the triangle area. Hence, derive the expression $2A = x_{21}y_{31} - x_{31}y_{21}$ in Eq. 5.2-5.   
+(e) Is it also true that $2A = x_{32}y_{12} - x_{12}y_{32}$ ? Explain.   
+5.7 Let the function $\phi = 27\xi_1\xi_2\xi_3$ be defined over the triangle in Fig. 5.2-1.   
+(a) Sketch this function in isometric view (analogous to Fig. 1.1-3).   
+(b) Integrate $\phi$ over the triangle area $A$ .   
+5.8 Let a function $\phi$ vary linearly over face 1-2-3 of a tetrahedron. Values of $\phi$ at corner nodes on this face are $\phi_1, \phi_2$ , and $\phi_3$ . The integral of $\phi$ over face 1-2-3 is $A_{123}(\phi_1 + \phi_2 + \phi_3)/3$ . Derive this result by formal integration.   
+5.9 Show that Eq. 5.2-10 is produced by Eq. 5.2-8.   
+5.10 For a triangle having the dimensions shown, use formulas for area moments and products of inertia to numerically verify Eq. 5.2-9 for the case $r + s = 2$ .
+
+![](images/page-180_bcd0bcec67613c302c64b75e5f9a32626c404e18a659cf576b71f12895247a7e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+3
+6
+x
+1
+3
+2
+4
+4
+</details>
+
+Problem 5.10
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+<!-- source-page: 181 -->
+
+5.11 For $r + s = 1$ and again for $r + s = 2$ , write integration formulas analogous to Eq. 5.2-9 if the origin of coordinates need not be at the centroid of the triangle. (Let the centroid have coordinates $x = x_{c}$ and $y = y_{c}$ .)
+
+# Section 5.3
+
+5.12 For a cubic triangle (10 nodes), write an equation analogous to Eq. 5.3-2.
+
+5.13 From Eq. 5.3-2, derive the shape functions of Eqs. 5.3-5.
+
+# Section 5.4
+
+5.14 (a) For a linear triangle, the equation that corresponds to Eq. 5.3-3 is $\phi = b_{1} + b_{2}x + b_{3}y$ . From this, and without using area coordinates, obtain shape functions $N_{1}, N_{2}$ , and $N_{3}$ in terms of $x, y$ , and Cartesian coordinates $x_{i}$ and $y_{i}$ of the vertex nodes.   
+(b) Show that the $N_{i}$ of part (a) yield the $N_{i}$ of Eq. 5.3-4.   
+(c) Obtain [B] of Eq. 5.4-6 from the $N_{i}$ of part (a).
+
+5.15 Verify the results given in Eqs. 5.4-9. Start with Eqs. 5.4-8.
+
+# Section 5.5
+
+5.16 Write out the first three rows of matrix [H] in Eq. 5.5-8. Then show that the product [Q][H] in Eq. 5.5-9 yields the row matrix $\mathbf{B}_x$ defined by Eq. 5.5-4 (check at least the first row of the product).
+
+5.17 Consider use of the quadratic triangle for a scalar field problem (similar to Eqs. 5.4-1 through 5.4-3 for the linear triangle). Redefine matrices [Q] and [H] of the quadratic triangle as may be necessary in order to obtain a result for a scalar field problem analogous to the latter form of Eq. 5.5-10. If material property $k$ and thickness $t$ are constant, determine $c$ and [M] in the form $[\mathbf{k}] = c[\mathbf{H}]^T[\mathbf{M}][\mathbf{H}]$ , where integrations have been completed to yield [M], and $c$ represents constants.
+
+5.18 Impose nodal displacements associated with u and v of Eqs. 5.4-8 on the element of Fig. 5.4-1, but consider that the element is a quadratic triangle (six nodes; Eqs. 5.5-2). For simplicity, consider the special case v = 0. Does the quadratic element display the correct strains?
+
+5.19 Let an edge of a quadratic triangle lie parallel to the $x$ axis. Determine the consistent nodal loads that result from the following distributed loads on this edge.
+
+(a) Uniform traction in the $y$ direction.   
+(b) Shear stress parallel to the edge, varying parabolically from a maximum at midedge to zero at the adjacent vertices.   
+(c) A traction in the $y$ direction that varies linearly from $+\overline{\sigma}$ at one vertex to $-\overline{\sigma}$ at the adjacent vertex.
+
+5.20 (a) Evaluate nodal loads $\{\mathbf{r}_e\}$ that result from heating of an isotropic quadratic triangle. Let the temperature vary linearly over the element and be defined by corner temperatures $T_{1}, T_{2}$ , and $T_{3}$ . Express $\{\mathbf{r}_e\}$ in terms of $E$ , $\alpha, \nu$ , and corner coordinates and temperatures.
+
+(b) Show that the nodal forces of part (a) are self-equilibrating—that is, show $\Sigma (r_e)_i = 0$ .
+
+<!-- source-page: 182 -->
+
+# Section 5.6
+
+5.21 The strain-displacement relation for a tetrahedron can be written in the form $\{\epsilon\} = [\mathbf{Q}][\mathbf{H}]\{\mathbf{d}\}$ (analogous to Eqs. 5.5-9). Write out matrices [Q] and [H] for a quadratic tetrahedron. Use the symbol $a_{ij}$ to denote coefficients of the inverse of the matrix in Eq. 5.2-11, but do not bother to compute the inverse.
+
+5.22 Let a uniform pressure $p$ act normal to one face of a quadratic tetrahedron. What are the consistent nodal loads?
+
+<!-- source-page: 183 -->
+
+# THE ISOPARAMETRIC FORMULATION
+
+The “isoparametric” formulation can be used to produce many types of useful elements. Plane isoparametric elements are emphasized in the present chapter. Numerical integration, used in element formulation, is described and its possible pitfalls are discussed.
+
+# 6.1 INTRODUCTION
+
+The isoparametric formulation makes it possible to generate elements that are nonrectangular and have curved sides. These shapes have obvious uses in grading a mesh from coarse to fine, in modeling arbitrary shapes, and in modeling curved boundaries (Fig. 6.1-1). The isoparametric family includes elements for plane, solid, plate, and shell problems. There are also special elements for fracture mechanics and elements for nonstructural problems.
+
+In formulating isoparametric elements, natural coordinate systems must be used (systems $\xi\eta$ and $\xi\eta\zeta$ in Fig. 6.1-2). Displacements are expressed in terms of natural
+
+![](images/page-183_fc47260ab05b923756610c742cdae2cb4260c6871351aea4fa3bc51f24abcbc5.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+3D wireframe model of a curved structural component with grid pattern (no text or symbols)
+</details>
+
+Figure 6.1-1. Turbine blade, modeled by solid elements. (Courtesy of NASA Lewis Research Center, Cleveland, Ohio.)
+
+<!-- source-page: 184 -->
+
+![](images/page-184_75a0cc63d5ed0ad2c11dd0b7c946b719500e1e2527a01e8aee0a5a9378d9b7fa.jpg)  
+{a}
+
+![](images/page-184_eb79fae73dc1a64c95ca6af5e0cd35d183e258e7c0f6ec4eda7d7d64086f9d48.jpg)  
+(b)
+
+![](images/page-184_1567c598c91986044b27d587eb8b93e1054181261403a42a99d41f62d892c7c2.jpg)  
+{c}
+
+![](images/page-184_2becd155863ad78e6e02f838bb3afab158974ee2ccf9a94f227cfac4d708b384.jpg)  
+{d}
+
+![](images/page-184_0d166d8c7e5c92d8e7fc509147cb79339be495e02a02cd613487372c81cde742.jpg)  
+{e}   
+Figure 6.1-2. Example isoparametric elements. (a) Quadratic plane element. (b) Cubic plane element. (c) A “degraded” cubic element. The left and lower sides can be joined to linear and quadratic elements. (d) Quadratic solid element with some linear edges. (e) A quadratic plane triangle.
+
+coordinates, but must be differentiated with respect to global coordinates $x, y$ , and $z$ . Accordingly, a transformation matrix, called [J], must be invoked. In addition, integrations must be done numerically rather than analytically if elements are nonrectangular. Closed-form integrations are possible in some special cases, but expressions tend to be lengthy, tedious to work out, and therefore more subject to errors of algebra and coding than numerical integration.
+
+The term “isoparametric” means “same parameters” and is explained as follows. Because either displacements or coordinates can be interpolated from nodal values,
+
+1. Nodal d.o.f. $\{\mathbf{d}\}$ define displacements $\lfloor u \quad v \quad w \rfloor$ of a point in the element; that is, $\left| u \quad v \quad w \right|^T = [\mathbf{N}]\{\mathbf{d}\}$ .   
+2. Nodal coordinates $\{\mathbf{c}\}$ define coordinates $\lfloor x\quad y\quad z\rfloor$ of a point in the element; that is, $\lfloor x\quad y\quad z\rfloor^T = [\tilde{\mathbf{N}}]\{\mathbf{c}\}$ .
+
+Shape function matrices [N] and [N] are functions of $\xi$ , $\eta$ , and $\zeta$ . An element is isoparametric if [N] and [N] are identical. If [N] is of higher degree than [N], the element is called subparametric, but if [N] is of lower degree than [N], the element is called superparametric.
+
+Isoparametric elements were developed by Taig in 1958 [1.10], but no work was published until 1966 [6.1].
+
+# 6.2 AN ISOPARAMETRIC BAR ELEMENT
+
+As a simple introduction to isoparametric elements, consider a straight, three-node element, Fig. 6.2-1a. Coordinate $\xi$ is a natural or intrinsic coordinate: ends of the bar lie at $\xi = \pm 1$ , regardless of the physical length $L$ of the bar. Moreover, $\xi$ is attached to the bar and remains an axial coordinate regardless of how the bar
+
+![](images/page-184_c01d711ea83316c7ad7a5a897a7bddbe38327951de3b81d87e60f4f5baca264d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ξ = -1
+ξ = 0
+ξ = +1
+1
+3
+2
+x,u
+L
+</details>
+
+(a)
+
+![](images/page-184_07e1cce519e02827be42382192b7abedaf65c6c8bcbf191a88aee0d96044d8fd.jpg)
+
+![](images/page-184_17b4f99446afa285a60268b869fd46d457c535c4ec680fa9299e8af9ce21a85a.jpg)
+
+![](images/page-184_1ff7de3f6fda789152817873ed9552ce9e7aee86f587b4b4a4dcd73aeb8805c9.jpg)  
+(b)   
+Figure 6.2-1. (a) Three-node (quadratic) bar element with natural coordinate $\xi$ . (b) The three shape functions.
+
+<!-- source-page: 185 -->
+
+is oriented in global coordinates xyz. For convenience, not necessity, $\xi$ and x are collinear in the present example. Node 3 is at $\xi = 0$ , but need not be at the physical center of the bar.
+
+Coordinate $\xi$ in Fig. 6.2-1 differs from the natural coordinates $\xi_{1}$ and $\xi_{2}$ used in Section 5.1; $\xi$ is chosen in preference to $\xi_{1}$ and $\xi_{2}$ because it is more closely related to coordinates $\xi \eta$ used for plane quadrilateral elements in subsequent discussions.
+
+We begin with an assumed field, written in terms of the natural coordinate $\xi$ , for both coordinate $x$ and axial displacement $u$ :
+
+$$
+x = \left\lfloor 1 \quad \xi \quad \xi^ {2} \right\rfloor \left\{ \begin{array}{l} a _ {1} \\ a _ {2} \\ a _ {3} \end{array} \right\} \quad \text { and } \quad u = \left\lfloor 1 \quad \xi \quad \xi^ {2} \right\rfloor \left\{ \begin{array}{l} a _ {4} \\ a _ {5} \\ a _ {6} \end{array} \right\} \tag {6.2-1}
+$$
+
+where the $a_{i}$ are generalized coordinates. One can now determine shape functions by following the formal substitution procedure detailed in Eqs. 3.8-5 through 3.8-8. Another way to determine shape functions is to use Lagrange's interpolation formula, Eq. 3.12-3, replacing $x$ 's by $\xi$ 's. Or, finally, one can proceed largely by inspection. For example, note that the linear ramps $\frac{1}{2}(1 - \xi)$ and $\frac{1}{2}(1 + \xi)$ have magnitude $\frac{1}{2}$ at $\xi = 0$ (Fig. 3.12-1a). We require that $N_{1}$ and $N_{2}$ both vanish at $\xi = 0$ . Accordingly, if $N_{3} = 1 - \xi^{2}$ is known, we obtain $N_{1} = \frac{1}{2}(1 - \xi) - \frac{1}{2} N_{3}$ and $N_{2} = \frac{1}{2}(1 + \xi) - \frac{1}{2} N_{3}$ . By any of these procedures, we arrive at
+
+$$
+x = \left\lfloor \mathrm{N} \right\rfloor \left\lfloor x _ {1} \quad x _ {2} \quad x _ {3} \right\rfloor^ {T} \quad \text { and } \quad u = \left\lfloor \mathrm{N} \right\rfloor \left\lfloor u _ {1} \quad u _ {2} \quad u _ {3} \right\rfloor^ {T} \tag {6.2-2}
+$$
+
+where
+
+$$
+\lfloor \mathrm{N} \rfloor = \left[ \frac {1}{2} (- \xi + \xi^ {2}) \quad \frac {1}{2} (\xi + \xi^ {2}) \quad 1 - \xi^ {2} \right] \tag {6.2-3}
+$$
+
+The element is isoparametric because the same $\{N\}$ is used for interpolation of both x and u. To evaluate x or u at any point on the bar, we substitute the $\xi$ coordinate of that point into Eq. 6.2-2.
+
+Construction of the element stiffness matrix requires that the strain-displacement relation be known. Axial strain $\epsilon_{x}$ is
+
+$$
+\epsilon_ {x} = \frac {d u}{d x} = \left(\frac {d}{d x} [ N ]\right) \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \\ u _ {3} \end{array} \right\}, \quad \text { where } \quad \frac {d}{d x} = \frac {d \xi}{d x} \frac {d}{d \xi} \tag {6.2-4}
+$$
+
+The chain rule for $d / dx$ must be invoked because $\lfloor \mathbf{N} \rfloor$ is expressed in terms of $\xi$ rather than in terms of $x$ . Unfortunately, $d\xi / dx$ is not immediately available. We must first calculate its inverse, $dx / d\xi$ , from the first of Eqs. 6.2-2. Let $J = dx / d\xi$ . Then
+
+$$
+J = \frac {d}{d \xi} [ N ] \left\{ \begin{array}{l} x _ {1} \\ x _ {2} \\ x _ {3} \end{array} \right\} = \left\lfloor \frac {1}{2} (- 1 + 2 \xi) \quad \frac {1}{2} (1 + 2 \xi) \quad - 2 \xi \right] \left\{ \begin{array}{l} x _ {1} \\ x _ {2} \\ x _ {3} \end{array} \right\} \tag {6.2-5}
+$$
+
+J is called a Jacobian. It can be regarded as a scale factor that describes the physical length dx associated with a reference length $d\xi$ ; that is, dx = J $d\xi$ . The element stiffness matrix is
+
+<!-- source-page: 186 -->
+
+$$
+[ \mathbf {k} ] = \int_ {0} ^ {L} \left\lfloor \mathbf {B} \right] ^ {T} A E \left\lfloor \mathbf {B} \right\rfloor d x = \int_ {- 1} ^ {1} \left\lfloor \mathbf {B} \right] ^ {T} A E \left\lfloor \mathbf {B} \right\rfloor J d \xi \tag {6.2-6}
+$$
+
+where, from Eqs. 6.2-4 and 6.2-5,
+
+$$
+\left\lfloor \mathrm{B} \right] = \frac {1}{J} \frac {d}{d \xi} \left\lfloor \mathrm{N} \right] = \frac {1}{J} \left\lfloor \frac {1}{2} (- 1 + 2 \xi) \quad \frac {1}{2} (1 + 2 \xi) \quad - 2 \xi \right\rfloor \tag {6.2-7}
+$$
+
+Only if node 3 is at the middle of the bar does $J$ reduce to the constant value $J = L / 2$ . The specific form of $J$ depends on the numerical values assigned to $x_{1}$ , $x_{2}$ , and $x_{3}$ in Eq. 6.2-5. In general, $J$ is a function of $\xi$ . Accordingly, $[\mathbf{B}]$ contains $\xi$ in both numerator and denominator of every term. Therefore, Eq. 6.2-6 cannot be conveniently integrated in closed form. In practice, numerical integration is used instead.
+
+The preceding example illustrates some of the concepts and manipulations associated with isoparametric elements, but shows none of their versatility. For this we must consider plane and solid elements.
+
+# 6.3 PLANE BILINEAR ISOPARAMETRIC ELEMENT
+
+The following development generalizes the four-node element of Fig. 4.2-4 from a rectangle to arbitrary quadrilateral shape. For a rectangular element of side lengths $2a$ and $2b$ , with $x = 0$ and $y = 0$ at the element center, isoparametric coordinates $\xi$ and $\eta$ can be regarded as dimensionless Cartesian coordinates $\xi = x/a$ and $\eta = y/b$ . This special case can be used as a study aid in the following discussion.
+
+Isoparametric coordinates in a plane are shown in Fig. 6.3-1a. For a four-node element, axes $\xi$ and $\eta$ pass through midpoints of opposite sides. Axes $\xi$ and $\eta$ need not be orthogonal, and neither need be parallel to the $x$ axis or the $y$ axis. Sides of the element are at $\xi = \pm 1$ and at $\eta = \pm 1$ . Coordinates $x$ and $y$ within the element are defined by
+
+![](images/page-186_1408262f5da585ccc2a81b0d9237b64e775821384cdf4e47f0b520e044c1fb49.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ξ = -1
+ξ = -1/2
+η
+ξ = 1/2
+ξ = 1
+η = 1
+3
+η = 1/2
+y, v
+ξ
+η = -1/2
+1
+2
+η = -1
+x, u
+</details>
+
+(a)
+
+![](images/page-186_d0705c2f7bb02fc42603bc6493a736728820a23dba74f20fd1c995e6f5c28a2e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+η
+← 1 → ← 1 →
+4 3
+1 2
+1
+ξ
+1
+</details>
+
+(b)   
+Figure 6.3-1. (a) Four-node plane isoparametric element in $xy$ space. (b) Plane isoparametric element in $\xi \eta$ space.
+
+<!-- source-page: 187 -->
+
+$$
+x = \sum N _ {i} x _ {i} \quad \text { and } \quad y = \sum N _ {i} y _ {i} \tag {6.3-1}
+$$
+
+Summations run from 1 to 4. Individual shape functions are
+
+$$
+N _ {1} = \frac {1}{4} (1 - \xi) (1 - \eta) \quad N _ {2} = \frac {1}{4} (1 + \xi) (1 - \eta) \tag {6.3-2}
+$$
+
+$$
+N _ {3} = \frac {1}{4} (1 + \xi) (1 + \eta) \quad N _ {4} = \frac {1}{4} (1 - \xi) (1 + \eta)
+$$
+
+These $N_{i}$ are clearly similar to those in Eq. 3.12-10; indeed they are identical for the special case $\xi = x / a$ and $\eta = y / b$ . Equations 6.3-2 can be established by any of the usual methods: formal substitution, Lagrange's interpolation formula, or inspection and trial.
+
+For a given element geometry, the orientation of $\xi \eta$ axes with respect to $xy$ axes is dictated by Eqs. 6.3-2 and the node numbers assigned to the element at hand. For example, in Fig. 6.3-1a, a cyclic change in node numbers (2 changed to 1, 3 changed to 2, etc.) would place the $\xi$ axis where the $\eta$ axis is now shown and would place the $\eta$ axis in the present $-\xi$ direction. Figure 6.3-1b would not be changed: because of Eqs. 6.3-2, node 1 is always at $\xi = \eta = -1$ , node 2 always at $\xi = 1$ and $\eta = -1$ , and so on.
+
+The point $\xi = \eta = 0$ can be regarded as the center of the element, but it is not in general the centroid of the element area.
+
+Scalar Field Element. Let the field quantity be $\phi$ , where $\phi = \phi(x,y)$ or $\phi = \phi(\xi,\eta)$ . For example, in heat conduction analysis, $\phi$ represents temperature. The simplest element has one d.o.f. per node. Within a four-node element, $\phi$ is interpolated from nodal values $\{\phi_e\} = \lfloor \phi_1 \quad \phi_2 \quad \phi_3 \quad \phi_4 \rfloor^T$ ,
+
+$$
+\phi = \lfloor \mathrm{N} \rfloor \left\{\phi_ {e} \right\} \quad \text { or } \quad \phi = \sum N _ {i} \phi_ {i} \tag {6.3-3}
+$$
+
+where, to make the element isoparametric, the $N_{i}$ are taken from Eq. 6.3-2. The following derivatives of $\phi$ are needed in element formulation:
+
+$$
+\left\{ \begin{array}{l} \phi_ {, x} \\ \phi_ {, y} \end{array} \right\} = [ \mathbf {B} ] \left\{\phi_ {e} \right\}, \quad \text { where } \quad [ \mathbf {B} ] = \left[ \begin{array}{c c c c} N _ {1, x} & N _ {2, x} & N _ {3, x} & N _ {4, x} \\ N _ {1, y} & N _ {2, y} & N _ {3, y} & N _ {4, y} \end{array} \right] \tag {6.3-4}
+$$
+
+For the element characteristic matrix we refer to Eq. 3.10-9. With k a material property and t the element thickness,
+
+$$
+[ \mathbf {k} ] = \int \int [ \mathbf {B} ] ^ {T} k [ \mathbf {B} ] t d x d y = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} [ \mathbf {B} ] ^ {T} k [ \mathbf {B} ] t J d \xi d \eta \tag {6.3-5}
+$$
+
+in which J arises because of the change of coordinates. $^{1}$ In the latter form of Eq. 6.3-5, [B] is a function of $\xi$ and $\eta$ , and is determined as follows.
+
+Because $\phi$ is expressed in terms of $\xi$ and $\eta$ , not $x$ and $y$ , derivatives needed in Eq. 6.3-4 are not immediately available. We therefore begin by taking derivatives with respect to $\xi$ and $\eta$ instead:
+
+<!-- source-page: 188 -->
+
+$$
+\left\{ \begin{array}{l} \phi_ {, \xi} \\ \phi_ {, \eta} \end{array} \right\} = [ \mathbf {D} _ {N} ] \left\{\phi_ {e} \right\}, \quad \text { where } \quad [ \mathbf {D} _ {N} ] = \left[ \begin{array}{c c c c} N _ {1, \xi} & N _ {2, \xi} & N _ {3, \xi} & N _ {4, \xi} \\ N _ {1, \eta} & N _ {2, \eta} & N _ {3, \eta} & N _ {4, \eta} \end{array} \right] \tag {6.3-6}
+$$
+
+From Eqs. 6.3-2, $N_{1,\xi} = -(1 - \eta)/4$ , $N_{1,\eta} = -(1 - \xi)/4$ , and so on. Next we must relate $\phi_{, \xi}$ and $\phi_{, \eta}$ to $\phi_{, x}$ and $\phi_{, y}$ . The necessary relation is derived subsequently and has the form
+
+$$
+\left\{ \begin{array}{l} \phi_ {, x} \\ \phi_ {, y} \end{array} \right\} = [ \Gamma ] \left\{ \begin{array}{l} \phi_ {, \xi} \\ \phi_ {, \eta} \end{array} \right\} \tag {6.3-7}
+$$
+
+which provides the [B] matrix as
+
+$$
+[ \mathbf {B} ] = [ \boldsymbol {\Gamma} ] [ \mathbf {D} _ {N} ] \tag {6.3-8}
+$$
+
+Jacobian Matrix [J]. We now seek an expression for $[\Gamma]$ in Eq. 6.3-7. By the chain rule,
+
+$$
+\frac {\partial \phi}{\partial x} = \frac {\partial \phi}{\partial \xi} \frac {\partial \xi}{\partial x} + \frac {\partial \phi}{\partial \eta} \frac {\partial \eta}{\partial x} \quad \text {and} \quad \frac {\partial \phi}{\partial y} = \frac {\partial \phi}{\partial \xi} \frac {\partial \xi}{\partial y} + \frac {\partial \phi}{\partial \eta} \frac {\partial \eta}{\partial y} \tag {6.3-9}
+$$
+
+Thus $\Gamma_{11} = \xi_{,x}, \Gamma_{12} = \eta_{,x}, \Gamma_{21} = \xi_{,y}$ and $\Gamma_{22} = \eta_{,y}$ . Unfortunately, the partial derivatives of $\xi$ and $\eta$ with respect to $x$ and $y$ are not directly available from our equations. Therefore, we must write the inverse of Eq. 6.3-7 first, which is easily done. We write
+
+$$
+\begin{array}{l} \frac {\partial \phi}{\partial \xi} = \frac {\partial \phi}{\partial x} \frac {\partial x}{\partial \xi} + \frac {\partial \phi}{\partial y} \frac {\partial y}{\partial \xi} \\ \frac {\partial \phi}{\partial \eta} = \frac {\partial \phi}{\partial x} \frac {\partial x}{\partial \eta} + \frac {\partial \phi}{\partial y} \frac {\partial y}{\partial \eta} \end{array} \quad \text {or} \quad \left\{ \begin{array}{l} \phi , \xi \\ \phi , \eta \end{array} \right\} = [ J ] \left\{ \begin{array}{l} \phi , x \\ \phi , y \end{array} \right\} \tag {6.3-10}
+$$
+
+where [J] is called the Jacobian matrix:
+
+$$
+[ \mathbf {J} ] = \left[ \begin{array}{l l} x, _ {\xi} & y, _ {\xi} \\ x, _ {\eta} & y, _ {\eta} \end{array} \right] = \left[ \begin{array}{l l} \sum N _ {i, \xi} x _ {i} & \sum N _ {i, \xi} y _ {i} \\ \sum N _ {i, \eta} x _ {i} & \sum N _ {i, \eta} y _ {i} \end{array} \right] \tag {6.3-11}
+$$
+
+Equation 6.3-11 is valid for all plane isoparametric elements, where i ranges over the number of nodes (and shape functions) used to define element geometry. For the four-node element at hand, i runs from 1 to 4, so, from Eqs. 6.3-6 and 6.3-11,
+
+$$
+[ \mathbf {J} ] = [ \mathbf {D} _ {N} ] \left[ \begin{array}{l l} x _ {1} & y _ {1} \\ x _ {2} & y _ {2} \\ x _ {3} & y _ {3} \\ x _ {4} & y _ {4} \end{array} \right] \tag {6.3-12}
+$$
+
+in which, for the bilinear element,
+
+$$
+\left[ \mathbf {D} _ {N} \right] = \frac {1}{4} \left[ \begin{array}{c c c c} - (1 - \eta) & (1 - \eta) & (1 + \eta) & - (1 + \eta) \\ - (1 - \xi) & - (1 + \xi) & (1 + \xi) & (1 - \xi) \end{array} \right] \tag {6.3-13}
+$$
+
+<!-- source-page: 189 -->
+
+Matrix [Γ] is the inverse of [J],
+
+$$
+[ \Gamma ] = [ J ] ^ {- 1} = \frac {1}{J} \left[ \begin{array}{c c} J _ {2 2} & - J _ {1 2} \\ - J _ {2 1} & J _ {1 1} \end{array} \right] \tag {6.3-14}
+$$
+
+where J is the determinant of the Jacobian matrix
+
+$$
+J = \det [ J ] = J _ {1 1} J _ {2 2} - J _ {2 1} J _ {1 2} \tag {6.3-15}
+$$
+
+Jacobian J can be regarded as a scale factor that yields area dx dy from $d\xi d\eta$ . In general, J is a function of $\xi$ and $\eta$ , but for rectangles and parallelograms it is constant.
+
+All ingredients are now at hand for evaluation of $[k]$ according to the second form of Eqs. 6.3-5.
+
+Plane Stress Element. There are now two fields—namely, the displacements
+
+$$
+u = \sum N _ {i} u _ {i} \quad \text { and } \quad v = \sum N _ {i} v _ {i} \tag {6.3-16}
+$$
+
+where the $N_{i}$ are again the same as the functions used to define shape, Eqs. 6.3-1 and 6.3-2. Displacements u and v are x-parallel and y-parallel; they are not $\xi$ -parallel and $\eta$ -parallel.
+
+The strain-displacement relation is $\{\epsilon\} = [\mathbf{B}]\{\mathbf{d}\}$ , where $\{\mathbf{d}\} = [u_1 v_1 u_2 \cdots v_4]^T$ and [B] is the product of the rectangular matrices in the following three equations, which respectively state the strain-displacement relation (Eq. 1.5-6), an expanded form of Eq. 6.3-7, and an expanded form of Eq. 6.3-6:
+
+$$
+\{\epsilon \} = \left\{ \begin{array}{l} \epsilon_ {x} \\ \epsilon_ {y} \\ \gamma_ {x y} \end{array} \right\} = \left[ \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{array} \right] \left\{ \begin{array}{l} u _ {, x} \\ u _ {, y} \\ v _ {, x} \\ v _ {, y} \end{array} \right\} \tag {6.3-17}
+$$
+
+$$
+\left\{ \begin{array}{l} u _ {, x} \\ u _ {, y} \\ v _ {, x} \\ v _ {, y} \end{array} \right\} = \left[ \begin{array}{c c c c} \Gamma_ {1 1} & \Gamma_ {1 2} & 0 & 0 \\ \Gamma_ {2 1} & \Gamma_ {2 2} & 0 & 0 \\ 0 & 0 & \Gamma_ {1 1} & \Gamma_ {1 2} \\ 0 & 0 & \Gamma_ {2 1} & \Gamma_ {2 2} \end{array} \right] \left\{ \begin{array}{l} u _ {, \xi} \\ u _ {, \eta} \\ v _ {, \xi} \\ v _ {, \eta} \end{array} \right\} \tag {6.3-18}
+$$
+
+$$
+\left\{ \begin{array}{l} u, _ {\xi} \\ u, _ {\eta} \\ v, _ {\xi} \\ v, _ {\eta} \end{array} \right\} = \left[ \begin{array}{c c c c c c c c} N _ {1, \xi} & 0 & N _ {2, \xi} & 0 & N _ {3, \xi} & 0 & N _ {4, \xi} & 0 \\ N _ {1, \eta} & 0 & N _ {2, \eta} & 0 & N _ {3, \eta} & 0 & N _ {4, \eta} & 0 \\ 0 & N _ {1, \xi} & 0 & N _ {2, \xi} & 0 & N _ {3, \xi} & 0 & N _ {4, \xi} \\ 0 & N _ {1, \eta} & 0 & N _ {2, \eta} & 0 & N _ {3, \eta} & 0 & N _ {4, \eta} \end{array} \right] _ {8 \times 1} ^ {\{\mathbf {d} \}} \tag {6.3-19}
+$$
+
+Coefficients $\Gamma_{ij}$ are given by Eqs. 6.3-11 and 6.3-14.
+
+The element stiffness matrix, Eq. 4.1-5, is
+
+$$
+[ \mathbf {k} ] _ {8 \times 8} = \int \int_ {8 \times 3} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] _ {3 \times 3} [ \mathbf {B} ] _ {3 \times 8} t d x d y = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] [ \mathbf {B} ] t J d \xi d \eta \tag {6.3-20}
+$$
+
+<!-- source-page: 190 -->
+
+where t is the element thickness. Contributions to the element load vector $\{r_{e}\}$ , Eq. 4.1-6, include the terms
+
+$$
+\int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \left([ \mathbf {B} ] ^ {T} [ \mathbf {E} ] \left\{\boldsymbol {\epsilon} _ {0} \right\} - [ \mathbf {B} ] ^ {T} \left\{\boldsymbol {\sigma} _ {0} \right\} + [ \mathbf {N} ] ^ {T} \left\{\mathbf {F} \right\}\right) t J d \xi d \eta \tag {6.3-21}
+$$
+
+For convenience in computer programming, contributions to $\{r_{e}\}$ from surface tractions $\{\Phi\}$ are evaluated separately. For this, the manipulations associated with isoparametric coordinates may be unnecessary. For example, linearly varying traction on a straight edge is allocated to nodes as shown in Fig. 4.3-3.
+
+Remarks. Isoparametric elements are geometrically isotropic. Thus, for the element of Fig. 6.3-1, the numerical values of coefficients in [k] do not depend on whether element nodes are labeled 1-2-3-4, 2-3-4-1, 3-4-1-2, or 4-1-2-3. However, the cyclic order must be maintained and must run counterclockwise if $J$ is not to become negative over part or all of the element.
+
+From Eqs. 6.3-8 and 6.3-14 we see that $J$ appears in the denominator of each coefficient $B_{ij}$ . Again, $J$ is in general a function of $\xi$ and $\eta$ . Therefore, the denominator of each term to be integrated in Eqs. 6.3-5 and 6.3-20 will in general contain a polynomial of the form $a_1 + a_2\xi + a_3\eta + a_4\xi\eta$ . (A polynomial of higher degree appears for elements having more than four nodes.) Closed-form expressions for integrals of terms in Eq. 6.3-20 would be lengthy. Accordingly, integration is done numerically instead, usually by formulas known as Gauss quadrature.
+
+# 6.4 SUMMARY OF GAUSS QUADRATURE
+
+"Quadrature" is the name applied to evaluating an integral numerically, rather than analytically as is done in tables of integrals. There are many quadrature rules. The reader may have encountered the Newton–Cotes rules such as Simpson's rule. Here we discuss only the Gauss rules, as they are most appropriate for elements discussed in this chapter.
+
+One Dimension. An integral having arbitrary limits can be transformed so that its limits are from -1 to +1. With $f = f(x)$ , and with the substitution $x = \frac{1}{2}(1 - \xi)x_{1} + \frac{1}{2}(1 + \xi)x_{2}$ ,
+
+$$
+I = \int_ {x _ {1}} ^ {x _ {2}} f d x \quad \text { becomes } \quad I = \int_ {- 1} ^ {1} \phi d \xi \tag {6.4-1}
+$$
+
+Thus the integrand is changed from $f = f(x)$ to $\phi = \phi(\xi)$ , where $\phi$ incorporates the Jacobian of the transformation, $J = dx / d\xi = \frac{1}{2}(x_2 - x_1)$ . The latter form of Eq. 6.4-1 makes it possible to write convenient quadrature formulas.
+
+The foregoing linear transformation suffices to make arbitrary limit changes. We can always consider a convenient reference interval such as $-1$ to $+1$ . In practice the limit change is done automatically by the isoparametric transformation and $J$ is usually more complicated than $\frac{1}{2}(x_2 - x_1)$ .
+
+To approximate the integral in the simplest way, one can sample (evaluate) $\phi$ at the midpoint $\xi = 0$ and multiply by the length of the interval (Fig. 6.4-1a). Thus we approximate the shaded area by a rectangular area of height $\phi_{1}$ and length
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_020.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_020.md
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_020.md
@@ -0,0 +1,614 @@
+<!-- source-page: 191 -->
+
+![](images/page-191_0b6958739654498c06dbf7ac609c23c1a7ed1f7270352f6d5072de6da96382b0.jpg)
+
+<details>
+<summary>text_image</summary>
+
+φ
+φ = φ(ξ)
+1
+φ₁
+-1 0 +1 ξ
+</details>
+
+(a)
+
+![](images/page-191_6ffc39094b72a7f7f76834807a7e22e6f0cca9b0c02cefb445ca3b9d931e8bcc.jpg)
+
+<details>
+<summary>text_image</summary>
+
+φ
+←ξ₁→←ξ₂→
+1
+2
+φ₁
+φ₂
+-1 0 +1 ξ
+</details>
+
+(b)
+
+![](images/page-191_685f410cee1b05a0a4abbac88dd165bd9db3bf55e67bcdf190d54b159486bb55.jpg)
+
+<details>
+<summary>text_image</summary>
+
+φ
+←ξ₁ → ←ξ₃ →
+1 2 3
+φ₁ φ₂ φ₃
+-1 0 +1 ξ
+</details>
+
+(c)   
+Figure 6.4-1. Gauss quadrature to compute the shaded area under the curve $\phi = \phi(\xi)$ , using (a) one, (b) two, and (c) three sampling points (also called Gauss points).
+
+2, so that $I \approx 2\phi_1$ . This result is exact if $\phi = \phi(\xi)$ happens to describe a straight line of any finite slope.
+
+Generalization of the foregoing procedure leads to the quadrature formula
+
+$$
+I = \int_ {- 1} ^ {1} \phi d \xi \approx W _ {1} \phi_ {1} + W _ {2} \phi_ {2} + \dots + W _ {n} \phi_ {n} \tag {6.4-2}
+$$
+
+Thus, to approximate I, we evaluate $\phi = \phi(\xi)$ at each of several locations $\xi_{i}$ to obtain ordinates $\phi_{i}$ , multiply each $\phi_{i}$ by an appropriate weight $W_{i}$ , and add. In the preceding one-point example, where $I \approx 2\phi_{1}$ , we have n = 1 and $W_{1} = 2$ . Gauss was able to prescribe the locations $\xi_{i}$ and weights $W_{i}$ such that greatest accuracy is achieved for a given n.
+
+Sampling points are located symmetrically with respect to the center of the integration interval. Symmetrically paired points have the same weight $W_{i}$ . Data appear in Table 6.4-1. These data are sometimes called Gauss–Legendre coefficients because sampling point locations happen to be roots of Legendre polynomials. Much more extensive tabulations are available [6.2]. In programming,
+
+TABLE 6.4-1. SAMPLING POINTS AND WEIGHTS FOR GAUSS QUADRATURE OVER THE INTERVAL $\xi = -1$ TO $\xi = +1$ .
+<table><tr><td>Order n</td><td>Location  $\xi_{i}$  of Sampling Point</td><td>Weight Factor  $W_{i}$ </td></tr><tr><td>1</td><td>0.</td><td>2.</td></tr><tr><td>2</td><td> $\pm 0.57735 \ 02691 \ 89626 = \pm 1/\sqrt{3}$ </td><td>1.</td></tr><tr><td>3</td><td> $\pm 0.77459 \ 66692 \ 41483 = \pm \sqrt{0.6}$ </td><td> $0.55555 \ 55555 \ 55555 = \frac{5}{9}$ </td></tr><tr><td></td><td>0.</td><td> $0.88888 \ 88888 \ 88888 = \frac{8}{9}$ </td></tr><tr><td>4</td><td> $\pm 0.86113 \ 63115 \ 94053 = \pm \left[ \frac{3 + 2r}{7} \right]^{1/2}$ </td><td> $0.34785 \ 48451 \ 37454 = \frac{1}{2} - \frac{1}{6r}$ </td></tr><tr><td></td><td> $\pm 0.33998 \ 10435 \ 84856 = \pm \left[ \frac{3 - 2r}{7} \right]^{1/2}$ </td><td> $0.65214 \ 51548 \ 62546 = \frac{1}{2} + \frac{1}{6r}$ </td></tr><tr><td></td><td>where  $r = \sqrt{1.2}$ </td><td></td></tr></table>
+
+<!-- source-page: 192 -->
+
+we first check the correctness of tabulated data, then code the $\xi_{i}$ and $W_{i}$ with as many digits as the computer allows, in order to avoid unnecessary rounding error.
+
+Example. Consider the polynomial $\phi = a_{1} + a_{2}\xi + a_{3}\xi^{2} + a_{4}\xi^{3}$ , where the $\cdot a_{i}$ are constants. The exact integral is
+
+$$
+I = \int_ {- 1} ^ {1} \phi d \xi = 2 a _ {1} + \frac {2}{3} a _ {3} \tag {6.4-3}
+$$
+
+The approximate integral given by a one-point rule is
+
+$$
+I _ {1} \approx 2 a _ {1} \tag {6.4-4}
+$$
+
+The integral given by a two-point rule is, with $\xi_1 = -p$ , $\xi_2 = p$ , and $p = 1 / \sqrt{3}$ ,
+
+$$
+I _ {2} = 1. 0 \left(a _ {1} - a _ {2} p + a _ {3} p ^ {2} - a _ {4} p ^ {3}\right) + 1. 0 \left(a _ {1} + a _ {2} p + a _ {3} p ^ {2} + a _ {4} p ^ {3}\right) \tag {6.4-5}
+$$
+
+$$
+I _ {2} = 2 a _ {1} + \frac {2}{3} a _ {3}
+$$
+
+In the foregoing example we see an instance of a general rule: a polynomial of degree $2n - 1$ is integrated exactly by $n$ -point Gauss quadrature. Use of more than $n$ points will still produce the exact result. The degree of precision of a quadrature rule is the degree of the highest-order polynomial that is exactly integrated. Thus a second-order Gauss rule has degree of precision 3.
+
+If the function $\phi = \phi(\xi)$ is not a polynomial, Gauss quadrature is inexact, but becomes more accurate as more points are used. Here we refer to the accuracy of integration, not to the accuracy of the results of finite element analysis. We will see, for example, that some elements used for stress analysis are improved by using a Gauss rule of lower order than would be chosen if the goal were accurate integration. The question of what quadrature order is best is addressed in Sections 6.11 and 6.12.
+
+It is important to realize that the ratio of two polynomials is in general not a polynomial, and therefore will not be integrated exactly by Gauss quadrature.
+
+Two and Three Dimensions. Multidimensional Gauss rules, called Gaussian product rules, are formed by successive application of one-dimensional Gauss rules. In two dimensions, consider the function $\phi = \phi(\xi, \eta)$ . We elect to integrate first with respect to $\xi$ and then with respect to $\eta$ :
+
+$$
+I = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \phi (\xi , \eta) d \xi d \eta \approx \int_ {- 1} ^ {1} \left[ \sum_ {i} W _ {i} \phi (\xi_ {i}, \eta) \right] d \eta \tag {6.4-6}
+$$
+
+$$
+\approx \sum_ {j} W _ {j} \left[ \sum_ {i} W _ {i} \phi (\xi_ {i}, \eta_ {j}) \right] = \sum_ {i} \sum_ {j} W _ {i} W _ {j} \phi (\xi_ {i}, \eta_ {j})
+$$
+
+For the four-point rule depicted in Fig. 6.4-2a, $W_{i}W_{j} = 1$ , and Eq. 6.4-6 becomes
+
+$$
+I \approx \phi_ {1} + \phi_ {2} + \phi_ {3} + \phi_ {4} \tag {6.4-7}
+$$
+
+where $\phi_{i}$ is the numerical value of $\phi$ at the ith Gauss point. For the nine-point rule depicted in Fig. 6.4-2b, Eq. 6.4-6 yields
+
+<!-- source-page: 193 -->
+
+![](images/page-193_f1184a167bc853fe4a8beb961c16deaf0d35c2a4648b644437eb346137c74541.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ξ = -1/√3
+η
+ξ = +1/√3
+2
+4
+η = +1/√3
+ξ
+1
+3
+η = -1/√3
+</details>
+
+(a)
+
+![](images/page-193_7dcff2e4212d651d9bcc0a01869010d1251109a7582f56ff047875e9c0101925.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ξ = -√0.6
+η
+ξ = +√0.6
+η = +√0.6
+3
+6
+9
+2
+5
+8
+ξ
+1
+4
+7
+η = -√0.6
+</details>
+
+(b)   
+Figure 6.4-2. Gauss point locations in a quadrilateral element using (a) four points (order 2 rule), and (b) nine points (order 3 rule).
+
+$$
+I \approx \frac {2 5}{8 1} \left(\phi_ {1} + \phi_ {3} + \phi_ {7} + \phi_ {9}\right) + \frac {4 0}{8 1} \left(\phi_ {2} + \phi_ {4} + \phi_ {6} + \phi_ {8}\right) + \frac {6 4}{8 1} \phi_ {5} \tag {6.4-8}
+$$
+
+In three dimensions, the Gauss quadrature rule has the form
+
+$$
+I = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \phi (\xi , \eta , \zeta) d \xi d \eta d \zeta \approx \sum_ {i} \sum_ {j} \sum_ {k} W _ {i} W _ {j} W _ {k} \phi (\xi_ {i}, \eta_ {j}, \zeta_ {k}) \tag {6.4-9}
+$$
+
+In an equation such as Eq. 6.3-20, each coefficient of the integrand $[B]^{T}[E][B]tJ$ is in general a function of $\xi$ and $\eta$ . There are 64 coefficients (or, 36 different coefficients, owing to symmetry), and each must be integrated as $\phi$ is integrated in Eq. 6.4-6: by evaluation at specific points, multiplication by weight factors, and addition. It is not necessary to use the same Gauss rule in both directions, but doing so is most common.
+
+# 6.5 COMPUTER SUBROUTINES FOR THE BILINEAR ISOPARAMETRIC ELEMENT
+
+The stiffness matrix of Eq. 6.3-20 and the load vector of Eq. 6.3-21 are evaluated by the subroutines presented here. Gauss quadrature is used. Of course, the coding is not unique: other procedures may be more compact, more general, or more efficient [6.3,6.4]. Nevertheless, the subroutines show precisely what must be done. A thorough understanding of these subroutines makes it easier to understand the isoparametric formulation in general.
+
+The notation, assumptions, and procedures are as follows. Shape functions of Eqs. 6.3-2 and their derivatives can be written in the form
+
+$$
+N _ {i} = (1 + \xi \xi_ {i}) (1 + \eta \eta_ {i}) / 4 \tag {6.5-1a}
+$$
+
+$$
+N _ {i, \xi} = \xi_ {i} (1 + \eta \eta_ {i}) / 4 \tag {6.5-1b}
+$$
+
+$$
+N _ {i, \eta} = \eta_ {i} (1 + \xi \xi_ {i}) / 4 \tag {6.5-1c}
+$$
+
+<!-- source-page: 194 -->
+
+```fortran
+SUBROUTINE SHAPE (PXI,PET,XL,YL,EXI,EYI,ESI,THC,THK,DETJAC)
+IMPLICIT DOUBLE PRECISION (A-H,O-Z)
+DOUBLE PRECISION NXI(4),NET(4),JAC
+DIMENSION XII(4),ETI(5)
+DIMENSION DUP(1),XL(4),YL(4),EXI(4),EYI(4),ESI(4),THC(4)
+COMMON /Q4/ EN(4),JAC(2,2),B(3,9),E(3,3),SE(8,8),RE(8)
+EQUIVALENCE (XII(1),ETI(2)),(DUP(1),JAC(1))
+DATA ETI /-1.D0,-1.D0,+1.D0,+1.D0,-1.D0/
+C --- Find shape functions (EN) and their derivatives (NXI,NET).
+DO 10 L=1,4
+DUM1 = (1. + XII(L)*PXI)/4.
+DUM2 = (1. + ETI(L)*PET)/4.
+EN(L) = 4.*DUM1*DUM2
+NXI(L) = XII(L)*DUM2
+10 NET(L) = ETI(L)*DUM1
+C --- Clear arrays JAC and B (words in common are stored in sequence).
+DO 20 L=1,31
+20 DUP(L) = 0.
+C --- Find Jacobian JAC and its determinant. Replace JAC by its inverse.
+DO 30 L=1,4
+JAC(1,1) = JAC(1,1) + NXI(L)*XL(L)
+JAC(1,2) = JAC(1,2) + NXI(L)*YL(L)
+JAC(2,1) = JAC(2,1) + NET(L)*XL(L)
+30 JAC(2,2) = JAC(2,2) + NET(L)*YL(L)
+DETJAC = JAC(1,1)*JAC(2,2) - JAC(2,1)*JAC(1,2)
+DUM1 = JAC(1,1)/DETJAC
+JAC(1,1) = JAC(2,2)/DETJAC
+JAC(1,2) = -JAC(1,2)/DETJAC
+JAC(2,1) = -JAC(2,1)/DETJAC
+JAC(2,2) = DUM1
+C --- Form strain-displacement matrix [B] (zero entries are already set).
+DO 40 J=1,4
+L = 2*J
+K = L-1
+B(1,K) = JAC(1,1)*NXI(J) + JAC(1,2)*NET(J)
+B(2,L) = JAC(2,1)*NXI(J) + JAC(2,2)*NET(J)
+B(3,K) = B(2,L)
+40 B(3,L) = B(1,K)
+C --- Interpolate initial strains from corner values, and store as column
+C --- 9 of [B] (already cleared to zero). Similarly, get thickness THK.
+THK = 0.
+DO 50 L=1,4
+B(1,9) = B(1,9) + EN(L)*EXI(L)
+B(2,9) = B(2,9) + EN(L)*EYI(L)
+B(3,9) = B(3,9) + EN(L)*ESI(L)
+50 THK = THK + EN(L)*THC(L)
+RETURN
+END 
+```  
+Figure 6.5-1. Fortran subroutine SHAPE. For the plane stress element of Section 6.3, it calculates the shape functions and their derivatives, the Jacobian matrix, its inverse and determinant, matrix [B], and initial strains and element thickness, all at the point whose $\xi$ and $\eta$ coordinates are PXI and PET.
+
+where i is the number of the shape function, and
+
+$$
+\xi_ {i} = - 1., 1., 1., - 1. \quad \text { for } \quad i = 1, 2, 3, 4 \tag {6.5-2}
+$$
+
+$$
+\eta_ {i} = - 1., - 1., 1., 1. \quad \text { for } \quad i = 1, 2, 3, 4
+$$
+
+In Fig. 6.5-1, $\xi_{i}$ and $\eta_{i}$ are placed in arrays XII and ETI by DATA and EQUIVALENCE statements. $N_{i}, N_{i,\xi}$ and $N_{i,\eta}$ are computed and stored in arrays EN, NXI, and NET. Because of the EQUIVALENCE statement containing DUP(1), the loop on statement 20 neatly initializes arrays JAC and B to zero. Coordinates $\xi$ and $\eta$ in Eqs. 6.5-1 are called PXI and PET in Fig. 6.5-1 and are transmitted as formal parameters. PXI and PET are Gauss point coordinates if SHAPE is called by QUAD4 (Fig. 6.5-2), but other coordinates could be prescribed by a subsequent calling routine (as, for example, when evaluating [B] for use in stress calculation by means of Eq. 4.7-1).
+
+<!-- source-page: 195 -->
+
+```fortran
+SUBROUTINE QUAD4 (NGAUSS,XL,YL,EXI,EYI,ESI,THC,PLACE,WGT,
+    1    BODYFX,BODYFY)
+    IMPLICIT DOUBLE PRECISION (A-H,O-Z)    .
+    DOUBLE PRECISION JAC
+C    E = material property matrix. Data at element nodes is as follows:
+C    XY,YL    = Cartesian coordinates.    (Used in subroutine SHAPE)
+C    EXI,EYI,ESI = initial strains (x,y,shear). (Used in subroutine SHAPE)
+C    THC    = thicknesses in z direction. (Used in subroutine SHAPE)
+C    The calling program must supply the following data in 3 by 3 arrays:
+C    [PLACE] =    0.    -.57735---    -.77459---    |    [WGT] =    2.    1.    .555---    |
+C    0.    +.57735---    0.    |    0.    1.    .888---    |
+C    0.    0.    +.77459---    |    0.    0.    .555---    |
+    COMMON /Q4/ EN(4),JAC(2,2),B(3,9),E(3,3),SE(8,8),RE(8)
+    DIMENSION XL(4),YL(4),EXI(4),EYI(4),ESI(4),THC(4),BTE(8,3),
+    1    PLACE(3,3),WGT(3,3)
+C --- Clear load vector {r} and upper triangle of stiffness matrix [k].
+    DO 10    K=1,8
+    RE(K) = 0.
+    DO 10    L=K,8
+    10    SE(K,L) = 0.
+C --- Start Gauss quadrature loop. Use NGAUSS by NGAUSS rule.
+    DO 90    NA = 1,NGAUSS
+    PXI = PLACE(NA,NGAUSS)
+    DO 80    NB = 1,NGAUSS
+    PET = PLACE(NB,NGAUSS)
+    CALL SHAPE (PXI,PET,XL,YL,EXI,EYI,ESI,THC,THK,DETJAC)
+    DV = WGT(NA,NGAUSS)*WGT(NB,NGAUSS)*THK*DETJAC
+C ---- Store [B]-transpose times [E] in 8 by 3 work array [BTE].
+    DO 30    J=1,4
+    L = 2*J
+    K = L-1
+C. ---- Do only multiplications that give a nonzero product.
+    DO 20    N=1,3
+    BTE(K,N) = B(1,K)*E(1,N) + B(3,K)*E(3,N)
+    20    BTE(L,N) = B(2,L)*E(2,N) + B(3,L)*E(3,N)
+C ---- Add contribution of body forces to nodal load array {r}.
+    RE(K) = RE(K) + EN(J)*BODYFX*DV
+    30    RE(L) = RE(L) + EN(J)*BODYFY*DV
+C ---- Loop on rows of [k] (array SE) and {r} (array RE).
+    DO 70    NROW=1,8
+C ---- Add contribution of initial strains to load array {r}.
+    DO 40    J=1,3
+    40    RE(NROW) = RE(NROW) + BTE(NROW,J)*B(J,9)*DV
+C ---- Loop to add contribution to element stiffness matrix [k].
+    DO 60    NCOL=NROW,8
+    DUM = 0.
+C ---- Loop for product [B]T*[E]*[B]. Zeros in [B] not skipped.
+    DO 50    J=1,3
+    50    DUM = DUM + BTE(NROW,J)*B(J,NCOL)
+    60    SE(NROW,NCOL) = SE(NROW,NCOL) + DUM*DV
+    70    CONTINUE
+    80    CONTINUE
+    90    CONTINUE
+C --- Fill in lower triangle of element stiffness matrix by symmetry.
+    DO 100    K=1,7
+    DO 100    L=K,8
+    100    SE(L,K) = SE(K,L)
+    RETURN
+    END 
+```  
+Figure 6.5-2. Fortran subroutine QUAD4. It generates $[k]$ and $\{r_{e}\}$ for the plane stress element of Section 6.3 by Gauss quadrature. We store $[k]$ in array SE and $\{r_{e}\}$ in array RE.
+
+Through statement 40, subroutine SHAPE follows exactly the development in Section 6.3. In the DO 50 loop, initial strains $(\epsilon_{x0} = \mathsf{EXI}, \epsilon_{y0} = \mathsf{EYI}, \gamma_{xy0} = \mathsf{ESI})$ and element thickness $(t = \mathsf{THC})$ are prescribed at the four nodes. Values of these quantities at coordinates PXI and PET are found by interpolation (write $t = \Sigma \cdot N_i t_i$ , analogous to Eqs. 6.3-1). Initial stresses $\{\sigma_0\}$ are not included but can be added as an exercise.
+
+The foregoing calculations in SHAPE must be carried out at every Gauss point used by QUAD4 (Fig. 6.5-2).
+
+<!-- source-page: 196 -->
+
+QUAD4 requires as input data the global nodal coordinates XL and YL, nodal thicknesses and initial strains, the material property matrix [E] (presumed full, as for a general material, and constant over the element), body forces $ F\_{x} = \text{BODYFX} $ and $ F\_{y} = \text{BODYFY} $, and the quadrature order NGAUSS. As indicated by comments in the listing, Gauss quadrature data must be coded before the subroutine is used. Although perhaps less obvious than subroutine SHAPE, subroutine QUAD4 is a straightforward application of Gauss quadrature to Eqs. 6.3-20 and 6.3-21. DV represents the product $ W\_{i}W\_{j}tJ $, which is a common multiplier of each coefficient to be integrated—that is, of each coefficient in [B]$ ^{T} $[E][B], in [B]$ ^{T} $[E]{\epsilon\_{0}} \), and in [N]$ ^{T} ${F}.
+
+Cost is reduced by using quadrature to generate only the upper triangle of [k], leaving the lower triangle to be completed by symmetry as the last step. Coding of the DO 20 loop is an attempt to exploit the sparsity of [B]. Additional economies have been proposed [6.3,6.4].
+
+# 6.6 QUADRATIC PLANE ELEMENTS
+
+One, two, or more nodes can be placed on each side of a four-node quadrilateral. The resulting elements are called quadratic, cubic, and so on. Here we discuss the quadratic quadrilateral element (Fig. 6.6-1). Quadratic triangular elements are discussed in Section 6.8.
+
+As with the bilinear element, sides of the quadratic element are at $\xi = \pm 1$ and at $\eta = \pm 1$ . Two of the side nodes are at $\xi = 0$ and two are at $\eta = 0$ . Axes $\xi$ and $\eta$ may be curved in a quadratic element. As shown in Fig. 6.6-1, sides of an undeformed element may be straight lines or quadratic curves. Similarly, displacements may be linear or quadratic. (A side capable of deforming quadratically need not actually do so; it would deform only linearly in passing a constant-strain patch test.)
+
+Shape functions of a quadratic element can be generated systematically [3.4], or by inspection and trial, as follows. In Fig. 6.6-2a, one obtains $N_5$ by interpolating quadratically in $\xi$ and linearly in $\eta$ , taking care that $N_5 = 1$ at node 5 and $N_5 = 0$ at all other nodes. Similarly, $N_8$ is obtained in Fig. 6.6-2b. Next, one observes that $N_{(c)}$ (which is $N_1$ of a bilinear element) has ordinate 0.5 at nodes 5 and 8. The function $N_{(c)} - \frac{1}{2} N_5 - \frac{1}{2} N_8$ is therefore zero at all nodes but node 1, where
+
+![](images/page-196_7de8a828ab0d92ddd9d7c682c54db5cf0146fdec07f9eb60567d50277428ed76.jpg)
+
+<details>
+<summary>text_image</summary>
+
+η
+7
+3
+4
+8
+6
+ξ
+1
+5
+2
+(a)
+</details>
+
+![](images/page-196_7da00936303b4fefa623507c070a3549b49cbfed5a6a085e8e645f56ce25d363.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+3
+4
+5
+6
+7
+8
+η
+ξ
+(b)
+</details>
+
+Figure 6.6-1. Quadratic plane elements. Those shown have (a) straight sides and midside nodes, and (b) some curved sides and off-center side nodes.
+
+<!-- source-page: 197 -->
+
+![](images/page-197_d9182987405e059f804ad40e1ad6524c9a3b00a7df57c7b863a3d491feadef99.jpg)
+
+![](images/page-197_0bddad7a3fb661d06e3446299deb7c52a9f08ee4e8668bc085a559430f6d967d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+8
+4
+7
+3
+φ₁ = 1
+5
+2
+6
+ξ
+η
+N₁
+(d)
+</details>
+
+(a) $N_{5} = \frac{1}{2} (1 - \xi^{2})(1 - \eta)$   
+(b) $N_{8} = \frac{1}{2} (1 - \xi)(1 - \eta^{2})$   
+(c) $N_{(c)} = \frac{1}{4} (1 - \xi)(1 - \eta)$   
+(d) $N_{1} = N_{(c)} - \frac{1}{2} N_{5} - \frac{1}{2} N_{8}$
+
+(a) $N_{5} = \frac{1}{2} (1 - \xi^{2})(1 - \eta)$
+
+(b) $N_{8} = \frac{1}{2} (1 - \xi)(1 - \eta^{2})$
+
+(c) $N_{(c)} = \frac{1}{4} (1 - \xi)(1 - \eta)$
+
+(d) $N_{1} = N_{(c)} - \frac{1}{2} N_{5} - \frac{1}{2} N_{8}$
+
+Figure 6.6-2. Selected shape functions for the quadratic element in Fig. 6.6-1, shown normal to square elements in $\xi \eta$ coordinates.
+
+it is unity; therefore, it is shape function $N_{1}$ . The complete set of shape functions is
+
+$$
+N _ {1} = \frac {1}{4} (1 - \xi) (1 -, \eta) - \frac {1}{2} \left(N _ {8} + N _ {5}\right) \quad N _ {5} = \frac {1}{2} (1 - \xi^ {2}) (1 - \eta)
+$$
+
+$$
+N _ {2} = \frac {1}{4} (1 + \xi) (1 - \eta) - \frac {1}{2} \left(N _ {5} + N _ {6}\right) \quad N _ {6} = \frac {1}{2} (1 + \xi) \left(1 - \eta^ {2}\right) \tag {6.6-1}
+$$
+
+$$
+N _ {3} = \frac {1}{4} (1 + \xi) (1 + \eta) - \frac {1}{2} \left(N _ {6} + N _ {7}\right). \quad N _ {7} = \frac {1}{2} \left(1 - \xi^ {2}\right) (1 + \eta)
+$$
+
+$$
+N _ {4} = \frac {1}{4} (1 - \xi) (1 + \eta) - \frac {1}{2} (N _ {7} + N _ {8}) \quad N _ {8} = \frac {1}{2} (1 - \xi) (1 - \eta^ {2})
+$$
+
+The foregoing element, and other isoparametric elements having boundary nodes only, are sometimes called “serendipity” elements. Addition of an internal node (node 9) at $\xi = \eta = 0$ in Fig. 6.6-1 makes the element a “Lagrange” quadratic element, Fig. 6.6-3a. Element sides may be straight or curved. The name “Lagrange” is used because the element shape functions can be obtained by taking
+
+![](images/page-197_832bb8e8b9d795ea94b8e23c00ed16c642210aa4617cc4000bb86689c5ba5228.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+η
+4
+7
+3
+8
+9
+6
+ξ
+1
+5
+2
+x,u
+</details>
+
+(a)
+
+![](images/page-197_404e8f3012938958ca72577b5c8fb9eefc54dcea4559c2f2e22eb63af713b532.jpg)
+
+<details>
+<summary>text_image</summary>
+
+η
+ξ
+φ₉ = 1
+N₉
+</details>
+
+(b)
+
+![](images/page-197_5c5c73b1818d702beb72d180ad0df13bc2da362a8d6da125839092978e3c0863.jpg)
+
+<details>
+<summary>text_image</summary>
+
+η
+ξ
+φ₅ = 1
+N₅
+</details>
+
+(c)   
+Figure 6.6-3. (a) Nine-node Lagrange element in Cartesian coordinates. (b,c) Shape functions $N_{9}$ and $N_{5}$ , shown normal to square elements in $\xi\eta$ coordinates.
+
+<!-- source-page: 198 -->
+
+products of one-dimensional Lagrange interpolants (Eqs. 3.12-6). Computed results are usually most accurate if node 9 is assigned the location given by $\xi = \eta = 0$ in Eqs. 6.6-1—that is,
+
+$$
+x _ {9} = \sum N _ {i} x _ {i} = - \frac {1}{4} \left(x _ {1} + x _ {2} + x _ {3} + x _ {4}\right) + \frac {1}{2} \left(x _ {5} + x _ {6} + x _ {7} + x _ {8}\right) \tag {6.6-2}
+$$
+
+and similarly for $y_9$ . Thus the geometry of the Lagrange element is completely defined by coordinates of the eight boundary nodes.
+
+The shape function associated with node 9 of the quadratic Lagrange element is
+
+$$
+N _ {9} = (1 - \xi^ {2}) (1 - \eta^ {2}) \tag {6.6-3}
+$$
+
+which may be called a “bubble function” because it resembles a bubble blown over a quadrilateral opening in a plate, as shown in Fig. 6.6-3b. The first eight shape functions of the Lagrange quadratic element can be obtained by modifying the $N_{i}$ of Eqs. 6.6-1 so that each is zero at $\xi = \eta = 0$ (compare $N_{5}$ in Fig. 6.6-2a with $N_{5}$ in Fig. 6.6-3c). The results are shown in Table 6.6-1, which is explained as follows.
+
+With all nine nodes included, all nine $N_{i}$ of Table 6.6-1 are used. If only node 9 is omitted, the $N_{i}$ reduce to those of Eqs. 6.6-1. If nodes 5 through 9 are omitted, the element becomes bilinear and the $N_{i}$ reduce to those of Eqs. 6.3-2. If nodes 6 through 9 are omitted, the element has three linear sides and one quadratic side, and $N_{3}$ and $N_{4}$ become bilinear shape functions. (A five-node element is a “transition” element that can be connected to both bilinear and biquadratic elements without incompatibility.) Alternative shape functions for the nine-node element are explained in connection with Eqs. 8.1-4 and 8.1-5.
+
+Let it be required to determine the characteristic matrix [k] of a scalar field element (see Eq. 6.3-5). To do so, we express $J$ and [B] in terms of $\xi$ and $\eta$ , then perform Gauss quadrature as described in Section 6.4. [J] is given by Eq. 6.3-11, in which index $i$ runs from 1 to $n$ , where $n$ is the number of nodes in the element; that is,
+
+TABLE 6.6-1. SHAPE FUNCTIONS OF A PLANE QUADRILATERAL THAT HAS FROM FOUR TO NINE NODES. EXAMPLE (NOTE THAT $N_{5}$ THROUGH $N_{8}$ CONTAIN $N_{9}$ ): $N_{1} = \frac{1}{4}(1 - \xi)(1 - \eta) - \frac{1}{4}(1 - \xi^{2})(1 - \eta) - \frac{1}{4}(1 - \xi)(1 - \eta^{2}) + \frac{1}{4}N_{9}$ . NODE 9 IS AT $\xi = \eta = 0$ . 
+<table><tr><td rowspan="2"></td><td colspan="5">Include Only If Node i Is Present in the Element</td></tr><tr><td>i = 5</td><td>i = 6</td><td>i = 7</td><td>i = 8</td><td>i = 9</td></tr><tr><td> $N_1 = \frac{1}{4}(1 - \xi)(1 - \eta)$ </td><td> $-\frac{1}{2}N_5$ </td><td></td><td></td><td> $-\frac{1}{2}N_8$ </td><td> $-\frac{1}{4}N_9$ </td></tr><tr><td> $N_2 = \frac{1}{4}(1 + \xi)(1 - \eta)$ </td><td> $-\frac{1}{2}N_5$ </td><td> $-\frac{1}{2}N_6$ </td><td></td><td></td><td> $-\frac{1}{4}N_9$ </td></tr><tr><td> $N_3 = \frac{1}{4}(1 + \xi)(1 + \eta)$ </td><td></td><td> $-\frac{1}{2}N_6$ </td><td> $-\frac{1}{2}N_7$ </td><td></td><td> $-\frac{1}{4}N_9$ </td></tr><tr><td> $N_4 = \frac{1}{4}(1 - \xi)(1 + \eta)$ </td><td></td><td></td><td> $-\frac{1}{2}N_7$ </td><td> $-\frac{1}{2}N_8$ </td><td> $-\frac{1}{4}N_9$ </td></tr><tr><td> $N_5 = \frac{1}{2}(1 - \xi^2)(1 - \eta)$ </td><td></td><td></td><td></td><td></td><td> $-\frac{1}{2}N_9$ </td></tr><tr><td> $N_6 = \frac{1}{2}(1 + \xi)(1 - \eta^2)$ </td><td></td><td></td><td></td><td></td><td> $-\frac{1}{2}N_9$ </td></tr><tr><td> $N_7 = \frac{1}{2}(1 - \xi^2)(1 + \eta)$ </td><td></td><td></td><td></td><td></td><td> $-\frac{1}{2}N_9$ </td></tr><tr><td> $N_8 = \frac{1}{2}(1 - \xi)(1 - \eta^2)$ </td><td></td><td></td><td></td><td></td><td> $-\frac{1}{2}N_9$ </td></tr><tr><td> $N_9 = (1 - \xi^2)(1 - \eta^2)$ </td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+<!-- source-page: 199 -->
+
+$$
+[ \mathbf {J} ] = [ \mathbf {D} _ {N} ] \left[ \begin{array}{l l} x _ {1} & y _ {1} \\ x _ {2} & y _ {2} \\ \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \\ x _ {n} & y _ {n} \end{array} \right] \tag {6.6-4}
+$$
+
+where
+
+$$
+\left[ \mathbf {D} _ {N} \right] = \left[ \begin{array}{l l l l l} N _ {1, \xi} & N _ {2, \xi} & N _ {3, \xi} & \dots & N _ {n, \xi} \\ N _ {1, \eta} & N _ {2, \eta} & N _ {3, \eta} & \dots & N _ {n, \eta} \end{array} \right] \tag {6.6-5}
+$$
+
+Hence, Eq. 6.3-14 gives $[\Gamma] = [\mathbf{J}]^{-1}$ and Eq. 6.3-15 gives $J = \operatorname{det}[\mathbf{J}]$ . Derivatives of $\phi$ are
+
+$$
+\left\{ \begin{array}{l} \phi_ {, x} \\ \phi_ {, y} \end{array} \right\} = [ \Gamma ] \left\{ \begin{array}{l} \phi_ {, \xi} \\ \phi_ {, \eta} \end{array} \right\} \quad \text { and } \quad \left\{ \begin{array}{l} \phi_ {, \xi} \\ \phi_ {, \eta} \end{array} \right\} = [ \mathbf {D} _ {N} ] \{\phi_ {e} \} \tag {6.6-6}
+$$
+
+where $\{\phi_e\}$ are nodal values of $\phi$ . From Eqs. 6.6-6, we obtain
+
+$$
+\left\{ \begin{array}{l} \phi_ {, x} \\ \phi_ {, y} \end{array} \right\} = [ \mathbf {B} ] \{\phi_ {e} \}, \quad \text { where } \quad [ \mathbf {B} ] = [ \Gamma ] [ \mathbf {D} _ {N} ] \tag {6.6-7}
+$$
+
+The characteristic matrix, to be integrated numerically, is
+
+$$
+\underset {n \times n} {[ \mathbf {k} ]} = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \underset {n \times 2} {[ \mathbf {B} ] ^ {T}} k [ \mathbf {B} ] t J d \xi d \eta \tag {6.6-8}
+$$
+
+Except that $n$ may now be greater than 4, the foregoing argument is identical to that in Section 6.3.
+
+Remarks. By adding yet more nodes, quadrilateral elements become successively cubic, quartic, and so on. Examination of the polynomial expansions for serendipity elements (which have no internal nodes) shows that serendipity elements “leave out the middle” of a Pascal triangle, Fig. 6.6-4. In contrast, Lagrange elements (which have one or more internal nodes) use a square block of terms, which allows Lagrange elements to have better accuracy.
+
+All isoparametric elements lose. accuracy when distorted from a rectangular
+
+![](images/page-199_caa260beeb7d60b8b19682ed7f0f5b5662683e73c4f86d737a28e53af2b53a60.jpg)
+
+<details>
+<summary>flowchart</summary>
+```mermaid
+graph TD
+    A["Constant term"] --> B["1"]
+    C["Linear terms"] --> B
+    D["Quadratic terms"] --> B
+    E["Cubic terms"] --> B
+    F["Quartic terms"] --> B
+    G["Serendipity"] --> B
+    H["Lagrange"] -.-> I["ξ³η²"]
+    I --> J["ξ²η²"]
+    J --> K["ξη²"]
+    K --> L["η²"]
+    L --> M["Linear element"]
+    L --> N["Quadratic element"]
+    L --> O["Cubic element"]
+    I --> P["ξ³η³"]
+    P --> Q["ξ²η³"]
+    Q --> R["ξη³"]
+    R --> S["η³"]
+    S --> T["Linear element"]
+    S --> U["Quadratic element"]
+    S --> V["Cubic element"]
+```
+</details>
+
+Figure 6.6-4. Polynomial coefficients in plane serendipity elements (boundary nodes only) and plane Lagrange elements (boundary and internal nodes).
+
+<!-- source-page: 200 -->
+
+shape. However, the nine-node element is much less sensitive than the eight-node element to nonrectangularity, to curvature of sides, and to placing side nodes away from midsides. Indeed, it has been found that a nonrectangular nine-node element can still represent a state of pure bending, provided that sides are straight and side nodes are at midsides as in Fig. 6.6-3a [4.6]. The eight-node element does not have this capability.
+
+In dynamic problems, it is often desirable to use lumped (diagonal) mass matrices. All popular lumping schemes result in positive nodal masses for the nine-node element. Such is not the case for the eight-node element, which may have some negative nodal masses (see Chapter 13).
+
+It is easy to alter the subroutines in Section 6.5 to accommodate a quadratic element. The major change is revision of the shape functions in subroutine SHAPE. Otherwise, one increases the size of some arrays and increases the range of most DO loops. The Jacobian matrix remains 2 by 2. In forming [B] from Eqs. 6.3-17 through 6.3-19, the rectangular matrix in Eq. 6.3-19 contains 16 or 18 columns, depending on whether node 9 is omitted or included.
+
+# 6.7 HEXAHEDRAL (SOLID)
+
+# ISOPARAMETRIC ELEMENTS
+
+In three dimensions, the isoparametric procedure closely resembles the two-dimensional development in Section 6.3. The general procedures used for isoparametric elements have little to do with the specific shape functions of a particular element.
+
+For element geometry and the field quantity $\phi$ of a solid isoparametric element, we write
+
+$$
+x = \sum N _ {i} x _ {i} \quad y = \sum N _ {i} y _ {i} \quad z = \sum N _ {i} z _ {i} \quad \phi = \sum N _ {i} \phi_ {i} \tag {6.7-1}
+$$
+
+where i ranges over the number of nodes in the element. Shape functions $N_{i}$ are functions of isoparametric coordinates $\xi$ , $\eta$ , and $\zeta$ . Faces of the element lie at $\xi = \pm 1$ , $\eta = \pm 1$ , and $\zeta = \pm 1$ . The Jacobian matrix is defined analogously to Eq. 6.3-11, but is now 3 by 3:
+
+$$
+[ \mathrm{J} ] = \left[ \begin{array}{l l l} x, _ {\xi} & y, _ {\xi} & z, _ {\xi} \\ x, _ {\eta} & y, _ {\eta} & z, _ {\eta} \\ x, _ {\zeta} & y, _ {\zeta} & z, _ {\zeta} \end{array} \right] = \sum \left[ \begin{array}{l l l} N _ {i, \xi} x _ {i} & N _ {i, \xi} y _ {i} & N _ {i, \xi} z _ {i} \\ N _ {i, \eta} x _ {i} & N _ {i, \eta} y _ {i} & N _ {i, \eta} z _ {i} \\ N _ {i, \zeta} x _ {i} & N _ {i, \zeta} y _ {i} & N _ {i, \zeta} z _ {i} \end{array} \right] \tag {6.7-2}
+$$
+
+With $[\Gamma] = [\mathrm{J}]^{-1}$ ,
+
+$$
+\left\lfloor \phi_ {, x} \quad \phi_ {, y} \quad \phi_ {, z} \right] ^ {T} = [ \Gamma ] \left\lfloor \phi_ {, \xi} \quad \phi_ {, \eta} \quad \phi_ {, \zeta} \right] ^ {T} \tag {6.7-3}
+$$
+
+For a scalar field problem in three dimensions, with $n$ the number of nodes per element, Eq. 6.3-5 becomes
+
+$$
+[ \mathbf {k} ] _ {n \times n} = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} [ \mathbf {B} ] ^ {T} k [ \mathbf {B} ] J d \xi d \eta d \zeta \tag {6.7-4}
+$$
+
+The Jacobian determinant $J = \det[\mathbf{J}]$ expresses the ratio of volume $dx \, dy \, dz$ to $d\xi \, d\eta \, d\zeta$ . In general, [J] and [B] are functions of $\xi, \eta,$ and $\zeta$ .
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_021.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_021.md
new file mode 100644
index 00000000..923f63e8
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_021.md
@@ -0,0 +1,377 @@
+<!-- source-page: 201 -->
+
+![](images/page-201_6ffe0049b7d3316a9cf8d3fa78b0bcfb1c9ad4451be8529be79c2e06e9d39b49.jpg)
+
+<details>
+<summary>text_image</summary>
+
+3
+η
+7
+4
+8
+2
+1
+ξ
+6
+5
+</details>
+
+Figure 6.7-1. Linear solid (eight-node brick) element. Also called a “trilinear” element.
+
+In structural mechanics we deal with displacements $u$ , $v$ , and $w$ , which are parallel to $x$ , $y$ , and $z$ directions, respectively. Matrix [B] is evaluated by expanded forms of Eqs. 6.3-17 through 6.3-19. As expanded, Eq. 6.3-19 has nine rows, Eq. 6.3-18 contains a 9 by 9 square matrix, and Eq. 6.3-17 states the strain-displacement relations (Eqs. 1.5-6) as
+
+$$
+\left\{ \begin{array}{l} \epsilon_ {x} \\ \epsilon_ {y} \\ \epsilon_ {z} \\ \gamma_ {x y} \\ \gamma_ {y z} \\ \gamma_ {z x} \end{array} \right\} = [ \mathbf {H} ] \left\{ \begin{array}{l} u _ {, x} \\ u _ {, y} \\ u _ {, z} \\ v _ {, x} \\ \vdots \\ w _ {, z} \end{array} \right\}, \quad \text { where } \quad [ \mathbf {H} ] _ {6 \times 9} = \left[ \begin{array}{c c c c c c c c c} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \end{array} \right] \tag {6.7-5}
+$$
+
+Shape functions of a linear solid element (also called an eight-node brick element), Fig. 6.7-1, are
+
+$$
+N _ {i} = \frac {1}{8} (1 \pm \xi) (1 \pm \eta) (1 \pm \zeta) \tag {6.7-6}
+$$
+
+in which $i = 1, 2, \ldots, 8$ and the choice of algebraic signs should be obvious to an adequately prepared reader.
+
+A quadratic “serendipity” solid element has corner nodes and a node on each edge, for a total of 20 nodes. A quadratic “Lagrange” solid element also includes midface nodes and a node at $\xi = \eta = \zeta = 0$ , for a total of 27 nodes. Shape functions of these quadratic elements are analogous to those in Eqs. 6.6-1 and Table 6.6-1 and may be found in [2.1]. Stiffness matrices of these elements are, respectively, 60 by 60 and 81 by 81. Quadratic elements may be good enough to permit the structure of Fig. 6.7-2 to be modeled by a single layer of elements, except near the fillet where stress gradients may be large.
+
+![](images/page-201_fc8d0f880bb10774857e7417bc5a9f73f1596b00dc5e435cdb15bf81ba24db7b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y, v
+z, w
+x, u
+</details>
+
+Figure 6.7-2. Model of one octant of a cylinder-to-cylinder intersection.
+
+<!-- source-page: 202 -->
+
+# 6.8 TRIANGULAR ISOPARAMETRIC ELEMENTS
+
+In Chapter 5, we discussed triangular elements that are restricted to have straight sides and evenly spaced side nodes (Fig. 6.8-1a). In the present section these restrictions are removed, to allow triangles such as that in Fig. 6.8-1b, where sides need not be straight and side nodes need not be at midsides. We will see that the formulation procedure is essentially the same as that used for quadrilateral isoparametric elements.
+
+Let $\phi$ be a scalar field, interpolated from nodal values $\phi_{i}$ . Similarly, $x$ and $y$ are coordinates interpolated from nodal values of $x_{i}$ and $y_{i}$ :
+
+$$
+\phi = \sum N _ {i} \phi_ {i} \quad x = \sum N _ {i} x _ {i} \quad y = \sum N _ {i} y _ {i} \tag {6.8-1}
+$$
+
+The element is isoparametric if the same shape functions $N_{i}$ are used in all three summations. For a triangle, the $N_{i}$ are expressed in terms of area coordinates $\xi_{1}, \xi_{2}$ , and $\xi_{3}$ , as explained in Section 5.3.
+
+Imagine that the element characteristic matrix $[\mathbf{k}]$ for a scalar field problem is required. Equations 6.6-4 through 6.6-7 may be used exactly as written, without even a change in symbols. We argue as follows.
+
+Area coordinates are not independent. They satisfy the constraint relation $\xi_1 + \xi_2 + \xi_3 = 1$ . Accordingly, only two need be given to uniquely locate a point. For example, if $\xi_1$ and $\xi_2$ are given and the constraint relation is invoked, Eqs. 6.8-1 can be evaluated. We therefore define
+
+$$
+\xi_ {1} = \xi
+$$
+
+$$
+\xi_ {2} = \eta \tag {6.8-2}
+$$
+
+$$
+\xi_ {3} = 1 - \xi - \eta
+$$
+
+To evaluate the shape function derivatives seen in Eq. 6.6-5, we invoke the chain rule:
+
+$$
+\frac {\partial N _ {i}}{\partial \xi} = \frac {\partial N _ {i}}{\partial \xi_ {1}} \frac {\partial \xi_ {1}}{\partial \xi} + \frac {\partial N _ {i}}{\partial \xi_ {2}} \frac {\partial \xi_ {2}}{\partial \xi} + \frac {\partial N _ {i}}{\partial \xi_ {3}} \frac {\partial \xi_ {3}}{\partial \xi} = \frac {\partial N _ {i}}{\partial \xi_ {1}} - \frac {\partial N _ {i}}{\partial \xi_ {3}} \tag {6.8-3}
+$$
+
+![](images/page-202_175bbbdaaeb19ce48a6d8795c1b19c517e514dc1ebd309765036ad1e79ee7175.jpg)
+
+<details>
+<summary>text_image</summary>
+
+3
+5
+2
+A₁
+A₂
+A₃
+6
+4
+1
+y,v
+x,u
+</details>
+
+(a)
+
+![](images/page-202_1d657eb8d80fa5b83244a637e0f4ec856107f82deecd92b77fb2701f9d3918c9.jpg)
+
+<details>
+<summary>text_image</summary>
+
+3
+5
+2
+6
+4
+1
+y,v
+x,u
+</details>
+
+(b)
+
+![](images/page-202_ac8e855277df7b274ab897c0983ca703629c665aa1265ff9c9328bd160a90f39.jpg)
+
+<details>
+<summary>text_image</summary>
+
+B
+D
+M
+C
+C
+A
+D
+C
+D
+B
+M
+B
+D
+</details>
+
+(c)   
+Figure 6.8-1. (a) Quadratic triangle with straight sides and midside nodes, showing subareas $A_{i}$ that define area coordinates $\xi_{i}$ . (b) Quadratic triangle of arbitrary shape. (c) Sampling points used in quadrature formulas of Table 6.8-1.
+
+<!-- source-page: 203 -->
+
+Similarly
+
+$$
+\frac {\partial N _ {i}}{\partial \eta} = \frac {\partial N _ {i}}{\partial \xi_ {2}} - \frac {\partial N _ {i}}{\partial \xi_ {3}} \tag {6.8-4}
+$$
+
+For example, using the quadratic shape functions of Eqs. 5.3-5, we obtain for Eq. 6.6-5 the expression
+
+$$
+\left[ \begin{array}{c} \mathbf {D} _ {N} \\ 2 \times 6 \end{array} \right] = \left[ \begin{array}{c c c c c c} 4 \xi_ {1} - 1 & 0 & - 4 \xi_ {3} + 1 & 4 \xi_ {2} & - 4 \xi_ {2} & 4 \left(\xi_ {3} - \xi_ {1}\right) \\ 0 & 4 \xi_ {2} - 1 & - 4 \xi_ {3} + 1 & 4 \xi_ {1} & 4 \left(\xi_ {3} - \xi_ {2}\right) & - 4 \xi_ {1} \end{array} \right] \tag {6.8-5}
+$$
+
+[J] is produced by letting $[D_{N}]$ premultiply a 6 by 2 array of nodal coordinates (let n = 6 in Eq. 6.6-4). Hence, $[B] = [J]^{-1}[D_{N}]$ , as in Eq. 6.6-7.
+
+Equation 6.6-8 is a valid expression for [k] of an n-node isoparametric triangle if the limits are 0 to 1 on the first integral and 0 to 1 - $\eta$ on the second. Alternatively, we can write the following form, which is more appropriate to subsequent numerical integration:
+
+$$
+\underset {n \times n} {[ \mathbf {k} ]} = \int_ {A} \underset {n \times 2} {[ \mathbf {B} ] ^ {T}} k [ \mathbf {B} ] t d A \tag {6.8-6}
+$$
+
+Here t is element thickness, A is the area of the triangle, and [B] is a function of area coordinates $\xi_{1}$ , $\xi_{2}$ , and $\xi_{3}$ .
+
+Numerical Integration for Triangles. Let $\phi$ be a function of area coordinates $\xi_{1}$ , $\xi_{2}$ , and $\xi_{3}$ . The quadrature rule is
+
+$$
+\int_ {A} \phi d A = \frac {1}{2} \sum_ {i = 1} ^ {n} W _ {i} J _ {i} \phi_ {i} \tag {6.8-7}
+$$
+
+in which $\phi_{i}$ is the value of $\phi$ at a specific point in the triangle, $W_{i}$ is the weight appropriate to this point, and $n$ is the number of sampling points used. In going from Eq. 6.8-6 to Eq. 6.8-7 we set $dA = J d\xi d\eta$ . The factor of $\frac{1}{2}$ appears in Eq. 6.8-7 because the area of a reference triangle in area coordinates is $\frac{1}{2}$ . For an undistorted triangle of unit area in Cartesian coordinates, $J = 2$ throughout. Hence, since $\Sigma W_{i} = 1$ (see data in Table 6.8-1), we obtain $\int dA = 1$ when $\phi = 1$ in Eq. 6.8-7.
+
+Equation 6.8-7 must be applied to each term in the integrand of Eq. 6.8-6.
+
+Equation 6.8-7 is similar in form to Eq. 6.4-6, but in Eq. 6.8-7 weights are expressed directly rather than as the product of one-dimensional weights. Data appear in Table 6.8-1 [6.5]. For points of multiplicity 3 (points B, C, D, and M) there are three points that have the same weight $W_{i}$ . Area coordinates are given for one of the three; the other two are given by cyclic permutation. For example, using the first three-point formula in Table 6.8-1, we evaluate function $\phi$ at the three locations
+
+$$
+\begin{array}{c c c} (1) & (2) & (3) \\ \xi_ {1} = \frac {2}{3} & \xi_ {2} = \frac {2}{3} & \xi_ {3} = \frac {2}{3} \\ \xi_ {2} = \xi_ {3} = \frac {1}{6} & \xi_ {3} = \xi_ {1} = \frac {1}{6} & \xi_ {1} = \xi_ {2} = \frac {1}{6} \end{array} \tag {6.8-8}
+$$
+
+<!-- source-page: 204 -->
+
+TABLE 6.8-1. GAUSS QUADRATURE FORMULAS FOR INTEGRATION. OVER A TRIANGLE ACCORDING TO EQ. 6.8-7. APPROXIMATE LOCATIONS OF ENTRIES IN THE "POINTS" COLUMN ARE SHOWN IN FIG. 6.8-1C. ns = POINTS OF MULTIPLICITY 6, NOT SHOWN IN FIG. 6.8-1C. 
+
+<table><tr><td>Points</td><td>Multiplicity</td><td colspan="4">Area Coordinates  $\xi_1, \xi_2, \xi_3$ </td><td>Weights  $W_i$ </td></tr><tr><td></td><td></td><td colspan="2">1-point formula</td><td colspan="2">degree of precision 1</td><td></td></tr><tr><td>A</td><td>1</td><td>0.33333 33333 33333</td><td>0.33333</td><td>33333 33333</td><td>0.33333 33333 33333</td><td>1.00000 00000 00000</td></tr><tr><td></td><td></td><td colspan="2">3-point formula</td><td colspan="2">degree of precision 2</td><td></td></tr><tr><td>B</td><td>3</td><td>0.66666 66666 66667</td><td>0.16666</td><td>66666 66667</td><td>0.16666 66666 66667</td><td>0.33333 33333 33333</td></tr><tr><td></td><td></td><td colspan="2">3-point formula</td><td colspan="2">degree of precision 2</td><td></td></tr><tr><td>M</td><td>3</td><td>0.50000 00000 00000</td><td>0.50000</td><td>00000 00000</td><td>0.00000 00000 00000</td><td>0.33333 33333 33333</td></tr><tr><td></td><td></td><td colspan="2">4-point formula</td><td colspan="2">degree of precision 3</td><td></td></tr><tr><td>A</td><td>1</td><td>0.33333 33333 33333</td><td>0.33333</td><td>33333 33333</td><td>0.33333 33333 33333</td><td>-0.56250 00000 00000</td></tr><tr><td>B</td><td>3</td><td>0.60000 00000 00000</td><td>0.20000</td><td>00000 00000</td><td>0.20000 00000 00000</td><td>0.52083 33333 33333</td></tr><tr><td></td><td></td><td colspan="2">6-point formula</td><td colspan="2">degree of precision 4</td><td></td></tr><tr><td>B</td><td>3</td><td>0.81684 75729 80459</td><td>0.09157</td><td>62135 09771</td><td>0.09157 62135 09771</td><td>0.10995 17436 55322</td></tr><tr><td>C</td><td>3</td><td>0.10810 30181 68070</td><td>0.44594</td><td>84909 15965</td><td>0.44594 84909 15965</td><td>0.22338 15896 78011</td></tr><tr><td></td><td></td><td colspan="2">7-point formula</td><td colspan="2">degree of precision 5</td><td></td></tr><tr><td>A</td><td>1</td><td>0.33333 33333 33333</td><td>0.33333</td><td>33333 33333</td><td>0.33333 33333 33333</td><td>0.22500 00000 00000</td></tr><tr><td>B</td><td>3</td><td>0.79742 69853 53087</td><td>0.10128</td><td>65073 23456</td><td>0.10128 65073 23456</td><td>0.12593 91805 44827</td></tr><tr><td>C</td><td>3</td><td>0.47014 20641 05115</td><td>0.47014</td><td>20641 05115</td><td>0.05971 58717 89770</td><td>0.13239 41527 88506</td></tr><tr><td></td><td></td><td colspan="2">12-point formula</td><td colspan="2">degree of precision 6</td><td></td></tr><tr><td>B</td><td>3</td><td>0.87382 19710 16996</td><td>0.06308</td><td>90144 91502</td><td>0.06308 90144 91502</td><td>0.05084 49063 70207</td></tr><tr><td>D</td><td>3</td><td>0.50142 65096 58179</td><td>0.24928</td><td>67451 70910</td><td>0.24928 67451 70910</td><td>0.11678 62757 26379</td></tr><tr><td>ns</td><td>6</td><td>0.63650 24991 21399</td><td>0.31035</td><td>24510 33784</td><td>0.05314 50498 44817</td><td>0.08285 10756 18374</td></tr><tr><td></td><td></td><td colspan="2">13-point formula</td><td colspan="2">degree of precision 7</td><td></td></tr><tr><td>A</td><td>1</td><td>0.33333 33333 33333</td><td>0.33333</td><td>33333 33333</td><td>0.33333 33333 33333</td><td>-0.14957 00444 67682</td></tr><tr><td>D</td><td>3</td><td>0.47930 80678 41920</td><td>0.26034</td><td>59660 79040</td><td>0.26034 59660 79040</td><td>0.17561 52574 33208</td></tr><tr><td>B</td><td>3</td><td>0.86973 97941 95568</td><td>0.06513</td><td>01029 02216</td><td>0.06513 01029 02216</td><td>0.05334 72356 08838</td></tr><tr><td>ns</td><td>6</td><td>0.63844 41885 69810</td><td>0.31286</td><td>54960 04874</td><td>0.04869 03154 25316</td><td>0.07711 37608 90257</td></tr></table>
+
+<!-- source-page: 205 -->
+
+and apply weight $W = \frac{1}{3}$ to each. Center point $A$ has multiplicity 1. Points $ns$ (not shown) have multiplicity 6. The formulas in Table 6.8-1 are symmetric in the area coordinates. These formulas were not derived by Gauss but are called Gaussian because sampling points are optimally placed rather than simply being located in a uniform pattern.
+
+The “degree of precision” in Table 6.8-1 refers to the degree of the highest-order complete polynomial in Cartesian coordinates that is integrated exactly by a formula. If $\phi$ in Eq. 6.8-7 is not a polynomial, numerical integration is not exact but becomes more accurate as more sampling points are used. Inexact integration is expected for elements such as that in Fig. 6.8-1b, because geometric distortion produces a [B] matrix whose terms are not polynomials but rather the ratio of two polynomials.
+
+Remarks. Like triangles, tetrahedra can be distorted from regular shapes such as that in Fig. 5.6-1. Triangles and tetrahedra decline in accuracy as sides become curved or side nodes become unevenly spaced. There is evidence that the ten-node tetrahedron is more sensitive to distortion than the six-node triangle.
+
+If elements have straight sides and evenly spaced side nodes, one has the option of writing explicit expressions for stiffness coefficients rather than using numerical integration. The amount of algebra required to explicitly formulate a stiffness matrix in area or volume coordinates can be large. Fortunately, symbol-processing programs are available, of which MACSYMA may be the best known. Operations on polynomials, such as multiplication, factoring, differentiation, and integration can be performed symbolically, with output in the form of Fortran statements if so desired. The potential for saving time and reducing errors is obvious $[6.6]$ .
+
+# 6.9 CONSISTENT ELEMENT NODAL LOADS $\{r_{e}\}$
+
+Consistent element nodal loads caused by initial strains $\{\epsilon_{0}\}$ , initial stresses $\{\sigma_{0}\}$ , body forces $\{F\}$ , and surface tractions $\{\Phi\}$ are computed according to Eq. 4.1-6. This matter is thoroughly discussed in Section 4.3. Results presented there remain valid if sides are straight and elements are rectangular; it does not matter that the elements may now be called isoparametric. For curved sides and general element shapes, results will differ and will be affected by the amount of geometric distortion.
+
+Loads $\{r_{e}\}$ caused by $\{\epsilon_{0}\}$ , $\{\sigma_{0}\}$ , and $\{F\}$ are given by Eq. 6.3-21 for plane quadrilateral elements having any number of nodes. For solid hexahedra a triple integral is used, in which tJ $d\xi d\eta$ is replaced by J $d\xi d\eta d\zeta$ . For triangular elements integration spans the triangle area A, tJ $d\xi d\eta$ is replaced by t dA, and formulas discussed in Section 6.8 are used.
+
+A result of interest for the quadratic triangle is easy to show. Let the triangle be of constant thickness and have straight sides and midside nodes, as in Fig. 6.8-1a, and let there be a uniform body force in (say) the +x direction. Taking shape functions from Eq. 5.3-5, and using Eq. 5.2-8 for integrations, we obtain the following x-direction nodal loads:
+
+$$
+\int_ {A} \left\lfloor \mathbf {N} \right] _ {6 \times 1} ^ {T} F _ {x} t d A = \frac {F _ {x} A t}{3} \left\lfloor 0 0 0 1 1 1 \right] ^ {T} \tag {6.9-1}
+$$
+
+<!-- source-page: 206 -->
+
+Thus the entire force $F_{x}At$ is apportioned equally to only the midside nodes, regardless of the shape of the triangle (so long as sides remain straight and side nodes remain at midsides). This division also applies to the load produced by a uniform pressure against area $A$ of the triangle. (Such a pressure loading may act on one six-node face of a ten-node tetrahedron, or on a six-node triangle formulated as a plate bending element.) If element sides are curved, the functions to be integrated are not polynomials: then numerical integration is required in Eq. 6.9-1, and the resulting nodal loads are not the same as those in Eq. 6.9-1.
+
+Procedures for evaluating consistent loads $\{\mathbf{r}_e\}$ caused by tractions on curved edges and warped surfaces are discussed in Section 5.8 of the second edition of this book.
+
+# 6.10 THE VALIDITY OF ISOPARAMETRIC ELEMENTS
+
+The critical test of element validity is the patch test, discussed in Section 4.6. In the present section we will stop short of the patch test, but will argue that the isoparametric formulation endows elements with all characteristics needed for convergence as discussed in Section 4.5. Elements discussed in the present chapter do in fact pass the patch test.
+
+Isoparametric elements are well suited to problems whose functional II contains first spatial derivatives (or whose governing differential equations contain second spatial derivatives). Examples include heat conduction, plane stress analysis, and stress analysis of solids. We wish to show that isoparametric elements (a) provide $C^{0}$ continuity (interelement continuity of the primary field variable), and (b) contain a complete linear polynomial in Cartesian coordinates [6.7].
+
+Continuity. Interelement continuity of a field variable $\phi$ can be demonstrated with the aid of Fig. 6.10-1. Along the common edge $ABC$ , adjacent elements display the same edge-tangent coordinate: $\eta_1 = \eta_2$ . In addition, along edge $ABC$ , shape functions of the two elements are identical functions of $\eta$ and operate on d.o.f. of nodes $A, B$ , and $C$ only. Thus, whether viewed from element 1 or from element 2, the field variable along $ABC$ is the same function. This argument also shows that elements match geometrically, because coordinates of $A, B$ , and $C$ define a unique quadratic curve.
+
+Completeness. Imagine that $\phi = \phi(x,y,z)$ is a polynomial field. If element nodes are attached to this field, so that nodal d.o.f. are $\phi_{i} = \phi(x_{i},y_{i},z_{i})$ , does shape function interpolation from nodal $\phi_{i}$ yield the original field $\phi$ throughout the element? The answer is yes if $\phi$ is linear and the element is isoparametric. Specifically, we propose to show that $\phi$ within an isoparametric element is a complete linear polynomial—that is, that
+
+$$
+\phi = a _ {1} + a _ {2} x + a _ {3} y + a _ {4} z \tag {6.10-1}
+$$
+
+when nodal d.o.f. $\phi_{i}$ are consistent with this field and $\phi$ within the element is evaluated by the interpolation
+
+$$
+\phi = \sum N _ {i} \phi_ {i} \tag {6.10-2}
+$$
+
+<!-- source-page: 207 -->
+
+![](images/page-207_d150ef926c34ecf7983fe803536560131fc87b90973b384f5fafb7a0df01dd9e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+C
+η₁
+ξ₁
+①
+B
+η₂
+ξ₂
+②
+A
+</details>
+
+Figure 6.10-1. Adjacent elements cited in compatibility arguments.
+
+To do so we first evaluate Eq. 6.10-1 at each node:
+
+$$
+\phi_ {i} = a _ {1} + a _ {2} x _ {i} + a _ {3} y _ {i} + a _ {4} z _ {i} \tag {6.10-3}
+$$
+
+Here $x_{i}, y_{i}$ , and $z_{i}$ are Cartesian coordinates of node i, and i ranges over all nodes of the element, however many nodes there may be. Equations 6.10-2 and 6.10-3 yield
+
+$$
+\phi = a _ {1} \sum N _ {i} + a _ {2} \sum N _ {i} x _ {i} + a _ {3} \sum N _ {i} y _ {i} + a _ {4} \sum N _ {i} z _ {i} \tag {6.10-4}
+$$
+
+But, if the element is isoparametric, coordinates are interpolated in the same way as $\phi$ —that is,
+
+$$
+x = \sum N _ {i} x _ {i} \quad y = \sum N _ {i} y _ {i} \quad z = \sum N _ {i} z _ {i} \tag {6.10-5}
+$$
+
+Accordingly, if $\Sigma N_{i}=1$ , Eq. 6.10-4 reduces to Eq. 6.10-1, and the proposition is proved.
+
+To show that $\Sigma N_{i}=1$ , we note that Eqs. 6.10-5 can be used regardless of the location of the origin of Cartesian coordinates with respect to the element. Imagine that there is a second system XYZ, translated a distance h along the x axis so that $x=X+h$ . Hence
+
+$$
+x = \sum N _ {i} x _ {i} = \sum N _ {i} X _ {i} + h \sum N _ {i} = X + h \sum N _ {i} = (x - h) + h \sum N _ {i} \tag {6.10-6}
+$$
+
+from which we obtain $h = h \sum N_i$ and therefore conclude that $\sum N_i = 1$ .
+
+Furthermore, from $\Sigma N_{i} = 1$ we obtain
+
+$$
+\sum N _ {i, \xi} = \sum N _ {i, \eta} = \sum N _ {i, \zeta} = 0 \tag {6.10-7}
+$$
+
+Equations 6.10-7 can be helpful in checking derivations and coding. Area coordinates for a triangle contain a redundant coordinate, which must be eliminated by means of Eqs. 6.8-2 in order for Eqs. 6.10-7 to be valid.
+
+The foregoing arguments show that isoparametric elements have properties necessary to the passing of a patch test. They say nothing about accuracy in a coarse mesh, convergence rate with mesh refinement, or how accuracy declines as element geometry is distorted from a compact and regular shape (as in going from Fig. 6.6-1a to Fig. 6.6-1b). As sides become curved and as side nodes become unevenly spaced, an element tends to lose its ability to represent quadratic and higher polynomials in xyz coordinates. For example, an element that contains a
+
+<!-- source-page: 208 -->
+
+![](images/page-208_65268c887c686faeeaa9de5d1e0f35e709b0139ce64597b7695b383a4bcd3ef1.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+L
+4
+7
+3
+8
+6
+5
+1
+2
+x
+</details>
+
+Figure 6.10.2. Superparametric element, in which $\phi = \phi(\phi_1, \phi_2, \phi_3, \phi_4)$ .
+
+complete quadratic polynomial in $\xi\eta\zeta$ coordinates may not contain a complete quadratic polynomial in xyz coordinates, depending on the type of distortion from a regular shape. The argument of Eqs. 6.10-1 to 6.10-5 shows only that in spite of geometric distortion, a complete linear polynomial in xyz coordinates will remain.
+
+Subparametric and Superparametric Elements. Such elements are defined in Section 6.1. For example, with straight sides and midside nodes, the element of Fig. 6.6-1a is subparametric, as its shape is then defined by the coordinates of nodes 1 through 4. The field quantity $\phi$ is still defined by the $\phi_i$ of all eight nodes. For such an element, the foregoing completeness argument remains valid. Specifically, for the element of Fig. 6.6-1a, summations in Eqs. 6.10-4 and 6.10-5 run from 1 to 8, but in Eqs. 6.10-5 the interpolations reduce to those of the bilinear element, Eqs. 6.3-1 and 6.3-2, provided that $x_5 = (x_1 + x_2)/2$ , $y_5 = (y_1 + y_2)/2$ , and so on. Thus the linear field of Eq. 6.10-1 is present when geometry is defined by only the four corner nodes.
+
+Superparametric elements are usually not valid (however, with certain restrictions, valid superparametric elements for beams, plates, and shells are possible). Consider Fig. 6.10-2. Let the shape be defined by all eight nodes, that is, in a more general way than is the field quantity $\phi$ , which is defined by $\phi_i$ at nodes 1, 2, 3, and 4 only. Imagine that $\phi_1 = \phi_4 = 0$ while $\phi_2 = \phi_3 = c$ , a constant. Thus we expect the element to display the constant gradient $\phi_{,x} = c / L$ . However, gradient calculation according to Eq. 6.6-7 makes $\phi_{,x}$ a function of $\xi$ rather than constant, owing to [J] in Eq. 6.6-4, in which $x_6 < L$ . The superparametric element of Fig. 6.10-2 would fail a patch test.
+
+# 6.11 APPROPRIATE ORDER OF QUADRATURE
+
+For numerically integrated elements, we define “full integration” as a quadrature rule sufficient to provide the exact integrals of all terms $k_{ij}$ in the element stiffness matrix if the element is undistorted (e.g., if a quadratic element has straight sides and midside nodes). The same “full integration” rule will not exactly integrate all $k_{ij}$ if sides are curved or if side nodes are offset from the midpoints, for then J is not constant throughout the element.
+
+For example, in Eq. 6.2-6, [B] is linear in $\xi$ and $J$ is constant if node 3 is centered. Therefore, the integrand contains terms up to $\xi^2$ , which are integrated exactly by two Gauss points. Accordingly, for this element, even if node 3 is not centered, two-point Gauss quadrature is considered "full integration."
+
+Use of full integration is the only sure way to avoid pitfalls such as mesh instabilities, which are discussed in Section 6.12.
+
+<!-- source-page: 209 -->
+
+However, a lower-order quadrature rule, called “reduced integration,” may be desirable for two reasons. First, since the expense of generating a matrix $[k]$ by numerical integration is proportional to the number of sampling points, using fewer sampling points means lower cost. Second, a low-order rule tends to soften an element, thus countering the overly stiff behavior associated with an assumed displacement field. Softening comes about because certain higher-order polynomial terms happen to vanish at Gauss points of a low-order rule, so that these terms make no contribution to strain energy. In other words, with fewer sampling points, some of the more complicated displacement modes offer less resistance to deformation. In sum, our argument is that reduced integration may be able to simultaneously reduce cost, reduce accuracy in the evaluation of integral expressions, and increase the accuracy of a finite element analysis. Reduced integration should not be used if cost reduction is the only motivation.
+
+The number of Gauss points has a lower limit because in the limit of mesh refinement, element volume must be integrated exactly. We argue as follows. As a mesh is indefinitely refined, a constant-strain condition is approached in each element, provided that the element is valid in the patch test sense. Thus strain energy density $U_{0}$ becomes constant throughout each element. Strain energy in an element, for plane and solid problems, respectively, is
+
+$$
+U _ {e} = \int \int U _ {0} t J d \xi d \eta \quad \text { or } \quad U _ {e} = \int \int \int U _ {0} J d \xi d \eta d \zeta \tag {6.11-1}
+$$
+
+If $U_{0}$ is constant, then $U_{e}$ will be correct if volume $dV = tJ \, d\xi \, d\eta$ (or $dV = J \, d\xi \, d\eta \, d\zeta$ ) is correctly integrated. In practice, we prefer to use exact volume integration for any shape and size of element.
+
+From Eq. 4.1-10 we see that integrals in Eq. 6.11-1 produce terms in the element stiffness matrix [k]. Accordingly, if [k] is produced by an integration rule adequate to compute element volume exactly, the element will be able to provide the correct strain energy in a constant-strain deformation mode.
+
+Thus, for an element of arbitrary geometry, the minimum quadrature requirement is a rule that exactly integrates tJ (plane case) or J (solid case). In a plane bilinear element of constant thickness, tJ is linear in $\xi$ and in $\eta$ , so one Gauss point is required. In a plane quadratic element of constant thickness, tJ contains $\xi^{3}$ and $\eta^{3}$ , so a 2 by 2 Gauss rule is required. The eight-node solid also requires an order 2 Gauss rule (8 points).
+
+However, with rare practical exceptions, indefinitely repeated subdivision of a mesh yields elements that become straight-sided parallelograms of constant thickness. Thus t and J cease to be functions of the coordinates and, in the limit, a single Gauss point yields the correct element volume.
+
+For an isoparametric element based on an assumed displacement field, the best quadrature rule is usually the lowest-order rule that computes volume correctly and does not produce instability. Numerical testing of any proposed rule is mandatory. Solution accuracy may be mesh-dependent and problem-dependent, but usually one quadrature rule will be clearly superior to others. For bilinear and eight-node plane elements, and for the eight-node linear solid element, an order 2 Gauss rule is favored (four and eight points for plane and solid elements, respectively). The quadratic serendipity solid, having eight corner nodes and twelve edge nodes, can be integrated with an order 3 rule (27 points), but a special 14-point rule may be preferred, especially if the element is made very thin in one direction [6.8-6.10].
+
+<!-- source-page: 210 -->
+
+# 6.12 ELEMENT AND MESH INSTABILITIES
+
+An instability may also be called a spurious singular mode. In structural mechanics, an instability may be known as a mechanism, a kinematic mode, an hourglass mode, or a zero-energy mode. The term “zero-energy mode” refers to a nodal displacement vector $\{D\}$ that is not a rigid-body motion but nevertheless produces zero strain energy $\{D\}^{T}[K]\{D\}/2$ . Instabilities arise because of shortcomings in the element formulation process, such as use of a low-order Gauss quadrature rule. In the present context, an instability has nothing to do with buckling problems of structures.
+
+A structure that appears adequately constrained may yet have an upper limit that makes [K] singular. Or, unstable elements may combine to form a structure that is stable but unduly susceptible to certain load patterns, so that computed displacements are excessive.
+
+To explain the term “zero-energy mode” further and show that the term may arise, we substitute the relation $\{\epsilon\} = [B]\{d\}$ into the expression for strain energy in an element, $U_{e}$ . From Eq. 4.1-10 and the standard expression for [k], Eq. 4.1-5, we obtain
+
+$$
+U _ {e} = \frac {1}{2} \{\mathbf {d} \} ^ {T} [ \mathbf {k} ] \{\mathbf {d} \} = \frac {1}{2} \{\mathbf {d} \} ^ {T} \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] [ \mathbf {B} ] d V \{\mathbf {d} \} = \frac {1}{2} \int_ {V _ {e}} \{\boldsymbol {\epsilon} \} ^ {T} [ \mathbf {E} ] \{\boldsymbol {\epsilon} \} d V \tag {6.12-1}
+$$
+
+When $[\mathbf{k}]$ is formed by numerical integration, it contains only the information that can be sensed at the sampling points of the quadrature rule. If it happens that strains $\{\epsilon\} = [\mathbf{B}]\{\mathbf{d}\}$ are zero at all sampling points for a certain mode $\{\mathbf{d}\}$ , then $U_{e}$ will vanish for that $\{\mathbf{d}\}$ , and, according to Eq. 6.12-1, $[\mathbf{k}]$ will be a zero-stiffness matrix in the sense that strain energy $U_{e} = \{\mathbf{d}\}^{T}[\mathbf{k}]\{\mathbf{d}\} /2$ is zero for this particular $\{\mathbf{d}\}$ . We expect that $U_{e} = 0$ if $\{\mathbf{d}\}$ is a rigid-body motion. If $U_{e} = 0$ when $\{\mathbf{d}\}$ is a rigid-body motion, then an instability is present.
+
+not a rigid-body motion, then an instability is present. An element that displays a mechanism is said to be rank deficient. That is, the rank of [k] is less than the number of element d.o.f. minus the number of rigid-body modes.
+
+An instability in an existing [k] can be detected by means of an eigenvalue test (Section 18.8). In the present section we give examples of instabilities and briefly discuss their prevention.
+
+Examples. Consider the four-node plane (bilinear) element, whose stiffness matrix is 8 by 8. Eight independent displacement modes $\{\mathbf{d}\}$ can be identified (Fig. 6.12-1). The first three are rigid-body modes, for which $U_{e} = 0$ , as is correct, regardless of the quadrature rule used. The next three modes, numbers 4, 5, and 6, are constant-strain modes, for which $U_{e} > 0$ , regardless of the quadrature rule used. Modes 7 and 8 are bending modes. An order 1 rule, whose single Gauss point is at the element center, does not sense these modes, as $\epsilon_{x} = \epsilon_{y} = \gamma_{xy} = 0$ at the center. Accordingly, $U_{e} = 0$ for modes 7 and 8, and the element displays two mechanisms. These two spurious modes disappear if the Gauss rule is order 2 or greater.
+
+2 or greater.
+The foregoing mechanisms can appear in a mesh of elements as well as in a single element (Fig. 6.12-2). In Fig. 6.12-2d, modes 7 and 8 of Fig. 6.12-1 are combined with a rigid-body rotation of each element. The mechanisms of Fig.
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+<!-- source-page: 211 -->
+
+![](images/page-211_14743e993c8cf298801d9d476d48b61692597260c00adbeccb5882490e99abc7.jpg)  
+Figure 6.12-1. Independent displacement modes of a bilinear element.
+
+6.12-2 are called hourglass modes because of their physical shape. Each of these distortions, as well as each straining mode in Fig. 6.12-1, would be considered the same mode if nodal d.o.f. were reversed—that is, if $\{d\}$ of that mode were replaced by $-\{d\}$ .
+
+Elements need not be rectangular in order to display a mechanism. Imagine, for example, that displacements are $u = a_{1}\xi\eta$ and $v = a_{2}\xi\eta$ , where $a_{1}$ and $a_{2}$ are constants. Then, at $\xi = \eta = 0$ , we have $u_{, \xi} = u_{, \eta} = v_{, \xi} = v_{, \eta} = 0$ ; hence, according to Eqs. 6.3-17 and 6.3-18, $\epsilon_{x} = \epsilon_{y} = \gamma_{xy} = 0$ at the Gauss point of an order 1 rule, regardless of the shape of the element.
+
+Consider next the quadratic plane element, having either eight or nine nodes, and integrated with a 2 by 2 Gauss rule (Fig. 6.12-3). Displacements in the nine-node element of Fig. 6.12-3b are [6.11]
+
+$$
+\begin{array}{l} u = 3 \xi^ {2} \eta^ {2} - \xi^ {2} - \eta^ {2} \\ v = 0 \end{array} \tag {6.12-2}
+$$
+
+At the Gauss points of a 2 by 2 rule—that is, where $\xi$ and $\eta$ are $\pm 1/\sqrt{3}$ —one finds $u_{, \xi} = u_{, \eta} = v_{, \xi} = v_{, \eta} = 0$ . Therefore, according to Eqs. 6.3-17 and 6.3-18, strains are zero at these points, for any geometric shape of the undeformed ele
+
+![](images/page-211_560d3468fe5eebe589347c87a556ed2e8824b6d1327c7bafea74636e9adb6960.jpg)  
+Figure 6.12-2. (a) Mesh of four bilinear elements, showing Gauss points of an order 1 rule in each element (squares). (b,c,d) Possible mechanisms (“hourglass” modes).
+
+<!-- source-page: 212 -->
+
+![](images/page-212_52d358e24ce55995a4fe74ccedc77c782a6c73d639e3c2c134c4e34100ac08db.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+4
+7
+3
+8
+η
+ξ
+9
+6
+1
+5
+2
+</details>
+
+Eight- or nine-node elements   
+(a)
+
+![](images/page-212_7254d1be884df8114f5a310351d3a4a5f1b0e71fd11be6cdc91070721f3130bf.jpg)
+
+<details>
+<summary>text_image</summary>
+
+c,u
+</details>
+
+Nine-node element only   
+(b)
+
+![](images/page-212_ae1df7ef479082fa6f9ea45dbba9b9d6750c9fe256c7616518d93fe471df0fad.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Pure geometric diagram with curved and straight lines forming a grid-like structure (no text or symbols)
+</details>
+
+Nine-node element only   
+(c)
+
+![](images/page-212_04d8e03a77aea30ce4f1e85d90e2fd59670c6c7643632043bdd08897f7523886.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Pure geometric diagram of a symmetrical curved shape with dashed and solid lines, no text or symbols present
+</details>
+
+Eight- and nine-node elements   
+(d)   
+Figure 6.12-3. Possible mechanisms (“hourglass” modes) in quadratic elements integrated by an order 2 rule. Gauss points are shown by squares.
+
+ment. A similar $v$ field is possible (Fig. 6.12-3c). Thus we have identified two mechanisms.
+
+mechanisms.
+The foregoing two mechanisms are not possible in the eight-node element because the $\xi^{2}\eta^{2}$ term is not present (see Eqs. 6.6-1). However, yet another mechanism is possible in both eight-node and nine-node elements (Fig. 6.12-3d). Its displacement field is simple to state for a square element; it is
+
+$$
+u = \xi (3 \eta^ {2} - 1) \quad \text { and } \quad v = \eta (1 - 3 \xi^ {2}) \tag {6.12-3}
+$$
+
+Again, strains are zero at the Gauss points of a 2 by 2 rule. This mechanism is usually not of great concern because two adjacent elements cannot both have such a mode, as may be seen by trying to connect two deformed elements. Thus an instability present in individual elements is not present in the mesh.
+
+Summing up, Fig. 6.12-3 identifies three element instabilities in quadrature elements arising from a 2 by 2 Gauss quadrature rule. The element stiffness matrix has rank 12 for both eight-node and nine-node elements (rank equals order less the number of rigid-body and instability modes). None of these instabilities exists if the Gauss rule is 3 by 3 or greater.
+
+if the Gauss rule is 5 by 5 of greater. A mesh that has no mechanisms may yet behave badly because restraints on the mechanisms are weak. Consider Fig. 6.12-4a. Elements may be the four-node elements of Fig. 6.12-2 or the nine-node elements of Fig. 6.12-3, respectively integrated by one-point and four-point rules. Load P is concentrated and applied centrally rather than being distributed across the right end. Mechanisms are not possible because all nodes at the left support are fixed. However, this restraint
+
+![](images/page-212_235c621c3b39bfb6f3c8331b476b4d706739c5b667d94e7a79dd91647279d6f4.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+L_T
+</details>
+
+(a)
+
+![](images/page-212_b5bd6ce63dc3b709c12711a4e078e8571827b0473f83db3c4558b94041f97338.jpg)
+
+<details>
+<summary>text_image</summary>
+
+E₁>>E₂
+E₁
+E₂
+P
+</details>
+
+(b)   
+Figure 6.12-4. Problems that involve “near mechanisms,” if reduced integration is used. In (b), elements that were initially rectangular are here shown deformed.
+
+<!-- source-page: 213 -->
+
+becomes weaker with increasing distance from the support. Near the load, distortions of the type shown in Figs. 6.12-2b and 6.12-3b become pronounced. Indeed, for a 2 by 24 mesh, the computed displacement of load P may be over 500 times the displacement predicted by the elementary formula PL/AE [6.12].
+
+A similar situation is depicted in Fig. 6.12-4b [4.6]. A 2 by 2 Gauss rule is used to integrate [k] of each element. The stiff element, shown shaded, is weakly restrained by soft elements connected to the fixed boundary, allowing the mode of Fig. 6.12-3d (with signs of {d} reversed) to become pronounced, although not unbounded.
+
+Elements for solids, for plate bending, and for nonstructural problems can also suffer from instabilities. Methods for detecting and controlling these modes are similar to methods used for plane elements.
+
+A conservative analyst will avoid using any element that contains a possible instability because its dangers may not be foreseen.
+
+Control of Instabilities. Various control methods have been proposed. Their goal is to eliminate instability by providing restraint, but without simultaneously stiffening the element's response to "legitimate" modes that are already working well. In what follows we summarize an effective method, with particular reference to a rectangular bilinear element. The method adds "hourglass stiffness" to an element integrated by one-point quadrature. The resulting element is inexpensive to formulate and works very well.
+
+For simplicity, consider only the $x$ -direction nodal displacements $\{\mathbf{d}_x\}$ of the eight modes shown in Fig. 6.12-1. For modes 1 and 8, $\{\mathbf{d}_x\} = \{\mathbf{0}\}$ . An arbitrary combination of modes 2 through 6 is
+
+$$
+\left\{\mathbf {d} _ {x} \right\} = a _ {2} \left\{ \begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \end{array} \right\} + a _ {3} \left\{ \begin{array}{c} 1 \\ 1 \\ - 1 \\ - 1 \end{array} \right\} + a _ {4} \left\{ \begin{array}{c} - 1 \\ 1 \\ 1 \\ - 1 \end{array} \right\} + a _ {5} \left\{ \begin{array}{c} 1 \\ - 1 \\ - 1 \\ 1 \end{array} \right\} + a _ {6} \left\{ \begin{array}{c} - 1 \\ - 1 \\ 1 \\ 1 \end{array} \right\} \tag {6.12-4}
+$$
+
+where the $a_{i}$ are constants. Mode 7 is
+
+$$
+\{\mathbf {d} _ {x} \} _ {7} = a _ {7} \left[ \begin{array}{l l l l} 1 & - 1 & 1 & - 1 \end{array} \right] ^ {T} \tag {6.12-5}
+$$
+
+To provide mode 7 with the stiffness it lacks under one-point quadrature, we form the “stabilization matrix”
+
+$$
+[ \mathbf {k} ] _ {7} = \{\mathbf {d} _ {x} \} _ {7} \{\mathbf {d} _ {x} \} _ {7} ^ {T} \tag {6.12-6}
+$$
+
+A similar matrix $[k]_{8}$ , containing a constant $a_{8}$ , serves to restrain mode 8. To the stiffness matrix computed by one-point quadrature, we now add $[k]_{7}$ and $[k]_{8}$ . It is possible to choose values of $a_{7}$ and $a_{8}$ such that a rectangular element displays the exact strain energy in states of pure bending.
+
+Note that mode 7 is orthogonal to all other modes—that is,
+
+$$
+\{\mathbf {d} _ {x} \} _ {i} ^ {T} \{\mathbf {d} _ {x} \} _ {i} = \{\mathbf {0} \} \quad \text { for } \quad i = 1, 2, 3, 4, 5, 6, 8 \tag {6.12-7}
+$$
+
+Orthogonality prevents $[k]_{7}$ from stiffening modes other than mode 7. That this is so may be seen by computing nodal forces $\{\bar{r}\}_{i}$ associated with matrix $[k]_{7}$ ,
+
+<!-- source-page: 214 -->
+
+$$
+\{\overline {{{\mathbf {r}}}} \} _ {i} = [ \mathbf {k} ] _ {7} \{\mathbf {d} _ {x} \} _ {i} = \{\mathbf {d} _ {x} \} _ {7} \{\mathbf {d} _ {x} \} _ {7} ^ {T} \{\mathbf {d} _ {x} \} _ {i} = \{\mathbf {d} _ {x} \} _ {7} (0) = \{\mathbf {0} \} \tag {6.12-8}
+$$
+
+for $i = 1$ through 6 and for $i = 8$ .
+
+for $r = 1$ through 0 and for $r = 0$ . The foregoing control method can be generalized to elements having more than four nodes and to elements of arbitrary shape [6.11, 6.13, 13.49, 13.52-13.54].
+
+# 6.13 REMARKS ON STRESS
+
+# COMPUTATION
+
+Element stresses follow from Eq. 4.7-1, with the substitution $\{\epsilon\} = [\mathbf{B}]\{\mathbf{d}\}$ :
+
+$$
+\{\pmb {\sigma} \} = [ \mathbf {E} ] ([ \mathbf {B} ] \{\mathbf {d} \} - \{\pmb {\epsilon} _ {0} \}) + \{\pmb {\sigma} _ {0} \} \tag {6.13-1}
+$$
+
+Here, in isoparametric elements, [B] is a function of the natural coordinates and $\{\sigma\}$ contains stresses referred to the global coordinate system xyz. Where in the element should stresses be calculated? For isoparametric elements, it often happens that stresses (especially shear stresses) are most accurate at Gauss points of a quadrature rule one order less than that required for full integration of the element stiffness matrix.
+
+Consider Fig. 6.13-1. Sides of a bilinear element remain straight during deformation. A typical element, deformed by bending moment but with rigid-body motion removed, is shown in Fig. 6.13-1b. Displacements in the element are $u = -a_1\xi \eta$ and $v = 0$ , where $a_1$ is a positive constant. Thus shear strain $\gamma_{xy}$ is proportional to $\xi$ . On the neutral surface of bending, $\gamma_{xy}$ displays the sawtooth pattern seen in Fig. 6.13-1c. Only at $\xi = 0$ in each element is $\gamma_{xy}$ correctly computed (as zero) under pure bending deformation. In a general problem of plane stress analysis, where bending can occur in both directions (modes 7 and 8 of Fig. 6.12-1 simultaneously), the best computation point for $\gamma_{xy}$ in a bilinear element is at $\xi = \eta = 0$ . This is the Gauss point location of an order 1 rule, which is one order less than the order 2 rule of full integration.
+
+A similar circumstance occurs with the eight-node and nine-node quadratic elements. In the beam of Fig. 6.13-2, the exact $\gamma_{xy}$ is constant along the x axis. In the quadratic element, $\gamma_{xy}$ along the x axis displays the parabolic distributions shown. However, one finds that the quadratic element displays the correct $\gamma_{xy}$ at the Gauss points of a 2 by 2 quadrature rule. In other problems of stress analysis, normal strains can also display parabolic variations, and again the most accurate strains are to be found at the Gauss points of an order 2 rule.
+
+![](images/page-214_76c70f81b425499340a79aaeff55f598e64b514213d512b63dfd9493649e5b76.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+x,u
+P
+L L
+P
+</details>
+
+(a).
+
+![](images/page-214_67fd1551b5bf217f5f8938bff13481efc53da8dab462bfde95212108c5b00383.jpg)  
+(b)
+
+![](images/page-214_e1c74f743868b57c7ed069708f7ecb905af0246aa03422bba5b2ab0e06e033c6.jpg)
+
+<details>
+<summary>text_image</summary>
+
+γxy
+Finite element
+Exact
+0
+L
+2L
+x
+</details>
+
+(c)   
+Figure 6.13-1. (a) Beam loaded in bending. (b) Bending distortion of a typical bilinear element. (c) Shear strain along the x axis.
+
+<!-- source-page: 215 -->
+
+![](images/page-215_a9bfd5fa91e3b1e3e5dac3c2cda8fba30dfea0dd5c9b864e0039dabb085c17ea.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+x
+L
+L
+V
+</details>
+
+(a)
+
+![](images/page-215_115b7f208029f45ccd1cecd7cac75dd61cc087df506329b19182022528456635.jpg)
+
+<details>
+<summary>text_image</summary>
+
+γxy
+Finite element
+Exact
+0
+L
+2L
+x
+</details>
+
+(b)   
+Figure 6.13-2. (a) Beam loaded by transverse tip force V. (b) Shear strain along the x axis.
+
+In elements based on displacement fields, one expects stresses to be less accurate than displacements, as explained in Section 3.5. However, in the foregoing examples, stresses are “superaccurate” or “superconvergent” at the Gauss points because there they have the same degree of accuracy as displacements. Indeed, in unusual situations it may happen that stresses are more accurate than displacements. For example, in Fig. 6.12-4a, stresses may be substantially correct at Gauss points (of an order 1 or order 2 rule, for four- and nine-node elements, respectively), although displacements are grossly in error. This is possible because the modes that permit excessive displacements produce zero strain at the Gauss points.
+
+The theory of locating error-minimal points for stress computation is explained elsewhere [6.14,6.15]. One discovers that these points are Gauss points: at $\xi = \eta = 0$ in bilinear (plane) and trilinear (solid) elements, and where $\xi, \eta$ , and $\zeta$ are $\pm 1/\sqrt{3}$ in eight- or nine-node quadratic (plane) and 20- or 21-node quadratic (solid) elements. These conclusions are rigorously true for rectangular elements. For distorted elements, Gauss points may not be optimal locations but they remain very good choices.
+
+Stresses at Gauss points can be interpolated or extrapolated to other points in the element. The result obtained is usually more accurate than the result of evaluating Eq. 6.13-1 directly at the point of interest. The interpolation–extrapolation process is explained as follows.
+
+Imagine that stresses have been computed at the four Gauss points of a plane element (points 1, 2, 3, and 4 in Fig. 6.13-3). We now wish to interpolate or extrapolate these stress values to other points in the element. In Fig. 6.13-3, coordinate r is proportional to $\xi$ and s is proportional to $\eta$ . At (say) point 3, r = s = 1 and $\xi = \eta = 1/\sqrt{3}$ . Therefore, the factor of proportionality is $\sqrt{3}$ ; that is,
+
+![](images/page-215_273a30b10bddd3bbd22cfbcc3a6d5a59024fe283baa43229b3c5e8528f0b2cd1.jpg)
+
+<details>
+<summary>text_image</summary>
+
+D
+η
+r = 1 ξ = 1
+G
+C
+η = 1
+4
+s
+3
+s = 1
+H
+r
+F
+ξ
+1
+*P
+2
+A
+E
+B
+</details>
+
+Figure 6.13-3. Natural coordinate systems used in extrapolation of stresses from Gauss points.
+
+<!-- source-page: 216 -->
+
+$$
+r = \xi \sqrt {3} \quad \text { and } \quad s = \eta \sqrt {3} \tag {6.13-2}
+$$
+
+Stresses at any point P in the element are found by the usual shape functions,
+
+$$
+\sigma_ {P} = \sum N _ {i} \sigma_ {i} \quad \text { for } \quad i = 1, 2, 3, 4 \tag {6.13-3}
+$$
+
+where $\sigma$ is $\sigma_x, \sigma_y$ , or $\tau_{xy}$ . The $N_i$ are the bilinear shape functions given by Eq. 6.3-2, but now written in terms of $r$ and $s$ rather than $\xi$ and $\eta$ ; that is,
+
+$$
+N _ {i} = \frac {1}{4} (1 \pm r) (1 \pm s) \tag {6.13-4}
+$$
+
+In Eq. 6.13-3, the $N_{i}$ are evaluated at the r and s coordinates of point P. For example, let point P coincide with corner A. To calculate stress $\sigma_{xA}$ at corner A from $\sigma_{x}$ values at the four Gauss points, we substitute $r = s = -\sqrt{3}$ into the shape functions, and obtain
+
+$$
+\sigma_ {x A} = 1. 8 6 6 \sigma_ {x 1} - 0. 5 0 0 \sigma_ {x 2} + 0. 1 3 4 \sigma_ {x 3} - 0. 5 0 0 \sigma_ {x 4} \tag {6.13-5}
+$$
+
+For solids, an interpolation-extrapolation formula similar to Eq. 6.13-3 is based on stresses at eight Gauss points and the trilinear $N_{i}$ of Eq. 6.7-6.
+
+In Section 4.7 we advised that usually the temperature field used for thermal stress analysis should have the same competence as the element strain field. Accordingly, if element stresses are based on Gauss point values, thermal strains $\{\epsilon_0\}$ in Eq. 6.13-1 should also be based on Gauss point values.
+
+# 6.14 EXAMPLES. EFFECT OF ELEMENT GEOMETRY
+
+Simple test problems show how accuracy is affected by element distortion, changes in Gauss quadrature rule, and changes in element aspect ratio. Our examples are two-dimensional, but the trends displayed pertain to three-dimensional elements as well.
+
+Example Problems. Table 6.14-1 illustrates the behavior of the bilinear element when its [k] is formed by four-point Gauss quadrature. Results are expressed as
+
+TABLE 6.14-1. STRESSES AND DEFLECTIONS IN CANTILEVER BEAMS OF CONSTANT THICKNESS UNDER TRANSVERSE TIP LOAD P. LENGTH = 10, DEPTH = 2, $\nu = 0.25$ . VALUES BY BEAM THEORY = 1.000, OF WHICH 3% OF $v_{A}$ IS DUE TO TRANSVERSE SHEAR DEFORMATION. 
+<table><tr><td colspan="2"><img src="images/15fa0771efd72f6e2f835ba4463a863d67340a7104f3a8a4472744f8b0484e4e.jpg"/></td><td colspan="2"><img src="images/574a2d63da4438867763ad37e1aeb4b75afd6163bf611fe4651e53fffcd1a63c.jpg"/></td><td colspan="2"><img src="images/af099a7768148d0937be92137cee24bd0a7b408c94c6dc65812355a9f7884a68.jpg"/></td></tr><tr><td> $\sigma_{xB}$ </td><td> $v_A$ </td><td> $\sigma_{xC}$ </td><td> $v_A$ </td><td> $\sigma_{xC}$ </td><td> $v_A$ </td></tr><tr><td>0.096</td><td>0.091</td><td>0.727</td><td>0.682</td><td>0.301</td><td>0.494</td></tr></table>
+
+<!-- source-page: 217 -->
+
+the ratio of computed value to the value given by beam theory. We see that square elements are better than elongated elements, and that geometric distortion stiffens the element and makes answers less accurate.
+
+The principal failing of the bilinear element is that under pure bending loads, for which $\gamma_{xy}$ should be zero, the element displays substantial values of $\gamma_{xy}$ except at its center, as noted in connection with Fig. 6.13-1. This defect, known as parasitic shear, makes the element too stiff in bending. An improved form of the element discussed in Section 8.3.
+
+Table 6.14-2 illustrates the behavior of eight-node and nine-node versions of the quadratic element [6.12]. All nodes at the left end are fixed. Load P on the right end is allotted to nodes in the proportion 1–8–1, which is consistent with a parabolic distribution of shear stress. Side nodes are midway along the sides. Point B is a Gauss point of a 2 by 2 quadrature rule. Results are expressed as the ratio of computed value to the value given by beam theory.
+
+When elements are rectangular, we see that eight-node and nine-node elements have comparable accuracy. Both become stiffer if the quadrature rule used to generate [k] is changed from 2 by 2 to 3 by 3.
+
+Next in Table 6.14-2, elements are made trapezoidal by moving nodes C and D horizontally to positions L/4 and 3L/4, where L is the length of the beam. The final mesh in Table 6.14-2 introduces one curved interelement boundary by moving node E left of center an amount L/20. We see that 2 by 2 is the preferred integration rule, and that the eight-node element is much more sensitive to geometric distortion than the nine-node element. In one case, stress $\sigma_{xB}$ in the eight-node element is not even of the correct sign.
+
+The obvious lesson is that an ideal element is compact, straight-sided, and has equal corner angles. Of course, elements must be distorted to some extent in modeling an actual structure, but gratuitous distortion is to be avoided. In particular, if an element side is curved to model the curved boundary of a structure, other element sides that form interelement boundaries should be straight.
+
+Quadratic triangles (Sections 5.5 and 6.8) have approximately the same accuracy as the nine-node element. For example, in the second case in Table 6.14-2, let
+
+TABLE 6.14-2. STRESSES AND DEFLECTIONS IN TWO-ELEMENT CANTILEVER BEAMS OF CONSTANT THICKNESS UNDER TRANSVERSE TIP LOAD P. LENGTH = 100, DEPTH = 10, $\nu = 0.30$ . VALUES BY BEAM THEORY = 1.000 (IN WHICH THE TRANSVERSE-SHEAR CONTRIBUTION TO $v_{A}$ IS NEGLECTED). SKETCHES ARE NOT TO SCALE. 
+
+<table><tr><td>Element Type</td><td>Gauss Rule</td><td> $\sigma_{xB}$ </td><td> $v_A$ </td><td> $\sigma_{xB}$ </td><td> $v_A$ </td><td> $\sigma_{xB}$ </td><td> $v_A$ </td></tr><tr><td>8 node</td><td>2 × 2</td><td>1.000</td><td>0.968</td><td>0.051</td><td>0.362</td><td>-0.048</td><td>0.430</td></tr><tr><td>8 node</td><td>3 × 3</td><td>1.129</td><td>0.930</td><td>0.048</td><td>0.161</td><td>0.050</td><td>0.221</td></tr><tr><td>9 node</td><td>2 × 2</td><td>1.000</td><td>1.006</td><td>1.125</td><td>1.109</td><td>0.958</td><td>0.955</td></tr><tr><td>9 node</td><td>3 × 3</td><td>1.141</td><td>0.954</td><td>0.687</td><td>0.791</td><td>0.705</td><td>0.737</td></tr></table>
+
+<!-- source-page: 218 -->
+
+each quadrilateral be divided along its shorter diagonal, to which a midside node is added. Thus we produce four straight-sided quadratic triangles, which yield $v_{A} = 0.796$ .
+
+Geometric Distortion: Examples and Tests. Elements in Fig. 6.14-1 have very poor geometry. Such elements should not be used. But if used, and if surrounded by elements of acceptable geometry, stresses will be poor in and very near the distorted element, but reasonable at some distance away because of Saint-Venant's principle.
+
+In Fig. 6.14-1, Gauss points of a 2 by 2 rule lie at centers of the small black quadrilaterals. Dashed lines are lines of constant $\xi$ and constant $\eta$ . Dashed lines would be parallel if the element were rectangular (or if the elements were sketched in $\xi \eta$ space, where each element is square). The first element in Fig. 6.14-1 could be a bilinear element or a quadratic element with midside nodes. The remaining three elements are distortions of quadratic rectangles with midside nodes, respectively created by moving one corner node, one side node, and two side nodes.
+
+In Fig. 6.14-1a there is a singularity at node 3, where the Jacobian determinant $J$ is zero. Elsewhere in the element, $J > 0$ . If the $\xi \eta$ system were made left-handed, by numbering nodes clockwise around the element but leaving shape
+
+![](images/page-218_c4a4f0dfdaa83bec581b6ba4badae4ad84fd88390d5e589f79122daf167d6bf7.jpg)
+
+<details>
+<summary>text_image</summary>
+
+4
+η
+3
+ξ
+1
+2
+</details>
+
+(a)
+
+![](images/page-218_8311dd6091d29621841c8723becadbf8fc506c9e7daa9fe119dbdbee0149180d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+4
+7
+3
+η
+8
+1
+ξ
+6
+5
+2
+</details>
+
+(b)
+
+![](images/page-218_d26a8405f550d0af708f814eeed145d044ae5723a171955eaf255c71130e5a61.jpg)
+
+<details>
+<summary>text_image</summary>
+
+4
+8
+1
+5
+2
+η
+3,7
+ξ
+6
+</details>
+
+(c)
+
+![](images/page-218_44e56cef08a53dd50a1a6de7a5fb1598f9116d189c39bc63e844a83cbf742fc9.jpg)
+
+<details>
+<summary>text_image</summary>
+
+4
+8
+η
+ξ
+6,7
+1
+5
+2
+3
+</details>
+
+(d)   
+Figure 6.14-1. Badly shaped elements, showing Gauss points bounded by lines of constant $\xi$ and constant $\eta$ . (a) Interior angle at node 3 is $180^{\circ}$ . (b) Node 1 moved to the center of the original rectangle. (c) Node 7 moved from top midside to corner 3. (d) Two side nodes moved to center of the original rectangle.
+
+<!-- source-page: 219 -->
+
+functions and the $xy$ system unchanged, we would find $J < 0$ within the element, and all diagonal coefficients of the element [k] would be negative.
+
+In the latter three elements of Fig. 6.14-1, part of each element falls outside the intended element boundaries and J < 0 at one of the Gauss points. These distortions do not prevent the elements from passing constant-strain patch tests, but drastically reduce the ability of the elements to represent more complicated states of deformation [6.16].
+
+A user-oriented finite element program performs tests on the geometry of elements. Clearly it is easy to check that interior corner angles of quadrilaterals are not far from $90^{\circ}$ , that side nodes are not far from the midpoint of a straight line between adjacent corner nodes, and that J is positive at each Gauss point and not greatly different from the value of J at other Gauss points. It is extra trouble, but perhaps advisable, to check that J is also positive at vertex nodes [6.16].
+
+# PROBLEMS
+
+# Section 6.2
+
+6.1 (a) Determine $[N]$ of Eq. 6.2-3 by following the formal procedure suggested below Eq. 6.2-1.   
+(b) Determine [N] of Eq. 6.2-3 by use of Lagrange's formula, as suggested below Eq. 6.2-1.   
+6.2 (a) Show that if $x_{3}$ is at the middle of a bar of length $L$ (Fig. 6.2-1a), then $J = L / 2$ in Eq. 6.2-5. Let node 1 have the arbitrary value $x_{1}$ .   
+(b) How far from the center of the bar can node 3 be placed if, according to Eq. 6.2-4, strain $\epsilon_{x}$ is to remain positive at the ends of the bar for arbitrary values of $u_{1}, u_{2}$ , and $u_{3}$ ?   
+6.3 Determine the element stiffness matrix [k] if $x_{1} = 0$ , $x_{2} = L$ , and $x_{3} = L / 2$ in Fig. 6.2-1a. Let $A$ and $E$ be constant and do integrations explicitly.   
+6.4 Omit node 3 in Fig. 6.2-1a, so that the bar becomes a linear element with end nodes only. Derive the 2 by 2 stiffness matrix [k] by using the natural coordinate $\xi$ .   
+6.5 The bar shown is fixed at both ends. It is modeled by one three-node element, whose shape functions are given by Eq. 6.2-3. Show that if the bar is uniform and loaded axially by its own weight, the exact stress distribution is obtained.
+
+![](images/page-219_2c70774f5a6eb862af5ef9c0329cc51ccb5d2d932741ee73e5ce70a1169f0190.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L/2
+L/2
+1
+3
+2
+</details>
+
+Problem 6.5
+
+# Section 6.3
+
+6.6 With reference to Fig. 6.3-1a, let $x = \lfloor 1 \xi \eta \xi \eta \rfloor \lfloor a_1 a_2 a_3 a_4 \rfloor^T$ .
+
+(a) Hence, write [A] in the relation $\left\lfloor x_1 \quad x_2 \quad x_3 \quad x_4 \right\rfloor^T = [A] \left\lfloor a_1 \quad a_2 \quad a_3 \quad a_4 \right\rfloor^T$ .
+
+<!-- source-page: 220 -->
+
+(b) By inspection of Eqs. 6.3-2, write $[\mathbf{A}]^{-1}$ in the relation $x = \left\lfloor 1 \quad \xi \quad \eta \quad \xi \eta \right\rfloor$ $[\mathbf{A}]^{-1} \left| x_1 \quad x_2 \quad x_3 \quad x_4 \right|^T$ .   
+(c) Check your answers by seeing if $[\mathbf{A}][\mathbf{A}]^{-1} = [\mathbf{I}]$ .
+
+6.7 Sketch a quadrilateral, with corners properly lettered and $\xi \eta$ axes properly oriented, if shape functions are written as
+
+$$
+N _ {A} = \frac {1}{4} (1 - \xi) (1 + \eta) \quad N _ {C} = \frac {1}{4} (1 - \xi) (1 - \eta)
+$$
+
+$$
+N _ {B} = \frac {1}{4} (1 + \xi) (1 + \eta) \quad N _ {D} = \frac {1}{4} (1 + \xi) (1 - \eta)
+$$
+
+6.8 The choice of natural coordinates made in Fig. 6.3-1 is not unique. As an alternative one could adopt natural coordinates r and s, as shown in the sketch for this problem. Write shape functions of the bilinear element in terms of r and s.
+
+![](images/page-220_ea7fceae2f68e5de6a16792c493c75df76cff37f7ac86c367a2855fb2346356f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+s
+s = 1
+4
+3
+r = 1
+1
+2
+r
+y
+x
+</details>
+
+Problem 6.8
+
+![](images/page-220_c1e43436ccdc7671e5971d9c050cdff5fd01e0bd6db2da1a8934809c1dae2dc4.jpg)
+
+<details>
+<summary>text_image</summary>
+
+B
+A
+D
+C
+</details>
+
+Problem 6.9
+
+6.9 For the element shown, sketch the lines $\xi = -0.5, 0.0$ , and 0.5 and the lines $\eta = -0.5, 0.0$ , and 0.5. The $N_{i}$ are given by Eq. 6.3-2. Let $\xi = \eta = -1$ at (a) point $A$ , (b) point $B$ , (c) point $C$ , and (d) point $D$ .   
+6.10 Show that $y, \eta = J\xi_{,x}$ . Also write the remaining three similar relationships among the $\xi$ and $\eta$ derivatives of $x$ and $y$ and the $x$ and $y$ derivatives of $\xi$ and $\eta$ .   
+6.11 Sketch a bilinear element for which $J$ is a function of $\xi$ but not of $\eta$ .   
+6.12 Both elements shown are square and two units on a side. Both are improperly numbered. For each, determine [J] and $J$ , using the $N_{i}$ of Eq. 6.3-2. What do the given numberings imply about the $\xi \eta$ axes of the first element and the actual shape of the second element?
+
+![](images/page-220_0fe090bf04d648eba3f75e5928ae774f9e7f0374c26f172f2c1e3f8a2347414a.jpg)  
+(a)
+
+![](images/page-220_79752919eda5fa5d29df9af25b406cf29f30a66e6b7c931f49fa3ba277b70beb.jpg)  
+(b)   
+Problem 6.12
+
+6.13 Evaluate [J] and $J$ for each of the four elements shown. Also compute the ratio of element area to the area of a square two units on a side. How is this ratio related to $J$ , and why?
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_023.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_023.md
new file mode 100644
index 00000000..bd8eada9
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_023.md
@@ -0,0 +1,493 @@
+<!-- source-page: 221 -->
+
+![](images/page-221_28ea43a6a3cbd7fa8ebce998cd478aeb9556f0ac4cfb38a056be23ec18acacce.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+3
+4
+3
+1
+2
+3
+4
+1
+</details>
+
+(a)
+
+![](images/page-221_ec49e2893b0316ee7efd18da8e3728ef90e80baae508aa1964830ed8cb2c2b91.jpg)
+
+<details>
+<summary>text_image</summary>
+
+3
+3
+3
+3
+2
+2
+4
+1
+</details>
+
+(b)
+
+![](images/page-221_08761855d3bf3140493fc33c71aeaa8a9082b79fcc1dd14ef8fe65ccc5542f91.jpg)  
+(t)
+
+![](images/page-221_f16c78d359f5aaeba23cb792398f8a7abd0b050b55f0257ed6405128583ab646.jpg)
+
+<details>
+<summary>text_image</summary>
+
+3
+y
+3
+4
+3
+1
+2
+2
+2
+y
+</details>
+
+{d}   
+Problem 6.13
+
+# Section 6.4
+
+6.14 Derive the locations and weights of an order 2 Gauss rule by requiring that it integrate exactly the polynomial $\phi = a_{1} + a_{2}\xi + a_{3}\xi^{2} + a_{4}\xi^{3}$ in the range $-1 \leqslant \xi \leqslant 1$ . Assume that weights and points are symmetric with respect to the axis $\xi = 0$ .   
+6.15 Use one-, two-, and three-point Gauss quadrature to integrate each of the following functions. Compare these answers with the exact answers.
+
+(a) $\phi = \cos \pi x / 2$ between $x = -1$ and $x = 1$ .   
+(b) $\phi = (2 - x) / (2 + x)$ between $x = -1$ and $x = 1$ .   
+(c) $\phi = 1 / (x^2 - 3x + 4)$ between $x = -1$ and $x = 1$ .   
+(d) $\phi = 1 / x$ between $x = 1$ and $x = 7$ .
+
+6.16 Write an expression for $I$ , analogous to Eq. 6.4-8, for (a) a 2 by 3 quadrature rule, and (b) a 3 by 4 quadrature rule.   
+6.17 For any of the quadrature rules in Table 6.4-1, weights $W_{i}$ sum to 2 in one dimension, weight products $W_{i}W_{j}$ sum to 4 in two dimensions, and weight products $W_{i}W_{j}W_{k}$ sum to 8 in three dimensions. Why? Check this behavior in Eq. 6.4-8.   
+6.18 Use a 2 by 2 Gauss rule to approximate $I$ over the rectangular region shown.
+
+![](images/page-221_b813b7ab1e79f3688052ae704daea1dfc922fa86748dc9f4780a8b38ff5cdea1.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+← 6 →
+I = ∏(3+x²)/2+y² dx dy
+4
+x
+</details>
+
+Problem 6.18
+
+6.19 In Problem 6.8, what are the $r$ and $s$ coordinates of the Gauss points of an order 2 rule? And what are the corresponding weights $W_{i}$ ? (Integration is from 0 to 1 for both $r$ and $s$ .)
+
+6.20 (a) Determine the element stiffness matrix [k] of a two-node uniform bar element of length L by use of an order 2 Gauss quadrature rule. Check your result against Eq. 2.4-5.
+
+(b) Repeat part (a), but let the cross-sectional area vary linearly from $A_{1}$ at node 1 to $A_{2}$ at node 2.   
+(c) Repeat part (b), but use one-point Gauss quadrature.
+
+6.21 Determine the 3 by 3 element stiffness matrix [k] if node 3 in Fig. 6.2-1a is at the middle of the bar and AE is constant. Use an order 2 Gauss quadrature rule.
+
+<!-- source-page: 222 -->
+
+# Section 6.5
+
+6.22 Subroutine QUAD4 (Fig. 6.5-2) would be more efficient if DV were removed from statement 60 and placed elsewhere. Where? And why?   
+6.23 Using Figs. 6.5-1 and 6.5-2 as a guide, write Fortran statements that will generate the stiffness matrix of the bar element in Fig. 6.2-1 by three-point Gauss quadrature. Let cross-sectional area $A$ be linearly interpolated from known values at node 1 and node 2. Node 3 is not necessarily at the midpoint.   
+6.24 Using Figs. 6.5-1 and 6.5-2 as a guide, write Fortran statements that will generate the stiffness matrix of a uniform beam element (Fig. 4.2-2 and Eq. 4.2-5) by two-point Gauss quadrature.
+
+# Section 6.6
+
+6.25 For the element described by Eqs. 6.6-1, write the eight-term displacement function $\phi = a_{1} + a_{2}\xi + \cdots + a_{8}\xi\eta^{2}$ . Then write the 8 by 8 matrix [A] that arises in exchanging the $a_{i}$ for nodal d.o.f. (in the manner of Eq. 3.8-6).   
+6.26. The element of Fig. 6.6-1a can be called subparametric. Why?   
+6.27 (a) Sketch a rectangular eight-node element for which $J$ is a function of $\eta$ but not of $\xi$ .  
+(b) Sketch a nonrectangular eight-node element with midside nodes for which $J$ is a function of $\eta$ but not of $\xi$ .   
+6.28 What changes would be needed in Figs. 6.5-1 and 6.5-2 to convert these subroutines so that they apply to the eight-node element described by Eqs. 6.6-1? Do not code the shape functions and their derivatives, but otherwise describe the changes precisely.   
+6.29 Identify the defects associated with connecting four-node and eight-node elements in the pattern shown.
+
+![](images/page-222_4c2d3218ac0f4b452f09b52b74fa2b289585ed59ec8a5984734dcc617024ca7e.jpg)
+
+<details>
+<summary>flowchart</summary>
+
+```mermaid
+graph TD
+    A["①"] --> B["②"]
+    B --> C["③"]
+    C --> D["④"]
+    D --> E["End"]
+    A --> C
+    B --> C
+    C --> D
+```
+</details>
+
+Problem 6.29
+
+6.30 For the quadratic Lagrange element (nine nodes), sketch shape function $N_{1}$ in the manner of Fig. 6.6-2d. Decide whether $N_{1}$ is positive or negative in each quadrant by evaluating $N_{1}$ at $\xi = \pm \frac{1}{2}$ and $\eta = \pm \frac{1}{2}$ .   
+6.31 Consider the nine-node element whose shape functions are given by Table 6.6-1. Any of nodes 5 through 9 can be omitted. In similar fashion, could node 1 be omitted, as shown, so as to produce a valid element with two straight edges, whose displacements are governed by nodes 2, 4, 5, and 8? Suggestion: Consider the horizontal (or the vertical) displacement at the lower left corner.
+
+<!-- source-page: 223 -->
+
+![](images/page-223_e50d25fa3f519e7137165ed16337b001c4b23b93a604fa13434939faedd80bd3.jpg)  
+Problem 6.31
+
+![](images/page-223_de72fbc455a5920bbd883e904d2061b29978a77c86eeab2bb483333ca56def0c.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Geometric line drawing of a polyhedron with interconnected vertices and edges (no text or symbols)
+</details>
+
+Problem 6.32
+
+6.32 As an alternative to the nine-node element whose shape functions appear in Table 6.6-1, a nine-node element can be formed by combining eight linear triangles, as shown. What are comparative advantages and disadvantages of these two alternatives?
+
+# Section 6.7
+
+6.33 What changes would be needed in Figs. 6.5-1 and 6.5-2 to convert these subroutines so that they apply to the eight-node solid of Fig. 6.7-1? Describe the changes precisely, including the coding of new shape functions and their derivatives. Assume that a subroutine can be called to invert [J] and compute J.
+
+# Section 6.8
+
+6.34 Show that the bilinear element (Fig. 6.3-1a) becomes a constant-strain triangle if nodes 1 and 4 coalesce. For simplicity, use the particular geometry shown in the sketch.
+
+![](images/page-223_3e6e3ff63b9f91c34f9c5f8755abe3e35f39d8a5899e2666976846934e3b3afd.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+a
+3
+b
+1,4
+2
+x
+</details>
+
+Problem 6.34
+
+6.35 Use the arguments of Eqs. 6.8-1 to 6.8-5 to evaluate [B] for a linear (three-node) triangle. Compare your result with Eqs. 5.2-5 and 5.4-3.
+
+6.36 If $\phi$ is constant, the product $[\mathbf{B}]\{\phi_e\}$ must be zero. Use Eq. 6.8-5 to show that this is so.
+
+6.37 As an alternative to Eqs. 6.8-2 through 6.8-4, one can eliminate (say) $\xi_{3}$ from shape functions $N_{i}$ by use of the constraint relation $\xi_{1} + \xi_{2} + \xi_{3} = 1$ , then take the derivatives $\partial N_{i} / \partial \xi_{1}$ and $\partial N_{i} / \partial \xi_{2}$ . Verify that this procedure also yields Eq. 6.8-5.
+
+6.38 Consider a six-noded triangle. Imagine that we wish to move the side nodes to the positions shown, where $a$ and $b$ are dimensionless fractions of edge length (so that $a + b = 1$ ). We can accomplish this positioning in isoparametric fashion, using Eqs. 5.3-5, but the element displacement field is then not a complete quadratic in Cartesian coordinates. Show that we can accomplish the positioning in subparametric fashion, thus retaining the quad-
+
+<!-- source-page: 224 -->
+
+![](images/page-224_2596bc72085123d388a9b6ec0a882b5f125748bb722188d96d2776fcfd73a6b1.jpg)
+
+<details>
+<summary>text_image</summary>
+
+3
+6 a b
+b
+a b
+1 4 2
+5
+</details>
+
+Problem 6.38
+
+ratic field. Suggestion: Abandon uniform side-node spacing on the reference element, and apply Eq. 5.3-2 to element geometry. Obtain shape functions $N_{i}$ , and show that they reduce to Eqs. 5.3-5 for $a = b = \frac{1}{2}$ .
+
+6.39 (a) If $\phi = 1$ in Eq. 6.8-7, one concludes that $\Sigma W_{i} = 1$ . Verify this property in Table 6.8-1.
+
+(b) All points listed in Table 6.8-1 should satisfy the constraint relation $\xi_{1} + \xi_{2} + \xi_{3} = 1$ . Verify that this is so for the 13-point formula.
+
+6.40 (a) Integrate the function $\phi = (1 + \xi_1\xi_2)^{-1}$ , using each of the first four integration formulas in Table 6.8-1. Let $A = 1$ and $J = 2$ .
+
+(b) Integrate the function $\phi = \xi_1\xi_2\xi_3$ by use of the appropriate rule in Table 6.8-1. Verify your result by use of Eq. 5.2-8. Let $A = 1$ and $J = 2$ .
+
+6.41 Evaluate Eq. 5.2-10 for the triangle shown. Evaluate this same integral by use of the first three-point formula in Table 6.8-1, and compare results.
+
+6.42 In the wedge-shaped elements shown, let $\zeta$ be a coordinate that has values $+1$ and $-1$ , respectively, on the top and bottom triangular faces. Write shape functions $N_{i}$ for (a) the 6-node element, and (b) the 15-node element.
+
+![](images/page-224_7494ced0e78564f65a1a42737cca83f4d9e126442317ae165c2fba0e5659dc0a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+(-1, 5)
+(-3, 0)
+1
+3
+2
+(4, -5)
+x
+y
+</details>
+
+Problem 6.41
+
+![](images/page-224_6a1f9d2aca73f47732c281e7671daa5aa4f580bb0c75ab8d425cb2e38d49c633.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+3
+4
+5
+6
+</details>
+
+![](images/page-224_c3b4a35e813c6bed3546ecbe75dbbff4d119c7faabfe934a4a4c147abfa8e4ee.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+9
+3
+7
+8
+2
+10
+15
+11
+4
+6
+13
+5
+14
+</details>
+
+Problem 6.42
+
+# Section 6.9
+
+6.43 Verify the results given in Eq. 6.9-1.
+
+6.44 Let the centroid of the triangle of Fig. 6.8-1a lie at $x = y = 0$ . Imagine that pressure $p = cx$ , where $c$ is a constant, acts normal to area $A$ of the triangle. Evaluate the consistent nodal loads produced by $p$ , in terms of $c, A, x_1, x_2,$ and $x_3$ .
+
+6.45 Let a uniform traction act on the surface of a nine-node quadratic element of rectangular shape. In the consistent load vector $\{\mathbf{r}_e\}$ , what fraction of the total force appears at each node? (See Fig. 4.3-5 for the corresponding eight-node case.)
+
+<!-- source-page: 225 -->
+
+# Section 6.10
+
+6.46 Use shape functions of Table 6.6-1 to demonstrate the interelement compatibility argument made in Section 6.10 with reference to Fig. 6.10-1.   
+6.47 Verify that $\Sigma N_{i} = 1$ for the $N_{i}$ of (a) Eqs. 5.1-5, (b) Eqs. 5.1-6, (c) Eqs. 5.3-5, (d) Eqs. 6.3-2, and (e) Eqs. 6.6-1.   
+6.48 Verify that Eqs. 6.10-7 are satisfied for the $N_{i}$ of (a) Eqs. 5.1-5, (b) Eqs. 5.1-6, (c) Eqs. 5.4-1, and (d) Eqs. 6.3-2.   
+6.49 Let $u_{1} = 0$ and $\dot{u}_{2} > 0$ in the three-node bar element shown. For $0 < x < L$ , interpolate axial displacement $u$ linearly, so that the element is superparametric. Calculate axial strain $\epsilon_{x} = \lfloor \mathbf{B} \rfloor \{\mathbf{u}\}$ , where $\{\mathbf{u}\} = \lfloor u_{1} - u_{2} \rfloor^{T}$ and $J$ (the denominator of $\lfloor \mathbf{B} \rfloor$ ) is given by Eq. 6.2-5. Hence, show that the element fails unless $x_{3} = L / 2$ .
+
+![](images/page-225_46503d78f3a6fe7cb9e1843cc8cb9cee2bb4e033cfae16a7341b8a2437bc6684.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+l
+1
+3
+2
+x,u
+x₃
+</details>
+
+Problem 6.49
+
+6.50 Write out the formula $x = \sum N_i x_i$ for the element of Fig. 6.6-1a. Hence, show the correctness of the completeness argument in Section 6.10 for this subparametric element.
+
+# Section 6.11
+
+6.51 Let $u_{1}, u_{2}$ , and $u_{3}$ be prescribed in the three-node bar of Fig. 6.2-1. Let $AE$ be constant. What order of Gauss rule is needed to calculate strain energy in the element if (a) $x_{3} = L / 2$ , and (b) $x_{3} \neq L / 2$ ?   
+6.52 If element thickness $t$ can vary and is computed as $t = \sum N_{i}t_{i}$ from nodal values $t_{i}$ , what order of Gauss quadrature is needed to compute the exact volume of (a) a bilinear element (four nodes), and (b) a quadratic element (eight nodes)?   
+6.53 Show that the volume of a trilinear solid element is correctly computed by an order 2 Gauss rule.   
+6.54 Let the following elements be rectangular in geometry, with side nodes evenly spaced and thicknesses constant. What order of Gauss quadrature is needed to obtain the exact stiffness matrix—that is, to integrate each $k_{ij}$ exactly?   
+(a) Plane bilinear element (four nodes).   
+(b) Plane quadratic element (eight nodes).   
+(c) Solid trilinear element (eight nodes).   
+(d) Plane quadratic triangle (six nodes).
+
+6.55 Repeat Problem 6.54 if element thickness $t$ is variable and interpolated from nodal values, $t = \Sigma N_{i}t_{i}$ .
+
+6.56 A 2 by 2 Gauss rule is used to form [k] for each element in Table 6.14-1. What will the qualitative change in deflection $v_{A}$ in each of the three cases be if the Gauss rule is changed to 3 by 3? Why?
+
+<!-- source-page: 226 -->
+
+# Section 6.12
+
+6.57 The plane structure shown is built of four bilinear elements, each integrated by one-point Gauss quadrature.
+
+(a) Sketch the possible mechanisms of the structure.
+
+(b) If you ask a computer program to solve for the displacement of load $P$ , what do you think will happen?
+
+(c) Add one roller support that will prevent an instability.
+
+![](images/page-226_e3098d20c6925f34865e24299059702db3199eb309aacd9702432e421d9a5d8c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+</details>
+
+Problem 6.57
+
+6.58 Verify that Eqs. 6.12-3 yield zero strains at Gauss points of an order 2 rule.
+
+6.59 For each of the following elements, write (if possible) a vector $\{\mathbf{d}\}$ of nodal d.o.f. that represents an instability mode under one-point Gauss quadrature.
+
+(a) The three-node bar element of Fig. 6.2-1.
+
+(b) The standard four-d.o.f. beam element, Fig. 4.2-2.
+
+6.60 Imagine that the bilinear element of Fig. 6.12-1 is integrated with a 2 by 1 Gauss rule. What is the rank of the element stiffness matrix?
+
+6.61 (a) Consider the quadratic serendipity solid element (a hexahedron having eight corner nodes and twelve side nodes). If integrated by 'an order 2 Gauss rule, what do you anticipate will be the rank of its stiffness matrix? State your reason.
+
+(b) Can these elements be put together so that the mesh has a mechanism?
+
+6.62 There exists a six-point quadrature rule for hexahedra that uses a sampling point at the middle of each face [6.9]. What mechanisms are possible for a rectangular eight-node element whose stiffness matrix is formed by this rule? Can a mesh of elements also display these mechanisms?
+
+6.63 Consider the upper right-hand element in Fig. 6.12-2d. Let the undeformed element be square, two units on a side.
+
+(a) Show that strains are zero at the center of the element. Let all nonzero $d_{i}$ in $\{\mathbf{d}\}$ have magnitude $c$ .
+
+(b) Show that $\{\mathbf{d}\}$ of the deformed element can be obtained by combining modes 7 and 8 of Fig. 6.12-1 with a rigid-body rotation.
+
+6.64 (a) Let a vector $\{\mathbf{d}\}$ contain nodal d.o.f. of an arbitrary plane element, with all $u_{i}$ in the upper half and all $v_{i}$ in the lower half. Consider the following similarly-partitioned vectors of 0's, 1's, and nodal coordinates $x_{i}$ and $y_{i}$ :
+
+$$
+\left\{ \begin{array}{l} \mathbf {1} \\ \mathbf {0} \end{array} \right\}, \left\{ \begin{array}{l} \mathbf {x} \\ \mathbf {0} \end{array} \right\}, \left\{ \begin{array}{l} \mathbf {y} \\ \mathbf {0} \end{array} \right\}, \left\{ \begin{array}{l} \mathbf {0} \\ \mathbf {1} \end{array} \right\}, \left\{ \begin{array}{l} \mathbf {0} \\ \mathbf {x} \end{array} \right\}, \left\{ \begin{array}{l} \mathbf {0} \\ \mathbf {y} \end{array} \right\}
+$$
+
+Identify these vectors, singly or in combination, with three rigid-body modes and three constant-strain modes.
+
+<!-- source-page: 227 -->
+
+(b) Let the element be square, two units on a side and have nine nodes. Write $\{\mathbf{d}\}$ for the mode of Eq. 6.12-3b, and show that it is orthogonal to the modes of part (a).
+
+6.65 Let the trilinear solid element of Fig. 6.7-1 be rectangular and two units on a side, so that $\xi = x$ , $\eta = y$ and $\zeta = \dot{z}$ .
+
+(a) What is the rank of the element stiffness matrix if it is integrated by use of a single Gauss point?   
+(b) Consider only $x$ -direction displacements $u$ . Let $\{\mathbf{d}_x\}$ represent the $u_i$ of the eight nodes. Write a $\{\mathbf{d}_x\}$ of arbitrary magnitude for each zero-energy mode that involves only the $u_i$ .   
+(c) Similarly, write a $\{\mathbf{d}_x\}$ for a total of four independent rigid-body and constant-strain modes that involve only the $u_{i}$ .   
+(d) Show that the modes of part (b) are orthogonal to those of part (c).
+
+6.66 Determine $a_{7}$ in Eq. 6.12-5 so that a rectangular element of uniform thickness has the exact strain energy in mode 7 (a pure bending mode). Express your answer in terms of the elastic modulus and element dimensions.
+
+# Section 6.13
+
+6.67 Use the bending-deformation mode of Fig. 6.13-1b and the $N_{i}$ of Eq. 6.3-2 to show that $\gamma_{xy} = -c\xi$ on $\eta = 0$ , where c is a positive constant. For simplicity, assume that elements are square.
+
+6.68 (a) Verify the numerical factors in Eq. 6.13-5.
+
+(b) Apply Eq. 6.13-3 to nodes $B$ , $C$ , and $D$ in Fig. 6.13-3. (Obtain numerical factors, as in Eq. 6.13-5.)   
+(c) Apply Eq. 6.13-3 to nodes $E, F, G$ , and $H$ in Fig. 6.13-3. (Obtain numerical factors, as in Eq. 6.13-5.)
+
+6.69 Write a formula analogous to Eq. 6.13-3 that uses $\sigma_{i}$ at the eight Gauss points of an order 2 rule in a solid element. Use it to write expressions for $\sigma_{x}$ , analogous to Eq. 6.13-5, at (a) node 8 in Fig. 6.7-1, and (b) the point where axis $\xi$ pierces the right-hand face in Fig. 6.7-1.   
+6.70 In the bilinear element (four nodes), stresses calculated directly at nodes agree exactly with stress extrapolated to nodes from four Gauss points, if the element is a parallelogram. Results disagree if the element is an arbitrary quadrilateral. Why?   
+6.71 Imagine that the bilinear element is not of constant thickness. What role does the thickness variation play in stress calculation according to Eq. 6.13-1? Suggest an ad hoc adjustment for thickness variation that might improve the accuracy of computed stresses.
+
+# Section 6.14
+
+6.72 In Fig. 6.14-1b, locate the points described by the following coordinates.
+
+(a) $\xi = -1$ and $\xi = -1 / \sqrt{3}$ .   
+(b) $\eta = 0$ and $\eta = -1$ .
+
+6.73 Consider an isosceles triangle, created by moving nodes 3 and 4 of a rectangular bilinear element (Fig. 4.2-4) so that they coincide on the $\eta$ axis.
+
+(a) Sketch the element and the Gauss points of a 2 by 2 rule, in the manner of Fig. 6.14-1.   
+(b) If $J$ is computed at each Gauss point, what is the ratio $J_{\max} / J_{\min}$ ?
+
+<!-- source-page: 228 -->
+
+# Summary Questions
+
+6.74 Each of the structures shown may be analyzed as two-dimensional. Greatest stresses and greatest deflections are desired. Imagine that an initial (coarse mesh) analysis is to be undertaken, so that errors will be roughly 10% or less. For each structure, sketch a suitable mesh, first using linear elements and then using quadratic elements. State your assumptions and approximations regarding how loads and supports are specified, the use of symmetry, the quadrature rule needed, treatment of stress concentrations, and so on.
+
+![](images/page-228_ad814f0980b72df31ae1fa9f1f3c3ad830e95b5d4d315531b779fa7ada01bb88.jpg)
+
+<!-- source-page: 229 -->
+
+# COORDINATE TRANSFORMATION
+
+Uses of coordinate transformations in structural mechanics are described, with emphasis on transformation of stiffness properties.
+
+# 7.1 INTRODUCTION
+
+Coordinate transformation permits the calculation of elastic property matrix $[E']$ and stiffness matrix $[k']$ in one coordinate system with subsequent transformation to matrices $[E]$ and $[k]$ in another coordinate system. Other uses of coordinate transformation include condensation techniques in structural dynamics and imposition of constraints. Constraints are discussed in detail in Chapter 9.
+
+The form $[Q] = [T]^{7}[Q'][T]$ appears repeatedly. Here $[Q']$ is the matrix to be transformed and $[T]$ is the transformation matrix. The transformed matrix $[Q]$ is symmetric if $[Q']$ is symmetric. Matrix $[T]$ may be rectangular or square. If square it may not be orthogonal. The specific form of $[T]$ depends on the problem at hand.
+
+One often has the option of taking $[Q']$ as either an element matrix or the corresponding structure matrix. Computer programming is usually easiest when transformations are done before elements are assembled, even though we must then transform several small matrices instead of one large one.
+
+Formal matrix multiplication to produce $[T]^{T}[Q^{\prime}][T]$ is often wasteful because [T] is often sparse. Sparsity should be exploited, or terms in the product should be hand-calculated and then coded.
+
+Caution. Transformations modify stiffness matrices. Errors and inconsistencies in stiffness matrices can lead to numerical difficulties and seriously degrade accuracy. It matters little if errors in [T] produce only a slightly different geometry than intended. But damage is done if errors in [T] act to falsify equilibrium equations. To avoid damage, we should state and manipulate transformation matrices and constraint equations with as much precision as is granted to stiffness coefficients $K_{ij}$ .
+
+# 7.2 TRANSFORMATION OF VECTORS
+
+Consider a vector V whose scalar components in the x, y, and z directions are u, v, and w (Fig. 7.2-1). Components of V in the $x'$ , $y'$ and $z'$ directions are $u'$ , $v'$ , and $w'$ . We wish to express $u'$ , $v'$ , and $w'$ in terms of u, v, w, and the cosines of
+
+<!-- source-page: 230 -->
+
+![](images/page-230_90870196b706addc6c49a6a755582402bfd61d9ba0d858a65a0a7c84f3c19e29.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y'
+y
+v'
+v
+V
+x'
+x
+u
+y
+w'
+w
+z
+z'
+</details>
+
+Direction cosines of axes: 
+
+<table><tr><td></td><td>x</td><td>y</td><td>z</td></tr><tr><td> $x'$ </td><td> $\ell_1$ </td><td> $m_1$ </td><td> $n_1$ </td></tr><tr><td> $y'$ </td><td> $\ell_2$ </td><td> $m_2$ </td><td> $n_2$ </td></tr><tr><td> $z'$ </td><td> $\ell_3$ </td><td> $m_3$ </td><td> $n_3$ </td></tr></table>
+
+Figure 7.2-1. Coordinate systems xyz and $x'y'z'$ , with table of direction cosines of angles between axes; for example, $\ell_{1}$ is the cosine of the angle between axes x and $x'$ . Components of a vector V can be expressed in either coordinate system.
+
+angles between axes $x'y'z'$ and xyz. Vector V can be regarded as a position vector, a nodal force or moment vector, or a nodal displacement or rotation vector. $^{1}$
+
+Component $u'$ can be regarded as the sum of components of displacements $u$ , $v$ , and $w$ parallel to the $x'$ axis. That is, $u' = \ell_1 u + m_1 v + n_1 w$ . Components $v'$ and $w'$ can be written similarly. In matrix format, these relations are
+
+$$
+\left\{ \begin{array}{l} u ^ {\prime} \\ v ^ {\prime} \\ w ^ {\prime} \end{array} \right\} = [ \Lambda ] \left\{ \begin{array}{l} u \\ v \\ w \end{array} \right\}, \quad \text { where } \quad [ \Lambda ] = \left[ \begin{array}{l l l} \ell_ {1} & m _ {1} & n _ {1} \\ \ell_ {2} & m _ {2} & n _ {2} \\ \ell_ {3} & m _ {3} & n _ {3} \end{array} \right] \tag {7.2-1}
+$$
+
+Matrix $[\Lambda]$ is called the rotation matrix. Equation 7.2-1 relates vectorial components in two systems. Matrix $[\Lambda]$ is orthogonal (i.e., its inverse is equal to its transpose). Therefore, the inverse of the transformation in Eq. 7.2-1 is
+
+$$
+\left\{ \begin{array}{l} u \\ v \\ w \end{array} \right\} = [ \Lambda ] ^ {T} \left\{ \begin{array}{l} u ^ {\prime} \\ v ^ {\prime} \\ w ^ {\prime} \end{array} \right\} \tag {7.2-2}
+$$
+
+Vector V in Fig. 7.2-1 may represent a displacement whose components can be expressed as $\{d\} = \left[u \quad v \quad w\right]^{T}$ or as $\{d'\} = \left[u' \quad v' \quad w'\right]^{T}$ . Or, V may represent a force whose components can be expressed as $\{r\} = \left[f_{x} \quad f_{y} \quad f_{z}\right]^{T}$ or as $\{r'\} = \left[f_{x}' \quad f_{y}' \quad f_{z}'\right]^{T}$ . Accordingly, components of displacements and forces obey the transformation rules
+
+$$
+\{\mathbf {d} ^ {\prime} \} = [ \Lambda ] \{\mathbf {d} \} \quad \text { and } \quad \{\mathbf {d} \} = [ \Lambda ] ^ {T} \{\mathbf {d} ^ {\prime} \} \tag {7.2-3}
+$$
+
+$$
+\{\mathbf {r} ^ {\prime} \} = [ \Lambda ] \{\mathbf {r} \} \quad \text { and } \quad \{\mathbf {r} \} = [ \Lambda ] ^ {T} \{\mathbf {r} ^ {\prime} \} \tag {7.2-4}
+$$
+
+$^{1}$ Provided that rotation is small, as is usual. Finite (large) rotations do not combine vectorially.
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_024.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_024.md
new file mode 100644
index 00000000..f6c325e5
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_024.md
@@ -0,0 +1,565 @@
+<!-- source-page: 231 -->
+
+Similarly, $[\Lambda]$ and $[\Lambda]^T$ can be used to transform (small) rotations $\{\mathbf{d}\} = \left\lfloor \theta_x \quad \theta_y \quad \theta_z \right\rfloor^T$ and moments $\{\mathbf{r}\} = \left\lfloor M_x \quad M_y \quad M_z \right\rfloor^T$ .
+
+In Eqs. 7.2-3 and 7.2-4 we require only that xyz and $x'y'z'$ each be a set of three mutually perpendicular directions. Neither system need be a Cartesian system. For example, $x'y'z'$ might be a cylindrical system (in which x = r, $y = \theta$ , and z = z).
+
+General Transformations. If $\{\mathbf{d}'\} = [\mathbf{T}]\{\mathbf{d}\}$ , where $[\mathbf{T}]$ is not necessarily orthogonal and may not even be square, it remains true that $\{\mathbf{r}\} = [\mathbf{T}]^T\{\mathbf{r}'\}$ . The proof is as follows. We argue that since $\{\mathbf{r}\}$ and $\{\mathbf{r}'\}$ describe the same resultant force, work done by the force during a prescribed virtual displacement must be independent of the coordinate system in which the work is computed. Let $\{\delta \mathbf{d}\}$ and $\{\delta \mathbf{d}'\}$ be two ways to describe the same virtual displacement (i.e., $\{\delta \mathbf{d}'\} = [\mathbf{T}]\{\delta \mathbf{d}\}$ ). Writing the work equality and using the relation $\{\delta \mathbf{d}'\}^T = \{\delta \mathbf{d}\}^T [\mathbf{T}]^T$ , we obtain
+
+$$
+\{\delta \mathbf {d} \} ^ {T} \{\mathbf {r} \} = \{\delta \mathbf {d} ^ {\prime} \} ^ {T} \{\mathbf {r} ^ {\prime} \} \quad \text { or } \quad \{\delta \mathbf {d} \} ^ {T} \{\mathbf {r} \} = \{\delta \mathbf {d} \} ^ {T} [ \mathbf {T} ] ^ {T} \{\mathbf {r} ^ {\prime} \} \tag {7.2-5}
+$$
+
+from which
+
+$$
+\{\delta \mathbf {d} \} ^ {T} (\{\mathbf {r} \} - [ \mathbf {T} ] ^ {T} \{\mathbf {r} ^ {\prime} \}) = 0, \quad \text { and   therefore } \quad \{\mathbf {r} \} = [ \mathbf {T} ] ^ {T} \{\mathbf {r} ^ {\prime} \} \tag {7.2-6}
+$$
+
+The latter equation may be written because the equation before it must be true for any virtual displacement $\{\delta d\}$ . Only when [T] is orthogonal does the second of Eqs. 7.2-3 result from $\{d'\} = [T]\{d\}$ and the first of Eqs. 7.2-4 result from Eq. 7.2-6.
+
+# 7.3 TRANSFORMATION OF STRESS, STRAIN, AND MATERIAL PROPERTIES
+
+Transformation of stresses $\{\sigma\}$ and strains $\{\epsilon\}$ in two dimensions leads to the familiar Mohr's circle calculations. In this section we consider the problem in three dimensions. We also consider the transformation of material properties [E]. Analogous transformations related to plate bending appear in Section 11.1.
+
+Strains. Strain transformations are essentially transformations of displacement derivatives. That is, to relate $\epsilon_{x}^{\prime}$ in coordinates $x^{\prime}y^{\prime}z^{\prime}$ to $\epsilon_{x}$ in coordinates xyz, we must relate $\partial u^{\prime}/\partial x^{\prime}$ to $\partial u/\partial x$ and to other derivatives of u, v, and w. From Eq. 7.2-1,
+
+$$
+\frac {\partial u ^ {\prime}}{\partial x ^ {\prime}} = \ell_ {1} \frac {\partial u}{\partial x ^ {\prime}} + m _ {1} \frac {\partial v}{\partial x ^ {\prime}} + n _ {1} \frac {\partial w}{\partial x ^ {\prime}}, \text { and   so   on } \tag {7.3-1}
+$$
+
+By chain rule differentiation, with $\partial x / \partial x' = \ell_1$ , $\partial y / \partial x' = m_1$ , and $\partial z / \partial x' = n_1$ ,
+
+$$
+\frac {\partial u}{\partial x ^ {\prime}} = \ell_ {1} \frac {\partial u}{\partial x} + m _ {1} \frac {\partial u}{\partial y} + n _ {1} \frac {\partial u}{\partial z} \tag {7.3-2}
+$$
+
+<!-- source-page: 232 -->
+
+By this process we obtain
+
+$$
+\left[ \begin{array}{c c c} \frac {\partial u ^ {\prime}}{\partial x ^ {\prime}} & \frac {\partial u ^ {\prime}}{\partial y ^ {\prime}} & \frac {\partial u ^ {\prime}}{\partial z ^ {\prime}} \dots \frac {\partial w ^ {\prime}}{\partial z ^ {\prime}} \end{array} \right] ^ {T} = \left[ \begin{array}{c c c} \ell_ {1} \Lambda & m _ {1} \Lambda & n _ {1} \Lambda \\ \ell_ {2} \Lambda & m _ {2} \Lambda & n _ {2} \Lambda \\ \ell_ {3} \Lambda & m _ {3} \Lambda & n _ {3} \Lambda \end{array} \right] \left[ \begin{array}{c c c c c} u _ {, x} & u _ {, y} & u _ {, z} & \dots & w _ {, z} \end{array} \right] ^ {T} \tag {7.3-3}
+$$
+
+where $[\Lambda]$ is given by Eq. 7.2-1. The 9 by 9 square matrix in Eq. 7.3-3 is orthogonal.
+
+A state of strain can be expressed as $\{\epsilon'\}$ in $x'y'z'$ coordinates or as $\{\epsilon\}$ in xyz coordinates. One now introduces the strain-displacement relations (Eqs. 1.5-6) into Eq. 7.3-3. After straightforward but tedious expansion and gathering of terms, one obtains the relation between $\{\epsilon'\}$ and $\{\epsilon\}$ as
+
+$$
+\{\epsilon^ {\prime} \} = [ \mathrm{T} _ {\epsilon} ] \{\epsilon \} \tag {7.3-4}
+$$
+
+where
+
+$$
+\left[ \mathrm{T} _ {\epsilon} \right] = \left[ \begin{array}{c c c c c c} \ell_ {1} ^ {2} & m _ {1} ^ {2} & n _ {1} ^ {2} & \ell_ {1} m _ {1} & m _ {1} n _ {1} & n _ {1} \ell_ {1} \\ \ell_ {2} ^ {2} & m _ {2} ^ {2} & n _ {2} ^ {2} & \ell_ {2} m _ {2} & m _ {2} n _ {2} & n _ {2} \ell_ {2} \\ \ell_ {3} ^ {2} & m _ {3} ^ {2} & n _ {3} ^ {2} & \ell_ {3} m _ {3} & m _ {3} n _ {3} & n _ {3} \ell_ {3} \\ \hline 2 \ell_ {1} \ell_ {2} & 2 m _ {1} m _ {2} & 2 n _ {1} n _ {2} & \ell_ {1} m _ {2} + \ell_ {2} m _ {1} & m _ {1} n _ {2} + m _ {2} n _ {1} & n _ {1} \ell_ {2} + n _ {2} \ell_ {1} \\ 2 \ell_ {2} \ell_ {3} & 2 m _ {2} m _ {3} & 2 n _ {2} n _ {3} & \ell_ {2} m _ {3} + \ell_ {3} m _ {2} & m _ {2} n _ {3} + m _ {3} n _ {2} & n _ {2} \ell_ {3} + n _ {3} \ell_ {2} \\ 2 \ell_ {3} \ell_ {1} & 2 m _ {3} m _ {1} & 2 n _ {3} n _ {1} & \ell_ {3} m _ {1} + \ell_ {1} m _ {3} & m _ {3} n _ {1} + m _ {1} n _ {3} & n _ {3} \ell_ {1} + n _ {1} \ell_ {3} \end{array} \right] \tag {7.3-5}
+$$
+
+Strains in $\{\epsilon'\}$ and $\{\epsilon\}$ are ordered as in Eqs. 1.5-6, and the engineering definition of shear strain is used (e.g., $\gamma_{xy} = u_{,y} + v_{,x}$ ). Partitioning seen in Eq. 7.3-5 is used in what follows.
+
+Stresses. A stress transformation relates stresses $\{\sigma\}$ in xyz coordinates to stresses $\{\sigma'\}$ in $x'y'z'$ coordinates. To determine the form of this transformation, we consider internal virtual work per unit volume, done by stresses during a prescribed virtual displacement. This work must be the same whether it is computed in the xyz system or in the $x'y'z'$ system. Therefore, writing the work equality and using Eq. 7.3-4, we obtain
+
+$$
+\{\delta \boldsymbol {\epsilon} \} ^ {T} \{\boldsymbol {\sigma} \} = \{\delta \boldsymbol {\epsilon} ^ {\prime} \} ^ {T} \{\boldsymbol {\sigma} ^ {\prime} \} \quad \text { or } \quad \{\delta \boldsymbol {\epsilon} \} ^ {T} \{\boldsymbol {\sigma} \} = \{\delta \boldsymbol {\epsilon} \} ^ {T} [ \mathbf {T} _ {\epsilon} ] ^ {T} \{\boldsymbol {\sigma} ^ {\prime} \} \tag {7.3-6}
+$$
+
+Equation 7.3-6 must be true for any virtual strain state $\{\delta\epsilon\}$ . Hence
+
+$$
+\{\boldsymbol {\sigma} \} = [ \mathbf {T} _ {\epsilon} ] ^ {T} \{\boldsymbol {\sigma} ^ {\prime} \} \quad \text { or } \quad \{\boldsymbol {\sigma} ^ {\prime} \} = [ \mathbf {T} _ {\epsilon} ] ^ {- T} \{\boldsymbol {\sigma} \} \tag {7.3-7}
+$$
+
+Coefficients in $\{\sigma\}$ and $\{\sigma'\}$ are ordered as in Eq. 1.7-1.
+
+The inverse-transpose matrix in Eq. 7.3-7 is easy to compute. After assigning labels $T_{11}$ , $T_{12}$ , $T_{21}$ , and $T_{22}$ to the partitions in Eq. 7.3-5, one discovers that
+
+$$
+\text { if } \quad [ \mathbf {T} _ {\epsilon} ] = \left[ \begin{array}{l l} \mathbf {T} _ {1 1} & \mathbf {T} _ {1 2} \\ \mathbf {T} _ {2 1} & \mathbf {T} _ {2 2} \end{array} \right] \quad \text { then } \quad [ \mathbf {T} _ {\epsilon} ] ^ {- T} = \left[ \begin{array}{l l} \mathbf {T} _ {1 1} & 2 \mathbf {T} _ {1 2} \\ \frac {1}{2} \mathbf {T} _ {2 1} & \mathbf {T} _ {2 2} \end{array} \right] \tag {7.3-8}
+$$
+
+<!-- source-page: 233 -->
+
+![](images/page-233_613255452b3ccdc240c7c5ef27fac0c8ec79bbd92e814f0811142869a5c393ab.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y',v'
+y,v
+β
+β
+x',u'
+x,u
+</details>
+
+<table><tr><td></td><td>x</td><td>y</td><td>z</td></tr><tr><td> $x'$ </td><td> $\ell_1 = \cos \beta$ </td><td> $m_1 = \sin \beta$ </td><td> $n_1 = 0$ </td></tr><tr><td> $y'$ </td><td> $\ell_2 = -\sin \beta$ </td><td> $m_2 = \cos \beta$ </td><td> $n_2 = 0$ </td></tr><tr><td> $z'$ </td><td> $\ell_3 = 0$ </td><td> $m_3 = 0$ </td><td> $n_3 = 1$ </td></tr></table>
+
+Figure 7.3-1. The two-dimensional case. Coordinate systems $xy$ and $x'y'$ , with table of direction cosines between axes.
+
+Thus $[T_{\epsilon}]^{-T}$ is obtained from $[T_{\epsilon}]$ by shifting factors of 2 in $[T_{\epsilon}]$ symmetrically about the diagonal.
+
+Material Properties. A single stress-strain relation can be written as $\{\sigma\} = [\mathbf{E}]\{\epsilon\}$ in the xyz coordinate system or as $\{\sigma'\} = [\mathbf{E}']\{\epsilon'\}$ in the $x'y'z'$ coordinate system. Imagine that $[\mathbf{E}']$ is known and $[\mathbf{E}]$ is desired. By substitution from Eqs. 7.3-4, 7.3-7, and the relation $\{\sigma'\} = [\mathbf{E}']\{\epsilon'\}$ ,
+
+$$
+\{\boldsymbol {\sigma} \} = [ \mathbf {T} _ {\epsilon} ] ^ {T} \{\boldsymbol {\sigma} ^ {\prime} \} = [ \mathbf {T} _ {\epsilon} ] ^ {T} [ \mathbf {E} ^ {\prime} ] \{\boldsymbol {\epsilon} ^ {\prime} \} = [ \mathbf {T} _ {\epsilon} ] ^ {T} [ \mathbf {E} ^ {\prime} ] [ \mathbf {T} _ {\epsilon} ] \{\boldsymbol {\epsilon} \} \tag {7.3-9}
+$$
+
+from which
+
+$$
+[ \mathbf {E} ] = [ \mathbf {T} _ {\epsilon} ] ^ {T} [ \mathbf {E} ^ {\prime} ] [ \mathbf {T} _ {\epsilon} ] \tag {7.3-10}
+$$
+
+This transformation concerns conditions at a point. Therefore, it is not necessary that xyz and $x'y'z'$ be Cartesian systems. For example, one coordinate system might be Cartesian and the other cylindrical.
+
+Plane Problems. A two-dimensional problem is a special case in which $n_3 = 1$ and $\ell_3 = m_3 = n_1 = n_2 = 0$ (see Fig. 7.3-1). In the $xy$ plane, $\{\epsilon\} = \left\lfloor \epsilon_x \quad \epsilon_y \quad \gamma_{xy} \right\rfloor^T$ , $\{\sigma\} = \left\lfloor \sigma_x \quad \sigma_y \quad \tau_{xy} \right\rfloor^T$ , [E] is 3 by 3, and
+
+$$
+\left[ \mathbf {T} _ {\epsilon} \right] = \left[ \begin{array}{c c c} c ^ {2} & s ^ {2} & c s \\ s ^ {2} & c ^ {2} & - c s \\ - 2 c s & 2 c s & c ^ {2} - s ^ {2} \end{array} \right] \quad \text { and } \quad \left[ \mathbf {T} _ {\epsilon} \right] ^ {- T} = \left[ \begin{array}{c c c} c ^ {2} & s ^ {2} & 2 c s \\ s ^ {2} & c ^ {2} & - 2 c s \\ - c s & c s & c ^ {2} - s ^ {2} \end{array} \right] \tag {7.3-11}
+$$
+
+where $c = \cos \beta$ and $s = \sin \beta$ . Hence, one can recognize Eqs. 7.3-7 as the familiar Mohr's circle relations used in elementary mechanics of materials.
+
+# 7.4 TRANSFORMATION OF STIFFNESS MATRICES
+
+In two coordinate systems such as xyz and $x'y'z'$ , the element stiffness relation can be written as
+
+$$
+[ \mathbf {k} ] \{\mathbf {d} \} = \{\mathbf {r} \} \quad \text { or   as } \quad [ \mathbf {k} ^ {\prime} ] \{\mathbf {d} ^ {\prime} \} = \{\mathbf {r} ^ {\prime} \} \tag {7.4-1}
+$$
+
+<!-- source-page: 234 -->
+
+The stiffness matrix of a given element can be expressed as either [k] or [k']. The two matrices differ because they operate on different nodal d.o.f.—namely, {d} and {d'. We imagine here that [k'] is known and [k] is desired. The necessary transformation is now derived.
+
+A review of the argument associated with Eqs. 7.2-5 and 7.2-6 shows that no special form need be assumed for the matrix that relates $\{\mathbf{d}'\}$ and $\{\mathbf{d}\}$ . It is required only that the relation be known. We will call the relational matrix [T]. The argument of Eqs. 7.2-5 and 7.2-6 is that
+
+$$
+\text { if } \quad \{\mathbf {d} ^ {\prime} \} = [ \mathbf {T} ] \{\mathbf {d} \} \quad \text { then } \quad \{\mathbf {r} \} = [ \mathbf {T} ] ^ {T} \{\mathbf {r} ^ {\prime} \} \tag {7.4-2}
+$$
+
+Examples will follow. For now we remark only that $\{d\}$ and $\{d'\}$ need not be the same size and need not even contain the same kind of d.o.f.
+
+Hence, the stiffness transformation is easy to derive. By substitution of Eqs. 7.4-2 into Eq. 7.4-1,
+
+$$
+[ \mathbf {k} ] \{\mathbf {d} \} = \{\mathbf {r} \} = [ \mathbf {T} ] ^ {T} \{\mathbf {r} ^ {\prime} \} = [ \mathbf {T} ] ^ {T} [ \mathbf {k} ^ {\prime} ] \{\mathbf {d} ^ {\prime} \} = [ \mathbf {T} ] ^ {T} [ \mathbf {k} ^ {\prime} ] [ \mathbf {T} ] \{\mathbf {d} \} \tag {7.4-3}
+$$
+
+from which
+
+$$
+[ \mathbf {k} ] = [ \mathbf {T} ] ^ {T} [ \mathbf {k} ^ {\prime} ] [ \mathbf {T} ] \tag {7.4-4}
+$$
+
+Equation 7.4-4 does not change the orientation of the element in fixed global coordinates or alter element properties; rather, this transformation alters the formal expression of element properties to agree with a change of d.o.f. from $\{\mathbf{d}'\}$ to $\{\mathbf{d}\}$ .
+
+For future reference, we note that mass and damping matrices used in dynamics transform in the same way. That is, $[m] = [T]^{T}[m'][T]$ and $[c] = [T]^{T}[c'][T]$ .
+
+# 7.5 EXAMPLES: TRANSFORMATION OF STIFFNESS MATRICES
+
+Plane Truss Element. Imagine that the stiffness matrix of the bar in Fig. 7.5-1a in local coordinates $x'y'$ is called $[k']$ and is known. From it, $[k]$ is to be determined, where $[k]$ is the stiffness matrix of the bar referred to global coordinates xy. Thus $[k']$ and $[k]$ describe the same bar but use different d.o.f. to do so. We have
+
+$$
+\left[ \mathbf {k} ^ {\prime} \right] = \frac {A E}{L} \left[ \begin{array}{c c c c} 1 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 \\ - 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \quad \text { and } \quad \left\{ \begin{array}{l} u _ {1} ^ {\prime} \\ v _ {1} ^ {\prime} \\ u _ {2} ^ {\prime} \\ v _ {2} ^ {\prime} \end{array} \right\} = \left[ \begin{array}{c c c c} c & s & 0 & 0 \\ - s & c & 0 & 0 \\ 0 & 0 & c & s \\ 0 & 0 & - s & c \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \\ u _ {2} \\ v _ {2} \end{array} \right\} \tag {7.5-1}
+$$
+
+where $c = \cos \beta$ and $s = \sin \beta$ . The square matrix of sines and cosines is [T]. It is built from matrices [A] of Eq. 7.2-1, specialized to two dimensions (Fig. 7.3-1). We find that [k] = [T] $^{7}$ [k′][T] is the stiffness matrix given by Eq. 2.4-3, as expected.
+
+However, the foregoing procedure involves unnecessary effort. Terms in rows
+
+<!-- source-page: 235 -->
+
+![](images/page-235_b43c79992990e50616d83a277e59da420f5363f032cfa233006ae28719b795a0.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+y',v'
+L
+2
+x',u'
+β
+A,E
+1
+x,u
+</details>
+
+(n)
+
+![](images/page-235_ef663ebd9f9e8c0c0abefd9126720fb0cd0922355b60b23448789654888a89ef.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+y',v'
+L
+x',u'
+1
+A,E
+z',w'
+z,w
+r,u
+</details>
+
+(b)   
+Figure 7.5-1. A uniform two-force (bar or truss) element in local and global reference frames. (a) Two-dimensional case. (b) Three-dimensional case.
+
+2 and 4 of [T], which pertain to $v_1'$ and $v_2'$ , are always multiplied by zero. This is physically reasonable, as axis $x'$ completely defines the orientation of the element. Rather than use Eqs. 7.5-1, it is more efficient to use for [k'] the 2 by 2 matrix in Eq. 2.4-5. Thus
+
+$$
+[ \mathbf {k} ^ {\prime} ] = \frac {A E}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right] \quad \text { and } \quad \left\{ \begin{array}{l} u _ {1} ^ {\prime} \\ u _ {2} ^ {\prime} \end{array} \right\} = [ \mathbf {T} ] \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \\ u _ {2} \\ v _ {2} \end{array} \right\} \tag {7.5-2}
+$$
+
+where, with $c = \cos \beta$ and $s = \sin \beta$ ,
+
+$$
+[ \mathbf {T} ] _ {2 \times 4} = \left[ \begin{array}{l l l l} c & s & 0 & 0 \\ 0 & 0 & c & s \end{array} \right] \tag {7.5-3}
+$$
+
+With $[k']$ and $[T]$ thus defined, the operation $[k] = [T]^{T}[k'][T]$ again produces the expected 4 by 4 matrix of Eq. 2.4-3.
+
+Space Truss Element. With $[k']$ again defined as in Eq. 7.5-2, we wish to obtain from it the 6 by 6 matrix $[k]$ for the element in Fig. 7.5-1b, which operates on nodal displacements parallel to x, y, and z axes. Vectors of local and global d.o.f. for this element are
+
+$$
+\{\mathbf {d} ^ {\prime} \} = \left[ \begin{array}{l l} u _ {1} ^ {\prime} & u _ {2} ^ {\prime} \end{array} \right] ^ {T} \quad \text { and } \quad \{\mathbf {d} \} = \left[ \begin{array}{l l l l l l} u _ {1} & v _ {1} & w _ {1} & u _ {2} & v _ {2} & w _ {2} \end{array} \right] ^ {T} \tag {7.5-4}
+$$
+
+The transformation is $\{\mathbf{d}'\} = [\mathbf{T}]\{\mathbf{d}\}$ , where
+
+$$
+\left[ \begin{array}{l} \mathbf {T} \\ 2 \times 6 \end{array} \right] = \left[ \begin{array}{c c c c c c} \ell_ {1} & m _ {1} & n _ {1} & 0 & 0 & 0 \\ 0 & 0 & 0 & \ell_ {1} & m _ {1} & n _ {1} \end{array} \right] \tag {7.5-5}
+$$
+
+and $\ell_1, m_1$ , and $n_1$ are direction cosines of axis $x'$ . The desired result is $[\mathbf{k}] = [\mathbf{T}]^T[\mathbf{k}'][\mathbf{T}]$ .
+
+Plane Frame Element. This element is a plane beam but with axial deformation permitted. We first write the stiffness matrix $[k']$ in local coordinates $x'y'$ , Fig.
+
+<!-- source-page: 236 -->
+
+![](images/page-236_9957e2a5b5558d68e3a4f03c661adf51044bc3e06bd70ffa4d8d2e0ab8e0d6bb.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y'
+L
+v2'
+u2'
+x'
+r1'
+A,E,I
+θ2'
+u1'
+θ1'
+</details>
+
+$$
+Z = A E / L \quad K = 1 2 E I / L ^ {3}
+$$
+
+$$
+A = 4 E I / L \quad M = 6 E I / L ^ {2}
+$$
+
+$$
+B = 2 E I / L
+$$
+
+![](images/page-236_2e1e96dfcb250cf86968e743a928b379c862c73087d618c919e5f3c7b5919477.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+L
+v2
+u2
+v1
+A,E,I
+θ2
+β
+u1
+x
+θ1
+c = cosβ
+s = sinβ
+</details>
+
+$$
+F = Z c ^ {2} + K s ^ {2} \quad H = - M s
+$$
+
+$$
+G = (Z - K) c s \quad Q = M c
+$$
+
+$$
+P = Z s ^ {2} + K c ^ {2}
+$$
+
+$$
+\left[ \begin{array}{c c c c c c} Z & 0 & 0 & - Z & 0 & 0 \\ 0 & K & M & 0 & - K & M \\ 0 & M & A & 0 & - M & B \\ - Z & 0 & 0 & Z & 0 & 0 \\ 0 & - K & - M & 0 & K & - M \\ 0 & M & B & 0 & - M & A \end{array} \right]
+$$
+
+$[\mathbf{k}^{\prime}]$ (for primed d.o.f.)
+
+(a)
+
+$$
+\left[ \begin{array}{c c c c c c} F & G & H & - F & - G & H \\ G & P & Q & - G & - P & Q \\ H & Q & A & - H & - Q & B \\ - F & - G & - H & F & G & - H \\ - G & - P & - Q & G & P & - Q \\ H & Q & B & - H & - Q & A \end{array} \right]
+$$
+
+[k] (for unprimed d.o.f.)
+
+(b)
+
+Figure 7.5-2. The stiffness matrix of a uniform plane frame element.
+
+7.5-2a. The element can both stretch and bend in the $xy$ (or the $x'y'$ ) plane. Element stiffness matrix $[\mathbf{k}']$ operates on the d.o.f.
+
+$$
+\left\{\mathbf {d} ^ {\prime} \right\} = \left\lfloor u _ {1} ^ {\prime} v _ {1} ^ {\prime} \theta_ {1} ^ {\prime} u _ {2} ^ {\prime} v _ {2} ^ {\prime} \theta_ {2} ^ {\prime} \right] ^ {T} \tag {7.5-6}
+$$
+
+D.o.f. $u_{1}^{\prime}$ and $u_{2}^{\prime}$ are associated with axial stiffness AE/L. The remaining d.o.f. are associated with bending. Axial and bending effects do not interact (unless a large axial load produces “beam–column” action). We therefore create the 6 by 6 matrix $[k^{\prime}]$ of the frame element by taking terms from the beam element matrix (Eq. 4.2-5) and the truss element matrix (Eq. 7.5-1). The resulting $[k^{\prime}]$ appears in Fig. 7.5-2a.
+
+To generate [k] in global coordinates $xy$ we apply Eq. 7.4-4. The transformation matrix is
+
+$$
+\left[ \begin{array}{l} \mathbf {T} \\ 6 \times 6 \end{array} \right] = \left[ \begin{array}{l l} \mathbf {T} _ {n} & \mathbf {0} \\ \mathbf {0} & \mathbf {T} _ {n} \end{array} \right], \quad \text { where } \quad \left[ \begin{array}{l} \mathbf {T} _ {n} \end{array} \right] = \left[ \begin{array}{l l l} \cos \beta & \sin \beta & 0 \\ - \sin \beta & \cos \beta & 0 \\ 0 & 0 & 1 \end{array} \right] \tag {7.5-7}
+$$
+
+The “1” appears in $[T_{n}]$ because the rotation vectors do not change in direction: $\theta_{1}^{\prime} = \theta_{1}$ and $\theta_{2}^{\prime} = \theta_{2}$ . The [k] that results from transformation, and the d.o.f. on which it operates, are shown in Fig. 7.5-2b.
+
+# 7.6 INCLINED SUPPORT
+
+Consider a structure that has translational nodal d.o.f. directed along the coordinate axes xyz. It may happen that a certain node is allowed to move only in a
+
+<!-- source-page: 237 -->
+
+![](images/page-237_31afd1fa4addcaa9dfda99b64f044ee914b359e7ed7702fcf74757a108ae1bbb.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+s,V
+4
+2
+3
+1
+x,u
+β
+r,U
+</details>
+
+Figure 7.6-1. A plane truss or plane frame in which node 3 is allowed to move in only the r direction.
+
+plane that is not parallel to a coordinate plane. In other words, displacement is prohibited in a direction that is not parallel to x, y, or z axes. A way to treat this boundary condition is now illustrated by using a plane structure.
+
+In Fig. 7.6-1, the inclined support requires that $v_{3} = -u_{3} \tan \beta$ (or, in terms of other d.o.f., it requires that $V_{3} = 0$ while $U_{3}$ is unrestrained). It is easier to deal with the constraint $V_{3} = 0$ than with the constraint $v_{3} = -u_{3} \tan \beta$ . The procedure described in the following replaces $u_{3}$ and $v_{3}$ by $U_{3}$ and $V_{3}$ without changing other d.o.f. of the structure. One then uses a standard method to set $V_{3} = 0$ (see Section 2.10). $U_{3}$ remains active and is computed as part of the solution vector $\{D\}$ in the usual way.
+
+Before any support conditions are imposed in Fig. 7.6-1, the structure stiffness equations $[K]\{D\} = \{R\}$ , partitioned by node, are
+
+$$
+\left[ \begin{array}{c c c c} \mathbf {K} _ {1 1} & \mathbf {K} _ {1 2} & \mathbf {K} _ {1 3} & \mathbf {K} _ {1 4} \\ \mathbf {K} _ {2 1} & \mathbf {K} _ {2 2} & \mathbf {0} & \mathbf {K} _ {2 4} \\ \mathbf {K} _ {3 1} & \mathbf {0} & \mathbf {K} _ {3 3} & \mathbf {K} _ {3 4} \\ \mathbf {K} _ {4 1} & \mathbf {K} _ {4 2} & \mathbf {K} _ {4 3} & \mathbf {K} _ {4 4} \end{array} \right] \left\{ \begin{array}{l} \mathbf {D} _ {1} \\ \mathbf {D} _ {2} \\ \mathbf {D} _ {3} \\ \mathbf {D} _ {4} \end{array} \right\} = \left\{ \begin{array}{l} \mathbf {R} _ {1} \\ \mathbf {R} _ {2} \\ \mathbf {R} _ {3} \\ \mathbf {R} _ {4} \end{array} \right\} \tag {7.6-1}
+$$
+
+where, depending on whether the structure is a plane truss or a plane frame,
+
+$$
+\{\mathbf {D} _ {i} \} = \left[ \begin{array}{l l} u _ {i} & v _ {i} \end{array} \right] ^ {T} \quad \text { or } \quad \{\mathbf {D} _ {i} \} = \left[ \begin{array}{l l l} u _ {i} & v _ {i} & \theta_ {i} \end{array} \right] ^ {T} \tag {7.6-2}
+$$
+
+To replace $u_{3}$ and $v_{3}$ by $U_{3}$ and $V_{3}$ , we write the transformation relation
+
+$$
+\left\{ \begin{array}{l} u _ {3} \\ v _ {3} \end{array} \right\} = [ \mathbf {T} _ {3} ] \left\{ \begin{array}{l} U _ {3} \\ V _ {3} \end{array} \right\}. \quad \text { or } \quad \left\{ \begin{array}{l} u _ {3} \\ v _ {3} \\ \theta_ {3} \end{array} \right\} = [ \mathbf {T} _ {3} ] \left\{ \begin{array}{l} U _ {3} \\ V _ {3} \\ \theta_ {3} \end{array} \right\} \tag {7.6-3}
+$$
+
+where, with $c = \cos \beta$ and $s = \sin \beta$ ,
+
+$$
+\left[ \mathrm{T} _ {3} \right] = \left[ \begin{array}{c c} c & s \\ - s & c \end{array} \right] \quad \text { or } \quad \left[ \mathrm{T} _ {3} \right] = \left[ \begin{array}{c c c} c & s & 0 \\ - s & c & 0 \\ 0 & 0 & 1 \end{array} \right] \tag {7.6-4}
+$$
+
+for plane truss and plane frame, respectively. The transformation matrix [T] for the entire structure is a unit matrix except for $[T_{3}]$ on the diagonal. With [I] a 2 by 2 or a 3 by 3 unit matrix, [T] is
+
+$$
+[ \mathbf {T} ] = \left[ \begin{array}{l l l l} \mathbf {I} & \mathbf {I} & \mathbf {T} _ {3} & \mathbf {I} \end{array} \right] \tag {7.6-5}
+$$
+
+<!-- source-page: 238 -->
+
+After Eq. 7.6-1 is transformed, $\lfloor U_3 \quad V_3 \rfloor^T$ or $\lfloor U_3 \quad V_3 \quad \theta_3 \rfloor^T$ replaces $\{\mathbf{D}_3\}$ , $[\mathbf{T}_3]^T\{\mathbf{R}_3\}$ replaces $\{\mathbf{R}_3\}$ , and the structure stiffness matrix becomes
+
+$$
+[ \mathbf {T} ] ^ {T} [ \mathbf {K} ] [ \mathbf {T} ] = \left[ \begin{array}{c c c c} \mathbf {K} _ {1 1} & \mathbf {K} _ {1 2} & \mathbf {K} _ {1 3} \mathbf {T} _ {3} & \mathbf {K} _ {1 4} \\ \mathbf {K} _ {2 1} & \mathbf {K} _ {2 2} & \mathbf {0} & \mathbf {K} _ {2 4} \\ \mathbf {T} _ {3} ^ {T} \mathbf {K} _ {3 1} & \mathbf {0} & \mathbf {T} _ {3} ^ {T} \mathbf {K} _ {3 3} \mathbf {T} _ {3} & \mathbf {T} _ {3} ^ {T} \mathbf {K} _ {3 4} \\ \mathbf {K} _ {4 1} & \mathbf {K} _ {4 2} & \mathbf {K} _ {4 3} \mathbf {T} _ {3} & \mathbf {K} _ {4 4} \end{array} \right] \tag {7.6-6}
+$$
+
+Transformed arrays [K] and {R} can be transformed again if there is another skew support. Conceivably, all nodes of the truss or frame could be skew and all translational d.o.f. in {D} could have different directions. If n successive transformations are used so that $\{D'\} = [T_{1}]\{D\}$ , $\{D''\} = [T_{2}]\{D'\}$ , and so on, original d.o.f. $\{D^{n}\}$ are related to final d.o.f. {D} by the equation
+
+$$
+\{\mathbf {D} ^ {n} \} = [ \mathbf {T} _ {n} ] [ \mathbf {T} _ {n - 1} ] \cdot \cdot \cdot [ \mathbf {T} _ {1} ] \{\mathbf {D} \}. \tag {7.6-7}
+$$
+
+In the preceding explanation, transformation is done at the structure level. This approach requires that we construct and use [T] in a manner consistent with whatever compact storage format has been adopted for the structure stiffness matrix. It also requires that a d.o.f. to be suppressed (e.g., $V_{3}$ in Fig. 7.6-1) remain present until transformation is complete. If, instead, the separate element matrices are transformed before assembly, the scheme of Figs. 2.10-4 and 2.10-5 can be used to exclude from $\{D\}$ the d.o.f. to be suppressed. The required transformation matrix for a plane frame element, with all six of its d.o.f. included, is
+
+$$
+[ \mathbf {T} ] _ {6 \times 6} = \left[ \begin{array}{l l} \mathbf {T} _ {3} & \mathbf {0} \\ \mathbf {0} & \mathbf {I} \end{array} \right] \quad \text { or } \quad [ \mathbf {T} ] _ {6 \times 6} = \left[ \begin{array}{l l} \mathbf {I} & \mathbf {0} \\ \mathbf {0} & \mathbf {T} _ {3} \end{array} \right] \tag {7.6-8}
+$$
+
+depending on which node of the element coincides with the affected node of the frame. This transformation must be applied to every element that is attached to the affected node (node 3 in Fig. 7.6-1).
