김경종
14 KiB
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
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
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
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
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
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
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
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
| [ ] | Rectangular or square matrix. |
| { }, [ ], [ ] | Column, row, and diagonal matrices. |
| [ ]T | Matrix, transpose. |
| [ ]-1, [ ]-T | Matrix inverse and inverse transpose; that is, ([ ]-1)T ≡ ([ ]T)-1. |
| || || | Norm of a matrix or a vector. |
| . | Time differentiation; for example, i = du/dt, ii = d2u/dt2. |
| , | Partial differentiation if the following subscript(s) is literal; for example, w,x = ∂w/∂x, w,xy = ∂2w/∂x ∂y. |
| { ∂Π/∂a} | Represents [ ∂Π/∂a1 ∂Π/∂a2 · · · ∂Π/∂an]T, where Π is a scalar function of a1, a2, . . . , an. |
LATIN SYMBOLS
| A | Area or cross-sectional area. |
| [A] | Relates {d} to {a}; {d} = [A]{a}. |
| {a} | Generalized coordinates. |
| B | Bulk modulus, $B = E/(3 - 6\nu)$ . |
| [B] | Spatial derivative(s) of the field variable(s) are [B]{d}. |
| $C^{m}$ | Field continuity of degree m (Section 3.11). |
| [C] | Damping matrix. Constraint matrix. |
| d.o.f. | Degree(s) of freedom. |
| D | Displacement. Flexural rigidity of a plate or shell. |
| {D}, {d} | Nodal d.o.f. of structure and element, respectively. |
| E | Modulus of elasticity. |
| [E] | Matrix of elastic stiffnesses (Section 1.7). |
| {F} | Body forces per unit volume. |
| G | Shear modulus. |
| I | Moment of inertia of cross-sectional area. |
| [I] | Unit matrix (also called identity matrix). |
| J | Determinant of [J] (called the Jacobian). |
| [J] | The Jacobian matrix. |
| k | Spring stiffness. Thermal conductivity. |
| [K], [k] | Structure and element conventional stiffness matrices. |
| $[K_{\sigma}]$ , $[k_{\sigma}]$ | Structure and element stress stiffness matrices. |
| L, $L_{T}$ | Length of element, length of structure. |
| $\ell$ , m, n | Direction cosines. |
| $n_{\text{eq}}$ | Number of equations. |
| [M], [m] | Structure and element mass matrices. |
| [N], [N] | Shape (or basis, or interpolation) functions. |
| O | Order; for example, $O(h^{2}) =$ a term of order $h^{2}$ . |
| [0], {0} | Null matrix, null vector. |
| {P} | Externally applied concentrated loads on structure nodes. |
| q | Distributed load (surface or line). |
| {R} | Total load on structure nodes; {R} = {P} + $\Sigma$ {re}. |
| {re} | Loads applied to nodes by element, for example, by temperature change or distributed load (Eq. 4.1-6). |
| S, Se | Surface, element surface. |
| T | Temperature. |
| t | Thickness. Time. |
| [T] | Transformation matrix. |
| U, U0 | Strain energy, strain energy per unit volume. |
| u, v, w | Displacements, for example, in directions x, y, z. |
| {u} | Vector of displacements; {u} = [u v w]T. |
| V, Ve | Volume, element volume. |
| x, y, z | Cartesian coordinates. |
GREEK SYMBOLS
| $\alpha$ | Coefficient of thermal expansion, penalty number. |
| $[\Gamma]$ | Jacobian inverse; $[\Gamma] = [J]^{-1}$ . |
| $\{ \epsilon \}, \{ \epsilon_0 \}$ | Strains, initial strains. |
| $[\kappa], \{ \kappa \}$ | Matrix of thermal conductivities, vector of curvatures. |
| $\lambda$ | Eigenvalue. Lagrange multiplier. |
| $\nu$ | Poisson's ratio of an isotropic material. |
| $\xi, \eta, \zeta$ | Isoparametric coordinates. |
| $\xi_1, \xi_2, \xi_3$ | Area coordinates. |
| $\Pi$ | A functional; for example, $\Pi_p =$ potential energy. |
| $\rho$ | Mass density. |
| $\{ \sigma \}, \{ \sigma_0 \}$ | Stresses, initial stresses. |
| $\phi$ | A dependent variable. Meridian angle of a shell. |
| $\{ \Phi \}$ | Surface tractions. |
| $\omega$ | Circular frequency in radians per second. |