+
+# 7.7 JOINING DISSIMILAR ELEMENTS TO ONE ANOTHER
+
+An element match termed “dissimilar” is depicted in Fig. 7.7-1a. The left end of a plane frame element is to be attached at an arbitrary location along an edge of a plane four-node quadrilateral element. Node 5 of the frame element does not coincide with a node of the quadrilateral. Moreover, rotational d.o.f. appear at nodes 5 and 6, but nodes 1 through 4 have only translational d.o.f. A method of connecting these two elements is now described.
+
+The frame element stiffness relation is $[\mathbf{k}']\{\mathbf{d}'\} = \{\mathbf{r}'\}$ , where
+
+$$
+\{\mathbf {d} ^ {\prime} \} = \left\lfloor u _ {5} v _ {5} \theta_ {5} u _ {6} v _ {6} \theta_ {6} \right\rfloor^ {T} \tag {7.7-1}
+$$
+
+We seek modified matrices $[\mathbf{k}]$ and $\{\mathbf{r}\}$ for the frame element, where
+
+$$
+[ \mathbf {k} ] = [ \mathbf {T} ] ^ {T} [ \mathbf {k} ^ {\prime} ] [ \mathbf {T} ] \quad \text { and } \quad \{\mathbf {r} \} = [ \mathbf {T} ] ^ {T} \{\mathbf {r} ^ {\prime} \} \tag {7.7-2}
+$$
+
+<!-- source-page: 239 -->
+
+![](images/page-239_866bf3b47390512ac555ebd229d096aea260e7a0a028d9078bcda87a247bfd7f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+4
+3
+a
+L
+b
+2
+6
+y, v
+1
+x, u
+β
+</details>
+
+[a]
+
+![](images/page-239_ab1a48da7b49cd101319a403887bbfae220bd16e360329beff41f309fda11e33.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L₂
+c
+4
+y, v
+3
+a
+6
+b
+L₁
+d
+5
+1
+2
+x, u
+</details>
+
+{b}   
+Figure 7.7-1. (a) A standard plane frame element connected to a four-node plane element. (b) A two-force (bar) element connected to a four-node plane element.
+
+New d.o.f. of the frame element are to be
+
+$$
+\{\mathbf {d} \} = \left[ \begin{array}{l l l l l l l} u _ {2} & v _ {2} & u _ {3} & v _ {3} & u _ {6} & v _ {6} & \theta_ {6} \end{array} \right] ^ {T} \tag {7.7-3}
+$$
+
+In the expression $\{\mathbf{d}'\} = [\mathbf{T}]\{\mathbf{d}\}$ , transformation matrix [T] is written by saying that translational motion of node 5 is linearly interpolated along edge 2-3 from translational d.o.f. at nodes 2 and 3, and that rotation $\theta_{5}$ is the same as the rotation of edge 2-3. Thus, with $c = \cos \beta$ and $s = \sin \beta$ ,
+
+$$
+[ \mathbf {T} ] _ {6 \times 7} = \left[ \begin{array}{l l} \mathbf {T} _ {3} & \mathbf {0} \\ \mathbf {0} & \mathbf {I} \end{array} \right], \quad \text { where } \quad [ \mathbf {T} _ {3} ] = \frac {1}{L} \left[ \begin{array}{c c c c} a & 0 & b & 0 \\ 0 & a & 0 & b \\ c & s & - c & - s \end{array} \right] \tag {7.7-4}
+$$
+
+and $[I]=\left[\begin{matrix}1&1&1\end{matrix}\right]$ . We see that $[k]$ is a 7 by 7 matrix. Quadrilateral and frame elements can now be assembled to one another or assembled into the rest of the structure. Node 5 and its d.o.f. do not appear in the assembled structure. Node 5 may be called a “slave” node because its d.o.f. are completely determined by d.o.f. of “master” nodes 2 and 3.
+
+As a second example, consider the problem of Fig. 7.7-1b. A two-force member, perhaps a reinforcing bar in concrete, is to be connected to points arbitrarily located along edges of a plane four-node element. Matrices $\{r'\}$ and $[k']$ of the bar are associated with d.o.f. $u_{5}$ , $v_{5}$ , $u_{6}$ , and $v_{6}$ . By transformation, $\{r'\}$ and $[k']$ are to be converted to $\{r\}$ and $[k]$ , which are associated with d.o.f. $u_{i}$ and $v_{i}$ of the four corner nodes of the quadrilateral, i = 1, 2, 3, 4. Thus
+
+$$
+\left\{\mathbf {r} ^ {\prime} \right\} _ {4 \times 1} ^ {\cdot} \text { becomes } \left\{\mathbf {r} \right\} _ {8 \times 1} \text { and } \left[ \mathbf {k} ^ {\prime} \right] _ {4 \times 4.} \text { becomes } \left[ \mathbf {k} \right] _ {8 \times 8} \tag {7.7-5}
+$$
+
+The displacement transformation for d.o.f. $\{d'\}$ of the bar is $\{d'\} = [T]\{d\}$ , and the required transformation matrix [T] is 4 by 8. With displacements linearly interpolated along edges of the quadrilateral, [T] contains terms like those in the first two rows of $[T_{3}]$ in Eq. 7.7-4. After transformation, $\{r\}$ and $[k]$ can be directly added to the corresponding arrays of the quadrilateral or assembled into the structure. Nodes 5 and 6 and their d.o.f. are then not explicitly present.
+
+One can say that d.o.f. of the frame and bar elements in Fig. 7.7-1 are constrained to follow d.o.f. of the quadrilateral. $^{2}$
+
+<!-- source-page: 240 -->
+
+# 7.8 RIGID LINKS. RIGID ELEMENTS
+
+Rigid members impose relationships among d.o.f. This circumstance is sometimes called a multipoint constraint. $^{3}$ In the present section we consider rigid members as an application of coordinate transformation.
+
+Rigid Links. Imagine that a plate is to be reinforced by a beam (Fig. 7.8-1). Nodes of the beam do not coincide with nodes of the plate. (If nodes were coincident, the beam-plate connection would be easy; one would simply assemble elements in the usual way.) Even with an offset beam, it is still possible to connect beam and plate in such a way that d.o.f. of only the plate appear in the assembled structure. The procedure for doing so is now described.
+
+The procedure invokes a transformation that makes beam d.o.f. at nodes 3 and 4 “slave” to “master” d.o.f. at nodes 1 and 2 in the plate. This is accomplished by adding imaginary, weightless, rigid links—one between nodes 1 and 3 and another between nodes 2 and 4. We assume that the beam has bending stiffness (associated with d.o.f. $w_{3}$ , $\theta_{3}$ , $w_{4}$ , and $\theta_{4}$ ) and axial stiffness (associated with d.o.f. $u_{3}$ and $u_{4}$ ). These six d.o.f. must be incorporated in the transformation relation. At the left end, the transformation is
+
+$$
+\left\{ \begin{array}{l} u _ {3} \\ w _ {3} \\ \theta_ {3} \end{array} \right\} = [ \mathbf {T} _ {\ell} ] \left\{ \begin{array}{l} u _ {1} \\ w _ {1} \\ \theta_ {1} \end{array} \right\}, \quad \text { where } \quad [ \mathbf {T} _ {\ell} ] = \left[ \begin{array}{l l l} 1 & 0 & b \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] \tag {7.8-1}
+$$
+
+A similar transformation is written at the right end by replacing subscripts 1 by 2 and 3 by 4. We see that d.o.f. $u_{3}$ and $u_{4}$ are activated by $\theta_{1}$ and $\theta_{2}$ . Thus, because of the rigid links, axial stiffness of the beam is seen as bending stiffness by the plate d.o.f.
+
+Let $\{\mathbf{r}'\}$ and $[\mathbf{k}']$ be beam element matrices associated with d.o.f. at nodes 3 and 4 (see Fig. 7.5-2a for $[\mathbf{k}']$ ). Transformed arrays $\{\mathbf{r}\}$ and $[\mathbf{k}]$ , associated with d.o.f. at plate nodes 1 and 2, are
+
+$$
+\begin{array}{l} \{\mathbf {r} \} = [ \mathbf {T} ] ^ {T} \{\mathbf {r} ^ {\prime} \} \\ [ \mathbf {k} ] = [ \mathbf {T} ] ^ {T} [ \mathbf {k} ^ {\prime} ] [ \mathbf {T} ] \end{array} \quad \text { where } \quad \begin{array}{l} [ \mathbf {T} ] = \left[ \begin{array}{c c} \mathbf {T} _ {\ell} & \mathbf {0} \\ \mathbf {0} & \mathbf {T} _ {\ell} \end{array} \right] \\ 6 \times 6 \end{array} \tag {7.8-2}
+$$
+
+![](images/page-240_800c0e7d20ebb55b0ae9d6d10c8a1c1755f4efff276ee99d20fc9f067b2cfb5a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+y,v
+Plate
+Beam
+x,u
+</details>
+
+(a)
+
+![](images/page-240_b3502b2f6a9a47de95b2a99a46c9be3789f5b7121ddeee82cda8f679f174a1dc.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+Plate
+1
+2
+x,u
+3
+Beam
+4
+b
+b
+L
+x
+</details>
+
+(b)
+
+![](images/page-240_1f324bd0a54add731a9ca8675e5e776075b9cef585cb62208d21c0e4251a6d82.jpg)  
+(c)   
+Figure 7.8-1. (a) A reinforcing beam joined to one edge of a plate element. (b) Side view. (c) Typical node i (i = 1, 2, 3, 4), showing d.o.f. considered in the coordinate transformation.
+
+$^{3}$ Constraints are discussed in detail in Chapter 9.
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_025.md
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+<!-- source-page: 241 -->
+
+Clearly this procedure can be extended to deal with a stiffener that is arbitrarily oriented in space, and with rigid links that are not perpendicular to the stiffener.
+
+The foregoing transformation introduces an error that can cause displacements to be significantly overestimated [7.1]. The error can be attributed to incomplete coupling between beam and plate. Axial displacement in the beam should be
+
+$$
+u _ {\text { beam }} = u _ {\text { plate }} + b \theta_ {\text { plate }} \tag {7.8-3}
+$$
+
+Imagine, for example, that all d.o.f. of the plate in Fig. 7.8-1b are zero but $w_{2}$ . Then $w_{plate}$ is cubic in x and $\theta_{plate}$ is quadratic in x. Hence, according to Eq. 7.8-3, $u_{beam}$ should be quadratic in x. However, Eq. 7.8-1 yields $u_{3} = u_{4} = 0$ ; hence, $u_{beam} = 0$ . Thus, for this deformation mode, beam and plate bending stiffnesses are simply added rather than being combined in a way that recognizes a common neutral axis. For a test case in which a uniform cantilever was loaded by a transverse tip force, with n plate elements along the length, tip displacement was overestimated by 69% for n = 1, 17% for n = 2, and 4.3% for n = 4 [7.1]. The error tends toward zero as each element approaches a state of constant curvature.
+
+A method that eliminates the error was suggested by Miller [7.2]. He introduces axial displacement d.o.f. at x = L/2, say $u_{5}$ in the plate and $u_{6}$ in the beam. Axial displacement in the beam is now quadratic in x, as is desired. The axial stiffness portion of $[k']$ is 3 by 3 and is associated with $u_{3}$ , $u_{4}$ , and $u_{6}$ (see Section 6.2). The transformation is essentially that of Eq. 7.8-2, augmented by
+
+$$
+u _ {6} = u _ {5} + b \left(\frac {d w}{d x}\right) _ {x = L / 2} \tag {7.8-4}
+$$
+
+where plate rotation $dw/dx$ depends on $w_1$ , $\theta_1$ , $w_2$ , and $\theta_2$ . Transformation causes the 7 by 7 beam element stiffness matrix to operate on d.o.f. $u_1$ , $w_1$ , $\theta_1$ , $u_2$ , $w_2$ , $\theta_2$ , and $u_5$ . This matrix is then combined with the plate element stiffness matrix (whose row and column corresponding to $u_5$ are null). Finally, condensation removes $u_5$ , thus producing a combined [k] that operates on the usual plate element d.o.f.
+
+Another difficulty, encountered in dynamic problems, is that the transformation converts a diagonal beam mass matrix $[m']$ to a nondiagonal mass matrix [m]. Ad hoc adjustments of [m] can make it diagonal again.
+
+Rigid Elements. A rigid element might be used to model part of a linkage mechanism that couples elastic bodies. Or a particular element might be of much higher modulus than surrounding elements. In the latter case, errors of the type discussed in Section 18.2 are likely, and it is better to make the element perfectly rigid rather than very stiff.
+
+Imagine that the triangle of Fig. 7.8-2 is to be idealized as perfectly rigid.
+
+![](images/page-241_f36ba2874ec1328db65a401d48d9804347c2b49f0ddceacc323d57c4180f0075.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+a
+2
+b
+3
+1
+x,u
+</details>
+
+Figure 7.8-2. A plane triangle. Other elements of the structure are connected to it but are not shown.
+
+<!-- source-page: 242 -->
+
+Therefore, its motion is completely described by three d.o.f., say $u_1, v_1$ , and $u_2$ . These d.o.f.-are related to the original six d.o.f. by the transformation
+
+$$
+\{\mathbf {d} ^ {\prime} \} = [ \mathrm{T} ] \{\mathbf {d} \} \quad \text { or } \quad \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \\ u _ {2} \\ v _ {2} \\ u _ {3} \\ v _ {3} \end{array} \right\} = \left[ \begin{array}{c c c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ - a / b & 1 & a / b \\ 1 & 0 & 0 \\ - a / b & 1 & a / b \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \\ u _ {2} \end{array} \right\} \tag {7.8-5}
+$$
+
+in which $u_{3} = u_{1}$ and $v_{2} = v_{3} = v_{1} - \theta a$ , where $\theta = (u_{1} - u_{2}) / b$ is a small rigid-body rotation. Transformation according to Eq. 7.8-5 is applied to all elements of the structure that contain any of the d.o.f. $v_{2}, u_{3}$ , and $v_{3}$ . Thus, $v_{2}, u_{3}$ , and $v_{3}$ no longer appear as d.o.f. in $\{\mathbf{D}\}$ . The particular stiffness coefficients of triangle 1-2-3 do not matter; they are overridden by the rigid-body constraint.
+
+The choice $\{\mathbf{d}\} = [u_1 - v_1 - u_2]^T$ is not unique, and would be unacceptable if node numbers were rearranged so that $y_2 - y_1 = b = 0$ . Not only would there be a division by zero in Eq. 7.8-5, but the use of $u_1$ and $u_2$ as independent d.o.f. would contradict the assumption that the triangle is rigid.
+
+# PROBLEMS
+
+# Section 7.2
+
+7.1 If a vector $\mathbf{V}$ has length $L$ , then $\mathbf{V} \cdot \mathbf{V} = L^2$ regardless of the coordinate system in which $\mathbf{V}$ resides. Hence, using Eq. 7.2-1, show that $\Sigma \ell_i = 1$ , $\Sigma \ell_i m_i = 0$ , and so on (six such relations altogether).   
+7.2 (a) Let $x' = -x$ and $y' = -y$ . What is [A] in Eq. 7.2-1 if both coordinate systems are right-handed?   
+(b) Similarly, what is $[\Lambda]$ if $x' = y$ and $z' = z$ ?   
+7.3 (a) If $z = z'$ and $x'$ is located at a counterclockwise angle $\theta$ from $x$ , what is [A] in Eq. 7.2-1?   
+(b) For this $[\Lambda]$ , show that $[\Lambda]^{-1} = [\Lambda]^T$ .
+
+# Section 7.3
+
+7.4 Let $[\mathbf{E}']$ be 3 by 3, as for a plane stress problem. Show that Eq. 7.3-10 yields $[\mathbf{E}'] = [\mathbf{E}]$ if the material is isotropic.   
+7.5 Let an orthotropic material have principal directions $x', y'$ , and $z$ (i.e., axes $z'$ and $z$ coincide). Write the 6 by 6 matrix $[T_{\epsilon}]$ for this situation. Express your answer in terms of $\sin \beta$ and $\cos \beta$ .   
+7.6 Consider a plane problem for which the 3 by 3 matrix $[\mathbf{E}']$ is diagonal, with $E_{11}' = E_a$ , $E_{22}' = E_b$ , and $E_{33}' = G$ . What is [E] for an arbitrary angle $\beta$ in Fig. 7.3-1? As a partial check on your answer, try the case $\beta = \pi/2$ .   
+7.7 Is $[\mathbf{T}_{\epsilon}]$ of Eq. 7.3-11 an orthogonal matrix?
+
+<!-- source-page: 243 -->
+
+# Section 7.4
+
+7.8 For a given distortion, strain energy in an element (Eq. 4.1-10) must be independent of the coordinate system in which it is computed. Use this argument to derive Eq. 7.4-4.
+
+# Section 7.5
+
+7.9 Two forms of $[k']$ for a truss element are given in Section 7.5, one by Eq. 7.5-1 and the other by Eq. 7.5-2. Verify that appropriate transformation of each form produces the stiffness matrix of Eq. 2.4-3.
+
+7.10 (a) Verify the [k] determined in Problem 2.16 by coordinate transformation of Eq. 2.4-3.   
+(b) Obtain the same result by coordinate transformation of $[\mathbf{k}^{\prime}]$ in Eq. 7.5-1.   
+(c) Obtain the same result by coordinate transformation of $[\mathbf{k}^{\prime}]$ in Eq. 7.5-2.
+
+7.11 Write a compact set of Fortran statements that will generate $[\mathbf{k}]$ of a space truss element (see Eq. 7.5-5).
+
+7.12 Verify that [k] in Fig. 7.5-2 follows from $[\mathbf{k}^{\prime}]$ by application of Eqs. 7.4-4 and 7.5-7.
+
+7.13 A plane grillage is a plane network of straight members that carries loads normal to its plane. Thus the grillage resembles a plane frame, but carries lateral loads. A typical member resists bending and torsional deformation and has six d.o.f., as shown. Write, in terms of angle $\alpha$ in the xy plane, the transformation matrix that would be used to convert $[k']$ to $[k]$ , where $[k]$ operates on d.o.f. w (lateral deflection), $\theta_{x}$ (rotation about the x axis), and $\theta_{y}$ (rotation about the y axis) at each node.
+
+![](images/page-243_663f3cb0a13143d85cd5110f7ac59978b6f4913de998130571b4d32f18d88a98.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,z' (lateral)
+θy'1
+w1
+1
+θx'1
+α
+x'
+y
+w2
+2
+θx'2
+θy'2
+y'
+</details>
+
+Problem 7.13
+
+7.14 A space beam can resist axial load, twisting about its axis, bending about either principal axis of its cross section, and transverse loads. Assume that the beam is straight, uniform, and has six d.o.f. at each end.
+
+(a) Let the beam lie on the $x'$ axis and let $y'$ and $z'$ be parallel to principal axes of the cross section. Write the 12 by 12 stiffness matrix $[\mathbf{k}']$ that operates on d.o.f. $\{\mathbf{d}'\} = \left[u_1' v_1' w_1' \theta_{x1}' \cdots \theta_{y2}' \theta_{z2}'\right]^T$ . Let $\theta$ vectors point in the positive coordinate directions.
+
+(b) Now consider that the beam is arbitrarily oriented in xyz coordinates.
+
+<!-- source-page: 244 -->
+
+Stiffness matrix [k] is desired, where [k] operates on d.o.f. {d} that are parallel to x, y, and z axes. Write the transformation matrix [T].
+
+7.15 The size, shape, and orientation in space of a plane triangular element are defined by the known global coordinates of its three corner nodes.
+
+(a) Let node 3 be at the origin of local coordinates $x'y'z'$ . In addition, let nodes 3 and 1 define the $x'$ axis and let the plane of the element define the $x'y'$ plane. Describe how to compute the direction cosines of Fig. 7.2-1 from the given information.
+
+(b) If the element is a constant-strain triangle, write the matrix [T] that will produce the 9 by 9 global matrix [k] from the 6 by 6 local matrix $[\mathbf{k}^{\prime}]$ .
+
+7.16 The bar shown is rigid and is supported by two linear springs of stiffness $k_{1}$ and $k_{2}$ . Only vertical displacement is permitted.
+
+(a) Write the stiffness matrix that operates on d.o.f. $v_{1}$ and $v_{2}$ . Then transform this matrix so that you obtain a stiffness matrix that operates on $v_{1}$ and $\theta_{1}$ , where $\theta_{1}$ is a small rotation of the bar with respect to the horizontal and about the left end.
+
+(b) Redefine $v_{2}$ so that it is the vertical displacement at the midpoint of the bar. Do not change $v_{1}$ . Then repeat part (a).
+
+![](images/page-244_b096b51f2e7c4ccbffbc8bb8655cf488745bc6dd55413ed3af03aa67db116f37.jpg)
+
+<details>
+<summary>text_image</summary>
+
+v₁
+Rigid bar
+v₂
+k₁
+k₂
+L
+</details>
+
+Problem 7.16a
+
+7.17 Work Problem 7.16 in reverse. That is, start with the final stiffness matrix that operates on $v_{1}$ and $\theta_{1}$ . By transformation, obtain from it the matrix in Problem 7.16a that operates on $v_{1}$ and $v_{2}$ . Similarly, obtain the matrix in Problem 7.16b that operates on $v_{1}$ and the midpoint $v_{2}$ .
+
+7.18 Let a standard bar element of axial stiffness $k = AE / L$ be restricted to motion along its axis. Its [k] is 2 by 2 and operates on nodal d.o.f. $u_{1}$ and $u_{2}$ . Transform [k] so that it operates on nodal d.o.f. $u_{1}$ and $u_{r}$ , where $u_{r}$ is the displacement of node 2 relative to node 1.
+
+7.19 A three-node bar element and its shape functions are shown in Fig. 6.2-1. Imagine that d.o.f. $u_{3}$ is to be replaced by $u_{r}$ , where $u_{r}$ is the displacement at $\xi = 0$ relative to the displacement at $\xi = 0$ dictated by $u_{1}$ and $u_{2}$ . Thus, $u_{3} = u_{r} + \frac{1}{2}(u_{1} + u_{2})$ . Write the transformation matrix and use it to determine the new shape functions.
+
+# Section 7.6
+
+7.20 Let loads $F_{x}$ and $F_{y}$ act at node 3 in Fig. 7.6-1. Verify that the operation $[\mathbf{T}]^{T}\{\mathbf{r}'\}$ transforms $F_{x}$ and $F_{y}$ to the correct $r$ and $s$ components.
+
+7.21 Let Fig. 7.6-1 represent a plane truss for which axial stiffness $k = AE / L$ is the same for each bar. Also let the three interior angles in each panel be $45^{\circ}$ , $45^{\circ}$ , and $90^{\circ}$ . Apply a downward load $P$ at node 4 and set $u_{4} = 0$ . If $\beta = \arctan 0.75$ , what is the force in bar 3-1 in terms of $P$ ?
+
+<!-- source-page: 245 -->
+
+![](images/page-245_bc4916dfad39c57b6edb0b9489ab1f4a395b1d4fabd8e125391befebbf8348b2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L
+P
+A, E, I
+β
+U
+</details>
+
+Problem 7.22
+
+7.22 The right end of the cantilever beam slides without friction on a rigid wall, as shown. Represent the cantilever as a single element with axial, transverse, and rotational d.o.f. at the right end.
+
+(a) Transform and impose the boundary conditions. Thus, obtain a 2 by 2 matrix [K] that operates on tangential displacement U and rotation $\theta$ at the right end.   
+(b) In addition, let the condition $\theta = 0$ be imposed. Solve for $U$ .
+
+7.23 Imagine that, at a certain node of a space truss, motion is to be prohibited along a line whose direction cosines are $\ell_1, \ell_2$ , and $\ell_3$ . Motion is permitted in all directions normal to the line. Original nodal d.o.f. are displacements in coordinate directions $x, y$ , and $z$ .
+
+(a) Explain precisely how to define suitable new directions for d.o.f. at the node, and write the transformation matrix at the node (analogous to $[\mathbf{T}_3]$ in Eq. 7.6-4).   
+(b) Check your result for the special case $\ell_2 = 1$ .
+
+# Section 7.7
+
+7.24 The element shown is of arbitrary quadrilateral shape and is formulated as a bilinear element (Section 6.3). A constant-strain triangle (six d.o.f.; lettered nodes) is to be attached, so that lettered nodes lie at $\xi = \pm 0.5$ and $\eta = \pm 0.5$ . Write [T] in the relation $\{\mathbf{d}'\} = [\mathbf{T}]\{\mathbf{d}\}$ , where $\{\mathbf{d}'\}$ and $\{\mathbf{d}\}$ contain d.o.f. of lettered nodes (“slaves”) and numbered nodes (“masters”) respectively.
+
+![](images/page-245_e2df980a901e653768cf82f686e03351d7e3119efd9b52c5ab749cb48e1903f1.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+3
+4
+A
+B
+C
+ξ
+η
+</details>
+
+Problem 7.24
+
+7.25 At node 5 of the frame element in Fig. 7.7-1a, let forces $F_{x}$ and $F_{y}$ and moment $M_{s}$ (counterclockwise) be applied. How are these loads distributed to nodes 2 and 3 by the transformation of Eq. 7.7-4? Do the new loads exert the same force and moment resultants as the original loads?   
+7.26 Write the transformation matrix for the problem described in connection with Eq. 7.7-5. What is [k'] for this problem?   
+7.27 Plane element 1 in the sketch is bilinear (Section 6.3). It has the usual two d.o.f. per node. Plane frame element 2 has the usual three d.o.f. per node $(u_{i}, w_{i}, \theta_{i})$ . Consider the element stiffness matrices $[k_{1}]$ and $[k_{2}]$ .
+
+<!-- source-page: 246 -->
+
+![](images/page-246_9e4f99836231fb8cacb3651b68ee24dbafef165f94ba49fc75b5f20b912eb0a2.jpg)  
+Problem 7.27
+
+(a) Write a transformation matrix $[\mathbf{T}_1]$ that could be used to convert $[\mathbf{k}_1]$ so that it operates on the d.o.f. of element 2.   
+(b) Write a transformation matrix $[\mathbf{T}_2]$ that could be used to convert $[\mathbf{k}_2]$ so that it operates on the d.o.f. of element 1.   
+(c) Should $[\mathbf{T}_1][\mathbf{T}_2]$ and $[\mathbf{T}_2][\mathbf{T}_1]$ be unit matrices? Find an argument that says so.   
+(d) Evaluate the products $[\mathbf{T}_1][\mathbf{T}_2]$ and $[\mathbf{T}_2][\mathbf{T}_1]$ . How can the results be explained?
+
+# Section 7.8
+
+7.28 Element $ij$ is a plane frame element (see sketch). Imagine that d.o.f. at nodes $i$ and $j$ are to be made slave to d.o.f. at nodes 1 and 2 via rigid links $i1$ and $j2$ . Write the 6 by 6 transformation matrix [T].
+
+![](images/page-246_1172e42cd030d4efe8885fdd374931a40c2521e171bd2d110641faba1175db01.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+a₁
+b₁
+i
+β
+x,u
+j
+2
+a₂
+b₂
+</details>
+
+Problem 7.28
+
+7.29 Consider a frame element $ij$ , arbitrarily oriented in space. D.o.f. at node $i$ are $\{u_i \quad v_i \quad w_i \quad \theta_{xi} \quad \theta_{yi} \quad \theta_{zi}\}$ , where rotational d.o.f. vectors point in positive coordinate directions. D.o.f. at node $j$ are similar. The element is to be made slave to d.o.f. at some other nodes (nodes 1 and 2, say) via rigid links $i1$ and $j2$ , which are arbitrarily oriented. Write the 12 by 12 transformation matrix [T].
+
+7.30 (a) Evaluate Eq. 7.8-4. That is, express $u_{6}$ in terms of $b, u_{5}, w_{1}, \theta_{1}, w_{2}$ , and $\theta_{2}$ (see Eq. 4.2-3 and Fig. 3.13-2).   
+(b) Write the 7 by 7 transformation matrix [T] in $\{\mathbf{d}'\} = [\mathbf{T}]\{\mathbf{d}\}$ , where $\{\mathbf{d}'\} = \left[u_3 \quad w_3 \quad \theta_3 \quad u_4 \quad w_4 \quad \theta_4 \quad u_6\right]^T$ .
+
+![](images/page-246_43f33aa2c5fbea0abcf32f5a44482836ed14ba20a790fa66d4904969dd40c619.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+b
+P
+</details>
+
+![](images/page-246_1d4f01398f9f7cfd6e75325ed74dd5ce45090e48ba4a5df7704f859ffd27afde.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Q
+P
+2
+b
+3
+1
+a
+</details>
+
+Problem 7.31
+
+<!-- source-page: 247 -->
+
+7.31 Obtain nodal loads $\{r\} = [T]^{T}\{r'\}$ for the elements and loads shown. Use [T] from Eq. 7.8-1 and Eq. 7.8-5 for the respective elements. Sketch $\{r\}$ , and argue why $\{r\}$ is reasonable or unreasonable.   
+7.32 Rewrite Eq. 7.8-5 if the triangle is of arbitrary shape, with nodal coordinates $x_{i}$ and $y_{i}$ ( $i = 1, 2, 3$ ).   
+7.33 Rewrite Eq. 7.8-5 if the “master” d.o.f. are changed from $u_{1}$ , $v_{1}$ , and $u_{2}$ to $u_{2}$ , $u_{3}$ , and $v_{3}$ .
+
+<!-- source-page: 248 -->
+
+# TOPICS IN STRUCTURAL MECHANICS
+
+Miscellaneous elements, procedures, and remarks are presented. Some topics are of general interest while others pertain to structural mechanics.
+
+# 8.1 D.O.F. WITHIN ELEMENTS. CONDENSATION
+
+Occasionally, the basic building block of a finite element mesh is a macroelement—that is, a “patch” that consists of two or more elements coupled together. A macroelement can be regarded as a small structure. Its component elements are called subelements. Two examples appear in Fig. 8.1-1. Both macroelements are built of triangular subelements. In both cases the user of a computer program need define only the boundary nodes (which are numbered in the sketch). The program itself can automatically locate the internal nodes, generate and combine matrices of the subelements, and produce a stiffness matrix and load vector associated with only the boundary nodes. There is no limit to the number of subelements or the number of internal d.o.f. This observation leads to substructuring, discussed in Section 8.14.
+
+D.o.f. of internal nodes are coupled only to d.o.f. of other internal nodes and to d.o.f. of nodes on the macroelement boundary. There is no coupling of internal d.o.f. to d.o.f. of nodes outside the macroelement. Accordingly, equations associated with internal d.o.f. can be processed separately from other equations of the structure. Separate processing can be both efficient and convenient for users, as will be seen subsequently. In the present section we emphasize the processing procedures.
+
+Condensation. Condensation is the process of reducing the number of d.o.f. by substitution, for example, by starting a Gauss elimination solution of equations for unknowns but stopping before the stiffness matrix has been fully reduced. Condensation by elimination is also called static condensation. Condensation in dynamics is usually called reduction and introduces an approximation. Static condensation, described as follows, is strictly a manipulation and introduces no approximation.
+
+Let the equations $[k]\{d\} = \{r\}$ represent a portion of the entire structure. This portion might be a macroelement built of subelements or a single element that has “nodeless” d.o.f. (such an element will be described in the following). Let d.o.f. $\{d\}$ be partitioned so that $\{d\} = \left[d_{r} - d_{c}\right]^{T}$ , where $\{d_{r}\}$ are boundary d.o.f. to be retained and $\{d_{c}\}$ are internal d.o.f. to be eliminated by condensation. Thus $[k]\{d\} = \{r\}$ becomes
+
+$$
+\left[ \begin{array}{l l} \mathbf {k} _ {r r} & \mathbf {k} _ {r c} \\ \mathbf {k} _ {c r} & \mathbf {k} _ {c c} \end{array} \right] \left\{ \begin{array}{l} \mathbf {d} _ {r} \\ \mathbf {d} _ {c} \end{array} \right\} = \left\{ \begin{array}{l} \mathbf {r} _ {r} \\ \mathbf {r} _ {c} \end{array} \right\} \tag {8.1-1}
+$$
+
+<!-- source-page: 249 -->
+
+![](images/page-249_68771f474d975568aaa4d230b3f33acf86e51a0ec7c4c56166e6a1c3e5288b63.jpg)
+
+<details>
+<summary>text_image</summary>
+
+4
+3
+1
+2
+(a)
+</details>
+
+![](images/page-249_83a651ef5c4395a623c501a23d0e6d073abbc86cc9f3bf402af9c0156f54892b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+3
+2
+1
+(b)
+</details>
+
+Figure 8.1-1. Elements having internal nodes. Boundary nodes are numbered; internal nodes are not. (a) A quadrilateral built of four triangles. (b) A triangle built of three triangles.
+
+The lower partition is solved for $\{\mathbf{d}_c\}$ :
+
+$$
+\{\mathbf {d} _ {c} \} = - [ \mathbf {k} _ {c c} ] ^ {- 1} ([ \mathbf {k} _ {c r} ] \{\mathbf {d} _ {r} \} - \{\mathbf {r} _ {c} \}) \tag {8.1-2}
+$$
+
+Next, $\{\mathbf{d}_c\}$ is substituted into the upper partition of Eqs. 8.1-1. Thus
+
+$$
+\underbrace {([ \mathbf {k} _ {r r} ] - [ \mathbf {k} _ {r c} ] [ \mathbf {k} _ {c c} ] ^ {- 1} [ \mathbf {k} _ {c r} ])} _ {\text { condensed   [k] }} \{\mathbf {d} _ {r} \} = \underbrace {\{\mathbf {r} _ {r} \} - [ \mathbf {k} _ {r c} ] [ \mathbf {k} _ {c c} ] ^ {- 1} \{\mathbf {r} _ {c} \}} _ {\text { condensed   } \{\mathbf {r} \}} \tag {8.1-3}
+$$
+
+The element is now treated in standard fashion; that is, the condensed [k] and the condensed $\{r\}$ are assembled into the structure, boundary conditions are imposed, and structural d.o.f. $\{D\}$ are computed. Thus $\{d_{r}\}$ becomes known, and $\{d_{c}\}$ (which may be needed in stress calculation) follows from Eq. 8.1-2. Computation of $\{d_{c}\}$ is called recovery of internal d.o.f. Computer algorithms for condensation and recovery are discussed in Section 8.2.
+
+Equation 8.1-3 is Gauss elimination, carried out on d.o.f. $\{d_{c}\}$ only (compare with Eq. B.2-2, Appendix B). Completion of the elimination process, and solution for $\{d_{r}\}$ , awaits assembly of all remaining elements of the structure. Thus condensation is simply the first set of eliminations in a solution of the structure equations $\{K\}\{D\} = \{R\}$ . The same solution vector $\{D\}$ would result if internal d.o.f. were eliminated later. The advantage of eliminating them first, at the element level and before assembly, is that the order of the structure stiffness matrix is reduced because d.o.f. $\{d_{c}\}$ are not carried into the global set of equations.
+
+The partitioning used in Eq. 8.1-1 is a conceptual convenience rather than a computational necessity. D.o.f. to be condensed can appear anywhere in $\{d\}$ , and can be processed serially rather than simultaneously. After a d.o.f. $d_{k}$ is condensed, rows $i \neq k$ and columns $j \neq k$ comprise the condensed [k].
+
+Nodeless D.o.f. Internal d.o.f. need not be associated with a node. The 18 d.o.f. plane element of Table 6.6-1 can be restated in terms of nodeless internal d.o.f., as we now describe. For i = 1 to 8, let shape functions $N_{i}$ be those of the 16 d.o.f. element, Eq. 6.6-1. Then displacements in the 18 d.o.f. element are
+
+$$
+u = \sum_ {i = 1} ^ {8} N _ {i} u _ {i} + N _ {9} a _ {1} \quad \text { and } \quad v = \sum_ {i = 1} ^ {8} N _ {i} v _ {i} + N _ {9} a _ {2} \tag {8.1-4}
+$$
+
+<!-- source-page: 250 -->
+
+where, as in Eq. 6.6-3,
+
+$$
+N _ {9} = (1 - \xi^ {2}) (1 - \eta^ {2}) \tag {8.1-5}
+$$
+
+Mode $N_9$ is called a “bubble function” mode, and $a_1$ and $a_2$ are nodeless d.o.f. to be condensed, $\{d_c\} = [a_1 - a_2]^T$ . Physically, $a_1$ and $a_2$ represent the displacement components at $\xi = \eta = 0$ relative to the displacement components $\Sigma N_i u_i$ and $\Sigma N_i v_i$ at $\xi = \eta = 0$ dictated by d.o.f. at the eight boundary nodes. It is not necessary to assign such a physical meaning, or to calculate actual displacements at $\xi = \eta = 0$ , because d.o.f. $a_1$ and $a_2$ are not connected to other elements—that is, $a_1$ and $a_2$ in an element are not d.o.f. of another element as well.
+
+In processing, nodeless d.o.f. are treated no differently than any other d.o.f. Thus, for the 18 d.o.f. element, it does not matter whether the $N_{i}$ are given by Table 6.6-1 or by Eqs. 8.1-4: if formulation procedures of preceding chapters are used consistently, then, from either starting point, identical 16 by 16 condensed matrices [k] and 16 by 1 consistent load vectors $\{\mathbf{r}_{e}\}$ appear after condensation of the two internal d.o.f. ( $u_{9}$ and $v_{9}$ or $a_{1}$ and $a_{2}$ ). Before condensation, loads in $\{\mathbf{r}_{e}\}$ associated with $a_{1}$ and $a_{2}$ may appear to be too large, even though correct, because $a_{1}$ and $a_{2}$ are not actual displacements.
+
+When $a_1$ and $a_2$ are used as internal d.o.f., element geometry (e.g., the Jacobian matrix [J] of Eq. 6.6-4) is defined by the eight $N_i$ of Eqs. 6.6-1 and the coordinates of the eight boundary nodes. Use of Eq. 6.6-2 and the $N_i$ of Table 6.6-1 would yield the same geometry but with slightly more computational effort.
+
+Releases. A “release” is a lack of complete connection between nodes that would usually be fully connected. The plane frame of Fig. 8.1-2 is a case in point. The structure displacement vector $\{D\}$ contains three d.o.f. per node. At node A, where there is a hinge, the two frames are not to share the same nodal rotation $\theta_{A}$ , as this would imply a rigid connection rather than a hinge. One way to model the hinge is to condense $\theta$ at A in (say) the left frame, then fill the row and column just condensed with zeros so that no rotational stiffness at A will be contributed to the structure by the left frame. Thus assembly makes the left and right frames share only $u_{A}$ and $v_{A}$ , and $\theta_{A}$ in $\{D\}$ now represents the rotation at A in the right frame. Rotation at A in the left frame is treated as an internal d.o.f. to be recovered after $\{D\}$ is known.
+
+Another way to treat the hinge at A in Fig. 8.1-2 is to define two separate nodes at A, one in the left frame and one in the right but having the same location. Thus there are a total of six d.o.f. at A. Next, one joins only translational d.o.f. of the two nodes by means of a constraint technique (see Chapter 9). A similar treatment could be used at an interface between elastic bodies that may slide on one another.
+
+![](images/page-250_6bb72226bb638db1b0c3b398abfcb5783b228c5ec07b63558843522b736f5e0c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+A
+x,u
+</details>
+
+Figure 8.1-2. Two frames with a hinge connection at A.
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_026.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_026.md
new file mode 100644
index 00000000..325b384c
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_026.md
@@ -0,0 +1,497 @@
+<!-- source-page: 251 -->
+
+```lua
+C--- Do condensation operations on lower triangle of SE.
+DO 30 K=1, NUM
+LL = NSIZE - K
+KK = LL + 1
+DO 20 L=1, LL
+DUM = SE(KK, L)/SE(KK, KK)
+DO 10 M=1, L
+10 SE(L, M) = SE(L, M) - SE(KK, M) * DUM
+RE(L) = RE(L) - RE(KK) * DUM
+20 CONTINUE
+30 CONTINUE
+C--- Fill in the upper triangle of SE by symmetry.
+DO 40 K=1, LL
+DO 40 L=1, K
+40 SE(L, K) = SE(K, L) 
+```  
+Figure 8.2-1. A Fortran condensation algorithm that accepts a symmetric matrix, stored in full rather than banded format, does Gauss elimination from the bottom with diagonal pivots and yields the condensed [k] and {r} of Eq. 8.1-3.
+
+# 8.2 CONDENSATION AND RECOVERY ALGORITHMS
+
+Our algorithm is a straightforward coding of Gauss elimination to produce the condensed arrays [k] and {r} of Eqs. 8.1-3 from the uncondensed arrays of Eq. 8.1-1. Let there be NSIZE d.o.f. in the uncondensed system, and let it be required to condense the last NUM of these d.o.f. In Fig. 8.2-1 condensation is done in place, so that after completion of the condensation algorithm, the condensed [k] resides in the upper left NSIZE-NUM rows and columns of array SE, and the condensed {r} resides in the upper NSIZE-NUM rows of array RE.
+
+After structural equations have been solved, d.o.f. $\{d_{r}\}$ reside in the structure solution vector $\{D\}$ . It remains to recover $\{d_{c}\}$ in Eq. 8.1-2. This can be done by back-substitution, thus completing the Gauss elimination solution that was begun by condensation. Figure 8.2-2 gives the Fortran coding. In Fig. 8.2-2, arrays SM and RM contain the last NUM rows of arrays SE and RE exactly as they stand after completion of the condensation routine. In Fig. 8.2-1 these rows follow the condensed matrices, and are numbered NUM+1 through NSIZE. In Fig. 8.2-2 these rows are duplicated in arrays SM and RM, where they occupy rows 1 through NUM. In Fig. 8.2-2 the first NSIZE-NUM rows of array DE contain the known d.o.f. $\{d_{r}\}$ . Internal d.o.f. $\{d_{c}\}$ are computed and stored in the last NUM rows of array DE.
+
+In the foregoing algorithms, mass storage would probably be used to store the last NUM rows of arrays SE and RE from Fig. 8.2-1. Node point coordinates, and $\{\epsilon_{0}\}$ and $\{\sigma_{0}\}$ , would also be stored. After $\{D\}$ is known, these data would be recalled and Fig. 8.2-2 used to compute $\{d_{c}\}$ . At each point where stresses are needed, [B] can be reconstructed from the node point coordinates. Finally, strains are $\{\epsilon\} = [B]\left[d_{r} \cdot d_{c}\right]^{T} - \{\epsilon_{0}\}$ and stresses are $\{\sigma\} = [E]\{\epsilon\} + \{\sigma_{0}\}$ .
+
+```txt
+DO 60 J=1, NUM
+JJ = NSIZE - NUM + J
+DUM = 0.
+K = JJ - 1
+DO 50 L=1, K
+50 DUM = DUM + SM(J, L)*DE(L)
+60 DE(JJ) = (RM(J) - DUM)/SM(J, JJ) 
+```  
+Figure 8.2-2. Recovery of previously condensed d.o.f. $\{d_{c}\}$ when $\{d_{r}\}$ and $\{r_{c}\}$ are known.
+
+<!-- source-page: 252 -->
+
+Alternative Method. In an alternative method [8.1], explicit recovery of $\{d_{c}\}$ is avoided. Instead, as part of the process of generating the element stiffness matrix, an element stress matrix [S] and stress vector $\{\rho\}$ are also generated. After element d.o.f. $\{d_{r}\}$ are known, element stresses $\{\sigma\}$ are computed by the equation
+
+$$
+\{\pmb {\sigma} \} = [ \mathbf {S} ] \{\mathbf {d} _ {r} \} + \{\pmb {\rho} \} \tag {8.2-1}
+$$
+
+in which previously condensed d.o.f. $\{\mathbf{d}_c\}$ do not appear.
+
+The most significant differences between the two methods are as follows. The first method, Figs. 8.2-1 and 8.2-2, explicitly recovers $\{d_{c}\}$ and reconstructs a (somewhat sparse) matrix [B] at each stress point. The alternative method generates, condenses, and stores an array [S] that is smaller than [B] but not sparse. Both methods yield the same stresses. The relative cost of the two methods depends on billing charges for computing and for mass storage, the size of $\{d_{r}\}$ in relation to $\{d_{c}\}$ , the number of load conditions, the number of stress points, and other less important factors. The method of Figs. 8.2-1 and 8.2-2 tends to be cheaper if $\{d_{c}\}$ is small in relation to $\{d_{r}\}$ , if the number of load cases is small, or if the number of stress points is large. Further comparison appears in [8.2].
+
+# 8.3 PARASITIC SHEAR. INCOMPATIBLE ELEMENTS
+
+Parasitic Shear. Bilinear elements, discussed in Section 6.3, are attractive because they are simple and have only corner nodes. Unfortunately, they are too stiff in bending, whether the element is a rectangle or an arbitrary quadrilateral. We illustrate the point with reference to the rectangular element in Fig. 8.3-1. Here $\xi$ and $\eta$ are dimensionless Cartesian coordinates, $\xi = x/a$ and $\eta = y/b$ . Let bending moment $M_{1}$ be applied, so that nodal displacements $\overline{u}$ arise in response, as shown in Fig. 8.3-1b. According to Eqs. 6.3-2 and 6.3-16, the element deformation field is
+
+$$
+u = \xi \eta \overline {{{u}}} \qquad \text { and } \qquad v = 0 \tag {8.3-1}
+$$
+
+Thus top and bottom edges $\eta = \pm 1$ remain straight, and strains in the element are
+
+![](images/page-252_9d8abd8b5d3331c861c5a9bd1f9bf8209163cbffa1df74cfc31e9842e48092a0.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+a a
+4 3
+η
+ξ
+b
+x,u
+b
+1 2
+(a)
+</details>
+
+![](images/page-252_67f12419aaf77d72333e887ffe9f36c133273c56144b7587e5349830968fb5e7.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ū
+ū
+η
+ξ
+M₁
+M₁
+ū
+ū
+(b)
+</details>
+
+![](images/page-252_025a60e3f4cb9daa289f35ebf5b48b6ed19c26b070eab4c49efb7750b6babaab.jpg)
+
+<details>
+<summary>text_image</summary>
+
+M₂
+η
+ξ
+M₂
+(c)
+</details>
+
+Figure 8.3-1. (a) A rectangular bilinear element. (b) The bilinear element deformed by bending moment $M_{1}$ . (c) Correct deformed geometry for pure bending under bending moment $M_{2}$ .
+
+<!-- source-page: 253 -->
+
+$$
+\epsilon_ {x} = \eta \frac {\overline {{{u}}}}{a} \quad \epsilon_ {y} = 0 \quad \gamma_ {x y} = \xi \frac {\overline {{{u}}}}{b} \tag {8.3-2}
+$$
+
+The correct shape under pure bending, Fig. 8.3-1c, is
+
+$$
+u = \xi \eta \overline {{u}} \quad \text { and } \quad v = (1 - \xi^ {2}) \frac {a \overline {{u}}}{2 b} + (1 - \eta^ {2}) \nu \frac {b \overline {{u}}}{2 a} \tag {8.3-3}
+$$
+
+where $\nu$ is Poisson's ratio. From Eqs. 8.3-3, the correct strains under pure bending are
+
+$$
+\epsilon_ {x} = \eta \frac {\overline {{{u}}}}{a} \quad \epsilon_ {y} = - \nu \eta \frac {\overline {{{u}}}}{a} \quad \gamma_ {x y} = 0 \tag {8.3-4}
+$$
+
+Upon comparing Eqs. 8.3-2 and 8.3-4, we see that if bending displacements $\overline{u}$ are imposed, the correct behavior gives rise to storage of strain energy caused by normal strains alone, but the bilinear element stores strain energy caused by normal strain $\epsilon_{x}$ and a spurious shear strain $\gamma_{xy}$ . Thus, for the same deformation, $M_{1} > M_{2}$ in Fig. 8.3-1. Specifically, by computing the ratio of strain energies in the two cases, we obtain
+
+$$
+\frac {M _ {1}}{M _ {2}} = \frac {1}{1 + \nu} \left[ \frac {1}{1 - \nu} + \frac {1}{2} \left(\frac {a}{b}\right) ^ {2} \right] \tag {8.3-5}
+$$
+
+The unwanted shear strain that produces $M_{1} > M_{2}$ is called parasitic shear. Equation 8.3-5 shows that its effect is disastrous if a/b is large; that is, for large a/b the mesh “locks.” Locking is discussed in more detail in Section 9.4.
+
+Incompatible Elements. Upon comparing Eqs. 8.3-1 and 8.3-3, we see that the bilinear element errs by omitting from v the displacement modes associated with $(1 - \xi^{2})$ and $(1 - \eta^{2})$ . In the bending mode where $v = \overline{v}\xi\eta$ , similar modes are omitted from u. The eight-node trilinear solid element (Section 6.7) suffers from the same defects. These elements, whether rectangular or not, can be improved by adding the missing modes as internal freedoms. We write [8.3]
+
+$$
+\begin{array}{c c}\text {   Eight - node   solid   element   } \rightarrow\\\text {   Four - node   plane   element   } \rightarrow\\u = \sum N _ {i} u _ {i} + (1 - \xi^ {2}) a _ {1} + (1 - \eta^ {2}) a _ {2}&+ (1 - \zeta^ {2}) a _ {7}\\v = \sum N _ {i} v _ {i} + (1 - \xi^ {2}) a _ {3} + (1 - \eta^ {2}) a _ {4}&+ (1 - \zeta^ {2}) a _ {8}\\w = \sum N _ {i} w _ {i} + (1 - \xi^ {2}) a _ {5} + (1 - \eta^ {2}) a _ {6}&+ (1 - \zeta^ {2}) a _ {9}\end{array}\tag {8.3-6}
+$$
+
+where the $a_{i}$ are nodeless d.o.f. For the plane element, $i = 1, 2, 3, 4$ and the $N_{i}$ are given by Eq. 6.3-2. For the solid element, $i = 1, 2, \ldots, 8$ and the $N_{i}$ are given by Eq. 6.7-6. The plane element associated with Eqs. 8.3-6 is usually called the Q6 element. If rectangular, it models pure bending exactly regardless of element aspect ratio. In programming this element—for example, by modifying Figs. 6.5-1 and 6.5-2—one adds four columns to array B in order to accommodate the
+
+<!-- source-page: 254 -->
+
+four additional d.o.f., expands other arrays and loop indexes as required, but computes the Jacobian matrix as before (as though the element had only the basic nodal d.o.f.).
+
+The Q6 element is incompatible or nonconforming. For example, as suggested by Fig. 8.3-2, the mode $u = (1 - \eta^{2})a_{2}$ might be activated in one element but not in its neighbors to the left and right, thus producing a gap on one side and an overlap on the other. But incompatible elements are still valid if incompatibilities disappear and a constant-strain state is approached as the mesh is refined. That is, the element is valid if it passes the patch test. If an element of general shape is to pass the patch test, a modified integration scheme is needed; it will be described subsequently.
+
+Incompatible elements often yield results of high quality. Interelement gaps and overlaps tend to soften a structure. Softening counters the inherent overstiffness of an assumed-displacement approximation. A good balance of the two effects leads to good results with a coarse mesh. However, the upper-bound nature of the approximation is lost: there is no guarantee that a mesh of incompatible elements will be stiffer than the actual structure. Moreover, in problems that should be independent of Poisson's ratio $\nu$ , a coarse mesh of incompatible elements may display a dependence on $\nu$ .
+
+After formulation of element matrices, condensation removes the $a_{i}$ of Eq. 8.3-6. Thus, for the four-node plane element, $a_{1}$ through $a_{4}$ are eliminated, leaving an 8 by 8 condensed matrix [k]. For the special case of a rectangular element, as in Fig. 8.3-1, this condensed [k] is the same as the 8 by 8 [k] produced directly by the displacement field [8.4],
+
+$$
+\left\{ \begin{array}{l} u \\ v \end{array} \right\} = \left[ \begin{array}{c c c c c c c c} N _ {1} & N _ {x} & N _ {2} & - N _ {x} & N _ {3} & N _ {x} & N _ {4} & - N _ {x} \\ N _ {y} & N _ {1} & - N _ {y} & N _ {2} & N _ {y} & N _ {3} & - N _ {y} & N _ {4} \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \\ \vdots \\ u _ {4} \\ v _ {4} \end{array} \right\} \tag {8.3-7}
+$$
+
+where $N_{1}$ through $\dot{N}_{4}$ are given by Eqs. 6.3-2, and
+
+$$
+N _ {x} = (1 - \xi^ {2}) \nu \frac {a}{8 b} + (1 - \eta^ {2}) \frac {b}{8 a} \tag {8.3-8}
+$$
+
+$$
+N _ {y} = (1 - \xi^ {2}) \frac {a}{8 b} + (1 - \eta^ {2}) \nu \frac {b}{8 a}
+$$
+
+![](images/page-254_14e380d65c44644c669eab33e0adaac7a43f35e2c5a7abf11c89f55498bd3a80.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y, v
+η
+4
+3
+a₂
+ξ
+a₂
+1
+2
+x, u
+</details>
+
+![](images/page-254_cee72192b491d195443efa77789f0fc5c5ef4672aaf85e062c4b713adce2bbb0.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y, v
+a₃
+4
+η
+3
+ξ
+1
+2
+a₃
+x, u
+</details>
+
+Figure 8.3-2. Dashed lines show edge displacements associated with the incompatible modes $u = (1 - \eta^{2})a_{2}$ and $v = (1 - \xi^{2})a_{3}$ in the plane element described by Eqs. 8.3-6.
+
+<!-- source-page: 255 -->
+
+Yet a third way to obtain the same [k] (for a rectangular element) is to begin with the stress field
+
+$$
+\sigma_ {x} = \beta_ {1} + \beta_ {4} y \quad \sigma_ {y} = \beta_ {2} + \beta_ {5} x \quad \tau_ {x y} = \beta_ {3} \tag {8.3-9}
+$$
+
+where the $\beta_{i}$ are constants. Using these stress modes, one can invoke the “hybrid” method (Section 8.5), or one can compute the associated displacement field by integration and then proceed in the usual way. Indeed, the latter approach was used very early in finite element history [1.8; see also the discussion cited in Ref. 8.8].
+
+Modified Integration Scheme. The plane and solid elements described by Eq. 8.3-6 pass patch tests only if they are either rectangular or parallelograms and parallelepipeds. A modified integration scheme, here described, corrects this failing [8.3]. Thus modified, the plane element is known as the QM6 element.
+
+In the augmented strain-displacement relation $\{\epsilon\} = [\mathbf{B}]\{\mathbf{d}\}$ , let $[\mathbf{B}_a]$ represent the latter columns of [B], that is, the portion of [B] associated with nodeless d.o.f. $a_i$ . Hence, from Eq. 4.1-6, the contribution of the $a_i$ to the consistent element nodal load vector $\{\mathbf{r}_e\}$ is
+
+$$
+\left\{\mathbf {r} _ {e a} \right\} = - \int_ {V _ {c}} \left[ \mathbf {B} _ {a} \right] ^ {T} \left\{\boldsymbol {\sigma} _ {0} \right\} d V \tag {8.3-10}
+$$
+
+Imagine that instead of representing initial stresses, $\{\sigma_{0}\}$ represents element stresses produced by nodal displacements $\{d_{r}\}$ on the element boundary. The basic isoparametric element, with neither the $a_{i}$ nor $\{r_{ea}\}$ present, is able to pass a patch test. In other words, when $\{\sigma_{0}\}$ is constant and produced by the “essential” d.o.f. $\{d_{r}\}$ , certain “correct” nodal loads associated with $\{\sigma_{0}\}$ are applied by an element to its nodes. These loads should not be disturbed if incompatible modes are added. Accordingly, in a patch test no additional nodal loads should be associated with the $a_{i}$ . This means that $\{r_{ea}\}$ of Eq. 8.3-10 must vanish when $\{\sigma_{0}\}$ is constant. When $\{\sigma_{0}\}$ is constant, we see that $\{r_{ea}\}$ will be zero if, for plane and solid elements, respectively,
+
+$$
+\int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \left[ \mathbf {B} _ {a} \right] ^ {T} t J d \xi d \eta = 0 \quad \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \left[ \mathbf {B} _ {a} \right] ^ {T} J d \xi d \eta d \zeta = 0 \tag {8.3-11}
+$$
+
+where t is element thickness and J is the Jacobian determinant. For parallelograms of constant thickness and parallelepipeds, t and J are constant and $[B_{a}]$ contains first powers of $\xi$ and $\eta$ (and $\zeta$ for solids), so that Eq. 8.3-11 is satisfied automatically. For elements of general shape, t, J, and $[B_{a}]$ are more complicated functions; Eqs. 8.3-11 are not satisfied and the patch test is failed. But we can “artificially” satisfy Eqs. 8.3-11 as follows. In forming $[B_{a}]$ and integrating, instead of using the correct $[J]^{-1}$ and J at the Gauss quadrature points, use the constant values $[J_{0}]^{-1}$ and $J_{0}$ , where $[J_{0}]$ and $J_{0}$ are the Jacobian matrix and its determinant at $\xi = \eta = \zeta = 0$ . In addition, for plane elements, use $t_{0}$ rather than t if t varies. (If Figs. 6.5-1 and 6.5-2 are adapted to the QM6 element, these adjustments can all be confined to Fig. 6.5-1, in which only columns 9 through 12 of the augmented [B] are affected.)
+
+Elements Q6 and QM6 work almost as well as the quadratic element of Eqs.
+
+<!-- source-page: 256 -->
+
+![](images/page-256_7e99e8148aa15a73f67327989780fada1ec302f0db8d40a28ef4798afb162996.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y, v
+48
+y, v
+B
+C
+16
+A
+44
+Mesh N = 2
+Mesh N = 4
+x, u
+x, u
+</details>
+
+<table><tr><td></td><td>Mesh</td><td>N=2</td><td>N=4</td></tr><tr><td rowspan="3">QM6</td><td> $v_C$ </td><td>0.884</td><td>0.967</td></tr><tr><td> $\sigma_A$ </td><td>0.840</td><td>0.978</td></tr><tr><td> $\sigma_B$ </td><td>0.788</td><td>0.926</td></tr><tr><td rowspan="3">Bilinear</td><td> $v_C$ </td><td>0.498</td><td>0.769</td></tr><tr><td> $\sigma_A$ </td><td>0.558</td><td>0.830</td></tr><tr><td> $\sigma_B$ </td><td>0.457</td><td>0.753</td></tr></table>
+
+Figure 8.3-3. A plane structure with a uniformly distributed load along the right edge and with $E = 1.0$ , $\nu = 1/3$ . "Bilinear" refers to the element of Eqs. 6.3-2, and $v_{C} =$ deflection at $C$ , $\sigma_{A} =$ maximum stress at $A$ , $\sigma_{B} =$ minimum stress at $B$ , all reported as the ratio of computed value to best-known answer [8.3].
+
+6.6-1, but only if rectangular. Nevertheless, the QM6 element is considerably more accurate than the bilinear element, as seen in Fig. 8.3-3. (The same problem is solved, using different elements, in Fig. 8.6-2.)
+
+Stresses are calculated by using all element d.o.f., including the $a_{i}$ , to evaluate element strains. Calculated stresses are usually more accurate if element nodal loads in $\{r_{e}\}$ that are associated with incompatible modes are set to zero during recovery of the associated d.o.f. $a_{i}$ .
+
+# 8.4 ROTATIONAL D.O.F. IN PLANE ELEMENTS
+
+The d.o.f. considered are rotations at corner nodes. Such a d.o.f. may also be called a drilling freedom. Its vector representation is normal to the plane of the element. The usual translational d.o.f. at nodes are retained. Thus, a plane triangle with only corner nodes has nine d.o.f. [8.5].
+
+A good reason for use of drilling d.o.f. is found in the modeling of shells as an assembly of flat elements (discussed in detail in Section 12.3). Each element can model bending and stretching actions. Typically, in a general-purpose computer program, d.o.f. allowed at each structure node consist of three displacements and three rotations. Thus drilling d.o.f. are present among structural d.o.f. {D} whether or not they are present among element d.o.f. {d}. If flat shell elements connected to a certain node all happen to be coplanar, but elements do not include drilling d.o.f., then the drilling d.o.f. in {D} at that node is not resisted, and [K] is singular. This difficulty is neatly avoided by including drilling d.o.f. in {d}. Simultaneously, all of the six d.o.f. available at a node are exploited.
+
+In what follows we presume that drilling d.o.f. are associated with parabolic displaced shapes of element sides. In Fig. 8.4-1a, drilling d.o.f. $\omega_{i}$ and $\omega_{j}$ appear at nodes i and j of a typical element side of length L. At midside, $\omega_{i}$ and $\omega_{j}$ produce the edge-normal displacement $\delta$ :
+
+$$
+\delta = \frac {L}{8} \left(\omega_ {j} - \omega_ {i}\right) \tag {8.4-1}
+$$
+
+Thus, if $\omega_{i} = \omega_{j}$ , the edge remains straight. If $\omega_{i} \neq \omega_{j}$ , the side assumes a parabolic shape. If $\omega_{i} = -\omega_{j}$ , one can regard $\delta$ as the midspan deflection of a simply
+
+<!-- source-page: 257 -->
+
+![](images/page-257_7415196cbb66da1316cb2f8c46daf79f30dcc4196e656fd18a195b6fa5107cce.jpg)
+
+<details>
+<summary>text_image</summary>
+
+α
+ωj
+L/2
+δ
+α
+y,v
+L/2
+s
+ωi
+x,u
+</details>
+
+(a)
+
+![](images/page-257_981831ffaee1daf9a49f599bdd015a2a491c4c1ba83d4378e121050d24f54d27.jpg)
+
+<details>
+<summary>text_image</summary>
+
+v₂
+ω₂
+u₂
+2
+ω₃
+3
+u₃
+v₃
+1
+ω₁
+u₁
+v₁
+</details>
+
+(b)
+
+![](images/page-257_5f22ccd4b3c554d2829a4661384dfc36ea4fd2c602eb7d6a43794eee3ca4e639.jpg)
+
+<details>
+<summary>flowchart</summary>
+
+```mermaid
+graph TD
+    v2 --> u2
+    v2 --> v5
+    v2 --> 2
+    v5 --> u5
+    2 --> 5
+    5 --> 3
+    3 --> u3
+    3 --> v3
+    v3 --> u6
+    2 --> 4
+    4 --> u4
+    4 --> 6
+    6 --> 1
+    1 --> u1
+    1 --> v1
+    6 --> v6
+    v6 --> u6
+    6 --> v5
+    v5 --> u5
+    6 --> v4
+    v4 --> u4
+```
+</details>
+
+(c)   
+Figure 8.4-1. (a) Side displacement produced by drilling freedoms $\omega_{i}$ and $\omega_{j}$ . (b) A nine-d.o.f. plane triangle. (c) The linear-strain triangle.
+
+supported beam of length L under a pure bending moment that produces end rotations $|\omega_{i}| = |\omega_{j}|$ . Extending this beam analogy, imagine that supports of the beam have translational displacement components $u_{i}, v_{i}, u_{j}$ , and $v_{j}$ . Thus edge displacement is the sum of two parts: (a) a straight shape associated with nodal translational d.o.f., and (b) the parabolic shape associated with $\delta$ . The x and y components of $\delta$ are $\delta \cos \alpha$ and $\delta \sin \alpha$ . Therefore, the total displacements u and v of a typical point on the edge are
+
+$$
+\left\{ \begin{array}{l} u \\ v \end{array} \right\} = \frac {L - s}{L} \left\{ \begin{array}{l} u _ {i} \\ v _ {i} \end{array} \right\} + \frac {s}{L} \left\{ \begin{array}{l} u _ {j} \\ v _ {j} \end{array} \right\} + \frac {(L - s) s}{2 L} \left(\omega_ {j} - \omega_ {i}\right) \left\{ \begin{array}{l} \cos \alpha \\ \sin \alpha \end{array} \right\} \tag {8.4-2}
+$$
+
+Similar expressions may be written for all other sides of the element. Hence, one can devise shape functions $N_{i}$ for a triangle, a quadrilateral, and so on. In the triangle of Fig. 8.4-1b, contributions to u and v from nodal translations are interpolated linearly over the element, and contributions associated with drilling d.o.f. are interpolated using products of area coordinates, such as $\xi_{1}\xi_{2}$ for $\delta$ on side 1–2. Finally the usual process of element formulation is pursued (Eqs. 4.1-5 and 4.1-6).
+
+One can also begin with an element that has straight sides and midside nodes, and convert it to an element that has corner nodes only, each with two translational d.o.f. and one drilling freedom. The conversion process relates midside $\delta$ 's to nodal $\omega$ 's and constrains each edge-tangent displacement component to vary linearly with edge-tangent coordinate s. The element may have any number of sides. As applied to a linear-strain triangle, Fig. 8.4-1c, the transformation procedure is outlined as follows.
+
+Consider, for example, side 1–4–2 of the linear-strain triangle, Fig. 8.4-1c. D.o.f. at node 4 are related to d.o.f. at nodes 1 and 2 of the new element, Fig. 8.4-1b, by evaluating Eq. 8.4-2 with s = L/2. With i = 1, j = 2, L cos $\alpha = y_{2} - y_{1}$ , and L sin $\alpha = x_{1} - x_{2}$ , Eq. 8.4-2 yields
+
+$$
+\left\{ \begin{array}{l} u _ {4} \\ v _ {4} \end{array} \right\} = \frac {1}{2} \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \end{array} \right\} + \frac {1}{2} \left\{ \begin{array}{l} u _ {2} \\ v _ {2} \end{array} \right\} + \frac {\omega_ {2} - \omega_ {1}}{8} \left\{ \begin{array}{l} y _ {2} - y _ {1} \\ x _ {1} - x _ {2} \end{array} \right\} \tag {8.4-3}
+$$
+
+<!-- source-page: 258 -->
+
+After doing the same for d.o.f. at nodes 5 and 6, we can relate d.o.f. in Figs. 8.4-1b and 8.4-1c by the transformation
+
+$$
+\left[ u _ {1} v _ {1} u _ {2} \dots u _ {6} v _ {6} \right] ^ {T} = \left[ \mathrm{T} \right] _ {1 2 \times 9} \left[ u _ {1} v _ {1} \omega_ {1} u _ {2} v _ {2} \omega_ {2} u _ {3} v _ {3} \omega_ {3} \right] ^ {T} \tag {8.4-4}
+$$
+
+If desired, we can obtain the new element by transforming the stiffness matrix $[k']$ of the linear-strain triangle:
+
+$$
+\begin{array}{l} \left[ \mathbf {k} \right] = [ \mathbf {T} ] ^ {T} \left[ \mathbf {k} ^ {\prime} \right] [ \mathbf {T} ] \\ 9 \times 9 \end{array} \tag {8.4-5}
+$$
+
+More efficient coding will result if $[k]$ is formulated directly from shape functions $[N]$ appropriate to the nine-d.o.f. element. These may be obtained by transforming shape functions $[N']$ of the linear-strain triangle:
+
+$$
+\left\{ \begin{array}{l} u \\ v \end{array} \right\} = \left[ \begin{array}{l} \mathbf {N} ^ {\prime} \\ 2 \times 1 2 \end{array} \right] \left\{ \begin{array}{l} \mathbf {d} ^ {\prime} \\ 1 2 \times 1 \end{array} \right\} = \left[ \begin{array}{l} \mathbf {N} ^ {\prime} \end{array} \right] \left[ \begin{array}{l} \mathbf {T} \end{array} \right] \left\{ \begin{array}{l} \mathbf {d} \\ 9 \times 1 \end{array} \right\} = \left[ \begin{array}{l} \mathbf {N} \\ 2 \times 9 \end{array} \right] \left\{ \begin{array}{l} \mathbf {d} \end{array} \right\} \tag {8.4-6}
+$$
+
+Shape functions $N_{i}$ in Eq. 8.4-6 agree with those obtained by the procedure described below Eq. 8.4-2.
+
+Deformations are everywhere zero if all nodal rotations in the mesh have the same value. This mechanism can be suppressed by prescribing the value of one nodal rotation in the mesh, for example, by setting $\omega_{1} = 0$ .
+
+The true rotation at a node i is defined as
+
+$$
+\theta_ {i} = \frac {1}{2} (v _ {, x} - u _ {, y}) _ {i} \tag {8.4-7}
+$$
+
+For rigid-body rotation in the plane of the element, $\theta_{i} = \omega_{i}$ . Otherwise, equality may not prevail [8.5]. Interelement continuity of true rotations is not in general provided, nor is it necessary for proper convergence of the finite element solution.
+
+A plane quadrilateral with drilling d.o.f. is described in [8.37]. It incorporates various improvements, including a device to control the aforementioned mechanism. This device is summarized as follows. Let the strain energy in each element be augmented by $U_{\omega}$ ,
+
+$$
+U _ {\omega} ^ {\prime} = \frac {1}{2} V _ {e} G (2 \gamma) \left(\theta_ {0} - \frac {1}{n} \sum_ {i = 1} ^ {n} \omega_ {i}\right) ^ {2} \tag {8.4-8}
+$$
+
+where $V_{e} = \text{element volume}$ , G = shear modulus, $\gamma = \text{dimensionless constant}$ ( $\gamma = 10^{-6}$ is recommended), $\theta_{0} = (v_{,x} - u_{,y})_{0}/2$ is the rotation at the element center, and n = number of nodes where rotational d.o.f. $\omega_{i}$ are used. By inserting shape functions, we can write the parenthetic expression in Eq. 8.4-8 in matrix format; that is, $(\cdots) = \lfloor Q \rfloor \{d\}$ , where $\lfloor Q \rfloor$ is a row matrix. Hence
+
+$$
+U _ {\omega} = \frac {1}{2} \left\{\mathbf {d} \right\} ^ {T} \left[ \mathbf {k} _ {\omega} \right] \left\{\mathbf {d} \right\} \quad \text {   in   which   } \quad \left[ \mathbf {k} _ {\omega} \right] = 2 \gamma V _ {e} G \left[ \mathbf {Q} \right] ^ {T} \left[ \mathbf {Q} \right] \tag {8.4-9}
+$$
+
+where $[k_{\omega}]$ is a rank 1 “stabilization matrix” that is added to the existing element stiffness matrix. Matrix $[k_{\omega}]$ has no effect on the ability of the element to represent constant-strain states and rigid-body modes.
+
+Numerical examples are reported in Fig. 8.6-2 and in Refs. 8.5, 8.6, and 8.37.
+
+<!-- source-page: 259 -->
+
+# 8.5 ASSUMED-STRESS HYBRID FORMULATION
+
+The assumed-stress hybrid method is a way of formulating a stiffness matrix by use of independent assumptions of (a) an equilibrium stress field within the element, and (b) interelement-compatible displacement modes on the element boundary. Mathematically, the method can be stated as a modified complementary energy principle. The principle of stationary complementary energy states that: among all stress fields that satisfy the differential equations of equilibrium, the stress field that also satisfies compatibility conditions makes the complementary energy stationary with respect to small variations of stress. For a linearly elastic material, strain energy per unit volume can be written as
+
+$$
+\begin{array}{l} U _ {0} = \frac {1}{2} \{\boldsymbol {\epsilon} \} ^ {T} [ \mathrm{E} ] \{\boldsymbol {\epsilon} \} \quad \text { or   as } \quad U _ {0} = \frac {1}{2} \{\boldsymbol {\sigma} \} ^ {T} [ \mathrm{E} ] ^ {- 1} \{\boldsymbol {\sigma} \} \tag {8.5-1} \\ (\text { potential   energy }) \quad \text {(complementary   energy)} \\ \end{array}
+$$
+
+Starting with an expression for $U_{0}$ , one can write various functionals. The functional for potential energy is $\Pi_{p}$ , Eq. 4.1-1, which yields the stiffness matrix of an element based on an assumed displacement field. Analogously, one can write a complementary energy functional that yields the stiffness matrix of an assumed-stress hybrid element [8.7]. Rather than discuss the functional, we consider the following more direct method, which is the method by which assumed-stress hybrid elements were first derived [8.8]. Although the hybrid method is general, the discussion that follows is oriented toward plane problems without body forces.
+
+One begins with a stress field that satisfies the differential equations of equilibrium, Eqs. 1.6-2 and 1.6-4. Symbolically,
+
+$$
+\{\sigma \} = [ \mathbf {P} ] \{\beta \} \tag {8.5-2}
+$$
+
+where, for plane problems, $\{\sigma\} = \left[\sigma_x - \sigma_y - \tau_{xy}\right]^T$ , and $\{\beta\}$ contains constants $\beta_i$ that are yet to be determined. Equations 8.3-9 are a $5-\beta$ example of such an equilibrium stress field. From Eqs. 8.5-1 and 8.5-2, the complementary strain energy in an element of volume $V_e$ is
+
+$$
+U = \int_ {V _ {e}} U _ {0} d V = \frac {1}{2} \{\boldsymbol {\beta} \} ^ {T} [ \mathbf {H} ] \{\boldsymbol {\beta} \} \tag {8.5-3}
+$$
+
+where
+
+$$
+[ \mathbf {H} ] = \int_ {V _ {e}} [ \mathbf {P} ] ^ {T} [ \mathbf {E} ] ^ {- 1} [ \mathbf {P} ] d V \tag {8.5-4}
+$$
+
+Let $\{\Phi\}$ represent tractions at the element boundary $S_{e}$ , obtained by evaluating Eq. 8.5-2 on the boundary. Also let boundary displacements $\{\mathbf{u}_{b}\}$ be interpolated from nodal d.o.f. $\{\mathbf{d}\}$ . (An example will follow.) Thus
+
+$$
+\{\Phi \} = [ \mathbf {R} ] \{\beta \} \quad \text { and } \quad \{\mathbf {u} _ {b} \} = [ \mathbf {L} ] \{\mathbf {d} \} \tag {8.5-5}
+$$
+
+The total complementary energy in the element is U minus work done by tractions $\{\Phi\}$ in moving through displacements $\{u_{b}\}$ ; that is,
+
+<!-- source-page: 260 -->
+
+$$
+\Pi_ {c} = U - \int_ {S _ {e}} \{\Phi \} ^ {T} \{\mathbf {u} _ {b} \} d S = \frac {1}{2} \{\boldsymbol {\beta} \} ^ {T} [ \mathbf {H} ] \{\boldsymbol {\beta} \} - \{\boldsymbol {\beta} \} ^ {T} [ \mathbf {G} ] \{\mathbf {d} \} \tag {8.5-6}
+$$
+
+where
+
+$$
+[ \mathbf {G} ] = \int_ {S _ {e}} [ \mathbf {R} ] ^ {T} [ \mathbf {L} ] d S \tag {8.5-7}
+$$
+
+Making $\Pi_{c}$ stationary with respect to variations of stress, we write
+
+$$
+\frac {\partial \Pi_ {c}}{\partial \beta_ {i}} = 0 \quad \text { for } \quad i = 1, 2, \dots , n \quad \text { or } \quad \left\{\frac {\partial \Pi_ {c}}{\partial \beta} \right\} = \{0 \} \tag {8.5-8}
+$$
+
+from which
+
+$$
+[ \mathbf {H} ] \{\boldsymbol {\beta} \} = [ \mathbf {G} ] \{\mathbf {d} \} \quad \text { or } \quad \{\boldsymbol {\beta} \} = [ \mathbf {H} ] ^ {- 1} [ \mathbf {G} ] \{\mathbf {d} \} \tag {8.5-9}
+$$
+
+At this point one can say that we have asked for the stress field within an element when displacements on its boundary are prescribed, and answered by finding values of $\{\beta\}$ that define the best stress field $\{\sigma\}$ that is contained in the approximation $\{\sigma\} = [P]\{\beta\}$ . Substitution of $\{\beta\}$ from Eq. 8.5-9 into Eq. 8.5-3 yields
+
+$$
+U = \frac {1}{2} \{\mathbf {d} \} ^ {T} [ \mathbf {k} ] \{\mathbf {d} \}, \quad \text { where } \quad [ \mathbf {k} ] = [ \mathbf {G} ] ^ {T} [ \mathbf {H} ] ^ {- 1} [ \mathbf {G} ] \tag {8.5-10}
+$$
+
+in which $[k]$ is recognized as a stiffness matrix because its form matches that of Eq. 4.1-10.
+
+Example. Consider a typical straight edge $ij$ of a plane element, Fig. 8.5-1a. Tractions $\Phi_x$ and $\Phi_y$ are related to stresses $\sigma_x$ , $\sigma_y$ , and $\tau_{xy}$ by direction cosines $\ell$ and $m$ of the outward normal to the edge. From Eq. 1.6-3,
+
+$$
+\begin{array}{l} \Phi_ {x} = \ell \sigma_ {x} + m \tau_ {x y} \quad \text { where } \quad \ell = \cos \alpha = (y _ {j} - y _ {i}) / L _ {i j} \tag {8.5-11} \\ \Phi_ {y} = \ell \tau_ {x y} + m \sigma_ {y} \quad \text { where } \quad m = \sin \alpha = (x _ {i} - x _ {j}) / L _ {i j} \\ \end{array}
+$$
+
+For the particular case of a 5- $\beta$ rectangular element, Eqs. 8.3-9 and Fig. 8.5-1b, arrays [P] and [R] are
+
+$$
+[ \mathbf {P} ] = \left[ \begin{array}{l l l l l} 1 & 0 & 0 & y & 0 \\ 0 & 1 & 0 & 0 & x \\ 0 & 0 & 1 & 0 & 0 \end{array} \right] \quad \text { and } \quad [ \mathbf {R} ] = \left[ \begin{array}{l l l l l} 0 & 0 & - 1 & 0 & 0 \\ 0 & - 1 & 0 & 0 & - x \\ 1 & 0 & 0 & y & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \cdot & x \\ - 1 & 0 & 0 & - y & 0 \\ 0 & 0 & - 1 & 0 & 0 \end{array} \right] \tag {8.5-12}
+$$
+
+where [R] is obtained from [P] and Eq. 8.5-11. The first two rows of [R] pertain to $\Phi_x$ and $\Phi_y$ along side 1-2 (where $\ell = 0$ and $m = -1$ ), the third and fourth rows pertain to $\Phi_x$ and $\Phi_y$ along side 2-3 (where $\ell = 1$ and $m = 0$ ), and so on. Matrix [L] of Eq. 8.5-5 is 8 by 8 for the element of Fig. 8.5-1b, and relates $u$ and $v$ displacement components along all four edges to nodal d.o.f. {d}. That is,
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+<!-- source-page: 261 -->
+
+![](images/page-261_3c6c04802ce3c41a73718cf80ad6ba51b4748cb65991750bfe075837aac66803.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+α
+j
+Φy
+σx ← τxy → Φx
+σy
+Lij
+i
+s
+α
+x,u
+</details>
+
+(a)
+
+![](images/page-261_43022d31d3ec4f4b25bd9411f9ba2adf68f1d9a4ffa1577a3c2b03517174ea66.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+a
+a
+4
+3
+b
+x,u
+b
+1
+2
+</details>
+
+(b)   
+Figure 8.5-1. (a) Stresses and surface tractions at a typical edge ij. (b) A rectangular element.
+
+$$
+\left[ \begin{array}{l l l l l} u _ {1 2} & v _ {1 2} & u _ {2 3} \cdot \cdot \cdot u _ {4 1} & v _ {4 1} \end{array} \right] ^ {T} = [ \mathrm{L} ] \left[ \begin{array}{l l l l} u _ {1} & v _ {1} \cdot \cdot \cdot u _ {4} & v _ {4} \end{array} \right] ^ {T} \tag {8.5-13}
+$$
+
+If $\mu$ and $\nu$ along an edge are linearly interpolated from nodal d.o.f. on that edge, the rows of [L] are
+
+$$
+\text { row   1: } \quad \frac {a - x}{2 a}, 0, \frac {a + x}{2 a}, 0, 0, 0, 0, 0 \tag {8.5-14}
+$$
+
+$$
+\text { row   2: } \quad 0, \frac {a - x}{2 a}, 0, \frac {a + x}{2 a}, 0, 0, 0, 0
+$$
+
+and so on. All ingredients are now at hand: the element stiffness matrix is obtained by application of Eqs. 8.5-4, 8.5-7, and 8.5-10. In integration of Eq. 8.5-4, $dV$ becomes $t \, dA = t \, dx \, dy$ , where $t$ is the element thickness. In integration of Eq. 8.5-7, $dS = t \, ds$ , where $ds = dx$ or $ds = dy$ for sides parallel to $x$ and $y$ axes, respectively. Terms that contain $x$ are integrated from $-a$ to $+a$ and terms that contain $y$ are integrated from $-b$ to $+b$ .
+
+Remarks. After $\{d\}$ is known, element stresses are recovered by use of Eqs. 8.5-2 and 8.5-9. Thus
+
+$$
+\{\sigma \} = [ \mathrm{P} ] [ \mathrm{H} ] ^ {- 1} [ \mathrm{G} ] \{\mathrm{d} \} \tag {8.5-15}
+$$
+
+Assumed-stress hybrid elements can be joined to displacement-based elements because both use displacement quantities as nodal d.o.f. The user of a computer program may be unaware that some of its elements are hybrid elements.
+
+Assumed-stress hybrid elements become stiffer as $\{\beta\}$ grows—that is, as more terms are added to the stress expansion. They usually become more flexible as element edges are permitted more complicated displacement patterns. No bound can be set: we cannot say in general that a mesh of hybrid elements will be too stiff or too flexible. If $\{\sigma\} = [P]\{\beta\}$ is not a complete polynomial, the element will not be geometrically isotropic. For example, the stress field described by [P] of Eq. 8.5-12 is not complete. A complete linear stress field in the plane is
+
+<!-- source-page: 262 -->
+
+$$
+\left\{ \begin{array}{l} \sigma_ {x} \\ \sigma_ {y} \\ \tau_ {x y} \end{array} \right\} = \left[ \begin{array}{c c c c c c c} 1 & 0 & 0 & y & 0 & x & 0 \\ 0 & 1 & 0 & 0 & x & 0 & y \\ 0 & 0 & 1 & 0 & 0 & - y & - x \end{array} \right] \left\{ \begin{array}{l} \beta_ {1} \\ \beta_ {2} \\ \vdots \\ \beta_ {7} \end{array} \right\} \tag {8.5-16}
+$$
+
+This field contains seven $\beta_{i}$ rather than nine in order to satisfy the differential equations of equilibrium. If used with linear edge displacements (e.g., Eq. 8.5-14), the $7-\beta$ element has eight displacement d.o.f. but is stiffer than a $5-\beta$ element having the same eight displacement d.o.f.
+
+It is possible to relax the requirement that stresses satisfy the differential equations of equilibrium a priori. Procedures suggested by Wolf [8.9] and Pian [8.10] use a functional in which displacements within the element act as Lagrange multipliers of the differential equations of equilibrium. Equilibrium is then satisfied in an average sense rather than at every point.
+
+# 8.6 A PLANE HYBRID TRIANGLE WITH ROTATIONAL D.O.F.
+
+Hybrid element theory (Section 8.5) and “drilling” d.o.f. (Section 8.4) are combined in the element here described [8.11]. It is a triangle built of three subtriangles. The final macroelement is a nine-d.o.f. triangle (Fig. 8.6-1). The same element can be obtained by standard displacement theory, as explained in the following.
+
+Basic Triangular Subelement. Stresses are assumed constant: $\sigma_{x} = \beta_{1}$ , $\sigma_{y} = \beta_{2}$ , and $\tau_{xy} = \beta_{3}$ . Thus [P] is a unit matrix, and Eq. 8.5-4 yields
+
+$$
+\left[ \mathbf {H} \right] _ {3 \times 3} = A t [ \mathbf {E} ] ^ {- 1} \quad \text { and } \quad \left[ \mathbf {H} \right] ^ {- 1} = \frac {1}{A t} [ \mathbf {E} ] \tag {8.6-1}
+$$
+
+where $A =$ subelement area and $t =$ subelement thickness.
+
+where $A =$ subcontinent area and $\mathbf{A} = \mathbf{A}_0$ . Matrix [R] is 6 by 3 and is formed from boundary tractions $\Phi_x = \ell \beta_1 + m\beta_3$ and $\Phi_y = \ell \beta_3 + m\beta_2$ . Direction cosines $\ell$ and $m$ are given by Eq. 8.5-11. For
+
+![](images/page-262_86d66a92dd6e8061e389096936e4dd33ede70497dbb5ad0cb6310c737d785d72.jpg)  
+(a)   
+Figure 8.6-1. (a) Triangular macroelement built of three subtriangles. D.o.f. of a typical node i are shown. (b) Node labels for the subtriangles.
+
+<!-- source-page: 263 -->
+
+example, terms in the first two rows of [R] pertain to side 1–2 of the subelement and are
+
+$$
+\text { row   1: } \quad (y _ {2} - y _ {1}) / L _ {1 2}, 0, (x _ {1} - x _ {2}) / L _ {1 2}
+$$
+
+$$
+\text { row   2: } \quad 0, (x _ {1} - x _ {2}) / L _ {1 2}, (y _ {2} - y _ {1}) / L _ {1 2} \tag {8.6-2}
+$$
+
+Boundary displacements and nodal d.o.f. are related via matrix [L]. The relationship resembles Eq. 8.5-13, but involves one less side, one less node, and three d.o.f. per node. Rows of [L] are written by application of Eq. 8.4-2. For example, the first row of [L] expresses the relation
+
+$$
+u _ {1 2} = \frac {L _ {1 2} - s _ {1 2}}{L _ {1 2}} u _ {1} + \frac {s _ {1 2}}{L _ {1 2}} u _ {2} + \frac {y _ {2} - y _ {1}}{L _ {1 2}} \frac {(L _ {1 2} - s _ {1 2}) s _ {1 2}}{2 L _ {1 2}} (\omega_ {2} - \omega_ {1}) \tag {8.6-3}
+$$
+
+where $s_{12}$ is an edge-tangent coordinate, Fig. 8.6-1b.
+
+When matrix [G] of Eq. 8.5-7 is computed, $dS = t ds_{ij}$ and limits of integration are from 0 to $L_{ij}$ , where $L_{ij}$ is $L_{12}$ , $L_{23}$ , or $L_{31}$ for the respective element sides. Let nodal d.o.f. have the ordering $\{\mathbf{d}\} = \left[u_1 \quad v_1 \quad \omega_1 \quad u_2 \quad v_2 \quad \omega_2 \quad u_3 \quad v_3 \quad \omega_3\right]^T$ . Then, in partitioned form, [G] is
+
+$$
+\left[ \begin{array}{l} \mathbf {G} \\ 3 \times 9 \end{array} \right] = \left[ \begin{array}{l l l} \mathbf {G} _ {A} & \mathbf {G} _ {B} & \mathbf {G} _ {C} \end{array} \right] \tag {8.6-4}
+$$
+
+where, with $y_{ij} = y_i - y_j$ and $x_{ij} = x_i - x_j$ ,
+
+$$
+\left[ \mathrm{G} _ {A} \right] = \frac {t}{1 2} \left[ \begin{array}{c c c} 6 y _ {2 3} & 0 & y _ {3 1} ^ {2} - y _ {1 2} ^ {2} \\ 0 & 6 x _ {3 2} & x _ {1 3} ^ {2} - x _ {2 1} ^ {2} \\ 6 x _ {3 2} & 6 y _ {2 3} & 2 \left(x _ {1 3} y _ {3 1} - x _ {2 1} y _ {1 2}\right) \end{array} \right] \tag {8.6-5}
+$$
+
+Submatrices $[G_{B}]$ and $[G_{C}]$ are obtained from $[G_{A}]$ by advancing the subscripts by 1 and 2 respectively around the loop 1–2–3; for example, $x_{32}$ in $[G_{A}]$ becomes $x_{13}$ in $[G_{B}]$ and $x_{21}$ in $[G_{C}]$ .
+
+The foregoing triangular subelement can also be generated from an assumed displacement field: one uses the 2 by 9 matrix [N] of Eq. 8.4-6, and integrates $[B]^{T}[E][B]$ by using a single Gauss point at the centroid of the element (or subelement, in the present context).
+
+Macroelement. The composite element—that is, the macroelement, Fig. 8.6-1a—is formed by combining three of the foregoing subelements. Node 4 is at the centroid of the macroelement, that is, at
+
+$$
+x _ {4} = \frac {1}{3} (x _ {1} + x _ {2} + x _ {3}) \quad \text { and } \quad y _ {4} = \frac {1}{3} (y _ {1} + y _ {2} + y _ {3}) \tag {8.6-6}
+$$
+
+D.o.f. at node 4 can be eliminated by static condensation, leaving a three-node element having nine d.o.f.
+
+However, such an element is a bit too flexible in many problems. Accordingly, prior to condensation, the rotational d.o.f. at node 4 is constrained to be the average of rotational d.o.f. at nodes 1, 2, and 3. This is accomplished by first transforming the 12 by 12 matrix [k] so that it operates on nodal d.o.f. {d} in which $\omega_{4r}$ replaces $\omega_{4}$ , where
+
+<!-- source-page: 264 -->
+
+![](images/page-264_2cae6b9bd448d5e60fa1d415da3280cb3213864fc84f3e8302ecc46349de88f5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+48
+B
+16
+C
+A
+44
+Mesh N = 2
+x,u
+y,v
+B
+C
+A
+Mesh N = 4
+x,u
+</details>
+
+<table><tr><td></td><td>Mesh</td><td>N=2</td><td>N=4</td></tr><tr><td rowspan="3">Equation8.4-5</td><td> $v_{C}$ </td><td>0.840</td><td>0.950</td></tr><tr><td> $\sigma_{A}$ </td><td>0.682</td><td>0.878</td></tr><tr><td> $\sigma_{B}$ </td><td>0.730</td><td>0.856</td></tr><tr><td rowspan="3">Figure8.6-1</td><td> $v_{C}$ </td><td>0.900</td><td>0.969</td></tr><tr><td> $\sigma_{A}$ </td><td>0.746</td><td>0.909</td></tr><tr><td> $\sigma_{B}$ </td><td>0.917</td><td>0.882</td></tr></table>
+
+Figure 8.6-2. A plane structure with a uniformly distributed load along the right edge and with $E = 1.0$ , $\nu = 1/3$ . Results displayed are obtained from elements that have drilling d.o.f. “Equation 8.4-5” refers to the nine-d.o.f. displacement-based element obtainable from the linear-strain triangle [8.5]. “Figure 8.6-1” refers to the composite element with constant-stress subtriangles and $\omega_4 = (\omega_1 + \omega_2 + \omega_3)/3$ . Here $v_C =$ deflection at $C$ , $\sigma_A =$ maximum stress at $A$ , $\sigma_B =$ minimum stress at $B$ , all reported as the ratio of computed value to best-known answer.
+
+$$
+\omega_ {4 r} = \omega_ {4} - \frac {1}{3} (\omega_ {1} + \omega_ {2} + \omega_ {3}) \tag {8.6-7}
+$$
+
+Here $\omega_{4r}$ is the rotational d.o.f. at node 4 relative to the average of the three vertex rotational d.o.f. The constraint is $\omega_{4r} = 0$ , which is enforced by striking out the row and column of the transformed [k] associated with $\omega_{4r}$ . This leaves an 11 by 11 stiffness matrix, from which $u_{4}$ and $v_{4}$ are eliminated by static condensation.
+
+The resulting nine-d.o.f. element is geometrically isotropic and has rank 5, corresponding to three rigid-body motions and the mechanism in which all nodal rotations are the same. The patch test is passed. Numerical results are reported as the “Fig. 8.6-1” entries in Fig. 8.6-2. The same problem is solved, using different elements, in Fig. 8.3-3.
+
+# 8.7 USER-DEFINED ELEMENTS. ELASTIC KERNEL
+
+The user of a computer program may wish to supply an element of a type not provided in the program. If all stiffness coefficients $k_{ij}$ of the new element are supplied directly, rather than as the output of a tested algorithm, there is substantial risk of introducing numerical error. In the following we explain a procedure whereby only some of the $k_{ij}$ need be supplied, and we comment on why this procedure is effective in avoiding the possible error.
+
+First, the element stiffness equation $[\mathbf{k}]\{\mathbf{d}\} = \{\mathbf{r}\}$ is partitioned,
+
+$$
+\left[ \begin{array}{l l} \mathbf {k} _ {R R} & \mathbf {k} _ {R E} \\ \mathbf {k} _ {R E} ^ {T} & \mathbf {k} _ {E E} \end{array} \right] \left\{ \begin{array}{l} \mathbf {d} _ {R} \\ \mathbf {d} _ {E} \end{array} \right\} = \left\{ \begin{array}{l} \mathbf {r} _ {R} \\ \mathbf {r} _ {E} \end{array} \right\} \tag {8.7-1}
+$$
+
+in which d.o.f. $\{d_{R}\}$ are used to define rigid-body motion of the element and d.o.f. $\{d_{E}\}$ are used to define straining modes (an example follows). For any element, there is more than one way to partition $\{d\}$ into $\{d_{R}\}$ and $\{d_{E}\}$ . We wish to describe how the stiffness matrix $[k_{EE}]$ of an element that is fully (but not redundantly) restrained from rigid-body motion ( $\{d_{R}\} = \{0\}$ ) can be converted to a complete stiffness matrix [k], ready for assembly into the structure.
+
+<!-- source-page: 265 -->
+
+The lower partition of Eq. 8.7-1 is solved for $\{d_{E}\}$ . Also, this expression for $\{d_{E}\}$ is substituted into the upper partition. Thus
+
+$$
+\left[ \begin{array}{c c} \left(\mathbf {k} _ {R R} - \mathbf {k} _ {R E} \mathbf {k} _ {E E} ^ {- 1} \mathbf {k} _ {R E} ^ {T}\right) & \mathbf {k} _ {R E} \mathbf {k} _ {E E} ^ {- 1} \\ - \mathbf {k} _ {E E} ^ {- 1} \mathbf {k} _ {R E} ^ {T} & \mathbf {k} _ {E E} ^ {- 1} \end{array} \right] \left\{ \begin{array}{l} \mathbf {d} _ {R} \\ \mathbf {r} _ {E} \end{array} \right\} = \left\{ \begin{array}{l} \mathbf {r} _ {R} \\ \mathbf {d} _ {E} \end{array} \right\} \tag {8.7-2}
+$$
+
+Matrix $[\mathbf{k}_{EE}]$ is symmetric and is invertible because rigid-body motion is prevented. If there is no elastic distortion, then $\{\mathbf{r}_E\} = \{\mathbf{0}\}$ and, therefore,
+
+$$
+- \left[ \mathbf {k} _ {E E} \right] ^ {- 1} \left[ \mathbf {k} _ {R E} \right] ^ {T} \left\{\mathbf {d} _ {R} \right\} = \left\{\mathbf {d} _ {E} \right\} \tag {8.7-3}
+$$
+
+With $\{r_{E}\}=\{0\}$ , only rigid-body motion is possible. We also know that in rigid-body motion nodal d.o.f. are related strictly by kinematics, expressed by a matrix [T] of element dimensions:
+
+$$
+\{\mathbf {d} _ {E} \} = [ \mathbf {T} ] \{\mathbf {d} _ {R} \} \tag {8.7-4}
+$$
+
+Typically $\{d_{E}\}$ contains more d.o.f. than $\{d_{R}\}$ , in which case [T] contains more rows than columns. Because Eqs. 8.7-3 and 8.7-4 must be true for any $\{d_{R}\}$ , we thus conclude, by comparison, that
+
+$$
+[ \mathbf {k} _ {R E} ] ^ {T} = - [ \mathbf {k} _ {E E} ] [ \mathbf {T} ] \tag {8.7-5}
+$$
+
+Next imagine that $\{\mathbf{r}_E\} \neq \{\mathbf{0}\}$ . Then, because $\{\mathbf{d}_R\}$ contains only enough d.o.f. to prevent rigid-body motion, $\{\mathbf{r}_R\}$ can be computed from $\{\mathbf{r}_E\}$ entirely by equations of statics, independently of $\{\mathbf{d}_R\}$ . Accordingly, the coefficient of $\{\mathbf{d}_R\}$ in the upper partition of Eq. 8.7-2 must vanish. From this and Eq. 8.7-5 we obtain
+
+$$
+[ \mathbf {k} _ {R R} ] = [ \mathbf {k} _ {R E} ] [ \mathbf {k} _ {E E} ] ^ {- 1} [ \mathbf {k} _ {R E} ] ^ {T} = [ \mathbf {T} ] ^ {T} [ \mathbf {k} _ {E E} ] [ \mathbf {T} ] \tag {8.7-6}
+$$
+
+The stiffness matrix that operates on all nodal d.o.f., $\{\mathbf{d}\} = \lfloor \mathbf{d}_R - \mathbf{d}_E\rfloor^T$ , is therefore
+
+$$
+[ \mathbf {k} ] = \left[ \begin{array}{c c} \mathbf {T} ^ {T} \mathbf {k} _ {E E} \mathbf {T} & - \mathbf {T} ^ {T} \mathbf {k} _ {E E} \\ - \mathbf {k} _ {E E} \mathbf {T} & \mathbf {k} _ {E E} \end{array} \right] \tag {8.7-7}
+$$
+
+Matrix $[\mathbf{k}_{EE}]$ is called the elastic kernel.
+
+The user must supply $[k_{EE}]$ (or supply and then invert the flexibility matrix $[k_{EE}]^{-1}$ ) and [T] to the computer program. Equation 8.7-7 then produces a [k] that requires no force to produce rigid-body motion. If the user were required to prescribe the entire [k], the individual $k_{ij}$ would have to be of full computer-word accuracy to avoid the possibility of introducing serious errors (Section 18.2). By use of Eq. 8.7-7, slight errors in the $(k_{EE})_{ij}$ produce only slight defects in elastic response; they do not cause rigid-body motion to be misrepresented.
+
+Example. Consider the standard beam element of Fig. 4.2-2. Let the element be fixed at its left end, so that $\{\mathbf{d}_R\} = \lfloor w_1 - \theta_1 \rfloor^T$ . Then $\{\mathbf{d}_E\} = \lfloor w_2 - \theta_2 \rfloor^T$ . From Eq. 4.2-4, the curvature
+
+$$
+w _ {, x x} = \left\lfloor \frac {6}{L ^ {2}} - \frac {1 2 x}{L ^ {3}} \quad - \frac {2}{L} + \frac {6 x}{L ^ {2}} \right\rfloor \left\{\mathbf {d} _ {E} \right\} \tag {8.7-8}
+$$
+
+<!-- source-page: 266 -->
+
+is used to construct $[\mathbf{k}_{EE}]$ , which is found to be the lower right 2 by 2 submatrix in Eq. 4.2-5. Equation 8.7-4 becomes
+
+$$
+\left\{ \begin{array}{l} w _ {2} \\ \theta_ {2} \end{array} \right\} = [ \mathrm{T} ] \left\{ \begin{array}{l} w _ {1} \\ \theta_ {1} \end{array} \right\}, \quad \text { where } \quad [ \mathrm{T} ] = \left[ \begin{array}{l l} 1 & L \\ 0 & 1 \end{array} \right] \tag {8.7-9}
+$$
+
+Equation 8.7-7 then yields [k] of Eq. 4.2-5.
+
+# 8.8 HIGHER DERIVATIVES AS NODAL D.O.F.
+
+For the following discussion we define “essential” d.o.f. as the particular nodal d.o.f. needed to achieve the minimally-acceptable degree of interelement compatibility. These are the familiar nodal d.o.f. used in Chapters 1 through 7: for example, $u_{i}$ and $v_{i}$ for bars and plane elements, $w_{i}$ and $\theta_{i}$ for beam elements. We define a “higher derivative” as one that is not needed to define interelement compatibility. Thus, in the stretching of a bar or in plane stress, all derivatives of u and v would be considered “higher.” In the bending of a beam or a thin plate, higher derivatives are second and greater derivatives of lateral displacement. When used as nodal d.o.f., higher derivatives are also called “extra” or “excessive.”
+
+Elements with higher-derivative d.o.f. have certain advantages. They are based on fields having many generalized coordinates, so they provide good accuracy in a coarse mesh. Strains (or curvatures) needed in the calculation of stresses (or bending moments) appear in $\{\mathbf{D}\}$ . Thus, being primary unknowns, strains may be computed more accurately than the conventionally computed strains $\{\epsilon\} = [\mathbf{B}]\{\mathbf{d}\}$ , which invoke difference operations on essential d.o.f. $\{\mathbf{d}\}$ . Moreover, the extra d.o.f. are available at nodes, the very place where conventional strains $\{\epsilon\} = [\mathbf{B}]\{\mathbf{d}\}$ are likely to be least accurate.
+
+However, elements having higher-derivative d.o.f. are sometimes awkward to use. At an elastic–plastic boundary, or where there is an abrupt change in stiffness or material properties, continuity of higher derivative d.o.f. must not be enforced. For example, if two beam elements of different stiffness are joined, they have the same moment but different curvature at the node they share. A maneuver appropriate to such a circumstance is to release the curvature d.o.f. in one of the elements before assembly (Section 8.1). But, by doing so, we reduce the benefit of these d.o.f. where it is most needed—near a high-stress gradient.
+
+Release of higher-derivative d.o.f. is again required and the benefit of these d.o.f. is again reduced if elements with derivative d.o.f. must be used in combination with elements that have only essential nodal d.o.f. Indeed, many computer programs allow up to six d.o.f. per node (three translations and three rotations) and so may be unable to accommodate a higher-order element without basic changes.
+
+Boundary conditions may become awkward because the physical meaning of higher-derivative d.o.f. and their associated nodal loads is obscure. For example, if a plane element includes derivatives $u_{,x}, u_{,y}, v_{,x}$ , and $v_{,y}$ as nodal d.o.f., a stress-free boundary dictates a constraint relation among these d.o.f. but does not dictate the numerical value of any of them.
+
+In summary, higher-derivative d.o.f. tend to make the finite element method
+
+<!-- source-page: 267 -->
+
+awkward in application to problems for which it is most powerful—to structures built of different element types and involving thickness changes, stiffeners, and parts that join with sharp angles instead of smooth curves.
+
+# 8.9 FRACTURE MECHANICS.
+
+# SINGULARITY ELEMENTS
+
+Fracture mechanics deals with the conditions under which a body can fail owing to the propagation of an existing crack of macroscopic size $[8.12]$ . In analysis, one might ask for the load that will produce failure when a crack of known size is present, or for the allowable size of a crack when a known load must be sustained.
+
+Consider an arbitrarily loaded body that contains a crack. By isolating material in the immediate neighborhood of a crack tip, one can identify the three possible deformation modes shown in Fig. 8.9-1. These modes may appear singly or in arbitrary combination. Formulas exist for stresses and displacements in the immediate neighborhood of a crack tip. For example, if the crack is Mode I and the material is linearly elastic and isotropic, the y-direction stress and displacement are.
+
+$$
+\sigma_ {y} = \frac {K _ {\mathrm{I}}}{(2 \pi r) ^ {1 / 2}} \left(\cos \frac {\theta}{2}\right) \left[ 1 + \sin \frac {\theta}{2} \sin \frac {3 \theta}{2} \right] \tag {8.9-1}
+$$
+
+$$
+v = \frac {(2 \pi r) ^ {1 / 2}}{8 G \pi} K _ {\mathrm{I}} \left[ (2 \kappa + 1) \sin \frac {\theta}{2} - \sin \frac {3 \theta}{2} \right] \tag {8.9-2}
+$$
+
+where $G =$ shear modulus, and with $\nu =$ Poisson's ratio,
+
+$$
+\kappa = 3 - 4 \nu \quad (\text { plane   strain }) \quad \text { or } \quad \kappa = \frac {3 - \nu}{1 + \nu} \quad (\text { plane   stress }) \tag {8.9-3}
+$$
+
+Here $K_{I}$ is called the stress intensity factor for Mode I. It can be defined as
+
+$$
+K _ {\mathrm{I}} = \lim _ {r \rightarrow 0} \left[ \sigma_ {y} (2 \pi r) ^ {1 / 2} \right] \quad \text { for } \quad \theta = 0 \tag {8.9-4}
+$$
+
+![](images/page-267_9453f9f491a58a18f15132360e4f79846f9860cb579f273679e86a6df57b3ea2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y, v
+r
+θ
+x, u
+a, w
+a
+</details>
+
+Mode I (opening)
+
+![](images/page-267_5a65bd426fbd7336a4a1ed18da2725ade032d79beed479963e8b20e9f1008d43.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y, v
+r
+θ
+x, u
+z, w
+a
+</details>
+
+Mode II (sliding)
+
+![](images/page-267_71f0a42e2ab6b331c9985aa63a1ef15c0580aa38606c0faa5e1aae161ed5a8e8.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y, v
+r
+θ
+x, u
+z, u
+a
+</details>
+
+Mode III (tearing)   
+Figure 8.9-1. Deformation modes in the immediate neighborhood of a crack tip.
+
+<!-- source-page: 268 -->
+
+![](images/page-268_757f1eb1420e233affa87a36044a93ae5d522e902c308550bd0ba3f2c8ba3223.jpg)
+
+<details>
+<summary>text_image</summary>
+
+σ
+2a
+σ
+2c
+</details>
+
+Figure 8.9-2. Flat plate with a central crack of width 2a. Tensile stress $\sigma$ is uniform well away from the crack.
+
+A stress intensity factor is not a stress concentration factor: a stress intensity factor pertains to a singularity in the stress field, whereas a stress concentration factor pertains to geometries that do not produce infinite stresses. Nevertheless, two factors are analogous in that results are known and tabulated for several geometries and loadings $[8.13]$ . For example, in Fig. 8.9-2,
+
+$$
+K _ {1} = \sigma (\pi a) ^ {1 / 2} \frac {1 - 0 . 5 (a / c) + 0 . 3 2 6 (a / c) ^ {2}}{\left[ 1 - (a / c) \right] ^ {1 / 2}} \tag {8.9-5}
+$$
+
+There is a value of $K_{\mathrm{I}}$ denoted by $K_{\mathrm{IC}}$ and called fracture toughness. $K_{\mathrm{IC}}$ can be regarded as a material constant for which data are known. If $K_{\mathrm{I}} = K_{\mathrm{IC}}$ in Eq. 8.9-5, a "critical" condition exists—that is, fracture impends. Thus, given $K_{\mathrm{IC}}$ , $a$ , and $c$ , one can solve for the critical value of $\sigma$ . Or, given $K_{\mathrm{IC}}$ , $\sigma$ , and $c$ , one can solve for the critical value of crack length $2a$ . In Eq. 8.9-5, note that $\sigma$ is stress on the gross cross-sectional area $2ct$ , not stress on the net area $2(c - a)t$ .
+
+For complicated geometries and loadings, formulas such as Eq. 8.9-5 are not tabulated. A substitute relation can be determined numerically. Specifically, one can apply an arbitrarily chosen reference load to a finite element model and solve for $K_{I}$ by methods described in connection with Eqs. 8.9-8. The critical load is then equal to $K_{IC}/K_{I}$ times the reference load. (The same loads may also produce nonzero values of $K_{II}$ and $K_{III}$ . Unfortunately, for such mixed-mode conditions, the failure load cannot be accurately predicted by existing methods.)
+
+Quarter-Point Elements (QPE). The stress field of Eq. 8.9-1 displays a stress singularity of order $r^{-1/2}$ . An element having side nodes can be made to display a $r^{-1/2}$ stress (or strain) singularity by appropriate definition of its geometry. Consider, for example, the three-node bar element discussed in Section 6.2. This element is shown again in Fig. 8.9-3, now with node 3 moved to the quarter point. With $x_{1}=0$ , $x_{2}=L$ , and $x_{3}=L/4$ , Eq. 6.2-2 yields
+
+![](images/page-268_f7390580290ef3abe15a393848121e366809205a4d24164940a311eda174028b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ξ = -1
+ξ = 0
+ξ = +1
+1
+3
+2
+x,u
+L
+4
+3L
+4
+</details>
+
+Figure 8.9-3. Three-node quarter-point bar element.
+
+$^{1}$ Provided that t and a are both at least $2.5(K_{1c}/Y)^{2}$ , where t = specimen thickness and Y = yield strength in a tension test. For smaller values of t and a, fracture toughness is a function of t and a.
+
+<!-- source-page: 269 -->
+
+$$
+x = \frac {L}{4} (1 + \xi) ^ {2} \quad \text { or } \quad \xi = 2 \left(\frac {x}{L}\right) ^ {1 / 2} - 1 \tag {8.9-6}
+$$
+
+Equations 6.2-5 and 6.2-7 yield J and $[B]$ , from which $\xi$ may be eliminated by means of Eq. 8.9-6. One obtains
+
+$$
+[ \mathbf {B} ] = \left\lfloor \left(\frac {2}{L} - \frac {3}{2 (L x) ^ {1 / 2}}\right), \left(\frac {2}{L} - \frac {1}{2 (L x) ^ {1 / 2}}\right), \left(- \frac {4}{L} + \frac {2}{(L x) ^ {1 / 2}}\right) \right\rfloor \tag {8.9-7}
+$$
+
+Accordingly, stress $\sigma_x = E[\mathbf{B}]\{\mathbf{d}\}$ varies as $x^{-1/2}$ —that is, as $r^{-1/2}$ along the line $\theta = 0$ . Stress becomes infinite at $x = 0$ for displacements other than rigid-body motion.
+
+The six-node plane triangle discussed in Section 5.5 can display the $r^{-1/2}$ singularity in its strain field if its side nodes are moved to quarter points near the crack tip, a shown in Fig. 8.9-4 [8.14]. The quarter-point sides should be straight and the side node opposite the crack tip should be at midside.
+
+Another effective singularity element can be formed from a four-sided quadratic element (Fig. 6.6-1a) by collapsing one side to produce a triangle: for example, in Fig. 6.6-1a, nodes 1, 4, and 8 can be assigned the same coordinates and the same displacements. Nodes 5 and 7 are moved to the quarter points near the collapsed side. The side opposite (side 2–6–3 in this example) must be kept straight to avoid significant errors.
+
+Rectangular QPE's—for example, the element of Fig. 6.6-1a with node 1 at the crack tip and nodes 5 and 8 at the quarter points—display the $r^{-1/2}$ singularity only along two sides and the diagonal [8.15]. They are less accurate than triangular QPE's, which display the $r^{-1/2}$ singularity along all rays emanating from the crack tip.
+
+In a QPE, the singularity is precisely at the vertex—that is, at r = 0 in Fig. 8.9-4b. If side nodes are closer to midsides, the singularity moves away from the element (and becomes infinitely distant if side nodes are at midsides). Side nodes need not be precisely at quarter points, as a small error of order e in position produces an error of order $e^{2}$ in the stress intensity factor [8.15]. Indeed, one may
+
+![](images/page-269_203adb05db24ad39ab1c05f2052bef818e0b13c9d1500b340d40e7a76101ff69.jpg)
+
+<details>
+<summary>text_image</summary>
+
+σ
+a
+y
+x
+l
+σ
+</details>
+
+(a)
+
+![](images/page-269_e16951655cceadc267c279d0b28340e17b73d3688113c7216c78421eda648ae6.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+C2
+B2
+r
+x,u
+C1
+B1
+l/4
+l
+</details>
+
+(b)   
+Figure 8.9-4. (a) Plate with edge crack of length a. Only those elements around the crack tip are shown. (b) Mesh of QPE's around the crack tip.
+
+<!-- source-page: 270 -->
+
+deliberately use an “almost” QPE. In three-dimensional analysis, curvature of the crack front may place the singularity inside a QPE. This trouble may be avoided by placing side nodes a bit closer to midside rather than at quarter points.
+
+In a QPE, strains are represented as a constant plus a term proportional to $r^{-1/2}$ , as may be seen in Eq. 8.9-7. Accordingly, if $\ell / a$ in Fig. 8.9-4 is small, the region of the structure in which the singular stress field is represented decreases. But if $\ell / a$ is large, the nonsingular variations of stress are represented by only the constant term over a larger region of the structure. The best value of $\ell / a$ in a mesh with a fixed number of elements is problem-dependent. Many analyses have used $\ell / a \approx 0.1$ , but the value of $\ell / a$ is not critical in a body of arbitrary geometry if the mesh is adequate to represent stresses in the body were the crack not present [8.16,8.17]. An additional recommendation is that at least four (in a Mode I problem) or eight (in a mixed-mode problem) elements surround the crack tip [8.18]. There is disagreement as to what quadrature rule is best in generating [k] of a QPE.
+
+A common way to calculate stress intensity factors from a finite element analysis is the crack-opening displacement method. From displacements of nodes on $\theta = \pm \pi$ in Fig. 8.9-4 [8.17], it can be shown that QPE's yield the Mode I and Mode II stress intensity factors
+
+$$
+K _ {\mathrm{I}} = \frac {2 G}{\kappa + 1} \left(\frac {\pi}{2 \ell}\right) ^ {1 / 2} \left[ \left(4 v _ {B 2} - v _ {C 2}\right) - \left(4 v _ {B 1} - v _ {C 1}\right) \right] \tag {8.9-8a}
+$$
+
+$$
+K _ {\mathrm{H}} = \frac {2 G}{\kappa + 1} \left(\frac {\pi}{2 \ell}\right) ^ {1 / 2} \left[ \left(4 u _ {B 2} - u _ {C 2}\right) - \left(4 u _ {B 1} - u _ {C 1}\right) \right] \tag {8.9-8b}
+$$
+
+where $\kappa$ is given by Eq. 8.9-3.
+
+QPE's have the appeal of simplicity. Elements having side nodes are available in most programs, and they become QPE's when input data locates their side nodes at quarter points. Stress intensity factors are easily computed from Eqs. 8.9-8. However, other methods are available, as follows.
+
+Other Methods. By various methods, including hybrid methods, the strain or stress field used to formulate an element can be made to contain a singularity without invoking special placement of nodes. Indeed, a stress intensity factor can become a d.o.f. in {D}. As alternatives to Eqs. 8.9-8, one can use the virtual crack extension method [8.19] or the J integral.
+
+The simplest alternative is not to use singularity elements at all. Stress intensity factors can be obtained by using ordinary elements to surround the crack tip. Singularity elements merely make possible greater accuracy for a given computational effort.
+
+# 8.10 ELASTIC FOUNDATIONS
+
+Sometimes one elastic structure is supported by another, but stress analysis is required for only the first of the two. Then it suffices to model the effect of the second structure on the first. We need not model the second structure in such detail that stresses within it can be determined. Examples include a rail on a roadbed or a pavement slab on soil. The rail or the slab must be analyzed; the
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@@ -0,0 +1,450 @@
+<!-- source-page: 271 -->
+
+supporting effect of the roadbed or soil must be modeled. Elastic support can be represented by a foundation stiffness matrix, $[k_{f}]$ for a foundation element and $[K_{f}] = \Sigma [k_{f}]$ for the entire foundation structure. If $[K_{s}]$ is the stiffness matrix of the supported structure, then $[K_{s}] + [K_{f}]$ is the net stiffness matrix of the supported structure on its elastic foundation.
+
+In what follows we presume that $[K_{f}]$ is to operate on interface d.o.f. only—that is, on d.o.f. of only the nodes shared by the supported structure and its foundation [8.20,8.21]. Physically, the jth column of $[K_{f}]$ represents forces (and perhaps moments as well) that must be applied to nodes on the surface of the foundation to cause the jth foundation d.o.f. to have unit value while all other d.o.f. on the interface are zero.
+
+For an elastic solid foundation model, $[K_{f}]$ is a full matrix, as loads must be applied to all interface d.o.f. when only one of these d.o.f. is activated. An approximate foundation model, simple and inexpensive yet often adequate, is the Winkler foundation model.
+
+A Winkler foundation, Fig. 8.10-1a, deflects only where load is applied. Adjacent foundation material is utterly unaffected. A Winkler foundation of modulus $\beta$ applies a vertical pressure $\beta w$ when deflected vertically an amount w. In this regard the foundation acts exactly like a liquid of density $\beta$ . However, we assume that, unlike the pressure of buoyancy, foundation pressure can act either upward or downward. If instead part of a structure lifts off the foundation, the problem is nonlinear, as then contact forces and contact geometry are both unknown at the outset.
+
+We define a foundation element as the area on the foundation surface that makes contact with an element (or element face) of the supported structure. Thus a rectangular plate element in a supported paving slab would define a rectangular foundation element of identical shape and size. To determine $[k_{f}]$ for a Winkler foundation element, we can use the following strain energy argument. Let dA be an increment of the area A of the foundation element. Then deflection w normal to A produces a force increment $dF = \beta w \, dA$ . By analogy with a linear spring, whose strain energy is $F\Delta/2$ when deflected an amount $\Delta$ , the strain energy increment in the foundation is $dU = dF(w/2) = \beta w^{2} \, dA/2$ . If w is governed by d.o.f. $\{d\}$ of the aforementioned plate element, then $w = \lfloor N \rfloor\{d\}$ , where $\lfloor N \rfloor$ is the lateral-displacement shape function matrix of the plate element. Hence
+
+$$
+U = \frac {1}{2} \int \beta w ^ {2} d A = \frac {1}{2} \int w ^ {T} \beta w d A = \frac {1}{2} \left\{\mathbf {d} \right\} ^ {T} \left[ \mathbf {k} _ {f} \right] \left\{\mathbf {d} \right\} \tag {8.10-1}
+$$
+
+in which
+
+$$
+[ \mathbf {k} _ {f} ] = \int \beta [ \mathbf {N} ] ^ {T} [ \mathbf {N} ] d A \tag {8.10-2}
+$$
+
+![](images/page-271_bb2bed0af262a0ee42fa110847f40a887684b1c6265139df9eb37f57d2cef573.jpg)  
+Figure 8.10-1. Deflections of elastic foundations. Uniform pressure p is applied directly to the foundation; no structure is interposed. (a) Winkler foundation model. (b) Elastic solid foundation model.
+
+<!-- source-page: 272 -->
+
+is a foundation stiffness matrix that operates on the same d.o.f. as the plate element in contact with the foundation.
+
+If the supported element were a beam rather than a plate, then [N] in Eq. 8.10-1 would contain the cubic shape functions of a beam and $dA = b \, dx$ , where $b$ is the width of the beam. However, cubic functions are not exact because the beam element is not loaded only by forces and moments at its end nodes; it is also loaded by distributed foundation pressure. Use of the exact deflected shape [8.22] leads to stiffness coefficients in the combined matrix $[\mathbf{k}_{\text{beam}}] + [\mathbf{k}_f]$ . These coefficients involve lengthy expressions, but they are much more accurate in a coarse mesh than coefficients based on a cubic polynomial.
+
+If the supported element is neither a plate nor a beam, but (say) an eight-node hexahedron, then $\{\mathbf{d}\}$ in Eq. 8.10-1 would not contain nodal rotation d.o.f. It would contain only the $w$ d.o.f. of the four corner nodes of the quadrilateral contact area, and the $N_{i}$ would be those of Eq. 6.3-2. Indeed, one could ignore nodal rotation d.o.f. even if the supported element is a plate. Then $[\mathbf{k}_f]$ becomes more sparse and does not resist nodal rotations. Ultimately one can imagine for the supported structure element only a rigid-body lateral translation $w$ , and divide the foundation resisting force $\beta A w$ equally among element nodes in contact with the foundation. Thus, for a supported element that has $n$ contacting nodes, $[\mathbf{k}_f]$ becomes a diagonal matrix whose $n$ nonzero coefficients are $k_{fii} = \beta A / n$ . By this "lumping" procedure the foundation is reduced to a set of linear springs at the contacting nodes.
+
+The name scalar element is given to a linear or torsional spring that connects a node to a support. A scalar element can resist only a deformation along its axis or a twist about its axis.
+
+# 8.11 MEDIA OF INFINITE EXTENT
+
+Many physical problems deal with an unbounded medium. Examples include a wing moving through air, diffraction of water around an island, and a load supported by the ground (Fig. 8.11-1a). In all these problems a finite element model must be terminated somewhere short of infinity. Simple truncation at a rigid boundary, Fig. 8.11-1b, is usually adequate in static problems. However, it is
+
+![](images/page-272_5fb8fa9fab03a00fa6879f27ed5c0a118de57863ccdba4364570ab044c4093c9.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+r
+r
+Infinite
+extent
+Symmetric
+</details>
+
+(a)
+
+![](images/page-272_77c344f2eff02a343790e740806c1056946f66dd1a47e2c81905fdd74303cedb.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+Rigid
+boundary
+</details>
+
+(b)
+
+![](images/page-272_f9a41ca75a2a25868995de5eb097f790cbe76997278abc81b48765b6a936a2b6.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P ← a ← b
+Infinite
+elements
+(shaded)
+</details>
+
+(c)   
+Figure 8.11-1. (a) Load P on plane or axially symmetric body of infinite extent below the x axis. (b) Large mesh of conventional elements. (c) Smaller mesh, bounded by infinite elements.
+
+<!-- source-page: 273 -->
+
+unclear where the rigid boundary should be placed, and the analysis may be expensive because many elements are used. In dynamic problems a rigid boundary reflects a wave, regardless of the size of the mesh; therefore, the model misrepresents reality.
+
+Various methods for numerical analysis of unbounded field problems, both static and dynamic, have been devised $[8.23]$ . In what follows we summarize a particular kind of “infinite element” for static analysis that is simple and effective $[8.23-8.25]$ . In Fig. 8.11-1c, infinite elements permit satisfactory results to be obtained from fewer elements than would otherwise be required.
+
+Infinite Elements. In stress analysis, infinite elements are analogous to an elastic foundation in that they provide correct or approximately correct support conditions for a region of interest that is modeled by a mesh of standard elements. Stresses in the infinite elements are usually not of interest and may not be accurate.
+
+In formulating an infinite element, one makes use of two sets of shape functions. These are the standard shape functions [N] and either one of the following: (1) “decay” shape functions $[N_{d}]$ , which approach zero as a coordinate approaches infinity, or (2) “growth” shape functions [M], which grow without limit as a coordinate approaches infinity. In the first method [N] is applied to geometry and $[N_{d}]$ to the field variable, so that the element remains of finite size while the field variable decays. In the second method [N] is applied to the field variable and [M] to geometry, so that the element grows to infinite size. The second method yields what are called “mapped” infinite elements. They are easy to implement and are described as follows.
+
+In order to illustrate concepts and introduce procedures, we consider a one-dimensional element—namely, element 1–2–3 in Fig. 8.11-2 [8.25]. Distance a between nodes 1 and 2 may be considered a characteristic length of the element. Point 0, a distance a to the left of node 1, is not a node; it is a “pole” whose significance is discussed subsequently. Geometry of the element is interpolated according to
+
+$$
+x = M _ {1} x _ {1} + M _ {2} x _ {2}, \quad \text { where } \quad \begin{array}{l} M _ {1} = - \frac {2 \xi}{1 - \xi} \\ M _ {2} = \frac {1 + \xi}{1 - \xi} \end{array} \tag {8.11-1}
+$$
+
+![](images/page-273_1a65fad1cb52fd1db37af2261766bf019539f6888dc8a35f31b9700a222042ad.jpg)
+
+<details>
+<summary>text_image</summary>
+
+x → r
+← x₀ → ← a → ← a →
+0 • 1 2 ∥ 3
+← x₁ →
+← x₂ →
+← x₃ = x ∥
+</details>
+
+(a)
+
+![](images/page-273_a34c77da74615884a06d07fb8bd95a665e2b330ca33783aa4597fc9a43799075.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ξ = -1
+ξ = 0
+ξ = +1
+1
+2
+3
+M₂ = (1 + ξ)/1 - ξ
+1
+0
+M₁ = -2ξ/1 - ξ
+</details>
+
+(b)   
+Figure 8.11-2. (a) One-dimensional infinite element in physical space. (b) The same element in natural-coordinate space.
+
+<!-- source-page: 274 -->
+
+which yields $x = x_{1}$ at $\xi = -1$ and $x = x_{2}$ at $\xi = 0$ . As for $x_{3}$ , from Eq. 8.11-1,
+
+$$
+x _ {3} = \lim _ {\xi \rightarrow 1} \frac {- 2 \xi x _ {1} + (1 + \xi) x _ {2}}{1 - \xi} = \infty \tag {8.11-2}
+$$
+
+Accordingly, the mapping of Eq. 8.11-1 automatically places node 3 at infinity, and node 3 need not be explicitly present in Eq. 8.11-1. A field variable $\phi$ can be interpolated by standard shape functions. For the present three-node line element, from Eq. 6.2-2, the field interpolation $\phi = \lfloor N \rfloor \{\phi_e\}$ is the usual quadratic
+
+$$
+\phi = \left\lfloor - \frac {\xi + \xi^ {2}}{2} \quad (1 - \xi^ {2}) \quad \frac {\xi + \xi^ {2}}{2} \right\rfloor \left\{ \begin{array}{l} \phi_ {1} \\ \phi_ {2} \\ \phi_ {3} \end{array} \right\} \tag {8.11-3}
+$$
+
+Typically, $\phi_{3}$ is set to a constant value (usually zero) as a boundary condition. Formulation of the element stiffness matrix, Eq. 6.2-6, proceeds in standard fashion except that mapping functions $M_{1}$ and $M_{2}$ of Eq. 8.11-1 are used to form the Jacobian $J$ . Specifically, in Eq. 6.2-6 we require the strain-displacement matrix [B] and the Jacobian $J$ , which for the infinite line element are
+
+$$
+\left\lfloor \mathbf {B} \right\rfloor = \frac {1}{J} \left\lfloor \frac {d}{d \xi} \mathbf {N} \right\rfloor \quad \text { and } \quad J = M _ {1, \xi} x _ {1} + M _ {2, \xi} x _ {2} \tag {8.11-4}
+$$
+
+where $J = dx / d\xi$ is obtained from Eq. 8.11-1 and $\lfloor \mathbf{N}\rfloor$ is given by Eq. 8.11-3.
+
+To show how the foregoing infinite element represents field quantity $\phi$ , we first solve Eq. 8.11-1 for $\xi$ . With $x = x_{0} + r$ and other dimensions shown in Fig. 8.11-2a,
+
+$$
+\xi = \frac {x - x _ {2}}{x - 2 x _ {1} + x _ {2}} = 1 - \frac {2 a}{r} \tag {8.11-5}
+$$
+
+Substitution of Eq. 8.11-5 into Eq. 8.11-3 yields
+
+$$
+\phi = \phi_ {3} + \left(- \phi_ {1} + 4 \phi_ {2} - 3 \phi_ {3}\right) \frac {a}{r} + \left(2 \phi_ {1} - 4 \phi_ {2} + 2 \phi_ {3}\right) \frac {a ^ {2}}{r ^ {2}} \tag {8.11-6}
+$$
+
+We see that as $r$ approaches infinity, $\phi$ approaches $\phi_3$ (which may be set to zero as a boundary condition). The constant value $\phi = c$ prevails if $\phi_1 = \phi_2 = \phi_3 = c$ , but linear variations of $\phi$ with $r$ are not represented. In general, the two parenthetic expressions in Eq. 8.11-6 do not vanish, so $\phi$ becomes infinite at point 0 because $r = 0$ at point 0. Point 0 is therefore a pole or singular point about which field quantity $\phi$ decays. This suggests that in a problem such as that of Fig. 8.11-1c, in which there is indeed a singularity at $r = 0$ , one should use $a = b$ .
+
+It is not necessary that the mapping and the field interpolation rely on identical sets of nodes. For example, we can use the three-node mapping of Fig. 8.11-2 and Eq. 8.11-1, but replace Eq. 8.11-3 by a linear field interpolation between nodes 1 and 3,
+
+$$
+\phi = \left\lfloor \frac {1 - \xi}{2} \quad \frac {1 + \xi}{2} \right\rfloor \left\{ \begin{array}{l} \phi_ {1} \\ \phi_ {3} \end{array} \right\} \tag {8.11-7}
+$$
+
+This is perhaps the simplest possible infinite element.
+
+<!-- source-page: 275 -->
+
+Equation 8.11-7 offers the following physical interpretation. Let $\phi$ be axial displacement u and let node 3 be fixed. Then, from Eqs. 8.11-1 and 8.11-7, axial strain is
+
+$$
+\epsilon_ {x} = \frac {1}{J} \left\lfloor \frac {d}{d \xi} \mathbf {N} \right\rfloor \left\{ \begin{array}{l} u _ {1} \\ 0 \end{array} \right\} = \frac {(1 - \xi) ^ {2}}{2 a} \left(- \frac {1}{2}\right) u _ {1} = - \frac {u _ {1}}{2 a} \frac {(1 - \xi) ^ {2}}{2} \tag {8.11-8}
+$$
+
+We see that for an imagined element of physical length 2a between nodes 1 and 3, axial strain decays parabolically from $\epsilon_{x} = -u_{1}/a$ at end $\xi = -1$ to $\epsilon_{x} = 0$ at end $\xi = +1$ , rather than being the constant value $\epsilon_{x} = -u_{1}/2a$ throughout as would be the case for a standard two-node element of length 2a.
+
+For analysis of plane and axially symmetric bodies, one needs infinite elements that are mathematically two-dimensional. Such an element is shown in Fig. 8.11-3. It extends to infinity in the $\xi$ direction and is directly analogous to the element of Fig. 8.11-2. If the field variable $\phi$ is set to zero at element nodes 5 and 6, one need not use $N_{5}$ and $N_{6}$ in element formulation, and d.o.f. $\phi_{5}$ and $\phi_{6}$ need not appear in $\{\mathbf{D}\}$ . However, nodal d.o.f. $\phi_{i}$ on outer edges of infinite elements may be left unspecified, as d.o.f. to be determined, if unrestrained outer boundaries do not imply the possibility of rigid-body motion. An axially symmetric plane problem, in which only axially symmetric deformations are allowed, is a case in point.
+
+Computer programming of mapped infinite elements is straightforward. In terms of Figs. 6.5-1 and 6.5-2, the essential change is alteration of the loop on statement 30 in Fig. 6.5-1: mapping functions [M] must be used to generate the Jacobian matrix, its inverse, and its determinant. Throughout the remainder of the subroutine one uses shape functions [N] and shape function derivatives (appropriate to the number of element nodes used for the field variable) in the manner already programmed.
+
+Boundary Element Method (BEM). The BEM is an alternative to the finite element method (FEM). BEM can be applied to bounded or unbounded domains, but seems best suited to the latter. Like FEM, BEM uses nodes and elements, but only on the boundary. Thus, as compared with FEM, dimensionality is reduced by one; for example, a solid analyzed by BEM uses a two-dimensional mesh that covers only its surface. BEM and FEM can be coupled, so that BEM might replace infinite elements as the supporting medium for a structure modeled by FEM. BEM accurately models response in the domain bounded by its mesh (unlike infinite
+
+![](images/page-275_cf5e74acc4f69dfaab08d7c4ca2715f9259054c51f82436dc9d119bba3a8883c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Decay origin
+a
+1
+2
+3
+4
+η
+ξ
+5
+6
+</details>
+
+Mapping Functions
+
+$$
+M _ {1} = \frac {- 2 \xi}{1 - \xi} \frac {1 - \eta}{2}
+$$
+
+$$
+M _ {2} = \frac {- 2 \xi}{1 - \xi} \frac {1 + \eta}{2}
+$$
+
+$$
+M _ {3} = \frac {1 + \xi}{1 - \xi} \frac {1 - \eta}{2}
+$$
+
+$$
+M _ {4} = \frac {1 + \xi}{1 - \xi} \frac {1 + \eta}{2}
+$$
+
+Shape Functions
+
+$$
+N _ {1} = \frac {1}{4} (- \xi + \xi^ {2}) (1 - \eta)
+$$
+
+$$
+N _ {2} = \frac {1}{4} (- \xi + \xi^ {2}) (1 + \eta)
+$$
+
+$$
+N _ {3} = \frac {1}{2} (1 - \xi^ {2}) (1 - \eta)
+$$
+
+$$
+N _ {4} = \frac {1}{2} (1 - \xi^ {2}) (1 + \eta)
+$$
+
+$$
+N _ {5} = \frac {1}{4} (\xi + \xi^ {2}) (1 - \eta)
+$$
+
+$$
+N _ {6} = \frac {1}{4} (\xi + \xi^ {2}) (1 + \eta)
+$$
+
+Figure 8.11-3. A two-dimensional infinite element. Several additional elements are described in [8.25].
+
+<!-- source-page: 276 -->
+
+elements, which provide support but do not offer internal accuracy). However, the computational expense of BEM increases quickly if the response at several interior locations is needed. Although [K] of FEM is usually large, sparse, and symmetric, the analogous matrix of BEM is small, full, and unsymmetric. With an increase in the ratio of surface to volume, BEM becomes a less attractive alternative to FEM, because a mesh must be supplied for each boundary (each surface, hole, joint plane, or other discontinuity).
+
+The theory of BEM is not easily explained. The mathematics required is more advanced than that needed for FEM. The interested reader will find several texts, conference proceedings, journal articles, and surveys [8.26,8.27].
+
+# 8.12 FINITE ELEMENTS AND FINITE DIFFERENCES
+
+Both the finite element method and the older finite difference method discretize a continuum, and both generate simultaneous algebraic equations to be solved for nodal d.o.f. Otherwise, the methods are superficially different. Finite difference stencils overlap one another and sometimes have nodes outside the structure boundary. Finite elements do not overlap and have no nodes outside the structure boundary. Finite differences are usually explained as a way to solve differential equations; finite elements are usually explained as a way to minimize a functional.
+
+But a finite difference model can be derived from a functional [8.28]. For example, if $\Pi_p$ is the functional and $\{\mathbf{D}\}$ are nodal d.o.f., we can write finite difference expressions for the derivatives in $\Pi_p$ and generate algebraic equations from the stationary condition $\{\partial \Pi_p / \partial \mathbf{D}\} = \{\mathbf{0}\}$ . This procedure is called the finite difference energy method. It produces a symmetric coefficient matrix if the finite element method produces a symmetric coefficient matrix for the same physical problem.
+
+Thus the finite difference and finite element methods differ only in the choice of d.o.f. and in the location of nodes. Indeed, we can say that finite elements are a device for generating finite difference equations. Sometimes the two methods produce identical equations.
+
+Both methods have about the same accuracy. Computer cost is often less when finite differences are used. Inevitably, cost comparisons depend on the type of problem and program organization as well as on the analysis method.
+
+The finite difference energy method is well suited to shells of revolution $[8.28]$ . It is also suited to “pure” continua, where there is only one medium, such as a homogeneous solid or fluid. It is not well suited to a structure with a complicated boundary shape or to a structure that must be modeled by a mixture of materials or a mixture of forms, such as a vehicle that combines bar, beam, plate, and shell components. For such a problem the finite element method has no rival.
+
+# 8.13 REANALYSIS METHODS
+
+Imagine that an initial solution has been obtained. Then the structure is altered: by changing member sizes, changing materials, or otherwise altering the finite
+
+<!-- source-page: 277 -->
+
+element mesh. Loads on the structure are not changed. $^{2}$ Symbolically, we have
+
+$$
+\text { Initial   system: } \quad [ \mathbf {K} ] \{\mathbf {D} \} = \{\mathbf {R} \} \tag {8.13-1}
+$$
+
+$$
+\text { Altered   system: } \quad [ \mathbf {K} ^ {*} ] \{\mathbf {D} ^ {*} \} = \{\mathbf {R} \} \tag {8.13-2}
+$$
+
+where
+
+$$
+[ \mathbf {K} ^ {*} ] = [ \mathbf {K} ] + [ \Delta \mathbf {K} ] \quad \text { and } \quad \{\mathbf {D} ^ {*} \} = \{\mathbf {D} \} + \{\Delta \mathbf {D} \} \tag {8.13-3}
+$$
+
+D.o.f. $\{D^{*}\}$ are desired. The obvious approach is complete re-solution: that is, solve Eq. 8.13-2. Alternatives, called reanalysis methods, intend to obtain $\{D^{*}\}$ with less computational effort than complete re-solution, by using information available from the previously obtained solution of Eq. 8.13-1. In vibration analysis, the analogous problem is to obtain modified frequencies without redoing the eigenvalue extraction.
+
+Many methods of reanalysis have been proposed [8.29]. They have been categorized as follows [8.30].
+
+1. Direct methods require a finite and predictable number of steps. They produce the exact $\{D^{*}\}$ and work best when only a small portion of the structure is altered. If [K] has semibandwidth b, and less than roughly b rows of [K] are altered, then a direct method of reanalysis may be more efficient than complete re-solution.   
+2. Iterative methods converge from $\{D\}$ toward $\{D^{*}\}$ at a rate that is case-dependent. Iterative methods work best when alterations are small. Large differences between $[K]$ and $[K^{*}]$ make the iterations converge slowly or even diverge.   
+3. Approximate methods are usually based on a truncated series expansion or on a reduced set of structural equations. They are best suited to problems where exact results are not needed, for example, in intermediate stages of design or optimization.
+
+To do reanalysis, one must choose among the many methods, and revise and enlarge the computer program. The option of complete re-solution is easier and may also be more efficient in many problems.
+
+However, the option of substructuring should be noted. Substructuring (Section 8.14) is done for various reasons. One of its benefits is that the effect of alterations in a single substructure is efficiently computed. In this regard, substructuring is a direct method of reanalysis.
+
+# 8.14 SUBSTRUCTURING
+
+Mathematically, a substructure is a partially solved portion of the complete set of structural equations. Physically, a substructure is one of two or more parts
+
+<!-- source-page: 278 -->
+
+into which a structure or a finite element mesh is divided. Multilevel substructuring is possible (Fig. 8.14-1). Substructuring has other names in other contexts: blocking or dissection when used by numerical analysts, and diakoptics or tearing when used by electrical engineers. We will describe the procedure in structural terms [8.31,8.32].
+
+Procedure. In brief, a substructure is a “superelement,” that is, a single element with many nodes on its boundary and many interior d.o.f. The name “macroelement” is also appropriate. The process is that of condensation and recovery, as described in Sections 8.1 and 8.2. Indeed, elements in Fig. 8.1-1 are substructures having few d.o.f. After the division of a structure into substructures has been selected, static analysis proceeds as follows.
+
+1. Evaluate [k] and {r} for each substructure, where [k] and {r} pertain to all d.o.f. of the substructure. Eliminate internal d.o.f. by condensation; that is, apply Eq. 8.1-3. The condensed [k] and {r} pertain to only the boundary d.o.f. {d} of the substructure, which may be called “attachment” d.o.f.   
+2. Assemble substructures by connecting attachment nodes (i.e., nodes shared by substructures). Thus generate structural equations $[K_{m}]\{D_{m}\} = \{R_{m}\}$ , in which $\{D_{m}\}$ contains the attachment d.o.f. of all substructures. (Attachment nodes on mating boundaries of adjacent substructures must match in physical placement and in orientation of their d.o.f.) Solve for $\{D_{m}\}$ .   
+3. For each substructure, extract from $\{D_{m}\}$ the attachment d.o.f. $\{d_{r}\}$ of that substructure. Use Eq. 8.1-2 to compute interior d.o.f. $\{d_{c}\}$ . Now all d.o.f. of the substructure are known. Hence, stress calculation proceeds in the usual way.
+
+Clearly this is a finite element process in which elements have many internal d.o.f. and are given the name “substructures.” It differs from a standard finite element process in that one does not form a single stiffness matrix that operates on all
+
+![](images/page-278_12a714a51140df0f1192e4de8fbb94bc11f84f2941d0167317895430327d9dec.jpg)
+
+<details>
+<summary>text_image</summary>
+
+A
+6
+A
+2'
+1
+2
+3'
+4
+5
+3
+2
+1
+3
+4
+5
+(a)
+(b)
+</details>
+
+Figure 8.14-1. (a) An aircraft divided into substructures 1, 2, 2', and so on. (b) Division of substructure 2' into further substructures.
+
+<!-- source-page: 279 -->
+
+d.o.f. of the structure. Instead, there are several stiffness matrices, one for each substructure, and there is information about how to connect them so as to form $[K_{m}]$ .
+
+Remarks. The names “masters” and “slaves” are sometimes used for retained and condensed d.o.f., respectively. In static analysis, all attachment d.o.f. $\{D_{m}\}$ are masters and all d.o.f. interior to the substructures are slaves. In dynamic analysis, most interior d.o.f. are slaves but some may be masters. (Condensation in dynamics is discussed in Section 13.7.) Thus, in dynamic analysis, master d.o.f. $\{D_{m}\}$ may not consist entirely of attachment d.o.f. shared by substructures. In static analysis no approximation is introduced by substructuring. In dynamic analysis some loss of accuracy is produced by substructuring.
+
+In both static and dynamic analysis one desires that boundaries between substructures be small, so that the ratio of masters to slaves is small and $[K_{m}]$ is kept to more manageable size. Accordingly, some structures are more amenable to substructuring than others; for example, a long cylinder can be more effectively substructured than a sphere.
+
+No two substructures need be alike. However, there is special advantage if a structure contains many repetitions of the same form, particularly if the ratio of masters to slaves is small. Consider Fig. 8.14-2a. After the first analysis step has produced the condensed substructure [k] that operates on d.o.f. along boundaries AB and CD, this [k] need only be replicated with different node numberings to form the structure matrix $[K_{m}] = \Sigma [k]$ . (Indeed, substructures within ABCD can be identified: the condensed [k] of the shaded substructure in Fig. 8.14-2b can be reflected about vertical and horizontal centerlines of ABCD, after which condensation of d.o.f. along these centerlines produces the condensed [k] of ABCD.) Thus substructuring is computationally efficient, as internal d.o.f. are processed only once, in forming the condensed [k] of the typical substructure. Without substructuring, the analysis would take longer, even though the total number of d.o.f. is unchanged.
+
+Other advantages of substructuring include the following. There is a managerial advantage in breaking a large problem into smaller and more tractable parts. Different substructures can be studied simultaneously by different design groups. The work of one group can be almost independent of the others if the interaction between substructures is small. Design changes or analysis of nonlinearities, if
+
+![](images/page-279_572712b642cf4c108406603c11e24661427bc7a62d6eedbbdd44d08097e08f05.jpg)
+
+<details>
+<summary>text_image</summary>
+
+A
+D
+B
+C
+Typical
+substructure
+</details>
+
+(a)
+
+![](images/page-279_54a3d13303569152bbdf6580e6755649b92da859300778abb4e0294aa5b02597.jpg)
+
+<details>
+<summary>text_image</summary>
+
+A
+H
+D
+G
+E
+J
+F
+L
+I
+B
+K
+C
+</details>
+
+(b)   
+Figure 8.14-2. (a) Typical repeating substructure ABCD in an I beam with holes in its web. (b) A possible substructure of ABCD is shown shaded.
+
+<!-- source-page: 280 -->
+
+confined to a single substructure, leave matrices of all other substructures unchanged. The results of substructure analyses can be checked separately and revised if necessary before substructures are combined to form the complete structure.
+
+Disadvantages of substructuring include the following. Substructuring replaces one long computer run by several shorter runs. Although this can be an advantage, it is a disadvantage if turnaround is slow. The computer program is more complicated because of increased file handling and data transfers, the need for efficient data structures, and user conveniences. Thus bookkeeping and overhead expense increase. If only one analysis is to be performed and there are no repeating substructures, it is cheaper to analyze the structure entire than to use substructuring. (In practice, design changes are expected, so it is unlikely that there will be but a single analysis.)
+
+In vibration analysis it is possible to compute modes and frequencies of a structure from modes of its component substructures (see Section 13.8).
+
+# 8.15 STRUCTURAL SYMMETRY
+
+Figure 8.15-1a represents a thin square plate under lateral load. Imagine that the plate is homogeneous, isotropic, uniformly loaded, and has all four edges simply supported. Axes x, y, s, and t are all axes of symmetry. Accordingly, in static analysis, one need not analyze the entire plate: analysis of a single quadrant or a single octant—for example, one of those shown shaded in Fig. 8.15-1a—tells all that there is to know. As compared with analysis of the entire plate, data preparation time and computational expense are reduced.
+
+How can symmetry be recognized? To be symmetric, a structure must have symmetry of shape, material properties, and support conditions. Symmetry can be classed as reflective with respect to an axis or to a plane, or rotational with respect to an axis. A symmetric structure is one for which one or more reflections and/or rotations brings the structure to a configuration indistinguishable from the
+
+![](images/page-280_0cfd736ad632794b426736c43bf7a7eff478d2b721d52a3eac629ddfa717e3c2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+t
+a
+y
+a
+s
+D
+C
+x
+a
+A
+B
+</details>
+
+(a)
+
+![](images/page-280_d34586c689723bee1b6fb9507d97b8b2c2b9ed56987582834634f4dea26553a6.jpg)
+
+<details>
+<summary>text_image</summary>
+
+q
+= q/2
+Original
+loading
+Symmetric
+component
++ q/2
+Antisymmetric
+component
+L/2 L/2
+</details>
+
+(b)   
+Figure 8.15-1. (a) A laterally loaded square plate with simply supported edges. (b) A uniform beam whose loading is broken into symmetric and antisymmetric components.
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_029.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_029.md
new file mode 100644
index 00000000..049481b7
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_029.md
@@ -0,0 +1,495 @@
+<!-- source-page: 281 -->
+
+original configuration with respect to shape, material properties, and support conditions. In the plate of Fig. 8.15-1a, each dashed line is an axis of reflective symmetry. An axis normal to the plate through its center is an axis of rotational symmetry, because successive $90^{\circ}$ rotations bring the structure into coincidence with itself. Other examples of rotational symmetry include solids of revolution (Chapter 10) and cyclic symmetry (Section 8.16).
+
+A symmetric structure may carry symmetric or antisymmetric loads. Antisymmetry of loads exists if a single reflection of the structure, with its loads, followed by reversal of all loads, results in self-coincidence. An example appears in Fig. 8.15-1b, in which the plane of reflection is normal to the beam and intersects its center.
+
+Symmetric loads, when acting on symmetric structures, produce symmetric effects in linear static analysis $[8.33]$ . To exploit this rule, we analyze a part of the structure with appropriate boundary conditions. At a plane of geometric symmetry, displacement boundary conditions for symmetric loading are
+
+1. No translational motion is perpendicular to a plane of geometric symmetry.   
+2. Rotation vectors have no component in a plane of geometric symmetry.
+
+At a plane of geometric symmetry, displacement boundary conditions for anti-symmetric loading are
+
+1. Translational motion has no component in a plane of geometric symmetry.   
+2. Rotation vectors have no component perpendicular to a plane of geometric symmetry.
+
+For symmetric and antisymmetric loads acting on symmetric structures, the resulting displacements and stresses are respectively symmetric and antisymmetric.
+
+Consider, for example, the plate of Fig. 8.15-1a. Let finite elements (not shown) have as d.o.f. lateral translation w and rotations $\theta_{y}$ and $\theta_{x}$ about x and y axes, respectively (e.g., $\theta_{y} = w_{,y}$ ). Under uniform load, the loading has symmetry with respect to x and y axes. Quadrant ABCD can be analyzed with boundary conditions w = 0 along AB and AD, $\theta_{x} = 0$ along AB and BC, and $\theta_{y} = 0$ along AD and CD. If the load is uniform but alternately up and down over the four quadrants in checkerboard fashion, the loading has antisymmetry with respect to x and y axes. Quadrant ABCD can be analyzed with boundary conditions w = 0 along AB, BC, CD and DA, $\dot{\theta}_{y} = 0$ along AD and BC, and $\theta_{x} = 0$ along AB and CD.
+
+Remarks. In vibration analysis, symmetry must be exploited with caution, as symmetry of geometry does not imply symmetry of all vibration modes. Similarly, caution is needed if buckling or other nonlinear behavior arises, as symmetries present in an initial linear analysis may subsequently disappear.
+
+It may be expedient to express a load as the sum of symmetric and antisymmetric components (Fig. 8.15-1b). Thus, rather than analyzing the entire structure once, one analyses half the structure twice [8.34].
+
+Additional types of symmetries, known as skew-symmetric and skew-antisymmetric, can be identified and exploited [8.35].
+
+Before a large model is analyzed, a model with few d.o.f. can be studied to check that anticipated symmetries indeed exist, and perhaps discover unanticipated symmetries.
+
+<!-- source-page: 282 -->
+
+# 8.16 CYCLIC SYMMETRY
+
+A structure such as an impeller in a centrifugal pump is not a solid of revolution and cannot be analyzed as such. Nor is there a plane of reflective symmetry. Yet one can recognize a repetition of geometry and loading (Fig. 8.16-1). This circumstance is called cyclic symmetry, sectorial symmetry, or rotational periodicity. It is possible to analyze a representative substructure rather than the entire structure. The procedure is as follows [8.36].
+
+For a typical substructure AABB, Fig. 8.16-1b, let $[K]\{D\} = \{R\}$ represent the substructure equations, where $\{D\}$ includes all substructure d.o.f. In partitioned form, these equations are
+
+$$
+\left[ \begin{array}{l l l} \mathbf {K} _ {I I} & \mathbf {K} _ {I A} & \mathbf {K} _ {I B} \\ \mathbf {K} _ {I A} ^ {T} & \mathbf {K} _ {A A} & \mathbf {K} _ {A B} \\ \mathbf {K} _ {I B} ^ {T} & \mathbf {K} _ {A B} ^ {T} & \mathbf {K} _ {B B} \end{array} \right] \left\{ \begin{array}{l} \mathbf {D} _ {I} \\ \mathbf {D} _ {A} \\ \mathbf {D} _ {B} \end{array} \right\} = \left\{ \begin{array}{l} \mathbf {R} _ {I} \\ \mathbf {R} _ {A} \\ \mathbf {0} \end{array} \right\} + \left\{ \begin{array}{l} \mathbf {0} \\ \mathbf {F} _ {A} \\ \mathbf {F} _ {B} \end{array} \right\} \tag {8.16-1}
+$$
+
+where $\{D_{A}\}$ and $\{D_{B}\}$ contain d.o.f. on interface boundaries AA and BB, respectively, and $\{D_{I}\}$ contains all remaining d.o.f. (from nodes within the substructure and noninterface nodes on boundaries r = a and r = b). Loads $\{F_{A}\}$ and $\{F_{B}\}$ result from elastic deformations and are applied along AA and BB by neighboring substructures. Loads $\{R_{I}\}$ and $\{R_{A}\}$ represent imposed loads, typically caused by rotation and uneven (but cyclically symmetric) heating. Loads $\{R_{B}\}$ are absent because imposed loads on interface boundaries must appear on only one interface boundary; if mistakenly placed on both, the substructure receives twice the load intended.
+
+All repeating substructures are identical. Therefore, subject to a subsequent caution,
+
+$$
+\{\mathbf {D} _ {B} \} = \{\mathbf {D} _ {A} \} \quad \text { and } \quad \{\mathbf {F} _ {B} \} = - \{\mathbf {F} _ {A} \} \tag {8.16-2}
+$$
+
+![](images/page-282_b8bc8f2fce4387bd40aedcfb95cdd44cb11f9a60198c6be71196f12be9638a81.jpg)  
+(a)   
+Figure 8.16-1. (a) A hypothetical pump impeller, viewed along its axis of rotation. Vanes such as DD, seen here in edge view, are mounted on a circular disk. (b) A typical repeating substructure.
+
+<!-- source-page: 283 -->
+
+Using transformation procedures explained in Chapter 7, we write
+
+$$
+\left\{ \begin{array}{l} \mathbf {D} _ {I} \\ \mathbf {D} _ {A} \\ \mathbf {D} _ {B} \end{array} \right\} = [ \mathbf {T} ] \left\{ \begin{array}{l} \mathbf {D} _ {I} \\ \mathbf {D} _ {A} \end{array} \right\}, \quad \text { where } \quad [ \mathbf {T} ] = \left[ \begin{array}{l l} \mathbf {I} & \mathbf {0} \\ \mathbf {0} & \mathbf {I} \\ \mathbf {0} & \mathbf {I} \end{array} \right] \tag {8.16-3}
+$$
+
+and [I] is a unit matrix. The transformations $[T]^{T}[K][T]$ and $[T]^{T}\{R\}$ , applied to [K] and $\{R\}$ of Eq. 8.16-1, yield
+
+$$
+\left[ \begin{array}{c c} \mathbf {K} _ {I I} & \mathbf {K} _ {I A} + \mathbf {K} _ {I B} \\ \mathbf {K} _ {I A} ^ {T} + \mathbf {K} _ {I B} ^ {T} & \mathbf {K} _ {A A} + \mathbf {K} _ {A B} + \mathbf {K} _ {A B} ^ {T} + \mathbf {K} _ {B B} \end{array} \right] \left\{ \begin{array}{l} \mathbf {D} _ {I} \\ \mathbf {D} _ {A} \end{array} \right\} = \left\{ \begin{array}{l} \mathbf {R} _ {I} \\ \mathbf {R} _ {A} \end{array} \right\} \tag {8.16-4}
+$$
+
+in which $\{F_{A}\}$ and $\{F_{B}\}$ do not appear because of Eq. 8.16-2. Solution for nodal d.o.f. and stresses now proceeds in the usual way.
+
+Caution. The number and location of nodes along AA and BB must correspond exactly, and d.o.f. at corresponding nodes must have the same orientation with respect to the interface boundary. For example, in Fig. 8.16-1b, if i and j are corresponding nodes (e.g., both the kth node on their respective boundaries), then we must have $r_{i} = r_{j}$ , and d.o.f. at i and j must have directions such as those shown, where u and v are respectively tangent and normal to each interface boundary. Directions u (radial) and v (tangential) in polar coordinates with point C as pole are also acceptable, but directions u and v in Cartesian coordinates are not acceptable.
+
+Equation 8.16-4 can be produced automatically by the assembly process, thus avoiding the transformation defined by Eq. 8.16-3. The trick is to assign the same node number to corresponding nodes along AA and BB; for example, nodes i and j cited in the preceding paragraph would both be given the number (say) 125. Thus, the additions seen in Eq. 8.16-4 are produced automatically when elements are assembled. One must of course use actual node point coordinates in the formulation of element matrices.
+
+# PROBLEMS
+
+# Section 8.1
+
+8.1 Apply Eq. 8.1-3 to the problem in Fig. 2.11-1. Specifically, eliminate $u_{2}$ and $u_{3}$ in Fig. 2.11-1b, and obtain the condensed equation $2u_{4} = 8$ seen in Fig. 2.11-1d.   
+8.2 What values of the 18 d.o.f. in Eq. 8.1-4 are associated with rigid-body translation of magnitude $\bar{u}$ in the +x direction?   
+8.3 A 9 by 9 transformation matrix [T] can be applied to the shape function matrix [N] associated with Table 6.6-1 to yield the shape function matrix associated with Eq. 8.1-4. Write this matrix [T] and the equation that relates the two shape function matrices.   
+8.4 A three-node bar element with a central node 3 is shown, along with its stiffness matrix, which operates on d.o.f. $\lfloor u_1 \quad u_2 \quad u_3 \rfloor$ .   
+(a) Determine the 2 by 2 matrix [k] produced by condensation of $u_{3}$ .
+
+<!-- source-page: 284 -->
+
+![](images/page-284_b827f48f3229f8dfcfef9eeec4845834543b94acdc6e4df4e6235f3d4426ec07.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1 3 2 x,u
+L/2 L/2
+</details>
+
+$$
+[ \mathbf {k} ] = \frac {A E}{3 L} \left[ \begin{array}{r r r} 7 & 1 & - 8 \\ 1 & 7 & - 8 \\ - 8 & - 8 & 1 6 \end{array} \right]
+$$
+
+Problem 8.4
+
+(b) Apply a uniformly distributed axial load of intensity $q$ . From the consistent load vector $\{\mathbf{r}_e\}$ , obtain a condensed load vector associated with $u_{1}$ and $u_{2}$ .
+
+8.5 A stiffness matrix and a consistent load vector can be formulated for the three-node bar element of Problem 8.4 by use of the displacement field
+
+$$
+u = \frac {L - x}{L} u _ {1} + \frac {x}{L} u _ {2} + x (L - x) a _ {1}
+$$
+
+where $a_1$ is a nodeless d.o.f.
+
+(a) Determine the 3 by 3 stiffness matrix [k] dictated by the given u field. Let the element be uniform.   
+(b) Under what circumstances do you think the added mode $x(L - x)a_1$ will improve the results given by the basic linear element? In what stage of a finite element stress analysis does $a_1$ have an effect?
+
+8.6 Imagine that d.o.f. $\theta_{1}$ and $\theta_{2}$ of the standard four-d.o.f. beam element (Fig. 4.2-2 and Eq. 4.2-5) are to be eliminated. What do you think the 2 by 2 condensed matrix [k] will be? Verify your prediction.   
+8.7 Let a plane frame element be joined to a rotational spring at each end, with respective spring stiffnesses $k_{1}$ and $k_{2}$ (moment per radian). Let $\beta_{1}$ and $\beta_{2}$ be structure node rotations. Rotational d.o.f. $\theta_{1}$ and $\theta_{2}$ of the frame element are to be connected to structure nodes through the rotational springs, so that in Fig. 4.2-2 $\theta_{1} \neq \beta_{1}$ and $\theta_{2} \neq \beta_{2}$ unless $k_{1}$ and $k_{2}$ approach infinity. Translational d.o.f. are to be connected directly, as usual. Beginning with an 8 by 8 stiffness matrix that operates on d.o.f. $\lfloor u_{1} \quad w_{1} \quad \theta_{1} \quad u_{2} \quad w_{2} \quad \theta_{2} \quad \beta_{1} \quad \beta_{2} \rfloor^{T}$ , describe how to determine a 6 by 6 matrix [k] that operates on d.o.f. $\lfloor u_{1} \quad w_{1} \quad \beta_{1} \quad u_{2} \quad w_{2} \quad \beta_{2} \rfloor^{T}$ and is a function of $A, E, I, L, k_{1}$ , and $k_{2}$ .   
+8.8 Addition to an element of internal d.o.f., such as $a_1$ and $a_2$ in Eq. 8.1-4, can be regarded as a device that permits better approximation of equilibrium equations within the element, without affecting interelement compatibility. Accordingly, do you think the constant-strain triangle (Section 5.4) would be improved by addition of the bubble function modes $u = \xi_1\xi_2\xi_3a_1$ and $v = \xi_1\xi_2\xi_3a_2$ ? Why or why not?   
+8.9 Consider the frame of Fig. 8.1-2. Imagine that, before assembly, rotation $\theta_{A}$ is condensed in all four elements that meet at node $A$ . What do you think will be the effect of these condensations, both physically and in the numerical process?   
+8.10 Cantilever beams AB and BC are identical and are connected by a hinge at B, as shown. Use condensation, as described in connection with Fig. 8.1-2, to evaluate the rotation in both beams at B. Verify your result by elementary beam theory.
+
+<!-- source-page: 285 -->
+
+![](images/page-285_59afcec885e1a91730628737a990a9f78d6adb5c246a2aa3d982b6b3cb325a2b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L
+L
+P
+A
+①
+B
+②
+C
+</details>
+
+Problem 8.10
+
+8.11 In Problem 8.10, how would you determine the value of P needed to produce a prescribed amount of relative rotation between the beams at B?
+
+# Section 8.2
+
+8.12 Modify Figs. 8.2-1 and 8.2-2 to allow for NL load cases rather than only one.
+
+# Section 8.3
+
+8.13 (a) For the elements shown in Figs. 8.3-1b and 8.3-1c, compute the ratio of element strain energies, $U_{1} / U_{2}$ . (b) Use this result to verify the correctness of Eq. 8.3-5.   
+8.14 Consider the two beams built of rectangular elements in Table 6.14-1 (one-element case and the first five-element case). If one assumes that Eq. 8.3-5 is approximately true for these beams, what end deflections would be expected? Compare these results with those in Table 6.14-1.   
+8.15 (a) Do the $a_i$ of Eqs. 8.3-6 represent relative or absolute motions? (b) If, after computation of nodal d.o.f. in a mesh of QM6 elements, the nodeless d.o.f. $a_i$ are omitted from stress computation, what consequences do you expect? Consider, for example, the rectangular-element test cases in Table 6.14-1.   
+8.16 The sketch shows a cantilever beam modeled by QM6 elements. For the loading shown, will exact values of stresses $\sigma_{x}$ be computed? Why or why not?
+
+![](images/page-285_f8e07a5eb15a3fd31fd6162d674a37d6263f8aaa9145628648103b51c2dc4b60.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P/2'
+P/2
+</details>
+
+Problem 8.16
+
+![](images/page-285_4de182050cdc252179f39c367bd765a0042eea54b3ffffe175ee78cc30654adb.jpg)
+
+<details>
+<summary>flowchart</summary>
+
+```mermaid
+graph TD
+    A["1"] --> B["2"]
+    B --> C["3"]
+    C --> A
+    style A fill:#fff,stroke:#000
+    style B fill:#fff,stroke:#000
+    style C fill:#fff,stroke:#000
+```
+</details>
+
+Problem 8.17
+
+8.17 All three elements in the beam shown are plane QM6 elements. Examine displacements along sides of element 2 under the moment loading shown. Hence, show that pure bending is not modeled exactly by nonrectangular QM6 elements.   
+8.18 For the nodal d.o.f. $\overline{u}$ applied in Fig. 8.3-1, show that Eq. 8.3-7 yields Eqs. 8.3-3.   
+8.19 Imagine that Figs. 6.5-1 and 6.5-2 are to be modified so that they will apply to the QM6 element. Clearly state what changes and additions are required, and supply new coding where needed in Fig. 6.5-1.
+
+<!-- source-page: 286 -->
+
+# Section 8.4
+
+8.20 Show that $\delta$ in Eq. 8.4-1 can be regarded as a beam midspan deflection, as noted below Eq. 8.4-1.
+
+8.21 Following the procedure suggested below Eq. 8.4-2, write shape functions for the nine-d.o.f. triangle of Fig. 8.4-1b.
+
+8.22 (a) Establish the contents of matrix [T] in Eq. 8.4-4.
+
+(b) Use this [T] to determine shape functions $N_{i}$ of a nine-d.o.f. triangle from shape functions $N_{i}^{\prime}$ of a linear-strain triangle.
+
+8.23 Imagine that lateral deflection $w$ of a uniform beam element is defined by three nodal values, as shown.
+
+(a) Establish the 3 by 4 transformation matrix [T] that will convert this element to one that operates on the standard d.o.f. $w_{1}$ , $\theta_{1}$ , $w_{2}$ , and $\theta_{2}$ .
+
+(b) Hence, establish the new shape functions $N_{1}, N_{2}, N_{3}$ , and $N_{4}$ .
+
+(c) What property does the element have that may pose a difficulty?
+
+![](images/page-286_862f601a083ad3bf0286ca5d58549970e2937b4b0a3b87c76188404499307255.jpg)
+
+<details>
+<summary>text_image</summary>
+
+w₁
+w₃
+w₂
+1
+3
+2
+L/2
+L/2
+x
+</details>
+
+$$
+w = \frac {2 x ^ {2} - 3 L x + L ^ {2}}{L ^ {2}} w _ {1} + \frac {2 x ^ {2} - L x}{L ^ {2}} w _ {2} + \frac {4 x (L - x)}{L ^ {2}} w _ {3}
+$$
+
+Problem 8.23
+
+# Section 8.5
+
+8.24 Complete the steps of generating [k] for the element of Fig. 8.5-1b, as follows. Use [P] and [R] from Eq. 8.5-12.
+
+(a) Complete matrix [L], begun in Eqs. 8.5-14.   
+(b) Generate matrix [G], Eq. 8.5-7.   
+(c) Generate matrix [H], Eq. 8.5-4. For simplicity, let $\nu = 0$ , so that $[\mathbf{E}] = \mathrm{E}[1, 1, \frac{1}{2}]$ .   
+(d) Generate [k], Eq. 8.5-10, again for $\nu = 0$ .
+
+8.25 Use the assumed-stress hybrid method to evaluate [k] for the six-d.o.f. plane triangle shown. Use $\{\pmb{\beta}\} = [\beta_1 \beta_2 \beta_3]^T$ . For simplicity, take Poisson's ratio as zero. (This [k] should agree with the [k] obtained in Problem 4.11c.)
+
+![](images/page-286_4143f267a2b34c8d8ca0bd60ef4f91614970e60c89b43778e4c88f5d2fe725d0.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+a
+3
+b
+1
+2
+x,u
+</details>
+
+Problem 8.25
+
+8.26 (a) Write the equation $\{\sigma\} = [\mathbf{P}]\{\boldsymbol{\beta}\}$ for a plane element if $\sigma_x = \beta_1 + \beta_4x$ , $\sigma_y = \beta_2 + \beta_5y$ , and $\tau_{xy} = \beta_3$ . Do you think such an element would be a good one?
+
+<!-- source-page: 287 -->
+
+(b) If $\beta_{4} = \beta_{5} = 0$ in part (a), so that $\{\pmb{\beta}\} = [\beta_{1}, \beta_{2}, \beta_{3}]^{T}$ , what defect would you expect to see in the stiffness matrix of a plane eight-d.o.f. rectangular element?
+
+8.27 The beam element shown is to include the effects of transverse shear deformation. If bending moment $M$ is taken as $M = \beta_1 + \beta_2x$ , then the shear force $V = \beta_2$ satisfies the equilibrium equation $dM/dx = V$ . With $\{\sigma\} = [M, V]^T$ , strain energy in the element is $U = \frac{1}{2} \int \{\sigma\}^T \left[ \frac{1}{EI} \cdot \frac{f}{AG} \right] \{\sigma\} dx$ , where $f$ is a “form factor” ( $f = 1.2$ for a rectangular cross section). [R] relates nodal moments and shear forces to $\{\beta\}$ , [L] is a unit matrix, and [R] $^T$ [L] requires no integration. Derive [k] and show that it reduces to Eq. 4.2-5 as shear modulus $G$ becomes large [4.11].
+
+![](images/page-287_03c8b4330e5d3d5540f278ba2798f9f960b0c57ecfa4c96065f6195b06873a9a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+M₁ → x
+V₁
+L
+M₂
+V₂
+</details>
+
+![](images/page-287_269f04277b10f067a19cc408db012f96ee60ed057163c198d2fe84e1263af690.jpg)
+
+<details>
+<summary>text_image</summary>
+
+w₁
+θ₁
+1
+w₂
+θ₂
+2
+L
+</details>
+
+Problem 8.27
+
+# Section 8.7
+
+8.28 (a) Following the example of Eqs. 8.7-8 and 8.7-9, determine [k] for a standard four-d.o.f. beam element. However, use $\{\mathbf{d}_R\} = \lfloor w_1 - w_2 \rfloor^T$ .
+
+(b) Imagine that the leading diagonal coefficient of $[\mathbf{k}_{EE}]$ in part (a) is in error by an amount $e$ . Show that $[\mathbf{k}]$ still represents rigid-body motion correctly.
+
+8.29 The two-spring structure shown is allowed axial nodal displacements $u_{1}, u_{2}$ , and $u_{3}$ . If $\{\mathbf{d}_R\} = u_1$ , write $[\mathbf{k}_{EE}]$ , and from it determine $[\mathbf{k}]$ .
+
+8.30 (a) A bar element of axial stiffness $k = AE / L$ is permitted only axial nodal displacements $u_{1}$ and $u_{2}$ . Write $[\mathbf{k}_{EE}]$ , and from it determine $[\mathbf{k}]$ .
+
+(b) Repeat part (a), but let there be four d.o.f. $\{\mathbf{d}\} = \lfloor u_1 \quad v_1 \quad u_2 \quad v_2 \rfloor^T$ , as in Eq. 2.4-3, so that plane motion is possible.
+
+8.31 A flat elastic disk has inside and outside radii $r_1$ and $r_2$ , as shown. Nodal d.o.f. are circumferential displacements $v_1$ and $v_2$ . When the disk is fixed at $r = r_1$ , the ratio of torque $T_2$ on edge $r = r_2$ to the resulting angle of twist $\theta_2$ is a number $C$ . Determine the stiffness matrix [k] that operates on d.o.f. $v_1$ and $v_2$ , in terms of $C, r_1$ , and $r_2$ . Verify that [k]{d} = {0} if {d} represents rigid-body motion.
+
+![](images/page-287_3076967950ee5ffdd1e2abc832439fd34d54c81ff7165b260d3cdfa441f86095.jpg)  
+Problem 8.29
+
+![](images/page-287_cd76e73dc8d5da46433ede1158f80622f1c72006d2fae11406ce813f4597b7e7.jpg)
+
+<details>
+<summary>text_image</summary>
+
+r₂
+r₁
+v₁
+v₂
+T₂,θ₂
+</details>
+
+Problem 8.31
+
+<!-- source-page: 288 -->
+
+# Section 8.8
+
+8.32 The structure shown is built of plane elements. D.o.f. (at each corner node) consist of $u, v, u_{,x}, v_{,x}, u_{,y}$ , and $v_{,y}$ . Pressure $p$ acts along edge $AB$ . Edge $BC$ is fixed. What boundary conditions should be imposed on nodal d.o.f. along edges $AB, BC, CD$ , and $DA$ ? Assume that the material is isotropic. What is different if the material is anisotropic?
+
+![](images/page-288_e5c9dafe443a766bc9960da44ea35b6cb2a4beea5cb67435b4e3fde2414047e1.jpg)
+
+<details>
+<summary>text_image</summary>
+
+t
+y
+A
+p
+B
+C
+x
+s
+α
+D
+</details>
+
+Problem 8.32
+
+# Section 8.9
+
+8.33 A long bar, 100 mm wide and 20 mm thick, is loaded in tension by an axial force P.
+
+(a) If the yield strength is $Y = 1150 \mathrm{MPa}$ and $K_{\mathrm{IC}} = 77 \mathrm{MPa} \sqrt{\mathrm{m}}$ , and a central crack $15 \mathrm{~mm}$ long is present, what force $P$ will fracture the bar?   
+(b) If the yield strength is Y = 1410 MPa and $K_{IC} = 50 MPa \sqrt{m}$ , what is the critical crack length if the force P determined in part (a) is applied?
+
+8.34 Rather than use Eq. 8.9-8a to determine $K_{\mathrm{I}}$ , one can determine $K_{\mathrm{I}}$ from displacements of points $B1$ and $B2$ alone in Fig. 8.9-4. Derive the appropriate formula.   
+8.35 Consider the bar element of Fig. 8.9-3, but place node 3 at the third point rather than at the quarter point. At what value of $x / L$ is a stress singularity indicated?   
+8.36 Let quarter-point elements be used to solve a certain crack problem (e.g., Fig. 8.9-4). Imagine that the problem is solved again, this time using quarter-point elements of smaller size. Now the computed results are found to be less accurate than before. Explain how this is possible.
+
+# Section 8.10
+
+8.37 The beam element shown has the usual d.o.f. $\{\mathbf{d}\} = \left[w_1 \quad \theta_1 \quad w_2 \quad \theta_2\right]^T$ . The element has width $b$ and rests on a Winkler foundation of modulus $\beta$ . Determine the foundation matrix $[\mathbf{k}_f]$ defined by each of the following approximations.
+
+(a) Deflection $w$ is cubic in $x$ , as in the standard beam element.   
+(b) Deflection $w$ is quadratic in $x$ (see Eq. 8.4-2).   
+(c) Deflection $w$ is linear in $x$ (and independent of $\theta_1$ and $\theta_2$ ).   
+(d) Deflection $w$ is constant.
+
+<!-- source-page: 289 -->
+
+![](images/page-289_1d8af2d8182e89e0bfd9cf0db3d10a85a0049d8fbb85e5bd37454d0d76611b85.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+L
+1
+2
+x
+</details>
+
+Problem 8.37
+
+8.38 The sketches represent top views of triangular elements that rest on a Winkler foundation of modulus $\beta$ . Assume that vertical deflection w depends only on nodal values of w. Determine an expression for $[k_{f}]$ of
+
+(a) the three-node element (Eqs. 5.3-4).   
+(b) the six-node element (Eqs. 5.3-5), if sides are straight and side nodes are at midsides.
+
+![](images/page-289_336fe2e12bf59ab13dfb8fef7da60a5aa0bbfb58b9dfbf0260ede02607e4f475.jpg)  
+(a)
+
+![](images/page-289_14a5f64b4d12a2f07db7a67c3ee14af7c53d7536fdfaa41ced419c9ad8cab56c.jpg)  
+(b)   
+Problem 8.38
+
+8.39 Imagine that separation is possible between a beam and its Winkler elastic foundation. Outline a solution algorithm for such a problem. In this exercise, do not be concerned with computational efficiency.   
+8.40 Imagine that a Winkler elastic foundation, which has translational modulus $\beta$ , is augmented by a rotational modulus $\alpha$ (whose units are force divided by length). For an element on such a composite foundation, what formula for $[k_{f}]$ replaces Eq. 8.10-2? (A symbolic result is desired, with terms defined, rather than specifics of a particular element.)
+
+# Section 8.11
+
+8.41 Show that Eqs. 8.11-5 and 8.11-6 indeed result from the manipulations described.   
+8.42 Use $\phi$ from Eq. 8.11-7 and the mapping of Eq. 8.11-1 to determine the following:
+
+(a) $\phi$ as a function of $r$ (analogous to Eq. 8.11-6).   
+(b) $J$ as a function of $\xi$ and $a$ .   
+(c) Element matrix [k]. Use exact integration.   
+(d) Element matrix [k]. Use a two-point Gauss rule.
+
+8.43 (a) For the infinite element shown, let field variable $\phi$ depend on nodal values $\phi_{1}, \phi_{2}, \phi_{5}$ , and $\phi_{6}$ only (not on $\phi_{3}$ and $\phi_{4}$ ). Write mapping functions and shape functions, in the manner of Fig. 8.11-3.
+
+<!-- source-page: 290 -->
+
+![](images/page-290_a25e9fdf31935ff19d028181f2aaed2c168112d6ee9cc6629bd76137fbb69c94.jpg)
+
+<details>
+<summary>text_image</summary>
+
+a
+x
+y
+2b
+1
+2
+3
+η
+4
+ξ
+5
+6
+</details>
+
+Problem 8.43
+
+![](images/page-290_6af44eef5646fe66b301fbc07da09e451d0be0e1c12d88313e0b9a88552c88de.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+3
+4
+5
+η
+6
+ξ
+</details>
+
+Problem 8.44
+
+(b) Let sides 1–3–5 and 2–4–6 be parallel. Evaluate [J] and J.   
+(c) If $\phi_5 = \phi_6 = 0$ , what 2 by 2 element characteristic matrix [k] operates on $\phi_1$ and $\phi_2$ ? Again let sides 1-3-5 and 2-4-6 be parallel. (See Eq. 6.3-5, and let $t =$ element thickness and $k =$ material characteristic, both uniform over the element.)
+
+8.44 Write mapping functions for the infinite plane element shown.
+
+# Section 8.13
+
+8.45 If $\{\mathbf{R}\}$ is unchanged and structural alterations are minor, then $\{\Delta \mathbf{D}\} \approx -[\mathbf{K}]^{-1}([\Delta \mathbf{K}]\{\mathbf{D}\})$ . Derive this expression for $\{\Delta \mathbf{D}\}$ . What are advantages and disadvantages of this method?   
+8.46 Equation 8.13-2 can be cast in the iterative form $\{\mathbf{K}\} \{\mathbf{D}^{*}\}_{i+1} = \{\mathbf{R}\} - [\Delta \mathbf{K}\} \{\mathbf{D}^{*}\}_{i}$ . Consider the application of this equation to single-d.o.f. problems as follows.
+
+(a) Let $K = 0.5, K^{*} = 0.8$ , and $R = 2$ . Starting with $D_0^* = D = 4.0$ , compute $D_0^*$ (i.e., apply five iterative cycles).   
+(b) For what range of values of $\Delta K / K$ does this iterative method converge?
+
+# Section 8.14
+
+8.47 In Fig. 8.14-2b, imagine that the reduced [k] for substructure AEFGH is known. How can one transform this [k] so that it pertains to substructure HGIJD, ready for assembly with substructure AEFGH? For brevity, consider only translational d.o.f. $u_{i}$ and $v_{i}$ at the lettered corners.
+
+# Section 8.15
+
+8.48 Let the plate of Fig. 8.15-1a be uniformly loaded. Imagine that octant ABC is modeled by square elements, as shown in the sketch for this problem, so that some elements straddle the symmetry axis AC. What boundary conditions should be applied to these elements, for example, to typical element 1–2–3–4? State these conditions with reference to (a) st axes, and (b) xy axes.
+
+![](images/page-290_d5e7b9af3a394628944c0c5195051305c13b193f97a2f6052b2ecb41aa588262.jpg)
+
+<details>
+<summary>text_image</summary>
+
+A
+1
+2
+3
+4
+C
+B
+</details>
+
+Problem 8.48
+
+![](images/page-290_bcf0b323c6b0b2cbae2748efc9e69c552c7ac6cecebd7e80ae37a42efd799560.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+y, v
+x, u
+</details>
+
+Problem 8.49
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_030.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_030.md
new file mode 100644
index 00000000..3b839df1
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_030.md
@@ -0,0 +1,460 @@
+<!-- source-page: 291 -->
+
+8.49 For the frame shown, geometry is symmetric but loads P are antisymmetric with respect to the y axis.
+
+(a) By considering a reflection and a load reversal, show that vertical support reactions are equal in magnitude and horizontal support reactions are zero.   
+(b) If half the frame is analyzed, what support condition should be used at the point where the y axis crosses the frame?
+
+8.50 By superposing results from symmetric and antisymmetric loadings in Fig. 8.15-1b, and using formulas from beam theory, determine (a) the deflection at midspan, and (b) the rotation at the left end.
+
+# Section 8.16
+
+8.51 The three-node truss shown carries radial loads P and contains three identical bars, each of axial stiffness $k = AE/L$ . Use cyclic symmetry methods to determine the radial displacement of a typical node.
+
+![](images/page-291_1852a61fe9e484211ad73b00f49bffe19d5af9c15fa4b6c62eafc64addd01dd0.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+3
+1
+2
+P
+L
+</details>
+
+Problem 8.51
+
+![](images/page-291_c79f4419117ce9ce6dc63f7c72355debd0e273dc30662f9b026b5b11467c9b4c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+4
+3
+1
+2
+x,u
+</details>
+
+Problem 8.52
+
+8.52 The plane structure shown consists of four identical triangular areas, which form a square outer boundary and a square opening. Imagine that there exists an 8 by 8 stiffness matrix $[k']$ for one triangle, which operates on d.o.f. $u_{i}$ and $v_{i}$ at nodes i = 1, 2, 3, 4.
+
+(a) Construct a transformation matrix $[\mathbf{T}_a]$ for the operation $[\mathbf{k}] = [\mathbf{T}_a]^T [\mathbf{k}'][\mathbf{T}_a]$ , in which $[\mathbf{k}]$ operates on d.o.f. suitable for exploitation of cyclic symmetry.   
+(b) Construct a 4 by 8 constraint transformation matrix [T] appropriate to this exercise (see Eq. 8.16-3).   
+(c) What is the appropriate transformation matrix if parts (a) and (b) are to be done as a single transformation?
+
+8.53 A long, uniform beam is supported and loaded in a repetitive pattern, as shown. Use cyclic symmetry methods to determine nodal d.o.f. $w_{1}$ , $\theta_{1}$ , and $\theta_{2}$ in terms of P, a, E, and I.
+
+![](images/page-291_0756ed9af76c33e05659cba609d3c3a682b7f08e717f0ed7fe581b436630e1da.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+P
+1
+2
+P
+x
+a
+2a
+a
+2a
+</details>
+
+Problem 8.53
+
+<!-- source-page: 292 -->
+
+# CONSTRAINTS
+
+Constraints enforce a relationship among d.o.f. Procedures for imposing a constraint include transformation, Lagrange multipliers, and penalty functions. Naturally arising constraints, constraint counting, and integration rules for incompressible materials are also discussed.
+
+# 9.1 CONSTRAINTS. TRANSFORMATIONS
+
+Constraints. A constraint either prescribes the value of a d.o.f. (as in imposing a support condition) or prescribes a relationship among d.o.f. In common terminology, a single-point constraint sets a single d.o.f. to a known value (often zero), and a multipoint constraint imposes a relationship between two or more d.o.f. Thus support conditions in the three-bar truss of Fig. 2.2-1 invoke three single-point constraints. Rigid links and rigid elements, discussed in Section 7.8, each invoke a multipoint constraint.
+
+Figure 9.1-1 shows an example in which constraints could be imposed. In a typical frame, axial deformation of a member can usually be ignored; only bending deformation is significant. Accordingly, in Fig. 9.1-1, one could impose the single-point constraints $v_{A} = 0$ and $v_{B} = 0$ , and the multipoint constraint $u_{A} = u_{B}$ , after which the active d.o.f. consist of only $\theta_{A}$ , $\theta_{B}$ , and either $u_{A}$ or $u_{B}$ . (Failure to impose the constraint $u_{A} = u_{B}$ invites numerical difficulty; see Section 18.2.) Special-purpose computer programs for tall buildings may incorporate constraints of this type by allowing only three d.o.f. per floor, these being the rotation $\theta_{z}$ of a floor about a vertical z axis and the horizontal displacement components u and v.
+
+For each equation of constraint, one d.o.f. can be eliminated from the vector of structural d.o.f. $\{D\}$ . However, doing so may involve appreciable manipulation and typically increases the bandwidth (or the frontwidth) of the structural equations. The Lagrange multiplier method of treating constraints, discussed subsequently, adds to the number of equations but requires less manipulation.
+
+Transformation Equations. Constraint equations that couple d.o.f. in $\{D\}$ can be written in the form
+
+$$
+[ \mathbf {C} ] \{\mathbf {D} \} = \{\mathbf {Q} \} \tag {9.1-1}
+$$
+
+where $[C]$ and $\{Q\}$ contain constants. There are more d.o.f. in $\{D\}$ than constraint equations, so $[C]$ has more columns than rows. We now consider the common case $\{Q\} = \{0\}$ . Let Eq. 9.1-1 be partitioned so that
+
+$$
+\left[ \begin{array}{l l} \mathbf {C} _ {r} & \mathbf {C} _ {c} \end{array} \right] \left\{ \begin{array}{l} \mathbf {D} _ {r} \\ \mathbf {D} _ {c} \end{array} \right\} = \{\mathbf {0} \} \tag {9.1-2}
+$$
+
+<!-- source-page: 293 -->
+
+![](images/page-293_65295c1951311b5d36fe493d3790a6d063f2bd5711c275deb6bfe701e363efa6.jpg)
+
+<details>
+<summary>text_image</summary>
+
+u_A
+v_A
+θ_A
+A
+B
+v_H
+θ_H
+u_B
+C
+D
+</details>
+
+Figure 9.1-1. A three-element plane frame, fixed at nodes C and D. D.o.f. at nodes A and B are shown.
+
+where $\{D_{r}\}$ and $\{D_{c}\}$ are, respectively, d.o.f. to be retained and d.o.f. to be eliminated or “condensed out.” Because there are as many d.o.f. $\{D_{c}\}$ as there are independent equations of constraint in Eq. 9.1-2, matrix $[C_{c}]$ is square and nonsingular. Solution for $\{D_{c}\}$ yields
+
+$$
+\{\mathbf {D} _ {c} \} = [ \mathbf {C} _ {r c} ] \{\mathbf {D} _ {r} \}, \quad \text { where } \quad [ \mathbf {C} _ {r c} ] = - [ \mathbf {C} _ {c} ] ^ {- 1} [ \mathbf {C} _ {r} ] \tag {9.1-3}
+$$
+
+We now write as one relation the identity $\{\mathbf{D}_r\} = \{\mathbf{D}_r\}$ and Eq. 9.1-3:
+
+$$
+\left\{ \begin{array}{l} \mathbf {D} _ {r} \\ \mathbf {D} _ {c} \end{array} \right\} = [ \mathbf {T} ] \{\mathbf {D} _ {r} \}, \quad \text { where } \quad [ \mathbf {T} ] = \left[ \begin{array}{l} \mathbf {I} \\ \mathbf {C} _ {r c} \end{array} \right] \tag {9.1-4}
+$$
+
+With the transformation matrix [T] now defined, the familiar transformations $\{\mathbf{R}\} = [\mathbf{T}]^T\{\mathbf{R}'\}$ and $[\mathbf{K}] = [\mathbf{T}]^T[\mathbf{K}'][\mathbf{T}]$ of Eqs. 7.4-2 and 7.4-4 can be applied to the structural equations $[\mathbf{K}']\{\mathbf{D}'\} = \{\mathbf{R}'\}$ , which are partitioned as
+
+$$
+\left[ \begin{array}{l l} \mathbf {K} _ {r r} & \mathbf {K} _ {r c} \\ \mathbf {K} _ {c r} & \mathbf {K} _ {c c} \end{array} \right] \left\{ \begin{array}{l} \mathbf {D} _ {r} \\ \mathbf {D} _ {c} \end{array} \right\} = \left\{ \begin{array}{l} \mathbf {R} _ {r} \\ \mathbf {R} _ {c} \end{array} \right\} \tag {9.1-5}
+$$
+
+The condensed system is
+
+$$
+\left[ \mathbf {K} _ {r r} + \mathbf {K} _ {r c} \mathbf {C} _ {r c} + \mathbf {C} _ {r c} ^ {T} \mathbf {K} _ {c r} + \mathbf {C} _ {r c} ^ {T} \mathbf {K} _ {c c} \mathbf {C} _ {r c} \right] \left\{\mathbf {D} _ {r} \right\} = \left\{\mathbf {R} _ {r} + \mathbf {C} _ {r c} ^ {T} \mathbf {R} _ {c} \right\} \tag {9.1-6}
+$$
+
+After Eq. 9.1-6 is solved for $\{\mathbf{D}_r\}$ , Eq. 9.1-3 yields $\{\mathbf{D}_c\}$ . If $\{\mathbf{Q}\} \neq \{\mathbf{0}\}$ in Eq. 9.1-1, additional terms appear on the right-hand side of Eq. 9.1-6.
+
+If Eq. 9.1-2 simply sets certain d.o.f. $\{D_{c}\}$ to zero, then $[C_{r}]=[0]$ and $[C_{c}]=[I]$ , hence $[C_{rc}]=[0]$ , and Eq. 9.1-6 is equivalent to discarding rows and columns associated with $\{D_{c}\}$ . Otherwise, the choice of which d.o.f. to place in $\{D_{c}\}$ is not unique, so the choice of $[C_{c}]$ is not unique. One might then define $[C_{c}]$ to be the last c linearly independent columns of [C].
+
+It is possible to avoid the reordering, partitioning, and matrix multiplications implied by Eq. 9.1-6 by applying individual constraint equations serially and retaining all d.o.f. of $\{D_{r}\}$ and $\{D_{c}\}$ in the transformed equations [6.1]. The transformed coefficient matrix may not be positive definite.
+
+<!-- source-page: 294 -->
+
+![](images/page-294_3bc94e311a8aa808af39a4ea89432040473c1b56859eb7c9530f574fe38c04b5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+1
+2
+3
+x₁u
+P
+P
+P
+L
+L
+L
+</details>
+
+Figure 9.1-2. Three identical bar elements, each of axial stiffness $k = AE/L$ .
+
+$$
+\left[ \begin{array}{c c c} 2 k & - k & 0 \\ - k & 2 k & - k \\ 0 & - k & k \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \\ u _ {3} \end{array} \right\} = \left\{ \begin{array}{l} P \\ P \\ P \end{array} \right\} \tag {9.1-7}
+$$
+
+Imagine that the constraint $u_{2} = u_{3}$ is to be imposed. With the choice $D_{c} = u_{3}$ , Eqs. 9.1-2 and 9.1-3 become
+
+$$
+[ 0 \quad 1 \quad | - 1 ] \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \\ u _ {3} \end{array} \right\} = 0 \quad \text { and } \quad [ \mathbf {C} _ {r c} ] = [ 0 \quad 1 ] \tag {9.1-8}
+$$
+
+The transformation matrix of Eq. 9.1-4 and the reduced system of Eq. 9.1-6 are
+
+$$
+[ \mathbf {T} ] = \left[ \begin{array}{l l} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{array} \right] \quad \text { and } \quad \left[ \begin{array}{c c} 2 k & - k \\ - k & k \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \end{array} \right\} = \left\{ \begin{array}{l} P \\ 2 P \end{array} \right\} \tag {9.1-9}
+$$
+
+Equation 9.1-9 yields $u_{1} = 3P / k$ and $u_{2} = 5P / k$ . Hence, Eq. 9.1-8 yields $u_{3} = 5P / k$ .
+
+In Section 8.16, we note that two different nodes can be forced to have the same d.o.f. in $\{D\}$ by giving them the same node number. (Actual nodal coordinates are still used in the generation of element matrices.) Thus a node whose d.o.f. would all appear in $\{D_{c}\}$ can be assigned a node number associated with $\{D_{r}\}$ instead of using the transformation, Eq. 9.1-6. Any externally applied loads on d.o.f. $\{D_{c}\}$ must be transferred to d.o.f. in $\{D_{r}\}$ . In applying this method to the foregoing example problem, one assigns the number 2 to the rightmost two nodes. This causes addition of the four coefficients in [k] of the right element, for a sum of zero at node 3, effectively removing the right element (but not its load) from the structure, and producing Eq. 9.1-9 upon assembly of the remaining two elements.
+
+The condensed system in Eq. 9.1-6 is different from the system obtained by static condensation, Eq. 8.1-3. In Eq. 8.1-3, condensed d.o.f. are related to retained d.o.f. by equilibrium equations already present in the system $[K]\{D\}=\{R\}$ . In Eq. 9.1-6, condensed d.o.f. $\{D_{c}\}$ are related to retained d.o.f. $\{D_{r}\}$ by supplementary equations of constraint that replace certain equilibrium equations. Accordingly, constraints may appear to falsify certain equilibrium equations. Figure 9.1-3 is a case in point. The original system, and the system that results from the constraint $v_{1}=v_{2}$ , are respectively
+
+$$
+\left[ \begin{array}{l l} k & 0 \\ 0 & k \end{array} \right] \left\{ \begin{array}{l} v _ {1} \\ v _ {2} \end{array} \right\} = \left\{ \begin{array}{l} P \\ 0 \end{array} \right\} \quad \text { and } \quad (2 k) v _ {1} = P \tag {9.1-10}
+$$
+
+<!-- source-page: 295 -->
+
+![](images/page-295_e24a1f94d608087b1443498e6d1a5c5e0343679494935713ce4ea960308706d3.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+L
+P
+Rigid bar
+1
+2
+x
+k
+k
+(a)
+</details>
+
+![](images/page-295_50c597b81e29657da55b9559b2e6a71968cb67a5026540cdf22dceb734a3e7c5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+1
+P/2
+2
+P/2
+(b)
+</details>
+
+Figure 9.1-3. (a) A rigid bar supported by two springs. (b) External and elastic forces applied to the bar if the constraint $v_{1} = v_{2}$ is imposed. Forces of constraint are not shown.
+
+Hence, $v_{1} = v_{2} = P/2k$ , and forces carried by the springs are $kv_{1} = kv_{2} = P/2$ . Net forces applied to the bar, Fig. 9.1-3b, satisfy equilibrium of y-direction forces but not moment equilibrium. Of course, the condensed structure is not that of Fig. 9.1-3b; it is a single spring of stiffness 2k, loaded by force P.
+
+# 9.2 LAGRANGE MULTIPLIERS
+
+Lagrange's method of undetermined multipliers is used to find the maximum or minimum of a function whose variables are not independent but have some prescribed relation. In structural mechanics the function is potential energy $\Pi_p$ and the variables are d.o.f. in $\{\mathbf{D}\}$ . System unknowns become $\{\mathbf{D}\}$ and the Lagrange multipliers.
+
+The theory is easy to describe. We write the constraint equation (Eq. 9.1-1) as the homogeneous equation $[C]\{D\} - \{Q\} = \{0\}$ and multiply its left-hand side by a row vector $\{\lambda\}^{T}$ that contains as many Lagrange multipliers $\lambda_{i}$ as there are constraint equations. Next we add the result to the potential expression, Eq. 4.1-7:
+
+$$
+\Pi_ {p} = \frac {1}{2} \{\mathbf {D} \} ^ {T} [ \mathbf {K} ] \{\mathbf {D} \} - \{\mathbf {D} \} ^ {T} \{\mathbf {R} \} + \{\lambda \} ^ {T} ([ \mathbf {C} ] \{\mathbf {D} \} - \{\mathbf {Q} \}) \tag {9.2-1}
+$$
+
+The expression in parentheses is zero, so we have added nothing to $\Pi_{p}$ . Next we make $\Pi_{p}$ stationary by writing the equations $\{\partial\Pi_{p}/\partial\mathbf{D}\}=\{\mathbf{0}\}$ and $\{\partial\Pi_{p}/\partial\boldsymbol{\lambda}\}=\{\mathbf{0}\}$ , following differentiation rules stated in Appendix A. The result is
+
+$$
+\left[ \begin{array}{l l} \mathbf {K} & \mathbf {C} ^ {T} \\ \mathbf {C} & \mathbf {0} \end{array} \right] \left\{ \begin{array}{l} \mathbf {D} \\ \boldsymbol {\lambda} \end{array} \right\} = \left\{ \begin{array}{l} \mathbf {R} \\ \mathbf {Q} \end{array} \right\} \tag {9.2-2}
+$$
+
+The lower partition of Eqs. 9.2-2 is Eq. 9.1-1, the equation of constraint. Equations 9.2-2 are solved for both $\{D\}$ and $\{\lambda\}$ . The $\lambda_{i}$ may be interpreted as forces of constraint (see the following example problem).
+
+Strict partitioning—that is, $\{D\}$ followed by $\{\lambda\}$ in Eq. 9.2-2—increases bandwidth to the maximum. If instead the $D_{i}$ and $\lambda_{i}$ are interlaced, bandwidth can be much less, although not as small as when the $\lambda_{i}$ are absent. However, in a Gauss elimination solution with pivoting on the diagonal, a zero pivot appears if a constraint equation is processed before any of the d.o.f. to which it is coupled. Otherwise, the null submatrix fills in and the solution proceeds normally if the stiffness matrix [K] is by itself positive definite.
+
+<!-- source-page: 296 -->
+
+The Lagrange multiplier method is more attractive than the transformation method of Section 9.1 if there are few constraint equations that couple many d.o.f. However, Lagrange multipliers are active at the structure level, but transformation equations can be applied at either the structure level or element by element. The latter has the appeal of disposing of constraints at an early stage, when the matrices are small and more manageable.
+
+Example. Again we solve the example problem of Fig. 9.1-2. The constraint equation is Eq. 9.1-8. Equation 9.2-2 assumes the form
+
+$$
+\left[ \begin{array}{c c c c} 2 k & - k & 0 & 0 \\ - k & 2 k & - k & 1 \\ 0 & - k & k & - 1 \\ 0 & 1 & - 1 & 0 \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \\ u _ {3} \\ \lambda \end{array} \right\} = \left\{ \begin{array}{l} P \\ P \\ P \\ 0 \end{array} \right\} \tag {9.2-3}
+$$
+
+The solution of Eq. 9.2-3 is
+
+$$
+\left\lfloor u _ {1} \quad u _ {2} \quad u _ {3} \quad \lambda \right\rfloor = \left\lfloor \frac {3 P}{k} \quad \frac {5 P}{k} \quad \frac {5 P}{k} \quad - P \right\rfloor \tag {9.2-4}
+$$
+
+The result $\lambda = -P$ can be regarded as the force of constraint applied through the now rigid link 2-3. The algebraic sign of $\lambda$ is not significant: had we written [C] = [0 - 1 1] in Eq. 9.1-8, we would obtain $\lambda = +P$ but the same values of $u_{1}, u_{2}$ , and $u_{3}$ .
+
+# 9.3 PENALTY FUNCTIONS
+
+If the constraint equation $[\mathbf{C}]\{\mathbf{D}\} = \{\mathbf{Q}\}$ , Eq. 9.1-1, is written in the form
+
+$$
+\{\mathbf {t} \} = [ \mathbf {C} ] \{\mathbf {D} \} - \{\mathbf {Q} \} \tag {9.3-1}
+$$
+
+then $\{\mathbf{t}\} = \{\mathbf{0}\}$ implies satisfaction of the constraints. The usual potential $\Pi_p$ of a structural system can be augmented by a penalty function $\{\mathbf{t}\}^T[\alpha]\{\mathbf{t}\} / 2$ , where $[\alpha]$ is a diagonal matrix of "penalty numbers" $\alpha_i$ . Thus
+
+$$
+\Pi_ {p} = \frac {1}{2} \{\mathbf {D} \} ^ {T} [ \mathbf {K} ] \{\mathbf {D} \} - \{\mathbf {D} \} ^ {T} \{\mathbf {R} \} + \frac {1}{2} \{\mathbf {t} \} ^ {T} [ \alpha ] \{\mathbf {t} \} \tag {9.3-2}
+$$
+
+If $\{\mathbf{t}\} = \{\mathbf{0}\}$ the constraints are satisfied and we have added nothing to $\Pi_p$ . If $\{\mathbf{t}\} \neq \{\mathbf{0}\}$ the penalty of constraint violation becomes more prominent as $[\alpha]$ increases.
+
+Next we substitute Eq. 9.3-1 into Eq. 9.3-2 and write the minimum condition $\{\partial \Pi_p / \partial \mathbf{D}\} = \{\mathbf{0}\}$ . Thus, from Eqs. 9.3-1 and 9.3-2,
+
+$$
+\left([ \mathbf {K} ] + [ \mathbf {C} ] ^ {T} [ \alpha ] [ \mathbf {C} ]\right) \{\mathbf {D} \} = \{\mathbf {R} \} + [ \mathbf {C} ] ^ {T} [ \alpha ] \{\mathbf {Q} \} \tag {9.3-3}
+$$
+
+in which $[\mathbf{C}]^T [\alpha ][\mathbf{C}]$ can be called the penalty matrix. If $[\alpha ] = [\mathbf{0}]$ , the constraints are ignored., As $[\alpha ]$ grows, $\{\mathbf{D}\}$ changes in such a way that the constraint equations are more nearly satisfied. The analyst is responsible for selecting appropriate numerical values of the $\alpha_{i}$ .
+
+Preferably, for a reason that will subsequently be explained, penalty numbers
+
+<!-- source-page: 297 -->
+
+$\alpha_{i}$ are dimensionless. Equations 9.3-1 can easily be written in such a way that the $\alpha_{i}$ are dimensionless if d.o.f. coupled by the constraint equation are all of the same type, for example, all translations or all rotations. If d.o.f. are different types, some types can be redefined to agree with the others (e.g., $L\theta_{i}$ can replace $\theta_{i}$ ); however, the labor of making such a change may outweigh its benefits.
+
+The method of imposing a prescribed zero or nonzero d.o.f. $D_{i}$ by adding large numbers to $K_{ij}$ and $R_{i}$ is a penalty method. For example, in Fig. 2.10-6 the penalty matrix added to [K] contains a single coefficient—namely, the large spring stiffness $k_{s}$ . In this example, $\alpha$ is dimensionless if we define $k_{s} = \alpha k$ and the constraint as $\sqrt{k} v_{1} = 0$ , where $k$ is a spring stiffness whose magnitude is approximately the same as a typical $K_{ij}$ already present in [K].
+
+Example. Imagine that the constraint $u_{1} = u_{2}$ is to be imposed on the structure of Fig. 9.3-1.
+
+There is no unique way to write the constraint relation. We will write [C] in such a way that the penalty numbers are dimensionless. Thus for Eq. 9.3-1 we elect to write
+
+$$
+[ \mathbf {C} ] = \left\lfloor \sqrt {k} - \sqrt {k} \right\rfloor \quad \{\mathbf {D} \} = \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \end{array} \right\} \quad \{\mathbf {Q} \} = \{\mathbf {0} \} \tag {9.3-4}
+$$
+
+where $k = AE / L$ . With but one constraint, $[\alpha] = \alpha$ , a scalar. Equation 9.3-3 becomes
+
+$$
+\left(\left[ \begin{array}{c c} 2 k & - k \\ - k & k \end{array} \right] + \alpha \left[ \begin{array}{c c} k & - k \\ - k & k \end{array} \right]\right) \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \end{array} \right\} = \left\{ \begin{array}{l} P \\ P \end{array} \right\} \tag {9.3-5}
+$$
+
+which has the solution
+
+$$
+u _ {1} = \frac {2 P}{k} \quad \text { and } \quad u _ {2} = \frac {3 + 2 \alpha}{1 + \alpha} \frac {P}{k} \tag {9.3-6}
+$$
+
+If $\alpha = 0$ , then $u_{2} = 3P / k$ , as expected. As $\alpha$ becomes large, $u_{2}$ approaches the value $2P / k$ , which is correct for the constrained system. Note that $u_{1} - u_{2} = -P / k(1 + \alpha)$ and $\{\mathbf{t}\} = t = \sqrt{k}(u_{1} - u_{2})$ , so that the coefficient of $\alpha$ in the penalty function approaches zero as $\alpha$ approaches infinity.
+
+The symbolic manipulations that produce Eq. 9.3-6 from Eq. 9.3-5 obscure a difficulty that may arise if the manipulations are done numerically. The second square matrix in Eq. 9.3-5 is recognized as the stiffness of a bar element that spans nodes 1 and 2 (i.e., a “constraint” bar in parallel with the bar already there).
+
+![](images/page-297_dc72799c5458e18706948fa3986351c73c08750f4ad0ebb5a70a0af42b247dae.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+1
+2
+x,u
+P
+P
+L
+L
+</details>
+
+Figure 9.3-1. Two identical bar elements, each of axial stiffness $k = AE/L$ .
+
+<!-- source-page: 298 -->
+
+As $\alpha$ grows, the structure becomes the error-prone case of a stiff region supported by a flexible region (see Section 18.2).
+
+If constraints do not couple all d.o.f. in $\{D\}$ , then $[C]$ has more columns than rows, and $[C]^{T}[\alpha][C]$ is certain to be a singular matrix. In some important problems, discussed in Section 9.4, constraints do couple all d.o.f. in $\{D\}$ , and singularity of $[C]^{T}[\alpha][C]$ is not guaranteed. However, we want this matrix to be singular, as the following argument illustrates.
+
+For simplicity let all $\alpha_{i}$ in $[\alpha]$ be the same number, say $\alpha$ . In addition, let $\{\mathbf{Q}\} = \{\mathbf{0}\}$ in Eq. 9.3-3. Then, as $\alpha$ becomes large, Eq. 9.3-3 becomes
+
+$$
+[ \mathbf {C} ] ^ {T} [ \mathbf {C} ] \{\mathbf {D} \} \approx \frac {1}{\alpha} \{\mathbf {R} \} \tag {9.3-7}
+$$
+
+Equation 9.3-7 shows that if $[C]^{T}[C]$ is nonsingular, then as $\alpha$ grows the solution vector $\{D\}$ approaches zero. In other words, the mesh “locks.” Only if $[C]^{T}[C]$ is singular can $\{D\}$ be nonzero. Then the number of independent nonzero $\{D\}$ 's that satisfy Eq. 9.3-7 is equal to the difference between the order of $[C]^{T}[C]$ and its rank. The practical significance of this argument is discussed in subsequent sections.
+
+In comparison with Lagrange multipliers, penalty functions have the advantage of introducing no new variables. However, the penalty matrix may significantly increase the bandwidth (or wave front) of the structural equations, depending on how d.o.f. are numbered and what d.o.f. are coupled by the constraint equation. Implementation of a penalty function can be as easy as assigning a high modulus to an element already in the structure. Penalty functions have the disadvantage that penalty numbers must be chosen in an allowable range: large enough to be effective but not so large as to provoke numerical difficulties $[9.2,9.3]$ .
+
+# 9.4 NATURALLY ARISING PENALTY FORMULATIONS. NUMERICAL INTEGRATION AND CONSTRAINTS
+
+In Section 9.3, constraints are imposed by explicitly adding a penalty matrix to an existing stiffness matrix [K]. In some situations [K] already contains a contribution that can be identified as a penalty matrix. Thus some applications of penalty methods arise naturally in the sense that large stiffnesses with respect to particular deformations can be interpreted as penalty numbers. Here we discuss two of these applications—transverse shear in beams and material incompressibility—which can easily lead to “locking” difficulties unless care is taken in the choice of element type and integration rule. Physical interpretation of the constraints makes it possible to understand the reasons for locking and leads to guidelines that can be used to avoid the difficulty. The guideline presented subsequently is called “constraint counting.” It does not provide a rigorous guarantee of success but can be quite effective in practice. In the present section we identify the constraints precisely, and in the next section we show how to count them.
+
+Mindlin Beam Element. The beam element shown in Fig. 9.4-1a allows transverse shear deformation. The element has four d.o.f., as is usual. However, rotational d.o.f. $\theta_{1}$ and $\theta_{2}$ are not values of dw/dx at the nodes, as is the case with the standard beam element of Eq. 4.2-5. A plane initially normal to the midsurface
+
+<!-- source-page: 299 -->
+
+![](images/page-299_85c68ad30dde89d39b0acfad0cc55cfc4e476ee65965d2f4aa78637c4bf96db2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+w₁
+θ₁
+1
+x,u
+t
+2
+w₂
+θ₂
+L
+</details>
+
+(a)
+
+![](images/page-299_d3ea266a3d5dafb5ecc4bdab3a5ba49669f53219d9dbe86b397a0cd105c0213d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+w
+θ
+u = -zθ
+w_{r,r}
+z
+w
+x,u
+</details>
+
+(b)   
+Figure 9.4-1. (a) A “Mindlin” beam element. (b) Displacements and rotations.
+
+remains plane but not necessarily normal. In Fig. 9.4-1a, $\theta$ represents the rotation of a line that was initially normal to the undeformed longitudinal axis of the beam. We will interpolate $\theta$ and w independently, so that the beam can represent transverse shear deformation. This element is called a Mindlin beam element [9.4]. It can model constant bending moment but not linearly varying bending moment. The Mindlin beam element generalizes to Mindlin plate elements, discussed in Section 11.3. The standard beam element, also known as an Euler beam element, can model linearly varying bending moment but does not account for transverse shear deformation. $^{1}$
+
+Axial normal strain $\epsilon_{x}$ and transverse shear strain $\gamma_{zx}$ in a Mindlin beam element are, from Fig. 9.4-1b,
+
+$$
+\epsilon_ {x} = u _ {, x} = - z \theta_ {, x} \quad \text { and } \quad \gamma_ {z x} = w _ {, x} - \theta \tag {9.4-1}
+$$
+
+where $\theta$ is a small angle of rotation. Thus $\gamma_{zx}$ is taken as constant over the depth t. Nonzero stresses are assumed to consist only of axial normal stress $\sigma_{x}$ and transverse shear stress $\tau_{zx}$ . Accordingly, strain energy in the Mindlin beam element is $U = U_{b} + U_{s}$ , where $U_{b}$ and $U_{s}$ are strain energies of bending and shear, respectively:
+
+$$
+U _ {b} = \frac {1}{2} \int_ {V _ {e}} \frac {\sigma_ {x} ^ {2}}{E} d V = \frac {1}{2} \int_ {V _ {e}} E \epsilon_ {x} ^ {2} d V = \frac {1}{2} \frac {E b t ^ {3}}{1 2} \int_ {0} ^ {L} \theta_ {, x} ^ {2} d x \tag {9.4-2a}
+$$
+
+$$
+U _ {s} = \frac {1}{2} \int_ {V _ {e}} \frac {\tau_ {z x} ^ {2}}{G} d V = \frac {1}{2} \int_ {V _ {e}} G \gamma_ {z x} ^ {2} d V = \frac {1}{2} \frac {G b t}{1 . 2} \int_ {0} ^ {L} (w _ {, x} - \theta) ^ {2} d x \tag {9.4-2b}
+$$
+
+Here E = elastic modulus, G = shear modulus, b is the width of the beam, and 1.2 is the “form factor” that accounts for a parabolic distribution of $\tau_{zx}$ over a rectangular cross section. $^{2}$
+
+$^{1}$ The standard beam element, Eq. 4.2-5, can be modified to account for transverse shear deformation, while retaining the ability to model both constant and linearly varying bending moment [4.2,4.11].   
+$^{2}$ Let a beam have a rectangular cross section of dimensions b by t. If P is the transverse shear force, then $\tau_{zx} = (3P/2bt^{3})(t^{2} - 4z^{2})$ , where z = 0 at the neutral axis. Equation 9.4-2b then yields $U_{s} = 1.2(P^{2}L/btG)/2$ . This result suggests the view that a uniform stress $\tau_{zx} = P/bt$ acts over a modified area A = bt/1.2, so that the same $U_{s}$ results. A uniform stress $\tau_{zx} = G\gamma_{zx}$ is provided by Eq. 9.4-1.
+
+<!-- source-page: 300 -->
+
+Displacement w and rotation $\theta$ are interpolated linearly between nodes:
+
+$$
+w = \frac {L - x}{L} w _ {1} + \frac {x}{L} w _ {2} \quad \text { and } \quad \theta = \frac {L - x}{L} \theta_ {1} + \frac {x}{L} \theta_ {2} \tag {9.4-3}
+$$
+
+In the usual way, from Eqs. 9.4-2 and 9.4-3 we obtain a 4 by 4 element stiffness matrix $[k]=[k_{b}]+[k_{s}]$ , where $[k_{b}]$ resists bending strain $\epsilon_{x}$ and $[k_{s}]$ resists shear strain $\gamma_{zx}$ . With this notation, for a structure,
+
+$$
+([ \mathbf {K} _ {b} ] + [ \mathbf {K} _ {s} ]) \{\mathbf {D} \} = \{\mathbf {R} \} \tag {9.4-4}
+$$
+
+D.o.f. $w_{i}$ and $\theta_{i}$ in $\{\mathbf{D}\}$ are coupled by terms in $[\mathbf{K}_s]$ but not by terms in $[\mathbf{K}_b]$ .
+
+Deflections $\{\mathbf{D}\}$ of a thin beam should be governed by only $[\mathbf{K}_b]$ because transverse shear deformation is negligible. In other words, if $Gbt / 1.2$ becomes much larger than $Ebt^3 / 12$ in Eq. 9.4-2, then $[\mathbf{K}_s]$ , which arises from $U_s$ , should enforce the constraint $\gamma_{zx} = 0$ . But, as a beam becomes slender, $[\mathbf{K}_s]$ grows in relation to $[\mathbf{K}_b]$ . So $[\mathbf{K}_s]$ acts as a penalty matrix that causes Eq. 9.4-4 to yield $\{\mathbf{D}\} = \{\mathbf{0}\}$ — unless $[\mathbf{K}_s]$ is singular. In other words, unless $[\mathbf{K}_s]$ is singular, the computed deflection of a very slender beam is almost zero. A singular $[\mathbf{K}_s]$ can enforce the $\gamma_{zx} = 0$ constraint without locking. As discussed subsequently, a singular $[\mathbf{K}_s]$ can be achieved by reduced integration.
+
+As a simple example of locking, let the beam element of Fig. 9.4-1a be fixed at the left end and loaded by a transverse force P at the right end. Then $w = w_{2}x/L$ , $\theta = \theta_{2}x/L$ , and $\Pi_{p} = U - Pw_{2}$ . For simplicity let $\nu = 0$ , so that E = 2G. Then the equilibrium equations $\partial\Pi_{p}/\partial w_{2} = 0$ and $\partial\Pi_{p}/\partial\theta_{2} = 0$ yield
+
+$$
+w _ {2} = \frac {1 2 (t / L) ^ {2} + 2 0}{1 2 (t / L) ^ {2} + 5} \left(1. 2 \frac {P L}{G A}\right) \tag {9.4-5}
+$$
+
+where A = bt. For a very short beam, we obtain $w_{2} \approx 1.2PL/GA$ , which is the correct expression for the portion of end deflection that is due to transverse shear deformation. For a slender beam, L >> t, we obtain $w_{2} \approx 4.8PL/GA$ . This value includes no bending deformation and is far too small; that is, the beam locks as L/t increases.
+
+Incompressible Materials. As Poisson's ratio $\nu$ approaches 0.5, a material becomes incompressible. Values of $\nu$ near 0.5 occur in rubberlike materials and in materials that flow, such as fluids and plastic solids. Unless the problem is one of plane stress, the value $\nu = 0.5$ is forbidden because denominators become zero in material property matrices [E] (Eqs. 1.7-3 and 1.7-4, for example). It is tempting to approximate incompressibility by using (say) $\nu = 0.49$ . But, near $\nu = 0.5$ , stresses are strongly dependent on $\nu$ . Also, structural equations become ill conditioned as $\nu$ approaches 0.5, for reasons explained next.
+
+The shear modulus G and bulk modulus B of an isotropic material are
+
+$$
+G = \frac {E}{2 (1 + \nu)} \quad B = \frac {E}{3 (1 - 2 \nu)} \tag {9.4-6}
+$$
+
+In terms of $G$ and $B$ , the material property matrix [E] of Eq. 1.7-3 is
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_031.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_031.md
new file mode 100644
index 00000000..8fbbf15a
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_031.md
@@ -0,0 +1,410 @@
+<!-- source-page: 301 -->
+
+$$
+[ \mathbf {E} ] = G \left[ \begin{array}{c c c c c c} 4 / 3 & - 2 / 3 & - 2 / 3 & 0 & 0 & 0 \\ - 2 / 3 & 4 / 3 & - 2 / 3 & 0 & 0 & 0 \\ - 2 / 3 & - 2 / 3 & 4 / 3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right] + B \left[ \begin{array}{c c c c c c} 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right] \tag {9.4-7}
+$$
+
+or, abbreviated, $[E] = G[E_{G}] + B[E_{B}]$ . The element stiffness matrix (Eq. 4.1-5) becomes
+
+$$
+[ \mathbf {k} ] = G \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} _ {G} ] [ \mathbf {B} ] d V + B \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} _ {B} ] [ \mathbf {B} ] d V \tag {9.4-8}
+$$
+
+Therefore, structural equations have the form
+
+$$
+(G [ \mathbf {K} _ {G} ] + B [ \mathbf {K} _ {B} ]) \{\mathbf {D} \} = \{\mathbf {R} \} \tag {9.4-9}
+$$
+
+As $\nu$ approaches 0.5, bulk modulus $B$ approaches infinity. Therefore, $B[\mathbf{K}_B]$ acts as a penalty matrix that enforces the constraint of incompressibility. As $\nu$ approaches 0.5, numerical trouble becomes more likely, and finally the mesh "locks"—unless $[\mathbf{K}_B]$ is singular.
+
+Remarks. In the notation of Eq. 9.3-3, Eqs. 9.4-4 and 9.4-9 correspond to homogeneous constraints, $\{Q\} = \{0\}$ . Therefore, Eqs. 9.4-4 and 9.4-9 arise naturally from minimization of a potential $\Pi_{p}$ that can be stated in the form
+
+$$
+\Pi_ {p} = \beta \int_ {V} (F + \alpha H) d V + P \tag {9.4-10}
+$$
+
+Here F and H are proportional to strain energy densities, $\beta$ is a common factor of F and H, $\alpha$ is the penalty number, and P represents work done by loads of all types (P need not be detailed here). There is no unique way to choose $\beta$ , but if possible it should be chosen in such a way that $\alpha$ is dimensionless. A dimensionless $\alpha$ makes it easier to select a numerical value of $\alpha$ such that ill-conditioning is avoided (see guidelines at the end of this section).
+
+For the Mindlin beam, the correspondence between Eqs. 9.4-2 and 9.4-10 can be as follows. Replace dV by dx in Eq. 9.4-10, and let
+
+$$
+F = \frac {E}{2} \theta_ {, x} ^ {2} \quad H = \frac {E}{2 L _ {T} ^ {2}} (w _ {, x} - \theta) ^ {2} \tag {9.4-11}
+$$
+
+$$
+\beta = \frac {b t ^ {3}}{1 2} \alpha = \frac {1 0 L _ {T} ^ {2} G}{t ^ {2} E} = \frac {5}{1 + \nu} \left(\frac {L _ {T}}{t}\right) ^ {2}
+$$
+
+where t is the beam depth (thickness) and $L_{T} = \Sigma L_{i}$ is the total length of the beam after elements of length $L_{i}$ have been assembled. $^{3}$ We know that beam deflections depend more strongly on $L_{T}$ and t than on the beam width b. Accordingly, b is excluded from F, H, and $\alpha$ , so that the effect of the $L_{T}/t$ ratio is apparent.
+
+<!-- source-page: 302 -->
+
+Since the factor $5 / (1 + \nu)$ varies little for valid choices of $\nu$ , we see that $\alpha$ depends strongly on $(L_T / t)^2$ . When $(L_T / t)^2$ is large, the penalty number $\alpha$ is large. (Note that when $L_T / t$ approaches infinity so does $L_i / t$ , so the foregoing arguments could be restated using element length rather than $L_T$ .)
+
+The incompressible case, Eq. 9.4-9, can be obtained from Eq. 9.4-10 if
+
+$$
+F = \frac {E}{4 (1 + \nu)} \{\boldsymbol {\epsilon} \} ^ {T} \left[ \mathrm{E} _ {G} \right] \{\boldsymbol {\epsilon} \} \quad H = \frac {E}{2} \{\boldsymbol {\epsilon} \} ^ {T} \left[ \mathrm{E} _ {B} \right] \{\boldsymbol {\epsilon} \} \tag {9.4-12}
+$$
+
+$$
+\beta = 1 \quad \alpha = \frac {1}{3 (1 - 2 \nu)}
+$$
+
+The penalty number $\alpha$ becomes large as $\nu$ approaches 0.5.
+
+Constraints and Quadrature Points. We wish to show that the number of penalty function constraints is proportional to the number of sampling points used to integrate element matrices.
+
+Numerical integration of stiffness matrices (such as $[K_{b}]$ , $[K_{s}]$ , $[K_{G}]$ , and $[K_{B}]$ in Eqs. 9.4-4 and 9.4-9) corresponds to evaluation of energy $\Pi_{p}$ by numerical integration. Thus Eq. 9.4-10 can be written as
+
+$$
+\Pi_ {p} \approx \beta \sum_ {I = 1} ^ {\text {numel}} \left[ \sum_ {i = 1} ^ {n} \left(F _ {I} J _ {I}\right) _ {i} W _ {i} + \alpha \sum_ {j = 1} ^ {m} \left(H _ {I} J _ {I}\right) _ {j} T _ {j} \right] + P \tag {9.4-13}
+$$
+
+where $(F_{I}J_{I})_{i}$ and $(H_{I}J_{I})_{j}$ are values of functions F and H times the Jacobian J, evaluated at the ith or jth sampling point, and $W_{i}$ and $T_{j}$ are positive weights (or weight products) appropriate to the integration rule. If n = m and sampling points i and j are the same, the integration scheme is uniform; otherwise it is selective. The integration scheme is called full if enough sampling points are used to provide exact integration of all stiffness coefficients of an undistorted element (e.g., a rectangular element). If fewer sampling points are used—that is, if either n or m is reduced—the integration scheme is called reduced and $\Pi_{p}$ is said to be under-integrated. In what follows we assume that kinematic modes of deformation are either impossible or are suppressed by boundary conditions (kinematic modes are discussed in Section 6.12).
+
+Now consider the effect of letting $\alpha$ become very large in Eq. 9.4-10. If the correct $\Pi_{\rho}$ is to be closely approximated, a large $\alpha$ must be associated with a zero or near-zero value of the second summation in Eq. 9.4-13. Since $(J_{I})_{j} > 0$ and $T_{j} > 0$ , the desired condition is that $(H_{I})_{j} = 0$ . From equations such as Eqs. 9.4-11 and 9.4-12, we see that $H = 0$ implies satisfaction of the constraint (zero shear strain and zero volume change in the respective cases). Thus, in these examples, each integration point used to evaluate the penalty matrix imposes a constraint, and the total number of constraints in the structure is the number of elements times the number of penalty integration points per element.
+
+The preceding discussion suggests that locking difficulties may be avoided by applying reduced integration to terms that yield a penalty matrix. Thus, for example, we would use a one-point rule to evaluate $U_{s}$ of Eq. 9.4-2b when the shape functions of Eq. 9.4-3 are used (for which two-point integration would be exact). Further discussion and other examples appear in Section 9.5.
+
+In the foregoing examples, material property matrices used in generating the
+
+<!-- source-page: 303 -->
+
+penalty matrix have rank one (E in Eq. 9.4-11, $[E_{B}]$ in Eq. 9.4-12). In other problems, the analogous matrices may have rank greater than one. Then the number of penalty constraints per element may be greater than the number of integration points [9.8]. This happens in certain plate-bending formulations, where an integration point constrains two transverse shear strains, namely $\gamma_{yz}$ and $\gamma_{zx}$ (see Section 11.3). In some problems it is conceivable that the number of constraints will exceed the number of d.o.f. This circumstance implies that some of the constraints are redundant.
+
+Guideline for Choice of $\alpha$ . If computer words carry approximately p decimal digits, experience has shown that $\alpha$ should not exceed $10^{p/2}$ if ill-conditioning and numerical difficulty are to be avoided. If this guideline is followed, coefficients of [K] in Eq. 9.3-3 influence the latter p/2 digits in computer words used to store the complete matrix $[K + C^{T}\alpha C]$ . Typically $10^{p/2}$ is $10^{3}$ to $10^{4}$ in single precision and $10^{6}$ to $10^{7}$ in double precision. If material properties yield a larger value of $\alpha$ , it is best to lower $L_{T}/t$ (Eq. 9.4-11) or $\nu$ (Eq. 9.4-12) artificially so that the guideline is satisfied.
+
+The foregoing choice of $\alpha$ is made after one has chosen an integration rule that avoids locking of the mesh. Even without locking, the penalty matrix enforces constraints, and it is to avoid numerical difficulty associated with these remaining constraints that one takes care in the choice of $\alpha$ .
+
+# 9.5 CONSTRAINT COUNTING
+
+We seek a guideline for choosing a suitable numerical integration formula in problems where penalty constraints arise naturally. Specifically, if the number of constraints is proportional to the number of sampling points used to integrate the penalty matrix, how many points per element should be used?
+
+In what follows we continue to assume that all weight factors in quadrature rules are positive. Otherwise, it is conceivable that terms of a summation will cancel one another. This would confuse the counting rule.
+
+Mesh Locking. Let the beam in Fig. 9.5-1a be built of the shear-flexible beam elements of Fig. 9.4-1. Support conditions suppress w and $\theta$ d.o.f. at the fixed
+
+![](images/page-303_fd0fd86272cb10d263f7476ce7833d75fe77519624cd5262b96cd64ea8a00ec7.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L
+L
+L
+t
+P
+①
+②
+Nel
+</details>
+
+(a)
+
+![](images/page-303_7f51eb6f88960d514009af908d9af09e8ddc5ba1781c4b0cd93002f4375944ea.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Nes
+N2
+es
+Nes
+</details>
+
+(b)   
+Figure 9.5-1. (a) Cantilever beam built of $N_{el}$ elements. (b) An $N_{es}$ by $N_{es}$ mesh of bilinear elements. The case $N_{es} = 4$ is shown.
+
+<!-- source-page: 304 -->
+
+end, leaving $2N_{e\ell}$ active d.o.f. in $\{D\}$ . Now imagine that $[K_{s}]$ of Eq. 9.4-4 is generated by use of two sampling points per element (thus, $[K_{s}]$ is integrated exactly). Then, if the beam is slender, there are $2N_{e\ell}$ penalty constraints. All d.o.f. $\{D\}$ are now used to satisfy the constraint $w_{,x} - \theta \approx 0$ , and the computed deflection of load P is nearly zero. This conclusion is unchanged by changing the supports. With no supports, rigid-body motion of the “locked” beam becomes possible.
+
+The same situation prevails if the beam is built of bilinear plane elements (four nodes, eight d.o.f., as in Table 6.14-1) integrated with a 2 by 2 Gauss rule. Now the beam contains $4N_{e\ell}$ active d.o.f. However, owing to the “parasitic shear” discussed in Section 8.3, there are $4N_{e\ell}$ constraints if the beam is thin. Again the mesh locks as the length-to-thickness ratio becomes large.
+
+In Fig. 9.5-1b there are two d.o.f. per node and therefore $2N_{es}^{2}$ active d.o.f. Let the material be nearly incompressible; that is,
+
+$$
+\epsilon_ {V} = u _ {, x} + v _ {, y} + w _ {, z} \approx 0 \tag {9.5-1}
+$$
+
+where $\epsilon_{V}$ is the volumetric strain. If the plane strain condition $w_{,z} = 0$ prevails, then the penalty constraint $u_{,x} + v_{,y} \approx 0$ is enforced at each integration point. With a 2 by 2 integration rule there are $4N_{es}^{2}$ constraints, and the mesh is locked.
+
+In the foregoing examples, the ratio of number of d.o.f. to number of penalty constraints is 1/1 in Fig. 9.5-1a and 1/2 in Fig. 9.5-1b. Note that these same ratios can be determined from a single element. Consider the addition to the mesh of a single element, such as the element shaded in Fig. 9.5-1b. It brings two additional d.o.f. to the mesh, and under a 2 by 2 Gauss rule it also brings four additional constraints, for a d.o.f.-to-constraint ratio of 1/2, as previously determined. For the mesh as a whole, the 1/2 ratio would still be approximately correct if support conditions are changed, provided that $N_{es}$ is large. Accordingly, we will henceforth presume that meshes contain many elements, and do our “constraint counting” by examining the additional d.o.f. and constraints that are brought to a mesh by adding a single element (or perhaps a single “macroelement” built of subelements).
+
+Desirable Constraint Ratios. We define the constraint ratio $r$ as the ratio of the number of active d.o.f. in $\{\mathbf{D}\}$ to the number of penalty constraints. Locking occurs if $r \leq 1$ . If $r$ is slightly greater than unity, the mesh does not lock, but poor results are likely because most d.o.f. are occupied in satisfying penalty constraints; few d.o.f. are left to model the elastic behavior of the system. Extensive numerical testing has shown that near-optimal constraint ratios are $r = 2/1$ for two-dimensional problems and $r = 3/1$ for three-dimensional problems. In each case these ratios correspond to the number of differential equations of equilibrium (two and three for plane and solid problems, respectively) divided by the number of constraint conditions on the system of governing differential equations (one constraint; Eq. 9.5-1 for incompressibility or $\gamma_{zx} = 0$ for Mindlin beams).
+
+A favorable constraint ratio is usually achieved by underintegration or by selective reduced integration—that is, by use of m < n in Eq. 9.4-13. In the context of Eq. 9.4-9 this means using a lower-order Gauss quadrature rule for $[K_{B}]$ than for $[K_{G}]$ .
+
+If quadrature points are unsymmetrically distributed, such an element may not be geometrically isotropic. This should be of no consequence, because lack of geometric isotropy is annoying only in a coarse mesh, and a rather fine mesh is needed to produce accurate results if the material is nearly incompressible.
+
+<!-- source-page: 305 -->
+
+Further Examples. With the Mindlin beam element of Fig. 9.4-1, it suffices to use uniform integration with $m = n = 1$ — that is, a single Gauss point for both $[\mathbf{k}_b]$ and $[\mathbf{k}_s]$ . This integration is exact for $[\mathbf{k}_b]$ because $\theta_{,x}$ is not a function of $x$ . However, $w_{,x} - \theta$ is linear in $x$ , so $[\mathbf{k}_s]$ is underintegrated by a single Gauss point. Thus two d.o.f. and one constraint are added to the mesh by each element, for an ideal constraint ratio of $r = 2/1$ . As an exercise, one can show that in place of Eq. 9.4-5, one now obtains a far more accurate result for a tip-loaded cantilever element. Additional discussion of transverse shear constraints can be found in [9.7,9.8] and in Section 11.3.
+
+Figure 9.5-2 shows elements that might be used for nearly incompressible media (Eq. 9.4-9). Plane and solid elements have respectively two and three d.o.f. per node. For the bilinear and trilinear elements, it is appropriate to use one-point integration to obtain $[K_{B}]$ : thus, for Figs. 9.5-2a and 9.5-2b, we obtain r = 2/1 and r = 3/1, respectively, which are the optimal ratios. In Fig. 9.5-2c, depending on whether the element has eight or nine nodes, uniform 3 by 3 integration gives r = 6/9 or r = 8/9, respectively, and selective reduced integration with a 2 by 2 rule for the penalty matrix gives r = 6/4 or r = 8/4. The latter is optimal.
+
+For convenience, triangles in Fig. 9.5-2d are considered as a two-element patch that brings eight d.o.f. to a mesh. No integration rule for the penalty matrix is entirely satisfactory: three points per triangle gives r = 8/6 (too low), and one point per triangle gives r = 8/2 (too high). Indeed, were the triangles to have vertex nodes only, even one point per triangle would be too high: thus r = 2/2, which means that a mesh would lock. These difficulties with triangles for plane problems suggest that tetrahedra for solid problems would not work well. However, the question of what approach is best for solid problems is not yet settled.
+
+In distorted meshes, one may find that reduced integration is not adequate to represent element volume exactly. This difficulty disappears with mesh refinement if subdivision causes elements to become parallelograms or parallelepipeds. The difficulty can be avoided by use of the consistent penalty method, which has additional attributes to recommend it (see Section 9.6).
+
+# 9.6 ADDITIONAL TECHNIQUES FOR INCOMPRESSIBLE MEDIA
+
+Deviatoric–Dilatational Splitting. It is convenient to regard stresses $\{\sigma\}$ as being composed of a deviatoric state $\{\sigma_{D}\}$ (which produces no change of volume) and a dilatational state $\{\sigma_{V}\}$ (which produces no change of shape). Thus, from Eq. 9.4-7,
+
+$$
+\{\boldsymbol {\sigma} \} = \{\boldsymbol {\sigma} _ {D} \} + \{\boldsymbol {\sigma} _ {V} \} = G [ \mathrm{E} _ {G} ] \{\boldsymbol {\epsilon} \} + B [ \mathrm{E} _ {B} ] \{\boldsymbol {\epsilon} \} \tag {9.6-1}
+$$
+
+![](images/page-305_68b7db65f6e7af2cb606d995529dc44aa30fc0fd413a8725def3d7b6c56013d7.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Simple geometric diagram showing a square with a vertical hatched border (no text or symbols)
+</details>
+
+(a)
+
+![](images/page-305_d49ffa6f8b1e7e3bf51497b1f672ec183b9faa4d36d009824c6ced95290abc04.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Simple line drawing of a 3D cube with shaded edges and vertices (no text or symbols)
+</details>
+
+(b)
+
+![](images/page-305_7e2b37236b6681807acc6244805649e5f8c3d1d1c093510b3bec72339bf3b8e8.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Simple geometric diagram of a square frame with hatched sides and a small circle at the center (no text or symbols)
+</details>
+
+(c)
+
+![](images/page-305_837bd743a7abda5b0afefc3c6b32819f3ffda4666b65186bf5eb922ddf568e78.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Simple geometric diagram showing a square with diagonal line and marked points, no text or symbols present
+</details>
+
+(d)   
+Figure 9.5-2. (a) Plane bilinear element. (b) Solid trilinear element. (c) Plane quadratic element. (d) Two plane triangular elements.
+
+<!-- source-page: 306 -->
+
+We discover that $\{\sigma_{V}\}$ contains three equal normal stresses and no shear stresses. Specifically, in terms of the volumetric strain $\epsilon_{V}$ stated in Eq. 9.5-1, nonzero stresses in $\{\sigma_{V}\}$ are
+
+$$
+\sigma_ {x V} = \sigma_ {y V} = \sigma_ {z V} = \lambda , \quad \text { where } \quad \lambda = B \epsilon_ {V} \tag {9.6-2}
+$$
+
+in which $\lambda$ is called the hydrostatic pressure function and B is the bulk modulus, defined in Eq. 9.4-6. Equations 9.6-1 and 9.6-2 yield $\lambda = (\sigma_{x} + \sigma_{y} + \sigma_{z})/3$ . (In the theory of plasticity, where $\{\sigma_{D}\}$ and $\{\sigma_{V}\}$ are also used, $\lambda$ is called the mean stress.) With $\lambda$ from Eq. 9.6-2, Eq. 9.6-1 assumes the form
+
+$$
+\{\boldsymbol {\sigma} \} = \left\{\boldsymbol {\sigma} _ {D} \right\} + \lambda \left[ \begin{array}{l l l l l l} 1 & 1 & 1 & 0 & 0 & 0 \end{array} \right] ^ {T} \tag {9.6-3}
+$$
+
+where $\{\sigma_D\} = G[\mathbf{E}_G]\{\epsilon\}$ and $G[\mathbf{E}_G]$ is stated in Eq. 9.4-7. For a completely incompressible material $\lambda$ is a “volumetric stress,” which can be regarded as a system of pressures that keeps the body from dilatating.
+
+One may argue that if Poisson's ratio approaches 0.5 while all other parameters of the problem are held fixed, the incompressible pressure field is, at each material point, the limit of the slightly compressible pressure $\lambda$ computed from Eq. 9.6-2 [9.9]. The penalty method provides a way to "perturb" the exactly incompressible solution slightly and thus obtain $\lambda$ as a good approximation of the exactly incompressible pressure.
+
+Pressure Calculation in the Penalty Method. In the penalty method, one can calculate pressure $\lambda$ by evaluating $\epsilon_{V}$ from the displacement field and then using Eq. 9.6-2. The error in $\lambda$ associated with use of a penalty constraint rather than an exact constraint is of order $10^{-p/2}$ if $\alpha$ is chosen as $10^{p/2}$ , as suggested at the end of Section 9.4. $^{5}$ However, this error estimate is valid only if $\lambda$ is calculated at locations where the constraint is enforced—that is, at the Gauss points used to evaluate element matrices $[k_{B}]$ . For the penalty method, then, the (reduced) volumetric integration points of Eq. 9.4-13 play a three-part role: that of “pressure points,” that of constraint points, and that of volumetric integration points.
+
+If pressures are desired at other points, techniques of Section 6.13 can be used to extrapolate pressures to displacement nodes or to any other points in an element. Expressions analogous to Eq. 6.13-5 can be developed for any of the elements discussed in Section 9.5. However, for many of those elements, pressures are susceptible to an instability akin to the hourglassing discussed in Section 6.12. Unfortunately, this is particularly true of elements such as the bilinear and quadratic elements of Fig. 9.5-2 when the standard one-point or four-point reduced formulas are selectively applied to the volumetric terms of Eq. 9.4-13. One way to avoid this is by postprocessing the computed pressures using an “averaging” or “smoothing” scheme to smooth out the penalty pressures. This can be rigorously justified in terms of error analysis [9.11-9.13], and is not difficult to implement [9.7,9.14].
+
+<!-- source-page: 307 -->
+
+Consistent Penalty Method. The consistent penalty method provides an alternative way to calculate $[K_{B}]$ . Reduced integration is not required, and incompressibility constraints are imposed at certain “pressure points” rather than at integration points. Thus, as compared with the preceding penalty method, constraint points are divorced from integration points, and the choice of integration rule for $[K_{B}]$ is not dictated by constraint counting. Constraint counting is still used, but now to achieve a balance between the number of pressure points and the number of displacement d.o.f. If pressure points are well chosen, the aforementioned pressure instabilities will not arise. A full description of the method is beyond the scope of this book: it is a special case of the “[B] method” of [9.7], and details appear in [9.10]. The following summary is offered.
+
+Let $\lambda_{i}$ represent hydrostatic pressure at the pressure points, such as the three shown in Fig. 9.6-1. Pressure $\lambda$ over the element is interpolated from these $\lambda_{i}$ . Thus $\lambda$ is interpolated independently of d.o.f. at element nodes. The formulation procedure leads to equations like Eq. 9.2-2, except that (a) $\{\mathbf{Q}\} = \{\mathbf{0}\}$ , and (b) the lower-right submatrix [0] is replaced by a square nonsingular matrix that represents slight compressibility.
+
+An important feature of pressure points is that they are not shared by adjacent elements, even if pressure points are placed on element boundaries. Thus the pressure points $\lambda_{i}$ are internal d.o.f. that can be eliminated by condensation before elements are assembled (see Eq. 8.1-1, and let $\{d_{c}\} = \{\lambda\}$ ). The result of condensation and assembly is a set of equations like Eq. 9.4-9, obtained with computational efficiency comparable to that of the reduced-integration penalty method, and with an improved $[K_{B}]$ .
+
+The nine-node, three-pressure-point element of Fig. 9.6-1, in consistent penalty form and with $\alpha = 10^{p/2}$ , is the best element known for two-dimensional incompressible elasticity and fluid flow. That the constraint ratio is r = 8/3, rather than the optimal r = 2/1, evidently causes no ill effects. There are rigorous error bounds for the element, showing that the pressure is as accurate as strains in compressible elasticity [9.10]. There are no difficulties with spurious pressure modes.
+
+In three-dimensional problems, the situation is not nearly as clear. There appears to be no three-dimensional analogue of the two-dimensional quadratic element with three pressure points. At present, we recommend the eight-node brick of Fig. 9.5-2b with one pressure point, using the consistent formulation or the selective/reduced formulation (if the elements are not severely distorted by isoparametric transformations). For undistorted elements these two formulations are identical [9.7,9.10]. Unfortunately, this element has spurious modes, and pressure smoothing is required. We recommend the pressure-smoothing scheme described in [9.14]. Development of good elements for incompressible media in three dimensions is an active area of current finite element research.
+
+![](images/page-307_53dfa83e8093cbb5d83f95293c7c8c37175e5758cad88585e999122f71e5b55a.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Simple geometric diagram with a rectangle and four marked points (no text or symbols)
+</details>
+
+Figure 9.6-1. A plane element with nine nodes and three pressure points, used in a consistent penalty method [9.10].
+
+<!-- source-page: 308 -->
+
+# PROBLEMS
+
+# Section 9.1
+
+9.1 In Fig. 9.1-1, let members CA, AB, and BD be identical. Also assume that conditions $v_{A} = v_{B} = 0$ have already been imposed, so that [K] operates on d.o.f. $u_{A}, \theta_{A}, u_{B}$ , and $\theta_{B}$ . Write [K], then condense it to a 3 by 3 matrix by imposing the constraint $u_{A} = u_{B}$ , so that $\{D\}$ becomes $\{D\} = \left[u_{A} \quad \theta_{A} \quad \theta_{B}\right]^{T}$ .   
+9.2 Let the quadratic element of Eqs. 6.6-1 have straight sides and midside nodes. Imagine that the displacement of each side node is to be the average of displacements of the two adjacent corner nodes. Write the appropriate form of Eq. 9.1-4. (You may wish to consider the $u_{i}$ separately from the $v_{i}$ .) What kind of element will be produced by applying these constraints? Verify your prediction.   
+9.3 Write Eq. 7.8-5 in the form of Eq. 9.1-2. How many equations of constraint are there?   
+9.4 Write the specific form of Eq. 9.1-2 appropriate to Fig. 8.1-2; that is, write the equation that joins the two frames at $A$ with a hinge connection. For simplicity, include in your equations only the three d.o.f. of each frame at $A$ (six d.o.f. altogether). Identify matrices $[\mathbf{C}_r]$ and $[\mathbf{C}_c]$ in your formulation.   
+9.5 The bar element shown has axial stiffness $k = AE / L$ and axially directed d.o.f. $u_{1}$ and $u_{2}$ . Using the procedure of Section 9.1, solve for $u_{1}$ if the constraint $u_{2} = 0$ is imposed.   
+9.6 (a) Let $\{\mathbf{Q}\}$ be nonzero in Eq. 9.1-1. Hence, derive the equation analogous to Eq. 9.1-6. (b) Let the constraint $u_{2} = \overline{u}$ be applied in Problem 9.5. Use the method of part (a) to determine $u_{1}$ .   
+9.7 Element 2 is to be connected to element 1, as shown. Explain in detail how to treat the stiffness matrix of element 2 before assembly so that node 3 is constrained to lie on linear edge 1–2. (D.o.f. of nodes 1 and 2, but not of node 3, are to appear in the structural equations.)
+
+![](images/page-308_72e51e335d0c8ab8d2378b813dc8c88ca9e260eb2660cd34b0be34813300bc3f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P → 1 A,E 2 → x,u
+</details>
+
+Problem 9.5
+
+![](images/page-308_59cdf4c865c5c6759f33517137bb74cd07fd63429d3d19379adf67eb279d0c94.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+L
+a
+①
+②
+3
+2
+</details>
+
+Problem 9.7
+
+9.8 Three nodes lie on an $x$ axis at coordinates $x_1, x_2,$ and $x_3$ . Write a relation in the form of Eq. 9.1-1 that constrains their $x$ -direction displacements to be directly proportional to $x$ .   
+9.9 If Fig. 9.1-2 were to represent three identical beam elements under lateral load, what constraint among d.o.f. would be enforced by the “same node number” device (described below Eq. 9.1-9) applied to nodes 2 and 3? Would [k] of the right element contribute to stiffness of the structure? If so, in what way?
+
+<!-- source-page: 309 -->
+
+9.10 The two rigid links AB and BC are connected by a hinge at B and are supported by identical springs at A, B, and C, as shown. Write the structural equations that use $v_{A}$ , $v_{B}$ , and $v_{C}$ as d.o.f. Then impose the constraint that the hinge does not allow relative rotation between the two links, and solve for $v_{A}$ , $v_{B}$ , and $v_{C}$ .
+
+![](images/page-309_443650618ad77eec140d48aede3bd37111d38a61acfbf5aafe0a10c7a8b06c8d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+L
+L
+A
+B
+C
+P
+x
+k
+k
+k
+</details>
+
+Problem 9.10
+
+9.11 The two-element uniform cantilever beam shown is built of standard beam elements (Eq. 4.2-5). Imagine now that element 2-3 is to be made rigid. Impose this constraint, eliminate $w_{3}$ and $\theta_{3}$ , solve for $w_{2}$ and $\theta_{2}$ , and compare these results with the prediction of elementary beam theory.
+
+9.12 Two identical standard beam elements (Eq. 4.2-5) are joined at node 2 and are simply supported at nodes 1, 2, and 3, as shown. Node 3 is loaded by moment $M_0$ .
+
+(a) Impose the constraint $\theta_{2} = \theta_{3}$ and solve for all three rotational d.o.f. in terms of $M_0, L, E$ , and $I$ .   
+(b) Sketch a free-body diagram of the constrained two-element structure.
+
+![](images/page-309_a95e7cf695635b815317e714b40a28a2f158d7c41ed866b00d80958da60cbb83.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+3
+P
+a
+a
+</details>
+
+Problem 9.11
+
+![](images/page-309_1fc31c6b8e5f7c8b1794b18c6d84380e25499075259176b52f273fe1e2175b7d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+3
+M₀
+L
+L
+</details>
+
+Problem 9.12
+
+# Section 9.2
+
+9.13 The following question is strictly mathematical, and serves as a review of the Lagrange multiplier method. What is the area of the largest rectangle that can be inscribed in the ellipse $(x / a)^2 + (y / b)^2 - 1 = 0$ ?   
+9.14 Solve the problem of Fig. 9.1-3 (i.e., impose $v_{1} = v_{2}$ ) by use of a Lagrange multiplier.   
+9.15 A bar of axial stiffness $k = AE / L$ lies along the $x$ axis and is allowed only axial displacements $u$ . Its right end carries a force $P = 3$ in the $+x$ direction. Its left end (node 1) is to be displaced two units to the right. Impose displacement $u_{1} = 2$ and solve for $u_{2}$ by use of a Lagrange multiplier.   
+9.16 To model the uniform cantilever beam shown, use a single standard beam element (Eq. 4.2-5). Use the Lagrange multiplier method to determine the deflection of load P under the following constraint conditions.
+
+(a) The right end is to remain tangent to a straight line between nodes 1 and 2.
+
+(b) The right end is to rotate half as much as the midpoint of the beam, but in the opposite direction.
+
+<!-- source-page: 310 -->
+
+![](images/page-310_fc6adef540bd6645c63c15d53e8d6d055ed647e56ea88aed10e57aad5c33c8ac.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+P
+L
+</details>
+
+Problem 9.16
+
+# Section 9.3
+
+9.17 (a) Derive Eq. 9.3-3 from Eqs. 9.3-1 and 9.3-2. (b) Revise the argument associated with Eq. 9.3-7: do not make the simplifying assumptions $\lceil \alpha \rfloor = \lceil \mathbf{I} \rceil \alpha$ and $\{\mathbf{Q}\} = \{\mathbf{0}\}$ .
+
+9.18 Solve the problem of Fig. 9.1-3 (i.e., impose $v_{1} = v_{2}$ ) by use of a penalty number.
+
+9.19 Repeat Problem 9.15, but use a penalty number $\alpha$ instead of a Lagrange multiplier. For $k = 1$ , tabulate $u_{1}$ and $u_{2}$ for the values $\alpha = 1, 4, 10$ , and 100.
+
+9.20 Use the penalty method to solve
+
+(a) Problem 9.16a.   
+(b) Problem 9.16b.
+
+# Section 9.4
+
+9.21 Derive Eq. 9.4-5.
+
+9.22 Use one-point quadrature to evaluate $U_{s}$ in Eq. 9.4-2. Hence, use the equations $\partial\Pi_{p}/\partial w_{2}=0$ and $\partial\Pi_{p}/\partial\theta_{2}=0$ to derive expressions for $w_{2}$ and $\theta_{2}$ for the problem posed in connection with Eq. 9.4-5. Compare these results with exact values.
+
+9.23 (a) Let a Mindlin beam element (Fig. 9.4-1a) be simply supported at nodes 1 and 2 and loaded in pure bending. Use Eqs. 9.4-2 to show that exact integration gives an element strain energy U consistent with an effective moment of inertia $I_{e} = I(1 + GL^{2}/1.2Et^{2})$ , where $I = bt^{3}/12$ for the rectangular cross section.
+
+(b) Show that if $U_{s}$ is integrated by one-point quadrature, the resulting $U$ is consistent with the exact moment of inertia.
+
+9.24 Use Eqs. 9.4-2 and 9.4-3 to evaluate the stiffnesses $[k_{b}]$ and $[k_{s}]$ of the Mindlin beam element in the following ways, and determine the rank of each matrix.
+
+(a) Evaluate $[\mathbf{k}_b]$ by one-point Gauss quadrature.   
+(b) Evaluate $[\mathbf{k}_s]$ by one-point Gauss quadrature. Call the result $[\mathbf{k}_s]_1$ .   
+(c) Evaluate $[\mathbf{k}_s]$ by two-point Gauss quadrature. Call the result $[\mathbf{k}_s]_2$ .
+
+9.25 (a) Use $[\mathbf{k}_b]$ and $[\mathbf{k}_s]_1$ from Problem 9.24 to model a one-element cantilever beam fixed at node 1 (the left end). Load node 2 by moment $M$ only. Solve for $w_2$ and $\theta_2$ . Investigate what happens as $L$ becomes much larger than $t$ .
+
+(b) Repeat part (a) but use $[\mathbf{k}_s]_2$ rather than $[\mathbf{k}_s]_1$ .
+
+9.26 In Problem 9.25, compute the ratio of $w_{2}$ obtained from use of $[\mathbf{k}_s]_1$ to $w_{2}$ obtained from use of $[\mathbf{k}_s]_2$ . If $\nu = 0$ , for what value of $L / t$ is $w_{2}$ only $10\%$ in error?
+
+9.27 A rectangular bilinear element (four nodes) is subjected to the constraint $\iint (\epsilon_x + \epsilon_y) dx dy = 0$ . Show that the same constraint is produced by setting $\epsilon_x + \epsilon_y = 0$ in a one-point Gauss quadrature rule.
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_032.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_032.md
new file mode 100644
index 00000000..24224a03
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_032.md
@@ -0,0 +1,473 @@
+<!-- source-page: 311 -->
+
+# Section 9.5
+
+9.28 Give a detailed explanation of why the constraint ratio r is the same for a one-element test mesh as it is for a square test mesh with $N_{es}$ elements per side.
+
+9.29 The sketch shows a mesh of plane constant-strain triangular elements. The material is incompressible, and plane strain conditions are enforced. All nodes i in contact with the supports are fixed ( $u_{i} = v_{i} = 0$ ).
+
+(a) What is the constraint ratio? Will the mesh lock?   
+(b) Can the constraint ratio be increased by use of reduced integration?   
+(c) What is the constraint ratio if the number of elements is greatly increased?   
+(d) Consider elements 1 and 2, and show that the incompressibility constraint implies $u_{j} = v_{j} = 0$ at node $j$ . Extend this argument to show that the mesh is completely locked.   
+(e) Try some other arrangements of constant-strain triangles to see whether the same conclusion holds.   
+(f) Revise the support conditions so that part (a) of this problem will yield the constraint ratio $1/1$ .
+
+9.30 What are the constraint ratios $r$ (in a mesh of many elements) for the plane and solid Q6 elements of Section 8.3 if integration is done using an order 2 Gauss rule and the material is nearly incompressible?
+
+9.31 The elements shown are (a) a cubic triangle, and (b) a quartic triangle. The stiffness matrix of the cubic triangle can be exactly integrated by the six-point formula of Table 6.8-1 (when material properties and element thickness are constant). Similarly, there is a ten-point formula that integrates the quartic element stiffness matrix exactly. In a refined mesh, for an incompressible material, what is the constraint ratio for each of these exactly integrated elements?
+
+![](images/page-311_0c342161b2bad7130dabe226fc05698d10afb48117c3407a003a321803da41db.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y,v
+j
+①
+②
+x,u
+</details>
+
+Problem 9.29
+
+![](images/page-311_9dd769e79ea35e9f5cfa0f5583e2b701e218b97c0687f6a40cd3c76ffbba9e62.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Simple geometric diagram of a triangle with internal dots (no text or labels)
+</details>
+
+(a)
+
+![](images/page-311_a4945282c8826977c65184d28d3bc9ad8d583dc824115a8e63e6645b06bafbbf.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Simple geometric diagram of a triangle with internal dots (no text or symbols)
+</details>
+
+(b)   
+Problem 9.31
+
+# Section 9.6
+
+9.32 (a) Show that $\lambda = (\sigma_x + \sigma_y + \sigma_z) / 3$ , as stated below Eq. 9.6-2.
+
+(b) Show that $\beta \alpha H$ in Eq. 9.4-10 can be expressed as $B\epsilon_V^2 / 2$ .
+
+9.33 Consider the element shown, which is proposed as a higher-order solid element for incompressible media. It has 20 nodes on the edges and a “bubble” shape function added at $\xi = \eta = \zeta = 0$ . (Shape functions $N_{1}$ through $N_{20}$ are not needed in this problem but may be found in Ref. 2.1, p. 201.)
+
+(a) Give an expression for the shape function $N_{21}(\xi, \eta, \zeta)$ that assures inter-element compatibility.   
+(b) What is the constraint ratio when four internal pressure points (at the nodes of a tetrahedron) are used? (Note: This is an attempt to generalize
+
+<!-- source-page: 312 -->
+
+![](images/page-312_29f13de6aab217094ec5ffc4da4efd86e459194d370aa5f188e686ec73e9dd9e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+η
+ξ
+ξ
+</details>
+
+Problem 9.33   
+Fig. 9.6-1 to a solid element. Little is currently known of the behavior of this element.)
+
+<!-- source-page: 313 -->
+
+# SOLIDS OF REVOLUTION
+
+Analysis methods for axially symmetric bodies are described. Loads may be with or without axial symmetry. Loads without axial symmetry are treated by superposition, using Fourier series.
+
+# 10.1 INTRODUCTION
+
+A solid of revolution is generated by revolving a plane figure about an axis, and is most easily described in cylindrical coordinates r, $\theta$ , and z (Fig. 10.1-1). The geometry is axially symmetric, and if material properties and loads are also axially symmetric, the problem is mathematically two-dimensional. That is, if geometry, support conditions, loads $\{R\}$ , and material property matrix [E] are all independent of $\theta$ , and if the material either is isotropic or has $\theta$ as a principal material direction, then static displacements and stresses are independent of $\theta$ : circumferential displacement v is zero, material points have only u (radial) and w (axial) displacement components, and the nonzero stresses are those shown in Fig. 10.1-1a. The analysis procedure for static problems having axial symmetry is very similar to the procedure used for static problems of plane stress or plane strain. (In a vibration or buckling problem, symmetric and unsymmetric modes should be expected, even if geometry, support conditions, and elastic properties are all axially symmetric. A vibration or buckling analysis that assumes $\theta$ independence would miss all modes that are not axially symmetric.)
+
+If the solid is axially symmetric but the loading is not, displacements and stresses are three-dimensional rather than axially symmetric. A Fourier series method can then be used. The given loading is expressed as the sum of several component loadings, and an analysis is done for each load component. According to the principle of superposition, the original problem is solved by adding the solutions of the component problems. Each component analysis remains mathematically two-dimensional. Thus the original three-dimensional problem is exchanged for a series of two-dimensional problems. The exchange is usually worthwhile because three-dimensional problems are expensive to set up and run.
+
+A finite element model of a solid of revolution has nodal circles rather than nodal points (Fig. 10.1-1). So does a shell of revolution, which is an effective model if the body is thin-walled (Section 12.4). If a body of revolution (having nodal circles) must be attached to a solid body (having nodal points), there is some difficulty in making the connection.
+
+Finite element analysis for axially symmetric solids was first published in 1965 [10.1]. Computer programs are readily available [10.2]. Indeed, minor additions
+
+<!-- source-page: 314 -->
+
+![](images/page-314_306b8a28f5c2f4f51011a2e8452ba39686127bcf45005a200eb2938617f26915.jpg)
+
+<details>
+<summary>text_image</summary>
+
+r,u
+z,w
+θ,v
+σz
+τzr
+σθ
+(a)
+β
+y'
+x'
+3
+4
+z,w
+y'
+β
+4
+3
+x'
+2
+1
+Axis
+r,u
+Nodal circle 2
+(b)
+</details>
+
+Figure 10.1-1. An axially symmetric finite element of rectangular cross section. (a) Isometric view, showing stresses produced by axially symmetric loading. (b) Cross section containing the z axis. Hatching suggests an orthotropic material whose principal directions are $x'$ , $y'$ , and $\theta$ .
+
+to a program for plane problems makes the program capable of analyzing solids of revolution as well.
+
+# 10.2 ELASTICITY RELATIONS FOR AXIAL SYMMETRY
+
+If the analysis problem is axially symmetric, then (see Fig. 10.1-1)
+
+$$
+v = 0 \text { and } \tau_ {r \theta} = \tau_ {\theta z} = \gamma_ {r \theta} = \gamma_ {\theta z} = 0 \tag {10.2-1}
+$$
+
+Equations 10.2-1 prevail if geometry, support conditions, and loading are all axially symmetric, $\theta$ is a principal material direction, and $\beta$ in Fig. 10.1-1b is independent of $\theta$ . Thus the material may be isotropic. Or, if orthotropic, principal material axes $x'$ and $y'$ must not change direction with $\theta$ , and the third principal material axis must not form a helix. Accordingly, the most general stress–strain relation allowed has the form
+
+$$
+\left\{ \begin{array}{l} \sigma_ {r} \\ \sigma_ {\theta} \\ \sigma_ {z} \\ \tau_ {z r} \end{array} \right\} = \left[ \begin{array}{c c c c} E _ {1 1} & E _ {1 2} & E _ {1 3} & E _ {1 4} \\ \varepsilon_ {2 1} & E _ {2 2} & E _ {2 3} & E _ {2 4} \\ E _ {3 1} & \varepsilon_ {3 2} & E _ {3 3} & E _ {3 4} \\ \text {symm} _ {4 1} & \varepsilon_ {4 2} & \varepsilon_ {4 3} & E _ {4 4} \end{array} \right] \left(\left\{ \begin{array}{l} \epsilon_ {r} \\ \epsilon_ {\theta} \\ \epsilon_ {z} \\ \gamma_ {z r} \end{array} \right\} - \{\epsilon_ {0} \}\right) \tag {10.2-2}
+$$
+
+in which $\{\epsilon_{0}\}$ represents initial strains, and the trivial relations $\tau_{r\theta}=0$ and $\tau_{\theta z}=0$ are simply not written. If, in addition to $\theta$ , r and z are also principal material directions ( $\beta=0$ in Fig. 10.1-1), then $E_{14}=E_{24}=E_{34}=0$ . For the special case of isotropy and thermal loading, Eq. 10.2-2 becomes
+
+$$
+\left\{ \begin{array}{l} \sigma_ {r} \\ \sigma_ {\theta} \\ \sigma_ {z} \\ \tau_ {z r} \end{array} \right\} = \frac {(1 - \nu) E}{(1 + \nu) (1 - 2 \nu)} \left[ \begin{array}{c c c c} 1 & f & f & 0 \\ & 1 & f & 0 \\ & & 1 & 0 \\ \text {symm.} & & & \mathrm{g} \end{array} \right] \left(\left\{ \begin{array}{l} \epsilon_ {r} \\ \epsilon_ {\theta} \\ \epsilon_ {z} \\ \gamma_ {z r} \end{array} \right\} - \left\{ \begin{array}{l} \alpha T \\ \alpha T \\ \alpha T \\ 0 \end{array} \right\}\right) \tag {10.2-3a}
+$$
+
+<!-- source-page: 315 -->
+
+in which
+
+$$
+f = \frac {\nu}{1 - \nu} \quad \text { and } \quad g = \frac {1 - 2 \nu}{2 (1 - \nu)} \tag {10.2-3b}
+$$
+
+and $\alpha =$ coefficient of thermal expansion and T = temperature relative to a reference temperature at which the body is free of stress. Thus $E_{44} = G$ , the shear modulus.
+
+The strain-displacement relations are
+
+$$
+\epsilon_ {r} = u _ {, r} \quad \epsilon_ {\theta} = \frac {2 \pi (r + u) - 2 \pi r}{2 \pi r} = \frac {u}{r} \tag {10.2-4}
+$$
+
+$$
+\epsilon_ {z} = w _ {, z} \quad \gamma_ {z r} = u _ {, z} + w _ {, r}
+$$
+
+In matrix format, Eqs. 10.2-4 are
+
+$$
+\left\{ \begin{array}{l} \epsilon_ {r} \\ \epsilon_ {\theta} \\ \epsilon_ {z} \\ \gamma_ {z r} \end{array} \right\} = [ \partial ] \left\{ \begin{array}{l} u \\ w \end{array} \right\}, \quad \text { where } \quad [ \partial ] = \left[ \begin{array}{c c} \partial / \partial r & 0 \\ 1 / r & 0 \\ 0 & \partial / \partial z \\ \partial / \partial z & \partial / \partial r \end{array} \right] \tag {10.2-5}
+$$
+
+Or, in alternative format, the same relations are
+
+$$
+\left\{ \begin{array}{l} \epsilon_ {r} \\ \epsilon_ {\theta} \\ \epsilon_ {z} \\ \gamma_ {z r} \end{array} \right\} = [ \mathbf {H} ] \left\{ \begin{array}{l} u _ {, r} \\ u _ {, z} \\ w _ {, r} \\ w _ {, z} \\ u \end{array} \right\}, \quad \text {where} \quad [ \mathbf {H} ] = \left[ \begin{array}{c c c c c} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 / r \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 \end{array} \right] \tag {10.2-6}
+$$
+
+# 10.3 FINITE ELEMENTS FOR AXIAL SYMMETRY
+
+One may follow the standard formulation procedure, which is contained in Eqs. 4.1-5 and 4.1-6. Consider, for example, an eight-d.o.f. element of rectangular cross section, shown in Fig. 10.3-1. Its displacement field $\{u\} = [N]\{d\}$ is
+
+$$
+\left\{ \begin{array}{l} u \\ w \end{array} \right\} = \left[ \begin{array}{c c c c c c c c} N _ {1} & 0 & N _ {2} & 0 & N _ {3} & 0 & N _ {4} & 0 \\ 0 & N _ {1} & 0 & N _ {2} & 0 & N _ {3} & 0 & N _ {4} \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ w _ {1} \\ u _ {2} \\ \vdots \\ w _ {4} \end{array} \right\} \tag {10.3-1}
+$$
+
+where shape functions $N_{1}$ through $N_{4}$ are stated in Eq. 3.12-10, except that z replaces y and $r - r_{m}$ replaces x, where $r_{m}$ is the mean radius $(r_{1} + r_{2} + r_{3} + r_{4})/4$ . Thus
+
+$$
+N _ {i} = \frac {[ a \pm (r - r _ {m}) ] (b \pm z)}{4 a b} \tag {10.3-2}
+$$
+
+<!-- source-page: 316 -->
+
+![](images/page-316_181ddbdedc6769dc95c0551f767d5a2f4088fbc882233952cdff2cdab7852169.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Axis of
+revolution
+z
+y
+a
+a
+4
+3
+b
+x
+1
+2
+b
+r_m
+x = r - r_m
+r
+</details>
+
+Figure 10.3-1. Geometry of an element of rectangular cross section.
+
+in which $z = 0$ at the element center. The element stiffness matrix is
+
+$$
+[ \mathbf {k} ] _ {8 \times 8} = \int_ {A} \int_ {- \pi} ^ {\pi} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] [ \mathbf {B} ] r d \theta d A \tag {10.3-3}
+$$
+
+where $A =$ cross-sectional area of the element and $dA = drdz$ . From Eqs. 10.2-5 and 10.3-1, [B] = [∂][N]. However, since $\partial/\partial r = \partial/\partial x$ and $r = r_m + x$ , we can also write
+
+$$
+[ \mathbf {B} ] = \left[ \begin{array}{c c} \partial / \partial x & 0 \\ 1 / (r _ {m} + x) & 0 \\ 0 & \partial / \partial z \\ \partial / \partial z & \partial / \partial x \end{array} \right] [ \mathbf {N} ], \quad \text { where } \quad N _ {i} = \frac {(a \pm x) (b \pm z)}{4 a b} \tag {10.3-4}
+$$
+
+and where $x = 0$ at $r = r_m$ , the element center.
+
+Equation 10.3-4 yields the same [B] as does $[\mathbf{B}] = [\partial][\mathbf{N}]$ with the $N_{i}$ given by Eq. 10.3-2. Equation 10.3-4 shows that, as compared with a plane problem, the only change in [B] is the added row that computes $\epsilon_{\theta} = u / r$ (compare Eqs. 10.3-4 and 4.2-14). Moreover, [k] is the same size; for example, it is 8 by 8 for the foregoing four-node bilinear element, whether the problem is plane or axially symmetric.
+
+symmetric. If the element is isoparametric, shape functions $N_{i}$ are functions of $\xi$ and $\eta$ , and we must use the usual coordinate transformation to relate derivatives,
+
+$$
+\left\{ \begin{array}{l} u, _ {r} \\ u, _ {z} \\ w, _ {r} \\ w, _ {z} \\ u \end{array} \right\} = [ \Gamma ] _ {5 \times 5} \left\{ \begin{array}{l} u, _ {\xi} \\ u, _ {\eta} \\ w, _ {\xi} \\ w, _ {\eta} \\ u \end{array} \right\}, \quad \text { where } \quad [ \Gamma ] = \left[ \begin{array}{c c c} \mathrm{J} ^ {- 1} & 0 & 0 \\ 0 & \mathrm{J} ^ {- 1} & 0 \\ 0 & 0 & 1 \end{array} \right] \tag {10.3-5}
+$$
+
+Jacobian matrix [J] is unchanged: it is still 2 by 2 and is as described in Section 6.3. The $\xi$ and $\eta$ derivatives in Eq. 10.3-5 are related to nodal d.o.f. by the equation
+
+$$
+\left[ \begin{array}{l l l l l} u _ {, \xi} & u _ {, \eta} & w _ {, \xi} & w _ {, \eta} & u \end{array} \right] ^ {T} = \left[ \begin{array}{l} \mathbf {Q} \end{array} \right] \left[ \begin{array}{l l l l} u _ {1} & w _ {1} & u _ {2} \dots w _ {4} \end{array} \right] ^ {T} \tag {10.3-6}
+$$
+
+<!-- source-page: 317 -->
+
+The first four rows of [Q] appear in Eq. 6.3-19. The fifth row is $[N_{1}$ , 0, $N_{2}$ , 0, $N_{3}$ , 0, $N_{4}$ , 0]. Shape functions $N_{i}$ of a four-node isoparametric element are given by Eqs. 6.3-2. From Eqs. 10.2-6, 10.3-5, and 10.3-6, [B] = [H][ $\Gamma$ ][Q]. The element stiffness matrix of an isoparametric element is
+
+$$
+[ \mathbf {k} ] = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \int_ {- \pi} ^ {\pi} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] [ \mathbf {B} ] r d \theta J d \xi d \eta \tag {10.3-7}
+$$
+
+where
+
+$$
+r = \sum N _ {i} r _ {i} \quad \text { or } \quad r = r _ {m} + \sum N _ {i} x _ {i} \tag {10.3-8}
+$$
+
+In numerical integration, either of Eqs. 10.3-8 may be used to determine $r$ at quadrature points.
+
+Element nodal loads $\{r_{e}\}$ —for example, from heating or from centrifugal force—are calculated in straightforward fashion from Eq. 4.1-6. Here $dV = r \, d\theta \, dA$ or $dV = r \, d\theta \, J \, d\xi \, d\eta$ , and $dS = r \, d\theta \, d\ell$ , where $d\ell$ is an increment of meridional length.
+
+A uniform line load q (units N/m) on a nodal circle of radius $r_{i}$ produces the nodal load $2\pi r_{i}q$ . The net static force is also $2\pi r_{i}q$ if q acts axially, but is zero if q acts radially. Nevertheless, radial load q produces deformations and stresses.
+
+Remarks. The preceding formulation produces $2\pi$ as a multiplier of every $K_{ij}$ and every $R_{i}$ in the structural equations $[K]\{D\} = \{R\}$ . This superfluous multiplier can be avoided by letting integrals for $[k]$ and $\{r_{e}\}$ have theta limits of from zero to one radian. With this approach, externally applied loads must also pertain to a one-radian segment.
+
+Some coefficients in the integrands of Eqs. 10.3-3 and 10.3-7 have $1/r$ as a multiplier. With Gauss quadrature these terms remain finite because there are no Gauss points at $r = 0$ . If $[\mathbf{k}]$ is formed explicitly, we can produce “core” elements by using a displacement field for which $u = 0$ for $r = 0$ and evaluating indeterminate forms $0/0$ that appear in the formulation by the L'Hôpital rule. If numerical integration is used instead, for acceptable accuracy core elements may require more integration points in the radial direction than are used for elements distant from the axis of revolution.
+
+During stress computation, the indeterminate form $\epsilon_{\theta}=u/r=0/0$ arises for points on the z axis. We can avoid this trouble by calculating $\epsilon_{\theta}$ slightly alway from the axis or by extrapolating strains at Gauss points to the axis. Another option is to exploit the theoretical requirement that $\epsilon_{r}=\epsilon_{\theta}$ at r=0. Thus, for stress computation at r=0, we merely replace the $\epsilon_{\theta}$ row of [B] by the $\epsilon_{r}$ row.
+
+Because of axial symmetry, z-direction translation is the only possible rigid-body motion. It can be restrained by prescribing w on a single nodal circle. The radial displacement u = 0 should be prescribed at all nodes that lie on the z axis.
+
+Valid elements for solids of revolution must pass a weak patch test (see Section 4.6). Consider, for example, elements Q6 and QM6 (Section 8.3). These elements develop a spurious radial bulge because internal d.o.f. are activated. The bulge creates spurious shear strain $\gamma_{zr}$ except at $\xi = \eta = 0$ . Nevertheless, the element is valid because the bulge tends to vanish as element cross-sectional dimensions become small in comparison with the mean radius of the element. In general use,
+
+<!-- source-page: 318 -->
+
+stresses predicted by the QM6 element may be more reliable if $\gamma_{zr}$ is evaluated only at $\xi = \eta = 0$ .
+
+# 10.4 FOURIER SERIES
+
+The response of an axially symmetric body to asymmetric loads can be analyzed by superposing component analyses, each of which represents the response attributable to one component of the total load. The method relies on Fourier series, which is summarized as follows without reference to bodies of revolution.
+
+Fourier series represent functions that are periodic. A Fourier series for a dependent variable $\phi = \phi(\theta)$ can be written
+
+$$
+\phi = \sum_ {n = 0} ^ {\infty} p _ {n} \cos n \theta + \sum_ {n = 1} ^ {\infty} q _ {n} \sin n \theta \tag {10.4-1}
+$$
+
+where $n$ is an integer. The period of $\phi$ is $2\pi$ , for example, from $\theta = -\pi$ to $\theta = \pi$ .
+
+Sine terms are called odd or antisymmetric, as $\phi(\theta) = -\phi(-\theta)$ . Cosine terms are called even or symmetric, as $\phi(\theta) = \phi(-\theta)$ . Coefficients $p_n$ and $q_n$ are functions of $n$ but not of $\theta$ . The following integrals, where $m$ and $n$ are integers, are useful:
+
+$$
+\int_ {- \pi} ^ {\pi} \sin m \theta \sin n \theta d \theta = \left\{ \begin{array}{l l} \pi & \text { for } m = n \neq 0 \\ 0 & \text { for } m \neq n \text { and for } m = n = 0 \end{array} \right. \tag {10.4-2a}
+$$
+
+$$
+\int_ {- \pi} ^ {\pi} \cos m \theta \cos n \theta d \theta = \left\{ \begin{array}{l l} 2 \pi & \text { for } m = n = 0 \\ \pi & \text { for } m = n \neq 0 \\ 0 & \text { for } m \neq n \end{array} \right. \tag {10.4-2b}
+$$
+
+$$
+\int_ {- \pi} ^ {\pi} \sin m \theta \cos n \theta d \theta = 0 \quad \text { for   all } m \text { and } n \tag {10.4-2c}
+$$
+
+Imagine that a certain periodic function $\phi = \phi(\theta)$ is known, but not expressed as a Fourier series. An equivalent Fourier series representation of $\phi$ requires that $p_{n}$ and $q_{n}$ in Eq. 10.4-1 be determined. To do so, we integrate the function, then multiply it by the single term $\cos n\theta$ and integrate, then multiply it by the single term $\sin n\theta$ and integrate. Equation 10.4-1 is similarly integrated, making use of Eqs. 10.4-2. Thus equations that determine $p_{0}, p_{n}$ , and $q_{n}$ are
+
+$$
+\int_ {- \pi} ^ {\pi} \phi d \theta = 2 \pi p _ {0} \quad \int_ {- \pi} ^ {\pi} \phi \cos n \theta d \theta = \pi p _ {n} \quad \int_ {- \pi} ^ {\pi} \phi \sin n \theta d \theta = \pi q _ {n} \tag {10.4-3}
+$$
+
+Integrals in Eq. 10.4-3 may be evaluated analytically, numerically, or even graphically.
+
+Example. The square wave in Fig. 10.4-1 is to be represented by a Fourier series. Here $\theta = \pi x / L$ , and $\phi = \phi_0$ can be regarded as a uniformly distributed load of intensity $\phi_0$ on a span of length $L$ . In Eqs. 10.4-3 we use $\phi = -\phi_0$ for $-L < x < 0$ and $\phi = +\phi_0$
+
+<!-- source-page: 319 -->
+
+![](images/page-319_d681411c349eac6a083a824c676518d50cea3fe078dc9ea46fe51e7e6ae5e4d9.jpg)
+
+<details>
+<summary>text_image</summary>
+
+φ
+φ₀
+-L 0 L
+-φ₀
+</details>
+
+(a)
+
+![](images/page-319_47670fb32130941bf2471238c4b1e07cef4991ff334cccd02a9fc4fec9a6881b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+φ
+φ₀
+L
+x
+</details>
+
+(b)
+
+![](images/page-319_7b58cb20fbffdb6232481feea1ecb41c8351dc6093d88879c23d1e0040da3c8f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+φ
+φ₀
+L
+x
+</details>
+
+{c}
+
+![](images/page-319_7cb1c567d00217742215b544928eb266896e578982d8965d864d85244a6219b2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+φ
+φ₀
+x
+L
+</details>
+
+{d}   
+Figure 10.4-1. (a) Square wave and its representation by truncated Fourier series, using terms (b) n = 1, (c) n = 1 and 3, and (d) n = 1, 3, and 5.
+
+for $0 < x < L$ . Results of the respective integrations in Eq. 10.4-3 are 0, 0, and 0 ( $n$ even) or $4\phi_0 / n$ ( $n$ odd). Hence, $p_0 = 0$ , $p_n = 0$ , and $q_n = 0$ ( $n$ even). For $n$ odd,
+
+$$
+q _ {n} = \frac {4 \phi_ {0}}{n \pi} \quad \text { and } \quad \phi = \sum_ {n = 1, 3, \dots} \frac {4 \phi_ {0}}{n \pi} \sin \frac {n \pi x}{L} \tag {10.4-4}
+$$
+
+As suggested by Fig. 10.4-1, the square wave can be modeled arbitrarily closely by using enough series terms.
+
+Example. Concentrated loads P produce an interesting result. In the coordinates of Fig. 10.4-1a, consider a load P downward at x = -L/2 and a load P upward at x = +L/2. Here $\theta = \pi x / L$ , and P can be regarded as a concentrated center load on a beam that extends from x = 0 to x = L. Equations 10.4-3 yield $p_{0} = 0$ , $p_{n} = 0$ , and
+
+$$
+q _ {n} = \frac {2 P}{L} \sin \frac {n \pi}{2} \quad \phi = \sum_ {n = 1, 2, 3, \dots} \left(\frac {2 P}{L} \sin \frac {n \pi}{2}\right) \sin \frac {n \pi x}{L} \tag {10.4-5}
+$$
+
+This series for $\phi$ does not converge. However, when used to load the aforementioned beam, convergent results are obtained for displacement and stress.
+
+A Simple Application in Stress Analysis. The beam problem depicted in Fig. 10.4-2 illustrates features of series analysis that also appear in series analysis of solids (and shells) of revolution. For a beam, the equilibrium and moment-curvature relations are
+
+$$
+V _ {, x} = \phi \quad M _ {, x} = V \quad E I w _ {, x x} = M \tag {10.4-6}
+$$
+
+![](images/page-319_747723f5fd82fb8d1b855dc98224dc946f307a90c9feb2aecb725dde0b4f0f0a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+x
+L
+</details>
+
+{a}
+
+![](images/page-319_c88dcc574022c4a4c9dff12ce217f03bd1cec60c91bb8b652e44b8446971c81f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+φ
+qₙ
+qₙ
+L
+x
+</details>
+
+{b}   
+Figure 10.4-2. (a) Simply supported beam. (b) The sine wave loading $\phi = q_{n} \sin(n\pi x/L)$ , where $q_{n}$ is the amplitude of $\phi$ . The case n = 5 is depicted.
+
+<!-- source-page: 320 -->
+
+where V is transverse shear force, M is bending moment, and $\phi$ is distributed load. When combined, Eqs. 10.4-6 yield, for constant bending stiffness EI,
+
+$$
+E I w _ {, x x x x} = \phi \tag {10.4-7}
+$$
+
+Consider the loading of Fig. 10.4-2b, which corresponds to one term of the second summation in the Fourier series of Eq. 10.4-1:
+
+$$
+\phi = q _ {n} \sin \frac {n \pi x}{L} \tag {10.4-8}
+$$
+
+Here $q_{n}$ does not depend on x. Assume that the displacement is the admissible function
+
+$$
+w = w _ {n} \sin \frac {n \pi x}{L} \tag {10.4-9}
+$$
+
+where $w_{n}$ does not depend on $x$ . We substitute $w$ and $\phi$ into Eq. 10.4-7 and obtain
+
+$$
+\left[ E I \left(\frac {n \pi}{L}\right) ^ {4} w _ {n} - q _ {n} \right] \sin \frac {n \pi x}{L} = 0 \tag {10.4-10}
+$$
+
+This can be true for all $x$ only if the bracketed expression vanishes. Hence
+
+$$
+w _ {n} = \frac {q _ {n}}{E I} \left(\frac {L}{n \pi}\right) ^ {4} \tag {10.4-11}
+$$
+
+Substitution of Eq. 10.4-11 into Eq. 10.4-9 defines the correct and unique solution for w, since it satisfies all requirements of equilibrium, compatibility, and boundary conditions (Section 1.6). Note that a sine wave of loading produces corresponding sine waves of deflection and bending moment, regardless of n. That is, the various harmonics are uncoupled—the nth wave does not interact with the mth wave.
+
+Now that $w_{n}$ is known, Eq. 10.4-9 defines w for any x and for any n. If two or more sine wave loadings act simultaneously, each associated with a different n, the net deflection and the net bending moment are determined by superposition; that is,
+
+$$
+w = \sum_ {n} q _ {n} \frac {L ^ {4}}{E I n ^ {4} \pi^ {4}} \sin \frac {n \pi x}{L} \quad \text { and } \quad M = - \sum_ {n} q _ {n} \frac {L ^ {2}}{n ^ {2} \pi^ {2}} \sin \frac {n \pi x}{L} \tag {10.4-12}
+$$
+
+where $M = EIw_{,xx}$ . A particular loading requires a particular $q_{n}$ . For example, to analyze the effect of a uniformly distributed loading of intensity $\phi_{0}$ , we substitute $q_{n} = 4\phi_{0}/n\pi$ from Eq. 10.4-4 into Eqs. 10.4-12. For a concentrated load at midspan, we substitute $q_{n}$ from Eq. 10.4-5.
+
+What we have done is find the response of the beam to a single load component from a single equation (Eq. 10.4-11) that says nothing about how $w$ varies with $x$ . We pay for this simplicity by having to solve the equation several times, once for each Fourier component of loading.
+
+In dealing with bodies of revolution, we replace Eq. 10.4-11 by a set of simultaneous algebraic equations, which must be solved for each Fourier component of loading.
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_033.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_033.md
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@@ -0,0 +1,453 @@
+<!-- source-page: 321 -->
+
+# 10.5 LOADS WITHOUT AXIAL
+
+# SYMMETRY: INTRODUCTION
+
+In solids (and shells) of revolution under loads without axial symmetry, we can arrange to compute structural response to each Fourier harmonic of loading from a set of comparatively simple equations that makes no reference to how deformations vary with circumferential coordinate $\theta$ . This set of equations must be solved several times, once for each harmonic of loading. Response attributable to the given loading is found by superposing the separate analyses and, in general, displays deformations that vary with $\theta$ as well as with r and z. The superposition method is much less expensive than a single three-dimensional analysis if only a few Fourier harmonics are needed to represent the load. As examples, wind loading requires few harmonics but a concentrated force requires many.
+
+With $\theta$ a principal material direction, the most general stress-strain relation $\{\sigma\} = [\mathbf{E}]\{\epsilon\}$ has the form
+
+$$
+\left\{ \begin{array}{l} \sigma_ {r} \\ \sigma_ {\theta} \\ \sigma_ {z} \\ \tau_ {z r} \\ \tau_ {r \theta} \\ \tau_ {\theta z} \end{array} \right\} = \left[ \begin{array}{c c c c c c} E _ {1 1} & E _ {1 2} & E _ {1 3} & E _ {1 4} & 0 & 0 \\ & E _ {2 2} & E _ {2 3} & E _ {2 4} & 0 & 0 \\ & & E _ {3 3} & E _ {3 4} & 0 & 0 \\ & & & E _ {4 4} & 0 & 0 \\ \text { symmetric } & & & & E _ {5 5} & E _ {5 6} \\ & & & & & E _ {6 6} \end{array} \right] \left\{ \begin{array}{l} \epsilon_ {r} \\ \epsilon_ {\theta} \\ \epsilon_ {z} \\ \gamma_ {z r} \\ \gamma_ {r \theta} \\ \gamma_ {\theta z} \end{array} \right\} \tag {10.5-1}
+$$
+
+If $r$ and $z$ are also principal material directions, or if the material is isotropic, then $E_{14} = E_{24} = E_{34} = E_{56} = 0$ . If the material is isotropic, one uses Eq. 10.2-3 and the values $E_{44} = E_{55} = E_{66} = G$ , where $G$ is the shear modulus, $G = 0.5E / (1 + \nu)$ .
+
+Let loading of the body be expressed as Fourier series: for example, the radially directed body force is $F_{r} = \Sigma \overline{F}_{rn} \cos n\theta$ , where $\overline{F}_{rn}$ is an amplitude, like $p_{n}$ in Eq. 10.4-1. Thus
+
+$$
+\left[ \begin{array}{l l l l l} F _ {r} & F _ {z} & \Phi_ {r} & \Phi_ {z} & T \end{array} \right] = \sum_ {n} \left[ \begin{array}{l l l l l} \overrightarrow {F} _ {r n} & \overrightarrow {F} _ {z n} & \overrightarrow {\Phi} _ {r n} & \overrightarrow {\Phi} _ {z n} & \overrightarrow {T} _ {n} \end{array} \right] \cos n \theta \tag {10.5-2a}
+$$
+
+$$
+\left[ \begin{array}{l l} F _ {\theta} & \Phi_ {\theta} \end{array} \right] = \sum_ {n} \left[ \begin{array}{l l} \overline {{F}} _ {\theta n} & \overline {{\Phi}} _ {\theta n} \end{array} \right] \sin n \theta \tag {10.5-2b}
+$$
+
+where T is temperature, and the $F'$ s and $\Phi$ 's are, respectively, body forces per unit volume and surface tractions in the r, $\theta$ , and z directions. Equations 10.5-2 represent a state of symmetry with respect to the plane $\theta = 0$ (antisymmetric Fourier terms are considered subsequently).
+
+We will show that when loads are described by Eqs. 10.5-2, displacements are described by
+
+$$
+\text { Radial   displacement } = u = \sum_ {n} \overline {{u}} _ {n} \cos n \theta \tag {10.5-3a}
+$$
+
+$$
+\text { Circumferential   displacement } = v = \sum_ {n} \overline {{v}} _ {n} \sin n \theta \tag {10.5-3b}
+$$
+
+$$
+\text { Axial   displacement } = w = \sum_ {n} \overline {{w}} _ {n} \cos n \theta \tag {10.5-3c}
+$$
+
+<!-- source-page: 322 -->
+
+All three displacements are needed because the physical problem is three-dimensional. In Eqs. 10.5-2 and 10.5-3, $n$ is an integer, and all barred quantities are functions of $r, z$ , and $n$ but not of $\theta$ . Thus the barred terms are amplitudes.
+
+The strain-displacement relations in cylindrical coordinates are $\{\epsilon\} = [\partial]\{\mathbf{u}\}$ ; that is,
+
+$$
+\left\{ \begin{array}{l} \epsilon_ {r} \\ \epsilon_ {\theta} \\ \epsilon_ {z} \\ \gamma_ {z r} \\ \gamma_ {r \theta} \\ \gamma_ {\theta z} \end{array} \right\} = \left[ \begin{array}{c c c} \partial / \partial r & 0 & 0 \\ 1 / r & \partial / (r \partial \theta) & 0 \\ 0 & 0 & \partial / \partial z \\ \partial / \partial z & 0 & \partial / \partial r \\ \partial / (r \partial \theta) & (\partial / \partial r - 1 / r) & 0 \\ 0 & \partial / \partial z & \partial / (r \partial \theta) \end{array} \right] \left\{ \begin{array}{l} u \\ v \\ w \end{array} \right\} \tag {10.5-4}
+$$
+
+These relations are independent of material properties and of whether or not $u, v,$ and $w$ are described by series.
+
+Consider now a typical single harmonic of displacement, say the nth. If we substitute Eqs. 10.5-3 into Eq. 10.5-4 and the resulting strains into Eq. 10.5-1, we find that stresses of the nth harmonic have the form
+
+$$
+\left\lfloor \sigma_ {r n} \quad \sigma_ {\theta n} \quad \sigma_ {z n} \quad \tau_ {z r n} \right\rfloor = \left\lfloor \overline {{{\sigma}}} _ {r n} \quad \overline {{{\sigma}}} _ {\theta n} \quad \overline {{{\sigma}}} _ {z n} \quad \overline {{{\tau}}} _ {z r n} \right\rfloor \cos n \theta \tag {10.5-5a}
+$$
+
+$$
+\left[ \tau_ {r \theta n} \quad \tau_ {\theta z n} \right] = \left[ \bar {\tau} _ {r \theta n} \quad \bar {\tau} _ {\theta z n} \right] \sin n \theta \tag {10.5-5b}
+$$
+
+where the barred quantities are functions of $r$ , $z$ , and $n$ but not of $\theta$ . If Eqs. 10.5-2 and 10.5-5 are substituted into the three differential equations of equilibrium, Eqs. 1.6-4, we find that these three equations assume the forms
+
+$$
+Q _ {1} \cos n \theta = 0 \quad Q _ {2} \cos n \theta = 0 \quad Q _ {3} \sin n \theta = 0 \tag {10.5-6}
+$$
+
+where the $Q_{i}$ are functions of r, z, and n but not of $\theta$ . Equations 10.5-6 are analogous to Eq. 10.4-10. Equations 10.5-6 must prevail for all $\theta$ , so $Q_{1} = Q_{2} = Q_{3} = 0$ . As will be seen in Section 10.6, in a finite element context the equations $Q_{1} = Q_{2} = Q_{3} = 0$ produce the equilibrium equations of the nth harmonic
+
+$$
+[ \mathbf {K} ] _ {n} \{\mathbf {D} \} _ {n} - \{\mathbf {R} \} _ {n} = \{\mathbf {0} \} \tag {10.5-7}
+$$
+
+Equations 10.5-7 are analogous to Eq. 10.4-11. Their solution yields the nodal d.o.f. $\{\mathbf{D}\}_{n} = [\overline{u}_{1n} - \overline{v}_{1n} - \overline{w}_{1n} - \overline{u}_{2n} \ldots]^T$ , which are displacement amplitudes of the nodal circles in the $n$ th harmonic. Matrix $[\mathbf{K}]_n$ depends on $n$ . The load coefficients in $\{\mathbf{R}\}_{n}$ correspond to the $q_n$ of Eq. 10.4-1.
+
+We see that n circumferential waves of loading are associated with n circumferential waves of stress and of displacement. The Fourier harmonics are not coupled. Different numerical values of n present different problems that do not interact. Thus the need for a division into finite elements in the circumferential direction is replaced by the need to superpose separate solutions for a structure divided into finite elements in only its cross section. A single mesh is used for all the separate solutions. In most practical problems only a few load harmonics need be analyzed. A computer program can automatically cycle through a user-specified number of harmonics and superpose the separate solutions.
+
+<!-- source-page: 323 -->
+
+Remarks. The preceding discussion invokes only loads and displacements that have $\theta = 0$ as a plane of symmetry. In general, antisymmetric terms are also present. Thus Eqs. 10.5-2 are augmented to read
+
+$$
+\left\lfloor F _ {r} \quad F _ {z} \quad \Phi_ {r} \quad \Phi_ {z} \quad T \right] = \sum_ {n} \left\lfloor \overline {{\mathrm{L}}} _ {c n} \right\rfloor \cos n \theta + \sum_ {n} \left\lfloor \overline {{\overline {{\mathrm{L}}}}} _ {s n} \right\rfloor \sin n \theta \tag {10.5-8a}
+$$
+
+$$
+\left[ \begin{array}{l l} F _ {\theta} & \Phi_ {\theta} \end{array} \right] = \sum_ {n} \left[ \overline {{\mathbf {L}}} _ {s n} \right] \sin n \theta + \sum_ {n} \left[ \overline {{\overline {{\mathbf {L}}}}} _ {c n} \right] \cos n \theta \tag {10.5-8b}
+$$
+
+where $\left\lfloor\overline{L}_{cn}\right\rfloor$ and $\left\lfloor\overline{L}_{sn}\right\rfloor$ represent the symmetric load amplitudes, already present in Eqs. 10.5-2, and $\left\lfloor\overline{L}_{sn}\right\rfloor$ and $\left\lfloor\overline{L}_{cn}\right\rfloor$ represent additional antisymmetric load amplitudes. Similarly, the symmetric displacement field, Eqs. 10.5-3, is augmented by antisymmetric terms and becomes
+
+$$
+u = \sum_ {n} \bar {u} _ {n} \cos n \theta + \sum_ {n} \bar {\bar {u}} _ {n} \sin n \theta \tag {10.5-9a}
+$$
+
+$$
+v = \sum_ {n} \bar {v} _ {n} \sin n \theta - \sum_ {n} \bar {\bar {v}} _ {n} \cos n \theta \tag {10.5-9b}
+$$
+
+$$
+w = \sum_ {n} \overline {{w}} _ {n} \cos n \theta + \sum_ {n} \overline {{\overline {{w}}}} _ {n} \sin n \theta \tag {10.5-9c}
+$$
+
+The motivation for the arbitrarily chosen negative sign in the v series is explained in the subsection that follows Eq. 10.6-9.
+
+Axially symmetric problems are represented by the n = 0 terms of the single-barred series. For $n = 1, 2, 3, \ldots$ , loads and displacements of the single-barred series represent symmetry about the plane $\theta = 0$ . For $n = 0, 2, 4, 6, \ldots$ , loads and deformations have both $\theta = 0$ and $\theta = \pi/2$ as planes of symmetry. Example symmetric loads appear in Fig. 10.5-1c.
+
+Antisymmetric problems (e.g., Fig. 10.5-1f) are represented by the double-barred series. Pure torque is represented by the n = 0 terms of the double-barred series. Thus, for example, we can study the twist of shafts of variable diameter. In the torsion problem u and w are everywhere zero, so a finite element solution based on a stress function is also possible [10.3].
+
+When n = 0, for any node i, nodal d.o.f. $\overline{v}_{ni}$ , $\overline{u}_{ni}$ , and $\overline{w}_{ni}$ have no stiffness associated with them, so these d.o.f. must be suppressed to avoid a singular stiffness matrix.
+
+The simplest displacement boundary condition is zero displacement on a nodal circle. This requires that displacement amplitudes on the circle be zero in every harmonic. If loads are symmetric about both the $\theta = 0$ and $\theta = \pi/2$ planes, then u and v are zero at r = 0 in all harmonics. Nonzero and asymmetric displacement conditions can be represented as Fourier series and the separate amplitude coefficients used as prescribed displacements in the separate analyses.
+
+Additional constraints on the Fourier displacement amplitudes can be deduced from the condition that strains remain finite at r = 0 [10.4]. If these constraints are not imposed, numerical integration makes some stiffness coefficients significantly larger than others. This circumstance is not likely to be troublesome in static analysis, but it may require a very small time step if explicit integration is applied to transient problems.
+
+<!-- source-page: 324 -->
+
+![](images/page-324_ceecfa2babad6bfdbd9930fe631ed12997b36b502974f365a33950a6455e1931.jpg)
+
+<details>
+<summary>text_image</summary>
+
+cos nθ
+n=1
+θ
+r
+</details>
+
+(a)
+
+![](images/page-324_70f006e7bee8cbcd0d11274b36505d5bf37f9fcb8ec72ef46f028c2e087d8bef.jpg)
+
+<details>
+<summary>text_image</summary>
+
+cos nθ
+n = 2
+θ
+r
+</details>
+
+(b)
+
+![](images/page-324_8800d068c634f11c82bc089138a33d24febd4e80cbd5ae1499e42d7366883209.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P₁
+P₂
+θ
+r
+q
+P₁
+P₂
+</details>
+
+(c)
+
+![](images/page-324_1feabd3e12a3bdae37672d0e8dbf010c6d9ac62b17a13a6d15c5e9d2762e8b30.jpg)
+
+<details>
+<summary>text_image</summary>
+
+sin nθ
+n=1
+θ
+r
+</details>
+
+(d)
+
+![](images/page-324_45ff3c89d134a389d45e041d24063cf5b629f9c43ab34be2ffb6bc7d13be1fd4.jpg)
+
+<details>
+<summary>text_image</summary>
+
+sin nθ
+n = 2
+θ
+r
+</details>
+
+(e)
+
+![](images/page-324_e6f65aafbd89b635de2c285c7dcdfb36f934d9ad6b465f39ef430b5b7fffdd32.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P₄
+P₃
+θ
+r
+q
+P₃
+P₄
+</details>
+
+(1)   
+Figure 10.5-1. $(a,b)$ Example cosine terms. (c) Possible symmetric loads in an $r\theta$ plane. $(d,e)$ Example sine terms. (f) Possible antisymmetric loads in an $r\theta$ plane.
+
+# 10.6 LOADS WITHOUT AXIAL
+
+# SYMMETRY: ELEMENT MATRICES
+
+Within an element, one can interpolate amplitudes $\overline{u}_{n}$ , $\overline{v}_{n}$ , and $\overline{w}_{n}$ of Eqs. 10.5-3 from nodal amplitudes $\overline{u}_{in}$ , $\overline{v}_{in}$ , and $\overline{w}_{in}$ . Consider, for example, the four-node element in Fig. 10.1-1b. Displacement amplitudes $\{\overline{u}\}_{n}$ in harmonic n are $\{\overline{u}\}_{n} = [\overline{N}]\{\overline{d}\}_{n}$ , in which nodal displacement amplitudes $\{\overline{d}\}_{n}$ pertain to the nth harmonic and are independent of $\theta$ . Written out, this relation is
+
+$$
+\left\{ \begin{array}{l} \overline {{u}} _ {n} \\ \overline {{v}} _ {n} \\ \overline {{w}} _ {n} \end{array} \right\} = \underbrace {\left[ \begin{array}{c c c} N _ {1} & 0 & 0 \\ 0 & N _ {1} & 0 \\ 0 & 0 & N _ {1} \end{array} \right]} _ {1} \underbrace {\left[ \begin{array}{c c c} N _ {2} & 0 & 0 \\ 0 & N _ {2} & 0 \\ 0 & 0 & N _ {2} \end{array} \right]} _ {2} \underbrace {\left[ \begin{array}{c c c} \dots & \dots & \dots \\ \dots & \dots & \dots \\ \dots & \dots & \dots \\ 3 & 4 \end{array} \right]} _ {4} \{\overline {{\mathbf {d}}} \} _ {n} \tag {10.6-1a}
+$$
+
+in which element nodal displacement amplitudes are
+
+$$
+\{\overline {{{\mathbf {d}}}} \} _ {n} = \left[ \begin{array}{l l l l l l l} \overline {{{u}}} _ {1 n} & \overline {{{v}}} _ {1 n} & \overline {{{w}}} _ {1 n} & \overline {{{u}}} _ {2 n} & \overline {{{v}}} _ {2 n} & \overline {{{w}}} _ {2 n} & \dots \end{array} \right] ^ {T} \tag {10.6-1b}
+$$
+
+For a rectangular four-node element, shape functions $N_{i}$ are as stated in Eq. 10.3-2 (or in Eq. 10.3-4). For an isoparametric four-node element, shape functions $N_{i}$ are as stated in Eqs. 6.3-2. An element with more nodes will display more partitions in Eq. 10.6-1a and more nodal amplitudes in Eq. 10.6-1b. The same shape functions $N_{i}$ can be used for all harmonics.
+
+The same interpolation is used for the double-barred series in Eqs. 10.5-9: thus, in Eqs. 10.6-1, single-barred quantities become double-barred quantities. In what follows we discuss the single-barred series. The double-barred series is treated similarly.
+
+<!-- source-page: 325 -->
+
+Summation of the various Fourier harmonics yields displacements in an element is stated by Eqs. 10.5-3. This same displacement field can be written in matrix format by attaching $\cos n\theta$ to rows 1 and 3 in Eq. 10.6-1a and $\sin n\theta$ to row 2, then summing the various harmonics. Thus
+
+$$
+\left\{ \begin{array}{l} u \\ v \\ w \end{array} \right\} = \sum_ {n} \left\{ \begin{array}{l} \bar {u} _ {n} \cos n \theta \\ \bar {v} _ {n} \sin n \theta \\ \bar {w} _ {n} \cos n \theta \end{array} \right\} = \sum_ {n} \underbrace {\left[ \begin{array}{c c c c} N _ {1} \cos n \theta & 0 & 0 & \dots \\ 0 & N _ {1} \sin n \theta & 0 & \dots \\ 0 & 0 & N _ {1} \cos n \theta & \dots \end{array} \right]} _ {[ \mathrm{N} ] _ {n}} \{\overline {{{\mathbf {d}}}} \} _ {n} \tag {10.6-2}
+$$
+
+or, with the summation written out,
+
+$$
+\left\{ \begin{array}{l} u \\ v \\ w \end{array} \right\} = \underbrace {\left[ \begin{array}{l l l l} \mathrm{N} _ {n = 0} & \mathrm{N} _ {n = 1} & \mathrm{N} _ {n = 2} & \dots \end{array} \right] \{\overline {{\mathbf {d}}} \}} _ {[ \mathrm{N} ]} \tag {10.6-3a}
+$$
+
+in which $\{\overline{\mathbf{d}}\}$ lists amplitudes from all harmonics, that is,
+
+$$
+\{\overline {{{\mathbf {d}}}} \} = \left\lfloor \{\overline {{{\mathbf {d}}}} \} _ {n = 0} \quad \{\overline {{{\mathbf {d}}}} \} _ {n = 1} \quad \{\overline {{{\mathbf {d}}}} \} _ {n = 2} \quad \dots \right\rfloor^ {T} \tag {10.6-3b}
+$$
+
+Note that $[N]_{n}$ depends on n only because of the cos $n\theta$ and sin $n\theta$ terms. By applying the operator matrix $[\partial]$ in Eq. 10.5-4, one obtains the strain–displacement relation:
+
+$$
+\{\epsilon \} = [ \partial ] \left\{ \begin{array}{l} u \\ v \\ w \end{array} \right\} = \underbrace {[ \partial ] \left[ \begin{array}{l l l l} \mathrm{N} _ {n = 0} & \mathrm{N} _ {n = 1} & \mathrm{N} _ {n = 2} & \dots \end{array} \right] \{\overline {{\mathbf {d}}} \}} _ {[ \mathbf {B} ] = [ \mathbf {B} _ {n = 0} \quad \mathbf {B} _ {n = 1} \quad \dots ]} \tag {10.6-4}
+$$
+
+For example, one determines that the contribution of the $n$ th harmonic to strains, $\{\pmb{\epsilon}\}_{n} = [\mathbf{B}]_{n}\{\mathbf{d}\}_{n}$ , is
+
+$$
+\left\{ \begin{array}{l} \epsilon_ {r n} \\ \epsilon_ {\theta n} \\ \epsilon_ {z n} \\ \gamma_ {z r n} \\ \gamma_ {r \theta n} \\ \gamma_ {\theta z n} \end{array} \right\} = \underbrace {\left[ \begin{array}{c c c c} N _ {1 , r} \cos n \theta & 0 & 0 & \dots \\ \frac {N _ {1}}{r} \cos n \theta & \frac {n N _ {1}}{r} \cos n \theta & 0 & \dots \\ 0 & 0 & N _ {1 , z} \cos n \theta & \dots \\ N _ {1 , z} \cos n \theta & 0 & N _ {1 , r} \cos n \theta & \dots \\ - \frac {n N _ {1}}{r} \sin n \theta & \left(N _ {1 , r} - \frac {N _ {1}}{r}\right) \sin n \theta & 0 & \dots \\ 0 & N _ {1 , z} \sin n \theta & - \frac {n N _ {1}}{r} \sin n \theta & \dots \end{array} \right]} _ {1} \underbrace {\left\{ \begin{array}{l} \bar {u} _ {1 n} \\ \bar {v} _ {1 n} \\ \bar {w} _ {1 n} \\ \vdots \end{array} \right\}} _ {2, 3, 4,..} \tag {10.6-5}
+$$
+
+<!-- source-page: 326 -->
+
+where the partitioning corresponds to that used in Eq. 10.6-1a. If the element is isoparametric, the usual transformation of derivatives must be included (see e.g. Eq. 6.3-7):
+
+$$
+N _ {i, r} = \Gamma_ {1 1} N _ {i, \xi} + \Gamma_ {1 2} N _ {i, \eta} \quad \text { and } \quad N _ {i, z} = \Gamma_ {2 1} N _ {i, \xi} + \Gamma_ {2 2} N _ {i, \eta} \tag {10.6-6}
+$$
+
+Shape functions $N_{i}$ depend on r and z. Therefore, we see from Eq. 10.6-5 that [B] is a function of r, z, n, and $\theta$ . The element stiffness matrix is given by Eq. 10.3-3 or Eq. 10.3-7. Let there be J nodes per element and M harmonics included in the summation. Then the integrand matrix $[B]^{T}[E][B]$ is a full matrix of size 3JM by 3JM. It is composed of an M by M array of 3J by 3J submatrices. The off-diagonal submatrices contain $\sin m\theta$ sin $n\theta$ or $\cos m\theta$ cos $n\theta$ in every term, where m and n are different integers. According to Eqs. 10.4-2, these terms integrate to zero. We are left with only M on-diagonal submatrices, each 3J by 3J and containing $\sin^{2} n\theta$ or $\cos^{2} n\theta$ in every term. After integration according to Eqs. 10.4-2, the common factor $\pi$ (or $2\pi$ for n = 0) appears in every term. Integration with respect to r and z (or $\xi$ and $\eta$ ) is done as though the problem were axially symmetric. After integration is complete, terms in each 3J by 3J submatrix have the form $A + Bn^{2}$ or the form Cn, where A, B, and C depend on material properties and element geometry but are independent of n and $\theta$ . Accordingly, the various expressions here symbolized by A, B, and C need be generated only once, regardless of the number of harmonics used. Element equations, and structural equations after assembly of elements, have the respective forms
+
+$$
+\left[ \begin{array}{c c c c} \mathbf {k} _ {0} & & & \\ & \mathbf {k} _ {1} & & \\ & & \ddots & \\ & & & \ddots \end{array} \right] \left\{ \begin{array}{l} \overline {{\mathbf {d}}} _ {0} \\ \overline {{\mathbf {d}}} _ {1} \\ \cdot \\ \cdot \\ \cdot \end{array} \right\} = \left\{ \begin{array}{l} \overline {{\mathbf {r}}} _ {0} \\ \overline {{\mathbf {r}}} _ {1} \\ \cdot \\ \cdot \\ \cdot \end{array} \right\} \quad \text {and} \quad \left[ \begin{array}{c c c c} \mathbf {K} _ {0} & & & \\ & \mathbf {K} _ {1} & & \\ & & \ddots & \\ & & & \ddots \end{array} \right] \left\{ \begin{array}{l} \overline {{\mathbf {D}}} _ {0} \\ \overline {{\mathbf {D}}} _ {1} \\ \cdot \\ \cdot \\ \cdot \end{array} \right\} = \left\{ \begin{array}{l} \overline {{\mathbf {R}}} _ {0} \\ \overline {{\mathbf {R}}} _ {1} \\ \cdot \\ \cdot \\ \cdot \end{array} \right\} \tag {10.6-7}
+$$
+
+where each $[k]_{n}$ is of size 3J by 3J, and subscripts 0, 1, and so on indicate the number of the Fourier harmonic. The M separate harmonics are not coupled. In practice, matrices are not built for all harmonics at once (as Eqs. 10.6-7 seem to imply). Instead, as suggested by Eq. 10.5-7, separate harmonics of loading are analyzed serially, with results stored for subsequent superposition.
+
+Element loads (Eq. 4.1-6) include contributions such as
+
+$$
+\{\mathbf {r} \} = \int_ {V _ {e}} [ \mathbf {N} ] _ {3 J M \times 3} ^ {T} \left\{\mathbf {F} \right\} d V + \int_ {V _ {e}} [ \mathbf {B} ] _ {3 J M \times 6} ^ {T} \left[ \mathbf {E} \right] _ {6 \times 6} \left\{\boldsymbol {\epsilon} _ {0} \right\} d V \tag {10.6-8}
+$$
+
+where [N] and [B] are given by Eqs. 10.6-3a and 10.6-4. Body forces {F} are, from Eq. 10.5-8,
+
+$$
+\{\mathbf {F} \} = \left\{ \begin{array}{l} \overline {{{F}}} _ {r 0} + \overline {{{F}}} _ {r 1} \cos \theta + \overline {{{F}}} _ {r 2} \cos 2 \theta + \dots \\ 0 + \overline {{{F}}} _ {\theta 1} \sin \theta + \overline {{{F}}} _ {\theta 2} \sin 2 \theta + \dots \\ \overline {{{F}}} _ {z 0} + \overline {{{F}}} _ {z 1} \cos \theta + \overline {{{F}}} _ {z 2} \cos 2 \theta + \dots \end{array} \right\} \tag {10.6-9}
+$$
+
+<!-- source-page: 327 -->
+
+Initial strains $\{\epsilon_{0}\}$ are written similarly. Integration of Eq. 10.6-8 according to Eqs. 10.4-2 shows that $\{\bar{r}\}_{0}$ in Eq. 10.6-7 contains only the zero-harmonic (axially symmetric) load terms, $\{\bar{r}\}_{1}$ contains only the first-harmonic load terms, and so on. Thus again we see the uncoupling of harmonics.
+
+In the pure torsion harmonic n = 0, nodal d.o.f. $\overline{u}_{ni}$ and $\overline{w}_{ni}$ have no stiffness associated with them. In the axially symmetric harmonic n = 0, nodal d.o.f. $\overline{v}_{ni}$ have no stiffness associated with them. These d.o.f. must be suppressed in Eqs. 10.6-7.
+
+Stresses in an element are computed in the usual way, that is, by the equation $\{\sigma\} = [E]([B]\{d\} - \{\epsilon_{0}\})$ , in which [B] is as stated in Eq. 10.6-4. Thus stresses from the various harmonics are superposed.
+
+Antisymmetric Harmonics. When the double-barred terms in Eqs. 10.5-8 and 10.5-9 are used, the preceding arguments are almost unchanged. One finds that $\sin n\theta$ and $\cos n\theta$ are interchanged in Eqs. 10.6-2, 10.6-5, and 10.6-9. In addition, algebraic signs change in the last two rows of $[\mathbf{B}]_n$ in Eq. 10.6-5. However, one finds that stiffness matrices $[\mathbf{k}]_0$ , $[\mathbf{k}]_1$ , and so on, in Eq. 10.6-7 are identical to those obtained in the symmetric case. This convenience is the motivation for the arbitrarily chosen negative sign in Eqs. 10.5-9: if the sign were positive instead, the $[\mathbf{k}]_i$ for a given $i$ would differ between symmetric and antisymmetric cases.
+
+More General Elastic Properties. If $\theta$ is not a principal material direction, [E] in Eq. 10.5-1 becomes a full matrix, and each stress in Eq. 10.5-5 depends on both $\sin n\theta$ and $\cos n\theta$ . Symmetric and antisymmetric terms are now coupled in each harmonic, but different harmonics are uncoupled. Thus Eqs. 10.6-7 are still valid, but each $[\mathbf{k}]_n$ is now $6J$ by $6J$ in size. Details of these arguments appear in [10.5].
+
+The problem is more difficult if elastic properties depend on $\theta$ . One physical cause of this circumstance is the combination of temperature-dependent moduli and a $\theta$ -dependent temperature field. A Fourier series attack can again be used, but all harmonics are coupled [10.6,10.7].
+
+# 10.7 RELATED PROBLEMS
+
+The Fourier series treatment described in Sections 10.5 and 10.6 is also known as the semianalytical method and the separation of variables method. When used for plates, it is called the finite strip method.
+
+Besides its application to plates and to solids and shells of revolution, the Fourier series method can be applied to prismatic solids [10.8,10.9]. Then the name finite prism method may be used. The solid, and its elements, are prismatic (Fig. 10.7-1). The displacement field is again Eq. 10.5-9, except that $\pi y/L$ replaces $\theta$ . If only the single-barred series are used, deformation and loading are symmetric about the xz plane, with v = 0 at y = 0 and at $y = \pm L$ . As usual, arbitrary loads and displacements are treated by determining their Fourier coefficients and making a separate analysis for each, then superposing results. Problems such as that of Fig. 10.7-1 may require 9 to 19 Fourier coefficients.
+
+In a physical sense, what has been done in Fig. 10.7-1 is to take a toroidal solid that extends from $-\pi$ to $\pi$ and straighten it out to form a prismatic solid that extends from -L to L. The straightening can be “faked” by moving the z axis
+
+<!-- source-page: 328 -->
+
+![](images/page-328_71de6ee73434ccea2dfb532b20b9ec7e1969aa36cf7d612fdb9caa3560e6a700.jpg)  
+Figure 10.7-1. A point load P over a long tunnel. A suitably large portion of the surrounding earth or rock is modeled by finite elements, one of which is shown and shaded. (a) Front view. (b) Right-side view.
+
+in Fig. 10.7-1a far to the left and making it an axis of revolution. Then the almost-prismatic solid can be analyzed by a computer program for solids of revolution.
+
+The problem of a curved beam bent by a moment $M_0$ is axially symmetric in geometry and material properties. But it is not obvious that an axisymmetric analysis can deal with a moment loading. Reference 10.10 describes how. The trick is to use a thermal load to simulate the strains produced by $M_0$ . Note that if the curved beam is a thin-walled pipe elbow, its cross section becomes oval in response to $M_0$ and therefore is more flexible than a pipe whose cross section remains circular. Pipe elbow elements that include this effect have been developed [10.11].
+
+Some geometries are “almost” axially symmetric, for example, the geometry or material properties have modest departures from $\theta$ independence, or an axially symmetric body is attached to a body that has no symmetry. Aspects of such problems are discussed in [10.7,10.12,10.13].
+
+# PROBLEMS
+
+# Section 10.2
+
+10.1 If a problem is to be mathematically two-dimensional, $\theta$ independence is required of all dependent variables. Explain by example why this requires that $\theta$ be a principal direction of an orthotropic material. Suggestion: Consider axial load on a cylinder.
+
+# Section 10.3
+
+10.2 Imagine that Fig. 6.12-1 depicts displacements and deformations of the square cross section of an axially symmetric four-node element. The axis of revolution is to the left of each cross section. For parts (a) and (b), identify each of the eight modes: that is, is it a rigid-body mode, a zero-energy deformation mode, or a straining mode?
+
+(a) Let [k] be generated by one-point Gauss quadrature.
+
+(b) Let [k] be generated by four-point Gauss quadrature.
+
+10.3 Arguments are presented in Section 6.11 regarding the number of Gauss points needed for correct volume calculation and correct convergence of computed results. Reference there is to plane elements. How should these
+
+<!-- source-page: 329 -->
+
+arguments be amended if reference is to axially symmetric elements instead?
+
+10.4 Revise Figs. 6.5-1 and 6.5-2 to deal with an axially symmetric problem; that is, describe precisely what changes and additions are necessary.   
+10.5 Consider an axially symmetric element whose cross section is a three-node triangle. The displacement field has the form seen in Eq. 4.2-9. Determine stiffness matrix $[k_{a}]$ , which is defined by Eq. 4.1-18, to the extent of writing the integrand in detail and computing the product $[B_{a}]^{T}[E][B_{a}]$ . Use [E] from Eq. 10.2-2.   
+10.6 The sketch shows the cross section of a flat element shaped like a metal washer. D.o.f. are radial displacements $u_{1}$ and $u_{2}$ at nodal circles 1 and 2. The material is isotropic.
+
+(a) Formulate matrices [N] and [B].   
+(b) Let $\nu = 0$ , and generate [k] for a one-radian segment by explicit integration.   
+(c) Let $L = r_2 - r_1$ and $r_m = (r_1 + r_2)/2$ . Simplify integration by assuming that $r = r_m$ . Hence, determine [k] (for a nonzero Poisson's ratio). For what geometry is this [k] a good approximation?   
+(d) For $\nu = 0$ , show that the [k]'s of parts (b) and (c) agree for $r_m >> L$ .   
+(e) From part (b), obtain [k] for the special case $\nu = r_1 = u_1 = 0$ .   
+(f) From part (c), obtain [k] for the special case $\nu = r_1 = u_1 = 0$ .
+
+![](images/page-329_7d53d68bf407db71b6d90782c09a9b54cd591a06e1096d529b1dba389f1cd7f9.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z
+1
+t
+2
+r, u
+r₁
+r₂
+</details>
+
+Problem 10.6
+
+10.7 The four-node axisymmetric element shown has a rectangular cross section of dimensions $2a$ and $2b$ . The material is homogeneous, isotropic, and has mass density $\rho$ . Evaluate element nodal loads $\{\mathbf{r}_e\}$ produced by the following actions.
+
+(a) Uniform radial pressure $p_1$ (tensile).   
+(b) Uniform radial pressure $p_{2}$ (tensile).   
+(c) Line load $q_{1}$ (units N/m), which acts at $x = y = 0$ .   
+(d) Line load $q_{2}$ (units N/m), which acts at $x = 0$ , $y = b$ .
+
+![](images/page-329_b5bdfc2eab766463472f95be79c1b100db00526164c0d4b5b6636e9633fd22f2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+4
+q2
+3
+y
+q1
+x
+1
+2
+p1
+a
+a
+p2
+r,u
+b
+b
+rm
+</details>
+
+Problem 10.7
+
+<!-- source-page: 330 -->
+
+(e) Uniform heating an amount $T$ . Stop after setting up a triple integral over $d\theta dx dy$ .
+
+(f) Rotation at constant angular velocity $\omega$ . Stop after setting up a triple integral over $d\theta dx dy$ .
+
+10.8 The sketch represents three nodes on a $z =$ constant face of an axisymmetric quadratic element. Node 7 is at midside. Determine the consistent nodal load vector for these three nodes if $z$ -direction surface traction $\Phi_z$ is applied as follows.
+
+(a) $\Phi_{z}$ is the constant value $p$ over the face.
+
+(a) $\Phi_z$ is the constant value $p_1$ with $\Phi_z = (\xi^2 - \xi)p_4/2 + (1 - \xi^2)p_7 + (\xi^2 + \xi)p_3/2$ , which is a parabolic variation based on nodal values $p_4, p_7$ , and $p_3$ .
+
+![](images/page-330_60a634b4ccec59ec82eaaa66da6ea0463bbfadfc44ba8bc88195b1be97c120b0.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z₁w
+r₃
+r₇
+r₄
+4
+7
+3
+r
+ξ
+</details>
+
+Problem 10.8
+
+10.9 In the sketch for Problem 10.8, imagine that $r_4 = 0$ , and that $w_4 > 0$ and $w_7 = w_3 = 0$ . Is such a deformation mode reasonable? What do you conclude about $\gamma_{zr}$ at $r = 0$ ?
+
+10.10 Show that $\epsilon_r = \epsilon_\theta$ at $r = 0$ in an axially symmetric problem.
+
+# Section 10.4
+
+10.11 (a) Use Eqs. 10.4-2 to verify the right-hand sides of Eqs. 10.4-3.
+
+(b) Verify the expression for $q_{n}$ in Eq. 10.4-4.
+
+(b) Verify the expression for $q_{n}$ is given by (c) Verify the expression for $q_{n}$ in Eq. 10.4-5. Suggestion: Imagine that $P$ is generated by a distributed load of large intensity acting over a small length.
+
+(d) Determine a Fourier series that represents two radial loads $P$ on a disk, one outward at $\theta = -\pi / 2$ and another outward at $\theta = +\pi / 2$ .
+
+10.12 Use Eqs. 10.4-12 to compute the center deflection and center bending moment in a uniform simply supported beam. Use one, then two, then three series terms, each time computing the percentage error of the approximation.
+
+(a) Consider a uniformly distributed upward load.
+
+(b) Consider a concentrated downward load at midspan.
+
+10.13 A uniform, simply supported beam is loaded by a linearly varying distributed load that has intensity $q_{L}$ at $x = L$ , as shown.
+
+(a) Determine a Fourier series representation of the load.
+
+(a) Determine a Fourier series of $x = L/2$ , hence, determine the deflection and the bending moment at x = L/2, using 1, 2, and 3 series terms. Compute the percentage error of each of these approximations.
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_034.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_034.md
new file mode 100644
index 00000000..10ce8872
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_034.md
@@ -0,0 +1,475 @@
+<!-- source-page: 331 -->
+
+![](images/page-331_cf9329e2c64997f9c0e8aa85b100e5a7f8498b1ea910357e0a73c34804888df7.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+qL
+x
+L
+</details>
+
+Problem 10.13
+
+# Section 10.5
+
+10.14 Assume that a certain axially symmetric pressure vessel can be adequately modeled by four-node elements (as in Fig. 10.1-1b), arranged in a 20 by 1 mesh, so that one element spans the wall thickness of the vessel and 20 elements span the axial dimension. Also assume that the loading is described by Eq. 10.5-2, with harmonics $n = 1, 2, 3, 4, 5$ , and 6. Alternatively, one could contemplate a fully three-dimensional analysis with a 20 by 1 by $m$ mesh of eight-node solid elements, where $m$ is the number of elements around the circumference.
+
+(a) Estimate $m$ so that the three-dimensional model would be of adequate accuracy. Assume that three elements per half-wave of displacement are acceptable.   
+(b) Make an estimate of the cost ratio of the three-dimensional solution to the series solution. Base your estimate on the expense of generating stiffness matrices with an order 2 Gauss rule.   
+(c) Repeat part (b), but base your estimate on the expense of solving equations with a banded equation solver (see Appendix B).
+
+10.15 If $\{\epsilon\} = \{\mathbf{0}\}$ , Eqs. 10.5-4 have the solution
+
+$$
+\left\{ \begin{array}{l} u \\ v \\ w \end{array} \right\} = \left[ \begin{array}{c c c c c c} 0 & \cos \theta & z \cos \theta & 0 & \sin \theta & z \sin \theta \\ 0 & - \sin \theta & - z \sin \theta & r & \cos \theta & z \cos \theta \\ 1 & 0 & - r \cos \theta & 0 & 0 & - r \sin \theta \end{array} \right] \left\{ \begin{array}{l} a _ {1} \\ a _ {2} \\ \cdot \\ \cdot \\ \cdot \\ a _ {6} \end{array} \right\}
+$$
+
+where the $a_{i}$ are constants. A displacement field must contain these terms if there is to be rigid-body motion without strain.
+
+(a) Show that this field does in fact yield $\{\epsilon\} = \{0\}$ .   
+(b) Compare this field with Eqs. 10.5-9: identify columns of the rectangular matrix as to the value of $n$ and as to belonging to the single-barred or the double-barred series.   
+(c) For each of the six columns of the rectangular matrix, describe the physical meaning of the displacement mode it represents.
+
+10.16 Specialize Eqs. 10.5-9 to represent the following rigid-body motions.
+
+(a) Axial translation.   
+(b) Translation perpendicular to the z axis in the plane $\theta = 0$ .   
+(c) Translation perpendicular to the $z$ axis in the plane $\theta = \pi /2$ .   
+(d) Rotation about the z axis.   
+(e) Rotation about the line $\theta = z = 0$ .   
+(f) Rotation about the line $\theta = \pi / 2, z = 0$ .
+
+<!-- source-page: 332 -->
+
+10.17 Seven forces are applied to one end of a cylindrical bar of outer radius c, as shown. Forces $P_{4}$ form the couple $2P_{4}c$ . Stresses at midheight z = h/2 could be calculated by elementary formulas $\sigma = Mc/I$ , and so on. However, imagine that, as an exercise, finite elements are to be used instead, so loads must be expressed in the form of Eqs. 10.5-8. For each of the six different loadings, use Eqs. 10.5-8 to write expressions for surface tractions on end z = h that are statically equivalent to the original loading. (This can be done using only terms for which n = 0 and/or n = 1). Express your answers in terms of c, the $P_{i}$ , $\sin \theta$ , and $\cos \theta$ .
+
+![](images/page-332_cdd972b1538025fb4c9613e7f60eba9a19221a914616a2818d1ee207c0ccd10b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P6
+P1
+P4
+P4
+P5
+c
+P2
+P3
+θ=0
+h
+z
+</details>
+
+Problem 10.17
+
+10.18 A flat plate containing a circular hole of radius R is loaded by axial force P and bending moment M, as shown. The region enclosed by the dashed line of radius c is to be isolated and analyzed as a solid of revolution. If t = plate thickness and $c >> R$ , what load terms from Eqs. 10.5-8 should be used in analysis? Express your answers in terms of P, M, h, t, n, and $\theta$ .
+
+(a) Consider $P$ only (let $M = 0$ ).   
+(b) Consider $M$ only (let $P = 0$ ).
+
+![](images/page-332_bf6e90ba7f1f020d4c0d0db8aae88ca23dc1bfc93d8f5f25e865da3db5934750.jpg)
+
+<details>
+<summary>text_image</summary>
+
+M
+P
+h
+c
+r
+2R
+θ
+M
+P
+</details>
+
+Problem 10.18
+
+# Section 10.6
+
+10.19 Assume that [E] has the form shown in Eq. 10.5-1. Use the first partition of $[B]_{n}$ , as stated in Eq. 10.6-5, to demonstrate the following about the matrix product $[P] = [B]^{T}[E][B]$ . (Note: It is not necessary to write out details of terms extraneous to the question posed.)
+
+<!-- source-page: 333 -->
+
+(a) Show that [P] contains $\sin^2 n\theta$ or $\cos^2 n\theta$ in each term of an on-diagonal submatrix.   
+(b) Show that [P] contains either $\sin m\theta$ $\sin n\theta$ or $\cos m\theta$ $\cos n\theta$ , where $m \neq n$ , in each term of an off-diagonal submatrix.   
+(c) Show that each term of an on-diagonal submatrix has the form $A + Bn^2$ or the form $Cn$ .
+
+10.20 (a) Write the form of $[\mathbf{B}]_n$ in Eq. 10.6-5 appropriate to antisymmetric terms (the double-barred series in Eq. 10.5-9).   
+(b) Show that $[k]_{0}$ , $[k]_{1}$ , and so on, are identical to the corresponding matrices obtained for symmetric terms. Suggestion: See the note in Problem 10.19.   
+(c) Show that the conclusion reached in part (b) would not be true if the negative sign in Eq. 10.5-9b were changed to positive.
+
+10.21 Consider the flat element analyzed in Problem 10.6. However, now allow circumferential displacement v as well as radial displacement u, so that loads without axial symmetry can be treated. Element nodal d.o.f. are now $u_{1}$ , $v_{1}$ , $u_{2}$ , and $v_{2}$ . Let $\theta = 0$ be a plane of symmetry. Formulate $[B]_{n}$ for this element (analogous to $[B]_{n}$ in Eq. 10.6-5, but including all partitions).
+
+10.22 The element described in Problem 10.21 can be used to solve problems of disks and rings under concentrated loads (see sketch). Why do Fourier harmonics for displacement and stress form convergent series, although the series for concentrated loads are not convergent?
+
+![](images/page-333_eae4cccf2cfa3730874a81f3b6356b1456c50c87a939cfbbc8613303251f3eab.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+θ
+r
+P
+θ
+r
+P
+</details>
+
+Problem 10.22
+
+<!-- source-page: 334 -->
+
+# 11 CHAPTER
+
+# BENDING OF FLAT PLATES
+
+Concepts and equations related to the bending of flat plates are reviewed. Elements for thin plates and plates having transverse shear deformation are discussed. Test problems for plate elements are presented.
+
+# 11.1 PLATE-BENDING THEORY
+
+Loads, Stresses, and Moments. A flat plate, like a straight beam, supports transverse loads by bending action. Figure 11.1-1a shows stresses that act on cross sections of a plate whose material is homogeneous and linearly elastic. Normal stresses $\sigma_{x}$ and $\sigma_{y}$ vary linearly with z and are associated with bending moments $M_{x}$ and $M_{y}$ . Shear stress $\tau_{xy}$ also varies linearly with z and is associated with twisting moment $M_{xy}$ . Normal stress $\sigma_{z}$ is considered negligible in comparison with $\sigma_{x}$ , $\sigma_{y}$ , and $\tau_{xy}$ . Transverse shear stresses $\tau_{yz}$ and $\tau_{zx}$ vary quadratically with z. Lateral load q includes surface load and body force, both in the z direction. Unless stated otherwise, “plate bending” means that external loads have no components parallel to the xy plane and that $\sigma_{x} = \sigma_{y} = \tau_{xy} = 0$ on the midsurface z = 0. Excepting stress $\tau_{xy}$ , the foregoing stress patterns are a direct extension of beam theory from one dimension to two.
+
+Stresses in Fig. 11.1-1 produce the following bending moments M and transverse shear forces Q:
+
+$$
+M _ {x} = \int_ {- t / 2} ^ {t / 2} \sigma_ {x} z d z \quad M _ {y} = \int_ {- t / 2} ^ {t / 2} \sigma_ {y} z d z \quad M _ {x y} = \int_ {- t / 2} ^ {t / 2} \tau_ {x y} z d z \tag {11.1-1a}
+$$
+
+$$
+Q _ {x} = \int_ {- t / 2} ^ {t / 2} \tau_ {z x} d z \quad Q _ {y} = \int_ {- t / 2} ^ {t / 2} \tau_ {y z} d z \tag {11.1-1b}
+$$
+
+The M's are moments per unit length and the Q's are forces per unit length. Differential total moments and forces are $M_{x}$ dy, $Q_{x}$ dy, and so on, as shown in Fig. 11.1-1b. Stresses $\sigma_{x}$ , $\sigma_{y}$ , and $\tau_{xy}$ are largest at the surfaces $z = \pm t/2$ , where they have the respective magnitudes $6M_{x}/t^{2}$ , $6M_{y}/t^{2}$ , and $6M_{xy}/t^{2}$ . At arbitrary values of z,
+
+$$
+\sigma_ {x} = \frac {M _ {x} z}{t ^ {3} / 1 2} \quad \sigma_ {y} = \frac {M _ {y} z}{t ^ {3} / 1 2} \quad \tau_ {x y} = \frac {M _ {x y} z}{t ^ {3} / 1 2} \tag {11.1-2}
+$$
+
+<!-- source-page: 335 -->
+
+![](images/page-335_7fcbd796e867ab07df79b1ba68aec0f034c4095b5283c76603de455b6f8e0fd5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z
+dy
+dx
+y
+x
+q
+τyz
+σy
+τxy
+τzx
+σx
+τxy
+l
+</details>
+
+(a)
+
+![](images/page-335_95551c174ead4bae5356d1bd9474b850ba77f854781634484b8863a33ac2f1f5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+q dx dy
+dy
+M_y dx
+dx
+M_xy dx
+Q_y dx
+M_x dy
+Q_x dy
+M_xy dy
+x
+y
+</details>
+
+(b)   
+Figure 11.1-1. (a) Stresses that act on a differential element of a homogeneous, linearly elastic plate. The distributed lateral load is q (force per unit area). (b) The same differential element, viewed normal to the plate. Forces $\odot$ and $\otimes$ act in the positive and negative z directions, respectively.
+
+as may be verified by substituting Eqs. 11.1-2 into Eqs. 11.1-1a. Transverse shear stresses are usually small in comparison with $\sigma_{x}$ , $\sigma_{y}$ , and $\tau_{xy}$ . They have greatest magnitude at z = 0, where $\tau_{yz} = 1.5Q_{y}/t$ and $\tau_{zx} = 1.5Q_{x}/t$ .
+
+Deformations (Kirchhoff Theory). Points on the midsurface z = 0 move in only the z direction as the plate deforms in bending. A line that is straight and normal to the midsurface before loading is assumed to remain straight and normal to the midsurface after loading (see line OP in Fig. 11.1-2). Thus transverse shear deformation is assumed to be zero. A point not on the midsurface has displacement components u and v in the x and y directions, respectively. From Fig. 11.1-2, with $w_{,x}$ and $w_{,y}$ small angles of rotation,
+
+$$
+\begin{array}{l} \epsilon_ {x} = u _ {, x} \quad = - z w _ {, x x} \\ u = - z w, x \quad \text { hence } \quad \epsilon_ {y} = v, y \quad = - z w, y y \tag {11.1-3} \\ \gamma_ {x y} = u _ {, y} + v _ {, x} = - 2 z w _ {, x y} \\ \end{array}
+$$
+
+![](images/page-335_718c91c13fb68459c7e29c3cd42ca38391a80c7ae39433224116191afa96957b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+dx
+t/2
+P
+0
+z
+x,u
+t/2
+</details>
+
+(a)
+
+![](images/page-335_c26e97b1e47aa88d3883eecc94f2637d06f2040f4ac9ff856d0815957c6ee2c0.jpg)
+
+<details>
+<summary>text_image</summary>
+
+w
+w_x
+u = -z w_x
+P
+z
+w_x
+Midsurface
+w
+0
+x,u
+w_x
+</details>
+
+(b)   
+Figure 11.1-2. (a) Differential element of a thin plate before loading. (b) After loading: deformations associated with Kirchhoff plate theory. Point P displaces w units up and $zw_{,x}$ units leftward because of midsurface displacement w and small rotation $w_{,x}$ .
+
+<!-- source-page: 336 -->
+
+These are the strain-displacement relations of Kirchhoff plate theory, which is applicable to a thin plate.
+
+Deformations (Mindlin Theory). A line that is straight and normal to the midsurface before loading is assumed to remain straight but not necessarily normal to the midsurface after loading. Thus, transverse shear deformation is allowed. The motion of a point not on the midsurface is not governed by slopes $w_{,x}$ and $w_{,y}$ as in Kirchhoff theory. Rather, its motion depends on rotations $\theta_{x}$ and $\theta_{y}$ of lines that were normal to the midsurface of the undeformed plate (Fig. 11.1-3). Thus, with $\theta_{x}$ and $\theta_{y}$ small angles of rotation,
+
+$$
+\begin{array}{l} u = - z \theta_ {x} \quad \epsilon_ {x} = - z \theta_ {x, x} \quad \begin{array}{l} \gamma_ {x y} = - z \left(\theta_ {x, y} + \theta_ {y, x}\right) \\ \gamma_ {y z} = w _ {, y} - \theta_ {y} \end{array} \tag {11.1-4} \\ v = - z \theta_ {y} \quad \epsilon_ {y} = - z \theta_ {y, y} \quad \gamma_ {z x} = w _ {, x} - \theta_ {x} \\ \end{array}
+$$
+
+The foregoing expressions for strain are obtained by straightforward application of Eqs. 1.5-4 and 1.5-5. Equations 11.1-4 are the strain-displacement relations of Mindlin plate theory. This theory accounts for transverse shear deformation and is therefore especially suited to the analysis of thick plates and sandwich plates.
+
+Moment–Curvature Relations (Kirchhoff Theory). We begin with stress–strain relations. Let x and y be principal directions of an orthotropic material. Stress $\sigma_{z}$ is considered negligible in comparison with $\sigma_{x}$ , $\sigma_{y}$ , and $\tau_{xy}$ . Transverse shear strains are also considered ineligible, so stress–strain relations that involve them need not be written. What remains is the plane stress–strain relation $\{\sigma\} = [E](\{\epsilon\} - \{\epsilon_{0}\})$ ; that is [11.1],
+
+$$
+\left\{ \begin{array}{l} \sigma_ {x} \\ \sigma_ {y} \\ \tau_ {x y} \end{array} \right\} = \left[ \begin{array}{c c c} E _ {x} ^ {\prime} & E ^ {\prime \prime} & 0 \\ E ^ {\prime \prime} & E _ {y} ^ {\prime} & 0 \\ 0 & 0 & G \end{array} \right] \left(\left\{ \begin{array}{l} \epsilon_ {x} \\ \epsilon_ {y} \\ \gamma_ {x y} \end{array} \right\} - \left\{ \begin{array}{l} \alpha_ {x} T \\ \alpha_ {y} T \\ 0 \end{array} \right\}\right) \tag {11.1-5}
+$$
+
+where initial strains $\{\epsilon_{0}\}$ are presumed caused by thermal expansion with principal expansion coefficients $\alpha_{x}$ and $\alpha_{y}$ . For an isotropic material, with E = elastic modulus and $\nu =$ Poisson's ratio,
+
+![](images/page-336_be52a7f8b7c0b5fbf2cba7ff318c72540011433d35a5050742c7454236833dbc.jpg)
+
+<details>
+<summary>text_image</summary>
+
+θx
+u = -zθx
+w
+P
+z
+Midsurface
+w
+0
+x,u
+w1x
+</details>
+
+Figure 11.1-3. Differential plate element after loading, analogous to Fig. 11.1-2b, but with transverse shear deformation allowed $w_{,x} \neq \theta_{x}$ , so that $\gamma_{zx} = w_{,x} - \theta_{x} \neq 0$ .
+
+<!-- source-page: 337 -->
+
+$$
+E _ {x} ^ {\prime} = E _ {y} ^ {\prime} = \frac {E ^ {\prime \prime}}{\nu} = \frac {E}{1 - \nu^ {2}} \quad \text { and } \quad G = \frac {E}{2 (1 + \nu)} \tag {11.1-6}
+$$
+
+The moment-curvature relation is obtained by substitution of Eqs. 11.1-3 into Eq. 11.1-5 and the result into Eqs. 11.1-1a. This process yields
+
+$$
+\{\mathbf {M} \} = - [ \mathbf {D} _ {K} ] (\{\kappa \} - \{\kappa_ {0} \}) \tag {11.1-7}
+$$
+
+where moments and curvatures are
+
+$$
+\{\mathbf {M} \} = \left[ \begin{array}{l l l} M _ {x} & M _ {y} & M _ {x y} \end{array} \right] ^ {T} \quad \text { and } \quad \{\boldsymbol {\kappa} \} = \left[ \begin{array}{l l l} w _ {, x x} & w _ {, y y} & 2 w _ {, x y} \end{array} \right] ^ {T} \tag {11.1-8}
+$$
+
+In $[\mathbf{D}_K]$ we have $D_{K13} = D_{K31} = D_{K23} = D_{K32} = 0$ and the nonzero terms
+
+$$
+D _ {K 1 1} = \frac {E _ {x} ^ {\prime} t ^ {3}}{1 2} \quad D _ {K 1 2} = D _ {K 2 1} = \frac {E ^ {\prime \prime} t ^ {3}}{1 2} \quad D _ {K 2 2} = \frac {E _ {y} ^ {\prime} t ^ {3}}{1 2} \quad D _ {K 3 3} = \frac {G t ^ {3}}{1 2} \tag {11.1-9}
+$$
+
+If the material is isotropic, then
+
+$$
+\left[ \mathbf {D} _ {K} \right] = \left[ \begin{array}{c c c} D & \nu D & 0 \\ \nu D & D & 0 \\ 0 & 0 & (1 - \nu) D / 2 \end{array} \right], \quad \text { where } \quad D = \frac {E t ^ {3}}{1 2 (1 - \nu^ {2})} \tag {11.1-10}
+$$
+
+D is called “flexural rigidity” and is analogous to bending stiffness EI of a beam. Indeed, if the plate has unit width and $\nu = 0$ , then $D = EI = Et^{3}/12$ .
+
+As a particular example of initial curvatures $\{\kappa_{0}\}$ , consider a temperature gradient $T = -2zT_{0}/t$ . This is a linear temperature variation from $T_{0}$ at z = -t/2 to $-T_{0}$ at z = t/2. Thus the process that yields Eq. 11.1-7 gives the initial curvatures
+
+$$
+\{\kappa_ {0} \} = \left[ \begin{array}{l l l} 2 \alpha_ {x} T _ {0} / t & 2 \alpha_ {y} T _ {0} / t & 0 \end{array} \right] ^ {T} \tag {11.1-11}
+$$
+
+Equation 11.1-7 shows that actions in the x and y directions are coupled, even for an isotropic plate. In Fig. 11.1-4a, $w_{,yy}$ is constant and $w_{,xx} = w_{,xy} = 0$ in the central portion, but $M_{x}$ is nonzero because of the Poisson effect. But $M_{x} = 0$ at the free edges $x = \pm a$ , so these edges curl a bit (Fig. 11.1-4b). Only if $a \approx t$ , so that $M_{x} \approx 0$ throughout, does the plate act like a beam, displaying the familiar anticlastic surface. The pure twist of Fig. 11.1-4c is associated with moments $-M_{xy}$ alone ( $M_{x} = M_{y} = 0$ ) if the plate is isotropic.
+
+![](images/page-337_7c3fc602d430ae00ae9e7ba8fc12815d17c6a395e3c4113c47171addbdfaf671.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+a
+My
+a
+y
+My
+x
+</details>
+
+(a)
+
+![](images/page-337_ffb06e652f505d1ae46ad53cfe716e4edce2f07dbc47db2039c1108c2cf24501.jpg)  
+(b)
+
+![](images/page-337_456e07ec793fe169d2ee880a4699da67359efb9799a73a3a3b6e53149ef75027.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+y
+x
+</details>
+
+(c)   
+Figure 11.1-4. (a) Bending to a cylindrical surface by moments $M_{y}$ on the edges y = constant. (b) Cross section cut by the xz plane. (c) The w = xy state of pure twist: $w_{xx} = w_{yy} = 0$ , $w_{xy} > 0$ .
+
+<!-- source-page: 338 -->
+
+Moment-Curvature Relations (Mindlin Theory). Again let $x$ and $y$ be principal material directions. The moment-curvature relations of Mindlin plate theory are obtained by essentially the same procedure as used to obtain Eq. 11.1-7. However, we must use Eqs. 11.1-4 instead of Eqs. 11.1-3 and include the shear stress-strain relations $\tau_{yz} = G_{yz}\gamma_{yz}$ and $\tau_{zx} = G_{zx}\gamma_{zx}$ . The resulting moment-curvature relation is abbreviated as $\{\mathbf{M}\} = -[\mathbf{D}_M](\{\boldsymbol{\kappa}\} - \{\boldsymbol{\kappa}_0\})$ . Written out, this relation is
+
+$$
+\left\{ \begin{array}{l} M _ {x} \\ M _ {y} \\ M _ {x y} \\ Q _ {y} \\ Q _ {x} \end{array} \right\} = - \underbrace {\left[ \begin{array}{c c c c c} & & 0 & 0 \\ \left[ \mathbf {D} _ {K} \right] & & 0 & 0 \\ 3 \times 3 & & 0 & 0 \\ 0 & 0 & 0 & G _ {y z} t & 0 \\ 0 & 0 & 0 & 0 & G _ {z x} t \end{array} \right]} _ {[ \mathbf {D} _ {M} ]} \left(\underbrace {\left\{ \begin{array}{c} \theta_ {x , x} \\ \theta_ {y , y} \\ \theta_ {x , y} + \theta_ {y , x} \\ \theta_ {y} - w _ {, y} \\ \theta_ {x} - w _ {, x} \end{array} \right\}} _ {\{\boldsymbol {\kappa} \}} - \{\boldsymbol {\kappa} _ {0} \}\right) \tag {11.1-12}
+$$
+
+where $[\mathbf{D}_K]$ is the same as in Eq. 11.1-7. The shear stiffness terms $G_{yz}t$ and $G_{zx}t$ in Eq. 11.1-12 may be replaced by $G_{yz}t / 1.2$ and $G_{zx}t / 1.2$ to permit the parabolic distributions of $\tau_{yz}$ and $\tau_{zx}$ (shown in Fig. 11.1-1a) to be replaced by uniform distributions, as explained in Section 9.4. If represented as rotation vectors by the right-hand rule, $\theta_x$ and $\theta_y$ point in the $-y$ and $+x$ directions, respectively. Initial curvatures $\{\kappa_0\}$ are those of Kirchhoff theory, augmented by zeros in positions 4 and 5.
+
+Initial curves are not possible. If the plate is isotropic, then $G_{yz} = G_{zx} = G$ and Eqs. 11.1-10 apply to submatrix $[\mathbf{D}_K]$ in Eq. 11.1-12. For an isotropic sandwich plate, Fig. 11.1-5, with thin facings, $G$ the shear modulus of the core, and $E$ and $\nu$ the elastic modulus and Poisson ratio of each facing,
+
+$$
+D _ {M 1 1} = D _ {M 2 2} = \frac {D _ {M 1 2}}{\nu} = \frac {D _ {M 2 1}}{\nu} = \frac {E h (c + h) ^ {2}}{2 \left(1 - \nu^ {2}\right)} \tag {11.1-13}
+$$
+
+$$
+D _ {M 3 3} = \frac {E h (c + h) ^ {2}}{4 (1 + \nu)} \quad D _ {M 4 4} = D _ {M 5 5} = \frac {G (c + h) ^ {2}}{c}
+$$
+
+and all other entries in the 5 by 5 matrix $[\mathbf{D}_M]$ of Eq. 11.1-12 are zero [11.2]. If principal material directions are $x'$ and $y'$ rather than $x$ and $y$ , as in Fig. 7.3-1, then coordinate transformation is required. Arrays in Eq. 11.1-7 transform as $\{\mathbf{M}\} = [\mathbf{T}_\epsilon]^T\{\mathbf{M}'\}$ , $\{\boldsymbol{\kappa}'\} = [\mathbf{T}_\epsilon]\{\boldsymbol{\kappa}\}$ , and $[\mathbf{D}_K] = [\mathbf{T}_\epsilon]^T [\mathbf{D}_K'] [\mathbf{T}_\epsilon]$ , where $[\mathbf{T}_\epsilon]$ is given by Eq. 7.3-11. Coefficients in the southeast corner of $[\mathbf{D}_M]$ in Eq. 11.1-12 are $D_{K44} = m_1^2 G_{z'x'}t + m_2^2 G_{y'z'}t$ , $D_{K55} = \ell_1^2 G_{z'x'}t + \ell_2^2 G_{y'z'}t$ , and $D_{K45} = D_{K54} = \ell_1m_1G_{z'x'}t + \ell_2m_2G_{y'z'}t$ , where the $\ell$ 's and $m$ 's are given in Fig. 7.3-1. Remarks. A plate can be loaded by distributed lateral load of intensity $q$ and by initial curvatures $\{\kappa_0\}$ , as just discussed. Concentrated forces and line loads may
+
+![](images/page-338_2d65e2041eb4b8310080c9ff128ef0610b776030944df7f5c27623f0f6996c73.jpg)
+
+<details>
+<summary>text_image</summary>
+
+facing
+core
+facing
+h
+c
+h
+</details>
+
+Figure 11.1-5. Cross section of a sandwich plate. Typically the core resists little but transverse shear strains, so almost all bending stiffness is provided by membrane action in thin facings.
+
+<!-- source-page: 339 -->
+
+also be present. Edge moments M and transverse shears Q may be applied as known loads or as support reactions. Except for line loads, these loads are analogous to loads present in beam theory. Nodal equivalents of loads q and $\{\kappa_{0}\}$ can be computed by means of Eq. 4.1-6.
+
+In finite element analysis of plates, whether by Kirchhoff or Mindlin theory, d.o.f. at a node i are typically one lateral displacement $(w_{i})$ and two rotations $(w_{,xi}$ and $w_{,yi}$ or $\theta_{xi}$ and $\theta_{yi})$ . At a free edge none of the three d.o.f. is restrained. At a clamped edge all d.o.f. are restrained. Further discussion of boundary conditions appears in Section 11.5.
+
+Full compatibility of interelement displacements requires that, in any $z =$ constant layer, displacements $u, v,$ and $w$ be the same in adjacent elements where the elements meet. Accordingly, from Eqs. 11.1-3, compatible Kirchhoff elements are $C^1$ elements, as they must display interelement continuity of $w, w_{xx}$ , and $w_{yy}$ . Note that along (say) a $y$ -parallel interelement boundary, continuity of $w$ ensures continuity of $w_{yy}$ but not continuity of the boundary-normal slope $w_{xx}$ . From Eqs. 11.1-4, compatible Mindlin elements are $C^0$ elements, as the fields, $w, \theta_x,$ and $\theta_y$ (but not their derivatives) must be interelement-continuous. Note that along (say) a $y$ -parallel interelement boundary, continuity of $\theta_x$ does not imply continuity of $w_{xx}$ unless the plate is so thin that $\gamma_{zx} = 0$ .
+
+We have tacitly assumed that material properties are either independent of z or symmetric with respect to the midsurface z = 0. If not, bending may produce forces in the xy plane so that the midsurface is not a surface where $\sigma_{x} = \sigma_{y} = \tau_{xy} = 0$ . This effect is pronounced in two-layer laminated plates [11.3].
+
+Appreciable in-plane forces may also arise if deflections w are more than a few tenths of the plate thickness. This happens even when supports apply no in-plane forces, because the deflected shape of the plate requires stretching or shortening in the midsurface (unless deflections are small or the deflected shape is cylindrical or conical). In-plane forces act to support part of the load. Thus the stiffness of a plate effectively increases as deflection increases, which makes the problem nonlinear. In some problems linear theory may overestimate displacements by 50% if deflection w equals thickness t [11.1].
+
+# 11.2 FINITE ELEMENTS FOR PLATES
+
+A great many finite elements for plates have been proposed: an incomplete survey lists 154 references and 88 different elements $[11.4]$ . In what follows we briefly consider some options in element formulation. Further details may be found in Ref. 11.5 and in papers cited by Ref. 11.4.
+
+Kirchhoff Elements. Kirchhoff theory is applicable to thin plates, in which transverse shear deformation is neglected. Strain energy in the plate is determined entirely by in-plane strains $\epsilon_{x}$ , $\epsilon_{y}$ and $\gamma_{xy}$ . In turn, strains are determined entirely by the lateral displacement field $w = w(x, y)$ , as shown by Eqs. 11.1-3.
+
+The starting point for formulating an element stiffness matrix is the strain energy term of Eq. 4.1-1,
+
+$$
+U ^ {\prime} = \int_ {V} \frac {1}{2} \{\boldsymbol {\epsilon} \} ^ {T} [ \mathbf {E} ] \{\boldsymbol {\epsilon} \} d V, \quad \text { where } \quad \{\boldsymbol {\epsilon} \} ^ {T} = \left[ - z w _ {, x x} - z w _ {, y y} - 2 z w _ {, y y} \right] \tag {11.2-1}
+$$
+
+<!-- source-page: 340 -->
+
+![](images/page-340_80875b896dede70e3f2364bcd4a33fad4f1078ab96659983f1649854f995537b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+1
+w₃
+4
+a
+y
+2
+w, x3
+3
+b
+b
+w, y3
+x
+</details>
+
+Figure 11.2-1. Twelve-d.o.f. rectangular Kirchhoff plate element, with typical d.o.f. shown at node 3.
+
+and [E] is given by Eq. 11.1-5. With $dV = dz \, dA$ , where $dA = dx \, dy$ is an increment of midsurface area $A$ , integration through thickness $t$ yields
+
+$$
+U = \int_ {A} \frac {1}{2} \{\boldsymbol {\kappa} \} ^ {T} [ \mathbf {D} _ {K} ] \{\boldsymbol {\kappa} \} d A, \quad \text { where } \quad \{\boldsymbol {\kappa} \} ^ {T} = \left[ w _ {, x x} \quad w _ {, y y} \quad 2 w _ {, x y} \right] \tag {11.2-2}
+$$
+
+and $[D_{K}]$ is the moment–curvature relation of Eq. 11.1-7. An interpolation of w from element nodal d.o.f. $\{d\}$ is devised, then differentiated to yield curvatures $\{\kappa\}$ . For an element having N nodes,
+
+$$
+w = \left\lfloor \mathbf {N} \right\rfloor_ {1 \times 3 N} \{\mathbf {d} \} \quad \text { hence } \quad \{\kappa \} = \left[ \mathbf {B} \right] _ {3 \times 3 N} \{\mathbf {d} \} \tag {11.2-3}
+$$
+
+D.o.f. of a Kirchhoff element are $\{\mathbf{d}\} = \left[w_1 \quad w_{,x1} \quad w_{,y1} \ldots w_N \quad w_{,xN} \quad w_{,yN}\right]^T$ . Finally, by substitution of Eq. 11.2-3 into Eq. 11.2-2, the element stiffness matrix [k] appears:
+
+$$
+U = \frac {1}{2} \{\mathbf {d} \} ^ {T} [ \mathbf {k} ] \{\mathbf {d} \}, \quad \text { where } \quad \underset {3 N \times 3 N} {[ \mathbf {k} ]} = \int_ {A} [ \mathbf {B} ] ^ {T} [ \mathbf {D} _ {K} ] [ \mathbf {B} ] d A \tag {11.2-4}
+$$
+
+For example, consider the twelve-d.o.f. rectangular element of Fig. 11.2-1. Typical d.o.f. $w_{,x3}$ and $w_{,y3}$ are slopes (i.e., rotations) of the plate midsurface at node 3. Their vector representations, shown in Fig. 11.2-1, are determined according to the right-hand rule. Lateral displacement w of this element has the form [11.5]
+
+$$
+w = \left[ 1, x, y, x ^ {2}, x y, y ^ {2}, x ^ {3}, x ^ {2} y, x y ^ {2}, y ^ {3}, x ^ {3} y, x y ^ {3} \right] \{\mathrm{a} \} \tag {11.2-5}
+$$
+
+This element does not preserve interelement continuity of boundary-normal slopes. Vector $\{a\}$ contains twelve generalized coordinates, which must be exchanged for the twelve nodal d.o.f. $\{d\}$ by the usual process (e.g., Eqs. 3.13-3). Thus Eqs. 11.2-3 are established, and [k] follows from Eq. 11.2-4.
+
+Early efforts to formulate triangular Kirchhoff elements in the same way met with unexpected difficulties. A nine-term field for w is appropriate to the element of Fig. 11.2-2a. Unfortunately, as seen in Eq. 11.2-5, a complete cubic contains 10 terms. Candidate nine-term fields include
+
+$$
+w = \left[ 1, x, y, x ^ {2}, y ^ {2}, x ^ {3}, x ^ {2} y, x y ^ {2}, y ^ {3} \right] \{\mathbf {a} \} \tag {11.2-6a}
+$$
+
+$$
+w = \left\lfloor 1, x, y, x ^ {2}, x y, y ^ {2}, x ^ {3}, x ^ {2} y + x y ^ {2}, y ^ {3} \right\rfloor \{\mathrm{a} \} \tag {11.2-6b}
+$$
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_035.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_035.md
new file mode 100644
index 00000000..4ca98eac
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_035.md
@@ -0,0 +1,470 @@
+<!-- source-page: 341 -->
+
+![](images/page-341_2ba1266f6cfe85937278e012fd7ce05576c647f0dde3b1a8ea2a601a7bc25e30.jpg)
+
+<details>
+<summary>flowchart</summary>
+
+```mermaid
+graph TD
+    w1 -->|w1x1| 1
+    w1 -->|w1y1| 2
+    w2 -->|w2x2| 2
+    w2 -->|w2y2| 3
+    w3 -->|w3x3| 3
+    w3 -->|w3y3| 3
+    1 --> w1
+    2 --> w3
+    3 --> w3
+```
+</details>
+
+(a)
+
+![](images/page-341_f89591d14d17e0ac45022d326b64346d5f069dd506b6ebb1b40d14e8e3839ccc.jpg)
+
+<details>
+<summary>flowchart</summary>
+
+```mermaid
+graph TD
+    w1["1"] -->|w_{n6}| 6["6"]
+    w1 -->|w_{n4}| 4["4"]
+    w2["2"] -->|w_{n5}| 5["5"]
+    w2 -->|w_{n4}| 4
+    w3["3"] -->|w_{n5}| 5
+    w3 -->|w_{n4}| 4
+    w1 -->|w_{n6}| 6
+```
+</details>
+
+(b)   
+Figure 11.2-2. Triangular Kirchhoff plate elements. (a) Nine-d.o.f. element. (b) Six-d.o.f., constant-curvature element. The symbol $\odot$ indicates an arrow directed out of the paper.
+
+Equation 11.2-6a omits the $xy$ term. The resulting element is therefore incapable of passing a constant-twist patch test, and is considered unacceptable. Equation 11.2-6b leads to an element that lacks geometric isotropy, has poor convergence properties, and for certain element shapes has a singular transformation matrix $[\mathbf{A}]$ in the relation $\{\mathbf{d}\} = [\mathbf{A}]\{\mathbf{a}\}$ .
+
+Eventually such difficulties were overcome, often by using subelements, by abandoning strict Kirchhoff plate theory, or by using variational principles other than stationary potential energy. An interesting result of these efforts is the element of Fig. 11.2-2b, which appears to be the simplest possible Kirchhoff element. It can represent only rigid-body motion (constant w, $w_{,x}$ , or $w_{,y}$ ) and constant-curvature states (constant $w_{,xx}$ , $w_{,yy}$ , or $w_{,xy}$ ). Its six d.o.f. are translations w at corners and normal slopes $w_{,n}$ at midsides, where n is a direction normal to the side.
+
+After nodal d.o.f. have been obtained by solution of the structural equations, element curvatures $\{\kappa\}$ are obtained from Eq. 11.2-3, then bending moments from Eq. 11.1-7, and finally stresses from Eqs. 11.1-2.
+
+Mindlin Elements. Nodal d.o.f. consist of lateral deflections $w_{i}$ and rotations $\theta_{xi}$ and $\theta_{yi}$ of midsurface normals. The corresponding deflections and rotations within an element are obtained by independent shape function interpolations:
+
+$$
+w = \sum N _ {i} w _ {i} \quad \theta_ {x} = \sum N _ {i} \theta_ {x i} \quad \theta_ {y} = \sum N _ {i} \theta_ {y i} \tag {11.2-7}
+$$
+
+Equations 11.1-4 and 11.2-7 yield strains. We see that the field for w is coupled to the fields for $\theta_{x}$ and $\theta_{y}$ only via the shear strains $\gamma_{yz}$ and $\gamma_{zx}$ . Using the strains of Eq. 11.1-4, one can write an expression for strain energy and from it obtain an element stiffness matrix. Mindlin elements are considered in more detail in Section 11.3.
+
+Discrete Kirchhoff Elements. Initially, Eqs. 11.2-7 are again used to express displacements and rotations within an element. However, more d.o.f. are included than are to appear in the final element. Strains of importance in Kirchhoff theory
+
+<!-- source-page: 342 -->
+
+$(\epsilon_x, \epsilon_y, \text{and } \gamma_{xy} \text{ only})$ are evaluated from Eqs. 11.1-4. Accordingly, element strains and strain energy depend on $\theta_x$ and $\theta_y$ but are independent of $w$ . Next, using $\gamma_{yz} = \gamma_{zx} = 0$ in Eqs. 11.1-4, we impose the “Kirchhoff constraints” $w_{,y} = \theta_y$ and $w_{,x} = \theta_x$ at certain points. These points are sufficient in number to eliminate the “excess” d.o.f. Thus $w$ becomes coupled to the rotations $\theta_x$ and $\theta_y$ , and only nodal d.o.f. used for interelement connections remain. Discrete Kirchhoff elements are considered further in Section 11.4.
+
+Finite Strips. The finite strip method [11.7] exploits the “semianalytical” method described in Section 10.5. Imagine that the plate of Fig. 11.2-3 is simply supported along edges y = 0 and y = b. Lateral displacement w can be taken as
+
+$$
+w = \sum_ {n} \left[ \mathrm{N} \right] \{\overline {{{\mathrm{d}}}} \} _ {n} \sin \frac {n \pi y}{b} \tag {11.2-8}
+$$
+
+where $[N]$ represents standard cubic shape functions, Fig. 3.13-2, and $\{\overline{d}\}_{n}$ contains amplitudes of displacement and rotation along nodal lines 1–1 and 2–2 for mode n. Thus Eq. 11.2-8 expresses the superposition of several solutions, each associated with a single Fourier harmonic of loading. This superposition method replaces division of the plate into elements in the y direction. The finite strip method can also be applied to folded plates and to box beams, either straight or curved. Advantages include modest needs for computer resources and input data. Disadvantages include an inability to cope with arbitrary shapes and arbitrary boundary conditions.
+
+Nodal Loads. Consistent element nodal loads $\{r_{e}\}$ , Eq. 4.1-6, include terms such as
+
+$$
+\int_ {A} \left\lfloor \mathbf {N} \right] ^ {T} q d A \quad \int \left\lfloor \mathbf {N} _ {, x} \right] ^ {T} \overline {{M}} _ {x} d y \quad \text { and } \quad \int \left\lfloor \mathbf {N} _ {, x} \right] ^ {T} \overline {{M}} _ {x y} d x \tag {11.2-9}
+$$
+
+where $q$ is the intensity of distributed lateral load, and $\overline{M}_x$ and $\overline{M}_{xy}$ are prescribed boundary values of moment loads. The respective shape function matrices in Eqs. 11.2-9 are associated with lateral displacement, rotation about the $y$ axis along a $y$ -parallel edge that carries $\overline{M}_x$ , and rotation about the $y$ axis along an $x$ -parallel edge that carries $\overline{M}_{xy}$ .
+
+As a simpler but less accurate alternative to Eq. 4.1-6, a distributed load on a plate element can be “lumped” by assigning equal fractions of the total force on the element to its translational d.o.f. Correct answers are approached with mesh refinement.
+
+For Kirchhoff elements, the first of Eqs. 11.2-9 produces nodal moments as
+
+![](images/page-342_cd30fb1cd463ad24f56df3abf9eb86277ea523ca58f695c8a6ee84363dd87b78.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+b
+1
+y
+2
+L
+1
+2
+x
+</details>
+
+Figure 11.2-3. A thin rectangular plate divided into finite strips. A typical strip is shaded.
+
+<!-- source-page: 343 -->
+
+well as nodal forces, because $w = [N]\{d\}$ describes a field dependent on nodal rotations as well as nodal translations. For Mindlin elements, from the first of Eqs. 11.2-7, $w = [N]\{w\}$ describes a field dependent on only nodal translations, so Eq. 11.2-9 yields only nodal forces. If elements have no internal nodes, a one-element, simply supported, uniformly loaded Mindlin plate would then yield zero deflection, which is unreasonable. Engineers may therefore be willing to forgo mathematical consistency and use in Eq. 11.2-9 a substitute field that provides nodal moments, or they may adopt some other ad hoc strategy in order to improve accuracy in a coarse mesh.
+
+In Kirchhoff plate theory, nodal loads produced by initial curvatures $\{\kappa_{0}\}$ (e.g., Eq. 11.1-11) are
+
+$$
+\left\{\mathbf {r} _ {e} \right\} = \int_ {A} \left[ \mathbf {B} \right] ^ {T} \left[ \mathbf {D} _ {K} \right] \left\{\kappa_ {0} \right\} d A \tag {11.2-10}
+$$
+
+In Mindlin plate theory, the last two entries in the product $[D_{M}]\{\kappa_{0}\}$ are zero. Accordingly, [B] can be truncated to three rows, so that [B]{d} yields only $\theta_{x,x}$ , $\theta_{y,y}$ , and $\theta_{x,y} + \theta_{y,x}$ (see Eq. 11.3-4). Then Eq. 11.2-10 yields nodal moments produced by $\{\kappa_{0}\}$ , but no z-direction nodal forces.
+
+# 11.3 MINDLIN PLATE ELEMENTS
+
+Mindlin plate elements account for bending deformation and for transverse shear deformation. Accordingly, they may be used to analyze thick plates as well as thin plates. When used for thin plates, however, they may be less accurate than Kirchhoff elements, which do not allow transverse shear deformation.
+
+Typical Mindlin plate elements are shown in Fig. 11.3-1. For convenience of notation, but not because of any demand of finite element plate theory, we will assume that all three d.o.f. shown in Fig. 11.3-1c are present at every node. Rotations $\theta_{x}$ and $\theta_{y}$ are shown by two-headed arrows according to the right-hand rule. These are rotations of a line that was normal to the midsurface of the undeformed plate. Note that $\theta_{x} \neq w_{,x}$ and $\theta_{y} \neq w_{,y}$ unless we approach the thin-plate limit, in which case $\gamma_{zx} = \gamma_{yz} = 0$ (see Fig. 11.1-3). A special form of Mindlin plate element is the Mindlin beam element, which the reader may wish to review (see Eqs. 9.4-1 and 9.4-2 and Fig. 9.4-1).
+
+Stiffness Matrix. The starting point for formulating an element stiffness matrix is an expression for strain energy U. With A the area of the plate midsurface,
+
+$$
+U = \frac {1}{2} \int_ {A} \int_ {- t / 2} ^ {t / 2} \{\boldsymbol {\epsilon} \} ^ {T} [ \mathrm{E} ] \{\boldsymbol {\epsilon} \} d z d A \tag {11.3-1}
+$$
+
+where $\{\pmb{\epsilon}\}^T = \left\lfloor \epsilon_x \quad \epsilon_y \quad \gamma_{xy} \quad \gamma_{yz} \quad \gamma_{zx} \right\rfloor$ , and the individual strains are stated in terms of displacements by Eqs. 11.1-4. Integration through the thickness yields
+
+$$
+U = \frac {1}{2} \int_ {A} \{\boldsymbol {\kappa} \} ^ {T} [ \mathbf {D} _ {M} ] \{\boldsymbol {\kappa} \} d A \tag {11.3-2}
+$$
+
+<!-- source-page: 344 -->
+
+![](images/page-344_35ac3a2cc98b1859188182d602ccbdd6c7acd46e4fb2371dd42025aa334d18f2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+4
+η
+ξ
+3
+1
+2
+x
+</details>
+
+(a)
+
+![](images/page-344_3ccc1f74078795fbdf2741c3103915c238cf4cfde24c1011e9c91be8b12636bc.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+4
+7
+η
+3
+8
+9
+ξ
+6
+1
+5
+2
+x
+</details>
+
+(b)
+
+![](images/page-344_800c1feba10d4a28327bea09f5899f98a02d548d416366f6855d5351184ae24e.jpg)
+
+<details>
+<summary>flowchart</summary>
+
+```mermaid
+graph TD
+    w_i((w_i)) -->|θ_yi| x
+    w_i -->|θ_xi| i
+    i --> y
+    x --> y
+```
+</details>
+
+(c)   
+Figure 11.3-1. (a) Bilinear element, top view. (b) Quadratic element, top view. (c) Notation and sign convention for d.o.f. at a typical node i. The symbol ⊙ indicates an arrow directed out of the paper.
+
+where $[D_{M}]$ and $\{\kappa\}$ are defined in Eq. 11.1-12. If the d.o.f. of Fig. 11.3-1c are present at every node, the same shape functions $N_{i}$ are used to interpolate w, $\theta_{x}$ , and $\theta_{y}$ from nodal values of these quantities, that is,
+
+$$
+\left\{ \begin{array}{l} w \\ \theta_ {x} \\ \theta_ {y} \end{array} \right\} = \sum_ {i = 1} ^ {N} \left[ \begin{array}{c c c} N _ {i} & 0 & 0 \\ 0 & N _ {i} & 0 \\ 0 & 0 & N _ {i} \end{array} \right] \left\{ \begin{array}{l} w _ {i} \\ \theta_ {x i} \\ \theta_ {y i} \end{array} \right\} \quad \text { or } \quad \left\{\mathbf {u} \right\} = \left[ \begin{array}{l} \mathbf {N} \\ 3 \times 3 N \end{array} \right] \left\{\mathbf {d} \right\} \tag {11.3-3}
+$$
+
+where $N$ is the number of nodes per element, and $\{\mathbf{d}\} = \left[w_1 \quad \theta_{x1} \quad \theta_{y1} \ldots w_N \quad \theta_{xN} \quad \theta_{yN}\right]^T$ . Curvatures $\{\kappa\}$ stated in Eq. 11.1-12 are
+
+$$
+\{\boldsymbol {\kappa} \} = \left\{ \begin{array}{c} \theta_ {x, x} \\ \theta_ {y, y} \\ \theta_ {x, y} + \theta_ {y, x} \\ \theta_ {y} - w _ {, y} \\ \theta_ {x} - w _ {, x} \end{array} \right\} = [ \partial ] \{\mathbf {u} \}, \quad \text { where } \quad [ \partial ] = \left[ \begin{array}{c c c} 0 & \partial / \partial x & 0 \\ 0 & 0 & \partial / \partial y \\ 0 & \partial / \partial y & \partial / \partial x \\ - \partial / \partial y & 0 & 1 \\ - \partial / \partial x & 1 & 0 \end{array} \right] \tag {11.3-4}
+$$
+
+Equations 11.3-3 and 11.3-4 yield
+
+$$
+\{\boldsymbol {\kappa} \} = \underset {5 \times 3 N} {[ \mathbf {B} ]} \{\mathbf {d} \}, \quad \text { where } \quad [ \mathbf {B} ] = [ \partial ] [ \mathbf {N} ] = \left[ \begin{array}{c c c c} 0 & N _ {1, x} & 0 & \dots \\ 0 & 0 & N _ {1, y} & \dots \\ 0 & N _ {1, y} & N _ {1, x} & \dots \\ - N _ {1, y} & 0 & N _ {1} & \dots \\ - N _ {1, x} & N _ {1} & 0 & \dots \end{array} \right] \tag {11.3-5}
+$$
+
+And finally, from Eqs. 11.3-2 and 11.3-5,
+
+$$
+U = \frac {1}{2} \{\mathbf {d} \} ^ {T} [ \mathbf {k} ] \{\mathbf {d} \}, \quad \text { where } \quad \underset {3 N \times 3 N} {[ \mathbf {k} ]} = \int_ {A} [ \mathbf {B} ] ^ {T} [ \mathbf {D} _ {M} ] [ \mathbf {B} ] d A \tag {11.3-6}
+$$
+
+If the plate is rectangular, shape functions $N_{i}$ can be expressed in terms of x and y. Then dA = dx dy. If the plate is of more general shape, as shown in Fig. 11.3-1, the $N_{i}$ can be expressed in terms of isoparametric coordinates $\xi$ and $\eta$ . Then dA = J d $\xi$ d $\eta$ , where J is the Jacobian determinant. For bilinear and quadratic elements respectively, shape functions $N_{i}$ are given by Eqs. 6.3-2 and Table
+
+<!-- source-page: 345 -->
+
+6.6-1. Shape function derivatives, needed in [B], are determined by the usual transformation (e.g, Eq. 6.3-7):
+
+$$
+N _ {i, x} = \Gamma_ {1 1} N _ {i, \xi} + \Gamma_ {1 2} N _ {i, \eta} \quad \text { and } \quad N _ {i, y} = \Gamma_ {2 1} N _ {i, \xi} + \Gamma_ {2 2} N _ {i, \eta} \tag {11.3-7}
+$$
+
+Mindlin plate elements can be regarded as special forms of solid elements. For example, the bilinear plate element of Fig. 11.3-1a closely resembles the trilinear solid element of Fig. 6.7-1, if the solid element is made thin in one direction. The solid element has twice as many d.o.f. as the plate element. In addition, nodes of the solid that lie on a midsurface-normal line define the thickness-direction strain, $\epsilon_{z}$ , which is ignored in plate-bending theory. If $\epsilon_{z}$ were included in the formulation, nodes that span the thickness would be coupled by stiffness coefficients that become very large in comparison with bending stiffnesses as the plate becomes thin. The discrepancy may lead to numerical difficulty of a type discussed in Section 18.2. In summary, considerations of economy and robustness indicate that solid elements should not be used to model plates.
+
+Quadrature Rule and Locking. One can regard the stiffness matrix of a Mindlin plate element as being composed of a bending stiffness $[k_{b}]$ and a transverse shear stiffness $[k_{s}]$ . From Eq. 11.3-6, with $[B] = [B_{b}] + [B_{s}]$ ,
+
+$$
+[ \mathbf {k} ] = \underbrace {\int_ {A} [ \mathbf {B} _ {b} ] ^ {T} [ \mathbf {D} _ {M} ] [ \mathbf {B} _ {b} ] d A} _ {[ \mathbf {k} _ {b} ]} + \underbrace {\int_ {A} [ \mathbf {B} _ {s} ] ^ {T} [ \mathbf {D} _ {M} ] [ \mathbf {B} _ {s} ] d A} _ {[ \mathbf {k} _ {s} ]} \tag {11.3-8}
+$$
+
+$[B_{b}]$ is associated with in-plane strains $\epsilon_{x}, \epsilon_{y}$ , and $\gamma_{xy}$ and is obtained by setting rows 4 and 5 of [B] to zero. $[B_{s}]$ is associated with transverse shear strains $\gamma_{yz}$ and $\gamma_{zx}$ , and is obtained by setting rows 1, 2, and 3 of [B] to zero. The cross product terms $[B_{b}]^{T}[D_{M}][B_{s}]$ and $[B_{s}]^{T}[D_{M}][B_{b}]$ are zero because of the distribution of zeros in $[B_{b}]$ , $[B_{s}]$ , and $[D_{M}]$ . Bending stiffness $[k_{b}]$ mobilizes only the $[D_{K}]$ portion of $[D_{M}]$ , and transverse shear stiffness $[k_{s}]$ mobilizes only the $G_{yz}t$ and $G_{zx}t$ terms in $[D_{M}]$ .
+
+The splitting of [k] into components $[k_{b}]$ and $[k_{s}]$ is also seen in Eq. 9.4-4 of Section 9.4, where it is argued that each integration point used to evaluate $[k_{s}]$ imposes one constraint on transverse shear strain $\gamma_{zx}$ of the two-node Mindlin beam element, and may produce locking of the mesh if the beam is thin and too many integration points are used to evaluate $[k_{s}]$ . Similar considerations apply to Mindlin plate elements. However, each integration point used for $[k_{s}]$ brings two constraints to a Mindlin plate element, one associated with $\gamma_{yz}$ and the other with $\gamma_{zx}$ . Locking of Mindlin plate elements caused by too many transverse shear constraints can be avoided by adopting a reduced or selective integration rule to generate [k]. Or, one can redefine the transverse shear interpolation; see [11.17].
+
+Various Mindlin plate elements are possible. Some are summarized in Table 11.3-1. Typical behavior is reported in Fig. 11.3-2. “Full integration” is sufficient to avoid element mechanisms. The stiffness matrix of a rectangular element with midside nodes is integrated exactly by full integration.
+
+The bilinear element responds properly to pure bending with either reduced or selective integration. With full (2 by 2) integration and pure bending, parasitic shear strains appear at the Gauss points (Fig. 11.3-3a). As the element becomes
+
+<!-- source-page: 346 -->
+
+TABLE 11.3-1. DATA FOR SELECTED MINDLIN PLATE ELEMENTS. 
+
+<table><tr><td rowspan="2">Element Type</td><td colspan="3">Integration Rule</td><td rowspan="2">ShearConstraints</td><td rowspan="2">Number ofMechanisms</td></tr><tr><td>Type</td><td> $[k_b]$ </td><td> $[k_s]$ </td></tr><tr><td rowspan="3"></td><td>Bilinear:</td><td>Reduced</td><td> $1 \times 1$ </td><td> $1 \times 1$ </td><td>2</td></tr><tr><td>4 nodes,</td><td>Selective</td><td> $2 \times 2$ </td><td> $1 \times 1$ </td><td>2</td></tr><tr><td>12 d.o.f.</td><td>Full</td><td> $2 \times 2$ </td><td> $2 \times 2$ </td><td>4</td></tr><tr><td rowspan="3"></td><td>Quadratic:</td><td>Reduced</td><td> $2 \times 2$ </td><td> $2 \times 2$ </td><td>8</td></tr><tr><td>9 nodes,</td><td>Selective</td><td> $3 \times 3$ </td><td> $2 \times 2$ </td><td>8</td></tr><tr><td>27 d.o.f.</td><td>Full</td><td> $3 \times 3$ </td><td> $3 \times 3$ </td><td>18</td></tr><tr><td rowspan="3"></td><td>Serendipity:</td><td>Reduced</td><td> $2 \times 2$ </td><td> $2 \times 2$ </td><td>8</td></tr><tr><td>8 nodes,</td><td>Selective</td><td> $3 \times 3$ </td><td> $2 \times 2$ </td><td>8</td></tr><tr><td>24 d.o.f.</td><td>Full</td><td> $3 \times 3$ </td><td> $3 \times 3$ </td><td>18</td></tr><tr><td rowspan="3"></td><td>Heterosis:</td><td></td><td></td><td></td><td></td></tr><tr><td>9 nodes,</td><td>Selective</td><td> $3 \times 3$ </td><td> $2 \times 2$ </td><td>8</td></tr><tr><td>26 d.o.f.</td><td>(Ref. 11.8.</td><td colspan="3">D.o.f. at the center node are  $\theta_x$  and  $\theta_y$  only.)</td></tr></table>
+
+![](images/page-346_f306f707d3ae5e28ab30163fe1ad3ad62655d0465e5354c43cf2aa98c83306a9.jpg)
+
+<details>
+<summary>line</summary>
+
+| L_T/t | (computed w_t) ÷ (theoretical w_t) |
+|-------|-----------------------------------|
+| 10    | 1.15                              |
+| 20    | 1.05                              |
+| 30    | 1.02                              |
+| 50    | 1.00                              |
+| 100   | 0.98                              |
+| 200   | 0.96                              |
+| 300   | 0.94                              |
+| 500   | 0.90                              |
+| 1000  | 0.85                              |
+| 10^6  | 0.80                              |
+</details>
+
+Figure 11.3-2. Center deflection of a uniformly loaded clamped square plate of side length $L_{T}$ and thickness t. An 8 by 8 mesh is used in all cases. Thin plates correspond to large $L_{T}/t$ . Transverse shear deformation becomes significant for small $L_{T}/t$ . Integration rules, from Table 11.3-1, are reduced (R), selective (S), and full (F) [11.9].
+
+<!-- source-page: 347 -->
+
+thin, its stiffness is due almost entirely to parasitic shear. Thus, if fully integrated, a bilinear Mindlin plate element exhibits almost no bending deformation: that is, the mesh “locks.”
+
+The nine-node quadratic element, when integrated with any number of Gauss points, can properly represent pure bending. As seen in Fig. 11.3-3b, because lateral deflection w can vary quadratically, zero-shear conditions impose the constraint of pure bending rather than no bending. The latter element in Fig. 11.3-3b displays linearly varying bending. Here w = 0, not w cubic in x as for a standard beam or a Kirchhoff plate, because the quadratic plate element can display only a quadratic variation of w. Accordingly, linearly varying bending is represented by w = 0 and a quadratic variation of $\theta_{x}$ . It can be shown that this state is properly represented only when $[k_{s}]$ is integrated with a 2 by 2 rule (Ref. 12.6; see also Problem 11.20).
+
+The “serendipity” element has eight nodes and uses the quadratic shape functions of Eqs. 6.6-1. It is an unreliable element: as shown by Fig. 11.3-2, its accuracy is acceptable only for small values of $L_{T}/t$ , regardless of quadrature rule.
+
+The upper curves in Fig. 11.3-2 become horizontal, indicating apparent convergence, but may diverge for very large values of $L_{T}/t$ . This difficulty has nothing to do with locking. Rather, it is caused by the penalty matrix $[k_{s}]$ becoming numerically so large that it overwhelms matrix $[k_{b}]$ , as discussed in Section 9.4. Divergence can be avoided by basing $[k_{s}]$ on a value of t that is arbitrarily increased, if necessary, so that
+
+$$
+\frac {5}{1 + \nu} \left(\frac {L _ {T}}{t}\right) ^ {2} <   1 0 ^ {p / 2} \tag {11.3-9}
+$$
+
+where p is the approximate number of decimal digits per computer word (see Eq. 9.4-11). The actual value of t is used to compute matrix $[k_{b}]$ . With this adjustment, transverse shear deformation is misrepresented only when it is so small as not to matter anyway.
+
+The “heterosis” element [11.8] is the best of the elements summarized in Table 11.3-1. It does not exhibit locking, erratic convergence characteristics, or mech-
+
+![](images/page-347_edde00b89c927fda0f959f55171320da539b589d036cbc5d99372a12355b5557.jpg)  
+Figure 11.3-3. (a) Edge view of bilinear element, shown deformed, in pure bending. (b) Edge view of quadratic element, shown deformed, in pure bending and in linearly varying bending.
+
+<!-- source-page: 348 -->
+
+![](images/page-348_ccac1f2de3b92772a421234a4d37c8c63eb6f7e3f0d34d6f7d533541ef303fef.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+x
+</details>
+
+In-plane twist mode $w = 0, \theta_{,r} = -y, \theta_{y} = x$   
+(a)
+
+![](images/page-348_3733bca2b015b7213d21c8dbc36c7715e74398bb59349e632d6253eccb8b82c5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+x
+</details>
+
+w-hourglass mode $w = xy, \theta_{x} = \theta_{y} = 0$   
+(b)
+
+![](images/page-348_2a81a1c34e6922504e191612e829927fb78ac37ecc6a035604d5b8ea06420d94.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+x
+</details>
+
+$\theta_{x}$ -hourglass mode $w = 0,\theta_{x} = xy,\theta_{y} = 0$   
+(c)
+
+![](images/page-348_6f39a431ee9bc9791f139110a49327c4f85545d4f95748a36b620df7bba27076.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+x
+</details>
+
+$\theta_y$ -hourglass mode $w = 0, \theta_x = 0, \theta_y = xy$   
+(d)   
+Figure 11.3-4. Mechanisms of the four-node bilinear Mindlin element with reduced integration (one point for all terms) [11.4].
+
+anisms. Other good Mindlin plate elements can be formulated by hybrid methods [9.8].
+
+Mechanisms. An element having one or more mechanisms is not a foolproof element. Occasional numerical disaster is possible in the hands of a user who is unaware or inattentive. Mechanisms of elements in Table 11.3-1 resemble those of the corresponding plane elements, discussed in Section 6.12.
+
+Mechanisms of the bilinear element with reduced integration are shown in Fig. 11.3-4. With selective integration, only two of these mechanisms remain possible: the in-plane twist mode and the w-hourglass mode. The in-plane twist mode, Fig. 11.3-4a, is not communicable between adjacent elements, so that a mesh of two or more elements cannot have this mechanism. Control of mechanisms is discussed in [6,11-6,13,13,49,13,52-13,54].
+
+Under both reduced and selective integration, the quadratic and serendipity elements have the mechanism described by Eqs. 6.12-3 (for Mindlin plates, substitute $-z\theta_{x}$ for u and $-z\theta_{y}$ for v). This mechanism cannot be communicated between elements. Three additional mechanisms are possible in the nine-node quadratic element under reduced integration. One comes from Eq. 6.12-2, and another from Eq. 6.12-2 with u and v interchanged. The third mechanism is u = v = 0, $w = 3\xi^{2}\eta^{2} - \xi^{2} - \eta^{2}$ . The latter mechanism is not possible in the heterosis element because the $\xi^{2}\eta^{2}$ term is not present in the w field.
+
+Stress Computation. When element nodal d.o.f. $\{d\}$ are known, Eq. 11.3-5 yields curvatures $\{\kappa\}$ , Eq. 11.1-12 yields moments and shears, and Eqs. 11.1-2 yields stresses. Transverse shear stresses may be greatly in error except at Gauss points of the selective integration rule appropriate to $[k_{s}]$ . (Even at these points, accuracy may be poor unless thickness t used in $[k_{s}]$ has been adjusted according to Eq. 11.3-9.) Thus, in the bilinear element, transverse shear stresses $\tau_{yz}$ and $\tau_{zx}$ should be calculated at the element center, and these values assumed to prevail throughout the element. With the remaining elements in Table 11.3-1, it usually is good strategy to calculate stresses at Gauss points of a 2 by 2 rule, then extrapolate to other locations in the element as required. Extrapolation of stresses is discussed in Section 6.13.
+
+# 11.4 A TRIANGULAR DISCRETE KIRCHHOFF ELEMENT
+
+The element to be discussed was published in 1969 [11.10]. It was reexamined over ten years later and found to remain among the best elements for analysis of
+
+<!-- source-page: 349 -->
+
+thin plates [11,11]. The element is currently known as DKT, for discrete Kirchhoff triangle. Explicit expressions [11,12] and Fortran coding [11,13] for the element are available. Details of element formulation involve lengthy expressions. The following is a summary. Its essential step, that is, the enforcement of zero transverse shear strain at specific points, is also used in the formulation of other discrete Kirchhoff elements.
+
+The starting point is a straight-sided element with corner and midside nodes (Fig. 11.4-1a). Rotations $\theta_{x}$ and $\theta_{y}$ of a midsurface-normal line are each interpolated from nodal rotations $\theta_{xi}$ and $\theta_{yi}$ , where i runs from 1 to 6, using a complete quadratic polynomial:
+
+$$
+\theta_ {x} = \sum N _ {i} \theta_ {x i} \quad \text { and } \quad \theta_ {y} = \sum N _ {i} \theta_ {y i} \tag {11.4-1}
+$$
+
+Here the $N_{i}$ are given by Eqs. 5.3-5. There are a total of twelve d.o.f. in Eq. 11.4-1. Lateral deflection w along each edge is assumed to be cubic in an edge-tangent coordinate s. Thus, along side 2–3 for example, the rotation $w_{ss}$ at midside node 5 is
+
+$$
+w _ {, s 5} = - \frac {3}{2 L _ {2 3}} w _ {2} - \frac {1}{4} w _ {, s 2} + \frac {3}{2 L _ {2 3}} w _ {3} - \frac {1}{4} w _ {, s 3} \tag {11.4-2}
+$$
+
+Two similar equations are written for the remaining two sides. When nodal values of $w_{,s}$ are replaced by nodal values of $w_{,x}$ and $w_{,y}$ by coordinate transformation, there are a total of nine d.o.f. in these three equations for $w_{,s}$ (vertex-node values of w, $w_{,x}$ , and $w_{,y}$ ). Accordingly, in Eq. 11.4-1 and the three rotation equations such as 11.4-2, there are a total of 21 d.o.f.
+
+We seek a nine-d.o.f. element that has the nodal d.o.f. shown in Fig. 11.4-1b. Accordingly, the twelve d.o.f. $\theta_{xi}$ and $\theta_{yi}$ at nodes 1 through 6 must be expressed in terms of $w_{i}$ , $w_{,xi}$ , and $w_{,yi}$ at only the corner nodes. Constraints used for this purpose are as follows.
+
+1. Transverse shear strains $\gamma_{yz}$ and $\gamma_{zx}$ vanish at corners 1, 2, and 3. Thus, from Eqs. 11.1-4,
+
+$$
+\theta_ {x i} = w _ {, x i} \quad \text { and } \quad \theta_ {y i} = w _ {, y i} \quad \text { for } \quad i = 1, 2, 3 \tag {11.4-3}
+$$
+
+![](images/page-349_c6ca60ee62a1b13adb34ea044bb51a8f6e14f4364a2d872da3a6193a29e4dd15.jpg)
+
+<details>
+<summary>text_image</summary>
+
+s
+3
+i
+w_i
+θ_yi
+θ_xi
+y, v
+6
+ξ_2
+ξ_1
+ξ_3
+1
+4
+2
+n
+z, w
+x, u
+</details>
+
+(a)
+
+![](images/page-349_dff8f047fbf8c59d13f9b35e9817d796ef30fd5bb8df651b8ef74b8dcf3543d2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+1
+3
+2
+z
+x
+w_i
+w_yi
+w_xi
+</details>
+
+(b)   
+Figure 11.4-1. Development of a discrete Kirchhoff triangle. (a) Initial element and its d.o.f. Area coordinates $\xi_{1}$ , $\xi_{2}$ , and $\xi_{3}$ are shown. (b) Final nine-d.o.f. element and its d.o.f.
+
+<!-- source-page: 350 -->
+
+2. Transverse shear strain $\gamma_{sz}$ vanishes at nodes 4, 5, and 6, where s is an edge-tangent coordinate. Thus
+
+$$
+\theta_ {s i} = w _ {, s i} \quad \text { for } \quad i = 4, 5, 6 \tag {11.4-4}
+$$
+
+Use of Eq. 11.4-4 requires coordinate transformation operations.
+
+3. Normal slopes vary linearly along each edge. Thus
+
+$$
+\theta_ {n 4} = \frac {1}{2} (w, _ {n 1} + w, _ {n 2}) \quad \theta_ {n 5} = \frac {1}{2} (w, _ {n 2} + w, _ {n 3}) \quad \theta_ {n 6} = \frac {1}{2} (w, _ {n 3} + w, _ {n 1}) \tag {11.4-5}
+$$
+
+After the foregoing constraints have been applied, the twelve nodal d.o.f. $\theta_{xi}$ and $\theta_{yi}$ in Eqs. 11.4-1 are expressed in terms of the nine nodal d.o.f. $w_{i}$ , $w_{,xi}$ , and $w_{,yi}$ at the corners. Symbolically, these relations are
+
+$$
+\left[ \begin{array}{l l l l l l l l} \theta_ {x 1} & \theta_ {y 1} & \theta_ {x 2} & \dots & \theta_ {y 6} \end{array} \right] ^ {T} = \underset {1 2 \times 9} {[ \mathbf {T} ]} \left[ \begin{array}{l l l l l l l} w _ {1} & w _ {, x 1} & w _ {, y 1} & w _ {2} & \dots & w _ {, y 3} \end{array} \right] ^ {T} \tag {11.4-6}
+$$
+
+To generate the element stiffness matrix, we can begin with Eqs. 11.4-1. Strains are stated as in Mindlin plate theory, Eqs. 11.1-4, but now $\gamma_{yz}$ and $\gamma_{zx}$ are ignored. Thus one considers only the strains $\epsilon_{x} = -z\theta_{x,x}$ , $\epsilon_{y} = -z\theta_{y,y}$ , and $\gamma_{xy} = -z(\theta_{x,y} + \theta_{y,x})$ , that is,
+
+$$
+\{\epsilon \} = - z [ \partial ] \left\{ \begin{array}{l} \theta_ {x} \\ \theta_ {y} \end{array} \right\}, \quad \text { where } \quad [ \partial ] = \left[ \begin{array}{c c} \partial / \partial x & 0 \\ 0 & \partial / \partial y \\ \partial / \partial y & \partial / \partial x \end{array} \right] \tag {11.4-7}
+$$
+
+Equations 11.4-1 and 11.4-7 yield the strains $\{\epsilon\} = \left[\epsilon_x \quad \epsilon_y \quad \gamma_{xy}\right]^T$ as
+
+$$
+\{\boldsymbol {\epsilon} \} = - z [ \partial ] \underbrace {\left[ \begin{array}{c c c c c c c} N _ {1} & 0 & N _ {2} & 0 & \dots & N _ {6} & 0 \\ 0 & N _ {1} & 0 & N _ {2} & \dots & 0 & N _ {6} \end{array} \right]} _ {[ \mathbf {B} _ {\theta} ]} \underbrace {\left[ \begin{array}{c c c c c} \theta_ {x 1} & \theta_ {y 1} & \dots & \theta_ {y 6} \end{array} \right] ^ {T}} _ {\{\mathbf {d} _ {\theta} \}} \tag {11.4-8}
+$$
+
+Equations 5.2-7 must be used in forming $[B_{\theta}]$ . After integration through the thickness, the strain energy expression $U = \frac{1}{2} \int \{\epsilon\}^{T}[E]\{\epsilon\} \, dV$ becomes
+
+$$
+U = \frac {1}{2} \left\{\mathbf {d} _ {\theta} \right\} ^ {T} \left[ \mathbf {k} _ {\theta} \right] \left\{\mathbf {d} _ {\theta} \right\}, \quad \text { where } \quad \left[ \mathbf {k} _ {\theta} \right] _ {1 2 \times 1 2} = \int_ {A} \left[ \mathbf {B} _ {\theta} \right] ^ {T} \left[ \mathbf {D} _ {K} \right] \left[ \mathbf {B} _ {\theta} \right] d A \tag {11.4-9}
+$$
+
+Matrix $[D_{K}]$ is the rigidity matrix of Kirchhoff plate theory, for example, Eq. 11.1-10. Matrix $[k_{\theta}]$ operates on the twelve rotational d.o.f. used in Eq. 11.4-1. It can be converted to a 9 by 9 matrix [k], which operates on the standard Kirchhoff d.o.f. at the corners, by applying Eq. 11.4-6:
+
+$$
+[ \mathbf {k} ] = [ \mathbf {T} ] ^ {T} [ \mathbf {k} _ {\theta} ] [ \mathbf {T} ] \tag {11.4-10}
+$$
+
+Transformations of this type are explained in Section 7.4. However, more efficient coding results if Eq. 11.4-6 is substituted into Eq. 11.4-8, so that the strain-displacement relation contains nine columns rather than twelve. Thus [k] is produced directly.
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_036.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_036.md
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+<!-- source-page: 351 -->
+
+TABLE 11.4-1. CENTER DEFLECTIONS OF SQUARE PLATES, COMPUTED BY THE DKT ELEMENT [11.10]. A TYPICAL MESH IS SHOWN IN FIG. 11.4-2. RESULTS ARE REPORTED AS THE RATIO OF COMPUTED DEFLECTION TO EXACT DEFLECTION ACCORDING TO THIN-PLATE THEORY USING $\nu = 0.3$ [11.1]. SIMPLY SUPPORTED CASES USE CLASSICAL BOUNDARY CONDITIONS (Eqs. 11.5-1). 
+
+<table><tr><td rowspan="2">Mesh Size</td><td colspan="2">Uniformly Loaded</td><td colspan="2">Concentrated Center Load</td></tr><tr><td>Simply Supported</td><td>Clamped</td><td>Simply Supported</td><td>Clamped</td></tr><tr><td> $N_{es} = 1$ </td><td>1.025</td><td>1.500</td><td>1.076</td><td>1.012</td></tr><tr><td> $N_{es} = 2$ </td><td>0.999</td><td>1.228</td><td>1.008</td><td>1.046</td></tr><tr><td> $N_{es} = 4$ </td><td>1.001</td><td>1.069</td><td>1.003</td><td>1.019</td></tr><tr><td> $N_{es} = 8$ </td><td>1.001</td><td>1.021</td><td>1.001</td><td>1.007</td></tr></table>
+
+One can visualize the foregoing DKT plate element as a stack of plane linear-strain triangles, pinned together by a rigid thickness-direction rod at each vertex, and with additional constraints that impose Eqs. 11.4-4 and 11.4-5 at midsides.
+
+If the DKT plate element is homogeneous and of constant thickness, [k] is integrated exactly by a three-point quadrature rule. Explicit formulas for [k] are also available [11.12,11.13]. After element d.o.f. {d} are known, element strains { $\epsilon$ } are computed by successive application of Eqs. 11.4-6 and 11.4-8. Hence, stresses are $\{\sigma\} = [\mathrm{E}](\{\epsilon\} - \{\epsilon_{0}\})$ .
+
+The behavior of the DKT element is reported in Table 11.4-1 [11.10]. Distributed loads were lumped by assigning one-third the total element load to translational d.o.f. at each vetex. Additional results may be found in Refs. 11.11 and 11.12, which show that the element yields accurate bending moments, performs well even at large aspect ratios, and satisfactorily solves the “twisted ribbon” test case.
+
+Fortran coding for the DKT element appears in Fig. 11.4-3. Required input consists of nodal x and y coordinates (X1,X2,X3,Y1,Y2,Y3) and rigidity matrix $[D_{K}]$ (the 3 by 3 array D). The stiffness matrix is delivered in array SE. Displacement w is positive in the +z direction. The arrangement of nodal d.o.f. used in this subroutine is
+
+$$
+\{\mathbf {d} \} = \left[ w _ {1} \quad w _ {, y 1} \quad - w _ {, x 1} \quad w _ {2} \quad w _ {, y 2} \quad - w _ {, x 2} \quad w _ {3} \quad w _ {, y 3} \quad - w _ {, x 3} \right] ^ {T} \tag {11.4-11}
+$$
+
+Thus $w_{,y}$ and $-w_{,x}$ are represented by rotation vectors in the $+x$ and $+y$ directions, respectively.
+
+![](images/page-351_4737bd524bbcefd5185b94ff94345c6d9b6560befd7e37108fa90d78674dbdd1.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Mesh Nes = 2
+</details>
+
+Figure 11.4-2. Typical mesh on one quadrant of a square plate (used for the results reported in Table 11.4-1). There is symmetry about the centerlines.
+
+<!-- source-page: 352 -->
+
+![](images/page-352_73fafdd54dbede4e41c70c215e9a972a7597f3bad750e2c6caf33296c47813a1.jpg)
+
+<details>
+<summary>text_image</summary>
+
+SUBROUTINE DKT (D,X1,Y1,X2,Y2,X3,Y3,SE)
+IMPLICIT DOUBLE PRECISION (A-H,O-Z)
+DIMENSION D(3,3),DD(9,9),QQ(9,9),PP(3,3),PT(2,3),RS(2,3),Q(3)
+DIMENSION GG(10,9),KOD(2,9),B(3),C(3),ALS(3),PX(3,3),SE(9,9)
+DATA KOD /1,1,2,3,3,2,4,4,5,6,6,5,7,7,8,9,9,8/
+DATA PP /12.D0,4.D0,4.D0,4.D0,2.D0,1.D0,4.D0,1.D0,2.D0/
+B(1)=Y2-Y3
+B(2)=Y3-Y1
+B(3)=Y1-Y2
+C(1)=X3-X2
+C(2)=X1-X3
+C(3)=X2-X1
+DET=(B(1)*C(2)-B(2)*C(1))*24.
+DO 10 I=1,3
+DO 10 J=1,3
+10 PX(I,J)=PP(I,J)/DET
+DO 25 I=1,3
+DO 25 J=1,3
+DO 25 K1=1,3
+II=(I-1)*3+K1
+DO 25 K2=1,3
+JJ=(J-1)*3+K2
+25 DD(II,JJ)=D(I,J)*PX(K1,K2)
+DO 30 I=1,3
+ALS(I)=B(I)*B(I)+C(I)*C(I)
+PT(1,I)=6.*C(I)/ALS(I)
+PT(2,I)=6.*B(I)/ALS(I)
+RS(1,I)=3.*C(I)*C(I)/ALS(I)
+RS(2,I)=3.*B(I)*B(I)/ALS(I)
+30 Q(I)=3.*B(I)*C(I)/ALS(I)
+DO 720 I=1,10
+DO 720 J=1,9
+720 GG(I,J)=0.
+DO 730 I=1,2
+II=(I-1)*5
+P1=PT(I,1)
+P2=PT(I,2)
+P3=PT(I,3)
+R1=RS(I,1)
+R2=RS(I,2)
+R3=RS(I,3)
+GG(II+1,KOD(I,1))=P3
+GG(II+2,KOD(I,1))=-P2
+GG(II+3,KOD(I,1))=-P3
+GG(II+4,KOD(I,1))=P2-P3
+GG(II+5,KOD(I,1))=P2
+GG(II+1,KOD(I,2))=-Q(3)
+GG(II+2,KOD(I,2))=-Q(2)
+GG(II+3,KOD(I,2))=Q(3)
+GG(II+4,KOD(I,3))=-1.-R3
+GG(II+5,KOD(I,3))=R3
+GG(II+6,KOD(I,4))=-P3
+GG(II+7,KOD(I,4))=P1+P3
+GG(II+8,KOD(I,5))=-Q(3)
+GG(II+9,KOD(I,6))=Q(3)
+GG(II+10,KOD(I,7))=-Q(3)-Q(1)
+GG(II+11,KOD(I,8))=-1.-R3
+GG(II+12,KOD(I,9))=R3-R1
+GG(II+13,KOD(I,10))=R3
+GG(II+14,KOD(I,11))=R3-R1
+GG(II+15,KOD(I,12))=-P2
+GG(II+16,KOD(I,13))=-P2
+GG(II+17,KOD(I,14))=-Q(2)
+GG(II+18,KOD(I,15))=-Q(2)-Q(1)
+GG(II+19,KOD(I,16))=-Q(2)
+GG(II+20,KOD(I,17))=-P1-P2
+GG(II+21,KOD(I,18))=-Q(2)
+GG(II+22,KOD(I,19))=-1.-R2
+GG(II+23,KOD(I,20))=-R2-R1
+GG(II+24,KOD(I,21))=-R2
+GG(II+25,KOD(I,22))=-R2
+730 CONTINUE
+DO 850 I=1,9
+QQ(1,I)=B(2)*GG(1,I)+B(3)*GG(2,I)
+QQ(2,I)=2.*B(2)*GG(3,I)+B(3)*GG(4,I)
+QQ(3,I)=B(2)*GG(4,I)+2.*B(3)*GG(5,I)
+QQ(4,I)=-C(2)*GG(6,I)-C(3)*GG(7,I)
+QQ(5,I)=-2.*C(2)*GG(8,I)-C(3)*GG(9,I)
+QQ(6,I)=-C(2)*GG(9,I)-2.*C(3)*GG(10,I)
+QQ(7,I)=C(2)*GG(1,I)+C(3)*GG(2,I)
+1 -B(2)*GG(6,I)-B(3)*GG(7,I)
+QQ(8,I)=2.*C(2)*GG(3,I)+C(3)*GG(4,I)
+1 -2.*B(2)*GG(8,I)-B(3)*GG(9,I)
+QQ(9,I)=C(2)*GG(4,I)+2.*C(3)*GG(5,I)
+1 -B(2)*GG(9,I)-2.*B(3)*GG(10,I)
+850 CONTINUE
+DO 855 I=1,9
+DO 855 J=1,9
+GG(I,J)=0.
+DO 855 K=1,9
+855 GG(I,J)=GG(I,J)+DD(I,K)*QQ(K,J)
+DO 960 L=1,9
+DO 960 J=L,9
+DUM=0.
+DO 900 K=1,9
+900 DUM=DUM+QQ(K,L)*GG(K,J)
+SE(L,J)=DUM
+960 SE(J,L)=DUM
+RETURN
+END
+</details>
+
+Figure 11.4-3. Fortran statements that generate the 9 by 9 stiffness matrix of a DKT plate element [adapted from Ref. 11.13]. See text for notation and order of d.o.f.
+
+# 11.5 BOUNDARY CONDITIONS AND TEST CASES
+
+Boundary Conditions. Conditions at the edge of a plate are classed as clamped, free, or simply supported. Typically, no single condition prevails along the entire plate boundary. In the notation of Fig. 11.5-1, plate boundary conditions are as follows.
+
+<!-- source-page: 353 -->
+
+![](images/page-353_b0c867ea91e65539fbf10f587f5bd74227aee61a2c82af47c32d492cffd0e6aa.jpg)
+
+<details>
+<summary>text_image</summary>
+
+s
+θₙ or wₙ
+θₛ or wₛ
+y
+β
+x
+w
+n
+</details>
+
+(n)
+
+![](images/page-353_2066ca4995acd45ab66f69b6cb6c8849e13dbe72c95d4c6df1bba70572645f44.jpg)
+
+<details>
+<summary>text_image</summary>
+
+s
+Mn
+n
+Mhs
+Qn
+y
+β
+x
+</details>
+
+(b)
+
+$$
+w _ {i, x} = w _ {i, n} \cos \beta - w _ {i, s} \sin \beta
+$$
+
+$$
+w _ {, y} = w _ {, n} \sin \beta + w _ {, s} \cos \beta
+$$
+
+(c)   
+Figure 11.5-1. Coordinates n and s are edge-normal and edge-tangent, respectively. (a) Rotations and lateral displacement. (b) Moments and transverse shear force. (c) Transformation relations for rotations. 
+<table><tr><td>clamped</td><td>free</td><td>simply supported (finite element)</td><td>simply supported (classical theory)</td></tr><tr><td> $w = 0$ </td><td> $Q_n = 0^1$ </td><td> $w = 0$ </td><td> $w = 0$ </td></tr><tr><td> $\theta_n = 0$ </td><td> $M_n = 0$ </td><td> $M_n = 0$ </td><td> $M_n = 0$ </td></tr><tr><td> $\theta_s = 0$ </td><td> $M_{ns} = 0$ </td><td> $M_{ns} = 0$ </td><td> $\theta_s = 0$ </td></tr></table>
+
+Since transverse shear strain is taken as zero in classical thin-plate theory, Eqs. 11.5-1 are modified for Kirchhoff and discrete Kirchhoff elements by replacing $\theta_{n}$ by $w_{,n}$ and $\theta_{s}$ by $w_{,s}$ .
+
+A clamped edge prevents all motion along the edge. Along a free edge, nodal d.o.f. are unspecified, and remain part of the vector $\{D\}$ of unknown d.o.f. Conditions along a simply supported or “hinged” edge have been found troublesome and require more explanation, as follows.
+
+In classical thin-plate theory, since $\gamma_{zs}=0$ , the boundary condition w=0 necessarily implies the boundary condition $w_{,s}=0$ as well. Thus, for a thin simply supported plate, we would expect good numerical results using the “classical theory” conditions in Eqs. 11.5-1, whether the element type is Kirchhoff, discrete Kirchhoff, or Mindlin. Such is indeed the case if boundaries intersect at right angles, as at the four corners of a rectangular plate. However, if the plate is skew, most elements give poor results. For example, if the plate of Fig. 11.5-2a is modeled by a uniform 14 by 14 mesh, the center displacement may be underestimated by more than 20%. If only the boundary conditions are changed, to the “finite element” simply supported conditions in Eqs. 11.5-1, the error may decline to less than 3%. Apparently, classical simply supported conditions overconstrain the mesh when interior corner angles exceed $\pi/2$ . The finite element simply supported conditions produce a plate model that is point-supported at its boundary nodes. Although this may appear to allow too little constraint, especially in a coarse mesh, good results are obtained in practice [11.14].
+
+In Eqs. 11.5-1, the various quantities may have prescribed values other than zero. For example, a line load $(Q_{n} \neq 0)$ could be prescribed along an edge of a
+
+<!-- source-page: 354 -->
+
+![](images/page-354_92a8210f74458cc9739091d5f2b4e66e0ddd98b1041ab7fe9f328dcd42e4ff96.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+a
+x
+a sin (β/2)
+β
+</details>
+
+(a)
+
+![](images/page-354_a0bf237427c03501d6e5880ae37a6058e9b07dba89b34eda037803f55a82f5cd.jpg)
+
+<details>
+<summary>line</summary>
+
+| x/a    | M_x / qa² (×10⁻³) | M_y / qa² (×10⁻³) |
+| ------ | ----------------- | ----------------- |
+| 0.00   | 20.0              | -10.0             |
+| 0.05   | 10.0              | 0.0               |
+| 0.10   | 10.0              | 10.0              |
+| 0.15   | 15.0              | 10.0              |
+| 0.20   | 18.0              | 11.0              |
+| 0.25   | 20.0              | 11.5              |
+</details>
+
+(b)   
+Figure 11.5-2. (a) Skew plate with equal side lengths (rhombic plate). (b) Bending moments along the x axis in a simply supported rhombic plate [11.15].
+
+plate and represented by concentrated lateral forces at nodes along the edge. A nonzero w along the edge may be prescribed instead of $Q_{n}$ . Thus the edge becomes simply supported or clamped, depending on whether $M_{n} = 0$ or $\theta_{n} = 0$ .
+
+Test Cases. In Section 4.5 we argue that an element must be able to display a constant-strain state. Plate elements are no exception. Each layer z = constant of a plate element must be able to display constant $\epsilon_{x}$ , $\epsilon_{y}$ , and $\gamma_{xy}$ . Hence, in patch-testing a Kirchhoff or discrete Kirchhoff plate element, we look for constant curvatures $w_{,xx}$ and $w_{,yy}$ and for constant twist $w_{,xy}$ . In patch-testing a Mindlin plate element, we look for constant curvatures $\theta_{x,x}$ and $\theta_{y,y}$ , constant twist $\theta_{x,y} + \theta_{y,x}$ , and constant transverse shear strains $w_{,y} - \theta_{y}$ and $w_{,x} - \theta_{x}$ . Figure 11.5-3 depicts a patch test for constant $w_{,xx}$ (or for constant $\theta_{x,x}$ ). One must enforce $w_{,y} = 0$ (or $\theta_{y} = 0$ ) at nodes 1 through 4 in order to prevent curling of the edges (unless $\nu = 0$ ).
+
+Popular test cases include square plates, with various support conditions and loads (see Table 11.4-1). Rectangular plates may be used as well, to test the effect of element aspect ratio. Circular plates can be used to test nonrectangular elements. Exact results are available for many such problems [11.1].
+
+The simply supported rhombic plate, Fig. 11.5-2a, is a difficult test case. The obtuse corners are singular points, where moments are theoretically infinite. Some element types fail to show that $M_{x}$ and $M_{y}$ are of opposite sign near these corners.
+
+![](images/page-354_7ea9fcb9bd5c0035333b2c230a4b92070e5da07c049bade7007bd9e3a9370055.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+4
+x
+5
+3
+Mₙ
+H
+1
+2
+Mₙ
+</details>
+
+Figure 11.5-3. Patch test for constant curvature, with $M_{x} = 2M_{a}/H$ , where $M_{a}$ is a moment load on a node.
+
+<!-- source-page: 355 -->
+
+![](images/page-355_72f1883ec6cebd61351ecf314659770e278d21a638a4ccf2ff82cecfaa4e0f11.jpg)
+
+<details>
+<summary>line</summary>
+
+| Element | L  | w3 (×10³) |
+|---------|----|-----------|
+| Element X | 0  | 0         |
+| Element X | 2  | 5         |
+| Element X | 4  | 10        |
+| Element X | 6  | 15        |
+| Element X | 8  | 20        |
+| Element X | 10 | 25        |
+| Element X | 12 | 30        |
+| Element DKT (both loadings) | 0  | 0         |
+| Element DKT (both loadings) | 2  | 5         |
+| Element DKT (both loadings) | 4  | 10        |
+| Element DKT (both loadings) | 6  | 15        |
+| Element DKT (both loadings) | 8  | 20        |
+| Element DKT (both loadings) | 10 | 25        |
+| Element DKT (both loadings) | 12 | 30        |
+</details>
+
+Figure 11.5-4. “Twisted ribbon” test case [11.12,11.16]. Here $E = 10^{7}$ , $\nu = 0.25$ , t = 0.05, $w_{3} = \text{deflection of corner indicated}$ .
+
+[11.14-11.16]. Difficulties attendant to use of classical simply supported boundary conditions for this problem have already been noted.
+
+The “twisted ribbon,” Fig. 11.5-4, is a test that shows the effect of aspect ratio $[11.12,11.16]$ . The twisting moment may be applied by corner forces or by corner couples, as shown. Usually the entire plate is modeled by one rectangular element or by two triangular elements. “Benchmark” values were obtained from a mesh of 16 rectangular Kirchhoff elements having 16 d.o.f. each. Many types of element fail this test, such as the one identified as “element X,” which is too stiff at large aspect ratio, and may fail even to produce a displacement of the correct algebraic sign. For element DKT, the two loadings product slightly different results, but the difference is scarcely noticeable when plotted. If triangulation is made along diagonal 2–4 rather than diagonal 1–3 as shown, elements DKT and X both product somewhat different results than shown in Fig. 11.5-4.
+
+# PROBLEMS
+
+# Section 11.1
+
+11.1 (a) Verify the stress formulas of Eqs. 11.1-2.   
+(b) Similarly, verify the formulas for $\tau_{yz}$ and $\tau_{zx}$ given below Eqs. 11.1-2.   
+11.2 In elementary mechanics of materials, one derives equations for normal and shear stress at an arbitrary angle $\theta$ in the $xy$ plane, such as $\sigma_{n} = \frac{1}{2} (\sigma_{x} + \sigma_{y}) + \frac{1}{2} (\sigma_{x} - \sigma_{y})\cos 2\theta +\tau_{xy}\sin 2\theta$ . What analogous expressions relate bending and twisting moments $M_{n}$ and $M_{ns}$ to $M_{x},M_{y}$ , and $M_{xy}$ ? Suggestion: Use Eqs. 11.1-2.   
+11.3 (a) In Fig. 11.1-1b presume that the $M$ 's and $Q$ 's are functions of $x$ and $y$ , so that (for example) $M_x dy$ acts along the edge $x = 0$ and $(M_x + M_{x,x})$
+
+<!-- source-page: 356 -->
+
+$dx$ ) dy acts along the parallel edge. Show that the equilibrium equations are $Q_{x,x} + Q_{y,y} = -q, M_{x,x} + M_{xy,y} = Q_x$ , and $M_{xy,x} + M_{y,y} = Q_y$ .
+
+(b) Hence, show that $M_{x,xx} + 2M_{xy,xy} + M_{y,yy} + q = 0$ .   
+(c) Use the result of part (b), and Eq. 11.1-7 for isotropic conditions, to show that $\nabla^4 w = q / D$ , where $\nabla^4$ is the biharmonic operator.
+
+11.4 A rectangular plate of thickness t has dimensions a and b, as shown. The plate is simply supported along edges AB and CD. Edges BC and DA remain free. If a uniform downward pressure p is applied to the upper surface, what are the principal stresses at the middle of the lower surface, and what is the deflection at the center of the plate?
+
+![](images/page-356_0b5e2930ca8572190e58bed3ac7f4af1023b5078b55028fd739f890298a988ee.jpg)
+
+<details>
+<summary>text_image</summary>
+
+A
+D
+y
+a
+B
+C
+b
+x
+</details>
+
+Problem 11.4
+
+11.5 (a) Write Eq. 11.1-5 for an isotropic material. Then derive terms in $[\mathbf{D}_K]$ , using the procedure given above Eq. 11.1-7.   
+(b) Verify the correctness of Eq. 11.1-11.   
+11.6 Consider an isotropic thin square plate, with edges parallel to $x$ and $y$ axes, loaded only along its edges. Describe the edge loads if the lateral deflection is (a) $w = c_{1}(x^{2} + y^{2})$ , and (b) $w = c_{2}(y^{2} - x^{2})$ , where $c_{1}$ and $c_{2}$ are constants.   
+11.7 (a) In a sandwich plate, Fig. 11.1-5, the average transverse shear strain $\gamma$ is related to the core shear strain $\gamma_{c}$ by $(c + h)\gamma = c\gamma_{c}$ . Derive this expression. Assume that the facings are much stiffer than the core.
+
+(b) Derive the expression for $D_{M44}$ in Eqs. 11.1-13. Suggestion: Consider strain energy.   
+(c) Work from the bending stiffness $EI$ of a sandwich beam and derive the expression for $D_{M11}$ in Eqs. 11.1-13 (except for the $1 - \nu^2$ factor).
+
+11.8 One sometimes wonders how wide a beam can be before it should be regarded as a plate. How would you decide? Or what would you do if unable to decide?
+
+# Section 11.2
+
+11.9 (a) Consider the rectangular plate element of Fig. 11.2-1 and the displacement field of Eq. 11.2-5. Show that interelement compatibility of normal slopes is lacking. For example, show that $w_{,y}$ along $y = b$ does not depend only on $w_{,y}$ at nodes 3 and 4.
+
+(b) Similarly, what can be said about interelement compatibility of $w$ and $w_{xx}$ along edge 3-4?   
+(c) In Eq. 11.2-5, the terms $a_{11}x^3y$ and $a_{12}xy^3$ might be replaced by $a_{11}x^4$ and $a_{12}y^4$ , but it is not wise to do so. Why?
+
+11.10 Imagine that the lateral displacement $w$ of a triangular thin-plate element
+
+<!-- source-page: 357 -->
+
+is taken as a complete quintic (21 terms). For each element shown, and without calculation, allocate d.o.f. to the nodes in a way that seems acceptable. Consider higher-order d.o.f. as needed. Is interelement compatibility achieved? Consider compatibility conditions on the edge x = 0.
+
+![](images/page-357_c08c11c954b6197a400defd74343d91799b053ed6b17637a83c347d8ef960236.jpg)  
+Problem 11.10
+
+![](images/page-357_d6abd55350eb52f927404739ab6dd1a52757c683480b1ff586382a193c31d7b2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Axis
+w1
+w2
+w,r1
+1
+l
+2
+w,r2
+r
+r1
+L
+r2
+</details>
+
+Problem 11.11
+
+11.11 The sketch represents the cross section of an annular (axisymmetric) element for analysis of thin plates ( $t << L$ ). Geometry, loads, and deformations are all axially symmetric. The material is isotropic. Derive the element stiffness matrix [k], to the extent of fully defining [B], [D $_{K}$ ], and all other terms used in your formula for [k], but do not integrate.   
+11.12 Imagine that the Mindlin beam element of Fig. 9.4-1 is uniform, uniformly loaded, and simply supported at nodes 1 and 2. With one-point quadrature for transverse shear terms, the stiffness matrix that operates of d.o.f. $\theta_{1}$ and $\theta_{2}$ is
+
+$$
+[ \mathrm{k} ] = \frac {E b t ^ {3}}{1 2 L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right] + \frac {G b t L}{4 . 8} \left[ \begin{array}{l l} 1 & 1 \\ 1 & 1 \end{array} \right]
+$$
+
+where b = element width and t = element depth. Determine the rotation at node 2, and compare it with the exact value, if nodal loads produced by the distributed load are calculated (a) consistently, and (b) from a cubic field for w.
+
+11.13 Imagine that the plate of Fig. 11.2-3 is square, simply supported, and modeled by a single finite strip (i.e., $L = b$ ). The lateral load is distributed and is described by $q = q_0 \sin (\pi x / b) \sin (\pi y / b)$ . Determine the center deflection of the finite strip. Suggestions: See Problem 11.3c. Arbitrarily elect to evaluate the equation at the center of the plate. Note, however, that this procedure would not be used in a finite strip computer program.
+
+11.14 Let the rectangular plate element of Fig. 11.2-1 carry a uniformly distributed load q in the +z direction. Determine the resulting nodal moment loads by assuming that the plate element acts like a beam clamped at both ends, spanning first the dimension 2a and then the dimension 2b. Show these loads, properly directed, on a sketch of the plate.
+
+11.15 (a) A constant moment $\overline{M}_{xy}$ is applied along edge 2--3 of the element in Fig. 11.2-1. What nodal loads result? Assume that $w$ along this edge is cubic in $y$ , and governed by $w$ and $w_{yy}$ at nodes 2 and 3.   
+(b) What would be your answer if the element is instead a Mindlin element, Eqs. 11.2-7, with shape functions $N_{1}$ through $N_{4}$ ?
+
+<!-- source-page: 358 -->
+
+# Section 11.3
+
+11.16 Imagine that the four-node plate element of Fig. 11.3-1a is to be obtained by specialization of the eight-d.o.f. solid element of Fig. 6.7-1. Describe the steps and substitutions that convert Eqs. 6.7-1 to the equations $w = \sum N_i w_i$ , $u = -z \sum N_i \theta_{xi}$ , and $v = -z \sum N_i \theta_{yi}$ , where $i = 1, 2, 3, 4$ .   
+11.17 Show that Eq. 11.3-8 follows from Eq. 11.3-6 when $[\mathbf{D}_M]$ , $[\mathbf{B}_b]$ , and $[\mathbf{B}_s]$ are defined as stated in the text.   
+11.18 A uniform load $q$ acts upward on a rectangular plate element of side lengths $2a$ and $2b$ . What are the consistent nodal loads for each of the four elements listed in Table 11.3-1?   
+11.19 A square plate under concentrated center load is to be analyzed. The boundary is clamped, meaning that all boundary d.o.f. are set to zero. Let the model consist of a single element, which occupies one quadrant. After imposing boundary conditions, how many unknown d.o.f. are left for each of the elements in Table 11.3-1?   
+11.20 The sketch shows more detail of the latter portion of Fig. 11.3-3b. Lateral displacement w is zero at the ends and at the center. Imagine that nothing varies with y; that is, beam action is to be modeled. Under bending moment that varies linearly with x, thin-beam theory shows that end sections rotate an amount $\theta_{b}$ and the middle rotates an amount $\theta_{b}/2$ in the opposite direction. On this must be superposed rotations $\theta_{s}$ caused by the constant transverse shear force. Thus
+
+$$
+\gamma_ {z x} = w _ {, x} - \theta_ {x} = \left(\frac {\theta_ {b}}{2} - \theta_ {s}\right) - \frac {3 \theta_ {b}}{2} \left(\frac {x}{a}\right) ^ {2}
+$$
+
+(a) Derive this expression for $\gamma_{zx}$ , using quadratic shape functions and nodal values of $w$ and $\theta_x$ at $A$ , $M$ , and $C$ .   
+(b) Determine the values of $x / a$ for which $\gamma_{zx}$ is correctly represented, even when $a >> t$ .
+
+![](images/page-358_04dfe9b1a83f24202c03c68d096fb1d68bfcb784103c5c714841d760c52ac633.jpg)
+
+<details>
+<summary>text_image</summary>
+
+θb + θs
+θ
+θb/2 - θs
+θb + θs
+A
+M
+t
+C
+a
+a
+x
+</details>
+
+Problem 11.20
+
+11.21 Consider a rectangular quadratic element (Table 11.3-1). Under what circumstances or deformation states will $\gamma_{yz}$ and $\gamma_{zx}$ be correctly evaluated along the line $\xi = 0$ ?   
+11.22 (a) Sketch a rectangular element in its deformed state if displacements are described by $u = v = 0$ , $w = 3\xi^2\eta^2 - \xi^2 - \eta^2$ .   
+(b) Show that this deformation mode yields zero strains at the Gauss points of a 2 by 2 rule.
+
+<!-- source-page: 359 -->
+
+(c) What kind of loads and support conditions would activate this mode, either for a single element or for a mesh of elements?
+
+11.23 Sketch a 2 by 3 mesh of rectangular bilinear elements. Superposed on this sketch, show the mesh deformed into the w-hourglass mode of Fig. 11.3-4.
+
+11.24 Use Eq. 11.3-4 to evaluate $\{\kappa\}$ for each of the modes in Fig. 11.3-4. Show that only the first two $\{\kappa\}$ 's are null for selective integration, and that all four $\{\kappa\}$ 's are null for reduced integration.
+
+# Section 11.4
+
+11.25 (a) Derive Eq. 11.4-2.
+
+(b) In Eq. 11.4-2, express $w_{,s5}$ in terms of $w$ , $w_{,x}$ , and $w_{,y}$ at nodes 2 and 3. Let the $n$ axis in Fig. 11.4-1a make an angle $\alpha$ with the $x$ axis. What is $\alpha$ in terms of the $x$ and $y$ coordinates of nodes 2 and 3?   
+(c) Similarly, in Eq. 11.4-5, what is $\theta_{n5}$ in terms of $w$ , $w_{,x}$ , and $w_{,y}$ at nodes 2 and 3 and angle $\alpha$ ?
+
+11.26 What can be said about interelement compatibility of the DKT element?   
+11.27 Why are three Gauss points adequate for exact integration of a DKT element?   
+11.28 In Table 11.4-1, consider the clamped, uniformly loaded test case with mesh N = 1. Do you think the computed result would be more accurate if nodal moments were included in the element load vectors? Why?   
+11.29 Imagine that the serendipity element of Table 11.3-1 is to be given a “discrete Kirchhoff” treatment by explicitly enforcing zero transverse shear strain at the Gauss points of a 2 by 2 rule. How may d.o.f. do these constraints eliminate? What d.o.f. do you think it appropriate to retain in $\{d\}$ , and why?
+
+# Section 11.5
+
+11.30 Demonstrate the free-edge condition $Q_{n} + M_{ns,s} = 0$ , which is stated in the footnote for Eqs. 11.5-1. Suggestion: Consider couple forces $M_{ns} \Delta s$ and $(M_{ns} + M_{ns,s} \Delta s) \Delta s$ in adjacent “cells” of length $\Delta s$ along the edge.   
+11.31 Consider the mesh shown in Fig. 11.4-2. The mesh models one quadrant of a symmetrically loaded and symmetrically supported square plate. Under each of the following support conditions, how many unknown d.o.f. remain in $\{D\}$ after boundary conditions have been imposed? Of these, which do you expect will have the same magnitude?
+
+(a) Clamped.   
+(b) Simply supported (finite element conditions).   
+(c) Simply supported (classical theory conditions).
+
+<!-- source-page: 360 -->
+
+# SHELL'S
+
+The physics of shell behavior is summarized. Advantages and disadvantages of various displacement fields are illustrated by means of singly curved (arch) elements. Element formulations are presented for general shells and for shells of revolution.
+
+# 12.1 SHELL GEOMETRY AND BEHAVIOR. SHELL ELEMENTS
+
+A shell forms a curved surface in space. Usually a shell is thin in comparison with its span. Geometrically, a shell is described by its thickness t and the shape of the shell midsurface. At every point on the midsurface, one can draw two small arcs that lie in the midsurface, and orient the arcs so as to fit the largest and smallest curvatures of the midsurface at that point. These arcs will be mutually perpendicular. They define the principal radii of curvature at that point. In general, principal radii vary from point to point. The centers of curvature lie on a normal to the midsurface.
+
+Examples of shell geometry appear in Fig. 12.1-1. The cylinder and cone are singly curved. In addition, they are developable, which means that if slit lengthwise they can be unrolled to form flat sheets without having to stretch their midsurfaces. The sphere and hyperboloid are doubly curved and are not developable. Radii of curvature are constant in the cylinder and sphere but are variable in the cone and hyperboloid.
+
+Each shell in Fig. 12.1-1 happens to be a shell of revolution, meaning that the midsurface is generated by rotating a straight or curved generating line about an axis of revolution in the plane of the generator. A meridian is the intersection of the midsurface with a plane that contains the axis of revolution. A parallel is the intersection of the midsurface with a plane perpendicular to the axis.
+
+In general, a shell simultaneously displays bending stresses and membrane stresses. Bending stresses in a shell correspond to bending stresses in a plate (Fig. 11.1-1a) and produce bending and twisting moments (Eqs. 11.1-1a). Membrane stresses correspond to stresses in a plane stress problem: they act tangent to the midsurface, and produce midsurface-tangent forces per unit length. These are the membrane forces $N_{x}$ , $N_{y}$ , and $N_{xy}$ given by
+
+$$
+N _ {x} = \int_ {- t / 2} ^ {t / 2} \sigma_ {x} d z \quad N _ {y} = \int_ {- t / 2} ^ {t / 2} \sigma_ {y} d z \quad N _ {x y} = \int_ {- t / 2} ^ {t / 2} \tau_ {x y} d z \tag {12.1-1}
+$$
+
+where $x$ and $y$ are orthogonal coordinates in the midsurface and $z$ is a direction
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_037.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_037.md
new file mode 100644
index 00000000..c9e2efc6
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_037.md
@@ -0,0 +1,434 @@
+<!-- source-page: 361 -->
+
+![](images/page-361_c49274aaa232765d28dd2b86edb040838fdf46d59bd2a1c561202bd09a313055.jpg)  
+$R_{\theta} = \text{const.}$   
+$R_{s} = x$   
+Cylinder   
+$R_{s} = x$   
+Cone   
+$R_{\theta} = R$   
+$R_{s} = R$   
+Sphere   
+$R_{\theta} > 0$   
+$R_{5} < 0$   
+Hyperboloid
+
+Figure 12.1-1. Shells of revolution, showing principal radii of curvature. $R_{s}$ is the radius of curvature of the meridian.
+
+normal to the midsurface. Stresses in the shell are composed of a membrane component $\sigma_{m}$ and a bending component $\sigma_{b}$ ; for example, normal stress in the x direction is
+
+$$
+\sigma_ {x} = \sigma_ {m x} + \sigma_ {b x} = \frac {N _ {x}}{t} + \frac {M _ {x} z}{t ^ {3} / 1 2} \tag {12.1-2}
+$$
+
+Thus it is assumed that stresses vary linearly through the thickness. Stress on the midsurface z = 0 is zero if $N_{x} = N_{y} = N_{xy} = 0$ .
+
+A shell can carry a large load if membrane action dominates over bending, just as a thin wire can carry a large load in tension but only a small load in bending. Practically, it is not possible to have only membrane action in a shell. Bending action is also present if concentrated loads are applied, if supports apply moments or transverse forces, or if a radius of curvature changes abruptly. As an example of the latter, consider closing a cylindrical pressure vessel with a cap that is hemispherical or ellipsoidal: where the cylinder meets the cap, radius $R_{s}$ of the meridian changes abruptly from infinite to finite. Typically, bending action is quite localized—that is, bending stresses are large only quite near the load or discontinuity that produces them.
+
+Classical shell theory produces equations that are very difficult to solve. The governing equations in terms of displacements are complicated; they have relatively simple forms only if many approximations are made. Authorities do not agree on what approximations are acceptable, so various shell theories have been proposed (e.g., those of Donnell, Flügge, Sanders, Vlasov, etc.). Like Kirchhoff plate theory, shell theories are limited to small deflections unless higher-order terms are added to account for membrane strains associated with rotation of the shell midsurface.
+
+Classical shell theory is concerned with thin shells, in which transverse shear deformation is considered negligible. In practice one may also encounter thick shells. Then one must account for transverse shear deformation, and perhaps also for the effects of thickness-direction normal stress.
+
+Shell Elements. Finite elements for shells have been among the most difficult elements to devise. Three approaches to the problem have been pursued:
+
+<!-- source-page: 362 -->
+
+1. Flat elements, formed by combining a plane membrane element with a plate bending element.   
+2. Curved elements, formulated by use of a classical shell theory.   
+3. Mindlin-type elements, similar to Mindlin plate elements described in Section 11.3. Such elements can be regarded as special forms of solid elements, made thin in one direction.
+
+Flat triangular elements model a shell as a faceted surface. Flat elements are easy to formulate. They pass patch tests and do not exhibit strain under rigid-body motion. However, although membrane-bending coupling is present throughout an actual curved shell, it is absent in individual flat elements. In the past, flat elements have not been particularly accurate, but have been useful because of the difficulties of other approaches.
+
+A curved element is necessarily more complicated than a flat element, first because its geometry is more complicated. Then, regardless of what classical shell theory is used, its approximations and complexities are incorporated in the element. Some curved elements cannot display rigid-body motion without strain, either because of defects in the shell theory or because of shortcomings in the element displacement field. Typically, the user of a curved element must supply data in addition to nodal coordinates in order to describe element geometry. Some curved shell elements include derivatives of membrane strain and curvature among their nodal d.o.f.
+
+Mindlin-type or “degenerated solid” elements can be curved and appear to occupy a middle ground between flat elements and curved elements formulated by use of shell theory, both in accuracy and in ease of use. Elements for thin
+
+![](images/page-362_a5a3cb07a89d68bef3d784a629705f8766838947bcf711be62e4770619920539.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+3D wireframe model of a mechanical component with intersecting surfaces (no text or symbols)
+</details>
+
+Figure 12.1-2. Intersecting pipes, modeled by quadrilateral and triangular shell elements. (Courtesy of Algor Interactive Systems, Inc., Pittsburgh, Pennsylvania.)
+
+<!-- source-page: 363 -->
+
+shells and for thick shells are available. As with Mindlin plate elements, possible difficulties with locking must be addressed.
+
+Any of the foregoing three approaches might be used to provide elements for a particular shell, such as the shell shown in Fig. 12.1-2.
+
+The membrane stiffness of a thin shell is much larger than its bending stiffness. This is reflected in a large discrepancy between the associated stiffness coefficients in [K], regardless of how the shell elements are formulated. Numerical errors of the type discussed in Section 18.2 are possible.
+
+At present it is not clear whether the most cost-effective thin-shell elements will be flat or curved. In what follows, emphasis is placed on flat elements, elements for shells of revolution, and Mindlin elements. Curved elements for thin shells of general shape that are based on a classical shell theory are beyond the scope of this text.
+
+Test Cases for General Shell Elements. If assigned a flat geometry, shell elements can be used to solve problems of plane stress and plate bending. Accordingly, one can begin with patch tests and other commonly used test problems for plane problems and plates (e.g., Fig. 4.6-1, Tables 6.14-1, 6.14-2, and 11.4-1, and proposed test problems in Ref. 12.15). Singly curved shell elements can be tested on arch problems (e.g., Fig. 12.2-1a) and on cylindrical shell problems (e.g., Fig. 12.4-4). A good element will have good accuracy on these initial tests and will not have mechanisms. Various additional problems have been used as test cases for general shell elements, including a pinched cylinder, a pinched hemisphere, and a slit cylinder under twisting load. Details may be found in [12.8,12.14–12.16].
+
+# 12.2 CIRCULAR ARCHES AND ARCH ELEMENTS
+
+The study of arch elements provides insight into various aspects of shell element behavior. In what follows we consider an arch of constant mean radius R, loaded in its own plane.
+
+Equations for Thin Circular Arches. We assume that the arch is thin—that is, that $R \gg t$ in Fig. 12.2-1. A point on the arch midline has s-direction (tangential) displacement u and z-direction (radial) displacement w. Let $\epsilon_{s}$ represent tangential
+
+![](images/page-363_d184f5b06a3c634ba6ca41c5fe46f0c84a69859129a778fc8e77d29081e87d31.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+t
+R
+</details>
+
+{a}
+
+![](images/page-363_86ef3dfd60ddf83b3c3289e82fa24373c2316898b984e2ba78eca405dbad37d2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z,w
+u
+t
+s
+1
+s = - L/2
+R
+s = L/2
+2
+</details>
+
+(b)   
+Figure 12.2-1. (a) Semicircular arch with clamped ends and concentrated center load. (b) Arch element of arc length L. Coordinates s and z are respectively tangent and normal to the arch midline.
+
+<!-- source-page: 364 -->
+
+strain at an arbitrary point, a distance z from the midline. With the aid of Fig. 12.2-2, and with R constant, we write
+
+$$
+\epsilon_ {s} = \frac {d}{d s} \left(\delta_ {a} + \delta_ {c}\right) + \frac {w}{R} = u _ {, s} + \frac {w}{R} + z \left(\frac {u _ {, s}}{R} - w _ {, s s}\right) \tag {12.2-1}
+$$
+
+If the approximation $\delta_{a} \approx u$ is introduced, the term $zu_{,s} / R$ disappears. In alternative notation, Eq. 12.2-1 is
+
+$$
+\epsilon_ {s} = \epsilon_ {m} + z \kappa , \quad \text { where } \quad \left\{ \begin{array}{l l} \epsilon_ {m} = u _ {, s} + \frac {w}{R} & \text {(12.2 - 2a)} \\ \kappa = \frac {u _ {, s}}{R} - w _ {, s s} & \text {(12.2 - 2b)} \end{array} \right.
+$$
+
+Membrane strain $\epsilon_{m}$ appears along the arch midline and is associated with s-direction (membrane) force. Curvature change $\kappa$ is associated with bending moment and is considered positive when the radius of curvature decreases. Transverse shear deformation $\gamma_{zs}$ is considered zero for a thin arch.
+
+Strain energy U in an arch element is composed of a membrane contribution $U_{m}$ and a bending contribution $U_{b}$ . For an element of arc length L,
+
+$$
+U = U _ {m} + U _ {b} = \int_ {- L / 2} ^ {L / 2} \frac {E A}{2} \epsilon_ {m} ^ {2} d s + \int_ {- L / 2} ^ {L / 2} \frac {E I}{2} \kappa^ {2} d s \tag {12.2-3}
+$$
+
+where $E =$ elastic modulus, $A =$ cross-sectional area, and $I =$ moment of inertia of $A$ about the neutral axis of bending ( $I = bt^3/12$ for a rectangular cross section of width $b$ ). The expression for $U$ can be derived by integration of strain energy density $E\epsilon_s^2/2$ through the arch thickness $t$ . A term linear in $z$ disappears, and the terms shown in Eq. 12.2-3 remain. Note that membrane stiffness $EA$ becomes much larger than bending stiffness $EI$ as arch thickness $t$ becomes small.
+
+For rigid-body motion, $\epsilon_{m} = \kappa = 0$ . The displacement field for rigid-body motion is therefore
+
+$$
+u = b _ {1} \cos \phi + b _ {2} \sin \phi + b _ {3} \tag {12.2-4a}
+$$
+
+$$
+w = b _ {1} \sin \phi - b _ {2} \cos \phi \tag {12.2-4b}
+$$
+
+![](images/page-364_6532315fa5d09d3c100af0a74a456b291df33d61723928a3f9dd9826e4c578d6.jpg)
+
+<details>
+<summary>text_image</summary>
+
+t
+z
+R
+δa
+u
+</details>
+
+$$
+\frac {\delta_ {n}}{R + z} = \frac {u}{R}
+$$
+
+(a)   
+![](images/page-364_bb1d5594e168e69fb32ed899272f79672d9ace771f7f44de2c59e8a1daf03760.jpg)
+
+<details>
+<summary>text_image</summary>
+
+t
+R
+z
+w
+w
+</details>
+
+$$
+\epsilon = \frac {w}{R + z} \approx \frac {w}{R}
+$$
+
+(b)   
+![](images/page-364_0428c6872ad25e7d1c09131e6e4e1c3e48302a914d8145db22c5094bf0dda671.jpg)
+
+<details>
+<summary>text_image</summary>
+
+δc
+z
+R
+w1s
+w1s
+l
+</details>
+
+$$
+\delta_ {c} = - z w _ {1 s}
+$$
+
+{c}
+
+Figure 12.2-2. Axial, radial, and rotational motions of a thin arch (R >> t), used to formulate an expression for axial strain. Angle $w_{ss}$ is presumed small.
+
+<!-- source-page: 365 -->
+
+where $\phi = s / R$ and the $b_{i}$ are constants. Here $b_{1}$ and $b_{2}$ represent translations in mutually perpendicular directions and $b_{3}$ represents a rotation about the center of curvature.
+
+A thin arch will bend but will have very little membrane strain. Accordingly, from Eq. 12.2-2, we obtain the inextensibility condition:
+
+$$
+\epsilon_ {m} = 0 \quad \text { implies } \quad u _ {, s} + \frac {w}{R} = 0 \tag {12.2-5}
+$$
+
+This condition is satisfied in the limit of thinness as L/t becomes infinite.
+
+Straight Arch Elements. The use of straight elements to model an arch is analogous to the use of flat elements to model a shell. A straight arch element is identical to a plane frame element. To obtain it, one merely combines a standard two-d.o.f. bar element (Eq. 2.4-5) with a standard four-d.o.f. beam element (Eq. 4.2-5). Thus, using d.o.f. in Fig. 12.2-3a, the element stiffness equation $[k]\{d\} = \{r\}$ is
+
+$$
+\left[ \begin{array}{c c} {[ \mathrm{k} _ {\text {bar}} ]} & {[ 0 ]} \\ {2 \times 2} & {2 \times 4} \\ {[ 0 ]} & {[ \mathrm{k} _ {\text {beam}} ]} \\ {4 \times 2} & {4 \times 4} \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \\ w _ {1} \\ \beta_ {1} \\ w _ {2} \\ \beta_ {2} \end{array} \right\} = ^ {\prime} \{\mathbf {r} \} \tag {12.2-6}
+$$
+
+in which $[k_{bar}]$ and $[k_{beam}]$ come respectively from $U_{m}$ and $U_{b}$ in Eq. 12.2-3. Nodal rotations $\beta_{1}$ and $\beta_{2}$ are nodal values of $w_{ss}$ .
+
+For assembly with other elements having different orientation, a common set of structural d.o.f. is needed. The choice is not unique. One possibility is to use tangential and radial translational d.o.f. $D_{s}$ and $D_{r}$ at each node, as shown in Fig. 12.2-3b. Coordinate transformation (Section 7.4) is used to replace nodal values of u and w by nodal values of $D_{s}$ and $D_{r}$ . D.o.f. $\beta_{1}$ and $\beta_{2}$ are unchanged by this transformation. (A similar transformation is described in connection with Eq. 12.4-3.)
+
+The displacement field that produces Eq. 12.2-6 can be written
+
+$$
+u = a _ {1} + a _ {2} s \tag {12.2-7a}
+$$
+
+$$
+w = a _ {3} + a _ {4} s + a _ {5} s ^ {2} + a _ {6} s ^ {3} \tag {12.2-7b}
+$$
+
+![](images/page-365_194a1c080ede206c0c40647406c5caac4e62c06f8f94c642ac837b4e8a70cf73.jpg)
+
+<details>
+<summary>text_image</summary>
+
+w₁
+β₁
+t
+w₂
+β₂
+u₁
+u₂
+L
+2
+L
+2
+s
+</details>
+
+(a)
+
+![](images/page-365_2bcc0fcfebabe3ab9b2217bf2f76c9587a2d4f3b096144c737d3765ec6606028.jpg)
+
+<details>
+<summary>text_image</summary>
+
+D_{r1} D_{s1} D_{r2}
+β1 β2
+1 2 D_{s2}
+3 β3
+D_{r3}
+(b) D_{s3}
+</details>
+
+Figure 12.2-3. (a) Straight element, showing d.o.f. in the local coordinate system. (b) Possible choice of global d.o.f, where $D_{r}$ and $D_{s}$ are, respectively, radial and tangential.
+
+<!-- source-page: 366 -->
+
+where $u$ and $w$ are respectively parallel and normal to the straight element, $s$ is a (straight) axial coordinate, and the $a_i$ are constants. Rigid-body motion is accounted for by $a_1, a_3$ , and $a_4$ . Since $R$ is infinite for a straight element, Eqs. 12.2-2 yield $\epsilon_m = a_2$ and $\kappa = -2a_5 - 6a_6s$ . Thus constant-strain states are possible, and these states are not coupled to rigid-body motion. The inextensibility condition, Eq. 12.2-5, is satisfied when $a_2 = 0$ . When an additional element is attached to elements already in place, three new d.o.f. are added to the mesh (e.g., the d.o.f. at node 3 in Fig. 12.2-3b), but only one inextensibility constraint is added ( $a_2 = 0$ ). Therefore the mesh will not lock, regardless of the values of $R, L,$ and $t$ . (Locking concepts are discussed in Sections 9.4 and 9.5.)
+
+Accordingly, the straight element of Eqs. 12.2-7 is free of major defects. It can be rigorously shown that the straight element is a valid model for a curved arch, and provides convergence to exact answers as the mesh is refined [12.1]. A drawback is that membrane and bending actions are not coupled within a single straight element, as evidenced by the off-diagonal null matrices in Eq. 12.2-6. Another drawback is that a distributed load must be modeled by nodal forces only. If nodal moment loads are also applied, as the consistent formulation of Eq. 4.1-6 would dictate, and if elements are of unequal lengths, then spurious bending moments appear. (These moments would be correct if the actual structure were a polygon of straight bars.) And, during stress computation, stresses caused by nodal displacements should not be adjusted to account for distributed load on the element (as discussed following Eq. 4.7-3).
+
+In a coarse mesh, accuracy is improved by taking element length L as the arc length between nodes, rather than the chord length as is suggested by Fig. 12.2-3b. For example, let R/t = 40 for the arch of Fig. 12.2-1a, and let the entire arch be modeled by four straight elements, each of length $L = \pi R/4$ . The error in the computed displacement of load P is approximately 2%. Use of the chord length $L = 2R \sin 22.5^{\circ}$ gives an error of approximately 10%.
+
+Curved Arch Elements. A great many types of curved arch element have been proposed, partly because a good curved element proved to be more elusive than first anticipated. Often, curved elements were far less accurate than straight elements: they were much too stiff, especially when applied to an arch that is thin and has a large rise-to-span ratio. Originally the difficulties were blamed on a lack of rigid-body motion capability. Subsequently the difficulties were attributed primarily to membrane locking [12.17], which is described as follows.
+
+The simplest curved element has the geometry shown in Fig. 12.2-1b and the displacement field of Eqs. 12.2-7, where u and w are now circumferential and radial displacements. Coordinate s follows the arch midline. Element d.o.f. are nodal values of u, w, and $w_{,s}$ . (In contrast to straight elements, no coordinate transformation is needed to match d.o.f. prior to assembly of elements.) From Eqs. 12.2-2 and 12.2-7, membrane strain $\epsilon_{m}$ is
+
+$$
+\epsilon_ {m} = \left(a _ {2} + \frac {a _ {3}}{R}\right) + \frac {a _ {4}}{R} s + \frac {a _ {5}}{R} s ^ {2} + \frac {a _ {6}}{R} s ^ {3} \tag {12.2-8}
+$$
+
+Now imagine that thickness $t$ of the arch approaches zero. The inextensibility condition, $\epsilon_{m} = 0$ for all $s$ , demands that
+
+<!-- source-page: 367 -->
+
+$$
+a _ {2} + \frac {a _ {3}}{R} = a _ {4} = a _ {5} = a _ {6} = 0 \tag {12.2-9}
+$$
+
+The condition $a_{2} + a_{3}/R = 0$ implies $\epsilon_{m} = 0$ at s = 0, the element center. This constraint causes no difficulty. The remaining conditions, $a_{4} = a_{5} = a_{6} = 0$ , imply that $w_{,s} = w_{,ss} = w_{,sss} = 0$ . These are severe and spurious constraints. They produce what is called membrane locking; that is, they make the model much too stiff. Numerical evidence supports this conclusion: for example, for the arch of Fig. 12.2-1a with R/t = 40, four identical elements predict a center displacement that is less than 5% of its correct value [12.2]. Equation 12.2-8 shows that the contribution of w terms to $\epsilon_{m}$ decreases as R becomes large. That is, the tendency to lock decreases as elements become more shallow, and disappears if elements become straight.
+
+One can also conclude that the mesh will lock by doing “constraint counting,” which is discussed in Section 9.5. Energy $U_{m}$ in Eq. 12.2-3 acts as a penalty function that enforces the inextensibility constraint as a curved element becomes thin—that is, as the ratio A/I becomes large. If numerically integrated, $U_{m}$ (and the membrane stiffness matrix it produces) enforces one constraint for every integration point used to evaluate $U_{m}$ . Each constraint effectively removes one d.o.f. from its intended role of modeling deformations. Accordingly, if there are three or more integration points per element, all d.o.f. of the structure are used to satisfy support conditions and constraints. No d.o.f. are left to model the bending deformations, so a mesh of curved elements tends to lock.
+
+This insight suggests the simple remedy of selective integration. One can use a two-point rule to evaluate the stiffness matrix that comes from $U_{b}$ , but use a one-point rule to evaluate the stiffness matrix that comes from $U_{m}$ . The single Gauss point is at the element center, s = 0. Accordingly, from Eq. 12.2-8, the constraint enforced is
+
+$$
+a _ {2} + \frac {a _ {3}}{R} = 0 \tag {12.2-10}
+$$
+
+Now only one d.o.f. per element is used in satisfying the constraint, and the mesh does not lock. Such elements are quite accurate: they have about the same accuracy as the aforementioned straight elements [12.2].
+
+Exactly integrated curved elements can also behave well if element displacement fields are properly designed. For example, consider the fields [12.3]
+
+$$
+u = a _ {1} + a _ {2} \phi + a _ {3} \phi^ {2} + a _ {4} \phi^ {3} + a _ {5} \phi^ {4} + a _ {6} \phi^ {5} \tag {12.2-11a}
+$$
+
+$$
+w = - a _ {2} - 2 a _ {3} \phi - 3 a _ {4} \phi^ {2} - 4 a _ {5} \phi^ {3} - 5 a _ {6} \phi^ {4} \tag {12.2-11b}
+$$
+
+where $\phi = s/R$ . These displacements satisfy the inextensibility condition $\epsilon_{m} = 0$ for all s. Since $\epsilon_{m} = 0$ , the resulting stiffness matrix comes entirely from the $U_{b}$ term in Eq. 12.2-3. Computed results are almost exact. We see that if displacement fields are to have enough d.o.f. for a six-d.o.f. element and are also to display $\epsilon_{m} = 0$ , the membrane field must be at least quintic and the lateral displacement field must be one degree lower.
+
+To allow the preceding element to display membrane strain, one can add a
+
+<!-- source-page: 368 -->
+
+constant $a_7$ to Eq. 12.2-11b. Thus [k] becomes 7 by 7 and includes a contribution from $U_m$ in Eq. 12.2-3. D.o.f. $a_7$ is internal to each element. Again, results are excellent [12.3].
+
+We note that the rigid-body motion field of Eqs. 12.2-4 is not explicitly included in any of the preceding curved elements. One can say that such motion is implicitly approximated in Eqs. 12.2-11 because $\sin \phi$ and $\cos \phi$ can be expanded as power series, whose initial terms are contained in Eqs. 12.2-11. Another element that implicitly approximates rigid-body motion is based on a displacement field of the form [12.4]
+
+$$
+u = a _ {1} + a _ {2} s + a _ {7} \left(1 - \frac {4 s ^ {2}}{L ^ {2}}\right) \tag {12.2-12a}
+$$
+
+$$
+w = a _ {3} + a _ {4} s + a _ {5} s ^ {2} + a _ {6} s ^ {3} \tag {12.2-12b}
+$$
+
+The mode associated with d.o.f. $a_{7}$ is internal or “nodeless”; it vanishes at the element ends $s = \pm L/2$ . D.o.f. $a_{7}$ can be removed by condensation prior to assembly of elements. As compared with Eqs. 12.2-7, Eqs. 12.2-12 reduce spurious strain energy associated with rigid-body motion by factors of 122,000 and 18,000 for curved elements that subtend arcs of $12^{\circ}$ and $20^{\circ}$ , respectively [12.4].
+
+A curved element that explicitly includes rigid-body motion capability has been suggested. Known as the Cantin–Clough element, it is based on fields of the form [12.2]
+
+$$
+u = a _ {1} \cos \phi + a _ {2} \sin \phi + a _ {3} + a _ {4} s \tag {12.2-13a}
+$$
+
+$$
+w = a _ {1} \sin \phi - a _ {2} \cos \phi + a _ {5} s ^ {2} + a _ {6} s ^ {3} \tag {12.2-13b}
+$$
+
+where $\phi = s / R$ . Rigid-body motion, Eqs. 12.2-4, is displayed by the element when $a_4 = a_5 = a_6 = 0$ . From Eqs. 12.2-2 and 12.2-13,
+
+$$
+\epsilon_ {m} = a _ {4} + \frac {a _ {5}}{R} s ^ {2} + \frac {a _ {6}}{R} s ^ {3} \quad \text { and } \quad \kappa = \frac {a _ {4}}{R} - 2 a _ {5} - 6 a _ {6} s \tag {12.2-14}
+$$
+
+Inextensibility requires that $a_4 = a_5 = a_6 = 0$ , which in turn yields $\kappa = 0$ . Accordingly, we expect to encounter locking difficulties. Indeed, for the thin-arch problem of Fig. 12.2-1a, the central deflection is almost $50\%$ low when 20 Can-tin-Clough elements are used for the entire arch [12.2].
+
+From the foregoing examples we conclude that membrane locking is much more detrimental to thin curved elements than is a lack of explicit rigid-body motion capability.
+
+Mindlin Arch Elements. Displacements of a Mindlin arch element are described by tangential and normal displacements of the midline and by rotation of a normal to the midline—that is, by $u, w,$ and $\beta$ . Thus rotation $\beta$ may differ from rotation $w_{,s}$ . As with Mindlin beam and plate elements discussed in Section 9.4 and Chapter 11, Mindlin arch elements can account for transverse shear deformation, which here is $\gamma_{zs}$ . Only if $\gamma_{zs} = 0$ does $\beta = w_{,s}$ .
+
+Strain energy in a Mindlin arch element is $U = U_{m} + U_{b} + U_{s}$ , in which the respective contributions to U are due to membrane strain $\epsilon_{m}$ , curvature change
+
+<!-- source-page: 369 -->
+
+$\kappa$ , and transverse shear strain $\gamma_{zs}$ [12.5]. For an element of length $L$ and constant radius $R$ ,
+
+$$
+U _ {m} = \int_ {- L / 2} ^ {L / 2} \frac {E A}{2} \epsilon_ {m} ^ {2} d s, \quad \text {where} \quad \epsilon_ {m} = u _ {, s} + \frac {w}{R} \tag {12.2-15a}
+$$
+
+$$
+U _ {b} = \int_ {- L / 2} ^ {L / 2} \frac {E I}{2} \kappa^ {2} d s, \quad \text { where } \quad \kappa = \frac {u _ {, s}}{R} - \beta_ {, s} \tag {12.2-15b}
+$$
+
+$$
+U _ {s} = \int_ {- L / 2} ^ {L / 2} \frac {G A}{2} \gamma_ {z s} ^ {2} d s, \quad \text { where } \quad \gamma_ {z s} = w _ {, s} - \beta \tag {12.2-15c}
+$$
+
+To account for a parabolic variation of $\gamma_{zs}$ through the thickness, we may replace G with 5G/6. A two-node element can be based on linear interpolations. With $a_{i} = \text{constants and } N_{1} = 0.5 - s/L$ , $N_{2} = 0.5 + s/L$ , linear fields are
+
+$$
+u = a _ {1} + a _ {2} s \quad \text { or } \quad u = N _ {1} u _ {1} + N _ {2} u _ {2} \tag {12.2-16a}
+$$
+
+$$
+w = a _ {3} + a _ {4} s \quad \text { or } \quad w = N _ {1} w _ {1} + N _ {2} w _ {2} \tag {12.2-16b}
+$$
+
+$$
+\beta = a _ {5} + a _ {6} s \quad \text { or } \quad \beta = N _ {1} \beta_ {1} + N _ {2} \beta_ {2} \tag {12.2-16c}
+$$
+
+Arguments concerning locking in this element are very similar to arguments made in connection with Eqs. 12.2-8 through 12.2-10 and are summarized as follows. As thickness t approaches zero, all strain energy should be in bending; that is, $\epsilon_{m}$ and $\gamma_{zs}$ should each vanish for all s, which implies, for a curved element,
+
+$$
+a _ {2} + \frac {a _ {3}}{R} = a _ {4} = a _ {5} = a _ {6} = 0 \tag {12.2-17}
+$$
+
+or which implies, for a straight element $(R = \infty)$ ,
+
+$$
+a _ {2} = a _ {4} - a _ {5} = a _ {6} = 0 \tag {12.2-18}
+$$
+
+In either case, an element added to a thin arch brings three d.o.f. with it, but all d.o.f. are occupied in satisfying constraints, and the mesh locks as t approaches zero. However, reduced integration can produce a workable element. If $U_{m}$ and $U_{s}$ are integrated with a one-point rule, only two constraints are imposed per element, one each on $\epsilon_{m}$ and $\gamma_{zs}$ , and the mesh does not lock.
+
+A curved element based on Eqs. 12.2-16 does not explicitly contain the rigid-body motion field, Eqs. 12.2-4. A straight element $(R = \infty)$ can display rigid-body motion; that is, it displays $\epsilon_{m} = \kappa = \gamma_{zs} = 0$ if $a_{2} = a_{6} = 0$ and $a_{4} = a_{5}$ .
+
+A three-node Mindlin element, having nodes at $s = -L / 2$ , $s = 0$ , and $s = +L / 2$ , would be called a quadratic element. With $\xi = s / (L / 2)$ , its displacement field has the form
+
+$$
+u = a _ {1} + a _ {2} \xi + a _ {3} \xi^ {2} \tag {12.2-19a}
+$$
+
+$$
+w = a _ {4} + a _ {5} \xi + a _ {6} \xi^ {2} \tag {12.2-19b}
+$$
+
+$$
+\beta = a _ {7} + a _ {8} \xi + a _ {9} \xi^ {2} \tag {12.2-19c}
+$$
+
+<!-- source-page: 370 -->
+
+As the arch becomes extremely thin, the inextensibility condition $\epsilon_{m} = 0$ for all $s$ implies
+
+$$
+\frac {2 a _ {2}}{L} + \frac {a _ {4}}{R} = \frac {4 a _ {3}}{L} + \frac {a _ {5}}{R} = a _ {6} = 0 \tag {12.2-20}
+$$
+
+The constraint $a_{6}=0$ implies membrane locking, as it prevents a thin arch from displaying a constant value of $w_{,ss}$ . Reduced integration offers a remedy [12.5,12.6]. Membrane strain can be written in the form
+
+$$
+\epsilon_ {m} = \left(\frac {2 a _ {2}}{L} + \frac {a _ {4}}{R} + \frac {a _ {6}}{3 R}\right) + \left(\frac {4 a _ {3}}{L} + \frac {a _ {5}}{R}\right) \xi + \frac {a _ {6}}{R} \left[ \xi^ {2} - \frac {1}{3} \right] \tag {12.2-21}
+$$
+
+If integration of membrane energy $U_{m}$ is performed by two-point Gauss quadrature, for which Gauss points are at $\xi = \pm 1/\sqrt{3}$ , then the bracketed expression vanishes. Thus no constraint is placed on $a_{6}$ ; rather, $\epsilon_{m} = 0$ implies only the vanishing of the two parenthetic expressions in Eq. 12.2-21.
+
+A similar argument can be applied to the condition $\gamma_{zs} = 0$ , which should prevail in an extremely thin arch. The conclusion is that $\gamma_{zs} = 0$ enforces the constraint $\beta_{ss} = 0$ in a quadratic element. This is not a locking condition, but it degrades element performance. Again the remedy is to use reduced integration: a two-point Gauss rule should be used to evaluate shear energy $U_{s}$ . (This advice was previously given in connection with a quadratic plate element, Fig. 11.3-3b.)
+
+The quadratic arch element does not explicitly contain the rigid-body motion capability described by Eqs. 12.2-4. However, Eqs. 12.2-4 pertain to a circular arch. Imagine now that the element shape is defined by the coordinates of its three nodes. Thus the arch element has parabolic shape. Then, since displacement fields are also second degree, the parabolic element is of the isoparametric family, and arguments given in Section 6.10 demonstrate that the capability for rigid-body motion is present. Similarly, a quadratic shell element, adapted from a quadratic isoparametric solid element, is able to display rigid-body motion without strain. (Some cautions about thickness-direction integration should be noted; see Ref. 12.7.)
+
+Remarks. During stress computation in numerically integrated elements, one should evaluate strains at the Gauss points of the reduced quadrature rule appropriate to the element: for example, for Mindlin elements, at the center in linear elements and at $\xi = \pm 1/\sqrt{3}$ in quadratic elements. Large spurious strains may appear at other locations, as shown in Fig. 6.13-2. Figure 6.13-2 depicts transverse shear strain, but membrane strain can display similar behavior. Accurate stress computation in extremely thin elements may require a restriction on thickness t, as noted in the following paragraph.
+
+Use of reduced integration avoids the imposition of spurious constraints, but not all constraints. The constraints that remain may cause numerical difficulty if the element is extremely thin. Difficulty is avoided by simply not allowing thickness t to fall below a certain limit in the computation of stiffness matrix coefficients associated with strain energies $U_{m}$ and $U_{s}$ . This matter is discussed at the end of Section 9.4.
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_038.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_038.md
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_038.md
@@ -0,0 +1,537 @@
+<!-- source-page: 371 -->
+
+# 12.3 FLAT ELEMENTS FOR SHELLS
+
+A curved shell can be approximated as a faceted surface, formed by connecting flat triangular elements together at vertex nodes. If we elect to use three translational and three rotational d.o.f. per node, each triangular element has 18 d.o.f. Let a typical element lie in the xy plane of a local coordinate system xyz. Nodal d.o.f. are shown in Fig. 12.3-1. Let these d.o.f. be called $\{d'\}$ and be arranged in the order $^{1}$ .
+
+$$
+\{\mathbf {d} ^ {\prime} \} = \left[ \begin{array}{l l l l l l} \mathbf {u} _ {i} & \mathbf {v} _ {i} & \theta_ {z i} & \mathbf {w} _ {i} & \theta_ {x i} & \theta_ {y i} \end{array} \right] ^ {T} \tag {12.3-1}
+$$
+
+where $\left[u_{i}\right]=\left[u_{1}\quad u_{2}\quad u_{3}\right]$ , $\left[v_{i}\right]=\left[v_{1}\quad v_{2}\quad v_{3}\right]$ , and so on. A plane element that includes “drilling freedoms” $\theta_{z}$ among its d.o.f. is described in Section 8.6. Its 9 by 9 stiffness matrix, which we now call $[k_{m}]$ , models membrane action in the shell. To this element we add a nine-d.o.f. triangular plate element, such as element DKT of Section 11.4. Its 9 by 9 stiffness matrix, which we now call $[k_{b}]$ , models bending action in the shell. For the composite flat shell element, whose stiffness matrix in local xyz coordinates is called $[k']$ , we have
+
+$$
+\left[ \mathbf {k} ^ {\prime} \right] \left\{\mathbf {d} ^ {\prime} \right\} = \left[ \begin{array}{c c} {\left[ \mathbf {k} _ {m} \right]} & {\left[ \mathbf {0} \right]} \\ {9 \times 9} & {9 \times 9} \\ {- - } & {- - } \\ {\left[ \mathbf {0} \right]} & {\left[ \mathbf {k} _ {b} \right]} \\ {9 \times 9} & {9 \times 9} \end{array} \right] \left\{ \begin{array}{l} \mathbf {u} _ {i} \\ \mathbf {v} _ {i} \\ \boldsymbol {\theta} _ {z i} \\ \mathbf {w} _ {i} \\ \boldsymbol {\theta} _ {x i} \\ \boldsymbol {\theta} _ {y i} \end{array} \right\} \tag {12.3-2}
+$$
+
+This stiffness matrix is clearly analogous to that of a straight arch element, Eq. 12.2-6. Coordinate transformation of $[k']$ is required prior to assembly of elements so that a common set of d.o.f. is used at each structure node shared by two or
+
+![](images/page-371_df75091ef67a5e395a6569549cbf096521a964ee5327f3dedb2845ec4a4ab28c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z
+θz1
+u1
+v1
+y
+1
+θz2
+2
+v2
+3
+θz3
+u2
+v3
+u3
+x
+</details>
+
+{α}
+
+![](images/page-371_512c83b92b2a4298d3e83d47b211516a311cb21c479d1b9449b79db8cf66373e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z
+w₁
+θᵧ₁
+θₓ₁
+y
+1
+w₂
+2
+3
+w₃
+θₓ₃
+θᵧ₂
+θᵧ₃
+θₓ₂
+θᵧ₂
+x
+(t)
+</details>
+
+(b)   
+Figure 12.3-1. Triangular element in a local xy plane. (a) D.o.f. associated with membrane action. (b) D.o.f. associated with bending action.
+
+$^{1}$ This ordering of d.o.f. is used only for convenience of notation. If this ordering is used, coefficients in [k] must be appropriately arranged.
+
+<!-- source-page: 372 -->
+
+more elements. Membrane and bending actions are not coupled within a single element, as evidenced by the 9 by 9 null matrices in Eq. 12.3-2. Nevertheless, the element works well enough to be competitive with curved elements [12.8]. (In Ref. 12.8, $[k_{m}]$ is identical to the “basic triangular subelement” discussed in Section 8.6.)
+
+If $[\mathbf{k}_m]$ pertains to the constant-strain triangle (Eq. 5.4-7), which does not use $\theta_z$ d.o.f., Eq. 12.3-2 is replaced by
+
+$$
+\left[ \mathbf {k} ^ {\prime} \right] \left\{\mathbf {d} ^ {\prime} \right\} = \left[ \begin{array}{c c c} \left[ \mathbf {k} _ {m} \right] & [ \mathbf {0} ] & \\ 6 \times 6 & 6 \times 3 & [ \mathbf {0} ] \\ - & - & - \\ [ \mathbf {0} ] & [ \mathbf {0} ] & 9 \times 9 \\ 3 \times 6 & 3 \times 3 & \\ - & - & - \\ & [ \mathbf {0} ] & [ \mathbf {k} _ {b} ] \\ & 9 \times 9 & 9 \times 9 \end{array} \right] \left\{ \begin{array}{l} \mathbf {u} _ {i} \\ \mathbf {v} _ {i} \\ \boldsymbol {\theta} _ {z i} \\ \mathbf {w} _ {i} \\ \boldsymbol {\theta} _ {x i} \\ \boldsymbol {\theta} _ {y i} \end{array} \right\} \tag {12.3-3}
+$$
+
+Again there is no coupling on the element level between $[k_{m}]$ and $[k_{b}]$ . In addition, no stiffness is associated with $\theta_{z}$ d.o.f. This means that the structure stiffness matrix will be singular if all elements connected to any node happen to be coplanar. This potential difficulty can be avoided by modifying element matrices $[k^{\prime}]$ as follows. Replace the on-diagonal null matrix in Eq. 12.3-3 by the 3 by 3 matrix in the following equation, which causes element-normal nodal rotations $\theta_{z}$ to produce corresponding moments $M_{z}$ [12.9],
+
+$$
+\left\{ \begin{array}{l} M _ {z 1} \\ M _ {z 2} \\ M _ {z 3} \end{array} \right\} = \alpha E V \left[ \begin{array}{c c c} 1. 0 & - 0. 5 & - 0. 5 \\ - 0. 5 & 1. 0 & - 0. 5 \\ - 0. 5 & - 0. 5 & 1. 0 \end{array} \right] \left\{ \begin{array}{l} \theta_ {z 1} \\ \theta_ {z 2} \\ \theta_ {z 3} \end{array} \right\} \tag {12.3-4}
+$$
+
+where $E$ is elastic modulus, $V$ is element volume, and $\alpha$ is a number such as 0.3 or less [12.9]. The added matrix provides each $\theta_z$ d.o.f. with a fictitious stiffness but offers no resistance to the mode $\theta_{z1} = \theta_{z2} = \theta_{z3}$ or to any other rigid-body motion.
+
+Another way to avoid singularity is to eliminate rotation about a normal to the shell from the list of global d.o.f. at each node. Thus the element has 15 d.o.f. rather than 18.
+
+Numerical tests show that the element of Eq. 12.3-2 is more accurate than the element of Eq. 12.3-3, and that reducing the latter element to 15 d.o.f. further degrades its performance [12.8].
+
+Difficulties with spurious bending moments, noted in connection with straight arch elements, can also occur when flat elements are used to model a shell.
+
+# 12.4 SHELLS OF REVOLUTION
+
+A shell of revolution resembles a solid of revolution in that elements are symmetric with respect to an axis and node points are cross sections of nodal circles. An element meridional cross section resembles an arch element, whose behavior and pitfalls are discussed in Section 12.2.
+
+Geometry and notation are shown in Fig. 12.4-1. Circumferential displacement v may be nonzero because initially we make no restriction that loads and dis-
+
+<!-- source-page: 373 -->
+
+![](images/page-373_e520ce070ea87f910480ca440b9a507b0abd986fff2934b5f09c2694efa47410.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Nodal circle 1
+w
+v,θ
+σ_H
+σ_x
+τ_SH
+L
+u
+Nodal circle 2
+</details>
+
+(a)
+
+![](images/page-373_984364a30eab90660ea83aae92900e6e32f4e01e0c621b369fcf0bf48ad7bd67.jpg)
+
+<details>
+<summary>text_image</summary>
+
+s = - \frac{L}{2}
+w
+u
+φ
+s
+s = \frac{L}{2}
+φ
+r₁
+φ₁
+1
+P
+Rₛ
+r
+2
+r₂
+φ₂
+</details>
+
+(b)   
+Figure 12.4-1. (a) Shell of revolution finite element. (b) Meridian of the element. Displacements u, v, and w are mutually orthogonal. $R_{s}$ is the radius of curvature of the meridian.
+
+placements must be axially symmetric. The following geometric relations can be written:
+
+$$
+R _ {\theta} = \frac {r}{\cos \phi} \quad R _ {s} = - \frac {d s}{d \phi} \quad \sin \phi = \frac {d r}{d s} \quad \cos \phi = - \frac {d z}{d s} \tag {12.4-1}
+$$
+
+$R_{\theta}$ and $R_{s}$ are principal radii of curvature. $R_{s}$ is considered negative if its center lies “outside” the shell (Fig. 12.1-1d). The center of an arc $R_{\theta}$ d $\theta$ lies on the axis of revolution, but the center of an arc $R_{s}$ d $\phi$ may not. Both centers lie on the same normal to the shell. Both radii can vary with s. Distance r and angle $\phi$ can be expressed in terms of s by integrating the second and third of Eqs. 12.4-1. Thus, if $R_{s}$ is assumed constant over the element [12.10],
+
+$$
+\phi = \phi_ {1} - \frac {1}{R _ {s}} \left(\frac {L}{2} + s\right) \quad \text { and } \quad r = r _ {1} + R _ {s} (\cos \phi - \cos \phi_ {1}) \tag {12.4-2}
+$$
+
+The equation for r fails if the meridian is straight, but then r can be linearly interpolated between $r_{1}$ and $r_{2}$ in terms of s. Stresses shown in Fig. 12.4-1a may vary with s and with $\theta$ . They may also vary with distance z from the shell mid-surface because of bending action.
+
+Loads without axial symmetry can be treated by superposition; that is, an analysis is performed for each Fourier harmonic of loading and the results of all harmonics are superposed. In essence, the method is that used for solids of revolution, as described in Section 10.5.
+
+Assembly of elements must allow for the possibility that meridians of adjacent elements may meet at a cusp—for example, where a cylindrical shell is joined to a conical cap. Accordingly, it is appropriate to transform from element d.o.f. (Fig. 12.4-2a) to a convenient set of global d.o.f. (Fig. 12.4-2b). In general, one should infer that all these d.o.f. represent amplitudes of nodal displacements and rota-
+
+<!-- source-page: 374 -->
+
+![](images/page-374_32319ea7a2b556b5a4655ec502288dea0cdb2b3714ad84ff27f1eeab115f89de.jpg)
+
+<details>
+<summary>text_image</summary>
+
+φ₁
+w₁
+φ₁
+u₁
+1
+β₁
+w₂
+φ₂
+2
+β₂
+u₂
+φ₂
+(a)
+</details>
+
+![](images/page-374_574c17421da417a30f3a9246d36ebd12fe8c7a7c808dc0e76c83c8e6db92b142.jpg)
+
+<details>
+<summary>text_image</summary>
+
+W₁
+β₁
+1
+U₁
+2
+W₂
+β₂
+U₂
+(b)
+</details>
+
+Figure 12.4-2. D.o.f. at nodes, in (a) local, and (b) global directions. D.o.f. $v_{1}$ and $v_{2}$ , not shown, are perpendicular to the paper.
+
+tions, as in a problem without axial symmetry analyzed by Fourier series. If axial symmetry prevails, circumferential nodal displacements $v_{i}$ are zero, and the remaining nodal displacements and rotations are $\theta$ -independent motions. The transformation of d.o.f. at a typical node i is
+
+$$
+\left\{ \begin{array}{l} u _ {i} \\ w _ {i} \\ v _ {i} \\ \beta_ {i} \end{array} \right\} = \left[ \begin{array}{c c c c} \sin \phi_ {i} & - \cos \phi_ {i} & 0 & 0 \\ \cos \phi_ {i} & \sin \phi_ {i} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \left\{ \begin{array}{l} U _ {i} \\ W _ {i} \\ v _ {i} \\ \beta_ {i} \end{array} \right\} \tag {12.4-3}
+$$
+
+D.o.f. $v_{i}$ and $\beta_{i}$ require no transformation. By writing Eq. 12.4-3 for node 1 and then for node 2, one constructs an 8 by 8 transformation matrix [T], which yields an element matrix $[k] = [T]^{T}[k^{\prime}][T]$ , ready for assembly into the structure. If the problem is axially symmetric, d.o.f. $v_{i}$ do not appear, and [T] is 6 by 6.
+
+Axial Symmetry. If material properties, support conditions, and loads are all independent of $\theta$ , then displacements and stresses are also independent of $\theta$ . Thus $v = \tau_{s\theta} = 0$ in Fig. 12.4-1. (Here we exclude pure torsion, for which $v \neq 0$ and $\tau_{s\theta} \neq 0$ , and vibration and buckling, whose displacement modes may lack axial symmetry despite axial symmetry of material properties, supports, and loads.)
+
+For axial symmetry, the strain-displacement relations are
+
+$$
+\epsilon_ {m s} = \frac {d u}{d s} + \frac {w}{R _ {s}} \quad \epsilon_ {m \theta} = \frac {u \sin \phi + w \cos \phi}{r} \tag {12.4-4}
+$$
+
+$$
+\kappa_ {s} = \frac {d}{d s} \left(\frac {u}{R _ {s}}\right) - \frac {d ^ {2} w}{d s ^ {2}} \quad \kappa_ {\theta} = \frac {\sin \phi}{r} \left(\frac {u}{R _ {s}} - \frac {d w}{d s}\right)
+$$
+
+where $\epsilon_{ms}$ and $\epsilon_{m\theta}$ are membrane strains of the shell midsurface, and $\kappa_{s}$ and $\kappa_{\theta}$ are curvature changes of the midsurface. Subscripts s and $\theta$ refer to meridional and circumferential directions, respectively. Transverse shear strain $\gamma_{zs}$ is assumed to be negligible because the shell is thin. The formulation of a thin shell element proceeds in a way very similar to the formulation of a thin arch element. First, one writes displacement fields for u and w that depend on nodal values of u, w, and rotation $\beta = w_{,s}$ . These fields are substituted into Eqs. 12.4-4, and the results into the strain energy expression
+
+<!-- source-page: 375 -->
+
+$$
+U = \frac {1}{2} \int_ {- L / 2} ^ {L / 2} \{\epsilon \} ^ {T} \left[ \begin{array}{c c} \mathbf {E} _ {M} & \mathbf {0} \\ \mathbf {0} & \mathbf {D} _ {K} \end{array} \right] \{\epsilon \} 2 \pi r d s \tag {12.4-5}
+$$
+
+where, with $C = Et / (1 - \nu^2)$ and $D = Et^3 / 12(1 - \nu^2)$ for an isotropic material,
+
+$$
+\left[ \mathbf {E} _ {M} \right] = C \left[ \begin{array}{l l} 1 & \nu \\ \nu & 1 \end{array} \right] \quad \left[ \mathbf {D} _ {K} \right] = D \left[ \begin{array}{l l} 1 & \nu \\ \nu & 1 \end{array} \right] \quad \{\epsilon \} = \left\{ \begin{array}{l} \epsilon_ {m s} \\ \epsilon_ {m \theta} \\ K _ {s} \\ K _ {\theta} \end{array} \right\} \tag {12.4-6}
+$$
+
+Equation 12.4-5 reduces to the form $U = \{d\}^{T}[k]\{d\}/2$ , from which one identifies the element stiffness matrix [k]. After numerical values of nodal d.o.f. have been calculated, strains and curvature changes follow from Eqs. 12.4-4. Membrane forces and bending moments in the shell are
+
+$$
+\left\{ \begin{array}{l} N _ {s} \\ N _ {\theta} \end{array} \right\} = \frac {E t}{1 - \nu^ {2}} \left[ \begin{array}{l l} 1 & \nu \\ \nu & 1 \end{array} \right] \left\{ \begin{array}{l} \epsilon_ {m s} \\ \epsilon_ {m \theta} \end{array} \right\} \quad \text { and } \quad \left\{ \begin{array}{l} M _ {s} \\ M _ {\theta} \end{array} \right\} = \frac {E t ^ {3}}{1 2 (1 - \nu^ {2})} \left[ \begin{array}{l l} 1 & \nu \\ \nu & 1 \end{array} \right] \left\{ \begin{array}{l} \kappa_ {s} \\ \kappa_ {\theta} \end{array} \right\} \tag {12.4-7}
+$$
+
+Stresses a distance z from the midsurface are obtained by use of Eq. 12.1-2, with x = s for meridional stress and $x = \theta$ for circumferential stress (see Fig. 12.4-3b).
+
+Equations 12.4-4 simplify if $R_{s}$ is infinite; for example, if the element is conical. Use of conical elements to model a doubly curved shell is analogous to use of straight elements to model an arch. If $R_{s}$ is infinite and $\phi = 0$ , the element becomes cylindrical.
+
+A Mindlin Axisymmetric Shell Element. This element is similar to the Mindlin beam element discussed in Section 9.4. We consider a conical element, Fig. 12.4-3. Thus $\phi$ is independent of s within a single element and $R_{s}$ is infinite. As before, u and w are midsurface displacements, respectively parallel and normal to a meridian. Let $\beta$ represent the rotation of a line that was normal to the midsurface of the undeformed shell. Transverse shear strain is $\gamma_{zs} = (dw/ds) -$
+
+![](images/page-375_2c0e25f78221f2f6ce53c2678d820ef1c34ea4a399ef551e676c5d70be6f696e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+z
+t
+φ
+ξ = 2s/L
+2
+u1
+w1
+L/2
+β1
+s
+ξ
+w2
+u2
+β2
+(a)
+</details>
+
+![](images/page-375_0555c4b4b9bd752d531d3fd248d7b434b522ef03a5af3ff8d397b0d14ad9a432.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ds
+M_H
+r dθ
+Q_s
+M_s
+N_s
+(b)
+</details>
+
+Figure 12.4-3. (a) Cross section of a conical shell element. Distance z is measured from the midsurface. (b) A differential element, bounded by meridians and parallels, and viewed normal to the shell. Membrane forces $N_{s}$ and $N_{\theta}$ , transverse shear force $Q_{s}$ , and bending moments $M_{s}$ and $M_{\theta}$ are shown.
+
+<!-- source-page: 376 -->
+
+$\beta$ . If $\gamma_{zs}$ is small, then $(dw/ds) \approx \beta$ . Thus, with the addition of $\gamma_{zs}$ , and with $R_s$ infinite, Eqs. 12.4-4 become
+
+$$
+\{\boldsymbol {\epsilon} \} = \left\{ \begin{array}{l} \epsilon_ {m s} \\ \epsilon_ {m \theta} \\ \kappa_ {s} \\ \kappa_ {\theta} \\ \gamma_ {z s} \end{array} \right\} = \left[ \begin{array}{c c c} d / d s & 0 & 0 \\ (\sin \phi) / r & (\cos \phi) / r & 0 \\ 0 & 0 & - d / d s \\ 0 & 0 & - (\sin \phi) / r \\ 0 & d / d s & - 1 \end{array} \right] \left\{ \begin{array}{l} u \\ w \\ \beta \end{array} \right\} \tag {12.4-8}
+$$
+
+where, as in Eq. 12.2-15b, $\beta$ has taken the place of dw/ds.
+
+Equations 12.4-5 and 12.4-6 are again applicable if $\{\epsilon\}$ is taken from Eq. 12.4-8 and the material property matrix in Eq. 12.4-5 is augmented by a shear stiffness term $5Gt/6$ —that is, if the material property matrix is
+
+$$
+\left[ \begin{array}{c c c} \mathbf {E} _ {M} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {D} _ {K} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & 5 G t / 6 \end{array} \right] \tag {12.4-9}
+$$
+
+in which the factor $5/6$ accounts for replacement of the true parabolic variation of $\gamma_{\mathrm{zf}}$ through the thickness by a uniform transverse shear strain.
+
+In what follows we describe a specific element that uses linear interpolations for $u$ and $\beta$ and a quadratic interpolation for $w$ . Membrane locking is avoided because the meridian is straight. Shear locking is avoided by making $\gamma_{zs}$ constant over the element, thus invoking only one penalty constraint as the element becomes thin. In the transverse shear constraint, the coefficient $5Gt/6$ in Eq. 12.4-9 plays the role of a penalty number and may require adjustment according to guidelines offered at the end of Section 9.4. The element was suggested by Tessler [12.11].
+
+The displacement field is taken as
+
+$$
+\left\{ \begin{array}{l} u \\ w \\ \beta \end{array} \right\} = \left[ \begin{array}{c c c c c c c} N _ {1} & N _ {2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & N _ {1} & N _ {2} & N _ {3} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & N _ {1} & N _ {2} \end{array} \right] \{\mathbf {d} \} \tag {12.4-10}
+$$
+
+where $\{d\} = \left[u_{1} \quad u_{2} \quad w_{1} \quad w_{2} \quad w_{3r} \quad \beta_{1} \quad \beta_{2}\right]^{T}$ , in which $w_{3r}$ is the transverse displacement at the element center s = 0, measured relative to the average value $(w_{1} + w_{2})/2$ . With $\xi = 2s/L$ , the shape functions are
+
+$$
+N _ {1} = \frac {1}{2} (1 - \xi) \quad N _ {2} = \frac {1}{2} (1 + \xi) \quad N _ {3} = 1 - \xi^ {2} \tag {12.4-11}
+$$
+
+Shear locking could be avoided by use of one Gauss point at s = 0 to integrate stiffness terms associated with $\gamma_{zs}$ . Then internal d.o.f. $w_{3r}$ would be eliminated by static condensation after a 7 by 7 stiffness matrix [k] is formulated. However, the same result can be produced more efficiently by eliminating $w_{3r}$ at the outset via a requirement that $\gamma_{zs}$ be constant over the element. It is possible to satisfy this requirement because the w field is one degree higher than the $\beta$ field. A constant $\gamma_{zs}$ implies that
+
+$$
+\frac {d \gamma_ {z s}}{d s} = \frac {d ^ {2} w}{d s ^ {2}} - \frac {d \beta}{d s} = 0 \tag {12.4-12}
+$$
+
+<!-- source-page: 377 -->
+
+Equations 12.4-8 through 12.4-12 yield $w_{3r} = (\beta_{1} - \beta_{2})L/8$ . With this substitution, Eq. 12.4-10 becomes
+
+$$
+\left\{ \begin{array}{l} u \\ w \\ \beta \end{array} \right\} = \left[ \begin{array}{c c c c c c} N _ {1} & N _ {2} & 0 & 0 & 0 & 0 \\ 0 & 0 & N _ {1} & N _ {2} & \overline {{{N}}} _ {3} & - \overline {{{N}}} _ {3} \\ 0 & 0 & 0 & 0 & N _ {1} & N _ {2} \end{array} \right] \{\mathbf {d} \} \tag {12.4-13}
+$$
+
+where $\{\mathbf{d}\} = \lfloor u_1, u_2, w_1, w_2, \beta_1, \beta_2 \rfloor^T$ , and, with $\xi = 2s / L$ ,
+
+$$
+N _ {1} = \frac {1}{2} (1 - \xi) \quad N _ {2} = \frac {1}{2} (1 + \xi) \quad \overline {{{N}}} _ {3} = \frac {L}{8} (1 - \xi^ {2}) \tag {12.4-14}
+$$
+
+We see that w is quadratic in s and that the w associated with $\overline{N}_{3}$ and $\beta_{1} = -\beta_{2}$ is identical to the lateral displacement of a beam whose end rotations are of equal magnitude but opposite sign. One can easily show that Eq. 12.4-13 yields a $\gamma_{zs}$ that is the same as $\gamma_{zs}$ at s = 0 from Eq. 12.4-10, which is the result desired.
+
+Equations 12.4-8 and 12.4-13 are used to formulate the element stiffness matrix. If the element is cylindrical and its stiffness matrix is to be integrated exactly, membrane contributions require three Gauss points (because $\epsilon_{\theta}$ is quadratic in s), curvature contributions require two Gauss points, and the transverse-shear contribution requires one Gauss point. Numerical tests of the element show very good accuracy (e.g., Fig. 12.4-4).
+
+Another Option for Cylindrical Shells. A cylindrical shell that is not circular or not symmetrically loaded can be analyzed by applying Fourier series to a substitute toroidal shell. If the given shell is slightly bent to form a toroidal shell of large radius (like an inner tube for a bicycle tire), the axial coordinate of the cylindrical shell becomes the circumferential coordinate of the toroidal shell. Series analysis of the toroidal shell creates several repetitions of geometry and loading around the circumference. A typical repetition provides a satisfactory model of the original cylindrical shell if enough series terms are used.
+
+![](images/page-377_09cfa04682ef8fc521e960ce977c545f48a1acea4110cbb754657674304dd95b.jpg)
+
+<details>
+<summary>line</summary>
+| s (in.) | M_s (in-lb/in.) |
+| ------- | -------------- |
+| 5       | 0              |
+| 6       | 0              |
+| 7       | 0              |
+| 8       | 5              |
+| 9       | 20             |
+| 10      | 0              |
+</details>
+
+Figure 12.4-4. Meridional bending moment $M_{s}$ in a cylindrical shell with open ends under uniform internal radial pressure [12.11]. The shell from s = 5 in. to s = 10 in. is spanned by eight identical elements. Finite element results are shown at element midpoints.
+
+<!-- source-page: 378 -->
+
+# 12.5 ISOPARAMETRIC GENERAL SHELL ELEMENTS
+
+Summary. A shell of general shape can be modeled by three-dimensional solid elements that (typically) have a thickness dimension considerably smaller than their other dimensions, as in Fig. 12.5-1a. But even for a very thick shell, three nodes along thickness-direction lines supply more d.o.f. than needed. Elimination of the middle nodes yields the element of Fig. 12.5-1b, in which thickness-direction strain $\epsilon_{3}$ is modeled as constant through the thickness. Here the subscript 3 indicates a direction normal to the shell midsurface. As the element becomes even thinner, stiffness coefficients associated with $\epsilon_{3}$ become far larger than other stiffness coefficients. This circumstance invites numerical difficulties. The difficulty can be avoided by constraining adjacent thickness-direction nodes to have the same thickness-direction displacement. Thus five d.o.f. are associated with each pair of thickness-direction nodes in Fig. 12.5-1b. These five d.o.f. can be attached to a single node whose d.o.f. are three translations and two rotations. These five d.o.f. define the motion of a thickness-direction line that remains straight but not necessarily normal to the shell midsurface after deformation. Thus the final element has midsurface nodes only. In formulating the element stiffness matrix, one uses a material property matrix [E] that corresponds to the plane stress condition $\sigma_{3} = 0$ .
+
+The element need not have eight nodes, as in Fig. 12.5-1c; other popular forms have four or nine nodes. All are Mindlin-type elements, and are therefore able to account for transverse shear deformation. They can be regarded as more general forms of the plate elements discussed in Section 11.3. As is the case with other plate and shell elements, Mindlin shell elements may encounter problems attributable to mechanisms and locking. These problems may be provoked (or avoided) by the choice of element geometry, type and number of d.o.f., shape functions, and numerical integration scheme. Pertinent references include [12.6,12.7, 12.9,12.12-12.14].
+
+Some details are presented in what follows. No particular element or number of nodes is assumed.
+
+Geometry. At a typical node $i$ , Fig. 12.5-2a, one can write a thickness-direction vector $\mathbf{V}_{3i}$ ,
+
+$$
+\mathbf {V} _ {3 i} = t _ {i} \left\{ \begin{array}{l} \ell_ {3 i} \\ m _ {3 i} \\ n _ {3 i} \end{array} \right\}, \quad \text { where } \quad \left\{ \begin{array}{l} \ell_ {3 i} \\ m _ {3 i} \\ n _ {3 i} \end{array} \right\} = \frac {1}{t _ {i}} \left\{ \begin{array}{l} x _ {j} - x _ {k} \\ y _ {j} - y _ {k} \\ z _ {j} - z _ {k} \end{array} \right\} \tag {12.5-1}
+$$
+
+![](images/page-378_91c34548347924532427f706a9e9a8df8209d230fb042d188ae3b082aaf7c1f8.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ξ
+ξ
+η
+</details>
+
+(a)
+
+![](images/page-378_94a95ae346c130c2bd1cff120d0218ec522b8f5d61d8dfce12c6a832625681a6.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ξ
+η
+ξ
+</details>
+
+(b)
+
+![](images/page-378_e644a407d1e6609a56c46b2d781a6a233a607ee85a2e5a5a9b7c6bbedb8d9d00.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ξ
+ξ̃
+η
+</details>
+
+(c)   
+Figure 12.5-1. (a) A 20-node, 60-d.o.f. solid element. (b) Elimination of four mid-edge nodes yields a 16-node, 48-d.o.f. element. (c) Further constraint yields an 8-node, 40-d.o.f. shell element.
+
+<!-- source-page: 379 -->
+
+![](images/page-379_619102b4fcae987d7b9093a236230c38a7b5fed570294ea61dc6f5dcfe1d9739.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ζ = +1
+j
+V3i
+P
+i
+ti
+k
+ζ = 0
+ζ = -1
+</details>
+
+(a)
+
+![](images/page-379_2d15d4649226e5e1e6a43a8d27bc95cec2612c244c50ab78693f8ee1162b4412.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z
+w_i
+V_{3i}
+i
+v_i
+y
+\beta_i
+v_{2i}
+u_i
+V_{1i}
+\alpha_i
+x
+</details>
+
+(b)
+
+![](images/page-379_ba6bfc2027aedc0b36a0e095ec73e814983639255cda0ea23b2154f178a573ba.jpg)
+
+<details>
+<summary>text_image</summary>
+
+j
+αi (ζ t_i/2)
+P
+βi (ζ t_i/2)
+ζ t_i/2
+i
+V_{2i}
+V_{1i}
+</details>
+
+(c)   
+Figure 12.5-2. (a) Typical node i, with thickness-direction vector $V_{3i}$ . (b) Orthogonal vectors at node i. Directions of rotational d.o.f. $\alpha_{i}$ and $\beta_{i}$ are given by the right-hand rule. Translational d.o.f. $u_{i}$ , $v_{i}$ , and $w_{i}$ are in the Cartesian directions x, y, and z. (c) Displacements of an arbitrary point P on $V_{3i}$ owing to small nodal rotations.
+
+in which $\ell_{3i}$ , $m_{3i}$ , and $n_{3i}$ are direction cosines of the line kij. The Cartesian coordinates of an arbitrary point in the element are
+
+$$
+\left\{ \begin{array}{l} x \\ y \\ z \end{array} \right\} = \sum N _ {i} \left\{ \begin{array}{l} x _ {i} \\ y _ {i} \\ z _ {i} \end{array} \right\} + \sum N _ {i} \zeta \frac {t _ {i}}{2} \left\{ \begin{array}{l} \ell_ {3 l} \\ m _ {3 i} \\ n _ {3 i} \end{array} \right\} \tag {12.5-2}
+$$
+
+where $x_{i} = (x_{j} + x_{k})/2$ , and so on, and shape functions $N_{i}$ are functions of $\xi$ and $\eta$ but are independent of $\zeta$ . For example, the $N_{i}$ are given by Eqs. 6.3-2 for a four-node element, by Eqs. 6.6-1 for an eight-node element, and by Table 6.6-1 for a nine-node element. To define element geometry, one can either supply the Cartesian coordinates of all nodes j and k, or supply $x_{i}$ , $y_{i}$ , $z_{i}$ , $t_{i}$ , and the direction cosines of $V_{3i}$ for all nodes i.
+
+Vectors $V_{1i}$ and $V_{2i}$ in Fig. 12.5-2 are perpendicular to each other and to $V_{3i}$ . Thus $V_{1i}$ and $V_{2i}$ are tangent to the midsurface, but they are not required to have a particular relation to Cartesian coordinate directions. $V_{1i}$ and $V_{2i}$ are used to define the directions of nodal rotation d.o.f. $\alpha_{i}$ and $\beta_{i}$ , which are shared by all elements that share node i. (It is possible that the set of $\alpha_{i}$ and $\beta_{i}$ directions will differ from node to node.) One can define $V_{1i}$ as a principal material direction if the material is orthotropic. Or, one can define a midsurface-tangent vector $e_{1i}$ whose components are $\Delta x = \Sigma N_{i,\xi} x_{i} \Delta \xi$ , $\Delta y = \Sigma N_{i,\xi} y_{i} \Delta \xi$ , and $\Delta z = \Sigma N_{i,\xi} z_{i} \Delta \xi$ , each evaluated at the node i in question, and with $\Delta \xi$ a small number. A similar vector $e_{2i}$ can be defined using an increment $\Delta \eta$ . Then $e_{3i} = e_{1i} \times e_{2i}$ and $V_{3i} = t_{i} e_{3i} / e_{3i}$ . Finally, $V_{1i} = e_{1i}$ and $V_{2i} = V_{3i} \times V_{1i}$ . Direction cosines of $V_{1i}$ and $V_{2i}$ are given by dividing each vector by its magnitude. To avoid input data errors, if the three vectors are supplied as data rather than being calculated within the program, one should ensure that $V_{1i}$ , $V_{2i}$ , and $V_{3i}$ are precisely orthogonal.
+
+For later use, we define the following matrix of direction cosines.
+
+$$
+[ \boldsymbol {\mu} _ {i} ] = \left[ - \frac {\mathbf {V} _ {2 i}}{V _ {2 i}} \quad \frac {\mathbf {V} _ {1 i}}{V _ {1 i}} \right] = \left[ \begin{array}{l l} - \ell_ {2 i} & \ell_ {1 i} \\ - m _ {2 i} & m _ {1 i} \\ - n _ {2 i} & n _ {1 i} \end{array} \right] \tag {12.5-3}
+$$
+
+where $V_{1i}$ and $V_{2i}$ are the magnitudes of $\mathbf{V}_{1i}$ and $\mathbf{V}_{2i}$ .
+
+<!-- source-page: 380 -->
+
+The 3 by 3 Jacobian matrix [J], defined by Eq. 6.7-2, contains terms such as
+
+$$
+x _ {, \xi} = \sum N _ {i, \xi} \left(x _ {i} + \zeta t _ {i} \ell_ {3 i} / 2\right)
+$$
+
+$$
+x _ {, \eta} = \sum N _ {i, \eta} \left(x _ {i} + \zeta t _ {i} \ell_ {3 i} / 2\right) \tag {12.5-4}
+$$
+
+$$
+x _ {, \zeta} = \sum N _ {i} \left(t _ {i} \ell_ {3 i} / 2\right)
+$$
+
+Displacements and Strains. The displacement of a point P on vector $V_{3i}$ , Fig. 12.5-2, consists of the displacement of node i plus the displacement relative to node i created by rotation of $V_{3i}$ . The relative displacement components, shown in Fig. 12.5-2c, must be resolved into x, y, and z components before being added to the displacements of node i. Thus, for example, point P has the x-direction displacement
+
+$$
+u _ {P} = u _ {i} - \alpha_ {i} \left(\zeta \frac {t _ {i}}{2}\right) \ell_ {2 i} + \beta_ {i} \left(\zeta \frac {t _ {i}}{2}\right) \ell_ {1 i} \tag {12.5-5}
+$$
+
+in which nodal rotations $\alpha_{i}$ and $\beta_{i}$ are presumed small. Displacements of an arbitrary point in the element are
+
+$$
+\left\{ \begin{array}{l} u \\ v \\ w \end{array} \right\} = \sum N _ {i} \left\{ \begin{array}{l} u _ {i} \\ v _ {i} \\ w _ {i} \end{array} \right\} + \sum N _ {i} \zeta \frac {t _ {i}}{2} [ \boldsymbol {\mu} _ {i} ] \left\{ \begin{array}{l} \alpha_ {i} \\ \beta_ {i} \end{array} \right\} \tag {12.5-6}
+$$
+
+Following standard isoparametric procedure, we express strains in terms of displacement derivatives,
+
+$$
+\left[ \begin{array}{l l l l l l} \epsilon_ {x} & \epsilon_ {y} & \epsilon_ {z} & \gamma_ {x y} & \gamma_ {y z} & \gamma_ {z x} \end{array} \right] ^ {T} = [ \mathbf {H} ] \left[ \begin{array}{l l l l l l} u _ {, x} & u _ {, y} & u _ {, z} & v _ {, x} \dots w _ {, z} \end{array} \right] ^ {T} \tag {12.5-7}
+$$
+
+$$
+\left\{ \begin{array}{c} u _ {, x} \\ u _ {, y} \\ u _ {, z} \\ v _ {, x} \\ \vdots \\ w _ {, z} \end{array} \right\} = \left[ \begin{array}{c c c} \mathbf {J} ^ {- 1} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {J} ^ {- 1} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \mathbf {J} ^ {- 1} \end{array} \right] \left\{ \begin{array}{c} u _ {, \xi} \\ u _ {, \eta} \\ u _ {, \zeta} \\ v _ {, \xi} \\ \vdots \\ w _ {, \zeta} \end{array} \right\} \tag {12.5-8}
+$$
+
+where [H] is stated in Eq. 6.7-5 and $[J]^{-1}$ is the inverse of the 3 by 3 Jacobian matrix [J]. All six strains are included in Eq. 12.5-7 because the shell midsurface has no particular orientation with respect to Cartesian coordinates xyz. The condition $\sigma_{3}=0$ will be introduced subsequently via the stress–strain relation. From Eq. 12.5-6 we obtain
+
+$$
+\left\{ \begin{array}{c} u, _ {\xi} \\ u, _ {\eta} \\ u, _ {\zeta} \\ v, _ {\xi} \\ \vdots \\ w, _ {\zeta} \end{array} \right\} = \sum \left[ \begin{array}{c c c c c} N _ {i, \xi} & 0 & 0 & - \zeta t _ {i} N _ {i, \xi} \ell_ {2 i} / 2 & \zeta t _ {i} N _ {i, \xi} \ell_ {1 i} / 2 \\ N _ {i, \eta} & 0 & 0 & - \zeta t _ {i} N _ {i, \eta} \ell_ {2 i} / 2 & \zeta t _ {i} N _ {i, \eta} \ell_ {1 i} / 2 \\ 0 & 0 & 0 & - t _ {i} N _ {i} \ell_ {2 i} / 2 & t _ {i} N _ {i} \ell_ {1 i} / 2 \\ 0 & N _ {i, \xi} & 0 & - \zeta t _ {i} N _ {i, \xi} m _ {2 i} / 2 & \zeta t _ {i} N _ {i, \xi} m _ {1 i} / 2 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & - t _ {i} N _ {i} n _ {2 i} / 2 & t _ {i} N _ {i} n _ {1 i} / 2 \end{array} \right] \left\{ \begin{array}{c} u _ {i} \\ v _ {i} \\ w _ {i} \\ \alpha_ {i} \\ \beta_ {i} \end{array} \right\} \tag {12.5-9}
+$$
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_039.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_039.md
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@@ -0,0 +1,398 @@
+<!-- source-page: 381 -->
+
+Combination of Eqs. 12.5-7, 12.5-8, and 12.5-9 yields
+
+$$
+\left[ \begin{array}{l l l l l l} \epsilon_ {x} & \epsilon_ {y} & \epsilon_ {z} & \gamma_ {x y} & \gamma_ {y z} & \gamma_ {z x} \end{array} \right] ^ {T} = \sum \left[ \mathbf {B} _ {i} \right] \left[ \begin{array}{l l l l l} u _ {i} & v _ {i} & w _ {i} & \alpha_ {i} & \beta_ {i} \end{array} \right] ^ {T} \tag {12.5-10}
+$$
+
+The complete strain-displacement matrix [B] is built of as many 6 by 5 blocks $[\mathbf{B}_i]$ as there are nodes in the element.
+
+Stiffness Matrix [k]. The stress–strain relation can be stated as
+
+$$
+\{\boldsymbol {\sigma} \} = [ \mathrm{E} ] \{\boldsymbol {\epsilon} \} \quad \text { or   as } \quad \{\boldsymbol {\sigma} ^ {\prime} \} = [ \mathrm{E} ^ {\prime} ] \{\boldsymbol {\epsilon} ^ {\prime} \} \tag {12.5-11}
+$$
+
+where $\{\sigma\}$ contains stresses in Cartesian directions xyz and $\{\sigma'\}$ contains stresses in local directions normal and tangent to the shell midsurface. The latter relation is $^{2}$
+
+$$
+\left\{ \begin{array}{l} \sigma_ {1} \\ \sigma_ {2} \\ \sigma_ {3} \\ \tau_ {1 2} \\ \tau_ {2 3} \\ \tau_ {3 1} \end{array} \right\} = \underbrace {\left[ \begin{array}{c c c c c c} E _ {1 1} & E _ {1 2} & 0 & 0 & 0 & 0 \\ E _ {1 2} & E _ {2 2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & G _ {1 2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 5 G _ {2 3} / 6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 5 G _ {3 1} / 6 \end{array} \right]} _ {[ \mathbf {E} ^ {\prime} ]} \left\{ \begin{array}{l} \epsilon_ {1} \\ \epsilon_ {2} \\ \epsilon_ {3} \\ \gamma_ {1 2} \\ \gamma_ {2 3} \\ \gamma_ {3 1} \end{array} \right\} \tag {12.5-12}
+$$
+
+where directions 1 and 2 are tangent to the midsurface and direction 3 is normal to it. These directions are presumed to be principal material directions if the material is orthotropic. The factors of 5/6 account for a parabolic variation of transverse shear strain through the thickness. Note that Eq. 12.5-12 is contrived to make the transverse normal stress $\sigma_{3}$ equal to zero. [E] is obtained from $[E']$ by the coordinate transformation $[E] = [T_{\epsilon}]^{T}[E'][T_{\epsilon}]$ (see Eq. 7.3-10). This transformation must be carried out at each Gauss point used in generating [k] by numerical integration. Direction cosines needed in $[T_{\epsilon}]$ are the direction cosines of vectors $V_{1}$ , $V_{2}$ , and $V_{3}$ at the Gauss point. In turn, these vectors can be found by shape function interpolation from nodal values,
+
+$$
+\mathbf {V} _ {1} = \sum N _ {i} \mathbf {V} _ {1 i} \quad \mathbf {V} _ {2} \doteq \sum N _ {i} \mathbf {V} _ {2 i} \quad \mathbf {V} _ {3} = \sum N _ {i} \mathbf {V} _ {3 i} \tag {12.5-13}
+$$
+
+in which the $N_{i}$ are evaluated at the Gauss point in question.
+
+The element stiffness matrix is
+
+$$
+\underset {5 N \times 5 N} {[ \mathbf {k} ]} = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \underset {5 N \times 6} {[ \mathbf {B} ] ^ {T}} \underset {6 \times 6} {[ \mathbf {E} ]} \underset {6 \times 5 N} {[ \mathbf {B} ]} \det [ \mathbf {J} ] d \xi d \eta d \zeta \tag {12.5-14}
+$$
+
+where N is the number of nodes per element. If material properties are independent of $\zeta$ , and if small errors are acceptable [12.7], then thickness-direction integration can be done explicitly. In doing so one discards terms in [J] that depend on $\zeta$ , under the assumption that these terms are negligible if the element is not sharply
+
+<!-- source-page: 382 -->
+
+curved. Next, [B] is split into a part $[B_{0}]$ that is independent of $\zeta$ and a part $\zeta[B_{1}]$ that is linear in $\zeta$ , so that $[B] = [B_{0}] + \zeta[B_{1}]$ . Thus terms linear in $\zeta$ integrate to zero and Eq. 12.5-14 becomes
+
+$$
+[ \mathbf {k} ] = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} (2 [ \mathbf {B} _ {0} ] ^ {T} [ \mathbf {E} ] [ \mathbf {B} _ {0} ] + \frac {2}{3} [ \mathbf {B} _ {1} ] ^ {T} [ \mathbf {E} ] [ \mathbf {B} _ {1} ]) \det [ \mathbf {J} ] d \xi d \eta \tag {12.5-15}
+$$
+
+in which [J] remains 3 by 3 but is evaluated on the midsurface, $\zeta = 0$ .
+
+Difficulties arising from shear locking, membrane locking, and mechanisms can be dealt with by selective and reduced integration and other strategies $[12.13]$ . As an element becomes thin, the penalty matrix associated with transverse shear must not be allowed to overwhelm the rest of the stiffness matrix (see the remarks that close Section 9.4).
+
+Element nodal loads (Eq. 4.1-6) come from the usual sources. Those associated with initial strains are, since $\{\epsilon_{0}^{\prime}\} = [T_{\epsilon}]\{\epsilon_{0}\}$ ,
+
+$$
+\int_ {V _ {\epsilon}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] \{\boldsymbol {\epsilon} _ {0} \} d V = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} [ \mathbf {B} ] ^ {T} [ \mathbf {T} _ {\epsilon} ] ^ {T} [ \mathbf {E} ^ {\prime} ] \{\boldsymbol {\epsilon} _ {0} ^ {\prime} \} \det [ \mathbf {J} ] d \xi d \eta d \zeta \tag {12.5-16}
+$$
+
+Finally, element stresses referred to local directions 1–2–3 are
+
+$$
+\{\sigma^ {\prime} \} = [ \mathrm{E} ^ {\prime} ] ([ \mathrm{T} _ {\epsilon} ] [ \mathrm{B} ] \{\mathrm{d} \} - \{\epsilon_ {0} ^ {\prime} \}) \tag {12.5-17}
+$$
+
+Stresses at Gauss points may be more accurate than stresses computed elsewhere in the element, as noted in Section 6.13.
+
+Usually, elements share a common tangent plane at each interelement boundary. Thus d.o.f. $\alpha_{i}$ and $\beta_{i}$ are midsurface-tangent vectors in all elements that share node i. This ideal circumstance would disappear if elements were to form a ridge line where they meet, as in a folded plate. Then $V_{3i}$ could be defined as an average shell normal vector, with $\alpha_{i}$ and $\beta_{i}$ normal to $V_{3i}$ , but accuracy loss would be expected.
+
+# PROBLEMS
+
+# Section 12.1
+
+12.1 In terms of $R_{s}$ and $R_{\theta}$ , how would you qualitatively describe the shape of an American football?   
+12.2 According to elementary stress formulas and Eqs. 12.1-1, what are $N_x$ , $N_y$ , and $N_{xy}$ for a cylindrical tank under internal pressure?   
+12.3 The cylindrical tank shown contains a step change in thickness and is capped by a hemispherical shell. Loads consist of axial force $Q$ and internal pressure $p$ .
+
+(a) Explain why load Q cannot be supported by membrane action only.
+
+(b) Free the built-in support condition at CC and divide the vessel into three parts by making circumferential cuts along AA and BB. Then show by a sketch the displaced shape of each part produced by pressure p alone.
+
+(c) Show by a sketch the loads (applied by one part to another) needed to restore continuity of displacements.
+
+<!-- source-page: 383 -->
+
+![](images/page-383_4a51a104d460eca9f7426f3100918a5471097d75829b0abe0bd66ee406d5a967.jpg)
+
+<details>
+<summary>text_image</summary>
+
+A
+B
+C
+p
+Q
+t
+A
+B
+C
+t
+2t
+</details>
+
+Problem 12.3
+
+12.4 Imagine that a garden hose has an elliptical cross section. Explain why internal pressure causes bending stresses to appear, and sketch their approximate variation over the outside perimeter of the cross section. Also explain why, if internal pressure is to be carried by membrane stresses alone, the cross section must first become circular.
+
+# Section 12.2
+
+12.5 Write an expression for strain $\epsilon_s$ if the arch radius $R$ is a function of $s$ .
+
+12.6 (a) Derive Eqs. 12.2-4. Suggestion: Combine Eqs. 12.2-2 and integrate the resulting differential equation.
+
+(b) Consider a quarter-circle arch that occupies the first quadrant of a Cartesian reference frame. By three separate sketches, show the displacement fields associated with $b_{1}, b_{2}$ , and $b_{3}$ in Eqs. 12.2-4.
+
+12.7 Write the coordinate transformation of nodal d.o.f. for a straight element; that is, define all terms completely in terms of $L$ and $R$ . Assume that local d.o.f. are ordered as shown in Eq. 12.2-6, and that global d.o.f. in Fig. 12.2-3b have the order $\left[ D_{s1} \quad D_{r1} \quad \beta_1 \quad D_{s2} \quad D_{r2} \quad \beta_2 \right]^T$ .
+
+12.8 Model a complete circular ring by four straight elements of equal length. The model is therefore a square, as shown. Compute the relative separation of loads $P$ , accounting for bending stiffness only. Take the element length as (a) chord length $L = \sqrt{2} R$ , and (b) arc length $L = \pi R / 2$ .
+
+![](images/page-383_70438c382be0f6d0faed6b534570619b075e916c2970fef997c01e38d6f56cfe.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+R
+P
+L
+P
+L
+P
+L
+</details>
+
+Problem 12.8
+
+12.9 For a curved arch, integrate $U_{m}$ of Eq. 12.2-3 using the displacement field of Eqs. 12.2-7. Hence, show that the condition $U_{m} = 0$ implies Eqs. 12.2-9.
+
+12.10 Using the following displacement fields for a curved arch, establish the relation between nodal d.o.f. and the $a_{i}$ ; that is, find [A] in the relation
+
+$$
+\left\lfloor u _ {1} \quad w _ {1} \quad \beta_ {1} \quad u _ {2} \quad w _ {2} \quad \beta_ {2} \right\rfloor^ {T} = [ \mathrm{A} ] \left\lfloor a _ {1} \quad a _ {2} \quad a _ {3} \quad a _ {4} \quad a _ {5} \quad a _ {6} \right\rfloor^ {T}
+$$
+
+where $\beta = w_{,s}$ .
+
+(a) Use Eqs. 12.2-7.   
+(b) Use Eqs. 12.2-11.
+
+<!-- source-page: 384 -->
+
+12.11 Would the mode associated with d.o.f. $a_7$ in Eq. 12.2-12a be of any benefit to (a) the straight element of Fig. 12.2-3a, or (b) the curved element based on Eqs. 12.2-7? Consider both full and selective integration in your explanation of part (b).
+
+12.12 In the stiffness matrix of the element associated with Eqs. 12.2-11, the diagonal coefficients associated with nodal deflections $w_{1}$ and $w_{2}$ are each
+
+$$
+k = \frac {E I}{R ^ {3}} \left(\frac {2 4}{\lambda^ {3}} - \frac {1 9 2}{3 5 \lambda} + \frac {1 6 \lambda}{3 5}\right)
+$$
+
+where $\lambda = L / 2R$ for an element of arc length $L$ . Use this information to solve for the deflection of load $P$ in Fig. 12.2-1a from a two-element model.
+
+12.13 In formulating an element stiffness matrix [k] from Eqs. 12.2-11, energy $U_{m}$ makes no contribution to [k]. Why cannot $U_{m}$ simply be discarded in formulating other curved arch elements, for example, those associated with Eqs. 12.2-7 and 12.2-13?
+
+12.14 (a) Verify Eqs. 12.2-17 and 12.2-18.
+
+(b) Determine the analogous equations of constraint that pertain to use of reduced integration for $U_{m}$ and $U_{s}$ .
+
+12.15 Investigate the $\gamma_{zs} = 0$ condition and the effect of reduced integration on the three-node Mindlin element (analogous to Eqs. 12.2-20 and 12.2-21).
+
+# Section 12.3
+
+12.16 Consider a doubly curved shell, modeled by flat triangular elements, such as those of Eqs. 12.3-2 and 12.3-3. What can you say about interelement compatibility of edge displacements?
+
+12.17 Write an equation analogous to Eq. 12.3-4 but appropriate to a flat element that has four nodes. Do you think this equation should be used if the element is warped rather than flat?
+
+12.18 Imagine that each side of a rectangular box is modeled by a mesh of flat shell elements. Internal pressure is applied. Along the edges where sides intersect, what d.o.f. can probably be set to zero, and why?
+
+# Section 12.4
+
+12.19 (a) Derive Eqs. 12.4-2.
+
+(b) Rewrite Eqs. 12.4-2 in a form appropriate to a shell with a straight meridian.
+
+![](images/page-384_56cd286f26e748c5dccccac83f4cff8ccc4c6c309a0bb70ca344bc84a7f28371.jpg)
+
+<details>
+<summary>text_image</summary>
+
+T
+r₁
+s
+L
+t
+φ
+</details>
+
+Problem 12.20
+
+<!-- source-page: 385 -->
+
+12.20 The conical shell shown is fixed at its base and loaded by torque T at the top. Use mechanics of materials concepts, not finite elements, to answer the following questions.
+
+(a) What is shear stress $\tau_{s\theta}$ , in terms of $T$ , $r_1$ , $t$ , $\phi$ , and $s$ ?   
+(b) What is the angle of rotation of the top of the truncated cone relative to the bottom, in terms of $T$ , $r_1$ , $t$ , $\phi$ , $L$ , and shear modulus $G$ ?
+
+12.21 Specialize Eqs. 12.4-4 to the following cases.
+
+(a) A cylindrical shell, with $s$ the only independent variable.   
+(b) A flat plate, with $r$ the only independent variable.   
+(c) A sphere, with $\phi$ the only independent variable.
+
+12.22 Consider a thin cylindrical shell of radius R whose midsurface axial strain $\epsilon_{m}$ is unrestrained. By considering the energy associated with membrane strains $\epsilon_{ms}$ and $\epsilon_{m\theta}$ , show that displacement w is in effect resisted by an elastic foundation of modulus $Et/R^{2}$ (as well as being resisted by bending stiffness).   
+12.23 Consider a cylindrical shell, thin-walled and symmetrically loaded, but without axial loads. Thus $N_{s} = 0$ , $\epsilon_{ms} = -\nu\epsilon_{m\theta}$ , and displacement u need not be considered. Generate the 4 by 4 element stiffness matrix. Use a cubic w field.   
+12.24 Show that $\gamma_{zs}$ at $s = 0$ from Eq. 12.4-10 is the same as $\gamma_{zs}$ for all $s$ from Eq. 12.4-13.   
+12.25 Show that application of Eq. 12.4-12 converts Eqs. 12.4-10 and 12.4-11 to Eqs. 12.4-13 and 12.4-14.   
+12.26 Verify the remark made in the sentence following Eq. 12.4-14.   
+12.27 Let a constant pressure p be applied to the inside surface of the element described by Eq. 12.4-13. Evaluate the consistent element nodal load vector.   
+12.28 The cylindrical shell shown is modeled by two Mindlin elements, prevented from rotation $\beta$ at nodal circles 1 and 3, and loaded by a uniform radial pressure on the inside surface. There are no end caps. Consistently computed nodal loads are applied. Will computed results display nonzero meridional curvature $\kappa_{s}$ in the following situations? Answer without doing calculations.
+
+(a) $L_{1} = L_{2}$ , and $w$ varies linearly between nodes.   
+(b) $L_{1} \neq L_{2}$ , and $w$ varies linearly between nodes.   
+(c) $L_{1} = L_{2}$ , and the element of Eqs. 12.4-13 is used.   
+(d) $L_{1} \neq L_{2}$ , and the element of Eqs. 12.4-13 is used.
+
+![](images/page-385_49704cfde9ac8eaa95ca74b7cc57cf42c4d90fcb7a2ed8d325adf15dde53d6b9.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+3
+L₁
+L₂
+</details>
+
+Problem 12.28
+
+<!-- source-page: 386 -->
+
+# Section 12.5
+
+12.29 Write an equation analogous to Eq. 12.5-2 but applicable to the element of Fig. 12.5-1b. (Each shape function should depend on $\xi$ and $\eta$ and should be multiplied by a linear function of $\zeta$ .)
+
+12.30 (a) Define terms in the Jacobian matrix [J] in terms of $\zeta$ , $N_{i}$ , $N_{i,\xi}$ , $N_{i,\eta}$ , nodal coordinates, and components of $\mathbf{V}_{3i}$ .
+
+(b) Specialize your result for the case of an element that is flat, of constant thickness, and whose midsurface coincides with the xy plane.
+
+12.31 (a) Imagine that instead of rotational d.o.f. $\alpha_{i}$ and $\beta_{i}$ in Fig. 12.5-2, we elect to use rotational d.o.f. $\beta_{xi}$ , $\beta_{yi}$ , and $\beta_{zi}$ , which are small rotations about axes $x$ , $y$ , and $z$ . Write the appropriate form of Eq. 12.5-6 and define the terms in the direction cosine matrix $[\mu_i]$ you use.
+
+(b) Check that your formula agrees with Eq. 12.5-6 for the special cases of all vectors $\mathbf{V}_3$ parallel to the $x$ axis, then the $y$ axis, and finally the $z$ axis.
+
+(c) The element now has six d.o.f. per node rather than five. What possible difficulty does this element present?
+
+12.32 (a) How can the transformation of $[\mathbf{E}']$ to $[\mathbf{E}]$ be made more computationally efficient? (Exploit the null row and the null column in $[\mathbf{E}']$ .)
+
+(b) Write Eq. 12.5-14 in a form that uses $[E']$ rather than $[E]$ .
+
+12.33 Let a typical $[B_{i}]$ in Eq. 12.5-10 have the form $[B_{i}] = [H][\overline{B}_{i}]$ , where $[H]$ is defined by Eq. 6.7-5 and $[\overline{B}_{i}]$ is a 9 by 5 matrix. Express $[\overline{B}_{i}]$ as a function of $\zeta$ , $t_{i}$ , $N_{i}$ , $N_{i,\xi}$ , $N_{i,\eta}$ , direction cosines, and the $\Gamma_{ij}$ in $[\Gamma] = [J]^{-1}$ .
+
+12.34 Two elements of the type shown by Fig. 12.5-1c are to be connected side by side. However, they do not share a common tangent plane; for example, they may be perpendicular, like adjacent sides of a box. How should the connection be accomplished?—that is, how should d.o.f. along the connection line be treated?
+
+12.35 Consider a membrane shell (a shell that has no bending stiffness). Review the formulations presented in Section 12.5, and state how they may be simplified or specialized to deal with a membrane shell.
+
+12.36 The sketch represents an end of an isoparametric bar element, whose geometry is defined by the position of nodes along its centerline and vectors $V_{2i}$ and $V_{3i}$ that span its rectangular cross section.
+
+(a) Write an equation of geometry analogous to Eq. 12.5-2.   
+(b) Write an equation of displacement analogous to Eq. 12.5-6.
+
+![](images/page-386_f93142e0f5508925d1e15066ca2e471f933284448c43d5484a6cf29c2d57da52.jpg)
+
+<details>
+<summary>text_image</summary>
+
+V₃ᵢ
+i
+V₂ᵢ
+V₁ᵢ
+</details>
+
+Problem 12.36
+
+<!-- source-page: 387 -->
+
+# FINITE ELEMENTS IN DYNAMICS AND VIBRATIONS
+
+The use of the finite element method for the dynamic analysis of structures is described. Mass and damping matrices are derived. Modal and direct time integration methods of analysis are discussed.
+
+# 13.1 INTRODUCTION
+
+If the frequency of excitation applied to a structure is less than roughly one-third of the structure's lowest natural frequency of vibration, then the effects of inertia can be neglected and the problem is quasistatic. That is, the equations $[K]\{D\} = \{R\}$ are sufficiently accurate even though loads $\{R\}$ , and hence displacements $\{D\}$ , vary (slowly) with time. Loads $\{R\}$ may result from surface loads and/or body forces. Forces that result from constant or almost constant acceleration are treated in the same manner as gravity forces—that is, by the integral that contains $\{F\}$ in Eq. 4.1-6.
+
+Inertia becomes important if excitation frequencies are higher than noted above or if the structure vibrates freely. The mass matrix, written as [m] for an element and [M] for a structure, accounts for inertia and is a discrete representation of the continuous distribution of mass in a structure. The effects of damping, if important, are accounted for by damping matrices [c] and [C].
+
+Problems of dynamics can be categorized as either wave propagation problems or structural dynamics problems. In wave propagation problems the loading is often an impact or an explosive blast. The excitation, and hence the structural response, are rich in high frequencies. In such problems we are usually interested in the effects of stress waves. Thus the time duration of analysis is usually short and is typically of the order of a wave traversal time across a structure. A problem that is not a wave propagation problem, but for which inertia is important, is called a structural dynamics problem. In this category, the frequency of excitation is usually of the same order as the structure's lowest natural frequencies of vibration.
+
+Problems of structural dynamics can be subdivided into two broad classifications. In one, we ask for natural frequencies of vibration and the corresponding mode shapes. Usually, we wish to compare natural frequencies of the structure with frequencies of excitation. In design, it is usually desirable to assure that these frequencies are well separated. In the other classification, we ask how a structure moves with time under prescribed loads and/or motions of its supports; that is, we ask for a time-history analysis. Two popular methods of time-history
+
+<!-- source-page: 388 -->
+
+analysis are modal methods and direct integration methods. (“Time history” is a commonly used term referring to the record of the variation of a quantity over some interval of time.)
+
+Structural dynamics has an extensive literature and good textbooks $[13.1, 13.2, 13.45, 13.46]$ . Methods of structural dynamics are largely independent of finite element analysis because these methods presume the availability of stiffness, mass, and damping matrices but do not demand that they arise from a finite element discretization. Indeed, many popular methods were developed before the advent of the finite element method by using matrices resulting from finite difference discretizations. Today, however, matrices are most often obtained from finite element discretizations, and the analysis tools are tailored to fit finite element models.
+
+# 13.2 DYNAMIC EQUATIONS. MASS AND DAMPING MATRICES
+
+Equations that govern the dynamic response of a structure or medium will be derived by requiring the work of external forces to be absorbed by the work of internal, inertial, and viscous forces for any small kinematically admissible motion (i.e., any small motion that satisfies both compatibility and essential boundary conditions). For a single element, this work balance becomes
+
+$$
+\begin{array}{l} \int_ {V _ {e}} \{\delta \mathbf {u} \} ^ {T} \{\mathbf {F} \} d V + \int_ {S _ {e}} \{\delta \mathbf {u} \} ^ {T} \{\boldsymbol {\Phi} \} d S + \sum_ {i = 1} ^ {n} \{\delta \mathbf {u} \} _ {i} ^ {T} \{\mathbf {p} \} _ {i} \\ = \int_ {V _ {e}} \left(\left\{\delta \boldsymbol {\epsilon} \right\} ^ {T} \left\{\boldsymbol {\sigma} \right\} + \left\{\delta \mathbf {u} \right\} ^ {T} \rho \left\{\ddot {\mathbf {u}} \right\} + \left\{\delta \mathbf {u} \right\} ^ {T} \kappa_ {d} \left\{\dot {\mathbf {u}} \right\}\right) d V \tag {13.2-1} \\ \end{array}
+$$
+
+where $\{\delta u\}$ and $\{\delta e\}$ are respectively small arbitrary displacements and their corresponding strains, $\{F\}$ are body forces, $\{\Phi\}$ are prescribed surface tractions (which typically are nonzero over only a portion of surface $S_{e}$ ), $\{p\}_{i}$ are concentrated loads that act at a total of n points on the element, $\{\delta u\}_{i}^{T}$ is the displacement of the point at which load $\{p\}_{i}$ is applied, $\rho$ is the mass density of the material, $\kappa_{d}$ is a material-damping parameter analogous to viscosity, and volume integration is carried out over the element volume $V_{e}$ .
+
+Using usual notation, we have for the displacement field $\{\mathbf{u}\}$ (which is a function of both space and time) and its first two time derivatives
+
+$$
+\{\mathbf {u} \} = [ \mathbf {N} ] \{\mathbf {d} \} \quad \{\dot {\mathbf {u}} \} = [ \mathbf {N} ] \{\dot {\mathbf {d}} \} \quad \{\ddot {\mathbf {u}} \} = [ \mathbf {N} ] \{\ddot {\mathbf {d}} \} \tag {13.2-2}
+$$
+
+In Eqs. 13.2-2, shape functions [N] are functions of space only and nodal d.o.f. {d} are functions of time only. Thus Eqs. 13.2-2 represent a local separation of variables. Combination of Eqs. 13.2-1 and 13.2-2 yields
+
+$$
+\begin{array}{l} \{\delta \mathbf {d} \} ^ {T} \left[ \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} \{\boldsymbol {\sigma} \} d V + \int_ {V _ {e}} \rho [ \mathbf {N} ] ^ {T} [ \mathbf {N} ] d V \{\ddot {\mathbf {d}} \} + \int_ {V _ {e}} \kappa_ {d} [ \mathbf {N} ] ^ {T} [ \mathbf {N} ] d V \{\dot {\mathbf {d}} \} \right. \\ - \int_ {V _ {e}} [ \mathbf {N} ] ^ {T} \{\mathbf {F} \} d V - \int_ {S _ {e}} [ \mathbf {N} ] ^ {T} \{\boldsymbol {\Phi} \} d S - \sum_ {i = 1} ^ {n} \{\mathbf {p} \} _ {i} \Bigg ] = 0 \tag {13.2-3} \\ \end{array}
+$$
+
+<!-- source-page: 389 -->
+
+in which it has been assumed that the locations of concentrated loads $\{p\}_{i}$ are coincident with node point locations. Since $\{\delta d\}$ is arbitrary, Eq. 13.2-3 can be written as
+
+$$
+[ \mathbf {m} ] \{\dot {\mathbf {d}} \} + [ \mathbf {c} ] \{\dot {\mathbf {d}} \} + \left\{\mathbf {r} ^ {\text {int}} \right\} = \left\{\mathbf {r} ^ {\text {ext}} \right\} \tag {13.2-4}
+$$
+
+where the element mass and damping matrices are defined as
+
+$$
+[ \mathbf {m} ] = \int_ {V _ {e}} \rho [ \mathbf {N} ] ^ {T} [ \mathbf {N} ] d V \tag {13.2-5}
+$$
+
+$$
+[ \mathbf {c} ] = \int_ {V _ {e}} \kappa_ {d} [ \mathbf {N} ] ^ {T} [ \mathbf {N} ] d V \tag {13.2-6}
+$$
+
+and the element internal force $^{1}$ and external load vectors are defined as
+
+$$
+\{\mathbf {r} ^ {\text {int}} \} = \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} \{\boldsymbol {\sigma} \} d V \tag {13.2-7}
+$$
+
+$$
+\left\{\mathbf {r} ^ {\text {ext}} \right\} = \int_ {V _ {e}} [ \mathbf {N} ] ^ {T} \left\{\mathbf {F} \right\} d V + \int_ {S _ {e}} [ \mathbf {N} ] ^ {T} \left\{\boldsymbol {\Phi} \right\} d S + \sum_ {i = 1} ^ {n} \left\{\mathbf {p} \right\} _ {i} \tag {13.2-8}
+$$
+
+Equation 13.2-4 is a system of coupled, second-order, ordinary differential equations in time and is called a finite element semidiscretization because although displacements $\{d\}$ are discrete functions of space, they are still continuous functions of time. Methods of dynamic analysis focus on how to solve this equation. Modal methods, discussed in Section 13.6, attempt to uncouple the equations, each of which can then be solved independently of others. Direct integration methods, discussed in Sections 13.9 to 13.13, discretize Eq. 13.2-4 in time to obtain a sequence of simultaneous algebraic equations.
+
+Structure matrices [M], [C], and $\{R^{int}\}$ are constructed by the conceptual expansion of element matrices [m], [c], and $\{r^{int}\}$ to “structure size” followed by addition of overlapping coefficients, exactly as explained in Sections 2.5 to 2.7. However, as discussed in subsequent sections, the exact manner in which $\{R^{int}\}$ is computed is often intimately mated with the dynamic analysis procedure.
+
+When Eqs. 13.2-5 and 13.2-6 are evaluated using the same shape functions [N] as used in the displacement field interpolation (Eqs. 13.2-2), the results are called consistent mass and consistent damping matrices. These matrices are symmetric. On the element level, they are generally full, but on the structure level, they have the same sparse topology as the structure stiffness matrix. When $\rho$ and $\kappa_{d}$ are nonzero, consistent matrices [m] and [c] are positive definite. That is, using the mass matrix for example, the kinetic energy $\frac{1}{2}\{\dot{d}\}^{T}[m]\{\dot{d}\}$ is positive for any nonzero $\{\dot{d}\}$ .
+
+Consistent damping matrix [c] is easily evaluated for a Newtonian fluid; its terms are given by Rayleigh [13.3]. In structures we are less interested in viscous damping than in dry friction and hysteresis loss. These energy loss mechanisms are not well understood, and from a practical standpoint Eq. 13.2-6 does not
+
+<!-- source-page: 390 -->
+
+correctly represent structural damping. In Section 13.4, we present some popular ad hoc damping schemes for structural dynamics.
+
+The internal force vector, Eq. 13.2-7, represents loads at nodes caused by straining of material. Equations 13.2-4 and 13.2-7 are valid for both linear and nonlinear material behavior; that is, in Eq. 13.2-7, $\{\sigma\}$ could be a nonlinear function of strain or strain rate. For linearly elastic material behavior, $\{\sigma\} = [E][B]\{d\}$ and Eq. 13.2-7 becomes
+
+$$
+\{\mathbf {r} ^ {\text {int}} \} = [ \mathbf {k} ] \{\mathbf {d} \} \tag {13.2-9}
+$$
+
+where the usual definition of the stiffness matrix holds—that is,
+
+$$
+[ \mathbf {k} ] = \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] [ \mathbf {B} ] d V \tag {13.2-10}
+$$
+
+When Eq. 13.2-10 is used, Eq. 13.2-4 becomes
+
+$$
+[ \mathbf {m} ] \{\ddot {\mathbf {d}} \} + [ \mathbf {c} ] \{\dot {\mathbf {d}} \} + [ \mathbf {k} ] \{\mathbf {d} \} = \left\{\mathbf {r} ^ {\text {ext}} \right\} \tag {13.2-11}
+$$
+
+which can be interpreted as saying that external loads are equilibrated by a combination of inertial, damping, and elastic forces. For the assembled structure, from Eq. 13.2-11,
+
+$$
+[ \mathbf {M} ] \{\ddot {\mathbf {D}} \} + [ \mathbf {C} ] \{\dot {\mathbf {D}} \} + [ \mathbf {K} ] \{\mathbf {D} \} = \left\{\mathbf {R} ^ {\text {ext}} \right\} \tag {13.2-12}
+$$
+
+where $\{R^{ext}\}$ corresponds to loads $\{R\}$ of a static problem, but is in general a function of time. Or, returning to Eq. 13.2-4, equations of the assembled structure can be written in the alternative form
+
+$$
+[ \mathbf {M} ] \{\ddot {\mathbf {D}} \} + [ \mathbf {C} ] \{\dot {\mathbf {D}} \} + \left\{\mathbf {R} ^ {\text {int}} \right\} = \left\{\mathbf {R} ^ {\text {ext}} \right\} \tag {13.2-13}
+$$
+
+which does not require that the material be linearly elastic.
+
+# 13.3 MASS MATRICES, CONSISTENT AND DIAGONAL
+
+A mass matrix is a discrete representation of a continuous distribution of mass. A consistent element mass matrix is defined by Eqs. 13.2-5—that is, by $[m] = \int \rho[N]^{T}[N] \, dV$ . It is termed “consistent” because $[N]$ represents the same shape functions as are used to generate the element stiffness matrix [13.4]. A simpler and historically earlier formulation is the lumped mass matrix, which is obtained by placing particle masses $m_{i}$ at nodes i of an element, such that $\Sigma m_{i}$ is the total element mass. Particle “lumps” have no rotary inertia unless rotary inertia is arbitrarily assigned, as is sometimes done for the rotational d.o.f. of beams and plates. A lumped mass matrix is diagonal but a consistent mass matrix is not. The two formulations have different merits, and various considerations enter into deciding which one, or what combination of them, is best suited to a particular analysis procedure.
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+<!-- source-page: 391 -->
+
+Examples. Consider the uniform bar element shown in Fig. 13.3-1a. Shape functions to be used in Eq. 13.2-5 are given by Eq. 3.8-4 with s replaced by x. Mass increment $\rho \, dV$ in Eq. 13.2-5 can be written as $(m/L) \, dx$ , where $m = \rho AL$ is the total mass of the element. The consistent and lumped mass matrices are, respectively,
+
+$$
+[ \mathbf {m} ] = \frac {m}{6} \left[ \begin{array}{l l l l} 2 & 0 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ 1 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 \end{array} \right] \quad [ \mathbf {m} ] = \frac {m}{2} \left[ \begin{array}{l l l l} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \tag {13.3-1}
+$$
+
+The lumped mass is obtained by placing half of the total element mass m as a particle at each node. The particle mass m/2 appears four times in [m] because four nodal acceleration vectors are resisted by inertia.
+
+For the uniform beam element of Fig. 13.3-1b, shape functions are given in Fig. 3.13-2, and Eq. 13.2-5 yields the consistent mass matrix
+
+$$
+[ \mathbf {m} ] = \frac {m}{4 2 0} \left[ \begin{array}{c c c c} 1 5 6 & 2 2 L & 5 4 & - 1 3 L \\ 2 2 L & 4 L ^ {2} & 1 3 L & - 3 L ^ {2} \\ 5 4 & 1 3 L & 1 5 6 & - 2 2 L \\ - 1 3 L & - 3 L ^ {2} & - 2 2 L & 4 L ^ {2} \end{array} \right] \tag {13.3-2}
+$$
+
+where $m = \rho AL$ is the total element mass. The (diagonal) lumped mass matrix of the beam is given by
+
+$$
+[ \mathbf {m} ] = \frac {m}{2} \left[ \begin{array}{l l l l} 1 & \alpha L ^ {2} / 2 1 0 & 1 & \alpha L ^ {2} / 2 1 0 \end{array} \right] \tag {13.3-3}
+$$
+
+where the second and fourth diagonal terms account for rotary inertia. Sometimes rotary inertia is neglected ( $\alpha = 0$ ). If included, it is often selected as the mass moment of inertia I of a uniform slender bar of length L/2 and mass m/2 = $\rho AL/2$ spinning about one end; that is, $I = (m/2)(L/2)^{2}/3$ , for which $\alpha = 17.5$ . If an element is tapered or has nonuniform density, its mass and stiffness matrices change.
+
+A plane frame element has three d.o.f. per node. Its mass matrix is formed by expanding and then combining the bar and beam mass matrices. An element arbitrarily oriented in xy coordinates requires the coordinate transformation described in Section 7.4.
+
+Further examples appear in Fig. 13.3-2.
+
+Equations 13.3-1 and 13.3-3 illustrate ad hoc mass matrix lumping that is guided by intuition and physical insight. The lumped mass matrices obtained are effective
+
+![](images/page-391_e820a2282e5805a10f5715e4948cbf34af705227d8e7c6d797d07932473030ba.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L
+2
+w₁
+u₁
+1
+x
+L
+w₂
+u₂
+</details>
+
+![](images/page-391_be3807959585a0de312e3aabf6207da4765fd9dd271d1de0f0ba4800a9f431c6.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L
+2
+w₁
+w₂
+θ₂
+1
+2
+θ₁
+x
+L
+</details>
+
+Figure 13.3-1. Uniform bar and beam elements with the respective $\{d\}$ vectors $\left[u_{1} \quad w_{1} \quad u_{2} \quad w_{2}\right]^{T}$ and $\left[w_{1} \quad \theta_{1} \quad w_{2} \quad \theta_{2}\right]^{T}$ . Dashed lines show the conceptual shapes and tributary lengths used to obtain the lumped-mass coefficients associated with d.o.f. at node 1.
+
+<!-- source-page: 392 -->
+
+Constant-strain triangle. With a linear displacement field in each direction, and $\{d\} = \left[u_{1} \quad u_{2} \quad u_{3} \quad v_{1} \quad v_{2} \quad v_{3} \quad w_{1} \quad w_{2} \quad w_{3}\right]^{T}$ ,
+
+![](images/page-392_5143484f0f255c9b2f3133e2ba487c95a030f2b181856726be2b3e51306af57e.jpg)
+
+$$
+[ \mathbf {m} ] _ {9 \times 9} = [ \mathbf {Q} \quad \mathbf {Q} \quad \mathbf {Q} ], \quad \text { where } \quad [ \mathbf {Q} ] = \frac {\rho A t}{1 2} \left[ \begin{array}{l l l} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{array} \right]
+$$
+
+Bilinear rectangle. With a bilinear displacement field in each direction, and $\{d\} = \left[u_{1} \quad u_{2} \quad u_{3} \quad u_{4} \quad v_{1} \quad v_{2} \quad v_{3} \quad v_{4} \quad w_{1} \quad w_{2} \quad w_{3} \quad w_{4}\right]^{T}$ ,
+
+$$
+\left[ \begin{array}{l l} 4 & 3 \\ 1 & 2 \end{array} \right] \quad [ \mathrm{m} ] _ {1 2 \times 1 2} = [ \mathrm{Q} \quad \mathrm{Q} \quad \mathrm{Q} ], \quad \text { where } \quad [ \mathrm{Q} ] = \frac {\rho A t}{3 6} \left[ \begin{array}{l l l l} 4 & 2 & 1 & 2 \\ 2 & 4 & 2 & 1 \\ 1 & 2 & 4 & 2 \\ 2 & 1 & 2 & 4 \end{array} \right]
+$$
+
+Figure 13.3-2. Consistent mass matrices for plane elements allowed to move in three dimensions. A = surface area, $\rho$ = uniform mass density, and t = uniform thickness.
+
+and widely used. However, for higher-order elements (e.g., quadratic-displacement plane elements) or elements of irregular shape, intuition can be risky. Accordingly, systematic schemes for lumping are necessary.
+
+HRZ Lumping Scheme. The HRZ scheme $[13.5,13.6]$ is an effective method for producing a diagonal mass matrix. It can be recommended for arbitrary elements. The idea is to use only the diagonal terms of the consistent mass matrix, but to scale them in such a way that the total mass of the element is preserved. Specifically, the procedural steps are as follows.
+
+1. Compute only the diagonal coefficients of the consistent mass matrix.   
+2. Compute the total mass of the element, m.   
+3. Compute a number $s$ by adding the diagonal coefficients $m_{ii}$ associated with translational d.o.f. (but not rotational d.o.f., if any) that are mutually parallel and in the same direction.   
+4. Scale all the diagonal coefficients by multiplying them by the ratio $m / s$ , thus preserving the total mass of the element.
+
+As examples, for the bar and beam elements shown in Fig. 13.3-1, the preceding four steps yield, respectively, the diagonal matrices
+
+$$
+[ \mathbf {m} ] = \frac {m}{2} \left[ \begin{array}{l l l l} 1 & 1 & 1 & 1 \end{array} \right] \tag {13.3-4}
+$$
+
+$$
+[ \mathbf {m} ] = \frac {m}{7 8} \left[ \begin{array}{l l l l} 3 9 & L ^ {2} & 3 9 & L ^ {2} \end{array} \right] \tag {13.3-5}
+$$
+
+Further examples appear in Fig. 13.3-3.
+
+Test cases to date show that for flexural and low-order finite elements, the
+
+<!-- source-page: 393 -->
+
+![](images/page-393_3b0992b1e4244e63a76e2d41d1edd630db9a03ce722fce1af53ab05830dc15b1.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1/36 (3/76)
+8/36 (16/76)
+Serendipity element
+(a)
+</details>
+
+![](images/page-393_35b6b68cf94945772251ac6e98d1196f6d86d8cf0354e369e76a7e1bf69f587c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1/36 (1/36)
+• 16/36 (16/36)
+4/36 (4/36)
+Lagrange element
+(b)
+</details>
+
+Figure 13.3-3. Diagonal mass matrices of plane rectangular elements obtained using the four-step HRZ scheme [13.5]. Each element has constant thickness. Numbers shown are fractions of the total element mass at each node. First number: based on 2 by 2 Gauss quadrature. Second number (in parentheses): based on 3 by 3 Gauss quadrature. (a) Serendipity element. (b) Lagrange element.
+
+accuracy of this form of diagonal mass matrix is excellent, often surpassing that of the consistent mass matrix (see Table 13.3-1). For higher-order plane elements, such as the six-node quadratic triangle and the eight-node serendipity quadrilateral, this scheme may be less accurate for transient analysis than an optimally lumped mass matrix $[13.7,13.8]$ .
+
+Optimal Lumping. Mass lumping can be thought of as the result of applying an appropriate quadrature rule to evaluate $\int\rho[N]^{T}[N]dV$ . If the integration points of a quadrature rule coincide with nodal locations of an element having translational d.o.f. only, then no off-diagonal terms are generated and the mass matrix is diagonal. If the element also has rotational d.o.f., then lumping by quadrature produces block-diagonal matrices that are of lesser practical usefulness because they are not diagonal. In the sequel, we shall be concerned with lumping by quadrature for elements with translational d.o.f. only.
+
+Let $p$ be the degree of the highest-degree complete polynomial in [N] and $m$ the highest-order derivative in the strain energy expression (e.g., $m = 1$ for elasticity and $m = 2$ for bending). Fix [13.9] has shown that a quadrature rule
+
+TABLE 13.3-1. PERCENTAGE ERRORS OF COMPUTED NATURAL FREQUENCIES OF A SIMPLY SUPPORTED THICK SQUARE PLATE [13.5]. HALF THE PLATE WAS MODELED BY A 4 BY 2 MESH OF EIGHT-NODE 24-D.O.F. ELEMENTS (TABLE 11.3-1). THE AD HOC LUMPED MASS MATRIX HAS EQUAL MASS PARTICLES AT EACH NODE. CONSISTENT MASS MATRIX RESULTS DO NOT GUARANTEE AN UPPER BOUND BECAUSE THE STIFFNESS MATRIX IS BASED ON REDUCED INTEGRATION. 
+
+<table><tr><td colspan="2">Mode</td><td colspan="3">Type of Mass Matrix Used</td></tr><tr><td>m</td><td>n</td><td>Consistent (%)</td><td>HRZ Lumping (%)</td><td>Ad Hoc Lumping (%)</td></tr><tr><td>1</td><td>1</td><td>-0.11</td><td>+0.32</td><td>+0.32</td></tr><tr><td>2</td><td>1</td><td>-0.40</td><td>+0.45</td><td>-0.45</td></tr><tr><td>2</td><td>2</td><td>-0.35</td><td>-2.75</td><td>-4.12</td></tr><tr><td>3</td><td>1</td><td>+5.18</td><td>+0.05</td><td>-5.75</td></tr><tr><td>3</td><td>2</td><td>+4.68</td><td>-2.96</td><td>-10.15</td></tr><tr><td>3</td><td>3</td><td>+13.78</td><td>-5.18</td><td>-19.42</td></tr><tr><td>4</td><td>2</td><td>+16.88</td><td>+1.53</td><td>+31.70</td></tr></table>
+
+<!-- source-page: 394 -->
+
+with degree of precision at least $2(p - m)$ will yield comparable accuracy and cause no loss of convergence rate beyond what is already inherent in the finite element method with consistent mass matrices [13.68]. Diagonal mass matrices integrated in this way are said to be optimally lumped.
+
+As an example of optimal lumping, consider the three-node quadratic-displacement bar element shown in Fig. 6.2-1, for which p = 2 and m = 1. Let the mass density $\rho$ and cross-sectional area A be uniform and the nodes uniformly spaced. The minimum order of integration for optimal mass matrix lumping is $n = 2(2 - 1) = 2$ . An integration rule having at least this degree of precision and having integration points at nodal locations is the three-point Newton–Cotes formula [13.12], which is identical to Simpson's rule. (This integration rule will exactly integrate a cubic polynomial.)
+
+$$
+\int_ {a} ^ {b} f (x) d x = (b - a) \left[ \frac {1}{6} f (x = a) + \frac {1}{6} f (x = (a + b) / 2) + \frac {1}{6} f (x = b) \right] \tag {13.3-6}
+$$
+
+From Eq. 13.2-5, a single term of the mass matrix is
+
+$$
+m _ {i j} = \rho A \int N _ {i} N _ {j} d x = \rho A \int_ {- 1} ^ {1} N _ {i} N _ {j} J d \xi \tag {13.3-7}
+$$
+
+The Jacobian $J$ is $J = L / 2$ , so Eq. 13.3-6 yields
+
+$$
+\begin{array}{l} m _ {i j} = \rho A L [ \frac {1}{6} N _ {i} (\xi = - 1) N _ {j} (\xi = - 1) \\ + \frac {4}{6} N _ {i} (\xi = 0) N _ {j} (\xi = 0) + \frac {1}{6} N _ {i} (\xi = 1) N _ {j} (\xi = 1) ] \tag {13.3-8} \\ \end{array}
+$$
+
+Thus $m_{ij} = 0$ for $i \neq j$ , and, with node 3 at the middle of the bar,
+
+$$
+[ \mathbf {m} ] = \frac {\rho A L}{6} \left[ \begin{array}{l l l} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array} \right] \tag {13.3-9}
+$$
+
+Equation 13.3-9 also agrees with the mass matrix obtained by HRZ lumping. In addition, the portion of element mass allocated to each d.o.f. is the same as the nodal allocation of a uniform traction load on the edge of a quadratic element as shown by Fig. 4.3-4.
+
+Nodes of Lagrangian elements coincide with integration points of the Lobatto quadrature rule [13.10]. Lobatto integration weights are always positive, hence optimal lumping for Lagrangian elements results in positive definite diagonal mass matrices; that is, each node in the element has a positive mass lump associated with it. Results for the quadratic Lagrange element are shown in Fig. 13.3-4. Additional results appear in [13.7]. In cubic and higher-order Lagrange elements, nodal masses are positive but nodes in the reference element must be at special positions. In general it is not possible to construct an integration rule of a required accuracy that simultaneously permits arbitrary specification of integration point locations and has positive weighting. Thus optimally lumped diagonal mass matrices for triangular and serendipity quadrilateral elements (particularly the quadratic and higher-order elements) frequently have some zero or negative nodal masses as shown in Fig. 13.3-4.
+
+For low-order elements, such as the linear-displacement bar, the constant-strain
+
+<!-- source-page: 395 -->
+
+![](images/page-395_c5d5320a8df0e1154b61d122ee0821ed153b1d9771757be62fa0c322a346069a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+0 A/3
+-A/12 A/3
+4A/9
+A/36 A/9
+</details>
+
+Figure 13.3-4. Optimal mass matrix lumping for some common two-dimensional elements [13.7]. The triangular element is an equilateral reference element in area coordinates. Rectangular elements are isoparametric reference elements. Elements are uniform and have mass proportional to element area A.
+
+triangle, and the bilinear quadrilateral, ad hoc lumping usually gives the same result as optimal lumping. Also, for the quadratic Lagrange element, HRZ lumping and optimal lumping produce the same diagonal mass matrix. For cubic and higher-order Lagrange elements, HRZ lumping and optimal lumping may be different; however, these elements are rarely used in dynamics. For quadratic and higher-order triangular and serendipity quadrilateral elements, HRZ lumping and optimal lumping are markedly different: numerical tests show that the displacement, velocity, and acceleration time-history results for HRZ lumped models can be less accurate than for optimally lumped models in some problems $[13.7, 13.8]$ .
+
+Remarks. With any mass matrix, the product $[m]\{\ddot{d}\}$ or $[m]\{\ddot{d}\}$ must yield the correct total force on an element according to Newton's law F = ma when $\{\ddot{d}\}$ represents a rigid-body translational acceleration. The rationale is that for convergence to correct results, this kind of motion must be correctly represented because it is the only motion experienced by an element when a mesh is indefinitely refined.
+
+Consistent mass matrices [m] and [M] are positive definite (i.e., the kinetic energy $\frac{1}{2}\{\dot{d}\}^{T}[m]\{\dot{d}\}>0$ for all $\{\dot{d}\}\neq\{\mathbf{0}\}$ ). A lumped mass matrix is positive semi-definite or indefinite if zero or negative masses, respectively, appear on the diagonal. The zeros may or may not make some operations awkward, depending on the algorithm, and negative masses usually (but not always [13.7]) require some special treatment.
+
+If the mesh layout correctly represents the structure volume, elements are compatible and not softened by low-order integration rules, and mass matrices are consistent, then computed natural frequencies are upper bounds to the exact values. If any of these restrictions is violated, such as by the use of any lumping scheme, a bound cannot be guaranteed [13.11]. The upper bound is computed with an error of order $h^{2(q-1)}$ , where h and q are defined in Section 18.6 [13.13]. The upper-bound property is illustrated by the numerical example in Section 13.5.
+
+We cannot say that either lumped or consistent mass matrices are best for all problems. Consistent matrices are more accurate for flexural problems, such as beams and shells, but negligibly so if the wavelength of the mode spans more than about four elements [13.14]. Lumped matrices usually yield natural frequencies that are less than the exact values. For a bar element, MacNeal [13.15] finds that [m] and [m] of Eq. 13.3-1 yield natural frequency errors of order $h^{2}$ in opposite directions and that an [m] that is the average of the two yields a natural frequency error of only order $h^{4}$ . Similar improvement is possible with beam elements [13.46].
+
+<!-- source-page: 396 -->
+
+In wave propagation problems using linear-displacement field elements, lumped masses give greater accuracy because of fewer spurious oscillations. For higher-order elements, diagonal mass matrices obtained by ad hoc lumping may be deficient in accuracy compared to optimal lumping.
+
+As for efficiency, lumped mass matrices are simpler to form, occupy less storage space, and require less computational effort. Indeed, some methods of dynamic analysis are practicable only with lumped mass matrices. Usually it is more important to have a diagonal mass matrix in time-history analysis than in vibration analysis, as time-history analysis is usually much more expensive.
+
+A practitioner contemplating the use of lumped matrices for higher-order elements having translational d.o.f. only has basically four choices: (1) use lower-order elements instead; (2) use Lagrangian elements with optimal lumping, which always results in positive nodal masses; (3) use non-Lagrangian elements with optimal lumping, which may result in a nonpositive definite mass matrix with diagonal coefficients that can be positive, zero, or negative; or (4) use ad hoc lumping which sacrifices optimal accuracy but maintains positive nodal masses. While option 3 is an accurate and workable computational alternative, the most expedient analyses will result from options 1 and 2. Option 4 may give inaccurate results and should be avoided.
+
+# 13.4 DAMPING
+
+Damping in structures is not viscous; rather, it is due to mechanisms such as hysteresis in the material and slip in connections. These mechanisms are not well understood. Moreover, they are awkward to incorporate into the equations of structural dynamics, or they make the equations computationally difficult. Therefore, the actual damping mechanism is usually approximated by viscous damping. Comparisons of theory and experiment show that this approach is sufficiently accurate in most cases.
+
+The treatment of damping in computational analyses can be categorized as (1) phenomenological damping methods, in which the actual physical dissipative mechanisms such as elastic–plastic hysteresis loss, structural joint friction, or material microcracking are modeled, or (2) spectral damping methods, in which viscous damping is introduced by means of specified fractions of critical damping $[13.16]$ . (Critical damping, for which the damping ratio is $\xi = 1$ , marks the transition between oscillatory and nonoscillatory response.) Phenomenological methods require detailed models for the dissipative mechanisms and almost always result in nonlinear analyses; hence, they are seldom used. With spectral damping approaches, experimental observations of the vibratory response of structures are used to assign a fraction of critical damping as a function of frequency, or more commonly, a single damping fraction for the entire frequency range of a structure $[13.16]$ . The damping ratio $\xi$ depends on the material and the stress level. In steel piping, $\xi$ ranges from about 0.5% at low stress levels to about 5% at high stress levels. In bolted or riveted steel structures, and in reinforced or prestressed concrete, $\xi$ has the approximate range 2% to 15%.
+
+A popular spectral damping scheme, called Rayleigh or proportional damping, is to form damping matrix [C] as a linear combination of the stiffness and mass matrices, that is,
+
+$$
+[ \mathbf {C} ] = \alpha [ \mathbf {K} ] + \beta [ \mathbf {M} ] \tag {13.4-1}
+$$
+
+<!-- source-page: 397 -->
+
+where $\alpha$ and $\beta$ are called, respectively, the stiffness and mass proportional damping constants. Matrix [C] given by Eq. 13.4-1 is an orthogonal damping matrix because it permits modes to be uncoupled by eigenvectors associated with the undamped eigenproblem (Section 13.6). The relationship between $\alpha$ , $\beta$ , and the fraction of critical damping $\xi$ at frequency $\omega$ is given by the following equation, proof of which is left as an exercise:
+
+$$
+\xi = \frac {1}{2} \left(\alpha \omega + \frac {\beta}{\omega}\right) \tag {13.4-2}
+$$
+
+Damping constants $\alpha$ and $\beta$ are determined by choosing the fractions of critical damping ( $\xi_{1}$ and $\xi_{2}$ ) at two different frequencies ( $\omega_{1}$ and $\omega_{2}$ ) and solving simultaneous equations for $\alpha$ and $\beta$ . Thus
+
+$$
+\alpha = 2 \left(\xi_ {2} \omega_ {2} - \xi_ {1} \omega_ {1}\right) / \left(\omega_ {2} ^ {2} - \omega_ {1} ^ {2}\right) \tag {13.4-3a}
+$$
+
+$$
+\beta = 2 \omega_ {1} \omega_ {2} (\xi_ {1} \omega_ {2} - \xi_ {2} \omega_ {1}) / (\omega_ {2} ^ {2} - \omega_ {1} ^ {2}) \tag {13.4-3b}
+$$
+
+Shown in Fig. 13.4-1 is the fraction of critical damping versus frequency. Damping attributable to $\alpha[K]$ increases with increasing frequency, whereas damping attributable to $\beta[M]$ increases with decreasing frequency. For structures that may have rigid-body motion, it is important that the mass-proportional damping not be excessive. Positive values of $\beta$ less than about 0.1 per time unit are usually acceptable [13.17].
+
+Usually, $\omega_{1}$ and $\omega_{2}$ are chosen to bound the design spectrum. Thus $\omega_{1}$ is taken as the lowest natural frequency of the structure, and $\omega_{2}$ is the maximum frequency of interest in the loading or response. For example, in seismic analyses, 30 Hz is often used as the upper frequency because the spectral content of seismic design spectra are insignificant above that frequency.
+
+More general proportional damping schemes are possible in which the fraction
+
+![](images/page-397_d262abe22f891e6ddcb55608f8e5b609063299e4410e8e7c15bd88c019bc27bd.jpg)
+
+<details>
+<summary>line</summary>
+| Frequency | Stiffness-proportional damping: ξ = αω/2, β = 0 | Mass-proportional damping: ξ = β/2ω, α = 0 |
+| --------- | ----------------------------------------------- | ------------------------------------------ |
+| ω₁        | ξ₁                                            | ξ₂                                         |
+| ω₂        | ξ₂                                            | ξ₂                                         |
+</details>
+
+Figure 13.4-1. Fraction of critical damping versus frequency for Rayleigh damping. Contribution of stiffness and mass proportional damping to total damping is also shown.
+
+<!-- source-page: 398 -->
+
+of critical damping at any desired number of frequencies can be specified [13.18]. However, these schemes usually produce fully populated damping matrices, and hence are rarely used in practice.
+
+# 13.5 NATURAL FREQUENCIES AND MODE SHAPES
+
+The Eigenvalue Problem. An undamped structure, with no external loads applied to unrestrained d.o.f., undergoes harmonic motion (caused perhaps by initial conditions) in which each d.o.f. moves in phase with all other d.o.f. Thus
+
+$$
+\{\mathbf {D} \} = \{\overline {{\mathbf {D}}} \} \sin \omega t \quad \text { and } \quad \{\ddot {\mathbf {D}} \} = - \omega^ {2} \{\overline {{\mathbf {D}}} \} \sin \omega t \tag {13.5-1}
+$$
+
+where $\{\overline{D}\}=$ amplitudes of nodal d.o.f. vibration and $\omega=circular\ frequency$ (radians per second). The cyclic frequency (in Hertz) is $f=\omega/2\pi$ and the period is T=1/f (seconds). Both $\omega$ and f are called simply “frequency” and pertain to undamped motion unless stated otherwise.
+
+Combining Eqs. 13.5-1 with Eq. 13.2-12, and with [C] and $\{\mathbf{R}^{\mathrm{ext}}\}$ both zero, we obtain
+
+$$
+([ \mathbf {K} ] - \lambda [ \mathbf {M} ]) \{\overline {{\mathbf {D}}} \} = \{\mathbf {0} \}, \quad \text { where } \quad \lambda = \omega^ {2} \tag {13.5-2}
+$$
+
+This is the basic statement of the vibration problem. Equation 13.5-2 is called a generalized eigenproblem or simply an eigenproblem. When the matrix $[K] - \lambda[M]$ is nonsingular, Eq. 13.5-2 has only the trivial solution $\{\overline{D}\} = \{0\}$ . We are interested in nontrivial solutions and hence wish to determine the eigenvalues (or characteristic numbers, or latent roots) $\lambda$ that satisfy
+
+$$
+\det ([ \mathbf {K} ] - \lambda [ \mathbf {M} ]) = 0 \tag {13.5-3}
+$$
+
+Associated with each eigenvalue $\lambda_{i}$ is an eigenvector $\{\overline{D}\}_{i}$ , which is sometimes called a normal (or natural, or characteristic, or principal) mode. The lowest nonzero $\omega_{i}$ is called the fundamental vibration frequency. Appendix C lists some important properties of eigenvalues and eigenvectors such as orthogonality and linear independence.
+
+If [K] and [M] are $n_{\mathrm{eq}}$ by $n_{\mathrm{eq}}$ matrices, then, under conditions usually satisfied in structural analysis, Eq. 13.5-2 has $n_{\mathrm{eq}}$ eigenvalues and $n_{\mathrm{eq}}$ eigenvectors (see Appendix C). All eigenvalues are positive if [K] and [M] are both positive definite, as is usually the case when [K] has all rigid-body modes constrained and when [M] is either a consistent mass matrix or a lumped mass matrix with strictly positive diagonal coefficients. A partly or completely unsupported structure has positive semidefinite [K] and has one zero eigenvalue associated with each possible rigid-body motion. When [M] is lumped—that is, a diagonal matrix—some of its coefficients $M_{ii}$ may be zero (which is commonly the case if rotational d.o.f. are present in $\{\overline{\mathbf{D}}\}$ but not associated with rotary inertia) or negative (which is a rare situation that is due to optimal lumping). Typically each zero $M_{ii}$ is associated with an infinite eigenvalue, and each negative $M_{ii}$ is associated with a negative eigenvalue, which gives rise to an imaginary frequency. Some algorithms for eigenvalue extraction require positive definite mass matrices and hence will not
+
+<!-- source-page: 399 -->
+
+work for lumped matrices having some zero or negative diagonal coefficients. When there are negative diagonal coefficients, the reader is referred to algorithms described in Appendix C and in [13.7]. When a diagonal coefficient $M_{ii}$ is zero, the associated d.o.f. can be removed from $\{\overline{D}\}$ by condensation prior to eigenanalysis.
+
+Rayleigh Quotient. Let [K] be symmetric and [M] be positive definite and symmetric. If we premultiply Eq. 13.5-2 by $\{\overline{D}\}^{T}$ and solve for $\lambda$ , we obtain the Rayleigh quotient
+
+$$
+\lambda = \frac {\{\overline {{{\mathbf {D}}}} \} ^ {T} [ \mathbf {K} ] \{\overline {{{\mathbf {D}}}} \}}{\{\overline {{{\mathbf {D}}}} \} ^ {T} [ \mathbf {M} ] \{\overline {{{\mathbf {D}}}} \}} \tag {13.5-4}
+$$
+
+If $\{\overline{D}\}$ approximates the ith eigenvector with first-order error, then $\lambda$ approximates the corresponding eigenvalue with second-order error. Thus a casual estimate of $\{\overline{D}\}$ may result in an accurate estimate of $\lambda$ . The Rayleigh quotient is, in fact, an extreme value when $\{\overline{D}\}$ varies in the neighborhood of an exact eigenvector [13.18,13.19]; accordingly, the extraction of an eigenvalue can be approached as an optimization problem. Values of the Rayleigh quotient are bounded by the largest and smallest eigenvalues of the mesh. That is, for any vector $\{v\}$ ,
+
+$$
+\lambda_ {\min} \leq \frac {\{\mathbf {v} \} ^ {T} [ \mathbf {K} ] \{\mathbf {v} \}}{\{\mathbf {v} \} ^ {T} [ \mathbf {M} ] \{\mathbf {v} \}} \leq \lambda_ {\max} \tag {13.5-5}
+$$
+
+where $\lambda_{min}$ and $\lambda_{max}$ are the smallest and largest eigenvalues of Eq. 13.5-2. Equation 13.5-5 presumes that [M] is positive definite. If [M] is indefinite, as may result from optimal lumping or lumping with zero rotary inertia, the Rayleigh quotient may be positive or negative infinity for some choices of $\{v\}$ .
+
+In certain direct integration algorithms, it is necessary to have prior knowledge of the largest eigenvalue of the mesh, $\lambda_{max}$ . An upper bound to $\lambda_{max}$ can be obtained by considering the eigenproblem for a single, unsupported element
+
+$$
+\det ([ \mathbf {k} ] - \lambda^ {\prime} [ \mathbf {m} ]) = 0 \tag {13.5-6}
+$$
+
+The largest eigenvalue of Eq. 13.5-6 can often be obtained by hand calculation. If we denote the largest eigenvalue of Eq. 13.5-6 among all elements by $\lambda_{\mathrm{max}}^{\prime}$ , then
+
+$$
+\lambda_ {\max} \leq \lambda_ {\max} ^ {\prime} \tag {13.5-7}
+$$
+
+The reason is that the largest eigenvalue is a maximum of the Rayleigh quotient, Eq. 13.5-5. Suppose we consider an alternative Rayleigh quotient instead, in which no constraints of nodal-value sharing are imposed. Then the resulting maximum eigenvalue would be $\lambda_{max}^{\prime}$ of Eq. 13.5-6. The actual $\lambda_{max}$ could be obtained from the alternative Rayleigh quotient by imposing a large number of linear constraints that specify the nodal-value sharing of the original mesh. Optimization theory shows that imposing such constraints can only lower a maximum, and thus $\lambda_{max} \leq \lambda_{max}^{\prime}$ . Applying the same argument to $\lambda_{min}$ shows that $\lambda_{min} \geq \lambda_{min}^{\prime}$ , which is consistent with our physical intuition that imposing constraints raises the fundamental frequency.
+
+<!-- source-page: 400 -->
+
+Example. We consider an elementary example of a matrix eigenproblem and its solution. Consider the uniform one-dimensional unsupported bar with mass density $\rho$ , elastic modulus E, and cross-sectional area A shown in Fig. 13.5-1. With consistent [m], Eq. 13.5-2 becomes
+
+$$
+\left(\frac {A E}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right] - \omega^ {2} \frac {\rho A L}{6} \left[ \begin{array}{c c} 2 & 1 \\ 1 & 2 \end{array} \right]\right) \left\{ \begin{array}{l} \overline {{u}} _ {1} \\ \overline {{u}} _ {2} \end{array} \right\} = \left\{ \begin{array}{l} 0 \\ 0 \end{array} \right\} \tag {13.5-8}
+$$
+
+For nontrivial amplitudes $\{\overline{d}\} = \left[\overline{u}_{1}, \overline{u}_{2}\right]^{T}$ to exist, the determinant of the expression in parentheses must vanish. Thus
+
+$$
+\omega^ {2} (\omega^ {2} \rho A L - 1 2 A E / L) = 0 \tag {13.5-9}
+$$
+
+from which $\omega_{1}=0$ and $\omega_{2}=(2/L)\sqrt{3E/\rho}=(3.464/L)\sqrt{E/\rho}$ . The easiest way to determine the eigenvector associated with any eigenvalue $\lambda_{i}$ is to set one d.o.f. in the eigenvector $\{\overline{d}\}_{i}$ to an arbitrary nonzero number $(\overline{d}_{1}=1$ , for example), substitute the known value of $\lambda_{i}$ , and solve for the remaining amplitudes in $\{\overline{d}\}_{i}$ . (If, by coincidence, the d.o.f. amplitude that was assumed to be nonzero is in fact zero, then the resulting system of equations will be singular and no solution will exist for the remaining amplitudes. Then the eigenvector can usually be found by assuming a nonzero value for one of the other amplitudes.) Thus, from Eq. 13.5-8,
+
+$$
+\text { for   } \omega_ {1} = 0 \quad \{\overline {{\mathbf {d}}} \} _ {1} = \left\lfloor 1 - 1 \right\rfloor^ {T} \tag {13.5-10a}
+$$
+
+$$
+\text { for } \omega_ {2} = (3. 4 6 4 / L) \sqrt {E / \rho} \quad \{\overline {{{\mathbf {d}}}} \} _ {2} = \left\lfloor 1 - 1 \right\rfloor^ {T} \tag {13.5-10b}
+$$
+
+The first eigenvector describes a rigid-body translation in the x direction. The second describes an axial straining mode (for which the exact fundamental frequency of a continuous unsupported bar of length L is $(\pi/L)\sqrt{E/\rho}$ ). Hence, the single-element consistent-mass model overpredicts the exact fundamental frequency by about 10%, thus illustrating the upper-bound property noted in Section 13.3.
+
+If, instead of the consistent mass matrix, the lumped mass matrix $[m] = (\rho AL/2)[1 \_ 1]$ is used in Eq. 13.5-4, the computed frequencies are $\omega_{1} = 0$ and $\omega_{2} = (2/L)\sqrt{E/\rho}$ . Thus we see that the upper-bound property is destroyed by lumping. Eigenvectors for the lumped-mass case are the same as those of Eqs. 13.5-10.
+
+In many design situations, we wish to know if severe dynamic excitation of a structure is likely. Therefore we compare the frequency spectrum of the structure with that of the time-dependent loading. If a natural frequency of the structure is
+
+![](images/page-400_86afe26a77ad26248b61f47248f0f1564511c5a354c21f50002f7b53556cfbd4.jpg)
+
+<details>
+<summary>text_image</summary>
+
+u₁
+1
+x
+L
+(a)
+2
+u₂
+</details>
+
+![](images/page-400_5640c49673f8627fa4f723813e2824b5c9d480fa4fcccf00a0b9c4363b5f3213.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Exact and one element
+1.0
+ω₁ = 0
+Exact
+0
+L
+x
+One element
+ω₂ > 0
+(b)
+</details>
+
+Figure 13.5-1. (a) Unsupported two-d.o.f. uniform bar. (b) Vibration modes $\omega_{1}=0$ (rigid-body translation) and $\omega_{2}>0$ (axial straining mode).
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_041.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_041.md
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index 00000000..164dc919
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_041.md
@@ -0,0 +1,336 @@
+<!-- source-page: 401 -->
+
+close to an excitation frequency, then severe vibration and “beating” are likely. This usually necessitates alteration of the structure’s natural frequencies by resizing or by adding members or dampers. If frequencies of the structure and the excitation are well separated, the structure still vibrates, but the amplitude of the response is likely to be tolerable.
+
+In static analysis, symmetry can be exploited, for example, by analyzing half of the entire structure. In vibration analysis, symmetry of structure and supports does not imply symmetry of all vibration modes. By imposing symmetry one would exclude all antisymmetric modes, which are probably as important as symmetric modes.
+
+One should be aware that stabilization methods used to suppress mechanisms in underintegrated elements may be associated with relatively low stiffness. The associated nonphysical vibration modes may contaminate that portion of the vibration spectrum of greatest interest $[13.70]$ .
+
+# 13.6 TIME-HISTORY ANALYSIS. MODAL METHODS
+
+In a time-history or dynamic response problem, we solve Eq. 13.2-12 for $\{D\}$ , $\{\dot{D}\}$ , and $\{\ddot{D}\}$ as functions of time. When $[M]$ , $[C]$ , and $[K]$ are known and are time-independent, the problem is linear. When initial values of $\{D\}$ and $\{\dot{D}\}$ are prescribed, Eq. 13.2-12 is called an initial value problem. If material behavior is nonlinear, then the internal force vector $\{R^{int}\}$ replaces $[K]\{D\}$ in Eq. 13.2-12, which is then called a nonlinear initial value problem (Eq. 13.2-13). For reasons that will become clear, in this section we consider only linear problems, for which $\{R^{int}\} = [K]\{D\}$ .
+
+The modal or mode superposition method of analysis transforms Eq. 13.2-12 so that $\{D\}$ and its time derivatives are replaced by $\{Z\}$ and its time derivatives, where $\{Z\}$ is a vector of modal amplitudes, or, in other words, a vector of generalized d.o.f. If damping is orthogonal, as Rayleigh damping is, then the transformed equations are uncoupled and each can be solved independently of all others. Solutions of these modal equations are superposed to yield the solution of the original problem [13.1,13.18,13.20,13.21]. Other solution methods for dynamic response are considered in Section 13.9 et seq.
+
+Modal analysis is effective because of special properties that eigenvectors possess. These properties are orthogonality and linear independence (see Appendix C). Eigenvectors $\{\overline{\mathbf{D}}\}$ in Eq. 13.5-2 are orthogonal with respect to the (symmetric) mass and stiffness matrices. That is,
+
+$$
+\{\overline {{{\mathbf {D}}}} \} _ {i} ^ {T} [ \mathbf {M} ] \{\overline {{{\mathbf {D}}}} \} _ {j} = 0 \quad \text { and } \quad \{\overline {{{\mathbf {D}}}} \} _ {i} ^ {T} [ \mathbf {K} ] \{\overline {{{\mathbf {D}}}} \} _ {j} = 0, \quad \text { where } \quad i \neq j \tag {13.6-1}
+$$
+
+If eigenvectors are normalized with respect to the mass matrix, then the quadratic product of the stiffness matrix yields the undamped natural frequencies. That is, by Eq. 13.5-4 with $\lambda = \omega^{2}$ ,
+
+$$
+\text { if } \quad \{\overline {{\mathbf {D}}} \} _ {i} ^ {T} [ \mathbf {M} ] \{\overline {{\mathbf {D}}} \} _ {i} = 1 \quad \text { then } \quad \{\overline {{\mathbf {D}}} \} _ {i} ^ {T} [ \mathbf {K} ] \{\overline {{\mathbf {D}}} \} _ {i} = \omega_ {i} ^ {2} \tag {13.6-2}
+$$
+
+(A vector is normalized by dividing each of its terms by the same constant.) If $[\phi]$ is the modal matrix, that is, an $n_{eq}$ by $n_{eq}$ matrix whose columns are the normalized eigenvectors, then Eq. 13.6-2 yields
+
+<!-- source-page: 402 -->
+
+$$
+[ \phi ] ^ {T} [ \mathbf {M} ] [ \phi ] = [ \mathbf {I} ] \quad \text { and } \quad [ \phi ] ^ {T} [ \mathbf {K} ] [ \phi ] = [ \omega^ {2} ] \tag {13.6-3}
+$$
+
+where $\left[\omega^{2}\right]$ is a diagonal matrix of the squared natural frequencies. It is called the spectral matrix.
+
+Using the property of linear independence of eigenvectors, we can express any vector $\{\mathbf{D}\}$ as a linear combination of eigenvectors. Thus
+
+$$
+\{\mathbf {D} \} = [ \phi ] \{\mathbf {Z} \} \tag {13.6-4}
+$$
+
+where the $Z_{i}$ in vector $\{Z\}$ state the proportion of each eigenvector in the transformation. Equation 13.6-4 represents a convenient d.o.f. transformation in which the $Z_{i}$ are modal amplitudes and, similar to $\{D\}$ , are functions of time. No approximation is introduced by the transformation if all modes of the original system are retained in $[\phi]$ .
+
+Mode Displacement Method. In the mode displacement method of modal analysis, we premultiply Eq. 13.2-12 by $[\phi]^{T}$ and combine the result with Eq. 13.6-4. Thus, in view of Eqs. 13.6-3, the coefficients of $\{\ddot{Z}\}$ and $\{Z\}$ become the respective diagonal matrices [I] and $[\omega^{2}]$ . If [C] is given by Eq. 13.4-1, or by other orthogonal forms [13.1,13.18], then $[\xi] = [\phi]^{T}[C][\phi]$ is also a diagonal matrix and the transformed equations are completely uncoupled. Using customary notation, we write the coefficients in a diagonal matrix $[\xi]$ as $2\xi_{i}\omega_{i}$ , where $\xi_{i}$ is the fraction of critical damping in mode i. Thus
+
+$$
+\ddot {Z} _ {i} + 2 \xi_ {i} \omega_ {i} \dot {Z} _ {i} + \omega_ {i} ^ {2} Z _ {i} = p _ {i}, \quad \text { where } \quad p _ {i} = \{\boldsymbol {\phi} \} _ {i} ^ {T} \{\mathbf {R} ^ {\text { ext }} \} \tag {13.6-5}
+$$
+
+in which $\{\phi\}_{i}$ is the $i$ th column of $[\phi]$ . There are as many uncoupled ordinary differential equations as there are d.o.f. In each equation, $p_{i}$ is a known function of time. Initial values for Eq. 13.6-5 are obtained from known initial values of $\{\mathbf{D}\}$ and $\{\dot{\mathbf{D}}\}$ as follows. Premultiply both sides of Eq. 13.6-4 by $[\phi]^{T}[\mathbf{M}]$ and take note of Eqs. 13.6-3. Thus
+
+$$
+\{\mathbf {Z} (t = 0) \} = [ \phi ] ^ {T} [ \mathbf {M} ] \{\mathbf {D} (t = 0) \} \quad \text { and } \quad \{\dot {\mathbf {Z}} (t = 0) \} = [ \phi ] ^ {T} [ \mathbf {M} ] \{\dot {\mathbf {D}} (t = 0) \} \tag {13.6-6}
+$$
+
+When $\{\mathbf{Z}\}$ has been determined as a function of time from Eq. 13.6-5, $\{\mathbf{D}\}$ is obtained from Eq. 13.6-4.
+
+There are many ways to time-integrate Eqs. 13.6-5. In fact, Eq. 13.6-5 can be solved exactly for $Z_{i}(t)$ . If $p_{i}$ is piecewise linear in time, then exact solutions can be written for $Z_{i}$ and $\dot{Z}_{i}$ in terms of $e^{-\xi\omega t}\sin\omega t$ and $e^{-\xi\omega t}\cos\omega t$ . However, for more general loading, an exact solution can be tedious and it is more effective to use a direct integration method, as discussed in Section 13.9 et seq.
+
+At first glance, calculating $[\phi]$ in Eq. 13.6-4 seems to be a prohibitive computational expense. But for many problems, the higher-frequency modes participate little in the structural response and therefore only a small number of low-frequency modes need be used. Thus, only the first m equations of Eq. 13.6-5, where typically $m \ll n_{eq}$ , are solved and the transformation Eq. 13.6-4 is approximated by
+
+$$
+\{\mathbf {D} \} \approx \sum_ {i = 1} ^ {m} \{\boldsymbol {\phi} \} _ {i} Z _ {i} \quad \text { where } \quad m <   n _ {\mathrm{eq}} \tag {13.6-7}
+$$
+
+<!-- source-page: 403 -->
+
+in which $\{\phi\}_{i}$ is the ith normalized eigenvector. A measure of the error at time t is denoted by $e(t)$ . It can be quantified [13.18] by
+
+$$
+e (t) \equiv \frac {\left\| \left\{\mathbf {R} ^ {\text {ext}} \right\} - [ \mathbf {M} ] \{\ddot {\mathbf {D}} \} - [ \mathbf {C} ] \{\dot {\mathbf {D}} \} - [ \mathbf {K} ] \{\mathbf {D} \} \right\|}{\left\| \left\{\mathbf {R} ^ {\text {ext}} \right\} \right\|} \tag {13.6-8}
+$$
+
+where $\{D\}$ and its time derivatives are obtained from Eq. 13.6-7 and $\parallel$ denotes any vector norm. It is assumed in writing Eq. 13.6-8 that $\{R^{ext}\} \neq \{0\}$ at the particular instant $e(t)$ is computed. For an accurate analysis, $e(t)$ should be small (1% or less, as a conjecture) for the entire duration of analysis.
+
+In many structural dynamics problems, more modes participate in the quasi-static response than in the dynamic response. The mode displacement method may have difficulty in reproducing quasistatic deflection shapes of structures when m is small. A modification of the mode displacement method, discussed next, removes this deficiency. Additional comments appear in Section 13.14.
+
+Mode Acceleration Method. In the mode acceleration method, we perform modal transformation on only the inertial and viscous terms of Eq. 13.2-12 [13.2, 13.22, 13.23, 13.24]. This yields
+
+$$
+[ \mathbf {M} ] [ \phi ] \{\ddot {\mathbf {Z}} \} + [ \mathbf {C} ] [ \phi ] \{\dot {\mathbf {Z}} \} + [ \mathbf {K} ] \{\mathbf {D} \} = \left\{\mathbf {R} ^ {\text {ext}} \right\} \tag {13.6-9}
+$$
+
+If no rigid-body modes are possible and $[K]$ is properly formed, then $[K]^{-1}$ exists and Eq. 13.6-9 can be solved for $\{D\}$ . From Eq. 13.6-9,
+
+$$
+\{\mathbf {D} \} = [ \mathbf {K} ] ^ {- 1} \left\{\mathbf {R} ^ {\text {ext}} \right\} - [ \mathbf {K} ] ^ {- 1} ([ \mathbf {M} ] [ \phi ] \{\ddot {\mathbf {Z}} \} + [ \mathbf {C} ] [ \phi ] \{\dot {\mathbf {Z}} \}) \tag {13.6-10}
+$$
+
+Equation 13.6-3 yields $[K]^{-1}[\phi]^{-T} = [\phi]\left[\omega^{2}\right]^{-1}$ . Hence, if [C] is orthogonal, Eq. 13.6-10 can be written as
+
+$$
+\{\mathbf {D} \} = [ \mathbf {K} ] ^ {- 1} \left\{\mathbf {R} ^ {\text { ext }} \right\} - [ \phi ] \left[ \omega^ {2} \right] ^ {- 1} \left(\left\{\ddot {\mathbf {Z}} \right\} + [ \xi ] \left\{\dot {\mathbf {Z}} \right\}\right) \tag {13.6-11}
+$$
+
+where $[\xi]$ is a diagonal matrix with ith diagonal coefficient $2\xi_{i}\omega_{i}$ (proof of Eq. 13.6-11 is left as an exercise). In Eq. 13.6-11, [K] is the original $n_{eq}$ by $n_{eq}$ stiffness matrix of Eq. 13.2-12. Here $[\phi]$ is $n_{eq}$ by $n_{eq}$ , $[\omega^{2}]$ and $[\xi]$ are $n_{eq}$ by $n_{eq}$ diagonal matrices, and $\{Z\}$ is an $n_{eq}$ by 1 vector of modal amplitudes. As with mode displacement analysis, only a small number of modes, m, is usually required for accurate results. Hence, the matrix multiplication indicated in the second term of Eq. 13.6-11 is carried out over only the lowest m eigenvectors and Eq. 13.6-11 is approximated by
+
+$$
+\{\mathbf {D} \} \approx [ \mathbf {K} ] ^ {- 1} \left\{\mathbf {R} ^ {\text {ext}} \right\} - \sum_ {i = 1} ^ {m} \left\{\phi \right\} _ {i} \left(\frac {1}{\omega_ {i} ^ {2}} \ddot {Z} _ {i} + \frac {2 \xi_ {i}}{\omega_ {i}} \dot {Z} _ {i}\right), \quad \text {where} \quad m <   n _ {\mathrm{eq}} \tag {13.6-12}
+$$
+
+The first term on the right-hand side of Eq. 13.6-12 represents the quasistatic response, and the second term represents the dynamic correction attributable to inertia and viscous effects. To implement the mode acceleration method, we solve Eq. 13.6-5 for $\dot{Z}_{i}$ and $\ddot{Z}_{i}, i = 1, 2, \ldots, m$ exactly as we would in a mode displacement analysis (note that we do not need $Z_{i}$ ). Then Eq. 13.6-12 is used for superposition (rather than Eq. 13.6-7 as in the mode displacement method) to obtain $\{D\}$ as a function of time.
+
+<!-- source-page: 404 -->
+
+Obviously, the mode acceleration algorithm is adept at reproducing the quasistatic structure deflection shape. If the magnitude of the external load (but not its distribution) varies with time, then $\{R^{ext}\}=s(t)\{S\}$ , where $\{S\}$ is time-independent and describes the distribution of external load, and $s(t)$ is a time-dependent scalar that describes the amplitude of the external load. We call such loading proportional loading. Thus the quasistatic displacement $[K]^{-1}\{S\}$ need only be scaled by $s(t)$ at each instant in time. Usually, fewer modes are required in mode acceleration than in a mode displacement analysis of equivalent accuracy [13.24]. On the other hand, for nonproportional loading the mode acceleration method requires the solution of simultaneous equations at each step of the solution (only forward and back substitution after [K] has been factored). If $\{R^{ext}\}=\{0\}$ , both methods yield identical results for the same m.
+
+If rigid-body modes are possible, [K] is singular and the mode acceleration method cannot be employed in the straightforward manner indicated by Eqs. 13.6-11 and 13.6-12. Discussion of this matter appears in [13.2].
+
+Superposition of Ritz Vectors. A disadvantage of the mode displacement and mode acceleration methods is that computation of eigenvectors is expensive. For large structures it is often the most costly part of a dynamic response analysis. Using Ritz vectors (which may also be called basis vectors), we make a transformation analogous to Eqs. 13.6-4 and 13.6-7. Ritz vectors do not have all of the desirable properties of eigenvectors but are much more economical to compute.
+
+Time-history analysis by superposition of Ritz vectors is a type of Rayleigh–Ritz analysis. A Ritz vector d.o.f. transformation
+
+$$
+\{\mathbf {D} \} = [ \mathbf {W} ] \{\mathbf {y} \} \tag {13.6-13}
+$$
+
+is used where [W] is an $n_{eq}$ by m matrix whose columns are Ritz vectors $\{w\}_{i}$ and $\{y\}$ is a vector of generalized coordinates. Thus [W] = $[w_{1} \quad w_{2} \quad \ldots \quad w_{m}]$ and $\{y\}$ is an m by 1 vector whose terms $y_{i}$ state the proportion of $\{w\}_{i}$ in the transformation. The number m of Ritz vectors is chosen by the analyst. There are many ways to obtain the Ritz vectors in Eq. 13.6-13. Some of these procedures are rather arbitrary. If the Ritz vectors are the lowest m eigenvectors of Eq. 13.5-2 and are normalized according to Eq. 13.6-2, then Eq. 13.6-13 reduces to Eq. 13.6-7 of the mode displacement method. It is not necessary that Ritz vectors be close approximations of eigenvectors. Ideally, however, Ritz vectors are linear combinations of the lowest eigenvectors. A reliable method for generating good Ritz vectors (without solving an eigenproblem) is to determine [W] by solving
+
+$$
+[ \mathbf {K} ] [ \mathbf {W} ] = [ \mathbf {R} ] \tag {13.6-14}
+$$
+
+where $[K]$ is the $n_{eq}$ by $n_{eq}$ stiffness matrix of the complete structure and $[R]$ is an $n_{eq}$ by m matrix whose columns are linearly independent load patterns selected by the analyst to excite important (i.e., lower) displacement modes of the structure. In writing Eq. 13.6-14, we assume that $[K]$ is nonsingular. For treating problems with rigid-body modes, a modified method is required.
+
+The use of Ritz vectors is not limited to problems of dynamics. Ritz vectors constitute a “reduced basis” that serves to reduce the cost of repetitive calculation cycles, which appear in modal methods of dynamics, in nonlinear static problems, and in design optimization.
+
+<!-- source-page: 405 -->
+
+Ritz Vectors in Time-History Analysis. Often, the load patterns used in Eq. 13.6-14 are obtained from actual load patterns applied to the structure. In dynamic analysis, a question that immediately arises is what load patterns are most effectively used in Eq. 13.6-14 when the loads that shake a structure, $\{R^{ext}\}$ , are time-dependent. A method called superposition of Ritz vectors effectively addresses this question [13.25]. It is necessary that the external load be representable as a superposition of proportional loads—that is,
+
+$$
+\{\mathbf {R} ^ {\mathrm{ext}} \} = \sum_ {j = 1} ^ {\ell} s _ {j} (t) \{\mathbf {S} \} _ {j} \tag {13.6-15}
+$$
+
+where $\{S\}_{j}$ describes the distribution of the jth load component, $s_{j}(t)$ is a scalar that describes the time-dependent amplitude of $\{S\}_{j}$ , and $\ell$ is the number of $\{S\}_{j}$ needed for superposition. For many practical problems $\ell$ is small.
+
+The dynamic response $\{\mathbf{D}\}_{j}$ to load component $s_j(t)\{\mathbf{S}\}_{j}$ is computed independently for each $j$ . Then the total structural response is obtained by superposition
+
+$$
+\{\mathbf {D} \} = \sum_ {j = 1} ^ {\ell} \{\mathbf {D} \} _ {j} \tag {13.6-16}
+$$
+
+In the response analysis for each $\{\mathbf{D}\}_{j}$ , a Ritz vector d.o.f. transformation
+
+$$
+\{\mathbf {D} \} _ {j} = [ \mathbf {W} ] _ {j} \{\mathbf {y} \} _ {j} \tag {13.6-17}
+$$
+
+is used where $[W]_{j}$ is an $n_{eq}$ by m matrix of Ritz vectors. An algorithm for generating [M]-orthogonal Ritz vectors, when given a load pattern $\{S\}_{j}$ , is shown in Table 13.6-1. In this algorithm, the first Ritz vector $\{w\}_{1}$ is proportional to the
+
+TABLE 13.6-1. COMPUTATIONAL PROCEDURE FOR GENERATION OF [M]-ORTHOGONAL RITZ VECTORS [13.25]. 
+<table><tr><td colspan="2">1. Obtain  $n_{eq}$  by  $n_{eq}$  mass and stiffness matrices, [M] and [K].</td></tr><tr><td colspan="2">2. Factor the stiffness matrix; for example, [K] = [L][L] $^T$ .</td></tr><tr><td colspan="2">3. For each proportional load component  $\{S\}_{j}, j = 1, 2, \ldots, \ell$ , compute [W] $_j$  = [w $_1$  w $_2$  ... w $_m$ ] as follows:</td></tr><tr><td colspan="2">3.1 Solve for the first Ritz vector,  $\{w\}_{1}$ :</td></tr><tr><td colspan="2"> $[K]\{w^{*}\}_{1} = \{S\}_{j}$  solve for  $\{w^{*}\}_{1}$  $\{w\}_{i}^{T}[M]\{w\}_{1} = 1$  normalize  $\{w^{*}\}_{1}$  to yield  $\{w\}_{1}$ </td></tr><tr><td colspan="2">3.2 Solve for additional Ritz vectors,  $\{w\}_{i}$ ;  $i = 2, 3, \ldots, m$ :</td></tr><tr><td colspan="2"> $[K]\{w^{*}\}_{i} = [M]\{w\}_{i-1}$  solve for  $\{w^{*}\}_{i}$  $\{w^{**}\}_{i} = \{w^{*}\}_{i} - \sum_{k=1}^{i-1} \{w\}_{k}^{T}[M]\{w^{*}\}_{i}\{w\}_{k}$  [M]-orthogonalize  $\{w^{*}\}_{i}$  to yield  $\{w^{**}\}_{i}$  $\{w\}_{i}^{T}[M]\{w\}_{i} = 1$  normalize  $\{w^{**}\}_{i}$  to yield  $\{w\}_{i}$ </td></tr><tr><td colspan="2">3.3 Assemble Ritz vectors  $\{w\}_{i}$  into  $n_{eq}$  by  $m$  matrix [W] $_j$ .</td></tr><tr><td colspan="2">3.4 Next load component;  $j \leftarrow j + 1$ , go to Step 3.1.</td></tr></table>
+
+<!-- source-page: 406 -->
+
+quasistatic deflection shape of the structure under loads $\{\mathbf{S}\}_{j}$ . To obtain the second Ritz vector, we impose $\{\mathbf{w}\}_{1}$ as a nodal acceleration vector; hence, $\{\mathbf{w}\}_{2}$ is proportional to the quasistatic deflection shape for inertial loads $[\mathbf{M}]\{\mathbf{w}\}_{1}$ . To obtain the third Ritz vector, we impose $\{\mathbf{w}\}_{2}$ as a nodal acceleration, and so on. If loads $\{\mathbf{S}\}_{j}$ are zero, as for a structure that moves freely after initial velocities are prescribed, the procedure of Table 13.6-1 must be modified. A possible modification is to simply prescribe $\{\mathbf{w}\}_{1}$ (e.g., as null except for unity corresponding to a d.o.f. expected to have significant displacement), then go to Step 3.2.
+
+Once a mass-matrix-orthogonal $[\mathbf{W}]_j$ is obtained, Eqs. 13.2-12, 13.6-16, and 13.6-17 are combined and premultiplied by $[\mathbf{W}]_j^T$ to yield
+
+$$
+\{\ddot {\mathbf {y}} \} _ {j} + [ \mathbf {C} ] _ {j} \{\dot {\mathbf {y}} \} _ {j} + [ \mathbf {K} ] _ {j} \{\mathbf {y} \} _ {j} = \{\mathbf {P} \} _ {j} \tag {13.6-18}
+$$
+
+where the transformed stiffness matrix, damping matrix, and load vector are
+
+$$
+[ \mathbf {K} ] _ {j} = [ \mathbf {W} ] _ {j} ^ {T} [ \mathbf {K} ] [ \mathbf {W} ] _ {j} \tag {13.6-19a}
+$$
+
+$$
+[ \mathbf {C} ] _ {j} = [ \mathbf {W} ] _ {j} ^ {T} [ \mathbf {C} ] [ \mathbf {W} ] _ {j} \tag {13.6-19b}
+$$
+
+$$
+\{\mathbf {P} \} _ {j} = s _ {j} (t) [ \mathbf {W} ] _ {j} ^ {T} \{\mathbf {S} \} _ {j} \tag {13.6-19c}
+$$
+
+After the $\{\mathbf{y}\}_{j}$ are known (as functions of time) for all $j$ , Eqs. 13.6-16 and 13.6-17 yield $\{\mathbf{D}\} = \{\mathbf{D}(t)\}$ . Note that we must generate a $[\mathbf{W}]_j$ and also solve Eqs. 13.6-18 as many times as there are load components in Eq. 13.6-15.
+
+Matrices $[\mathbf{K}]_j$ and $[\mathbf{C}]_j$ have dimension $m$ by $m$ . Therefore, if $m \ll n_{\mathrm{eq}}$ , Eqs. 13.6-18 represent a much smaller system of simultaneous ordinary differential equations than the original system, Eq. 13.2-12. In general, matrices $[\mathbf{K}]_j$ and $[\mathbf{C}]_j$ are full (but $[\mathbf{M}]_j$ is a unit matrix according to the procedure of Table 13.6-1). Equations 13.6-18 are usually solved for $\{\mathbf{y}\}_j$ and its time derivatives by a direct integration method. Optionally, the Ritz vectors can be made to be also stiffness-matrix-orthogonal using the procedure of [13.25]. This entails greater expense in forming $[\mathbf{W}]_j$ , but then $[\mathbf{K}]_j$ is diagonal (as is $[\mathbf{C}]_j$ if damping is orthogonal). Equations 13.6-18 are then completely uncoupled and can be solved independently, either exactly or approximately.
+
+Several features of Ritz mode superposition are apparent from Table 13.6-1. Each Ritz vector is obtained by solving a system of simultaneous algebraic equations. Each solution is relatively inexpensive once [K] has been factored. As with the mode acceleration method, static structure deflection shapes are accurately modeled. However, since [K] must be nonsingular, the method cannot treat problems with rigid-body modes unless modified. When the number of loads $\ell$ necessary for the superposition in Eq. 13.6-13 becomes large, the method loses its attractiveness since an independent set of Ritz vectors $[\mathbf{W}]_j$ must be determined and an independent system of ordinary differential equations, Eqs. 13.6-18, must be solved for each load component. However, for many problems only a few load components are necessary. The reduction in the size of the original problem can be remarkable. For example, for earthquake shaking of structures, the number of Ritz vectors necessary for accurate response analysis is almost always less than 50, whereas equations in the original system may number several thousand [13.25].
+
+<!-- source-page: 407 -->
+
+Nonlinear Problems. In material-nonlinear problems, [K] and [C] are time-dependent. Usually [M] does not change with time. Modal techniques employ superposition and hence are often assumed to be inapplicable to nonlinear problems. However, nonlinear problems can be accommodated if all nonlinearities are treated as pseudoloads and incorporated with the external load $\{R^{ext}\}$ . This approach requires that modes be superposed at each time step to obtain $\{D\}$ (and if necessary $\{\dot{D}\}$ ) so that the material constitutive law can be evaluated. Pseudoloads are then calculated and transformed back to modal equations, the solution of which is then incremented in time by one step. Strictly speaking, when response is nonlinear $[\phi]$ does not represent normal modes of vibration. Equation 13.6-4 should then be viewed as simply a coordinate transformation [13.26,13.27]. For structural dynamics problems with nonlinearities that are mild and localized in the mesh, mode displacement superposition has been used, sometimes effectively. For severe nonlinearities, convergence of the pseudoload approach is poor. For most nonlinear problems, direct time integration methods are preferable to superposition methods.
+
+# 13.7 MASS CONDENSATION. GUYAN REDUCTION
+
+In static analyses, problems with 10,000 d.o.f. or more are common. In dynamic analyses in which we ask for natural frequencies and mode shapes, problems with only 1000 d.o.f. can be difficult. The major difficulty is the expense of computing eigenvalues and eigenvectors.
+
+However, condensation can be employed to reduce the number of d.o.f., which reduces the expense of computing eigenvalues and eigenvectors and the expense of subsequent calculations. Condensation is detrimental to accuracy, but negligibly so if properly used. However, one may not wish to use condensation if $[M]$ has been obtained by optimal lumping, as the use of optimal lumping implies that particular effort is being made to ensure accuracy.
+
+The literature is vast so we present only important ideas $[13.1,13.18,13.28]$ . In Section 13.8, the component mode synthesis method is described, which has significant condensation features.
+
+We present a condensation algorithm known as Guyan reduction, mass condensation, or eigenvalue economization [13.29]. Equation 13.5-2 is partitioned and written as
+
+$$
+\left(\left[ \begin{array}{l l} \mathbf {K} _ {m m} & \mathbf {K} _ {m s} \\ \mathbf {K} _ {m s} ^ {T} & \mathbf {K} _ {s s} \end{array} \right] - \lambda \left[ \begin{array}{l l} \mathbf {M} _ {m m} & \mathbf {M} _ {m s} \\ \mathbf {M} _ {m s} ^ {T} & \mathbf {M} _ {s s} \end{array} \right]\right) \left\{ \begin{array}{l} \overline {{\mathbf {D}}} _ {m} \\ \overline {{\mathbf {D}}} _ {s} \end{array} \right\} = \left\{ \begin{array}{l} 0 \\ 0 \end{array} \right\} \tag {13.7-1}
+$$
+
+where the m “master” d.o.f. $\{\overline{D}_{m}\}$ are to be retained and the s “slave” d.o.f. $\{\overline{D}_{s}\}$ are to be removed by condensation. The principal assumption in Guyan reduction is that for the lowest-frequency modes, inertia forces on slave d.o.f. are much less important than elastic forces transmitted by the master d.o.f. (subsequently, this assumption will be used as a guideline for selecting whether a d.o.f. should be a master or a slave). In other words, slave d.o.f. are assumed to move quasi-statically in response to the motion of master d.o.f. Thus, in order to obtain a relation between $\{D_{s}\}$ and $\{D_{m}\}$ , we temporarily ignore all mass but $[M_{mm}]$ . From the lower partition of Eq. 13.7-1,
+
+<!-- source-page: 408 -->
+
+$$
+\left\{\overline {{\mathbf {D}}} _ {s} \right\} _ {s \times 1} = - \left[ \mathbf {K} _ {s s} \right] _ {s \times s} ^ {- 1} \left[ \mathbf {K} _ {m s} \right] _ {s \times m} ^ {T} \left\{\overline {{\mathbf {D}}} _ {m} \right\} _ {m \times 1} \tag {13.7-2}
+$$
+
+Accordingly, with $n_{\mathrm{eq}} = m + s$ and $[\mathbf{I}]$ an $m$ by $m$ identity matrix,
+
+$$
+\left\{ \begin{array}{l} \overline {{\mathbf {D}}} _ {m} \\ \overline {{\mathbf {D}}} _ {s} \end{array} \right\} = \underset {n _ {\mathbf {e q}} \times m} {\left[ \mathbf {T} \right]} \left\{ \begin{array}{l} \overline {{\mathbf {D}}} _ {m} \\ m \times 1 \end{array} \right\}, \quad \text {where} \quad [ \mathbf {T} ] = \left[ \begin{array}{c} \mathbf {I} \\ - \mathbf {K} _ {s s} ^ {- 1} \mathbf {K} _ {m s} ^ {T} \end{array} \right] \tag {13.7-3}
+$$
+
+Substitution of Eq. 13.7-3 into Eq. 13.7-1 and premultiplication by $[T]^{T}$ yields the condensed eigenproblem
+
+$$
+\left(\left[ \mathbf {K} _ {r} \right] - \lambda \left[ \mathbf {M} _ {r} \right]\right) \left\{\overline {{\mathbf {D}}} _ {m} \right\} = \{\mathbf {0} \} \tag {13.7-4}
+$$
+
+where the reduced matrices are symmetric and are given by
+
+$$
+\left[ \mathbf {K} _ {r} \right] _ {m \times m} = [ \mathbf {T} ] ^ {T} [ \mathbf {K} ] [ \mathbf {T} ] \quad \text { and } \quad \left[ \mathbf {M} _ {r} \right] _ {m \times m} = [ \mathbf {T} ] ^ {T} [ \mathbf {M} ] [ \mathbf {T} ] \tag {13.7-5}
+$$
+
+Note that even if [K] is banded and [M] is diagonal, $[K_{r}]$ and $[M_{r}]$ are in general full and $[M_{r}]$ is a combination of both mass and stiffness coefficients. If damping matrix [C] and external loads $\{R^{ext}\}$ appear in the equation of motion, then condensed damping matrix $[C_{r}] = [T]^{T}[C][T]$ and condensed external loads $\{R_{r}^{ext}\} = [T]^{T}\{R^{ext}\}$ appear in the reduced equation of motion
+
+$$
+[ \mathbf {M} _ {r} ] \{\ddot {\mathbf {D}} _ {m} \} + [ \mathbf {C} _ {r} ] \{\dot {\mathbf {D}} _ {m} \} + [ \mathbf {K} _ {r} ] \{\mathbf {D} _ {m} \} = \left\{\mathbf {R} _ {r} ^ {\text {ext}} \right\} \tag {13.7-6}
+$$
+
+where $\{\mathbf{D}_m\}$ represents the displacements of master d.o.f.
+
+If [M] is diagonal and slave d.o.f. carry no mass, then $[K_{r}]$ is the same matrix as produced by “static condensation,” $[M_{r}]$ contains the nonzero $M_{ii}$ of [M], and condensation produces no loss of accuracy. (Static condensation is discussed in Section 8.1.)
+
+In vibration problems, when eigenvalues $\lambda_{i}$ and eigenvectors $\{\overline{D}_{m}\}_{i}$ of the reduced system are known, slave modes $\{\overline{D}_{s}\}_{i}$ can be recovered by use of Eq. 13.7-2. However, it is more accurate to recover $\{\overline{D}_{s}\}_{i}$ from the lower partition of Eq. 13.7-1 in which advantage is taken of the previously neglected slave node masses. Thus
+
+$$
+\{\overline {{{\mathbf {D}}}} _ {s} \} _ {i} = - \left[ \mathbf {K} _ {s s} - \lambda_ {i} \mathbf {M} _ {s s} \right] ^ {- 1} \left[ \mathbf {K} _ {m s} ^ {T} - \lambda_ {i} \mathbf {M} _ {m s} ^ {T} \right] \{\overline {{{\mathbf {D}}}} _ {m} \} _ {i} \tag {13.7-7}
+$$
+
+Reference 13.34 includes another form of this expression that is more computationally efficient.
+
+Remarks. Why create a detailed finite element model if we subsequently intend to discard many d.o.f. by condensation? Because coarse finite element discretizations usually do not have sufficient detail for accurate stiffness and mass representations. Furthermore, accurate stress computations usually require fine discretizations [13.30].
+
+Because reduction destroys any preexisting band structure and sparsity, m must be considerably smaller than $n_{eq}$ for Guyan reduction to be cost-effective. If original matrices [M] and [K] have unusually small bandwidths, it may be prudent
+
+<!-- source-page: 409 -->
+
+to avoid condensation because a gain in efficiency can only be obtained by taking $m \ll n_{eq}$ , in which case accuracy may suffer.
+
+If applied only to [K], Eq. 13.7-3 becomes the static condensation algorithm, Eq. 8.1-3. There are computational advantages to generating [T] in terms of flexibility instead of stiffness [13.28,13.31,13.32].
+
+We see that [T] plays the same role as the matrix $[W]_{j}$ of Ritz vectors in Eqs. 13.6-19. Indeed, [T] can be regarded as a matrix of Ritz vectors and Guyan reduction as a Rayleigh–Ritz method. However, [T] and $[W]_{j}$ are not the same: [T] is obtained more efficiently, without reference to the applied loading, but is not mass-matrix-orthogonal. $[W]_{j}$ is more adept at reproducing quasistatic structure deflection shapes, and since it is obtained from the applied loading, it can usually have fewer columns than [T] for a given level of accuracy.
+
+If [M] is lumped so that $\left[M_{mm}\right]$ contains all the nonzero masses, and $[M_{ss}]$ is null, then Eq. 13.7-2 follows without the necessity of our principal assumption. This suggests another approach to condensation: the analyst can lump mass at only the d.o.f. to be retained as masters. However, this approach requires experience and is less accurate than condensation with a more populated mass matrix.
+
+In going from Eq. 13.7-1 to Eq. 13.7-4, eigenvalues are raised because constraints are imposed [13.33]. This behavior, and the considerable accuracy possible when $m \ll n_{\mathrm{eq}}$ , are shown in Fig. 13.7-1. The higher eigenvalues of the original system are absent from the reduced system because several d.o.f. have been discarded. After $\{\overline{\mathbf{D}}_s\}$ has been recovered by use of Eq. 13.7-7, an eigenvalue can be improved by substituting the eigenvector $\{\mathbf{D}\} = \{\overline{\mathbf{D}}_m, \overline{\mathbf{D}}_s\}$ into the Rayleigh quotient of the full system, Eq. 13.5-4.
+
+Master d.o.f. should be those for which inertia is most important. Such d.o.f. have a large mass-to-stiffness ratio. Thus rotational d.o.f. rarely appear as masters. A master d.o.f. should be retained at each node that carries a time-varying applied load or has a time-varying prescribed displacement. Master d.o.f. should not be clustered in one area of the mesh. If they are, some vibration modes may be almost linearly dependent. This is not of major concern, but, if ignored, can lead to the disconcerting appearance of negative eigenvalues in the highest modes of the reduced eigenproblem $[13.35]$ .
+
+The selection of master and slave d.o.f. can be automated as follows [13.36]. Diagonal coefficients of [K] and [M] are scanned, and the d.o.f. $i$ for which $K_{ii} / M_{ii}$
+
+![](images/page-409_eadc408584ad870b3266cfedd8ad330e193fec3904a637a1e7e2234218c6e9b5.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Grid pattern with diagonal lines and circular markers, no text or symbols present
+</details>
+
+Full system, 90 d.o.f. $\omega_{1} = 3.469$
+
+(one displacement $\omega_{2} = 8.535$
+
+and two rotations $\omega_{3} = 21.450$
+
+at each node) $\omega_{4} = 27.059$
+
+Reduced system, 6 mas- $\omega_{1} = 3.473$
+
+ter d.o.f. (lateral $\omega_{2} = 8.604$
+
+displacements at $\omega_{3} = 22.690$
+
+nodes circled) $\omega_{4} = 29.490$
+
+Figure 13.7-1. First four vibration frequencies of a thin, square cantilever plate [13.37]. The analysis uses triangular plate elements and consistent mass matrices.
+
+<!-- source-page: 410 -->
+
+is largest is selected as the first slave. In case of a tie, the first d.o.f. encountered is taken as a slave. Then [K] and [M] are condensed (by one order). The condensed matrices are now scanned, the largest $K_{rii}/M_{rii}$ is selected as the next slave, and another condensation is performed. This process repeats until a user-specified number of d.o.f. remain. These are the masters, chosen in a near-optimal way.
+
+The number of masters can be chosen automatically by specifying a cut-off frequency, $\omega_{c}$ [13.44]. This frequency is taken to be about three times the highest frequency of interest in the excitation and/or structural response. Modes of the structure having frequencies greater than $\omega_{c}$ are quasistatic with respect to the excitation and can therefore be neglected in the dynamic response, although they may participate in the quasistatic response. Automatic selection of masters and slaves proceeds as described above except that condensation is terminated when the largest ratio $K_{rii}/M_{rii}$ is less than $\omega_{c}^{2}$ . Another procedure that may be more efficient is given in [13.38].
+
+One may combine manual selection of some master d.o.f. with automatic selection of the rest. A motivation would be to retain as masters those d.o.f. subjected to time-varying loads. It would be very inconvenient to eliminate these d.o.f. as slaves.
+
+The number of master d.o.f., m, may be as low as 1/10 or 1/20 the total number of d.o.f. If there are two or three times as many master d.o.f. as eigenvalues to be computed, the highest computed eigenvalue may err by less than 10%. These estimates are rough and strongly problem-dependent [13.36,13.37].
+
+Mass condensation yields good results if the choice of master d.o.f. is good and if the ratio of master d.o.f. to total d.o.f. $(m/n_{\mathrm{eq}})$ is not too small. As alternatives, the subspace iteration and Lanczos methods of eigenproblem solution employ condensation techniques that may be more reliable than Eq. 13.7-3.
+
+Example: Beam Vibration. Nonzero d.o.f. in the problem of Fig. 13.7-2 are $w_{1}$ and $\theta_{2}$ . Accordingly, using the stiffness and mass matrices of Eqs. 4.2-5 and 13.3-2, respectively, we obtain the eigenproblem
+
+$$
+\left(\frac {E I}{L ^ {3}} \left[ \begin{array}{l l} 1 2 & 6 L \\ 6 L & 4 L ^ {2} \end{array} \right] - \frac {\omega^ {2} m}{4 2 0} \left[ \begin{array}{c c} 1 5 6 & - 1 3 L \\ - 1 3 L & 4 L ^ {2} \end{array} \right]\right) \left\{ \begin{array}{l} \overline {{w}} _ {1} \\ \overline {{\theta}} _ {2} \end{array} \right\} = \left\{ \begin{array}{l} 0 \\ 0 \end{array} \right\} \tag {13.7-8}
+$$
+
+where $m = \rho AL$ is the mass of the beam. The eigenvalues of this two-d.o.f. system are
+
+$$
+\omega_ {1} ^ {2} = 6. 1 3 6 2 \frac {E I}{m L ^ {3}} \quad \text { and } \quad \omega_ {2} ^ {2} = 7 5 8. 1 7 \frac {E I}{m L ^ {3}} \tag {13.7-9}
+$$
+
+From continuous beam theory, the two lowest frequencies are
+
+$$
+\omega_ {1} ^ {2} = 6. 0 8 8 1 \frac {E I}{m L ^ {3}} \quad \text { and } \quad \omega_ {2} ^ {2} = 4 9 3. 1 3 \frac {E I}{m L ^ {3}} \tag {13.7-10}
+$$
+
+![](images/page-410_d805a905aeb23511e9470a4051d5c2c5a14379e38bc90cc759df3854d30b31df.jpg)
+
+<details>
+<summary>text_image</summary>
+
+w₁
+θ₂
+1
+2
+L
+</details>
+
+Figure 13.7-2. A uniform beam. The left end is allowed to displace but not to rotate. The right end is simply supported.
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@@ -0,0 +1,358 @@
+<!-- source-page: 411 -->
+
+and, as expected, are lower than the frequencies of Eqs. 13.7-9. With $\overline{w}_1 = 1$ , the amplitude of $\overline{\theta}_2$ in mode 1, from Eq. 13.7-8 and beam theory, respectively, is
+
+$$
+\bar {\theta} _ {2} = - 1. 5 7 0 4 / L (\text {approximate}) \quad \bar {\theta} _ {2} = - 1. 5 7 0 8 / L (\text {exact}) \tag {13.7-11}
+$$
+
+To condense the size of the eigenproblem of Eq. 13.7-8, we employ Guyan reduction and select $\overline{w}_{1}$ as master and $\overline{\theta}_{2}$ as slave. Using Eqs. 13.7-3 and 13.7-5, with $K_{ss} = 4EI/L$ and $K_{ms} = 6EI/L^{2}$ , we obtain
+
+$$
+[ \mathrm{T} ] = \left[ \begin{array}{c} 1 \\ - 3 / 2 L \end{array} \right] \quad \text { and } \quad \left(\frac {3 E I}{L ^ {3}} - \omega^ {2} \frac {2 0 4 m}{4 2 0}\right) \overline {{w}} _ {1} = 0 \tag {13.7-12}
+$$
+
+from which $\omega_{1}^{2}=6.1765EI/mL^{3}$ . This is the only eigenvalue obtainable. As expected, it is higher than $\omega_{1}^{2}$ in Eq. 13.7-9. With $\overline{w}_{1}=1$ , recovery of $\overline{\theta}_{2}$ from Eqs. 13.7-2 and 13.7-7, respectively, yields
+
+$$
+\overline {{{\theta}}} _ {2} = - 1. 5 0 0 0 / L \quad \text { and } \quad \overline {{{\theta}}} _ {2} = - 1. 5 7 0 9 / L \tag {13.7-13}
+$$
+
+An improved estimate of $\omega_{1}^{2}$ can be found by substituting $\{\overline{\mathbf{D}}\} = \lfloor 1 - 1.5709 / L \rfloor^{T}$ into the Rayleigh quotient (Eq. 13.5-4), with [K] and [M] taken from Eq. 13.7-8. The resulting eigenvalue is the same as $\omega_{1}^{2}$ in Eq. 13.7-9.
+
+A lumped-mass solution for the fundamental vibration frequency is easily obtained for this problem by placing a mass particle $m_1 = m / 2$ at node 1. From beam theory, $w_1 = P_1 L^3 / 3EI$ , so $K = P_1 / w_1 = 3EI / L^3$ . Then $\omega^2 = K / m_1 = 6EI / mL^3$ . Note that this value is not an upper bound.
+
+# 13.8 COMPONENT MODE SYNTHESIS
+
+General Remarks. In component mode synthesis, or simply modal synthesis, a structure is subdivided into components or substructures, each of which is analyzed independently for natural frequencies and, more importantly, for mode shapes. The component mode shapes are then “assembled” to give displacement shapes or load patterns (either interpretation is possible) of the original structure. These shapes or patterns are not eigenvectors of the original structure, but give rise to Ritz vectors that are subsequently used to transform the original displacement d.o.f. to generalized d.o.f. Thus the size of the system matrices is reduced. Eigenanalysis and/or time-history analysis of the reduced system equations can be performed much more economically and usually with surprising accuracy $[13.39]$ .
+
+Component mode synthesis can be regarded as an alternative to Guyan reduction. More importantly, modal synthesis has the managerial advantage of allowing different design groups to work on different parts of a large structure, as discussed in Section 8.14 with respect to static substructuring. There are many methods of component mode synthesis and an extensive literature $[13.2,13.40]$ . In what follows we present only fundamentals.
+
+We will use a “reduced basis” [W] to replace d.o.f. $\{D\}$ of the entire (or assembled) structure by a vector of generalized coordinates $\{y\}$ that contains fewer d.o.f. than $\{D\}$ . The relation between $\{D\}$ and $\{y\}$ is stated by Eq. 13.6-13, here repeated:
+
+$$
+\left\{\mathbf {D} \right\} _ {n _ {\mathrm{eq}} \times 1} = \left[ \mathbf {W} \right] _ {n _ {\mathrm{eq}} \times m} \left\{\mathbf {y} \right\} _ {m \times 1} \tag {13.6-13}
+$$
+
+<!-- source-page: 412 -->
+
+in which m is considerably less than $n_{eq}$ . Matrix [W] is an array of Ritz vectors. The question of how to establish [W] occupies much of our subsequent discussion. Equations 13.2-12 and 13.6-13 yield the reduced dynamic problem
+
+$$
+[ \mathbf {M} _ {r} ] \{\ddot {\mathbf {y}} \} + [ \mathbf {C} _ {r} ] \{\dot {\mathbf {y}} \} + [ \mathbf {K} _ {r} ] \{\mathbf {y} \} = \{\mathbf {R} _ {r} \} \tag {13.8-1}
+$$
+
+where $\{\mathbf{R}_r\} = [\mathbf{W}]^T\{\mathbf{R}^{\mathrm{ext}}\}$ and the reduced $m$ by $m$ mass, damping, and stiffness matrices are
+
+$$
+[ \mathbf {M} _ {r} ] = [ \mathbf {W} ] ^ {T} [ \mathbf {M} ] [ \mathbf {W} ] \quad [ \mathbf {C} _ {r} ] = [ \mathbf {W} ] ^ {T} [ \mathbf {C} ] [ \mathbf {W} ] \quad [ \mathbf {K} _ {r} ] = [ \mathbf {W} ] ^ {T} [ \mathbf {K} ] [ \mathbf {W} ] \tag {13.8-2}
+$$
+
+With $\{\mathbf{y}\} = \{\overline{\mathbf{y}}\}$ sin $\omega t$ and $\lambda = \omega^2$ , the reduced eigenproblem without damping is
+
+$$
+\left(\left[ \mathbf {K} _ {r} \right] - \lambda \left[ \mathbf {M} _ {r} \right]\right) \{\overline {{{\mathbf {y}}}} \} = \{\mathbf {0} \} \tag {13.8-3}
+$$
+
+Of the several methods of establishing an effective [W], we will summarize two. In the first, which we label CMS1 for short, [W] contains deflection vectors obtained by solving Eq. 13.6-14—that is, $[W] = [K]^{-1}[R]$ . Load patterns in [R] are assembled eigenvectors of the substructures—that is, the component modes. A modification is needed if [K] is singular, as for an unsupported structure; see [13.2]. In the second method we discuss, here labeled CMS2 for short, [W] contains the assembled component modes themselves. This method has no difficulty with an unsupported structure. In both of these methods, [W] also contains supplementary vectors related to the motion of d.o.f. shared by substructures. The supplementary vectors may be called rigid-body modes, constraint modes, or attachment modes, depending on how the shared d.o.f. are treated. In illustrations that follow we assume that shared d.o.f. are fixed for the substructure analyses that precede synthesis.
+
+Implementation. To illustrate method CMS1, consider a structure that is divided into $\ell$ substructures. Also assume that substructure k is attached to substructures k - 1 and $k + 1$ only; $k = 2, 3, \ldots, \ell - 1$ . Denote matrices for substructure k by $[K_k]$ and $[M_k]$ where all attachment d.o.f. (i.e., d.o.f. that are common to attaching substructures) are fixed. Then, for each substructure, the eigenproblem
+
+$$
+([ \mathbf {K} _ {k} ] - \lambda [ \mathbf {M} _ {k} ]) \{\overline {{{\mathbf {D}}}} _ {k} \} = \{\mathbf {0} \} \tag {13.8-4}
+$$
+
+is solved for normal modes of vibration $\{\overline{D}_{k}\}$ . These modes form the columns of the substructure modal matrix $[\phi_{k}]$ , which is $n_{k}$ by $n_{\phi}$ . Here $n_{k}$ is the number of interior d.o.f. of substructure k (i.e., the total number of d.o.f. of the substructure minus the number of attachment d.o.f.) and $n_{\phi}$ (usually $n_{\phi} \ll n_{k}$ ) is the number of normal nodes to be determined (the same for each substructure). Ritz vectors [W] for the entire structure can be obtained by solving Eq. 13.6-14 with
+
+$$
+\left[ \mathbf {R} \right] _ {n _ {\mathrm{eq}} \times m} = \left[ \begin{array}{c c c c} \phi_ {1} & \mathbf {0} & \mathbf {0} & \dots \\ \mathbf {0} & \mathbf {I} _ {1, 2} & \mathbf {0} & \dots \\ \phi_ {2} & \mathbf {0} & \mathbf {0} & \dots \\ \mathbf {0} & \mathbf {0} & \mathbf {I} _ {2, 3} & \dots \\ \vdots & \vdots & \vdots \\ \phi_ {\ell} & \mathbf {0} & \mathbf {0} & \dots \end{array} \right] \tag {13.8-5}
+$$
+
+<!-- source-page: 413 -->
+
+where $\left[I_{k,k+1}\right]$ is a unit matrix with number of rows equal to the number of attachment d.o.f. between substructures k and $k+1$ , and $m=n_{\phi}+n_{a}$ , where $n_{a}$ is the total number of attachment d.o.f. in the synthesized model. The first $n_{\phi}$ columns of Eq. 13.8-5 represent assembled substructure normal modes (i.e., vertically stacked columns of substructure normal modes $[\phi_{k}]$ ) that are imposed as nodal loads in Eq. 13.6-14. However, since attachment d.o.f. for each substructure were fixed, the Ritz modes resulting from these load patterns are not capable of exciting attachment d.o.f. Thus, the first $n_{\phi}$ load patterns in Eq. 13.8-5 are supplemented by additional loads $\left[I_{k,k+1}\right]$ which physically correspond to successively applying a unit load to each attachment d.o.f. of the assembled structure while keeping all other d.o.f. load-free. The resulting Ritz vectors are called attachment modes and correspond to the displacements of all d.o.f. of the assembled structure due to the unit applied loads.
+
+Method CMS2 includes what is perhaps the most popular method of component mode synthesis, the Craig-Bampton method $[13.43]$ . Here the assembled substructure normal modes obtained by fixing attachment d.o.f. are used directly as Ritz vectors. These are then supplemented by constraint modes, which are deflection shapes of the assembled structure obtained by successively applying a unit displacement to each attachment d.o.f. while keeping all other attachment d.o.f. fixed.
+
+In all methods of component mode synthesis, the reduced eigenproblem, Eq. 13.8-3, is obtained by imposing constraints on the original large system. Thus, frequencies computed from Eq. 13.8-3 are upper bounds to those of the original system equations. Occasionally, it is possible to have all Ritz vectors orthogonal to one of the eigenvectors of the assembled structure. Then that eigenvector will be missing from the reduced eigenproblem, Eq. 13.8-3. In practice, the likelihood of missing lower-spectrum eigenvectors is reduced by using appropriate supplementary Ritz vectors such as attachment modes, constraint modes, and so on.
+
+In the following example problem, method CMS1 is more accurate than method CMS2. Method CMS2 is more prevalent in practice than method CMS1. However, when one considers that the original stiffness matrix $[K]$ must be factored to compute constraint modes (and/or other supplementary Ritz vectors), the additional cost required to obtain Ritz vectors by Eqs. 13.6-14 and 13.8-5 is small, and therefore method CMS1 merits consideration.
+
+Example. Consider axial vibrations of the structure shown in Fig. 13.8-1, whose stiffness and lumped mass matrices are
+
+$$
+[ \mathbf {K} ] = \frac {A E}{L} \left[ \begin{array}{r r r r r} 1 & - 1 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 \\ 0 & - 1 & 3 & - 2 & 0 \\ 0 & 0 & - 2 & 4 & - 2 \\ 0 & 0 & 0 & - 2 & 4 \end{array} \right] \quad [ \mathbf {M} ] = \frac {\rho A L}{2} \left[ \begin{array}{r r r r r} 1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 0 & 4 \end{array} \right] \tag {13.8-6}
+$$
+
+Two substructures are created, one consisting of elements 1 and 2 and the other of elements 3, 4, and 5. With node 3 fixed, matrices for substructure 1 are
+
+$$
+[ \mathbf {K} _ {1} ] = \frac {A E}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 2 \end{array} \right] \quad \left[ \mathbf {M} _ {1} \right] = \frac {\rho A L}{2} \left[ \begin{array}{c c} 1 & 0 \\ 0 & 2 \end{array} \right] \tag {13.8-7}
+$$
+
+<!-- source-page: 414 -->
+
+![](images/page-414_6b303084104b5563dd27f78335bc005a5037809a6cfd09c2dcf6e265bcf1d91a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+A, E
+2A, E
+① ② ③ ④ ⑤
+1 2 3 4 5
+L L L L L L
+</details>
+
+Figure 13.8-1. Bar with variable cross section modeled by five uniform elements of length L.
+
+and for substructure 2
+
+$$
+[ \mathbf {K} _ {2} ] = \frac {A E}{L} \left[ \begin{array}{c c} 4 & - 2 \\ - 2 & 4 \end{array} \right] \quad [ \mathbf {M} _ {2} ] = \frac {\rho A L}{2} \left[ \begin{array}{c c} 4 & 0 \\ 0 & 4 \end{array} \right] \tag {13.8-8}
+$$
+
+In the sequel, we assume that AE/L = 1 and $\rho AL/2 = 1$ . For substructure 1, the eigenproblem Eq. 13.8-4 is solved with the matrices of Eq. 13.8-7 to yield
+
+$$
+\lambda_ {1} = \frac {2 - \sqrt {2}}{2} \quad \lambda_ {2} = \frac {2 + \sqrt {2}}{2} \quad \{\overline {{\mathbf {D}}} _ {1} \} _ {1} = \left\{ \begin{array}{c} 1 \\ \sqrt {2} / 2 \end{array} \right\} \quad \{\overline {{\mathbf {D}}} _ {1} \} _ {2} = \left\{ \begin{array}{c} 1 \\ - \sqrt {2} / 2 \end{array} \right\} \tag {13.8-9}
+$$
+
+where eigenvectors are normalized so that the first coefficient has unit amplitude. Solving Eq. 13.8-4 using the second substructure's matrices, Eq. 13.8-8, we obtain
+
+$$
+\lambda_ {1} = \frac {1}{2} \quad \lambda_ {2} = \frac {3}{2} \quad \{\overline {{{\mathbf {D}}}} _ {2} \} _ {1} = \left\{ \begin{array}{l} 1 \\ 1 \end{array} \right\} \quad \{\overline {{{\mathbf {D}}}} _ {2} \} _ {2} = \left\{ \begin{array}{l} 1 \\ - 1 \end{array} \right\} \tag {13.8-10}
+$$
+
+For method CMS1, we must evaluate Eq. 13.8-5, which becomes
+
+$$
+[ \mathbf {R} ] = \left[ \begin{array}{c c c} 1 & 1 & 0 \\ \sqrt {2} / 2 & - \sqrt {2} / 2 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & - 1 & 0 \end{array} \right] \tag {13.8-11}
+$$
+
+The first two columns of Eq. 13.8-11 are the assembled component normal modes while the third column has the effect of “releasing” node 3. The resulting Ritz vectors, from Eqs. 13.6-14, 13.8-6, and 13.8-11, are
+
+$$
+[ \mathbf {W} ] = \left[ \begin{array}{l l l} 6. 7 6 8 & 2. 2 3 2 & 1. 5 \\ 5. 7 6 8 & 1. 2 3 2 & 1. 5 \\ 4. 0 6 1 & 0. 9 3 9 3 & 1. 5 \\ 3. 2 0 7 & 0. 7 9 2 9 & 1. 0 \\ 1. 8 5 4 & 0. 1 4 6 4 & 0. 5 \end{array} \right] \quad \text {(method CMS1)} \tag {13.8-12}
+$$
+
+Reduced stiffness and mass matrices, from Eqs. 13.8-2 and 13.8-12, are
+
+$$
+\left[ \mathbf {K} _ {r} \right] = \left[ \begin{array}{l l l} 1 5. 9 1 & 4. 0 4 3 & 4. 0 6 1 \\ 4. 0 4 3 & 2. 0 0 7 & 0. 9 3 9 3 \\ 4. 0 6 1 & 0. 9 3 9 3 & 1. 5 0 0 \end{array} \right] \quad \left[ \mathbf {M} _ {r} \right] = \left[ \begin{array}{l l l} 2 1 6. 7 & 5 2. 0 2 & 6 2. 2 6 \\ 5 2. 0 2 & 1 3. 2 7 & 1 4. 7 4 \\ 6 2. 2 6 & 1 4. 7 4 & 1 8. 5 0 \end{array} \right] \tag {13.8-13}
+$$
+
+<!-- source-page: 415 -->
+
+TABLE 13.8-1. NATURAL FREQUENCIES FOR COMPONENT MODE ANALYSIS OF THE STRUCTURE SHOWN IN FIG. 13.8-1. 
+
+<table><tr><td>Procedure</td><td> $\omega_1$ </td><td> $\omega_2$ </td><td> $\omega_3$ </td></tr><tr><td>Original structure (five d.o.f.)</td><td>0.2651</td><td>0.6156</td><td>1.000</td></tr><tr><td>Method CMS1 (three d.o.f.)</td><td>0.2656</td><td>0.8794</td><td>1.182</td></tr><tr><td>Method CMS2 (three d.o.f.)</td><td>0.2719</td><td>0.9927</td><td>1.268</td></tr></table>
+
+Natural frequencies of the reduced eigenproblem, Eq. 13.8-3, are reported in Table 13.8-1. Also reported are the first three frequencies of the original structure, whose matrices are given by Eq. 13.8-6.
+
+For method CMS2, we take as Ritz vectors the assembled component normal modes, plus constraint modes [13.43]. Thus
+
+$$
+[ \mathbf {W} ] = \left[ \begin{array}{c c c} 1 & 1 & 1 \\ \sqrt {2} / \dot {2} & - \sqrt {2} / 2 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 2 / 3 \\ 1 & - 1 & 1 / 3 \end{array} \right] \quad (\text { method   CMS2 }) \tag {13.8-14}
+$$
+
+Here the third column of Eq. 13.8-14 (i.e., $\{\mathbf{w}_3\}$ ) is the deflection shape obtained by solving $[\mathbf{K}]\{\mathbf{w}_3\} = \{\mathbf{p}\}$ , where
+
+$$
+\left\{\mathrm{w} _ {3} \right\} = \left\lfloor u _ {1} \quad u _ {2} \quad 1 \quad u _ {4} \quad u _ {5} \right] ^ {T} \tag {13.8-15a}
+$$
+
+$$
+\{\mathbf {p} \} = \left[ \begin{array}{l l l l l} 0 & 0 & p _ {3} & 0 & 0 \end{array} \right] ^ {T} \tag {13.8-15b}
+$$
+
+and [K] is the original structure stiffness matrix of Eq. 13.8-6. As there is but one attachment d.o.f. in this simple problem, the third column of Eq. 13.8-12 and the third column of Eq. 13.8-14 describe the same displacement pattern. Natural frequencies obtained by solving the reduced eigenproblem, Eq. 13.8-3, are given in Table 13.8-1.
+
+As expected, both methods overestimate frequencies obtained by using all d.o.f. of the structure. The discrepancy is greater for higher frequencies. However, higher frequencies are usually of little interest, as structural response is dominated by lower frequencies. The several lowest frequencies are usually estimated accurately by a modest number of component modes.
+
+# 13.9 TIME-HISTORY ANALYSIS. DIRECT INTEGRATION METHODS
+
+In direct integration methods or step-by-step' methods, a finite difference approximation is used to replace the time derivatives appearing in Eq. 13.2-12 or 13.2-13 (i.e., $\{\ddot{\mathbf{D}}\}$ and $\{\dot{\mathbf{D}}\}$ ) by differences of displacement $\{\mathbf{D}\}$ at various instants of time. Finite difference methods for approximately solving initial value problems have been well studied and have a rich literature [13.47]. Over the past two decades, methods that are particularly effective for transient finite element equations have flourished and continue to be an active area of research [13.48,13.49]. Direct integration is an alternative to modal methods (Section 13.6). For many structural dynamics and wave propagation problems, including those with com-
+
+<!-- source-page: 416 -->
+
+plicated nonlinearities, direct integration is more expedient. Many methods of direct integration are popular and the choice of method is strongly problem-dependent. In this section explicit and implicit methods are introduced. In Sections 13.10 and 13.11 these methods are discussed in detail.
+
+In direct integration, the approach is to write the equation of motion, Eq. 13.2-12, at a specific instant of time,
+
+$$
+\boxed {[ \mathbf {M} ] \{\ddot {\mathbf {D}} \} _ {n} + [ \mathbf {C} ] \{\dot {\mathbf {D}} \} _ {n} + [ \mathbf {K} ] \{\mathbf {D} \} _ {n} = \{\mathbf {R} ^ {\text {ext}} \} _ {n}} \tag {13.9-1}
+$$
+
+where subscript n denotes time $n \Delta t$ and $\Delta t$ is the size of the time increment or time step. The absence of time step subscripts on matrices [M], [C], and [K] in Eq. 13.9-1 implies linearity. For problems with material nonlinearity, [K] is a function of displacements and therefore of time as well. Accordingly, from Eq. 13.2-13,
+
+$$
+[ \mathbf {M} ] \{\ddot {\mathbf {D}} \} _ {n} + [ \mathbf {C} ] \{\dot {\mathbf {D}} \} _ {n} + \left\{\mathbf {R} ^ {\text {int}} \right\} _ {n} = \left\{\mathbf {R} ^ {\text {ext}} \right\} _ {n} \tag {13.9-2}
+$$
+
+$\{R^{int}\}_{n}$ is the internal force vector at time $n \Delta t$ due to straining of material. It is obtained by assembling element internal force vectors, $\{r^{int}\}_{n}$ , given by Eq. 13.2-7 using $\{\sigma\}_{n}$ . For nonlinear problems, $\{R^{int}\}_{n}$ is a nonlinear function of $\{D\}_{n}$ and possibly time derivatives of $\{D\}_{n}$ . For linear problems, $\{R^{int}\}_{n} = [K]\{D\}_{n}$ . In Eq. 13.9-2, [M] and [C] are taken as time-independent, although for some problems these may be nonlinear also.
+
+In the following sections, [M] is assumed to be positive definite. Moreover, unless otherwise stated, [K] is positive semidefinite; that is, [K] may permit rigid-body motion.
+
+Difference methods for direct integration of Eqs. 13.9-1 and 13.9-2 can be categorized as explicit or implicit. Explicit methods have the form
+
+$$
+\{\mathbf {D} \} _ {n + 1} = f \left(\left\{\mathbf {D} \right\} _ {n}, \left\{\dot {\mathbf {D}} \right\} _ {n}, \left\{\ddot {\mathbf {D}} \right\} _ {n}, \left\{\mathbf {D} \right\} _ {n - 1}, \dots\right) \tag {13.9-3}
+$$
+
+and hence permit $\{D\}_{n+1}$ to be determined in terms of completely historical information consisting of displacements and time derivatives of displacements at time $n \Delta t$ and before. Implicit methods have the form
+
+$$
+\{\mathbf {D} \} _ {n + 1} = f (\{\dot {\mathbf {D}} \} _ {n + 1}, \{\ddot {\mathbf {D}} \} _ {n + 1}, \{\mathbf {D} \} _ {n}, \dots) \tag {13.9-4}
+$$
+
+and hence computation of $\{\mathbf{D}\}_{n+1}$ requires knowledge of the time derivatives of $\{\mathbf{D}\}_{n+1}$ , which are unknown. Explicit and implicit methods have markedly different properties. This has important practical implications.
+
+Methods that have the general form of Eqs. 13.9-3 and 13.9-4 are called multistep methods. When the right-hand sides of Eqs. 13.9-3 and 13.9-4 contain information dating back to time $n \Delta t$ only, the methods are called single-step methods. When the right-hand sides of Eqs. 13.9-3 and 13.9-4 contain information dating back to time $(n - 1) \Delta t$ , they are called two-step methods. Single-step methods are easy to start from initial conditions. Multistep methods require special starting procedures that may be awkward or may introduce inaccuracies. Poor starting procedures (not described in this book) may degrade the accuracy of the entire analysis [13.69].
+
+<!-- source-page: 417 -->
+
+# 13.10 EXPLICIT DIRECT INTEGRATION METHODS
+
+A popular method, which is characteristic of explicit methods in general, is the central-difference method. It approximates velocity and acceleration by
+
+$$
+\{\dot {\mathbf {D}} \} _ {n} = \frac {1}{2 \Delta t} \left(\left\{\mathbf {D} \right\} _ {n + 1} - \left\{\mathbf {D} \right\} _ {n - 1}\right) \tag {13.10-1}
+$$
+
+$$
+\{\ddot {\mathbf {D}} \} _ {n} = \frac {1}{\Delta t ^ {2}} \left(\{\mathbf {D} \} _ {n + 1} - 2 \{\mathbf {D} \} _ {n} + \{\mathbf {D} \} _ {n - 1}\right) \tag {13.10-2}
+$$
+
+Equations 13.10-1 and 13.10-2 are obtained by expanding $\{\mathbf{D}\}_{n+1}$ and $\{\mathbf{D}\}_{n-1}$ in Taylor series about time $n\Delta t$ :
+
+$$
+\{\mathbf {D} \} _ {n + 1} = \{\mathbf {D} \} _ {n} + \Delta t \{\dot {\mathbf {D}} \} _ {n} + \frac {\Delta t ^ {2}}{2} \{\ddot {\mathbf {D}} \} _ {n} + \frac {\Delta t ^ {3}}{6} \{\dddot {\mathbf {D}} \} _ {n} + \dots \tag {13.10-3}
+$$
+
+$$
+\{\mathbf {D} \} _ {n - 1} = \{\mathbf {D} \} _ {n} - \Delta t \{\dot {\mathbf {D}} \} _ {n} + \frac {\Delta t ^ {2}}{2} \{\ddot {\mathbf {D}} \} _ {n} - \frac {\Delta t ^ {3}}{6} \{\dddot {\mathbf {D}} \} _ {n} + \dots \tag {13.10-4}
+$$
+
+Subtracting Eq. 13.10-4 from Eq. 13.10-3 yields Eq. 13.10-1 while adding Eqs. 13.10-3 and 13.10-4 yields Eq. 13.10-2. In both cases, terms containing $\Delta t^{2}$ and higher powers are omitted from Eqs. 13.10-1 and 13.10-2. Hence, the central-difference formulas, Eqs. 13.10-1 and 13.10-2, are said to be second-order accurate. In other words, the error is $O(\Delta t^{2})$ which implies that halving the time step should approximately quarter the error.
+
+Combining Eqs. 13.10-1 and 13.10-2 with Eq. 13.9-1 provides
+
+$$
+\begin{array}{l} \left[ \frac {1}{\Delta t ^ {2}} \mathbf {M} + \frac {1}{2 \Delta t} \mathbf {C} \right] \left\{\mathbf {D} \right\} _ {n + 1} \\ = \left\{\mathbf {R} ^ {\text {ext}} \right\} _ {n} - [ \mathbf {K} ] \left\{\mathbf {D} \right\} _ {n} + \frac {1}{\Delta t ^ {2}} [ \mathbf {M} ] \left(2 \left\{\mathbf {D} \right\} _ {n} - \left\{\mathbf {D} \right\} _ {n - 1}\right) + \frac {1}{2 \Delta t} [ \mathbf {C} ] \left\{\mathbf {D} \right\} _ {n - 1} \tag {13.10-5} \\ \end{array}
+$$
+
+# Remarks.
+
+1. Equation 13.10-5 is a system of linear algebraic equations. If [M] and [C] are diagonal, then the equations are uncoupled and $\{\mathbf{D}\}_{n+1}$ can be obtained without solving simultaneous equations.
+
+2. For small finite element models, [K] can be formed and stored in the computer's core memory and the internal force $\{\mathbf{R}^{\mathrm{int}}\}_{n} = [\mathbf{K}]\{\mathbf{D}\}_{n}$ at each time step can be obtained by matrix multiplication. However, it is more common, even for linear problems, to compute the internal force vector at each time step by summation of element contributions (i.e., element-by-element). Element contributions are given by Eq. 13.2-7. Because the element [k] need not be formed or stored, explicit methods can treat large three-dimensional models with comparatively modest computer storage requirements.
+
+<!-- source-page: 418 -->
+
+3. Starting the method from $n = 0$ requires $\{\mathbf{D}\}_{-1}$ , which can be computed from known initial conditions $\{\mathbf{D}\}_0$ and $\{\dot{\mathbf{D}}\}_0$ and Eq. 13.10-4:
+
+$$
+\{\mathbf {D} \} _ {- 1} = \{\mathbf {D} \} _ {0} - \Delta t \{\dot {\mathbf {D}} \} _ {0} + \frac {\Delta t ^ {2}}{2} \{\ddot {\mathbf {D}} \} _ {0} \tag {13.10-6}
+$$
+
+where terms with $\Delta t^3$ and higher powers are omitted. $\{\ddot{\mathbf{D}}\}_0$ is obtained from the equation of motion, Eq. 13.9-1, at time zero:
+
+$$
+\{\ddot {\mathbf {D}} \} _ {0} = [ \mathbf {M} ] ^ {- 1} \left(\left\{\mathbf {R} ^ {\text {ext}} \right\} _ {0} - [ \mathbf {K} ] \{\mathbf {D} \} _ {0} - [ \mathbf {C} ] \{\dot {\mathbf {D}} \} _ {0}\right) \tag {13.10-7}
+$$
+
+4. To compute $\{\mathbf{D}\}_{n+1}$ requires $\{\mathbf{R}^{\mathrm{int}}\}_{n}$ . For nonlinear material constitutive laws that are functions of strain (but not of strain rate), $\{\mathbf{R}^{\mathrm{int}}\}_{n}$ is easy to evaluate because $\{\mathbf{D}\}_{n}$ , and hence the strain at time $n\Delta t$ , is known. For this reason, explicit methods are well suited to treatment of material nonlinearity.
+
+5. Equation 13.10-5 is conditionally stable and requires $\Delta t$ such that
+
+$$
+\Delta t \leq 2 / \omega_ {\max} \tag {13.10-8}
+$$
+
+where $\omega_{max}$ is the highest natural frequency of $\det([K] - \omega^{2}[M]) = 0$ . If Eq. 13.10-8 is not satisfied, computations will be unstable. This is indicated by an obviously erroneous time-history solution that grows unbounded, perhaps by orders of magnitude per time step. $^{2}$ (In nonlinear problems, instabilities may be more difficult to detect.)
+
+6. A feature of Eq. 13.10-5 is that stability (i.e., maximum allowable time step size) is not affected by damping. For the central difference method to be economically competitive with implicit methods, both [M] and [C] must be diagonal. For reasons discussed in Section 13.13, explicit integration is usually more accurate with lumped mass matrices than with consistent mass matrices. However, it is difficult to model damping by spectral methods if [C] must be diagonal.
+
+7. It is conceivable that we may know the displacement and velocity of a structure at a given instant (i.e., initial conditions) and wish to integrate backward in time (i.e., use a negative $\Delta t$ ) to determine the configuration of the structure at some instant in the past. The central-difference method is conditionally stable for such applications and requires $-2/\omega_{max} \leq \Delta t \leq 2/\omega_{max}$ . Henceforth, we will be concerned with positive $\Delta t$ only.
+
+Alternative Form for Nondiagonal [C]. A form of the central-difference method that does not require diagonal [C] is obtained by approximating the velocity and acceleration by
+
+$$
+\{\dot {\mathbf {D}} \} _ {n - 1 / 2} = \frac {1}{\Delta t} \left(\left\{\mathbf {D} \right\} _ {n} - \left\{\mathbf {D} \right\} _ {n - 1}\right) \tag {13.10-9}
+$$
+
+$^{2}$ The case $\Delta t = 2/\omega_{max}$ may be called “limiting stability.” If $\Delta t = 2/\omega_{max}$ , the numerical solution may diverge in certain cases, but only in arithmetic fashion. If $\Delta t > 2/\omega_{max}$ , divergence is exponential. Further discussion of these matters appears in Section 13.13.
+
+<!-- source-page: 419 -->
+
+$$
+\begin{array}{l} \{\ddot {\mathbf {D}} \} _ {n} = \frac {1}{\Delta t} \left(\{\dot {\mathbf {D}} \} _ {n + 1 / 2} - \{\dot {\mathbf {D}} \} _ {n - 1 / 2}\right) \\ = \frac {1}{\Delta t ^ {2}} \left(\{\mathbf {D} \} _ {n + 1} - 2 \{\mathbf {D} \} _ {n} + \{\mathbf {D} \} _ {n - 1}\right) \tag {13.10-10} \\ \end{array}
+$$
+
+The equation of motion, Eq. 13.9-1, is modified by lagging the velocity by one-half time step:
+
+$$
+[ \mathbf {M} ] \{\ddot {\mathbf {D}} \} _ {n} + [ \mathbf {C} ] \{\dot {\mathbf {D}} \} _ {n - 1 / 2} + [ \mathbf {K} ] \{\mathbf {D} \} _ {n} = \left\{\mathbf {R} ^ {\text {ext}} \right\} _ {n} \tag {13.10-11}
+$$
+
+Combination of Eqs. 13.10-9 through 13.10-11 yields
+
+$$
+\frac {1}{\Delta t ^ {2}} [ \mathbf {M} ] \{\mathbf {D} \} _ {n + 1} = \left\{\mathbf {R} ^ {\text {ext}} \right\} _ {n} - [ \mathbf {K} ] \{\mathbf {D} \} _ {n} + \frac {1}{\Delta t ^ {2}} [ \mathbf {M} ] \left(\left\{\mathbf {D} \right\} _ {n} + \Delta t \left\{\dot {\mathbf {D}} \right\} _ {n - 1 / 2}\right) - [ \mathbf {C} ] \left\{\dot {\mathbf {D}} \right\} _ {n - 1 / 2} \tag {13.10-12}
+$$
+
+If [M] is lumped, then computation of $\{D\}_{n+1}$ does not require the solution of simultaneous equations. There are no restrictions on the form of [C]. The method can be started using the initial displacement $\{D\}_{0}$ and the approximation $\{\dot{D}\}_{-1/2} \approx \{\dot{D}\}_{0}$ . Alternatively, $\{\dot{D}\}_{-1/2}$ can be approximated by the forward difference formula with negative $\Delta t$ —that is,
+
+$$
+\{\dot {\mathbf {D}} \} _ {- 1 / 2} = \{\dot {\mathbf {D}} \} _ {0} - \frac {\Delta t}{2} \{\ddot {\mathbf {D}} \} _ {0} \tag {13.10-13}
+$$
+
+where $\{\ddot{\mathbf{D}}\}_{0}$ is obtained from Eq. 13.10-7. Although the central-difference formulas, Eqs. 13.10-9 and 13.10-10, are second-order accurate, we can only guarantee first-order accuracy in the time integration of Eq. 13.10-11 when $[\mathbf{C}] \neq [\mathbf{0}]$ because viscous forces $[\mathbf{C}]\{\dot{\mathbf{D}}\}_{n-1/2}$ lag by half a time step. However, for practical structures (which are not heavily damped), Eqs. 13.10-5 and 13.10-12 have almost the same accuracy.
+
+The stability condition for Eq. 13.10-12 is
+
+$$
+\Delta t \leq \frac {2}{\omega_ {\max}} \left(\sqrt {1 + \xi^ {2}} - \xi\right) \tag {13.10-14}
+$$
+
+where $\xi$ is the fraction of critical damping at the highest undamped natural frequency, $\omega_{max}$ . For proportional damping, $\xi$ at frequency $\omega_{max}$ can be computed from Eq. 13.4-2. Equation 13.10-14 is more restrictive than Eq. 13.10-8.
+
+Implementation. A computational procedure for central-difference integration of undamped equations of motion with possible material nonlinearity is given in Table 13.10-1 (modification of this scheme to include damping is left as an exercise). In the element internal force evaluation, $\int [B]^{T}\{\sigma\}_{n} dV$ requires the same order of quadrature as used for the element stiffness matrix $\int [B]^{T}[E][B] dV$ . Therefore, guidelines given in Section 6.11 for stiffness matrix calculation are also applicable to internal force calculation. Note that internal forces must be computed
+
+<!-- source-page: 420 -->
+
+TABLE 13.10-1. COMPUTATIONAL PROCEDURE FOR DIRECT INTEGRATION BY THE CENTRAL-DIFFERENCE METHOD, USING THE FORM GIVEN IN Eq. 13.10-12 BUT WITHOUT DAMPING.
+
+1. Set initial conditions $\{\mathbf{D}\}_{0} = \{\mathbf{D}(t = 0)\}$ and $\{\dot{\mathbf{D}}\}_{-1/2} = \{\dot{\mathbf{D}}(t = 0)\}$ , $n = 0$ .   
+2. Assemble [M].   
+3. Compute internal force $\{\mathbf{R}^{\mathrm{int}}\}_n$ as follows:   
+3.1 Loop over elements $e = 1, 2, \ldots$ ;   
+3.2 Compute element internal force $\{\mathbf{r}_e^{\mathrm{int}}\}_{n} = \int_{V_e} [\mathbf{B}]^T \{\boldsymbol{\sigma}\}_{n} dV$ ;   
+3.3 Assemble element internal force, $\{\mathbf{r}_e^{\mathrm{int}}\}_n$ , into global internal force $\{\mathbf{R}^{\mathrm{int}}\}_n$ ;   
+3.4 Next element; go to Step 3.1.   
+4. Update displacement by Eq. 13.10-12:
+
+$$
+\{\mathbf {D} \} _ {n + 1} = \Delta t ^ {2} [ \mathbf {M} ] ^ {- 1} (\{\mathbf {R} ^ {\text { ext }} \} _ {n} - \{\mathbf {R} ^ {\text { int }} \} _ {n}) + \{\mathbf {D} \} _ {n} + \Delta t \{\dot {\mathbf {D}} \} _ {n - 1 / 2}
+$$
+
+5. Update velocity [at time $(n + \frac{1}{2})\Delta t]$ by Eq. 13.10-9:
+
+$$
+\{\dot {\mathbf {D}} \} _ {n + 1 / 2} = \frac {1}{\Delta t} (\{\mathbf {D} \} _ {n + 1} - \{\mathbf {D} \} _ {n})
+$$
+
+6. Output if desired; $n \leftarrow n + 1$ , go to Step 3.
+
+at each time step. This is the most expensive part of the per-time-step cost of an explicit method. Hence, there is considerable motivation to use reduced quadrature to evaluate internal forces. For example, explicit transient analysis with the four-node bilinear quadrilateral element with one-point quadrature will be roughly one-fourth as expensive as analysis using four-point quadrature. In three dimensions, the savings are even more dramatic. However, when reduced integration is used, additional precautions must be taken to prevent mesh instabilities (discussed in Section 6.12). Flanagan and Belytschko [13.52] present an effective scheme using hourglass control, or a stabilization matrix, that adds artificial stiffness to the element to suppress the zero-energy modes (see also [13.49, 13.53, 13.54]).
+
+Stability: Estimation of $\omega_{max}$ . The central-difference method, as well as explicit methods in general, is conditionally stable. If $\Delta t$ is too large, the method fails. If $\Delta t$ is much smaller than necessary, computations are too expensive. Therefore, it is necessary to determine, or accurately bound, $\omega_{max}$ in Eq. 13.10-8 or Eq. 13.10-14. Equation 13.5-7 demonstrates that $\omega_{max}$ for the assembled finite element model is bounded by the maximum frequency of the constituent unassembled and unsupported elements. This frequency can often be computed by hand calculation.
+
+Consider the uniform linear-displacement bar element shown in Fig. 13.5-1. With lumped masses, the highest frequency of this element is given in the Example of Section 13.5 as
+
+$$
+(\omega_ {\max}) _ {e} = \frac {2 c}{L} \tag {13.10-15}
+$$
+
+where the dilatational wave speed, acoustic wave speed, or simply wave speed, $c = \sqrt{E/\rho}$ is the speed at which information travels in the bar. From Eq. 13.10-8, we must use $\Delta t \leq 2/\omega_{max}$ . Therefore, if Eq. 13.10-15 represents the maximum element frequency among all elements in a mesh, then stable integration
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_043.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_043.md
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_043.md
@@ -0,0 +1,492 @@
+<!-- source-page: 421 -->
+
+by the central difference method (Eq. 13.10-5 or Eq. 13.10-12 with [C] = [0]) requires
+
+$$
+\Delta t \leq \frac {L}{c} \tag {13.10-16}
+$$
+
+which is called the CFL condition after Courant, Friedrichs, and Lewy [13.12,13.50]. The physical interpretation of this condition is that $\Delta t$ must be small enough that information does not propagate across more than one element per time step. This observation is true only of linear-displacement elements with lumped mass. Note that if a consistent mass matrix is used, the example of Section 13.5 gives $(\omega_{\mathrm{max}})_{e} = 2\sqrt{3} (c/L)$ so that $\Delta t \leq (L/c)/\sqrt{3}$ ; this is more restrictive than the time step criterion for an element with lumped masses, Eq. 13.10-16. This result is typical. Thus, in addition to providing uncoupled equations and generally more accurate results than consistent mass matrices in explicit integration, lumped mass matrices provide for larger stable time steps.
+
+Higher-order elements yield higher maximum frequencies than lower-order elements. For this reason and for reasons discussed in Section 13.14, one may wish to avoid higher-order elements when doing explicit integration. Similarly, one should avoid penalty constraints, as large penalty numbers make $\omega_{max}$ very large.
+
+For plane and solid finite elements, it is usually difficult to analytically calculate $(\omega_{\mathrm{max}})_{e}$ . Rather, it is possible to use Eq. 13.10-16 with L replaced by an effective element diameter, $L_{e}$ . Effective element diameters are shown in Fig. 13.10-1 for some low-order displacement field elements. For higher-order displacement field elements, effective diameters are difficult to calculate. As an alternative to calculation, the highest frequency of an element (or of a global system of equations) can be accurately bounded by Gerschgorin's theorem [13.12], which, for lumped mass matrices, states that
+
+$$
+\omega_ {\max} ^ {2} \leq \max \left(k _ {i i} + \sum_ {\substack {j = 1 \\ j \neq i}} ^ {n _ {e}} | k _ {i j} |\right) / m _ {i i} \quad \text {where} \quad i = 1, 2, \dots , n _ {e} \tag{13.10 - 17}
+$$
+
+Here $n_{e}$ is the number of d.o.f. per element. Equation 13.10-17 can be applied to the assembled structure if $k_{ii}$ , $k_{ij}$ , and $m_{ii}$ are replaced by structure coefficients $K_{ii}$ , $K_{ij}$ , and $M_{ii}$ , and $n_{e}$ becomes $n_{eq}$ , the number of global d.o.f. Equation
+
+![](images/page-421_827c1b238433158a93ff68c5ff048a2670fbfd60f253916c5f9e1548ffb6bf7f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+√2L
+L
+L
+L_e = √2L
+</details>
+
+![](images/page-421_1c1ca18d94e2b621223b4edeaa662ff8172489a003bea1ee5238424debdcc5ca.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L
+L
+L
+Lₑ = L
+</details>
+
+![](images/page-421_6b341949dc98a3e4db254daa47b3c17b117dcca20f20133bc24ec2c587d43fc8.jpg)
+
+<details>
+<summary>text_image</summary>
+
+h₁
+h₂
+Lₑ = min(h₁, h₂)
+</details>
+
+Figure 13.10-1. Effective element diameters for some common two-dimensional elements. Results for three-node elements are exact. Results for the four-node element are exact if the element is rectangular ( $h_{1} = h_{2}$ ) and conservative otherwise. Inscribed circles touch at least three sides of the element.
+
+<!-- source-page: 422 -->
+
+13.10-17 fails if $m_{ii} = 0$ (or if $M_{ii} = 0$ ), as may happen with some mass allocation schemes. The difficulty of $m_{ii} = 0$ (but $M_{ii} \neq 0$ ) is addressed in [13.51].
+
+A useful definition of the relative size of time step used in direct integration is the Courant number, $C_{n}$ , defined as
+
+$$
+C _ {n} = \frac {\Delta t _ {\text { actual }}}{\Delta t _ {\text { stable }}} \tag {13.10-18}
+$$
+
+where $\Delta t_{actual}$ is the actual time step used and $\Delta t_{stable}$ is the largest time step permitted for stable explicit integration by a specified explicit method (e.g., Eqs. 13.10-8 or 13.10-14). For reasons discussed in Section 13.13, we recommend a maximum $C_{n}$ of about 0.95 to 0.98.
+
+Usually $\Delta t_{stable}$ in Eq. 13.10-18 is not precisely known because $\omega_{max}$ for the finite element mesh is not precisely known. One can determine a precise value of $\omega_{max}$ only by a somewhat inconvenient eigenvalue calculation. Instead, practitioners obtain a close upper bound to $\Delta t_{stable}$ by approximating $\omega_{max}$ by the largest unconstrained frequency among all elements, $(\omega_{\mathrm{max}})_{e}$ , as obtained perhaps from Eq. 13.10-15 or (as an upper bound) from Eq. 13.10-17.
+
+Example: Wave Propagation. Consider the uniform steel bar and tip loading shown in Fig. 13.10-2. The bar is undamped, initially at rest, and is modeled by 40 equal-length linear-displacement elements, so that $L = L_{T}/40 = 0.5$ in. The highest element frequency is given by Eq. 13.10-15 as
+
+$$
+(\omega_ {\max}) _ {e} = \frac {2}{0 . 5} \sqrt {\frac {E}{\rho}} = 8. 0 5 3 9 (1 0 ^ {5}) \mathrm{rad/sec} \tag {13.10-19}
+$$
+
+This value is very close to the maximum frequency of the finite element mesh, which is $\omega_{\max}=8.0523(10^{5})$ rad/sec. Therefore, stable integration by the central difference method requires, according to Eq. 13.10-14 with $\xi=0$ , that $\Delta t\leq2/(\omega_{\max})_{e}=2.483(10^{-6})$ sec. The computational procedure of Table 13.10-1 is implemented in the Fortran program shown in Fig. 13.10-3.
+
+The stress time-history response at the midpoint of element 20 (at x = 9.75 in.) is shown in Fig. 13.10-4 for $\Delta t = 2.4(10^{-6})$ sec ( $C_{n} = 0.966$ ) and an analysis duration equivalent to two wave traversals along the bar (83 time steps). The exact solution is also shown. Many important features of numerical solution of wave propagation problems are displayed. One is that stress at x = 9.75 in. is zero until sufficient time has elapsed for the stress wave to propagate from the instantaneously loaded tip to the bar midpoint. This is a characteristic feature of a class of partial differential equations called hyperbolic, among which the equation of motion is an example. The increase in mean compressive stress from 100 to 200 psi at 0.15 msec is due to wave reflection from the built-in end.
+
+![](images/page-422_fa3d81f96a830ba5e0babf071e4ce4932a05c1f59b5d09a99cf0ed3814c5403b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P(t)
+x
+L_T
+</details>
+
+![](images/page-422_0a4e0677a736dfb01412f2af75bd224070da45738fde5c5c6ec7e3c9d2d04659.jpg)
+
+<details>
+<summary>line</summary>
+| t | P(t), lb |
+|---|---|
+| 0 | 100 |
+| 1 | 100 |
+| 2 | 100 |
+| 3 | 100 |
+| 4 | 100 |
+| 5 | 100 |
+| 6 | 100 |
+| 7 | 100 |
+| 8 | 100 |
+| 9 | 100 |
+| 10 | 100 |
+| 11 | 100 |
+| 12 | 100 |
+| 13 | 100 |
+| 14 | 100 |
+| 15 | 100 |
+| 16 | 100 |
+| 17 | 100 |
+| 18 | 100 |
+| 19 | 100 |
+| 20 | 100 |
+| 21 | 100 |
+| 22 | 100 |
+| 23 | 100 |
+| 24 | 100 |
+| 25 | 100 |
+| 26 | 100 |
+| 27 | 100 |
+| 28 | 100 |
+| 29 | 100 |
+| 30 | 100 |
+| 31 | 100 |
+| 32 | 100 |
+| 33 | 100 |
+| 34 | 100 |
+| 35 | 100 |
+| 36 | 100 |
+| 37 | 100 |
+| 38 | 100 |
+| 39 | 100 |
+| 40 | 100 |
+| 41 | 100 |
+| 42 | 100 |
+| 43 | 100 |
+| 44 | 100 |
+| 45 | 100 |
+| 46 | 100 |
+| 47 | 100 |
+| 48 | 100 |
+| 49 | 100 |
+| 50 | 100 |
+| 51 | 100 |
+| 52 | 100 |
+| 53 | 100 |
+| 54 | 100 |
+| 55 | 100 |
+| 56 | 100 |
+| 57 | 100 |
+| 58 | 100 |
+| 59 | 100 |
+| 60 | 100 |
+| 61 | 100 |
+| 62 | 100 |
+| 63 | 100 |
+| 64 | 100 |
+| 65 | 100 |
+| 66 | 100 |
+| 67 | 100 |
+| 68 | 100 |
+| 69 | 100 |
+| 70 | 100 |
+| 71 | 100 |
+| 72 | 100 |
+| 73 | 100 |
+| 74 | 100 |
+| 75 | 100 |
+| 76 | 100 |
+| 77 | 100 |
+| 78 | 100 |
+| 79 | 100 |
+| 80 | 100 |
+| 81 | 100 |
+| 82 | 100 |
+| 83 | 100 |
+| 84 | 100 |
+| 85 | 100 |
+| 86 | 100 |
+| 87 | 100 |
+| 88 | 100 |
+| 89 | 100 |
+| 90 | 100 |
+| 91 | 100 |
+| 92 | 100 |
+| 93 | 100 |
+| 94 | 100 |
+| 95 | 100 |
+| 96 | 100 |
+| 97 | 100 |
+| 98 | 100 |
+| 99 | 100 |
+| 100 | 100 |
+</details>
+
+Figure 13.10-2. One-dimensional uniform bar with instantaneous tip loading. The bar is initially at rest. $A = 1.0 \text{ in.}^{2}$ , $E = 30(10^{6}) \text{ psi}$ , $\rho = 7.4(10^{-4}) \text{ lb-sec}^{2}/\text{in.}^{4}$ , $L_{T} = 20 \text{ in.}$ Load $P_{0} = 100 \text{ lb}$ is applied at $t = 0$ .
+
+<!-- source-page: 423 -->
+
+```prolog
+C---- Program for 1-dimensional wave propagation through NELE equal-
+C---- length finite elements using the central difference method
+    IMPLICIT DOUBLE PRECISION (A-H,O-Z)
+    DIMENSION X(101),RM(101),D(101),V(101),FEXT(101),FINT(101)
+    DATA X,RM,FEXT/101*0.,101*0.,101*0./
+
+C
+C    NELE = number of elements    ELELEN = length of element
+C    CSA = cross-sectional area    DENSTY = mass density
+C    E = elastic modulus    DELT = time step
+C    NSTEP = number of time steps    IFREQ = output interval
+
+C
+C---- Input mesh data
+    READ(5,*)NELE,ELELEN,CSA,DENSTY,E,DELT,NSTEP,IFREQ
+    NUMNOD=NELE+1
+
+C---- Generate nodal coordinates and form lumped mass matrix
+    DO 10 I=1,NUMNOD
+    10 X(I)=FLOAT(I-1)*ELELEN
+    DO 20 K=1,NELE
+    RM(K)=RM(K)+DENSTY*CSA*ELELEN/2.
+    20 RM(K+1)=RM(K+1)+DENSTY*CSA*ELELEN/2.
+
+C---- Set at-rest initial conditions and tip-load force
+    DO 30 I=1,NUMNOD
+    D(I)=0.
+    30 V(I)=0.
+    FEXT(1)=100.
+
+C---- Integration loop
+    DO 100 N=1,NSTEP
+
+C---- Get internal force for last time step
+    CALL INTFOR(D,FINT,X,E,CSA,NELE)
+
+C---- Update displacements using the central difference method and
+C---- enforce zero-displacement boundary condition at built-in end
+    DO 40 I=1,NUMNOD
+    DOLD=D(I)
+    D(I)=DELT**2*(FEXT(I)-FINT(I))/RM(I)+D(I)+DELT*V(I)
+    IF(I.EQ,NUMNOD) D(I)=0.
+
+40 V(I)=(D(I)-DOLD)/DELT
+
+C---- Output if desired
+    SIG20=E*((D(21)-D(20))/ELELEN)
+    IF(MOD(N,IFREQ).EQ.0)
+    WRITE(7,1000)N,FLOAT(N)*DELT,D(20),V(20),SIG20
+
+100 CONTINUE
+    STOP
+
+1000 FORMAT(I5,4E15.4)
+    END
+
+    SUBROUTINE INTFOR(D,FINT,X,E,CSA,NELE)
+    IMPLICIT DOUBLE PRECISION (A-H,O-Z)
+    DIMENSION D(1),FINT(1),X(1)
+    NUMNOD=NELE+1
+
+C---- Zero internal force vector
+    DO 10 I=1,NUMNOD
+    10 FINT(I)=0.
+
+C---- Loop over elements
+    DO 20 K=1,NELE
+    RL=X(K+1)-X(K)
+C---- Compute strain and stress
+    STRAIN=(D(K+1)-D(K))/RL
+    STRESS=STRAIN*E
+
+C---- Assemble contribution into internal force vector
+    F1=-STRESS*CSA
+    F2=+STRESS*CSA
+    FINT(K)=FINT(K)+F1
+    FINT(K+1)=FINT(K+1)+F2
+
+20 CONTINUE
+    RETURN
+    END 
+```  
+Figure 13.10-3. Fortran program for direct integration by the central-difference method using the procedure in Table 13.10-1.
+
+<!-- source-page: 424 -->
+
+![](images/page-424_f7aac51d8ac8b54fc3828a460d567d9ba2f7c306862d302754b6505133d4f11e.jpg)
+
+<details>
+<summary>line</summary>
+
+| Time (milliseconds) | Exact (psi) | Central difference (psi) |
+| ------------------- | ----------- | ------------------------ |
+| 0.00                | 0           | 0                        |
+| 0.04                | -100        | -100000                  |
+| 0.06                | -100        | -100000                  |
+| 0.08                | -100        | -100000                  |
+| 0.10                | -100        | -100000                  |
+| 0.12                | -100        | -100000                  |
+| 0.14                | -100        | -100000                  |
+| 0.16                | -200        | -200000                  |
+| 0.18                | -200        | -200000                  |
+| 0.20                | -200        | -200000                  |
+</details>
+
+Figure 13.10-4. Stress time history at x = 9.75 in. for a 40-element model of the 20-in. bar shown in Fig. 13.10-2 using $\Delta t = 2.4(10^{-6})$ sec ( $C_{n} = 0.966$ ). Inset shows instability that results from taking $\Delta t$ too large ( $C_{n} = 1.007$ ).
+
+Figure 13.10-4 also shows severe spurious oscillations, or noise, created by the algorithm, as the stress wave passes. Spurious oscillations are high-frequency motions that the mesh cannot accurately resolve and are similar to the Gibbs phenomenon in fitting a finite Fourier series to a piecewise-continuous function. These spurious oscillations are often a nuisance. Stiffness-proportional damping is sometimes employed to attenuate noise although it is usually better practice to employ a different integration method, such as discussed in Section 13.12, which automatically dissipates high-frequency motion.
+
+To demonstrate the instability in explicit integration that results when the maximum stable time step is exceeded, we repeated the analysis using $\Delta t = 2.5(10^{-6})$ sec ( $C_n = 1.007$ ). The results, shown in the inset of Fig. 13.10-4, are typical of instability in linear analysis and demonstrate a solution that increases wildly with each time step until eventually the computer program aborts.
+
+When $\Delta t$ is slightly less than the stability limit, the numerical solution may display spurious “beating” in which the amplitude of response repeatedly grows and decays. To illustrate beating, we repeated the foregoing analysis using $\Delta t = 2.481(10^{-6})$ sec ( $C_{n} = 0.999$ ). Results are shown in Fig. 13.10-5. We see that average results are good but spurious oscillations are severe. For some problems, particularly problems having few d.o.f., beating can be more detrimental to accuracy than shown by this example.
+
+Concluding Remarks. When an explicit method is used for a single ordinary differential equation, accuracy is markedly time-step-size-dependent. It may therefore appear that the stability criterion is of only academic interest since $\Delta t$ must be considerably smaller to achieve satisfactory accuracy. However, this supposition is not true of systems of finite element equations (about 20 equations or more) in which it is typically observed that excellent accuracy can be obtained using a time step size just under the stability limit. The reason is that the equations of motion are a stiff system of ordinary differential equations, which has a broad frequency spectrum. The stability criterion, Eq. 13.10-8, is based upon the highest frequency, or shortest time scale phenomenon that the mesh can possibly repro-
+
+<!-- source-page: 425 -->
+
+![](images/page-425_bbe6f35acb3e5b722f36c8eb2365d694e288c7864bdb4cb03c9eec5c4441d417.jpg)
+
+<details>
+<summary>line</summary>
+
+| Time (milliseconds) | Exact | Central difference, C_N = 0.999 |
+| ------------------- | ----- | ------------------------------- |
+| 0.00                | -100  | -100                            |
+| 0.02                | -100  | -100                            |
+| 0.04                | -100  | -100                            |
+| 0.06                | -100  | -100                            |
+| 0.08                | -100  | -100                            |
+| 0.10                | -100  | -100                            |
+| 0.12                | -100  | -100                            |
+| 0.14                | -100  | -100                            |
+| 0.16                | -200  | -200                            |
+| 0.18                | -200  | -200                            |
+| 0.20                | -200  | -200                            |
+</details>
+
+Figure 13.10-5. Stress time history at x = 9.75 in. for a 40-element model of the bar shown in Fig. 13.10-2 using $\Delta t = 2.481(10^{-6})$ sec ( $C_{n} = 0.999$ ). The response demonstrates “beating” from using a time step close to the stability limit.
+
+duce. It is true that motions of the mesh that occur on this time scale are not accurately resolved by taking $\Delta t$ close to the stability limit. Fortunately, these motions contribute little to the structural response, which is dominated by much lower-frequency, long-time-scale phenomena that are accurately resolved. In other words, we do not require high-frequency phenomena to be accurately resolved; all that we ask is that they be stably integrated. The implication is that in explicit methods, a $\Delta t$ satisfying stability criteria is usually satisfactory to guarantee accuracy. Even so, the maximum allowable $\Delta t$ is often smaller than one would like, because so many time steps may be needed to span the duration of an analysis.
+
+# 13.11 IMPLICIT DIRECT INTEGRATION METHODS
+
+Most of the useful implicit methods are unconditionally stable and have no restriction on the time step size other than as required for accuracy. A popular unconditionally stable implicit method is called the trapezoidal rule or the average acceleration method. (It is also known as the Crank–Nicolson method when applied to parabolic partial differential equations, such as the heat conduction equation.) The trapezoidal rule relates displacements, velocities, and accelerations by
+
+$$
+\{\mathbf {D} \} _ {n + 1} = \{\mathbf {D} \} _ {n} + \frac {\Delta t}{2} \left(\{\dot {\mathbf {D}} \} _ {n} + \{\dot {\mathbf {D}} \} _ {n + 1}\right) \tag {13.11-1}
+$$
+
+$$
+\{\dot {\mathbf {D}} \} _ {n + 1} = \{\dot {\mathbf {D}} \} _ {n} + \frac {\Delta t}{2} \left(\{\ddot {\mathbf {D}} \} _ {n} + \{\ddot {\mathbf {D}} \} _ {n + 1}\right) \tag {13.11-2}
+$$
+
+<!-- source-page: 426 -->
+
+TABLE 13.11-1. COMPUTATIONAL PROCEDURE FOR DIRECT INTEGRATION BY THE TRAPEZOIDAL-RULE METHOD.
+
+1. Form [K], [C], and [M].   
+2. Set initial conditions $\{\mathbf{D}\}_{0} = \{\mathbf{D}(t = 0)\}$ and $\{\dot{\mathbf{D}}\}_{0} = \{\dot{\mathbf{D}}(t = 0)\}$ ; use Eq. 13.9-1 to compute $\{\ddot{\mathbf{D}}\}_{0} = [\mathbf{M}]^{-1}(\{\mathbf{R}^{\mathrm{ext}}\}_{0} - [\mathbf{C}]\{\dot{\mathbf{D}}\}_{0} - [\mathbf{K}]\{\mathbf{D}\}_{0})$ ; $n = 0$ .   
+3. Form effective stiffness matrix, Eq. 13.11-6, and factor it.   
+4. Form effective load vector, Eq. 13.11-7.   
+5. Solve $\{\mathbf{K}^{\mathrm{eff}}\} \{\mathbf{D}\}_{n + 1} = \{\mathbf{R}^{\mathrm{eff}}\}_{n + 1}$ for $\{\mathbf{D}\}_{n + 1}$ by forward- and back-substitution.   
+6. Update velocity $\{\dot{\mathbf{D}}\}_{n+1}$ and acceleration $\{\ddot{\mathbf{D}}\}_{n+1}$ by Eqs. 13.11-3 and 13.11-4.   
+7. Output if desired; $n \leftarrow n + 1$ , go to Step 4.
+
+Equations 13.11-1 and 13.11-2 can be obtained using Taylor series (this is left as an exercise) in which it is seen that they are second-order accurate. Alternatively, Eqs. 13.11-1 and 13.11-2 can be solved for $\{\dot{D}\}_{n+1}$ and $\{\ddot{D}\}_{n+1}$ to provide
+
+$$
+\{\dot {\mathbf {D}} \} _ {n + 1} = \frac {2}{\Delta t} \left(\{\mathbf {D} \} _ {n + 1} - \{\mathbf {D} \} _ {n}\right) - \{\dot {\mathbf {D}} \} _ {n} \tag {13.11-3}
+$$
+
+$$
+\{\ddot {\mathbf {D}} \} _ {n + 1} = \frac {4}{\Delta t ^ {2}} \left(\{\mathbf {D} \} _ {n + 1} - \{\mathbf {D} \} _ {n}\right) - \frac {4}{\Delta t} \{\dot {\mathbf {D}} \} _ {n} - \{\ddot {\mathbf {D}} \} _ {n} \tag {13.11-4}
+$$
+
+Combination of Eqs. 13.11-3 and 13.11-4 with the equation of motion, Eq. 13.9-1 at time $(n + 1)\Delta t$ , yields
+
+$$
+[ \mathbf {K} ^ {\text { eff }} ] \{\mathbf {D} \} _ {n + 1} = \{\mathbf {R} ^ {\text { eff }} \} _ {n + 1} \tag {13.11-5}
+$$
+
+where the effective stiffness matrix and effective load vector are, respectively,
+
+$$
+[ \mathbf {K} ^ {\mathrm{eff}} ] = \frac {4}{\Delta t ^ {2}} [ \mathbf {M} ] + \frac {2}{\Delta t} [ \mathbf {C} ] + [ \mathbf {K} ] \tag {13.11-6}
+$$
+
+$$
+\begin{array}{l} \{\mathbf {R} ^ {\text { eff }} \} _ {n + 1} = \{\mathbf {R} ^ {\text { ext }} \} _ {n + 1} + [ \mathbf {M} ] \left(\frac {4}{\Delta t ^ {2}} \{\mathbf {D} \} _ {n} + \frac {4}{\Delta t} \{\dot {\mathbf {D}} \} _ {n} + \{\ddot {\mathbf {D}} \} _ {n}\right) \\ + [ \mathbf {C} ] \left(\frac {2}{\Delta t} \{\mathbf {D} \} _ {n} + \{\dot {\mathbf {D}} \} _ {n}\right) \tag {13.11-7} \\ \end{array}
+$$
+
+A flow chart for this algorithm is given in Table 13.11-1.
+
+# Remarks.
+
+1. Equation 13.11-5 is a system of coupled linear algebraic equations even if [M] and [C] are diagonal. For linear problems, $[K^{eff}]$ need be formed and factored only once. After the initial expense of factorization, time stepping can be performed for only the cost of forward- and back-substitution.   
+2. If [M] is positive definite, then $[\mathbf{K}^{\mathrm{eff}}]$ is nonsingular even if [K] permits rigid-body displacements.   
+3. The method is easily started from initial conditions $\{\mathbf{D}\}_{0}$ and $\{\dot{\mathbf{D}}\}_{0}$ and Eq. 13.10-7.
+
+<!-- source-page: 427 -->
+
+4. For problems having material nonlinearity, [K] and hence $[K^{eff}]$ are functions of $\{D\}_{n+1}$ , and possibly its time derivative, which are unknowns. Accordingly, [K] must be predicted using an estimate for $\{D\}_{n+1}$ . Equation 13.11-5 is then solved for an improved $\{D\}_{n+1}$ ; hence, the prediction of [K] is improved, and so on. For severe nonlinearity, convergence may be difficult and expensive.
+
+5. In terms of computational efficiency, Eq. 13.11-6 shows that there is no merit to using a lumped mass matrix. In Section 13.13 we show that implicit integration with consistent mass matrices is usually more accurate than with lumped mass matrices.
+
+Choice of Time Step $\Delta t$ . Since the method is numerically stable for any $\Delta t$ , time step selection is based on accuracy considerations alone. Compared with explicit methods, the per-time-step cost of an implicit method is high. Thus implicit methods are economically attractive only when $\Delta t$ can be much larger than would be used in an explicit method. Unconditional stability (which emphatically does not imply unconditional accuracy) coupled with the economic need for large $\Delta t$ tempts many analysts into using time steps that are too large. To select a time step that will provide accurate results, one must identify the highest frequency of interest in the loading or response of a structure. Let this frequency be called $\omega_{u}$ . As an approximation, structure modes with frequency higher than about $3\omega_{u}$ participate quasistatically in the response while modes with frequency lower than $3\omega_{u}$ also participate dynamically. With second-order accurate time integration methods (most popular methods are second-order accurate), a minimum of 20 time steps per period of $\omega_{u}$ should provide very good accuracy for modes that participate dynamically in the response; that is, use $\Delta t < (2\pi/\omega_{u})/20 \approx 0.3/\omega_{u}$ , unless a smaller $\Delta t$ is required because of convergence difficulties in nonlinear analysis. As an additional assurance of an accurate solution, analysis should be repeated using a smaller time step than used in the first analysis. In linear problems of a type well suited to implicit integration, a Courant number $^{3}$ $C_{n}$ of about 20 is typical although occasionally it can be as high as 100 and still yield accurate results.
+
+# 13.12 OTHER IMPLICIT AND EXPLICIT METHODS. MIXED METHODS
+
+The central-difference and trapezoidal-rule methods do not provide automatic dissipation of high-frequency numerical noise, as is sometimes desirable. In what follows, we discuss some methods that often have dissipation (also called artificial viscosity or numerical damping).
+
+The Houbolt method [13.55] is obtained by cubic Lagrange interpolation of $\{D\}$ at times $(n - 2)\Delta t$ through $(n + 1)\Delta t$ . Exact differentiation of the interpolant yields
+
+$$
+\{\dot {\mathbf {D}} \} _ {n + 1} = \frac {1}{6 \Delta t} (1 1 \{\mathbf {D} \} _ {n + 1} - 1 8 \{\mathbf {D} \} _ {n} + 9 \{\mathbf {D} \} _ {n - 1} - 2 \{\mathbf {D} \} _ {n - 2}) \tag {13.12-1}
+$$
+
+$^{3}$ The definition of $C_{n}$ , Eq. 13.10-18, employs $\Delta t_{stable}$ for explicit integration. This definition remains useful in the present context even though most implicit methods are stable for any time step.
+
+<!-- source-page: 428 -->
+
+$$
+\{\ddot {\mathbf {D}} \} _ {n + 1} = \frac {1}{\Delta t ^ {2}} (2 \{\mathbf {D} \} _ {n + 1} - 5 \{\mathbf {D} \} _ {n} + 4 \{\mathbf {D} \} _ {n - 1} - \{\mathbf {D} \} _ {n - 2}) \tag {13.12-2}
+$$
+
+This method is implicit and unconditionally stable but provides artificial damping that is too high for low-frequency response. The Houbolt method was once common in general-purpose transient codes but has been supplanted by methods with better algorithmic damping properties and now is more of historical interest.
+
+The Newmark family of methods [13.56] is very popular and is given by
+
+$$
+\{\mathbf {D} \} _ {n + 1} = \{\mathbf {D} \} _ {n} + \Delta t \{\dot {\mathbf {D}} \} _ {n} + \frac {\Delta t ^ {2}}{2} [ (1 - 2 \beta) \{\ddot {\mathbf {D}} \} _ {n} + 2 \beta \{\ddot {\mathbf {D}} \} _ {n + 1} ] \tag {13.12-3}
+$$
+
+$$
+\{\dot {\mathbf {D}} \} _ {n + 1} = \{\dot {\mathbf {D}} \} _ {n} + \Delta t [ (1 - \gamma) \{\ddot {\mathbf {D}} \} _ {n} + \gamma \{\ddot {\mathbf {D}} \} _ {n + 1} ] \tag {13.12-4}
+$$
+
+where $\beta$ and $\gamma$ are chosen by the analyst to control stability and accuracy. Substitution of Eqs. 13.12-3 and 13.12-4 into Eq. 13.9-1 at time $(n + 1)\Delta t$ yields equations similar to Eq. 13.10-5 for explicit Newmark methods ( $\beta = 0$ ) and to Eqs. 13.11-5, 13.11-6, and 13.11-7 for implicit Newmark methods ( $\beta > 0$ ). It can be shown that the stability of this algorithm is [13.49]:
+
+unconditional stability when
+
+$$
+2 \beta \geq \gamma \geq \frac {1}{2} \tag {13.12-5}
+$$
+
+conditional stability when
+
+$$
+\gamma \geq \frac {1}{2}, \quad \beta <   \frac {1}{2}, \quad \text { and } \quad \Delta t \leq \frac {\xi (\gamma - \frac {1}{2}) + \sqrt {\gamma / 2 - \beta + \xi^ {2} (\gamma - \frac {1}{2}) ^ {2}}}{\omega_ {\max} (\gamma / 2 - \beta)} \tag {13.12-6}
+$$
+
+The method is unstable for $\gamma < \frac{1}{2}$ . (As a special case, we note that Eq. 13.12-6 yields infinite $\Delta t$ when $\gamma = \frac{1}{2}$ and $\beta = \frac{1}{4}$ —that is, the trapezoidal rule without damping.) According to our definitions of implicit and explicit methods, Newmark's method is implicit unless $\gamma = \beta = 0$ , which is unstable for any $\Delta t$ and therefore cannot be used. However, most analysts refer to Newmark's method with $\beta = 0$ and $\gamma \geq \frac{1}{2}$ as being explicit in which it is noted that, for practical purposes, [C] must be null or diagonal to avoid the solution of simultaneous equations. An explicit version of the Newmark method, called a predictor-corrector algorithm, that permits nondiagonal [C] is described in [13.61]. Implementation of the explicit and implicit Newmark methods are essentially the same as the computational procedures given in Tables 13.10-1 and 13.11-1, respectively.
+
+A variety of useful techniques obtained from the Newmark family is listed in Table 13.12-1. When $\gamma = \frac{1}{2}$ , the methods have no algorithmic damping and are second-order accurate. An exception is the Fox-Goodwin method with $[C] = [0]$ , which is fourth-order accurate. Taking $\gamma > \frac{1}{2}$ introduces artificial damping, but also reduces the accuracy of the Newmark methods to first order. For implicit Newmark methods, taking
+
+$$
+\beta = \frac {1}{4} (\gamma + \frac {1}{2}) ^ {2} \tag {13.12-7}
+$$
+
+maximizes the high-frequency dissipation for a given value of $\gamma > \frac{1}{2}$ [13.46].
+
+<!-- source-page: 429 -->
+
+TABLE 13.12-1. SUMMARY OF NEWMARK METHODS: $u =$ UNDAMPED, $d =$ DAMPED. $\Omega_{\mathrm{crit}} = \omega_{\max}\Delta t_{\max}$ . STABILITY REQUIRES $\Delta t\leq \Omega_{\mathrm{crit}} / \omega_{\max}$ . 
+
+<table><tr><td></td><td>Method</td><td> $\beta$ </td><td> $\gamma$ </td><td> $\Omega_{\text{crit}}$ </td><td>Accuracy</td></tr><tr><td rowspan="4">Implicit</td><td>Artificially damped</td><td> $>\gamma/2$ </td><td> $>1/2$ </td><td> $\infty^{(u,d)}$ </td><td> $O(\Delta t)$ </td></tr><tr><td>Average acceleration (trapezoidal rule)</td><td>1/4</td><td>1/2</td><td> $\infty^{(u,d)}$ </td><td> $O(\Delta t^{2})$ </td></tr><tr><td>Linear acceleration</td><td>1/6</td><td>1/2</td><td> $2\sqrt{3} \approx 3.464^{(u)}$ Eq. (A) $^{(d)}$ </td><td> $O(\Delta t^{2})$ </td></tr><tr><td>Fox-Goodwin (royal road)</td><td>1/12</td><td>1/2</td><td> $\sqrt{6} \approx 2.449^{(u)}$ Eq. (A) $^{(d)}$ </td><td> $O(\Delta t^{4})^{(u)}$  $O(\Delta t^{2})^{(d)}$ </td></tr><tr><td rowspan="2">Explicit</td><td>Central difference [M], [C] diagonal</td><td>0</td><td>1/2</td><td> $2^{(u)}$ Eq. (A) $^{(d)}$ </td><td> $O(\Delta t^{2})$ </td></tr><tr><td>Artificially damped [M], [C] diagonal</td><td>0</td><td> $>1/2$ </td><td>Eq. (A) $^{(u,d)}$ </td><td> $O(\Delta t)$ </td></tr></table>$$
+\Omega_ {\mathrm{crit}} = \frac {\xi (\gamma - \frac {1}{2}) + \sqrt {\gamma / 2 - \beta + \xi^ {2} (\gamma - \frac {1}{2}) ^ {2}}}{\gamma / 2 - \beta} \tag {A}
+$$
+
+Interestingly, the presence of damping in the explicit Newmark method raises the stability limit. This is in contrast to Eqs. 13.10-8 and 13.10-14 for other forms of the central-difference method in which no change and a decrease in stability limit, respectively, are observed. Thus, if the fraction $\xi$ of critical damping at $\omega_{\mathrm{max}}$ is not known, a conservative $\Delta t$ is obtained by taking $\xi = 0$ in Eq. 13.12-6. However, practical necessity dictates that [C] be diagonal in the explicit Newmark method, thus the lagged central-difference algorithm, Eq. 13.10-12 (or a predictor-corrector algorithm [13.61]) is preferable for problems in which physically realistic spectral damping is to be modeled.
+
+The Newmark linear-acceleration method appears to be ideal for problems such as earthquake shaking response analysis in which piecewise linear-acceleration records are typically used as excitation. Unfortunately, this implicit method is only conditionally stable. The Wilson- $\theta$ method [13.18,13.28,13.46,13.57] is also a linear-acceleration method, but is unconditionally stable.
+
+A disadvantage of the Newmark methods is that algorithmic damping can only be obtained at the expense of reduced accuracy. The $\alpha$ -method, proposed by Hilber, Hughes, and Taylor [13.58], does not have this weakness, and with appropriate choice of parameters retains second-order accuracy and provides effective high-frequency dissipation. The method uses the Newmark formulas, Eqs. 13.12-3 and 13.12-4, with the modified equation of motion
+
+$$
+\begin{array}{l} [ \mathbf {M} ] \{\ddot {\mathbf {D}} \} _ {n + 1} + (1 + \alpha) [ \mathbf {C} ] \{\dot {\mathbf {D}} \} _ {n + 1} - \alpha [ \mathbf {C} ] \{\dot {\mathbf {D}} \} _ {n} + (1 + \alpha) [ \mathbf {K} ] \{\mathbf {D} \} _ {n + 1} \\ - \alpha [ \mathbf {K} ] \{\mathbf {D} \} _ {n} = (1 + \alpha) \{\mathbf {R} _ {n + 1} ^ {\mathrm{ext}} - \alpha \{\mathbf {R} _ {n} ^ {\mathrm{ext}} \} \tag {13.12-8} \\ \end{array}
+$$
+
+If the parameters are selected so that $-\frac{1}{3} \leq \alpha \leq 0$ , $\gamma = (1 - 2\alpha)/2$ , and $\beta = (1 - \alpha)^{2}/4$ , the method is implicit, unconditionally stable, and second-order accurate [13.46]. When these guidelines are used, with $\alpha = 0$ , the method reduces to the trapezoidal rule, which has no dissipation. Decreasing $\alpha$ increases the amount of numerical damping.
+
+<!-- source-page: 430 -->
+
+Mixed Methods. A current trend in time integration analysis is to create algorithms that combine explicit and implicit methods, so as to capitalize on the strong points of each. Such schemes are called mixed integration methods. Belystschko and Mullen [13.59,13.60] developed a method that uses a nodal partition to separate nodes into implicit and explicit groups. D.o.f. in the explicit and implicit groups are then integrated by explicit and implicit methods respectively. The motivation for such an approach is that structures often contain spatial subdomains having markedly different time scales. For example, in fluid–structure interaction problems, the time scales associated with the fluid are usually much longer than those associated with the structure. By using an implicit method for the structure and an explicit method for the fluid, we exploit the strong points of each integrator. Hughes and Liu [13.61,13.62] developed an implicit–explicit method similar to Refs. [13.59,13.60] but more implementationally attractive. It uses element partitions rather than nodal partitions. Operator-splitting methods are mixed methods in which a nonlinear material constitutive law is split into parts that give rise to time-dependent and time-independent terms that are integrated explicitly and implicitly, respectively [13.63,13.64]. Element-by-element implicit methods have been developed in which a conventional implicit method is used except that Eqs. 13.11-5 are approximately solved using element-level calculations only [13.65,13.66,13.67]. Thus these methods appear to have the good stability characteristics of implicit methods with the low per-time-step cost of explicit methods. At present, however, these methods are not sufficiently robust and success is very problem-dependent.
+
+# 13.13 STABILITY ANALYSIS. ACCURACY OF DIRECT INTEGRATION METHODS
+
+In Sections 13.10 and 13.11 many stability and accuracy properties of direct integration methods are stated. In the present section we substantiate some of these properties and offer further suggestions for use of the methods. References [13.18,13.46,13.49] contain extensive discussions.
+
+Stability. When we examine stability, it is sufficient to consider the homogeneous form of the equation of motion obtained by taking $\{R^{ext}\} = \{0\}$ . The idea is that if a solution procedure is stable with no external loading, then it will also be stable if $\{R^{ext}\}$ is nonzero but bounded. A number of methods for assessing stability are possible. They fall into two broad categories. In the first, called spectral or Fourier stability, one examines the effects of a time integration method on a single equation of motion obtained by modally uncoupling the original structure equations. In the second, called energy stability, one deals with the original structure matrices and establishes the conditions under which a norm of the solution at time $n \Delta t$ can be bounded by a norm of the solution at time zero. Spectral stability usually provides more insight and sometimes more precise results. Energy methods sometimes provide results that are slightly more conservative than spectral methods, but can be applied to complicated problems to which spectral methods may be inapplicable. References on energy stability techniques include [13.46,13.49, 13.61,13.63].
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_044.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_044.md
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+<!-- source-page: 431 -->
+
+Spectral Stability: Central-Difference Method. To illustrate spectral stability, we consider the central-difference method applied to the undamped, uncoupled, homogeneous equation of motion (Eq. 13.6-5 with subscript i dropped for clarity):
+
+$$
+\ddot {Z} + \omega^ {2} Z = 0 \tag {13.13-1}
+$$
+
+The solution of Eq. 13.13-1 is purely oscillatory and is given by Eq. 13.13-15. Using the central-difference method, Eq. 13.10-2, we can write a difference approximation of Eq. 13.13-1 at time $n \Delta t$ as
+
+$$
+Z _ {n + 1} + \left(\omega^ {2} \Delta t ^ {2} - 2\right) Z _ {n} + Z _ {n - 1} = 0 \tag {13.13-2}
+$$
+
+Because the central-difference method is a two-step method, the exact solution of the difference approximation, Eq. 13.13-2, can be shown to consist of a linear combination of two parts,
+
+$$
+Z _ {n} = C _ {1} \lambda_ {1} ^ {n} + C _ {2} \lambda_ {2} ^ {n} \quad \text { when } \lambda_ {1} \neq \lambda_ {2} \tag {13.13-3a}
+$$
+
+$$
+Z _ {n} = C _ {1} \lambda_ {1} ^ {n} + n \Delta t C _ {2} \lambda_ {1} ^ {n} \quad \text { when } \lambda_ {1} = \lambda_ {2} \tag {13.13-3b}
+$$
+
+where
+
+$$
+\lambda_ {1} = e ^ {\mu_ {1} \Delta t} \quad \text { and } \quad \lambda_ {2} = e ^ {\mu_ {2} \Delta t} \tag {13.13-4}
+$$
+
+in which $\mu_{1}$ and $\mu_{2}$ (and hence $\lambda_{1}$ and $\lambda_{2}$ ) are generally complex. Constants $C_{1}$ and $C_{2}$ can be determined from initial conditions.
+
+The essence of the stability argument is embodied in Eqs. 13.13-3. Assuming for now that $\lambda_{1} \neq \lambda_{2}$ , Eq. 13.13-3a provides an unbounded solution for $Z_{n}$ if $|\lambda_{1}| > 1$ or if $|\lambda_{2}| > 1$ . This is instability. If $|\lambda_{1}| \leq 1$ and $|\lambda_{2}| \leq 1$ , then $Z_{n}$ will decay or remain steady with time, thus providing stable computation.
+
+If $\lambda_{1} = \lambda_{2}$ , then even though $|\lambda_{1}| = |\lambda_{2}| = 1$ , $Z_{n}$ will not be bounded and will grow in arithmetic fashion due to the second term of Eq. 13.13-3b. If so, computation will be unstable. Although this situation arises infrequently in practical finite element models, it is well to be aware of its possible occurrence.
+
+For methods other than central difference, it is possible to have $\lambda_{1} = \lambda_{2}$ , $|\lambda_{1}| < 1$ , and $|\lambda_{2}| < 1$ , in which case $Z_{n}$ of Eq. 13.3-3b will be bounded.
+
+To determine the criterion for $\Delta t$ that provides a stable computation, we proceed as follows. We first assume that $\lambda_1 \neq \lambda_2$ and select initial conditions such that $C_1$ or $C_2$ is zero. Equation 13.13-3a then becomes
+
+$$
+Z _ {n} = C \lambda^ {n}, \quad \text { where } \quad \lambda = e ^ {\mu \Delta t} \tag {13.13-5}
+$$
+
+and where subscripts have been dropped from C and $\lambda$ . Combining Eq. 13.13-5 with Eq. 13.13-2 and dividing by $C\lambda^{n}$ , we obtain the characteristic equation
+
+$$
+\lambda^ {2} + \left(\omega^ {2} \Delta t ^ {2} - 2\right) \lambda + 1 = 0 \tag {13.13-6}
+$$
+
+Solving Eq. 13.13-6 provides the two solutions for $\lambda$ ,
+
+$$
+\lambda_ {1, 2} = \frac {1}{2} \left(2 - \omega^ {2} \Delta t ^ {2} \pm \omega \Delta t \sqrt {\omega^ {2} \Delta t ^ {2} - 4}\right) \tag {13.13-7}
+$$
+
+<!-- source-page: 432 -->
+
+Although it is possible to determine the $\omega \Delta t$ that satisfy $|\lambda_1| \leq 1$ and $|\lambda_2| \leq 1$ directly from Eq. 13.13-7, it is more expedient to note that the solutions of a quadratic equation $a\lambda^2 + b\lambda + c = 0$ satisfy $\lambda_1\lambda_2 = c / a$ , where $c / a = 1$ for Eq. 13.13-6. If the radicand of Eq. 13.13-7 is positive, then both $\lambda_1$ and $\lambda_2$ are real and $\lambda_1\lambda_2 = 1$ indicates that the absolute value of one $\lambda$ is less than unity while the absolute value of the other $\lambda$ is greater than unity. Thus instability results since one $\lambda$ does not satisfy $|\lambda| \leq 1$ . If the radicand of Eq. 13.13-7 is negative, then $\lambda_1$ and $\lambda_2$ are complex conjugates and each is of unit modulus by virtue of $\lambda_1\lambda_2 = 1$ . Thus stable computation always results if the radicand of Eq. 13.13-7 is negative. If the radicand of Eq. 13.13-7 is zero, then $\lambda_1 = \lambda_2 = -1$ . Since the characteristic equation has repeated solutions, Eq. 13.13-3b is the appropriate expression for $Z_n$ in which it is observed that with $|\lambda_1| = 1$ , the solution will diverge in arithmetic fashion due to the second term in Eq. 13.13-3b. Hence, stable computation requires
+
+$$
+\Delta t <   \frac {2}{\omega} \tag {13.13-8}
+$$
+
+This result depends on the form of differential equation and the type of difference approximation. If one of these is changed, say by the addition of damping to Eq. 13.13-1, then in general the stability criterion is also changed.
+
+Note that Eq. 13.13-1 is just one of many uncoupled equations and that our intent is to solve the system of coupled equations by direct integration. Thus it is necessary to evaluate Eq. 13.13-6 for each of the $n_{eq}$ frequencies of the model and select the $\Delta t$ that is most restrictive. This yields
+
+$$
+\Delta t <   \frac {2}{\omega_ {\max}} \tag {13.13-9}
+$$
+
+where $\omega_{\mathrm{max}}$ is the highest of the $n_{\mathrm{eq}}$ natural frequencies.
+
+Equation 13.13-9 usually appears in the literature as $\Delta t \leq 2 / \omega_{\max}$ . However, as discussed earlier, using $\Delta t = 2 / \omega_{\max}$ in central-difference integration of the undamped equations of motion yields instability and hence should be avoided. In practical problems, $\omega_{\max}$ is rarely known and, as discussed in Section 13.10, is usually bounded by the maximum element frequency among all elements, $(\omega_{\max})_e$ . In practical problems, the case $(\omega_{\max})_e = \omega_{\max}$ is rare, so that $(\omega_{\max})_e > \omega_{\max}$ almost always prevails. Accordingly, the time step $\Delta t \leq 2 / (\omega_{\max})_e$ almost always provides stable computation.
+
+Spectral Stability: Trapezoidal Rule. To analyze the stability of the undamped, homogeneous equation of motion when integrated by the trapezoidal rule, we begin by summing Eq. 13.13-1 at times $(n - 1)\Delta t$ and $(n + 1)\Delta t$ with twice Eq. 13.13-1 at time $n\Delta t$ . This provides
+
+$$
+\ddot {Z} _ {n + 1} + 2 \ddot {Z} _ {n} + \ddot {Z} _ {n - 1} + \omega^ {2} \left(Z _ {n + 1} + 2 Z _ {n} + Z _ {n - 1}\right) = 0 \tag {13.13-10}
+$$
+
+To express the trapezoidal rule in terms of accelerations and displacements only, we subtract Eq. 13.11-1 at time $n \Delta t$ from Eq. 13.11-1 at time $(n + 1) \Delta t$ and then combine with Eq. 13.11-2 written at times $(n + 1) \Delta t$ and $n \Delta t$ to eliminate velocities. This provides the trapezoidal formula for second derivatives
+
+<!-- source-page: 433 -->
+
+$$
+Z _ {n + 1} - 2 Z _ {n} + Z _ {n - 1} = \frac {\Delta t ^ {2}}{4} \left(\ddot {Z} _ {n + 1} + 2 \ddot {Z} _ {n} + \ddot {Z} _ {n - 1}\right) \tag {13.13-11}
+$$
+
+Combining Eqs. 13.13-10 and 13.13-11 to eliminate accelerations provides
+
+$$
+(1 + h) Z _ {n + 1} + (2 h - 2) Z _ {n} + (1 + h) Z _ {n - 1} = 0, \quad \text { where } \quad h = \frac {\omega^ {2} \Delta t ^ {2}}{4} \tag {13.13-12}
+$$
+
+Assuming that $\lambda_1 \neq \lambda_2$ , we combine Eqs. 13.13-12 and 13.13-5 and divide by $\lambda^{n-1}$ to obtain the characteristic equation
+
+$$
+(1 + h) \lambda^ {2} + (2 h - 2) \lambda + 1 + h = 0 \tag {13.13-13}
+$$
+
+Solutions of Eq. 13.13-13 for $\lambda$ are
+
+$$
+\lambda_ {1, 2} = \frac {1 - h \pm 2 \sqrt {- h}}{1 + h}, \quad \text { where } \quad h = \frac {\omega^ {2} \Delta t ^ {2}}{4} \tag {13.13-14}
+$$
+
+In accordance with the discussion following Eq. 13.13-7, $\lambda_{1}\lambda_{2}=1$ for the trapezoidal rule. In addition, note that the radicand of Eq. 13.13-14 is always negative. Therefore, $\lambda_{1}$ and $\lambda_{2}$ are complex conjugates (hence distinct) and both of unit modulus. Thus $|\lambda|\leq1$ is satisfied regardless of the value of h. Therefore, trapezoidal-rule integration of the undamped equations of motion is unconditionally stable. The same is true when damping is included, although we have not proved it.
+
+Amplitude and Period Error. To analyze the errors in using the central-difference method and the trapezoidal rule, which are representative of explicit and implicit methods in general, we again focus attention on a single, uncoupled homogeneous equation, Eq. 13.13-1. The exact solution is harmonic and can be written as
+
+$$
+Z _ {n} ^ {\text { exact }} = \widetilde {C} _ {1} (\cos \omega t + i \sin \omega t) + \widetilde {C} _ {2} (\cos \omega t - i \sin \omega t) \tag {13.13-15}
+$$
+
+where $\tilde{C}_{1}$ and $\tilde{C}_{2}$ are determined from initial conditions, and $i = \sqrt{-1}$ . The approximate solution obtained by direct integration may display amplitude error and period error. Amplitude error can be either amplitude increase, which is the same as instability, or amplitude decay, which is more commonly called artificial damping or viscosity. Period error can be either period elongation or period contraction. These errors are shown in Fig. 13.13-1.
+
+Because $\lambda_{1}$ and $\lambda_{2}$ are complex conjugates (in both the central-difference and trapezoidal-rule methods), $\mu_{1}$ and $\mu_{2}$ in Eq. 13.13-4 are also complex conjugates and can be written as
+
+$$
+\mu_ {1} = a + i b \quad \text { and } \quad \mu_ {2} = a - i b \tag {13.13-16}
+$$
+
+where $a$ and $b$ are real. By use of Eq. 13.13-16, Eq. 13.13-4 can be written as
+
+$$
+\lambda_ {1} ^ {n} = e ^ {a n \Delta t} e ^ {i b n \Delta t} = e ^ {a n \Delta t} (\cos b n \Delta t + i \sin b n \Delta t) \tag {13.13-17a}
+$$
+
+<!-- source-page: 434 -->
+
+![](images/page-434_e48197fe02ec36f9b706c132f5231e159b698cbea29010dabd4537e08f590967.jpg)
+
+<details>
+<summary>line</summary>
+
+| Time Segment              | Displacement, Z(t) |
+| ------------------------- | ------------------ |
+| Exact harmonic response   | Minimum            |
+| Period contraction         | Maximum            |
+| Possible direct integration solution | Minimum            |
+| Amplitude decay           | Maximum            |
+</details>
+
+Figure 13.13-1. Possible errors in direct integration.
+
+$$
+\lambda_ {2} ^ {n} = e ^ {a n \Delta t} e ^ {- i b n \Delta t} = e ^ {a n \Delta t} (\cos b n \Delta t - i \sin b n \Delta t) \tag {13.13-17b}
+$$
+
+in which the latter forms are obtained by use of DeMoivre's theorem. Combination of Eqs. 13.13-17 and 13.13-3a provides the exact solution to the central-difference approximation:
+
+$$
+Z _ {n} = C _ {1} e ^ {a n \Delta t} (\cos b n \Delta t + i \sin b n \Delta t) + C _ {2} e ^ {a n \Delta t} (\cos b n \Delta t - i \sin b n \Delta t) \tag {13.13-18}
+$$
+
+Comparison of Eqs. 13.13-15 and 13.13-18 shows that amplitude error will result unless a = 0 and period error will result unless $b = \omega$ .
+
+We define period error, $P$ , by
+
+$$
+P = \frac {2 \pi / b}{2 \pi / \omega} = \frac {\omega}{b} \tag {13.13-19}
+$$
+
+where $2\pi/\omega$ and $2\pi/b$ are respectively the period of the actual system and the period of the system created by the time integration algorithm. The following types of error are possible:
+
+$$
+P > 1 \quad \text { period   elongation }
+$$
+
+$$
+P = 1 \quad \text { no   period   error }
+$$
+
+$$
+P <   1 \quad \text { period   contraction }
+$$
+
+To determine b, we note from Eq. 13.13-17a with n = 1 that
+
+$$
+\frac {\mathrm{Im} (\lambda)}{\mathrm{Re} (\lambda)} = \frac {\sin b \Delta t}{\cos b \Delta t} = \tan b \Delta t \tag {13.13-20}
+$$
+
+<!-- source-page: 435 -->
+
+For the central-difference method, Eqs. 13.13-7 and 13.13-20 give the period, b, of the numerically integrated solution as
+
+$$
+b = \frac {1}{\Delta t} \tan^ {- 1} \frac {\omega \Delta t \sqrt {4 - \omega^ {2} \Delta t ^ {2}}}{2 - \omega^ {2} \Delta t ^ {2}} \tag {13.13-21}
+$$
+
+Combination of Eqs. 13.13-19 and 13.13-21 yields the period error of the central-difference method as
+
+$$
+P = \omega \Delta t \left[ \tan^ {- 1} \frac {\operatorname{Im} (\lambda)}{\operatorname{Re} (\lambda)} \right] ^ {- 1} = \omega \Delta t \left[ \tan^ {- 1} \frac {\omega \Delta t \sqrt {4 - \omega^ {2} \Delta t ^ {2}}}{2 - \omega^ {2} \Delta t ^ {2}} \right] ^ {- 1} \tag {13.13-22}
+$$
+
+in which the arctangent function is required to yield a positive angle. Equation 13.13-22 represents period contraction and is plotted in Fig. 13.13-2. For the trapezoidal rule, Eqs. 13.13-14, 13.13-19, and 13.13-20 give the period error
+
+$$
+P = \omega \Delta t \left[ \tan^ {- 1} \frac {4 \omega \Delta t}{4 - \omega^ {2} \Delta t ^ {2}} \right] ^ {- 1} \tag {13.13-23}
+$$
+
+in which the arctangent function is required to yield a positive angle. This represents period elongation and is plotted in Fig. 13.13-2.
+
+Figure 13.13-2 suggests guidelines for mass-matrix selection in direct integration. Natural frequencies obtained by use of consistent mass matrices are overestimated. Hence the periods of modes are underestimated, or contracted. When consistent mass matrices are used with the trapezoidal rule, which has period elongation, period errors in the direct integration of the equations of motion are partially compensatory. This observation is true of implicit methods in general. Use of lumped mass matrices usually underestimates natural frequencies and hence overestimates, or elongates, periods. When lumped mass matrices are used with the central-difference method, period errors are partially compensatory. This observation is also true of explicit methods in general.
+
+![](images/page-435_651eda027a0ab4d9bb83d405e422b0c3a50a227823b45d3113a40c30d3a9dd5a.jpg)
+
+<details>
+<summary>line</summary>
+| ωΔt | Period error, P |
+| --- | --- |
+| 0 | 1 |
+| 2 | 2/π |
+| 6 | >2 |
+</details>
+
+Figure 13.13-2. Period errors for the central-difference and trapezoidal-rule methods.
+
+<!-- source-page: 436 -->
+
+We emphasize that Fig. 13.13-2 pertains to the mode whose frequency is $\omega$ . A multi-d.o.f. structure has many modes. For structural analysis one selects a $\Delta t$ such that $\omega \Delta t$ is small for all modes of practical interest.
+
+The following amplitude errors are possible:
+
+$$
+a > 0 \quad \text { amplitude   growth   (instability) }
+$$
+
+$$
+a = 0 \quad \text { no   amplitude   error }
+$$
+
+$$
+a <   0 \quad \text { amplitude   decay   (artificial   damping) }
+$$
+
+Neither the central-difference nor the trapezoidal-rule method has amplitude error. This can be seen immediately by noting that, when integration is stable, Eq. 13.13-3a applies, and $\left|\lambda_{1}\right|=\left|\lambda_{2}\right|=1$ (although $\lambda_{1}\neq\lambda_{2}$ ). Hence, Eq. 13.13-3a shows that, although $Z_{n}$ may be periodic, it will not grow or decay.
+
+To show in a more formal manner that the central-difference method does not have amplitude error, we consider the sum $\lambda_1 + \lambda_2$ as given by Eqs. 13.13-7 and 13.13-17 with $n = 1$ . This provides
+
+$$
+e ^ {a \Delta t} \cos b \Delta t = 1 - \frac {1}{2} \omega^ {2} \Delta t ^ {2} \tag {13.13-24}
+$$
+
+Combining Eq. 13.13-24 with Eq. 13.13-21 provides
+
+$$
+a \Delta t = \ln \frac {1 - \omega^ {2} \Delta t ^ {2} / 2}{\cos \left[ \tan^ {- 1} \frac {\omega \Delta t \sqrt {4 - \omega^ {2} \Delta t ^ {2}}}{2 - \omega^ {2} \Delta t ^ {2}} \right]} \tag {13.13-25}
+$$
+
+Evaluation of Eq. 13.13-25 for $0 \leq \omega \Delta t < 2$ shows that $a \Delta t = 0$ , thus verifying that the central-difference method has no amplitude error. However, this does not imply that the amplitude of response as predicted by the central-difference method will agree with the exact response. This is particularly true when using stable time steps that are very close to the stability limit. To show this, consider Eqs. 13.13-15 and 13.13-18. When $\lambda_{1}$ and $\lambda_{2}$ are distinct, $C_{1}$ and $C_{2}$ in Eq. 13.13-18 are close approximations of $\widetilde{C}_{1}$ and $\widetilde{C}_{2}$ in Eq. 13.13-15. When $\Delta t$ is very close to the stability limit, $\lambda_{1}$ and $\lambda_{2}$ are almost equal and the expressions within the parentheses of Eq. 13.13-18 are almost linear combinations of one another. In this situation, $C_{1}$ and $C_{2}$ may differ greatly from $\widetilde{C}_{1}$ and $\widetilde{C}_{2}$ . In fact, it is for this reason that the “beating” phenomenon of Fig. 13.10-5 is displayed when $\Delta t$ is very close to the stability limit. What is implied by the statement of zero-amplitude error is that the envelope of the numerical solution has a mean value that does not grow or decay in comparison with the exact solution.
+
+Following the same procedure used for the central-difference method, we obtain, for the trapezoidal rule,
+
+$$
+a \Delta t = \ln \frac {4 - \omega^ {2} \Delta t ^ {2}}{(4 + \omega^ {2} \Delta t ^ {2}) \cos \left[ \tan^ {- 1} \left(\frac {4 \omega \Delta t}{4 - \omega^ {2} \Delta t ^ {2}}\right) \right]} \tag {13.13-26}
+$$
+
+which gives $a \Delta t = 0$ for all values of $\omega \Delta t$ , and therefore implies zero-amplitude error.
+
+<!-- source-page: 437 -->
+
+# 13.14 CONCLUDING REMARKS ON TIME-HISTORY ANALYSIS
+
+Choice of Method. The choice of method for time-history analysis is strongly problem-dependent. The efficiency of a given method depends on whether the problem is of a wave propagation or a structural dynamics type, the time span for which analysis is required, whether response is linear or nonlinear, and the topology of the finite element mesh.
+
+In wave propagation problems the excitation is usually rich in high-frequency components. Time scales of interest are short and of the order of the acoustic wave traversal time across a structure. Usually we are interested in observing the passage of stress waves through elements and the transients produced. In structural dynamics problems, the excitation and response are characterized by low-frequency, long-time-scale components. Analysis duration is usually long compared to that normally required for a wave propagation problem.
+
+Modal superposition methods are economical when only a small portion of the total number of vibration modes of a model need be used for superposition. Wave propagation problems would require a very large number of modes to be included. Therefore, superposition methods are generally not appropriate for wave propagation problems. In structural dynamics problems, excitation and structural response are dominated by low-frequency components; hence superposition methods can be very effective.
+
+If material response becomes nonlinear or deformations become large, a structure's eigenvalues and eigenvectors change. Because of the expense of solving eigenproblems, it is not prudent to continuously update eigenpairs during nonlinear response. Rather, most modal methods for nonlinear problems treat nonlinearities by pseudoload techniques in which loads that account for nonlinearities are transferred to the right-hand side of the equation of motion. These methods are not robust. Convergence is strongly problem-dependent and is often poor. Generally speaking, direct integration methods are preferable for nonlinear problems.
+
+With explicit methods of direct integration, stability typically requires that the time step be small enough that information does not propagate across more than one element per time step (e.g., the CFL condition). Explicit methods are ideal for wave propagation problems in which behavior at the stress wave front is of engineering importance. Here the stability restriction is not a serious disadvantage because a small $\Delta t$ is necessary for accuracy. Other factors in favor of explicit time integration are easy implementation, accurate treatment of general nonlinearities, and the capability of treating very large problems with only modest computer storage requirements. For structural dynamics problems, time scales and analysis durations are usually long and accuracy considerations alone would permit a $\Delta t$ much larger than the upper limit of $\Delta t$ for stable explicit integration. Although explicit methods are often used for structural dynamics problems, they are not as well suited to this class of problems as they are to wave propagation problems.
+
+The only advantage of implicit methods over explicit methods is that they allow a much larger $\Delta t$ because they are unconditionally stable (conditionally stable implicit methods are not often used). Implicit methods are expensive for wave propagation problems since accuracy requires a small $\Delta t$ . For long-duration structural dynamics problems, implicit methods are usually more effective than explicit methods, although this depends on mesh topology and severity of nonlinearities.
+
+<!-- source-page: 438 -->
+
+Compared with explicit methods, implicit methods are more difficult to implement, particularly for nonlinear problems, and they require considerably more computer storage.
+
+Choice of Element and Mesh. When discretizing a structure or a medium, an analyst can choose from simple, low-order elements such as the linear-displacement bar, quadratic beam, and bilinear quadrilateral, or from higher-order elements such as the quadratic Lagrange and serendipity quadrilaterals. In wave propagation problems, discontinuities of strain propagate throughout the model. Lower-order displacement elements are more adept at modeling these discontinuities than are higher-order elements, which tend to produce more numerical noise. Structural dynamics problems tend to have strain fields that vary smoothly with time. Hence, higher-order elements can be used to more advantage than in wave propagation problems. Higher-order elements can also be used effectively in eigenvalue problems.
+
+Guidelines discussed in Chapter 19 for construction of finite element meshes for quasistatic problems are also useful for dynamic problems. Thus, static or dynamic stress analysis requires a finer mesh (particularly near stress raisers) than does quasistatic deflection analysis or calculation of lower vibration frequencies. A dynamic problem may require more elements than the analogous quasistatic problem. For example, in the problem of Fig. 13.10-2, only one linear-displacement bar element is necessary for the quasistatic solution, whereas many more elements are necessary to capture the essential features of the dynamic problem or to calculate accurate natural frequencies and mode shapes.
+
+Element sizes should not change abruptly. If they do, the mass matrix will be a poor discrete representation of the actual continuous mass distribution of the structure. This gives rise to artificial wave reflections and additional numerical noise when waves cross boundaries between elements of markedly different size.
+
+With explicit methods, a lumped mass matrix is preferred for reasons of economy and accuracy. With implicit methods, a consistent mass matrix is preferred for accuracy and is only slightly detrimental to economy.
+
+# PROBLEMS
+
+# Section 13.1
+
+13.1 A single d.o.f. spring–mass system has natural frequency $\omega_{1} = \sqrt{k/m}$ . It is excited by a force $P_{0} \sin \omega_{2} t$ . What is the limiting value of the ratio $\omega_{2}/\omega_{1}$ such that the amplitude of motion differs from the static displacement by less than 10%?
+
+# Section 13.2
+
+13.2 Show that under constant acceleration $\{\ddot{d}\}$ , nodal “loads” $[m]\{\ddot{d}\}$ with $[m]$ given by Eq. 13.2-5 are the same as the body force nodal loads given by Eq. 4.1-6.
+
+# Section 13.3
+
+13.3 (a) Can a diagonal coefficient in a consistent mass matrix ever be negative? Explain.
+
+<!-- source-page: 439 -->
+
+(b) Imagine that a beam is vibrated so that nodes of the vibration mode coincide with nodes of the finite element mesh. Would you prefer the consistent mass matrix of Eq. 13.3-2 or the lumped mass matrix of Eq. 13.3-3 with $\alpha = 0$ ? Is Eq. 13.3-3 with $\alpha \neq 0$ an acceptable alternative?
+
+13.4 (a) Derive the consistent mass matrix given by Eq. 13.3-1.
+
+(b) Imagine that the cross-sectional area of the bar shown in the figure varies linearly from area $A_{0}$ at the left end to area $\gamma A_{0}$ at the right end, where $\gamma$ is a constant. Determine the consistent mass matrix associated with d.o.f. $u_{1}$ and $u_{2}$ .
+
+(c) Show that [m] and [m] in Eq. 13.3-1 each yield the correct nodal forces under a rigid-body translational acceleration in the bar's axial direction.
+
+![](images/page-439_aae2c496f61b831b31d8ff0f53578b64b58bbbe561f17e5e8996fa8b6aac05cd.jpg)
+
+<details>
+<summary>text_image</summary>
+
+u₁
+u₂
+2
+x
+L
+</details>
+
+Problem 13.4
+
+![](images/page-439_f1b94810fe230a943fedc454c0d9a8df5c4ac76cf35db01e0d3fbba6b9535dc0.jpg)
+
+<details>
+<summary>text_image</summary>
+
+u₁
+2
+u₂
+3
+u₃
+1
+x
+L
+2
+L
+2
+</details>
+
+Problem 13.7
+
+13.5 (a) Derive the consistent mass matrix given by Eq. 13.3-2.
+
+(b) Show that Eq. 13.3-2 yields the correct nodal forces and moments under a rigid-body translational acceleration transverse to the axis of the beam.
+
+13.6 Determine the consistent mass matrix of the constant-strain triangle (Fig. 4.2-3). The element has uniform density and thickness. Arrange d.o.f. in the order $\{\mathbf{d}\} = \left[u_1 u_2 u_3 v_1 v_2 v_3\right]^T$ .
+
+13.7 (a) Derive the consistent mass matrix that operates on d.o.f. $u_{1}, u_{2}$ and $u_{3}$ for the uniform quadratic-displacement bar shown.
+
+(b) Using heuristic arguments, derive an ad hoc lumped matrix for this element that agrees with HRZ lumping (Eq. 13.3-9).
+
+13.8 Determine $\alpha$ in Eq. 13.3-3 such that the beam element has the correct kinetic energy $I\omega^2 / 2$ under rigid-body rotation about its center. Then consider modeling a simply supported beam with one element and computing the natural frequencies of vibration. What conclusions can you draw from this problem?
+
+13.9 For the HRZ mass-lumping procedure:
+
+(a) Verify Eqs. 13.3-4 and 13.3-5.   
+(b) Verify the nodal masses shown in parentheses for the element shown in Fig. 13.3-3a.   
+(c) Verify the nodal masses shown in parentheses for the element shown in Fig. 13.3-3b.   
+(d) Determine the lumped mass matrix $\lceil \mathbf{m} \rceil$ for the constant-strain triangle element of Problem 13.6. Consider $x$ - and $y$ -direction accelerations separately.
+
+13.10 For optimal lumping by quadrature verify the nodal masses for the following elements shown in Fig. 13.3-4.
+
+(a) The quadratic-displacement (six-node) triangle.   
+(b) The quadratic-displacement Lagrange (nine-node) quadrilateral.
+
+<!-- source-page: 440 -->
+
+13.11 For a uniform beam element having the usual four d.o.f. (see, e.g., Fig. 13.3-1b), consider the lateral-displacement field
+
+$$
+w = \left\lfloor (1 - \xi), (\xi - \xi^ {2}) L / 2, \xi , (- \xi + \xi^ {2}) L / 2 \right\rfloor \left\lfloor w _ {1} \quad \theta_ {1} \quad w _ {2} \quad \theta_ {2} \right\rfloor^ {T}
+$$
+
+where $\xi = x / L$ .
+
+(a) Show that $w$ is linear in $x$ if $\theta_1 = \theta_2$ . Also show that this field yields the correct displacement $w$ and curvature $w_{,xx}$ under pure bending.   
+(b) Use Eq. 13.2-5 to evaluate the mass matrix.   
+(c) Obtain a diagonal mass matrix by applying the HRZ procedure.
+
+# Section 13.4
+
+13.12 (a) Determine the Rayleigh proportional damping constants $\alpha$ and $\beta$ for fractions of critical damping of 3% and 20% at frequencies of 5 and 15 Hz, respectively.
+
+(b) For the values of $\alpha$ and $\beta$ determined in part (a), draw a graph similar to Fig. 13.4-1. Comment on the fraction of critical damping experienced by frequencies below 5 Hz and above 15 Hz. Is caution called for?
+
+13.13 Consider a particle that is allowed to free-fall from at-rest initial conditions under its own weight due to gravity. If the particle has mass-proportional damping, then the equation governing its velocity v is $\dot{v} + \beta v = g$ , where g is the acceleration due to gravity. (Remark: The same equation governs the velocity of a particle allowed to sink in a viscous fluid where $\beta$ is related to a fluid viscosity.)
+
+(a) Determine the analytic solution for v.   
+(b) Consider the ratio of the damped velocity to the undamped velocity (i.e., $v_{\text{undamped}} = gt$ ) for values of $t$ of about 1 second. Does this ratio offer guidelines on what values of $\beta$ are permissible without excessively damping rigid-body modes?
+
+# Section 13.5
+
+13.14 (a) Show that the Rayleigh quotient, Eq. 13.5-4, can be regarded as stating an equality between the maximum strain and kinetic energies associated with mode $\{\overline{\mathbf{D}}\}$ .
+
+(b) Imagine that redesign produces small changes in [M] and [K]. Hence, the natural frequency $\lambda_{i}$ of each mode $\{\overline{\mathbf{D}}\}_{i}$ is slightly changed, by an amount $\Delta \lambda_{i}$ . Using the Rayleigh quotient and neglecting terms of higher order, derive an expression for $\Delta \lambda_{i}$ in terms of $\lambda_{i}$ , $\{\overline{\mathbf{D}}\}_{i}$ , [M], [ΔK], and [ΔM].
+
+13.15 (a) Prove that the lower inequality of Eq. 13.5-5 is true. Suggestion: Express $\{\mathbf{v}\}$ as a linear combination of eigenvectors (each normalized as in Eq. 13.6-2), factor the common term $\lambda_{\min}$ out of the numerator, and argue that when $\{\mathbf{v}\} \neq \{\overline{\mathbf{D}}_{\min}\}$ the numerator is too large and hence overestimates $\lambda_{\min}$ .
+
+(b) Prove that the upper inequality of Eq. 13.5-5 is true.
+
+13.16 Consider the following stiffness and mass matrices:
+
+$$
+[ \mathbf {K} ] = \left[ \begin{array}{c c} 2 & - 2 \\ - 2 & 5 \end{array} \right] \quad [ \mathbf {M} ] = \left[ \begin{array}{c c} 1 & 0 \\ 0 & 1 \end{array} \right]
+$$
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_045.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_045.md
new file mode 100644
index 00000000..f35c9911
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_045.md
@@ -0,0 +1,409 @@
+<!-- source-page: 441 -->
+
+Exact eigenvalues and eigenvectors for axial vibration are $\lambda_1 = 1$ , $\lambda_2 = 6$ , $\{\overline{\mathbf{D}}\}_1 = [2 - 1]^T$ , and $\{\overline{\mathbf{D}}\}_2 = [1 - 2]^T$ . Consider approximate eigenvectors $[1.7 - 1.0]^T$ and $[1.2 - 2.0]^T$ and show that the Rayleigh quotient provides an accurate estimate of $\lambda_1$ and $\lambda_2$ .
+
+13.17 Consider axial vibrations of a uniform bar of length L and mass $m = \rho AL$ , free at one end and fixed at the other. Using two-node bar elements, model the bar first by one element, then by two elements of equal length L/2. In each case, compute the lowest natural frequency using
+
+(a) the consistent mass matrix [m].   
+(b) the lumped mass matrix [m].   
+(c) the average mass matrix ([m] + [m])/2.
+
+The exact lowest natural frequency is $\omega_{\mathrm{I}} = (\pi /2L)\sqrt{E / \rho}$ .
+
+13.18 In Problem 13.17 consider the convergence rates of $\lambda_{\mathrm{min}}$ as the mesh is refined. Are they in accord with the rates predicted in Section 13.3?
+
+13.19 The stiffness, consistent mass, and optimally lumped mass matrices for the unsupported, uniform, three-node quadratic bar element shown are
+
+$$
+\frac {A E}{3 L} \left[ \begin{array}{r r r} 7 & - 8 & 1 \\ - 8 & 1 6 & - 8 \\ 1 & - 8 & 7 \end{array} \right], \quad \frac {\rho A L}{3 0} \left[ \begin{array}{r r r} 4 & 2 & - 1 \\ 2 & 1 6 & 2 \\ - 1 & 2 & 4 \end{array} \right], \quad \frac {\rho A L}{6} \left[ \begin{array}{r r r} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{array} \right]
+$$
+
+(a) For axial vibration, determine the three natural frequencies and mode shapes using the consistent mass matrix.   
+(b) Repeat part (a) using the optimally lumped mass matrix.   
+(c) What is the physical significance of the lowest frequency and mode for parts (a) and (b)?   
+(d) The exact frequencies of an unsupported continuous bar of length $L$ are $\omega_{n} = (n\pi / L) \sqrt{E / \rho}$ ; $n = 0, 1, 2, \ldots$ . What are the percentage errors of the frequencies computed in parts (a) and (b)?
+
+![](images/page-441_c3b2221a3b79a9e3bf78ea9ee11825792879061506091a13bf926def7fb154ba.jpg)
+
+<details>
+<summary>text_image</summary>
+
+u₁
+2 → u₂
+3 → u₃
+1
+x
+L
+2
+L
+2
+</details>
+
+Problem 13.19
+
+13.20 Fix one end of the three-node bar treated in Problem 13.19. Determine the two natural frequencies and mode shapes of axial vibration.
+
+(a) Use the consistent mass matrix.   
+(b) Use the optimally lumped mass matrix.   
+(c) Use ad hoc lumping in which particles of mass $\rho AL/3$ are placed at each node.   
+(d) For parts (a), (b), and (c), estimate the lowest frequency by means of the Rayleigh quotient and the assumed displacement mode $u_{1} = 0$ , $u_{2} = 1$ , and $u_{3} = 2$ .
+
+13.21 Model a simply supported uniform beam of length 2L by a single element. Determine the natural frequencies of vibration, where possible, by using
+
+<!-- source-page: 442 -->
+
+the mass matrices cited here. Use the element stiffness matrix given in Eq. 4.2-5. The exact fundamental frequency is $\omega_{1} = (\pi^{2}/4L^{2})\sqrt{EI/\rho A}$ .
+
+(a) The consistent mass matrix [m] for a bar element given by the first of Eqs. 13.3-1 with d.o.f. $u_{1}$ and $u_{2}$ discarded.   
+(b) The consistent mass matrix [m] for a beam element given by Eq. 13.3-2.   
+(c) The lumped mass matrix $\lceil \mathbf{m} \rfloor$ are given by Eq. 13.3-3 with $\alpha = 0$ .   
+(d) Repeat part (c) with $\alpha = 17.5$ .   
+(e) The lumped mass matrix $\lceil m \rceil$ given by Eq. 13.3-5.   
+(f) The matrix [m] given by Problem 13.11(b).
+
+13.22 Consider a uniform cantilever beam of length $L$ , modeled by a single beam element. Repeat parts (a) through (f) of Problem 13.21. The exact fundamental frequency is $\omega = (3.516 / L^2) \sqrt{EI / \rho A}$ .
+
+13.23 A uniform beam is clamped at both ends. In order to exploit geometric symmetry, only the left half of the beam is modeled for vibration analysis. How would you ensure that analysis of the left half yields all natural frequencies of the original clamped-clamped beam?
+
+# Section 13.6
+
+13.24 (a) Verify Eqs. 13.6-1.
+
+(b) Explain a procedure for normalizing a vector (as, for example, $\{\overline{\mathbf{D}}\}_i$ is normalized with respect to [M] in Eq. 13.6-2).   
+(c) Show that Eqs. 13.6-1 are true for the modes stated in Eqs. 13.5-10.   
+(d) Show that Eqs. 13.6-1 are true for the exact modes stated in Problem 13.16.
+
+13.25 Verify Eq. 13.4-2. Suggestion: Compare the uncoupled equation of motion with Rayleigh damping to the single d.o.f. equation $\ddot{Z} + 2\xi\omega\dot{Z} + \omega^{2}Z = p$ .
+
+13.26 Consider the stiffness and lumped mass matrices given in Problem 13.16. Derive the system of uncoupled ordinary differential equations for a mode displacement response analysis.
+
+13.27 (a) Let $k = m = 1$ in the spring-mass system shown. Let this system be set in axial motion with initial conditions $u_{1} = u_{2} = \dot{u}_{1} = 0, \dot{u}_{2} = 1$ . Compute $u_{1}$ and $u_{2}$ at times $t = 1, 2, 3, 4$ , and 5 by use of the mode displacement method. Include both modes of the original system in $[\phi]$ . (b) What is the greatest percentage error in $u_{1}$ and $u_{2}$ if only the lowest mode is used, so that $[\phi]$ becomes a column vector?
+
+![](images/page-442_1dc7b49c447819f8327d83d09780c05fdfe9e6fff6511ea6e30ee31242d10c70.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k
+m
+k
+m
+1
+2
+→ u₁
+→ u₂
+</details>
+
+Problem 13.27
+
+13.28 Verify Eq. 13.6-11. Suggestion: Insert $[\phi]^{-T}[\phi]^T$ after $[\mathbf{K}]^{-1}$ in the second term of Eq. 13.6-10. Then use orthogonality and the relation $[\phi]^{-1}[\mathbf{K}]^{-1} = [\omega^2]^{-1}[\phi]^T$ to obtain Eq. 13.6-11.
+
+13.29 Consider the two-spring, two-mass system described in Problem 13.27. Let this system be undeformed and at rest at time $t = 0$ . For each of the
+
+<!-- source-page: 443 -->
+
+following two loadings, use the mode displacement method and then the mode acceleration method to determine $u_{1} = u_{1}(t)$ and $u_{2} = u_{2}(t)$ . Retain only the lowest mode $\{\phi\}_{1}$ in the transformation. Compute numerical values of $u_{1}$ and $u_{2}$ at times t = 2, 4, 6, 8, and 10.
+
+(a) Node 1 is not loaded. A force $F_{2} = 1$ is applied to node 2 at $t = 0$ .   
+(b) Forces $F_{1} = 1$ and $F_{2} = -1$ are applied to nodes 1 and 2 respectively at $t = 0$ .
+
+13.30 Show that the mode acceleration method reduces to the mode displacement method if the structure moves freely—that is, with $\{R^{ext}\} = \{0\}$ .
+
+13.31 Consider applying a Ritz vector analysis to the system of two springs and two masses described in Problem 13.27.
+
+(a) Externally applied loads are zero so arbitrarily assign $\{\mathbf{w}^{*}\}_{1} = [1 - 0]^{T}$ in Table 13.6-1. Hence, establish a 2 by 2 array [W] of Ritz vectors and the transformed system of Eq. 13.6-18.   
+(b) Repeat part (a), now using $\{\mathbf{w}^{*}\}_{1} = [0 - 1]^{T}$ .   
+(c) If $\{\mathbf{w}^{*}\}_{1}$ is arbitrarily taken as $[1 - 2]^{T}$ , and no additional vectors are used, what is the resulting form of Eq. 13.6-18? What fundamental frequency $\omega$ does this equation yield? By what other name do you know this method of calculating $\omega$ ?
+
+# Section 13.7
+
+13.32 Mass condensation, starting from Eq. 13.7-1, would be more accurate if the assumption $[M_{ms}] = [M_{ss}] = [0]$ were not made. What is an objection to this approach?
+
+13.33 (a) Show that Eq. 13.7-5 yields $[\mathbf{K}_r] = [\mathbf{K}_{mm}] - [\mathbf{K}_{ms}][\mathbf{K}_{ss}]^{-1}[\mathbf{K}_{ms}]^T$ . Where has the relation been seen before?
+
+(b) Derive a similar expression for $[M_{r}]$ from Eq. 13.7-5.
+
+13.34 If $\lambda[\mathbf{M}]\{\overline{\mathbf{D}}\}$ in Eq. 13.5-2 is regarded as a vector of inertia loads $\{\mathbf{R}\}$ , and $[\mathbf{K}]$ is inverted to become the flexibility matrix $[\mathbf{F}]$ , we can write $[\mathbf{F}]\{\mathbf{R}\} = \{\mathbf{D}\}$ .
+
+(a) Partition this equation into $m$ master and $s$ slave d.o.f. as in Eq. 13.7-1 and let $\{\mathbf{R}_s\} = \{\mathbf{0}\}$ . Derive the transformation
+
+$$
+\left\{ \begin{array}{l} \overline {{\mathbf {D}}} _ {m} \\ \overline {{\mathbf {D}}} _ {s} \end{array} \right\} = [ \mathbf {T} ] \{\overline {{\mathbf {D}}} _ {m} \}, \quad \text { where } \quad [ \mathbf {T} ] = \left[ \begin{array}{l} \mathbf {I} \\ \mathbf {F} _ {m s} ^ {T} \mathbf {F} _ {m m} ^ {- 1} \end{array} \right]
+$$
+
+(b) Show that this transformation is mathematically the same as that of Eq. 13.7-3.   
+(c) How can $[\mathbf{F}_{mm}]$ be computed from [K] and what is its physical meaning?   
+(d) Why is the transformation of part (a) likely to be more computationally efficient than the form used in Eq. 13.7-3?
+
+13.35 Consider the two-d.o.f. unsupported bar of Fig. 13.5-1. What is the reduced stiffness matrix that results from taking $u_{2}$ as a slave d.o.f.? Is this result reasonable?
+
+13.36 Consider application of the procedure for automatic selection of master d.o.f. described in Section 13.7 to the structure shown. Only axial motion is permitted. Recall from Section 2.11 that condensation of a d.o.f. places
+
+<!-- source-page: 444 -->
+
+![](images/page-444_38f8daead9108a5018c63ba13f0417e2b87395ed2d8bc531453f733d39e01c46.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k
+m
+k
+m
+k
+m
+k
+m
+k
+m
+1
+2
+3
+4
+5
+x, u
+</details>
+
+Problem 13.36
+
+the two adjacent springs in series. To simplify this problem, but for no sound theoretical reason, assume that condensation of a mass m effectively adds mass m/2 to the two adjacent masses, so that [M] remains diagonal. For the structure shown,
+
+(a) choose two masters by making three left-to-right sweeps.   
+(b) choose two masters by making three right-to-left sweeps.   
+(c) determine the frequencies $\omega_{1}$ and $\omega_{2}$ in parts (a) and (b) and compare results. The exact first two frequencies for the five-d.o.f. structure are $0.2846\sqrt{k/m}$ and $0.8308\sqrt{k/m}$ .
+
+13.37 (a) The frequency $\omega_1^2 = 6.1765EI / mL^3$ is computed below Eq. 13.7-12 in the example that closes Section 13.7. Improve this estimate, if possible, by using $\overline{w}_1 = 1$ and the first $\overline{\theta}_2$ of Eq. 13.7-13 in the Rayleigh quotient (Eq. 13.5-4). Use [K] and [M] from Eq. 13.7-8.
+
+(b) Repeat part (a) using the second $\bar{\theta}_2$ of Eq. 13.7-13.
+
+13.38 (a) In the example problem that closes Section 13.7, is the choice of $\overline{w}_1$ as master and $\overline{\theta}_2$ as slave consistent with the rule of largest $M_{ii} / K_{ii}$ ?
+
+(b) Make the other choice, $\bar{\theta}_{2}$ as master and $\overline{w}_{1}$ as slave, and compute the frequency and mode shape (analogous to Eqs. 13.7-12 and 13.7-13).   
+(c) Improve the estimate of $\omega_{1}$ from part (b) by using its mode shape in the Rayleigh quotient, Eq. 13.5-4, with [K] and [M] taken from Eq. 13.7-8.
+
+13.39 Apply mass condensation to the system shown. Only axial motion is permitted. Let $k = 1$ and $m = 2$ . Determine the fundamental vibration frequency of the reduced system and compare it with the exact value for the original system. Determine the fundamental mode of the reduced system, using first Eq. 13.7-3 and then Eq. 13.7-7. Finally, obtain improved estimates of $\omega_{1}$ by using each of these modes in the Rayleigh quotient.
+
+![](images/page-444_c41c30bb872cc06d118dffc2dade27dfc7b2f0fad2fdaf3bb4cdb8b1fe0e7b45.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k
+m
+k
+m
+1
+2
+→ u₁
+→ u₂
+</details>
+
+Problem 13.39
+
+13.40 Many methods of solving large eigenproblems require factoring either the stiffness matrix or a combination of the stiffness and mass matrices (e.g., the determinant search and subspace iteration methods). Factoring requires approximately $n_{eq}b^{2}/2$ operations (i.e., multiplications) where $n_{eq}$ is the number of equations and b is the semibandwidth. For full matrices, the number of operations is about $n_{eq}^{3}/6$ . Consider a system of 5000 equations with b = 500. If this system of equations is partitioned into m master and s slave d.o.f., what must m be so that factoring the condensed (full) system is no more expensive than factoring the original (banded) system? What if b = 100 instead?
+
+<!-- source-page: 445 -->
+
+# Section 13.8
+
+13.41 Repeat the example that closes Section 13.8 using only the first vector of assembled component normal modes plus the additional mode that accounts for having node 3 fixed in the component mode analyses (i.e., omit the second column of Eq. 13.8-12 and the second column of Eq. 13.8-14).
+
+13.42 Consider axial vibration of the spring–mass system shown with k = 1 and m = 1. Natural frequencies are $\omega_{1} = 0.9246$ , $\omega_{2} = 1.574$ , and $\omega_{3} = 2.381$ . Create one substructure consisting of the springs to the left of node 2 and another substructure consisting of the springs to the right of node 2. Using component mode synthesis, determine the two natural frequencies of the reduced structure.
+
+(a) Use method CMS1, following the example in the text.   
+(b) Repeat part (a) except omit the last Ritz vector obtained by applying a unit load to node 2 (only one frequency can be determined).   
+(c) Use method CMS2, following the example in the text.   
+(d) Repeat part (c) except omit the last Ritz vector obtained by applying a unit displacement to node 2 (only one frequency can be determined).
+
+![](images/page-445_b368c7de4409b7fd9852d2c01da57b90b6acf2fc179b83cd76b4de3637a5cd7b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k m k m 2k m 2k
+1 2 3
+</details>
+
+Problem 13.42
+
+13.43 Repeat Problem 13.42, but let each of the four springs have stiffness $k = 1$ . Natural frequencies and mode shapes of the original structure are
+
+$$
+\begin{array}{l} \omega_ {1} = 0. 7 6 5 4 \quad \{\overline {{{\mathbf {D}}}} \} _ {1} = \left[ \begin{array}{l l l} 1 & \sqrt {2} & 1 \end{array} \right] ^ {T} \\ \omega_ {2} = 1. 4 1 4 \quad \{\overline {{{\mathbf {D}}}} \} _ {2} = \left[ - 1 0 1 \right] ^ {T} \\ \omega_ {3} = 1. 8 4 8 \quad \{\overline {{{\mathbf {D}}}} \} _ {3} = \left\lfloor 1 - \sqrt {2} 1 \right] ^ {T} \\ \end{array}
+$$
+
+In some cases, a frequency of the original structure may be missing from the reduced structure. Why?
+
+# Section 13.9
+
+13.44 The forward and backward Euler direct integration methods are defined by
+
+$$
+\begin{array}{l} \{\mathbf {D} \} _ {n + 1} = \{\mathbf {D} \} _ {n} + \Delta t \{\dot {\mathbf {D}} \} _ {n} \quad \text { forward   Euler } \\ \{\mathbf {D} \} _ {n + 1} = \{\mathbf {D} \} _ {n} + \Delta t \{\dot {\mathbf {D}} \} _ {n + 1} \quad \text { backward   Euler } \\ \end{array}
+$$
+
+Are these methods explicit or implicit?
+
+# Section 13.10
+
+13.45 Using the Taylor series expansion, determine the order of accuracy of the direct integration methods defined in Problem 13.44.
+
+13.46 Verify Eq. 13.10-12.
+
+13.47 Show that Eq. 13.10-12 reduces to Eq. 13.10-5 if damping is zero.
+
+13.48 (a) Apply the Gerschgorin bound, Eq. 13.10-17, with element coefficients replaced by structure coefficients and $n_e$ replaced by $n_{\mathrm{eq}}$ , to obtain a
+
+<!-- source-page: 446 -->
+
+bound on the maximum mesh frequency in terms of $E$ , $\rho$ , and $L$ for the model and boundary conditions shown in Fig. 13.10-2. Then divide the bar into 40 elements of equal length and evaluate the bound numerically. (b) Show that the bound obtained in part (a) agrees precisely with the element bound, Eq. 13.10-15. (Note: Usually the Gerschgorin and element bounds do not agree.)
+
+13.49 Consider a model consisting of one linear-displacement bar finite element, with lumped mass, and one end fixed. Do the Gerschgorin bound, Eq. 13.10-17, and the element bound, Eqs. 13.5-7 and 13.10-15, show good agreement with the exact frequency of this model?   
+13.50 Consider a uniform free-free bar (i.e., an unsupported bar) modeled by equal-length, linear-displacement finite elements. For such a situation, the maximum frequency of the entire model and the maximum unconstrained element frequency are the same. Why?   
+13.51 In the example of Section 13.10, the element bound was within one-thousandth of a percent of the maximum mesh frequency. Do you expect the agreement to improve or deteriorate as the number of elements in the mesh increases? Why?   
+13.52 A particle of unit mass is supported by a spring of unit stiffness. There is no damping and no external load. Thus $k = m = \omega = 1$ . At time $t = 0$ , the particle has zero displacement, zero acceleration, but unit velocity. Use the central-difference method, Eq. 13.10-5, to compute displacement versus time over five time steps. Use a $\Delta t$ of (a) 1.0, (b) $\sqrt{2.0}$ , (c) 2.0, and (d) 3.0. Does there appear to be an amplitude error? Why?   
+13.53 Repeat the example of Section 13.10 for the bar shown in Fig. 13.10-2 but with the right-hand end unsupported. Modify the Fortran program and compare the displacement, velocity, and stress time histories at the position x = 9.75 in. with those for the bar in Fig. 13.10-2. Experiment with time steps of different size. Also examine the time history solutions for position x = 4.75 in.   
+13.54 Repeat the example of Section 13.10 with a mesh of nonuniform length elements. For the first 15 in. of the bar, use 30 elements, each of length 0.5 in. For the remaining 5 in. of the bar, use five elements of length 1 in. Compare the velocity and stress time-histories at the position $x = 9.75$ in. with those of the example of Section 13.10 and comment on any differences.   
+13.55 For central-difference integration of equations of motion with Rayleigh damping, derive an equation analogous to Eqs. 13.10-5 and 13.10-12, and modify the computational procedure of Table 13.10-1. In deriving this algorithm, write the viscous forces as $\alpha[\mathbf{K}]\{\dot{\mathbf{D}}\}_{n-1/2} + \beta[\mathbf{M}]\{\dot{\mathbf{D}}\}_n$ . For the mass-proportional part of the damping, approximate $\{\dot{\mathbf{D}}\}_n$ by Eq. 13.10-1. Note that the stiffness-proportional part of the viscous forces can be obtained element-by-element by summation of the element contributions $\alpha \int [\mathbf{B}]^T\{\dot{\boldsymbol{\sigma}}\}_{n-1/2} dV$ (verify this). Based on Eqs. 13.10-8 and 13.10-14, can you suggest what the stability criterion for this scheme will be?   
+13.56 Using the damping algorithm developed in Problem 13.55, modify the Fortran program of Fig. 13.10-3 to include Rayleigh damping. Use stiffness-proportional damping to give 20% critical damping at the frequency given by Eq. 13.10-15 and zero mass-proportional damping. Repeat the example
+
+<!-- source-page: 447 -->
+
+problem of Section 13.10 and compare results. Experiment with different mass- and stiffness-proportional damping constants.
+
+# Section 13.11
+
+13.57 Derive Eq. 13.11-1 using Taylor series and show that $\{D\}_{n+1}$ is approximated with an error of $O(\Delta t^{2})$ . Suggestion: Write Taylor series for $\{D\}_{n+1}$ about time $n \Delta t$ and $\{D\}_{n}$ about time $(n + 1) \Delta t$ and then combine them to obtain Eq. 13.11-1 plus higher-order terms.   
+13.58 Verify Eqs. 13.11-3 through 13.11-7.   
+13.59 Why do you think the name “trapezoidal rule” is applied to Eqs. 13.11-1 and 13.11-2?   
+13.60 Repeat Problem 13.52, using the trapezoidal rule (Table 13.11-1). Use four time steps, of magnitude (a) $\Delta t = 2.0$ , and (b) $\Delta t = 1.0$ .   
+13.61 In seismic analysis of structures, 30 Hz is usually used as a cutoff frequency; that is, the excitation is composed of components with frequencies lower than 30 Hz. Using this cutoff frequency, what is the largest $\Delta t$ that should be used for an unconditionally stable implicit method to give accurate results?
+
+# Section 13.12
+
+13.62 Show that when $\beta = \frac{1}{4}$ and $\gamma = \frac{1}{2}$ , Eqs. 13.12-3 and 13.12-4 can be expressed as Eqs. 13.11-1 and 13.11-2.
+
+13.63 Why is the term “linear acceleration” used when $\beta = \frac{1}{6}$ in Table 13.12-1?
+
+13.64 (a) Use the explicit Newmark method, Eqs. 13.12-3 and 13.12-4 with $\beta = 0$ , to derive a computational procedure analogous to Table 13.10-1. State whether [M] and [C] must be diagonal.
+
+(b) Does this procedure have any advantages in comparison with the central-difference procedure of Eq. 13.10-5?
+
+13.65 (a) Using the results of Problem 13.64, modify the Fortran program of Fig. 13.10-3 to use the explicit Newmark method ( $\beta = 0, \gamma \geq \frac{1}{2}$ ).
+
+(b) Repeat the example of Section 13.10 using the algorithm of part (a) with $\gamma = 0.5$ (no artificial damping).
+
+(c) Repeat the example of Section 13.10 using the algorithm of part (a) with $\gamma = 0.6, 0.7, 0.8, 0.9, 1.0$ (increasing artificial damping).
+
+13.66 Use the implicit Newmark method, Eqs. 13.12-3 and 13.12-4 with $\beta > 0$ , to derive equations analogous to Eqs. 13.11-5 through 13.11-7.
+
+# Section 13.13
+
+13.67 Consider the uncoupled homogeneous equation of motion with damping but without inertia: $2\xi\omega\dot{Z} + \omega^{2}Z = 0$ . Using spectral stability, determine the stability criterion for direct integration of this equation by:
+
+(a) The central-difference method, Eq. 13.10-1.
+
+(b) The trapezoidal rule. Suggestion: Sum the equation of motion at times $n \Delta t$ and $(n + 1) \Delta t$ and combine with Eq. 13.11-1 to eliminate velocities.
+
+(c) The forward Euler method defined in Problem 13.44.
+
+(d) The backward Euler method defined in Problem 13.44.
+
+<!-- source-page: 448 -->
+
+13.68 Consider Problem 13.52 again, in which the central-difference method (Eq. 13.10-5) is applied to a spring-mass system for which $k = m = \omega = 1$ . Now use $\Delta t = \sqrt{3.96}$ and start the algorithm using $u_0 = 0$ and $u_{-1} = -1$ . Follow the motion for at least ten cycles, and observe that the computed amplitude displays "beating" but no net growth.
+
+13.69 (a) Derive Eq. 13.13-23.
+
+(b) Derive Eq. 13.13-26.
+
+13.70 (a) Numerically evaluate Eqs. 13.13-22 and 13.13-25 using $\omega \Delta t = 0, 1, \sqrt{2}$ , 2 for the period and amplitude errors of the central-difference method.
+
+(b) Numerically evaluate Eq. 13.13-23 and 13.13-26 using $\omega \Delta t = 0, 1, 2, 4$ for the period and amplitude errors of the trapezoidal rule.
+
+(c) Analytically show that Eqs. 13.13-25 and 13.13-26 reduce to forms that yield $a\Delta t = 0$ for all values of $\omega \Delta t$ .
+
+13.71 Consider the central-difference solutions obtained in Problem 13.52, parts (a), (b), and (c). What is the period error in each case? Check that the values you obtain agree with Eq. 13.13-22.
+
+13.72 Consider the trapezoidal-rule solutions obtained in Problem 13.60, parts (a) and (b). Using approximations as necessary, determine the period and the period error of each solution. Check that the values you obtain agree with Eq. 13.13-23.
+
+# Section 13.14
+
+13.73 The uncoupled equations produced by a modal analysis (Section 13.6) have a lower $\omega_{max}$ than $\omega_{max}$ of the full system. Hence, in integrating the uncoupled equations, what are the relative merits of explicit and implicit methods? How does the specific choice of modal method affect your answer?
+
+<!-- source-page: 449 -->
+
+# STRESS STIFFENING AND BUCKLING
+
+Bending stiffness is affected by membrane forces. Matrices that account for this effect are formulated and applied to problems such as buckling. The nature of the buckling problem is discussed and warnings given against oversimplification.
+
+# 14.1 INTRODUCTION
+
+Buckling of bars, frames, plates, and shells may occur as a structural response to membrane forces. Membrane forces act along member axes and tangent to plate and shell midsurfaces. The membrane force in a bar (or column) is the axial load. Membrane forces in a shell are defined by Eqs. 12.1-1.
+
+Buckling occurs when a member or a structure converts membrane strain energy into strain energy of bending with no change in externally applied load. A critical condition, at which buckling impends, exists when it is possible that the deformation state may change slightly in a way that makes the loss in membrane strain energy numerically equal to the gain in bending strain energy. In a slender bar of length L, axial stiffness AE/L is much greater than bending stiffness $EI/L^{3}$ . Similarly, in a thin-walled structure such as a shell, membrane stiffness is typically orders of magnitude greater than bending stiffness. Accordingly, small membrane deformations can store a large amount of strain energy, but comparatively large lateral deflections and cross-section rotations are needed to absorb this energy in bending deformations.
+
+One can also take the view that membrane forces alter the bending stiffness of a structure. Thus buckling occurs when compressive membrane forces are large enough to reduce the bending stiffness to zero for some physically possible deformation mode. If the membrane forces are reversed—that is, made tensile rather than compressive—bending stiffness is effectively increased. This effect is called stress stiffening.
+
+The effects of membrane forces are accounted for by a matrix $[k_{\sigma}]$ that augments the conventional stiffness matrix [k]. Matrix $[k_{\sigma}]$ has been given various names, as follows: initial stress stiffness matrix, differential stiffness matrix, geometric stiffness matrix, and stability coefficient matrix. In what follows we give $[k_{\sigma}]$ the name stress stiffness matrix. Matrix $[k_{\sigma}]$ is defined by an element's geometry, displacement field, and state of stress. Thus, $[k_{\sigma}]$ is independent of elastic properties. (However, by introducing the stress–strain relation, $[k_{\sigma}]$ can alternatively be written in terms of elastic properties and strains or deformations.) The structure matrix $[K_{\sigma}]$ is built by summing overlapping terms of element matrices $[k_{\sigma}]$ , in the same way that the conventional [K] is built by summing overlapping terms of element matrices [k].
+
+Analysis of a Beam-Column. Consider the simply supported beam shown in Fig.
+
+<!-- source-page: 450 -->
+
+![](images/page-450_0ab964ce596b4e7e060ffdb863cf3a60ab0d493a1c9be07a9d347c2a7785248b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+L
+2
+w_c
+P
+P
+x
+q
+L
+</details>
+
+$\{n\}$
+
+![](images/page-450_8413934eb8f24909a132a6f85bd7862944e59ba888cb1dde3549688e142c7fe5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ds = (1 + w_x^2)^(1/2) dx
+w_x dx
+dx
+</details>
+
+(b)   
+Figure 14.1-1. (a) A uniform beam on simple supports. (b) Geometric relations for a differential element of length dx.
+
+14.1-1. Axial force P, positive in tension, is regarded as being imposed at the outset, for example, by cooling the bar while not allowing its ends to move toward one another. We will use energy concepts and a single d.o.f. to illustrate the stiffening effect of axial force P and to derive the buckling load $P_{cr} = -\pi^{2}EI/L^{2}$ , where the negative sign indicates compression.
+
+Strain energy in bending is given by the standard expression that involves the square of curvature $w_{,xx}$ :
+
+$$
+U _ {b} = \frac {1}{2} \int_ {0} ^ {L} E I w _ {, x x} ^ {2} d x \tag {14.1-1}
+$$
+
+Let a small lateral displacement $w = w(x)$ take place. Thus each differential length dx is changed to a new length ds, where ds > dx because the distance between supports is not allowed to change. From Fig. 14.1-1b,
+
+$$
+d s = (1 + w _ {, x} ^ {2}) ^ {1 / 2} d x \approx \left(1 + \frac {w _ {, x} ^ {2}}{2}\right) d x \tag {14.1-2}
+$$
+
+where the latter approximation comes from the first two terms of the binomial expansion. The approximation is valid if $w_{;x}^{2}<<1$ , which restricts this development to small rotations. Axial membrane strain in the bar is therefore
+
+$$
+\epsilon_ {m} = \frac {d s - d x}{d x} \approx \frac {w _ {, x} ^ {2}}{2} \tag {14.1-3}
+$$
+
+In the linear theory of elasticity we ignore terms of order $w_{,x}^{2}$ . But here we seek the consequences of retaining the more important of the higher-order terms that linear theory neglects. We are taking a physical approach to formulating these terms. They may also be obtained by a systematic procedure of linearization, as will be touched upon in Section 14.4.
+
+During a small lateral displacement $w = w(x)$ , axial force P in the bar remains practically constant. As each elemental length dx lengthens an amount $\epsilon_{m}$ dx, the force P it carries does work (and stores membrane strain energy) in the amount $P\epsilon_{m}$ dx. Thus the change in membrane energy is $^{1}$
+
+$^{1}$ The same expression would result from the assumptions of a roller support at the right end, constant P, and constant length ( $\int ds = L$ ), which would cause the right end to move a distance $u_{L} = \int \epsilon_{m} dx$ , so that force P at x = L gains potential in the amount $Pu_{L}$ .
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_046.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_046.md
new file mode 100644
index 00000000..87cfdff5
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_046.md
@@ -0,0 +1,449 @@
+<!-- source-page: 451 -->
+
+$$
+U _ {m} = P \int_ {0} ^ {L} \epsilon_ {m} d x = \frac {1}{2} \int_ {0} ^ {L} P w _ {, x} ^ {2} d x \tag {14.1-4}
+$$
+
+Let us assume that $w$ varies as half a sine wave, which happens to be the exact shape for Euler column buckling. Thus
+
+$$
+w = w _ {c} \sin \frac {\pi x}{L} \quad \text { yields } \quad \left\{ \begin{array}{l} U _ {b} = \frac {\pi^ {4} E I}{4 L ^ {3}} w _ {c} ^ {2} \\ U _ {m} = \frac {\pi^ {2} P}{4 L} w _ {c} ^ {2} \end{array} \right. \tag {14.1-5a}
+$$
+
+where $w_{c}$ is the center deflection of the beam.
+
+To do a buckling analysis, we presume that lateral load q is zero. Thus, during buckling, membrane energy is exchanged for bending energy without any input of external work. Therefore
+
+$$
+U _ {b} + U _ {m} = 0 \quad \text { yields } \quad P = - \frac {\pi^ {2} E I}{L ^ {2}} \tag {14.1-6}
+$$
+
+which is the classical Euler buckling load, independent of $w_{c}$ so long as $w_{c}$ is small.
+
+Now consider a deflection problem rather than a buckling problem. Imagine that distributed lateral load q, in the form of a half sine wave with amplitude $q_{c}$ , is applied to the beam in the positive z direction. With $w = w_{c} \sin(\pi x/L)$ and $q = q_{c} \sin(\pi x/L)$ , load q has potential
+
+$$
+\Omega = - \int_ {0} ^ {L} q w d x = - \frac {q _ {c} L}{2} w _ {c} \tag {14.1-7}
+$$
+
+The total potential is $\Pi_p = U_b + U_m + \Omega$ , and the equilibrium value of $w_c$ is given by $\partial \Pi_p / \partial w_c = 0$ . Thus, from Eqs. 14.1-5 and 14.1-7,
+
+$$
+(k + k _ {\sigma}) w _ {c} = \frac {q _ {c} L}{2} \quad \text { where } \quad \left\{ \begin{array}{l} k = \frac {\pi^ {4} E I}{2 L ^ {3}} \\ k _ {\sigma} = \frac {\pi^ {2} P}{2 L} \end{array} \right. \tag {14.1-8}
+$$
+
+The stiffness coefficient $(k + k_{\sigma})$ is the sum of conventional stiffness k and stress stiffness $k_{\sigma}$ . If P = 0, we obtain $w_{c} = q_{c}L^{4}/\pi^{4}EI$ . This result is exact (see Eq. 10.4-11). A tensile load $(P > 0)$ decreases the lateral deflection $w_{c}$ produced by transverse load q. This is the “stress stiffening” effect. If P is compressive $(P < 0)$ , then $w_{c}$ is increased, becoming infinite when $P = -\pi^{2}EI/L^{2}$ , which again defines the buckling load $P = P_{cr}$ . When $P = P_{cr}$ , the net stiffness $(k + k_{\sigma})$ is zero.
+
+Stress stiffness $k_{\sigma}$ can be written in terms of displacement rather than force. Imagine that end x = L of the bar is roller-supported and can have axial displacement $u_{L}$ . Expressing load P in terms of displacement $u_{L}$ , we have
+
+<!-- source-page: 452 -->
+
+![](images/page-452_a527554ddeb2f9d9cab9279f44f6107ce1eb9e1675ee2e45f3790e9c2695806a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+w_c
+F
+F
+x
+L
+</details>
+
+$\{a\}$
+
+![](images/page-452_c7f4e8fe69e963ba4179b9be4090d633634a70258c6a78214742172d783a2bd0.jpg)
+
+<details>
+<summary>line</summary>
+
+| w_c | F (Primary path) | F (Bifurcation point) | F (Secondary path (e = 0)) | F (Increasing e) |
+|-----|------------------|------------------------|-----------------------------|------------------|
+| 0   | 0                | 0                      | 0                           | 0                |
+| >0  | >0               | >0                     | >0                          | >0               |
+</details>
+
+(b)   
+Figure 14.1-2. Bar under compressive load F. In (b), e indicates the magnitude of imperfection.
+
+$$
+P = \frac {A E}{L} u _ {L} \quad \text { and } \quad k _ {\sigma} = \frac {\pi^ {2}}{2 L} \frac {A E}{L} u _ {L} = \frac {\pi^ {2} A E}{2 L ^ {2}} u _ {L} \tag {14.1-9}
+$$
+
+Equation 14.1-9 suggests that the following two-stage analysis is possible (although unnecessary in this simple example). In the first stage, one does a conventional static analysis (without $k_{\sigma}$ ) to determine $u_{L}$ produced by load P. Hence, from Eq. 14.1-9, $k_{\sigma}$ becomes known. One can now use the net stiffness ( $k + k_{\sigma}$ ) in Eq. 14.1-8 to determine the lateral displacement $w_{c}$ produced by lateral load q. Specifically, in this example we obtain $u_{L} = PL/AE$ and $k_{\sigma} = \pi^{2}P/2L$ , exactly as in Eq. 14.1-8. A two-stage analysis is accurate if displacements associated with the first stage are not coupled to displacements associated with the second stage. (If coupling is significant, a multistage analysis is required; see Sections 14.5 and 17.7.) The motivation for using two stages rather than one is that in most structures the distribution of membrane forces is not known a priori and must be determined by the first-stage calculation before the effect of an additional loading can be determined or a buckling analysis performed.
+
+Caution. Buckling theory presumes the existence of a bifurcation point. Consider, for example, Fig. 14.1-2. At the bifurcation (buckling) load, two equilibrium configurations are possible: the column could remain straight (primary path) or it could buckle (secondary path). A bifurcation point exists if the column is perfectly straight, perfectly uniform, perfectly free of end moments and lateral loads, and forces F are perfectly centered and perfectly axial. In reality there are always imperfections, whose magnitude we denote by e. If $e \neq 0$ , the column displays no bifurcation point and structures in general display “limit points.” A computed buckling load is then only an approximation of how much load a structure will carry. The approximation may be quite wrong, and may err on the unconservative (unsafe) side. Most “buckling” problems should be approached as nonlinear problems in which prebuckling deformations are taken into account. These concepts are discussed further in Sections 14.4, 14.5, and 14.7.
+
+These cautionary remarks do not obviate the usefulness of $[k_{\sigma}]$ in stress-stiffening and nonlinear analyses.
+
+# 14.2 STRESS STIFFNESS MATRICES FOR BEAMS AND BARS
+
+In this section, stress stiffness matrices $[k_{\sigma}]$ for prismatic members are derived from Eq. 14.1-4 by use of an assumed lateral displacement field $w = w(x)$ . Axial
+
+<!-- source-page: 453 -->
+
+force P in the member is presumed known in terms of loads applied to the structure, either a priori or by elastic analysis, depending on whether the structure in which the member resides is statically determinate or not. Attention is restricted to displacements in a plane. Beam and bar elements are placed on a local x axis. Matrices $[k_{\sigma}]$ for elements arbitrarily oriented in global coordinates can be obtained by straightforward use of the transformation $[T]^{T}[k_{\sigma}][T]$ (see Eq. 7.4-4). Matrices $[k_{\sigma}]$ for space frames and space trusses are derivable by use of expressions discussed in Section 14.4.
+
+To begin, we include conventional strain terms as well as $w_{,x}^{2}/2$ from Eq. 14.1-3 in the strain expression. In this way we show clearly which terms lead to the conventional stiffness matrix [k] and which to $[k_{\sigma}]$ .
+
+Plane Beam. The beam in Fig. 14.2-1 can have axial displacement $u = u(x)$ and lateral displacement $w = w(x)$ . Membrane strain is $\epsilon_{m} = u_{,x} + \frac{1}{2} w_{,x}^{2}$ , where the latter term comes from Eq. 14.1-3. At a distance z from the centroidal axis, the contribution of bending to the axial strain is $\epsilon = -z w_{,xx}$ , as derived in Eq. 11.1-3. The total axial strain of an arbitrarily located fiber is therefore
+
+$$
+\epsilon_ {x} = u _ {, x} + \frac {1}{2} w _ {, x} ^ {2} - z w _ {, x x} \tag {14.2-1}
+$$
+
+Each fiber carries uniaxial stress. Strain energy in the element is therefore
+
+$$
+U = \int_ {V _ {e}} \frac {1}{2} E \epsilon_ {x} ^ {2} d V = \int_ {0} ^ {L} \int_ {A} \frac {1}{2} E \epsilon_ {x} ^ {2} d A d x \tag {14.2-2}
+$$
+
+We substitute Eq. 14.2-1 into Eq. 14.2-2 and note that
+
+$$
+\int_ {A} d A = A \quad \int_ {A} z d A = 0 \quad \int_ {A} z ^ {2} d A = I \quad \int_ {A} E u _ {, x} d A = P \tag {14.2-3}
+$$
+
+where P is the axial force, positive in tension. If a term dependent on $w_{,x}^{4}$ is discarded as negligible in comparison with other terms, we obtain
+
+$$
+U = \int_ {0} ^ {L} \frac {A E}{2} u _ {, x} ^ {2} d x + \int_ {0} ^ {L} \frac {P}{2} w _ {, x} ^ {2} d x + \int_ {0} ^ {L} \frac {E I}{2} w _ {, x x} ^ {2} d x \tag {14.2-4}
+$$
+
+The first integral yields [k] for a bar element; it contains coefficients AE/L and is associated with d.o.f. $u_{1}$ and $u_{2}$ . The third integral yields [k] for a standard beam element; it contains coefficients such as $12EI/L^{3}$ and is associated with d.o.f. $w_{1}$ , $\theta_{1}$ , $w_{2}$ , and $\theta_{2}$ . The second integral yields $[k_{\sigma}]$ . This integral was previously seen
+
+![](images/page-453_a0a5e2d507c03d9e8a2bf54cb2e1f57be768b6087d11cd452032fb2b12fdb7a3.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+w₁
+θ₁
+A, E, I
+w₂
+θ₂
+u₁
+u₂
+x, u
+L
+</details>
+
+(a)
+
+![](images/page-453_37249590b96e611634c1b92838723cdbddefa69c7538580aaecb25e0ef873fd3.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+w₁
+A, E
+w₂
+u₁
+u₂
+x, u
+L
+</details>
+
+(b)   
+Figure 14.2-1. Plane elements and their d.o.f. (a) Beam. (b) Bar. A = cross-sectional area, E = elastic modulus, I = moment of inertia of A.
+
+<!-- source-page: 454 -->
+
+as Eq. 14.1-4. It describes work done, and strain energy stored, when lateral displacement w causes differential elements to stretch an amount $w_{,x}^{2} dx/2$ in the presence of a constant axial force P. The standard $[k_{\sigma}]$ for a beam is developed from the integral expression as follows.
+
+With nodal d.o.f. $\{\mathbf{d}\} = [w_1 \theta_1 w_2 \theta_2]^T$ and shape functions $[\mathbf{N}]$ , lateral displacement $w$ and its first derivative $w_{xx}$ are
+
+$$
+w = [ \mathbf {N} ] \{\mathbf {d} \} \quad \text { where } \quad [ \mathbf {N} ] = \left[ \begin{array}{l l l l} N _ {1} & N _ {2} & N _ {3} & N _ {4} \end{array} \right] \tag {14.2-5}
+$$
+
+$$
+w _ {, x} = [ \mathbf {G} ] \{\mathbf {d} \} \quad \text { where } \quad [ \mathbf {G} ] = \left\lfloor N _ {1, x} \quad N _ {2, x} \quad N _ {3, x} \quad N _ {4, x} \right\rfloor \tag {14.2-6}
+$$
+
+The second integral in Eq. 14.2-4 yields
+
+$$
+\int_ {0} ^ {L} \frac {P}{2} w _ {, x} ^ {2} d x = \frac {1}{2} \int_ {0} ^ {L} w _ {, x} ^ {T} P w _ {, x} d x = \frac {1}{2} \left\{\mathbf {d} \right\} ^ {T} [ \mathbf {k} _ {\sigma} ] \left\{\mathbf {d} \right\} \tag {14.2-7}
+$$
+
+where
+
+$$
+[ \mathbf {k} _ {\sigma} ] = \int_ {0} ^ {L} [ \mathbf {G} ] ^ {T} P [ \mathbf {G} ] d x \tag {14.2-8}
+$$
+
+Force P is constant in this member and can be removed from the integral. Using the standard $N_{i}$ given in Fig. 3.13-2, we obtain [14.2]
+
+$$
+\left[ \mathbf {k} _ {\sigma} \right] = \frac {P}{3 0 L} \left[ \begin{array}{c c c c} 3 6 & 3 L & - 3 6 & 3 L \\ 3 L & 4 L ^ {2} & - 3 L & - L ^ {2} \\ - 3 6 & - 3 L & 3 6 & - 3 L \\ 3 L & - L ^ {2} & - 3 L & 4 L ^ {2} \end{array} \right] \tag {14.2-9}
+$$
+
+where P is positive in tension. (By inserting rows and columns of zeros, $[k_{\sigma}]$ could be written as a 6 by 6 matrix that operates on the d.o.f. $\{d\} = [u_{1} w_{1} \theta_{1} u_{2} w_{2} \theta_{2}]^{T}$ . Coordinate transformation could follow; the resulting $[k_{\sigma}]$ could then be used for an arbitrarily oriented member of a plane frame.)
+
+Plane Bar. The foregoing arguments can be repeated, but with curvature $w_{,xx}$ and d.o.f. $\theta_{1}$ and $\theta_{2}$ omitted from Eq. 14.2-4. Nonzero terms in $[k_{\sigma}]$ are then associated with d.o.f. $w_{1}$ and $w_{2}$ , and matrix $[G]$ describes a rotation of the bar that is independent of x,
+
+$$
+w _ {, x} = \left\lfloor \mathbf {G} \right] \left\{ \begin{array}{l} w _ {1} \\ w _ {2} \end{array} \right\} \quad \text { where } \quad \left\lfloor \mathbf {G} \right\rfloor = \left\lfloor - \frac {1}{L} \quad \frac {1}{L} \right\rfloor \tag {14.2-10}
+$$
+
+Hence, Eq. 14.2-8 yields
+
+$$
+[ \mathbf {k} _ {\sigma} ] = \frac {P}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right] \quad \text { for } \quad \{\mathbf {d} \} = \left\lfloor w _ {1} \quad w _ {2} \right\rfloor^ {T} \tag {14.2-11a}
+$$
+
+<!-- source-page: 455 -->
+
+or
+
+$$
+\left[ \mathbf {k} _ {\sigma} \right] = \frac {P}{L} \left[ \begin{array}{c c c c} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & - 1 \\ 0 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 1 \end{array} \right] \quad \text {   for   } \quad \{\mathbf {d} \} = \left[ \begin{array}{c c c c} u _ {1} & w _ {1} & u _ {2} & w _ {2} \end{array} \right] ^ {T} \tag {14.2-11b}
+$$
+
+Remarks. Equations 14.2-11 are exact for small deflections of a bar that may rotate but does not bend. Equation 14.2-9, which allows bending, is approximate because a cubic lateral-displacement field is not exact when a beam carries axial load as well as loads that produce bending. As usual, accuracy is gained by dividing a given beam into two or more elements. A single beam-column element can be exact if the element formulation is based on the exact displacement field. Formulation of such an element yields a combined matrix $[\mathbf{k} + \mathbf{k}_{\sigma}]$ , which remains 4 by 4 but has coefficients that are more complicated than coefficients in Eq. 14.2-9.
+
+Note that if $\{d\}$ represents a small rigid-body rotation, the conventional stiffness matrix [k] yields zero forces; that is, $[k]\{d\} = \{0\}$ . Such is not the case for the stress stiffness matrix; that is, $[k_{\sigma}]\{d\} \neq \{0\}$ . This result does not imply that $[k_{\sigma}]$ is in error. Consider, for example, $[k_{\sigma}]$ of Eq. 14.2-11. If the element is given a small rotation $\theta$ , then axial strain $\epsilon_{x} = \theta^{2}/2$ appears and transverse “kickoff” forces of magnitude $P\theta$ appear at the nodes. These forces can be regarded as inseparable from buckling; that is, in the buckled state of a structure, the loading provided by kickoff forces from the various elements is exactly resisted by a deformation state whose associated rotations create the kickoff forces. If all higher-order terms were retained in the expression for $\epsilon_{x}$ , the rigid-body rotation of an element would not create nodal forces (see Section 14.4).
+
+When conventional stiffness and stress stiffness are both taken into account, the total or effective stiffness matrix is $[k] + [k_{\sigma}]$ for an element and $[K] + [K_{\sigma}]$ for a structure. Thus, in direct analogy to Eq. 14.1-8, one accounts for the stiffening or weakening effect of axial load on bending stiffness. The inclusion of $[K_{\sigma}]$ does not require the inclusion of extra d.o.f. in $\{D\}$ when the equation $([K] + [K_{\sigma}])\{D\} = \{R\}$ is used in place of $[K]\{D\} = \{R\}$ . Use of $[K] + [K_{\sigma}]$ to solve buckling problems is discussed in Section 14.5.
+
+# 14.3 STRESS STIFFNESS MATRIX OF A PLATE ELEMENT
+
+For a flat plate, just as for a bar or a beam, an expression for $[k_{\sigma}]$ can be obtained by examination of the work done by constant membrane forces as they act through displacements associated with small lateral deflections. Membrane forces, Fig. 14.3-1, are defined by
+
+$$
+N _ {x} = \int_ {- t / 2} ^ {t / 2} \sigma_ {x} d z \quad N _ {y} = \int_ {- t / 2} ^ {t / 2} \sigma_ {y} d z \quad N _ {x y} = \int_ {- t / 2} ^ {t / 2} \tau_ {x y} d z \tag {14.3-1}
+$$
+
+<!-- source-page: 456 -->
+
+![](images/page-456_5c9545f643f69765100e29393e4deeacb8ca074e9fb901018a643c1f253e9030.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+y
+Nx
+Ny
+Nx y
+</details>
+
+Figure 14.3-1. Differential element of a flat plate, showing membrane forces $N_{x}$ , $N_{y}$ , and $N_{xy}$ .
+
+where membrane stresses $\sigma_{x}, \sigma_{y}$ , and $\tau_{xy}$ are either known a priori or calculated by standard static stress analysis, using, for example, plane bilinear isoparametric elements.
+
+Membrane strains associated with small rotations $w_{,x}$ and $w_{,y}$ of the plate mid-surface are [11.1,14.4]
+
+$$
+\epsilon_ {x} = \frac {1}{2} w _ {, x} ^ {2} \quad \epsilon_ {y} = \frac {1}{2} w _ {, y} ^ {2} \quad \gamma_ {x y} = w _ {, x} w _ {, y} \tag {14.3-2}
+$$
+
+If membrane forces $N_{x}$ , $N_{y}$ , and $N_{xy}$ are assumed to be independent of the small lateral deflection $w = w(x, y)$ , then the work associated with the membrane forces and the strains of Eqs. 14.3-2 is
+
+$$
+U _ {\sigma} = \int_ {A} \left(\frac {1}{2} w _ {, x} ^ {2} N _ {x} + \frac {1}{2} w _ {, y} ^ {2} N _ {y} + w _ {, x} w _ {, y} N _ {x y}\right) d A \tag {14.3-3a}
+$$
+
+$$
+U _ {\sigma} = \frac {1}{2} \iint \left\{ \begin{array}{l} w _ {, x} \\ w _ {, y} \end{array} \right\} ^ {T} \left[ \begin{array}{l l} N _ {x} & N _ {x y} \\ N _ {x y} & N _ {y} \end{array} \right] \left\{ \begin{array}{l} w _ {, x} \\ w _ {, y} \end{array} \right\} d x d y = \frac {1}{2} \{\mathbf {d} \} ^ {T} [ \mathbf {k} _ {\sigma} ] \{\mathbf {d} \} \tag {14.3-3b}
+$$
+
+One must choose a displacement field $w = w(x, y)$ whose form is appropriate to the element shape and its d.o.f., for example, Eq. 11.2-5. From the displacement field one obtains rotations, that is,
+
+$$
+w \doteq \lfloor \mathbf {N} \rfloor \{\mathbf {d} \} \quad \text { yields } \quad \left\{ \begin{array}{l} w _ {, x} \\ w _ {, y} \end{array} \right\} = \left[ \begin{array}{l} \mathbf {G} \end{array} \right] \left\{ \begin{array}{l} \mathbf {d} \end{array} \right\} \tag {14.3-4}
+$$
+
+where n is the number of d.o.f. per element. Matrix [G] is in general a function of x and y. Equations 14.3-3b and 14.3-4 yield
+
+$$
+[ \mathbf {k} _ {\sigma} ] = \iint [ \mathbf {G} ] ^ {T} \left[ \begin{array}{l l} N _ {x} & N _ {x y} \\ N _ {x \bar {y}} & N _ {y} \end{array} \right] [ \mathbf {G} ] d x d y \tag {14.3-5}
+$$
+
+where integration extends over the element area. If the element is of the isoparametric family, shape functions [N] are expressed in terms of dimensionless coordinates $\xi$ and $\eta$ . Therefore, one must invoke [J], the Jacobian matrix of Eq. 6.3-11. Thus
+
+$$
+\left\{ \begin{array}{l} w, \xi \\ w, \eta \end{array} \right\} = [ \mathbf {G} _ {I} ] \{\mathbf {d} \} \quad \text { and } \quad \left\{ \begin{array}{l} w, x \\ w, y \end{array} \right\} = [ \mathbf {J} ] ^ {- 1} \left\{ \begin{array}{l} w, \xi \\ w, \eta \end{array} \right\} \tag {14.3-6}
+$$
+
+<!-- source-page: 457 -->
+
+where $[G_{I}]$ contains derivatives of the shape functions with respect to $\xi$ and $\eta$ . Equations 14.3-3b and 14.3-6 yield
+
+$$
+\left[ \mathbf {k} _ {\sigma} \right] = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} \left[ \mathbf {G} _ {I} \right] ^ {T} [ \mathbf {J} ] ^ {- T} \left[ \begin{array}{l l} N _ {x} & N _ {x y} \\ N _ {x y} & N _ {y} \end{array} \right] [ \mathbf {J} ] ^ {- 1} \left[ \mathbf {G} _ {I} \right] J d \xi d \eta \tag {14.3-7}
+$$
+
+where $J$ is the Jacobian determinant.
+
+Membrane forces $N_{x}$ , $N_{y}$ , and $N_{xy}$ may vary over an element. Then, in numerical integration to evaluate $[k_{\sigma}]$ , different membrane forces would be used at different sampling points.
+
+Note that $[k_{\sigma}]$ is determined independently of material properties, except to the extent that material properties may influence computed values of $N_{x}$ , $N_{y}$ , and $N_{xy}$ . Accordingly, a given $[k_{\sigma}]$ is equally applicable to both isotropic and anisotropic structures.
+
+# 14.4 A GENERAL FORMULATION FOR $[\mathbf{k}_{\sigma}]$
+
+In Sections 14.2 and 14.3, each type of element is approached as a special case. It is desirable to also have a general formula for $[k_{\sigma}]$ , analogous to the formula for the conventional [k] (Eq. 4.1-5), that may be specialized to particular geometries, and requires only that a specific displacement field be chosen. Such a formula is derived in the present section. $^{2}$
+
+The formula for $[k_{\sigma}]$ , Eq. 14.4-7, is “linearized” and is limited to small displacements. To demonstrate in a general way that this is so requires comparatively lengthy and complicated arguments. We will omit these arguments [2.1] and instead illustrate the nature of the approximation by means of a simple particular case (Eqs. 14.4-10 to 14.4-16).
+
+Green-Lagrange Strain. Various advanced texts [e.g., 2.1, 3.1, 9.9] discuss the expressions for stress and strain appropriate to problems that involve large deformations. The following equations define a strain measure commonly known as Green-Lagrange strain:
+
+$$
+\epsilon_ {x} = u _ {, x} + \frac {1}{2} \left(u _ {, x} ^ {2} + v _ {, x} ^ {2} + w _ {, x} ^ {2}\right) \tag {14.4-1a}
+$$
+
+$$
+\epsilon_ {y} = v _ {, y} + \frac {1}{2} \left(u _ {, y} ^ {2} + v _ {, y} ^ {2} + w _ {, y} ^ {2}\right) \tag {14.4-1b}
+$$
+
+$$
+\epsilon_ {z} = w _ {, z} + \frac {1}{2} \left(u _ {, z} ^ {2} + v _ {, z} ^ {2} + w _ {, z} ^ {2}\right) \tag {14.4-1c}
+$$
+
+$$
+\gamma_ {x y} = u _ {, y} + v _ {, x} + \left(u _ {, x} u _ {, y} + v _ {, x} v _ {, y} + w _ {, x} w _ {, y}\right) \tag {14.4-1d}
+$$
+
+$$
+\gamma_ {y z} = v _ {, z} + w _ {, y} + \left(u _ {, y} u _ {, z} + v _ {, y} v _ {, z} + w _ {, y} w _ {, z}\right) \tag {14.4-1e}
+$$
+
+$$
+\gamma_ {z x} = w _ {, x} + u _ {, z} + (u _ {, z} u _ {, x} + v _ {, z} v _ {, x} + w _ {, z} w _ {, x}) \tag {14.4-1f}
+$$
+
+<!-- source-page: 458 -->
+
+The initial terms in Eqs. 14.4-1 are the customary engineering definitions of normal and shear strain ( $\epsilon_{x} = u_{,x}$ , etc.). The added terms, in parentheses, become significant if displacement gradients are not small. Green–Lagrange strains are zero for a rigid-body rotation of any magnitude. In Eqs. 14.4-1, all displacement derivatives are computed in the original coordinate system, regardless of how large a rigid-body rotation may be superposed on the deformations. This is the “total Lagrangian” approach, in which all displacements are measured in a reference frame that is stationary rather than attached to the deforming structure. The stationary coordinates may also be called “material coordinates” and may be denoted in some papers by uppercase labels X, Y, and Z.
+
+Green-Lagrange normal strains correspond to defining the strain of a line segment by the equation
+
+$$
+\epsilon = \frac {1}{2} \left[ \left(\frac {d s ^ {*}}{d s}\right) ^ {2} - 1 \right] \tag {14.4-2}
+$$
+
+where $ds$ and $ds^*$ are respectively the initial and final lengths of the line segment. If $ds \approx ds^*$ , Eq. 14.4-2 reduces to the usual small-strain approximation, $\epsilon = (ds^* - ds)/ds$ .
+
+Formula for $[k_{\sigma}]$ . Imagine that initial stresses $\{\sigma_{0}\}$ prevail. If these stresses are assumed to remain constant as strains $\{\epsilon\}$ occur, the associated work is $^{3}$
+
+$$
+\int_ {V} \{\boldsymbol {\epsilon} \} ^ {T} \left\{\boldsymbol {\sigma} _ {0} \right\} d V \quad \text {where} \quad \left\{ \begin{array}{l l} \left\{\boldsymbol {\epsilon} \right\} ^ {T} = \left\lfloor \epsilon_ {x} \quad \epsilon_ {y} \dots \gamma_ {z x} \right\rfloor & (1 4. 4 - 3 a) \\ \left\{\boldsymbol {\sigma} _ {0} \right\} = \left\lfloor \sigma_ {x 0} \quad \sigma_ {y 0} \dots \tau_ {z x 0} \right\rfloor^ {T} & (1 4. 4 - 3 b) \end{array} \right.
+$$
+
+With $\{\epsilon\}$ given by Eqs. 14.4-1, the integrand $\{\epsilon\}^{T}\{\sigma_{0}\}$ first displays the terms $u_{,x}\sigma_{x0} + v_{,y}\sigma_{y0} + \cdots$ . These terms lead to nodal loads associated with $\{\sigma_{0}\}$ , as given by Eq. 4.1-6. What remains is
+
+$$
+U _ {\sigma} = \int_ {V} \left[ \frac {1}{2} \left(u _ {, x} ^ {2} + v _ {, x} ^ {2} + w _ {, x} ^ {2}\right) \sigma_ {x 0} + \dots + \left(u _ {, z} u _ {, x} + v _ {, z} v _ {, x} + w _ {, z} w _ {, x}\right) \tau_ {z x 0} \right] d V \tag {14.4-4}
+$$
+
+If we define
+
+$$
+\{\delta \} = \left[ \begin{array}{l l l l l l l l l} u _ {, x} & u _ {, y} & u _ {, z} & v _ {, x} & v _ {, y} & v _ {, z} & w _ {, x} & w _ {, y} & w _ {, z} \end{array} \right] ^ {T} \tag {14.4-5}
+$$
+
+then Eq. 14.4-4 can be written in the form
+
+$$
+U _ {\sigma} = \frac {1}{2} \int_ {V} \left\{\delta \right\} ^ {T} \left[ \begin{array}{l l l} \mathrm{s} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathrm{s} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \mathrm{s} \end{array} \right] \left\{\delta \right\} d V \quad \text {where} \quad [ \mathrm{s} ] = \left[ \begin{array}{l l l} \sigma_ {x 0} & \tau_ {x y 0} & \tau_ {z x 0} \\ \tau_ {x y 0} & \sigma_ {y 0} & \tau_ {y z 0} \\ \tau_ {z x 0} & \tau_ {y z 0} & \sigma_ {z 0} \end{array} \right] \tag {14.4-6}
+$$
+
+$^{3}$ This assumption restricts the subsequent development, Eqs. 14.4-3 to 14.4-7, to small strains and small rotations. More advanced arguments [2.1] show that a more elaborate definition of stress than engineering stresses $\{\sigma_{0}\}$ is required if one is to write a strain energy expression that is meaningful in analyses of large deformations.
+
+<!-- source-page: 459 -->
+
+This expression is analogous to Eqs. 14.2-7 and 14.3-3b, and yields $[k_{\sigma}]$ in an analogous way. Let the element displacement field be given by $\{u\} = [N]\{d\}$ , as usual, where $\{u\} = [u \quad v \quad w]^{T}$ and $\{d\}$ contains nodal d.o.f. Also let $\{\delta\} = [G]\{d\}$ , where [G] is obtained from shape functions [N] by appropriate differentiation and ordering of terms. Equation 14.4-6 becomes $U_{\sigma} = \{d\}^{T}[k_{\sigma}]\{d\}/2$ , where
+
+$$
+\left[ \mathbf {k} _ {\sigma} \right] = \int_ {V _ {e}} [ \mathbf {G} ] ^ {T} \left[ \begin{array}{l l l} \mathbf {s} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {s} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \mathbf {s} \end{array} \right] [ \mathbf {G} ] d V \tag {14.4-7}
+$$
+
+As an example, consider the bar of Fig. 14.2-1b, again with motion restricted to the xz plane. For this case all initial stresses are zero except for axial stress $\sigma_{x0}$ . We assume that u and w are linear in x and require that v = 0. Accordingly, with $N_{1} = (L - x)/L$ and $N_{2} = x/L$ , we write
+
+$$
+\begin{array}{l} u = N _ {1} u _ {1} + N _ {2} u _ {2} \\ w = N _ {1} w _ {1} + N _ {2} w _ {2} \end{array} \quad [ \mathbf {G} ] = \frac {1}{L} \left[ \begin{array}{c c c c} - 1 & 0 & 1 & 0 \\ 0 & - 1 & 0 & 1 \end{array} \right] \tag {14.4-8}
+$$
+
+Nonzero d.o.f. are $\{\mathbf{d}\} = \left[u_1 w_1 u_2 w_2\right]^T$ . Also, $\{\delta\} = \left[u_{,x} w_{,x}\right]^T$ . Equation 14.4-7 reduces to
+
+$$
+\left[ \mathbf {k} _ {\sigma} \right] = \int_ {0} ^ {L} \left[ \mathbf {G} \right] ^ {T} \left[ \begin{array}{c c} \sigma_ {x 0} & 0 \\ 0 & \sigma_ {x 0} \end{array} \right] [ \mathbf {G} ] \mathrm{A} d x = \frac {P}{L} \left[ \begin{array}{c c c c} 1 & 0 & - 1 & 0 \\ 0 & 1 & 0 & - 1 \\ - 1 & 0 & 1 & 0 \\ 0 & - 1 & 0 & 1 \end{array} \right] \tag {14.4-9}
+$$
+
+where $P = \sigma_{x0}A$ . This $[k_{\sigma}]$ is almost the same as that in Eq. 14.2-11, but contains four more nonzero terms. However, note that the additional nonzero terms occupy the same positions as the nonzero terms in the conventional stiffness matrix (see Eq. 2.5-3). Thus, in the net stiffness matrix $[k] + [k_{\sigma}]$ , we see coefficients $\pm(AE + P)/L$ (corresponding to d.o.f. $u_{1}$ and $u_{2}$ ) and $\pm P/L$ (corresponding to d.o.f. $w_{1}$ and $w_{2}$ ). Since AE >> P in any practical problem, the “extra” P/L terms in $[k_{\sigma}]$ can be discarded. In this way Eq. 14.4-9 reduces to Eq. 14.2-11.
+
+Full Nonlinearity: An Example. In the foregoing development it is not obvious what approximations are contained in $[k_{\sigma}]$ . In what follows we allow large rotations, and show by example that use of $[k_{\sigma}]$ implies complete linearization of the problem and negligible rotations prior to buckling.
+
+Consider the one-element elastic bar in Fig. 14.4-1. If forces are applied only at the ends, displacements vary linearly.
+
+$$
+u = \frac {x}{L} u _ {2} \quad \text { and } \quad w = \frac {x}{L} w _ {2} \tag {14.4-10}
+$$
+
+Measurements are made in the original coordinate system $xz$ ; that is, $x$ is not regarded as an axial coordinate that rotates as the bar rotates. For example, the bar becomes vertical if $u_2 = -L$ and $w_2 = \pm L$ ; nevertheless, one uses the horizontal coordinate $x$ in the fields $u = u_2x/L$ and $w = w_2x/L$ . Thus $u_2$ is the $x$ -direction component of the displacement of node 2; it is not the stretch of the bar unless $w_2 = 0$ . End 2 of the bar is located by coordinates $x = L$ and $z = 0$ , regardless of the values of $u_2$ and $w_2$ .
+
+<!-- source-page: 460 -->
+
+![](images/page-460_22cda13b43009698f59a2f08c0ea944b91d235dff9179cebbe4c346749d0c8af.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+1 A, E 2 x, u
+L
+</details>
+
+{a}
+
+![](images/page-460_b7e1f21fd1330b3d1026872f6a891089707aad1f1094fafb8e4c8f1525c73183.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+1
+θ
+L
+u₂
+Rₓ
+2
+Rₓ
+w₂
+x, u
+</details>
+
+(b)   
+Figure 14.4-1. (a) A bar element, hinged at node 1, prior to loading. (b) The displaced and deformed bar after forces $R_{x}$ and $R_{z}$ are applied to node 2.
+
+The bar moves in the $xz$ plane and carries uniaxial stress. If strains are small, the total potential of the bar is
+
+$$
+\Pi_ {p} = \frac {1}{2} \int_ {0} ^ {L} A E \epsilon_ {x} ^ {2} d x - R _ {x} u _ {2} - R _ {z} w _ {2} \tag {14.4-11}
+$$
+
+where $\epsilon_{x}$ is axial strain, as if the bar occupies its original x-parallel orientation. From Eqs. 14.4-1a and 14.4-10,
+
+$$
+\epsilon_ {x} = \frac {u _ {2}}{L} + \frac {1}{2} \left(\frac {u _ {2} ^ {2}}{L ^ {2}} + \frac {w _ {2} ^ {2}}{L ^ {2}}\right) \tag {14.4-12}
+$$
+
+The resulting expression for $\Pi_p$ still allows large rotation of the bar. Static equilibrium prevails when $\partial \Pi_p / \partial u_2 = 0$ and $\partial \Pi_p / \partial w_2 = 0$ . Results of these calculations can be written in the form
+
+$$
+\frac {A E}{L} \left(\left[ \begin{array}{l l} 1 & 0 \\ 0 & 0 \end{array} \right] + \frac {1}{2 L} \left[ \begin{array}{l l} 3 u _ {2} & w _ {2} \\ w _ {2} & u _ {2} \end{array} \right] + \frac {1}{2 L ^ {2}} \left[ \begin{array}{l l} u _ {2} ^ {2} & u _ {2} w _ {2} \\ u _ {2} w _ {2} & w _ {2} ^ {2} \end{array} \right]\right) \left\{ \begin{array}{l} u _ {2} \\ w _ {2} \end{array} \right\} = \left\{ \begin{array}{l} R _ {x} \\ R _ {z} \end{array} \right\} \tag {14.4-13}
+$$
+
+One finds that if $u_{2}$ and $w_{2}$ represent rigid-body rotation about node 1—that is, if $u_{2} = -L(1 - \cos \theta)$ and $w_{2} = L \sin \theta$ —then $R_{x} = 0$ and $R_{z} = 0$ for any rotation $\theta$ , no matter how large.
+
+In Eq. 14.4-13, $u_{2}$ and $w_{2}$ are total displacements. An analogous incremental form can be obtained from Eq. 14.4-13 by writing $R_{x} = R_{x}(u_{2}, w_{2})$ and $R_{z} = R_{z}(u_{2}, w_{2})$ , then forming expressions for $dR_{x}$ and $dR_{z}$ by differentiation. Thus
+
+$$
+\underbrace {\left(\frac {A E}{L} \left[ \begin{array}{l l} 1 & 0 \\ 0 & 0 \end{array} \right] \right.} _ {[ \mathrm{K} ]} + \underbrace {\frac {A E}{L ^ {2}} \left[ \begin{array}{l l} 3 u _ {2} & w _ {2} \\ w _ {2} & u _ {2} \end{array} \right]} _ {[ \mathrm{N} _ {1} ]} + \underbrace {\frac {A E}{2 L ^ {3}} \left[ \begin{array}{c c} 3 u _ {2} ^ {2} + w _ {2} ^ {2} & 2 u _ {2} w _ {2} \\ 2 u _ {2} w _ {2} & u _ {2} ^ {2} + 3 w _ {2} ^ {2} \end{array} \right]} _ {[ \mathrm{N} _ {2} ]} \Bigg) \left\{ \begin{array}{l} d u _ {2} \\ d w _ {2} \end{array} \right\} = \left\{ \begin{array}{l} d R _ {x} \\ d R _ {z} \end{array} \right\} \tag {14.4-14}
+$$
+
+This equation describes nodal force increments $dR_{x}$ and $dR_{z}$ associated with small nodal displacements $du_{2}$ and $dw_{2}$ , where $du_{2}$ and $dw_{2}$ are measured from a current
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+<!-- source-page: 461 -->
+
+reference configuration defined by $u_{2}$ and $w_{2}$ . Thus the reference configuration may display an angle $\theta$ far different from its initial value $\theta = 0$ .
+
+A correspondence with preceding equations may be recognized as follows. If displacements $u_{2}$ and $w_{2}$ in Eq. 14.4-14 are sufficiently small, the third matrix in Eq. 14.4-14, which depends quadratically on displacements, may be discarded in comparison with the other two matrices. If in addition we substitute $(AE/L)u_{2}=P$ , set $w_{2}=0$ , and presume that $u_{2}<<L$ , then there is a further simplification to
+
+$$
+\left(\frac {A E}{L} \left[ \begin{array}{l l} 1 & 0 \\ 0 & 0 \end{array} \right] + \frac {P}{L} \left[ \begin{array}{l l} 3 & 0 \\ 0 & 1 \end{array} \right]\right) \left\{ \begin{array}{l} d u _ {2} \\ d w _ {2} \end{array} \right\} = \left\{ \begin{array}{l} d R _ {x} \\ d R _ {z} \end{array} \right\} \tag {14.4-15}
+$$
+
+The second matrix in Eq. 14.4-15 is analogous to the lower right 2 by 2 submatrix in Eq. 14.4-9, except that the number 3 appears instead of the number 1. Equation 14.4-15 can be reconciled with Eq. 14.2-11 by means of the same arguments that are applied to Eq. 14.4-9; that is, by noting that in practice $AE \gg P$ . Thus the number 3 is discarded and the second matrix in Eq. 14.4-15 is recognized as $[K_{\sigma}]$ .
+
+In summary, by discarding the quadratic stiffness terms in Eq. 14.4-14, we obtain two forms of incremental equations:
+
+$$
+([ \mathbf {K} ] + [ \mathbf {N} _ {1} ]) \{d \mathbf {D} \} = \{d \mathbf {R} \} \quad \text { and } \quad ([ \mathbf {K} ] + [ \mathbf {K} _ {\sigma} ]) \{d \mathbf {D} \} = \{d \mathbf {R} \} \tag {14.4-16}
+$$
+
+Equations 14.4-16 are symbolic statements of approximations that can be applied to linearly elastic structures in general, not only to the elastic bar. Both equations are linearized—that is, made to depend on first powers of displacements—by discarding quadratic terms. Either equation may be used to compute a buckling load. Upon buckling, infinitesimal displacements $\{dD\}$ occur, measured with reference to a prebuckling configuration $\{D\}$ . In computing a buckling load, use of $[K_{\sigma}]$ implies that prebuckling rotations in $\{D\}$ are zero, whereas use of $[N_{1}]$ implies only that prebuckling rotations are small. For initially flat plates and initially straight columns, the two formulations yield the same buckling load. In practical problems where prebuckling rotations are not negligible, use of $[N_{1}]$ seems always to yield a smaller buckling load than does use of $[K_{\sigma}]$ , but there is no guarantee that either approach will always err on the high side or the low side of the actual collapse load [14.1].
+
+# 14.5 BIFURCATION BUCKLING
+
+A bifurcation buckling load is the load for which a reference configuration of the structure and an infinitesimally close (buckled) configuration are both possible equilibrium configurations. As a buckling displacement $\{dD\}$ takes place from a reference configuration $\{D\}$ , the load does not change. Accordingly, from Eqs. 14.4-16,
+
+$$
+([ \mathbf {K} ] + [ \mathbf {N} _ {1} ]) \{d \mathbf {D} \} = \{\mathbf {0} \} \quad \text { or } \quad ([ \mathbf {K} ] + [ \mathbf {K} _ {\sigma} ]) \{d \mathbf {D} \} = \{\mathbf {0} \} \tag {14.5-1}
+$$
+
+We ask for the level of deformation (symbolized by $[N_{1}]$ ) or the level of stress (symbolized by $[K_{\sigma}]$ ) such that a solution $\{dD\}$ other than $\{dD\} = \{0\}$ is possible.
+
+<!-- source-page: 462 -->
+
+Use of $[N_{1}]$ implies that prebuckling rotations are small but are not to be ignored. Use of $[K_{\sigma}]$ implies that prebuckling rotations are either ignored or are zero. The latter is “classical” buckling analysis, as commonly used for straight columns and flat plates.
+
+In what follows we emphasize classical buckling analysis, which uses $[K_{\sigma}]$ . One begins by applying to the structure a reference level of loading $\{R\}_{ref}$ and carrying out a standard linear static analysis to obtain membrane stresses in elements (e.g., to determine membrane stresses in a flat plate under thermal load). Hence, we generate a stress stiffness matrix $[K_{\sigma}]_{ref}$ appropriate to $\{R\}_{ref}$ . For another load level, with $\lambda$ a scalar multiplier,
+
+$$
+[ \mathbf {K} _ {\sigma} ] = \lambda [ \mathbf {K} _ {\sigma} ] _ {\text { ref }} \quad \text { when } \quad \{\mathbf {R} \} = \lambda \{\mathbf {R} \} _ {\text { ref }} \tag {14.5-2}
+$$
+
+Equations 14.5-2 imply that multiplying all loads $R_{i}$ in $\{R\}_{ref}$ by $\lambda$ also multiplies the intensity of the stress field by $\lambda$ but does not change the distribution of stresses. Then, since external loads do not change during an infinitesimal buckling displacement $\{dD\}$ ,
+
+$$
+([ \mathbf {K} ] + \lambda_ {\mathrm{cr}} [ \mathbf {K} _ {\sigma} ] _ {\text { ref }}) \{\mathbf {D} \} = ([ \mathbf {K} ] + \lambda_ {\mathrm{cr}} [ \mathbf {K} _ {\sigma} ] _ {\text { ref }}) \{\mathbf {D} + d \mathbf {D} \} = \lambda_ {\mathrm{cr}} \{\mathbf {R} \} _ {\text { ref }} \tag {14.5-3}
+$$
+
+Subtraction of the first equation from the second yields
+
+$$
+([ \mathbf {K} ] + \lambda_ {\mathrm{cr}} [ \mathbf {K} _ {\sigma} ] _ {\mathrm{ref}}) \{d \mathbf {D} \} = \{\mathbf {0} \} \tag {14.5-4}
+$$
+
+Equation 14.5-4 defines an eigenvalue problem whose lowest eigenvalue $\lambda_{cr}$ is associated with buckling. The critical or buckling load is, from Eq. 14.5-2,
+
+$$
+\{\mathbf {R} \} _ {\mathrm{cr}} = \lambda_ {\mathrm{cr}} \{\mathbf {R} \} _ {\mathrm{ref}} \tag {14.5-5}
+$$
+
+The eigenvector $\{d\mathbf{D}\}$ associated with $\lambda_{\mathrm{cr}}$ defines the buckling mode. The magnitude of $\{d\mathbf{D}\}$ is indeterminate. Therefore $\{d\mathbf{D}\}$ identifies shape but not amplitude.
+
+A physical interpretation of Eq. 14.5-4 as follows. Terms in parentheses in Eq. 14.5-4 comprise a total or net stiffness matrix $[K_{net}]$ . Since forces $[K_{net}]\{dD\}$ are zero, one can say that membrane stresses of critical intensity reduce the stiffness of the structure to zero with respect to buckling mode $\{dD\}$ .
+
+If the foregoing analysis were to use $[N_{1}]$ instead of $[K_{\sigma}]$ , the displacements needed to construct $[N_{1}]$ would be those obtained from static analysis under load $\{R\}_{ref}$ . Figure 14.5-1 gives an example of how well these two approaches to buckling analysis compare with the actual collapse load.
+
+![](images/page-462_32853a502f452bef44b0472a226c14c12171d71ea598ca6791214b63fa521e2f.jpg)
+
+<details>
+<summary>other</summary>
+| Method | H | H (H = 35.0) | H (H = 3.49) |
+| :--- | :--- | :--- | :--- |
+| [Kσ] used | 21.35 | 0.520 | |
+| [N1] used | 20.58 | 0.173 | |
+| Nonlinear | 20.47 | 0.100 | |
+</details>
+
+Figure 14.5-1. An elastic bar, hinged at both ends. The linearized buckling load $P_{cr}$ is compared with the collapse load determined by a more exact nonlinear analysis [14.1].
+
+<!-- source-page: 463 -->
+
+Computational methods for determining $\lambda_{cr}$ are numerous (see Appendix C). Eigenvalue extraction methods used to compute natural frequencies and modes of vibration (Section 13.5) can also be applied to buckling problems. An algorithm that requires inversion of $[K_{\sigma}]$ may fail, because $[K_{\sigma}]$ may not be a positive definite matrix. If only one $\lambda_{cr}$ is required, it may be wasteful to use a method that automatically extracts several eigenvalues. However, at times one may wish to know the several lowest eigenvalues and their associated buckling modes in order to gain insight into ways of stiffening or supporting the structure so as to make buckling less likely.
+
+By basing $[K]$ and $[K_{\sigma}]$ on the original, undeformed geometry, Eq. 14.5-4 ignores prebuckling nonlinearities that may actually be present; that is, the possible dependence of $[K]$ and $[K_{\sigma}]_{ref}$ on deformation is ignored. One way to account for such nonlinearity is to base $[K]$ and $[K_{\sigma}]_{ref}$ on the configuration just before buckling [14.1]. More specifically, one applies a trial level of load $\{R\}_{base}$ and performs a nonlinear static analysis. A result of this analysis is $[K_{t}]$ , the “tangent” stiffness matrix of the structure in its current deformed configuration. As compared with $[K]$ of the structure before loads are applied, $[K_{t}]$ is degraded in stiffness because of membrane stresses produced by $\{R\}_{base}$ . A small trial load increment $\{\Delta R\}$ is applied, and displacements produced by $[K_{t}]$ and $\{\Delta R\}$ are used to compute membrane stresses. Thus $[K_{\sigma}]_{ref}$ for the current configuration is established. The linear eigenvalue problem
+
+$$
+\left(\left[ \mathbf {K} _ {t} \right] + \Delta \lambda_ {\mathrm{cr}} \left[ \mathbf {K} _ {\sigma} \right] _ {\text {ref}}\right) \{d \mathbf {D} \} = \{\mathbf {0} \} \tag {14.5-6}
+$$
+
+is solved for $\Delta\lambda_{cr}$ . The computed value of $\Delta\lambda_{cr}$ reduces the net stiffness, which is the coefficient of $\{dD\}$ in Eq. 14.5-6, to zero with respect to the buckling mode. The predicted buckling load is
+
+$$
+\{\mathbf {R} \} _ {\mathrm{cr}} = \{\mathbf {R} \} _ {\text {base}} + \Delta \lambda_ {\mathrm{cr}} \{\Delta \mathbf {R} \} \tag {14.5-7}
+$$
+
+Equation 14.5-6 presumes that stresses change in intensity but not in distribution when the load increases an amount $\Delta\lambda_{cr}\{\Delta\mathbf{R}\}$ . This assumption becomes more nearly true as $\{R\}_{base}$ approaches $\{R\}_{cr}$ . By using a sequence of increasing loads $\{R\}_{base}$ , one can approach the correct buckling load arbitrarily closely. At convergence, $\Delta\lambda_{cr}=0$ and $\{R\}_{cr}=\{R\}_{base}$ .
+
+Example: Classical Linear Buckling. Consider the uniform column in Fig. 14.5-2. Let the column be modeled by one beam element. We take [k] from Eqs. 2.4-3 and 4.2-5, and $[k_{\sigma}]$ from Eq. 14.2-9. By inspection, we see that the axial load throughout the bar has magnitude P. We arbitrarily choose the reference value of P as -1.0, where the negative sign indicates compression, not that the load is directed leftward. Nonzero d.o.f. are $u_{2}$ , $w_{2}$ , and $\theta_{2}$ . Thus, for a single element, Eq. 14.5-4 becomes
+
+$$
+\left(c _ {2} \left[ \begin{array}{c c c} c _ {1} / c _ {2} & 0 & 0 \\ 0 & 1 2 & - 6 L \\ 0 & - 6 L & 4 L ^ {2} \end{array} \right] + \lambda_ {\mathrm{cr}} \frac {- 1}{3 0 L} \left[ \begin{array}{c c c} 0 & 0 & 0 \\ 0 & 3 6 & - 3 L \\ 0 & - 3 L & 4 L ^ {2} \end{array} \right]\right) \left\{ \begin{array}{l} u _ {2} \\ w _ {2} \\ \theta_ {2} \end{array} \right\} = \left\{ \begin{array}{l} 0 \\ 0 \\ 0 \end{array} \right\} \tag {14.5-8}
+$$
+
+where $c_{1} = AE/L$ and $c_{2} = EI/L^{3}$ . A solution other than $u_{2} = w_{2} = \theta_{2} = 0$ requires that the expression in parentheses have a zero determinant. Thus we write the char-
+
+<!-- source-page: 464 -->
+
+![](images/page-464_b0a2b70268758d290e8e25d2a66edda872db494a9f988f86370378ad50544d52.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+P
+x, u
+L
+</details>
+
+Figure 14.5-2. A uniform elastic bar, fixed at x = 0 and free at x = L. The exact $P_{cr}$ is $\pi^{2}EI/4L^{2} = 2.4674EI/L^{2}$ (in compression).
+
+acteristic polynomial and extract its lowest root (this method is suitable for hand calculation provided there are few d.o.f.). The lowest root in the present case is
+
+$$
+\lambda_ {\mathrm{cr}} = 2. 4 8 6 0 E I / L ^ {2} \quad \text { hence } \quad P _ {\mathrm{cr}} = \lambda_ {\mathrm{cr}} (- 1. 0) = - 2. 4 8 6 0 E I / L ^ {2} \tag {14.5-9}
+$$
+
+Matrix $[\mathbf{k}_{\sigma}]$ in Eq. 14.5-8 comes from Eq. 14.2-9. If instead we use $[\mathbf{k}_{\sigma}]$ from Eq. 14.2-11, we obtain $P_{\mathrm{cr}} = -3EI / L^2$ , which is less accurate than $P_{\mathrm{cr}}$ in Eq. 14.5-9. Yet both answers are upper bounds to the correct magnitude of $P_{\mathrm{cr}}$ , which is $2.4674EI / L^2$ . Bounds are discussed further in Section 14.6.
+
+The buckling mode can be computed from Eq. 14.5-8 by setting $\lambda = \lambda_{\mathrm{cr}}$ , choosing an arbitrary value for one of the d.o.f. (e.g., $\theta_{2} = 1$ ), and solving for the remaining d.o.f. Thus we obtain
+
+$$
+u _ {2} = 0 \quad w _ {2} = 0. 6 3 7 9 L \quad \theta_ {2} = 1 \tag {14.5-10}
+$$
+
+D.o.f. in Eq. 14.5-10 are buckling displacements $\{d\mathbf{D}\}$ , measured relative to the reference state in which $\lambda_{\mathrm{cr}}[\mathbf{K}_{\sigma}]_{\mathrm{ref}}$ is associated with initial membrane stresses. In this reference state, $u_{2} = -PL / AE$ and $w_{2} = \theta_{2} = 0$ . We see that $u_{2}$ plays no role in Eq. 14.5-8. Indeed, the buckling equation could have been written using $w_{2}$ and $\theta_{2}$ as the only d.o.f.
+
+Example: Condensation of D.O.F. In structural dynamics, one can use condensation to reduce the size of the eigenvalue problem (see Section 13.7). The same can be done in buckling problems, with $[K_{\sigma}]$ taking the place of [M]. One might elect to eliminate rotational d.o.f. In the preceding example, to eliminate $\theta_{2}$ from Eq. 14.5-8, Eq. 13.7-3 becomes
+
+$$
+[ \mathrm{T} ] = \left[ \begin{array}{c c} 1 & 0 \\ 0 & 1 \\ 0 & 3 / 2 L \end{array} \right] \quad \text { where } \quad \frac {3}{2 L} = - \left(\frac {L}{4 E I}\right) \left(- \frac {6 E I}{L ^ {2}}\right) \tag {14.5-11}
+$$
+
+The transformations of Eq. 13.7-5 convert Eq. 14.5-8 to
+
+$$
+\left(c _ {2} \left[ \begin{array}{c c} c _ {1} / c _ {2} & 0 \\ 0 & 3 \end{array} \right] + \lambda_ {\mathrm{cr}} \frac {- 1}{3 0 L} \left[ \begin{array}{c c} 0 & 0 \\ 0 & 3 6 \end{array} \right]\right) \left\{ \begin{array}{l} u _ {2} \\ w _ {2} \end{array} \right\} = \left\{ \begin{array}{l} 0 \\ 0 \end{array} \right\} \tag {14.5-12}
+$$
+
+from which $\lambda_{cr} = 2.5EI/L^{2}$ and $P_{cr} = -2.5EI/L^{2}$ . We see that condensation has slightly increased the magnitude of the computed buckling load.
+
+# 14.6 REMARKS ON $[K_{\sigma}]$ AND ITS USES
+
+Field on Which $[k_{\sigma}]$ is Based. A stress stiffness matrix is termed “consistent” if built from the same shape functions used to build the conventional stiffness matrix.
+
+<!-- source-page: 465 -->
+
+If the structure geometry is well modeled and if elements are compatible and not softened by low-order integration rules, then such a formulation yields an upper bound to the magnitude of the correct buckling load. The “correct” buckling load is the linear bifurcation load of the structure in its reference configuration; it is not necessarily the collapse load of the actual structure. Finite element analysis would yield the correct buckling load if $[k]$ and $[k_{\sigma}]$ were based on fields that include the buckled shape as a possible displacement mode. In the case of buckling of a perfect pin-ended column, this mode is sinusoidal rather than cubic, and the correct buckling load has magnitude $\pi^{2}EI/L^{2}$ .
+
+We can base [k] and $[k_{\sigma}]$ on different displacement fields. We recall from the convergence requirements of Section 4.5 that if a strain energy expression involves displacement derivatives of order m, the displacement field must provide inter-element continuity of displacement derivatives of order m - 1 as the mesh is refined. Energy integrals that yield $[k_{\sigma}]$ involve first derivatives of displacement, so continuity of displacement is all that is required. Thus, for example, we would expect to be able to determine $P_{cr}$ for a pin-ended column by using the conventional beam [k] but the $[k_{\sigma}]$ of Eq. 14.2-11. This is indeed the case but, for a given accuracy, we must divide the column into more elements than when we use the consistent $[k_{\sigma}]$ of Eq. 14.2-9.
+
+It is sometimes recommended that $[k_{\sigma}]$ for a complicated element be based on a simpler displacement field than that used to construct the conventional stiffness matrix, in order to increase computational efficiency with little loss in accuracy. The “best” $[k_{\sigma}]$ is probably intermediate to the consistent $[k_{\sigma}]$ and the simplest possible $[k_{\sigma}]$ . Numerical evidence suggests that computed buckling loads are increased when $[k_{\sigma}]$ is simplified. If $[k_{\sigma}]$ is generated by numerical integration, a simplified displacement field is effectively employed by adopting a reduced order of quadrature.
+
+A valid $[\mathbf{k}_{\sigma}]$ must not generate nodal loads during a rigid-body translation. Nodal loads do appear when an element rotates. Indeed, from Eq. 14.5-1 one can interpret buckling as a displacement state $\{d\mathbf{D}\}$ in which pseudo-loads $[\mathbf{K}_{\sigma}]\{d\mathbf{D}\}$ are equal in magnitude to the corresponding resistances $[\mathbf{K}]\{d\mathbf{D}\}$ .
+
+Stress stiffness matrices have been devised for many buckling problems, for example, for homogeneous and sandwich plates [14.5-14.7], torsional and torsional-flexural buckling of prismatic members [5.4,14.8], tapered bars and plates [14.6,14.9], and nonconservative problems [14.10].
+
+Applications of $[K_{\sigma}]$ . A shell of revolution usually has a nonaxisymmetric buckling mode even if geometry, supports, material properties, and loading are all axisymmetric. The buckling mode will probably display many waves in each hoop circle. It is commonly assumed that the buckling mode varies circumferentially as a single Fourier harmonic. Thus buckling analysis is similar to the displacement analysis described in Sections 10.5 and 10.6 [14.11]. First the shell is divided into elements such as those shown in Fig. 12.4-2. Then one selects a specific number n of circumferential waves and computes the corresponding $[K]_{n}$ and $[K_{\sigma}]_{n}$ . Next one solves the eigenvalue problem to obtain $\lambda_{cr}$ for n waves. The entire procedure is repeated for $n + 1$ waves, for $n + 2$ waves, and so on. Provided that the initial n is sufficiently small, the lowest of the sequence of $\lambda_{cr}$ values can be identified as the desired buckling parameter. It is not obvious which mode will govern, and many analyses may be needed: Ref. 14.12 mentions a case where buckling is associated with 39 circumferential waves. If many waves also appear in the meridional direction, many elements are needed even if the shell has simple geometry.
+
+<!-- source-page: 466 -->
+
+Dynamic analysis of undamped structures with membrane forces leads to the equations
+
+dynamic response: $[\mathbf{K} + \mathbf{K}_{\sigma}]\{\mathbf{D}\} + [\mathbf{M}]\{\ddot{\mathbf{D}}\} = \{\mathbf{R}\}$ (14.6-1)
+
+natural frequencies: $([\mathbf{K} + \mathbf{K}_{\sigma}] - \omega^{2}[\mathbf{M}])\{\overline{\mathbf{D}}\} = \{\mathbf{0}\}$ (14.6-2)
+
+where  $ [M] = mass matrix, \{\ddot{D}\} = accelerations of nodal d.o.f., \omega = circular frequency, and \{\overline{D}\} = amplitudes of nodal d.o.f. Tensile membrane forces increase the frequencies. Compressive forces decrease them and produce the root  \( \omega = 0 $  if buckling impends.
+
+A structure may have no conventional stiffness [K]. An example is a linkage of pin-connected bars, like a chain, with each link idealized as rigid. Similarly, some elastic structures may have a [K] that offers no resistance to certain loads. Examples include straight cables and flat membranes, which have no bending stiffness with which to resist lateral loads. Static problems of this type can be analyzed by the equation $[\mathbf{K}_{\sigma}]\{\mathbf{D}\} = \{\mathbf{R}\}$ , where $\{\mathbf{D}\}$ contains d.o.f. associated with small lateral deflection. Analogous dynamic problems, such as a plucked string or a vibrating membrane, can be analyzed by Eqs. 14.6-1 and 14.6-2 with [K] = [0].
+
+# 14.7 REMARKS ON BUCKLING AND BUCKLING ANALYSIS
+
+A real structure may collapse at a load quite different than that predicted by a linear bifurcation buckling analysis. The following remarks, extracted largely from Refs. 14.13 to 14.15, describe types of buckling behavior and caution against oversimplification in analysis. Throughout the discussion it is assumed that the material of the structure remains linearly elastic and that loads are gradually applied.
+
+Figure 14.7-1 illustrates some of the ways a structure may behave. Here $P$ is either the load or is representative of its magnitude, and $D$ is displacement of some d.o.f. of interest. In Fig. 14.7-1a, the primary or prebuckling path happens to be linear. At bifurcation, either of two adjacent and infinitesimally close equilibrium positions are possible. Thereafter, for $P > P_{\mathrm{cr}}$ , a real (imperfect) structure
+
+![](images/page-466_4d5c180a0c0452f4c9ad46bed463964414d1a06ed7084d0b6d6c46bea6dc5783.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+Primary path
+Limit point
+Secondary path
+(postbuckling)
+Pcr
+Bifurcation point
+D
+</details>
+
+(a)
+
+![](images/page-466_42652ad707078746a4f69eff1ca7cecee8fe15f757454b9673e7a7995dc6c922.jpg)
+
+<details>
+<summary>line</summary>
+
+| Point Type              | Description                     | P Value |
+| ----------------------- | --------------------------------- | ------- |
+| Bifurcation point       | Bifurcation point                 | P_cr    |
+| Limit point (on primary path) | Limit point (on primary path)    | P_cr    |
+| Postbuckling (secondary path) | Postbuckling (secondary path)   | P_cr    |
+| Actual (imperfect) structure | Actual (imperfect) structure    | P_cr    |
+</details>
+
+{b}   
+Figure 14.7-1. Possible load versus displacement behaviors of thin-walled structures.
+
+<!-- source-page: 467 -->
+
+follows the secondary path. The secondary (postbuckling) path rises, which means that the structure has postbuckling strength. In this case $P_{cr}$ characterizes a local buckling action that has little to do with overall strength. This structure finally collapses at a limit point, which is defined as a relative maximum on the P versus D curve for which there is no adjacent equilibrium position. Loose terminology may refer to the limit point load as a buckling load. The action at collapse becomes dynamic, because the slope of the curve becomes negative and the structure releases elastic energy, which is converted into kinetic energy.
+
+A different type of behavior is depicted in Fig. 14.7-1b. Here the perfect (idealized) structure has a nonlinear primary path. The postbuckling path falls, so there is no postbuckling strength. If the primary path is close to a falling secondary path, the structure is called imperfection sensitive, which means that the collapse load of the actual structure is strongly affected by small changes in direction of loads, manner of support, or changes in geometry. The actual structure, which has imperfections, displays a limit point rather than bifurcation, as shown by the dashed line.
+
+Figure 14.7-2 shows how the response may be affected by overall structure geometry. Figure 14.7-2c, for a deep spherical cap, also applies qualitatively to a cylindrical shell under axial compression. The deep cap and the cylindrical shell are imperfection sensitive: if the radius-to-thickness ratio is large, laboratory specimens buckle at roughly one half of the theoretical bifurcation load, even when heroic efforts are made to achieve geometric perfection.
+
+In the absence of prior knowledge about how a structure behaves, one must anticipate that a computed bifurcation buckling load may be far above or far below the actual collapse load, that imperfections may be influential, and that prebuckling nonlinearities may be important. Nonlinearities may arise because pressure loads change direction as the structure deforms, because the deformed shape is more (or less) susceptible to instability than the undeformed shape, or because of the effects of deformation on the membrane stress distribution. Nonlinearities may be accounted for as described in connection with Eqs. 14.5-6 and 14.5-7, or by a
+
+![](images/page-467_4b78c1bd928bba630525c8dbcbf42ef6de48ffa1cffeee6037db399ac1dec04c.jpg)  
+Figure 14.7-2. Pressure p versus center deflection D for thin-walled elastic structures. Dashed line: linear theory. Solid line: nonlinear theory and actual behavior. (a) Circular plate. (b) Shallow spherical cap. (c) Deep spherical cap.
+
+<!-- source-page: 468 -->
+
+nonlinear analysis that effectively plots load versus displacement and signals collapse when the total stiffness matrix of the structure in its current configuration becomes singular.
+
+Collapse analysis should be approached with caution and with expertise. Novices are cautioned against misuse of computer programs. For example, the axially compressed cylindrical shell is an attractive test case, yet the problem is analytically difficult because several eigenvalues are clustered and correspond to quite different eigenmodes. Even a program that can negotiate the difficulties will not produce the correct result if, misled by the geometric simplicity of the problem, the user has employed so few d.o.f. that the many waves of the actual buckling mode cannot be properly modeled $[14.14]$ .
+
+# PROBLEMS
+
+# Section 14.1
+
+14.1 A straight wooden column has an axial hole that fits closely but without friction around a metal rod, as shown. The rod is tensioned by tightening nuts that bear on the ends of the column. Assume that linear elasticity prevails and that the rod remains precisely centered in the column. Will the column buckle?
+
+![](images/page-468_9c8f655d905abccb26d4a1233ea0dd18eb239cf98e9d4726de8ea3a02d3517d5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L
+</details>
+
+Problem 14.1
+
+![](images/page-468_fc96d24d628d33632a96c1df0c681d6720215067588490d3ba70e558cae64113.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+M₀
+P
+k
+k
+L
+</details>
+
+Problem 14.2
+
+![](images/page-468_3366c0fc0db235c605a444e20a84514ecd668d84e78041037a2802cead147782.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+e
+k
+L
+</details>
+
+Problem 14.4
+
+14.2 A rigid bar is supported by two springs, each of stiffness k, and loaded by horizontal forces P, as shown. Use the view that loads P do work during a small rotation of the bar (noted in the footnote in Section 14.1), and determine:
+
+(a) the angle of rotation of the rigid bar, in terms of $M_0, k, P$ , and $L$ .   
+(b) the (compressive) value of $P$ for buckling (when $M_0 = 0$ ), in terms of $k$ and $L$ .
+
+14.3 For the problem described by Fig. 14.1-1, let $P_{cr} = -\pi^{2}EI/L^{2}$ and $w_{c0}$ represent the value of $w_{c}$ produced by q alone (when P = 0). Plot $w_{c}/w_{c0}$ versus $P/P_{cr}$ as P goes from $P = P_{cr}$ (compressive) to $P = 5|P_{cr}|$ (tensile).
+
+14.4 A rigid bar is pivoted at the lower end and held by a linear spring at the upper end, as shown. Load P is offset a distance e from the bar axis. Find $P_{cr}$ (for e = 0). Also, using small-angle approximations, express lateral deflection $\Delta$ of the top in terms of P, e, k, and L. Plot $\Delta/L$ versus $P/P_{cr}$ for e/L = 0, 0.01, and 0.02.
+
+<!-- source-page: 469 -->
+
+# Section 14.2
+
+14.5 Buckling of a tapered column is to be studied. Each element of the column is tapered. In which element matrices ([k] or $[\mathbf{k}_{\sigma}]$ ) does the effect of taper appear, and how is it to be included?
+
+14.6 Show that Eq. 14.2-9 is produced by Eq. 14.2-8 and the standard cubic beam shape functions.
+
+14.7 (a) Show that Eq. 14.2-4 results from Eqs. 14.2-1 through 14.2-3. (b) How must $u_{,x}$ and $w_{,x}$ be related if the term in $\epsilon_x^2$ that contains $w_{,x}^4$ is to be less than $5\%$ of $u_{,x}w_{,x}^2$ ? Hence, what limiting angle of rotation is indicated if $u_{,x}$ is 0.002?
+
+14.8 Construct a 4 by 4 matrix $[\mathbf{k}_{\sigma}]$ for a uniform beam element, analogous to Eq. 14.2-9, by using the quadratic displacement field $w = (1 - \xi)w_{1} + \xi w_{2} + (1 - \xi)\xi L(\theta_{1} - \theta_{2}) / 2$ , where $\xi = x / L$ .
+
+14.9 Derive each of the two $[k_{\sigma}]$ matrices in Eq. 14.2-11 by imposing the appropriate restrictions on $[k_{\sigma}]$ of Eq. 14.2-9.
+
+14.10 A member of a plane, pin-jointed truss makes an angle $\beta$ with the global x axis. Determine an expression for $[k_{\sigma}]$ , analogous to Eq. 14.2-11b, that operates on global d.o.f. $u_{1}$ , $w_{1}$ , $u_{2}$ , and $w_{2}$ . Express your answer in terms of P, L, and $\beta$ .
+
+14.11 For the system shown, establish a set of two equations that could be solved to determine $w_{2}$ and $\theta_{2}$ in member 1–2 in terms of P, Q, E, I, L, and c. The connection at node 2 transmits no moment.
+
+![](images/page-469_3df9666898caf3e530548d1b707082c48dbf18a4af1cfe0209ee7300fb32ee58.jpg)
+
+<details>
+<summary>text_image</summary>
+
+EI
+Q
+Rigid
+1
+2
+3
+P
+L
+c
+</details>
+
+Problem 14.11
+
+![](images/page-469_ad03afa7fa04a21cf52056892bdf6a8f99e484c06eb53e79768c36afbe7b459d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+L
+Rigid
+Q
+1
+2
+P
+x
+Linear spring
+k
+</details>
+
+Problem 14.12
+
+14.12 The bar shown is hinged at node 1 and may be considered rigid and weightless. In terms of $P, Q, k,$ and $L$ , what is deflection $w_2$ if $P$ is (a) zero, (b) $0.96kL$ in tension, and (c) $0.96kL$ in compression?
+
+14.13 Solve Problem 14.12(c), in which $P = -0.96kL$ , by the following iterative method. For $Q$ alone, $w_{2} = Q / k$ . Now, with $w_{2} > 0$ , $Q$ and $P$ exert a moment about node 1. Another analysis therefore yields a $w_{2}$ larger than before. The process repeats.
+
+14.14 The column shown is fixed at the left end. Axial load $P$ at the simply supported end has eccentricity $e$ from the centerline. Model the column by one element.
+
+(a) Determine the rotation $\theta_{2}$ in terms of $P, L, E, I$ , and $e$ .
+
+(b) If $e = 0$ , what load $P$ will make $\theta_2 \neq 0$ ? What is the percentage error of this result?
+
+<!-- source-page: 470 -->
+
+![](images/page-470_a1bc559bd14eddd50599d84acdb87962b6f2b76dd9a5f923e11c111d110e98c7.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+P
+1
+2
+e
+x
+L
+</details>
+
+Problem 14.14
+
+![](images/page-470_e4ab8db1b56203e653cbd15450247773445aa9aa8bbf5dad2ddfefeaae334d6a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+EI = 110 N · m²
+0.1 N
+1
+2
+P
+x
+3.0 m
+</details>
+
+Problem 14.15
+
+(a) Determine $w_{2}$ if $P = 0$ .   
+(b) Determine $w_{2}$ if $P = 30.0 \, \mathrm{N}$ in compression.   
+(c) Determine $w_{2}$ if P = 30.0 N in tension.
+
+14.15 The cantilever beam shown may be subjected to either a tensile or a compressive axial force P. Model the beam by one element. Use $w_{2}$ and $\theta_{2}$ as d.o.f. and take $[k_{\sigma}]$ from Eq. 14.2-9.   
+14.16 Repeat Problem 14.15, but use the $[\mathbf{k}_{\sigma}]$ developed in Problem 14.8.   
+14.17 In Problem 14.15(b), evaluate the bending moment at $x = 0$ by the calculation $M = EI[\mathbf{B}]\{\mathbf{d}\}$ , where $[\mathbf{B}]$ is based on a cubic field and $\{\mathbf{d}\} = [0 \ 0 \ w_2 \ \theta_2]^T$ . Compare this $M$ with that obtained by statics; that is, $M = 0.1L - Pw_2$ , where $P = -30.0 \ \text{N}$ in this case.   
+14.18 Let the 4 by 1 displacement vector $\{\mathbf{d}\}$ represent a small rigid-body rotation of a bar or beam about its left end. Determine the nodal forces $[\mathbf{k}_{\sigma}]\{\mathbf{d}\}$ for a bar (Eq. 14.2-11) and a beam (Eq. 14.2-9). Express your answers in terms of $P$ , $L$ , and $w_{2}$ .   
+14.19 Equation 4.1-6 has been used to compute consistent nodal loads on a beam (e.g., see Fig. 4.3-6). If the beam also carries an axial force, are these loads still correct? Explain.   
+14.20 In Chapter 13 we discussed a diagonal element mass matrix [m]. Why is an analogous diagonal stress stiffness matrix $\left[\mathbf{k}_{\sigma}\right]$ unacceptable if $\{\mathbf{d}\}$ contains only translational d.o.f.?
+
+# Section 14.4
+
+14.21 Show that Eq. 14.4-2 yields the small-strain approximation $\epsilon = (ds^{*} - ds)/ds$ if $ds \approx ds^{*}$ .
+
+14.22 (a) Imagine that the bar element of Fig. 14.2-1b is generalized to three dimensions. Thus the bar lies on the $x$ axis of a local coordinate system xyz that is arbitrarily oriented with respect to global coordinates. Nodal d.o.f. consist of three translations at each node (in local directions). What is the 6 by 6 matrix $[\mathbf{k}_{\sigma}]$ in local coordinates?
+
+(b) How would you establish the $[\mathbf{k}_{\sigma}]$ that operates on translational d.o.f. in global coordinate directions?
+
+14.23 Show that Eq. 14.3-5 results from appropriate specialization of Eq. 14.4-7 (or of Eq. 14.4-6).
+
+14.24 Consider the eight-node trilinear (solid) isoparametric element. How many rows and columns are there in [G] of Eq. 14.4-7? Express the $G_{ij}$ in terms of shape function derivatives and coefficients $\Gamma_{ij}$ of the inverse Jacobian matrix. For convenience, let $\{d\} = \left[u_{1} \quad u_{2} \quad \ldots \quad u_{8} \quad v_{1} \quad \ldots \quad w_{8}\right]^{T}$ .
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_048.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_048.md
new file mode 100644
index 00000000..bf53fa1d
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_048.md
@@ -0,0 +1,475 @@
+<!-- source-page: 471 -->
+
+14.25 (a) A three-node plane triangular element is constrained to move only in its plane. Write the formula for $[\mathbf{k}_{\sigma}]$ in terms of a 2 by 2 matrix [s] and matrix [B] of Eq. 5.4-3. Let $\{\mathbf{d}\} = \left[u_1 \quad u_2 \quad u_3 \quad v_1 \quad v_2 \quad v_3\right]^T$ .   
+(b) Let the same element be allowed only deflection $w = w(x, y)$ normal to its plane. What then is the formula for $[\mathbf{k}_{\sigma}]$ ? Let $\{\mathbf{d}\} = \begin{bmatrix} w_1 & w_2 & w_3 \end{bmatrix}^T$ .
+
+14.26 (a) Show that Eqs. 14.4-11 and 14.4-12 yield Eq. 14.4-13.
+
+(b) Derive Eq. 14.4-14 from Eq. 14.4-13.
+
+14.27 Show that Eq. 14.4-12 yields $\epsilon_{x} = 0$ for the following rigid-body rotations. In addition, make a sketch that shows the displaced position of the bar.
+
+(a) $u_{2} = -1, w_{2} = 3$ , and $L = 5$ .   
+(b) $u_{2} = -5, w_{2} = 5$ , and $L = 5$ .   
+(c) $u_{2} = -10, w_{2} = 0$ , and $L = 5$ .   
+(d) $u_{2} = -L(1 - \cos \theta), w_{2} = L \sin \theta.$
+
+14.28 For each of the rigid-body rotations stated in Problem 14.27, show that Eq. 14.4-13 yields $R_{x} = R_{z} = 0$ .
+
+14.29 If $\Delta$ represents an axial-displacement increment, the associated force in a uniform bar is $F = AE\Delta/L$ . Show that Eq. 14.4-14 yields this result, whether the reference configuration of the bar in Fig. 14.4-1b is defined by $\theta = 0$ or by $\theta = \pi/2$ . Assume that strains are small.
+
+# Section 14.5
+
+14.30 (a) Equation 14.5-8 has two roots, of which the lower, $\lambda = \lambda_{\mathrm{cr}}$ , is given by Eq. 14.5-9. What is the other root and the corresponding mode shape?   
+(b) Verify that use of $[k_{\sigma}]$ from Eq. 14.2-11 yields $P_{cr} = -3EI/L^{2}$ for the problem of Fig. 14.5-2. Again use a single element.
+
+14.31 Use the $[k_{\sigma}]$ derived in Problem 14.8 to determine the buckling load in Fig. 14.5-2. Use one element. The only d.o.f. needed are $w_{2}$ and $\theta_{2}$ .
+
+14.32 The two pin-connected bars shown may be considered rigid and weightless. The linear springs each have stiffness $k$ .
+
+(a) Determine the buckling load $P_{\mathrm{cr}}$ .
+
+(b) In the buckling mode, determine $w_{3}$ if $w_{2} = 1$ . Sketch this mode, and show that "kickoff" forces $[\mathbf{K}_{\sigma}][w_{2} - w_{3}]^{T}$ are equal in magnitude to forces in the deflected springs.
+
+![](images/page-471_f18cbcc4fb8f5e7bf0e4969a394213bfd253de84100a8b3189cc53f106bf7005.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+a
+a
+Rigid
+Rigid
+P
+1
+2
+3
+k
+k
+x
+</details>
+
+Problem 14.32
+
+![](images/page-471_4f639868a0648b9844f56caebe96c31ad48471124432bc0324f742a4c4c6a0c7.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z, w
+← a → ← a →
+EI EI P
+1 2 3 x
+F
+</details>
+
+Problem 14.33
+
+14.33 The uniform beam shown is fixed at the left end, simply supported at the right end, and divided into two identical beam elements, each of length $a$ .
+
+(a) Set up a three-equation system $[K + K_{\sigma}]\{D\} = \{R\}$ , then eliminate d.o.f. $\theta_{2}$ and $\theta_{3}$ by condensation, leaving $w_{2}$ as the only d.o.f.
+
+<!-- source-page: 472 -->
+
+(b) Determine $P_{\mathrm{cr}}$ (for the case $F = 0$ ) and compute its percentage error.   
+(c) Plot $w_{2} / a$ versus $P / P_{\mathrm{cr}}$ if $F = 0.10EI / a^2$ .
+
+14.34 Determine buckling loads of the uniform columns shown. Express $P_{\mathrm{cr}}$ in terms of $E, I,$ and $L$ . Use the $[\mathbf{k}_{\sigma}]$ of Eq. 14.2-9.
+
+(a) For Fig. (a), use $\theta_{1}$ and $\theta_{2}$ as d.o.f.   
+(a) For Fig. (a), use $\theta_{1}$ and $\theta_{2}$ as d. on.  
+(b) For Fig. (b), impose symmetry about the midpoint, so that $\theta_{1}$ and $w_{2}$ are the only d.o.f. needed. Node 2 is not a hinge.   
+(c) For Fig. (c), let the linear spring have stiffness $k = 2EI / L^3$ . The lower end is fully fixed. No axial d.o.f. is needed.  
+(d) For Fig. (d), let the rotational spring have stiffness $k = EI / L$ . The lower   
+(d) For Fig. (d), let the rotational spring have stiffness $k = EI / L$ . The lower end is fully fixed. No axial d.o.f. is needed.   
+(e) Repeat part (c) letting $k \to \infty$ (top free to rotate but cannot translate).  
+(f) Repeat part (d) letting $k \to \infty$ (top free to translate but cannot rotate).
+
+![](images/page-472_2269d99cbf6dfba2bc96cfa4fa1e19534728a2d9bf3d2ad63b307d170b3d0066.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L
+P 1 2 P
+(a)
+</details>
+
+![](images/page-472_7bffd520ca4d7c2eb66076434bcccb08730d2b81d680733dbae6844fb6881c38.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+2
+k
+L
+1
+</details>
+
+![](images/page-472_3660ef98a021562feeffe286db04dd16f7823d2ea14e6dca1d0d9a4ade4f1499.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+2 k
+L
+1
+</details>
+
+![](images/page-472_d6c76842b3435d01a47b7bb989fa2bd684262866cb400a597fa6b9eacc2dd01f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L/2
+L/2
+P 1 2 3 P
+(b)
+</details>
+
+(c)   
+{d}   
+Problem 14.34
+
+14.35 Repeat Problem 14.34, but use the $[\mathbf{k}_{\sigma}]$ derived in Problem 14.8.   
+14.36 Repeat Problem 14.34, to the extent possible, if $[\mathbf{k}_{\sigma}]$ is taken from Eq. 14.2-11.   
+14.37 (a) Consider use of the condensation technique (described by example in Eqs. 14.5-11 and 14.5-12). How will you decide which d.o.f. to eliminate?
+
+(b) Using condensation, solve Problem 14.15(b).   
+(c) Using condensation, solve Problem 14.15(c).   
+(d) Using condensation to eliminate $\theta_{1}$ , solve Problem 14.34(b).   
+(e) Using condensation to eliminate $\theta_{1}$ , solve Problem 14.35(b).
+
+14.38 The two slender bars shown, which are of lengths $L$ and $\alpha L$ but otherwise identical, are fixed at $A$ and $C$ and welded together at $B$ , where they are simply supported and loaded by axial force $P$ . The possibility of buckling is to be analyzed. Use the two-stage procedure suggested below Eq. 14.1-9: determine axial displacement at $B$ in terms of $P$ , hence compute axial bar forces and $[\mathbf{k}_{\sigma}]$ for the bars, and finally seek $P_{\mathrm{cr}}$ using $\theta_2$ as the only d.o.f.
+
+14.39 The two-element frame shown is fixed at $A$ and at $C$ , and has the same $EI$ throughout. The connection at $B$ is rigid. Assume that d.o.f. that define axial strain are unnecessary because $AE >> EI$ . Determine angle $\beta$ that minimizes the buckling load $P_{\mathrm{cr}}$ .
+
+<!-- source-page: 473 -->
+
+![](images/page-473_4d8d719b3e9442705a764ab070735cf1be416e6ba8f4a8d7f5db06f192e6c988.jpg)
+
+<details>
+<summary>text_image</summary>
+
+A
+P B
+C
+L αL
+</details>
+
+Problem 14.38
+
+![](images/page-473_41f861aed6b8259ebc93f9bf73728875492eb5f0dca65ba07b38f55e7ab331ac.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1.3a
+A
+B
+P
+β
+a
+C
+</details>
+
+Problem 14.39
+
+14.40 Imagine that a diagonal stress stiffness matrix is proposed for the standard beam element, for which $\{\mathbf{d}\} = \left[w_1 \quad \theta_1 \quad w_2 \quad \theta_2\right]^T$ . The form of $[\mathbf{k}_{\sigma}]$ is $[\mathbf{k}_{\sigma}] = [0 \quad c \quad 0 \quad c]$ , where $c$ is a constant.
+
+(a) Determine $c$ by requiring that Eq. 14.2-7 be satisfied when $w_{,x}$ is constant over the element.   
+(b) Use this $[k_{\sigma}]$ to determine $P_{cr}$ for a uniform pin-ended column. Consider a one-element model.   
+(c) Repeat part (b), but consider a two-element model: impose symmetry about the center of the column, so that nonzero d.o.f. of one-half the column are $\theta_{1}$ and $w_{2}$ .
+
+# Section 14.6
+
+14.41 The weightless string shown is horizontal, under constant tension $T$ , and carries two particles, each of mass $m$ .
+
+(a) Determine the static deflections of the particles caused by gravity.   
+(b) Determine the natural frequencies of vibration and the mode shapes.
+
+14.42 Repeat Problem 14.41, but double the mass of the left-hand particle (to mass 2m).
+
+14.43 A massless and flexible string of length 2L hangs from the ceiling. It carries two particles, each of mass m, one at the middle and the other at the lower end.
+
+(a) What is the horizontal deflection of a small horizontal force $Q$ applied to the lower end?   
+(b) What are the natural frequencies of vibration and the mode shapes?
+
+14.44 The string shown is under tension T and has mass $\rho$ per unit length. Use [M] and $[K_{\sigma}]$ matrices associated with a cubic lateral-displacement field. Omit the conventional stiffness matrix [K]. Solve for the natural frequencies and mode shapes of small-displacement lateral vibrations. (The exact fundamental frequency is $\omega_{1}^{2} = \pi^{2}T/4\rho a^{2}$ .)
+
+(a) Use one element. Nonzero d.o.f. are then $\theta_{1}$ and $\theta_{2}$ .   
+(b) Use two elements and impose symmetry about the center. Nonzero d.o.f. to be used are then $\theta_{end}$ and $w_{center}$ .
+
+![](images/page-473_fa67379e8b9de6b1aec8604c31e4edf0386ecd567c681119acb378e1047d6d31.jpg)
+
+<details>
+<summary>text_image</summary>
+
+T
+m m
+T
+L L L
+</details>
+
+Problem 14.41
+
+![](images/page-473_1a1674f26e9c9753c5dd9c8b2ec8c8b060d3b9e83fc9a01a341774b377398891.jpg)
+
+<details>
+<summary>text_image</summary>
+
+T
+2a
+</details>
+
+Problem 14.44
+
+<!-- source-page: 474 -->
+
+14.45 Solve Problem 14.44 using $[k_{\sigma}]$ from Eq. 14.2-11, and (a) [M] based on a cubic lateral-displacement field, and (b) a lumped [M]. (That is, apply each of these mass matrices to each part of Problem 14.44.)   
+14.46 Model a simply supported beam by a single element. Let $L = 1.0 \, \text{m}$ , $A = 0.0002 \, \text{m}^2$ , $EI = 300.0 \, \text{N} \cdot \text{m}^2$ , and $\rho = 2100.0 \, \text{kg/m}^3$ . Impose symmetry (and reduce the problem to a single d.o.f.) by setting $\theta_2 = -\theta_1$ .
+
+(a) Determine the fundamental frequency $\omega_{1}$ if there is no axial force.   
+(b) Determine the axial force that makes the frequency 347 rad/sec.   
+(c) Determine the frequency if the axial force is 1200 N in compression.
+
+<!-- source-page: 475 -->
+
+# WEIGHTED RESIDUAL METHODS
+
+The construction of approximate solutions of differential equations by means of weighted residual methods is summarized. The Galerkin method, which is the most popular weighted residual method, is used to produce finite element formulations.
+
+# 15.1 INTRODUCTION
+
+Thus far we have presented the finite element method as a Rayleigh–Ritz method—that is, as an approximation technique that is applied to a variational principle. A variational principle uses an integral expression, called a functional, that yields the governing differential equations and nonessential boundary conditions of a problem when operated upon by standard procedures of the calculus of variations. The principle of stationary potential energy is only one of many variational principles.
+
+In an area of physical science other than structural mechanics, a variational principle may be unobtainable. This happens if the differential equation of the problem contains derivatives of odd order. A case in point is fluid mechanics, where, for some types of flow, all that is available are differential equations and boundary conditions. Yet the finite element method can still be applied by means of a weighted residual method. Like the Rayleigh–Ritz method, a weighted residual method uses integral expressions that contain the differential equations of a physical problem. Functional and residual formulations are both known as “weak” forms of stating the governing equations of a problem. The differential equations themselves comprise the “strong” form. (The weak form enforces conditions in an average or integral sense, whereas the strong form enforces them at every point.)
+
+The following introductory treatment uses both structural and nonstructural problems to illustrate procedures.
+
+# 15.2 SOME WEIGHTED RESIDUAL METHODS
+
+This section presents an overview and uses the following notation:
+
+$u =$ dependent variable(s), for example, displacements of a point
+
+$x =$ independent variable(s), for example, coordinates of a point
+
+$f,g =$ functions of $x$ , or constants, or zero
+
+$D, B =$ differential operators
+
+<!-- source-page: 476 -->
+
+Thus the governing differential equations and nonessential boundary conditions of an arbitrary physical problem are symbolized as
+
+$$
+D u - f = 0 \quad \text { in   domain } V \tag {15.2-1a}
+$$
+
+$$
+B u - g = 0 \quad \text { on   boundary } S \text { of } V \tag {15.2-1b}
+$$
+
+For example, in beam bending Eq. 15.2-1a becomes $EIw_{,xxx} = q$ , where w is lateral deflection and q is distributed lateral load. Thus $D = EId^{4}/dx^{4}$ , u = w, and f = q. Equation 15.2-1b symbolizes two equations, namely, $EIw_{,xx} - M_{B} = 0$ and $EIw_{,xxx} - V_{B} = 0$ , where $M_{B}$ and $V_{B}$ are prescribed values of bending moment and transverse shear force at ends of the beam.
+
+In general, the exact solution $u = u(x)$ of Eq. 15.2-1a is unknown and is often difficult to determine. We seek instead an approximate solution, $\bar{u}$ . Typically $\bar{u}$ is a polynomial that satisfies essential boundary conditions and contains undetermined coefficients $a_{1}, a_{2}, \ldots, a_{n}$ . Thus $\bar{u} = \bar{u}(a, x)$ , and $\bar{u}$ is “admissible” as defined in Section 3.2. To obtain an approximate solution we must determine values of the $a_{i}$ such that u and $\bar{u}$ are “close” in some sense.
+
+If $\bar{u}$ is substituted into Eqs. 15.2-1, equality does not prevail because $\bar{u}$ is not exact. The discrepancy can be expressed as residuals $R_{D}$ and $R_{B}$ , which are functions of x and the $a_{i}$ :
+
+$$
+R _ {D} = R _ {D} (a, x) = D \bar {u} - f \quad (\text { interior   residual }) \tag {15.2-2a}
+$$
+
+$$
+R _ {B} = R _ {B} (a, x) = B \bar {u} - g \quad (\text {boundary residual}) \tag {15.2-2b}
+$$
+
+In some physical problems it may happen that all boundary conditions are of the essential class. Then $R_{B}$ need not enter; only $R_{D}$ is used in determining the $a_{i}$ of an approximation $\bar{u} = \bar{u}(a, x)$ whose form satisfies essential boundary conditions a priori.
+
+Residuals may vanish for some values of $x$ , but they are not zero for all $x$ unless $\tilde{u}$ is the exact solution, $\tilde{u} \equiv u$ . We presume that $\tilde{u}$ is a good approximation of $u$ if residuals are small. Small residuals can be achieved by various schemes, each of which is designed to produce algebraic equations that can be solved for the $n$ coefficients $a_i$ . Some popular schemes are summarized as follows. Their use is illustrated in Section 15.3.
+
+Collocation. For n different values of x, the residuals are set to zero. The method is also called point collocation.
+
+$$
+R _ {D} (a, x _ {i}) = 0 \quad \text { for } \quad i = 1, 2, \dots , j - 1 \tag {15.2-3a}
+$$
+
+$$
+R _ {B} (a, x _ {i}) = 0 \quad \text { for } \quad i = j, j + 1, \dots , n \tag {15.2-3b}
+$$
+
+Subdomain. Over n different regions of V and S, the integral of the residual is set to zero. The method is also called subdomain collocation.
+
+$$
+\int_ {V _ {i}} R _ {D} (a, x) d V = 0 \quad \text { for } \quad i = 1, 2, \dots , j - 1 \tag {15.2-4a}
+$$
+
+$$
+\int_ {S _ {i}} R _ {B} (a, x) d S = 0 \quad \text { for } \quad i = j, j + 1, \dots , n \tag {15.2-4b}
+$$
+
+<!-- source-page: 477 -->
+
+Least Squares. The $a_{i}$ are chosen to minimize a function I:
+
+$$
+\frac {\partial I}{\partial a _ {i}} = 0 \quad \text { for } \quad i = 1, 2, \dots , n \tag {15.2-5}
+$$
+
+Function I is formed by integrating squares of the residuals,
+
+$$
+I = \int_ {V} \left[ R _ {D} (a, x) \right] ^ {2} d V + \alpha \int_ {S} \left[ R _ {B} (a, x) \right] ^ {2} d S \tag {15.2-6}
+$$
+
+where $\alpha$ is an arbitrary scalar multiplier that may be used to achieve dimensional homogeneity and also serves as a penalty number. Larger values of $\alpha$ increase the importance of $R_{B}$ relative to $R_{D}$ . The method is also called continuous least squares.
+
+Least Squares Collocation. Equation 15.2-5 is still used, but I is redefined. It is now defined in terms of squared residuals at several points i, where i runs from 1 to m and $m \geq n$ :
+
+$$
+I = \sum_ {i = 1} ^ {j - 1} \left[ R _ {D} \left(a, x _ {i}\right) \right] ^ {2} + \alpha \sum_ {i = j} ^ {m} \left[ R _ {B} \left(a, x _ {i}\right) \right] ^ {2} \tag {15.2-7}
+$$
+
+Equation 15.2-5 now yields n equations for the $a_{i}$ , even when m > n. The method is also called point least squares and overdetermined collocation. If m = n, the method becomes simple collocation.
+
+Galerkin. We select “weight functions” $W_{i} = W_{i}(x)$ and set the weighted averages of residual $R_{D}$ to zero. Or, in mathematical terms, we say that $R_{D}$ is made orthogonal to the weight functions:
+
+$$
+R _ {i} = \int_ {V} W _ {i} (x) R _ {D} (a, x) d V = 0 \quad \text { for } \quad i = 1, 2, \dots , n \tag {15.2-8}
+$$
+
+In the Bubnov–Galerkin method, usually called simply the Galerkin method, weight functions $W_{i}$ are coefficients of the generalized coordinates $a_{i}$ . Thus $W_{i} = \partial \bar{u} / \partial a_{i}$ . In the Petrov–Galerkin method, other forms of $W_{i}$ are used.
+
+In Galerkin methods, boundary residual $R_{B}$ is used in combination with integration by parts, so as to introduce nonessential boundary conditions. The procedure is illustrated in subsequent examples.
+
+Remarks. The commonality shared by the foregoing methods is that they all can be loosely symbolized as
+
+$$
+\int_ {\Gamma} W _ {i} R d \Gamma = 0 \tag {15.2-9}
+$$
+
+where R represents $R_{D}$ and/or $R_{B}$ and $\Gamma$ represents V and/or S. In words, Eq. 15.2-9 says that over the region of interest, the weighted residual has an average value of zero (i.e., $W_{i}R$ has zero average error). The various weighted residual methods differ in how $W_{i}$ is defined [15.1]. In the collocation and subdomain
+
+<!-- source-page: 478 -->
+
+methods, the $W_{i}$ are unit delta or step functions that are nonzero at certain points or over certain regions. In least squares methods, $W_{i} = \partial R/\partial a_{i}$ . In the Galerkin method, $W_{i} = \partial \bar{u}/\partial a_{i}$ .
+
+In solving a problem, whether by the Rayleigh–Ritz method or a weighted residual method, we begin by establishing a trial family of solutions, $\bar{u} = \bar{u}(a, x)$ . The $a_{i}$ that define the best form of $\bar{u}$ are chosen by the stationary–functional conditions $\partial\Pi/\partial a_{i} = 0$ for the Rayleigh–Ritz method or by Eq. 15.2-9 for a weighted residual method.
+
+Galerkin's method yields a symmetric coefficient matrix if the system of differential equations and boundary conditions is self-adjoint [15.1]. If differential equations and a variational principle are both available, then the Galerkin method and the Rayleigh-Ritz method yield identical solutions when both use the same approximating function $\bar{u}$ .
+
+Least squares methods always produce a symmetric coefficient matrix. Other advantages of least squares include the avoidance of integration in least squares collocation and the “tuning” permitted by adjustment of $\alpha$ in Eqs. 15.2-6 and 15.2-7. However, there are several disadvantages. Despite $\alpha$ , the continuous least squares method may be too strongly influenced by less important residuals. (Separate $\alpha_{i}$ may be applied to the separate residuals in least squares collocation.) The coefficient matrix tends to be ill conditioned. Because the weights are $W_{i} = \partial R/\partial a_{i}$ , both R and the $W_{i}$ contain derivatives of the same order. This means that in continuous least squares, integration by parts cannot reduce the highest-order derivatives of $\bar{u}$ , and elements of higher order are therefore needed so as to achieve the necessary degree of interelement continuity. For example, a two-node, axially loaded bar element would require two d.o.f. at each node—namely, u and $\varepsilon u_{,x}$ . This awkwardness could be avoided by reformulating the problem in terms of differential equations of lower order, but again more d.o.f. are required than in a simple displacement formulation, and element d.o.f. are a mixture of force and displacement quantities. Mixed formulations have not been popular in structural mechanics. The least squares collocation method does not require an initial reduction to first-order differential equations [15.4].
+
+# 15.3 EXAMPLE SOLUTIONS
+
+We illustrate the methods summarized in Section 15.2 by means of the following problem. Let the governing differential equation and nonessential boundary condition be $^{1}$
+
+$$
+u _ {, x x} + c x = 0 \quad \text { for } \quad 0 <   x <   L _ {T} \tag {15.3-1a}
+$$
+
+$$
+u _ {, x} - b = 0 \quad \text { at } \quad x = L _ {T} \tag {15.3-1b}
+$$
+
+where u has units of length, c is a constant having units (length) $^{-2}$ , and b is a dimensionless constant. The essential boundary condition is u = 0 at x = 0. A physical interpretation is given in Fig. 15.3-1. Equation 15.3-1a describes this
+
+<!-- source-page: 479 -->
+
+![](images/page-479_dabe956ba917627df477aff7428b2a63fe53506152e19be7682f26236d8c8b33.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+L_T
+σ_0
+A, E
+q = q(x)
+x, u
+∂²u/∂x² + q/AE = 0 for 0 < x < L_T
+E∂u/∂x = σ_0 at x = L_T
+</details>
+
+Figure 15.3-1. A physical interpretation of Eq. 15.3-1: axial displacement u of a uniform bar under linearly varying axial load q and end load $\sigma_{0}A$ , where A is the cross-sectional area. E is the elastic modulus.
+
+problem if $q$ is the linear function $q = q_0x$ and $c = q_0 / AE$ . Numerical comparison of exact and approximate solutions appears in Table 15.3-1.
+
+The exact solution of the problem is $u = (3L_{T}^{2}cx - cx^{3} + 6bx)/6$ , but we pretend that we do not know it. Instead, we seek two-parameter approximate solutions. Let the trial function be
+
+$$
+\bar {u} = a _ {1} x + a _ {2} x ^ {2} \tag {15.3-2}
+$$
+
+in which “best” values of $a_{1}$ and $a_{2}$ are required. Note that $\tilde{u}$ satisfies the essential boundary condition u = 0 at x = 0. Substitution of Eq. 15.3-2 into Eqs. 15.3-1 yields the interior and boundary residuals
+
+$$
+R _ {D} = 2 a _ {2} + c x \quad \text { and } \quad R _ {B} = (a _ {1} + 2 a _ {2} L _ {T}) - b \tag {15.3-3}
+$$
+
+The nonessential boundary condition appears only at $x = L_{T}$ , so $R_{B}$ is evaluated at $x = L_{T}$ .
+
+As an alternative to Eq. 15.3-2, one can select a trial function $\bar{u}$ that satisfies Eq. 15.3-1b a priori. Then $R_{B}$ need not be used in subsequent manipulations. However, the present approach is more in accord with finite element applications of weighted residual methods.
+
+Collocation. We arbitrarily elect to evaluate $R_{D}$ at $x = L_{T}/3$ . Equations 15.3-3, with $R_{D} = 0$ and $R_{B} = 0$ , yield
+
+$$
+a _ {1} = b + c L _ {T} ^ {2} / 3 \quad \text { and } \quad a _ {2} = - c L _ {T} / 6 \tag {15.3-4}
+$$
+
+TABLE 15.3-1 RESULTS FOR THE PROBLEM OF EQ. 15.3-1 AND FIG. 15.3-1, FOR THE SPECIAL CASE $b = 0, c = L_T = 1$ . 
+<table><tr><td>Quantity and Location</td><td>Exact</td><td>Collo- cation</td><td>Subdomain and Least Squares</td><td>Least Squares Collocation</td><td>Galerkin</td></tr><tr><td>u at  $x = {L}_{T}/2$ </td><td>0.2292</td><td>0.1250</td><td>0.1875</td><td>0.2500</td><td>0.2292</td></tr><tr><td>u at  $x = {L}_{T}$ </td><td>0.3333</td><td>0.1667</td><td>0.2500</td><td>0.3333</td><td>0.3333</td></tr><tr><td> ${u}_{,x}$  at  $x = 0$ </td><td>0.5000</td><td>0.3333</td><td>0.5000</td><td>0.6667</td><td>0.5833</td></tr><tr><td> ${u}_{,x}$  at  $x = {L}_{T}/2$ </td><td>0.3750</td><td>0.1667</td><td>0.2500</td><td>0.3333</td><td>0.3333</td></tr><tr><td> ${u}_{,x}$  at  $x = {L}_{T}$ </td><td>0</td><td>0</td><td>0</td><td>0</td><td>0.0833</td></tr></table>
+
+<!-- source-page: 480 -->
+
+Subdomain. We elect to evaluate Eq. 15.2-4a by integrating over the entire span, x = 0 to $x = L_{T}$ . Thus Eq. 15.2-4a and $R_{B} = 0$ yield
+
+$$
+a _ {1} = b + c L _ {T} ^ {2} / 2 \quad \text { and } \quad a _ {2} = - c L _ {T} / 4 \tag {15.3-5}
+$$
+
+Least Squares. If $\alpha = 1 / L_T$ the two terms in Eq. 15.2-6 have the same units, and
+
+$$
+I = \int_ {0} ^ {L _ {T}} (2 a _ {2} + c x) ^ {2} d x + \frac {1}{L _ {T}} \left[ \left(a _ {1} + 2 a _ {2} L _ {T}\right) - b \right] ^ {2} \tag {15.3-6}
+$$
+
+in which the latter term is already “integrated” over the point $x = L_{T}$ . The equations $\partial I/\partial a_{1} = 0$ and $\partial I/\partial a_{2} = 0$ are found to yield Eqs. 15.3-5.
+
+Least Squares Collocation. We arbitrarily elect to evaluate $R_{D}$ at two points, and arbitrarily choose them to be $x = L_{T}/3$ and $x = L_{T}$ . Residual $R_{B}$ is as stated in Eq. 15.3-3. Again let $\alpha = 1/L_{T}$ . The three residuals can be written in the form
+
+$$
+\left\{ \begin{array}{l} R _ {D 1} \\ R _ {D 2} \\ R _ {B} \end{array} \right\} = \left[ \begin{array}{l l} 0 & 2 \\ 0 & 2 \\ 1 / L _ {T} & 2 \end{array} \right] \left\{ \begin{array}{l} a _ {1} \\ a _ {2} \end{array} \right\} - \left\{ \begin{array}{c} - c L _ {T} / 3 \\ - c L _ {T} \\ b / L _ {T} \end{array} \right\} \tag {15.3-7}
+$$
+
+If all three residuals are set to zero, Eqs. 15.3-7 form an overdetermined system (three equations in two unknowns). A least squares solution requires that we form $I = R_{D1}^{2} + R_{D2}^{2} + R_{B}^{2}$ and apply Eq. 15.2-5 for i = 1 and i = 2. We introduce symbols [Q], {a}, and {c} for arrays on the right-hand side of Eq. 15.3-7 and obtain the least squares solution for $a_{1}$ and $a_{2}$ as follows:
+
+$$
+I = \{\mathbf {R} \} ^ {T} \{\mathbf {R} \} \quad \text { where } \quad \{\mathbf {R} \} = [ \mathbf {Q} ] \{\mathbf {a} \} - \{\mathbf {c} \} \tag {15.3-8a}
+$$
+
+$$
+I = \{\mathbf {a} \} ^ {T} [ \mathbf {Q} ] ^ {T} [ \mathbf {Q} ] \{\mathbf {a} \} - 2 \{\mathbf {a} \} ^ {T} [ \mathbf {Q} ] ^ {T} \{\mathbf {c} \} + \{\mathbf {c} \} ^ {T} \{\mathbf {c} \} \tag {15.3-8b}
+$$
+
+$$
+\left\{\frac {\partial I}{\partial \mathbf {a}} \right\} = \{\mathbf {0} \} \quad \text { yields } \quad [ \mathbf {Q} ] ^ {T} [ \mathbf {Q} ] \{\mathbf {a} \} = [ \mathbf {Q} ] ^ {T} \{\mathbf {c} \} \tag {15.3-8c}
+$$
+
+In writing Eq. 15.3-8b we have used the relation $\{\mathbf{c}\}^T [\mathbf{Q}]\{\mathbf{a}\} = \{\mathbf{a}\}^T [\mathbf{Q}]^T \{\mathbf{c}\}$ , which is true because each of the two matrix triple products is a scalar. In Eq. 15.3-8c, which is to be solved for $\{\mathbf{a}\}$ , the coefficient matrix $[\mathbf{Q}]^T [\mathbf{Q}]$ is symmetric and of the same order as $\{\mathbf{a}\}$ . Applying Eq. 15.3-8c to Eq. 15.3-7, we obtain
+
+$$
+a _ {1} = b + 2 c L _ {T} ^ {2} / 3 \quad \text { and } \quad a _ {2} = - c L _ {T} / 3 \tag {15.3-9}
+$$
+
+Galerkin. From Eqs. 15.2-8 and 15.3-1a, the ith residual is
+
+$$
+R _ {i} = \int_ {0} ^ {L _ {T}} W _ {i} (\bar {u}, _ {x x} + c x) d x \tag {15.3-10}
+$$
+
+where i = 1,2 in the present example. It is standard practice in the Galerkin method to begin with integration by parts. A motivation is to reduce the order of differentiation in the integral. If derivatives of order 2m appear, the integral is defined if the integrand has continuous derivatives through order 2m - 1. In the
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+<!-- source-page: 481 -->
+
+present example, 2m - 1 = 1, which means that d.o.f. of a finite element model would have to include nodal values of $\tilde{u}$ and $\tilde{u}_{,x}$ in order to achieve the required continuity. Integration by parts reduces 2m - 1 to 1 - 1 = 0, so that d.o.f. of a finite element model need only include nodal values of $\tilde{u}$ . Integration by parts also serves to introduce nonessential boundary conditions, as follows.
+
+In one dimension, the formula for integration by parts is written in conventional notation as $\int u dv = uv - \int v du$ , where $u$ and $v$ represent functions of $x$ . In the present example, we apply integration by parts to the term $W_{i}\tilde{u}_{,xx}$ only. Thus $W_{i}\tilde{u}_{,xx} dx$ is regarded as $u dv$ in the formula for integration by parts, where $u = W_{i}$ and $dv = \tilde{u}_{,xx} dx = d(\tilde{u}_{,x})$ . Equation 15.3-10 becomes
+
+$$
+R _ {i} = \int_ {0} ^ {L _ {T}} \left(- W _ {i, x} \bar {u}, _ {x} + W _ {i} c x\right) d x + \left[ W _ {i} \bar {u}, _ {x} \right] _ {0} ^ {L _ {T}} \tag {15.3-11}
+$$
+
+where, from Eq. 15.3-2,
+
+$$
+W _ {1} = \frac {\partial \bar {u}}{\partial a _ {1}} = x \quad \text { and } \quad W _ {2} = \frac {\partial \bar {u}}{\partial a _ {2}} = x ^ {2} \tag {15.3-12}
+$$
+
+Hence, $W_{i}\bar{u}_{i,x} = 0$ at x = 0. And, from Eq. 15.3-1b, the nonessential boundary condition is $\bar{u}_{,x} = b$ at $x = L_{T}$ . Therefore, Eq. 15.3-11 becomes
+
+$$
+R _ {i} = \int_ {0} ^ {L _ {T}} \left(- W _ {i, x} \bar {u} _ {, x} + W _ {i} c x\right) d x + \left[ W _ {i} b \right] _ {x = L _ {T}} \tag {15.3-13}
+$$
+
+With Eqs. 15.3-12 and 15.3-13, the conditions $R_{1} = 0$ and $R_{2} = 0$ yield
+
+$$
+a _ {1} = b + 7 c L _ {T} ^ {2} / 1 2 \quad \text { and } \quad a _ {2} = - c L _ {T} / 4 \tag {15.3-14}
+$$
+
+Remarks. Table 15.3-1 summarizes results for the particular case $b = 0$ , $c = L_T = 1$ . The quantity $u_{,x}$ is proportional to stress for the problem of Fig. 15.3-1. In general, different collocation points yield different results, and a large number of collocation points is usually beneficial in least squares collocation. Solutions summarized in Table 15.3-1 are not the best possible.
+
+The example chosen, because of its simplicity, has traits that do not prevail in general, as follows. Collocation and subdomain methods each yield only one equation from $R_{D}$ (because only $a_{2}$ appears in $R_{D}$ ), and $\alpha$ has no effect on the solution in least squares methods (because $a_{1}$ happens not to appear in $R_{D}$ ).
+
+# 15.4 GALERKIN FINITE ELEMENT METHOD
+
+In this section, one-dimensional examples are used to illustrate the finite element form of the Galerkin method. The interpretation of terms that result from integration by parts, and the assembly of elements to form a structure, are explained in the first example.
+
+Uniform Bar, Axial Load. Equilibrium of axial forces in Fig. 15.4-1 requires that
+
+<!-- source-page: 482 -->
+
+![](images/page-482_7b256a9cb00aa36ec18d59c33f47668b95b9971e8f4d51b47fe16ed82aaecf44.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+A, E
+q = q(x)
+x, u
+q dx
+Aσₓ ← → A(σₓ + σₓ, x dx)
+dx ←
+</details>
+
+(a)
+
+![](images/page-482_f0410aebffa5d01534afbc0f1944cb1541b10f56cda5406edada6fe9362bb1ec.jpg)
+
+<details>
+<summary>text_image</summary>
+
+F_{j-1} \n L \n F_{j-1} \n a \n b \n el.j-1
+</details>
+
+![](images/page-482_93bc7d3d26bfeeae96d7661f3b767855d213047f39758cf81823a095a837d57e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Fj
+L
+Fj
+b
+c
+el. j
+</details>
+
+{b}   
+Figure 15.4-1. (a) Uniform elastic bar under distributed axial load q. A = cross-sectional area, E = elastic modulus. (b) Adjacent elements j = .1 and j. Node b is shared after assembly of elements.
+
+$A\sigma_{x,x} + q = 0$ . Also, $\sigma_{x} = E\epsilon_{x} = Eu_{,x}$ . Therefore, the governing differential equation in terms of axial displacement u is
+
+$$
+A E u _ {, x x} + q = 0 \tag {15.4-1}
+$$
+
+At an end where an axial force $F$ is applied, the nonessential boundary condition is
+
+$$
+A E u _ {, x} - F = 0 \tag {15.4-2}
+$$
+
+At a free end, $F = 0$ . Essential boundary conditions consist of prescribed values of $u$ .
+
+Let the bar be divided into $\text{numel}$ elements of length $L$ . Each element has the assumed displacement field
+
+$$
+\bar {u} = \lfloor \mathrm{N} \rfloor \{\mathrm{d} \} \quad \text { where } \quad \left\{ \begin{array}{l} \lfloor \mathrm{N} \rfloor = \left\lfloor \frac {L - x}{L} \frac {x}{L} \right\rfloor \\ \{\mathrm{d} \} = \left\lfloor u _ {1} - u _ {2} \right\rfloor^ {T} \end{array} \right. \tag {15.4-3}
+$$
+
+and $x = 0$ at the left end of the element. D.o.f. $u_{1}$ and $u_{2}$ are coefficients of modes in the approximating field. Thus $u_{1}$ and $u_{2}$ play the same role as parameters $a_{i}$ in Sections 15.2 and 15.3. Weights $W_{i}$ used in the Galerkin method are therefore
+
+$$
+W _ {i} = \frac {\partial \bar {u}}{\partial d _ {i}} = N _ {i} \quad \text { where } \quad \left\{ \begin{array}{l} N _ {1} = (L - x) / L \\ N _ {2} = x / L \end{array} \right. \tag {15.4-4}
+$$
+
+The Galerkin residual equation, Eq. 15.2-8, becomes
+
+$$
+\sum_ {j = 1} ^ {\text { numel }} \int_ {0} ^ {L} N _ {i} (A E \bar {u}, _ {x x} + q) d x = 0 \tag {15.4-5}
+$$
+
+where index i ranges over all shape functions. When elements are assembled, activation of a single d.o.f. activates shape functions in the adjacent elements; that is, linear ramps are activated in elements on either side of the node. Thus,
+
+<!-- source-page: 483 -->
+
+on the structural level, shape functions for all but the first and last nodes are “hat functions,” as shown in Fig. 15.4-2. We see that there are as many structural shape functions as there are d.o.f., and therefore as many Galerkin residual equations as there are d.o.f. If AE is constant, $^{2}$ integration by parts yields
+
+$$
+\int_ {0} ^ {L} N _ {i} A E \bar {u} _ {, x x} d x = \left[ N _ {i} A E \bar {u} _ {, x} \right] _ {0} ^ {L} - \int_ {0} ^ {L} N _ {i, x} A E \bar {u} _ {, x} d x \tag {15.4-6}
+$$
+
+From Eq. 15.4-2, the nonessential boundary condition is $AE\bar{u}_{,x} = F$ at element ends. Substituting this and Eq. 15.4-6 into Eq. 15.4-5, we obtain
+
+$$
+\sum_ {j = 1} ^ {\text {numel}} \int_ {0} ^ {L} \left(- N _ {i, x} A E \bar {u}, _ {x} + N _ {i} q\right) d x + \sum_ {j = 1} ^ {\text {numel}} \left[ N _ {i} F \right] _ {0} ^ {L} = 0 \tag {15.4-7}
+$$
+
+We adopt the notation
+
+$$
+\lfloor \mathbf {B} \rfloor = \lfloor \mathbf {N} _ {, x} \rfloor ; \quad \text { then } \quad \bar {u} _ {, x} = \lfloor \mathbf {B} \rfloor \{\mathbf {d} \} = \left\lfloor - \frac {1}{L} \quad \frac {1}{L} \right\rfloor \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \end{array} \right\} \tag {15.4-8}
+$$
+
+After rearrangement, Eq. 15.4-7 becomes
+
+$$
+\sum_ {j = 1} ^ {\text {numel}} \underbrace {\int_ {0} ^ {L} \left\lfloor \mathbf {B} \right] ^ {T} A E \left\lfloor \mathbf {B} \right\rfloor d x} _ {[ \mathbf {k} ] _ {j}} \{\mathbf {d} \} = \sum_ {j = 1} ^ {\text {numel}} \underbrace {\int_ {0} ^ {L} \left\lfloor \mathbf {N} \right] ^ {T} q d x} _ {\left\{\mathbf {r} _ {e} \right\} _ {j}} + \sum_ {j = 1} ^ {\text {numel}} \left[ \left\lfloor \mathbf {N} \right] ^ {T} F \right] _ {0} ^ {L} \tag {15.4-9}
+$$
+
+Except for the last summation, Eq. 15.4-9 clearly yields the standard formula $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{R}\}$ , that is,
+
+$$
+\left(\sum_ {j = 1} ^ {\text { numel }} [ \mathbf {k} ] _ {j}\right) \{\mathbf {D} \} = \sum_ {j = 1} ^ {\text { numel }} \{\mathbf {r} _ {e} \} _ {j} + \{\mathbf {P} \} \tag {15.4-10}
+$$
+
+where $\{D\}$ replaces $\{d\}$ because of the usual expansion of element matrices to “structure size.”
+
+We must explain how the last summation in Eq. 15.4-9 can be regarded as $\{\mathbf{P}\}$ ,
+
+![](images/page-483_fddf29fba8ea0627ff9f3d4dafda1d0bb7a480c3c01f7fecff3589467667b3ad.jpg)
+
+<details>
+<summary>text_image</summary>
+
+N₁ N₂
+1 1 1
+1 2 3
+i - 1 i i + 1 i + 2
+N₁ N₁ + 1
+1 1 1
+n - 1 n
+Lₜ
+</details>
+
+Figure 15.4-2. Shape functions active on the structural level after bar elements have been assembled. The number of d.o.f., and shape functions, is $n = \text{numel} + 1$ .
+
+<!-- source-page: 484 -->
+
+the vector of externally applied concentrated loads. At ends of a typical element, x = 0 and x = L, respectively,
+
+$$
+\left\lfloor \mathrm{N} \right\rfloor_ {0} = \left\lfloor 1 \quad 0 \right] \quad \text { and } \quad \left\lfloor \mathrm{N} \right\rfloor_ {L} = \left\lfloor 0 \quad 1 \right\rfloor \tag {15.4-11}
+$$
+
+For two adjacent elements $j - 1$ and $j$ , Fig. 15.4-1b, the last summation in Eq. 15.4-9 produces the terms
+
+$$
+\begin{array}{l} \text {node a} - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - \\ \text {node b} - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - \\ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - \end{array} + \left( \begin{array}{c} \left\{ \begin{array}{l} 0 \\ 1 \end{array} \right\} F _ {j - 1} \\ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - \left\{ \begin{array}{l} 1 \\ 0 \end{array} \right\} F _ {j} \end{array} \right) + \left\{ \begin{array}{l} 0 \\ 1 \end{array} \right\} F _ {j} \tag {15.4-12}
+$$
+
+When elements are assembled, as at node $b$ , the resultant axial force $F_{j-1} - F_j$ is produced. This resultant appears in parentheses in Eq. 15.4-12 and is identified as the externally applied load $P$ at node $b$ . Of course, $P$ may be zero; then $F_{j-1} = F_j$ . For the problem of Fig. 15.3-1, load vector $\{\mathbf{P}\}$ would contain only one nonzero entry—namely, $F = \sigma_0 A$ associated with the rightmost node. (Expression 15.4-12 is not used in actual computation; it serves only to explain the transition from Eq. 15.4-9 to Eq. 15.4-10.)
+
+Beam Dynamics. In the following example we omit most of the summation signs that indicate assembly of elements and also omit the detailed explanation of nodal loads (as in Eq. 15.4-12). Thus we emphasize the generation of element matrices by the Galerkin method.
+
+With $\rho$ the mass density, the mass per unit length is $\rho_{L} = \rho A$ , where $A$ is the cross-sectional area of the beam. The moment-curvature relation is $EIw_{,xx} = M$ . Also, $M_{,x} = V$ and $V_{,x} = q$ , where $EI =$ bending stiffness, $w =$ lateral displacement, $M =$ bending moment, $V =$ transverse shear force, and $q =$ transverse load per unit length. The effective inertia load is $q = -\rho_{L}\ddot{w}$ , where $(^{\prime}) = d() / dt$ . Putting all this together, we obtain the governing differential equation
+
+$$
+E I w _ {, x x x x} + \rho_ {L} \ddot {w} = 0 \tag {15.4-13}
+$$
+
+Nonessential boundary conditions are
+
+$$
+E I w _ {, x x} - M _ {B} = 0 \quad \text { and } \quad E I w _ {, x x x} - V _ {B} = 0 \tag {15.4-14}
+$$
+
+where $M_{B}$ and $V_{B}$ are prescribed values of bending moment and transverse shear force at ends of the beam. Essential boundary conditions consist of prescribed values of w and $w_{,x}$ . The assumed lateral-displacement field $\tilde{w} = \tilde{w}(x)$ and weight functions $W_{i} = W_{i}(x)$ are given by
+
+$$
+\bar {w} = \lfloor \mathbf {N} \rfloor \{\mathbf {d} \} \quad \text { and } \quad W _ {i} = N _ {i} \tag {15.4-15}
+$$
+
+where $\{d\} = \left[w_{1} \quad \theta_{1} \quad w_{2} \quad \theta_{2}\right]^{T}$ and the $N_{i}$ are the usual cubic shape functions (see Fig. 3.13-2). The Galerkin residual equation for a single element is
+
+$$
+\int_ {0} ^ {L} \left[ \mathbf {N} \right] ^ {T} \left(E I \tilde {w}, _ {x x x x} + \rho_ {L} \ddot {\tilde {w}}\right) d x = 0 \tag {15.4-16}
+$$
+
+<!-- source-page: 485 -->
+
+where L is the element length. With EI constant, two integrations by parts yield
+
+$$
+\int_ {0} ^ {L} \left\lfloor \mathrm{N} \right] ^ {T} E I \bar {w} _ {, x x x x} d x = \int_ {0} ^ {L} \left\lfloor \mathrm{N} _ {, x x} \right] ^ {T} E I \bar {w} _ {, x x} d x + \left[ \left\lfloor \mathrm{N} \right] ^ {T} E I \bar {w} _ {, x x x} - \left\lfloor \mathrm{N} _ {, x} \right] ^ {T} E I \bar {w} _ {, x x} \right] _ {0} ^ {L} \tag {15.4-17}
+$$
+
+Substitution of Eqs. 15.4-14 and 15.4-17 into Eq. 15.4-16 yields
+
+$$
+\int_ {0} ^ {L} \left(\left\lfloor \mathbf {N} _ {, x x} \right\rfloor^ {T} E I \tilde {w} _ {, x x} + \rho_ {L} \left\lfloor \mathbf {N} \right\rfloor^ {T} \ddot {\tilde {w}}\right) d x + \left[ \left\lfloor \mathbf {N} \right\rfloor^ {T} V _ {B} - \left\lfloor \mathbf {N} _ {, x} \right\rfloor^ {T} M _ {B} \right] _ {0} ^ {L} = 0 \tag {15.4-18}
+$$
+
+Terms $V_{B}$ and $M_{B}$ become part of the load vector $\{R\}$ . The argument is analogous to that used in Eqs. 15.4-11 and 15.4-12. From Eq. 15.4-15,
+
+$$
+\ddot {\bar {w}} = \lfloor \mathbf {N} \rfloor \{\ddot {\mathbf {d}} \} \quad \text { and } \quad \bar {w} _ {, x x} = \lfloor \mathbf {B} \rfloor \{\mathbf {d} \}, \quad \text { where } \quad \lfloor \mathbf {B} \rfloor = \lfloor \mathbf {N} _ {, x x} \rfloor \tag {15.4-19}
+$$
+
+Substituting Eq. 15.4-19 into Eq. 15.4-18 and assembling elements, we obtain
+
+$$
+\sum_ {j = 1} ^ {\text { numel }} \left(\int_ {0} ^ {L} [ \mathbf {B} ] ^ {T} E I [ \mathbf {B} ] d x \{\mathbf {d} \} + \int_ {0} ^ {L} \rho_ {L} [ \mathbf {N} ] ^ {T} [ \mathbf {N} ] d x \{\ddot {\mathbf {d}} \}\right) = \{\mathbf {R} \} \tag {15.4-20}
+$$
+
+which is the standard dynamic equation $[K]\{D\} + [M]\{\ddot{D}\} = \{R\}$ , where $[K]$ and $[M]$ are respectively the structure stiffness and mass matrices and $\{R\}$ represents time-varying loads.
+
+Heat Flow in a Bar. We consider steady-state heat conduction in a bar with insulated lateral surface, Fig. 15.4-3a. $^{3}$ Heat flux q is axial and obeys the Fourier heat conduction equation, $q = -kT_{,x}$ . Here k is the thermal conductivity of the material. The negative sign indicates that the direction of heat flow is opposite to the direction of temperature increase. In the steady-state condition, the net rate of heat flow out of a differential element is zero. Thus, from Fig. 15.4-3b, $d(Aq)/dx = 0$ . Combining the two equations $q = -kT_{,x}$ and $d(Aq)/dx = 0$ , we obtain
+
+![](images/page-485_e80b03e58be9b1143e7ea34c9ef23ca9b7bf9f6415c1d24d6d62669bc2a79061.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Lateral surface insulated
+Cross-sectional area A = A(x)
+T = T₀
+(prescribed)
+1
+2
+q_R
+(prescribed)
+L
+x
+L_T
+(a)
+Aq
+Aq + d(Aq)
+dx
+(b)
+</details>
+
+Figure 15.4-3 (a) Heat flow in a tapered bar. A typical element 1–2 is shown shaded. (b) Heat flow through a differential element.
+
+$^{3}$ With J = joule (1 N·m), units of the quantities are T, °C; k, J/m·s·°C; q, J/m $^{2}$ ·s.
+
+<!-- source-page: 486 -->
+
+$$
+\frac {d}{d x} (A k T _ {, x}) = 0 \tag {15.4-21}
+$$
+
+as the governing differential equation. We presume that $Ak$ may vary with $x$ . The approximating temperature field is $\bar{T} = \lfloor \mathbf{N} \rfloor \{\mathbf{T}_e\}$ , where, for a two-node element, nodal temperatures are $\{\mathbf{T}_e\} = \lfloor T_1 - T_2 \rfloor^T$ and the $N_i$ are given by Eq. 15.4-4. Let $q_B$ indicate a boundary heat flux, that is, a heat flux at either end of the element. We write the Galerkin residual equation, integrate by parts, and substitute the nonessential boundary condition $q_B = -k\bar{T}_{,x}$ . Thus
+
+$$
+\int_ {0} ^ {L} \left\lfloor \mathbf {N} \right] ^ {T} (A k \tilde {T}, _ {x}), _ {x} d x = - \int_ {0} ^ {L} \left\lfloor \mathbf {N}, _ {x} \right] ^ {T} A k \tilde {T}, _ {x} d x - \left[ \left\lfloor \mathbf {N} \right] ^ {T} A q _ {B} \right] _ {0} ^ {L} = 0 \tag {15.4-22}
+$$
+
+With $\tilde{T}_{,x} = \lfloor \mathbf{N}_{,x}\rfloor \{\mathbf{T}_e\}$ , the element equation becomes
+
+$$
+\int_ {0} ^ {L} \left\lfloor \mathbf {N}, _ {x} \right\rfloor^ {T} A k \left\lfloor \mathbf {N}, _ {x} \right\rfloor d x \left\{\mathbf {T} _ {e} \right\} = - \left\{ \begin{array}{l} 0 \\ 1 \end{array} \right\} A _ {2} q _ {2 r} + \left\{ \begin{array}{l} 1 \\ 0 \end{array} \right\} A _ {1} q _ {1 r} = \left\{ \begin{array}{c} A _ {1} q _ {1 r} \\ - A _ {2} q _ {2 r} \end{array} \right\} \tag {15.4-23}
+$$
+
+where $A_{1}q_{1r}$ and $A_{2}q_{2r}$ are respectively heat flow rates at nodes 1 and 2 of the element, each considered positive when heat flows to the right. At the left end of the bar in Fig. 15.4-3a, nodal temperature $T_{0}$ is prescribed instead of a flux $q_{0}$ .
+
+It is convenient to adopt the convention that boundary heat flux is considered positive when heat flows into the bar. $^{4}$ Using $q_{1}$ and $q_{2}$ to represent these fluxes, we have $q_{1} = q_{1r}$ and $q_{2} = -q_{2r}$ . Accordingly, for a two-node element in which A and k are constant, Eq. 15.4-23 becomes
+
+$$
+\frac {A k}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right] \left\{ \begin{array}{l} T _ {1} \\ T _ {2} \end{array} \right\} = \left\{ \begin{array}{l} A q _ {1} \\ A q _ {2} \end{array} \right\} \tag {15.4-24}
+$$
+
+If this uniform element models the entire bar in Fig. 15.4-3a, then $T_{1} = T_{0}$ and $q_{2} = -q_{R}$ , where the negative sign indicates an 'outward flux. Thus Eq. 15.4-24 yields $T_{2} = T_{0} - q_{R}L/k$ and $q_{1} = q_{R}$ . If the boundary conditions were reversed, so that $T_{2} = T_{R}$ is prescribed at the right end and inward flux $q_{1} = q_{0}$ is prescribed at the left end, we would obtain $T_{1} = T_{R} + q_{0}L/k$ and $q_{2} = q_{0}$ .
+
+# 15.5 INTEGRATION BY PARTS
+
+Integral or “weak” formulations make frequent use of integration by parts. Some useful formulas are now reviewed.
+
+Let i, j, and k be unit vectors in the coordinate directions. Also let $F_{1}$ , $F_{2}$ , and $F_{3}$ be independent functions of the coordinates, and
+
+$$
+\mathbf {F} = F _ {1} \mathbf {i} + F _ {2} \mathbf {j} + F _ {3} \mathbf {k} \quad \text { and } \quad \nu = \ell \mathbf {i} + m \mathbf {j} + n \mathbf {k} \tag {15.5-1}
+$$
+
+where function F is defined in a volume V, $\nu$ is a unit outward normal on the
+
+$^{4}$ This rule is generalized to multidimensional problems by the convention that boundary flux is considered positive in the direction of the inward surface normal vector.
+
+<!-- source-page: 487 -->
+
+surface S of V, and $\ell$ , m, and n are direction cosines of v. The divergence theorem states that
+
+$$
+\int_ {V} \nabla \cdot \mathbf {F} d V = \int_ {S} \mathbf {F} \cdot \boldsymbol {\nu} d S \tag {15.5-2}
+$$
+
+where $\nabla \cdot \mathbf{F}$ is the divergence of $\mathbf{F}$ , for example,
+
+rectangular coordinates: $\nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} +\frac{\partial F_2}{\partial y} +\frac{\partial F_3}{\partial z}$ (15.5-3a)
+
+cylindrical coordinates: $\nabla \cdot \mathbf{F} = \frac{1}{r} \frac{\partial}{\partial r} (rF_1) + \frac{1}{r} \frac{\partial F_2}{\partial \theta} + \frac{\partial F_3}{\partial z}$ (15.5-3b)
+
+In Eq. 15.5-2, F and its first partial derivatives must be continuous in V and on S, and integration must proceed over all boundaries, interior as well as exterior.
+
+Let $P$ and $Q$ be functions of the coordinates. Then, for example, $(PQ)_{,x} = P_{,x}Q + PQ_{,x}$ . Therefore
+
+$$
+\int_ {V} P Q _ {, x} d V = - \int_ {V} P _ {, x} Q d V + \int_ {V} (P Q) _ {, x} d V \tag {15.5-4}
+$$
+
+If we regard $PQ$ as $F_{1}$ in Eq. 15.5-3a and let $F_{2} = F_{3} = 0$ , Eq. 15.5-2 allows us to replace the last integral in Eq. 15.5-4 by a surface integral. Thus Eq. 15.5-4 becomes the following formula for integration by parts in rectangular coordinates:
+
+$$
+\int_ {V} P Q _ {, x} d V = - \int_ {V} P _ {, x} Q d V + \int_ {S} P Q \ell d S \tag {15.5-5}
+$$
+
+Analogous formulas for the $x$ and $y$ derivatives are easy to derive.
+
+The same procedure may be applied in cylindrical coordinates. For example,
+
+$$
+\int_ {V} \frac {1}{r} \frac {\partial}{\partial r} (r P Q) d V = \int_ {V} \left[ \frac {\partial P}{\partial r} Q + P \frac {1}{r} \frac {\partial}{\partial r} (r Q) \right] d V \tag {15.5-6}
+$$
+
+We let $F_{1} = PQ$ and $F_{2} = F_{3} = 0$ in Eq. 15.5-3b, solve for the last term in Eq. 15.5-6, and apply Eq. 15.5-2. Thus
+
+$$
+\int_ {V} P \frac {1}{r} \frac {\partial}{\partial r} (r Q) d V = - \int_ {V} \frac {\partial P}{\partial r} Q d V + \int_ {S} P Q \ell d S \tag {15.5-7}
+$$
+
+In similar fashion,
+
+$$
+\int_ {V} \frac {1}{r} P \frac {\partial Q}{\partial \theta} d V = - \int_ {V} \frac {1}{r} \frac {\partial P}{\partial \theta} Q d V + \int_ {S} P Q m d S \tag {15.5-8}
+$$
+
+Formulas for integration by parts in two dimensions can be obtained directly from the preceding formulas by setting $F_{3} = 0$ in Eqs. 15.5-3 and presuming that integration with respect to z has already been done across a unit thickness.
+
+<!-- source-page: 488 -->
+
+# 15.6 TWO-DIMENSIONAL PROBLEMS
+
+The Quasiharmonic Equation. The “quasiharmonic” equation describes heat conduction and various other physical problems, as explained in more detail in Chapter 16. Here, without specifying the physical problem, we illustrate the formulation of element matrices by the Galerkin method.
+
+Consider a plane region of unit thickness, volume V, and boundary S. The governing equation and the nonessential boundary condition are, respectively,
+
+$$
+\text { in } V, \frac {\partial}{\partial x} (k _ {x} \phi_ {, x}) + \frac {\partial}{\partial y} (k _ {y} \phi_ {, y}) + Q = 0 \tag {15.6-1}
+$$
+
+$$
+\text { on   } S, \quad \ell k _ {x} \phi_ {, x} + m k _ {y} \phi_ {, y} - q _ {B} = 0 \tag {15.6-2}
+$$
+
+where $\phi = \phi(x, y)$ is the dependent variable and $\ell$ and $m$ are direction cosines of an outward normal to $S$ . Known quantities $k_x, k_y$ , and $Q$ may be either constant or functions of $x$ and $y$ . In the nonessential boundary condition, $q_B$ is a prescribed boundary flux, positive when directed into $V$ . Essential boundary conditions, which in general prevail over only a portion of $S$ , consist of prescribed values of $\phi$ . For the special case $k_x = k_y = \text{constant}$ and $Q = 0$ , Eq. 15.6-1 becomes Laplace's equation $\nabla^2\phi = 0$ . A function $\phi$ that satisfies $\nabla^2\phi = 0$ is called harmonic.
+
+The approximating field $\tilde{\phi}$ is
+
+$$
+\tilde {\phi} = \lfloor \mathbf {N} \rfloor \{\phi_ {e} \} = \left\lfloor N _ {1} \quad N _ {2} \quad \dots \quad N _ {n} \right\rfloor \{\phi_ {e} \} \tag {15.6-3}
+$$
+
+where n is the number of nodes per element and $\{\phi_{e}\}$ is the vector of element nodal d.o.f. The Galerkin residual equation is
+
+$$
+\iint [ \mathbf {N} ] ^ {T} \left[ \frac {\partial}{\partial x} \left(k _ {x} \bar {\phi}, _ {x}\right) + \frac {\partial}{\partial y} \left(k _ {y} \bar {\phi}, _ {y}\right) + Q \right] d x d y = 0 \tag {15.6-4}
+$$
+
+Integration by parts (e.g., Eq. 15.5-5) yields
+
+$$
+\iint \left\lfloor \mathrm{N} \right] ^ {T} \frac {\partial}{\partial x} \left(k _ {x} \tilde {\phi} _ {, x}\right) d x d y = - \iint \left\lfloor \mathrm{N} _ {, x} \right] ^ {T} k _ {x} \tilde {\phi} _ {, x} d x d y + \int \left\lfloor \mathrm{N} \right] ^ {T} k _ {x} \tilde {\phi} _ {, x} \ell d S \tag {15.6-5a}
+$$
+
+$$
+\int \int \left\lfloor \mathbf {N} \right] ^ {T} \frac {\partial}{\partial y} \left(k _ {y} \bar {\phi} _ {, y}\right) d x d y = - \int \int \left\lfloor \mathbf {N} _ {, y} \right] ^ {T} k _ {y} \bar {\phi} _ {, y} d x d y + \int \left\lfloor \mathbf {N} \right] ^ {T} k _ {y} \bar {\phi} _ {, y} m d S \tag {15.6-5b}
+$$
+
+Substitution of Eqs. 15.6-5 and 15.6-2 into Eq. 15.6-4 yields
+
+$$
+\iint \left(- \left\lfloor \mathrm{N}, _ {x} \right] ^ {T} k _ {x} \tilde {\phi}, _ {x} - \left\lfloor \mathrm{N}, _ {y} \right] ^ {T} k _ {y} \tilde {\phi}, _ {y} + \left\lfloor \mathrm{N} \right] ^ {T} Q\right) d x d y + \int \left\lfloor \mathrm{N} \right] ^ {T} q _ {B} d S = 0 \tag {15.6-6}
+$$
+
+Finally, substitution of $\tilde{\phi}_{,x} = \lfloor \mathbf{N}_{,x}\rfloor \{\phi_e\}$ and $\tilde{\phi}_{,y} = \lfloor \mathbf{N}_{,y}\rfloor \{\phi_e\}$ into Eq. 15.6-6 yields
+
+<!-- source-page: 489 -->
+
+$$
+\begin{array}{l} \left[ \iint \left(\lfloor \mathrm{N}, _ {x} \rfloor^ {T} k _ {x} \lfloor \mathrm{N}, _ {x} \rfloor + \lfloor \mathrm{N}, _ {y} \rfloor^ {T} k _ {y} \lfloor \mathrm{N}, _ {y} \rfloor\right) d x d y \right] \left\{\phi_ {e} \right\} \\ = \iint \left\lfloor \mathbf {N} \right] ^ {T} Q d x d y + \int \left\lfloor \mathbf {N} \right] ^ {T} q _ {B} d S \tag {15.6-7} \\ \end{array}
+$$
+
+Or, in customary notation, $[\mathbf{k}]\{\phi_e\} = \{\mathbf{r}\}$ .
+
+Plane Elasticity. The foregoing manipulations are little changed in application to problems of plane stress or plane strain. We summarize as follows. The governing differential equations are the equilibrium equations, and the nonessential boundary conditions involve boundary tractions $\Phi_{x}$ and $\Phi_{y}$ , that is,
+
+$$
+\sigma_ {x, x} + \tau_ {x y, y} + F _ {x} = 0 \quad \text { and } \quad \ell \sigma_ {x} + m \tau_ {x y} = \Phi_ {x} \tag {15.6-8}
+$$
+
+$$
+\tau_ {x y, x} + \sigma_ {y, y} + F _ {y} = 0 \quad \ell \tau_ {x y} + m \sigma_ {y} = \Phi_ {y}
+$$
+
+where $F_{x}$ and $F_{y}$ are body forces per unit volume, and $\ell$ and m are direction cosines of an outward normal to the boundary. Essential boundary conditions consist of prescribed values of displacements u and v. The element displacement field is
+
+$$
+\{\tilde {\mathbf {u}} \} = \left\{ \begin{array}{l} \tilde {u} \\ \tilde {v} \end{array} \right\} = \left[ \begin{array}{l l} \lfloor \mathbf {N} \rfloor & \lfloor \mathbf {0} \rfloor \\ \lfloor \mathbf {0} \rfloor & \lfloor \mathbf {N} \rfloor \end{array} \right] \{\mathbf {d} \} \tag {15.6-9}
+$$
+
+where $\{\mathbf{d}\} = \lfloor u_1 \quad u_2 \quad \ldots \quad u_n \quad v_1 \quad v_2 \quad \ldots \quad v_n \rfloor^T$ and $\lfloor \mathbf{N} \rfloor = \lfloor N_1 \quad N_2 \quad \ldots \quad N_n \rfloor$ for an $n$ -node element (e.g., Eqs. 3.12-10 for a four-node rectangle). The arrangement of terms in Eq. 15.6-9 is adopted only for convenience of notation. As there are now two differential equations, there are two Galerkin residual equations, that is,
+
+$$
+\begin{array}{l} \iint \left\lfloor \mathbf {N} \right] ^ {T} \left(\tilde {\sigma} _ {x, x} + \tilde {\tau} _ {x y, y} + F _ {x}\right) d x d y = 0 \\ \text { and } \quad \cdot \int \int \left[ \mathbf {N} \right] ^ {T} (\tilde {\tau} _ {x y, x} + \tilde {\sigma} _ {y, y} + F _ {y}) d x d y = 0 \tag {15.6-10} \\ \end{array}
+$$
+
+where $\bar{\sigma}_{x}$ , $\bar{\sigma}_{y}$ , and $\bar{\tau}_{xy}$ are the approximate stress fields produced by Eq. 15.6-9, the strain–displacement relations, and the stress–strain relations. There are four terms in Eqs. 15.6-10 to be integrated by parts. For example, the first such integration yields
+
+$$
+\iint \left\lfloor \mathbf {N} \right\rfloor^ {T} \tilde {\sigma} _ {x, x} d x d y = - \iint \left\lfloor \mathbf {N}, _ {x} \right\rfloor^ {T} \tilde {\sigma} _ {x} d x d y + \int \left\lfloor \mathbf {N} \right\rfloor^ {T} \tilde {\sigma} _ {x} \ell d S \tag {15.6-11}
+$$
+
+By this process the nonessential boundary conditions are introduced. Next, we introduce the relations
+
+$$
+\{\boldsymbol {\sigma} \} = [ \mathrm{E} ] (\{\boldsymbol {\epsilon} \} - \{\boldsymbol {\epsilon} _ {0} \}) + \{\boldsymbol {\sigma} _ {0} \} \tag {15.6-12}
+$$
+
+$$
+\{\epsilon \} = [ \partial ] \{\bar {\mathbf {u}} \} = [ \mathbf {B} ] \{\mathbf {d} \} \tag {15.6-13}
+$$
+
+where $[\partial]$ is the differential operator defined in Eq. 1.5-6. The final result given by Eqs. 15.6-10 is the same as given by Eqs. 4.1-5 and 4.1-6.
+
+<!-- source-page: 490 -->
+
+# PROBLEMS
+
+# Section 15.3
+
+15.1 Derive the differential equation shown in Fig. 15.3-1.
+
+15.2 Verify that despite collocation at $x = L_{T} / 3$ , Eqs. 15.3-2 and 15.3-4 do not yield $\bar{u} = u$ at $x = L_{T} / 3$ . Does this indicate that something is wrong? Explain.
+
+15.3 Show that the Galerkin condition $R_{i} = 0$ in Eq. 15.3-13 is the same as the stationary-functional condition $\partial \Pi / \partial a_{i} = 0$ if functional $\Pi$ is given by
+
+$$
+\Pi = \int_ {0} ^ {L _ {T}} \left(\frac {1}{2} \tilde {u} _ {, x} ^ {2} - c x \tilde {u}\right) d x - [ b \tilde {u} ] _ {x = L _ {T}}
+$$
+
+15.4 Consider the differential equation $u_{,xx} + 4u = 12$ , with essential boundary conditions $u = 3$ at $x = 0$ and $u = 1$ at $x = 1$ . There are no nonessential boundary conditions. The exact solution is $u = 3 - 2.1995 \sin 2x$ . A one-parameter approximating polynomial that meets the essential boundary conditions is $\bar{u} = 3 - 2x + a(x^2 - x)$ . Determine parameter $a$ in the range $0 < x < 1$ by (a) collocation, (b) subdomain, (c) least squares, (d) least squares collocation, and (e) Galerkin methods. Choose points at $x = 0.5$ in part (a) and at $x = \frac{1}{3}$ and $x = \frac{2}{3}$ in part (d). In each case calculate the percentage error of $\bar{u}$ at $x = 0.5$ and at $x = 0.7$ .
+
+15.5 Consider the differential equation $u_{,x} + 2u - 16x = 0$ with the boundary condition $u = 0$ at $x = 0$ . The exact solution is $u = 4(e^{-2x} - 1) + 8x$ . A two-parameter approximating polynomial that satisfies the boundary condition is $\tilde{u} = a_1x + a_2x^2$ . Determine $a_1$ and $a_2$ in the range $0 < x < 1$ and compute percentage errors of $\tilde{u}$ at $x = 0.5$ and at $x = 0.7$ .
+
+(a) Use least-squares collocation, with collocation points at $x = 0.25$ , $x = 0.50$ , and $x = 0.75$ .
+
+(b) Use the Galerkin method.
+
+15.6 Solve the problem defined by Eq. 15.3-1 in each of the following ways. Compare your answers with those in Table 15.3-1.
+
+(a) Use collocation, with the sampling point at $x = L_T / 2$ .
+
+(b) Use subdomain and integrate over the span $0 \leq x \leq L_T / 2$ .
+
+(c) Use least squares collocation. Evaluate $R_{D}$ at $x = 0$ , $x = L_{T} / 2$ , and $x = L_{T}$ . Use $\alpha = 1 / L_{T}$ .
+
+(d) Omit the residual $R_{D2}$ in Eq. 15.3-7.
+
+(e) Show why, in this particular problem, the solution provided by least squares collocation is independent of $\alpha$ .
+
+15.7 In Fig. 15.3-1, let $\sigma_0 = 0$ and let $q$ be the uniform traction $q = q_0$ , but acting on the right half $L_T / 2 < x < L_T$ only. As in Section 15.3, determine two-parameter approximate solutions by the five methods illustrated. In collocation, choose the point $x = 2L_T / 3$ ; otherwise, proceed as in Section 15.3. Compare exact and approximate results for the case $q_0 = L_T = AE = 1$ .
+
+15.8 Consider a uniform, simply supported beam of length L. For each of the following loadings, use the Galerkin method to determine constant a in the approximating lateral displacement field $\bar{w} = ax(L - x)$ . Also compute the percentage error of the predicted midspan deflection.
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+<!-- source-page: 491 -->
+
+(a) Sinusoidal distributed lateral load $q = q_0 \sin(\pi x / L)$ over $0 < x < L$ .
+
+(b) Uniformly distributed lateral load $q = q_0$ over $0 < x < L$ .
+
+# Section 15.4
+
+15.9 A cable carries constant axial tension T and contacts an elastic foundation of modulus B (force per unit length per unit of deflection w). The left and right ends of the cable are loaded by the respective forces $F_{L}$ and $F_{R}$ , which act perpendicular to the elastic foundation. The sketch shows the cable in its deflected position.
+
+(a) Show that the governing differential equation is $Tw_{xx} - Bw = 0$ .
+
+(b) Use the Galerkin method to establish formulas for element matrices analogous to those in Eq. 15.4-9.
+
+![](images/page-491_d9135ed23b1ee3cf0b9ac182ca04f4f69ccbb22452de088f1a7bcad896ced205.jpg)
+
+<details>
+<summary>text_image</summary>
+
+F_L
+w
+x
+F_R
+T
+T
+</details>
+
+Elastic foundation   
+Problem 15.9
+
+15.10 The equation of motion of a string is $Tw_{,xx} - \rho_L\ddot{w} = 0$ , where $T =$ constant axial tension, $w =$ small lateral displacement, $x =$ axial coordinate, $\rho_L =$ mass per unit length, and $\ddot{w} = d^2 w / dt^2$ .
+
+(a) Formulate finite element matrices by the Galerkin method. Let the element have two d.o.f.
+
+(b) Consider a simply supported uniform string of length $2L$ . Model it by two elements, each of length $L$ , and of the type formulated in part (a). Solve for the fundamental frequency of vibration. (The exact answer is $\omega^2 = \pi^2 T / 4\rho_L L^2$ .)
+
+(c) Can the problem of part (b) be solved using a single element? Explain how or why not.
+
+15.11 Let $F =$ constant axial force, positive in tension, and $B =$ elastic foundation modulus (force per unit length per unit of deflection $w$ ). With $F$ and $B$ taken into account, the differential equation of a beam becomes $EIw_{,xxx} - q - Fw_{,xx} + Bw = 0$ . Formulate expressions for the element matrices associated with $F$ and $B$ in a form analogous to Eq. 15.4-20.
+
+15.12 For the beam problem, Eqs. 15.4-13 to 15.4-20, demonstrate the treatment of interelement moments and shear forces, in the fashion of Eq. 15.4-12.
+
+15.13 Let the end cross sections of element 1-2 in Fig. 15.4-3 have areas $A_{1}$ and $A_{2}$ , respectively. If $A$ between ends is a linear function of $x$ , what equation replaces Eq. 15.4-24? Assume that $k$ is constant.
+
+15.14 Starting with Eq. 15.4-23, demonstrate the assembly of element thermal "load" vectors. Use Eq. 15.4-12 as a guide.
+
+15.15 In the differential equations cited in (a) and (b) below, $g = g(x)$ and $u = u(x)$ . Let the approximating field for an element of length $L$ be $\bar{u} = \lfloor \mathbf{N} \rfloor \{\mathbf{d}\}$ , as usual. In each case, what is a formula for [k] in the equation $[\mathbf{k}]\{\mathbf{d}\} = \{\mathbf{r}\}$ in terms of $g$ and $\lfloor \mathbf{N} \rfloor$ ? (You may ignore boundary conditions in this exercise.) (a) $gu_{,x} = 0$ . (b) $gu_{,xx} = 0$ .
+
+<!-- source-page: 492 -->
+
+# Section 15.6
+
+15.16 Show that Eq. 15.6-7 can also be obtained from Eq. 15.6-3 and the stationary condition of the functional
+
+$$
+\Pi = \frac {1}{2} \int \int \left(k _ {x} \phi_ {, x} ^ {2} + k _ {y} \phi_ {, y} ^ {2} - 2 Q \phi\right) d x d y - \int q _ {B} \phi d S
+$$
+
+15.17 The Helmholz equation, $p_{,xx} + p_{,yy} + p_{,zz} + (\omega / c)^2 p = 0$ , governs acoustic modes of vibration in a cavity with rigid walls. Here $p = p(x, y, z)$ represents the amplitude of sinusoidally varying pressure, $\omega$ is the circular frequency, and $c$ is the speed of sound in the medium. The boundary condition is $p_{,n} = 0$ , where $n$ is a direction normal to the wall. Derive formulas for finite element matrices, using the assumed pressure amplitude field $p = |\mathbf{N}| \{\mathbf{p}_e\}$ .
+
+15.18 In cylindrical coordinates and with $k_{x} = k_{y} = k =$ constant, Eq. 15.6-1 becomes
+
+$$
+\frac {1}{r} \frac {\partial}{\partial r} \left(r \frac {\partial \phi}{\partial r}\right) + \frac {1}{r ^ {2}} \frac {\partial^ {2} \phi}{\partial \theta^ {2}} + \frac {\partial^ {2} \phi}{\partial z ^ {2}} + \frac {Q}{k} = 0
+$$
+
+For a solid of revolution, the nonessential boundary condition is $k(\ell \phi_r + n\phi_{rz}) - q_B = 0$ , where $\ell$ and $n$ are direction cosines of a normal to the surface of the solid. For this problem, formulate equations analogous to Eqs. 15.6-7.
+
+15.19 The differential equation for wind-driven circulation in a shallow lake is
+
+$$
+\psi_ {, x x} + \psi_ {, y y} + A \psi_ {, x} + B \psi_ {, y} + C = 0
+$$
+
+where $\psi$ is the stream function and $A, B$ , and $C$ are functions of $x$ and $y$ . With $h =$ depth, depthwise average velocities are $u = \psi_y / h$ and $v = -\psi_x / h$ . Coordinates $x$ and $y$ are tangent to the lake surface. The nonessential boundary condition is $\psi_n = 0$ on the shoreline, where $n$ is a direction normal to the shoreline. Derive a finite element formulation by the Galerkin method [15.8]. A symbolic result is desired, analogous to Eq. 15.6-7, not details of a particular element.
+
+15.20 Complete the development outlined in Eqs. 15.6-8 to 15.6-13; that is, verify that Eqs. 4.1-5 and 4.1-6 are produced.
+
+15.21 An isotropic, flat disk has unit thickness and inner and outer radii $r_{i}$ and $r_{0}$ . The disk is set spinning about its center at constant angular velocity $\omega$ . The differential equation of equilibrium is
+
+$$
+\frac {1}{r} \frac {d}{d r} \left(r \sigma_ {r}\right) - \frac {\sigma_ {\theta}}{r} + \rho \omega^ {2} r = 0
+$$
+
+where $\sigma_{r}$ = radial stress, $\sigma_{\theta}$ = circumferential stress, and $\rho$ = mass density. Generate formulas for element matrices. Express your results in terms of the shape function matrix and its derivatives, as in Eq. 15.6-7.
+
+<!-- source-page: 493 -->
+
+15.22 Consider an elastic, axially symmetric solid under axially symmetric loads. Use arguments analogous to those of Eqs. 15.6-8 to 15.6-13 to formulate finite element matrices. For simplicity, omit body forces and initial stresses and strains. Express your results in terms of the shape function matrix and its derivatives, as in Eq. 15.6-7.
+
+<!-- source-page: 494 -->
+
+# HEAT CONDUCTION AND SELECTED FLUID PROBLEMS
+
+Heat conduction equations are reviewed and used to generate finite element formulations. Thermal transients are discussed. Certain problems of acoustics and flow, primarily those whose differential equation is of the same form as the heat conduction equation, are also treated.
+
+# 16.1 INTRODUCTION TO HEAT CONDUCTION PROBLEMS
+
+Heat conduction analysis may be performed to determine material temperatures and rates of heat flow. The temperature distribution may also be needed in order to perform an analysis for thermally induced stress. Fortunately, it is possible to use a single mesh layout for both problems: a computer program can read a single data file, compute temperatures at nodes, then use these temperatures in stress analysis. Remarks about thermally induced stress, and when a temperature gradient may not produce stress, appear in Sections 1.7 and 4.7. Thermally induced nodal loads are accounted for by terms in Eq. 4.1-6.
+
+A finite element formulation of steady-state heat conduction produces equations of the form $[K]\{T\} = \{R\}$ , where $\{T\}$ contains nodal temperatures of the structure. Matrices $[K]$ and $\{R\}$ can be generated by the method of making a suitable functional stationary or by a weighted residual method.
+
+Quantities used in our discussion are as follows. The unit of heat or energy is $J = 1$ joule $= 1 \, \text{N} \cdot \text{m}$ .
+
+c = specific heat (J/kg·°C)
+h = heat transfer coefficient, or film coefficient (J/m²·s·°C)
+k = thermal conductivity (J/m·s·°C)
+Q = rate of internal heat generation per unit volume (J/m³·s)
+q = heat flux per unit area (J/m²·s)
+q_B = prescribed flux normal to a surface (J/m²·s)
+ρ = mass density (kg/m³)
+T = temperature (°C)
+T_f = fluid temperature (°C) [used with h]
+t = time (s)
+T = ∂T/∂t (°C/s)
+
+In the foregoing units, actual fluids and solids display numerical values in the approximate ranges $10^{2} < c < 10^{4}$ , $10 < h < 10^{5}$ , and $10^{-2} < k < 500$ .
+
+Heat transfer into a solid across a fluid boundary layer is given by q =
+
+<!-- source-page: 495 -->
+
+$h(T_{f} - T)$ , where T is the surface temperature of the solid and $T_{f}$ is the fluid temperature on the other side of the boundary layer. The heat transfer coefficient h depends on the nature of the fluid, the geometry of the surface, and the dynamics of fluid motion past the surface.
+
+Sources of O include resistance to electric current and chemical reactions.
+
+In this chapter, the foregoing material properties are taken as independent of temperature unless specifically stated otherwise. Realistically, k is a function of T. Also, $\partial k/\partial T$ may be positive or negative, depending upon the material and sometimes the temperature as well. Similarly, h may depend on temperature. Temperature dependence of k and/or h makes the heat conduction equations nonlinear. This problem is briefly considered in Section 16.5. Radiation is another important source of nonlinearity.
+
+The starting point for heat conduction analysis is the Fourier heat conduction equation, which is
+
+$$
+q = - k \frac {\partial T}{\partial x} \tag {16.1-1}
+$$
+
+This equation states that heat flux q in direction x is proportional to the gradient of temperature in direction x. The negative sign indicates that heat flow is opposite to the direction of temperature increase.
+
+# 16.2 A ONE-DIMENSIONAL EXAMPLE
+
+Consider a straight but tapered bar, Fig. 16.2-1. Heat flows across end surfaces. Due to convection, heat also flows across lateral surfaces with the flux rate $h(T_{f} - T)$ , where $T_{f}$ is the temperature of surrounding fluid. This flux is directed into the bar if $T_{f} > T$ . Heat is also assumed to be generated internally at rate Q per unit volume. We assume that temperature T in the bar varies only with x, and ask for a finite element formulation that will yield $T = T(x)$ in the steady-state condition.
+
+In the steady-state condition, the net rate of heat flow into any differential element is zero. With volume element dV = A dx and surface area increment dS = p dx, where $p = p(x)$ is the perimeter of the cross section, Fig. 16.2-1b yields
+
+![](images/page-495_13108f8df8046b913fbafa49eee0e208e737395f000ad7d8d7cb01c4a1491e14.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Cross-sectional area A = A(x)
+Perimeter p = p(x)
+T = T₀ (prescribed)
+Section A-A
+q = q_R (prescribed)
+dx
+Aq
+Aq + d(Aq)
+QA dx + h(T_f - T) p dx
+(a)
+(b)
+</details>
+
+Figure 16.2-1. (a) A bar of varying circular cross section. A typical element 1–2 is shaded. (b) Contributions to heat flow through a differential element.
+
+<!-- source-page: 496 -->
+
+$$
+A q - [ A q + d (A q) ] + Q A d x + h (T _ {f} - T) p d x = 0 \tag {16.2-1}
+$$
+
+Combining Eqs. 16.1-1 and 16.2-1 and dividing by dx, we obtain the governing differential equation
+
+$$
+\frac {d}{d x} (A k T _ {, x}) + Q A + h (T _ {f} - T) p = 0 \tag {16.2-2}
+$$
+
+Boundary conditions at the ends are
+
+$$
+T = T _ {0} \text {   at   } x = 0 \quad \text {   and   } \quad T _ {, x} = - q _ {R} / k \text {   at   } x = L _ {T} \tag {16.2-3}
+$$
+
+These boundary conditions are respectively essential and nonessential.
+
+A finite element formulation can be developed from the following functional:
+
+$$
+\Pi = \int \left[ \frac {1}{2} A k T _ {, x} ^ {2} + \frac {1}{2} h p T ^ {2} - (Q A + h p T _ {f}) T \right] d x \tag {16.2-4}
+$$
+
+The standard manipulations of calculus of variations show that Eq. 16.2-2 and the nonessential boundary condition are produced by the stationary condition $d\Pi = 0$ (see Eq. 3.7-6). We interpolate temperature T and temperature gradient $T_{,x}$ along an element from nodal temperatures $\{T_{e}\}$ :
+
+$$
+T = \lfloor \mathbf {N} \rfloor \{\mathbf {T} _ {e} \} \quad \text { and } \quad T _ {, x} = \lfloor \mathbf {N}, _ {x} \rfloor \{\mathbf {T} _ {e} \} \tag {16.2-5}
+$$
+
+For the two-node element shown in Fig. 16.2-1, $\{\mathbf{T}_e\} = \lfloor T_1 - T_2 \rfloor^T$ and $N_1 = (L - x) / L$ , $N_2 = x / L$ . Next, because $T = T^T$ and $T^2 = T^T T$ , Eqs. 16.2-4 and 16.2-5 yield, for a single element,
+
+$$
+\begin{array}{l} \Pi_ {e} = \frac {1}{2} \left\{\mathbf {T} _ {e} \right\} ^ {T} \underbrace {\int_ {0} ^ {L} \left\lfloor \mathbf {N} , _ {x} \right] ^ {T} A k \left\lfloor \mathbf {N} , _ {x} \right\rfloor d x} _ {[ \mathbf {k} ]} \left\{\mathbf {T} _ {e} \right\} + \frac {1}{2} \left\{\mathbf {T} _ {e} \right\} ^ {T} \underbrace {\int_ {0} ^ {L} \left\lfloor \mathbf {N} \right] ^ {T} h p \left\lfloor \mathbf {N} \right\rfloor d x} _ {[ \mathbf {h} _ {\mathrm{ls}} ]} \left\{\mathbf {T} _ {e} \right\} \\ - \{\mathbf {T} _ {e} \} ^ {T} \underbrace {\int_ {0} ^ {L} \lfloor \mathbf {N} \rfloor^ {T} Q A d x} _ {\{\mathbf {r} _ {Q} \}} - \{\mathbf {T} _ {e} \} ^ {T} \underbrace {\int_ {0} ^ {L} \lfloor \mathbf {N} \rfloor^ {T} h p T _ {f} d x} _ {\{\mathbf {r} _ {1 s} \}} \tag {16.2-6} \\ \end{array}
+$$
+
+The subscript is stands for “lateral surface.” Quantities such as A, p, and Q are in general functions of x, and may be interpolated from nodal values if so desired.
+
+For the entire finite element structure, $\Pi$ is given by the summation $\Pi = \Sigma \Pi_e$ , which implies the expansion of element arrays to “structure size,” with the global nodal temperature array $\{\mathbf{T}\}$ replacing the several element arrays $\{\mathbf{T}_e\}$ , and with $[\mathbf{K}] = \Sigma [\mathbf{k}]$ , and so on, following the same matrix assembly procedures described in Section 2.7. Next, $\Pi$ is made stationary with respect to temperatures $\{\mathbf{T}\}$ . Writing the stationary condition for an element rather than for the structure, we have
+
+$$
+\left\{\frac {\partial \Pi}{\partial \mathbf {T} _ {e}} \right\} = \{\mathbf {0} \} \quad \text { yields } \quad ([ \mathbf {k} ] + [ \mathbf {h} _ {\mathrm{ls}} ]) \{\mathbf {T} _ {e} \} = \{\mathbf {r} _ {Q} \} + \{\mathbf {r} _ {\mathrm{ls}} \} \tag {16.2-7}
+$$
+
+<!-- source-page: 497 -->
+
+After assembly of elements, the corresponding global equations can be written in the form
+
+$$
+([ \mathbf {K} ] + [ \mathbf {H} _ {\mathrm{ls}} ]) \{\mathbf {T} \} = \{\mathbf {R} \} \quad \text { where } \quad \{\mathbf {R} \} = \{\mathbf {P} \} + \sum \{\mathbf {r} _ {e} \} \tag {16.2-8}
+$$
+
+where, recalling structural terminology, $[K] + [H_{ls}]$ is analogous to a stiffness matrix, $\{T\}$ is analogous to nodal displacements, $\{r_{e}\} = \{r_{Q}\} + \{r_{ls}\}$ is analogous to nodal loads from elements (caused, e.g., by self-weight), and $\{P\}$ is analogous to externally applied loads. For the bar in Fig. 16.2-1, $\{P\}$ contains a single nonzero term—namely $-q_{R}A_{R}$ —which represents the prescribed heat flux at the right end (out of the bar, in this case). Our sign convention for boundary values of q is that they are considered positive when heat flows into the body or element. The boundary condition at the left end is imposed by assigning $T = T_{0}$ at the leftmost node, which is analogous to assigning a known nonzero value to a displacement d.o.f. in a structural problem.
+
+The foregoing formulation is also easy to obtain by the Galerkin method, as described in Section 15.4. Equation 15.4-24 states [k] for a two-node element.
+
+# 16.3 HEAT CONDUCTION IN A PLANE
+
+In this section we consider the mathematical formulation of heat flow in a plane of unit thickness. Symbols defined in Section 16.1 are used. Radiation heat transfer is not included.
+
+Governing Equation. For a thermally orthotropic material, Fig. 16.3-1, Eq. 16.1-1 yields the heat fluxes
+
+$$
+q _ {r} = - k _ {r} T _ {, r} \quad \text { and } \quad q _ {s} = - k _ {s} T _ {, s} \tag {16.3-1}
+$$
+
+where $k_{r}$ and $k_{s}$ are principal thermal conductivities in the principal material directions r and s. Temperature gradients in the various coordinate directions are given by chain rule differentiation, that is, by
+
+$$
+\left\{ \begin{array}{l} T _ {, r} \\ T _ {, s} \end{array} \right\} = [ \Lambda ] \left\{ \begin{array}{l} T _ {, x} \\ T _ {, y} \end{array} \right\} \quad \text { where } \quad [ \Lambda ] ^ {\cdot} = \left[ \begin{array}{l l} x _ {, r} & y _ {, r} \\ x _ {, s} & y _ {, s} \end{array} \right] = \left[ \begin{array}{l l} \cos \beta & \sin \beta \\ - \sin \beta & \cos \beta \end{array} \right] \tag {16.3-2}
+$$
+
+![](images/page-497_79d91043ccfbe139adfcb59138ca22197432dbb8c9566caa1d0ac8b69c29f029.jpg)
+
+<details>
+<summary>text_image</summary>
+
+s
+β
+y
+r
+β
+x
+</details>
+
+Figure 16.3-1. A layered material with principal directions r and s.
+
+![](images/page-497_d1c3fec72b43e8d713c292fc945badeef806dbdd4c5ff552ec9f677e07a13642.jpg)
+
+<details>
+<summary>text_image</summary>
+
+q_y + q_y, y dy
+q_x dy q_x + q_x, x dx
+dx
+q_y
+</details>
+
+Figure 16.3-2. Heat flux through sides of a differential element.
+
+![](images/page-497_48916a1d467d0af0ac98ac4eab1b23db35734657ad7571a3aec8e4aa558a0fe8.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+x
+v
+α
+S (boundary)
+</details>
+
+Figure 16.3-3. Plane region with outward normal vector $\nu$ on its boundary S.
+
+<!-- source-page: 498 -->
+
+Heat flux $q$ is a vector and transforms like displacement, that is, $\lfloor q_x \quad q_y \rfloor^T = [\Lambda]^T \lfloor q_r \quad q_s \rfloor^T$ . Combining this with Eqs. 16.3-1 and 16.3-2, we obtain
+
+$$
+\left\{ \begin{array}{l} q _ {x} \\ q _ {y} \end{array} \right\} = - [ \kappa ] \left\{ \begin{array}{l} T _ {, x} \\ T _ {, y} \end{array} \right\} \quad \text { where } \quad [ \kappa ] = \left[ \begin{array}{l l} k _ {x} & k _ {x y} \\ k _ {x y} & k _ {y} \end{array} \right] = [ \Lambda ] ^ {T} \left[ \begin{array}{l l} k _ {r} & 0 \\ 0 & k _ {s} \end{array} \right] [ \Lambda ] \tag {16.3-3}
+$$
+
+For a body of unit thickness, the rate of heat generation in a differential element dx dy is Q dx dy. If lateral surfaces of the body are insulated, heat flux across the boundary of a differential element is as depicted in Fig. 16.3-2. The net inward flow of heat is
+
+$$
+Q d x d y - \left(q _ {x, x} d x\right) d y - \left(q _ {y, y} d y\right) d x = \left(Q - q _ {x, x} - q _ {y, y}\right) d x d y \tag {16.3-4}
+$$
+
+In general, heat flow produces a time rate of change of stored energy—namely, $c\rho \, dx \, dy \, \dot{T}$ . Therefore, $Q - q_{x,x} - q_{y,y} = c\rho \dot{T}$ . Combining this with Eq. 16.3-3, we obtain
+
+$$
+\frac {\partial}{\partial x} \left(k _ {x} T _ {, x} + k _ {x y} T _ {, y}\right) + \frac {\partial}{\partial y} \left(k _ {x y} T _ {, x} + k _ {y} T _ {, y}\right) + Q = c \rho \dot {T} \tag {16.3-5}
+$$
+
+If a lateral surface $z =$ constant is not insulated, so that there is a convective transfer of heat across a lateral surface of the plane body, the heat flux $q = h(T_f - T)$ flows into the body across the lateral surface. If this transfer occurs on both lateral surfaces of the body, and if $h$ and $T_f$ are the same on both surfaces, then the term $2h(T_f - T)$ must be added to the left-hand side of Eq. 16.3-5. A portion of a lateral surface may be neither insulated nor subject to convection; instead, an inward flux $q_1$ may be prescribed. Then $q_1$ must be added to the left-hand side of Eq. 16.3-5 (or add $q_1 / \tau$ for a body of thickness $\tau$ ).
+
+If the medium is homogeneous and isotropic, then $k_{xy} = 0$ and $k_x = k_y = k$ , where $k$ is a constant. Thus Eq. 16.3-5 becomes
+
+$$
+k (T _ {, x x} + T _ {, y y}) + Q = c \rho \dot {T} \tag {16.3-6}
+$$
+
+If, in addition, $Q = 0$ and a steady state prevails $(\dot{T} = 0)$ , then we obtain Laplace's equation, $T_{,xx} + T_{,yy} = 0$ .
+
+Boundary Conditions. One may prescribe temperature on part (or all) of the boundary and heat flux on another part (or all). On any one part, temperature or flux is prescribed, not both. In general, the prescribed quantities may be functions of time. The following boundary conditions apply on boundary S, Fig. 16.3-3, not on the lateral surfaces cited below Eq. 16.3-5.
+
+The essential boundary condition is a prescription of temperature on part or all of S. Alternative names are first or Dirichlet boundary condition.
+
+The nonessential boundary condition is a prescription of heat flux (possibly zero) on part or all of S. Alternative names are second or Neumann boundary condition. For a thermally isotropic material, flux in direction $\nu$ of Fig. 16.3-3 is $q_{\nu} = -kT_{,\nu}$ . In addition, by the chain rule,
+
+$$
+T _ {, \nu} = T _ {, x} x _ {, \nu} + T _ {, y} y _ {, \nu} = T _ {, x} \ell_ {B} + T _ {, y} m _ {B} \tag {16.3-7}
+$$
+
+<!-- source-page: 499 -->
+
+where $\ell_{B}$ and $m_{B}$ are direction cosines of $\nu$ . Adopting the convention that prescribed boundary heat flux $q_{B}$ is positive when directed into the body, the non-essential boundary condition is therefore $q_{B} = -q_{\nu}$ , or
+
+$$
+q _ {B} = k \left(T, _ {x} \ell_ {B} + T, _ {y} m _ {B}\right) \tag {16.3-8}
+$$
+
+If the body is thermally orthotropic, we write $q_{\nu} = q_{x}\cos \alpha + q_{y}\sin \alpha = q_{x}\ell_{B} + q_{y}m_{B}$ and obtain $q_{x}$ and $q_{y}$ from Eq. 16.3-3. Therefore, with $q_{B} = -q_{\nu}$ ,
+
+$$
+q _ {B} = \left(k _ {x} T _ {, x} + k _ {x y} T _ {, y}\right) \ell_ {B} + \left(k _ {x y} T _ {, x} + k _ {y} T _ {, y}\right) m _ {B} \tag {16.3-9}
+$$
+
+If there is convection heat transfer across all or part of boundary S, then $q_{B}$ is replaced by $h(T_{f} - T)$ for this part of S. Convection heat transfer across a lateral surface is not considered part of the boundary condition as it does not appear on S.
+
+Functional. A functional for plane heat conduction is
+
+$$
+\begin{array}{l} \Pi = \iint \left(\frac {1}{2} \left\{ \begin{array}{l} T _ {, x} \\ T _ {, y} \end{array} \right\} ^ {T} [ \kappa ] \left\{ \begin{array}{l} T _ {, x} \\ T _ {, y} \end{array} \right\} - Q T + \rho c \dot {T} T\right) d x d y \\ - \int h \left(T _ {f} T - \frac {1}{2} T ^ {2}\right) d S - \int q _ {B} T d S \tag {16.3-10} \\ \end{array}
+$$
+
+in which the surface integrals are each evaluated on the portion of S subject to convection or prescribed flux. If there is also convection across a lateral surface, the term $h(T_{f}T - T^{2}/2)$ must be subtracted from the integrand of the double integral, once for each lateral surface involved. With T, $T_{,x}$ , and $T_{,y}$ subject to variation, one can show by calculus of variations that the condition $\delta\Pi = 0$ produces Eqs. 16.3-5 and 16.3-9.
+
+# 16.4 GENERAL SOLIDS AND SOLIDS OF REVOLUTION
+
+The equations of Section 16.3 can be written in a form that also serves for general solids and solids of revolution. The governing equation, Eq. 16.3-5, is
+
+$$
+\{\partial \} ^ {T} ([ \kappa ] \{\mathbf {T} _ {\partial} \}) + Q = c \rho \dot {T} \tag {16.4-1}
+$$
+
+where $\{\partial\}^{T}$ is a differential operator and $\{T_{\partial}\}$ contains temperature gradients (examples follow). Allowing for either prescribed flux or convection on S, the non-essential boundary condition, Eq. 16.3-9, is
+
+$$
+q _ {B} = \{\boldsymbol {\mu} \} ^ {T} [ \boldsymbol {\kappa} ] \{\mathbf {T} _ {\partial} \} \quad \text { or } \quad h (T _ {f} - T) = \{\boldsymbol {\mu} \} ^ {T} [ \boldsymbol {\kappa} ] \{\mathbf {T} _ {\partial} \} \tag {16.4-2}
+$$
+
+where $\{\mu\}$ contains direction cosines of a normal to boundary S. The functional, Eq. 16.3-10, is
+
+$$
+\Pi = \int_ {V} \left(\frac {1}{2} \left\{\mathbf {T} _ {\partial} \right\} ^ {T} [ \boldsymbol {\kappa} ] \left\{\mathbf {T} _ {\partial} \right\} - Q T + \rho c \dot {T} T\right) d V - \int_ {S} \left(q _ {B} T + h T _ {f} T - \frac {1}{2} h T ^ {2}\right) d S \tag {16.4-3}
+$$
+
+<!-- source-page: 500 -->
+
+where, for the plane problem with unit thickness discussed in Section 16.3, $dV = (1)dx dy = dx dy$ , and
+
+$$
+\{\partial \} = \left\{ \begin{array}{l} \partial / \partial x \\ \partial / \partial y \end{array} \right\} \quad \left\{\mathbf {T} _ {\partial} \right\} = \left\{ \begin{array}{l} T _ {, x} \\ T _ {, y} \end{array} \right\} \quad \left\{\boldsymbol {\mu} \right\} = \left\{ \begin{array}{l} \ell_ {B} \\ m _ {B} \end{array} \right\} \tag {16.4-4}
+$$
+
+and $[\kappa]$ is given by Eq. 16.3-3. In structural terms, $\{\mathbf{T}_{\partial}\}$ is analogous to strains $\{\epsilon\}$ and $[\kappa]$ is analogous to material property matrix [E].
+
+General Solids. Extension from two dimensions to three requires that we include $T_{xz}$ in $\{T_{\partial}\}$ , include $k_{z}$ , $k_{yz}$ , and $k_{zx}$ in $[\kappa]$ , and do volume integration over dV = dx dy dz. Equations 16.4-1, 16.4-2, and 16.4-3 apply, with
+
+$$
+\{\partial \} = \left\{ \begin{array}{l} \partial / \partial x \\ \partial / \partial y \\ \partial / \partial z \end{array} \right\} \quad \left\{\mathbf {T} _ {\partial} \right\} = \left\{ \begin{array}{l} T _ {, x} \\ T _ {, y} \\ T _ {, z} \end{array} \right\} \quad \left\{\boldsymbol {\mu} \right\} = \left\{ \begin{array}{l} \ell_ {B} \\ m _ {B} \\ n _ {B} \end{array} \right\} \tag {16.4-5}
+$$
+
+If principal thermal conductivities are written as the diagonal matrix $[k_r \quad k_s \quad k_t]$ , then $[\kappa]$ is $[\kappa] = [\Lambda]^T [k_r \quad k_s \quad k_t][\Lambda]$ , where $[\Lambda]$ is given by Eq. 7.2-1.
+
+Solids of Revolution. The coordinates are $r$ (radial), $\theta$ (circumferential), and $z$ (axial). As compared with the general solid, $[\kappa]$ for a solid of revolution is computed in the same way, $dr$ and $r \, d\theta$ replace $dx$ and $dy$ , $T_{,r}$ and $T_{,\theta}/r$ replace $T_{,x}$ and $T_{,y}$ , $dV = r \, dr \, d\theta \, dz$ , and $dS = r \, d\theta \, db$ , where $db$ is an increment of boundary length in an $rz$ plane. Direction cosines in $[\Lambda]$ pertain to angles between principal material axes and coordinate directions $r$ , $\theta$ , and $z$ . In Eqs. 16.4-1, 16.4-2, and 16.4-3, we use
+
+$$
+\{\partial \} = \left\{ \begin{array}{c} (1 / r) + \partial / \partial r \\ (1 / r) \partial / \partial \theta \\ \partial / \partial z \end{array} \right\} \quad \left\{\mathbf {T} _ {\partial} \right\} = \left\{ \begin{array}{c} T _ {, r} \\ T _ {, \theta} / r \\ T _ {, z} \end{array} \right\} \quad \left\{\boldsymbol {\mu} \right\} = \left\{ \begin{array}{c} \ell_ {B} \\ 0 \\ n _ {B} \end{array} \right\} \tag {16.4-6}
+$$
+
+A boundary-normal $\pmb{\nu}$ has no $\theta$ component, so $\ell_B = \cos (\nu, x)$ , $m_B = 0$ , and $n_B = \cos (\nu, z)$ .
+
+If the temperature field and the material properties are axially symmetric, then all derivatives with respect to $\theta$ vanish, and the problem is mathematically two-dimensional.
+
+A nonsymmetric temperature field can be treated by Fourier series, as explained for stress analysis in Section 10.5. Thus a three-dimensional problem is replaced by a series of two-dimensional problems. If, for example, $\theta$ is a principal material direction and the temperature field is symmetric with respect to the $\theta = 0$ plane, one can use $T = \sum \overline{T}_n \cos n\theta$ , where $\overline{T}_n$ is a function of $n, r$ , and $z$ but is independent of $\theta$ . Boundary conditions are written in terms of their Fourier series components, and $\overline{T}$ is determined for $n = 0$ , $n = 1$ , $n = 2$ , and so on. Superposition of these solutions yields the resultant temperature field.
+
+# 16.5 FINITE ELEMENT FORMULATION
+
+The reader may wish to review Section 3.10, in which element matrices are formulated for a special case of plane heat conduction. The same procedures
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+<!-- source-page: 501 -->
+
+apply to the more general functional of Eq. 16.4-3 and are summarized as follows.
+
+We write a temperature field T in terms of element nodal temperatures $\{T_{e}\}$ , and from it compute the required temperature gradients $\{T_{e}\}$ :
+
+$$
+T = \left\lfloor \mathrm{N} \right\rfloor \left\{\mathrm{T} _ {e} \right\} \quad \text { and } \quad \left\{\mathrm{T} _ {\partial} \right\} = [ \mathrm{B} ] \left\{\mathrm{T} _ {e} \right\} \quad \text { where } \quad [ \mathrm{B} ] = \{\partial \} [ \mathrm{N} ] \tag {16.5-1}
+$$
+
+but $(1 / r)$ is deleted from row 1 of $\{\partial\}$ if Eq. 16.4-6 is used. Since $T = T^T$ , $T^2 = T^T T$ , and $\dot{T} = [\mathbf{N}]\{\dot{\mathbf{T}}_e\}$ , Eq. 16.4-3 can be written, for one element,
+
+$$
+\Pi_ {e} = \frac {1}{2} \left\{\mathbf {T} _ {e} \right\} ^ {T} ([ \mathbf {k} ] + [ \mathbf {h} ]) \left\{\mathbf {T} _ {e} \right\} + \left\{\mathbf {T} _ {e} \right\} ^ {T} \left([ \mathbf {c} ] \left\{\dot {\mathbf {T}} _ {e} \right\} - \left\{\mathbf {r} _ {q} \right\} - \left\{\mathbf {r} _ {Q} \right\} - \left\{\mathbf {r} _ {h} \right\}\right) \tag {16.5-2}
+$$
+
+where
+
+$$
+[ \mathbf {k} ] = \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \boldsymbol {\kappa} ] [ \mathbf {B} ] d V \quad \{\mathbf {r} _ {q} \} = \int_ {S _ {e}} [ \mathbf {N} ] ^ {T} q _ {B} d S
+$$
+
+$$
+[ \mathbf {h} ] = \int_ {S _ {e}} [ \mathbf {N} ] ^ {T} h [ \mathbf {N} ] d S \quad \{\mathbf {r} _ {Q} \} = \int_ {V _ {e}} [ \mathbf {N} ] ^ {T} Q d V \tag {16.5-3}
+$$
+
+$$
+[ \mathbf {c} ] = \int_ {V _ {e}} [ \mathbf {N} ] ^ {T} \rho c [ \mathbf {N} ] d V \quad \{\mathbf {r} _ {h} \} = \int_ {S _ {e}} [ \mathbf {N} ] ^ {T} h T _ {f} d S
+$$
+
+By adding the $\Pi_{e}$ contributions of the elements, we obtain $\Pi$ of the assembled system. Assembly implies the usual expansion of element arrays to “structure size,” so that the global nodal temperature array $\{T\}$ replaces $\{T_{e}\}$ of each element, $[K] = \Sigma [k]$ , and so on. Equations that make $\Pi$ stationary are $\{\partial\Pi/\partial T\} = \{0\}$ , that is,
+
+$$
+([ \mathbf {K} ] + [ \mathbf {H} ]) \{\mathbf {T} \} + [ \mathbf {C} ] \{\dot {\mathbf {T}} \} = \{\mathbf {R} _ {q} \} + \{\mathbf {R} _ {Q} \} + \{\mathbf {R} _ {h} \} \tag {16.5-4}
+$$
+
+In a plane problem with convection heat transfer across a lateral surface, additional terms appear, as noted below Eq. 16.3-10. For convection across one lateral surface, we define
+
+$$
+[ \mathbf {h} _ {\mathrm{ls}} ] = \int \int \lfloor \mathbf {N} \rfloor^ {T} h \lfloor \mathbf {N} \rfloor d x d y \quad \text { and } \quad \{\mathbf {r} _ {\mathrm{ls}} \} = \int \int \lfloor \mathbf {N} \rfloor^ {T} h T _ {f} d x d y \tag {16.5-5}
+$$
+
+The terms $[H_{ls}]\{T\}$ and $\{R_{ls}\}$ must be added to the left- and right-hand sides, respectively, of Eq. 16.5-4.
+
+Remarks. If the medium is isotropic, $[\kappa]$ becomes the diagonal matrix $k[1\quad1\quad1]$ in general solids and solids of revolution, or $k[1\quad1]$ in plane problems and axisymmetric solids with axisymmetric temperature distribution.
+
+The simplest special form of Eq. 16.5-4 is $[K]\{T\} = \{0\}$ , which represents steady-state conditions without internal sources or sinks, some of the nodal temperatures $T_{i}$ in $\{T\}$ prescribed, and no heat flow across the boundary other than that implicitly associated with the prescribed $T_{i}$ .
+
+Prescribed nodal temperatures—and their spatial derivatives, if also used as nodal d.o.f. in $\{T\}$ —can be treated like prescribed nodal displacements in structural mechanics (Section 2.10). Thus, as in Fig. 2.10-7, the right-hand side of Eq. 16.5-4 is modified, and ones and zeros appear in the coefficient matrix. An alter-
+
+<!-- source-page: 502 -->
+
+native procedure, which corresponds to Fig. 2.10-6, is to add a large conductivity $K_{D}$ to the appropriate diagonal coefficient in [K] and augment the corresponding coefficient on the right-hand side by $K_{D}\overline{T}$ , where $\overline{T}$ is the prescribed nodal temperature. This treatment greatly increases the maximum eigenvalues of the system and may therefore cause trouble in a direct integration analysis of how temperature varies with time [16.1].
+
+Matrices defined by Eqs. 16.5-3 and matrices used in structural mechanics have the following analogies of form:
+
+[k] is analogous to a conventional stiffness matrix.   
+[h] is analogous to an elastic foundation stiffness matrix.   
+[c] is analogous to a mass matrix.   
+$\{r_{0}\}$ is analogous to nodal loads from body force.   
+$\{\mathbf{r}_q\}, \{\mathbf{r}_h\}$ are analogous to nodal loads from surface traction.
+
+Matrix [c] is analogous to mass matrix [m] in that both multiply time derivatives of nodal d.o.f. and both provide resistance to time rates of change. But [c] multiplies first derivatives and [m] multiplies second derivatives.
+
+Like [m], [c] may be formulated as a consistent matrix or as a lumped matrix. The simplification provided by lumping may be accompanied by loss of accuracy. For an element with $n$ nodal temperatures in $\{\mathbf{T}_e\}$ , a lumped form of [c] results from multiplying $\rho c$ by the element volume $V_e$ and assigning $\rho cV_e / n$ to each element node. Thus $\rho cV_e / n$ appears $n$ times in a diagonal matrix [c]. Similar ad hoc lumping can be used to simplify calculation of [h] or $\{\mathbf{r}_Q\}$ . An $\{\mathbf{r}_Q\}$ or $\{\mathbf{R}_Q\}$ vector that contains a single nonzero term represents a point source or a point sink at a node. Proportional and optimal capacity-lumping schemes can be employed, in direct analogy to mass-lumping schemes described in Section 13.3, and having the same trade-offs of advantage and disadvantage.
+
+If the element is formulated in isoparametric fashion, then $dV = J \, d\xi \, d\eta \, d\zeta$ and the usual coordinate transformations are invoked. If the problem is essentially plane but the body is of varying thickness, then $dV = \tau \, dx \, dy$ , where thickness $\tau$ is a function of $x$ and $y$ . If the body is a solid of revolution, then $dV = r \, dr \, d\theta \, dz$ or $dV = rJ \, d\xi \, d\eta \, d\theta$ , in which radius $r$ should not be taken as constant over an element unless the element is far from the axis of revolution.
+
+Matrices [h], $\{\mathbf{r}_q\}$ , and $\{\mathbf{r}_h\}$ are zero unless the element has an edge or face on boundary $S$ and either convection or prescribed flux is associated with that part of $S$ . Even then, these matrices receive nonzero contributions only from the element edge or face that forms part of $S$ .
+
+It often happens that $[\kappa]$ must be regarded as a function of temperature. For steady-state conditions, a simple but perhaps inefficient way to solve this nonlinear problem is as follows. Estimate numerical values of conductivities, generate [K], and solve for {T}. Use these temperatures to obtain improved estimates of conductivities, generate a new [K], and solve for the new {T}. Repeat until convergence. Refinements of this iterative process are of course possible [16.2]. Techniques discussed in Chapter 17 can also be used.
+
+<!-- source-page: 503 -->
+
+![](images/page-503_ccf91718a696169cb5e5d22d6bcd9bdf24a587b5fe367c5c1a62c40eeef0e353.jpg)
+
+<details>
+<summary>text_image</summary>
+
+(boundary)
+Tf
+L34
+y
+η
+ξ
+4
+P
+1
+2
+x
+</details>
+
+Figure 16.5-1. Bilinear isoparametric element at the boundary of a plane structure where convection occurs.
+
+$T = [N]\{T_e\} = N_1T_1 + N_2T_2 + N_3T_3 + N_4T_4,$ where the $N_i$ are given by Eq. 6.3-2. Temperature gradients $\{T_d\}$ are
+
+$$
+\left\{ \begin{array}{l} T _ {, x} \\ T _ {, y} \end{array} \right\} = [ \mathrm{J} ] ^ {- 1} \left\{ \begin{array}{l} T _ {, \xi} \\ T _ {, \eta} \end{array} \right\} = \underbrace {[ \mathrm{J} ] ^ {- 1} \left[ \begin{array}{c c c c} N _ {1 , \xi} & N _ {2 , \xi} & N _ {3 , \xi} & N _ {4 , \xi} \\ N _ {1 , \eta} & N _ {2 , \eta} & N _ {3 , \eta} & N _ {4 , \eta} \end{array} \right]} _ {[ \mathrm{B} ]} \left\{ \begin{array}{l} T _ {1} \\ T _ {2} \\ T _ {3} \\ T _ {4} \end{array} \right\} \tag {16.5-6}
+$$
+
+where Jacobian matrix [J] is defined by Eq. 6.3-11. The conductivity matrix [k] for an element of thickness $\tau$ becomes
+
+$$
+[ \mathbf {k} ] = \int_ {- 1} ^ {1} \int_ {- 1} ^ {1} [ \mathbf {B} ] ^ {T} [ \boldsymbol {\kappa} ] [ \mathbf {B} ] \tau J d \xi d \eta \tag {16.5-7}
+$$
+
+where $\tau$ may be a function of $\xi$ and $\eta$ .
+
+Convection matrix [h] receives a contribution from side 3–4 only. Along side 3–4, $N_{1} = N_{2} = 0$ , $N_{3} = (1 + \xi)/2$ , $N_{4} = (1 - \xi)/2$ , and $J = L_{34}/2$ . Therefore, if $\tau$ and h are independent of $\xi$ ,
+
+$$
+[ \mathbf {h} ] = \int_ {- 1} ^ {1} [ \mathbf {N} ] ^ {T} h [ \mathbf {N} ] \tau J d \xi = \frac {h \tau L _ {3 4}}{6} \left[ \begin{array}{l l l l} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 1 & 2 \end{array} \right] \tag {16.5-8}
+$$
+
+In similar fashion, for $\{r_{h}\}$ we obtain
+
+$$
+\{\mathbf {r} _ {h} \} = \int_ {- 1} ^ {1} \left| \mathbf {N} \right| ^ {T} h T _ {f} \tau J d \xi = T _ {f} \frac {h \tau L _ {3 4}}{2} \left\{ \begin{array}{l} 0 \\ 0 \\ 1 \\ 1 \end{array} \right\} \tag {16.5-9}
+$$
+
+If a heat input $Q_{P}$ (units J/s) is prescribed at point P, we obtain the resulting $\{r_{Q}\}$ vector from Eq. 16.5-3 by saying that Q = 0 except at point P,
+
+$$
+\{\mathbf {r} _ {Q} \} = \int_ {V _ {\epsilon}} \left[ \mathbf {N} \right] ^ {T} Q d V = \left[ \mathbf {N} \right] _ {P} ^ {T} \int_ {V _ {\epsilon}} Q d V = \left[ \mathbf {N} \right] _ {P} ^ {T} Q _ {P}. \tag {16.5-10}
+$$
+
+where $\left[N\right]_{P}$ is the value of $\left[N\right]$ at point P.
+
+<!-- source-page: 504 -->
+
+# 16.6 THERMAL TRANSIENTS
+
+In thermal problems, as in structural mechanics, a time-varying solution may be obtained by the modal method or by direct temporal integration. If material properties are not temperature-dependent, and the solution is dominated by a few of the lowest eigenmodes and is needed over a long time span, then the modal method is favored. If the problem is nonlinear, or the solution displays sharp transients (which require many eigenmodes for an accurate description) and is needed over a short time span, then direct integration is favored.
+
+By either method, the equations to be solved have the form
+
+$$
+[ \mathbf {K} _ {T} ] \{\mathbf {T} \} + [ \mathbf {C} ] \{\dot {\mathbf {T}} \} = \{\mathbf {R} \} \tag {16.6-1}
+$$
+
+This equation is the same as Eq. 16.5-4, in which $[K_{T}]$ contains all matrices that premultiply $\{T\}$ , and $\{R\}$ contains all vectors on the right-hand side. One seeks to determine $\{T\}$ as a function of time when $[K_{T}]$ and $[C]$ are known, $\{R\}$ is a known function of time, and the initial temperatures at time t = 0 are known. Unless stated otherwise, we presume that $[K_{T}]$ and $[C]$ are independent of $\{T\}$ .
+
+Modal Method. The procedure is very similar to that used for structural dynamics (Section 13.6). It is outlined as follows. One first considers the eigenproblem
+
+$$
+([ \mathbf {K} _ {T} ] - \lambda [ \mathbf {C} ]) \{\overline {{\mathbf {T}}} \} = \{\mathbf {0} \} \tag {16.6-2}
+$$
+
+If each eigenvector $\{\overline{\mathbf{T}}\}_i$ is normalized with respect to $[\mathbf{C}]$ , that is, if $\{\overline{\mathbf{T}}\}_i^T [\mathbf{C}]\{\overline{\mathbf{T}}\}_i = 1$ , then
+
+$$
+[ \phi ] ^ {T} [ \mathbf {C} ] [ \phi ] = [ \mathbf {I} ] \quad \text { and } \quad [ \phi ] ^ {T} [ \mathbf {K} _ {T} ] [ \phi ] = [ \lambda ] \tag {16.6-3}
+$$
+
+where $[\phi]$ is the modal matrix; that is, a matrix whose columns are the normalized eigenvectors $\{\overline{\mathbf{T}}\}_{1}, \{\overline{\mathbf{T}}\}_{2}$ , and so on, $[\mathbf{I}]$ is a unit matrix, and $[\lambda]$ is the (diagonal) spectral matrix $[\lambda] = [\lambda_1 - \lambda_2 \ldots \lambda_n]$ . Nodal temperatures are transformed to generalized temperatures $\{\mathbf{Z}\}$ by
+
+$$
+\{\mathbf {T} \} = [ \phi ] \{\mathbf {Z} \} \tag {16.6-4}
+$$
+
+where the $Z_{i}$ in $\{Z\}$ state the proportion of each eigenvector in the transformation. We substitute Eq. 16.6-4 into Eq. 16.6-1, premultiply by $[\phi]^{T}$ , and take note of Eqs. 16.6-3. Thus we obtain uncoupled equations, each having the form
+
+$$
+\dot {Z} _ {i} + \lambda_ {i} Z _ {i} = p _ {i} \quad \text { where } \quad p _ {i} = \{\phi \} _ {i} ^ {T} \{\mathbf {R} \} \tag {16.6-5}
+$$
+
+Here $\{\phi\}_{i}$ is the $i$ th column of $[\phi]$ , and $i$ runs from 1 to $m$ , where $m$ is typically much less than the total number of d.o.f.; that is, only the first few columns of $[\phi]$ are used. After Eqs. 16.6-5 are integrated, $\{\mathbf{Z}\} = \{\mathbf{Z}(t)\}$ is known, and Eq. 16.6-4 yields $\{\mathbf{T}\} = \{\mathbf{T}(t)\}$ .
+
+Variants of modal methods used in structural mechanics can also be applied to thermal transient analysis [16.3].
+
+Direct Integration. Consider two temperature states, separated by time increment
+
+<!-- source-page: 505 -->
+
+$\Delta t$ , and denoted by $\{T\}_{n}$ and $\{T\}_{n+1}$ . A temporal integration scheme, known as the generalized trapezoidal rule, is based on the assumption that the two temperature states have the relation
+
+$$
+\{\mathbf {T} \} _ {n + 1} = \{\mathbf {T} \} _ {n} + \{(1 - \beta) \dot {\mathbf {T}} _ {n} + \beta \dot {\mathbf {T}} _ {n + 1} \} (\Delta t) \tag {16.6-6}
+$$
+
+Like Newmark's method for the second-order equations of structural dynamics, Eq. 16.6-6 contains a factor $\beta$ that the analyst may select. Next we write Eq. 16.6-1 for time $t$ and again for time $t + \Delta t$ , then multiply the first equation by $1 - \beta$ and the second by $\beta$ . Thus
+
+$$
+(1 - \beta) \left([ \mathbf {K} _ {T} ] \{\mathbf {T} \} _ {n} + [ \mathbf {C} ] \{\dot {\mathbf {T}} \} _ {n}\right) = (1 - \beta) \{\mathbf {R} \} _ {n} \tag {16.6-7a}
+$$
+
+$$
+\beta \left(\left[ \mathbf {K} _ {T} \right] \{\mathbf {T} \} _ {n + 1} + [ \mathbf {C} ] \{\dot {\mathbf {T}} \} _ {n + 1}\right) = \beta \{\mathbf {R} \} _ {n + 1} \tag {16.6-7b}
+$$
+
+Equations 16.6-7 are added, then Eq. 16.6-6 is used to eliminate time derivatives of temperature. This step requires that [C] not change with time. The result is
+
+$$
+\begin{array}{l} \left(\frac {1}{\Delta t} [ \mathbf {C} ] + \beta [ \mathbf {K} _ {T} ]\right) \{\mathbf {T} \} _ {n + 1} = \left(\frac {1}{\Delta t} [ \mathbf {C} ] - (1 - \beta) [ \mathbf {K} _ {T} ]\right) \{\mathbf {T} \} _ {n} \\ + (1 - \beta) \{\mathbf {R} \} _ {n} + \beta \{\mathbf {R} \} _ {n + 1} \tag {16.6-8} \\ \end{array}
+$$
+
+From a known $\{T\}_{0}$ at t = 0, Eq. 16.6-8 yields $\{T\}_{1}$ at $t = \Delta t$ . Then, using $\{T\}_{1}$ , we determine $\{T\}_{2}$ at $t = 2(\Delta t)$ , and so on. If $\Delta t$ is not changed, the coefficient of $\{T\}_{n+1}$ need be generated and forward-reduced only once; the equations are then repeatedly solved for a sequence of right-hand sides.
+
+Depending on $\beta$ , time step $\Delta t$ in Eq. 16.6-8 may have an upper limit if the algorithm is to be numerically stable. If $\beta < \frac{1}{2}$ the largest $\Delta t$ for stability is [16.4]
+
+$$
+\Delta t _ {\mathrm{cr}} = \frac {2}{(1 - 2 \beta) \lambda_ {\max}} \tag {16.6-9}
+$$
+
+where $\lambda_{max}$ is the largest eigenvalue of Eq. 16.6-2. If $\beta \geq \frac{1}{2}$ the algorithm is unconditionally stable; that is, stability (but not accuracy) is guaranteed as $\Delta t$ becomes indefinitely large. Names associated with various schemes are as follows:
+
+$$
+\begin{array}{l} \beta = 0 \quad \text { forward   difference   or   Euler   (conditionally   stable) } \\ \beta = \frac {1}{2} \quad \text {   Crank - - Nicolson   or   trapezoidal   rule   (unconditionally   stable)   } \\ \beta = \frac {2}{3} \quad \text { Galerkin   (unconditionally   stable) } \\ \beta = 1 \quad \text { backward   difference   (unconditionally   stable) } \\ \end{array}
+$$
+
+If $\beta = 0$ , the algorithm is termed explicit. If $\beta > 0$ , it is termed implicit. If [C] is a diagonal matrix and $\beta = 0$ , the computational effort per time step is small but so is $\Delta t_{\mathrm{cr}}$ . Among implicit methods, the choice $\beta = \frac{1}{2}$ is popular, but sharp transients may excite annoying oscillations in the solution. Oscillations can be reduced by using a smaller value of $\Delta t$ or numerically damped by using a value of $\beta$ somewhat greater than $\frac{1}{2}$ . If the problem is nonlinear the only unconditionally stable form of Eq. 16.6-8 is $\beta = 1$ ; however, it is not particularly accurate [16.4].
+
+Various other direct integration algorithms are available [16.5]. Detailed dis-
+
+<!-- source-page: 506 -->
+
+cussion of selected time-stepping algorithms may be found in the latter sections of Chapter 13.
+
+# 16.7 RELATED PROBLEMS. FLUID FLOW
+
+Several physical phenomena are described by the same form of differential equation that describes heat conduction, Eq. 16.3-5 or 16.4-1. For steady-state conditions and a homogeneous and isotropic material, this equation has the form
+
+$$
+k \nabla^ {2} \phi + Q = 0 \tag {16.7-1}
+$$
+
+where $\nabla^{2}$ is the Laplacian operator (defined in Eq. 16.7-4). Phenomena described by Eq. 16.7-1, or by its less specialized form, include the following:
+
+Heat conduction ( $\phi = \text{temperature}$ )
+Viscous flow in a pipe ( $\phi = \text{axial velocity}$ )
+Groundwater flow ( $\phi = \text{hydraulic head}$ )
+Pressurized membrane ( $\phi = \text{membrane deflection}$ )
+Elastic torsion ( $\phi = \text{stress function or warping function}$ )
+Electric conduction ( $\phi = \text{voltage}$ )
+Electrostatics ( $\phi = \text{field potential}$ )
+Magnetostatics ( $\phi = \text{magnetic potential}$ )
+Potential flow ( $\phi = \text{velocity potential or stream function}$ )
+
+For the first four phenomena cited, k represents conductivity, viscosity, permeability, and surface tension, and Q represents internal heat generated, pressure gradient, flow associated with a source or a sink, and pressure, respectively.
+
+With appropriate definition of variables, material properties, and boundary conditions, problems in any of these areas can be addressed by use of the computational procedures already established for heat conduction analysis. Indeed, several of these problems can be solved by appropriate use of a computer program for structural analysis [16.6]. For example, if a viscous incompressible fluid flows so slowly that inertia effects are negligible, the two-dimensional problem is described by the equation $\nabla^4\psi = 0$ , where $\psi$ is the stream function defined in Eq. 16.7-3b [16.7]. This equation has the same form as the equation $\nabla^4 w = q / D$ that describes bending of a flat plate. It is highly recommended that use of any of these analogies be preceded by study of the specific problem area in question.
+
+Potential Flow. A particularly simple example of Eq. 16.7-1 is provided by plane potential flow—that is, irrotational flow of an incompressible and inviscid fluid. Let u and v represent flow velocities in the x and y directions, respectively. The conditions of irrotationality and continuity are, respectively,
+
+$$
+u _ {, y} - v _ {, x} = 0 \qquad \text { and } \qquad u _ {, x} + v _ {, y} = 0 \tag {16.7-2}
+$$
+
+For analysis, we can use either a potential function $\phi = \phi(x, y)$ or a stream function $\psi = \psi(x, y)$ , which are defined such that
+
+$$
+\text { Potential   function } \phi \text {:} u = \phi_ {, x} \quad \text { and } \quad v = \phi_ {, y} \tag {16.7-3a}
+$$
+
+<!-- source-page: 507 -->
+
+$$
+\text { Stream   function   } \psi : \quad u = \psi_ {, y} \quad \text { and } \quad v = - \psi_ {, x} \tag {16.7-3b}
+$$
+
+If n and s are arbitrarily oriented, right-handed, orthogonal coordinates in the xy plane, then $\phi_{,n}$ represents flow velocity in the positive n direction and $\psi_{,n}$ represents flow velocity in the negative s direction (see Fig. 16.7-1b).
+
+Substitution of Eqs. 16.7-3 into Eqs. 16.7-2 shows that one of the two equations becomes $0 \equiv 0$ while the other becomes Laplace's equation. Thus the problem is described by
+
+$$
+\nabla^ {2} \phi = 0 \quad \text { or   by } \quad \nabla^ {2} \psi = 0 \quad \text { where } \quad \nabla^ {2} = \frac {\partial^ {2}}{\partial x ^ {2}} + \frac {\partial^ {2}}{\partial y ^ {2}} \tag {16.7-4}
+$$
+
+A finite element formulation produces element matrices of the form
+
+$$
+[ \mathbf {k} ] \{\phi_ {e} \} = \{\mathbf {r} \} \quad \text { or } \quad [ \mathbf {k} ] \{\psi_ {e} \} = \{\mathbf {r} \} \tag {16.7-5}
+$$
+
+where $\{\phi_{e}\}$ and $\{\psi_{e}\}$ contain nodal values of $\phi$ and $\psi$ , respectively, [k] is obtained from Eqs. 16.5-3 with $[\kappa]$ a unit matrix and dV = (1) dx dy, and $\{r\}$ comes from $\{r_{q}\}$ of Eqs. 16.5-3 with $q_{B}$ representing the prescribed flow rate $\phi_{,n}$ or $\psi_{,s}$ in a direction n normal to boundary S.
+
+An example application is depicted in Fig. 16.7-1. Uniform flow at velocity $u_{0}$ enters at the left. Flow velocities in the neighborhood of the cylindrical obstacle are desired. Because of symmetry about horizontal and vertical centerlines, only one quadrant need be modeled. Other known symmetries of the flow pattern, when associated with Eqs. 16.7-3, dictate boundary conditions shown in Fig. 16.7-1. For example, v = 0 along AB, CD, and DE; therefore, $\phi_{,y} = 0$ along these
+
+![](images/page-507_e3497d26a95097cf0908f27ec1654c9f74925124ad8b8c4aab86a0ca9b0cb371.jpg)
+
+<details>
+<summary>text_image</summary>
+
+u₀
+Quadrant modeled
+H
+H
+</details>
+
+(a)
+
+![](images/page-507_66fb7b435c2c3dab762a246b31b59e6150e241cb4f69e66225c1bacb4066ea66.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+φ,y = 0
+φ,x = u₀
+E
+D
+φ = 0
+s
+C
+A
+B
+n
+φ,n = 0
+φ,y = 0
+x
+</details>
+
+(b)
+
+![](images/page-507_73b155d1cbcdc5c35b75e744518a3a592ad3f595753cc4ce810faab33ca1d989.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ψ = H
+ψ = u₀y
+ψ = 0
+ψ = 0
+ψ = x = 0
+ψ = 0
+x
+y
+A
+B
+C
+D
+</details>
+
+(c)   
+Figure 16.7-1. (a) Flow around a cylindrical obstacle in a rectangular channel. (b,c) Boundary conditions associated with potential function and stream function solutions, respectively, in the quadrant modeled.
+
+<!-- source-page: 508 -->
+
+lines. The fluid has no velocity normal to the cylinder; therefore $\phi_{,n} = 0$ along $BC$ . A nodal value of $\phi$ must be assigned in order to make the coefficient matrix nonsingular, but the numerical value is arbitrary because only derivatives of $\phi$ are of interest. If $\phi = 0$ is prescribed at $C$ , the value $\phi = 0$ is dictated all along $CD$ because of the condition $\phi_{,y} = 0$ along $CD$ . Analogous remarks apply to use of $\psi$ rather than $\phi$ . At points such as $D$ , both essential and nonessential boundary conditions appear. Here one may prefer to assign $\phi_D = 0$ or $\psi_D = H$ rather than assigning the load terms $r_D = 0$ and letting $\phi_D$ or $\psi_D$ remain an unknown to be calculated by solving the global equivalent of Eqs. 16.7-5. If the formulation uses $\phi$ (or $\psi$ ) and its $x$ and $y$ derivatives as nodal d.o.f., values of $\phi$ and a derivative of $\phi$ may be applied at a single node, although not to a single d.o.f. at that node. When all nodal d.o.f. are known, gradients are calculated (e.g., by Eq. 16.5-6 with $T$ replaced by $\phi$ or $\psi$ ), and Eq. 16.7-3 yields the required flow velocities.
+
+# 16.8 FLUID VIBRATION AND WAVES, PRESSURE FORMULATION
+
+The governing differential equation, which is Eq. 16.8-4 in the discussion that follows, is called the wave equation. It describes phenomena in which energy is propagated by waves and has applications in problems of sound propagation, the sloshing of liquid in a container, and fluid-structure interaction. Its finite element expression has nodal pressures as d.o.f.
+
+Formulation. We consider a fluid without viscosity. Part of the total pressure at any point may be a hydrostatic pressure. The remaining pressure, denoted by p, is associated with motion of the fluid. Pressure gradients $p_{,x}$ , $p_{,y}$ , and $p_{,z}$ may exist in the respective coordinate directions. Hence, if Newton's law F = ma is applied to a differential element of volume dx dy dz, we obtain
+
+$$
+p _ {, x} = - \rho \ddot {u} \quad p _ {, y} = - \rho \ddot {v} \quad p _ {, z} = - \rho \ddot {w} \tag {16.8-1}
+$$
+
+where u, v, and w are displacements in the x, y, and z directions, and $\rho$ is the mass density, which is assumed to be essentially constant despite compressibility of the fluid. We differentiate each of Eqs. 16.8-1 with respect to its own spatial coordinate, then add. Thus
+
+$$
+p _ {, x x} + p _ {, y y} + p _ {, z z} = - \rho (i i _ {, x} + \ddot {v} _ {, y} + \ddot {w} _ {, z}) \tag {16.8-2a}
+$$
+
+or
+
+$$
+\nabla^ {2} p = - \rho \frac {d ^ {2}}{d t ^ {2}} \left(\epsilon_ {x} + \epsilon_ {y} + \epsilon_ {z}\right) \tag {16.8-2b}
+$$
+
+where $\epsilon_{x} = u_{,x}$ , and so on. Bulk modulus $B$ is defined as
+
+$$
+B = - \frac {p}{d V / V} = - \frac {p}{\epsilon_ {x} + \epsilon_ {y} + \epsilon_ {z}} \tag {16.8-3}
+$$
+
+The negative sign appears because volume decreases under increasing pressure. Equations 16.8-2b and 16.8-3 yield
+
+<!-- source-page: 509 -->
+
+$$
+\nabla^ {2} p = \frac {\rho}{B} \ddot {p} \quad \text { or } \quad \nabla^ {2} p = \frac {1}{c ^ {2}} \ddot {p} \tag {16.8-4}
+$$
+
+where c is the speed of sound in the medium, $c = \sqrt{B/\rho}$ . Equation 16.8-4 is the wave equation.
+
+Equation 16.8-4 must be solved in a volume V, subject to boundary conditions on its surface S. The essential boundary condition is p = 0, which prevails on a free fluid surface if waves are negligible. The nonessential boundary condition, which prevails on a solid boundary, is
+
+$$
+\frac {\partial p}{\partial n} = - \rho \ddot {u} _ {n} \tag {16.8-5}
+$$
+
+where n = outward normal direction and $\ddot{u}_{n} = \text{acceleration of the boundary in direction } n$ . For a rigid boundary, $\ddot{u}_{n} = 0$ . If the effect of small-amplitude waves on a fluid surface is to be included, we write $p = \rho g w_{s}$ , where g = acceleration of gravity and $w_{s} = \text{surface elevation relative to the mean surface level}$ . Combining $p = \rho g w_{s}$ with $p_{,z} = -\rho \ddot{w}_{s}$ from Eqs. 16.8-1, we obtain
+
+$$
+\frac {\partial p}{\partial z} = - \frac {1}{g} \ddot {p} \tag {16.8-6}
+$$
+
+as the boundary condition on a fluid surface with small-amplitude waves.
+
+A functional for this problem is
+
+$$
+\Pi = \int_ {V} \left(\frac {p _ {, x} ^ {2} + p _ {, y} ^ {2} + p _ {, z} ^ {2}}{2} + \frac {\rho}{B} p \ddot {p}\right) d V + \int_ {S _ {s}} \rho \ddot {u} _ {n} p d S + \int_ {S _ {f}} \frac {1}{g} p \ddot {p} d S \tag {16.8-7}
+$$
+
+where $S_{s} =$ solid boundary and $S_{f} =$ fluid boundary with wave action. As in Eqs. 16.8-4, $\rho / B$ may be replaced by $1 / c^2$ . The stationary condition $\delta \Pi = 0$ yields Eqs. 16.8-4, 16.8-5, and 16.8-6.
+
+A finite element formulation follows the familiar pattern. Pressure p within an element is interpolated from nodal d.o.f. $\{P_{e}\}$ , where $\{P_{e}\}$ may contain nodal pressures only or may also contain spatial derivatives of nodal pressures. Thus
+
+$$
+p = \lfloor \mathrm{N} \rfloor \{\mathrm{P} _ {e} \} \quad \text { and } \quad \ddot {p} = \lfloor \mathrm{N} \rfloor \{\ddot {\mathrm{P}} _ {e} \} \tag {16.8-8}
+$$
+
+We define the following global matrices, where summation signs indicate assembly of element matrices. The notation is analogous to that in Eqs. 16.5-3.
+
+$$
+[ \mathbf {K} ] = \sum \int_ {V _ {e}} \left(\left[ \mathbf {N}, _ {x} \right] ^ {T} \left[ \mathbf {N}, _ {x} \right] + \left[ \mathbf {N}, _ {y} \right] ^ {T} \left[ \mathbf {N}, _ {y} \right] + \left[ \mathbf {N}, _ {z} \right] ^ {T} \left[ \mathbf {N}, _ {z} \right]\right) d V \tag {16.8-9a}
+$$
+
+$$
+[ \mathbf {C} ] = \sum \int_ {V _ {e}} \frac {\rho}{B} [ \mathbf {N} ] ^ {T} [ \mathbf {N} ] d V \tag {16.8-9b}
+$$
+
+$$
+[ \mathbf {H} ] = \sum \int_ {S _ {f}} [ \mathbf {N} ] ^ {T} [ \mathbf {N} ] \frac {1}{g} d S \quad \left\{\mathbf {R} _ {s} \right\} = \sum \int_ {S _ {s}} [ \mathbf {N} ] ^ {T} \rho \ddot {u} _ {n} d S \tag {16.8-9c}
+$$
+
+<!-- source-page: 510 -->
+
+Equation 16.8-7 becomes
+
+$$
+\Pi = \frac {1}{2} \{\mathbf {P} \} ^ {T} [ \mathbf {K} ] \{\mathbf {P} \} + \{\mathbf {P} \} ^ {T} ([ \mathbf {C} ] + [ \mathbf {H} ]) \{\ddot {\mathbf {P}} \} + \{\mathbf {P} \} ^ {T} \{\mathbf {R} _ {s} \} \tag {16.8-10}
+$$
+
+where $\{\mathbf{P}\}$ is the global array of nodal d.o.f. The stationary condition $\{\partial \Pi / \partial \mathbf{P}\} = \{\mathbf{0}\}$ yields the finite element formulation
+
+$$
+[ \mathbf {K} ] \{\mathbf {P} \} + ([ \mathbf {C} ] + [ \mathbf {H} ]) \{\ddot {\mathbf {P}} \} = - \{\mathbf {R} _ {s} \} \tag {16.8-11}
+$$
+
+One must often deal with a boundary at infinity, that is, a boundary so distant that reflected waves do not appear for the duration of the event of interest. Such a condition can be modeled by “infinite elements” or by other analytical procedures [16.9–16.11].
+
+Sloshing and Acoustic Modes. Let walls that support or contain a fluid be rigid, so that $\ddot{u}_{n}=0$ . Thus the forcing function becomes zero, and the fluid vibrates in one of its natural modes. The pressure becomes $p=\overline{p}\sin\omega t$ , where $\omega$ is a natural frequency and amplitude $\overline{p}$ is a function of the spatial coordinates but is independent of time t. Equation 16.8-4 becomes
+
+$$
+\nabla^ {2} \overline {{{p}}} + \omega^ {2} \frac {\rho}{B} \overline {{{p}}} = 0 \quad \text { or } \quad \nabla^ {2} \overline {{{p}}} + \omega^ {2} \frac {1}{c ^ {2}} \overline {{{p}}} = 0 \tag {16.8-12}
+$$
+
+Similarly, a finite element formulation appears from Eq. 16.8-11 if we set $\{\mathbf{R}_s\} = \{\mathbf{0}\}$ and $\{\mathbf{P}\} = \{\overline{\mathbf{P}}\}\sin \omega t$ :
+
+$$
+[ [ \mathbf {K} ] - \omega^ {2} ([ \mathbf {C} ] + [ \mathbf {H} ]) ] \{\overline {{{\mathbf {P}}}} \} = \{\mathbf {0} \} \tag {16.8-13}
+$$
+
+This eigenvalue problem can be solved to obtain natural frequencies $\omega_{i}$ and the corresponding pressure modes $\{\overline{\mathbf{P}}\}_{i}$ . If $[\mathbf{C}] = [\mathbf{0}]$ , Eq. 16.8-13 yields the slosh frequencies of an incompressible liquid in a rigid container. If $[\mathbf{H}] = [\mathbf{0}]$ , Eq. 16.8-13 yields the acoustic modes of a fluid in a cavity with rigid walls.
+
+As a special case, consider the one-dimensional problem of acoustic modes in a pipe with rigid walls that lies along an $x$ axis (Fig. 16.8-1). Thus, in Eq. 16.8-7, $p_{,y} = p_{,z} = 0$ and $\ddot{u}_n = 0$ . Volume $dV$ becomes $A dx$ , where $A = A(x)$ for a pipe of variable cross section. The integral over $S_f$ is discarded because there are no surface waves, and the integral over $S_s$ is zero because $\ddot{u}_n = 0$ . Thus, only special forms of Eqs. 16.8-9a and 16.8-9b are used in Eq. 16.8-13. As an alternative to specialization, one may begin with a functional for this particular problem. It is
+
+$$
+\Pi = \int \left(\overline {{p}} _ {, x} ^ {2} - \frac {\omega^ {2}}{c ^ {2}} \overline {{p}} ^ {2}\right) A d x \tag {16.8-14}
+$$
+
+where $\overline{p}$ is the pressure amplitude. By making the usual definitions and substitutions (analogous to Eqs. 16.8-8 and 16.8-9), we may develop a finite element formulation directly from Eq. 16.8-14 [16.12]. As for boundary conditions: at a closed end, $\overline{p}_{,x}=0$ and $\overline{p}$ is unknown; at an open end, $\overline{p}=0$ and $\overline{p}_{,x}$ is unknown.
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@@ -0,0 +1,420 @@
+<!-- source-page: 511 -->
+
+![](images/page-511_b4b0f3ac3eb058698ac8d6357a39eeba3762658938865cea8fd250d00ad73689.jpg)
+
+<details>
+<summary>text_image</summary>
+
+x
+A = A(x).
+</details>
+
+(a)
+
+![](images/page-511_a09866e3081da1eb02c7e23256d59fcc7503e0d9fad0dc0db116db16892a3dd5.jpg)
+
+![](images/page-511_ee8d87ba34895cb6c5edfc5e7cc8e3931e2c7fed2b5ea5cf627b9376d50045b7.jpg)  
+(b)   
+Figure 16.8-1. (a) A pipe of variable cross-sectional area A. (b) Possible finite elements for acoustic mode analysis.
+
+# 16.9 FLUID-STRUCTURE INTERACTION
+
+Remarks. A structure may be surrounded by fluid or may contain fluid. A phenomenon of fluid–structure interaction exists when fluid and structure both move and exert forces upon one another associated with the motion. Different categories of the phenomenon may be identified [16.10].
+
+1. Problems having large relative motion; for example, aerodynamic flutter. Behavior is dominated by flow characteristics of the fluid.   
+2. Problems having limited fluid displacement; for example, a reservoir behind a dam under earthquake load, or wave action on an offshore structure.
+
+In this section we are concerned with the second category.
+
+One analytical approach is to model the structure by finite elements as usual, then load its wetted surface by the pressures described by Eq. 16.8-11. This approach results in a system of equations that uses displacement d.o.f. of the structure and pressure d.o.f. at nodes of the wetted surface. The coefficient matrices are unsymmetric. With the simplifying assumptions of no surface waves and incompressibility, the pressure d.o.f. may be eliminated, leaving a smaller system of equations that includes only the structural d.o.f. and has symmetric coefficient matrices. In this smaller system the fluid is represented by a “virtual” mass matrix that is added to the mass matrix of the structure. However, the simplifying assumption of incompressibility may be crude, as seems to be the case for a relatively stiff submerged structure $[16.9]$ .
+
+Another analytical approach is to model the fluid by displacement-based elements, as if the fluid were a structure with special material properties. This “mock-fluid” approach is described in some detail in the following discussion. As compared with the pressure-load approach, the mock-fluid approach typically requires more d.o.f. but produces symmetric matrices with smaller bandwidth or wave front.
+
+Other approaches are also available, for example, the boundary element method.
+
+Mock-Fluid Elements. We will model the fluid by elements that have displacement d.o.f., as do solid elements. Displacements are assumed to be small. Our needs are for a compressibility stiffness matrix $[k_{B}]$ , a wave or slosh stiffness matrix $[k_{S}]$ , and a mass matrix [m]. This formulation appears to be reliable for static analysis and for calculation of natural frequencies of fluid in rigid or flexible containers, but does not always accurately predict natural frequencies of elastic structures surrounded by fluid [16.15].
+
+<!-- source-page: 512 -->
+
+The strain energy per unit volume associated with compressibility is $B\epsilon_V^2 / 2$ , where $B$ is the bulk modulus, and $\epsilon_V$ is the volumetric strain, $\epsilon_V = dV / V$ . The expression $B\epsilon_V^2 / 2$ can be obtained from the standard expression $\{\epsilon\}^T [\mathbf{E}]\{\epsilon\} / 2$ by recognizing that $\epsilon_V = \epsilon_x + \epsilon_y + \epsilon_z$ and using [E] from Eq. 9.4-7 with $G = 0$ . If displacements $\{\mathbf{u}\} = [u \ v \ w]^T$ are interpolated over an element in standard fashion—that is, $\{\mathbf{u}\} = [\mathbf{N}]\{\mathbf{d}\}$ —then
+
+$$
+\epsilon_ {V} = \left\lfloor \mathbf {B} _ {V} \right\rfloor \{\mathbf {d} \} \quad \text { where } \quad \left\lfloor \mathbf {B} _ {V} \right\rfloor = \left\lfloor \frac {\partial}{\partial x} \quad \frac {\partial}{\partial y} \quad \frac {\partial}{\partial z} \right\rfloor [ \mathbf {N} ] \tag {16.9-1}
+$$
+
+Let us now restrict attention to linear elements, such as the four-node plane element shown in Fig. 16.9-1. We elect to evaluate $\epsilon_{V}$ at only the center of the element and regard its value there as representative of the element as a whole. Thus, with $\epsilon_{V0} = [B_{V0}]\{d\}$ the center value of $\epsilon_{V}$ , the strain energy of compressibility in an element of volume $V_{e}$ is
+
+$$
+U = \frac {1}{2} \left\{\mathbf {d} \right\} ^ {T} \left[ \mathbf {k} _ {B} \right] \left\{\mathbf {d} \right\} \quad \text { where } \quad \left[ \mathbf {k} _ {B} \right] = B V _ {e} \left\lfloor \mathbf {B} _ {V 0} \right\rfloor^ {T} \left\lfloor \mathbf {B} _ {V 0} \right\rfloor \tag {16.9-2}
+$$
+
+It is not necessary to use Eq. 16.9-2 to generate $[k_{B}]$ . One can use instead the formula for [k] of a structural element (Eq. 4.1-5), integrate using a single Gauss point at the element center, and take [E] from Eq. 9.4-7 but with G = 0. Thus, with reduced integration, bulk modulus B provides the element with resistance only to volume change of the element as a whole, and $[k_{B}]$ of Eq. 16.9-2 is produced.
+
+Note that the constraint of near-incompressibility is enforced at each Gauss point used to evaluate $[k_{B}]$ . Accordingly, if $[k_{B}]$ were formed by full integration of $B[B_{V}]^{T}[B_{V}]$ , $[k_{B}]$ would exhibit undesirable stiffnesses. For example, the plane four-node element of Fig. 16.9-1a would yield zero strain energy only when $\{d\}$ represents rigid-body motion or a pure shear deformation. Thus the slosh mode in Fig. 16.9-2b would be resisted by bulk modulus B, which is clearly unreasonable.
+
+One may contemplate the use of elements of higher order than the linear elements discussed thus far. One would then probably use more than one Gauss point per element, yet underintegrate so as to avoid locking due to too many penalty constraints, as discussed in Sections 9.4 and 9.5. However, higher-order elements seem not to be used in practice.
+
+![](images/page-512_cde2a6182a4bdd71093fcefb9a3e1aa4fc831862c3023d87b0e919346b8853e2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+a
+x
+w4 w3
+u4 4 3 u3
+u1 1 2 u2
+w1 w2
+Fluid surface
+for slosh
+stiffness
+(a)
+</details>
+
+![](images/page-512_3c38db6a2a174c6971f7aebf79f8a3254b3265d91e1574eb1d461cc751ca40a0.jpg)
+
+<details>
+<summary>text_image</summary>
+
+The only mode resisted
+εv0 > 0
+εv0 = 0
+εv0 = 0
+εv0 = 0
+εv0 = 0
+(b)
+</details>
+
+Figure 16.9-1. (a) Plane rectangular element. (b) Deformation modes of a plane element, showing volumetric strain at the element center. Rigid-body modes are not shown.
+
+<!-- source-page: 513 -->
+
+A slosh stiffness matrix $[k_{S}]$ is needed only if there is a free surface with waves or an oscillating boundary between fluids of different density. The energy associated with small-amplitude waves on a surface S is
+
+$$
+U = \int_ {S} \frac {1}{2} \rho g w _ {s} ^ {2} d S \quad \text { where } \quad w _ {s} = \left[ \mathbf {N} _ {S} \right] \{\mathbf {d} \} \tag {16.9-3}
+$$
+
+and $\rho = \cdot$ mass density of the fluid, $g =$ acceleration of gravity, $w_{s} =$ surface displacement in the upward direction, and $\lfloor \mathbf{N}_S\rfloor$ operates only on element d.o.f. in surface $S$ . For example, for the plane element of Fig. 16.9-1a, $w_{s} = \lfloor \mathbf{N}_{S}\rfloor \{\mathbf{d}\} = w_{3}x / a + w_{4}(a - x) / a$ . Equation 16.9-3 becomes
+
+$$
+U = \frac {1}{2} \left\{\mathbf {d} \right\} ^ {T} \left[ \mathbf {k} _ {S} \right] \left\{\mathbf {d} \right\} \quad \text { where } \quad \left[ \mathbf {k} _ {S} \right] = \int_ {S} \left[ \mathbf {N} _ {S} \right] ^ {T} \left[ \mathbf {N} _ {S} \right] \rho g d S \tag {16.9-4}
+$$
+
+The element mass matrix [m] is formulated in standard fashion, as though the element were elastic (e.g., see Eq. 13.2-5).
+
+If desired, a viscous damping matrix [c] can be included [16.13]
+
+$$
+[ \mathbf {c} ] = \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} _ {G} ] [ \mathbf {B} ] \mu d V \tag {16.9-5}
+$$
+
+where [B] is the standard strain-displacement matrix of Eq. 4.1-3, $[E_{G}]$ is the first rectangular matrix on the right-hand side of Eq. 9.4-7 (exclusive of G), and $\mu$ is the dynamic coefficient of viscosity [13.3].
+
+As is evident from Eq. 16.9-2, $[k_{B}]$ is a matrix of rank one. Accordingly, the assembled structure may have several zero-energy modes, called circulation modes. They involve no volume change of the fluid and no surface waves. One such mode is shown in Fig. 16.9-2a. A way to slightly restrain circulation modes is to make G a small positive number in Eq. 9.4-7 and use the structural formulation to obtain $[k_{B}]$ , as explained in the paragraph that follows Eq. 16.9-2. (This is not the same as attributing viscosity to the fluid.) Another way to inhibit circulation is to augment $[k_{B}]$ by the penalty matrix
+
+$$
+[ \mathbf {k} _ {c} ] = \alpha \int_ {V _ {e}} [ \mathbf {B} _ {c} ] ^ {T} [ \mathbf {B} _ {c} ] d V \tag {16.9-6}
+$$
+
+where $\alpha$ is a penalty number and $\{d\}^{T}[k_{c}]\{d\}$ is the integral over the element of the square of the curl of the displacement field [16.14]. For a plane element in
+
+![](images/page-513_049711465006a88d86a3d491ffdb88b2e9d0a7981963fb9fb23449d65762e50f.jpg)  
+Figure 16.9-2 (a) A circulation mode in a plane three-element model. (b) A one-element model for slosh in a plane rectangular tank. (c) A two-element model for acoustic analysis of a pipe with an elastic end.
+
+<!-- source-page: 514 -->
+
+the $xz$ plane, $(\operatorname{curl} \mathbf{u})^2 = (u_{,z} - w_{,x})^2$ and $[\mathbf{B}_c] = \left\lfloor \frac{\partial}{\partial z} - \frac{\partial}{\partial x} \right\rfloor [\mathbf{N}]$ . If $\alpha = B$ , circulation modes disappear and triangular elements yield good results [16.14]. If circulation modes are unrestrained, they may appear with frequencies indistinguishable from other low frequencies of greater interest [16.14]. Note also that $[\mathbf{k}_B] >> [\mathbf{k}_S]$ (Eqs. 16.9-2 and 16.9-4). Accordingly, if both are used in an analysis, there is a risk that $[\mathbf{k}_S]$ will "fall off the end" of computer words, as discussed in Section 18.2.
+
+Example: Plane Slosh Analysis. Consider the one-element model shown in Fig. 16.9-2b. Displacements in the y (out of plane) direction are assumed to be zero. The element has eight d.o.f., but because of the rigid walls, only $w_{3}$ and $w_{4}$ are unrestrained. Rather than invoke all eight d.o.f. and then eliminate six, we elect to use only $w_{3}$ and $w_{4}$ at the outset. Now u = 0, and
+
+$$
+w = \frac {x z}{a b} w _ {3} + \frac {(a - x) z}{a b} w _ {4} \tag {16.9-7}
+$$
+
+Using $[\mathbf{k}_B], [\mathbf{k}_S]$ , and the consistent mass matrix $[\mathbf{m}]$ , we obtain, for a unit thickness,
+
+$$
+\left(\frac {B a}{4 b} \left[ \begin{array}{l l} 1 & 1 \\ 1 & 1 \end{array} \right] + \frac {\rho g a}{6} \left[ \begin{array}{l l} 2 & 1 \\ 1 & 2 \end{array} \right]\right) \left\{ \begin{array}{l} w _ {3} \\ w _ {4} \end{array} \right\} + \frac {\rho a b}{1 8} \left[ \begin{array}{l l} 2 & 1 \\ 1 & 2 \end{array} \right] \left\{ \begin{array}{l} \ddot {w} _ {3} \\ \ddot {w} _ {4} \end{array} \right\} = \left\{ \begin{array}{l} R _ {3} \\ R _ {4} \end{array} \right\} \tag {16.9-8}
+$$
+
+where $R_{3}$ and $R_{4}$ are time-varying loads. Typically $(Ba/4b) >> (\rho ga/6)$ , which implies the condition $w_{3} = -w_{4}$ . This is a constraint of zero volume change. Imposing this constraint, and setting $R_{3} = R_{4} = 0$ and $\left[w_{3} \quad w_{4}\right] = \left[\overline{w}_{3} \quad \overline{w}_{4}\right] \sin \omega t$ , we obtain an eigenvalue problem that yields $\omega^{2} = 3g/b$ . The theoretical result is $\omega^{2} = (\pi g/a) \tanh(\pi b/a)$ , which yields $\omega^{2} = 3.13g/b$ for a = b.
+
+Example: Acoustic Modes. Air in a uniform pipe is modeled by two two-node elements (Fig. 16.9-2c). At the left end there is a massless plug connected to an elastic spring of stiffness k. For this problem, there is no slosh stiffness, $\epsilon_{V} = \epsilon_{x}$ , and $[k_{B}]$ has the same form as the stiffness matrix of a two-node elastic bar. To $[k_{B}]$ we add the structure (spring) stiffness k. Thus, with [m] the consistent mass matrix and the boundary condition $u_{3} = 0$ already imposed, the structure equations are
+
+$$
+\left(\frac {B A}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 2 \end{array} \right] + \left[ \begin{array}{l l} k & 0 \\ 0 & 0 \end{array} \right]\right) \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \end{array} \right\} + \frac {\rho A L}{6} \left[ \begin{array}{l l} 2 & 1 \\ 1 & 4 \end{array} \right] \left\{ \begin{array}{l} \ddot {u} _ {1} \\ \ddot {u} _ {2} \end{array} \right\} = \left\{ \begin{array}{l} R _ {1} \\ R _ {2} \end{array} \right\} \tag {16.9-9}
+$$
+
+where A is the constant cross-sectional area. The fundamental acoustic frequency in a pipe with rigid ends is obtained by setting $u_{1} = 0$ , $R_{2} = 0$ , and $u_{2} = \bar{u}_{2} \sin \omega t$ . Thus
+
+$$
+\omega^ {2} = \frac {3 B}{\rho L ^ {2}} = \frac {3 c ^ {2}}{L ^ {2}} = \frac {1 2 c ^ {2}}{L _ {T} ^ {2}} \quad \text { and } \quad \omega = 3. 4 6 \frac {c}{L _ {T}} \tag {16.9-10}
+$$
+
+The exact result (for infinite k) is $\omega = \pi c / L_{T}$ .
+
+<!-- source-page: 515 -->
+
+# PROBLEMS
+
+# Section 16.2
+
+16.1 Obtain Eq. 16.2-7, with matrices defined as in Eq. 16.2-6, by use of the Galerkin method.   
+16.2 The three-node bar shown is uniform. Temperature $T_{1}$ is prescribed, $Q = 0$ , node 3 is insulated ( $q_{3} = 0$ ), and heat is transferred across the lateral surface by convection. In the units used in Section 16.1, let $k = 180$ , $h = 12$ , $A = 0.1$ , $p = 1$ , and $L = 2$ .
+
+(a) Use two two-node elements to determine $T_{2}$ and $T_{3}$ in terms of $T_{1}$ and $T_{f}$ .
+
+(b) Repeat part (a), but use a single three-node element of length 2L. Note that $[h_{ls}]$ has the form of a mass matrix, and use an “optimally lumped” form (see Eq. 13.3-9).
+
+![](images/page-515_66e66ec75ef773969db12ad1f5fb84c9e5f5a479114437b027c654f7a656f690.jpg)  
+Problem 16.2
+
+# Section 16.3
+
+16.3 Verify that $[\kappa]$ is as defined by Eq. 16.3-3 (resolve $q_x$ and $q_y$ into components $q_r$ and $q_s$ ).   
+16.4 Imagine that an essentially two-dimensional body has a gradual variation in thickness $\tau$ —that is, $\tau = \tau(x, y)$ . What changes must be made in the equations developed in Section 16.3?   
+16.5 Verify that Eq. 16.3-10 yields the correct governing equation and nonessential boundary conditions from the condition $\delta \Pi = 0$ . Include the term $2h(T_fT - T^2 /2)$ in $\Pi$ , so as to account for convection on two identical lateral surfaces.
+
+# Section 16.4
+
+16.6 Devise an example that shows why $\theta$ must be a principal material direction if $T$ in a solid of revolution is to be symmetric with respect to the $\theta = 0$ plane.   
+16.7 Using conventional scalar notation like that in Eq. 16.3-5, write Eqs. 16.4-1 and 16.4-2 for the following special cases.
+
+(a) Solid of revolution with an axially symmetric temperature field and isotropic material.   
+(b) The plane problem in polar coordinates $(T, z = 0)$ . Let $[\kappa]$ be full, as for an anisotropic material.
+
+16.8 Derive the governing equation of Problem 16.7(b) from first principles, analogous to the Cartesian coordinate derivation in Eqs. 16.3-4 and 16.3-5.
+
+<!-- source-page: 516 -->
+
+# Section 16.5
+
+16.9 The three-node triangle shown is to be used for heat conduction analysis. The body in question is homogeneous, plane, isotropic, and of unit thickness.
+
+(a) Evaluate [k] in terms of $k$ and nodal coordinates.   
+(a) Evaluate [k] in terms of h and nodal coordinates if only side 1–3 transfers   
+(b) Evaluate [11] in terms of heat by convection.   
+(c) Write the “lumped” forms of [h] and [c].   
+(d) Write the lumped $\{r_Q\}$ in terms of $Q$ and nodal coordinates if $Q$ is constant over the element.
+
+![](images/page-516_2c2c8282be7c4a0d499bbfbf5489c84e4f575744133f76f4edcc93cbc51893b9.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+3
+1
+2
+x
+</details>
+
+Problem 16.9
+
+![](images/page-516_f9e12d427dc04fe56fca66f9b242a5a95b52ab94720a71862d5483a9cafe5ee1.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z
+1
+2
+r
+r₁
+r₂
+</details>
+
+Problem 16.11
+
+16.10 Repeat Problem 16.9, but regard the body as a solid of revolution, so that $r$ replaces $x$ and axis of revolution $z$ replaces $y$ .   
+16.11 The solid of revolution element shown is homogeneous, isotropic, flat, and of unit thickness. Surfaces $z =$ constant are insulated so that heat flows only radially. Assume that $T$ varies linearly with $r$ in the element and let $Q$ be constant. Evaluate the matrices defined by Eqs. 16.5-3 in terms of $r_1$ , $r_2$ , and material properties $k, c$ , and so on.   
+16.12 The element shown is a quadratic triangle with straight sides and midside nodes. Evaluate $\{\mathbf{r}_Q\}$ due to heat input $Q_P$ (units J/s), if $Q_P$ is
+
+(a) Concentrated at node 1.
+
+(b) Concentrated at node 4.
+
+(c) Concentrated at the centroid of the triangle.
+
+(d) Uniformly distributed along a line between nodes 5 and 6. Use a simple approximation.
+
+(e) Improve your answer to part (d) by devising a better approximation.
+
+16.13 The uniform bar element shown is described by Eqs. 16.2-6 and 16.2-7. Conductivity $k$ is a function of temperature. Let $T_{a}$ be the average temperature in the element. In the units used in Section 16.1, let $k = 100 - 0.4T_{a}$ , $h = 30$ , $A = 0.1$ , $p = 1$ , $L = 2$ , and $T_{f} = 200$ . Heat flows across the lateral surface. If $T = 400^{\circ}\mathrm{C}$ at node 1 and the right end is insulated ( $q_{R} = 0$ in Fig. 16.2-1), what is the steady-state value of $T_{2}$ ? Use an iterative solution method.
+
+![](images/page-516_a488a5861fa9a79ad22e9ebd06dc259208d0845482393efa8b3a5ecec000a7db.jpg)  
+Problem 16.12
+
+![](images/page-516_1dd4c8ea1c10b0f64fd54aa8625e08ababf7b775e0ba21a2b78f0790f4210389.jpg)  
+Problem 16.13
+
+![](images/page-516_effabeb7bbae7d53a37ff827d1bb70aa24172e49f723637d1839d9b282f76b2d.jpg)  
+Problem 16.14
+
+<!-- source-page: 517 -->
+
+16.14 The uniform bar shown is modeled by two linear elements. Lateral surfaces are insulated. Heat flows into the bar at node 3 at the prescribed rate 20A, where A is the cross-sectional area of the bar and the units are those used in Section 16.1. The temperature at node 1 is kept at zero. Let conductivity in an element be given by $k = 2 + 0.04T_{a}$ , where $T_{a}$ is the average temperature in an element. Perform two steps of an iterative solution for $T_{2}$ and $T_{3}$ , starting with $T_{a} = 0$ in each element.
+
+# Section 16.6
+
+16.15 Prove Eqs. 16.6-3.
+
+16.16 The sketch shows the actual variation of temperature with time at a certain node. Imagine that we start at point A and use Eq. 16.6-6 to predict T at time $t_{n+1}$ . Consider $\beta = 0$ , $\beta = 0.5$ , and $\beta = 1.0$ . For each of these three values make a sketch that shows how the predicted temperature compares with $T_{B}$ .
+
+![](images/page-517_46840ef70dbdf2254d04b4852d5e972416a928bd839764aec69dc57067a59d03.jpg)
+
+<details>
+<summary>line</summary>
+
+| Point | Time (t) | Temperature (T) |
+|-------|----------|------------------|
+| A     | t_n      | T_A              |
+| B     | t_{n+1}  | T_B              |
+</details>
+
+Problem 16.16
+
+16.17 Show that Eq. 16.6-8 follows from Eqs. 16.6-6 and 16.6-7.
+
+16.18 Imagine that the bar of Fig. 16.2-1 is initially at zero temperature and then is subjected to a heat flux at the right end. The flux appears at time t = 0 and thereafter remains constant. The left end is kept at zero temperature. For a one-element model, physical constants are such that Eq. 16.6-1 becomes $6T + 2\dot{T} = 3$ , where T is the temperature of the node where heat flux is imposed. We are to compute $T = T(t)$ by use of Eq. 16.6-8.
+
+(a) To what $T$ should $T(t)$ converge as $t$ becomes large?
+
+(b) What is the exact solution for $T = T(t)$ ?
+
+(c) What is $\Delta t_{\mathrm{cr}}$ for Euler's method?
+
+In what follows, use Eq. 16.6-8 with the values of $\beta$ and $\Delta t$ given. Take five time steps in each case. Compare results with exact values and with the other direct integration results.
+
+(d-g) With $\Delta t = 0.1$ , take $\beta$ as (d) 0, (e) $\frac{1}{2}$ , (f) $\frac{2}{3}$ , (g) 1.0.
+
+(h-k) With $\Delta t = 1.0$ , take $\beta$ as (h) 0, (i) $\frac{1}{2}$ , (j) $\frac{2}{3}$ , (k) 1.0.
+
+# Section 16.7
+
+16.19 Equations 16.5-3 apply to the problem of groundwater flow if $T =$ hydraulic head (meters of water), $[\kappa] =$ permeability coefficients, $q =$ seepage velocity, and $Q =$ flow rate per unit volume. The sketch represents the plan view of a square array of four wells in a homogeneous, horizontal aquifer of infinite extent and constant thickness $\tau$ between impermeable strata above and below it. The hydraulic head at a large distance from the group of wells is $h_0$ . The hydraulic head in the neighborhood of a typical well is
+
+<!-- source-page: 518 -->
+
+![](images/page-518_691d55e3a6c2070fb8a3fa99d040c49d80c355f01ab564e8423f8b285027108e.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Simple abstract shape with four small dots inside, no text or symbols present
+</details>
+
+Problem 16.19
+
+![](images/page-518_dc4a2987f63ab1770c0385a187baadca7b977cc3750b92c791a41310a52a35cb.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Free surface
+(zero pressure)
+</details>
+
+Problem 16.21
+
+desired. Each well pumps at constant flow rate $Q_{P}$ m $^{3}$ /s. Outline how to set up a finite element solution for steady-state $T = T(x, y)$ in the aquifer, with attention to mesh layout, prescribed d.o.f., and load terms.
+
+16.20 Derive Eqs. 16.7-2.   
+16.21 The sketch depicts a steady flow that emerges from an enclosure, thereafter to flow with a “free surface” (open to the air). Bernoulli’s equation is $\frac{1}{2}(u^{2} + v^{2}) + (p/\rho) + gy = \text{constant}$ , where p = pressure, $\rho = mass density$ , g = acceleration of gravity, and y = elevation above a datum. Locating the free surface requires an iterative solution process. Outline the major steps of this process [16.8].
+
+# Section 16.8
+
+16.22 (a) Derive Eqs. 16.8-1.
+
+(b) Show that $dV / V = \epsilon_x + \epsilon_y + \epsilon_z$ in Eq. 16.8-3.   
+(b) Show that $\alpha \in \mathbb{R}_{\mathbb{X}}$ is a finite set of $\mathbb{R}_{\mathbb{X}}$ for $\mathbb{R}_{\mathbb{X}} = 0$ (c) Show that with $\Pi$ defined by Eq. 16.8-7, the condition $\delta \Pi = 0$ yields Eqs. 16.8-4, 16.8-5, and 16.8-6.
+
+16.23 The formulation of Eqs. 16.8-9 is to be applied with a plane rectangular element of unit thickness (see sketch).
+
+(a) Generate element matrices [k], [c], and [h], in terms of $a, b, \rho, B$ , and $g$ .   
+(b) Imagine that the sketch represents a one-element model of an incompressible liquid in a rectangular tank that is open at the top. Write the appropriate form of Eq. 16.8-13 for the case $a = b = h$ .   
+(c) To reduce the number of equations, set $p_1 = -p_2$ and $p_4 = -p_3$ , then solve for the frequency of the slosh mode, again with $a = b = h$ . (Do not expect high accuracy. The theoretical result is stated below Eq. 16.9-8.)
+
+16.24 (a) Show that the functional $\Pi = \int [\overline{p}_{;x}^2 +\overline{p}_{;y}^2 +\overline{p}_{;z}^2 -(\omega \overline{p} /c)^2 ]dV$ yields the latter form of Eq. 16.8-12 from the stationary condition $\delta \Pi = 0$
+
+(b) Derive Eq. 16.8-13 (with $[\mathbf{H}] = [\mathbf{0}])$ from the functional stated in part (a).
+
+16.25 (a) Specialize Eq. 16.8-7 to the one-dimensional acoustic mode problem, Fig. 16.8-1. Use this result to show that the condition $\delta \Pi = 0$ yields the governing differential equation $d(Ap_{xx}) / dx - (A / c^2)\ddot{p} = 0$ .
+
+(b) Using Eq. 16.8-14, obtain the equation given by the condition $\delta \Pi = 0$ . Check your result by specializing the differential equation given in part (a).
+
+<!-- source-page: 519 -->
+
+![](images/page-519_3106809f28d6310a97df76b6d6c1cfde71687d8a1448f5a71f282f3dd5dc8259.jpg)
+
+<details>
+<summary>text_image</summary>
+
+a
+4
+3
+1
+2
+b
+</details>
+
+Problem 16.23
+
+![](images/page-519_35aaed9810e29e6763d485b964601d222262fa47d6c66a20517e0ae112014fcb.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+3
+L/2
+L/2
+</details>
+
+Problem 16.27
+
+16.26 Consider a uniform pipe of length $L$ and having closed ends. Model the pipe by a single element. Determine the frequency of the fundamental acoustic mode by use of the following element formulations. The exact result is $\omega = \pi c / L$ .
+
+(a) Use an element with d.o.f. $p_1$ and $p_2$ at ends 1 and 2.   
+(b) Use a four d.o.f. element. The d.o.f. at each end are $p$ and $p_{,x}$ .
+
+16.27 The sketch represents a three-node model of air in a uniform pipe. The end at the left is open. Determine $\omega$ for the acoustic mode of lowest frequency. The exact result is $\omega = \pi c/2L$ .
+
+(a) Use a single quadratic element.   
+(b) Use two two-node (linear) elements.
+
+# Section 16.9
+
+16.28 The sketch represents a rectangular tank of liquid. Beam EFGH is fixed at E and H and is in contact with the liquid surface. Imagine that motion is confined to the xz plane.
+
+(a) If the liquid were absent, what would be the appearance of the first two vibration modes of the beam? Sketch them. Which has the higher frequency?   
+(b) If the liquid is restored, what now are the appearances of the first two vibration modes that involve bending of the beam? Sketch them.   
+(c) If beam and liquid are each divided into three elements, with displacement d.o.f. and nodes at points A–H, how many d.o.f. are nonzero after boundary conditions are imposed? Identify these d.o.f.   
+(d) Are beam and liquid elements compatible? If no, is this acceptable? If yes, how is it accomplished?
+
+![](images/page-519_1acdb419357ed0ce195849611e5f648854c2173a41ead5f3abffc51c73e08837.jpg)
+
+<details>
+<summary>text_image</summary>
+
+z
+E F G H
+A B C D
+x
+</details>
+
+Problem 16.28
+
+16.29 Imagine that a very small amount of liquid is to be analyzed, and that the effects of surface tension $\sigma$ (force per unit length) are to be included. Determine an expression for the appropriate stiffness matrix that operates on displacement d.o.f. {d}.
+
+<!-- source-page: 520 -->
+
+16.30 For the three-element fluid model shown in Fig. 16.9-2a, sketch all possible circulation modes. Show the direction of the circulation in each case. Note: A simple reversal of the direction of all fluid displacements is not considered a different mode.   
+16.31 (a) Write an expression for $[\mathbf{B}_c]$ in Eq. 16.9-6, in the form of an operator matrix times shape function matrix [N]. Assume that the problem is three-dimensional.   
+(b) Generate $[\mathbf{k}_c]$ of Eq. 16.9-6 for the two-d.o.f. element described by Eq. 16.9-7.   
+16.32 Verify the correctness of the rectangular matrices in Eq. 16.9-8.   
+16.33 Use a three-node, mock-fluid model to solve for the lowest acoustic frequency in a uniform pipe with one open end (see Problem 16.27).   
+(a) Use a single quadratic element.   
+(b) Use two two-node (linear) elements.
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_053.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_053.md
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index 00000000..5ddf5401
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_053.md
@@ -0,0 +1,472 @@
+<!-- source-page: 521 -->
+
+# AN INTRODUCTION TO SOME NONLINEAR PROBLEMS
+
+Solution methods for nonlinear equations are discussed. Selected problems are described and given a finite element formulation. Emphasis is given to nonlinearities arising from material properties and changes in geometry.
+
+# 17.1 INTRODUCTION
+
+In structural mechanics, a problem is nonlinear if the stiffness matrix or the load vector depends on the displacements. Nonlinearity in structures can be classed as material nonlinearity (associated with changes in material properties, as in plasticity) or as geometric nonlinearity (associated with changes in configuration, as in large deflections of a slender elastic beam). In heat transfer, nonlinearity may arise from temperature-dependent conductivity (which makes the coefficient matrix depend on temperature) and from radiation (which makes the radiative heat flux a nonlinear function of temperature). In general, for a time-independent problem symbolized as $[K]\{D\} = \{R\}$ , in linear analysis both $[K]$ and $\{R\}$ are regarded as independent of $\{D\}$ , whereas in nonlinear analysis $[K]$ and/or $\{R\}$ are regarded as functions of $\{D\}$ .
+
+The classifications “linear” and “nonlinear” are artificial in that physical reality presents various problems, some of which can be satisfactorily approximated by linear equations. We are fortunate that linear approximations are quite good for many problems of stress analysis and heat conduction. Nonlinear approximations are more difficult to formulate, and solving the resulting equations may cost 10 to 100 times as much as a linear approximation having the same number of d.o.f.
+
+Many physical situations present nonlinearities too large to be ignored. Stress–strain relations may be nonlinear in either a time-dependent or a time-independent way. A change in configuration may cause loads to alter their distribution and magnitude or cause gaps to open or close. Mating parts may stick or slip. Welding and casting processes cause the material to change in conductivity, modulus, and phase. The generation and shedding of vortices in fluid flow past a structure produces oscillatory loads on the structure. Pre-buckling rotations alter the effective stiffness of a shell and change its buckling load. Thus we see that nonlinear effects may vary in type and may be mild or severe.
+
+An analyst must understand the physical problem and must be acquainted with various solution strategies. A single strategy will not always work well, and may not work at all for some problems. Several attempts may be needed in order to obtain a satisfactory result.
+
+Nevertheless, nonlinear analyses are undertaken more often than in the past. In part, this is because computing costs have declined and capable software has
+
+<!-- source-page: 522 -->
+
+become available. In addition, more demands are placed on structures: they must function at higher temperatures and pressures, offer earthquake resistance, and provide crashworthiness. Forming and extrusion processes must be analyzed in an attempt to reduce production costs. Plastics, elastomers, and composites are used with increasing frequency as structural materials; they display material nonlinearity well below the limits of their useful strengths [17.1].
+
+A comprehensive discussion of nonlinearity, even in structural mechanics alone, would require at least one complete volume. The following introductory treatment contains a sampling of nonlinear problems and presents some of the basic procedures for solving the associated equations.
+
+# 17.2 SOME SOLUTION METHODS
+
+A representative time-independent nonlinear problem can be stated as $[K]\{D\} = \{R\}$ , where $\{R\}$ is known and $[K]$ is a function of $\{D\}$ that can be computed for a given $\{D\}$ . We are required to compute $\{D\}$ —for example, to compute the displacement state associated with known loads. In what follows we introduce some of the available computational methods. For simplicity, a one-dimensional problem is chosen as the principal example.
+
+Consider a nonlinear spring, Fig. 17.2-1. The source of the nonlinearity is unimportant in the present discussion. We imagine that the spring stiffness k is composed of a constant term $k_{0}$ and a term $k_{N}$ that depends on deformation. Displacement u is caused by load P and is given by the equation
+
+$$
+(k _ {0} + k _ {N}) u = P \quad \text { where } \quad k _ {N} = f (u) \tag {17.2-1}
+$$
+
+We ask for the value of u when P is given. In order to mimic a realistic problem, we assume that $k_{N}$ is known in terms of u, and therefore that P can be calculated in terms of u, but that an explicit solution for u in terms of P is not available. Instead, iterative methods are needed to determine u, as follows.
+
+Direct Substitution. Let a load $P_{A}$ be applied to a softening spring (for which $k_{N} < 0$ ). For the first iteration, assume that $k_{N} = 0$ . Therefore, as the first approximation of displacement $u_{A}$ produced by $P_{A}$ , we compute $u_{1} = P_{A}/k_{0}$ . Using $u_{1}$ we compute the new stiffness approximation $k_{0} + k_{N1} = k_{0} + f(u_{1})$ , and then the new displacement approximation $u_{2}$ . Thus we generate the sequence of approximations
+
+![](images/page-522_5d5884bc23b4695b0f15b51d05ce29cdd4f3ec99e12b9c6642dc4b88f70c2108.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k = k₀ + kₙ, where
+k₀ = constant
+kₙ = function of u
+(a)
+</details>
+
+![](images/page-522_9e574971f940910e52f692de3f112c84fbe849f416854ec40e869b93efafcfc8.jpg)
+
+<details>
+<summary>line</summary>
+| u    | Hardening (k_N > 0) | Softening (k_N < 0) |
+| ---- | ------------------- | ------------------- |
+| Low  | 0                   | 0                   |
+| Mid  | ~0.5                | ~0.2                |
+| High | >0.5                | ~0.3                |
+</details>
+
+Figure 17.2-1. (a) A nonlinear spring. (b) When u > 0, there is hardening if $k_{N} > 0$ and softening if $k_{N} < 0$ . When u = 0, we assume that $k_{N} = 0$ .
+
+<!-- source-page: 523 -->
+
+$$
+u _ {1} = k _ {0} ^ {- 1} P _ {A}, u _ {2} = \left(k _ {0} + k _ {N 1}\right) ^ {- 1} P _ {A}, \dots , u _ {i + 1} = \left(k _ {0} + k _ {N i}\right) ^ {- 1} P _ {A} \tag {17.2-2}
+$$
+
+These calculations are interpreted graphically in Fig. 17.2-2a. We see that the approximate stiffnesses $k_{0} + k_{Ni}$ can be regarded as secants of the actual curve, each emanating from P = u = 0. After several iterations, the secant stiffness is $k_{0} + k_{N} \approx P_{A}/u_{A}$ , and the correct solution $u = u_{A}$ is closely approximated.
+
+In an alternative form of direct substitution, nonlinear terms $k_{Nu}$ are taken to the right-hand side. Thus, instead of Eq. 17.2-2, we have the sequence
+
+$$
+u _ {1} = k _ {0} ^ {- 1} P _ {A}, u _ {2} = k _ {0} ^ {- 1} (P _ {A} - k _ {N 1} u _ {1}), \dots , u _ {i + 1} = k _ {0} ^ {- 1} (P _ {A} - k _ {N i} u _ {i}) \tag {17.2-3}
+$$
+
+(Equations 17.2-2 and 17.2-3 will not yield the same values of $u_{2}, u_{3}$ , etc., but upon convergence both will yield the result $u_{\infty} = u_{A}$ .) Equation 17.2-3 is interpreted graphically in Fig. 17.2-2b. The effective loads applied in the second and third iterations in Fig. 17.2-2b are
+
+$$
+P _ {A} - k _ {N 1} u _ {1} = P _ {A} + \left(k _ {0} u _ {1} - \left[ k _ {0} + k _ {N 1} \right] u _ {1}\right) = P _ {A} + \left(P _ {a} - P _ {1}\right) \tag {17.2-4a}
+$$
+
+$$
+P _ {A} - k _ {N 2} u _ {2} = P _ {A} + \left(k _ {0} u _ {2} - \left[ k _ {0} + k _ {N 2} \right] u _ {2}\right) = P _ {A} + \left(P _ {I} - P _ {2}\right) \tag {17.2-4b}
+$$
+
+It may be helpful to note that $P_{a} - P_{1} = P_{I} - P_{A}$ and $P_{I} - P_{2} = P_{II} - P_{A}$ . The sequence of pseudoloads $P_{a} - P_{1}, P_{I} - P_{2}, \ldots$ , must converge if Eq. 17.2-3 is to converge to $u = u_{A}$ . Failure to converge is more likely with hardening structures than with softening structures.
+
+If convergence difficulties arise, underrelaxation may help. Thus, rather than updating a calculated value $u_{i+1}$ to its full value, we update instead to
+
+$$
+u _ {i + 1} = u _ {i} + \beta (\Delta u _ {i + 1}) \tag {17.2-5a}
+$$
+
+or, changing the form by the substitution $\Delta u_{i+1} = u_{i+1} - u_i$ , we have
+
+$$
+u _ {i + 1} = \beta u _ {i + 1} + (1 - \beta) u _ {i} \tag {17.2-5b}
+$$
+
+![](images/page-523_ee1a25c1621a82296fbde07e4f96ccb44daa56e71633b7b573d502e73a3253bf.jpg)
+
+<details>
+<summary>line</summary>
+| Point | u    | P    |
+|-------|------|------|
+| a     | u1   | P_A  |
+| b     | u2   | P_A  |
+| A     | uA   | P_A  |
+| 1     | u1   | P1   |
+| 2     | u2   | P1   |
+</details>
+
+![](images/page-523_77732b8598057932909bf557e4f6c9ae2602d6d8da493530fb24a59d73d20c44.jpg)  
+Figure 17.2-2. Graphical interpretations of direct substitution. (a) According to Eq. 17.2-2. (b) According to Eq. 17.2-3. Lines aI and 1b are parallel.
+
+<!-- source-page: 524 -->
+
+where “=” means “is replaced by,” as in Fortran, and $\beta$ is a number in the range $0 < \beta < 1$ .
+
+Note that for a multi-d.o.f. structure, k is a stiffness matrix rather than a scalar. Thus Eq. 17.2-2 requires that a new matrix $[K_{0} + K_{N}]$ be formed and reduced in each iteration, whereas Eq. 17.2-3 requires but one formation and reduction of $[K_{0}]$ . However, Eq. 17.2-3 will require more iterative cycles than Eq. 17.2-2 in order to reach a prescribed accuracy.
+
+Newton-Raphson (N-R). Imagine that we have applied load $P_A$ and somehow determined the corresponding displacement $u_A$ . That is, from Eq. 17.2-1,
+
+$$
+(k _ {0} + k _ {N A}) u _ {A} = P _ {A} \quad \text { where } \quad k _ {N A} = f (u _ {A}) \tag {17.2-6}
+$$
+
+The load is now increased to a value $P_B$ and the corresponding displacement $u_B$ is sought. A truncated Taylor series expansion of $P = f(u)$ about $u_A$ is
+
+$$
+f (u _ {A} + \Delta u _ {1}) = f (u _ {A}) + \left(\frac {d P}{d u}\right) _ {A} \Delta u _ {1} \tag {17.2-7}
+$$
+
+where
+
+$$
+\frac {d P}{d u} = \frac {d}{d u} \left(k _ {0} u + k _ {N} u\right) = k _ {0} + \frac {d}{d u} \left(k _ {N} u\right) = k _ {t} \tag {17.2-8}
+$$
+
+and $k_{t}$ is called the tangent stiffness. We seek $\Delta u_{1}$ for which $f(u_{A} + \Delta u_{1}) = P_{B}$ . Thus, with $f(u_{A}) = P_{A}$ and $k_{t}$ evaluated at $A$ , Eq. 17.2-7 becomes
+
+$$
+P _ {B} = P _ {A} + (k _ {t}) _ {A} \Delta u _ {1} \quad \text { or } \quad (k _ {t}) _ {A} \Delta u _ {1} = P _ {B} - P _ {A} \tag {17.2-9}
+$$
+
+where $P_{B} - P_{A}$ can be interpreted as a load imbalance—that is, as the difference between the applied load $P_{B}$ and the force $P_{A} = (k_{0} + k_{NA})u_{A}$ in the spring when its stretch is $u_{A}$ . The solution process is depicted in Fig. 17.2-3. After computing
+
+![](images/page-524_d110b71c339bf70bebfad1c9fafb9c47ef57014d721f81b12a4116fd13646ae9.jpg)
+
+<details>
+<summary>line</summary>
+| Point | Pressure (P) | Pressure (P_A) | Pressure (P_B) | Pressure (P_B - P_A) | Pressure (P_B) | Pressure (P_A) | Pressure (P_B - P_A) | Pressure (P_B) | Pressure (P_B - P_A) | Pressure (Δu₁) | Pressure (Δu₂) |
+|-------|--------------|----------------|----------------|----------------------|----------------|----------------|----------------------|----------------|----------------------|----------------|----------------|
+| 1     | ~0.8         | ~0.6           | ~0.9           | ~0.7                 | ~0.7           | ~0.6           | ~0.5                 | ~0.8           | ~0.6                 | ~0.8           | ~0.6           |
+| 2     | ~0.9         | ~0.7           | ~0.95          | ~0.8                 | ~0.8           | ~0.7           | ~0.6                 | ~0.9           | ~0.7                 | ~0.9           | ~0.7           |
+| 3     | ~0.95        | ~0.75          | ~0.98          | ~0.85                | ~0.85          | ~0.75          | ~0.65                | ~0.95          | ~0.75                | ~0.95          | ~0.75          |
+| 4     | ~0.98        | ~0.8           | ~0.99          | ~0.9                 | ~0.9           | ~0.8           | ~0.7                 | ~0.98          | ~0.8                 | ~0.98          | ~0.8           |
+| 5     | ~0.99        | ~0.85          | ~0.995         | ~0.95                | ~0.95          | ~0.85          | ~0.75                | ~0.99          | ~0.85                | ~0.99          | ~0.85          |
+| 6     | ~0.995       | ~0.9           | ~0.998         | ~0.98                | ~0.98          | ~0.9           | ~0.8                 | ~0.995         | ~0.9                 | ~0.995         | ~0.9           |
+| 7     | ~0.998       | ~0.95          | ~0.999         | ~0.99                | ~0.99          | ~0.95          | ~0.85                | ~0.998         | ~0.95                | ~0.998         | ~0.95          |
+| 8     | ~0.999       | ~0.98          | ~0.9995        | ~0.995               | ~0.995         | ~0.98          | ~0.9                 | ~0.999         | ~0.98                | ~0.999         | ~0.98          |
+| 9     | ~0.9995      | ~0.99          | ~0.9998        | ~0.998               | ~0.998         | ~0.99          | ~0.95                | ~0.9995        | ~0.99                 | ~0.9995        | ~0.99          |
+| 10    | ~0.9998      | ~0.995         | ~0.9999        | ~0.999               | ~0.999         | ~0.995         | ~0.98                | ~0.9998        | ~0.995               | ~0.9998        | ~0.995         |
+</details>
+
+Figure 17.2-3. N-R solution for $u_{B}$ caused by $P_{B}$ , starting from point A.
+
+![](images/page-524_4921cadfb469f05f44b02533d793c05ad65b80d1090db3148ca8455a29b5b065.jpg)
+
+<details>
+<summary>line</summary>
+
+| Point | u    | P    | Label |
+|-------|------|------|-------|
+| A     | u_A  | P_A  | A     |
+| b     | u_1  | P_B  | a     |
+| b     | u_2  | P_B  | b     |
+| 1     | u_1  | P_A  | 1     |
+| 2     | u_2  | P_B  | 2     |
+</details>
+
+Figure 17.2-4. Modified N–R solution for $u_{B}$ caused by $P_{B}$ , starting from point A.
+
+<!-- source-page: 525 -->
+
+$\Delta u_{1}$ , we update the displacement estimate to $u_{1}=u_{A}+\Delta u_{1}$ . For the next iteration, we obtain a new tangent stiffness $(k_{t})_{1}$ by use of Eq. 17.2-8 with $u=u_{1}$ , and obtain a new load imbalance $P_{B}-P_{1}$ , where $P_{1}$ comes from Eq. 17.2-1 with $u=u_{1}$ . The updated displacement estimate is $u_{2}=u_{1}+\Delta u_{2}$ , where $\Delta u_{2}$ is obtained by solving $(k_{t})_{1}\Delta u_{2}=P_{B}-P_{1}$ .
+
+Remarks. Methods discussed in this section extend directly to multiple d.o.f., where $k = k_{0} + k_{N}$ becomes $[K] = [K_{0} + K_{N}]$ , P becomes $\{R\}$ , and u becomes $\{D\}$ . In one dimension, if the stiffness can be stated as $k = k_{0} + k_{N}$ , the tangent stiffness $k_{t}$ is easily obtained (Eq. 17.2-8). Such a simple expression is not available if there are multiple d.o.f. However, in practice, the physics of the problem usually allows us to calculate the tangent-stiffness matrix $[K_{t}]$ . Neither $[K]$ nor $[K_{t}]$ need be symmetric in a nonlinear problem, but in some situations symmetry prevails or can be achieved by manipulation.
+
+In a multi-d.o.f. context, N-R iteration involves repeated solution of the equations $[K_{t}]_{i}\{\Delta D\}_{i+1} = \{\Delta R\}_{i+1}$ , where tangent-stiffness matrix $[K_{t}]$ and load imbalance $\{\Delta R\}$ are updated after each cycle. The solution process seeks to reduce the load imbalance, and consequently $\{\Delta D\}$ , to zero.
+
+Modified Newton–Raphson. This method differs from the N–R method only in that the tangent stiffness either is not updated or is updated infrequently. Thus, in multi-d.o.f. problems, we avoid the expensive repetitions of forming and reducing the tangent-stiffness matrix $[K_{r}]$ . However, more iterative cycles are needed in order to reach a prescribed accuracy. The process is depicted one-dimensionally in Fig. 17.2-4.
+
+If $[K_{i}]$ is referred to the initial configuration, the modified N–R method becomes almost identical to the direct substitution method of Eq. 17.2-3. The only difference is that modified N–R computes $u_{i+1}$ by adding $\Delta u_{i+1}$ to $u_{i}$ , and Eq. 17.2-3 computes $u_{i+1}$ directly.
+
+Incremental Methods. The foregoing discussion is concerned with locating a single point on the curve of P versus u. If the entire curve is required, one can approximate it as a series of points by applying an iterative process repeatedly: for example, in Fig. 17.2-3, after convergence under load $P_{B}$ , increase the load to $P_{C}$ and again iterate until convergence, then increase the load to $P_{D}$ , and so on. If instead the load is increased in each computational cycle, the solution method may be called incremental rather than iterative.
+
+The simplest incremental method is Euler's method of solving a first-order differential equation. To explain Euler's method we write Eq. 17.2-1 as $P = f(u)$ , define $k_{t} = dP/du$ , and consider load increments $\Delta P$ . Starting from P = 0 at u = 0, we compute successively
+
+$$
+u _ {1} = 0 + \left(k _ {t}\right) _ {0} ^ {- 1} \Delta P _ {1} \quad \text { where } \quad \left(k _ {t}\right) _ {0} = k _ {t} \text { at } u = 0 \tag {17.2-10a}
+$$
+
+$$
+u _ {2} = u _ {1} + \left(k _ {t}\right) _ {1} ^ {- 1} \Delta P _ {2} \quad \text { where } \quad \left(k _ {t}\right) _ {1} = k _ {t} \text {   at   } u = u _ {1} \tag {17.2-10b}
+$$
+
+$$
+u _ {3} = u _ {2} + \left(k _ {t}\right) _ {2} ^ {- 1} \Delta P _ {3} \quad \text { where } \quad \left(k _ {t}\right) _ {2} = k _ {t} \text {   at   } u = u _ {2} \tag {17.2-10c}
+$$
+
+and in general $u_{i+1} = u_i + (k_i)_{i}^{-1} \Delta P_{i+1}$ . The process is depicted in Fig. 17.2-5.
+
+A disadvantage of the foregoing method is apparent in Fig. 17.2-5: the approximate solution drifts further from the exact solution with every step. Progressive
+
+<!-- source-page: 526 -->
+
+![](images/page-526_17509a02ac40305af033a3b7cf84f0e3ad79e26ab3b863b95fabf4a13b18525c.jpg)
+
+<details>
+<summary>line</summary>
+
+| Point | u    | P1   | P2   | P3   |
+|-------|------|------|------|------|
+| 1     | u1   | (k1)0|      |      |
+| 2     | u2   |      |      | (k1)2|
+| 3     | u3   |      |      |      |
+</details>
+
+Figure 17.2-5. Purely incremental solution of the equation $P = f(u)$ .
+
+![](images/page-526_3a9be1f303af0ad18868ff8ea1bf165149c31fdcb46f84ff65c119260d53b9ee.jpg)  
+Figure 17.2-6. Incremental solution of $P = f(u)$ with load corrections $(P_{i} - P_{iR})$ .
+
+drift can be eliminated by introducing the load imbalance as a corrective term. Load imbalance has the same meaning in the present context as in the N-R method. With this corrective term we obtain
+
+$$
+u _ {i + 1} = u _ {i} + (k _ {t}) _ {i} ^ {- 1} \left[ \Delta P _ {i + 1} + (P _ {i} - P _ {i R}) \right] \tag {17.2-11}
+$$
+
+where $P_{i}$ is the externally applied load at step i ( $P_{i} = \Sigma \Delta P_{i}$ summed through step i), and $P_{iR}$ is the resisting load of the spring, $P_{iR} = (k_{0} + k_{Ni})u_{i}$ from Eq. 17.2-1. This method has been called “incremental with one-step N–R correction.” It is depicted in Fig. 17.2-6. Computed points 1, 2, . . . , do not lie on the curve, but they do not progressively drift away from the curve as in Fig. 17.2-5.
+
+Quasi-Newton Methods. Inverse-Broyden. In Fig. 17.2-7, displacements $u_{1}$ and $u_{2}$ are computed by two cycles of modified N–R iteration. Then a secant to the curve is established through points 1 and 2, and a step is taken along the secant. The next step, not shown, would be along a secant through points 2 and 3. With more iterations, leading to convergence, the secant stiffness approaches the exact tangent stiffness at A. Steps in secant directions are not quite as profitable as steps in tangent directions, as in the N–R method, but secant-stiffness steps are much cheaper and are more stable than tangent-stiffness steps.
+
+One expression of the quasi-Newton concept is the inverse-Broyden method.
+
+![](images/page-526_c230d59fc0a8d0c6c70299cf0e0b6a442174622f76db0b37fdfa4edd207986e4.jpg)
+
+<details>
+<summary>line</summary>
+| Point | u    | P    |
+|-------|------|------|
+| a     | u1   | k0   |
+| b     | u2   | k0   |
+| c     | u3   | 3    |
+| 1     | u1   | 1    |
+| 2     | u2   | 2    |
+| 3     | u3   | 3    |
+| A     | uA   | 3    |
+</details>
+
+Figure 17.2-7. Two modified N-R iterations, followed by a secant step along a line through points 1 and 2.
+
+<!-- source-page: 527 -->
+
+The label “inverse” means that the inverse of the stiffness matrix is updated, not the stiffness matrix itself. A complete explanation is beyond the scope of this text. References include $[17.2–17.6]$ . In outline, the method operates as follows. Imagine that a given load $\{R\}_{A}$ has been applied; we now wish to determine the corresponding $\{D\}$ by iteration. We write
+
+$$
+\{\mathbf {D} \} _ {i + 1} = \{\mathbf {D} \} _ {i} + \{\Delta \mathbf {D} \} _ {i + 1} \quad \text { where } \quad \{\Delta \mathbf {D} \} _ {i + 1} = [ \mathbf {K} ] _ {i} ^ {- 1} \{\Delta \mathbf {R} \} _ {i + 1} \tag {17.2-12}
+$$
+
+where $\{\Delta R\}_{i+1}$ is the load imbalance (as in Eq. 17.2-9) and $[K]_{i}$ can be regarded as a secant-stiffness matrix. After several iterations i, both $\{\Delta R\}_{i+1}$ and $\{\Delta D\}_{i+1}$ become small, and $\{D\}_{i+1}$ is a good approximation of $\{D\}$ under loads $\{R\}_{A}$ . The second of Eqs. 17.2-12 is expanded as follows,
+
+$$
+[ \mathbf {K} ] _ {i} ^ {- 1} \{\Delta \mathbf {R} \} _ {i + 1} = [ \mathbf {K} ] _ {I} ^ {- 1} \{\Delta \mathbf {R} \} _ {i + 1} + \sum_ {k = 1} ^ {i} [ \mathbf {p} ] _ {k} ^ {T} [ \mathbf {v} ] _ {k} \{\Delta \mathbf {R} \} _ {i + 1} \tag {17.2-13}
+$$
+
+where $[K]_{i}^{-1}$ is an estimate of $[K]^{-1}$ at the outset of iteration at a given load level. Thus, in each iteration, $[K]_{i}^{-1}$ is improved by the addition of one more rank 1 matrix $[p]^{T}[v]$ . Update matrices $[p]^{T}[v]$ are in general unsymmetric, but with continued updating and eventual convergence, a symmetric $[K]_{i}^{-1}$ may be approached.
+
+The computational efficiency of the procedure, which may make it the method of choice, arises as follows. Neither $[K]_{i}$ nor $[K]_{i}^{-1}$ is ever written out as a square matrix. Matrix $[K]_{I}$ is forward-reduced, to act as $[K]_{I}^{-1}$ , only once. Thereafter, as i increases, only imbalances $\{\Delta R\}_{i+1}$ are treated, by forward-reduction and back-substitution. Also, each product $\left[p\right]_{k}^{T}\left[v\right]_{k}\{\Delta R\}_{i+1}$ requires one vector-times-vector multiplication and one vector-times-scalar multiplication. Thus the procedure is “vectorizable” on vector-processing computers.
+
+In nonstructural problems, the negative of $[K]$ may be called the Jacobian and $\{\Delta R\}$ called the residual. $^{1}$ In some problems, nothing may be known about $[K]_{I}$ . Then a unit matrix might be assumed, $[K]_{I} = [I]$ . However, the better the estimate of $[K]_{I}$ , the faster the convergence. Similarly, starting with a good estimate of the solution $\{D\}$ will reduce the residual and speed convergence. If more than roughly 30 to 60 iterations are used, depending on the machine precision, the method may fail because successive updates $[p]^{T}[v]$ become linearly dependent. Then one can discard the updates and start afresh with a new $[K]_{I}$ , even if it is again $[K]_{I} = [I]$ . A fresh start is also recommended when starting to iterate with a new $\{R\}$ . As a termination criterion, Eq. 17.2-16 below is recommended.
+
+A Fortran version of the algorithm appears in Fig. 17.2-8.
+
+If $[K_{i}]$ is known to be symmetric and positive definite, one may apply the “BFGS method,” which is a powerful quasi-Newton method related to the inverse-Broyden-method [17.4, 17.5].
+
+Termination. The efficiency of a nonlinear solution method can be measured by its “order of termination.” Let $e_{i}$ represent a measure of the error after the ith iterative cycle. If $e_{i}$ is sufficiently small, it is often possible to bound $e_{i+1}$ . Practical possibilities include the following, illustrated here for a single d.o.f.
+
+$^{1}$ The Jacobian is the negative of the tangent stiffness because of our choice of sign in writing the residual as external force minus internal force and our desire that tangent stiffness reduce to conventional stiffness in the linear case. Other conventions are possible.
+
+<!-- source-page: 528 -->
+
+```txt
+SUBROUTINE INVBDN (K,A,Q,U,DU,NEQ,MBAND,V,DV,NBRY,ROI,IER)
+C Subroutine to carry out ONE inverse-Broyden iteration.
+C K = current iteration number, A = initial estimate of Jacobian (e.g.
+C stiffness) matrix, Q = current residual, U = current solution, DU =
+C increment to update U, NEQ = number of unknowns, MBAND = semiband-
+C width of A, V and DV = saved Broyden vectors, NBRY = number of Broy-
+C den vectors allowed = number of iterations allowed, ROI = saved con-
+C stant, IER = 132 if error detected in current iteration.
+IMPLICIT DOUBLE PRECISION (A-H,O-Z)
+C--- In DIMENSION line, Q(1), U(1), etc. act as Q(NEQ), U(NEQ), etc.
+DIMENSION A(NEQ,1),Q(1),U(1),DU(1),V(NBRY,1),DV(NBRY,1),ROI(1)
+DATA EPS /1.D-12/
+IER = 0
+C--- Calling program has computed residual Q and REDUCED form of A.
+C--- Call SOLVER to do only reduction and back-substitution of Q.
+CALL SOLVER (A,Q,NEQ,MBAND,2)
+IF (K .GT. 1) GO TO 150
+DO 100 I=1,NEQ
+100 DU(I) = Q(I)
+GO TO 500
+C--- Update U using the K-1 Broyden vectors already established.
+150 KMI = K - 1
+DO 400 I=1,KMI
+CK = 0.D0
+DO 200 J=1,NEQ
+200 CK = CK + DV(I,J)*Q(J)
+CONS = CK*ROI(I)
+DO 300 J=1,NEQ
+300 Q(J) = Q(J) + CONS*(DV(I,J)-V(I,J))
+400 CONTINUE
+C--- Establish the Kth Broyden vectors. Use them to further update U.
+500 DENO = 0.D0
+DO 600 J=1,NEQ
+V(K,J) = Q(J) + DU(J)
+DV(K,J) = DU(J)
+600 DENO = DENO + DU(J)*V(K,J)
+C--- If DENO < EPS, we may have a singular Jacobian, or may have con-
+C--- verged but not recognized it due to faulty termination criterion.
+IF (DABS(DENO) .GE. EPS) GO TO 650
+IER = 132
+RETURN
+650 CK = 0.D0
+ROI(K) = 1.D0/DENO
+DO 700 J=1,NEQ
+700 CK = CK + DU(J)*Q(J)
+CONS = ROI(K)*CK - 1.D0
+DO 800 J=1,NEQ
+DU(J) = Q(J)*CONS
+800 U(J) = U(J) + DU(J)
+RETURN
+END 
+```  
+Figure 17.2-8. Fortran coding of the inverse-Broyden algorithm. Subroutine SOLVER appears in Appendix B as Fig. B.2-3 (note that SOLVER is for a band-symmetric matrix). The best equation solver available should be used and can easily be substituted.
+
+$$
+\text { linear: } \quad | e _ {i + 1} | \leq C _ {1} | e _ {i} | \tag {17.2-14a}
+$$
+
+$$
+\text { superlinear: } \quad | e _ {i + 1} | \leq C _ {2} | e _ {i} | | e _ {i - 1} | = C _ {3} | e _ {i} | ^ {1. 6} \tag {17.2-14b}
+$$
+
+$$
+\text { quadratic: } \quad | e _ {i + 1} | \leq C _ {4} e _ {i} ^ {2} \tag {17.2-14c}
+$$
+
+where $C_{1}$ through $C_{4}$ are constants. Of the methods we have discussed, only N–R exhibits quadratic termination. The inverse-Broyden method exhibits a generalized superlinear termination. Note that both N–R and inverse-Broyden methods will fail if $P = f(u)$ exhibits zero slope at the intended solution $u_{A}$ (i.e., if $f'(u_{A}) = 0$ , which is a “limit point”).
+
+A termination criterion for an iterative process can be of various forms. One that is usually good is as follows. We define [17.4]
+
+<!-- source-page: 529 -->
+
+$$
+\mathrm{CNORM} = \left(\sum \Delta D _ {j} ^ {2}\right) ^ {1 / 2} \left(\sum D _ {j} ^ {2}\right) ^ {- 1 / 2} \tag {17.2-15a}
+$$
+
+$$
+\mathrm{RNORM} = \left(\sum \Delta R _ {j} ^ {2}\right) ^ {1 / 2} \left(\sum R _ {j} ^ {2}\right) ^ {- 1 / 2} \tag {17.2-15b}
+$$
+
+where, in structural problems, $\Delta D_{j}$ , $D_{j}$ , $\Delta R_{j}$ , and $R_{j}$ mean respectively displacement increment, displacement, load increment, and load. Summations span all terms (NEQ in Fig. 17.2-8). Thus CNORM and RNORM are ratios of Euclidean norms. In a nonstructural problem, $(\Sigma R_{j}^{2})^{1/2}$ might become the product of a modulus and the square of a characteristic length, or the product of viscosity, the reciprocal of a characteristic time, and the square of a characteristic length. We terminate when
+
+$$
+\max (\text { CNORM }, \text { RNORM }) \leq t o l \tag {17.2-16}
+$$
+
+where tol is chosen to balance accuracy requirements against machine precision. Possible choices are $tol = 10^{-5}$ in 64-bit arithmetic and $tol = 10^{-3}$ in 32-bit arithmetic. For the inverse-Broyden method at least, it is recommended that both CNORM and RNORM be less than tol, as Eq. 17.2-16 requires. In the rare case that $\{D\} = \{0\}$ is a possible solution or iterate, the denominator of CNORM becomes very small. Then one can safely set the denominator to unity.
+
+Concluding Remarks. Hardening structures are usually more difficult to analyze than softening structures. Iterative processes are more likely to converge slowly or fail to converge (Fig. 17.2-9).
+
+There is no need to maintain strict separation between solution methods. For example, we could adopt a modified N-R strategy, but occasionally update the tangent-stiffness matrix. Often such an update is most effective when done immediately after one iteration at a new load level using the old stiffness. Apparently this approach would work well in Fig. 17.2-9b. Underrelaxation would also help in this example.
+
+Two-dimensional sketches in the present section, for P versus u, are representative of multi-d.o.f. problems if the structure carries a single load $R_{i} = P$ and
+
+![](images/page-529_e4b3d1b0e19a7a8c9050d638ba7a679d1cc01c2fbcd4f7c97527e289eeb428a9.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+(a)
+(b_t)a
+B
+(b_t)a
+P_B
+P_A
+0
+0
+u
+</details>
+
+{a}
+
+![](images/page-529_23ff9d80cada8d55b09192f920a8ce440c601552d2c3fdfecf1906e50cf38c12.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+k₀
+P_B
+B
+A
+k₀
+P_A
+0
+0
+u
+</details>
+
+(b)   
+Figure 17.2-9. Hardening P versus u curves, attacked by (a) N–R, and (b) modified N–R methods.
+
+<!-- source-page: 530 -->
+
+![](images/page-530_765989edc6af69de682c7cab4fe24299da5499dd144bb7c3fc177cb319781a07.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+u
+u₁
+P₁
+</details>
+
+{a}
+
+![](images/page-530_78e86874dd38e780507a8509935bbe75b1fe6a9b6aac3a79740e80b828d1482a.jpg)
+
+<details>
+<summary>line</summary>
+
+| Point | u     | P     |
+|-------|-------|-------|
+| A     | 0     | 0     |
+| B     | u_B   | 0     |
+</details>
+
+(b)   
+Figure 17.2-10. (a) Shallow arch under load P. (b) Load versus displacement plot, showing limit point at A.
+
+$D_{i} = u$ is its displacement. In effect, the structure then acts as a single nonlinear spring in resisting $R_{i}$ . If there is more than one d.o.f., a two-dimensional sketch provides an incomplete and possibly misleading representation.
+
+One can provide displacement increments rather than load increments—that is, use “displacement control” rather than “load control.” However, in a typical problem one does not know in advance how the $D_{i}$ in $\{D\}$ will be related, so the appropriate increments $\Delta D_{i}$ are unknown.
+
+In Fig. 17.2-10, displacement control yields the entire curve of P versus u. Under load control, the physical structure will experience a sudden “snap” from A to B when the limit point at A is reached. Computational methods of traversing limit points have been devised [17.7, 17.8].
+
+Another way to traverse limit points is called viscous relaxation. In Fig. 17.2-10, imagine that displacement $u_{1}$ corresponding to load $P_{1}$ is required, but that the entire curve for $0 < u < u_{1}$ is not required. One can imagine that the structure is immersed in a viscous fluid. When load $P_{1}$ is applied, the structure moves slowly and without snapping. When motion ceases, the static displacement $u_{1}$ is achieved. Computationally, we solve a dynamic problem using step-by-step integration in time, including the nonlinear stiffness matrix and a damping matrix but omitting the mass matrix [17.9,17.10]. The method is most useful when there are strong geometric nonlinearities.
+
+# 17.3 ONE-DIMENSIONAL
+
+# ELASTIC-PLASTIC ANALYSIS
+
+Plastic Action. A material is called nonlinear if stresses $\{\sigma\}$ and strains $\{\epsilon\}$ are related by a strain-dependent matrix rather than a matrix of constants. Thus the computational difficulty is that equilibrium equations must be written using material properties that depend on strains, but strains are not known in advance. Plastic flow is often a cause of material nonlinearity. In the present section we use the case of uniaxial stress to introduce the formulation and solution of elastic-plastic problems.
+
+Imagine that yielding has already occurred; then a strain increment $d\epsilon$ takes place (Fig. 17.3-1a). This strain increment can be regarded as composed of an elastic contribution $d\epsilon^{e}$ and a plastic contribution $d\epsilon^{p}$ , so that $d\epsilon = d\epsilon^{e} + d\epsilon^{p}$ . The corresponding stress increment $d\sigma$ can be written in various ways,
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_054.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_054.md
new file mode 100644
index 00000000..48eb78e9
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_054.md
@@ -0,0 +1,415 @@
+<!-- source-page: 531 -->
+
+![](images/page-531_207cf3b0fb8e1e7da7cb5a24f05842252d1f4a2d1869c453e165da66c29ceaaa.jpg)
+
+<details>
+<summary>text_image</summary>
+
+σ
+σγ
+E
+D
+εγ
+A
+B
+E
+C
+ε
+εp
+εe = σB/E
+dεp
+dεe
+dε
+</details>
+
+(a)
+
+![](images/page-531_d3e021927c47e7f3c517284aadf286d858ed760e520eef7944decf93b1250bcc.jpg)
+
+<details>
+<summary>text_image</summary>
+
+σ
+σB
+σY
+B
+E
+E
+2σY
+|σB - 2σY|
+Ei
+Ei
+σB
+Kinematic
+Isotropic
+-σ
+</details>
+
+{b}   
+Figure 17.3-1. (a) Stress-strain plot in uniaxial stress, idealized as two straight lines, where $\sigma_{Y}$ is the stress at first onset of yielding. (b) Kinematic and isotropic hardening rules.
+
+$$
+d \sigma = E (d \epsilon - d \epsilon^ {p}) \quad d \sigma = E _ {t} d \epsilon \quad \text { and } \quad d \sigma = H d \epsilon^ {p} \tag {17.3-1}
+$$
+
+where H is called the strain-hardening parameter. Substitution of the first and third of Eqs. 17.3-1 into the second yields
+
+$$
+H = \frac {E _ {t}}{1 - (E _ {t} / E)} \quad \text { or } \quad E _ {t} = E \left(1 - \frac {E}{E + H}\right) \tag {17.3-2}
+$$
+
+where $E_{t}$ is the tangent modulus. When written in this form, the expression for $E_{t}$ is similar to a more general expression used for multiaxial states of stress. If E is finite and $E_{t} = 0$ , then H = 0, and the material is called “elastic–perfectly plastic.”
+
+A summary of elastic–plastic action in uniaxial stress is as follows. The yield criterion states that yielding begins when $|\sigma|$ reaches $\sigma_{Y}$ , where in practice $\sigma_{Y}$ is usually taken as the tensile yield strength. Subsequent plastic deformation may alter the stress needed to produce renewed or continued yielding; this stress exceeds the initial yield strength $\sigma_{Y}$ if $E_{t} > 0$ . A flow rule can be written in multidimensional problems. It leads to a relation between stress increments $\{d\sigma\}$ and strain increments $\{d\epsilon\}$ . In uniaxial stress this relation is simply $d\sigma = E_{t} d\epsilon$ , which describes the increment of stress produced by an increment of strain. Note, however, that if the material has yet to yield or is unloading, then $d\sigma = E d\epsilon$ (e.g., in Fig. 17.3-1a, complete unloading from point B leads to point C and a permanent strain $e^{p}$ ). Finally, there is a hardening rule, which describes how the yield criterion is changed by the history of plastic flow. For example, imagine that unloading occurs from point B in Fig. 17.3-1a. With reloading from point C, response will be elastic until $\sigma > \sigma_{B}$ , when renewed yielding occurs. If we assume that yielding reappears when $|\sigma| > \sigma_{B}$ , whether $\sigma$ is tensile or compressive, we have adopted the “isotropic hardening” rule (Fig. 17.3-1b). However, for common metals, such a rule is in conflict with the observed behavior that yielding reappears at a stress of approximate magnitude $\sigma_{B} - 2\sigma_{Y}$ when loading is reversed. Accordingly, a better match to observed behavior is provided by the “kinematic hardening” rule, which (for uniaxial stress) says that a total elastic range of $2\sigma_{Y}$ is preserved.
+
+<!-- source-page: 532 -->
+
+The discussion in the foregoing paragraph does not require that postelastic response be idealized as a straight line. In other words, $E_{t}$ need not be constant.
+
+As a simple application of one-dimensional plasticity, imagine that a tapered bar is to be loaded by an axial force $P$ (Fig. 17.3-2). Material properties are those depicted in Fig. 17.3-1. The bar is modeled by two-d.o.f. bar elements, each of constant cross section. For elastic conditions, the element stiffness matrix is given by Eq. 2.4-5, where $E = d\sigma / d\epsilon$ when $|\sigma| < \sigma_{Y}$ . Upon yielding, the stress-strain relation becomes $E_{t} = d\sigma / d\epsilon$ . Accordingly, letting $E_{\mathrm{ep}}$ represent the "elastic-plastic" stiffness, we write the element tangent-stiffness matrix as
+
+$$
+[ \mathbf {k} _ {t} ] = \frac {A E _ {\mathrm{ep}}}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right] \tag {17.3-3}
+$$
+
+where $E_{\mathrm{ep}} = E$ if the yield criterion is not exceeded or if unloading is taking place, and $E_{\mathrm{ep}} = E_t$ if plastic flow is involved.
+
+In numerical solutions, material may make the transition from elastic to plastic within an iterative cycle of the solution process. For example, imagine that $d\epsilon$ spans $\epsilon_{D}$ to $\epsilon_{A}$ in Fig. 17.3-1a. The problem of “rounding the corner” can be addressed by combining $E$ and $E_{t}$ according to the fraction $m$ of the total step $d\epsilon$ that is elastic. Thus let
+
+$$
+E _ {\mathrm{ep}} = m E + (1 - m) E _ {t} \quad \text { where } \quad m = \frac {\epsilon_ {Y} - \epsilon_ {D}}{\epsilon_ {A} - \epsilon_ {D}} \tag {17.3-4}
+$$
+
+Alternatively, by substituting stresses for strains and using the fictitious stress $\sigma^{*} = E\epsilon_{A}$ , we can write $m$ in terms of stresses, as $m = (\sigma_{Y} - \sigma_{D}) / (\sigma^{*} - \sigma_{D})$ . Refinements of this scheme are possible [17.11].
+
+The present discussion excludes thermal strains and creep strains. In general, these effects may appear in combination with elastic–plastic action. Then the total strain increment $d\epsilon^{tot}$ is a combination of elastic, plastic, thermal, and creep strain components. The strain increment $d\epsilon = d\epsilon^{e} + d\epsilon^{p}$ used in elastic–plastic analysis excludes thermal and creep strains. Thus
+
+$$
+d \epsilon^ {e} + d \epsilon^ {p} = d \epsilon^ {\mathrm{tot}} - d \epsilon^ {T} - d \epsilon^ {C} \tag {17.3-5}
+$$
+
+![](images/page-532_19232cc33ea0dd043b85212a267d893e73075e27d28ce3204ad6d5e905476f31.jpg)
+
+<details>
+<summary>text_image</summary>
+
+D
+P
+L_T
+n
+P
+L
+</details>
+
+(a)
+
+![](images/page-532_e75f7be9e686dd9395e2bcad5e35ef39763af9252dbf384565a99f49e3fd96a6.jpg)
+
+<details>
+<summary>line</summary>
+| D       | P (Exact) | P (ΔP₁) | P (ΔP₂) | P (ΔP₃) |
+|---------|-----------|---------|---------|---------|
+| 0       | 0         | 0       | 0       | 0       |
+| D₁      | ~0.5      | ~0.5    | ~0.5    | ~0.5    |
+| D₂      | ~1.0      | ~1.0    | ~1.0    | ~1.0    |
+| D₃      | ~1.5      | ~1.5    | ~1.5    | ~1.5    |
+</details>
+
+(b)   
+Figure 17.3-2. (a) A tapered bar and a finite element model using uniform elements, of which element n is typical. (b) Progress of a tangent-stiffness solution if step 3 of the algorithm is omitted. D = displacement of load P.
+
+<!-- source-page: 533 -->
+
+Tangent-Stiffness Method. Consider the tapered bar depicted in Fig. 17.3-2a. It is desired to trace the quasistatic load versus displacement curve and determine element stresses by means of a finite element model and load increments $\Delta P$ . Increments are small but not infinitesimal, so that $d\epsilon$ becomes $\Delta\epsilon$ , and the numerical solution is not exact. A numerical representation of the stress–strain relation must be stored, so that $\sigma$ , E, and $E_{t}$ can be obtained for any $\epsilon$ . The algorithm outlined below requires that we also store, and update after each computational cycle, the nodal displacements $\{D\}$ , element strains $\epsilon$ , and element stresses $\sigma$ . With two-d.o.f. bar elements (Eq. 17.3-3), $\sigma$ and $\epsilon$ are constant over each element length L.
+
+1. For the first computational cycle $(i = 1)$ , assume $E_{\mathfrak{ep}} = E$ for all elements. Apply the first load increment, $\{\Delta \mathbf{R}\}_{1}$ .   
+2. Using the current strains, determine the current $E_{\mathrm{ep}}$ in each element. Use Eq. 17.3-3 to obtain $[\mathbf{k}_t]_n$ for each element $n$ . Obtain the current structure tangent stiffness $[\mathbf{K}_t]_{i-1} = \hat{\Sigma} [\mathbf{k}_t]_n$ . Solve $[\mathbf{K}_t]_{i-1}\{\Delta \mathbf{D}\}_i = \{\Delta \mathbf{R}\}_i$ for $\{\Delta \mathbf{D}\}_i$ . (For the bar of Fig. 17.3-2a, $\Delta P$ at the right end is the only nonzero entry in $\{\Delta \mathbf{R}\}_i$ .) From $\{\Delta \mathbf{D}\}_i$ , obtain current strain increments $\Delta \epsilon_i$ for each element.   
+3. Optional. If any elements make the elastic-to-plastic transition, use Eq. 17.3-4 to revise $E_{\mathrm{ep}}$ for each such element, and go back to step 2. Without changing the applied load $\{\Delta \mathbf{R}\}_i$ , repeat steps 2 and 3 until convergence, which may be defined as $\Delta \epsilon$ being less than a prescribed fraction of the accumulated total $\epsilon$ in every element. These operations represent secant-stiffness iterations (see Fig. 17.2-2a) within one of the load steps of the tangent-stiffness procedure.   
+4. Update: $\{\mathbf{D}\}_{i} = \{\mathbf{D}\}_{i-1} + \{\Delta \mathbf{D}\}_{i}$ , and for each element, $\epsilon_{i} = \epsilon_{i-1} + \Delta \epsilon_{i}$ and $\sigma_{i} = \sigma_{i-1} + \Delta \sigma_{i}$ , where $\Delta \sigma_{i} = (E_{\mathrm{ep}})_{i} \Delta \epsilon_{i}$ . For the first cycle ( $i = 1$ ), initial values (subscript $i - 1$ ) of displacement, strain, and stress are typically all zero if one starts from the unloaded configuration, but are nonzero if one starts from a state in which plastic action impends.   
+5. Apply the next load increment and return to step 2.   
+6. Stop when $\Sigma \{\Delta \mathbf{R}\}_{i}$ reaches the total applied load.
+
+Three cycles of the foregoing algorithm are depicted in Fig. 17.3-2b. Each cycle produces a line segment whose slope corresponds to the current stiffness. Drift from the exact path can be reduced by using smaller load increments, by exercising step 3 previously discussed, and by using “corrective loads,” which are discussed in Section 17.5. Step 3 can be avoided by using load increments $\{\Delta R\}_{i}$ that bring a single element to the verge of yielding as each load increment is added. This is easily accomplished by scaling the incremental tangent-stiffness solutions.
+
+The foregoing incremental procedure is essentially a Newton–Raphson method; that is, a new tangent-stiffness matrix is used in each computational cycle.
+
+Initial-Stiffness Method. Again we seek displacements and stresses in a structure in which plastic action occurs. One can apply the iterative method described by Eq. 17.2-3 and Fig. 17.2-2b. Thus the original elastic stiffness matrix is used at all times. The effects of plastic action are regarded as initial stresses that produce fictitious loads, which are combined with the load actually applied (accordingly, this procedure is often called the initial-stress method). This procedure avoids
+
+<!-- source-page: 534 -->
+
+the expense of repeatedly forming and factoring a tangent-stiffness matrix, but may converge slowly if plastic strains are large or widespread [17.12].
+
+In Fig. 17.3-3a, imagine that we seek the strain $\epsilon_{B}$ associated with stress $\sigma_{B}$ . We can obtain $\epsilon_{B}$ using only the elastic modulus $E$ by writing $\epsilon_{B} = \sigma_{C} / E$ . Here $\sigma_{C}$ is the fictitious stress $\sigma_{C} = \sigma_{B} + E \Delta \epsilon^{p}$ .
+
+In computation, $e^{p}$ can be obtained by accumulating the plastic strain increments $\Delta e^{p}$ produced in the iterative cycles. From Eqs. 17.3-1,
+
+$$
+\Delta \epsilon^ {p} = \frac {1}{H} \Delta \sigma = \frac {1}{H} E _ {t} \Delta \epsilon = \frac {E}{E + H} \Delta \epsilon = \left(1 - \frac {E _ {t}}{E}\right) \Delta \epsilon \tag {17.3-6}
+$$
+
+In computation, the progression to $\epsilon_{B}$ is made in a series of steps, as shown in Fig. 17.3-3b, using “supplementary loads” defined in Eq. 17.3-7. It is not necessary that the stress–strain relation be piecewise linear.
+
+The calculation procedure for a structure such as that in Fig. 17.3-2a is as follows.
+
+1. Compute the elastic stiffness matrix [K] (which is identically the initial tangent-stiffness matrix). Solve $[K]\{D\} = \{R\}$ for $\{D\}$ , where $\{R\}$ is proportional to the actual load but of arbitrary level. From this solution, scale $\{R\}$ so that it becomes $\{R_{Y}\}$ , which causes yielding to impend. Scale $\{D\}$ similarly and call the result $\{D\}_{old}$ . Subsequent load increments may be chosen as $\{\Delta R\} = 0.05 \{R_{Y}\}$ or as $\{\Delta R\} = (E_{T}/E)\{R_{Y}\}$ , whichever is greater [17.13]. Initialize supplementary loads $\{\Delta R_{s}\}$ to zero.   
+2. Solve the equations $[K]\{\Delta D\} = \{\Delta R\} + \{\Delta R_{s}\}$ for $\{\Delta D\}$ .   
+3. Update displacements: $\{\mathbf{D}\}_{\text{new}} = \{\mathbf{D}\}_{\text{old}} + \{\Delta \mathbf{D}\}$ .   
+4. In each element, calculate the strain increment $\Delta \epsilon$ associated with $\{\Delta \mathbf{D}\}$ . Update element stress by adding $\Delta \sigma$ to the existing stress $\sigma$ , using $\Delta \sigma = E \Delta \epsilon$ if $\sigma < \sigma_Y$ and $\Delta \sigma = E_t \Delta \epsilon$ if $\sigma > \sigma_Y$ . For elements that make the elastic-to-plastic transition by the addition of $\Delta \sigma$ , evaluate $m$ by Eq. 17.3-4 and recompute $\Delta \sigma$ as $\Delta \sigma = Em \Delta \epsilon$ .   
+5. For all elements that display plastic strains ( $|\sigma| > \sigma_{Y}$ in Fig. 17.3-3), calculate plastic strain increments according to Eq. 17.3-6. In this calculation, use $(1 - m)\Delta\epsilon$ rather than $\Delta\epsilon$ for elements that make the elastic-to-plastic
+
+![](images/page-534_67d31e9e8d47ae95e5cb76404085a765219da3a4c700e4454178be1d9c67a49b.jpg)
+
+<details>
+<summary>line</summary>
+| Point | ε     | σ     | Δε     |
+|-------|-------|-------|--------|
+| E     | 0     | 0     | Δε^e   |
+| B     | ε_B   | 0     | Δε^p   |
+| C     | ε_B   | σ_C  | Δε^p   |
+| E_t   | ε_B   | σ_B  | Δε^p   |
+</details>
+
+(a)
+
+![](images/page-534_c5d7b7b8a396fe6ccc70aeee03a3991dd62daec890c20c3814f3dc7c6ffa1b46.jpg)
+
+<details>
+<summary>line</summary>
+
+| Point | ε    | σ    | Label |
+|-------|------|------|-------|
+| 1     | ε₁   | σY   | E     |
+| 2     | ε₂   | σY   | 1     |
+| B     | ε_B  | σY   | B     |
+</details>
+
+(b)   
+Figure 17.3-3. (a) $E \Delta e^{p}$ is regarded as an initial stress. (b) Iterative approach to the solution point B.
+
+<!-- source-page: 535 -->
+
+transition (see Eq. 17.3-4). Generate the supplementary loads by summing element contributions:
+
+$$
+\left\{\Delta \mathbf {R} _ {s} \right\} = \sum \left\{\Delta \mathbf {r} _ {s} \right\} \quad \text { where } \quad \left\{\Delta \mathbf {r} _ {s} \right\} = \int_ {0} ^ {L} \left\lfloor \mathbf {B} \right] ^ {T} E \Delta e ^ {p} A d x \tag {17.3-7}
+$$
+
+Solve the equations $[K]\{\Delta D\} = \{\Delta R_{s}\}$ for $\{\Delta D\}$ . Return to step 3.
+
+6. Repeat steps 3 through 5 until convergence. Then apply another load increment $\{\Delta R\}$ and return to step 2.
+
+7. Stop when $\{R_{Y}\} + \Sigma\{\Delta R\}$ reaches the total applied load.
+
+# 17.4 SMALL-STRAIN PLASTICITY RELATIONS
+
+Multiaxial states of stress can be analyzed if the theory in Section 17.3 is generalized. The following is a summary. In our discussion we use the engineering definition of shear strain (e.g., $\gamma_{xy} = u_{,y} + v_{,x}$ ), not the tensor definition (e.g., $\epsilon_{xy} = (u_{,y} + v_{,x})/2$ ).
+
+General. Plasticity theory has three parts: a yield criterion, a flow rule, and a hardening rule. The general theory and its various special forms are contrived to fit experimental data.
+
+Yield Criterion. We define a yield function F, which is a function of stresses $\{\sigma\}$ and quantities $\{\alpha\}$ and $W_{p}$ associated with the hardening rule. Yielding occurs when
+
+$$
+F (\sigma , \alpha , W _ {p}) = 0 \tag {17.4-1}
+$$
+
+where $\{\alpha\}$ and $W_{p}$ are defined by Eq. 17.4-3. Specifically, if we evaluate F using given values of $\{\sigma\}$ , $\{\alpha\}$ , and $W_{p}$ , then the possible results are F < 0 and F = 0. Respectively, these results mean that the material is in the elastic range or is yielding. The result F > 0 is not physically possible, as it indicates a state of stress that does not satisfy the constitutive law (e.g., $\sigma_{C}$ in Fig. 17.3-3 is not physically possible). Similarly, the respective results dF < 0 and dF = 0 imply elastic unloading and continued yielding. The result dF > 0 is not possible in the plastic regime.
+
+Flow Rule. We define a plastic potential Q, which has units of stress and is a function of the stresses, $Q = Q(\sigma, \alpha, W_{p})$ . With $d\lambda$ a scalar that may be called a “plastic multiplier,” plastic strain increments are given by
+
+$$
+\left\{d \epsilon^ {p} \right\} = \left\{\frac {\partial Q}{\partial \sigma} \right\} d \lambda \tag {17.4-2}
+$$
+
+Thus $de_{x}^{e} = (\partial Q/\partial \sigma_{x}) \, d\lambda$ , and so on. The flow rule is called “associated” if Q = F and “nonassociated” otherwise. Associated flow rules are commonly used for ductile metals, but nonassociated rules are better suited to soil and granular materials.
+
+<!-- source-page: 536 -->
+
+Hardening Rule. In Eq. 17.4-1, $\{\alpha\}$ locates the center of the yield surface in stress space. Initially, before any plastic strains appear, $\{\alpha\} = \{0\}$ . In “kinematic hardening,” the center moves in the direction of plastic straining, so that $\{\alpha\}$ becomes nonzero. Parameter $W_{p}$ describes how the yield surface grows. In “isotropic hardening,” $W_{p}$ is nonzero but $\{\alpha\}$ is zero. Quantities $\{\alpha\}$ and $W_{p}$ are defined as
+
+$$
+\{\boldsymbol {\alpha} \} = \int C \left\{d \boldsymbol {\epsilon} ^ {p} \right\} \quad \text { and } \quad W _ {p} = \int \left\{\boldsymbol {\sigma} \right\} ^ {T} \left\{d \boldsymbol {\epsilon} ^ {p} \right\} \tag {17.4-3}
+$$
+
+where C can be assumed to be a material constant [17.13,17.14]. For purely kinematic hardening, C = H (Fig. 17.4-1). $W_{p}$ can be identified as plastic work per unit volume. (Use of $W_{p}$ in F implies a “work-hardening” model. Alternatively, $W_{p}$ can be replaced by an effective plastic strain $\epsilon_{ef}^{p}$ , which implies a “strain-hardening” model. Either model can be used to represent isotropic hardening.)
+
+An incremental stress–strain relation, analogous to the relation $\{\sigma\} = [E]\{\epsilon\}$ of elasticity but valid into the elastic–plastic regime, can be derived as follows. First, we differentiate Eq. 17.4-1:
+
+$$
+d F = 0 = \left\{\frac {\partial F}{\partial \sigma} \right\} ^ {T} \{d \sigma \} + \left\{\frac {\partial F}{\partial \alpha} \right\} ^ {T} \{d \alpha \} + \frac {\partial F}{\partial W _ {p}} d W _ {p} \tag {17.4-4}
+$$
+
+From Eqs. 17.4-3 we obtain $\{d\alpha\} = C\{d\epsilon^{p}\}$ and $dW_{p} = \{\sigma\}^{T}\{d\epsilon^{p}\}$ . In addition, in multidimensional analogy to Eq. 17.3-1, we have
+
+$$
+\{d \sigma \} = [ \mathbf {E} ] \{d \epsilon^ {e} \} = [ \mathbf {E} ] (\{d \epsilon \} - \{d \epsilon^ {p} \}) \tag {17.4-5}
+$$
+
+where $\{d\epsilon\}$ is assumed to contain no creep or thermal strains (see Eq. 17.3-5). Making these substitutions into Eq. 17.4-4, using Eq. 17.4-2 to eliminate $\{d\epsilon^{p}\}$ , and solving for the plastic multiplier $d\lambda$ , we obtain
+
+$$
+d \lambda = \{\mathbf {C} _ {\lambda} \} ^ {T} \{d \epsilon \} \tag {17.4-6}
+$$
+
+![](images/page-536_999c2d3d78ccfcd0996d7984de47f4220c55c549589b732e65779db0ff65c739.jpg)
+
+<details>
+<summary>text_image</summary>
+
+σₓ
+σᵧ
+H
+αₓ = Hεₓᵖ
+H
+σᵧ
+σᵧ
+εₓᵖ
+Kinematic
+H
+Isotropic
+H
+</details>
+
+Figure 17.4-1. Stress versus plastic strain in uniaxial stress.
+
+<!-- source-page: 537 -->
+
+where
+
+$$
+\left\{\mathbf {C} _ {\lambda} \right\} ^ {T} = \frac {\left\{\frac {\partial F}{\partial \boldsymbol {\sigma}} \right\} ^ {T} [ \mathbf {E} ]}{\left\{\frac {\partial F}{\partial \boldsymbol {\sigma}} \right\} ^ {T} [ \mathbf {E} ] \left\{\frac {\partial Q}{\partial \boldsymbol {\sigma}} \right\} - C \left\{\frac {\partial F}{\partial \boldsymbol {\alpha}} \right\} ^ {T} \left\{\frac {\partial Q}{\partial \boldsymbol {\sigma}} \right\} - \frac {\partial F}{\partial W _ {p}} \left\{\boldsymbol {\sigma} \right\} ^ {T} \left\{\frac {\partial Q}{\partial \boldsymbol {\sigma}} \right\}} \tag {17.4-7}
+$$
+
+Finally, substituting Eq. 17.4-2 into Eq. 17.4-5, we obtain
+
+$$
+\{d \sigma \} = [ \mathbf {E} ] \left(\{d \epsilon \} - \left\{\frac {\partial Q}{\partial \sigma} \right\} d \lambda\right) \quad \text { or } \quad \{d \sigma \} = [ \mathbf {E} _ {\mathrm{ep}} ] \{d \epsilon \} \tag {17.4-8}
+$$
+
+where
+
+$$
+[ \mathbf {E} _ {\mathrm{ep}} ] = [ \mathbf {E} ] - [ \mathbf {E} ] \left\{\frac {\partial Q}{\partial \sigma} \right\} \{\mathbf {C} _ {\lambda} \} ^ {T} \tag {17.4-9}
+$$
+
+Equation 17.4-9 can be regarded as a generalized form of tangent modulus $E_{t}$ (see Eq. 17.3-2). Matrix $[E_{ep}]$ is symmetric if F = Q. It is valid even if the material is elastic—perfectly plastic. It can be used to generate a tangent-stiffness matrix $[k_{t}]$ , which expresses the relation between increments of nodal displacement and the resulting increments of nodal load,
+
+$$
+[ \mathbf {k} _ {t} ] = \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} _ {\mathrm{ep}} ] [ \mathbf {B} ] d V \tag {17.4-10}
+$$
+
+where $[E_{ep}]$ is given by Eq. 17.4-9 if F = 0 and dF = 0, but is replaced by elastic coefficients [E] if F < 0 or if dF < 0.
+
+The von Mises Criterion, Kinematic Hardening. Equation 17.4-9 does not presuppose particular forms of F and Q. Commonly used forms are those of the von Mises yield criterion and its associated flow rule. These forms are popular for analysis of isotropic ductile metals.
+
+To begin, we must introduce deviatoric stresses {s}, which are associated with distortion of shape but produce no volume change. By definition,
+
+$$
+\{\mathbf {s} \} = \{\boldsymbol {\sigma} \} - \sigma_ {m} \left[ \begin{array}{l l l l l l} 1 & 1 & 1 & 0 & 0 & 0 \end{array} \right] ^ {T} \quad \text { where } \quad \sigma_ {m} = \frac {1}{3} \left(\sigma_ {x} + \sigma_ {y} + \sigma_ {z}\right) \tag {17.4-11}
+$$
+
+Stress $\sigma_{m}$ is the mean or average normal stress. Thus $s_x = \sigma_x - \sigma_{m}, \ldots, s_{zx} = \tau_{zx}$ . For convenience, we define portions $\{s_{\sigma}\}$ and $\{s_{\tau}\}$ of $\{s\}$ as follows:
+
+$$
+\{\mathbf {s} \} = \left\{ \begin{array}{l} \mathbf {s} _ {\sigma} \\ \mathbf {s} _ {\tau} \end{array} \right\} \quad \text { where } \quad \{\mathbf {s} _ {\sigma} \} = \left\{ \begin{array}{l} s _ {x} \\ s _ {y} \\ s _ {z} \end{array} \right\} \quad \text { and } \quad \{\mathbf {s} _ {\tau} \} = \left\{ \begin{array}{l} \tau_ {x y} \\ \tau_ {y z} \\ \tau_ {z x} \end{array} \right\} \tag {17.4-12}
+$$
+
+Similarly, $\{\alpha\}$ of Eq. 17.4-3 is split into portions $\{\alpha_{\sigma}\}$ and $\{\alpha_{\tau}\}$ . With $\sigma_{Y}$ the yield strength in a uniaxial tensile test, the yield function is
+
+<!-- source-page: 538 -->
+
+$$
+F = \left[ \frac {3}{2} \left(\left\{\mathbf {s} _ {\sigma} \right\} - \left\{\boldsymbol {\alpha} _ {\sigma} \right\}\right) ^ {T} \left(\left\{\mathbf {s} _ {\sigma} \right\} - \left\{\boldsymbol {\alpha} _ {\sigma} \right\}\right) + 3 \left(\left\{\mathbf {s} _ {\tau} \right\} - \left\{\boldsymbol {\alpha} _ {\tau} \right\}\right) ^ {T} \left(\left\{\mathbf {s} _ {\tau} \right\} - \left\{\boldsymbol {\alpha} _ {\tau} \right\}\right) \right] ^ {1 / 2} - \sigma_ {Y} \tag {17.4-13}
+$$
+
+in which the positive root of the bracketed expression is intended. As before, $\sigma_{Y}$ is taken as the initial yield strength (unchanged by subsequent plastic strains). For uniaxial stress $\sigma_{x}$ , with $\{\alpha\}$ initially zero, Eq. 17.4-13 reduces to $F = |\sigma_{x}| - \sigma_{Y}$ , so that $|\sigma_{x}| = \sigma_{Y}$ defines the onset of yielding.
+
+To obtain an “associated” theory, we take $Q = F$ . Thus, after some manipulation,
+
+$$
+\left\{d \boldsymbol {\epsilon} ^ {p} \right\} = \left\{\frac {\partial F}{\partial \boldsymbol {\sigma}} \right\} d \lambda = \left(\frac {3}{2 \sigma_ {Y}} \left\{ \begin{array}{c} \mathbf {s} _ {\sigma} - \boldsymbol {\alpha} _ {\sigma} \\ \mathbf {0} \end{array} \right\} + \frac {3}{\sigma_ {Y}} \left\{ \begin{array}{c} \mathbf {0} \\ \mathbf {s} _ {\tau} - \boldsymbol {\alpha} _ {\tau} \end{array} \right\}\right) d \lambda \tag {17.4-14}
+$$
+
+which is known as the Prandtl-Reuss relation. Similarly, one concludes that $\{\partial Q / \partial \alpha\} = -\{\partial F / \partial \sigma\}$ . Because isotropic hardening is omitted in this example, $F$ does not contain $W_{p}$ , so $\partial F / \partial W_{p} = 0$ in Eq. 17.4-7.
+
+All quantities necessary for the construction of an elastic–plastic solution algorithm are now at hand. An algorithm is outlined in Section 17.5.
+
+Similar but specialized relations may be written for elastic–plastic problems of plates, in which the material may carry in-plane loads as well as bending loads [17.15]. Without such specialized relations, a thickness-direction numerical integration is required in each computational cycle, which is quite expensive.
+
+If the postyield portion of the stress–strain relation is not to be idealized as a straight line, one must store the following data for an isotropic material: $E$ , $\nu$ , $\sigma_{Y}$ , and a functional or tabular representation of $H$ or $E_{t}$ versus $\epsilon_{\text{ef}}^{p}$ , where $\epsilon_{\text{ef}}^{p}$ is an effective plastic strain defined by
+
+$$
+\begin{array}{l} \epsilon_ {\mathrm{ef}} ^ {p} = \frac {\sqrt {2}}{3} \left[ \left(\epsilon_ {x} ^ {p} - \epsilon_ {y} ^ {p}\right) ^ {2} + \left(\epsilon_ {y} ^ {p} - \epsilon_ {z} ^ {p}\right) ^ {2} + \left(\epsilon_ {z} ^ {p} - \epsilon_ {x} ^ {p}\right) ^ {2} \right. \\ + \frac {3}{2} \left\{\left(\gamma_ {x y} ^ {p}\right) ^ {2} + \left(\gamma_ {y z} ^ {p}\right) ^ {2} + \left(\gamma_ {z x} ^ {p}\right) ^ {2} \right\} ^ {1 / 2} \tag {17.4-15} \\ \end{array}
+$$
+
+in which the positive root of the bracketed expression is intended. In the plastic range where Poisson's ratio is 0.5, uniaxial stress $\sigma_x$ produces $\epsilon_{\mathrm{ef}}^p = |\epsilon_x^p|$ , so that data from a tension test are easily plotted and converted to a numerical representation. In computations with multiaxial states of stress and strain, all terms in Eq. 17.4-15 may be needed to compute $\epsilon_{\mathrm{ef}}^p$ .
+
+Specialization to Uniaxial Stress. Let $\epsilon_{x}$ be the only nonzero stress in $\{\sigma\}$ . For kinematic hardening, with $\sigma_{Y}$ the initial yield strength,
+
+$$
+F = Q = \left[ \left(\sigma_ {x} - \alpha_ {x}\right) ^ {2} \right] ^ {1 / 2} - \sigma_ {Y} = \left| \sigma_ {x} - \alpha_ {x} \right| - \sigma_ {Y} \tag {17.4-16}
+$$
+
+Hence, with "sgn" denoting "the sign of,"
+
+$$
+\frac {\partial F}{\partial \sigma_ {x}} = \frac {\partial Q}{\partial \sigma_ {x}} = \operatorname{sgn} \left(\sigma_ {x} - \alpha_ {x}\right) \quad \text {and} \quad \frac {\partial F}{\partial \alpha_ {x}} = - \operatorname{sgn} \left(\sigma_ {x} - \alpha_ {x}\right) \tag {17.4-17}
+$$
+
+<!-- source-page: 539 -->
+
+In addition, from Fig. 17.4-1, $\alpha_{x} = H\epsilon_{x}^{p}$ ; that is, C = H. Accordingly, with the term containing $W_{p}$ in Eq. 17.4-7 set to zero, we obtain from Eqs. 17.4-2 and 17.4-6
+
+$$
+d \epsilon_ {x} ^ {p} = \frac {\partial Q}{\partial \sigma_ {x}} d \lambda = \frac {\partial Q}{\partial \sigma_ {x}} C _ {\lambda} d \epsilon_ {x} = \frac {E}{E + H} d \epsilon_ {x} \tag {17.4-18}
+$$
+
+which agrees with Eq. 17.3-6. From Eq. 17.4-9 we obtain
+
+$$
+E _ {\mathrm{ep}} = E - E \frac {E}{E + H} = E \left(1 - \frac {E}{E + H}\right) \tag {17.4-19}
+$$
+
+which agrees with Eq. 17.3-2.
+
+# 17.5 ELASTIC-PLASTIC ANALYSIS PROCEDURES
+
+In the present section we summarize the tangent-stiffness method and the initial-stiffness method. The same two algorithms are discussed in a one-dimensional context in Section 17.3. The loading history and the geometry, support conditions, and material properties are assumed to be known. We seek the deformations and stresses in the body as a function of load. With either solution method, the load is incremented in several steps. The tangent-stiffness method allows large but expensive steps, while the initial-stiffness method uses small but inexpensive steps. It is not always clear which method will be better in particular problems. Detailed discussion of algorithms may be found in $[17.11–17.23]$ .
+
+When the material behavior is nonlinear, material properties in an element are dictated by material properties at a finite number of sampling points in each element. Typically these points are quadrature stations of a numerical integration rule. At each point one must keep a record of strains and update the record in each computational cycle. The number of points must be small to reduce computational expense. Accordingly, some analysts prefer simple elements, which may require only one sampling point per element. A contrary argument is that many sampling points are needed to accurately capture the spread of yielding in individual elements. In simple terms, the choice is between many simple elements and a smaller number of more sophisticated elements.
+
+In what follows we will assume that strain increments $\{d\epsilon\}$ include elastic components $\{d\epsilon^{e}\}$ and plastic components $\{d\epsilon^{p}\}$ , but that thermal strains $\{d\epsilon^{T}\}$ and creep strains $\{d\epsilon^{C}\}$ have already been subtracted out.
+
+We presume that a tensile test of the material has been performed, and a numerical representation of its stress–strain curve is stored. We also presume that specific choices of yield criterion, flow rule, and hardening rule have been made. If we choose the von Mises yield criterion, the Prandtl–Reuss flow relations, either kinematic or isotropic hardening, and a bilinear stress–strain relation, then we need store only $E, \nu, \sigma_{Y}$ , and either $E_{t}$ or H for an isotropic material. Alternatively, to represent a more general stress–strain relation, either $E_{t}$ or H may be defined as a function of $\epsilon_{ef}^{p}$ (Eq. 17.4-15). Then, in computation, we must record
+
+<!-- source-page: 540 -->
+
+and update the value of $\epsilon_{\mathrm{cf}}^2$ at each sampling point, and use it to obtain the current value of $E_{t}$ or $H$ .
+
+Tangent-Stiffness Method. Loads $\{R\}$ on the structure are applied in increments $\{\Delta R\}_{1}$ , $\{\Delta R\}_{2}$ , and so on, so that $\{R\} = \Sigma \{\Delta R\}_{i}$ . The first load increment might be contrived to place only the most highly stressed sampling point on the verge of yield, but we will not make this assumption. Procedural steps are as follows.
+
+1. At the outset, $\{\epsilon\} = \{\sigma\} = \{\alpha\} = \{0\}$ , $W_{p} = 0$ , and $[\mathbf{E}_{\mathrm{ep}}] = [\mathbf{E}]$ for all sampling points. These values prevail in the first computational cycle ( $i = 1$ ). Apply the first load increment, $\{\Delta \mathbf{R}\}_{1}$ .   
+2. Use the current conditions $\{\pmb{\sigma}\}_{i-1}, \{\pmb{\alpha}\}_{i-1}$ , and $W_{pi-1}$ to evaluate $[\mathbf{E}_{\mathrm{ep}}]_{i-1}$ for each sampling point. Note that $[\mathbf{E}_{\mathrm{ep}}]_{i-1} = [\mathbf{E}]$ for sampling points that have yet to yield ( $F < 0$ for the current $\{\pmb{\sigma}\}_{i-1}, \{\pmb{\alpha}\}_{i-1}$ , and $W_{pi-1}$ ) or are unloading ( $dF < 0$ for the most recent changes in $\{\pmb{\sigma}\}, \{\pmb{\alpha}\}$ , and $W_p$ ). Evaluate $[\mathbf{k}_i]$ for each element $n$ . The structure tangent-stiffness matrix is formed by the usual assembly, $[\mathbf{K}_i]_{i-1} = \Sigma [\mathbf{k}_i]_n$ . Solve for structure displacement increments $\{\Delta \mathbf{D}\}_i$ and strain increments $\{\Delta \pmb{\epsilon}\}_i$ at element sampling points from the equations
+
+$$
+[ \mathbf {K} _ {t} ] _ {i - 1} \{\Delta \mathbf {D} \} _ {i} = \{\Delta \mathbf {R} \} _ {i - 1} \quad \text { and } \quad \{\Delta \boldsymbol {\epsilon} \} _ {i} = [ \mathbf {B} ] \{\Delta \mathbf {d} \} _ {i} \tag {17.5-1}
+$$
+
+For sampling points in the plastic range, compute increments as follows.
+
+From Eq. 17.4-6: $\Delta \lambda_{i} = \int \{\mathbf{C}_{\lambda}\}^{T}\{d\epsilon \} \approx \{\mathbf{C}_{\lambda}\}_{i - 1}^{T}\{\Delta \epsilon \}_{i}$ (17:5-2)
+
+From Eq. 17.4-2: $\{\Delta \pmb{\epsilon}^p\}_{i} = \int \left\{\frac{\partial Q}{\partial \pmb{\sigma}}\right\} d\lambda \approx \left\{\frac{\partial Q}{\partial \pmb{\sigma}}\right\}_{i - 1}\Delta \lambda_i$ (17.5-3)
+
+From Eq. 17.4-5: $\{\Delta \sigma\}_{i} = [\mathbb{E}](\{\Delta \epsilon\}_{i} - \{\Delta \epsilon^{p}\}_{i})$ (17.5-4)
+
+From Eq. 17.4-3: $\{\Delta \alpha\}_{i} = \int C\{d\epsilon^{p}\} \approx C\{\Delta \epsilon^{p}\}_{i}$ (17.5-5)
+
+From Eq. 17.4-3: $\Delta W_{pi} = \int \{\pmb{\sigma}\}^T \{d\pmb{\epsilon}^p\} \approx \{\pmb{\sigma}\}_i^T \{\Delta \pmb{\epsilon}^p\}_i$ (17.5-6)
+
+Typically, a solution will use $\{\alpha\}$ or $W_{p}$ but not both. For sampling points in the elastic range, Eqs. 17.5-2 through 17.5-6 are not used. Instead, one computes only $\{\Delta\sigma\} = [E]\{\Delta\epsilon\}$ .
+
+3a. Optional. For sampling points that make the elastic-to-plastic transition, compute the fraction m of the current increment that is elastic. For example, writing Eq. 17.4-13 in the form $F = Y - \sigma_{Y}$ ,
+
+$$
+m = \frac {\sigma_ {Y} - Y _ {i - 1}}{Y _ {i} - Y _ {i - 1}} \tag {17.5-7}
+$$
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_055.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_055.md
new file mode 100644
index 00000000..e4a56dd4
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_055.md
@@ -0,0 +1,370 @@
+<!-- source-page: 541 -->
+
+where $Y_{i}$ is based on current stresses, computed by use of elastic coefficients, and temporarily updated for this step only. The revised $[E_{ep}]_{i-1}$ is $m[E]$ plus $(1 - m)$ times the right-hand side of Eq. 17.4-9; this reduces to
+
+$$
+[ \mathrm{E} _ {\mathrm{ep}} ] _ {i - 1} = [ \mathrm{E} ] - (1 - m) [ \mathrm{E} ] \left\{\frac {\partial Q}{\partial \sigma} \right\} _ {i - 1} \left\{\mathrm{C} _ {\lambda} \right\} _ {i - 1} ^ {T} \tag {17.5-8}
+$$
+
+Repeat steps 2 and 3a until convergence but without making the updates final (step 4 below) until convergence. In applying Eq. 17.5-2, use $(1 - m)\{\Delta\epsilon\}_{i}$ rather than $\{\Delta\epsilon\}_{i}$ , as $\{C_{\lambda}\}_{i-1}$ is zero for the elastic portion of the increment.
+
+3b. Optional. Without changing the load or recalculating $\{\Delta D\}_{i}$ , one can evaluate Eqs. 17.5-2 through 17.5-6 more accurately by dividing the increment $\{\Delta \epsilon\}_{i}$ into subincrements. After each such subincremental cycle, one updates $\{\sigma\}_{i}$ , $\{\alpha\}_{i}$ , and so on (Eqs. 17.5-10). Note that $\{C_{\lambda}\}_{i-1}$ is zero in elastic subincrements if the sampling point makes the elastic-to-plastic transition within the current load step.
+
+3c. Optional (but recommended). “Corrective” loads are introduced to prevent progressive drift, as discussed in connection with Figs. 17.2-5 and 17.2-6. Thus the next load increment is not simply $\{\Delta R\}_{i+1}$ ; rather, it is
+
+$$
+\{\Delta \mathbf {R} \} _ {i + 1} + \{\Delta \mathbf {R} _ {c} \} _ {i} \quad \text { where } \quad \{\Delta \mathbf {R} _ {c} \} = \{\mathbf {R} \} _ {i} - \sum \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} \{\boldsymbol {\sigma} \} _ {i} d V \tag {17.5-9}
+$$
+
+and $\{R\}_{i}$ is the total externally applied load in cycle i. The summation spans all elements of the structure and expresses the loads that elements apply to nodes because they have stresses $\{\sigma\}_{i}$ . Stresses $\{\sigma\}_{i}$ are updated values (Eqs. 17.5-10).
+
+4. Update the solution:
+
+$$
+\{\mathbf {D} \} _ {i} = \{\mathbf {D} \} _ {i - 1} + \{\Delta \mathbf {D} \} _ {i} \quad \{\boldsymbol {\sigma} \} _ {i} = \{\boldsymbol {\sigma} \} _ {i - 1} + \{\Delta \boldsymbol {\sigma} \} _ {i} \tag {17.5-10}
+$$
+
+$$
+\{\alpha \} _ {i} = \{\alpha \} _ {i - 1} + \{\Delta \alpha \} _ {i} \quad (W _ {p}) _ {i} = (W _ {p}) _ {i - 1} + (\Delta W _ {p}) _ {i}
+$$
+
+5. Apply the next load increment and return to step 2.
+
+6. Stop when $\Sigma \{\Delta \mathbf{R}\}_i$ reaches the total applied load.
+
+Exercising steps 3a, 3b, and/or 3c permits load increments to be larger without increasing the error. Step 3c can be exercised repeatedly within a given load increment, either with or without updates in the structure matrix $[K_{r}]$ ; the effect is depicted in Figs. 17.2-3 and 17.2-4. If step 3c is used in this way there should be no test for unloading $(dF < 0)$ until these cycles are complete because intermediate cycles can give false indications. Use of step 3b within this cycling gives a three-level process: subincrements within iterations within load increments.
+
+Flow rules often allow little or no volume change. Thus, if plastic strains become extremely large, the response becomes nearly incompressible, and fully integrated elements may encounter numerical difficulties associated with locking of the mesh. Use of selective reduced integration is recommended.
+
+<!-- source-page: 542 -->
+
+![](images/page-542_aa122590261bc0123153581c7c905440955ee975c55a2115c8a36e9581c85775.jpg)
+
+<details>
+<summary>text_image</summary>
+
+R
+1
+2
+3
+E
+E
+E
+D
+</details>
+
+Figure 17.5-1. Load versus displacement plot for a representative d.o.f. D in a multi-d.o.f. model, showing convergence of the initial stiffness method.
+
+Initial-Stiffness Method. The step-by-step procedure remains as outlined in Section 17.3. We need make only a few modifications in step 5 to allow for the multidimensionality of stress. Specifically, in place of Eq. 17.3-6, we use Eqs. 17.5-1 through 17.5-6 to obtain plastic strain increments and update the yield criterion and the flow rule. Fraction m, if used, is given by Eq. 17.5-7. These calculations and updates are performed at each sampling point that sustains plastic strains. Equation 17.3-7 is replaced by
+
+$$
+\{\Delta \mathbf {R} _ {s} \} = \sum \{\Delta \mathbf {r} _ {s} \} \quad \text { where } \quad \{\Delta \mathbf {r} _ {s} \} = \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] \{\Delta \boldsymbol {e} ^ {p} \} d V \tag {17.5-11}
+$$
+
+Note that sampling points that unload in a computational cycle $(\Delta F < 0)$ make no contribution to $\{\Delta \mathbf{R}_s\}$ .
+
+In the one-dimensional problem of Fig. 17.3-2a, the applied load must be carried by every element. The initial-stiffness method then converges slowly (Fig. 17.3-3b). In conditions of multiaxial stress, the yielding of one element is accompanied by a transfer of load to other elements, and the initial-stiffness method converges in fewer iterations (Fig. 17.5-1).
+
+# 17.6 NONLINEAR DYNAMIC PROBLEMS
+
+When the frequency of excitation exceeds about one-third the structure's lowest natural frequency of vibration, inertia becomes important and the problem is dynamic rather than quasistatic. Of the methods of response history analysis discussed in Chapter 13, direct time integration methods are usually the most effective for nonlinear problems. In this section, explicit and implicit methods are briefly discussed with particular emphasis on nonlinearities due to plasticity. Most remarks, however, are true of material nonlinearities in general. Direct integration methods that also include geometric nonlinearities are discussed [17.24]. It is assumed that the reader is familiar with Sections 13.9 through 13.14.
+
+Explicit Methods. Treatment of nonlinearities by explicit methods is usually straightforward, accurate, and effective. All remarks of Section 13.10 are valid, but elaboration is necessary for nonlinear problems, as follows.
+
+Explicit methods require that the internal force of each element, $\{r^{int}\}_{n}$ , be calculated before the new displacement $\{D\}_{n+1}$ can be computed. Element-by-element calculation of $\{r^{int}\}_{n}$ using Eq. 13.2-7 requires that element stresses $\{\sigma\}_{n}$ be known. For linear problems, $\{\sigma\}_{n} = [E][B]\{D\}_{n}$ in which $\{D\}_{n}$ is known. For plasticity, stress increment $\{d\sigma\} \approx \{\Delta\sigma\}$ can be computed from the strain incre-
+
+<!-- source-page: 543 -->
+
+ment, $\{\Delta \epsilon\} = [\mathbf{B}](\{\mathbf{D}\}_n - \{\mathbf{D}\}_{n-1})$ , and the constitutive law, Eq. 17.4-8. Hence, the stress at time $n \Delta t$ is given by $\{\sigma\}_n = \{\sigma\}_{n-1} + \{\Delta \sigma\}$ and $\{\mathbf{r}^{\mathrm{int}}\}_n$ can be obtained from Eq. 13.2-7.
+
+As with linear problems, the accuracy of an explicit solution is usually assured when the time-step stability criterion is satisfied. In Chapter 13, stability criteria are cited for several explicit methods as applied to linear problems, for example, Eqs. 13.10-8, 13.10-14, and 13.12-6 with $\beta = 0$ . Extensive computational experience suggests that these criteria are also valid for nonlinear problems provided that one uses the instantaneous value of $\omega_{max}$ , which is a function of material properties, element geometry, and mesh geometry.
+
+Many materials display “softening” behavior in which the tangent modulus decreases with increasing stress or strain level. Examples of such materials are ductile and brittle solids. For these materials, it can often be shown that the instantaneous value of $\omega_{max}$ will not exceed the $\omega_{max}$ for linearly elastic response [17.25]. Thus, a time step that is stable for purely elastic response will also be stable for nonlinear response. Alternatively, it is possible to change $\Delta t$ during a problem solution by continuously monitoring $\omega_{max}$ through the use of an element frequency bound. However, for most problems the possible increase in $\Delta t$ is small, as $\omega_{max}$ often continues to be governed by elements of the mesh that experience little or no plastic deformation.
+
+A material that displays “stiffening” has a tangent modulus that exceeds its initial modulus. In such a material $\omega_{max}$ may increase, making the stability criterion more restrictive. In problems with geometric nonlinearity, $\omega_{max}$ may increase or decrease. In these situations it is usually necessary to monitor $\omega_{max}$ through the use of element bounds during the course of a computational solution [17.24].
+
+Numerical instability is usually easy to detect in linear problems because the solution grows without limit. In nonlinear problems, with elastic–plastic or other energy-dissipating materials, extra energy introduced into the system by the numerical instability may be dissipated by plastic work or some other irreversible mechanism so that it is possible for the instability to be arrested $[17.26]$ . An “arrested instability” is often difficult to detect because the solution, although in error by 10% to 100% or more, may appear to be reasonable.
+
+Energy Balance Check. In the analysis of nonlinear dynamic problems by explicit methods, it is usually advisable to perform an energy balance check to help assure stable and accurate computation. Ideally, the energy at time $(n + 1)\Delta t$ in a system should satisfy the equation
+
+$$
+W _ {n + 1} ^ {\text {int}} + T _ {n + 1} = W _ {n + 1} ^ {\text {ext}} \tag {17.6-1}
+$$
+
+where W represents work and T represents kinetic energy. Physically, Eq. 17.6-1 states that the work of external loads is converted to kinetic energy and to energy either stored elastically or dissipated by plastic deformations. The separate terms in Eq. 17.6-1 are explained as follows.
+
+The internal work, $W_{n+1}^{int}$ , represents the work done by nodal loads that are developed from straining of material and is given by
+
+$$
+W _ {n + 1} ^ {\text {int}} = W _ {n} ^ {\text {int}} + \int_ {n \Delta t} ^ {(n + 1) \Delta t} \dot {W} ^ {\text {int}} d t \tag {17.6-2}
+$$
+
+<!-- source-page: 544 -->
+
+Noting that $\dot{W}_n^{\mathrm{int}} = \{\dot{\mathbf{D}}\}_{n}^{T|}\mathbf{R}^{\mathrm{int}}\}_n$ and approximating the integral in Eq. 17.6-2 by the trapezoidal rule, we obtain
+
+$$
+W _ {n + 1} ^ {\text {int}} = W _ {n} ^ {\text {int}} + \frac {\Delta t}{2} \left(\left\{\dot {\mathbf {D}} \right\} _ {n} ^ {T} \left\{\mathbf {R} ^ {\text {int}} \right\} _ {n} + \left\{\dot {\mathbf {D}} \right\} _ {n + 1} ^ {T} \left\{\mathbf {R} ^ {\text {int}} \right\} _ {n + 1}\right) \tag {17.6-3}
+$$
+
+Equation 17.6-3 is appropriate for use with explicit methods that compute velocities at whole time steps (e.g., the Newmark method with $\beta = 0$ ). When velocities are known at half time steps, as in the central-difference method, then Eq. 17.6-3 can be written as
+
+$$
+W _ {n + 1} ^ {\text {int}} = W _ {n} ^ {\text {int}} + \frac {\Delta t}{2} \left\{\dot {\mathbf {D}} \right\} _ {n + 1 / 2} ^ {T} \left(\left\{\mathbf {R} ^ {\text {int}} \right\} _ {n} + \left\{\mathbf {R} ^ {\text {int}} \right\} _ {n + 1}\right) \tag {17.6-4}
+$$
+
+The external work, $W_{n+1}^{\mathrm{ext}}$ , represents the work of the externally applied loads and is given by
+
+$$
+W _ {n + 1} ^ {\mathrm{ext}} = W _ {n} ^ {\mathrm{ext}} + \int_ {n \Delta t} ^ {(n + 1) \Delta t} \{\dot {\mathbf {D}} \} ^ {T} \{\mathbf {R} ^ {\mathrm{ext}} \} d t \tag {17.6-5}
+$$
+
+Difference expressions for $W_{n+1}^{ext}$ can be obtained from Eqs. 17.6-3 and 17.6-4 by replacing superscript “int” by “ext.” The kinetic energy, $T_{n}$ , is given by
+
+$$
+T _ {n} = \frac {1}{2} \{\dot {\mathbf {D}} \} _ {n} ^ {T} [ \mathbf {M} ] \{\dot {\mathbf {D}} \} _ {n} \tag {17.6-6}
+$$
+
+or, if half-time-step velocities are known, by
+
+$$
+T _ {n} = \frac {1}{2} \left(T _ {n - 1 / 2} + T _ {n + 1 / 2}\right) \tag {17.6-7}
+$$
+
+To construct an energy balance, we note that, in general, Eq. 17.6-1 is not satisfied exactly. To measure the quality of a solution, we can use-
+
+$$
+W _ {n} ^ {\text {int}} + T _ {n} - \left| W _ {n} ^ {\text {ext}} \right| \leq e (W _ {n} ^ {\text {int}} + T _ {n} + \left| W _ {n} ^ {\text {ext}} \right|) \tag {17.6-8}
+$$
+
+where e is a tolerance and absolute magnitude bars are a precaution to avoid small negative values of $W^{ext}$ due to numerical errors. Terms within parentheses on the right-hand side of Eq. 17.6-8 represent the total energy in the system. The left-hand side is the energy error. A stable explicit computation should satisfy Eq. 17.6-8 with $e \leq 0.02$ [17.24]. If satisfaction of Eq. 17.6-8 requires $e \geq 0.05$ , even for models with hundreds of elements and using thousands of time steps, then instability should be suspected.
+
+<!-- source-page: 545 -->
+
+response will also be stable for nonlinear response. Hence, with $\rho = 7.4(10^{-4})$ lb-sec $^2$ /in. $^4$ , the highest element frequency is $(\omega_{\max})_e = 2\sqrt{E/\rho}/L = 8.054(10^5)$ rad/sec. Thus stable integration by the central-difference method requires $\Delta t \leq 2/(\omega_{\max})_e = 2.483(10^{-6})$ sec according to Eq. 13.10-14 with $\xi = 0$ .
+
+The computational procedure in Table 13.10-1 and the Fortran program in Fig. 13.10-3 were used except that Subroutine INTFOR in Fig. 13.10-3 was replaced by the code in Fig. 17.6-1 (which does not include the energy balance check of Eq. 17.6-8). Additional alterations to the main program in Fig. 13.10-3 include adding SIGMA(101) and SIGYLD(101) to the DIMENSION statement, adding the tangent modulus (ET) and initial yield strength ( $\sigma_{Y}$ in Fig. 17.3-1) to the READ statement, initializing the stress in each element to zero and the yield strength in each element to $\sigma_{Y}$ , modifying the tip load, the CALL INTFOR statement, and the WRITE statement.
+
+The Fortran code in Fig. 17.6-1 generates the internal force vector for a model with elastic–plastic behavior. The yield function, called YLDFUN in Fig. 17.6-1, is obtained from the discussion following Eq. 17.4-13 as $F = |\sigma_{x}| - \sigma_{Y}$ , where $\sigma_{x}$ in element I is called SIGMA(I) in Fig. 17.6-1. Because of isotropic hardening, the stress required for renewed or continued yielding exceeds $\sigma_{Y}$ ; this stress is called $\sigma_{B}$ in Fig. 17.3-1b and SIGYLD(I) in Fig. 17.6-1. The code uses modulus E when operating on the linearly elastic part of the stress–strain diagram shown in Fig. 17.3-1 and modulus ET when operating on the plastic part of the diagram. When states of stress make the transition from elastic
+
+```prolog
+SUBROUTINE INTFOR(D,DOLD,FINT,SIGMA,X,SIGYLD,E,ET,CSA,NELE)
+IMPLICIT DOUBLE PRECISION (A-H,O-Z)
+DIMENSION D(1),DOLD(1),FINT(1),SIGMA(1),X(1),SIGYLD(1)
+NUMNOD=NELE+1
+C---- Zero internal force vector.
+DO 10 I=1,NUMNOD
+10 FINT(I)=0.
+C---- Loop over elements.
+DO 30 K=1,NELE
+RL=X(K+1)-X(K)
+C---- Compute strain increment.
+EPSOLD=(DOLD(K+1)-DOLD(K))/RL
+EPSNEW=( D(K+1)- D(K))/RL
+EPSINC=EPSNEW-EPSOLD
+C---- Compute stress increment (DSIGMA) assuming elastic response.
+DSIGMA=E*EPSINC
+C---- Check if DSIGMA satisfies yield function .LT. 0.
+YLDFUN=DABS(SIGMA(K)+DSIGMA)-SIGYLD(K)
+IF (YLDFUN.LT.0.) GO TO 20
+C---- If material was plastic before strain increment, and
+C---- YLDFUN .GE. 0., then entire strain increment is plastic.
+IF (DABS(SIGMA(K)).EQ.SIGYLD(K)) THEN
+DSIGMA=ET*EPSINC
+GO TO 20
+ELSE
+C---- Compute the portion of the strain increment that is elastic.
+RATIO=(SIGYLD(K)-DABS(SIGMA(K))/DABS(DSIGMA)
+DSIGMA=RATIO*EPSINC*E + (1.-RATIO)*EPSINC*ET
+ENDIF
+C---- Update stress and account for isotropic hardening.
+20 CONTINUE
+SIGMA(K)=SIGMA(K)+DSIGMA
+IF (DABS(SIGMA(K)).GT.SIGYLD(K)) SIGYLD(K)=DABS(SIGMA(K))
+C---- Assemble contribution into internal force vector.
+F1=-SIGMA(K)*CSA
+F2=+SIGMA(K)*CSA
+FINT(K)=FINT(K)+F1
+FINT(K+1)=FINT(K+1)+F2
+30 CONTINUE
+RETURN
+END 
+```  
+Figure 17.6-1. Fortran program to compute the internal force vector for a mesh of linear displacement bar elements with elastic-plastic, isotropic-hardening material.
+
+<!-- source-page: 546 -->
+
+to plastic, the fraction of the strain increment that is elastic is computed according to the procedure described in step 3a of Section 17.5. In the Fortran code, the elastic fraction of the strain increment, $m$ in Eq. 17.5-7, is called RATIO.
+
+The stress time-history results for the midpoint of element 20 (at $x = 9.75$ in.) are shown in Fig. 17.6-2 for $\Delta t = 2.4(10^{-6}) \sec (C_n = 0.966)$ and 83 time steps. Two separate stress waves arrive at different times. These correspond to an elastic wave traveling at speed $c_e = \sqrt{E / \rho}$ , which arrives first, followed by a plastic wave traveling at speed $c_p = \sqrt{E_t / \rho} = \frac{1}{2} c_e$ . Thus, at $x = 9.75$ in., an elastic wave arrives at $t = 0.0484$ msec and a plastic wave arrives at $t = 0.0968$ msec.
+
+The solution is devoid of noise until after the passage of the plastic wave. The reason is that during the initial passage of the elastic stress wave, numerical noise causes stress excursions above the initial yield stress. These excursions are transmitted at the plastic wave speed, and hence do not arrive at element 20 until approximately 0.11 msec has elapsed.
+
+Implicit Methods. Compared with explicit methods, implicit methods for nonlinear problems are less attractive in every respect save one—that is, the possibility of using large time steps permitted by the excellent stability properties of popular implicit methods. Nonlinearities present the same difficulty to both static and implicit dynamic solution algorithms: stiffness is a function of displacements, which are not known in advance. For example, in Eq. 13.11-5, $[K^{eff}]$ is a function of $\{D\}_{n+1}$ , which is unknown. Methods for addressing this difficulty are analogous to the tangent-stiffness and initial-stiffness methods described in Section 17.3 for nonlinear quasistatic problems.
+
+Tangent-Stiffness (Implicit) Method. In the tangent-stiffness method, the internal force in the equations of motion, Eq. 13.9-2, is written as
+
+$$
+\{\mathbf {R} ^ {\mathrm{int}} \} _ {n + 1} = \{\mathbf {R} ^ {\mathrm{int}} \} _ {n} + [ \mathbf {K} _ {t} ] \{\Delta \mathbf {D} \} \tag {17.6-9}
+$$
+
+where
+
+$$
+\{\Delta \mathbf {D} \} = \{\mathbf {D} \} _ {n + 1} - \{\mathbf {D} \} _ {n} \tag {17.6-10}
+$$
+
+Combining Eqs. 17.6-9 and 17.6-10 with the equations of motion, Eq. 13.9-2, and the trapezoidal rule equations, Eqs. 13.11-3 and 13.11-4, we obtain
+
+$$
+[ \mathbf {K} ^ {\mathrm{eff}} ] \{\Delta \mathbf {D} \} = \{\mathbf {R} ^ {\mathrm{eff}} \} _ {n + 1} \tag {17.6-11}
+$$
+
+where
+
+$$
+[ \mathbf {K} ^ {\mathrm{eff}} ] = \frac {4}{\Delta t ^ {2}} [ \mathbf {M} ] + \frac {2}{\Delta t} [ \mathbf {C} ] + [ \mathbf {K} _ {t} ] \tag {17.6-12}
+$$
+
+and
+
+$$
+\{\mathbf {R} ^ {\text { eff }} \} _ {n + 1} = \{\mathbf {R} ^ {\text { ext }} \} _ {n + 1} - \{\mathbf {R} ^ {\text { int }} \} _ {n} + [ \mathbf {M} ] \left(\frac {4}{\Delta t} \{\dot {\mathbf {D}} \} _ {n} + \{\ddot {\mathbf {D}} \} _ {n}\right) + [ \mathbf {C} ] \{\dot {\mathbf {D}} \} _ {n} \tag {17.6-13}
+$$
+
+Note that $[K_{t}]$ must be predicted using $\{D\}_{n}$ (and possibly $\{\dot{D}\}_{n}$ if strain rate effects are important) and must be factored at least once each time step during nonlinear
+
+<!-- source-page: 547 -->
+
+![](images/page-547_c137846d8f333fa7618410ef5d814f7bca5873ee1fb6496dd08fe861130e87aa.jpg)
+
+<details>
+<summary>line</summary>
+
+| Time (milliseconds) | Exact | Central difference |
+| ------------------- | ----- | ------------------ |
+| 0.00                | 0     | 0                  |
+| 0.05                | -40   | -40                |
+| 0.10                | -80   | -80                |
+| 0.15                | -80   | -80                |
+| 0.20                | -80   | -80                |
+</details>
+
+Figure 17.6-2. Stress versus time at x = 9.75 in. for a 40-element model of a 20-in. bar with elastic-plastic material using $\Delta t = 2.4(10^{-6})$ sec ( $C_{n} = 0.966$ ). Figure 13.10-2 applies, except that $P_{0} = 80{,}000$ lb.
+
+response. If $[K_{t}]$ is not an accurate prediction of the true tangent-stiffness matrix from time $n \Delta t$ to time $(n + 1) \Delta t$ ; then the solution of Eq. 17.6-11 for $\{\Delta D\}$ will be in error. The error in nodal forces—that is, the residual—is given by the imbalance in the equation of motion (Eq. 13.9-2) as
+
+$$
+\{\mathbf {R} ^ {\text { err }} \} = \{\mathbf {R} ^ {\text { ext }} \} _ {n + 1} - [ \mathbf {M} ] \{\ddot {\mathbf {D}} \} _ {n + 1} - [ \mathbf {C} ] \{\dot {\mathbf {D}} \} _ {n + 1} - \{\mathbf {R} ^ {\text { int }} \} _ {n + 1} \tag {17.6-14}
+$$
+
+where $\{R^{int}\}_{n+1}$ is computed using Eq. 17.6-9 and an improved tangent-stiffness matrix (i.e., $[K_{t}]$ obtained using $\{D\}_{n+1} = \{D\}_{n} + \{\Delta D\}$ , which is obtained by solving Eq. 17.6-11). Alternatively, $\{R^{int}\}_{n+1}$ can be computed element-by-element in the same way as for explicit direct integration. If measures are not taken to control the growth of $\{R^{err}\}$ , the solution will diverge in a manner similar to the instability displayed by explicit methods when the stability criterion is violated [17.26]. A computational procedure for the trapezoidal rule with error control is given in Table 17.6-1. Note that the procedure for error control is essentially the pseudo-load approach, described later in this section, applied within a time step. If enough iterations are performed within each time step to guarantee that $W^{err}$ of Table 17.6-1 is bounded for the entire solution, then the trapezoidal rule algorithm is unconditionally stable, although not necessarily accurate [17.27]. To assure accuracy, $W^{err}$ should be small.
+
+Initial-Stiffness (Implicit) Method. In the initial-stiffness method, the initial tangent-stiffness matrix of the structure is used throughout the analysis and corrective loads due to nonlinearities are transferred to the right-hand side of the equations solved at each iteration. With this method, internal forces are given by
+
+$$
+\{\mathbf {R} ^ {\mathrm{int}} \} _ {n + 1} = [ \mathbf {K} ] \{\mathbf {D} \} _ {n + 1} - \{\Delta \mathbf {R} _ {s} \} _ {n + 1} \tag {17.6-15}
+$$
+
+<!-- source-page: 548 -->
+
+TABLE 17.6-1. COMPUTATIONAL PROCEDURE FOR DIRECT INTEGRATION OF MATERIAL NONLINEAR PROBLEMS BY THE TRAPEZOIDAL RULE METHOD WITH TANGENTIAL STIFFNESS. SUPERPOSED TILDE ( $^{-}$ ) DENOTES QUANTITIES OBTAINED FROM AN ESTIMATED $[K_{L}]$ .
+
+1. Form [C] and [M] (also form $[\mathbf{K}_t]$ if $\{\mathbf{D}(t = 0)\} \neq \{0\}$ ).   
+2. Set initial conditions $\{\mathbf{D}\}_{0} = \{\mathbf{D}(t = 0)\}$ and $\{\dot{\mathbf{D}}\}_{0} = \{\dot{\mathbf{D}}(t = 0)\}$ ; use Eq. 13.9-1 to compute $\{\ddot{\mathbf{D}}\}_{0} = [\mathbf{M}]^{-1}(\{\mathbf{R}^{\mathrm{ext}}\}_{0} - [\mathbf{C}]\{\dot{\mathbf{D}}\}_{0} - [\mathbf{K}]\{\mathbf{D}\}_{0})$ ; use Eq. 13.2-7 to compute $\{\mathbf{R}^{\mathrm{int}}\}_{0}$ ; $n = 0$ .   
+3. Form $[\mathbf{K}_t]$ and compute $[\mathbf{K}^{\mathrm{eff}}] = \frac{4}{\Delta t^2} [\mathbf{M}] + \frac{2}{\Delta t} [\mathbf{C}] + [\mathbf{K}_t].$   
+4. Form $\{\mathbf{R}^{\mathrm{eff}}\}_{n + 1} = \{\mathbf{R}^{\mathrm{ext}}\}_{n + 1} - \{\mathbf{R}^{\mathrm{int}}\}_{n} + [\mathbf{M}] \left( \frac{4}{\Delta t} \{\dot{\mathbf{D}}\}_{n} + \{\ddot{\mathbf{D}}\}_{n} \right) + [\mathbf{C}]\{\dot{\mathbf{D}}\}_{n}$ .   
+5. Solve $[\mathbf{K}^{\mathrm{eff}}]\{\Delta \bar{\mathbf{D}}\} = \{\mathbf{R}^{\mathrm{eff}}\}_{n + 1}$ for $\{\Delta \bar{\mathbf{D}}\}$ .   
+6. Compute $\{\tilde{\mathbf{D}}\}_{n + 1} = \frac{2}{\Delta t}\{\Delta \tilde{\mathbf{D}}\} -\{\dot{\mathbf{D}}\}_{n}$ and $\{\tilde{\mathbf{D}}\}_{n + 1} = \frac{4}{\Delta t^2}\{\Delta \tilde{\mathbf{D}}\} -\frac{4}{\Delta t}\{\dot{\mathbf{D}}\}_{n} - \{\ddot{\mathbf{D}}\}_{n}$ .   
+7. Compute $\{\hat{\mathbf{R}}^{\mathrm{int}}\}_{n + 1} = \sum \{\hat{\mathbf{r}}^{\mathrm{int}}\}_{n + 1} = \sum \int [\mathbf{B}]^T \{\hat{\sigma}\}_{n + 1} dV$ using $\{\hat{\mathbf{D}}\}_{n + 1} = \{\mathbf{D}\}_n + \{\Delta \hat{\mathbf{D}}\}$ .   
+8. Compute $\{\mathbf{R}^{\mathrm{ext}}\} = \{\mathbf{R}^{\mathrm{ext}}\}_{n+1} - [\mathbf{M}]\{\dot{\bar{\mathbf{D}}}\}_{n+1} - [\mathbf{C}]\{\dot{\bar{\mathbf{D}}}\}_{n+1} - \{\bar{\mathbf{R}}^{\mathrm{int}}\}_{n+1}$ .   
+9. If $W^{\mathrm{err}} = \Delta t\{\dot{\mathbf{D}}_{[n + 1]}^{T}\{\mathbf{R}^{\mathrm{err}}\} >$ tolerance, add $\{\mathbf{R}^{\mathrm{err}}\}$ to $\{\mathbf{R}^{\mathrm{ext}}\}_{n + 1}$ and go to Step 4.   
+10. If tolerance is satisfied, update histories $\{\mathbf{D}\}_{n+1} = \{\mathbf{D}\}_n + \{\Delta \mathbf{D}\}$ , $\{\mathbf{D}\}_{n+1} = \{\mathbf{D}\}_{n+1}$ , $\{\ddot{\mathbf{D}}\}_{n+1} = \{\ddot{\mathbf{D}}\}_{n+1}$ , $\{\mathbf{R}^{\mathrm{int}}\}_{n+1} = \{\hat{\mathbf{R}}^{\mathrm{int}}\}_{n+1}$ ; $n \leftarrow n + 1$ , go to Step 3.
+
+where [K] is the initial tangent-stiffness matrix and $\{\Delta R_{s}\}$ is a vector of nodal loads computed so that the material constitutive law is satisfied. Combining Eq. 17.6-15 with the equations of motion, Eq. 13.9-2, and the trapezoidal rule equations, Eqs. 13.11-3 and 13.11-4, we obtain
+
+$$
+[ \mathbf {K} ^ {\mathrm{eff}} ] \{\mathbf {D} \} _ {n + 1} = \{\mathbf {R} ^ {\mathrm{eff}} \} _ {n + 1} \tag {17.6-16}
+$$
+
+where
+
+$$
+[ \mathbf {K} ^ {\mathrm{eff}} ] = \frac {4}{\Delta t ^ {2}} [ \mathbf {M} ] + \frac {2}{\Delta t} [ \mathbf {C} ] + [ \mathbf {K} ] \tag {17.6-17}
+$$
+
+and
+
+$$
+\begin{array}{l} \left\{\mathbf {R} _ {n + 1} ^ {\text {eff}} = \left\{\mathbf {R} _ {n + 1} ^ {\text {ext}} + [ \mathbf {M} ] \left(\frac {4}{\Delta t ^ {2}} \left\{\mathbf {D} \right\} _ {n} + \frac {4}{\Delta t} \left\{\dot {\mathbf {D}} \right\} _ {n} + \left\{\ddot {\mathbf {D}} \right\} _ {n}\right) \right. \right. \\ + [ \mathbf {C} ] \left(\frac {2}{\Delta t} \{\mathbf {D} \} _ {n} + \{\dot {\mathbf {D}} \} _ {n}\right) + \{\Delta \mathbf {R} _ {s} \} _ {n + 1} \tag {17.6-18} \\ \end{array}
+$$
+
+From Eq. 17.5-11, the corrective loads can be written as
+
+$$
+\{\Delta \mathbf {R} _ {s} \} _ {n + 1} = \sum \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} \{\sigma^ {\text {err}} \} d V \tag {17.6-19}
+$$
+
+where $\{\sigma^{\mathrm{err}}\}$ is the linear-stress prediction less the actual stresses.
+
+<!-- source-page: 549 -->
+
+Before Eq. 17.6-16 can be solved for $\{\mathbf{D}\}_{n+1}$ , it is necessary to use information at time $n\Delta t$ , for example, from an estimate of $\{\mathbf{D}\}_{n+1}$ , so that $\{\Delta\mathbf{R}_s\}_{n+1}$ can be computed. If solution vector $\{\mathbf{D}\}_{n+1}$ is not in close agreement with the prior estimate (and usually it is not), then $\{\Delta\mathbf{R}_s\}_{n+1}$ must be recomputed using the improved $\{\mathbf{D}\}_{n+1}$ and the solution of Eq. 17.6-16 repeated. This process continues until convergence of $\{\mathbf{D}\}_{n+1}$ and $\{\Delta\mathbf{R}_s\}_{n+1}$ . As an example, consider a uniform linear-displacement bar element with the material shown in Fig. 17.3-3a. If $\epsilon_B$ is the strain obtained from the predicted $\{\mathbf{D}\}_{n+1}$ , then $\sigma_C = E\epsilon_B$ is the stress that would result if material behavior were linearly elastic. Stress $\sigma_C$ , which violates the material's constitutive law, is implied by the term $[\mathbf{K}]\{\mathbf{D}\}_{n+1}$ in Eq. 17.6-15. To satisfy the constitutive law, we compute a “corrective” stress $\sigma_C - \sigma_B$ . The corresponding corrective load in the element is then $A(\sigma_C - \sigma_B)$ and the corrective internal load vector for the element is $\{\Delta\mathbf{r}_s\}_{n+1} = A(\sigma_C - \sigma_B) | -1 - 1|^T$ .
+
+The principal advantage of this method is that $[K^{eff}]$ need be formed and factored only once. Corrective loads $\{\Delta R_{s}\}_{n+1}$ are estimated using only information available at time $n \Delta t$ . Since this estimate is rarely satisfactory, several iterations must usually be performed within a time step to improve $\{\Delta R_{s}\}_{n+1}$ so that the material constitutive law is satisfied. For problems with mild nonlinearities, this method can be more economical than the tangent-stiffness method. When nonlinearities are severe, widespread, or both, this method often demonstrates very poor convergence or divergence and may require a time step that scarcely exceeds a Courant number of unity.
+
+Remarks on Implicit Methods. Solution of nonlinear problems by implicit methods is not easy and there are many pitfalls. Convergence is usually the major difficulty. It is usually good practice to repeat a solution using a smaller time step. Obviously the results of analyses using a different time step should show good agreement if one is to have faith in them.
+
+Of the tangent-stiffness and initial-stiffness methods, the tangent-stiffness method has better convergence properties, but is often much more expensive. However, the initial-stiffness approach sometimes does not converge. A hybrid solution strategy consisting of the initial-stiffness approach with occasional stiffness matrix reformation can be effective and is often used in practice. This strategy can be particularly effective when combined with the inverse-Broyden method of Fig. 17.2-8 to update the initial and revised stiffness matrices.
+
+# 17.7 A PROBLEM HAVING GEOMETRIC NONLINEARITY
+
+Consider the plane cantilever beam shown in Fig. 17.7-1a. We seek the quasistatic deflections produced by loads P and $M_{L}$ . We assume that the beam is slender and that its material is linearly elastic at all times. For small deflections, linear theory is adequate; for example, the root moment is $M_{0} = PL_{T} + M_{L}$ because moment arm $L_{T}$ is almost independent of load. For larger deflections, the moment arm H of force P is less than $L_{T}$ , and $M_{0} = PH + M_{L}$ , where H depends on P and $M_{L}$ . In such a problem, the nonlinearity is called geometric nonlinearity. The name implies that deformations significantly alter the location or distribution of loads, so that equilibrium equations must be written with respect to the deformed geometry, which is not known in advance.
+
+<!-- source-page: 550 -->
+
+![](images/page-550_c725e2722dd41999d5f4c7599ed69d5003057ed846cc063dbd09cc07f8a883a0.jpg)
+
+<details>
+<summary>text_image</summary>
+
+H
+y, v
+M_L
+P
+L_T
+x, u
+</details>
+
+(a)
+
+![](images/page-550_f02f2aa2d29992dcac0a4152d073c693146491728caab8b3514849ca135b2750.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P > P_cr
+P < P_cr
+L_T
+</details>
+
+(b)   
+Figure 17.7-1. (a) Cantilever beam under tip loading. (b) Column under axial load.
+
+To solve this problem we can use plane frame elements—that is, straight elements that have two nodes, six d.o.f., axial stiffness AE, and bending stiffness EI. The equilibrium configuration under a given loading will be obtained by the Newton–Raphson method of Fig. 17.2-3. The particulars given in what follows allow us to solve geometrically nonlinear problems of various plane structures that can be modeled by plane frame elements. Other elements [17.31] and other solution algorithms can also be used, but the same physical concept applies to all: we seek a displacement state in which the deformed structure is in equilibrium with loads applied to it [17.28, 17.29].
+
+A Computational Algorithm. A structure—for example, one of those in Fig. 17.7-1—is divided into finite elements in the usual way. We can begin with zero initial displacements, $\{D\}_{0} = \{0\}$ , or with a better estimate for $\{D\}_{0}$ if one is available. A load $\{R\}$ is applied and the corresponding $\{D\}$ is sought by iteration as follows. Computational details are explained subsequently.
+
+1. Form the tangent-stiffness matrix $[K_{i}]_{i}$ of the structure in the current configuration $\{D\}_{i}$ (which is $\{D\}_{0}$ initially, $\{D\}_{1}$ after the first computational cycle, etc.).   
+2. Use displacements $\{D\}_{i}$ to form resisting loads $\{R_{R}\}_{i}$ , which are loads applied to structure nodes by the deformed elements. (In the first computational cycle, these loads are zero if $\{D\}_{0} = \{0\}$ .)   
+3. Solve for displacement increments $\{\Delta \mathbf{D}\}_{i+1}$ and update the configuration to $\{\mathbf{D}\}_{i+1}$ :
+
+$$
+\{\mathbf {D} \} _ {i + 1} = \{\mathbf {D} \} _ {i + 1} + \{\Delta \mathbf {D} \} _ {i + 1} \quad \text { where } \quad \{\Delta \mathbf {D} \} _ {i + 1} = [ \mathbf {K} _ {t} ] _ {i} ^ {- 1} (\{\mathbf {R} \} + \{\mathbf {R} _ {R} \} _ {i}) \tag {17.7-1}
+$$
+
+Net loads $\{R\} + \{R_{R}\}_{i}$ are an imbalance between externally applied and internally generated nodal loads. The load imbalance drives the structure toward a configuration that reduces the imbalance. In an equilibrium configuration, the imbalance is zero.
+
+4. Check for convergence; for example, see if $\|\Delta D_{i+1}\| < e\|D_{i+1}\|$ , where e is a small number chosen by the analyst. If not converged, return to step 1.
+
+Clearly, many alternatives are possible within this overall strategy. Modified Newton–Raphson cycles can be invoked by updating $[K_{t}]$ only occasionally. Or, the inverse-Broyden method can be used. A final load level $\{R\}$ can be approached
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_056.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_056.md
new file mode 100644
index 00000000..f45f92d8
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_056.md
@@ -0,0 +1,487 @@
+<!-- source-page: 551 -->
+
+in several stages, with convergence required in each stage before going on to the next. Indeed, the latter alternative, or underrelaxation, may be mandatory in some problems that are reluctant to converge (or tend to converge to the wrong result, which is possible if there is more than one equilibrium state that is mathematically possible).
+
+In view of the many physical and computational alternatives that can be tested, coding the foregoing algorithm is a good educational device. Numerous test cases are available [17.32, 17.33]. A particularly simple one is that of Fig. 17.7-1a under tip moment $M_{L}$ alone. The y-direction deflection of the tip is
+
+$$
+v _ {\text { tip }} = \frac {E I}{M _ {L}} \left(1 - \cos \frac {M _ {L} L _ {T}}{E I}\right) \tag {17.7-2}
+$$
+
+which is valid for all values of $M_{L}$ provided that the material remains linearly elastic.
+
+Some Computational Details. Figure 17.7-2 shows a typical element. For the structures of Fig. 17.7-1, initial angle $\alpha_0$ is zero for all elements. Global coordinates $xy$ are fixed in space. A local system $x'y'$ is attached to each element and moves with it: we attach the origin $x' = y' = 0$ to node 1 and direct the $x'$ axis through node 2. Thus, in a local system $x'y'$ , three nodal d.o.f. are always zero: $u_1 = v_1 = v_2 = 0$ . All element d.o.f. in global directions, $D_1$ through $D_6$ , are in general nonzero. We presume that elements are small enough that local rotations are small—that is, that $|\theta_1| << 1$ and $|\theta_2| << 1$ .
+
+In the deformed and displaced configuration, element length projections $x_{L}$ and $y_{L}$ on global axes xy and the orientation $\alpha$ of the local $x'$ axis are
+
+$$
+x _ {L} = x _ {0} + D _ {4 1}, \quad y _ {L} = y _ {0} + D _ {5 2} \quad \alpha = \arctan (y _ {L} / x _ {L}) \tag {17.7-3a}
+$$
+
+where
+
+$$
+D _ {4 1} = D _ {4} - D _ {1} \quad \text { and } \quad D _ {5 2} = D _ {5} - D _ {2} \tag {17.7-3b}
+$$
+
+If $\alpha_{0}=0$ and $\alpha$ is computed in Fortran as DATAN2(YL,XL), then $\alpha$ can reach $\pm\pi$ before it becomes ambiguously defined.
+
+![](images/page-551_08cb62df61a179a3305d31046202214b344f57ba244589c77f8cfa76f6282e03.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+y'
+L0
+D6
+x'
+2
+D4
+y0
+D3
+1
+α0
+D5
+D1
+D2
+x0
+(a)
+x
+</details>
+
+![](images/page-551_f0b54b48e4778d744b5523debcc42da285747725d43fa4deda1f75b07d437989.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+L
+u2
+θ2
+x'
+L0
+u1 = v1 = 0
+θ1
+v2 = 0
+2
+yL
+α
+1
+xL
+(b)
+x
+</details>
+
+Figure 17.7-2. (a) Plane frame element, shown before any deformation or motion, but identifying global d.o.f. $D_{1}$ through $D_{6}$ . (b) The same element after deformation and motion, showing d.o.f. in local coordinates $x'y'$ .
+
+<!-- source-page: 552 -->
+
+Element d.o.f. in the local system $x'y'$ are $\{\mathbf{d}'\} = \begin{bmatrix} 0 & 0 & \theta_1 & u_2 & 0 & \theta_2 \end{bmatrix}^T$ , where
+
+$$
+\theta_ {1} = D _ {3} - (\alpha - \alpha_ {0}) \quad \theta_ {2.} = D _ {6} - (\alpha - \alpha_ {0}) \tag {17.7-4a}
+$$
+
+$$
+u _ {2} = \frac {1}{L + L _ {0}} \left[ \left(2 x _ {0} + D _ {4 1}\right) D _ {4 1} + \left(2 y _ {0} + D _ {5 2}\right) D _ {5 2} \right] \tag {17.7-4b}
+$$
+
+where again $D_{41} = D_{4} - D_{1}$ and $D_{52} = D_{5} - D_{2}$ . The expression for $u_{2}$ in Eq. 17.7-4b is more accurate than the expression $u_{2} = L - L_{0}$ (in which $L^{2} = x_{L}^{2} + y_{L}^{2}$ and $L_{0}^{2} = x_{0}^{2} + y_{0}^{2}$ ) because Eq. 17.7-4b avoids finding the small difference between large numbers. Equation 17.7-4b is obtained by writing $L^{2} - L_{0}^{2}$ , substituting for $x_{L}^{2}$ and $y_{L}^{2}$ from Eq. 17.7-3, factoring $L^{2} - L_{0}^{2}$ , and solving for $u_{2} = L - L_{0}$ [17.30]. In the denominator, $L + L_{0} \approx 2L_{0}$ , as we presume that strains are small.
+
+In the local system $x'y'$ , loads $\{\mathbf{r}'\}$ applied to nodes by the distorted element can be obtained from Eq. 4.1-6, using for $\{\sigma_0\}$ the stresses produced by local d.o.f. $\{\mathbf{d}'\}$ , that is, $\{\sigma_0\} = [\mathbf{E}][\mathbf{B}]\{\mathbf{d}'\}$ . An alternative calculation is
+
+$$
+\left\{\mathbf {r} ^ {\prime} \right\} = - \left[ \mathbf {k} ^ {\prime} \right] \left\{\mathbf {d} ^ {\prime} \right\} \tag {17.7-5}
+$$
+
+in which the element stiffness matrix $[k']$ in the local system does not change as the local system moves and the element deforms. Here $[k']$ is given by Fig. 7.5-2a. If axial forces are significant, particularly in compression, one should add to this matrix a stress stiffness matrix $[k_{\sigma}]$ , for example, Eq. 14.2-11b (with zeros added to expand the matrix to size 6 by 6). In $[k_{\sigma}]$ , the element axial force P is given by $P = (AE/L)u_{2}$ . Thus P is deformation-dependent, and may be considered unknown at the outset of the iterative process.
+
+Referred to global coordinates $xy$ , the element stiffness matrix and element nodal load vector are
+
+$$
+[ \mathbf {k} ] = [ \mathbf {T} ] ^ {T} [ \mathbf {k} ^ {\prime} ] [ \mathbf {T} ] \quad \text { and } \quad \{\mathbf {r} \} = [ \mathbf {T} ] ^ {T} \{\mathbf {r} ^ {\prime} \} \tag {17.7-6}
+$$
+
+where, for the frame element, transformation matrix [T] is given by Eq. 7.5-7 (with $\beta$ replaced by $\alpha$ ). In general, each element has a different $\alpha$ and therefore requires a different [T]. The structure tangent-stiffness matrix and resisting load vector in global coordinates are formed by the usual assembly process, which is symbolized by
+
+$$
+[ \mathbf {K} _ {t} ] = \sum [ \mathbf {k} ] \quad \text { and } \quad \{\mathbf {R} _ {R} \} = \sum \{\mathbf {r} \} \tag {17.7-7}
+$$
+
+Note that coordinate transformation is not the essence of an analysis procedure for geometric nonlinearity, but only the vehicle adopted here to establish what is essential: the properties of the system in its current configuration.
+
+# 17.8 OTHER NONLINEAR PROBLEMS
+
+Nonlinear computational mechanics continues to be an active area of research. Some of the nonlinear structural problems we have not discussed in this book are as follows.
+
+<!-- source-page: 553 -->
+
+Materials that creep are analyzed by use of constitutive relations and computational algorithms very similar to those used for analysis of plastic deformations. Indeed, creep and plasticity analyses are often combined [17.12, 17.34].
+
+Forming processes such as extrusion and rolling involve large plastic deformations. Elastic response may be considered negligible, and the material analyzed as if it were a viscous fluid. “Springback” after the forming process is complete can be modeled if a viscoelastic material is invoked. Casting processes involve considerable heat transfer calculations and changes of phase $[17.35]$ .
+
+Problems of moving contact fields, such as the rolling contact of a tire on pavement, have been addressed by special algorithms $[17.36]$ . Other contact problems, either static or dynamic, include bearings, joints in rock, and gaps that may open or close. Special elements for such problems have been devised $[17.13, 17.37, 17.38]$ .
+
+Membranes may be deflected by pressure, as when a balloon is inflated. Such a problem typically involves large strains, large deflections, and large rotations. Pressure loads change in direction, as they continue to act normal to the membrane. Element nodal loads $\{r_{e}\}$ also change in magnitude as pressure increases and as the element surface area subjected to pressure increases [17.39]. If a membrane is initially flat and initially unstressed, it has no resistance to lateral load. To avoid a singular structure stiffness matrix in the first iterative cycle, one can add a fictitious initial stress for the first cycle only, thereafter to be replaced by the computed stress and corresponding stress stiffness matrix.
+
+Cable problems are somewhat similar to membrane problems. One can follow the standard approach of dividing each cable of a cable network into many elements (e.g., two-node bar elements). However, this approach is inefficient if deflections are large. A method that treats an entire cable as a single element appears to be much more economical [17.40, 17.41].
+
+A nonlinear vibration problem leads to an eigenvalue problem of the standard symbolic form—that is, $([K] - \omega^{2}[M])\{\overline{D}\} = \{0\}$ ; however, stiffness matrix [K] is a function of $\{\overline{D}\}$ , the nodal amplitudes [17.42].
+
+# PROBLEMS
+
+# Section 17.2
+
+17.1 Sketch the progress of the two forms of direct substitution, as in Fig. 17.2-2, but let the curves be concave up, as for a hardening structure.
+
+17.2 Load P acts on a nonlinear spring, as shown. Let $k = 0.2 - u$ and P = 0.006. Apply three cycles of direct substitution according to Eq. 17.2-3. Also, apply three cycles of modified Newton–Raphson iteration, using k at u = 0. Thus, show that both methods yield the same results.
+
+![](images/page-553_708c8980799863c356bf23d68590d6005a40f29e3f5152f7d64575a2fed62e30.jpg)  
+Problem 17.2
+
+17.3 In Problem 17.2, what is the expression for the tangent stiffness $k_{t}$ in terms of displacement $u$ ?
+
+<!-- source-page: 554 -->
+
+![](images/page-554_c96b368c91dbe1a29962816d656ba2616e797e2706be5e4068ec3c74c69ecab2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k = 24 N/m
+P
+0.02 m
+Rigid
+k = 10 N/m
+D
+100 mm
+200 mm
+</details>
+
+Problem 17.4
+
+17.4 The bar shown in the sketch is rigid. Contact with the right-hand spring is made only when the $0.02\mathrm{-m}$ gap closes. Set up an equation for displacement $D$ of load $P$ , then apply three cycles of a direct-substitution solution for $D$ when $P = 0.24\mathrm{N}$ , as follows:
+
+(a) Use the procedure in which stiffness $K$ is repeatedly updated but the right-hand side (i.e., load $P$ ) is unchanged.   
+(b) Use the procedure in which $K$ is unchanged from its initial value, but the effective load is repeatedly updated.
+
+17.5 The bar shown is of length $L$ when unstressed, where $L^2 = a^2 + c^2$ . When load $P$ is zero, displacement $D$ is also zero. The bar has axial stiffness $AE / L$ , rolls without friction at $B$ , and does not buckle as a column. Assume that the roller is constrained to remain in contact with the wall.
+
+(a) For $a >> c$ , show that $\Pi_p = U - PD = (AE/8a^3)(-2cD + D^2)^2 - PD$ .   
+(b) Show that the equilibrium values of $D$ are given by roots of the equation $P = (AE / 2a^3)(2cD - D^2)(c - D)$ .   
+(c) Determine expressions for the secant stiffness and the tangent stiffness.   
+(d) Show that limit points are at $D = c(1 \pm \sqrt{3}/3)$ .   
+(d) Show that limit points are at $D = 0$ , $F = 241$ , and $F_{\mathrm{D}} = 241$ . Let constants of this system be such that points $A$ and $F$ on the $P$ versus $D$ curve are at $P_A = 241$ , $D_A = 0.211$ , $P_F = 250$ , and $D_F = 1.080$ . After convergence at $P = 200$ , $P$ is increased to 250, and the following sequence of displacements $D$ is generated by a Newton-Raphson algorithm: 0.173, 0.219, 0.071, 0.143, 0.190, 0.249, 0.199, 0.294, 0.235, 0.175, 0.222, 0.108, 0.166, 0.210, 1.178, 1.096, 1.080, 1.080. Explain this path to convergence by sketching it on the $P$ versus $D$ plot.   
+(f) Similarly, sketch the path that would be taken by a modified Newton-Raphson algorithm.
+
+![](images/page-554_904410ad781ad85e5e2fbf80075f534c72cfd1537f3fdac6220c2228ecb9296d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+a
+B
+A
+P
+c
+D
+</details>
+
+(a)
+
+![](images/page-554_40dc126ca4cff78f9740c0d6f0114ea29ecf940fd143aeacaf0a6af1a3d69c81.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+A
+0
+B
+C
+E
+D
+F
+</details>
+
+(b)   
+Problem 17.5
+
+17.6 Let loads $P_{1}$ and $P_{2}$ be functions of displacements $D_{1}$ and $D_{2}$ , that is, $P_{1} = f_{1}(D_{1}, D_{2})$ and $P_{2} = f_{2}(D_{1}, D_{2})$ . Let $D_{A}$ and $D_{B}$ be exact values of $D_{1}$
+
+<!-- source-page: 555 -->
+
+and $D_{2}$ produced by loads $P_{A}$ and $P_{B}$ . Let $D_{A}^{*}$ and $D_{B}^{*}$ be approximations of $D_{A}$ and $D_{B}$ . Assume that $D_{A} = D_{A}^{*} + \Delta D_{A}$ and $D_{B} = D_{B}^{*} + \Delta D_{B}$ . Derive the following equations (analogous to Eq. 17.2-9):
+
+$$
+\left[ \begin{array}{c c} \partial P _ {1} / \partial D _ {1} & \partial P _ {1} / \partial D _ {2} \\ \partial P _ {2} / \partial D _ {1} & \partial P _ {2} / \partial D _ {2} \end{array} \right] _ {D _ {A} ^ {*}, D _ {B} ^ {*}} \left\{ \begin{array}{c} \Delta D _ {A} \\ \Delta D _ {B} \end{array} \right\} = \left\{ \begin{array}{c} P _ {A} - f _ {1} (D _ {A} ^ {*}, D _ {B} ^ {*}) \\ P _ {B} - f _ {2} (D _ {A} ^ {*}, D _ {B} ^ {*}) \end{array} \right\}
+$$
+
+17.7 The sketch shows a nonlinear load versus deflection curve. In the exercises of this problem we pretend that $P$ and $dP / dD$ can be found when $D$ is known, but that an explicit expression for $D$ in terms of $P$ is not available. Solve for $D$ , using the situations and methods indicated. Sketch the progress of each solution on a plot of $P$ versus $D$ .
+
+(a) What $D$ is predicted by five cycles of Newton-Raphson iteration if $P = 8$ , starting from $P = D = 0$ ?   
+(b) What $D$ is predicted by five cycles of modified Newton-Raphson iteration if $P = 8$ , starting from $P = 7.5$ , $D = 3$ ?   
+(c) Repeat part (b) but use three cycles and update the tangent stiffness after the first cycle.   
+(d) What $D$ is predicted by four purely incremental (Euler's method) steps of $\Delta P = 2$ , starting from $P = 1$ and going to $P = 9$ ? Given: $D = 0.11111$ at $P = 1$ .   
+(e) Repeat part (d) but include a force imbalance correction at every step.   
+(f) What $D$ is predicted by five cycles of the direct-substitution algorithm of Eq. 17.2-2 if $P = 8$ , starting from $P = D = 0$ ?   
+(g) What $D$ is predicted by five cycles of the secant method depicted in Fig. 17.2-7 if $P = 8$ , starting from $P = D = 0$ ?
+
+17.8 The introductory remarks of Problem 17.7 again apply, but now to the hardening curve sketched.
+
+(a) What $D$ is predicted by five cycles of Newton-Raphson iteration if $P = 0.8$ , starting from $P = D = 0$ ?   
+(b) What $D$ is predicted by five cycles of modified Newton-Raphson iteration if $P = 3$ , starting from $P = 1.5$ , $D = 6$ ?   
+(c) Repeat part (b) but apply an underrelaxation factor $\beta = 0.6$ to the increments $\Delta D$ .   
+(d) Repeat part (b) but use four cycles and update the tangent stiffness after the first cycle.   
+(e) What $D$ is predicted by three purely incremental (Euler's method) steps
+
+![](images/page-555_c721cee903ddbf98592171bbbadf3410236911f7ab8b3050c1ff8312896dad8a.jpg)
+
+<details>
+<summary>line</summary>
+| D       | P        |
+|---------|----------|
+| 0       | 0        |
+| >0      | 10D/D + 1|
+</details>
+
+Problem 17.7
+
+![](images/page-555_4f6e01c2fd11aed5bf724536b4a6203d8e80e400b0a1763db6e97428968c670e.jpg)
+
+<details>
+<summary>line</summary>
+
+| D | P |
+|---|---|
+| 0 | 0 |
+| 10 | D / (10 - D)² |
+| 20 | D / (10 - D)² |
+</details>
+
+Problem 17.8
+
+<!-- source-page: 556 -->
+
+of $\Delta P = 1$ , starting from $P = 1.5$ and going to $P = 4.5$ ? Given: $D = 6$ at $P = 1.5$ .
+
+(f) Repeat part (e) but include a force imbalance correction at every step.   
+(g) What $D$ is predicted by four cycles of the direct-substitution algorithm of Eq. 17.2-2 if $P = 4$ , starting from $P = D = 0$ ?   
+(h) Repeat part (g) but apply an underrelaxation factor $\beta = 0.3$ in Eq. 17.2-5b.   
+(i) What $D$ is predicted by five cycles of the secant method in Fig. 17.2-7 if $P = 0.8$ , starting from $P = D = 0$ ?   
+(j) Repeat part (i), but apply an underrelaxation factor $\beta = 0.3$ to the increments $\Delta D$ .
+
+17.9 Use four cycles of the secant method depicted in Fig. 17.2-7 to calculate iterates $u_{i}$ for the spring problem posed in Problem 17.2.
+
+17.10 The Newton-Raphson method to solve $f(x) = 0$ can be formulated by defining an iteration function
+
+$$
+g (x) = x - \frac {f (x)}{f ^ {\prime} (x)}
+$$
+
+and seeking a solution $x^{*}$ that satisfies $x^{*} = g(x^{*})$ by taking a given $x_{0}$ and iterating: $x_{i+1} = g(x_{i})$ .
+
+(a) Verify that $x^{*}$ satisfies $f(x^{*}) = 0$ provided that $f'(x^{*}) \neq 0$ .   
+(b) Use a three-term exact Taylor series for $g(x)$ , that is,
+
+$$
+g (x) = g \left(x ^ {*}\right) + g ^ {\prime} \left(x ^ {*}\right) \left(x - x ^ {*}\right) + \frac {1}{2} g ^ {\prime \prime} (\bar {x}) \left(x - x ^ {*}\right) ^ {2}
+$$
+
+where $\overline{x}$ lies between $x$ and $x^{*}$ , to verify that the Newton-Raphson method terminates quadratically.
+
+17.11 Use Eq. 17.2-1, $(k_0 + k_N)u = P$ , to show that the principle of superposition does not apply to a nonlinear problem.
+
+# Section 17.3
+
+17.12 The bar shown is to be modeled by a single element. Thus there is one d.o.f.—namely, the displacement $u_{2}$ of load P. Apply the tangent-stiffness method to determine $u_{2}$ for P = 3.0 kN and for P = 6.0 kN. Use two load increments of $\Delta P = 3.0$ kN. Start from P = 0. Use two cycles of step 3 of the algorithm. Plot $u_{2}$ versus P, showing the exact solution and the progress of the incremental solution.
+
+![](images/page-556_bee830e8fcb36519190e43789f9a7c9e9beb464b962113b80251a8809b30be52.jpg)
+
+<details>
+<summary>line</summary>
+| ε       | σ, MPa |
+| ------- | ------ |
+| 0.001   | 20     |
+| 0.001   | 0      |
+</details>
+
+Problem 17.12
+
+<!-- source-page: 557 -->
+
+17.13 Repeat Problem 17.12 but apply only the 3.0-kN load. Start from $P = 2.0$ kN, apply $\Delta P = 1.0$ kN, and carry out five cycles of the initial-stiffness method. Compute the percentage error in the resulting $u_{2}$ .   
+17.14 The two bars shown are fixed to rigid walls at their outer ends and are welded together where load $P$ is applied. The material behavior is shown in the sketch. Use the tangent-stiffness method and the successive load increments $\Delta P = 20$ , $\Delta P = 10$ , and $\Delta P = -30$ . Determine the corresponding values of displacement $D$ . Show results on a plot of $P$ versus $D$ .
+
+![](images/page-557_00527c97756d42998fccd3c28c20c8aa553cab22cc2628c8f4dadb69acf323c5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+10
+10
+D
+P
+A = 1.0 throughout
+</details>
+
+![](images/page-557_cb2ab24a68937f92c46477aca09a3970336710a8fa820e974297473a23a30e02.jpg)  
+Problem 17.14
+
+17.15 For the bar sketched in Problem 17.14, apply the single load increment $\Delta P = 10$ after the yield point value of load $P$ is reached. Determine the value of displacement $D$ predicted by five cycles of the initial-stiffness algorithm.   
+17.16 The horizontal bar in the sketch is perfectly rigid and is constrained to remain horizontal as load $P$ and displacement $v$ increase. The three vertical bars are elastic-perfectly plastic with $A = 1$ , $E = 1$ and $L = 2$ . These bars have the respective yield point loads $F_{1} = 2$ , $F_{2} = 4$ , and $F_{3} = 6$ . Use the tangent-stiffness method to generate the $P$ versus $v$ relation. Use three steps. Scale each step so that one bar begins to yield at the end of the step.
+
+![](images/page-557_2ab33b0cc4048d434cb6d9e9bc81c78b5dd583ffbbe2ef0fdf46ff2415492fda.jpg)
+
+<details>
+<summary>text_image</summary>
+
+①
+②
+③
+L
+P, v
+</details>
+
+Problem 17.16
+
+17.17 A weightless and rigid block $B$ is pushed down in a frictionless guide by force $P$ . The force versus deflection plot for each of two supporting bars is given in the sketch. Solve for displacement $D$ of block $B$ under a force $P = 24 \mathrm{~N}$ , as follows.
+
+(a) Determine the exact solution.   
+(b) Use the tangent-stiffness method. Let the first load increment be $\Delta P = 19\mathrm{N}$ .   
+(c) Use the initial-stiffness method. Apply one load increment $\Delta P = 5$ N starting from the exact solution at P = 19 N.
+
+<!-- source-page: 558 -->
+
+![](images/page-558_e4cf912b75b87ba2854aeba1e97bf4290816b66956282adcc1e3ec07502d5392.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+B
+100 mm
+D
+Force, N
+10
+9
+1
+2
+Deflection, mm
+0
+1
+4
+</details>
+
+Problem 17.17
+
+17.18 Assume that members of a truss carry only uniaxial stress, and that members in compression will buckle elastically at their critical loads without yielding. Tensile members are elastic-perfectly plastic. Outline a tangent-stiffness algorithm for computation of the displacements produced by applied loads. For simplicity, assume that load reversal does not occur in any member.   
+17.19 In both the initial-stiffness algorithm and the tangent-stiffness algorithm, factor m of Eq. 17.3-4 is used only to correct a trial solution after it is computed. Can the correction be anticipated instead? Explain.
+
+# Section 17.4
+
+17.20 Verify Eq. 17.4-7.   
+17.21 Show that Eq. 17.4-13 becomes $F = \sigma_{a} - \sigma_{Y}$ when a state of uniaxial stress $\sigma_{a}$ is defined in the following ways. Let $\{\alpha\} = \{\mathbf{0}\}$ .
+
+(a) $\sigma_{x} = \sigma_{a}, \sigma_{y} = \sigma_{z} = \tau_{xy} = \tau_{yz} = \tau_{zx} = 0.$   
+(b) $\sigma_{x} = \sigma_{y} = \tau_{xy} = \sigma_{a} / 2, \sigma_{z} = \tau_{yz} = \tau_{zx} = 0.$   
+(c) $\sigma_{x} = 0.8\sigma_{a}, \sigma_{y} = 0.2\sigma_{a}, \tau_{xy} = 0.4\sigma_{a}, \sigma_{z} = \tau_{yz} = \tau_{zx} = 0.$
+
+17.22 Verify that Eq. 17.4-14 follows from Eq. 17.4-13.   
+17.23 Imagine that plastic action in bending is to be modeled and that several sampling points are used in the thickness direction (d in the sketch).
+
+(a) Why should the sampling points pertain to a trapezoidal or Simpson quadrature rule rather than to a Gauss–Legendre quadrature rule?   
+(b) Imagine that the stress distribution shown prevails across the depth of a beam of rectangular cross section. What is the percentage error of the computed bending moment $M_c$ , if $M_c$ is integrated from the stress distribution using a two-point Gauss rule? Repeat the calculation using a three-point Gauss rule.   
+(c) Repeat part (b), but use trapezoidal rules instead of Gauss rules. Try three, five, seven, and then nine sampling points.
+
+![](images/page-558_4c7eddf10b07783d5b249982877974102f6924ff466e6e738f3a549bbe242cff.jpg)
+
+<details>
+<summary>text_image</summary>
+
+d
+σₐ
+σₐ
+d/4
+d/4
+d/4
+d/4
+</details>
+
+Problem 17.23
+
+<!-- source-page: 559 -->
+
+17.24 Imagine that a state of uniaxial stress causes plastic strains. Show that Eq. 17.4-15 yields the correct value of $\epsilon_{\mathrm{ef}}^{p}$ if
+
+(a) The stress is parallel to the $x$ axis.
+
+(b) The stress acts at 45 degrees to the $x$ and $y$ axes.
+
+# Section 17.5
+
+17.25 Describe the steps of a tangent-stiffness solution algorithm in which each load increment causes a single sampling point to be brought to the initiation of yielding.
+
+17.26 Consider a plane structure modeled by finite elements. The material is isotropic but brittle: it cracks when the tensile stress in any direction exceeds a value $\sigma_{t}$ . Outline a tangent-stiffness algorithm for predicting deformations caused by increasing load. How will the collapse load be detected by this algorithm?
+
+17.27 Imagine that corrective loads $\{\Delta R_{c}\}$ are to be computed for a mesh of elements having internal d.o.f. Should internal d.o.f. carry loads that result from $\{\sigma\}$ , or should these loads be omitted from internal d.o.f.? If carried, should they be distributed to remaining d.o.f. (that is, condensed) by means of elastic element stiffness equations?
+
+17.28 Imagine that a plane beam of rectangular cross section is modeled by plane finite elements. The material is linearly elastic, but elastic moduli in tension and compression are different. Outline an algorithm that will calculate the stresses produced by a pure bending load. What are the comparative merits of tangent-stiffness and initial-stiffness solutions?
+
+# Section 17.6
+
+17.29 Consider an elastic-perfectly plastic material with an associated flow rule. The constitutive law for such a material is given by Eqs. 17.4-7 through 17.4-9 with $C = 0$ , $\partial F / \partial W_p = 0$ , and $F = Q$ . Equation 17.4-9 can be written as $[\mathbf{E}_{\mathrm{ep}}] = [\mathbf{E}] + [\mathbf{E}_p]$ , where $[\mathbf{E}_p] = -[\mathbf{E}]\{\partial F / \partial \sigma\} \{\mathbf{C}_\lambda\}^T$ . Using Eq. 17.4-7, show that $[\mathbf{E}_p]$ is negative semidefinite.
+
+17.30 Starting with the basic definition of the rate of internal work for an element $e$ as
+
+$$
+\dot {W} _ {e} ^ {\mathrm{int}} = \int_ {V _ {e}} \{\dot {\epsilon} \} ^ {T} \{\sigma \} d V
+$$
+
+show that the rate of internal work for the entire structure is $\dot{W}^{\mathrm{int}} = \{\dot{\mathbf{D}}\}^T\{\mathbf{R}^{\mathrm{int}}\}$ .
+
+17.31 Starting with $\dot{W}_n^{\mathrm{int}} = \{\dot{\mathbf{D}}\}_n^T\{\mathbf{R}^{\mathrm{int}}\}_n$ , show that for linearly elastic material behavior, $W_n^{\mathrm{int}} = \frac{1}{2}\{\mathbf{D}\}_n^T[\mathbf{K}]\{\mathbf{D}\}_n$ . Note: $\frac{d}{dt} (\{\mathbf{D}\}^T[\mathbf{K}]\{\mathbf{D}\}) = 2\{\dot{\mathbf{D}}\}^T[\mathbf{K}]\{\mathbf{D}\}$ if [K] is symmetric.
+
+17.32 Verify that Eq. 17.6-3 results from Eq. 17.6-2.
+
+17.33 Of the energies $W^{int}$ , $W^{ext}$ , and T, and the energy rates $\dot{W}^{int}$ and $\dot{W}^{ext}$ , which are always nonnegative and which can be positive, zero, or negative? Assume that the material is linearly elastic.
+
+<!-- source-page: 560 -->
+
+17.34 Repeat the example of Section 17.6 using different time steps and longer analysis duration.   
+17.35 Repeat the example of Section 17.6 using the nonlinearly elastic stiffening material shown (in which loading and unloading are both on the same path). It will be necessary to write a subroutine similar to Fig. 17.6-1 for this constitutive law. Note that the transition between slopes E and $E_{t}$ may occur during both loading and unloading. Analyze response for approximately one wave traversal along the bar. Carefully address: (a) the largest stable time step; (b) the correctness of the stress wave shape, and (c) the wave arrival time(s).
+
+![](images/page-560_2d260b08a797f9685b27543a35d89fed37cf3e1ea1651a5b154ac027002752f9.jpg)
+
+<details>
+<summary>line</summary>
+
+| ε       | σ     |
+| ------- | ----- |
+| 0       | 0     |
+| E       | 30(10)⁶ psi |
+| E_t     | 4E    |
+</details>
+
+Problem 17.35
+
+17.36 Derive Eqs. 17.6-11 through 17.6-13.
+
+# Section 17.7
+
+17.37 (a) If $P = 1.884P_{\mathrm{cr}}$ in Fig. 17.7-1b, the tip of the column rotates $120^{\circ}$ from its unloaded position. To what other equilibrium configuration might the numerical process converge? Answer qualitatively, without calculation.
+
+(b) Repeat part (a) if force $P$ is supplemented by a vertically directed tip force $Q = 0.001P_{\mathrm{cr}}$ . Sketch your answer.
+
+17.38 Verify Eq. 17.7-2, and show that it agrees with the formula $M_L L_T^2 / 2EI$ of elementary beam theory if $M_L$ is small.
+
+17.39 (a) Derive the expression for $u_{2}$ , Eq. 17.7-4b.
+
+(b) Show that $u_{2} = 0$ for a rigid-body rotation $\alpha$ about node 1, starting from $\alpha_0 = 0$ .
+
+17.40 Imagine that $M_L = 0$ in Fig. 17.7-1a and that the right end of the cantilever beam has rotated about $45^\circ$ under the action of force $P$ . Let the beam be divided into several elements. For the rightmost element, qualitatively sketch loads $[\mathbf{k}']\{\mathbf{d}'\}$ of Eq. 17.7-5 in the local system $x'y'$ and loads $\{\mathbf{r}\}$ of Eq. 17.7-6 in the global system $xy$ . Show these loads in the directions they actually act. Which of these loads will be zero after the iterative numerical process has converged?
+
+17.41 A long uniform beam has bending stiffness $EI$ , weight $q$ per unit length, and rests on a flat horizontal surface, as shown. A vertical force $F$ , where $F < qL_T / 2$ , is applied to one end.
+
+(a) Analytically determine the length $L_{s}$ that lifts off the surface $(L_{s} < L_{T})$ . (b) Imagine that the beam is modeled by several elements. Outline a numerical algorithm for calculating length $L_{s}$ .
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_057.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_057.md
new file mode 100644
index 00000000..a91cd689
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_057.md
@@ -0,0 +1,431 @@
+<!-- source-page: 561 -->
+
+![](images/page-561_f8fd88990c5d742d159fcc24e2ddfd7bb2e00f4898a1cc98706ce7deeaf72383.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L_T
+F
+L_S
+</details>
+
+Problem 17.41
+
+![](images/page-561_7e16432682a62f9dbe226c42fdde2209f486533f2e3925f1c71d648787a7a8cc.jpg)
+
+<details>
+<summary>text_image</summary>
+
+F
+B
+a
+A
+C
+a
+a
+</details>
+
+Problem 17.42
+
+17.42 The angle frame shown has uniform bending stiffness EI. At C, a small frictionless roller contacts a vertical surface.
+
+(a) Solve for the vertical deflection at $C$ caused by force $F$ , using mechanics of materials methods. Assume that this deflection is small.   
+(b) Solve by a computer program, for example, based on the algorithm described in Section 17.7. What difficulty appears in the first iterative cycle, and how will you overcome it?
+
+17.43 The algorithm described in Section 17.7 can be applied to a plane body modeled by four-node quadrilaterals. Describe in detail how one might track the rigid-body motion of a quadrilateral and compute its d.o.f. $\{d'\}$ in a local coordinate system attached to the element. Assume that all displacements lie in the xy plane.
+
+17.44 When P and its displacement D are zero, the springs are unstressed (see sketch). Spring stiffness $k_{0}$ is constant.
+
+(a) Determine the secant stiffness (the ratio of $P$ to $D$ ) for small values of $D$ ( $D << L$ ).   
+(b) What is the tangent stiffness for small values of $D$ ?   
+(c) Determine an expression for the secant stiffness if $D$ may be moderate to large.   
+(d) Describe an iterative way to calculate $D$ for a given $P$ , with particular emphasis on the first step.
+
+17.45 The frame shown is built of slender members that may be assumed to remain linearly elastic at all times. Connections at $A$ and $B$ are frictionless pins.
+
+(a) Qualitatively sketch the anticipated relation between load $P$ and its vertical displacement $D$ .   
+(b) How would you generate the $P$ versus $D$ relation numerically? That is, what difficulties do you anticipate, and how might you avoid or overcome them?
+
+![](images/page-561_1cb9e3227ba584a57b8dca19006f3835e4543fc2665afde9c82b76348a0d2e39.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k₀ P k₀
+D
+L L
+</details>
+
+Problem 17.44
+
+![](images/page-561_211042940f3656d4dbc3f9c5f334a6c2858d2742aa7145df0f67a76c2dae1a9d.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+D
+C
+B
+A
+</details>
+
+Problem 17.45
+
+<!-- source-page: 562 -->
+
+# NUMERICAL ERRORS AND CONVERGENCE
+
+Sources of computational error are categorized. Methods of detecting and avoiding such errors are presented. The relation between element size and solution error is discussed. Tests of element quality are reviewed.
+
+# 18.1 INTRODUCTION. ERROR CLASSIFICATION
+
+Computed results are rarely exact. Some of the many reasons are as follows. We divide a structure into elements whose displacement fields exclude many of the physically possible deformation modes. The type, number, and shapes of elements may be chosen within a broad range of possibilities, and some choices are better than others. The computer represents numbers by a finite number of bits or digits.
+
+Numerical difficulties may arise even when the analyst makes no outright blunder in using a computer program. In this chapter we assume that there are no outright blunders, that is, that the program used is appropriate to the task at hand, that the program is free of bugs, that the choices of element types, shapes, and quadrature orders are suitable, and that elements pass patch tests and do not lock. Some choices remain, such as the specific type and number of elements, their arrangement, and the way in which relatively stiff regions are treated. How these choices are made can increase or decrease numerical error, which is caused by the inability of a computer to store and process numbers in infinite precision.
+
+There is no single definitive test of solution accuracy short of knowing the correct result by other means. A calculation that survives one error test may fail another. A usually reliable error test may fail in particular cases. For example, given a single-precision matrix [H], one may calculate $[X] = [H]^{-1}$ , then $[Y] = [X]^{-1}$ . If [Y] agrees with [H], one expects [X] to be correct. In fact, when [H] is a tenth-order Hilbert matrix $(H_{ij} = (i + j - 1)^{-1}$ , which is notoriously ill conditioned), one study found that [Y] = [H] to seven-digit accuracy, yet [X] had coefficients in error by three orders of magnitude [18.1].
+
+Granted that some loss of precision is possible or even likely in finite element calculations, one should not introduce gratuitous errors: numerical constants, such as $\pi$ and Gauss point coordinates and weights, should be written with as many accurate digits as the machine can accommodate.
+
+Terminology. The following terms are useful in subsequent discussion [18.2]. This abbreviated list shows at once that there are several aspects to the study of numerical errors.
+
+Modeling error refers to the difference between a physical system and its math-
+
+<!-- source-page: 563 -->
+
+ematical model. For example, an actual plate may be mathematically modeled by Kirchhoff plate theory. Numerical analysis is performed on the mathematical model. It is this analysis that is subject to the following errors.
+
+Discretization error refers to the error caused by representing the infinitely many d.o.f. of a continuous mathematical model by a finite number of d.o.f. in its discretized form. For example, the aforementioned Kirchhoff plate is divided into finite elements, thus introducing discretization error.
+
+Round-off error is caused by use of a finite number of bits or digits to represent real numbers. The last digit retained may be rounded or may be obtained by simple truncation (chopping). The round-off limit is the smallest floating point number $\epsilon$ such that, in the computer, $1.0 + \epsilon > 1.0$ . For example, one may find $\epsilon = 7(10^{-15})$ on some machine in double-precision arithmetic.
+
+Inherited error at any stage of calculation is the sum of previous discretization and round-off errors.
+
+Manipulation error refers to round-off error introduced by an algorithm. For example, an equation solver performs numerical operations such as $A_{22} - (A_{21}/A_{11})A_{12}$ , where the $A_{ij}$ are matrix coefficients. The division and multiplication are each followed by rounding of the result to computer-word length, and the subtraction may lose several significant digits if $A_{22}$ and $(A_{21}/A_{11})A_{12}$ are almost equal. For example, if $R = D_{1} - D_{2}$ , where $D_{1} = 1.23456$ and $D_{2} = 1.23455$ , then $R = 1.00000(10^{-5})$ , which contains but one significant digit in its six-digit mantissa and is therefore far less accurate than either $D_{1}$ or $D_{2}$ .
+
+In a finite element context, inherited error is present after disparate element stiffnesses are added to form structural stiffness coefficients, and manipulation error is produced by solving the equations. Alternative terms for these errors are truncation and rounding. Inherited error and manipulation error have the same source: the round-off limit is not zero.
+
+# 18.2 ILL-CONDITIONING
+
+Concepts from Algebra. Consider the set of equations
+
+$$
+\left[ \begin{array}{c c} 1. 0 0 & - 1. 0 0 \\ - 1. 0 0 & 1. 0 2 \end{array} \right] \left\{ \begin{array}{l} x \\ y \end{array} \right\} = \left\{ \begin{array}{c} 4. 0 0 \\ - 2. 0 0 \end{array} \right\} \quad \text { for   which } \quad \left\{ \begin{array}{l} x \\ y \end{array} \right\} = \left\{ \begin{array}{l} 1 0 4 \\ 1 0 0 \end{array} \right\} \tag {18.2-1}
+$$
+
+and the very similar set of equations
+
+$$
+\left[ \begin{array}{c c} 1. 0 0 & - 1. 0 0 \\ - 1. 0 0 & 1. 0 1 \end{array} \right] \left\{ \begin{array}{l} x \\ y \end{array} \right\} = \left\{ \begin{array}{c} 4. 0 0 \\ - 2. 0 0 \end{array} \right\} \quad \text { for   which } \quad \left\{ \begin{array}{l} x \\ y \end{array} \right\} = \left\{ \begin{array}{l} 2 0 4 \\ 2 0 0 \end{array} \right\} \tag {18.2-2}
+$$
+
+A 1% change in one coefficient has changed the results by a factor of two. These equation sets are both ill conditioned, which means that their solutions are sensitive to small changes in either the coefficient matrix or the vector of constants.
+
+The solution of Eq. 18.2-1 can be regarded as the intersection point in xy space of two straight lines, one representing the equation x - y = 4 and the other representing the equation $-x + 1.02y = -2$ . When plotted, the two lines are seen to be almost parallel. Accordingly, when the second equation is changed to
+
+<!-- source-page: 564 -->
+
+$-x + 1.01y = -2$ (Eq. 18.2-2), the second line is rotated slightly, and the intersection point changes markedly.
+
+In matrix terminology, the rows of an ill-conditioned matrix are almost linearly dependent. For 2 by 2 systems such as Eqs. 18.2-1 and 18.2-2, this means that the second row of the coefficient matrix is almost a scalar multiple of the first row.
+
+A Gauss elimination solution of these equations changes the second diagonal coefficient in Eq. 18.2-1 to 1.02 - 1.00 = 0.02. If only two digits were retained, all coefficients would be represented as 1.0 and the matrix would be singular. Gauss elimination would produce 1.0 - 1.0 = 0.0 as the second diagonal coefficient and the computation y = 2.0/0.0 would be attempted.
+
+Comparatively Flexible Support. In Fig. 18.2-1, the circumstance $k_{1} >> k_{2}$ promotes ill-conditioning but the circumstance $k_{2} >> k_{1}$ does not. This is easy to see by examining the structure equations [K] $\{D\} = \{R\}$ , which are
+
+$$
+\left[ \begin{array}{c c} k _ {1} & - k _ {1} \\ - k _ {1} & k _ {1} + k _ {2} \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \end{array} \right\} = \left\{ \begin{array}{l} P \\ 0 \end{array} \right\} \tag {18.2-3}
+$$
+
+The rows of [K] are almost linearly dependent if $k_{1} >> k_{2}$ , but not if $k_{2} >> k_{1}$ . In a Gauss elimination solution, we calculate the reduced coefficient $K_{22} = (k_{1} + k_{2}) - k_{1}$ , which yields an inaccurate result if $k_{1} >> k_{2}$ and $k_{2} / k_{1}$ is close to the round-off limit.
+
+To make the point numerically, imagine that $k_{1} = 40$ and $k_{2} = 0.0014$ . If the last digit of $k_{2}$ is to appear in the number $k_{1} + k_{2}$ , the computer word mantissa must store at least six digits and $k_{1}$ must be represented as 40.0000. (That $k_{1}$ is not physically known to six-digit accuracy does not matter.) If five digits were stored, Gauss elimination would produce a reduced $K_{22}$ of 0.0010 rather than the correct reduced value $K_{22} = 0.0014$ . If only four digits were stored, $k_{1} + k_{2}$ would be represented as 40.00, and [K] would be singular. That is, [K] would represent only the single spring $k_{1}$ , unsupported and free to translate as a rigid body.
+
+In the foregoing example, the modeling and discretization errors are zero, but round-off produces an inherited error in [K] that becomes obvious during subsequent manipulations.
+
+In summary, and in structural terminology, a major cause of ill-conditioning in practical finite element models is a large difference in stiffnesses, with the stiffer region being supported by the more flexible region. This circumstance shifts essential numerical information to the latter digits of stiffness coefficients $K_{ij}$ . These latter digits may be so few in number that the solution is worthless. Physically,
+
+![](images/page-564_09fd946fdf1e72a0119282780a6ddcaf1935b7f526246689ac0f73464a37649b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+u₁
+u₂
+P 1 k₁ 2 k₂ x, u
+</details>
+
+Figure 18.2-1. Two-d.o.f. structure with linear springs of stiffness $k_{1}$ and $k_{2}$ .
+
+![](images/page-564_541218134817c180e64254c605784ae418fb24a746883f252b198f1bb67c5fc4.jpg)
+
+<details>
+<summary>text_image</summary>
+
+v₁
+θ₁
+u₁ 1
+2
+v₂
+θ₂
+u₂
+</details>
+
+Figure 18.2-2. A plane frame with six nonzero d.o.f.
+
+<!-- source-page: 565 -->
+
+the stiffer region has one or more displacement states that are almost rigid-body motions within a more flexible supporting structure. The limiting case is a structure without any supports: it has only rigid-body motion in static analysis, and its stiffness matrix is singular.
+
+In nonstructural problems, similar difficulties may arise. For example, in heat conduction analysis one might encounter a region of high conductivity imbedded in a region of low conductivity.
+
+One way to avoid difficulty is to use longer computer words—that is, double precision rather than single precision. But if equations are generated by use of single-precision arithmetic and are ill conditioned, will it help to solve them by use of double-precision arithmetic? From our discussion, the answer is no. Information already discarded cannot be recovered by subsequent manipulation, however accurately done.
+
+Structures susceptible to ill-conditioning include thin shells, for which membrane stiffness is much greater than bending stiffness. For the same reason, the frame of Fig. 18.2-2 may be troublesome: the axial stiffness of horizontal member 1–2 greatly exceeds its bending stiffness, so horizontal load will produce $u_{1} \approx u_{2}$ . For such a problem it is usually best to enforce $u_{1} = u_{2}$ by application of a constraint. Thus, a source of ill-conditioning is removed.
+
+An Important Special Case. The three-member structure in Fig. 18.2-3a has springs of stiffness $\alpha k$ and k. If $\alpha$ is large, a stiff structure (the spring of stiffness $\alpha k$ ) is supported against rigid-body motion by a flexible structure (the springs of stiffness k). Structural equations of this system are
+
+$$
+\left[ \begin{array}{c c} k (1 + \alpha c ^ {2}) & k \alpha c s \\ k \alpha c s & k (1 + \alpha s ^ {2}) \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ v _ {1} \end{array} \right\} = \left\{ \begin{array}{l} P \\ 0 \end{array} \right\} \quad \text { where } \quad \begin{array}{l} c = \cos \beta \\ s = \sin \beta \end{array} \tag {18.2-4}
+$$
+
+In $u_{1}v_{1}$ space, the straight lines corresponding to these two equations have the respective slopes
+
+$$
+- \frac {1 + \alpha c ^ {2}}{\alpha c s} \quad \text { and } \quad - \frac {\alpha c s}{1 + \alpha s ^ {2}} \tag {18.2-5}
+$$
+
+If $\alpha >> 1$ , the two slopes are nearly the same, and ill-conditioning is therefore
+
+![](images/page-565_3e0bfa3611d3a70d54c85441daf8d2325665a168426e6bc5d8aa568f86a967b5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k
+P
+1
+k
+αk
+β
+y, v
+x, u
+</details>
+
+(a)
+
+![](images/page-565_377fb326e4947b7f419357239affbf09db2a35962f9303cebb32bc2ac5e5fd0a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y, v
+D
+E
+n
+A
+B
+x, u
+C
+t
+</details>
+
+(b)   
+Figure 18.2-3. (a) Three-spring structure, having d.o.f. $u_{1}$ and $v_{1}$ at node 1. (b) Skew supports at A and B in a plane truss.
+
+<!-- source-page: 566 -->
+
+indicated. However, there are two exceptions: for $c = 0$ ( $\beta = \pi/2$ ) and for $s = 0$ ( $\beta = 0$ ), spring $\alpha k$ contributes no off-diagonal terms to the structure stiffness matrix, and the equations are well conditioned for any value of $\alpha$ .
+
+The practical implication of the foregoing analysis is as follows. In Fig. 18.2-3b, let each node i have the usual d.o.f. $u_{i}$ and $v_{i}$ . A very stiff bar DE approximates the constraint $u_{D} = 0$ but does not produce ill-conditioning because $\beta = 0$ for bar DE. At node A, imagine that the constraint of zero n-direction motion is desired. One can approximate this constraint condition by inserting a very stiff bar AC. This invites ill-conditioning if d.o.f. at A are x- and y-direction displacements, but not if d.o.f. at A are n- and t-direction displacements (introduced by coordinate transformation prior to assembly of elements). However, if coordinate transformation has been invoked, one may as well simply set the n-direction displacement at A to zero (as is implied at node B).
+
+Avoiding Trouble. We have seen that large stiffness differences may be troublesome. The ideal, not always attainable in practice, is a model without large discrepancies in stiffness.
+
+Arbitrary adjustments in modeling may be appropriate. A comparatively stiff region may be modeled as perfectly rigid by use of a constraint transformation, as has been noted in connection with Fig. 18.2-2. On the other hand, perhaps a stiff region can be made more flexible. For example, in Fig. 18.2-2 one may be able to decrease the axial stiffness $AE / L$ of bar 1-2 by a factor of (say) 100: if the members are slender, bending will still dominate the solution, and $u_{1}$ and $u_{2}$ will still greatly exceed the relative motion $u_{2} - u_{1}$ .
+
+Another possibility in modeling is to use relative motions, rather than absolute motions, for troublesome d.o.f. For example, in Fig. 18.2-1 one could introduce the d.o.f. $u_{r} = u_{1} - u_{2}$ . Instead of Eqs. 18.2-3 we now have
+
+$$
+\left[ \begin{array}{l l} k _ {1} & 0 \\ 0 & k _ {2} \end{array} \right] \left\{ \begin{array}{l} u _ {r} \\ u _ {2} \end{array} \right\} = \left\{ \begin{array}{l} P \\ P \end{array} \right\} \tag {18.2-6}
+$$
+
+Equations 18.2-6 are well conditioned for all values of $k_{1}$ and $k_{2}$ . Equations 18.2-6 must be obtained by using $u_{r}$ as a d.o.f. in element formulation and assembly. No purpose would be served by introducing $u_{r}$ by coordinate transformation of the assembled equations, Eqs. 18.2-3, as the coefficient $k_{1} + k_{2}$ already contains the error we seek to avoid.
+
+After a good finite element model is prepared, subsequent manipulation errors may be negligible. However, it is recommended that double-precision arithmetic be used for all phases of a finite element analysis, unless the round-off limit is about $10^{-14}$ in single precision or the problem has very few d.o.f.
+
+# 18.3 THE CONDITION NUMBER
+
+A numerical measure of the ill-conditioning in a coefficient matrix $[K]$ is the condition number, denoted here by $C(\mathbf{K})$ . A large condition number warns that a finite element solution may contain appreciable error: the anticipated error may be realized for some systems and some loadings but not for others.
+
+<!-- source-page: 567 -->
+
+Definition and Interpretation. The spectral condition number of a matrix $[K]$ , which we will call simply the condition number, is defined as
+
+$$
+C (\mathbf {K}) = \frac {\lambda_ {\max}}{\lambda_ {\min}} \tag {18.3-1}
+$$
+
+where $\lambda_{\max}$ and $\lambda_{\min}$ are largest and smallest eigenvalues of [K]. (It is best to scale [K] before calculating $C(\mathbf{K})$ , as described in connection with Eqs. 18.3-3 to 18.3-6.)
+
+It can be shown [18.1] that for each power of ten in the ratio $\lambda_{max}/\lambda_{min}$ , the operations of equation solving lose about one digit of accuracy in the displacement mode associated with $\lambda_{min}$ . In a direct method of solving equations, such as Gauss elimination, the difficulty may materialize near the end of the forward-reduction phase, when the difference is computed between two numbers that are almost equal. Error contamination spreads during the back-substitution phase. The estimated accuracy loss is
+
+$$
+\text { accurate   digits   lost } \approx \log_ {1 0} \frac {\lambda_ {\max}}{\lambda_ {\min}} = \log_ {1 0} C (\mathbf {K}) \tag {18.3-2}
+$$
+
+For example, imagine that $C(\mathbf{K}) = 10^{5}$ . If computer words have seven-digit capacity, only two reliable digits may remain in the computed displacements. With fourteen-digit capacity, nine accurate digits may remain.
+
+The estimate given by Eq. 18.3-2 is based on the inherited error, caused by round-off, that exists at the outset of equation solving. The estimate does not include the manipulation error of the solution algorithm.
+
+For stress analysis purposes, the estimate in Eq. 18.3-2 may be pessimistic because it ignores the load vector. Consider Fig. 18.3-1. For both load cases shown, the percentage errors in computed values of $u_{1}$ and $u_{2}$ are larger than the percentage error in the computed relative displacement $u_{1} - u_{2}$ . In stress analysis one seeks the largest strains. The strain of largest magnitude is $u_{2}/L$ in Fig. 18.3-1a but is $(u_{1} - u_{2})/L$ in Fig. 18.3-1b. Accordingly, ill-conditioning may be of little consequence in Fig. 18.3-1b. In general, for stress analysis one should estimate the number accurate digits lost as $\log_{10}(\lambda_{\max}/\lambda_{k})$ , where $\lambda_{k}$ is the lowest eigenvalue of [K] whose associated eigenmode is not approximately orthogonal to the load vector. Usually $\lambda_{k} = \lambda_{min}$ , but not necessarily. Thus, in Fig. 18.3-1b, the loss of digits in the strain $(u_{1} - u_{2})/L$ is more accurately predicted by using
+
+![](images/page-567_c936f3a35b2c1a9c114552f6807df1a9508a62c13df744d43627577792d218d4.jpg)
+
+<details>
+<summary>text_image</summary>
+
+u₁ → u₂
+12P → 13P → AE/L
+6AE/L
+L → L
+</details>
+
+(a)
+
+![](images/page-567_806bb48c78042f788423627f376db77873afb9daefa04c8ca3041bcaa2779e77.jpg)
+
+<details>
+<summary>text_image</summary>
+
+u₁ → u₂
+12P → 13P ← AE/L
+6AE/L
+L ← L
+</details>
+
+(b)   
+Figure 18.3-1. Two different load sets on a two-d.o.f. bar structure. The respective elements have axial stiffnesses 6AE/L and AE/L.
+
+<!-- source-page: 568 -->
+
+the $\lambda_{k}$ associated with the eigenmode in which $u_{1}$ and $u_{2}$ are of opposite sign than by using $\lambda_{k} = \lambda_{\min}$ (in whose eigenmode $u_{1}$ and $u_{2}$ are of the same sign).
+
+Calculation and Scaling. Besides ignoring the load vector, Eq. 18.3-2 may overestimate error because $C(\mathbf{K})$ is “artificially” high. In Fig. 18.2-1 for example, $C(\mathbf{K})$ is large when $k_{1} >> k_{2}$ and when $k_{2} >> k_{1}$ , yet the case $k_{2} >> k_{1}$ is well conditioned. One can arrange for $C(\mathbf{K})$ to be large only when [K] is truly ill conditioned by scaling [K] before calculating $C(\mathbf{K})$ , as follows. One constructs a diagonal scaling matrix [S] from diagonal coefficients in [K], then transforms [K] to the scaled matrix $[K_{s}]$ :
+
+$$
+[ \mathrm{K} _ {s} ] = [ \mathrm{S} ] [ \mathrm{K} ] [ \mathrm{S} ] \quad \text { where } \quad S _ {i i} = \frac {1}{\sqrt {K _ {i i}}} \tag {18.3-3}
+$$
+
+Diagonal coefficients of $[\mathbf{K}_s]$ are unity. The extreme eigenvalues $\lambda_{\max}$ and $\lambda_{\min}$ of $[\mathbf{K}_s]$ are to be used in Eq. 18.3-2.
+
+The eigenvalue problem that yields $\lambda_{\mathrm{max}}$ and $\lambda_{\mathrm{min}}$ of the scaled matrix $[\mathbf{K}_s]$ is
+
+$$
+\left(\left[ \mathbf {K} _ {s} \right] - \lambda \lceil \mathbf {I} \rceil\right) \{\mathbf {D} \} = \{\mathbf {0} \} \tag {18.3-4}
+$$
+
+which may be regarded as a vibration problem with a unit mass matrix and natural frequencies $\omega_{i}^{2} = \lambda_{i}$ . An alternative form, which yields the same eigenvalues as Eq. 18.3-4, is produced by substituting $\{D\} = \left[S\right]^{-1}\{D_{1}\}$ and premultiplying by $[S]^{-1}$ . Thus [18.3]
+
+$$
+\left(\left[ \mathrm{S} \right] ^ {- 1} \left[ \mathrm{K} _ {s} \right] \left[ \mathrm{S} \right] ^ {- 1} - \lambda \left[ \mathrm{S} \right] ^ {- 1} \left[ \mathrm{S} \right] ^ {- 1}\right) \left\{\mathrm{D} _ {1} \right\} = \{0 \} \tag {18.3-5}
+$$
+
+which is the same as
+
+$$
+([ \mathbf {K} ] - \lambda [ K _ {1 1} K _ {2 2} \dots K _ {n n} ]) \{\mathbf {D} _ {1} \} = \{\mathbf {0} \} \tag {18.3-6}
+$$
+
+Therefore, $\lambda_{max}$ and $\lambda_{min}$ of the scaled matrix $[K_{s}]$ can be calculated using the unscaled [K] and a diagonal “mass” matrix that is simply the principal diagonal of [K]. Equation 18.3-6 shows why an isolated stiff region raises $C(\mathbf{K})$ : an isolated large “mass” $K_{ii}$ , if not held by supports, reduces the lowest “frequency” but has little effect on the highest “frequency.”
+
+Scaling is used only to avoid obtaining an unrealistically pessimistic result from Eq. 18.3-2 because of “artificial” ill-conditioning. Scaling need not be applied to the [K] used in solving equations. Scaling of [K] has no effect on the accuracy of a direct-solution algorithm such as Gauss elimination, assuming that the scaling process itself introduces no manipulation error, and provided that the choice of pivots and the sequence of eliminations are unchanged [18.4].
+
+Because Eq. 18.3-2 is only an approximation, $\lambda_{max}$ and $\lambda_{min}$ need not be computed accurately. A close upper bound on $\lambda_{max}$ of $[K_{s}]$ can be obtained from the Gerschgorin bound, Eq. 13.10-17. Thus we add the magnitudes of coefficients in each row of $[K_{s}]$ , then choose the largest such row sum, that is,
+
+$$
+\lambda_ {\max} \approx \max Q _ {i} \quad \text { where } \quad Q _ {i} = \sum_ {j = 1} ^ {n} \left| K _ {s i j} \right| \tag {18.3-7}
+$$
+
+<!-- source-page: 569 -->
+
+where n is the order of $[K_{s}]$ . Unfortunately, there is no corresponding simple estimate of $\lambda_{min}$ . The expense of computing $\lambda_{min}$ is comparable to the expense of solving equations, which means that Eq. 18.3-2 does not provide an inexpensive a priori estimate of solution accuracy.
+
+Causes of Ill-Conditioning. A finite element model tends to produce ill-conditioned equations if an element or a patch of elements can respond to loads with large rigid-body motion but little deformation. Examples include (a) high-modulus inclusions, (b) plate elements that allow transverse shear strain but have large transverse shear stiffness because they are thin, (c) elements of severe shape distortion or large aspect ratio, and (d) stiff supports such as spring $\alpha k$ in Fig. 18.2-3. (Analogous difficulties can appear in nonstructural problems.) In the foregoing cases one can reduce or eliminate the trouble by changing the model. In the respective examples, one can (a) make the inclusion rigid by applying constraints, (b) use thin-plate elements or arbitrarily decrease the shear stiffness, (c) remodel using more regular and compact element shapes, and (d) use differently directly d.o.f. at the offending node.
+
+Ill-conditioning of a stiffness matrix may also be caused by mixing elements of different size and by using a fine mesh. It has been found that [18.5]
+
+$$
+C (\mathbf {K}) = b \left(\frac {h _ {\max}}{h _ {\min}}\right) ^ {2 m - 1} N ^ {2 m / n} \tag {18.3-8}
+$$
+
+where $b =$ a positive constant independent of $h_{\max}$ and $h_{\min}$ ,
+
+$$
+h _ {\min} = \text { smallest   node   spacing   in   any   element   of   the   mesh },
+$$
+
+$$
+h _ {\max} = \text { greatest   node   spacing   in   any   element   of   the   mesh },
+$$
+
+$$
+N = \text { number   of   elements },
+$$
+
+$$
+2 m = \text { differential   equation   order },
+$$
+
+$$
+n = \text { dimensionality }.
+$$
+
+To elaborate, the differential equation that describes the physical problem uses dispalements as dependent variables, has 2m as the highest derivative of the dependent variable(s), and requires n independent variables. As examples, for an axially loaded bar, 2m/n = 2/1; for a beam, 2m/n = 4/1; in plane stress, 2m/n = 2/2; in thin-plate bending, 2m/n = 4/2; in three-dimensional solids, 2m/n = 2/3; for a thin shell, 2m/n = 4/3. Thus, in a beam problem, if the length ratio $h_{max}/h_{min}$ of elements is changed from 1/1 to 10/1, $C(\mathbf{K})$ increases by a factor of 1000. If the number of elements is doubled, $C(\mathbf{K})$ increases by a factor of 16. If both of these changes are made, $C(\mathbf{K})$ increases by a factor of 16,000.
+
+As a practical matter, we would like to know the dependencies of the condition number on important mesh and problem parameters. For the unscaled stiffness matrix, Fried [18.5] has shown that
+
+$$
+\left(\frac {1}{\lambda_ {1} c _ {\max}}\right) \frac {\max _ {l \leq \ell \leq N} \left(\Lambda_ {\ell} ^ {k}\right)}{\max _ {l \leq \ell \leq N} \left(\Lambda_ {\ell} ^ {n}\right)} \leq C (\mathbf {K}) \leq \left(\frac {c _ {\max}}{\lambda_ {1}}\right) \frac {\max _ {l \leq \ell \leq N} \left(\Lambda_ {\ell} ^ {k}\right)}{\min _ {l \leq \ell \leq N} \left(\lambda_ {\ell} ^ {n}\right)} \tag {18.3-9}
+$$
+
+<!-- source-page: 570 -->
+
+where, using unscaled element stiffness and mass matrices,
+
+$$
+\begin{array}{l} \Lambda_ {\ell} ^ {k}, \Lambda_ {\ell} ^ {m} = \text { maximum   eigenvalue   of   the   stiffness   and   mass   matrices }, \\ \text { respectively,   of   element } \ell , \end{array}
+$$
+
+$$
+\lambda_ {\ell} ^ {m} = \text { minimum   eigenvalue   of   the   mass   matrix   of   element } \ell ,
+$$
+
+$$
+\lambda_ {1} = \text { minimum   eigenvalue   of   the   continuous   problem },
+$$
+
+$$
+N = \text { number   of   elements   in   the   structure },
+$$
+
+$$
+c _ {\max} = \text { maximum   number   of   elements   meeting   at   a   single   node. }
+$$
+
+The mass matrix cited need not be the consistent mass matrix; it may be lumped provided that no $\lambda_{\ell}^{m}$ is zero because a d.o.f. is assigned zero mass. Alternative notation for $\lambda_{1}$ is $\omega_{1}^{2}$ , where $\omega_{1}$ is the fundamental vibration frequency of the actual structure. Mass density cancels in the denominators and so may be taken as unity. One can use estimates of $\Lambda_{\ell}^{k}$ and $\Lambda_{\ell}^{m}$ obtained from the Gershgorin bound, Eq. 13.10-17. Hence, if $\lambda_{1}$ is known, Eq. 18.3-9 provides estimated numerical bounds on $C(\mathbf{K})$ . We see from Eq. 18.3-9 that the bound on $C(\mathbf{K})$ becomes less certain as more elements are connected to a node. However, $c_{max}$ is typically 4, 6, or even 8, which indicates a probable uncertainty of about one digit in $\log_{10} C(\mathbf{K})$ .
+
+The condition number may be strongly affected by Poisson's ratio $\nu$ . Fried [18.18] has shown that $b$ in Eq. 18.3-8 is given by
+
+$$
+b = \frac {b _ {1}}{1 - 2 \nu} \tag {18.3-10}
+$$
+
+where $b_{1}$ is a constant that is independent of E and $\nu$ (provided that E and $\nu$ are constant throughout the mesh). Equation 9.4-12 shows that b is a penalty number $\alpha$ , that is, $\alpha = b/3b_{1}$ . Near the incompressibility limit of $\nu = 0.5$ , Eq. 18.3-10 can increase $C(\mathbf{K})$ by orders of magnitude. If computer words carry about p digits each, and if the value of $\alpha$ or b approaches $10^{p/2}$ as suggested at the end of Section 9.4, then we might expect an accuracy loss of almost p/2 digits in the solution process. Such a loss may be of concern if pressures in nearly incompressible media are computed by the penalty method (see Eqs. 9.5-1 and 9.6-2), for the following reason. Strain $\epsilon_{V}$ in Eq. 9.5-1 is small because strains $u_{,x}$ , $v_{,y}$ , and $w_{,z}$ almost cancel one another when added to produce $\epsilon_{V}$ , even when using computer words of p-digit accuracy. We now propose to calculate $\epsilon_{V}$ by adding three strains that are each accurate in only about the leading p/2 digits. If $\epsilon_{V}$ is to be computed with useful accuracy, a more rigorous analysis suggests that $\alpha$ should be approximately $10^{p/3}$ . In practice the estimate $\alpha \approx 10^{p/3}$ appears to be sufficiently pessimistic that a value of $\alpha$ approaching $10^{p/2}$ can usually be used.
+
+# 18.4 DIAGONAL DECAY ERROR TESTS
+
+Here we describe a simple and inexpensive test for round-off errors that appear during a direct method of solving equations, such as the Gauss elimination method. The test can warn of possible trouble and can be used to terminate execution if serious trouble is indicated [18.6].
+
+Assume that the coefficient matrix [K] is symmetric and positive definite. Then, as each equation is processed, that is, as each unknown is eliminated, a subtraction operation reduces the magnitude of diagonal coefficients $K_{ii}$ that correspond to d.o.f. i yet to be eliminated (however, each $K_{ii}$ remains positive). Thus, each $K_{ii}$
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+<!-- source-page: 571 -->
+
+tends to accumulate round-off errors. A simple example of the decay of diagonal coefficients appears in Fig. 2.11-1. Another example appears in connection with Eq. 18.2-3, where Gauss elimination of $u_{1}$ produces the reduced coefficient $K_{22} = (k_{1} + k_{2}) - k_{1}$ . There it is seen that the reduced $K_{22}$ may have no accurate digits if $k_{1} >> k_{2}$ and computer words are too short. The reduced $K_{22}$ , now perhaps greatly in error, is used as a pivot in eliminating $u_{2}$ , causing error to propagate throughout the solution.
+
+A simple test of accuracy is to check the amount of decay of each $K_{ii}$ before it acts as a pivot. Let $P_{ii}$ be the reduced value of $K_{ii}$ just before it is used as a pivot in eliminating the ith d.o.f. The diagonal decay ratio
+
+$$
+r _ {i} = \frac {K _ {i i}}{P _ {i i}} \tag {18.4-1}
+$$
+
+where $K_{ii}$ is the original diagonal coefficient, is an approximate measure of the number of digits of accuracy lost; for example, roughly six accurate digits have been lost if any $r_{i}$ is $10^{6}$ , leaving only p - 6 accurate digits if computer words contain p digits.
+
+Table 18.4-1 shows numerical results obtained from a beam problem on a computer that carries slightly more than eleven digits in double precision arithmetic. We see that it is best if node numbers progress from the free end to the fixed end. (Unfortunately, this arrangement is not practical for all finite element structures.) We also see that trouble may not be detected until reduction is almost complete: in the case where node N is free, trouble appears only in the next-to-last equation, when the equation solver detects that node N is unsupported.
+
+The example in Table 18.4-1 uses elements of equal length. Consider the case N = 1000, now with elements mixed in length but varying in length only between 0.99999 and 1.00001. Then the tip-to-root numbering gives w = 1.0269 while diagonal decay ratios are essentially unchanged from their values of 8.0 and 2.0. This example shows the possibility of substantial accuracy loss that is not detected by the diagonal decay ratio.
+
+If a large coefficient appears only on the diagonal of [K], without corresponding
+
+TABLE 18.4-1. EFFECT OF NUMBER OF BEAM ELEMENTS AND NODE NUMBERING ON ACCURACY, WHERE DIAGONAL DECAY RATIO $r_{i}$ IS DEFINED BY EQ. 18.4-1 AND w IS THE RATIO OF THE COMPUTED DEFLECTION OF LOAD P TO ITS DEFLECTION ACCORDING TO BEAM THEORY. 
+<table><tr><td></td><td colspan="4">N Elements $^{a}$ 2N Nonzero D.O.F.</td><td colspan="4">N Elements $^{a}$ 2N Nonzero D.O.F.</td></tr><tr><td>EI = 8(10) $^{6}$ Length = 1000</td><td colspan="2"><img src="images/0dbf71c59f3e7d14d874adc4909fab260b9e012534e3dda0b5d9daf8f82ffd71.jpg"/></td><td colspan="2"><img src="images/052622d02d4adc7a56e578854368c7e7a2a52b8a8ed14683931c1134d48903e5.jpg"/></td><td colspan="2"><img src="images/8cbe576ac40274def29b8eb853f02be872acef0b10cdcecf8ec4a5dfa92f8e48.jpg"/></td><td colspan="2"><img src="images/4da109ffc9aa367bce9fd035407654b9364e694ff85ca03391035853ba5d1058.jpg"/></td></tr><tr><td>Number of elements</td><td>w</td><td> $r_{2N-2}$ </td><td> $r_{2N-1}$ </td><td> $r_{2N}$ </td><td>w</td><td> $r_{2N-2}$ </td><td> $r_{2N-1}$ </td><td> $r_{2N}$ </td></tr><tr><td>N = 10</td><td>1.0000</td><td>8.0</td><td>2.0</td><td>8.0</td><td>1.0000</td><td>5.3</td><td>1(10) $^{3}$ </td><td>4(10) $^{1}$ </td></tr><tr><td>N = 100</td><td>1.0000</td><td>8.0</td><td>2.0</td><td>8.0</td><td>0.9993</td><td>7.7</td><td>1(10) $^{6}$ </td><td>4(10) $^{2}$ </td></tr><tr><td>N = 1000</td><td>1.0000</td><td>8.0</td><td>2.0</td><td>8.0</td><td>0.1197</td><td>8.0</td><td>2(10) $^{8}$ </td><td>2(10) $^{3}$ </td></tr></table>
+
+$^{a}$ Elements are of equal length (see text).
+
+<!-- source-page: 572 -->
+
+large off-diagonal coefficients, it does not lead to large diagonal decay. Thus, $k_{2} >> k_{1}$ is acceptable in Eq. 18.2-3, and $\beta = 0$ or $\beta = \pi / 2$ is acceptable in Fig. 18.2-3a.
+
+It is the decay of diagonals that is significant, not their smallness. Small pivots per se do not provoke large error. The causes of a large decay ratio $r_{i}$ are the causes of ill-conditioning: inadequate supports, mechanisms, isolated stiff regions, and so on. It is interesting that the condition number of [K] is the same for both beams in Table 18.4-1, roughly $10^{12}$ when N = 1000, yet the severe loss of accuracy predicted by this large $C(\mathbf{K})$ materializes for only one of the two beams.
+
+# 18.5 RESIDUALS
+
+Consider the equation
+
+$$
+\{\Delta \mathbf {D} \} = [ \mathbf {K} ] ^ {- 1} \{\Delta \mathbf {R} \} \quad \text { where } \quad \{\Delta \mathbf {R} \} = \{\mathbf {R} \} - [ \mathbf {K} ] \{\mathbf {D} \} \tag {18.5-1}
+$$
+
+where $\{\Delta R\}$ is called the residual. If the equations $[K]\{D\} = \{R\}$ could be solved exactly, that is, so that the solution algorithm does not contaminate $\{D\}$ with any round-off error, then $\{\Delta R\} = \{0\}$ . Actually, there is some contamination, and $\{\Delta D\}$ can be regarded as a measure of the probable round-off error in $\{D\}$ . A similar scalar error measure is
+
+$$
+e = \frac {\{\mathbf {D} \} ^ {T} \{\Delta \mathbf {R} \}}{\{\mathbf {D} \} ^ {T} \{\mathbf {R} \}} \tag {18.5-2}
+$$
+
+Physically, e is the ratio of work done by residual loads to work done by actual loads when both act through displacements $\{D\}$ . If $|e|$ is (say) $10^{-8}$ or less, round-off contamination is probably negligible [18.7]. However, if the equations $[K]\{D\} = \{R\}$ are ill conditioned, it is possible that inaccurate solutions will yield small residuals and a small e.
+
+A measure of error in vibration analysis, similar to Eq. 18.5-2, is
+
+$$
+e = \frac {\{\overline {{{\mathbf {D}}}} \} ^ {T} ([ \mathbf {K} ] \{\overline {{{\mathbf {D}}}} \} - \omega^ {2} [ \mathbf {M} ] \{\overline {{{\mathbf {D}}}} \})}{\{\overline {{{\mathbf {D}}}} \} ^ {T} [ \mathbf {K} ] \{\overline {{{\mathbf {D}}}} \}} \tag {18.5-3}
+$$
+
+in which the quantity in parentheses would be zero if frequency $\omega$ and mode shape $\{\overline{\mathbf{D}}\}$ were uncontaminated by round-off error.
+
+One can use Eq. 18.5-1 as an iterative improvement scheme. Thus the computed $\{\Delta \mathbf{D}\}$ is added to the existing $\{\mathbf{D}\}$ , a new $\{\Delta \mathbf{R}\}$ is formed, a new $\{\Delta \mathbf{D}\}$ is computed and added to the current $\{\mathbf{D}\}$ , and so on. If this method is to succeed, $|e|$ must be less than unity, and the $\{\mathbf{R}\}$ and $[\mathbf{K}]$ used to compute $\{\Delta \mathbf{R}\}$ must be represented with greater precision than the $[\mathbf{K}]^{-1}$ used to compute $\{\Delta \mathbf{D}\}$ . With $[\mathbf{K}]$ represented accurately and $[\mathbf{K}]^{-1}$ represented inaccurately, convergence is toward the exact solution of $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{R}\}$ , as is desired, rather than toward the exact solution of $\{\mathbf{D}\} = [\mathbf{K}]^{-1}\{\mathbf{R}\}$ . Iterative improvement reduces the rounding error introduced during equation solving but does not reduce the inherited error present at the outset of equation solving.
+
+If a set of equations is seriously ill conditioned, it is usually better to rework the finite element model so as to improve its condition than to make heroic attempts
+
+<!-- source-page: 573 -->
+
+to improve a poor solution by iteration. “If a thing is not worth doing, it is not worth doing well” [18.8].
+
+Interpretation of $\{\Delta R\}$ . A small residual does not guarantee that equations have been solved accurately. Of two approximate solutions, it is possible that the solution of lesser accuracy will yield the smaller residuals [18.8].
+
+In structural mechanics, a small $\{\Delta R\}$ indicates that applied loads $\{R\}$ are balanced by resisting loads $[K]\{D\}$ arising from the deformed structure. Checking that $\{\Delta R\} \approx \{0\}$ is sometimes called a “statics check” or an “equilibrium check.” Passing an equilibrium check is no guarantee that results are accurate. For example, one may compute deflections and stresses that are physically unrealistic because the mesh is too coarse, yet find $\{\Delta R\} \approx \{0\}$ (merely because the equation solver is working properly).
+
+In summary, small residuals are a necessary but not sufficient condition for accuracy. Reference 18.21 suggests a residual norm test: that $\|\Delta R\|<<\|\widetilde{R}\|$ if the solution is sufficiently accurate. Reference 18.21 also suggests that the solution is probably acceptable if either the condition number $C(\mathbf{K})$ alone or the residual norm test alone indicates trouble, but probably unacceptable if both $C(\mathbf{K})$ and the residual norm test indicate trouble.
+
+# 18.6 DISCRETIZATION ERROR:
+
+# ANALYSIS
+
+In representing a mathematical continuum by finite elements, we select the number, type, and shape of elements, the grading of the mesh, allocate distributed loads to nodes, and represent support conditions by fixing certain d.o.f.. The approximation inherent in this process is called discretization error. Unfavorable discretization can provoke subsequent numerical difficulty, for example, as described in Section 18.2. However, in the present section we are concerned only with discretization error itself, that is, with the discrepancy between the discretized model and the mathematical model, with the latter being taken as correct.
+
+A simple rule is that the volume of the discretized structure should be correct. Thus, if a circular region is modeled by a polygon of straight-sided elements, the polygon should neither inscribe nor circumscribe the circle; rather, element sides should intersect the circle so that the area of the polygon equals the area of the circle. Similarly, straight elements that model an arch should not be chords, but longer, so that the sum of element lengths equals the arch length.
+
+Error Analysis. Discretization error can sometimes be determined by an order of error analysis. Consider, for example, the axially loaded bar of Fig. 18.6-1a. From the differential element, the equation of axial equilibrium is $A\sigma_{x,x} + q = 0$ . Or, substituting the stress–strain relation $\sigma_{x} = Eu_{,x}$ , the equilibrium equation is
+
+$$
+u _ {, x x} + \frac {q}{A E} = 0 \tag {18.6-1}
+$$
+
+Let the discretized model consist of standard two-d.o.f. bar elements. It is not hard to show that if the load integral is evaluated consistently (i.e., by Eq. 4.1-6), and if $A$ and $E$ are constant, then finite element displacements are exact
+
+<!-- source-page: 574 -->
+
+at the nodes (see Problems 18.28 and 18.29). This does not mean that finite element displacements are exact everywhere: the exact solution is in general not piecewise linear, so there is error between the nodes.
+
+The behavior of the error in the bar model can be understood by doing a comparatively simple error analysis. The error of the bar model is in many ways either characteristic of, or very relevant to, the behavior of the error in more complicated and realistic problems. We cannot hope to obtain discretization error bounds in all practical problems. But many model problems can be analyzed, each reflecting salient features of a more practical problem. What we touch on here is most directly applicable to linear elasticity problems with smooth (but possibly varying) elastic coefficients, with distributed (not point) loading, in bodies whose geometry does not induce strong singularities (no cracks). The arguments and conclusions rest on a fundamental fact of finite element error analysis: for a sufficiently refined mesh, the error in the finite element solution can be bounded by the error in approximating the exact solution by shape function interpolation that is exact at nodes [18.9]. Thus, whether or not the finite element solution is exact at nodes (and in general it is not), the error in nodal interpolation tells the story. The proviso of “a sufficiently refined mesh” is intended to exclude models having large discretization error, such as a beam modeled by a few constant-strain triangles, whose computed nodal displacements are greatly in error.
+
+Accordingly, in the bar problem we analyze the error of linear interpolation between exact nodal displacements $u_{i}$ and $u_{i+1}$ . In the ith element, displacement error $e = e(x)$ is
+
+$$
+e (x) = u (x) - u _ {i} \left(1 - \frac {x - x _ {i}}{h _ {i}}\right) - u _ {i + 1} \left(\frac {x - x _ {i}}{h _ {i}}\right) \tag {18.6-2}
+$$
+
+where $u(x)$ is the exact solution. Also, $u_{i} = u(x_{i})$ and $u_{i+1} = u(x_{i+1})$ , where $x_{i}$ and $x_{i+1}$ are the nodal locations. Element length is $h_{i} = x_{i+1} - x_{i}$ . There is no need to require all element lengths to be equal. In the $i$ th element, error $e(x)$ is a smooth function. At nodes, there are discontinuities in the first derivative $e'(x)$ , as shown by Fig. 18.6-1c. We note that $e(x_{i}) = e(x_{i+1}) = 0$ .
+
+One can determine an upper bound on $e(x)$ by the following argument [18.9]. Let $z$ be an axial coordinate $x_{i} \leq z \leq x_{i+1}$ such that $e'(z) = 0$ . That there is such a point is obvious from Fig. 18.6-1b and follows rigorously from Rolle's theorem of elementary calculus. In what follows we assume that $u''(x)$ is continuous in the interval $x_{i} \leq x \leq x_{i+1}$ . The change in $e'$ can be computed by integrating $e''$ . Thus, with $s$ a dummy variable, $e'(z) = 0$ , and using Eq. 18.6-2,
+
+$$
+e ^ {\prime} (x) - e ^ {\prime} (z) = e ^ {\prime} (x) = \int_ {z} ^ {x} e ^ {\prime \prime} (s) d s = \int_ {z} ^ {x} u ^ {\prime \prime} (s) d s \tag {18.6-3}
+$$
+
+because the linear terms in Eq. 18.6-2 do not contribute to $e''(x)$ . In addition,
+
+$$
+\left| \int_ {z} ^ {x} u ^ {\prime \prime} (s) d s \right| \leq \int_ {z} ^ {x} | u ^ {\prime \prime} (s) | d s \leq \int_ {x _ {i}} ^ {x _ {i + 1}} | u ^ {\prime \prime} (s) | d s \leq h _ {i} \left(\max _ {x _ {i} \leq x \leq x _ {i + 1}} | u ^ {\prime \prime} (x) |\right) \tag {18.6-4}
+$$
+
+$^{1}$ In the remainder of this section we use the notation $e_{,x} = e'$ , $e_{,xx} = e''$ , and so on.
+
+<!-- source-page: 575 -->
+
+![](images/page-575_4d7e00c76a03e57d76fdc6568727521d685a9029e8d2391fe4ba7624f21387ae.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Exact solution, u = u(x)
+Finite element (piecewise linear)
+x_{i+1}
+x_i
+x_i
+dx
+σ_x A
+(σ_x + dσ_x)A
+q dx
+q = q(x)
+i
+i + 1
+x
+ith element
+</details>
+
+(α)
+
+![](images/page-575_28726aaf672350942a3835cd9277fc6e4a75a218d091a4965a489c3036e77764.jpg)
+
+<details>
+<summary>text_image</summary>
+
+e(x)
+x_i
+x_{i+1}
+h_i
+O(h_i^2)
+x
+x
+z
+s
+(b)
+</details>
+
+![](images/page-575_bb0d1c3b25d616a71490e4f49c589ea85b9c3cab23d9358e570429b03020062e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+e'(x) = \frac{d}{dx}e(x)
+O(h_i)
+(c)
+x
+</details>
+
+Figure 18.6-1. The behavior of the error in a uniform bar under distributed axial load q. The exact axial displacement is $u = u(x)$ and the error is $e = e(x)$ .
+
+Therefore, on the ith element the error in strains $e'(x)$ is bounded by
+
+$$
+\left| e ^ {\prime} (x) \right| \leq h _ {i} \left(\max _ {x _ {i} \leq x \leq x _ {l + 1}} \left| u ^ {\prime \prime} (x) \right|\right) \tag {18.6-5}
+$$
+
+We can also bound the displacement error $e(x)$ by observing that it must have greatest magnitude at x = z, where $e'(z) = 0$ . Now z must be closer to $x_i$ or to $x_{i+1}$ . Assume that z is closer to $x_i$ , and compute $e(x_i)$ by a three-term Taylor series, with exact remainder, expanded about x = z,
+
+$$
+e (x _ {i}) = e (z) + (x _ {i} - z) e ^ {\prime} (z) + \frac {1}{2} (x _ {i} - z) ^ {2} e ^ {\prime \prime} (s) \tag {18.6-6}
+$$
+
+where $s$ is the remainder evaluation point on element $i$ . But $e(x_{i}) = 0$ , $e'(z) = 0$ , and $e''(s) = u''(s)$ from Eq. 18.6-2, so
+
+<!-- source-page: 576 -->
+
+$$
+e (z) = - \frac {1}{2} \left(x _ {i} - z\right) ^ {2} u ^ {\prime \prime} (s) \tag {18.6-7}
+$$
+
+According to the assumption that $z$ is closer to $x_{i}$ than to $x_{i + 1}$ , we conclude that $|z - x_{i}| \leq h_{i} / 2$ . Therefore, the error in displacements, $e(x)$ , is bounded by
+
+$$
+e (x) \leq \frac {1}{8} h _ {i} ^ {2} \left(\max _ {x _ {i} \leq x \leq x _ {i + 1}} \left| u ^ {\prime \prime} (x) \right|\right) \tag {18.6-8}
+$$
+
+One may verify that the same result is obtained when $z$ is assumed to be closer to $x_{i+1}$ than to $x_i$ .
+
+We note that the existence of a $z$ for which the error in strain is zero is the rationale for the existence of the optimal stress calculation points discussed in Section 6.13.
+
+Important features of the foregoing discussion are as follows.
+
+1. The strain error is proportional to element size, and the displacement error is proportional to the square of element size.   
+2. The error estimates are proportional to derivatives one order higher than the degree of the shape functions (for the bar, second derivatives and linear shape functions are involved).   
+3. Displacements are most accurate at or near element nodes. Strains are most accurate in element interiors, that is, at or near Gauss points.
+
+Using the notation “O” for “order,” we say that Eq. 18.6-5 displays a discretization error $O(h)$ in strain and Eq. 18.6-8 displays a discretization error $O(h^{2})$ in displacement. Thus, if $h = \max(h_{i})$ is cut in half to produce two elements, h is halved; the error in strain is approximately halved and the error in displacement is approximately quartered.
+
+Remarks. We would like to generalize the foregoing conclusions to other elements and to strain energy error as well [18.10]. For this purpose we define symbols as follows:
+
+$h =$ approximate "characteristic length" of an element: length of a linear element; length of the longest line segment connecting two points in a plane or solid element
+
+$q - 1 =$ degree of highest complete polynomial in the element displacement field
+
+2m = order of highest derivative in the governing equilibrium equation expressed in terms of displacements
+
+For the bar of Fig. 18.6-1, $q - 1 = 1$ and $2m = 2$ . For the standard four-d.o.f. beam element, $q - 1 = 3$ and $2m = 4$ . In plane and solid problems with bilinear and trilinear elements discussed in Chapter 6, $q - 1 = 1$ and $2m = 2$ . For bending of thin plates with twelve d.o.f. elements based on Eq. 11.2-5, $q - 1 = 3$ and $2m = 4$ .
+
+Because an element can fit exactly a displacement field of degree q - 1, it therefore has error $O(h^{q})$ in representing the polynomial fields of degree q and higher that in general are present in the exact solution. The error in strains (and
+
+<!-- source-page: 577 -->
+
+therefore stresses) is proportional to the error in the $r$ th derivative of the displacement field; that is, the stress error is $O(h^{q-r})$ . For plane and solid elements, $r = 1$ . For beam and plate elements, for which stresses are dictated by curvatures, $r = 2$ . Thus the standard four-d.o.f. beam element has displacement error $O(h^{4})$ and stress error $O(h^{2})$ .
+
+In plane problems it is sometimes possible to use optimal points for stress calculation, as noted in this section and in Section 6.13. Then the stress error is $O(h^{q})$ rather than $O(h^{q-1})$ .
+
+A strain energy expression contains squares of the mth displacement derivatives. Therefore, the strain energy error is $O(h^{2q-2m})$ .
+
+Again we remark that the foregoing error estimates are based on the assumption that nodal loads are evaluated consistently, that is, by Eq. 4.1-6. If ad hoc load lumping is used, or if Eq. 4.1-6 is evaluated by too low a quadrature rule, the estimates may not apply. For example, the beam deflection data seen at the end of Section 4.3 use ad hoc load lumping and display displacement error $O(h^{2})$ rather than the expected $O(h^{4})$ .
+
+Singularities. It is a general principle in finite element error estimation that constants involved in an error bound are proportional to the qth partial derivative of the exact solution. If this derivative is infinite, the error bound does not rigorously apply. For example, in problems with cracks, displacement derivatives through the second are singular at the crack tip; therefore, even with linear elements (for which q - 1 = 1), our estimates do not apply. Other problems may have singularities in higher derivatives. We often do not know the order of the lowest singular derivative, and therefore we cannot estimate the factor by which error will be reduced by subdividing the mesh.
+
+However, although singularities may slow convergence, the presence of singularities in derivatives higher than the second does not rule out convergence. A polynomial of degree q - 1 necessarily contains also a polynomial of degree q - p for $1 < p \leq q$ . If the $(q - p + 1)$ -order derivative of the exact solution is the highest-order derivative that is nonsingular, then we can apply our previous estimates with q - 1 replaced by q - p to deduce that the error in displacements is $O(h^{q-p+1})$ , the error in strains (calculated from first derivatives) is $O(h^{q-p})$ , and so on. The implication of this is that for each derivative of order q or less that is singular in the exact solution, we risk losing a power of h accuracy in the finite element solution. Such an estimate may be pessimistic, and often is. But when it holds, elements of degree q - p will have the same order of error as elements of degree q - 1, and the lower-degree elements will have less computational expense.
+
+As an example, consider again the axially loaded bar, with a step change in loading within an element. Specifically, for the element that spans $x = x_{i}$ to $x = x_{i+1}$ , let $x_{i} < x_{q} < x_{i+1}$ and
+
+$$
+q = 0 \text {   for   } x \leq x _ {q} \quad \text {   and   } \quad q = 1 \text {   for   } x > x _ {q} \tag {18.6-9}
+$$
+
+Then, from Eq. 18.6-1 $u_{,xx}$ is defined for all $x$ , $\max |u''(x)| = 1/AE$ , and Eqs. 18.6-5 and 18.6-8 make sense. However, $u'''(x)$ is a delta function at $x = x_q$ , so $\max |u'''(x)|$ is undefined. Accordingly, the error estimates remain as stated in Eqs. 18.6-5 and 18.6-8, whether the element is linear, quadratic, or of yet higher degree: the $O(h^3)$ displacement accuracy one would normally expect from a quadratic
+
+<!-- source-page: 578 -->
+
+element is not available. Full accuracy of a quadratic element could be recovered by placing a node at the step change in load, for example, set $x_{i} = x_{q}$ or set $x_{i+1} = x_{q}$ .
+
+For $C^{0}$ elements in general, we do not expect to lose much accuracy when properties such as A, E, v, thickness, and so on, display sudden jumps at inter-element boundaries. But if properties jump within elements or if the exact solution displays “blow-up” singularities in higher derivatives, considerable accuracy may be lost.
+
+# 18.7 DISCRETIZATION ERROR:
+
+# ESTIMATION AND EXTRAPOLATION
+
+In the latter part of Section 18.6 we argued that error e in a computed quantity could be represented as $e = O(h^{q-r})$ , where r = 0 for displacement error, r = 1 for stress or strain error in elasticity, and so on. A restatement of this result is
+
+$$
+e \approx C h ^ {q - r} \tag {18.7-1}
+$$
+
+where h is again the “characteristic length” and C is a problem-dependent constant that is influenced by element aspect ratio (ratio of longest side to shortest side), size ratio of the largest and smallest elements in the mesh, element type, quadrature rule, the $(q - p + 1)$ -order derivative of the exact solution (as in Section 18.6), and other factors [18.19]. Error e in a computed quantity $\phi$ may indeed refer to the error at a point, but more generally is
+
+$$
+e = \left\| \phi - \phi^ {h} \right\| \tag {18.7-2}
+$$
+
+where $\phi$ is the exact value, $\phi^h$ is the computed value, and the norm symbol $(\| \cdot \|)$ indicates an averaging process; for example, $e$ may represent the root-mean-square of several pointwise values, or an integrated average over a portion of the body, or the square root of strain energy in the entire body, and so on. If a single value of $\phi$ is intended, then $e = |\phi - \phi^h|$ .
+
+The relative error is $e_r = \| \phi - \phi^h \| / \| \phi \|$ . A rough indicator of this error is
+
+$$
+e _ {r} \approx \rho_ {1} \rho_ {2} h ^ {q - r} \quad \text { where } \quad h = \frac {1}{N ^ {1 / n}} \tag {18.7-3}
+$$
+
+Here q = one plus the degree of the highest complete polynomial in the element displacement field, r is defined above Eq. 18.7-1, $\rho_{1}$ = largest element aspect ratio, $\rho_{2}$ = ratio of characteristic length of the largest element to characteristic length of the smallest element, h = “dimensionless length,” N = number of elements in the mesh, and n = spatial dimension (n = 1, 2, or 3 for line, plane, and solid problems, respectively). Thus h is the characteristic length of an element in a domain that has been scaled so that its length, area, or volume is unity. The analyst has some control over all quantities in Eq. 18.7-3 except n. Our interpretation of $e_{r}$ is as follows. If $e_{r}$ is about 0.1 times the acceptable percentage error, the results are probably reliable (e.g., $e_{r} = 1\%$ if 10% error is acceptable). If $e_{r}$ is considerably smaller than this, the mesh may already be finer than necessary. If $e_{r}$ approaches or exceeds unity, further study is required. “Further study” may
+
+<!-- source-page: 579 -->
+
+![](images/page-579_8728618baf3ded2a35e53c496ac6f056fabcbd430052e541ea42644b075e02cf.jpg)
+
+<details>
+<summary>radar</summary>
+
+| x | y | Value |
+|---|---|-------|
+| 1 | 28 | 10 |
+| 1 | 19 | 11 |
+| 1 | 10 | 12 |
+| 1 | 37 | 20 |
+| 1 | 46 | 36 |
+| 1 | 55 | 27 |
+| 1 | 64 | 18 |
+| 1 | 73 | 36 |
+| 1 | 82 | 27 |
+| 2 | 28 | 10 |
+| 2 | 19 | 11 |
+| 2 | 10 | 12 |
+| 2 | 37 | 20 |
+| 2 | 46 | 36 |
+| 2 | 55 | 27 |
+| 2 | 64 | 18 |
+| 2 | 73 | 36 |
+| 2 | 82 | 27 |
+| 3 | 28 | 10 |
+| 3 | 19 | 11 |
+| 3 | 10 | 12 |
+| 3 | 37 | 20 |
+| 3 | 46 | 36 |
+| 3 | 55 | 27 |
+| 3 | 64 | 18 |
+| 3 | 73 | 36 |
+| 3 | 82 | 27 |
+| 4 | 28 | 10 |
+| 4 | 19 | 11 |
+| 4 | 10 | 12 |
+| 4 | 37 | 20 |
+| 4 | 46 | 36 |
+| 4 | 55 | 27 |
+| 4 | 64 | 18 |
+| 4 | 73 | 36 |
+| 4 | 82 | 27 |
+| 5 | 28 | 10 |
+| 5 | 19 | 11 |
+| 5 | 10 | 12 |
+| 5 | 37 | 20 |
+| 5 | 46 | 36 |
+| 5 | 55 | 27 |
+| 5 | 64 | 18 |
+| 5 | 73 | 36 |
+| 5 | 82 | 27 |
+| 6 | 28 | 10 |
+| 6 | 19 | 11 |
+| 6 | 10 | 12 |
+| 6 | 37 | 20 |
+| 6 | 46 | 36 |
+| 6 | 55 | 27 |
+| 6 | 64 | 18 |
+| 6 | 73 | 36 |
+| 6 | 82 | 27 |
+| 7 | 28 | 10 |
+| 7 | 19 | 11 |
+| 7 | 10 | 12 |
+| 7 | 37 | 20 |
+| 7 | 46 | 36 |
+| 7 | 55 | 27 |
+| 7 | 64 | 18 |
+| 7 | 73 | 36 |
+| 7 | 82 | 27 |
+| 8 | 28 | 10 |
+| 8 | 19 | 11 |
+| 8 | 10 | 12 |
+| 8 | 37 | 20 |
+| 8 | 46 | 36 |
+| 8 | 55 | 27 |
+| 8 | 64 | 18 |
+| 8 | 73 | 36 |
+| 8 | 82 | 27 |
+| 9 | 28 | 10 |
+| 9 | 19 | 11 |
+| 9 | 10 | 12 |
+| 9 | 37 | 20 |
+| 9 | 46 | 36 |
+| 9 | 55 | 27 |
+| 9 | 64 | 18 |
+| 9 | 73 | 36 |
+| 9 | 82 | 27 |
+| 10 | 28 | 10 |
+| 10 | 19 | 11 |
+| 10 | 10 | 12 |
+| 10 | 37 | 20 |
+| 10 | 46 | 36 |
+| 10 | 55 | 27 |
+| 10 | 64 | 18 |
+| 10 | 73 | 36 |
+| 10 | 82 | 27 |
+</details>
+
+Figure 18.7-1. A “Laplacian” plane mesh. Coordinates $x_{i}$ and $y_{i}$ of each interior node are equal to the average of the coordinates of the four adjacent nodes, for example, $x_{11} = (x_{2} + x_{10} + x_{12} + x_{20})/4$ .
+
+mean mesh refinement, alteration of the mesh without changing the number of elements, or assessment of the physical reasonability of the results if computational limitations do not permit mesh refinement or rearrangement.
+
+However, Eq. 18.7-3 may be very pessimistic. One can easily devise a patch test in which results are exact yet $e_{r} > 1000$ . Clearly, the role of stress gradients is not addressed by Eq. 18.7-3. In preparing or refining a mesh, an analyst must draw upon experience and intuition about gradients present and how well they can (or should) be modeled: if a test quantity such as stress or strain energy changes appreciably across interelement boundaries or within individual elements, mesh refinement may be needed.
+
+Consider, for example, Fig. 18.7-1. Here N = 72, n = 2, and we estimate $\rho_{1} = 2$ and $\rho_{2} = 8$ . Therefore, if elements are bilinear (q = 2), we estimate $e_{0} \approx 0.22$ for displacements (r = 0) and $e_{1} \approx 1.9$ for stresses (r = 1). These estimates indicate that results may not be reliable. This may indeed be the case if we seek the effect of a local disturbance, say a concentrated force at node 36. On the other hand, if nodes along the left edge are fixed and nodes along the right edge are loaded by a uniform traction, displacements and stresses near node 81 would probably be reliable because the mesh is smoothly graded, finer where stress gradients are likely to be larger, and fine enough at the location of interest. A uniform mesh ( $\rho_{1} = \rho_{2} = 1$ ) with elements the size of the smallest element in Fig. 18.7-1 might improve accuracy slightly, but would display $e_{0} = O(10^{-3})$ and $e_{1} = O(10^{-2})$ , and would be much more costly. We conclude that by mesh grading and suiting the mesh to the problem, one can “beat the estimate.”
+
+Multimesh Extrapolation. It is possible to use Eq. 18.7-1 and the computed results from two or more different meshes to extrapolate to an improved result. Ideally, this result has zero error. What rigor there is in this process depends on completely regular mesh refinement. That is, in each refinement, nodes and interelement boundaries of the coarser mesh are preserved, both in number and location, while adding new nodes and new boundaries; $\rho_{1}$ and $\rho_{2}$ of Eq. 18.7-3 remain constant; element types and quadrature rules are not changed; corner nodes stay corner nodes and side nodes stay side nodes (Fig. 18.7-2). In addition, the quantity to
+
+<!-- source-page: 580 -->
+
+![](images/page-580_1a0b0546b5586169333b755622ba12e90692e3af3dc4e8a4a4c85bfafb7d25d3.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Pure grid diagram with dashed lines and dots, no text or symbols present
+</details>
+
+{a}
+
+![](images/page-580_f7747989b6891806b2f2f34e31e143199070a2ef9a4dd3ee0b1a9a499b991d16.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Pure grid diagram with dashed lines and dots, no text or symbols present
+</details>
+
+(b)   
+Figure 18.7-2. Dashed lines indicate new element boundaries introduced by regular mesh subdivision. New nodes introduced by subdivision are not shown. (a) Bilinear elements; divisions by 2 shown. (b) Quadratic elements; divisions by 3 shown.
+
+be extrapolated must be calculated at a location fixed in space and fixed in position relative to an element (e.g., at a corner node) in all refinements. If such regularity is lacking, extrapolation can still yield a useful estimate, but without a sound basis in theory.
+
+If only two meshes are used, and the convergence rate is unknown, we have little choice but to assume that convergence is linear ( $e = O(h)$ ; see Fig. 18.7-3a). With data from three meshes, we may conclude that convergence is indeed linear, or that quadratic (or higher) convergence is a possibility. A too-coarse mesh may fail to display a definite trend. Incompatible elements often display nonmonotonic convergence (curve $AD$ in Fig. 18.7-3b). With nonmonotonic convergence, two meshes yield no reliable prediction: clearly, two data points can be extrapolated to a result that is better, worse, or no different, depending on the points chosen. If convergence is monotonic and $p$ is the order of error, $e = O(h^p)$ , then an improved result $\phi^0$ is obtainable by Richardson extrapolation [18.11],
+
+$$
+\phi^ {0} = \frac {\phi_ {1} h _ {2} ^ {p} - \phi_ {2} h _ {1} ^ {p}}{h _ {2} ^ {p} - h _ {1} ^ {p}} \tag {18.7-4}
+$$
+
+where, as examples, $p = 1$ for $e = O(h)$ in Fig. 18.7-3a and $p = 2$ for $e = O(h^2)$ in Fig. 18.7-3b. Equation 18.7-4 usually gives an inexact $\phi^0$ because conditions
+
+![](images/page-580_7192bec69d7c79c4f5f939f61b8526c38acc7ceece20d99c41028178c70aafd9.jpg)
+
+<details>
+<summary>line</summary>
+| h    | φ (Curve A) | φ (Curve B) |
+| ---- | ----------- | ----------- |
+| 0    | φ₀          | φ₀          |
+| h₂   | e = O(h)    | e = O(h)    |
+| h₁   | φ₁          | φ₁          |
+</details>
+
+{a}
+
+![](images/page-580_6dc8719755a6a029593733d45ae1d1ff666cfc061c1b6bd86e5cae07be4315c6.jpg)
+
+<details>
+<summary>line</summary>
+
+| h²     | φ (Curve A) | φ (Curve B) | φ (Curve C) |
+|--------|-------------|-------------|-------------|
+| 0      | φ⁰          | φ²          | φ¹          |
+| h²     | φ²          | φ¹          | φ²          |
+| h²     | e = O(h²)   | φ¹          | φ²          |
+</details>
+
+{b}   
+Figure 18.7-3. Convergence of a quantity $\phi$ with mesh refinement. AB, linear convergence. AC, quadratic convergence, AD, nonmonotonic convergence (but quadratic convergence when $h < h_{D}$ ).
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_059.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_059.md
new file mode 100644
index 00000000..e67a391f
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_059.md
@@ -0,0 +1,423 @@
+<!-- source-page: 581 -->
+
+![](images/page-581_0b716371399bd58bd1ee868052db15c8a0a4ebec13271a8c952d7baa799beb2b.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L = 2.40
+P = 100
+0.10
+E = 10^4
+w
+0.40
+v = 0.25
+</details>
+
+Figure 18.7-4. End-loaded cantilever beam. Mesh N = 8 is shown. Mesh N = 32 is suggested by dashed lines. All elements are bilinear.
+
+necessary for exactness do not prevail. Rather than regarding $\phi^0$ as the exact value, it seems preferable to regard
+
+$$
+e = \frac {\phi_ {2} - \phi^ {0}}{\phi^ {0}} 100\% \tag{18.7 - 5}
+$$
+
+as an estimate of percentage errors in mesh $h_{2}$ .
+
+Example: Regular Refinement. The cantilever beam problem shown in Fig. 18.7-4 was solved using three different meshes and three different formulations for each mesh. Results are presented in Fig. 18.7-5 and in Table 18.7-1 using the following notation. Element stiffness matrices were integrated using 2 by 2 quadrature (“Full”) or one-point quadrature (“Reduced” and “H-G”), where “H-G” refers to hourglass control (see Eq. 6.12-4). No mechanism is possible in the reduced-integration case because of complete fixity at the left end. The problem was solved on a microcomputer whose capacity did not permit the use of a finer mesh than N = 128.
+
+Figure 18.7-5 suggests that $e = O(h)$ for the full-integration case and $e = O(h^{2})$ for the reduced-integration case. Nevertheless, all extrapolated values in Table 18.7-1 were computed using p = 2 in Eq. 18.7-4.
+
+![](images/page-581_7d4fe782ab86ac70f68d6e528d43eec6e9c01d24b659c4501c204bbd2a3ad261.jpg)
+
+<details>
+<summary>line</summary>
+| h       | w (Reduced) | w (H-G) | w (Full) |
+| ------- | ----------- | ------- | -------- |
+| 0.0884  | ~0.8        | ~0.8    | ~0.8     |
+| 0.1764  | ~1.0        | ~0.8    | ~0.6     |
+| 0.3536  | ~1.5        | ~0.8    | ~0.4     |
+</details>
+
+(a)
+
+![](images/page-581_ad2e4e4a41ba3e7facc57dddca564b2b1253612d4efeafb8074b70115f41680f.jpg)
+
+<details>
+<summary>line</summary>
+
+| h²     | w (Reduced) | w (H−G) | w (Full) |
+| ------ | ----------- | ------- | -------- |
+| 0.008  | 0           | 0       | 0        |
+| 0.031  | ~0.5        | ~0.5    | ~0.5     |
+| 0.125  | ~1.0        | ~0.5    | ~0.5     |
+</details>
+
+(b)   
+Figure 18.7-5. Plots of data from Table 18.7-1, for the problem of Fig. 18.7-4: (a) w versus h, (b) w versus $h^{2}$ .
+
+<!-- source-page: 582 -->
+
+TABLE 18.7-1. COMPUTED DEFLECTION W IN FIG. 18.7-4. BEAM THEORY, INCLUDING TRANSVERSE SHEAR DEFORMATION, GIVES $w = (PL^{3}/3EI) + (6PL/5AG) = 8.640 + 0.180 = 8.820$ . EXTRAPOLATION IS BASED ON p = 2 IN EQ. 18.7-4. 
+
+<table><tr><td colspan="2">Mesh Data</td><td colspan="3">Computed Results (w)</td><td colspan="3">Extrapolated Results (w)</td></tr><tr><td>N</td><td> $h = 1/N^{1/2}$ </td><td>Full</td><td>Reduced</td><td>H-G</td><td>Full</td><td>Reduced</td><td>H-G</td></tr><tr><td>8</td><td>0.3536</td><td>4.562</td><td>11.440</td><td>8.572</td><td>7.978</td><td>8.617</td><td>8.768</td></tr><tr><td>32</td><td>0.1768</td><td>7.124</td><td>9.323</td><td>8.719</td><td>8.695</td><td>8.788</td><td>8.788</td></tr><tr><td>128</td><td>0.0884</td><td>8.302</td><td>8.922</td><td>8.771</td><td></td><td></td><td></td></tr></table>
+
+Example: Irregular Refinement. The beam in Fig. 18.7-6 contains a large central hole. We seek the largest principal stress. Each mesh shown is composed of bilinear elements, for which a 2 by 2 integration rule was used. Load $P$ is uniformly distributed across the right end. The upper half of the beam was modeled, with all nodes on $x = 0$ fixed and all nodes on $y = 0$ allowed only vertical displacement. Even so, the microcomputer could not accommodate regular mesh refinement. The meshes shown violate most rules of regular refinement. Indeed, the meshes could be substantially improved because they are undesirably coarse close to the hole and unnecessarily fine far from the hole. Stresses were evaluated at element centers. What is plotted as $\sigma_{\max}$ is the largest element center stress from each mesh, regardless of the element in which it appears.
+
+For pure bending load, photoelastic data indicate that $\sigma_{max}$ appears at the top edge of the hole and is $\sigma_{max} = 98.4$ [18.20]. One can perhaps assume that the error in $\sigma_{max}$ is $O(h)$ . Hence, a least squares fit of a straight line to the three data points yields
+
+![](images/page-582_279372e3bfe3370909cb12f3b9889c8a2070550939c94fb6e38ed72402dc22a4.jpg)
+
+<details>
+<summary>other</summary>
+
+| Dimension         | Value  |
+| ----------------- | ------ |
+| Width (N)         | 52 mesh |
+| Radius (r)        | 0.20   |
+| Total Width (N)   | 94 mesh |
+| Thickness (N)     | 130 mesh |
+| Stress (σ_max)    | 93.3   |
+| Least Squares Fit | 67.18  |
+| Constant (σ_max) | 61.15  |
+| Constant (σ_max) | 51.37  |
+| Constant (σ_max) | 1      |
+| Constant (σ_max) | 1/√130 |
+| Constant (σ_max) | 1/√94 |
+| Constant (σ_max) | 1/√52 |
+| Constant (σ_max) | 1/√N |
+</details>
+
+Figure 18.7-6. Cantilever beam with a large central hole. Each mesh is symmetric about both centerlines. Results of irregular mesh refinement are shown. (The authors are grateful to S-C. Liang and D. Rusche for doing the computations.)
+
+<!-- source-page: 583 -->
+
+![](images/page-583_3ae5252f5d39aa0020d2283e19a15c2fd9aa474525475bb4fb7b568e0c8d5fa6.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Geometric diagram of two adjacent rectangles with diagonal lines (no text or symbols)
+</details>
+
+Original mesh
+
+![](images/page-583_c3f7005464337d64fbb9f7b75f7a91eacbe606e0d37f90c360e7ff9294f6389d.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Pure geometric grid pattern with diagonal lines and dots, no text or symbols present
+</details>
+
+h refinement
+
+![](images/page-583_27ce0654c164ee770e15bddffdc3f8921dbb1fa5f479c7c565e0a54149981867.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Pure geometric diagram of a rectangle divided into two sections by diagonal lines (no text or symbols)
+</details>
+
+p refinement   
+Figure 18.7-7. The $h$ and $p$ versions of refinement of a plane mesh.
+
+$\sigma_{max} = 93.3$ , as shown. This extrapolated result is remarkably good, and perhaps somewhat fortuitous. At the very least, the plotted results show that none of the three meshes by itself yields a reliable result.
+
+The h version and the p version. The h and p versions of the finite element method are different ways of adding d.o.f. to the model, so as to reduce discretization error in a subsequent analysis. The h version refers to decreasing the characteristic length (h) of elements, by dividing each existing element into two or more elements, but without changing the types of elements used. The p version refers to increasing the degree of the highest complete polynomial (p) in elements, by adding nodes to elements, adding d.o.f. (e.g., derivative d.o.f.) to nodes, or both, but without changing the number of elements used. In Fig. 18.7-7, the next stage of p refinement might be to add derivative d.o.f. without changing the number of nodes.
+
+A sequence of successively refined meshes produces convergence toward correct results. The process is known as h convergence or p convergence, depending on the method of adding d.o.f. A computer program is termed “adaptive” if addition of d.o.f. and reanalysis can be accomplished with a minimum of direction from the analyst. The program is called “self-adaptive” if it can automatically decide where additional d.o.f. are most needed in the model, prepare a suitable new model, reanalyze, and keep repeating the process until a preselected convergence tolerance is achieved $[18.12,18.13]$ . Adaptive h refinement can continue until limits imposed by computer capacity and numerical noise are reached. Adaptive p refinement can continue until the highest-order polynomial coded in the program is used.
+
+Error estimates and monotonic convergence of successive solutions are made possible by completely regular mesh refinement, which is known in the present context as a “hierarchical” procedure. Hierarchical refinement produces a mesh that allows, as special cases, all displacement modes that were possible in the unrefined mesh. Both refinements in Fig. 18.7-7 are hierarchical. With either the h version or the p version, clever programming can incorporate computations done in the preceding mesh rather than repeating them entire for the current mesh.
+
+# 18.8 TESTS OF ELEMENT QUALITY
+
+Eigenvalue Test. The eigenvalue test is one of several tests of element quality. The test can detect zero-energy deformation modes, lack of invariance, and absence of rigid-body motion capability. It can also be used to estimate the relative quality of competing elements $[18.14,18.15]$ . We will describe the calculations first and then comment on how to interpret results.
+
+<!-- source-page: 584 -->
+
+Let loads $\{\bar{r}\}$ applied to element nodes be proportional to element nodal displacements $\{d\}$ through a factor $\lambda$ :
+
+$$
+[ \mathbf {k} ] \{\mathbf {d} \} = \{\overline {{{\mathbf {r}}}} \} = \lambda \{\mathbf {d} \} \quad \text { or } \quad ([ \mathbf {k} ] - \lambda [ \mathbf {I} ]) \{\mathbf {d} \} = \{\mathbf {0} \} \tag {18.8-1}
+$$
+
+This is an eigenproblem. Eigenvalues $\lambda_{i}$ are called eigenvalues of [k]. There are as many $\lambda_{i}$ as there are d.o.f. in $\{\mathbf{d}\}$ . Not all $\lambda_{i}$ need be different. To each $\lambda_{i}$ there corresponds an eigenvector $\{\mathbf{d}\}_{i}$ . If each $\{\mathbf{d}\}_{i}$ is normalized so that $\{\mathbf{d}\}_{i}^{T}\{\mathbf{d}\}_{i}=1$ , premultiplication of Eq. 18.8-1 by $\{\mathbf{d}\}_{i}^{T}$ yields
+
+$$
+\{\mathbf {d} \} _ {i} ^ {T} [ \mathbf {k} ] \{\mathbf {d} \} _ {i} = \lambda_ {i} \quad \text { or } \quad 2 U _ {i} = \lambda_ {i} \tag {18.8-2}
+$$
+
+where $U_{i}$ is strain energy in the element when its nodal d.o.f. are the normalized displacements $\{d\}_{i}$ (see Eq. 3.3-9). Usually the element is unrestrained for the eigenvalue test so that [k] is the complete element stiffness matrix. In an existing computer program, one can conveniently compute the $\lambda_{i}$ as the squared natural vibration frequencies of an unsupported element that has unit mass attached to each d.o.f. in $\{d\}$ . Note that $\lambda_{i}$ is unchanged if the deformation is reversed—that is, if $\{d\}_{i}$ is replaced by $-\{d\}_{i}$ .
+
+Equation 18.8-2 shows that [k] should yield $\lambda_{i}=0$ when $\{d\}_{i}$ represents any rigid-body motion. There are three linearly independent rigid-body motions possible in the plane. Therefore three of the $\lambda_{i}$ should be zero for a plane element. Six should be zero for a general solid or shell element, but only one for a solid or shell of revolution element (if only axially symmetric states are permitted).
+
+Zero-energy modes (mechanisms) also yield zero eigenvalues. The associated $\{d\}_{i}$ may appear in combination with the $\{d\}_{i}$ of a rigid-body motion.
+
+In testing an element, we first check that [k] has as many $\lambda_{i}=0$ values as expected. Too few suggests that the element lacks a desired capability for rigid-body motion without strain. Too many suggests the presence of one or more mechanisms.
+
+Nonzero eigenvalues are real and positive if $[k]$ is symmetric and positive semidefinite. If eigenvalues change when the element is reoriented in global coordinates, the element is not geometrically isotropic. Similar modes, such as the flexural modes of Fig. 18.8-1, should be associated with equal eigenvalues if the material is isotropic.
+
+Eigenvalues of $[k]$ can sometimes be used to compare different formulations of a given element type, for example, isoparametric versus hybrid formulations of
+
+![](images/page-584_91808999ba234380268b65a73582f6f0b22607213df541f825a3852a1a7b6cb7.jpg)  
+Figure 18.8-1. Nonzero eigenvalues and corresponding eigenvectors (deformation modes) of a square bilinear element in plane strain [18.16]. $E = 1.0$ , $\nu = 0.3$ , side length $= 1.0$ .
+
+<!-- source-page: 585 -->
+
+a four-node plane element. Properly formulated elements of the same size, shape, and material properties should be equally stiff in their constant strain modes. Their stiffness matrices should therefore have eigenvalues in common. Stiffnesses, and therefore eigenvalues, may differ in other modes. For compatible elements based on assumed displacement fields, the $\lambda_{i}$ are either exact or are upper bounds on the correct strain energy. Therefore, when comparing elements of the same size, shape, [E], node placement, and number and type of d.o.f., the element with the lowest strain energy is best. The stiffness matrix of this element has the lowest trace (tr[k] equals the sum of the eigenvalues of [k]). This argument fails if any $\lambda_{i}$ is not an upper bound, which happens if the element contains a mechanism.
+
+Other Tests. Remarks. In the single-element test [18.17], the response of a single element to a certain loading is examined as one changes the element aspect ratio, skewness, or taper. For example, in Fig. 18.8-2, a tip-loaded cantilever beam can be repeatedly analyzed as L/H is varied from a small value to a large value. It may happen that of two competing element formulations A and B, both work well when L/H = 1 but A is much more accurate than B when L/H >> 1. Then A has proved superior in this particular test. The test is very easy to perform. When used with but one element formulation, rather than in comparing different formulations, the test provides information about element behavior that is useful in modeling.
+
+Another type of element test compares strain energies $[18.15]$ . For a prescribed deformation state, strain energy in the mathematical continuum is computed over the volume spanned by the element. Strain energy in the element is also computed by imposing nodal d.o.f. consistent with the prescribed deformation state. The two energies are compared. The test is repeated, using other deformation states.
+
+Other tests have already been noted, including the patch test (Section 4.6) and applying a finite element model to various problems whose solutions are already known. The latter test is neither definitive nor general, but one would be foolish not to use it.
+
+In testing a new element, or in becoming acquainted with an unfamiliar element, it is appropriate to use a variety of tests. No single test is likely to be decisive, except perhaps in discovering a fatally flawed element. Element behavior tends to be case-dependent, so that the ranking of competing elements is likely to be different in different test cases and is likely to depend on whether stress or displacement is taken as the indicator of quality. The ideal element, probably never to be discovered, passes patch tests, yields good results in a coarse mesh, converges rapidly with mesh refinement, is almost insensitive to shape distortion, is geometrically isotropic, has no zero- or low-energy deformation modes when assembled with other elements, can be joined to elements of different type, rests on simple theory, and is economical to formulate.
+
+![](images/page-585_6447ddd2f462c98441c3bd071ebf03da2e10aecf55f3a0ad4bda87ffb397533e.jpg)
+
+<details>
+<summary>text_image</summary>
+
+H
+L
+P
+</details>
+
+Figure 18.8-2. Single-element model of a cantilever beam under transverse tip load.
+
+<!-- source-page: 586 -->
+
+# 18.9 CONCLUDING REMARKS
+
+In this chapter we have discussed tests that can be applied to elements, to an assembly of elements, or to a set of equations. If an error or a potential difficulty is present, no single test is certain to detect it. Even if several tests are applied and passed, one should be aware that tests can detect errors but cannot prove their absence, and that a trouble-free finite element structure may not be a good model of physical reality.
+
+Physical situations in structural mechanics that make numerical error more likely include elements with great shape distortion or large aspect ratio, an element whose shear or membrane stiffness is much larger than its bending stiffness, stiff elements used to approximate a rigid region, and a very fine mesh. This is a list of guidelines rather than firm rules, as in each situation one can identify one or more special cases in which the anticipated trouble does not materialize.
+
+On most digital computers, one should use double-precision data for all constants and double-precision arithmetic for all phases of generating structural equations and solving them. Partial double precision in equation solving and double-precision solution of equations generated in single precision are dangerous practices. It is risky to generalize from a single example, yet it is interesting that practical problems with $1.5(10^{6})$ d.o.f. have been solved with good engineering accuracy.
+
+# PROBLEMS
+
+# Section 18.2
+
+18.1 (a) Solve Eqs. 18.2-1 graphically by plotting the slope of each line and determining the intersection point. (b) Repeat part (a), using Eqs. 18.2-2 instead.
+
+18.2 Consider the equations $x + y = 2$ and $x + 1.01y = 2.01$ . Show that the solution is sensitive in small changes in both the coefficient matrix and the vector of constants.
+
+18.3 Show that Eq. 18.2-3 can be written in the form of Eq. 9.3-5, with $k = k_{2}$ and penalty number $\alpha = (k_{1} / k_{2}) - 1$ . What happens when $\alpha$ becomes very large?
+
+18.4 In Eq. 18.2-4, determine the reduced coefficient $K_{22}$ produced by Gauss elimination. Show that this result predicts trouble if $\alpha$ is large.
+
+18.5 The two-element beam shown is uniform, fixed at the left end, and simply supported at the right end. Show that the structural equations become ill conditioned if the scalar multiplier $\alpha$ is very small.
+
+![](images/page-586_487a073f3fe1771e1802363122309b22ea6120004d4e9ac0fcf1c6fb0fe57d25.jpg)
+
+<details>
+<summary>text_image</summary>
+
+1
+2
+3
+M₀
+L
+αL
+</details>
+
+Problem 18.5
+
+![](images/page-586_6dc647cd4a468cbf306417ce08c934cb60a8d0f8cfd04178629f9c1c9331a6e6.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L
+P
+1
+2
+k
+</details>
+
+Problem 18.7
+
+<!-- source-page: 587 -->
+
+18.6 (a) Obtain Eqs. 18.2-6 by coordinate transformation of Eqs. 18.2-3. Show that large cancellation error may be present.
+
+(b) Obtain Eqs. 18.2-6 by applying coordinate transformation to [k] of the left element in Fig. 18.2-1, then assembling the two elements.
+
+18.7 The left end of the one-element cantilever beam shown rests on a soft spring of stiffness k. Rotation is prevented at the left end.
+
+(a) Solve for the deflection of load P using $w_{1}$ , $w_{2}$ , and $\theta_{2}$ as nonzero d.o.f. Show that the equations are ill conditioned if $k << EI/L^{3}$ .
+
+(b) Form an element stiffness matrix that operates on d.o.f. $w_{1}, \theta_{1}, w_{21}$ , and $\theta_{21}$ , where $w_{21}$ and $\theta_{21}$ are the “relative” d.o.f. $w_{21} = w_{2} - (w_{1} + L\theta_{1})$ and $\theta_{21} = \theta_{2} - \theta_{1}$ . Impose one boundary condition, include the soft spring, and again solve for the deflection of load $P$ . Show that the equations are not ill conditioned.
+
+18.8 For a general structure, how many d.o.f. can be “relative” and how many must be “absolute”? (See Problem 18.7(b) for an illustration of these terms.)
+
+# Section 18.3
+
+18.9 (a) How small can $C(\mathbf{K})$ be?
+
+(b) What is $C(\mathbf{K})$ if $[\mathbf{K}]$ is diagonal?
+
+(c) Physically, what is implied if $C(\mathbf{K})$ is infinite?
+
+18.10 (a) Let $k_{1} = 10k_{2}$ in Fig. 18.2-1. Compute the condition numbers of the unscaled matrix [K] and of the scaled matrix $[\mathbf{K}_s]$ .
+
+(b) Repeat part (a), now with $k_{2} = 10k_{1}$ .
+
+18.11 (a) The two-spring system shown has d.o.f. $u_{1}$ and $u_{2}$ . For what value of scalar $c$ is the condition number of the unscaled stiffness matrix a minimum, and what is its minimum value?
+
+(b) Repeat part (a), but use the scaled stiffness matrix.
+
+![](images/page-587_a47e741d8b7c0126312b9927e046142b4f41d4c93cf5422e63f6a21c5b878062.jpg)  
+Problem 18.11
+
+18.12 Let $\beta = 45^{\circ}$ in Fig. 18.2-3a. Compute $C(\mathbf{K})$ in terms of $\alpha$ using (a) the unscaled matrix [K], and (b) the scaled matrix $[\mathbf{K}_s]$ .
+
+18.13 (a) Evaluate Eq. 18.3-2 for the structure shown in Fig. 18.3-1.
+
+(b) Let $AE / L = 174$ in Fig. 18.3-1a. Let a hypothetical computer retain only three significant digits per word, so that coefficients in [K] are represented as $K_{11} = -K_{12} = -K_{21} = 1040$ and $K_{22} = 1220$ . If subsequent manipulations are exact, what are the percentage errors in the computed values of $u_{1}, u_{2}$ , and $(u_{1} - u_{2}) / L$ ? Does Eq. 18.3-2 seem to apply?
+
+(c) Repeat part (b) for the problem of Fig. 18.3-1b.
+
+18.14 Interchange the bars in Fig. 18.3-1; for example, let the left bar have stiffness 174 and the right bar have stiffness 6(174) = 1044.
+
+(a) Compute $C(\mathbf{K})$ , using first the unscaled [K] and then the scaled matrix $[\mathbf{K}_s]$ .
+
+(b) Repeat Problem 18.13(b), where now $K_{11} = -K_{12} = -K_{21} = 174$ and $K_{22} = 1220$ .
+
+<!-- source-page: 588 -->
+
+18.15 A uniform beam is modeled by a single standard beam element. What is the condition number of the scaled stiffness matrix if the beam is (a) simply supported, and (b) cantilevered? (There are two nonzero d.o.f. in each case.)   
+18.16 Why, in Eq. 18.3-8, can $N^{2m/n}$ be replaced by $h^{-2m}$ ? (Here it is easiest to imagine that $h$ , the span of a typical element, is uniform throughout the mesh.)   
+18.17 The bar shown is uniform and is built of standard two-node elements. In axial vibration, its fundamental frequency $\omega_{1}$ is given by $\omega_{1}^{2} = \pi^{2}E/4L_{T}^{2}\rho$ , where $\rho$ is the mass density.
+
+(a) Let all elements have the same length $L$ . Use Eq. 18.3-9 to bound $C(\mathbf{K})$ in terms of $L_T$ and $L$ .   
+(b) Evaluate this bound numerically, using two elements $(L = L_T / 2)$ . Also evaluate the exact $C(\mathbf{K})$ , and compare.   
+(c) Repeat part (b), but use two unequal elements, of lengths $L$ and $2L$ , respectively.
+
+![](images/page-588_37d7040f28c41063ea70cf419305d4e6c09317c79d244ff7a6debfa217ffab09.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L
+L_T
+</details>
+
+Problem 18.17
+
+# Section 18.4
+
+18.18 (a) Write [K] for the structure shown, in which each spring has stiffness $k = 100$ . By examination of the first few steps of Gauss elimination, deduce an expression for the reduced diagonal coefficient $K_{ii}$ in terms of $k$ and $i$ . Hence, what are the diagonal decay ratios after the 99th and 100th eliminations?
+
+(b) Repeat part (a), but sequence the node numbers from right to left, so that node 1 carries load $P$ .
+
+18.19 In each part of this problem, imagine that the analyst has forgotten to specify any displacement boundary conditions, so that the structure is unsupported. In what equation (first, second, . . ., last) of the system $[K]\{D\} = \{R\}$ will the diagonal decay test detect this trouble?
+
+(a) The train of springs in Problem 18.18.   
+(b) The beams of Table 18.4-1.   
+(c) A plane frame having three d.o.f. per node.   
+(d) A solid of revolution having two d.o.f. (radial and axial) per node.   
+(e) A plane structure having two d.o.f. per node.
+
+18.20 Compute the diagonal decay ratio in terms of $\alpha$ for the problem described by Fig. 18.2-3a. Show that it is not large if $\beta = 0$ or if $\beta = \pi / 2$ .
+
+18.21 With unit axial loads at nodes 2 and 3 in the structure shown, the exact
+
+![](images/page-588_4a4109a029f763c759a533070509d4222c6307d92e0e1a59d4390c437df54497.jpg)
+
+<details>
+<summary>text_image</summary>
+
+k k k k k
+1 2 99 100 101 P
+</details>
+
+Problem 18.18
+
+![](images/page-588_21a5eae78e7e10b3224ca500af56c9be94b0a8f46cb20b20d200cdf113cebe7f.jpg)  
+Problem 18.21
+
+<!-- source-page: 589 -->
+
+structure equations are $8006.6u_{2}-8000u_{3}=1$ and $-8000u_{2}+8000u_{3}=1$ . Compute the exact values of $u_{2}$ and $u_{3}$ . Also compute approximate values under the assumption that the computer rounds numbers to only four digits after each operation of Gauss elimination. Does the accuracy loss agree with that predicted by (a) the diagonal decay ratio and (b) the condition number of [K]?
+
+18.22 What Fortran statements should be added, and where, if a test for diagonal decay is to be incorporated in the equation solver provided in Fig. B.2-3?
+
+# Section 18.5
+
+18.23 Consider the two equations $1.78u_{1} + 1.06u_{2} = 2.88$ and $0.94u_{1} + 0.56u_{2} = 1.52$ . What residual vector $\{\Delta \mathbf{R}\}$ is given by the approximate solution $u_{1} = 1.88$ , $u_{2} = -0.44$ , and what is $e$ of Eq. 18.5-2? What is the exact solution? Are the equations ill conditioned?
+
+18.24 Consider the ill-conditioned equations $u_{1} + u_{2} = 2$ , $u_{1} + 1.0001u_{2} = 2.0001$ . What residuals and what e are given by the approximate solution $u_{1} = 2.0$ , $u_{2} = 0.0$ , and by the approximate solution $u_{1} = u_{2} = 1.1$ ? Which of the two approximate solutions is most nearly correct?
+
+18.25 A linear spring of stiffness 28 N/m is loaded by a 0.5-N force. Using the approximate value $k^{-1} \approx 0.040$ m/N, we compute the approximate displacement $u \approx 0.020$ m. Improve this result by the iterative method, associated with Eq. 18.5-1, using the correct k and the approximate $k^{-1}$ .
+
+18.26 Consider the two equations $(u_{1} / 3) - (u_{2} / 3) = 1$ and $-(u_{1} / 3) + (7u_{2} / 12) = 0$ . Let the stiffness matrix and its inverse be approximated as
+
+$$
+[ \mathbf {K} ] \approx \left[ \begin{array}{c c} 0. 3 & - 0. 3 \\ - 0. 3 & 0. 5 \end{array} \right] \quad \text { and } \quad [ \mathbf {K} ] ^ {- 1} \approx \left[ \begin{array}{c c} 8. 0 & 5. 0 \\ 5. 0 & 5. 0 \end{array} \right]
+$$
+
+(a) Use the foregoing approximate matrices in 18.5-1. Apply three cycles of iterative improvement. Do displacements $u_{1}$ and $u_{2}$ appear to be converging toward correct values?   
+(b) Use the approximate $[\mathbf{K}]^{-1}$ to compute $\{\Delta \mathbf{D}\} = [\mathbf{K}]^{-1}\{\Delta \mathbf{R}\}$ but the correct [K] to compute $\{\Delta \mathbf{R}\}$ . Again apply three cycles of iterative improvement and assess the results.
+
+18.27 The cantilever beam shown is built of four constant-strain triangles (described in Chapter 5). For what loadings at nodes A and B will computed results be exact? For which of these loadings will residuals be essentially zero? For each loading plot the qualitative variation of $\sigma_{x}$ along the x axis, according to your expectation of the finite element results.
+
+![](images/page-589_19af193ce34b754dacc839be50f9da580b5ec51a2f223fcb757eca9f92df9806.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+Lr
+B
+x
+A
+</details>
+
+Problem 18.27
+
+<!-- source-page: 590 -->
+
+# Section 18.6
+
+18.28 For the bar problem of Fig. 18.6-1, show that the finite element solution yields exact displacements at the nodes when $A$ and $E$ are constant and the load integral, $\int [\mathbf{N}]^T q dx$ , is evaluated consistently over length $h_i$ of each element. Assume that $q = q(x)$ is a continuous function for $x_i \leq x \leq x_{i+1}$ . Suggestion: First show that the exact solution is
+
+$$
+A E u (x) = - \int_ {0} ^ {x} \int_ {0} ^ {s} q (\ell) d \ell d s + x \int_ {0} ^ {L _ {T}} q (\ell) d \ell \tag {a}
+$$
+
+Then evaluate the nodal load vector consistently. Integrate each entry by parts to show that the ith entry of the nodal load vector is
+
+$$
+\begin{array}{l} \int_ {0} ^ {h _ {i - 1}} q (x + x _ {i - 1}) d x - \frac {1}{h _ {i - 1}} \int_ {0} ^ {h _ {i - 1}} \int_ {0} ^ {x} q (s + x _ {i - 1}) d s d x \\ + \frac {1}{h _ {i}} \int_ {0} ^ {h _ {i}} \int_ {0} ^ {x} q (s + x _ {i}) d s d x \tag {b} \\ \end{array}
+$$
+
+The last integral is absent in the load at $L_{T}$ . Since the response at the nodes, $x_{i}$ , to given nodal loads is exact for bar elements, deduce the response to these nodal loads exactly, and show that the result coincides with Eq. (a) at the nodes.
+
+18.29 From Problem 18.28 we may deduce that the error function $e(x)$ defined in Eq. 18.6-2 has the expansion given in Eq. 18.6-3 on element $i$ . Show that $e'(x_i) = (h_i/2)u''(\bar{x})$ for an $\bar{x}$ in the $i$ th element. Suggestion: From Eq. (a) and Eq. 18.6-2, we may easily deduce that
+
+$$
+e ^ {\prime} (x _ {i}) = \frac {1}{A E} \int_ {x _ {i}} ^ {L _ {T}} q (x) d x - \frac {u _ {i + 1} - u _ {i}}{h _ {i}} \tag {c}
+$$
+
+Express $u_{i+1}$ and $u_{i}$ using Eq. (a), since these values have been shown to be exact in problem 18.28. Apply the mean value theorem for integrals to the resulting expression.
+
+18.30 If a uniform beam carries a smooth and continuous distributed lateral load, the standard beam element yields exact values of nodal displacements and rotations. Therefore, over one element, error $e(x)$ in lateral displacement has the appearance shown. A Taylor series analysis shows that the leading term in an expansion like Eq. 18.6-3 for $e(x)$ is $\frac{1}{2}(x - x_i)^2 e''(\overline{x})$ for an $\overline{x}$ in the element. Show that $e(x) = O(h^4)$ . Suggestion: Use the $N_i$ of Fig. 3.13-2, and show that if $u(0), u'(0), u(h)$ , and $u'(h)$ are exact, then $e(h/2) = O(h^4)$ and $e'(h/2) = O(h^3)$ on the element. Then
+
+![](images/page-590_4d8318c357a1c1ed4aee2b0121bc74525f0e7934af1398a5ef3eb6e3bfd8515f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+ξ
+e(ξ)
+e(h/2)
+ξ = 0
+ξ = h
+</details>
+
+Problem 18.30
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_060.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_060.md
new file mode 100644
index 00000000..6ba9b711
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_060.md
@@ -0,0 +1,387 @@
+<!-- source-page: 591 -->
+
+$$
+e (x) = e (h / 2) + \xi e ^ {\prime} (h / 2) + (\xi^ {2} / 2) e ^ {\prime \prime} (h / 2) + \dots \tag {d}
+$$
+
+and $e(h) = e'(h) = 0$ implies that $e''(h/2) = O(h^2)$ , from which the result follows.
+
+18.31 Assume that the loops in Fig. 18.6-1b are portions of parabolas. Hence, construct a geometric argument that shows that error $e_{max}$ is quartered if element length h is halved. Similarly, show that error $e'_{max}$ is halved if h is halved.
+
+# Section 18.7
+
+18.32 Imagine that stresses are to be calculated using the plane meshes shown in Fig. 18.7-2. Including meshes suggested by dashed lines, a total of four different meshes are indicated. For each of the four, what relative stress error is estimated by Eq. 18.7-3?
+
+18.33 Derive Eq. 18.7-4.
+
+18.34 Why would it be inadvisable to apply Eq. 18.7-4 to results obtained by use of the DKT plate element described in Section 11.4?
+
+18.35 For meshes $N = 1, 2$ , and 4, the respective computed values of displacement at a certain point in a plane mesh are 4.16, 4.64, and 4.76 units. What is the deflection predicted by extrapolation?
+
+18.36 Recompute the six “extrapolated results” in Table 18.7-1, this time using p = 1 in Eq. 18.7-4.
+
+18.37 What value of $\sigma_{max}$ is predicted by linear extrapolation in Fig. 18.7-6? Use all three possible combinations of two data points. What is the average of these three values?
+
+18.38 Verify that $\sigma_{max} = 93.3$ is the value predicted at $N = \infty$ in Fig. 18.7-6 by linear regression (least squares fit of a straight line).
+
+18.39 Use Eq. 18.7-4, with the appropriate value of p, to predict the converged value of stress at point A in Fig. 4.7-3a and in Fig. 4.7-3b.
+
+(a) Use the “standard” results in Table 4.7-1.
+
+(b) Use the iterated results in Table 4.7-1.
+
+18.40 Use Eq. 18.7-4, with a value of $p$ appropriate to the result to be extrapolated, to predict the converged values of $v_{C}, \sigma_{A}$ , and $\sigma_{B}$ in the following example problems.
+
+(a) “QM6” results in Fig. 8.3-3.
+
+(b) “Bilinear” results in Fig. 8.3-3.
+
+(c) “Equation 8.4-5” results in Fig. 8.6-2.
+
+(d) "Figure 8.6-1" results in Fig. 8.6-2.
+
+18.41 In Problem 13.17, axial vibrations of a bar were analyzed using three different mass matrices and meshes N = 1 and N = 2 for each mass matrix. The squared fundamental frequency is $\omega^{2} = cAE/mL$ , where $c = \pi^{2}/4$ for the exact result. For the respective mass matrices, the computations yield the following approximate values of c:
+
+Mesh $N = 1$ 3.000 2.000 2.400
+
+Mesh $N = 2$ 2.597 2.343 2.463
+
+<!-- source-page: 592 -->
+
+Use Eq. 18.7-4 to extrapolate to $N = \infty$ for each of the three sequences of two values.
+
+# Section 18.9
+
+18.42 Various situations that may promote numerical error are noted in the second paragraph of Section 18.9. For each of the four, cite an exception, for which the difficulty does not materialize.   
+18.43 Imagine that you must model a very stiff beam on a very soft elastic foundation. What will you do to assure that deflections and bending moments of the beam are computed accurately?   
+18.44 With mesh refinement, accuracy may improve, decline, or not change. For each of these three possibilities, give examples of problems (or situations, or meshes) that would behave in this way.
+
+<!-- source-page: 593 -->
+
+# MODELING, PROGRAMS, AND PROGRAMMING
+
+Guidelines are discussed for producing a finite element model of an actual structure. Advice is given regarding the writing and acquisition of computer programs.
+
+# 19.1 MODELING
+
+General. Modeling is an art based on the ability to visualize physical interactions. All basic and applied knowledge of physical problems, finite elements, and solution algorithms contributes to modeling expertise. Little is published regarding modeling. Practitioners tend to learn by doing and by talking with others. The documentation of a large general-purpose program usually contains some modeling advice in connection with specific example problems [e.g., 19.1].
+
+The documentation cannot be ignored: before all else fails, study the directions! Quite possibly, this study will not resolve all questions regarding terminology, symbols, assumptions, default conditions, and so on. Small test problems, sometimes involving a single element, can be run to clarify matters. Small test problems can help resolve questions about the sensitivity of an element type to aspect ratio and other shape distortions, whether beam and plate elements allow for transverse shear deformation, how distributed loads are treated, sensitivity to ill-conditioning, whether an eigensolver has trouble with zero or repeated eigenvalues (for the latter test one could use a three-dimensional beam of circular cross section [19.16]). These exercises will also increase confidence in use of the program and improve understanding of how various elements behave.
+
+In modeling, the principal difficulty faced by a typical user of a computer program is not understanding the physical action and boundary conditions of the actual structure, and the limitations of applicable theory, well enough to prepare a satisfactory model. Another difficulty is not understanding the behaviors of various elements, and the program's options and limitations, well enough to make an intelligent choice among them. The result may be a poor specification of the problem to be solved, a model that fails to reflect important features of the physical problem, fine detail irrelevant to the problem, a solution based on inappropriate loading or support conditions, and a surplus of computed results which are not properly examined and questioned $[19.2]$ . Automatic mesh generators make it easy to use too much fine detail. Powerful graphic postprocessors may smooth stress discontinuities that should warn of a need for local refinement or may hide questionable results by attractive display. It is possible that most finite element analyses are so flawed that they are worthless, and many experts feel that the situation is not improving. The reader is advised to review Section 1.8 of this book, “Warning: The Computed Answer May Be Wrong.”
+
+<!-- source-page: 594 -->
+
+Advice contained in the following discussion should not be regarded as a set of inflexible rules. An experienced and competent analyst may find exceptions that can sometimes be exploited to advantage.
+
+Cost and Dimensionality. Occasionally one may elect to analyze a one- or two-dimensional model rather than a two- or three-dimensional model, particularly when doing an initial simplified analysis. A rough guide to relative costs can be obtained by comparing the equation-solving costs of one-, two-, and three-dimensional meshes of comparable elements. For example, let us fill a square with $N^{2}$ elements and a cube with $N^{3}$ elements. Thus each mesh has N elements per side. If N is large, elements have corner nodes only, and each node has a single d.o.f., then the plane mesh produces rough $N^{2}$ equations having a semibandwidth of roughly N, for which the equation-solving expense is roughly proportional to $N^{2}(N)^{2} = N^{4}$ . The corresponding numbers for the solid mesh are $N^{3}$ , $N^{2}$ , and $N^{3}(N^{2})^{2} = N^{7}$ . Thus the rough measure of cost increases by the factor $N^{7}/N^{4} = N^{3}$ in going from two dimensions to three. Even for a mesh only ten elements on a side, this is a thousandfold increase.
+
+Which Element is Best? The appropriate answer is another question: Best for what? Element performance is problem-dependent. An element or mesh that works well in one situation may work badly in another. The analyst must understand how various elements behave in various situations, and must understand the physics of the problem well enough to make an intelligent choice of elements and mesh.
+
+A rough guideline, which falls well short of being a rule, is that elements of intermediate complexity work well for many problems. Thus, one would usually avoid using a great many of the simplest elements or a very few of the most complicated elements.
+
+Start Simply. A problem of moderate or large complexity should not be swallowed whole. One might begin with rough approximations from “back of the envelope” calculations. Numerically, a good beginning that often provides considerable insight is a “stick model,” which is a model built of a few bar and beam elements. A stick model is simple to prepare, cheap to run, and gives approximate results. If a more refined model gives greatly different results, the analyst should seek the reason for the discrepancy.
+
+A stick model, or a coarse-mesh model, can be used to guide subsequent refinement. If symmetry is to be exploited, a simple model serves to check anticipated symmetries and perhaps discover additional symmetries. A two-dimensional model may serve as an early analysis step in a three-dimensional problem, and may sometimes make a three-dimensional analysis unnecessary.
+
+If a dynamic or nonlinear analysis is contemplated, a linear static analysis of the proposed final mesh might be done beforehand as a relatively cheap test that may disclose flaws in the model. Loads applied in the static analysis should be contrived to produce strain distributions and gradients comparable to those anticipated in the subsequent dynamic or nonlinear analysis.
+
+Structure to Model. Modeling is more than just laying out a mesh. A focus on mesh layout or the minutiae of modeling may be at the expense of grasping important physical aspects that strongly affect the actual behavior.
+
+<!-- source-page: 595 -->
+
+In attempting to simulate reality with a mathematical model, the analyst must come to grips with the physics of the problem. What are the loads? What are the boundary conditions? Which actions are important and which are unimportant? Is the problem quasistatic or dynamic? If dynamic, is damping important? If so, how should it be represented? Is buckling a possibility? Is the material isotropic? Do properties depend on temperature or strain rate? Is plastic flow involved? If so, is it localized or widespread? Are there other nonlinearities that demand attention? These and other questions suggested by the problem at hand must be addressed before one can decide what element types and what arrangement of specific elements will produce a good model, and, if the problem is dynamic or nonlinear, what solution algorithm will produce reliable results at acceptable cost.
+
+Broad guidelines include the following. Include all real structure in the model; do not omit parts on the untested assumption that they carry little load or little stress. If a curved boundary is modeled as a polygon or a faceted surface, do so in a way that preserves the correct volume of the structure. In the analysis of thermally induced stresses, arrange element sizes and types so that the complexity of the temperature field can be approximately matched by the complexity of the strain field. If the temperature field is discontinuous across an interelement boundary, make sure the program does not “smear” the temperature change across elements by interpolation from nodal temperatures. Use consistent nodal loads rather than ad hoc lumping of loads. Use a relatively coarse mesh where gradients are known to be low and a relatively fine mesh where gradients are known to be high.
+
+In a coarse mesh, different element arrangements can produce a different model than intended. Consider, for example, a rectangular plate with clamped edges (Fig. 19.1-1). Unshaded elements are completely inactive because all their d.o.f. are set to zero. Thus the plate is modeled by only the shaded elements. The model differs from the structure in size and shape, and symmetries of behavior may be lost.
+
+Anticipate the Results and Know the Goal. If results were known in advance, it would be comparatively easy to prepare an adequate model. Similarly, if the probable results can be anticipated, a cheaper and better model will result. If locations of high stress are known in advance, it may be possible to model remote locations rather crudely. If the severity of stress gradients is anticipated, one can estimate the proper element size. If the goal of analysis is only to assess deflections, not to compute stresses, than a comparatively coarse mesh may suffice.
+
+Expect to Revise. Ideally, the original model is adequate and only a single analysis is performed. Far more often, the first analysis discloses inadequacies of the
+
+![](images/page-595_ae568e91ed69afaa018abde6837ec2e850c42cb0a276560336f6cd60d18c8c49.jpg)  
+Figure 19.1-1. Four arrangements of eight triangular elements to model a rectangular plate with clamped edges.
+
+<!-- source-page: 596 -->
+
+![](images/page-596_4544d0f3e10357f85d48d21a03101477ea43e1cb9830fff4cca548e200971459.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L_T
+P
+A
+B
+</details>
+
+{a}
+
+![](images/page-596_9b31ff3967361bfd7dcc61d8c35162f5c73c819877a93299bebdccbf65e877d5.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L_T
+P
+A
+B
+</details>
+
+(b)
+
+![](images/page-596_6dee71d4fdfd89fa8f79a8e917e40ac9141a43a9413b74de89063930dd303880.jpg)
+
+<details>
+<summary>text_image</summary>
+
+L_T
+P
+C
+A
+B
+</details>
+
+{c}   
+Figure 19.1-2. Beam carrying load P, with elastic support at A and simple support at B. (a) Actual structure. (b) Simple model. (c) Model with elastic support at A.
+
+model, and one or more revisions are needed. Rather than regard the revisions as attempts to correct previous failures, it is better to regard them as expected steps in an investigation that proceeds from the overly simple to the adequate [19,14].
+
+For example, at the outset one may not know enough about stress gradients or the behavior of available elements to immediately generate an adequate model. Then, rather than attempting to overwhelm ignorance by a very refined initial model, it is usually easier and cheaper to start with a crude model and refine it in successive analyses until it is adequate. We may also consider gaining the necessary modeling insight by analysis of a different problem, related but simpler, and preferably one for which analytical or experimental results are known.
+
+Supports. Typically, in expositions of theory, structural supports are idealized as completely rigid or as ideally hinged (i.e., a simple support). Actual supports lie somewhere between completely fixed and ideally hinged. Consider, for example, Fig. 19.1-2. The left end of the beam is elastically supported. The simple model, Fig. 19.1-2b, might be analyzed twice, first with fixity at A and then with a hinge at A, in an attempt to bound the correct response. In Fig. 19.1-2c the actual support elasticity is modeled by elements. One might terminate the beam model at A and use constraint relations to couple the beam rotation $\theta_{A}$ to d.o.f. of the elastic support elements, as discussed in connection with Fig. 7.7-1a. Alternatively, the beam model might be extended into the support, as shown, with translational d.o.f. of the leftmost beam element coupled to translational d.o.f. of the elastic support elements. The associated incompatibilities of displacement along AC are ignored.
+
+Superficially innocuous changes in support conditions can substantially affect results. Consider Fig. 19.1-3. In Fig. 19.1-3a, strains $\epsilon_{y}$ associated with the Poisson effect are prohibited at the left end. In Fig. 19.1-3b, they are permitted, as is
+
+![](images/page-596_319b6f116be57614345d903c8959f2d9217cb2718f436bd03e2ba705da11f8c2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+P
+x
+</details>
+
+(a)
+
+![](images/page-596_5490ae400c95ff17643f049e5ff02342351a8580db13dc80d0834f2c66a0fdb7.jpg)
+
+<details>
+<summary>text_image</summary>
+
+y
+P
+x
+</details>
+
+(b)   
+Figure 19.1-3. Two different models of a propped cantilever beam.
+
+<!-- source-page: 597 -->
+
+![](images/page-597_fe46ecd7b5fc657f6b7666de601b2c824cfd6731dd1dd59eec1fce3268eb3cd7.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Structure
+</details>
+
+![](images/page-597_9552e6e0e6249c95b17d69896eaaf0f4ecad557bb544127463d339c7d2ace24a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Approximate
+model
+</details>
+
+(a)
+
+![](images/page-597_31f29417f480bfc4f8e132914ae8f0eb3597069db29f24aa6b69e4be0411834a.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Structure
+</details>
+
+![](images/page-597_76daea3840feef0c7fa022ce1be76072295095525c8f6a6e68eeaf52500c970c.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Approximate
+model
+</details>
+
+(b)   
+Figure 19.1-4. Models that somewhat misrepresent the structure. (a) Welded connection of three angles whose axes are not concurrent. (b) Cylindrical vessel with step change in thickness and midsurfaces offset.
+
+presumed by elementary beam theory. This change in support conditions may produce large changes in $\sigma_{y}$ near the fixed end if Poisson's ratio is nonzero. In Fig. 19.1-3a the hinge support acts to resist rotation of the right end; in Fig. 19.1-3b it does not because it lies on the neutral surface of bending.
+
+Joints and Other Modifiers of Stiffness. Junctions between members may not have as much stiffness as a simple model attributes to them. For example, in Fig. 19.1-4, offsets produce significant local bending action in the actual structures, but this action is lacking in the approximate models that do not represent the offsets.
+
+The effect of joints is often underestimated but may have appreciable effect on global behavior. Stiffeners, swages, corrugations, and perforations also have an effect upon stiffness that cannot be ignored $[19.2]$ . However, detailed modeling—for example, of individual weld lines or of individual spot welds—is usually not appropriate unless the joint itself is the object of study.
+
+Some bodies have geometric irregularities that can be “smeared.” An example is a boiler tube sheet, which is a flat plate pierced by a regular pattern of identical holes. Effective elastic moduli and flexural rigidities for tube sheets have been established by analytical and experimental methods $[19.15]$ . Thus overall response can be analyzed as though the tube sheet were homogeneous. However, stress analysis must acknowledge the presence of individual holes.
+
+A related structure is a plate with a corrugated core, Fig. 19.1-5. If effective stiffnesses are known, the plate may be analyzed as if it were a homogeneous orthotropic plate. One way to obtain the effective stiffnesses is to model a small
+
+![](images/page-597_f24f0523196e034c236661202bc43372338c5bbf22be05adcc3f99c44835d304.jpg)
+
+<details>
+<summary>text_image</summary>
+
+s
+</details>
+
+Figure 19.1-5. Cross section of a plate with thin facings and a corrugated core, viewed parallel to axes of the corrugations.
+
+<!-- source-page: 598 -->
+
+part of the structure in suitable fine detail (e.g., span s in Fig. 19.1-5), apply boundary displacements to this mesh consistent with a state of constant strain or constant curvature, compute the resulting boundary force or boundary moment, and finally obtain the required stiffness as the ratio of force to displacement (or of moment to rotation). If necessary, the same process can be applied to the aforementioned tube sheet to obtain its effective stiffness coefficients.
+
+Element Shapes, Connection, and Grading. An element performs best if its shape is compact and regular. An element tends to stiffen and lose accuracy as its aspect ratio increases, as its corner angles become markedly different from one another, as sides become curved, or as side nodes (if present) become nonuniformly spaced. Figure 19.1-6 shows element shapes that are usually undesirable.
+
+Different elements have different sensitivities to shape distortion. Accordingly, an all-purpose guideline must be vague: keep aspect ratios near unity, corner angles of quadrilaterals near $90^{\circ}$ , side nodes at midsides, and sides straight. Elements derived as planar may behave badly if warped to fit a curved surface [19.4,19.5].
+
+There are, of course, exceptions. Elements of large aspect ratio may be used in areas where the strain gradient is almost zero. Side nodes may be moved to quarter-points to produce crack-tip elements. Sides may be curved to fit a curved boundary (but sides of the element interior to the mesh should be straight).
+
+Poor elements (Fig. 19.1-6) and poor element connections (Fig. 19.1-7) may produce only locally poor results. Usually, if the surrounding mesh is satisfactory, spurious gradients caused by a local mesh error die out rather than propagate, in accord with Saint-Venant's principle. The same is usually true of errors caused by ad hoc (but statically equivalent) nodal lumping of distributed loads.
+
+If a mesh is graded rather than uniform, as is usually the case, grading should be done in a way that produces no great discrepancy in size between adjacent elements. Figure 19.1-8 shows three examples of mesh grading that use quadrilaterals. If triangles are also permitted, the range of possibilities increases. In general, adjacent elements should not differ greatly in stiffness. As a working rule, if $E$ and $V_{e}$ represent elastic modulus and element volume, the ratio $E / V_{e}$ should not change by more than a factor of roughly 3 in going from one element to the next.
+
+![](images/page-598_e051fdf1b6609f947e6d27f7a382a657ac650bb9207e2c2062869e89ba69f7df.jpg)
+
+<details>
+<summary>text_image</summary>
+
+a
+b
+a >> b
+</details>
+
+Large aspect ratio
+
+![](images/page-598_fff20f3c3f5986bcfef9fb42a83f9ebfba79452875031fd7da780520c2cb40ae.jpg)
+
+<details>
+<summary>text_image</summary>
+
+a >> b
+a
+b
+</details>
+
+Near-triangle
+
+![](images/page-598_45ae43701bf109934141e16ee1dc35da898f3853c05236cf1c7ca4b1a961569d.jpg)  
+Off-center node
+
+![](images/page-598_e7b0a366b9faf1f58e621f051631bd40d78ad1152e6e2dc48f23c6a95141814e.jpg)  
+Highly skewed
+
+![](images/page-598_294737cd65c2e6cc76d1ec9143c48ebadc41e00c5d71b1bb3de493217f4dbe3b.jpg)  
+Triangular quadrilateral
+
+![](images/page-598_d762e4177c42359f9a183611b269c917136b1b5625084ef4717e87bcab24b107.jpg)  
+Curved side   
+Figure 19.1-6. Elements having shape distortions that tend to promote poor results.
+
+<!-- source-page: 599 -->
+
+![](images/page-599_7dbd311c05508542972a75341d9577fc592595f7547257011bada1dc1c1c7dee.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Pure geometric diagram of a square with internal lines and dots, no text or symbols present
+</details>
+
+{a}
+
+![](images/page-599_69cd8dfe6afa98348c96dfded02a17870ddc52a5392b823b4a27b83218dc5d76.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Pure geometric diagram of two connected squares with dots at vertices (no text or symbols)
+</details>
+
+{b}
+
+![](images/page-599_37c61bf00570816da5ff29d68213c6f97835419c452aab92cb70d540e1f757f2.jpg)
+
+<details>
+<summary>text_image</summary>
+
+A B C
+</details>
+
+{c}   
+Figure 19.1-7. Poor element connections. (a) Two bilinear elements and one quadratic element. (b) Two quadratic elements. (c) Two quadratic elements, connected at A and B but not at C (as if to model a crack from B to C).
+
+Checking the Model. A model should be checked before results are computed; afterward, there will be even greater reluctance to do the job. Ideally, a model is checked by an analyst who was not directly involved in its preparation and is therefore more likely to be objective.
+
+Graphical display makes it comparatively easy to detect gross errors, such as a misplaced node or a missing element. Preprocessors offer color, shrink plots, rotation, sectioning, exploded views, and removal of hidden lines as aids in the checking process (Figs. 19.1-9 and 19.1-10).
+
+Tests and warnings should already be coded into the program and should be exercised by a “check run” that precedes actual solution. Such tests may include examination for overly distorted element shapes, checking that adequate supports are provided, and searching for poor element connections like those in Fig. 19.1-7. The check run may also estimate the time required for the actual solution. All error messages and warnings produced by the program should be investigated, whether they appear during the check run or later.
+
+Built-in error tests cannot be given all responsibility for the success of an analysis. The program cannot know whether the element type is appropriate, if supports are properly located, whether data have been supplied using consistent units, and so on. Responsibility resides with the user.
+
+Ill-Conditioning, Locking, and Instability. Great stiffness discrepancies between elements, poor choice of quadrature rule, and a Poisson ratio near 0.5 in plane strain and solid problems may provoke ill-conditioning, locking, or instability. These are dangerous difficulties because they can seriously degrade results rather than making results so peculiar that it becomes obvious that something is wrong.
+
+![](images/page-599_70c6e3bf507c2bd20689c35225f3d988066c8fe51192e55a45156b708441de1e.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Geometric diagram of a square divided into smaller triangles and smaller squares (no text or symbols)
+</details>
+
+(a)
+
+![](images/page-599_b72fdf00f761003b4cc21e950d34f136066806e6489f2c75af1f0bb47aec0d0e.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Geometric pattern of interconnected squares and triangles (no text or symbols)
+</details>
+
+(b)
+
+![](images/page-599_9da47b23995be0b92e5182e9c138cc5730eaa563a1ddfd5f5128dcf985d7d8d6.jpg)  
+(c)   
+Figure 19.1-8. (a, b) Transitions from coarse to finer mesh that avoid abrupt size changes. (c) Possible mesh of quadrilaterals on one quadrant of a circular plate.
+
+<!-- source-page: 600 -->
+
+![](images/page-600_b6b89d73b7ef5a8a500b279595b767dad4f4bbf491160a0d54bbd9a08558a164.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Technical line drawing of a mechanical component with a central protrusion (no text or symbols)
+</details>
+
+Figure 19.1-9. The intersection of two cylindrical shells. The “shrink plot” shows elements at about 75% of their actual sizes.
+
+Some important “don’ts” are as follows. Do not support a stiff element by flexible elements; instead, impose rigid-body constraints on the stiff element. Do not “fake” a skew support (Fig. 18.2-3). Do not let Poisson’s ratio approach 0.5 in plane strain and solid problems unless a special formulation is used. Do not let three-dimensional elements or Mindlin plate and shell elements become extremely thin. Do not use a minimal integration rule without being aware of possible mechanisms.
+
+Some of these difficulties can be detected by error tests in the coding, such as a test for the condition number of the structure stiffness matrix or a test for diagonal decay during equation solving. Such tests are usually a posteriori rather than a priori and may be optimistic or pessimistic.
+
+Stresses. At optimal stress points, computed stresses may be as accurate as displacements. Usually, however, stresses are less accurate than displacements. Accordingly, a finer mesh is needed for stress analysis than for displacement
+
+![](images/page-600_a8dea16fa11ff93934792179704f1c3198b00ed0e440a15668783a2f1813e0e3.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+3D wireframe model of mechanical components (no text or symbols)
+</details>
+
+Figure 19.1-10. An exploded view of a machine part. (Courtesy of Algor Interactive Systems Inc., Pittsburgh, Pennsylvania.)
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+<!-- source-page: 601 -->
+
+analysis, and stresses are not considered reliable if displacements are suspect. Analogously, in vibration analysis, mode shapes are not considered reliable if natural frequencies are suspect.
+
+A postprocessor can usually display stress contours, which typically are smoothed, perhaps by working from nodal average stresses. Too much smoothing can make stresses appear more accurate than they really are. It may be preferable to avoid averaging between elements. For example, one may plot contours of the von Mises effective stress, element by element and without any averaging of stresses at nodes shared by elements. These contours may be plotted as “stress bands” [19.3] by designating equally spaced stress intervals, locating the areas of each element that fall into each interval, and using a different color to plot each interval. Typically, in an adequately refined mesh, the bands are slightly discontinuous across interelement boundaries but a global contour pattern is evident upon visual inspection. If no such global pattern is apparent, the mesh is too coarse. If the bands appear perfectly continuous, the mesh is finer than necessary. In Fig. 19.1-11a, discontinuities are probably too pronounced for the solution to be considered acceptable. In Fig. 19.1-11b, discontinuities are sufficiently small that the solution may be considered acceptable.
+
+Stresses must not be averaged across a step change in modulus or a step change in thickness. Specifically, if $\epsilon_{t}$ is the mechanical strain tangent to a boundary between two different materials, then, in the absence of initial strains, the corresponding stress $\sigma_{t}$ is proportional to $E_{1}\epsilon_{t}$ on one side and to $E_{2}\epsilon_{t}$ on the other. And, if a bar has a step change in cross-sectional area from $A_{1}$ to $A_{2}$ and no load is applied at the step, axial stresses on either side of the step are in the ratio $A_{1}/A_{2}$ .
+
+Mesh Refinement. A need for refinement of all or part of the mesh may be indicated by visual inspection of discontinuities in the stress bands just cited. Analogous numerical indices may be coded. As an example, consider a four-node plane element that is not very good at modeling bending. Strain energy $U_{0}$ per unit volume might be selected as a reference quantity, and the ratio of the greatest change in $U_{0}$ across the element to the value of $U_{0}$ at the element center taken as an index. If the value of this index exceeds a prescribed tolerance, either for individual elements or for a patch of elements, a need for refinement is indicated. A similar numerical test, not limited to low-order elements, could be based on changes in a reference quantity between adjacent elements.
+
+![](images/page-601_04ff6c534657305b1a423445299cf11cde3db3dbcc6498076da70eaccbde79ef.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Abstract geometric pattern with interlocking curved and angular shapes (no text or symbols)
+</details>
+
+(a)
+
+![](images/page-601_efced564a903aa7093e984358cdffeb630964ea11041ba3e2d433d5bebf87594.jpg)
+
+<details>
+<summary>natural_image</summary>
+
+Abstract black-and-white pattern with curved and angular shapes (no text or symbols)
+</details>
+
+(b)   
+Figure 19.1-11. Hypothetical stress bands in 3 by 3 patches of plane rectangular elements, representing solutions that are (a) probably inadequate, and (b) probably adequate.
+
+<!-- source-page: 602 -->
+
+If a model tends to be ill conditioned, further refinement may make results worse rather than better (see Table 18.4-1).
+
+If refinement does little to change results, one has evidence (but not proof) that results are satisfactory. Similarly, if the mesh is already rather fine, one may try a coarser mesh, again to discover if there are significant changes. Analogously, in a response history analysis, one might see whether results are significantly changed by a change in the number of modes used in a modal method or by a change in the time step $\Delta t$ of direct integration.
+
+Local Analysis. If localized mesh refinement is necessary, we need not reanalyze the entire structure with the local refinement imbedded in it. The portion of the structure that contains the refined mesh can be analyzed separately. It is loaded by whatever prescribed loads may be present and, along the boundary where it has been cut free of the rest of the structure, loaded by the displacements computed in the preceding analysis of the entire structure. If refinement adds nodes along this boundary, interpolation is needed to obtain the prescribed displacements of the added nodes.
+
+In such an analysis, a correction may be advisable. Typically the original mesh is stiffer than the refined mesh. Therefore boundary displacements to be imposed on the local refinement may be underestimated, which results in underestimation of stresses in the local refinement. An ad hoc correction scheme is as follows. For the original local mesh and the refined local mesh, respectively, compute nodal loads $[K]\{D\}$ produced by prescribed boundary displacements $\{D\}$ . These loads, $\{R\}_{0}$ and $\{R\}_{r}$ , will differ because $[K]$ is changed by refinement. The ratio of their norms, $\|R_{0}\|/\|R_{r}\|$ , usually exceeds unity and can be used as an approximate corrective multiplier to stresses computed in the local refinement.
+
+Vibrations and Dynamics. If the dynamic load includes frequencies of interest up to $\omega_{u}$ , then the mesh should be able to accurately represent modes associated with frequencies up to about $3\omega_{u}$ , and a mode superposition analysis should include frequencies up to about $3\omega_{u}$ . The time step $\Delta t$ in a direct integration analysis should be approximately $0.3/\omega_{u}$ or less, and must provide numerical stability if the integration method is conditionally stable. If a reduced basis is used for eigenvalue computations, there should be roughly four times as many master d.o.f. as eigenvalues to be accurately computed.
+
+In direct integration there should be a match between the type of algorithm and the mass matrix; for example, lumped masses are best for an explicit algorithm. Abrupt changes in element size should be avoided, as such changes tend to produce spurious wave reflections and numerical noise.
+
+Nonlinear Problems. Typically one must make many more trial runs in order to solve a nonlinear problem than to solve a linear problem. Not only must blunders be discovered and removed, but solution strategy must be guided by what is learned in preceding attempts. Here, much more than in linear problems, it is wise to start simply and not attempt the complete solution all at once. One might elect to solve a linear form of the problem first, then add nonlinearities one by one. Thus blunders are more easily discovered, the effect of each nonlinearity is more apparent, useful information is gained from each trial, and the risk of failure with a large, complicated, and expensive model is reduced [2.1].
+
+Nonlinear analyses tend to be very expensive. It is therefore necessary that no
+
+<!-- source-page: 603 -->
+
+![](images/page-603_5fe4101dac2446f3bb9d66c064858404337996f90d7d9ab1350482322a7772cb.jpg)
+
+<details>
+<summary>text_image</summary>
+
+P
+</details>
+
+Figure 19.1-12. Flat plate containing a large hole.
+
+superfluous nonlinearities be introduced. For example, imagine that yielding near the edge of the hole in Fig. 19.1-12 is to be analyzed. It is likely that yielding will also appear at the point of load application and at the support points. Yielding at these points will be treated with due respect (and due expense) by the algorithm [19.14]. Perhaps these concentrated forces result from oversimplification of loads and supports. If so, or if yielding in these locations is indeed ignorable, one might prevent undesired yielding by assigning a high yield strength to elements adjacent to the concentrated loads.
+
+Miscellaneous Perils. Carelessness or lack of adequate understanding can lead to puzzling or misleading results. If the problem involves vibration, buckling, or nonlinear behavior, then axisymmetric geometry and axisymmetric loads do not guarantee axisymmetric response: unless symmetry is known to prevail, it should not be imposed by choice of boundary conditions. A quarter-point element for crack analysis can be too large or too small: thus, mesh refinement may make results worse. Incompatible and underintegrated elements may display a dependence on Poisson's ratio in problems that should be independent of Poisson's ratio. Anisotropy adversely affects accuracy $[8.34]$ . If plane elements are warped so that element nodes are not all coplanar, results may be erratic and very sensitive to changes in the mesh $[19.4,19.5]$ . If convergence with mesh refinement is not monotonic, extrapolation of results from two different meshes may give a worse result than is given by either mesh. Imperfections of load, geometry, supports, and mesh may be far more important in a buckling problem than in a static problem. Buckling, collapse, and nonlinear analyses demand more expertise than static stress analysis.
+
+Check the Results. Computed results should be checked for “self-consistency,” for example, by checking that intended supports do indeed have zero displacement and that any symmetries of the finite element model are represented in stress and displacement results. Computed results should be compared with whatever else is available that can be used for comparison. Examples include “back of the envelope” calculations, approximate analytical models, experimental data, textbook and handbook cases, preceding numerical analyses of similar problems, numerical analysis of a related but simpler problem, and results for the same problem predicted by a different program (which ideally should be based on a different numerical method). All these results should be regarded with some skepticism: analytical models incorporate idealizations, mistakes may be made in mathematics, textbooks and handbooks may contain errors, numerical solutions are subject to errors in coding and in data preparation, and experiments may be improperly performed and the results misinterpreted. When the inevitable disa-
+
+<!-- source-page: 604 -->
+
+greements appear, the reason for the discrepancy should be sought, and the amount of disagreement satisfactorily explained.
+
+The time spent in processing and checking output should equal or exceed the time previously spent in data preparation. As with model preparation, an objective critique of the work should be obtained from a very competent analyst who is not directly involved with the project [19.14].
+
+# 19.2 PROGRAMMING AND PROGRAMS
+
+Programming. The writing of finite element programs is done by researchers and software vendors, who do so of necessity, and by students, who do so as an aid to learning. Those who use finite elements as a tool should buy or lease a program rather than write one. The few who write programs should do so in a way that makes the code easy to maintain and improve. These points are discussed in more detail later in this section.
+
+Fortran is the language of all major finite element programs and is likely to remain so for the foreseeable future because of the large investment already made in Fortran software. (However, critical parts of a Fortran program may be coded in assembly language for the sake of efficiency.) Ideally, coding is guided by a previously prepared user's manual, in order to impose discipline on the developers and to produce a user-oriented product. The code should be built on a data base structure, and should be modular, that is, divided into logical subsets, each composed of one or more subroutines. The code should have mnemonic names for variables, monotonically increasing statement numbers, and adequate comment statements so that personnel other than the original programmer can read and maintain it. Much additional advice about good programming practice is available, especially in the computer science literature.
+
+Dynamic Storage Allocation. Different problems require different amounts of computer memory, so it is inefficient to use fixed dimensions for arrays. Moreover, in different problems, the fractions of the total memory used by different phases of a single analysis run may differ. These difficulties are overcome by dynamic storage allocation, in which the dimensions of arrays are set at the time of execution. This procedure is common in finite element programs, but is often unfamiliar to the student. It is explained as follows.
+
+In Fig. 19.2-1, array A contains most of the storage space to be used by the program. Various subroutines will use this same space but call it by other names. Imagine that Subroutine INPUT is to use one-dimensional arrays X and Y, which must each contain NUMNP entries, and the two-dimensional array ID, which must contain NDOF rows and NUMNP columns. The “pointers” N1, N2, and N3 identify the starting addresses in array A for arrays X, Y, and ID, respectively. $^{1}$ In Subroutine INPUT we find the statements
+
+SUBROUTINE INPUT (X,Y,ID,NUMNP,NDOF)
+
+DIMENSION X(1), Y(1), ID(NDOF, 1)
+
+'When both real and integer quantities appear in blank common, the compiler must assign the same word length to reals and integers if the addressing is to work properly.
+
+<!-- source-page: 605 -->
+
+```txt
+COMMON A(15000)
+LIM = 15000
+C ----(STATEMENTS NOT ESSENTIAL TO THIS EXAMPLE OMITTED HERE)
+    N1 = 1
+    N2 = N1 + NUMNP
+    N3 = N2 + NUMNP
+    N4 = N3 + NUMNP*NDOF - 1
+    IF (N4 .GT. LIM) CALL ERROR (N4)
+    CALL INPUT (A(N1), A(N2), A(N3), NUMNP, NDOF)
+C ----(STATEMENTS NOT ESSENTIAL TO THIS EXAMPLE OMITTED HERE)
+    N1 = 1
+    N2 = N1 + NEQ*MBAND
+    N3 = N2 + NEQ - 1
+    IF (N3 .GT. LIM) CALL ERROR (N3)
+    CALL BUILD (A(N1), A(N2), NEQ, MBAND)
+C ----(STATEMENTS NOT ESSENTIAL TO THIS EXAMPLE OMITTED HERE)
+END 
+```  
+Figure 19.2-1. Hypothetical main routine that uses dynamic dimensioning. Subroutine ERROR (not shown) terminates execution if the problem is too large.
+
+Arrays X, Y, and ID are stored in consecutively addressed cells of blank common. Since starting addresses have already been defined, it is not necessary to state the actual array size; this explains the 1's used in the dimension statement. Array ID is stored by columns, so that the address in blank common of ID(M,N) is N3 + NDOF\*(N-1) + M-1. Thus the dimension statement must say ID(NDOF,1) rather than ID(1,1), so that the background bookkeeping can locate the proper address. $^{2}$ (For a three-dimensional array, the first two dimensions must be provided; only the third can be 1.)
+
+Later in the program, imagine that arrays X, Y, and ID have been stored elsewhere—for example, on a disk file. The memory space they occupied is now available for other use. In Fig. 19.2-1 we imagine that the space is now to be used to store the structure stiffness matrix S and load vector R. Accordingly, new pointers are computed. In Subroutine BUILD we find the statements
+
+# SUBROUTINE BUILD (S,R,NEQ,MBAND) DIMENSION S(NEQ,1),R(1)
+
+To change the capacity of the entire program, we change only the first two statements in Fig. 19.2-1, which necessitates recompiling only one subroutine.
+
+Documentation. The quality of documentation is of major concern, whether we must prepare it or must read it in order to use a program. Unfortunately, documentation is often difficult to use, perhaps because it is often written by people so familiar with the program that they are unable to take the user's viewpoint. It is safest to assume when preparing documentation that most users know nothing about the program and that the remainder will often overlook what is obvious to the programmer. Good documentation is expensive to write and to maintain, but it is important: a good program may fail in the marketplace if its documentation is unreadable, and a useful program may be abandoned if its documentation does not keep up with program changes and enhancements [19.6,19.7].
+
+A general-purpose commercial finite element program is accompanied by thou-
+
+$^{2}$ Some compilers have an option that checks subscripts in executable statements against array bounds in DIMENSION statements. A logic error will then be signaled. To avoid this signal one can disable the checking option or replace each 1 in the foregoing DIMENSION statement by NUMNP.
+
+<!-- source-page: 606 -->
+
+sands of pages of documentation, typically divided as follows: theoretical manual (describes the analytical basis and limitations of algorithms), user's manual (describes available elements and data preparation), example problems manual (describes test cases, showing input data, output data, and comparison with theory), and programming or systems manual (describes computer science topics, incomprehensible to most engineers). More recent developments are introductory handbooks, which describe a simple and often-used subset of the program; online interactive help, which provides instant explanation of a command or an error message; and videotapes, which offer training lectures on aspects of program use [19.7].
+
+Desirable features in documentation, not always present, include the following: index, nomenclature, table of elements, summary description of element behavior, estimates of timing and cost, modeling suggestions, glossary of errors and error messages, limitations of major algorithms, and lists of element quirks and frequently made errors [19.7]. At worst, a manual tersely explains acronyms in terms of other acronyms, so that making use of its information resembles trying to determine the function of an unfamiliar machine by reading its parts list.
+
+Before buying an expensive program, it is wise to read some of its manuals. Documentation can usually be purchased separately from software, and far more cheaply.
+
+Costs. The computational expense of a finite element analysis varies widely. At one extreme, when the computation is done on a personal computer that would otherwise be idle, the expense is almost zero. Toward the other extreme, when a large nonlinear problem occupies most of a day's running time on a supercomputer, the expense approaches an engineer's annual salary. Overall, hundreds of millions of dollars are spent each year on finite element modeling and computer costs.
+
+Similarly, the cost of writing a program varies widely (but is usually high). In one study of programming efficiency, each of several programmers with from 2 to 11 years of experience coded a logic problem. The ratio of best to worst was 25/1 for coding time, 5/1 for code size, and 13/1 for running time. There was no correlation between productivity and experience [19.8]. Another study covered more than 400 military software projects that lasted from 1 month to 8 years in length and involved from 2 to 200 people at a time. Time expended on design, coding, testing, and documentation was included in the study. Productivity was measured in lines of code per person per month, which we abbreviate here as “lines.” Average productivity over the life of each project ranged from 5 to 5000 lines. Average productivity over all projects was 200 lines, with two-thirds of the projects having average productivities between 75 and 550 lines [19.9]. From these data, we conclude that it rarely makes sense to write a program if one can be purchased or leased instead.
+
+After release of a software system, it must be maintained. Bugs must be corrected and enhancements added to keep the program alive in the marketplace. Over the lifetime of a significant software system, maintenance costs far exceed development costs.
+
+Programs for engineering are far outsold by programs for business. Yet the programs for engineering cost much more to develop, maintain, and support [19.10]. Programs for personal computers are proliferating, however no new major software system has entered the market for many years. This is no surprise when
+
+<!-- source-page: 607 -->
+
+TABLE 19.2-1. CHARACTERISTICS OF STRUCTURAL MECHANICS SOFTWARE [19.11]. K = MULTIPLIER OF 1000. 
+
+<table><tr><td>Type of Program</td><td>Single Element Static</td><td>Multiple Element Static</td><td>Large General Purpose</td></tr><tr><td>Number of statements</td><td>500–2000</td><td>2K–10K</td><td>100K–600K</td></tr><tr><td>Development  $cost^a$ </td><td>$5K–$50K</td><td>$25K–$200K</td><td>$2000K–$10,000K</td></tr><tr><td>Pages of documentation</td><td>20–100</td><td>50–500</td><td>2000–7000</td></tr><tr><td>Machine words needed</td><td>10K–30K</td><td>20K–50K</td><td>50K–150K</td></tr><tr><td>Diagnostics</td><td>few–moderate</td><td>few–moderate</td><td>extensive</td></tr><tr><td>Cost per  $run^a$ </td><td>$1–$100</td><td>$1–$500</td><td>$10–$10K</td></tr></table>
+
+"When this table was written, a graduating engineer started work at roughly \$1000 per month.
+
+one considers the high start-up costs (Table 19.2-1). Furthermore, even with a quality product, the entrepreneur must persevere for years, and incur additional costs that may exceed the development costs, in order to penetrate a reluctant market: users develop loyalties to software with which they are familiar and tend to be oblivious to other options. User suspiciousness is not without foundation. Unless there is good reason to believe that software will be supported for many years, one cannot justify the expense of acquiring it or the much higher expense of developing competence in its use.
+
+Some programs of quality are available cheaply, usually because they are of limited scope or because no commitment of maintenance or user support is supplied with the program. Examples include certain versions of the BOSOR programs for thin shells of revolution and certain versions of the SAP program for linear static and dynamic structural analysis.
+
+Commercial Programs. Hundreds of finite element programs are available, from small to large. Large general-purpose analysis systems share the following traits [19,12].
+
+Generality. Many element types are provided, so that almost any conceivable structure, supports, and boundary conditions can be treated. Linear problems of statics are certainly included; linear dynamics and heat transfer are almost certain to be included. Certain nonlinear capabilities, magnetics, or other special features are probably included.
+
+Large Size. The source code (not available to the user) comprises 100,000 to 600,000 statements and is the result of a development effort of from fifty to several hundred man years.
+
+Worldwide Distribution. The system is installed at tens or hundreds of locations in various countries.
+
+Large User Community. There are thousands of users in industry, data centers, consulting firms, research establishments, and universities.
+
+User Support. In addition to documentation, the vendor offers hotline support, consulting, training courses, user conferences, and newsletters.
+
+Portability. The program is available on a variety of machines, from supercom-
+
+<!-- source-page: 608 -->
+
+puters to minicomputers. Some versions of the program may be available on personal computers.
+
+Maintenance. Updated versions of the program are released every year or two. Each new release is thoroughly tested, so that few bugs remain.
+
+In choosing a software vendor, companies may compile a list of candidates by study of industry journals and by talking with associates at other companies. The list is shortened by discarding products that do not meet the company's needs, seem either overpriced or suspiciously inexpensive, or are new and untried. Programs that remain on the list are tested to see if the promotional claims are met and if the product will solve the company's everyday problems. The first program tested may stand out: if it is complex, others will not seem comprehensive enough; if it is comparatively easy to use, others will seem too difficult [19.10]. Probably none of the systems will be as easy to use as one might wish. For this reason service to the user is very important. A software vendor must support the product in order to survive amidst rising user expectations.
+
+<!-- source-page: 609 -->
+
+# MATRICES: SELECTED DEFINITIONS AND MANIPULATIONS
+
+This appendix summarizes portions of matrix theory that are frequently used in finite element analysis. Symbols used are arbitrary and imply no particular physical meaning. Further explanation may be found in any of several references, for example [A.1].
+
+Multiplication. Let [C] be the product [A][B] ([A] premultiplies [B]; [B] postmultiplies [A]). Then [C] is
+
+$$
+\underset {m \times q} {[ \mathrm{C} ]} = \underset {m \times n} {[ \mathrm{A} ]} \underset {n \times q} {[ \mathrm{B} ]} \quad \text { where } \quad C _ {i j} = \sum_ {k = 1} ^ {n} A _ {i k} B _ {k j} \tag {A.1}
+$$
+
+where i ranges from 1 to m and j ranges from 1 to q. Matrices [A] and [B] must be conformable for multiplication. That is, the number of columns in [A] must equal the number of rows in [B]. Fortran statements that accomplish the multiplication in Eq. A.1 are
+
+$$
+\mathrm{DO} 2 0 \mathrm{I} = 1, \mathrm{M}
+$$
+
+$$
+D O 2 0 ^ {\cdot} J = 1, Q
+$$
+
+$$
+\text { SUM } = 0.
+$$
+
+$$
+D O 1 0 K = 1, N
+$$
+
+$$
+1 0 \text { SUM } = \text { SUM } + A (I, K) * B (K, J)
+$$
+
+$$
+2 0 \mathrm{C} (\mathrm{I}, \mathrm{J}) = \text { SUM }
+$$
+
+To accomplish the multiplication $[\mathbf{A}]^T [\mathbf{B}]$ (for which $[\mathbf{A}]^T$ and $[\mathbf{B}]$ must be conformable), one replaces $\mathsf{A}(\mathsf{l},\mathsf{K})$ by $\mathsf{A}(\mathsf{K},\mathsf{l})$ in statement 10.
+
+The transpose of a product is the product of the transposes in reverse order, that is
+
+$$
+([ \mathbf {A} ] [ \mathbf {B} ]) ^ {T} = [ \mathbf {B} ] ^ {T} [ \mathbf {A} ] ^ {T} \tag {A.2}
+$$
+
+Linear Dependence. A set of vectors $\{\mathbf{v}\}_1, \{\mathbf{v}\}_2, \ldots, \{\mathbf{v}\}_n$ is linearly dependent if, for some $j$ ,
+
+$$
+\sum_ {i \neq j} \alpha_ {i} \{\mathbf {v} \} _ {i} = \{\mathbf {v} \} _ {j} \tag {A.3}
+$$
+
+where the $\alpha_{i}$ are scalars. For example, if $3\{\mathbf{v}\}_{2} + 4\{\mathbf{v}\}_{4} = \{\mathbf{v}\}_{5}$ , then vector 5 is linearly dependent on vectors 2 and 4, regardless of $n$ . Columns (or rows) of a matrix may be regarded as vectors; thus Eq. A.3 pertains to a matrix that has one or more linearly dependent columns (or rows).
+
+Rank. Singularity. The rank of a matrix [A] is defined as the order of the largest nonzero determinant in [A]. An equivalent definition of rank is the maximum number of linearly
+
+<!-- source-page: 610 -->
+
+independent rows (or columns) in [A]. As examples, the following matrices have ranks 2, 2, and 1 respectively.
+
+$$
+\left[ \begin{array}{c c c} 1 & 3 & - 3 \\ 2 & 4 & - 2 \\ 3 & 5 & - 1 \end{array} \right] \quad \left[ \begin{array}{c c c c} 1 & - 1 & 0 & 2 \\ - 1 & 2 & - 1 & - 1 \\ 0 & - 1 & 1 & - 1 \end{array} \right] \quad \left[ \begin{array}{c c c} 2 & 4 & - 2 \\ 1 & 2 & - 1 \\ 3 & 6 & - 3 \end{array} \right]
+$$
+
+A matrix whose rank is less than its order is said to be rank-deficient. The rank of a null matrix is zero. A square matrix $[A] = \{a\}\{b\}^{T}$ has rank 1 or is null, regardless of order. A square matrix whose rank is less than its order is called singular. The rank of a matrix product is
+
+$$
+\operatorname{rank} ([ \mathbf {A} ] [ \mathbf {B} ]) \leq \min (\operatorname{rank} [ \mathbf {A} ], \operatorname{rank} [ \mathbf {B} ]) \tag {A.4}
+$$
+
+Quadratic Forms. If [A] is a real square matrix and $\{\mathbf{x}\}$ is a real vector of the same order, then scalar $F$ is called a quadratic form, where
+
+$$
+F = \{\mathbf {x} \} ^ {T} [ \mathbf {A} ] \{\mathbf {x} \} \tag {A.5}
+$$
+
+Now imagine that we calculate all possible values of $F$ as follows. Let coefficients $x_{i}$ in $\{x\}$ be allowed to assume independently any and all real values except for all $x_{i}$ simultaneously zero. Then matrix [A] is called
+
+positive definite if $F > 0$ for all $\{\mathbf{x}\}$
+
+positive semidefinite if $F \geq 0$ for all $\{\mathbf{x}\}$
+
+negative semidefinite if $F \leq 0$ for all $\{\mathbf{x}\}$
+
+negative definite if $F < 0$ for all $\{\mathbf{x}\}$
+
+and simply indefinite if $F$ can be either positive or negative. For example, the following two matrices are respectively positive definite and positive semidefinite:
+
+$$
+\left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 2 \end{array} \right] \quad \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right]
+$$
+
+If a square matrix is positive definite or negative definite, it is also nonsingular.
+
+Differentiation. Differentiation of a matrix is accomplished by differentiating each of its terms. For example, if $\lfloor x \rfloor = \lfloor 1 - y \rfloor$ , then $d\lfloor x \rfloor / dy = \lfloor 0 - 1 \rfloor$ .
+
+Let $\{\mathbf{x}\} = \left\lfloor x_1, x_2, \ldots, x_n\right]^T$ and let [A] be an arbitrary $n$ by $n$ square matrix that does not depend on the $x_i$ . Suppose that the quadratic form
+
+$$
+\phi = \frac {1}{2} \{\mathbf {x} \} ^ {T} [ \mathbf {A} ] \{\mathbf {x} \} \tag {A.6}
+$$
+
+is to be differentiated with respect to each of the $x_{i}$ . The result is conveniently stated as a vector,
+
+$$
+\left\{\frac {\partial \phi}{\partial \mathbf {x}} \right\} = \left\lfloor \frac {\partial \phi}{\partial x _ {1}} \quad \frac {\partial \phi}{\partial x _ {2}} \quad \dots \quad \frac {\partial \phi}{\partial x _ {n}} \right\rfloor^ {T} = \frac {1}{2} ([ \mathbf {A} ] + [ \mathbf {A} ] ^ {T}) \{\mathbf {x} \} \tag {A.7}
+$$
+
+as may be verified by writing $\phi$ in terms of the $x_{i}$ and the $A_{ij}$ , taking the derivatives, and gathering terms. If [A] is symmetric, then
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_062.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_062.md
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+<!-- source-page: 611 -->
+
+$$
+\left\{\frac {\partial \phi}{\partial \mathbf {x}} \right\} = [ \mathrm{A} ] \{\mathbf {x} \} \quad \text { and } \quad \frac {\partial^ {2} \phi}{\partial x _ {i} \partial x _ {j}} = A _ {i j} = A _ {j i} \tag {A.8}
+$$
+
+As a special case, if [A] is a unit matrix, then $\{\partial \phi / \partial x\} = \{x\}$ .
+
+Let $\{\mathbf{x}\} = \left\lfloor x_1 \quad x_2 \quad \ldots \quad x_n \right\rfloor^T$ , $\{\mathbf{y}\} = \left\lfloor y_1 \quad y_2 \quad \ldots \quad y_m \right\rfloor^T$ , and [A] be an arbitrary $n$ by $m$ matrix that does not depend on the $x_i$ . Suppose that the scalar $\psi = \{\mathbf{x}\}^T[\mathbf{A}]\{\mathbf{y}\}$ is to be differentiated with respect to each of the $x_i$ . The result is conveniently stated as a vector:
+
+$$
+\left\{\frac {\partial \psi}{\partial x} \right\} = \left\lfloor \frac {\partial \psi}{\partial x _ {1}} \quad \frac {\partial \psi}{\partial x _ {2}} \quad \dots \quad \frac {\partial \psi}{\partial x _ {n}} \right\rfloor^ {T} = [ A ] \{y \} \tag {A.9}
+$$
+
+As for differentiation with respect to the $y_{i}$ , we note that since $\psi$ is scalar,
+
+$$
+\psi = \psi^ {T} = \{\mathbf {y} \} ^ {T} [ \mathbf {A} ] ^ {T} \{\mathbf {x} \} \tag {A.10}
+$$
+
+therefore, if [A] does not depend on the $y_{i}$ ,
+
+$$
+\left\{\frac {\partial \psi}{\partial \mathbf {y}} \right\} = \left[ \frac {\partial \psi}{\partial y _ {1}} \quad \frac {\partial \psi}{\partial y _ {2}} \quad \dots \quad \frac {\partial \psi}{\partial y _ {m}} \right] ^ {T} = [ \mathbf {A} ] ^ {T} \{\mathbf {x} \} \tag {A.11}
+$$
+
+As a special case, if [A] is a unit matrix, then
+
+$$
+\psi = \{\mathbf {x} \} ^ {T} \{\mathbf {y} \} \quad \left\{\frac {\partial \psi}{\partial \mathbf {x}} \right\} = \{\mathbf {y} \} \quad \left\{\frac {\partial \psi}{\partial \mathbf {y}} \right\} = \{\mathbf {x} \} \tag {A.12}
+$$
+
+<!-- source-page: 612 -->
+
+# SIMULTANEOUS ALGEBRAIC EQUATIONS
+
+Selected algorithms for equation solving are described. Fortran coding is provided for some of these algorithms.
+
+# B.1 INTRODUCTION
+
+Methods of computational mechanics usually produce large systems of simultaneous algebraic equations. Solution algorithms for these problems can be categorized as either direct or iterative.
+
+Direct methods provide solutions within a fixed number of steps. The number of steps can be calculated a priori from knowledge of the size of the problem and the specific procedure elected. The solution obtained would be exact if infinitely precise arithmetic were possible. Storage requirements can be large, especially for three-dimensional finite element problems for which matrices are not narrowly banded and “fill-in” terms are created during processing.
+
+Iterative (or indirect) methods provide approximate solutions that improve with continued iteration. The number of iterations needed is not known a priori: it depends on the size of the problem, the specific algorithm elected, the convergence criterion, and the numerical conditioning of the problem. (For ill-conditioned problems, iterative methods converge slowly and sometimes diverge.) Storage requirements are less than for direct methods, especially when bandwidths are large, because no “fill-in” terms are created. In this circumstance an iterative method may also be faster than a direct method. Convergence takes fewer iterations if a good initial guess is available. However, iterative methods are usually not competitive with direct methods except in specific areas such as reanalysis, optimization, and large three-dimensional problems.
+
+In linear static analysis, equation solving typically accounts for roughly one-quarter of the “number-crunching” cost. In nonlinear analysis the fraction may be much higher.
+
+Equation solving (and eigenvalue extraction) requires many multiplications, and roughly an equal number of additions or subtractions. Addition and subtraction are done much faster than multiplication. Accordingly, one can estimate the relative speed of competing algorithms by counting the number of multiplications required for each.
+
+Some computers have vector processors that compute the product of two vectors almost as quickly as the product of two scalars. For such a machine we seek an algorithm that can be “vectorized.”
+
+Often a problem is too large for all data to be stored in primary (core) memory. Many algorithms use out-of-core storage. In a virtual memory machine, the swapping of data in and out of core need not be explicitly coded; it is handled automatically by the operating system. However, if data are badly structured, a virtual memory machine may thrash, that is, spend an inordinate amount of time swapping information in and out of core.
+
+Small problems, and well-conditioned problems, can be analyzed in single precision arithmetic (i.e., with approximately 32 bits per word). However, double-precision arithmetic is usually recommended for all calculations in a finite element analysis.
+
+There are many ways to store sparse matrices, preserve their sparsity, avoid multipli-
+
+<!-- source-page: 613 -->
+
+cations by 0's and 1's, and so on [B.1]. No one selection of options is best in all cases, as different problems yield matrices of different topology.
+
+Actual listings of many algorithms are published [B.2-B.6, for example]. Others, already compiled and callable as subroutines, appear in widely available software packages such as IMSL, NAg, and LINPAC. Further information on related software and procedures appears in Appendix C.
+
+# B.2 SOLUTION OF SIMULTANEOUS LINEAR ALGEBRAIC EQUATIONS BY GAUSS ELIMINATION
+
+We wish to solve the system of equations $^{1}$
+
+$$
+[ \mathbf {A} ] \{\mathbf {x} \} = \{\mathbf {c} \} \tag {B.2-1}
+$$
+
+where [A] is an $n_{\text{eq}}$ by $n_{\text{eq}}$ matrix and $\{x\}$ and $\{c\}$ are $n_{\text{eq}}$ by 1 vectors. It is assumed that [A] is nonsingular and has constant coefficients, $\{c\}$ is given, and $\{x\}$ is to be determined. The concept of Gauss elimination is to combine the rows of Eq. B.2-1 in such a way that coefficient matrix [A] is transformed into upper triangular form. This is the forward-reduction phase. The equations are then sufficiently uncoupled to enable $\{x\}$ to be determined by back-substitution [A.1, pp. 12-13].
+
+The foregoing technique is detailed in Fig. B.2-1, and Fortran coding for it is given in Fig. B.2-2. During elimination, the $i$ th equation, $1 \leq i < n_{\text{eq}}$ , is used to reduce to zero all entries below the diagonal in the $i$ th column. Thus, in the $i$ th elimination, one divides by the $i$ th diagonal coefficient $A_{ii}$ . This simple strategy would fail if $A_{ii}$ were very small or zero. However, if [A] is a stiffness matrix, $A_{ii}$ is sufficiently large unless the structure is nearly unstable or badly modeled.
+
+When Fig. B.2-2 is studied, it may help to compare the coding with the following equation, which shows the result of one elimination (K = 2 in Fig. B.2-2, with NEQ = 3).
+
+$$
+\left[ \begin{array}{c c c} A _ {1 1} & A _ {1 2} & A _ {1 3} \\ 0 & A _ {2 2} - \left(A _ {2 1} / A _ {1 1}\right) A _ {1 2} & A _ {2 3} - \left(A _ {2 1} / A _ {1 1}\right) A _ {1 3} \\ 0 & A _ {3 2} - \left(A _ {3 1} / A _ {1 1}\right) A _ {1 2} & A _ {3 3} - \left(A _ {3 1} / A _ {1 1}\right) A _ {1 3} \end{array} \right] \left\{ \begin{array}{l} x _ {1} \\ x _ {2} \\ x _ {3} \end{array} \right\} = \left\{ \begin{array}{l} c _ {1} \\ c _ {2} - \left(A _ {2 1} / A _ {1 1}\right) c _ {1} \\ c _ {3} - \left(A _ {3 1} / A _ {1 1}\right) c _ {1} \end{array} \right\} \tag {B.2-2}
+$$
+
+An operation count shows that, for a large system of $n_{\mathrm{eq}}$ equations, forward-reduction in Fig. B.2-2 requires about $n_{\mathrm{eq}}^3 / 3$ multiplications [A.1, pp. 14-16]. If [A] were symmetric, and Fig. B.2-2 were altered accordingly, about $n_{\mathrm{eq}}^3 / 6$ multiplications would be needed. In either case, back-substitution requires about $n_{\mathrm{eq}}^2 / 2$ multiplications. Clearly, for a large system of equations, most of the solution time is spent in doing forward-reduction.
+
+If needed, the determinant of [A] can be calculated as the product of reduced diagonal coefficients in [A]. Thus, after statement 10 in Fig. B.2-2, insert the statements
+
+$$
+\begin{array}{l} \mathrm{DET} = 1. 0 \mathrm{D} 0 \\ \mathrm{DO} 1 5 \mathrm{I} = 1, \mathrm{NEQ} \tag {B.2-3} \\ 1 5 \text {   DET   } = \text {   DET   } * \text { A(I,I)   } \\ \end{array}
+$$
+
+However, DET is usually large, and may overflow. In some cases DET may be very small and may underflow. Overflow (or underflow) is less likely if statements B.2-3 are replaced by the statements
+
+$^{1}$ In some technical papers the symbolism $\{x\} = [A]^{-1}\{c\}$ merely implies solution for unknowns $\{x\}$ , not that matrix inversion is the numerical procedure actually used.
+
+<!-- source-page: 614 -->
+
+# Forward-Reduction Phase
+
+$$
+\left[ \begin{array}{c} \text {For} k = 2, 3, \dots , n _ {\mathrm{eq}} \\ \text {For} i = k, k + 1, \dots , n _ {\mathrm{eq}} \\ \left[ \begin{array}{c} r _ {i, k - 1} = A _ {i, k - 1} / A _ {k - 1, k - 1} \\ c _ {i} = c _ {i} - r _ {i, k - 1} c _ {k - 1} \end{array} \right. \\ \left[ \begin{array}{c} \text {For} j = k, k + 1, \dots , n _ {\mathrm{eq}} \\ A _ {i j} = A _ {i j} - r _ {i, k - 1} A _ {k - 1, j} \end{array} \right. \end{array} \right.
+$$
+
+# Back-Substitution Phase
+
+$$
+\begin{array}{r l} & x _ {n _ {\mathrm{eq}}} = c _ {n _ {\mathrm{eq}}} / A _ {n _ {\mathrm{eq}} n _ {\mathrm{eq}}} \\ & \left[ \begin{array}{c} \text {For} k = n _ {\mathrm{eq}} - 1, n _ {\mathrm{eq}} - 2, \dots , 1 \\ x _ {k} = \frac {1}{A _ {k k}} \left(c _ {k} - \sum_ {j = k + 1} ^ {n _ {\mathrm{eq}}} A _ {k j} x _ {j}\right) \end{array} \right. \end{array}
+$$
+
+Figure B.2-1. Algorithm for Gauss elimination solution of the system of $n_{eq}$ simultaneous linear algebraic equations $[A]\{x\} = \{c\}$ , where $\{c\}$ is specified; $[A] = A_{ij}$ , $\{c\} = c_i$ , $\{x\} = x_i$ . In this figure, “=” means “is replaced by,” as in Fortran.
+
+$$
+\begin{array}{r l} & \text { DETLN } = 0. \text { DO } \\ & \text { DO } 1 5 \text { I } = 1, \text { NEQ } \\ & 1 5 \text { DETLN } = \text { DETLN } + \text { DLOG } (\text { A(I,I) }) \end{array} \tag {B.2-4}
+$$
+
+which calculate the natural logarithm of the determinant.
+
+Band-Symmetric Matrix [A]. Let [A] be symmetric and banded [A.1, pp. 52–56]. With b the semibandwidth, coefficients in [A] to the right of its diagonal can be stored in a rectangular array of size $n_{eq}$ by b, with all diagonal matrix coefficients $A_{ii}$ in column 1 of the array (see Fig. 2.8-3b). A Gauss elimination equation solver for this storage format appears in Fig. B.2-3. It requires about $n_{eq}b^{2}/2$ multiplications for forward-reduction of [A] and about $n_{eq}b$ multiplications each for forward-reduction of {c} and for back-substitution.
+
+Figure B.2-3 exploits the fact that the “uneliminated” portion of [A] (i.e., the southeast corner of [A], below the eliminated equations) remains symmetric at all stages of reduction (see Eq. B.2-2). Thus the equation solver may use $A_{ij}$ where one expects to see $A_{ji}$ .
+
+During elimination of a row of [A], coefficients in this row “cast a shadow” of width b below the row in which reside all coefficients that are modified by that elimination (Fig. B.2-4a). Accordingly, in Fig. B.2-3, the DO 780 and DO 750 statements carry calculations
+```prolog
+SUBROUTINE SOL (A,C,X,NEQ)
+IMPLICIT DOUBLE PRECISION (A-H,O-Z)
+DIMENSION A(NEQ,NEQ),C(NEQ),X(NEQ)
+C--- Forward reduction phase.
+DO 10 K=2,NEQ
+DO 10 I=K,NEQ
+R = A(I,K-1)/A(K-1,K-1)
+C(I) = C(I) - R*C(K-1)
+DO 10 J=K,NEQ
+10 A(I,J) = A(I,J) - R*A(K-1,J)
+C--- Back substitution phase (results stored in X).
+X(NEQ) = C(NEQ)/A(NEQ,NEQ)
+DO 30 K=NEQ-1,1,-1
+X(K) = C(K)
+DO 20 J=K+1,NEQ
+20 X(K) = X(K) - A(K,J)*X(J)
+30 X(K) = X(K)/A(K,K)
+RETURN
+END 
+```  
+Figure B.2-2. Fortran statements for the Gauss elimination algorithm given in Fig. B.2-1.
+
+<!-- source-page: 615 -->
+
+<table><tr><td colspan="2">SUBROUTINE SOLVER (A,C,NEQ,MBAND,IFLAG)</td></tr><tr><td colspan="2">IMPLICIT DOUBLE PRECISION (A-H,O-Z)</td></tr><tr><td colspan="2">DIMENSION A(NEQ,1),C(1)</td></tr><tr><td colspan="2">C--- Treat the case of one or more independent equations.</td></tr><tr><td colspan="2">IF (MBAND ,GT. 1) GO TO 690</td></tr><tr><td colspan="2">DO 680 N=1,NEQ</td></tr><tr><td colspan="2">680 C(N) = C(N)/A(N,1)</td></tr><tr><td colspan="2">RETURN</td></tr><tr><td colspan="2">690 NEQP = NEQ + 1</td></tr><tr><td colspan="2">NEQM = NEQ - 1</td></tr><tr><td colspan="2">GO TO (700,800), IFLAG</td></tr><tr><td colspan="2">C--- Forward reduction of the coefficient matrix [A].</td></tr><tr><td colspan="2">700 DO 790 N=1,NEQM</td></tr><tr><td colspan="2">LIM = MIN(MBAND,NEQP-N)</td></tr><tr><td colspan="2">DO 780 L=2,LIM</td></tr><tr><td colspan="2">DUM = A(N,L)/A(N,1)</td></tr><tr><td colspan="2">I = N + L - 1</td></tr><tr><td colspan="2">J = 0</td></tr><tr><td colspan="2">DO 750 K=L,LIM</td></tr><tr><td colspan="2">J = J + 1</td></tr><tr><td colspan="2">750 A(I,J) = A(I,J) - DUM*A(N,K)</td></tr><tr><td colspan="2">A(N,L) = DUM</td></tr><tr><td colspan="2">780 CONTINUE</td></tr><tr><td colspan="2">790 CONTINUE</td></tr><tr><td colspan="2">C--- Forward reduction of the constant vector {c}.</td></tr><tr><td colspan="2">800 DO 830 N=1,NEQM</td></tr><tr><td colspan="2">LIM = MIN(MBAND,NEQP-N)</td></tr><tr><td colspan="2">DO 820 L=2,LIM</td></tr><tr><td colspan="2">I = N + L - 1</td></tr><tr><td colspan="2">C(I) = C(I) - A(N,L)*C(N)</td></tr><tr><td colspan="2">820 CONTINUE</td></tr><tr><td colspan="2">830 C(N) = C(N)/A(N,1)</td></tr><tr><td colspan="2">C(NEQ) = C(NEQ)/A(NEQ,1)</td></tr><tr><td colspan="2">C--- Back substitution. Former unknowns {x} overwrite {c}</td></tr><tr><td colspan="2">DO 860 N=NEQM,1,-1</td></tr><tr><td colspan="2">LIM = MIN(MBAND,NEQP-N)</td></tr><tr><td colspan="2">DO 850 L=2,LIM</td></tr><tr><td colspan="2">K = N + L - 1</td></tr><tr><td colspan="2">C(N) = C(N) - A(N,L)*C(K)</td></tr><tr><td colspan="2">850 CONTINUE</td></tr><tr><td colspan="2">860 CONTINUE</td></tr><tr><td colspan="2">RETURN</td></tr><tr><td colspan="2">END</td></tr></table>
+
+Figure B.2-3. Subroutine for Gauss elimination solution of $[A]\{x\} = \{c\}$ for band-symmetric matrix [A] (see Fig. 2.8-3b). NEQ = number of equations, MBAND = semibandwidth. Except for the case MBAND = 1, start at statement 700 (IFLAG = 1) for the first $\{c\}$ and at statement 800 (IFLAG = 2) for any additional vectors $\{c\}$ .
+
+![](images/page-615_92fd037f283414c2532b73deff4a51f22a7f871d58516e03b080bf3640e17177.jpg)
+
+<details>
+<summary>text_image</summary>
+
+"Active triangle"
+Zero
+Row i
+b
+b
+n_eq
+Symmetric
+(a)
+b
+n_eq
+Zero
+(b)
+</details>
+
+Figure B.2-4. The “active triangle”; that is, the triangular patch of coefficients in the stored band affected by elimination of row i during forward reduction, in (a) full matrix format, and (b) band storage format.
+
+<!-- source-page: 616 -->
+
+to the right and to the bottom, respectively, of the active triangle. The MIN function selects $\mathrm{LIM} = \mathrm{MBAND}$ except when this choice would cause processing to extend into the "zero triangle" in Fig. B.2-4b. Then $\mathrm{LIM} = \mathrm{NEQ} + 1 - \mathrm{N}$ is chosen.
+
+The determinant of [A] can be calculated from Fig. B.2-3 according to statements B.2-3 or B.2-4, but with A(I,I) replaced by A(I,1).
+
+If a new $\{c\}$ is to be processed for the same [A], one can enter Fig. B.2-3 at statement 800. Alternatively, with modest reprogramming, all constant vectors can be placed in a rectangular array and processed in a single pass through the subroutine.
+
+A recommended modification of Fig. B.2-3 is to test for decay of diagonal coefficients (Section 18.4).
+
+A possible modification of Fig. B.2-3 would avoid “do-nothing” computations that occur if DUM happens to be zero. One simply tests DUM in Fig. B.2-3 by an IF statement as soon as it is computed, and transfers to statement 780 if DUM=0. For example, the triangular block of zeros in Fig. B.2-5 remains zero throughout reduction of the matrix, so no purpose is served by exercising the innermost DO loop of Fig. B.2-3 on these coefficients. However, Fig. B.2-3 can be vectorized by a sophisticated compiler. Then the IF test would be of little help, or even detrimental, as it may inhibit vectorization on many compilers.
+
+Another possible modification in Fig. B.2-3 would be to use a semibandwidth appropriate to the row being eliminated; that is, use MBAND(N) rather than taking MBAND as constant throughout the process. Then one must count semibandwidths of rows and be aware that each may change from its initial value when fill-in terms are created by forward-reduction.
+
+Remarks. The Gauss elimination procedure of Fig. B.2-3 is row-oriented and uses a constant semibandwidth b. Some other forms of Gauss elimination are column-oriented and exploit the differing heights above the diagonal exhibited by various columns (see Fig. 2.8-1b). These procedures are known as profile, skyline, or active column solvers [B.2–B.6]. Some topologies of coefficient matrix [A] are treated more efficiently by an active column solver than by a band solver. Figure B.2-5 is a case in point. An active column solver can avoid storage and processing of both blocks of zero coefficients. A band solver (Fig. B.2-3) stores and processes the triangular block of zeros, all to no useful purpose.
+
+The wave front or frontal method is an arrangement of Gauss elimination in which assembly of structural equations alternates with their solution. The sequence in which equations are processed is driven by element numbering rather than by node numbering. The first equations to be eliminated are those associated with element 1 only. Then the adjacent element, element 2, makes its contribution of stiffness coefficients to the system of equations. If any additional equations are fully summed—that is, if any additional d.o.f. are shared by elements 1 and 2 only—these equations are eliminated. The next elimination awaits contributions from one or more additional elements. This repetitive alternation between assembly and solution can be viewed as a “wave” that sweeps over the structure
+
+![](images/page-616_8b58092e28ce9191289497507c02aa46bd85a0db3306f79b9d68384d8b5d182f.jpg)
+
+<details>
+<summary>text_image</summary>
+
+Zero
+Zero
+Symmetric
+</details>
+
+Figure B.2-5. Matrix topology better suited to an active column solver than to a band solver.
+
+<!-- source-page: 617 -->
+
+in a pattern dictated by the element numbering. For efficiency, consecutive element numbers should run “across” the structure—that is, in the direction that spans the smallest number of nodes.
+
+If efficiently coded, and if presented with a structure whose nodes (or elements) are numbered for efficient processing, active column solvers and wavefront solvers appear to be equally efficient. Active column solvers appear to be easier to program and understand.
+
+<!-- source-page: 618 -->
+
+# EIGENVALUES AND EIGENVECTORS
+
+This appendix discusses the nature of the eigenproblem and suggests solution methods appropriate to particular forms that may be encountered.
+
+# C.1 THE EIGENPROBLEM
+
+Let [A] and [B] be $n$ by $n$ square matrices. In an eigenvalue problem one seeks values of a scalar $\lambda$ such that the matrix equation
+
+$$
+([ \mathbf {A} ] - \lambda [ \mathbf {B} ]) \{\mathbf {x} \} = \{\mathbf {0} \} \tag {C.1-1}
+$$
+
+has solutions other than the trivial solution $\{x\} = \{0\}$ . There are at most n solutions $\lambda_{i}$ , not necessarily all distinct. The $\lambda_{i}$ are called eigenvalues (other names include characteristic values, latent roots, proper values, and principal values). Corresponding to each $\lambda_{i}$ there is an $\{x\}_{i}$ , called an eigenvector (other names include characteristic vector, proper vector, principal vector, normal mode, natural mode, and principal mode). Equation C.1-1 is called a generalized eigenproblem or simply an eigenproblem. If [B] happens to be the identity matrix, Eq. C.1-1 is called a standard eigenproblem [A.1] and the associated $\lambda_{i}$ are called eigenvalues of [A].
+
+Eigenproblems arise in vibration analysis (where $\sqrt{\lambda_{i}} = \omega_{i}$ is a vibration frequency and $\{x\}_{i}$ is the vibration mode) and in buckling analysis (where $\lambda_{i}$ indicates the critical load and $\{x\}_{i}$ is the buckling mode). In other physical problems the eigenproblem must be solved to obtain information needed for the modal method of time-history analysis. Section 13.5 discusses eigenproblems to the extent needed to solve simple vibration problems.
+
+# C.2 THE STANDARD EIGENPROBLEM
+
+Consider an eigenproblem of the form
+
+$$
+([ \mathbf {A} ^ {\prime} ] - \lambda [ \mathbf {B} ]) \{\mathbf {x} ^ {\prime} \} = \{\mathbf {0} \} \tag {C.2-1}
+$$
+
+where $[A']$ is an arbitrary n by n square matrix and $[B]$ is diagonal and nonsingular. There are always n eigenvalues of Eq. C.2-1. One may define a diagonal matrix $[T]$ and a transformation of $\{x'\}$ by
+
+$$
+T _ {i i} = \frac {1}{\sqrt {B _ {i i}}} \quad \text { and } \quad \{\mathbf {x} ^ {\prime} \} = \lceil \mathbf {T} \rceil \{\mathbf {x} \} \tag {C.2-2}
+$$
+
+Premultiplication of Eq. C.2-1 by [T] and substitution from Eq. C.2-2 yields the standard eigenproblem
+
+$$
+([ \mathbf {A} ] - \lambda [ \mathbf {I} ]) \{\mathbf {x} \} = \{\mathbf {0} \} \tag {C.2-3}
+$$
+
+<!-- source-page: 619 -->
+
+where $[A] = \left[T\right]\left[A'\right]\left[T\right]$ . Equations C.2-1 and C.2-3 have the same eigenvalues $\lambda_{i}$ . An eigenvector $\{x'\}_{i}$ of the original system is recovered from the corresponding eigenvector $\{x\}_{i}$ by the operation $\{x'\}_{i} = \left[T\right]\{x\}_{i}$ . (The foregoing transformation is convenient if $[B]$ is the diagonal mass matrix of a structural dynamics problem.)
+
+Properties of Eq. C.2-3 that may be useful in engineering applications are as follows. Most of these results can be deduced from [A.1, pp. 243–360].
+
+1. If [A] is real and symmetric, the $\lambda_{i}$ are real.   
+2. If [A] is real, symmetric, and positive semidefinite, there are no negative $\lambda_{i}$ . The number of nonzero $\lambda_{i}$ equals the rank of [A].   
+3. If [A] is real, symmetric, and positive definite, all $\lambda_{i}$ are positive.   
+4. If $[\mathbf{A}]$ is real and positive definite but unsymmetric, the matrix $([{\mathbf{A}}] + [{\mathbf{A}}]^T)$ has positive eigenvalues.   
+5. If $\{\mathbf{x}\}_i$ is an eigenvector, so is $c\{\mathbf{x}\}_i$ , where $c$ is an arbitrary nonzero scalar.   
+6. If all $\lambda_{i}$ are distinct, all eigenvectors are distinct and linearly independent.   
+7. A $\lambda_{i}$ repeated $k$ times may or may not have $k$ independent associated eigenvectors.   
+8. Let [G] be a square matrix, nonsingular and the same order as [A] but otherwise arbitrary. A matrix [C], obtained by the “similarity transformation” [C] = [G]⁻¹[A][G], has the same eigenvalues as [A]. If eigenvectors of [C] are {x\_c}, eigenvectors of [A] are [G]{x\_c}.   
+9. If [A] is real and $[\mathbf{A}]^T [\mathbf{A}] = [\mathbf{A}][\mathbf{A}]^T$ (e.g., [A] is real and symmetric), then eigenvectors are orthogonal, that is, $\{\mathbf{x}\}_{j}^{T}\{\mathbf{x}\}_{j} = 0$ if $i\neq j$ .   
+10. If eigenvectors are scaled so that $\{\mathbf{x}\}_{i}^{T}\{\mathbf{x}\}_{i} = 1$ , then $\{\mathbf{x}\}_{i}^{T}[\mathbf{A}]\{\mathbf{x}\}_{i} = \lambda_{i}$ (see the Rayleigh quotient for the real symmetric case, Eq. 13.5-4 or Eq. C.3-9).   
+11. If eigenvectors are scaled so that $\{\mathbf{x}\}_{i}^{T}\{\mathbf{x}\}_{i} = 1$ and [A] is symmetric and nonsingular, then
+
+$$
+[ \mathbf {A} ] = \sum_ {i = 1} ^ {n} \lambda_ {i} \{\mathbf {x} \} _ {i} \{\mathbf {x} \} _ {i} ^ {T} \quad \text { and } \quad [ \mathbf {A} ] ^ {- 1} = \sum_ {i = 1} ^ {n} \frac {1}{\lambda_ {i}} \{\mathbf {x} \} _ {i} \{\mathbf {x} \} _ {i} ^ {T} \tag {C.2-4}
+$$
+
+where $n$ is the order of [A].
+
+# C.3 THE GENERAL EIGENPROBLEM
+
+Lumping and condensation procedures can result in eigenproblems of the form of Eq. C.1-1 in which [A] and particularly [B] may be of a more general nature than considered thus far. However, we will assume that [A] and [B] are symmetric. The following terminology and observations are useful in understanding these problems.
+
+1. The matrix of linear polynomials $A_{ij} - \lambda B_{ij}$ , that is,
+
+$$
+[ \mathbf {A} ] - \lambda [ \mathbf {B} ] \tag {C.3-1}
+$$
+
+is called a pencil (of [A] and [B]).
+
+2. Eigenvalues of the pencil arise from nontrivial solutions of Eq. C.1-1 and thus are values of $\lambda$ that make the determinant of the pencil vanish,
+
+$$
+\det ([ \mathbf {A} ] - \lambda [ \mathbf {B} ]) = 0 \tag {C.3-2}
+$$
+
+(as in Eq. 13.5-3, for example).
+
+<!-- source-page: 620 -->
+
+3. The polynomial of degree n or less
+
+$$
+q (\lambda) = \det ([ \mathbf {A} ] - \lambda [ \mathbf {B} ]) \tag {C.3-3}
+$$
+
+is called the characteristic polynomial of the pencil.
+
+4. In the case described by Eq. C.2-1, and consequently in the standard eigenproblem, $q(\lambda)$ is of degree n and has n eigenvalues as roots (counting multiplicity).   
+5. In general, $q(\lambda)$ may be of degree less than $n$ . Matrix [B] must be singular for this to occur. In the most extreme case it is possible (when [A] also is singular) to have $q(\lambda) = 0$ for all $\lambda$ . Fortunately this rarely happens in practical problems [C.1-C.3]. The roots of $q(\lambda)$ are called the finite eigenvalues. Infinite eigenvalues, if any, do not correspond to roots of $q(\lambda)$ .   
+6. Let [R] and [S] be nonsingular matrices of numbers (not involving $\lambda$ ). The transformed pencil
+
+$$
+[ \mathbf {R} ] [ \mathbf {A} ] [ \mathbf {S} ] - \lambda [ \mathbf {R} ] [ \mathbf {B} ] [ \mathbf {S} ] \tag {C.3-4a}
+$$
+
+has the same characteristic polynomial as Eq. C.3-1, that is,
+
+$$
+\det ([ \mathbf {R} ] [ \mathbf {A} ] [ \mathbf {S} ] - \lambda [ \mathbf {R} ] [ \mathbf {B} ] [ \mathbf {S} ]) = \det ([ \mathbf {A} ] - \lambda [ \mathbf {B} ]) = q (\lambda) \tag {C.3-4b}
+$$
+
+7. The pencil of Eq. C.3-1 in general may have fewer than $n$ independent eigenvectors. The case of Eq. C.2-1 ([B] diagonal, $B_{ii} > 0$ ) is one in which there is a complete set of independent eigenvectors [Q], whose $n$ columns are the independent solutions of Eq. C.1-1.   
+8. Whenever there is a complete set of eigenvectors [Q], the pencil of Eq. C.3-1 is called diagonalizable. The special case of [B] positive definite is discussed in [A.1, pp. 343-346].   
+9. In the most common cases of diagonalizable pencils (when [B] is positive definite) the eigenvectors are orthogonal with respect to [B], and may be normalized so that [Q] simultaneously diagonalizes [A] and [B],
+
+$$
+[ \mathbf {Q} ] ^ {T} [ \mathbf {A} ] [ \mathbf {Q} ] = [ \Lambda ] \tag {C.3-5a}
+$$
+
+$$
+[ \mathbf {Q} ] ^ {T} [ \mathbf {B} ] [ \mathbf {Q} ] = [ \mathbf {I} ] \tag {C.3-5b}
+$$
+
+where $[\Lambda]$ is a diagonal matrix with $\Lambda_{ii} = \lambda_{i}$ . A set of [B]-orthogonal vectors, [Q], that satisfies Eq. C.3-5b is called orthonormal with respect to [B]. As a special case, if [B] is a unit matrix [I], then [Q] contains orthogonal unit vectors.
+
+When [B] is positive definite, case 9 applies. This can be seen by transforming the pencil to standard form by using the Cholesky decomposition [A.1, p. 195],
+
+$$
+[ \mathbf {B} ] = [ \mathbf {L} ] [ \mathbf {L} ] ^ {T} \tag {C.3-6}
+$$
+
+and applying Eqs. C.3-4 with $[R] = [L]^{-1}$ and $[S] = [L]^{-T}$ to obtain the pencil
+
+$$
+[ \mathbf {L} ] ^ {- 1} [ \mathbf {A} ] [ \mathbf {L} ] ^ {- \tau} - \lambda [ \mathbf {I} ] \tag {C.3-7}
+$$
+
+Note that $[L]^{-1}[A][L]^{-T}$ is symmetric and there are n independent and orthonormal eigenvectors. If [X] is a matrix whose columns are these eigenvectors, then the matrix of eigenvectors of the original pencil, Eq. C.3-1, is
+
+$$
+[ \mathbf {Q} ] = [ \mathbf {L} ] ^ {- T} [ \mathbf {X} ] \tag {C.3-8}
+$$
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_063.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_063.md
new file mode 100644
index 00000000..e735e8fe
--- /dev/null
+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_063.md
@@ -0,0 +1,261 @@
+<!-- source-page: 621 -->
+
+which are orthonormal with respect to [B] in the sense of Eq. C.3-5b. Note that [L] generalizes the diagonal transformation matrix [T] of Eq. C.2-2 to the case in which [B] need not be diagonal. When [A] is positive definite, all $\lambda_{i} > 0$ . When [A] is positive semidefinite, $\lambda_{i} \geq 0$ . In structural problems, the $\lambda_{i} = 0$ are associated with rigid-body motions or mechanisms.
+
+Either ad hoc or optimal mass lumping may lead to a problem in which [A] is positive semidefinite and [B] is diagonal with some $B_{ii} = 0$ (i.e., some massless d.o.f.). Consideration of the generalized Rayleigh quotient (see Eqs. 13.5-4 and 13.5-5),
+
+$$
+\lambda_ {\min} \leq \frac {\{\mathbf {x} \} ^ {T} [ \mathbf {A} ] \{\mathbf {x} \}}{\{\mathbf {x} \} ^ {T} [ \mathbf {B} ] \{\mathbf {x} \}} \leq \lambda_ {\max} \tag {C.3-9}
+$$
+
+usually indicates that there are as many infinite eigenvalues, $\lambda_{\max} = \infty$ , as there are $B_{ii} = 0$ . It is possible to construct pathological forms of [A] and [B] that violate this rule. Pathological forms will not arise in finite element analysis if rigid-body motions (or mechanisms) are associated with positive kinetic energy [C.2]. It is up to the analyst to ensure that this condition prevails. If there are no pathological forms, [A] is positive semidefinite, and there are $m$ massless d.o.f. ( $B_{ii} = 0$ ), then there are $m$ infinite eigenvalues and $n - m$ finite eigenvalues associated with $n - m$ independent eigenvectors $\{\mathbf{x}\}$ . Thus $q(\lambda)$ has degree $n - m$ . The infinite eigenvalues are associated with $m$ independent eigenvectors $\{\mathbf{x}\}$ such that $[\mathbf{B}]\{\mathbf{x}\} = \{\mathbf{0}\}$ .
+
+Infinite eigenvalues can be observed in a more general context by considering the finite eigenvalues of the inverse eigenproblem, whose pencil is
+
+$$
+[ \mathbf {B} ] - \mu [ \mathbf {A} ] \tag {C.3-10}
+$$
+
+Many of the following statements are also true in a more general context, but if we limit our attention to the diagonalizable case, it is easy to see that the correspondence to the pencil of Eq. C.3-1 is that
+
+$$
+\frac {1}{\mu_ {i}} = \lambda_ {i} \tag {C.3-11}
+$$
+
+where $\mu_{i}=0$ corresponds to $\lambda_{i}=\infty$ . The complete set of eigenvectors is the same for both pencils. Equation C.3-11 defines an infinite eigenvalue even when [B] is nondiagonal. The pencils of Eqs. C.3-1 and C.3-10 may not be diagonalizable in general. When one is, the other clearly will be also, and this will be true whenever
+
+(a) Either [A] or [B] is positive definite.   
+(b) [Å] and [B] are positive semidefinite and there are no ill-disposed vectors (i.e. vectors such that [A]{x} = {0} and [B]{x} = {0}).   
+(c) [A] is semidefinite and [B] is an indefinite matrix arising from an optimal lumping formula, which leads to no indeterminate vectors [C.2] (i.e. vectors such that $\{\mathbf{x}\}^T [\mathbf{A}]\{\mathbf{x}\} = 0$ and $\{\mathbf{x}\}^T [\mathbf{B}]\{\mathbf{x}\} = 0$ ).
+
+Case (c) arises in optimal lumping in which some masses (quadrature weights) are negative, as explained in Section 13.3. No known optimal lumping scheme leads to indeterminate vectors. It should be noted that ill-disposed vectors are a special case of indeterminate vectors, and that in pathological cases indeterminate vectors are possible even without massless d.o.f., because an indefinite [B] always has vectors $\{\mathbf{x}\}$ such that $\{\mathbf{x}\}^7 [\mathbf{B}]\{\mathbf{x}\} = 0$ . Pencils that fall outside of restrictions (a) through (c) can be highly pathological but are almost never observed in finite element practice.
+
+In the most general diagonalizable cases—that is, (a) through (c) with [B] singular—we must revise our notion of orthogonality of [Q]. Simultaneous diagonalization of Eq. C.3-1 is possible, and
+
+<!-- source-page: 622 -->
+
+$$
+[ \mathbf {Q} ] ^ {T} [ \mathbf {A} ] [ \mathbf {Q} ] = [ \Lambda^ {+} ] \tag {C.3-12a}
+$$
+
+$$
+[ \mathbf {Q} ] ^ {T} [ \mathbf {B} ] [ \mathbf {Q} ] = [ \Omega ] \tag {C.3-12b}
+$$
+
+where $\Lambda_{ii}^{+} = \lambda_{i}$ for $0 \leq \lambda_{i} < \infty$ , $\Lambda_{ii}^{+} = 1$ for $\lambda_{i} = \infty$ . In case (c), $\lambda_{i} \leq 0$ with $|\lambda_{i}|$ “very large” are introduced for each negative mass [C.1] and $\Lambda_{ii}^{+} = |\lambda_{i}|$ for $\lambda_{i} < 0$ . $[\Omega]$ is also diagonal, and $\Omega_{ii} = 1$ for $0 \leq \lambda_{i} < \infty$ , $\Omega_{ii} = 0$ for $\lambda_{i} = \infty$ , and $\Omega_{ii} = -1$ for $\lambda_{i} < 0$ . In case (c), the negative signs can only be transferred from $[\Omega]$ to $[\Lambda^{+}]$ by a transformation of the pencil of the form of Eqs. C.3-4 in which $[R] \neq [Q]^{T}$ . The zeros on the diagonal of $[\Omega]$ cannot be avoided. Thus the eigenvectors with $0 \leq \lambda_{i} < \infty$ are orthonormal in the usual sense. The other eigenvectors can have “negative lengths” or be “orthogonal to themselves.” Their physical meaning has been sacrificed in such cases.
+
+# C.4 REMARKS ON SPECIAL FORMS
+
+We consider the general eigenproblem,
+
+$$
+([ \mathbf {A} ] - \lambda [ \mathbf {B} ]) \{\mathbf {x} \} = \{\mathbf {0} \} \quad \text { or } \quad ([ \mathbf {K} ] - \omega^ {2} [ \mathbf {M} ]) \{\mathbf {D} \} = \{\mathbf {0} \} \tag {C.4-1}
+$$
+
+where the latter form uses the symbolism adopted for vibration problems in Section 13.5. In what follows we assume that [A] and [B] are symmetric, and consider how to proceed under circumstances that may arise in finite element modeling.
+
+If [A] and [B] are positive definite, there is no special difficulty. One may proceed directly to solve the eigenproblem (see Section C.5).
+
+[B] will be positive semidefinite and diagonal if “lumping” is used with $B_{ii} = 0$ for some d.o.f. $x_{i}$ ; for example, if mass particles have no rotary inertia or are not attached to all nodes (this is not a recommended practice). One can eliminate the $x_{i}$ for which $B_{ii} = 0$ in a diagonal [B]. The procedure is discussed in Section 8.1. In the condensed system, matrices are of lower order than initially and [B] is positive definite. In the case where $B_{ii} = 0$ because of optimal lumping, condensation is not recommended, since it will lose the accuracy the scheme is designed to retain. Several algorithms discussed in the next section and in Ref. C.1 are either not affected if [B] fails to be positive definite or can be easily modified to work in this case.
+
+In the structural context, a positive semidefinite [A] implies an unsupported structure such as a spacecraft. This circumstance presents no obstacle to some algorithms for solving the eigenproblem. Other algorithms may require that [A] be nonsingular. This restriction can be met by using an eigenvalue shift. Thus we select a scalar $c$ and substitute $\lambda = \lambda' - c$ into Eq. C.1-1, to obtain
+
+$$
+([ \mathbf {A} + c \mathbf {B} ] - \lambda^ {\prime} [ \mathbf {B} ]) \{\mathbf {x} \} = \{\mathbf {0} \} \tag {C.4-2}
+$$
+
+in which $[A + cB]$ is positive definite if $\{x\}_{i}^{T}[B]\{x\}_{i} > 0$ for every $\{x\}_{i}$ for which $\{x\}_{i}^{T}[A]\{x\}_{i} = 0$ . We solve for shifted eigenvalues $\lambda_{i}^{\prime}$ , then obtain actual eigenvalues $\lambda_{i} = \lambda_{i}^{\prime} - c$ . Eigenvectors are not changed by shifting. If $|c|$ is too small, $[A + cB]$ is almost singular. If $|c|$ is too large, convergence of the eigensolver may be slow. A possible choice is c = 0.01r, where r is the ratio of the trace of [A] to the trace of [B]. Surprisingly, perhaps, shifting usually yields a positive definite $[A + cB]$ even when [B] is indefinite, as when [B] arises from an optimal lumping scheme [C.1]. If the problem is structural and [B] is diagonal, one can interpret Eq. C.4-2 physically by saying that a spring of stiffness $cB_{ii}$ has been added between each d.o.f. $x_{i}$ and ground, thus preventing any rigid-body motion.
+
+<!-- source-page: 623 -->
+
+# C.5 SOLUTION ALGORITHMS
+
+Algorithms for eigenvalue extraction are plentiful. The best choice for a particular eigenproblem depends on the order n of the matrices, their sparsity, the number of eigenvalues to be extracted, and where in the eigenspectrum the eigenvalues of interest are located. In what follows we indicate some choices commonly made in practice, but make little or no attempt to explain the workings of various algorithms. These details may be found in references such as $[C.4-C.12]$ . At the outset we note that the various methods are often used in combination rather than as stand-alone algorithms.
+
+Writing out the characteristic polynomial, $q(\lambda) = \det[A - \lambda B] = 0$ , and calculating its roots is a method suitable only for hand calculation, for example, when there are only one or two roots. If n is large, the method is costly and inaccurate.
+
+If all $n$ eigenvalues are required and $n$ is relatively small (roughly $n < 200$ ), the Jacobi method is a good choice.
+
+If [A] and [B] are narrowly banded and only a few eigenvalues are required (e.g., only $\lambda_{1}$ and $\lambda_{2}$ ), determinant search may be appropriate. Effectively, one seeks zeros of the characteristic polynomial by repeated factoring of [A - $\lambda$ B] for trial values of $\lambda$ . To ensure that no root is skipped, one can use the Sturm sequence property. $^{1}$
+
+The Sturm sequence property is also useful in calculating the eigenvalues that appear in a prescribed range. The associated eigenvectors must be calculated by a different method. A slightly modified Sturm sequence property holds for indefinite [B] resulting from optimal lumping [C.1].
+
+The inverse iteration or inverse power method computes the lowest eigenvalue. Or, when used with an eigenvalue shift, it computes the eigenvalue closest to the “shift point.” The associated eigenvector is automatically computed as part of the process. The inverse power method does not require a positive definite [B].
+
+The subspace iteration method uses k trial vectors that are iteratively improved as calculation proceeds. If m is the number of eigenvalues required, k may be taken as the smallest of the three numbers 2m, $m + 8$ , and n. Typically, k << n. There is a similarity between subspace iteration and mass condensation (Section 13.7).
+
+Householder reduction and Givens reduction transform a standard or general eigenproblem into one in which the symmetric [A] and [B] are tridiagonal, which then can be easily solved by one of the other methods cited here. The transformation involved can be chosen to respect band storage format.
+
+The Lanczos method is probably the most efficient algorithm available. Early difficulties with the method have been overcome. It is usually not available as a library package at computer installations, but appears destined to supplant other eigenproblem algorithms among the finite element community [C.5]. Like the Householder and Givens methods, the Lanczos method is a tridiagonal transformation method. It can be arranged so that it respects band or skyline/profile storage format. The resulting tridiagonal system is easily solved by other means.
+
+The QR and QZ methods can solve a general eigenproblem but are particularly suited to extracting all eigenvalues and eigenvectors of a tridiagonal system resulting from Givens, Householder, or Lanczos transformations. The QZ method is complicated, but is the only known method capable of handling cases where [A] and [B] are singular and do not satisfy restrictions (a) through (c) of Section C.3. Thus the QZ method is useful for testing small systems whose properties are unknown a priori.
+
+$^{1}$ We can use Subroutine SOLVER, Fig. B.2-3, to exploit the Sturm sequence property. Thus we select a numerical value of $\lambda$ , place [A] - $\lambda$ [B] in array A of Fig. B.2-3, and complete the loop on statement 790. At this time the number of eigenvalues exceeded by the selected $\lambda$ is equal to the number of negative diagonal coefficients (which appear in column 1 of array A in Fig. B.2-3).
+
+<!-- source-page: 624 -->
+
+Standard library packages (e.g., EISPACK, IMSL, NAg; Refs. C.10–C.12) provide several eigensolvers, usually tailored to exploit matrix properties such as symmetry or bandedness. The storage formats used, particularly in the more specialized algorithms, often disagree with storage formats for [A] and [B] in the program that generates these arrays.
+
+An explanation of many of the methods described here, aimed at an engineering audience, can be found in Chapter 5 of [C.7]. A guide to obtaining software from EISPACK, IMSL, NAg, and other packages is given in the Appendix of [C.7].
+
+<!-- source-page: 625 -->
+
+# REFERENCES
+
+# CHAPTER 1
+
+1.1 J. Ergatoudis, B. M. Irons, and O. C. Zienkiewicz, "Three-Dimensional Analysis of Arch Dams and Their Foundations," Research Report No. C/R/74/67, University of Wales, Swansea, 1968.   
+1.2 K. Wieghardt, "Über einen Grenzübergang der Elastizitätslehre und seine Andwendung auf die Statik hochgradig statisch unbestimmter Fachwerke," Verhandlungen des Vereins z. Beförderung des Gewerbefleisses, Abhandlungen, Vol. 85, 1906, pp. 139–176.   
+1.3 W. Riedel, “Beiträge zur Lösung des ebenen Problems eines elastichen Körpers mittels der Airyschen Spannungsfunktion,” Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 7, No. 3, 1927, pp. 169–188.   
+1.4 A. Hrennikoff, "Solution of Problems in Elasticity by the Framework Method," J. Appl. Mech., Vol. 8, No. 4, 1941, pp. A169-A175.   
+1.5 R. Courant, “Variational Methods for the Solution of Problems of Equilibrium and Vibrations,” Bulletin of the American Mathematical Society, Vol. 49, 1943, pp. 1–23.   
+1.6 S. Levy, “Structural Analysis and Influence Coefficients for Delta Wings,” J. Aero. Sci., Vol. 20, No. 7, 1953, pp. 449–454.   
+1.7 R. W. Clough, "The Finite Element Method After Twenty-Five Years: A Personal View," Computers & Structures, Vol. 12, No. 4, 1980, pp. 361–370.   
+1.8 M. J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp, “Stiffness and Deflection Analysis of Complex Structures,” J. Aero. Sci., Vol. 23, No. 9, 1956, pp. 805–823.   
+1.9 J. H. Argyris and S. Kelsey, Energy Theorems and Structural Analysis, Butterworths, London, 1960 (collection of papers published in Aircraft Engineering in 1954 and 1955).   
+1.10 J. Robinson, Early FEM Pioneers, Robinson & Associates, Dorset, England, 1985.
+
+# CHAPTER 2
+
+2.1 K. J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1982.   
+2.2 G. C. Everstine, “A Comparison of Three Resequencing Algorithms for the Reduction of Matrix Profile and Wavefront,” Int. J. Num. Meth. Engng., Vol. 14, No. 6, 1979, pp. 837–853.   
+2.3 S. W. Sloan, “An Algorithm for Profile and Wavefront Reduction of Sparse Matrices,” Int. J. Num. Meth. Engng., Vol. 23, No. 2, 1986, pp. 239–251.
+
+# CHAPTER 3
+
+3.1 H. L. Langhaar, Energy Methods in Applied Mechanics, John Wiley & Sons, New York, 1962.
+
+<!-- source-page: 626 -->
+
+3.2 K. Washizu, Variational Methods in Elasticity and Plasticity, 3rd Ed., Pergamon Press, Oxford, England, 1982.   
+3.3 P. Tong, "Exact Solution of Certain Problems by the Finite Element Method," AIAA Jnl., Vol. 7, No. 1, 1969, pp. 178-180.   
+3.4 A. A. Ball, “The Interpolation Function of a General Serendipity Rectangular Element,” Int. J. Num. Meth. Engng., Vol. 15, No. 5, 1980, pp. 773–778.
+
+# CHAPTER 4
+
+4.1 R. J. Melosh, "Basis for Derivation of Matrices for the Direct Stiffness Method," AIAA Jnl., Vol. 1, No. 7, 1963, pp. 1631-1637.   
+4.2 R. Narayanaswami and H. M. Adelman, "Inclusion of Transverse Shear Deformation in Finite Element Displacement Formulations," AIAA Jnl., Vol. 12, No. 11, 1974, pp. 1613–1614 (discussion: Vol. 13, No. 9, pp. 1253–1254; application to plates: Vol. 12, No. 12, pp. 1761–1763).   
+4.3 R. J. Melosh and D. W. Lobitz, "On a Numerical Sufficiency Test for Monotonic Convergence of Finite Element Models," AIAA Jnl., Vol. 13, No. 5, 1975, pp. 675–678.   
+4.4 W. E. Haisler and J. A. Stricklin, "Rigid-Body Displacements of Curved Elements in the Analysis of Shells by the Matrix Displacement Method," AIAA Jnl., Vol. 5, No. 8, 1967, pp. 1525-1527.   
+4.5 G. P. Bazeley, Y. K. Cheung, B. M. Irons, and O. C. Zienkiewicz, “Triangular Elements in Plate Bending—Conforming and Nonconforming Solutions,” Proc. First Conf. on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Ohio, 1965 (AFFDL-TR-66-80, Nov. 1966; AD-646-300, N.T.I.S.), pp. 547–576.   
+4.6 R. L. Taylor, J. C. Simo, O. C. Zienkiewicz, and A. C. H. Chan, "The Patch Test—A Condition for Assessing FEM Convergence," Int. J. Num. Meth. Engng., Vol. 22, No. 1, 1986, pp. 39–62.   
+4.7 I. U. Ojalvo, “Improved Thermal Stress Determination by Finite Element Methods,” AIAA Jnl., Vol. 12, No. 8, 1974, pp. 1131–1132.   
+4.8 J. Pittr and H. Hartl, "Improved Stress Evaluation Under Thermal Load for Simple Finite Elements," Int. J. Num. Meth. Engng., Vol. 15, No. 10, 1980, pp. 1507-1515.   
+4.9 G. Loubignac, G. Cantin, and G. Touzot, "Continuous Stress Fields in Finite Element Analysis," AIAA Jnl., Vol. 15, No. 11, 1977, pp. 1645–1647.   
+4.10 R. D. Cook and X. Huang, "Continuous Stress Fields by the Finite Element-Difference Method," Int. J. Num. Meth. Engng., Vol. 22, No. 1, 1986, pp. 229–240.   
+4.11 R. T. Severn, "Inclusion of Shear Deflection in the Stiffness Matrix for a Beam Element," J. of Strain Analysis, Vol. 5, No. 4, 1970, pp. 239-241.   
+4.12 R. E. Cornwell and R. D. Cook, "Improvement in Peak Stress Estimates Through Post-Processing," Finite Elements in Analysis and Design (to appear).   
+4.13 I. C. Taig, "Finite Element Analysis in Industry—Expertise or Proficiency?," in Accuracy, Reliability, and Training in FEM Technology, J. Robinson, ed., Pitman Press, England, 1984.
+
+# CHAPTER 5
+
+5.1 K. Bell, "A Refined Triangular Plate Bending Finite Element," Int. J. Num. Meth. Engrg., Vol. 1, No. 1, 1969, pp. 101-122.   
+5.2 N. M. Ferrers, An Elementary Treatise on Trilinear Coordinates, the Method of Reciprocal Polars and the Theory of Projections, Macmillan, London, 1861.
+
+<!-- source-page: 627 -->
+
+5.3 J. B. Mertie, "Transformation of Trilinear and Quadriplanar Coordinates to and from Cartesian Coordinates," The American Mineralogist, Vol. 49, Nos. 7/8, 1964, pp. 926–936.   
+5.4 R. H. Ghallagher, Finite Element Analysis: Fundamentals, Prentice-Hall, Englewood Cliffs, NJ, 1975.   
+5.5 C. A. Felippa, "Refined Finite Element Analysis of Linear and Nonlinear Two-Dimensional Structures," Ph.D. dissertation, University of California, Berkeley, 1966 (also available as PB-178-418 and PB-178-419, N.T.I.S.; computer programs in the latter).   
+5.6 G. Subramanian and C. Jayachandrabose, “Convenient Generation of Stiffness Matrices for the Family of Plane Triangular Elements,” Computers & Structures, Vol. 15, No. 1, 1982, pp. 85–89.
+
+# CHAPTER 6
+
+6.1 B. M. Irons, "Engineering Applications of Numerical Integration in Stiffness Methods," AIAA Jnl., Vol. 4, No. 11, 1966, pp. 2035-2037.   
+6.2 A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, NJ, 1966.   
+6.3 S. W. Sloan, “A Fast Stiffness Formulation for Finite Element Analysis of Two-Dimensional Solids,” Int. J. Num. Meth. Engng., Vol. 17, No. 9, 1981, pp. 1313–1323.   
+6.4 A. K. Gupta, "Efficient Numerical Integration of Element Stiffness Matrices," Int. J. Num. Meth. Engng., Vol. 19, No. 9, 1983, pp. 1410-1413.   
+6.5 D. A. Dunavant, "High Degree Efficient Symmetrical Gaussian Quadrature Rules for the Triangle," Int. J. Num. Meth. Engng., Vol. 21, No. 6, 1985, pp. 1129–1148.   
+6.6 A. K. Noor and C. M. Anderson, "Computerized Symbolic Manipulation in Nonlinear Finite Element Analysis," Computers & Structures, Vol. 13, Nos. 1-3, 1981, pp. 379-403.   
+6.7 I. Ergatoudis, B. M. Irons, and O. C. Zienkiewicz, "Curved Isoparametric, 'Quadrilateral' Elements for Finite Element Analysis," Int. J. Solids Structures, Vol. 4, No. 1, 1968, pp. 31–42.   
+6.8 P. C. Hammer and A. H. Stroud, "Numerical Evaluation of Multiple Integrals II," Math. Tables and Other Aids to Comp., Vol. 12, No. 64, 1958, pp. 272-280.   
+6.9 B. M. Irons, "Quadrature Rules for Brick Based Finite Elements," Int. J. Num. Meth. Engng., Vol. 3, No. 2, 1971, pp. 293-294.   
+6.10 T. K. Hellen, "Effective Quadrature Rules for Quadratic Solid Isoparametric Finite Elements," Int. J. Num. Meth. Engng., Vol. 4, No. 4, 1972, pp. 597–599.   
+6.11 B. Verhegghe and G. H. Powell, "Control of Zero-Energy Modes in 9-Node Plane Element," Int. J. Num. Meth. Engng., Vol. 23, No. 5, 1986, pp. 863–869.   
+6.12 R. D. Cook and Z. H. Feng, “Control of Spurious Modes in the Nine-Node Quadrilateral Element,” Int. J. Num. Meth. Engng., Vol. 18, No. 10, 1982, pp. 1576–1580.   
+6.13 W. K. Liu, J. S. J. Ong, and R. A. Uras, "Finite Element Stabilization Matrices: A Unification Approach," Comp. Meth. Appl. Mech. Engng., Vol. 53, No. 1, 1985, pp. 13–46.   
+6.14 E. Hinton and J. S. Campbell, "Local and Global Smoothing of Discontinuous Finite Element Functions Using a Least Squares Method," Int. J. Num. Meth. Engng., Vol. 8, No. 3, 1974, pp. 461-480.   
+6.15 J. Barlow, "Optimal Stress Locations in Finite Element Models," Int. J. Num. Meth. Engng., Vol. 10, No. 2, 1976, pp. 243-251 (discussion: Vol. 11, No. 3, p. 604).   
+6.16 A. Peano, "Inadmissible Distortion of Solid Elements and Patch Test Results," Comm. in Appl. Num. Meth., Vol. 3, No. 2, 1987, pp. 97-101.
+
+<!-- source-page: 628 -->
+
+# CHAPTER 7
+
+7.1 A. K. Gupta and P. S. Ma, "Error in Eccentric Beam Formulation," Int. J. Num. Meth. Engng., Vol. 11, No. 9, 1977, pp. 1473-1477.   
+7.2 R. E. Miller, "Reduction of the Error in Eccentric Beam Modelling," Int. J. Num. Meth. Engng., Vol. 15, No. 4, 1980, pp. 575-582.
+
+# CHAPTER 8
+
+8.1 E. L. Wilson, "The Static Condensation Algorithm," Int. J. Num. Meth. Engng., Vol. 8, No. 1, 1974, pp. 198–203.   
+8.2 R. D. Cook and V. N. Shah, “A Cost Comparison of Two Static Condensation-Stress Recovery Algorithms,” Int. J. Num. Meth. Engng., Vol. 12, No. 4, 1978, pp. 581–588.   
+8.3 R. L. Taylor, P. J. Beresford, and E. L. Wilson, “A Non-Conforming Element for Stress Analysis,” Int. J. Num. Meth. Engng., Vol. 10, No. 6, 1976, pp. 1211–1219.   
+8.4 M. Fröier, L. Nilsson, and A. Samuelsson, “The Rectangular Plane Stress Element by Turner, Pian and Wilson,” Int. J. Num. Meth. Engng., Vol. 8, No. 2, 1974, pp. 433–437.   
+8.5 D. J. Allman, "A Compatible Triangular Element Including Vertex Rotations for Plane Elasticity Analysis," Computers & Structures, Vol. 19, No. 1-2, 1984, pp. 1-8.   
+8.6 R. D. Cook, "On the Allman Triangle and a Related Quadrilateral Element," Computers & Structures, Vol. 22, No. 6, 1986, pp. 1065-1067.   
+8.7 P. Tong and T. H. H. Pian, “A Variational Principle and the Convergence of a Finite-Element Method Based on Assumed Stress Distribution,” Int. J. Solids Structures, Vol. 5, No. 5, 1969, pp. 463–472.   
+8.8 T. H. H. Pian, “Derivation of Element Stiffness Matrices by Assumed Stress Functions,” AIAA Jnl., Vol. 2, No. 7, 1964, pp. 1333–1336 (discussion: Vol. 3, No. 1, 1965, pp. 186–187).   
+8.9 J. P. Wolf, "Alternate Hybrid Stress Finite Element Models," Int. J. Num. Meth. Engng., Vol. 9, No. 3, 1975, pp. 601-615.   
+8.10 T. H. H. Pian and K. Sumihara, "Rational Approach for Assumed Stress Finite Elements," Int. J. Num. Meth. Engng., Vol. 20, No. 9, 1984, pp. 1685–1695.   
+8.11 R. D. Cook, “A Plane Hybrid Element with Rotational D.O.F. and Adjustable Stiffness,” Int. J. Num. Meth. Engng., Vol. 24, No. 8, 1987, pp. 1499–1508.   
+8.12 M. F. Kanninen and C. H. Popelar, Advanced Fracture Mechanics, Oxford University Press, New York, 1985.   
+8.13 D. P. Rooke and D. J. Cartwright, Compendium of Stress Intensity Factors, Her Majesty's Stationary Office, London, 1976.   
+8.14 R. S. Barsoum, "On the Use of Isoparametric Finite Elements in Linear Fracture Mechanics," Int. J. Num. Meth. Engng., Vol. 10, No. 1, 1976, pp. 25-37.   
+8.15 R. S. Barsoum, "Letter to the Editor," Int. J. Num. Meth. Engng., Vol. 18, No. 9, 1982, pp. 1420–1422.   
+8.16 L. P. Harrop, "The Optimum Size of Quarter-Point Crack Tip Elements," Int. J. Num. Meth. Engng., Vol. 18, No. 7, 1982, pp. 1101-1103.   
+8.17 N. A. B. Yahia and M. S. Shephard, "On the Effect of Quarter-Point Element Size on Fracture Criteria," Int. J. Num. Meth. Engng., Vol. 21, No. 10, 1985, pp. 1911–1924.   
+8.18 V. E. Saouma and D. Schwemmer, "Numerical Evaluation of the Quarter-Point Crack Tip Element," Int. J. Num. Meth. Engng., Vol. 20, No. 9, 1984, pp. 1629–1641.   
+8.19 D. M. Parks and E. M. Kamenetzky, "Weight Functions From Virtual Crack Extension," Int. J. Num. Meth. Engng., Vol. 14, No. 11, 1979, pp. 1693–1706.
+
+<!-- source-page: 629 -->
+
+8.20 A. D. Kerr, "Elastic and Viscoelastic Foundation Models," J. Appl. Mech., Vol. 31, No. 3, 1964, pp. 491-498.   
+8.21 M. S. Cheung, "A Simplified Finite Element Solution for the Plates on Elastic Foundation," Computers & Structures, Vol. 8, No. 1, 1978, pp. 139–145.   
+8.22 Z. Feng and R. D. Cook, "Beam Elements on Two-Parameter Elastic Foundations," J. of Engng. Mech., Vol. 109, No. 6, 1983, pp. 1390-1402.   
+8.23 O. C. Zienkiewicz, P. Bettess, T. C. Chaim, and C. Emson, “Numerical Methods for Unbounded Field Problems and a New Infinite Element Formulation,” in Computational Methods for Infinite Domain Media Structure Interaction, A. J. Kalinowski, ed., Am. Soc. Mech. Engrs., New York, 1981.   
+8.24 P. Bettess and J. A. Bettess, "Infinite Elements for Static Problems," Engineering Computations, Vol. 1, No. 1, 1984, pp. 4–16.   
+8.25 J. M. M. C. Marques and D. R. J. Owen, "Infinite Elements in Quasi-Static Materially Nonlinear Problems," Computers & Structures, Vol. 18, No. 4, 1984, pp. 739–751.   
+8.26 R. T. Fenner, “The Boundary Integral Equation (Boundary Element) Method in Engineering Stress Analysis,” J. of Strain Analysis, Vol. 18, No. 4, 1983, pp. 199–205.   
+8.27 J. Mackerle and T. Andersson, "Boundary Element Software in Engineering," Advances in Engng. Software, Vol. 6, No. 2, 1984, pp. 66–102 (lists 656 references).   
+8.28 S. J. Fenves et al., eds., Numerical and Computer Methods in Structural Mechanics, Academic Press, New York, 1973 (see papers by D. Bushnell, pp. 291–336, and by S. W. Key and R. D. Krieg, pp. 337–352).   
+8.29 J. S. Arora, "Survey of Reanalysis Techniques," J. Struct. Div., Proc. ASCE, Vol. 102, No. ST4, 1976, pp. 783-802 (lists 89 references).   
+8.30 U. Kirsch, Optimum Structural Design, McGraw-Hill, New York, 1981.   
+8.31 J. S. Przemienicki, "Matrix Structural Analysis of Substructures," AIAA Jnl., Vol. 1, No. 1, 1963, pp. 138–147.   
+8.32 A. K. Noor, H. A. Kamel and R. E. Fulton, “Substructuring Techniques—Status and Projections,” Computers & Structures, Vol. 8, No. 5, 1978, pp. 621–632.   
+3.33 P. G. Glockner, "Symmetry in Structural Mechanics," J. Struct. Div., Proc. ASCE, Vol. 99, No. ST1, 1973, pp. 71–89.   
+3.34 A. K. Noor and R. A. Camin, "Symmetry Considerations for Anisotropic Shells," Comp. Meth. Appl. Mech. Engng., Vol. 9, No. 3, 1976, pp. 317-335.   
+3.35 P. D. Mangalgiri, B. Dattaguru, and T. S. Ramamurthy, "Specification of Skew Conditions in Finite Element Formulation," Int. J. Num. Meth. Engng., Vol. 12, No. 6, 1978, pp. 1037–1041.   
+3.36 O. C. Zienkiewicz and F. C. Scott, “On the Principle of Repeatability and Its Application in Analysis of Turbine and Pump Impellers,” Int. J. Num. Meth. Engng., Vol. 4, No. 3, 1972, pp. 445–448.   
+.37 R. H. MacNeal and R. L. Harder, "A Refined Four-Noded Membrane Element with Rotational Degrees of Freedom," Computers & Structures, Vol. 28, No. 1, 1988, pp. 75–84.
+
+# CHAPTER 9
+
+9.1 J. F. Abel and M. S. Shephard, "An Algorithm for Multipoint Constraints in Finite Element Analysis," Int. J. Num. Meth. Engng., Vol. 14, No. 3, 1979, pp. 464-467.   
+9.2 O. C. Zienkiewicz, The Finite Element Method, 3rd Ed., McGraw-Hill, London, 1977.   
+9.3 C. A. Felippa, "Iterative Procedures for Improving Penalty Function Solutions of Algebraic Systems," Int. J. Num. Meth. Engng., Vol. 12, No. 5, 1978, pp. 821-836.   
+9.4 T. J. R. Hughes, R. L. Taylor, and W. Kanoknukulchai, "A Simple and Efficient
+
+<!-- source-page: 630 -->
+
+Finite Element for Plate Bending," Int. J. Num. Meth. Engng., Vol. 11, No. 10, 1977, pp. 1529–1543.   
+9.5 D. S. Malkus, Finite Element Analysis of Incompressible Solids, Ph.D. Dissertation, Boston University, Boston, 1976.   
+9.6 D. S. Malkus and T. J. R. Hughes, "Mixed Finite Element Methods-Reduced and Selective Integration Techniques: A Unification of Concepts," Comp. Meth. Appl. Mech. Engng., Vol. 15, No. 1, 1978, pp. 68–81.   
+9.7 T. J. R. Hughes, A Course in the Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1987.   
+9.8 R. L. Spilker and N. I. Munir, "A Hybrid Stress Quadratic Serendipity Displacement Mindlin Plate Element," Computers & Structures, Vol. 12, No. 1, 1980, pp. 11–21.   
+9.9 Y. C. Fung, Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, NJ, 1965.   
+9.10 M. Engelman, R. L. Sani, P. M. Gresho, and M. Bercovier, "Consistent vs. Reduced Integration Penalty Methods for Incompressible Media Using Several Old and New Elements," Int. J. for Num. Meth. in Fluids, Vol. 2, No. 1, 1982, pp. 25–42.   
+9.11 C. Johnson and J. Pitkäranta, "Analysis of Some Mixed Finite Element Methods Related to Reduced Integration," Mathematics of Computation, Vol. 38, No. 158, 1982, pp. 375-400.   
+9.12 E. T. Olsen, Stable Finite Elements for Non-Newtonian Flows; First Order Elements Which Fail the LBB Condition, Ph.D. Dissertation, Illinois Institute of Technology, Chicago, 1983.   
+9.13 J. Pitkäranta and R. Stenberg, "Error Bounds for the Approximation of the Stokes Problem Using Bilinear/Constant Elements on Irregular Quadrilateral Meshes," Report MAT-A222, Helsinki University of Technology, Espoo, Finland, 1984.   
+9.14 T. J. R. Hughes, W. K. Liu, and A. Brooks, "Finite Element Analysis of Incompressible Viscous Flows by the Penalty Function Formulation," J. of Comp. Phys., Vol. 30, No. 1, 1979, pp. 1–60.
+
+# CHAPTER 10
+
+10.1 E. L. Wilson, "Structural Analysis of Axisymmetric Solids," AIAA Jnl., Vol. 3, No. 12, 1965, pp. 2269–2274.   
+10.2 J. G. Close and R. M. Jones, SAAS III: Finite Element Analysis of Axisymmetric and Plane Solids with Different Orthotropic, Temperature-Dependent Material Properties in Tension and Compression, Aerospace Corp., San Bernardino, CA, 1971 (AD-729-188, N.T.I.S.).   
+10.3 O. C. Zienkiewicz and Y. K. Cheung, "Stresses in Shafts," The Engineer (London), Vol. 224, No. 5835, 1967, pp. 696–697.   
+10.4 T. Belytschko, "Finite Elements for Axisymmetric Solids Under Arbitrary Loadings with Nodes on Origin," AIAA Jnl., Vol. 10, No. 11, 1972, pp. 1532-1533 (discussion and closure: Vol. 11, No. 9, pp. 1357-1358).   
+10.5 J. Padovan, "Quasi-Analytical Finite Element Procedures for Axisymmetric Anisotropic Shells and Solids," Computers & Structures, Vol. 4, No. 3, 1974, pp. 467–483.   
+10.6 J. G. Crose, "Stress Analysis of Axisymmetric Solids with Asymmetric Properties," AIAA Jnl., Vol. 10, No. 7, 1972, pp. 866-871.   
+10.7 M. Sedaghat and L. R. Herrmann, "A Nonlinear, Semi-Analytical Finite Element Analysis for Nearly Axisymmetric Solids," Computers & Structures, Vol. 17, No. 3, 1983, pp. 389–401.   
+10.8 E. L. Wilson and P. C. Pretorius, "A Computer Program for the Analysis of
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+<!-- source-page: 631 -->
+
+Prismatic Solids," Report UC-SESM-70-21, Civil Engineering Department, University of California, Berkeley, 1970 (PB-196-462, N.T.I.S.).   
+10.9 O. C. Zienkiewicz and J. M. Too, "The Finite Prism in Analysis of Thick Simply Supported Bridge Boxes," Proc. Inst. Civil Engrs., Vol. 53, Part 2, 1972, pp. 147–172.   
+10.10 G. A. Greenbaum, L. D. Hofmeister, and D. A. Evenson, "Pure Moment Loading of Axisymmetric Finite Element Models," Int. J. Num. Meth. Engng., Vol. 5, No. 4, 1973, pp. 459–463.   
+10.11 K. J. Bathe and C. A. Almeida, "A Simple and Effective Pipe Elbow Element-Linear Analysis," J. Appl. Mech., Vol. 47, No. 1, 1980, pp. 93–100.   
+10.12 K. J. Han and P. L. Gould, "Line Node and Transitional Shell Element for Rotational Shells," Int. J. Num. Meth. Engng., Vol. 18, No. 6, 1982, pp. 879–895.   
+10.13 W. M. Chen and P. Tsai, "The Combined Use of Axisymmetric and Solid Elements for Three Dimensional Stress Analysis," Computers & Structures, Vol. 18, No. 4, 1984, pp. 689–694.
+
+# CHAPTER 11
+
+11:1 S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd Ed., McGraw-Hill, New York, 1959.   
+11.2 F. J. Plantema, Sandwich Construction, John Wiley & Sons, New York, 1966.   
+11.3 J. M. Whitney and A. W. Leissa, "Analysis of Heterogeneous Anisotropic Plates," J. Appl. Mech., Vol. 36, No. 2, 1969, pp. 261-266.   
+11.4 M. M. Hrabok and T. M. Hrudey, "A Review and Catalog of Plate Bending Finite Elements," Computers & Structures, Vol. 19, No. 3, 1984, pp. 479–495.   
+11.5 R. H. Gallagher, Finite Element Analysis: Fundamentals, Prentice-Hall, Englewood Cliffs, NJ, 1975.   
+11.6 R. J. Melosh, “Basis for Derivation of Matrices for the Direct Stiffness Method,” AIAA Jnl., Vol. 1, No. 7, 1963, pp. 1631–1637 (discussion: Vol. 2, No. 2, 1964, p. 403; Vol. 2, No. 6, 1964, p. 1161; Vol. 3, No. 6, 1965, pp. 1215–1216).   
+11.7 Y. K. Cheung, Finite Strip Method in Structural Analysis, Pergamon Press, Oxford, England, 1976.   
+11.8 T. J. R. Hughes and M. Cohen, "The 'Heterosis' Finite Element for Plate Bending," Computers & Structures, Vol. 9, No. 5, 1978, pp. 445–450.   
+11.9 E. Hinton and H. C. Huang, "A Family of Quadrilateral Mindlin Plate Elements with Substitute Shear Strain Fields," Computers & Structures, Vol. 23, No. 3, 1986, pp. 409–431.   
+11.10 J. A. Stricklin et al., "A Rapidly Converging Triangular Plate Element," AIAA Jnl., Vol. 7, No. 1, 1969, pp. 180–181.   
+11.11 J. L. Batoz, K. J. Bathe, and L. W. Ho, “A Study of Three-Node Triangular Plate Bending Elements,” Int. J. Num. Meth. Engng., Vol. 15, No. 12, 1980, pp. 1771–1812.   
+11.12 J. L. Batoz, “An Explicit Formulation for an Efficient Triangular Plate Bending Element,” Int. J. Num. Meth. Engng., Vol. 18, No. 7, 1982, pp. 1077–1089.   
+11.13 C. Jeyachandrabose and J. Kirkhope, “An Alternative Formulation for the DKT Plate Bending Element,” Int. J. Num. Meth: Engng., Vol. 21, No. 7, 1985, pp. 1289–1293.   
+11.14 T. J. R. Hughes, M. Cohen, and M. Haroun, "Reduced and Selective Integration Techniques in the Finite Element Analysis of Plates," Nucl. Engng. Design, Vol. 46, No. 1, 1978, pp. 203-222.   
+11.15 M. P. Rossow, "Efficient $C^0$ Finite-Element Solutions of Simply Supported Plates of Polygonal Shape," J. Appl. Mech., Vol. 44, No. 2, 1977, pp. 347-349.   
+11.16 T. J. R. Hughes and T. E. Tezduyar, "Finite Elements Based on Mindlin Plate
+
+<!-- source-page: 632 -->
+
+Theory with Particular Reference to the Four-Node Bilinear Isoparametric Element," J. Appl. Mech., Vol. 48, No. 3, 1981, pp. 587–596.   
+11.17 J. Donea and L. G. Lamain, "A Modified Representation of Transverse Shear in $C^0$ Quadrilateral Plate Elements," Comp. Meth. Appl. Mech. Engng., Vol. 63, No. 2, 1987, pp. 183-207.
+
+# CHAPTER 12
+
+12.1 F. Kikuchi and K. Tanizawa, "Accuracy and Locking-Free Property of the Beam Element Approximation for Arch Problems," Computers & Structures, Vol. 19, No. 1-2, 1984, pp. 103-110.   
+12.2 G. Prathap, "The Curved Beam/Deep Arch/Finite Ring Element Revisited," Int. J. Num. Meth. Engng., Vol. 21, No. 3, 1985, pp. 389–407.   
+12.3 H. R. Meck, "An Accurate Polynomial Displacement Function for Finite Ring Elements," Computers & Structures, Vol. 11, No. 4, 1980, pp. 265-269.   
+12.4 P. M. Mebane and J. A. Stricklin, "Implicit Rigid Body Motion in Curved Elements," AIAA Jnl., Vol. 9, No. 2, 1971, pp. 344–345.   
+12.5 G. Prathap and C. R. Babu, "An Isoparametric Quadratic Thick Curved Beam Element," Int. J. Num. Meth. Engng., Vol. 23, No. 9, 1986, pp. 1583–1600.   
+12.6 S. F. Pawsey and R. W. Clough, "Improved Numerical Integration of Thick Shell Finite Elements," Int. J. Num. Meth. Engng., Vol. 3, No. 4, 1971, pp. 575-586.   
+12.7 M. A. Crisfield, "Explicit Integration and the Isoparametric Arch and Shell Elements," Comm. in Appl. Num. Meth., Vol. 2, No. 2, 1986, pp. 181-187.   
+12.8 N. Carpenter, H. Stolarski, and T. Belytschko, "Improvements in 3-Node Triangular Shell Elements," Int. J. Num. Meth. Engng., Vol. 23, No. 9, 1986, pp. 1643–1667.   
+12.9 O. C. Zienkiewicz, The Finite Element Method, 3rd Ed., McGraw-Hill, London, 1977.   
+12.10 C. R. Babu and G. Prathap, "A Field-Consistent Two-Noded Axisymmetric Shell Element," Int. J. Num. Meth. Engng., Vol. 23, No. 7, 1986, pp. 1245–1261.   
+12.11 A. Tessler, "An Efficient, Conforming Axisymmetric Shell Element Including Transverse Shear and Rotary Inertia," Computers & Structures, Vol. 15, No. 5, 1982, pp. 567–574.   
+12.12 B. M. Irons and A. Razzaque, "Further Modification to Ahmad's Shell Element," Int. J. Num. Meth. Engng., Vol. 5, No. 4, 1973, pp. 588–589.   
+12.13 T. Belytschko et al., "Implementation and Application of a 9-Node Lagrange Shell Element With Spurious Mode Control," Computers & Structures, Vol. 20, No 1-3, 1985, pp. 121-128.   
+12.14 V. T. Nicholas and E. Citipitioglu, "A General Isoparametric Finite Element Program SDRC SUPERB," Computers & Structures, Vol. 7, No. 2, 1977, pp. 303–313   
+12.15 R. H. MacNeal and R. L. Harder, “A Proposed Standard Set of Problems to Tes Finite Element Accuracy,” Finite Elements in Analysis and Design, Vol. 1, No 1, 1985, pp. 3–20.   
+12.16 S. S. Murthy and R. H. Gallagher, "Patch Test Verification of a Triangular Thin Shell Element Based on Discrete Kirchhoff Theory," Comm. in Appl. Num. Meth. Vol. 3, No. 2, 1987, pp. 83–88.   
+12.17 H. Stolarski and T. Belytschko, "Membrane Locking and Reduced Integration for Curved Elements," J. Appl. Mech., Vol. 49, No. 1, 1982, pp. 172–176.
+
+# CHAPTER 13
+
+13.1 R. W. Clough and J. Penzien, Dynamics of Structures, McGraw-Hill, New York, 1975.
+
+<!-- source-page: 633 -->
+
+13.2 R. R. Craig, Jr., Structural Dynamics, John Wiley & Sons, New York, 1981.   
+13.3 J. W. S. Rayleigh, Theory of Sound, 2nd Ed., Vols. I and II, Dover Publications, New York, 1945 (originally published in 1894).   
+13.4 J. S. Archer, "Consistent Matrix Formulations for Structural Analysis Using Finite Element Techniques," AIAA Jnl., Vol. 3, No. 10, 1965, pp. 1910–1918.   
+13.5 E. Hinton, T. Rock, and O. C. Zienkiewicz, "A Note on Mass Lumping and Related Processes in the Finite Element Method," Earthquake Engng. Struct. Dynamics, Vol. 4, No. 3, 1976, pp. 245–249.   
+13.6. K. S. Surana, "Lumped Mass Matrices with Non-Zero Inertia for General Shell and Axisymmetric Shell Elements," Int. J. Num. Meth. Engng., Vol. 12, No. 11, 1978, pp. 1635–1650.   
+13.7 D. S. Malkus and M. E. Plesha, “Zero and Negative Masses in Finite Element Vibration and Transient Analysis,” Comp. Meth. Appl. Mech. Engng., Vol. 59, No. 3, 1986, pp. 281–306.   
+13.8 D. S. Malkus, M. E. Plesha, and M. R. Liu, “Reversed Stability Conditions in Transient Finite Element Analysis,” Comp. Meth. Appl. Mech. Engng., Vol. 68, No. 1, 1988, pp. 97–114.   
+13.9 G. J. Fix, "Effects of Quadrature Errors in Finite Element Approximation of Steady State Eigenvalue and Parabolic Problems," in Mathematical Foundations of the Finite Element Method, I. Babuska and A. K. Aziz, eds., Academic Press, New York, 1972, pp. 525-556.   
+13.10 Z. Kopal, Numerical Analysis, John Wiley & Sons, New York, 1955.   
+13.11 S. H. Crandall, Engineering Analysis, McGraw-Hill, New York, 1956.   
+13.12 E. Isaacson and H. B. Keller, Analysis of Numerical Methods, John Wiley & Sons, New York, 1966.   
+3.13 E. Dokumaci, "A Critical Examination of Discrete Models in Vibration Problems of Continuous Systems," J. Sound Vibration, Vol. 53, No. 2, 1977, pp. 153–164.   
+3.14 T. Belytschko, "A Survey of Numerical Methods and Computer Programs for Dynamic Structural Analysis, Nucl. Engng. Design, Vol. 37, No. 1, 1976, pp. 23-34.   
+3.15 R. H. MacNeal, ed., The NASTRAN Theoretical Manual (Level 16.0), NASA-SP-221(03), march 1976 (N79-27531, N.T.I.S.).   
+3.16 T. Belytschko and W. L. Mindle, "The Treatment of Damping in Transient Computations," in Damping Applications for Vibration Control, P. J. Torvik, ed., ASME AMD, Vol. 38, 1980, pp. 123–132.   
+3.17 M. E. Plesha, "Mixed Time Integration for the Transient Analysis of Jointed Media," Int. J. Num. An. Meth. Geomech., Vol. 10, No. 1, 1986, pp. 91-110.   
+3.18 K. J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1982.   
+3.19 L. Fox, An Introduction to Numerical Linear Algebra, Oxford University Press, New York, 1965.   
+3.20 W. C. Hurty and M. F. Rubinstein, Dynamics of Structures, Prentice-Hall, Englewood Cliffs, NJ, 1964.   
+3.21 O. C. Zienkiewicz, R. W. Lewis, and K. G. Stagg, Numerical Methods in Offshore Engineering, John Wiley & Sons, Chichester, England, 1978.   
+3.22 N. R. Maddox, "On the Number of Modes Necessary for Accurate Response and Resulting Forces in Dynamic Analysis," J. Appl. Mech., Vol. 42, No. 2, 1975, pp. 516–517.   
+3.23 O. E. Hansteen and K. Bell, "On the Accuracy of Mode Superposition Analysis in Structural Dynamics," Earthquake Engng. Struct. Dynamics, Vol. 7, No. 5, 1979, pp. 405–411.   
+3.24 R. E. Cornwell, R. R. Craig, Jr., and C. P. Johnson, “On the Application of the Mode-Acceleration Method to Structural Engineering Problems,” Earthquake Engng. Struct. Dynamics, Vol. 11, No. 5, 1983, pp. 679–688.   
+3.25 E. L. Wilson, M. Y. Yuan, and J. M. Dickens, "Dynamic Analysis by Direct Superposition of Ritz Vectors," Earthquake Engng. Struct. Dynamics, Vol. 10, No. 6, 1982, pp. 813–821.
+
+<!-- source-page: 634 -->
+
+13.26 V. N. Shah, G. J. Bohm, and A. N. Nahavandi, "Modal Superposition Method for Computationally Economical Nonlinear Structural Analysis," J. Pressure Vessel Tech., ASME, Vol. 101, No. 2, 1979, pp. 134–141.   
+13.27 J. A. Stricklin and W. E. Haisler, "Formulations and Solution Procedures for Nonlinear Structural Analysis," Computers & Structures, Vol. 7, No. 1, 1977, pp. 125–136.   
+13.28 K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1976.   
+13.29 R. J. Guyan, "Reduction of Stiffness and Mass Matrices," AIAA Jnl., Vol. 3, No. 2, 1965, p. 380.   
+13.30 I. U. Ojalvo, "Computer Methods for Determining Vibration Modes of Complex Structures," Shock and Vibration Digest, Vol. 5, No. 5, May, 1983, pp. 3–10.   
+13.31 C. A. Felippa, "Refined Finite Element Analysis of Linear and Nonlinear Two-Dimensional Structures," Ph.D. dissertation, University of California, Berkeley, 1966 (also available as PB-178-418 and PB-178-419, N.T.I.S.; computer programs in the latter).   
+13.32 J. D. Sowers, "Condensation of Free Body Mass Matrices Using Flexibility Coefficients," AIAA Jnl., Vol. 16, No. 3, 1978, pp. 272-273.   
+13.33 M. Geradin, "Error Bounds for Eigenvalue Analysis by Elimination of Variables," J. Sound Vibration, Vol. 19, No. 2, 1971, pp. 111-132.   
+13.34 R. L. Kidder, "Reduction of Structural Frequency Equations," AIAA Jnl., Vol. 11, No. 6, 1973, p. 892 (discussion and closure: Vol. 13, No. 5, 1975, pp. 701–703).   
+13.35 V. B. Watwood, T. Y. Chow, Z. Zudans, and W. H. Miller, "Combined Analysis and Evaluation of Piping Systems Using the Computer," Nucl. Engng. Design, Vol. 27, No. 3, 1974, pp. 334–342.   
+13.36 R. D. Henshell and J. H. Ong, "Automatic Masters for Eigenvalue Economization," Earthquake Engng. Struct. Dynamics, Vol. 3, No. 4, 1975, pp. 375–383.   
+13.37 R. G. Anderson, B. M. Irons, and O. C. Zienkiewicz, "Vibration and Stability of Plates Using Finite Elements," Int. J. Solids Structures, Vol. 4, No. 10, 1968, pp. 1031–1055.   
+13.38 K. W. Matta, "Selection of Degrees of Freedom for Dynamic Analysis," J. Pressure Vessel Tech., Vol. 109, No. 1, 1987, pp. 65–69.   
+13.39 W. C. Hurty, "Dynamic Analysis of Structural Systems Using Component Modes," AIAA Jnl., Vol. 3, No. 4, 1965, pp. 678–685.   
+13.40 R. R. Craig, Jr., "A Review of Time-Domain and Frequency Domain Component Mode Synthesis Methods," Int. J. Analytical and Experimental Modal Analysis, Vol. 2, No. 2, 1987, pp. 59–72.   
+13.41 R. H. MacNeal, "A Hybrid Method of Component Mode Synthesis," Computers & Structures, Vol. 1, No. 4, 1971, pp. 581–601.   
+13.42 S. Rubin, "An Improved Component-Mode Representation," AIAA/ASME/SAE 15th Structures, Structural Dynamics and Materials Conference, Las Vegas, Nevada, 1974, pp. 17-19.   
+13.43 R. R. Craig, Jr. and M. C. C. Bampton, "Coupling of Substructures for Dynamic Analysis," AIAA Jnl., Vol. 6, No. 7, 1968, pp. 1313–1319.   
+13.44 V. N. Shah and M. Raymund, "Analytical Selection of Masters for the Reduced Eigenvalue Problem," Int. J. Num. Meth. Engng., Vol. 18, No. 1, pp. 89–98, 1982.   
+13.45 M. Paz, Structural Dynamics: Theory and Computation, 2nd ed., Van Nostrand Reinhold, New York, 1984.   
+13.46 T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1987.   
+13.47 R. D. Richtmyer and K. W. Morton, Difference Methods for Initial Value Problems, Interscience, New York, 1967.   
+13.48 T. J. R. Hughes and T. Belytschko, "A Précis of Developments in Computational Methods for Transient Analysis," J. Appl. Mech., Vol. 50, No. 4b, 1983, pp. 1033–1041.
+
+<!-- source-page: 635 -->
+
+13.49 T. Belytschko and T. J. R. Hughes, eds., Computational Methods for Transient Analysis, North Holland, Amsterdam, 1983.   
+13.50 R. Courant, K. O. Friedrichs, and H. Lewy, “Uber die partiellen Differenzengleichungen der mathematischen Physik,” Math. Ann., Vol. 100, 1928, pp. 32–74. (Translated by P. Fox: “On the Partial Differential Equations of Mathematical Physics,” New York University, Courant Institute of Mathematical Sciences, Report NYO-7689, 1956.)   
+13.51 M. E. Plesha, "Eigenvalue Estimation for Dynamic Contact Problems," J. of Engng. Mech., Vol. 113, No. 3, 1987, pp. 457-462.   
+13.52 D. D. Flanagan and T. Belytschko, "A Uniform Strain Hexahedron and Quadrilateral with Orthogonal Hourglass Control," Int. J. Num. Meth. Engng., Vol. 17, No. 5, 1981, pp. 679–706. (Also see Errata, Vol. 19, No. 3, 1983, pp. 467–468.)   
+13.53 T. Belytschko, J. S. J. Ong, W. K. Liu, and J. M. Kennedy, "Hourglass Control in Linear and Nonlinear Problems," Comp. Meth. Appl. Mech. Engng., Vol. 43, No. 3, 1984, pp. 251–276.   
+13.54 W. K. Liu and T. Belytschko, "Efficient Linear and Nonlinear Heat Conduction with a Quadrilateral Element," Int. J. Num. Meth. Engng., Vol. 20, No. 5, 1984, pp. 931–948.   
+13.55 J. C. Houbolt, "A Recurrence Matrix Solution for the Dynamic Response of Elastic Aircraft," J. Aero. Sci., Vol. 17, No. 9, 1950, pp. 540–550.   
+13.56 N. M. Newmark, “A Method of Computation for Structural Dynamics,” J. Engng. Mech. Div., Proc. ASCE, Vol. 85, No. EM3, 1959, pp. 67–94.   
+13,57 E. L. Wilson, “A Computer Program for the Dynamic Stress Analysis of Underground Structures,” SESM Report No. 68-1, Division of Structural Engineering and Structural Mechanics, University of California, Berkeley, 1968.   
+13.58 H. M. Hilber, T. J. R. Hughes, and R. L. Taylor, “Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics,” Earthquake Engng. Struct. Dynamics, Vol. 5, No. 3, 1977, pp. 283–292.   
+13.59 T. Belytschko and R. Mullen, “Mesh Partitions of Explicit-Implicit Time Integration,” in Formulation and Computational Algorithms in Finite Element Analysis, K. J. Bathe, J. T. Oden, and W. Wunderlich, eds., MIT Press, Cambridge, MA, 1977, pp. 673–690.   
+13.60 T. Belytschko and R. Mullen, "Stability of Explicit-Implicit Mesh Partitions in Time Integration," Int. J. Num. Meth. Engng., Vol. 12, No. 10, 1978, pp. 1575–1586.   
+13.61 T. J. R. Hughes and W. K. Liu, "Implicit-Explicit Finite Elements in Transient Analysis: Stability Theory," J. Appl. Mech., Vol. 45, No. 2, 1978, pp. 371-374.   
+13.62 T. J. R. Hughes and W. K. Liu, "Implicit-Explicit Finite Elements in Transient Analysis: Implementation and Numerical Examples," J. Appl. Mech., Vol. 45, No. 2, 1978, pp. 375–378.   
+13.63 T. J. R. Hughes, K. S. Pister, and R. L. Taylor, "Implicit-Explicit Finite Elements in Nonlinear Transient Analysis," Comp. Meth. Appl. Mech. Engng., Vol. 17/18, Part I, 1979, pp. 159–182.   
+13.64 T. J. R. Hughes, W. K. Liu, and A. Brooks, "Finite Element Analysis of Incompressible Viscous Flows by the Penalty Function Formulation," J. of Comp. Phys., Vol. 30, No. 1, 1979, pp. 1-60.   
+13.65 M. Ortiz, P. M. Pinsky, and R. L. Taylor, "Unconditionally Stable Element-by-Element Algorithms for Dynamic Problems," Comp. Meth. Appl. Mech. Engng., Vol. 36, No. 2, 1983, pp. 223-239.   
+13.66 T. J. R. Hughes, I. Levit, and J. Winget, “Unconditionally Stable Element-by-Element Implicit Algorithm for Heat Conduction Analysis,” J. of Engng. Mech., Vol. 109, No. 2, 1983, pp. 576–585.   
+13.67 T. J. R. Hughes, I. Levit, and J. Winget, "An Element-by-Element Solution Algorithm for Problems of Structural and Solid Mechanics," Comp. Meth. Appl. Mech. Engng., Vol. 36, No. 2, 1983, pp. 241–254.   
+13.68 I. Fried and D. S. Malkus, "Finite Element Mass Matrix Lumping by Numerical
+
+<!-- source-page: 636 -->
+
+Integration With No Convergence Rate Loss," Int. J. Solids Structures, Vol. 11, No. 4, 1975, pp. 461–466.   
+13.69 G. M. Hulbert and T. J. R. Hughes, “An Error Analysis of Truncated Starting Conditions in Step-by-Step Time Integration: Consequences for Structural Dynamics,” Earthquake Engng. Struct. Dynamics, Vol. 15, No. 7, 1987, pp. 901–910.   
+13.70 R. A. Brockman, "Dynamics of the Bilinear Mindlin Plate Element," Int. J. Num. Meth. Engng., Vol. 24, No. 12, 1987, pp. 2343–2356.
+
+# CHAPTER 14
+
+14.1 S. C. Chang and J. J. Chen, "Effectiveness of Linear Bifurcation Analysis for Predicting the Nonlinear Stability Limits of Structures," Int. J. Num. Meth. Engng., Vol. 23, No. 5, 1986, pp. 831–846.   
+14.2 R. H. Gallagher and J. Padlog, "Discrete Element Approach to Structural Stability Analysis," AIAA Jnl., Vol. 1, No. 6, 1963, pp. 1437-1439.   
+14.3 E. Chwalla, "Second Order Theory," in Handbook of Engineering Mechanics, W. Flügge, ed., McGraw-Hill, New York, 1962.   
+14.4 S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability, 2nd Ed., McGraw-Hill, New York, 1961.   
+14.5 J. S. Przemieniecki, "Discrete-Element Methods for Stability Analysis," Aeronautical J., Vol. 72, No. 12, 1968, pp. 1077–1086.   
+14.6 R. A. Tinawi, "Anisotropic Tapered Elements Using Displacement Models," Int. J. Num. Meth. Engng., Vol. 4, No. 4, 1972, pp. 475-489.   
+14.7 R. D. Cook, "Finite Element Buckling Analysis of Homogeneous and Sandwich Plates," Int. J. Num. Meth. Engng., Vol. 9, No. 1, 1975, pp. 39–50.   
+14.8 R. S. Barsoum and R. H. Gallagher, "Finite Element Analysis of Torsional and Torsional-Flexural Stability Problems," Int. J. Num. Meth. Engng., Vol. 2, No. 3, 1970, pp. 335–352.   
+14.9 R. H. Gallagher and C. H. Lee, "Matrix Dynamic and Instability Analysis with Nonuniform Elements," Int. J. Num. Meth. Engng., Vol. 2, No. 2, 1970, pp. 265-275.   
+14.10 R. S. Barsoum, "Finite Element Method Applied to the Problem of Stability of a Non-Conservative System," Int. J. Num. Meth. Engng., Vol. 3, No. 1, 1971, pp. 63–87.   
+14.11 D. Bushnell, "Analysis of Ring-Stiffened Shells of Revolution Under Combined Thermal and Mechanical Loadings," AIAA Jnl., Vol. 9, No. 3, 1971, pp. 401-410.   
+14.12 D. R. Navaratna, T. H. H. Pian, and E. A. Witmer, "Analysis of Elastic Stability of Shells of Revolution by the Finite Element Method," AIAA Jnl., Vol. 6, No. 2, 1968, pp. 355–361.   
+14.13 A. D. Kerr and M. T. Soifer, "The Linearization of the Prebuckling State and Its Effect on the Determined Stability Loads," J. Appl. Mech., Vol. 36, No. 4, 1969, pp. 775–783.   
+14.14 D. Bushnell, "Buckling of Shells—Pitfall for Designers," AIAA Jnl., Vol. 19, No. 9, 1981, pp. 1183–1226.   
+14.15 D. Bushnell, "Computerized Analysis of Shells—Governing Equations," Computers & Structures, Vol. 18, No. 3, 1984, pp. 471–536.
+
+# CHAPTER 15
+
+15.1 S. H. Crandall, Engineering Analysis, McGraw-Hill, New York, 1956.
+
+15.2 B. A. Finlayson and L. E. Scriven, "The Method of Weighted Residuals—A Review," Applied Mechanics Reviews, Vol. 19, No. 9, 1966, pp. 735–748.
+
+<!-- source-page: 637 -->
+
+15.3 E. D. Eason, "A Review of Least-Squares Methods for Solving Partial Differential Equations," Int. J. Num. Meth. Engng., Vol. 10, No. 5, 1976, pp. 1021-1046.   
+15.4 W. L. Kwok, Y. K. Cheung, and C. Delcourt, “Application of Least Squares Collocation Technique in Finite Element and Finite Strip Formulation,” Int. J. Num. Meth. Engng., Vol. 11, No. 9, 1977, pp. 1391–1404.   
+15.5 M. F. N. Mohsen, "Some Details of the Galerkin Finite Element Method," Appl. Math. Modelling, Vol. 6, No. 3, 1982, pp. 165-170.   
+15.6 B. A. Finlayson, The Method of Weighted Residuals and Variational Principles, Academic Press, New York, 1972.   
+15.7 P. C. M. Lau and C. A. Brebbia, "The Cell Collocation Method in Continuum Mechanics," Int. J. Mech. Sci., Vol. 20, No. 2, 1978, pp. 83–95.   
+15.8 R. H. Gallagher, J. A. Liggett, and S. T. K. Chan, “Finite Element Shallow Lake Circulation Analysis,” J. Hydraulics Div., Proc. ASCE, Vol. 99, No. HY7, 1973, pp. 1083–1096.
+
+# CHAPTER 16
+
+16.1 E. L. Wilson, K. J. Bathe and F. E. Peterson, "Finite Element Analysis of Linear and Nonlinear Heat Transfer," Nucl. Engng. Design, Vol. 29, No. 1, 1974, pp. 110–124.   
+16.2 J. F. Lyness, D. R. J. Owen, and O. C. Zienkiewicz, "The Finite Element Analysis of Engineering Systems Governed by a Non-Linear Quasi-Harmonic Equation," Computers & Structures, Vol. 5, No. 1, 1975, pp. 65–79.   
+16.3 B. Nour-Omid, "Lanczos Method for Heat Conduction Analysis," Int. J. Num. Meth. Engng., Vol. 24, No. 1, 1987, pp. 251-262.   
+16.4 T. J. R. Hughes, "Unconditionally Stable Algorithms for Nonlinear Heat Conduction," Comp. Meth. Appl. Mech. Engng., Vol. 10, No. 2, 1977, pp. 135–139.   
+16.5 T. J. R. Hughes, I. Levit, and J. Winget, “Element-by-Element Implicit Algorithms for Heat Conduction,” J. of Engng. Mech., Vol. 109, No. 2, 1983, pp. 576–585.   
+16.6 G. C. Everstine, “Structural Analogies for Scalar Field Problems,” Int. J. Num. Meth. Engng., Vol. 17, No. 3, 1981, pp. 471–476.   
+16.7 P. Tong and Y. C. Fung, "Slow Particulate Flow in Channels and Tubes—Application to Biomechanics," J. Appl. Mech., Vol. 38, No. 4, 1971, pp. 721–728.   
+16.8 S. T. K. Chan, B. E. Larock, and L. R. Herrmann, "Free-Surface Ideal Fluid Flows by Finite Elements," J. Hydraulics Div., Proc. ASCE, Vol. 99, No. HY6, 1973, pp. 959–974.   
+16.9 O. C. Zienkiewicz, R. W. Lewis, and K. G. Stagg, Numerical Methods in Offshore Engineering, John Wiley & Sons, Chichester, England, 1978.   
+16.10 O. C. Zienkiewicz and P. Bettess, “Fluid-Structure Dynamic Interaction and Wave Forces. An Introduction to Numerical Treatment,” Int. J. Num. Meth. Engng., Vol. 13, No. 1, 1978, pp. 1–16.   
+16.11 R. J Astley and W. Eversman, "Finite Element Formulations for Acoustical Radiation," J. Sound Vibration, Vol. 88, No. 1, 1983, pp. 47–64.   
+16.12 A. Craggs, “A Note on the Theory and Application of a Simple Pipe Acoustic Element,” J. Sound Vibration, Vol. 85, No. 2, 1982, pp. 292–295.   
+16.13 R. D. Cook, “Comment on ‘Discrete Element Idealization of an Incompressible Liquid for Vibration Analysis’ and ‘Discrete Element Structural Theory of Fluids,’ ” AIAA Jnl., Vol. 11, No. 5, 1973, pp. 766–767.   
+16.14 M. A. Hamdi, Y. Ousset, and G. Verchery, "A Displacement Method for the Analysis of Vibrations of Coupled Fluid-Structure Systems," Int. J. Num. Meth. Engng., Vol. 13, No. 1, 1978, pp. 139–150.   
+16.15 L. G. Olson and K. J. Bathe, "A Study of Displacement-Based Fluid Finite Elements for Calculating Frequencies of Fluid and Fluid-Structure Systems," Nucl. Engng. Design, Vol. 76, No. 2, 1983, pp. 137–151.
+
+<!-- source-page: 638 -->
+
+# CHAPTER 17
+
+17.1 ANSYS News, Swanson Analysis Systems Inc., Houston, PA, Second Issue, 1982.   
+17.2 A. Ralston and P. Rabinowitz, A First course in Numerical Analysis, 1978, McGraw-Hill, New York, 1978.   
+17.3 D. M. Gay, "Some Convergence Properties of Bloyd's Method," (2004) Num. Anal., Vol. 16, No. 4, 1979, pp. 623–630.   
+17.4 FIDAP Theoretical Manual, 1986, 27, anston, IL 60201, 1986, pp. 7-1, 7-34.   
+17.5 J. E. Dennis and R. Schnabel, Numerical Methods for Constrained Optimization, Prentice-Hall, Englewood Cliffs, NJ, 1983. Introduction to the Finite Element Method with Applications to   
+17.6 R. E. White, An Introduction to the Times-Lin. Nonlinear Problems, John Wiley & Sons, New York, 1985.   
+17.7 M. A. Crisfield, "An Arc-Length Methods: Engng., Vol. 19, No. 9, 1983, pp. 1269–1289. tions," Int. J. Num. Meth. Engng., Vol. 19, No. 9, 1983, pp. 1269–1289.   
+17.8 E. Riks, "Progress in Collapse Analysis," 1, 1987, pp. 33–41.   
+17.9 R. L. Webster, "On the Static Analysis of Structural Structure," "Linearity," Computers & Structures, Vol. 11, Nos. 1/2, 1980, pp. 137–145.   
+17.10 O. C. Zienkiewicz and R. Lohner, "Accelerated Renalment Future Prospects for FEM," Int. J. Num. Meth. Engng., Vol. 21, No. 1, 1985, pp. 1–11.   
+17.11 S. W. Sloan, "Substepping Schemes for the Numerical Strategies for Stress-Strain Relations," Int. J. Num. Meth. Engng., Vol. 24, No. 5, 1987, pp. 893-911.   
+17.12 A. Levy and B. Pifko, "On Computational Strategies 18, 1981, pp. 747-771. ticity and Creep," Int. J. Num. Meth. Engng., Vol. 17, No. 5, 1981, pp. 747-771.   
+17.13 P. C. Kollmick, ANSYS Engineering, 1986. Analysis Systems, Inc., Houston, PA, March 1986.   
+17.14 D. R. J. Owen and E. Hinton, Finite Elements in Plasticity, Pineridge Press Ltd., Swansea, U.K., 1980.   
+17.15 H. R. Evans, D. O. Peska, and A. R. Tanerhan, "The Analysis of the Ultimate Load Behavior of Plated Structures by the Finite Element Method," Engineering Computations, Vol. 2, No. 4, 1985, pp. 271–284.   
+17.16 P. V. Marcal, "Large Deflection Analysis of Elastic AIAA Jnl., Vol. 8, No. 9, 1970, pp. 1627-1633.   
+17.17 D. Bushnell, "Large Deflection Elastic-Plastic Group," in Numerical Solution, of Nonlinear Structural Problems, R. F. Hartung, ed., ASME AMD, Vol. 38, 1973, pp. 103-138.   
+17.18 J. A. Stricklin, W. E. Haisler, and W. A. Von Riesemann, "Solution Procedures for Nonlinear Analysis by Combined Finite Element-Finite Difference Methods," Computers & Structures, Vol. 2, Nos. 5/6, 1972, pp. 955–974.   
+17.19 J. A. Stricklin, W. E. Haisler, and W. A. Von Klosehmann,putation and Solution Procedures for Material and/or Geometric Nonlinear Structural Analysis by the Finite Element Method," Report SC-CR-72-3102, Sandia Laboratories, Albuquerque, NM, July 1972.   
+17.20 L. D. Hofmeister, G. A. Greenbaum, and D. A. Evenson, "Large Strain, Elastic Plastic Finite Element Analysis," AIAA Jnl., Vol. 9, No. 7, 1971, pp. 1248–1254.   
+17.21 D. Bushnell, "BUSORS—Program for Building of Revolution Including Large Deflections and Creep," Computers & Structures, Vol. 6, No. 3, 1976, pp. 221–239.   
+17.22 G. C. Nayak and O. C. Zienkiewicz, "Elasto-Plastic Stress Analysis for Various Constitutive Relations Including Strain Softening," Int. J. Num. Meth. Engng., Vol. 5, No. 1, 1972, pp. 113–135.
+
+<!-- source-page: 639 -->
+
+17.23 A. Grill and K. Sorimachi, "The Thermal Loads in the Finite Element Analysis of Elasto-Plastic Stresses," Int. J. Num. Meth. Engng., Vol. 14, No. 4, 1979, pp. 499–505.   
+17.24 T. Belytschko and T. J. R. Hughes, eds., Computational Methods for Transient Analysis, North Holland, Amsterdam, 1983.   
+17.25 M. E. Plesha, "Eigenvalue Estimation for Dynamic Contact Problems," J. of Engng. Mech., Vol. 113, No. 3, 1987, pp. 457-462.   
+17.26 T. Belytschko, “A Survey of Numerical Methods and Computer Programs for Dynamic Structural Analysis,” Nucl. Engng. Design, Vol. 37, No. 1, 1976, pp. 23–34.   
+17.27 T. Belytschko and D. F. Schoeberle, "On the Unconditional Stability of an Implicit Algorithm for Nonlinear Structural Dynamics," J. Appl. Mech., Vol. 42, No. 4, 1975, pp. 865–869.   
+17.28 M. S. Gadala, M. A. Dokainish, and G. A. Oravas, "Formulation Methods of Geometric and Material Nonlinearity Problems," Int. J. Num. Meth. Engng., Vol. 20, No. 5, 1984, pp. 887–914.   
+17.29 K. Mattiasson, A. Bengtsson, and A. Samuelsson, “On the Accuracy and Efficiency of Numerical Algorithms for Geometrically Nonlinear Structural Analysis,” in Finite Element Methods for Nonlinear Problems, P. G. Bergan, K. J. Bathe, and W. Wunderlich, eds., Springer, Berlin, 1986.   
+17.30 T. Belytschko and B. J. Hsieh, "Non-Linear Transient Finite Element Analysis with Convected Coordinates," Int. J. Num. Meth. Engng., Vol. 7, No. 3, 1973, pp. 255–271.   
+17.31 D. W. Murray and E. L. Wilson, “Finite-Element Large Deflection Analysis of Plates,” J. Engr. Mech. Div., Proc. ASCE, Vol. 95, No. EM1, 1969, pp. 143–165.   
+17.32 R. Schmidt and D. A. DaDeppo, "A Survey of Literature on Large Deflection of Nonshallow Arches. Bibliography of Finite Deflections of Straight and Curved Beams, Rings, and Shallow Arches," J. Industrial Math. Soc., Vol. 21, Part 2, 1971, pp. 91–114.   
+17.33 J. H. Lau, "Large Deflections of Beams with Combined Loads," J. Engr. Mech. Div., Proc. ASCE, Vol. 108, No. EM1, 1982, pp. 180–185.   
+17.34 M. D. Snyder and K. J. Bathe, "A Solution Procedure for Thermo-Elastic-Plastic-Creep Problems," Nucl. Engng. Design, Vol. 64, No. 1, 1981, pp. 49–80.   
+17.35 J. F. T. Pittman, O. C. Zienkiewicz, R. D. Wood, and J. M. Alexander, Numerical Analysis of Forming Processes, John Wiley & Sons, Chichester, England, 1984.   
+17.36 J. Padovan, S. Tovichakchaikul, and I. Zeid, "Finite Element Analysis of Steadily Moving Contact Fields," Computers & Structures, Vol. 18, No. 2, 1984, pp. 191–200.   
+17.37 C. S. Desai, M. M. Zaman, J. G. Lightner, and H. J. Siriwardane, “Thin-Layer Element For Interfaces and Joints,” Int. J. Num. An. Meth. Geomech., Vol. 8, No. 1, 1984, pp. 19–43.   
+17.38 E. L. Wilson, “Finite Elements for Foundations, Joints and Fluids,” in Finite Elements in Geomechanics, G. Gudehus, ed., John Wiley & Sons, London, 1977, pp. 319–350.   
+17.39 K. Schweizerhof and E. Ramm, “Displacement Dependent Pressure Loads in Nonlinear Finite Element Analysis,” Computers & Structures, Vol. 18, No. 6, 1984, pp. 1099–1114.   
+17.40 A. H. Peyrot and A. M. Goulois, “Analysis of Cable Structures,” Computers & Structures, Vol. 10, No. 5, 1979, pp. 805–813.   
+17.41 H. B. Jayaraman and W. C. Knudson, “A Curved Element for the Analysis of Cable Structures,” Computers & Structures, Vol. 14, Nos. 3–4, 1981, pp. 325–333.   
+7.42 M. Sathyamoorthy, “Nonlinear Vibrations of Plates—A Review,” Shock and Vibration Digest, Vol. 15, No. 6, 1983, pp. 3–16.
+
+<!-- source-page: 640 -->
+
+# CHAPTER 18
+
+18.1 R. A. Rosanoff, J. F. Gloudeman, and S. Levy, "Numerical Conditioning of Stiffness Matrix Formulations for Frame Structures," Proc. Second Conf. on Matrix Meth. in Struct. Mech., Wright-Patterson AFB, Ohio, 1968, (AFFDL-TR-68-150, Dec. 1969; AD-703-685, N.T.I.S.), pp. 1029–1060.   
+18.2 S. Utku and R. J. Melosh, "Solution Errors in Finite Element Analysis," Computers & Structures, Vol. 18, No. 3, 1984, pp. 379-393.   
+18.3 S. Kelsey, K. N. Lee, and C. K. Mak, "The Condition of Some Finite Element Coefficient Matrices," in Computer Aided Engineering, G. M. L. Gladwell, ed., University of Waterloo Press, Waterloo, Ontario, 1971.   
+18.4 G. Forsythe and C. B. Moler, Computer Solution of Linear Algebraic Systems, Prentice-Hall, Englewood Cliffs, NJ, 1967.   
+18.5 I. Fried, "Condition of Finite Element Matrices Generated from Nonuniform Meshes, AIAA Jul., Vol. 10, No. 2, 1972, pp. 219-221.   
+18.6 B. M. Irons, "Roundoff Criteria in Direct Stiffness Solutions," AIAA Jmt., Vol. 6, No. 7, 1968, pp. 1308–1312.   
+18.7 R. H. MacNeal, ed., The NASTRAN Theoretical Manual (Level 10.0), NASA-SP-221(03), March 1976 (N79-27531, N.T.I.S.).   
+18.8 B. Noble, Applied Linear Algebra, Prentice-Hall, Englewood Cliffs, NJ, 1969.   
+18.9 G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973 (now copyrighted by Wellesley–Cambridge Press, Wellesley, MA, 1988).   
+18.10 I. Fried, "Accuracy and Condition of Curved (Isoparametric) Finite Elements," J. Sound Vibration, Vol. 31, No. 3, 1973, pp. 345–355.   
+18.11 S. H. Crandall, Engineering Analysis, McGraw-Hill, New York, 1956.   
+18.12 J. Robinson, "An Introduction to Hierarchical Displacement Elements and the Adaptive Technique," Finite Elements in Analysis and Design, Vol. 2, No. 4, 1986, pp. 377–388.   
+18.13 N. Kikuchi, "Adaptive Grid-Design Methods for Finite Element Analysis," Comp. Meth. Appl. Mech. Engng., Vol. 55, Nos. 1-2, 1986, pp. 129-160.   
+18.14 G. L. Rigby and G. M. McNeice, "A Strain Energy Basis for Studies of Element Stiffness Matrices," AIAA Jnl., Vol. 10, No. 11, 1972, pp. 1490–1493.   
+18.15 J. O. Dow, T. H. Ho, and H. D. Cabiness, "Generalized Finite Element Evaluation Procedure," J. of Struct. Engng., Vol. 111, No. 6, 1984, pp. 435-452.   
+18.16 W. P. Doherty, E. L. Wilson, and R. L. Taylor, "Stress Analysis of Axisymmetric Solids Utilizing Higher-Order Quadrilateral Finite Elements," Report UC-SESM-69-3, Civil Engineering Department, University of California, Berkeley, 1969, (PB-190-321, NTIS).   
+18.17 J. Robinson, "A Single Element Test," Comp. Meth. Appl. Mech. Engng., Vol. 7, No. 2, 1976, pp. 191–200.   
+18.18 I. Fried, "Influence of Poisson's Ratio on the Condition of the Finite Element Stiffness Matrix." Int. J. Solids Structures, Vol. 9, No. 3, 1973, pp. 323-329.   
+18.19 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, New York, 1978.   
+18.20 R. J. Roark and W. C. Young, Formulas for Stress and Strain, 5th ed., McGraw-Hill, New York, 1975.   
+18.21 Anon., A Finite Element Primer, Dept. of Trade and Industry, National Engineering Laboratory, Glasgow G75 OQU, U.K., 1986.
+
+# CHAPTER 19
+
+19.1 G. J. DeSalvo, ANSYS Engineering Analysis System Verification Manual, Swanson Analysis Systems Inc., Houston, PA, 1985.
diff --git a/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_065.md b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_065.md
new file mode 100644
index 00000000..4e276c23
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+++ b/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_065.md
@@ -0,0 +1,1722 @@
+<!-- source-page: 641 -->
+
+19.2 I. C. Taig, "Finite Element Analysis in Industry—Expertise or Proficiency?," in Accuracy, Reliability, and Training in FEM Technology (Proc. of Fourth World Congress), J. Robinson, ed., Robinson and Associates, Wimborne, England, 1984.   
+19.3 T. Sussman and K. J. Bathe, "Studies of Finite Element Procedures—Stress Band Plots and the Evaluation of Finite Element Meshes," Engineering Computations, Vol. 3, No. 3, 1986, pp. 178–191.   
+19.4 R. T. Haftka and J. C. Robinson, "Effect of Out-of-Planeness of Membrane Quadrilateral Finite Elements," AIAA Jnl., Vol. 11, No. 5, 1973, pp. 742–744.   
+19.5 R. H. MacNeal, ed., The NASTRAN Theoretical Manual (Level 16.0), NASA-SP-221(03), March 1976 (N79-27531, N.T.I.S.).   
+19.6 K. Christensen, “Writing Easy-to-Use Programs for Computers,” Mechanical Engineering, Sept. 1983, pp. 66–69.   
+19.7 H. H. Fong, "An Evaluation of Eight U.S. General Purpose Finite-Element Computer Programs," 23rd AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference, 1982, pp. 145–160.   
+19.8 P. Naur, B. Randell, and J. N. Buxton, Software Engineering: Concepts and Techniques, Petrocelli-Charter, New York, 1976.   
+19.9 J. R. Rice, Numerical Methods, Software, and Analysis, McGraw-Hill, New York, 1983.   
+19.10 R. Evans, "Guidelines for the Selection of Analysis Software," Mechanical Engineering, March 1987, pp. 42–43.   
+19.11 J. A. Swanson, "The Development of General Purpose Software, or What is a Software Supplier?," in Structural Mechanics Computer Programs: Surveys, Assessments, and Availability, W. Pilkey et al., eds., University Press of Virginia, Charlottesville, 1974, pp. 687–702.   
+19.12 E. Schrem, "Status and Trends in Finite Element Software," in State-of-the-Art Surveys on Finite Element Technology, A. K. Noor and W. Pilkey, eds., ASME, New York, 1983, pp. 325–340.   
+19.13 K. Bell, “Some Thoughts on Design, Development and Maintenance of Engineering Software,” Advances in Engng. Software, Vol. 8, No. 2, 1986, pp. 66–72.   
+19.14 Anon., A Finite Element Primer, Dept of Trade and Industry, National Engineering Laboratory, Glasgow G75 OQU, U.K., 1986.   
+19.15 T. Slot and W. J. O'Donnell, "Effective Elastic Constants for Thick Perforated Plates with Square and Triangular Penetration Patterns," J. Eng. Industry, Vol. 93, No. 4, 1971, pp. 935–942.   
+19.16 C. Meyer, ed., Finite Element Idealization, Am. Soc. of Civil Engrs., New York, 1987.
+
+# APPENDIX A
+
+A.1 G. Strang, Linear Albegra and Its Applications, 3rd Ed., Harcourt-Brace-Jovanovich, San Diego, 1988.
+
+# APPENDIX B
+
+B.1 I. S. Duff, "A Survey of Sparse Matrix Research," Proceedings of the IEEE, Vol. 65, No. 4, 1977, pp. 500–535 (cites 604 references).   
+B.2 W. F. Tinney and J. W. Walker, "Direct Solutions of Sparse Network Equations by Optimally Ordered Triangular Factorization," Proceedings of the IEEE, Vol. 55, No. 11, 1967, pp. 1801-1809.   
+B.3 D. P. Mondkar and G. H. Powell, "Towards Optimal In-Core Equation Solving," Computers & Structures, Vol. 4, No. 3, 1974, pp. 531-548.
+
+<!-- source-page: 642 -->
+
+B.4 E. L. Wilson and H. H. Dovey, "Solution or Reduction of Equilibrium Equations for Large Complex Structural Systems," Advances in Engng. Software, Vol. 1, No. 1, 1978, pp. 19–25.   
+B.5 C. A. Felippa, "Solution of Linear Equations with Skyline-Stored Symmetric Matrix," Computers & Structures, Vol. 5, No. 1, 1975, pp. 13–29.   
+B.6 E. Mendelssohn and M. Baruch, "Solution of Linear Equations with a Symmetrically Skyline-Stored Nonsymmetric Matrix," Computers & Structures, Vol. 18, No. 2, 1984, pp. 215-246.
+
+# APPENDIX C
+
+C.1 D. S. Malkus and M. E. Plesha, "Zero and Negative Masses in Finite Element Vibration and Transient Analysis," Comp. Meth. Appl. Mech. Engng., Vol. 59, No. 3, 1986, pp. 281–306.   
+C.2 D. S. Malkus and X. Qiu, "Divisor Structure of Finite Element Eigenproblems Arising from Negative and Zero Masses," Comp. Meth. Appl. Mech. Engng., Vol. 66, No. 3, 1988, pp. 365–368.   
+C.3 D. S. Malkus, M. E. Plesha, and M-R. Liu, "Reversed Stability Conditions in Transient Finite Element Analysis," Comp. Meth. Appl. Mech. Engng., Vol. 68, No. 1, 1988, pp. 97–114.   
+No. 1, 1988, pp. 97-114.
+C.4 K. J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1982.   
+C.5 T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1987.   
+C.6 H. Kardestuncer and D. H. Norrie, eds., The Finite Element Handbook, McGraw-Hill, New York, 1987.   
+C.7 G. Strang, Introduction to Applied Mathematics, Wellesley-Cambridge Press, Wellesley, MA, 1986.   
+C.8 J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, England, 1965.   
+C.9 J. R. Rice. Numerical Methods, Software, and Analysis, IMSL Reference Edition, McGraw-Hill, New York, 1983.   
+C.10 B. T. Smith et al., Matrix Eigensystem Routines—EISPACK Guide, Lecture Notes in Computer Science No. 6, 2nd Ed., Springer-Verlag, New York, 1976.   
+C.11 NAg Library Manual, Numerical Algorithms Group, Downers Group, C.12 IMSL Library Manual, International Mathematical and Statistical Library, Houston, TX 77036-5085.
+
+<!-- source-page: 643 -->
+
+# INDEX
+
+a-basis formulation, 112
+
+Accuracy:
+
+anisotropy, 23, 583
+
+arch elements, 346, 347, 348
+
+axial symmetry, 297
+
+bar element, 92–93, 553–556
+
+constant strain triangle, 156–157
+
+digits lost, 547
+
+dynamic analysis, 397, 404–405, 413–416
+
+element geometry, 196-199
+
+fluid-structure interaction, 491, 494
+
+higher-order elements, 246
+
+isoparametric elements, 187–188
+
+mass condensation, 388–390
+
+near-mechanisms, 192-193
+
+numerical integration, 172, 188–189
+
+plate elements, 325–328, 333
+
+Rayleigh-Ritz solution, 82–83
+
+singularity elements, 250
+
+stress calculation, 81, 133–134, 195
+
+substructuring, 259
+
+transformations, 209
+
+user-defined elements, 245
+
+see also Bounds; Convergence; Errors:
+
+Modeling advice
+
+Acoustic modes, 490, 494
+
+Admissible configuration, 71
+
+Alpha-method, 409
+
+Amplitude error, 413–416
+
+Analogies, 486
+
+Anisotropic material, 20–22, 23, 294, 583
+
+Antisymmetry, 260–261, 298
+
+Arches, 343–350
+
+Arrested instability, 523
+
+Artificial viscosity, 407–409
+
+Assembly of elements: algorithms, 43, 44, 47, 51
+
+discussed, 12, 38–44, 49, 59
+
+in Galerkin method, 463-464
+
+Asymmetric Loads:
+
+heat conduction, 480
+
+shell of revolution, 353, 357
+
+solid of revolution, 301–307
+
+Attachment d.o.f., 258, 392, 393
+
+Average acceleration method, 405
+
+Axial symmetry, see Symmetry
+
+Banded matrices:
+
+bandwidth computation, 45, 46, 53
+
+equation-solver, 595
+
+storage format, 46
+
+Bar elements:
+
+in space, 215
+
+nodal loads, 40–41
+
+plane, 8, 36–38, 113
+
+stress calculation, 55–56
+
+Basis functions, see Shape functions
+
+Basis vectors, 384
+
+Beam, curved, 308
+
+Beam-column, 429–432
+
+Beam elements:
+
+direct method, 9–11
+
+Mindlin, 278–280, 283, 285
+
+shape functions for, 101
+
+shear deformation, 114, 279
+
+standard (Euler), 113-114
+
+Beating, 404
+
+BFGS method, 507
+
+Bifurcation analysis, 432, 441–444
+
+Bilinear elements:
+
+axially symmetric, 295–297
+
+plane, 98–99, 115–116, 166–170, 173–176
+
+Blocking, 258
+
+Body forces:
+
+defined, 17
+
+stress calculation, 133
+
+Bounds:
+
+Gerschgorin, 401, 548
+
+hybrid elements, 241
+
+incompatible elements, 234
+
+on buckling load, 445
+
+on condition number, 548, 549
+
+on Rayleigh quotient, 379
+
+on stiffness, 82–83
+
+on vibration frequency, 375, 400-402
+
+Boundary conditions:
+
+acoustic modes, 490
+
+hear conduction, 477–482
+
+in Galerkin method, 461
+
+inclined support, 545–546
+
+on displacement, 48–53
+
+on stress, 18–19
+
+plate bending, 332–334
+
+solid of revolution, 297, 303
+
+types of, 83–84, 478
+
+with a fluid, 487–489
+
+with derivative d.o.f., 246
+
+Boundary element method, 255
+
+Box beam, 322
+
+Brick element, 180–181
+
+Broyden methods, 506–508
+
+Bubble function, 178, 230
+
+Buckling:
+
+analysis restrictions, 446–447
+
+columns, 431
+
+condensation of d.o.f., 444
+
+<!-- source-page: 644 -->
+
+Buckling (Continued)
+
+defined, 429
+
+eigenvalue problem, 442
+
+shells, 445, 447
+
+with nonlinearity, 441, 443
+
+Bulk modulus, 280
+
+$C^0$ elements, 96-99
+
+$C^1$ elements, 99-101
+
+$C^m$ continuity, 95
+
+Cables, 446, 533
+
+Calculus of variations, 83–87
+
+Cantin-Clough element, 348
+
+Central-difference method, 397–405, 411–412, 415–416, 525–527
+
+CFL condition, 401
+
+Change of phase, 533
+
+Characteristic matrix, defined, 7
+
+Checking of model and results, 24, 579, 583-584
+
+Circulation mode, 493
+
+Collapse analysis, 447
+
+Collocation method, 456, 459
+
+Compatibility, 124–126, 234, 319, 320
+
+Compatibility condition, 18
+
+Complementary energy, 239–240
+
+Completeness:
+
+isoparametric elements, 186–188
+
+of trial series, 82
+
+Component mode synthesis, 391–395
+
+Computer program steps, 4, 57
+
+Condensation:
+
+algorithm for, 231
+
+and constraints, 273–274
+
+in buckling analysis, 444
+
+in dynamics, 387–391
+
+static, 228–229
+
+Condition number, 546–550
+
+Conditional stability, 398, 399, 485, 523
+
+Conforming, see Compatibility
+
+Connections, modeling of, 577
+
+Conservative system, 70
+
+Consistent penalty method, 287
+
+Constraints:
+
+and quadrature points, 282
+
+counting, 283–285
+
+for joining of elements, 218–222
+
+Lagrange multipliers, 275–276
+
+modes, 392–393
+
+multipoint, 272
+
+naturally arising, 278–282, 325–328
+
+penalty functions, 276–278
+
+ratio, 284
+
+rigid elements, 220–222
+
+single-point, 272
+
+transformations, 272–274
+
+see also Boundary conditions
+
+Constant-strain triangle, 114–115, 155–157
+
+Contacts, moving, 533
+
+Continuity:
+
+degree of, 95
+
+integration by parts, 461
+
+least squares methods, 458
+
+Convergence:
+
+and path test, 129
+
+general discussion, 542–566
+
+monotonic, 560
+
+nonlinear problems, 502–510, 522, 529, 530
+
+requirements for, 82, 126–128
+
+with lumped loads, 124
+
+Coordinates (natural or intrinsic):
+
+for area, 149–150, 152
+
+for line, 147–148, 164
+
+for quadrilateral, 166–167
+
+for tetrahedra, 151–152
+
+Core element, 297
+
+Costs:
+
+and dimensionality, 574
+
+in nonlinear problems, 501
+
+program writing and running, 586–587
+
+Courant number, 402
+
+Cracks, 247–250, 533, 557
+
+Craig-Bampton method, 393
+
+Crank-Nicolson method, 405, 485
+
+Creep, 533
+
+Critical condition (buckling), 429, 432, 441-444
+
+Critical time step:
+
+explicit methods, 398–404, 408–409, 411–412
+
+thermal transients, 485
+
+with material nonlinearity, 523
+
+Curvatures:
+
+arch, 344
+
+beam, 113
+
+plate, 317, 318, 320, 324
+
+shell, 354
+
+Cyclic symmetry, 262–263
+
+Damping:
+
+numerical, 407–409, 510
+
+physical, 376–377
+
+Dead loads, see Body forces
+
+Degrees of freedom, defined, 2, 31, 69
+
+Derivatives, as nodal d.o.f., 246
+
+Determinant, calculation of, 593–594, 596
+
+Diagnostics, see Errors
+
+Diagonal decay test, 550–552
+
+Differential stiffness matrix, 429
+
+Differentiation of matrix forms, 590–591
+
+Direct method of forming [k], 7–11, 32–33, 36–37
+
+Direct integration (transients), 395–418, 484–485, 522–529
+
+Direct substitution, 502–504
+
+Dirichlet boundary condition, 478
+
+Discrete element, 31
+
+Discrete Kirchhoff elements, 321, 328–332
+
+Discretization error, 543, 553–563
+
+<!-- source-page: 645 -->
+
+Discretization into finite elements, 1-2
+
+Dissection, 258
+
+Dissipation, numerical, 407–409, 510
+
+Distortion, effect of, 187–188, 196–197, 578
+
+DKT element, 328–332
+
+D.o.f., see Degrees of freedom
+
+Drift, nonlinear solutions, 505–506, 513, 521
+
+Drilling d.o.f., 236–238, 242–244, 351–352
+
+Dynamic response analysis:
+
+accuracy, 381, 407, 413–418
+
+alpha method, 409
+
+basic equations, 368–370
+
+beating, 404, 405
+
+choice of procedure, 407, 417, 418
+
+direct integration, 395–418
+
+efficiency, 397, 400, 401
+
+explicit methods, 395–405, 408–409
+
+implicit methods, 396, 405–407, 407–410
+
+mixed methods, 410
+
+modal methods, 381–387
+
+multi-step methods, 396
+
+Newmark methods, 408–409
+
+nonlinearities, 387, 398, 417–418, 522–529
+
+numerical noise, 404, 418
+
+operator splitting, 410
+
+Ritz vectors, 385–386
+
+single-step methods, 396
+
+spurious oscillations, 404
+
+stability, 398, 399, 404, 408–409, 410–413
+
+Dynamic storage allocation, 584–585
+
+Eigenproblems:
+
+algorithms and programs, 603–604
+
+general discussion, 378, 598–604
+
+in heat conduction, 482, 484
+
+Rayleigh quotient. 379, 601
+
+Eigenvalue economization, 387
+
+Eigenvalue test for elements. 563–565
+
+Elastic kernel, 244–245
+
+Elastic support, 576
+
+Elasticity, theory of, 15-20
+
+Elastic-plastic problems:
+
+dynamic, 522–529
+
+elastic to plastic transition, 512, 520
+
+general formulation, 515–519
+
+initial-stiffness method, 513–515, 522
+
+material property matrix, 517, 521
+
+one dimensional, 510–515
+
+tangent stiffness method, 513, 520–522
+
+Energy and energy principles:
+
+arches and shells, 344, 349, 355
+
+bars and beams, 77, 279
+
+complementary energy, 239–240
+
+general expressions, 75, 109–110
+
+minimum and stationary, 71–73, 86–87
+
+nonlinear terms, 433, 436, 438
+
+plates, 320, 324, 330
+
+Energy balance check, 523–524
+
+Envelope, 46, 596
+
+Equation solving, 48, 53–54, 592–597
+
+Equilibrium:
+
+and constraints, 274–275
+
+between and within elements. 124–125
+
+check of, 553
+
+differential equations, 17–19
+
+in hybrid elements, 239, 242
+
+Errors:
+
+and load lumping, 123–124
+
+and mass matrix, 375–376, 415
+
+buckling analysis, 441, 447–448
+
+causes of, 24–25, 209, 566, 573, 579–580
+
+diagonal decay test, 550–552
+
+dynamic analysis, 381, 383, 413–416, 492, 523–524, 527
+
+in cyclic symmetry, 263
+
+nonlinear problems, 505, 513, 521, 523–524, 527
+
+numerical, analysis and discussion. 542–566
+
+of discretization, 553–563
+
+penalty formulations, 278, 283
+
+reduced by extrapolation, 559–562
+
+residual tests, 383, 527, 552–553
+
+stress computation, 81, 132–135, 194–195, 297, 580–581
+
+terminology, 542–543
+
+use of symmetry, 261, 293, 381, 583
+
+various elements, 156, 223, 297, 326–327
+
+very stiff elements, 221–222, 544–546
+
+with singularities, 557
+
+see also Accuracy; Bounds; Convergence; Modeling advice
+
+Euler equation, 84–85
+
+Euler integration, 485, 505
+
+Extra nodal d.o.f., 246
+
+Extrapolation:
+
+from multiple meshes, 559–562
+
+of Gauss point stresses, 195–196
+
+Factorization, see Equation solving
+
+Fills, defined, 47
+
+Film coefficient, 474–475
+
+Finite difference method, 256
+
+Finite element method:
+
+advantages and disadvantages. 4–5
+
+defined, 4, 90
+
+Finite prism method, 307
+
+Finite strip method, 307, 322
+
+Flexural rigidity, defined, 317
+
+Flow rule (plasticity), 515
+
+Fluid flow, 287, 486–488
+
+Fluid-structure interaction, 491–494
+
+Folded plate, 322, 362
+
+Forces, internal and external. 369
+
+Form factor, 279
+
+Forming processes, 533
+
+Forward reduction, see Equation solving
+
+Foundation, elastic, 250–252
+
+Fourier equation, 465, 475
+
+<!-- source-page: 646 -->
+
+Fourier series, 298–300
+
+Fox-Goodwin method, 409
+
+Fracture mechanics, 247–248
+
+Frame element, 114, 216, 345–346
+
+Frequency, terms for, 378
+
+Frontal method, 48, 596–597
+
+Full integration, 188, 282
+
+Functional:
+
+acoustical modes, 490
+
+defined, 69, 83
+
+element matrices from, 90–94
+
+for potential energy, 109
+
+heat conduction, 93, 476, 479
+
+possible unavailability, 455
+
+wave equation, 489
+
+Galerkin method, 87, 457–458, 460–466, 468–469
+
+Guass elimination, 53–54, 593–596
+
+Gauss quadrature, 170–173, 183–185
+
+Generalized coordinates, 79, 382, 384–386, 484
+
+Geometric stiffness matrix, 429
+
+Gerschgorin bound, 401, 548
+
+Global, 12
+
+Gravity load, see Body forces
+
+Green-Lagrange strain, 437
+
+Guyan reduction, 387–391
+
+Hardening rule, plasticity, 511, 516-518
+
+Hardening structure, 502
+
+Harmonic functions, 468
+
+Hat functions, 463
+
+Heat conduction:
+
+boundary conditions, 477–482
+
+Fourier equation, 465, 475
+
+functional for, 93, 476, 479
+
+general element expressions, 481
+
+multidimensional, 85, 93–94, 477–480
+
+nonlinear, 475, 482, 485
+
+one-dimensional, 465–466, 475–477
+
+quantities and units, 474
+
+Heterosis plate element, 326–328
+
+Higher-order element, 246
+
+Hilbert matrix, 542
+
+Hilber-Hughes-Taylor method, 409
+
+Hinge, 124, 230
+
+History of finite elements, 14
+
+Houbolt method, 407
+
+Hourglass mode, 190–194, 238
+
+see also Instabilities
+
+Hybrid elements, 235, 239–244
+
+Ill-conditioning, 52, 343, 358, 543–550
+
+Imperfections, and buckling, 432, 447
+
+Incompatible elements, 125, 233–236, 320
+
+Incompressibility:
+
+and ill-conditioning, 550, 580
+
+constraint counting, 285
+
+elastic-plastic problems, 521
+
+fluid elements, 492
+
+penalty constraints, 280–282, 285–287
+
+Indeterminacy, static. 75
+
+Inertia, rotary, 371, 378, 602
+
+see also Mass matrices
+
+Inertia forces, see Body forces
+
+Inextensibility condition, 345
+
+Infinite elements and media, 252–255
+
+Influence lines, 120
+
+Initial stress and strain:
+
+bar element, 22–23, 40–41, 56, 118
+
+in general, 56, 110, 170
+
+in plates, 317
+
+stress calculation, 56, 132–134, 196
+
+Initial stress stiffness matrix, 429
+
+Initial value problem, 381
+
+Instabilities, element and numerical:
+
+mechanisms, element or mesh, 130, 190-
+
+193, 238, 381, 492, 564
+
+numerical, in dynamics, 398, 404, 485, 523
+
+stabilization matrix, 193, 238, 352, 493
+
+see also Buckling
+
+Integration (spatial):
+
+and instabilities, 188–193
+
+by parts, 86, 87, 461, 466–467
+
+full, reduced, selective, 188, 189, 282, 326
+
+Gauss quadrature, 170–173, 183–185
+
+in natural coordinates, 149, 151, 152
+
+special quadrature rules, 189, 374
+
+see also Dynamic response analysis;
+
+Transients (thermal)
+
+Internal d.o.f., 177–178, 228–230, 233, 258,
+
+326,348
+
+Interpolation:
+
+defined, 95
+
+Hermitian, 100
+
+Lagrange's formula, 97–99
+
+modified, 325, 356–357
+
+Interpolation functions, see Shape functions
+
+Invariance, geometric, 128–129
+
+Inversion, of matrix, 54–55
+
+Isoparametric elements:
+
+defined, 164
+
+for shells, 358–362
+
+general discussion. 163–199
+
+incompatible modes, 232–236
+
+validity of, 186–188
+
+Isotropy:
+
+geometric (spacial), 128–129
+
+of material, 21–22
+
+Iterative improvement, 134–135, 552
+
+Jacobian and Jacobian matrix:
+
+for area, 168–169
+
+for line, 165
+
+for solid, 180
+
+other terminology, 507
+
+Jointed media, 533
+
+Joints, modeling of, 577
+
+<!-- source-page: 647 -->
+
+Kernel, elastic, 244–245
+
+Kickoff forces, 435
+
+Kinematic mode, 190–194, 238
+
+see also Instabilities
+
+Kirchhoff plate theory, 315, 316, 319–320
+
+Lagrange element, 99, 177–180, 197
+
+Lagrange's interpolation formula, 97–99
+
+Laplace's equation, 468, 478, 487
+
+Laplacian mesh, 559
+
+Least squares methods, 457, 458, 460
+
+Limit point, 446–447, 508, 510
+
+Linear-strain triangle, 156–159, 183, 197–198
+
+Load vector:
+
+bar element, 40-41, 118, 120
+
+beam elements, 123
+
+body of revolution, 306
+
+consistent, 91, 110, 119
+
+for arches, 346
+
+for plates, 322–323
+
+in Galerkin method, 464
+
+inconsistent (lumped), 119, 123
+
+initial strain and stress, 40, 56, 58–59, 110, 118
+
+isoparametric elements, 170, 185–186
+
+non-nodal point loads, 119
+
+quadratic elements, 121–122, 185
+
+sign of, 13
+
+work-equivalence, 119
+
+Loads:
+
+body of revolution, 297, 301–307
+
+corrective, 521, 529, 530
+
+deformation dependent, 447, 533
+
+fictitious (pseudoload), 387, 417, 503
+
+internal and external, 369
+
+on plates, 318–319
+
+types of, 76
+
+Lobatto quadrature rule, 374
+
+Locking:
+
+arch elements, 346–350
+
+bilinear element, 233
+
+constraint counting, 283–285
+
+elastic-plastic problems, 521
+
+fluid elements, 492
+
+incompressible material, 281
+
+Mindlin elements, 280, 325–327
+
+penalty constraints, 278, 280–283
+
+reduced integration, 282
+
+Lumping:
+
+error of, 123-124
+
+of foundation stiffness. 252
+
+of loads, 119
+
+of masses, 370–376, 378
+
+of thermal matrices, 482
+
+Macroelements, 228, 242, 258
+
+Mass condensation, 387
+
+Mass matrices:
+
+and eigenproblem, 378, 379, 601–602
+
+choice of (accuracy), 375–376, 415
+
+choice of (efficiency), 376, 398, 401, 407
+
+consistent, 370–372
+
+diagonal (lumped), 370–376
+
+Master and slave d.o.f., 220, 259, 387-390
+
+Material coordinates, 438
+
+Material properties:
+
+heat conduction, 474, 477–478
+
+temperature dependent, 23, 482
+
+see also Stress-strain relations
+
+Matrix manipulations, 589–591
+
+Mechanisms, 190–194, 238
+
+see also Instabilities
+
+Mechanisms (linkages), 446
+
+Membrane forces:
+
+calculation of, 432
+
+defined, 340
+
+in buckling, 429
+
+Membranes, 446, 533
+
+Meridian, defined, 340
+
+Mesh:
+
+and ill-conditioning, 549–550
+
+discretization errors, 553–563
+
+layout (modeling), 573–584
+
+local refinement, 582
+
+revision of, 256–257, 575–576
+
+Mindlin elements:
+
+arches, 348–350
+
+beam, 278–280
+
+plates, 321, 323–328
+
+shells, 355–362
+
+Mock-fluid elements, 491-494
+
+Modal methods, 381–387, 484
+
+Modal synthesis, 391–395
+
+Modeling advice, 24–25, 573–584
+
+Modification of structure, 256–257, 575–576
+
+Moment, bending vs. nodal, 114
+
+Moment-curvature relations, 114, 316–318
+
+Negative area or Jacobian, 150, 152, 170, 198–199
+
+Neumann boundary condition, 478
+
+Newmark methods, 408–409
+
+Newton-Raphson methods, 504–505, 509
+
+Node numbering schemes, 44–48
+
+Nodeless d.o.f., 229–230, 233, 348
+
+Noise, numerical, 404, 418, 526
+
+Nonconforming elements, 125, 233–236, 320
+
+Nonlinear problems:
+
+solution methods, 502–510
+
+Nonlinearities:
+
+discussion and methods, 501–533
+
+effect on buckling, 441, 447
+
+geometric, 439–441, 443, 529–533
+
+in dynamic problems, 387, 398, 417–418, 522–529
+
+in heat conduction, 475, 482, 485
+
+in plate bending, 319
+
+in strain-displacement relation, 437
+
+<!-- source-page: 648 -->
+
+Nonlinearities (Continued)
+
+incremental methods, 440–441, 505, 513, 520–522
+
+initial stiffness method, 513–515, 522, 527–529
+
+modeling advice, 582
+
+types of, 501
+
+Offsets, 220-221
+
+Operator-splitting methods, 410
+
+Orthogonality of eigenvectors, 381
+
+Parasitic shear:
+
+in beam and plate elements, 280, 327
+
+in plane elements, 194, 197, 232–233
+
+Patch test, 129–131, 297, 334
+
+Penalty constraints:
+
+choice of penalty number, 283
+
+discussed, 276–278
+
+for boundary conditions, 50–52
+
+in transient analysis, 401, 482
+
+Period error, 413-416
+
+Pipe elbow, 308
+
+Plane stress and plane strain, 21, 23
+
+Plasticity, see Elastic-plastic problems
+
+Plates and plate bending:
+
+boundary conditions, 332–333
+
+discussed, 314–335
+
+sandwich plate, 318
+
+test cases, 331, 334–335
+
+Point sources, 122, 481, 482
+
+Positive definite matrix, 75, 590
+
+Potential flow, 486–488
+
+Potential function (fluids), 486
+
+Prandtl-Reuss relation, 518
+
+Pressure calculation, 286–287
+
+Prestress, see Initial stress and strain
+
+Profile, 46, 596
+
+Programming and programs: commercial, 587–588, 593, 604
+
+costs of, 586–587
+
+documentation, 585–586
+
+Pseudoloads, 387, 417, 503
+
+Q6 or QM6 elements, 233–236, 297
+
+Quadratic forms, 590
+
+Quadrature, 170–173, 183–185
+
+Quality tests for elements, 563–565
+
+Quarter-point elements, 248–250
+
+Quasiharmonic equation, 468
+
+Quasistatic, defined, 367
+
+Quasi-Newton methods, 506–508
+
+Rank-deficiency, 190, 590
+
+see also Singularity; Instabilities
+
+Rayleigh quotient, 379, 391, 601
+
+Rayleigh-Ritz method: and Galerkin method, 458
+
+classical form, 78–83
+
+finite element form, 90–93
+
+properties of solution, 82–83
+
+Reactions, support, 35–36, 56–57
+
+Reanalysis methods, 256–257
+
+Recovery of condensed d.o.f., 229, 231, 388
+
+Reduced basis, 384–386, 391–395
+
+Reduction (in dynamics), 228, 387–391
+
+see also Equation solving
+
+Refinement of mesh:
+
+and extrapolation, 559–562
+
+$h$ and $p$ versions, 563
+
+modeling strategy, 581–582
+
+Reflected waves, 490
+
+Reinforcing beam, offset, 220–221
+
+Release of d.o.f., 230, 246
+
+Repetition of substructures, 259, 262–263
+
+Residual:
+
+as error test, 383, 527, 552–553
+
+in dynamics, 383, 507, 527
+
+weighted residual methods, 456
+
+Richardson extrapolation, 560
+
+Rigid body motion:
+
+and convergence, 127–129
+
+and stress stiffening, 435, 438, 440, 445
+
+arch elements, 344, 348, 349, 350
+
+in dynamics, 375, 380, 602
+
+in user-defined element, 245
+
+yields no forces, 34–35, 564
+
+Rigid elements, 220–222, 272
+
+Ritz vectors, see Reduced basis
+
+Rotational d.o.f. in plane elements, 236–238, 242–244
+
+Rotational periodicity, 262–263
+
+Rounding error, 543
+
+Scalar element, 252
+
+Secant methods, 506–508
+
+Sectorial symmetry, 262–263
+
+Semianalytical method, 307, 322
+
+Semidiscretization, defined, 369
+
+Separation of variables method, 307, 322
+
+Serendipity elements, 177–180, 197, 326–328, 373–375
+
+Shape functions:
+
+axial elements, 89, 90, 96–98, 165
+
+beam elements, 101, 280
+
+bubble, 178, 230
+
+derivation and properties, 89–90, 187
+
+plane quadrilaterals, 99, 167, 177, 178, 255
+
+plane triangles, 154
+
+solid elements, 117, 159, 181
+
+with drilling d.o.f., 238
+
+with incompatible modes, 229, 233, 234
+
+Shear deformation:
+
+beams and arches, 114, 278–280, 348
+
+modified shear interpolation, 325, 356–357
+
+plates and shells, 316, 355–356, 358
+
+Shells:
+
+buckling, 445, 447
+
+<!-- source-page: 649 -->
+
+general discussion, 340–362
+
+geometry of, 340, 353
+
+thick vs. thin, 341
+
+Simplex, 152
+
+Single-element test, 565
+
+Singularity:
+
+crack-tip elements, 248–250
+
+error analysis, 557–558
+
+infinite element, 254
+
+of stiffness matrix, 13, 34–35
+
+plate bending, 334
+
+see also Instabilities
+
+Sink, 122, 481, 482
+
+Skew support, 216-218, 545-546
+
+Skyline, 46, 596
+
+Slave d.o.f., 220, 259, 387-390
+
+Sloshing of fluid, 488, 490, 493–494
+
+Snapping instability, 510
+
+Softening structures, 502, 509
+
+Software, see Programming and programs
+
+Sound propagation, 488, 490
+
+Source, point, 122, 481, 482
+
+Sparsity:
+
+and constraints, 275, 278
+
+of matrix, 44–48, 592, 596, 603–604
+
+Spectral matrix, 382, 484
+
+Spectral stability, 411–413
+
+Spurious modes or oscillations, see Instabilities
+
+Stability coefficient matrix, 429
+
+Stability, see Buckling; Instabilities
+
+Stabilization matrix, 193, 238, 352, 493
+
+Static equivalence, 119
+
+Statics check, 553
+
+Stationary principles, 83–87
+
+see also Functional
+
+Stick model, 574
+
+Stiffener, offset, 220-221
+
+Stiffness matrices (conventional):
+
+arch element (straight), 345
+
+bar elements, 8, 37–38, 90, 264
+
+beam or frame elements, 9–11, 114, 216
+
+bilinear element, 116, 169
+
+change of coordinates, 213–218
+
+formulation procedures, 7–8, 32–33, 90, 110, 461–465
+
+Fortran subroutines, 174–175, 332
+
+general formula, 110
+
+hybrid element, 240
+
+physical meaning of, 9, 34
+
+plate elements, 320, 324, 325, 330, 332
+
+properties of, 34, 74
+
+shell elements, 351, 352, 362
+
+solid (brick) element, 116–117
+
+triangular elements, 115, 155, 158–159
+
+zero coefficients, 39–40, 48–49
+
+Strain, effective plastic, 518
+
+Strain-displacement matrix:
+
+bar and beam elements, 90, 112–114, 166, 249
+
+bilinear element, 116, 169
+
+body of revolution, 295–296, 305
+
+from shape functions, 110
+
+Mindlin plate, 324
+
+triangular elements, 155, 158
+
+Strain-displacement relations:
+
+arches, 344
+
+bodies of revolution; 295, 302, 354, 356
+
+general, 15–17
+
+plates, 315, 316
+
+Strain-hardening parameter, 511
+
+Stream function, 487
+
+Stresses calculated from elements:
+
+arches and shells, 346, 362
+
+best locations for, 93, 194–195, 297, 328, 350
+
+extrapolation from Gauss points, 195–196
+
+iterative improvement, 134–135
+
+oversmoothed contours, 132, 580–581
+
+"strain gage," 132–133
+
+stress concentration, 135–136
+
+superconvergence, 195
+
+with distributed loads, 133
+
+with initial stress, 56, 59, 132–134, 196
+
+with internal d.o.f., 232, 236
+
+Stress intensity factor, 248–250
+
+Stress stiffness matrices:
+
+bars, beams, and plates, 434–437
+
+defined, 429
+
+field compatibility, 444–445
+
+general expression, 437–439
+
+rigid body motion, 435, 445
+
+various elements, 445
+
+Stresses:
+
+accuracy of, 81
+
+at boundaries, 19, 125
+
+deviatoric and dilatational, 285–286, 517
+
+plates and shells, 314, 340–341
+
+Stress-strain relations:
+
+axial symmetry, 294, 301, 355
+
+change of coordinates, 213, 318, 361, 478
+
+in general, 20-23
+
+plates and shells, 316–318, 355, 361
+
+Strong form, 69, 84, 455
+
+Structural dynamics, defined, 367
+
+"Structure size," expansion to, 38, 41-44
+
+Subdomain method, 456, 460
+
+Subparametric elements, 164, 188
+
+Substructuring:
+
+dynamic, 391–395
+
+static, 257–260, 262–263
+
+Superelement, 258
+
+Superparametric elements, 164, 188
+
+Support reactions, 35–36, 56–57
+
+Supports, modeling of, 216–218, 545–546, 576
+
+see also Boundary conditions
+
+Surface tractions, 19
+
+Symbolic processing, 185
+
+Symmetry:
+
+axial, 293–297, 354–355
+
+<!-- source-page: 650 -->
+
+Symmetry (Continued)
+
+cyclic (sectorial), 262–263
+
+discussed, 260–261, 574
+
+Fourier series terms, 298
+
+possible misuse, 261, 293, 381
+
+System, defined, 70
+
+Tangent modulus or stiffness, 504, 511
+
+Tetrahedron, 152, 159
+
+Thermal stress, 22–23, 58–59, 132–134, 196
+
+see also Initial stress and strain
+
+Time-history analysis, defined, 367–368
+
+see also Dynamic response analysis; Transients (thermal)
+
+Torsion of shafts, 303, 307
+
+Total-Lagrangian approach, 438
+
+Transformation:
+
+and constraints, 272–274
+
+isoparametric, 168
+
+material properties; 213, 318, 478
+
+possible errors, 209
+
+various applications, 209–222
+
+Transients (thermal), 482, 484–485
+
+see also Dynamic response analysis
+
+Tranezoidal rule, 405–406, 412–413, 415–416, 485,528
+
+Triangular elements:
+
+linear, 144-115, 153-157
+
+quadratic, 156–159, 183, 197–198
+
+Triangular coordinates, 152
+
+Trilinear element, 116–117, 181
+
+Truss element, see Bar elements
+
+Twisted ribbon, 335
+
+Unbounded regions, 252–255, 490
+
+Underrelaxation, 503
+
+Unsymmetric, see Asymmetric loads
+
+User-defined element, 244–245
+
+Variational methods, 85–87
+
+Vibrations:
+
+nonlinear, 533
+
+of fluid, 488–494
+
+of structures, 367, 378–381
+
+with stress stiffening, 446
+
+Virtual work principle, 72, 87
+
+Viscous relaxation, 510
+
+von Mises criterion, 517–518
+
+Warped elements, 583
+
+Wave equation, 489
+
+Wave propagation, 367, 401–405, 417–418, 490
+
+Wavefront method, 48, 596–597
+
+Weak form, 69, 84, 455
+
+Weighted residual methods, 87, 455–470
+
+Winkler foundation, 251
+
+Work, see Energy and energy principles
+
+Yield criterion, 515
+
+Zero-energy mode, see Instabilities
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