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 |  موضوع: كتاب Fundamentals of Finite Element Analysis - Linear Finite Element Analysis الجمعة 06 سبتمبر 2024, 2:21 am | |
| 
أخواني في الله
أحضرت لكم كتاب
Fundamentals of Finite Element Analysis - Linear Finite Element Analysis
Ioannis Koutromanos
Department of Civil and Environmental Engineering
Virginia Polytechnic Intitute and State University
Blacksburg, VA, United States
With single-chapter contributions from:
James McClure
Advanced Research Computing
Virginia Polytechnic Institute and State University
Blacksburg, VA, United States
Christopher Roy
Department of Aerospace and Ocean Engineering
Virginia Polytechnic Institute and State University
Blacksburg, VA, United States
و المحتوى كما يلي :
Contents
Preface xiv
About the Companion Website xviii
1 Introduction 1
1.1 Physical Processes and Mathematical Models 1
1.2 Approximation, Error, and Convergence 3
1.3 Approximate Solution of Differential Equations and the Finite Element
Method 5
1.4 Brief History of the Finite Element Method 6
1.5 Finite Element Software 8
1.6 Significance of Finite Element Analysis for Engineering 8
1.7 Typical Process for Obtaining a Finite Element Solution for a Physical
Problem 12
1.8 A Note on Linearity and the Principle of Superposition 14
References 16
2 Strong and Weak Form for One-Dimensional Problems 17
2.1 Strong Form for One-Dimensional Elasticity Problems 17
2.2 General Expressions for Essential and Natural B.C. in One-Dimensional
Elasticity Problems 23
2.3 Weak Form for One-Dimensional Elasticity Problems 24
2.4 Equivalence of Weak Form and Strong Form 28
2.5 Strong Form for One-Dimensional Heat Conduction 32
2.6 Weak Form for One-Dimensional Heat Conduction 37
Problems 44
References 46
3 Finite Element Formulation for One-Dimensional Problems 47
3.1 Introduction—Piecewise Approximation 47
3.2 Shape (Interpolation) Functions and Finite Elements 51
3.3 Discrete Equations for Piecewise Finite Element Approximation 59
3.4 Finite Element Equations for Heat Conduction 66
3.5 Accounting for Nodes with Prescribed Solution Value (“Fixed” Nodes) 67
3.6 Examples on One-Dimensional Finite Element Analysis 68
3.7 Numerical Integration—Gauss Quadrature 91
3.8 Convergence of One-Dimensional Finite Element Method 100
vii3.9 Effect of Concentrated Forces in One-Dimensional Finite Element
Analysis 106
Problems 108
References 111
4 Multidimensional Problems: Mathematical Preliminaries 112
4.1 Introduction 112
4.2 Basic Definitions 113
4.3 Green’s Theorem—Divergence Theorem and Green’s Formula 118
4.4 Procedure for Multidimensional Problems 121
Problems 122
References 122
5 Two-Dimensional Heat Conduction and Other Scalar Field Problems 123
5.1 Strong Form for Two-Dimensional Heat Conduction 123
5.2 Weak Form for Two-Dimensional Heat Conduction 129
5.3 Equivalence of Strong Form and Weak Form 131
5.4 Other Scalar Field Problems 133
5.4.1 Two-Dimensional Potential Fluid Flow 133
5.4.2 Fluid Flow in Porous Media 137
5.4.3 Chemical (Molecular) Diffusion-Reaction 138
Problems 139
6 Finite Element Formulation for Two-Dimensional Scalar Field Problems 141
6.1 Finite Element Discretization and Piecewise Approximation 141
6.2 Three-Node Triangular Finite Element 148
6.3 Four-Node Rectangular Element 153
6.4 Isoparametric Finite Elements and the Four-Node Quadrilateral (4Q)
Element 158
6.5 Numerical Integration for Isoparametric Quadrilateral Elements 165
6.6 Higher-Order Isoparametric Quadrilateral Elements 176
6.7 Isoparametric Triangular Elements 178
6.8 Continuity and Completeness of Isoparametric Elements 181
6.9 Concluding Remarks: Finite Element Analysis for Other Scalar Field
Problems 183
Problems 184
References 188
7 Multidimensional Elasticity 189
7.1 Introduction 189
7.2 Definition of Strain Tensor 189
7.3 Definition of Stress Tensor 191
7.4 Representing Stress and Strain as Column Vectors—The Voigt Notation 193
7.5 Constitutive Law (Stress-Strain Relation) for Multidimensional Linear
Elasticity 194
7.6 Coordinate Transformation Rules for Stress, Strain, and Material Stiffness
Matrix 199
viii Contents7.7 Stress, Strain, and Constitutive Models for Two-Dimensional (Planar)
Elasticity 202
7.8 Strong Form for Two-Dimensional Elasticity 208
7.9 Weak Form for Two-Dimensional Elasticity 212
7.10 Equivalence between the Strong Form and the Weak Form 215
7.11 Strong Form for Three-Dimensional Elasticity 218
7.12 Using Polar (Cylindrical) Coordinates 220
References 225
8 Finite Element Formulation for Two-Dimensional Elasticity 226
8.1 Piecewise Finite Element Approximation—Assembly Equations 226
8.2 Accounting for Restrained (Fixed) Displacements 231
8.3 Postprocessing 232
8.4 Continuity—Completeness Requirements 232
8.5 Finite Elements for Two-Dimensional Elasticity 232
8.5.1 Three-Node Triangular Element (Constant Strain Triangle) 233
8.5.2 Quadrilateral Isoparametric Element 237
8.5.3 Example: Calculation of Stiffness Matrix and Equivalent Nodal Forces for
Four-Node Quadrilateral Isoparametric Element 245
Problems 251
9 Finite Element Formulation for Three-Dimensional Elasticity 257
9.1 Weak Form for Three-Dimensional Elasticity 257
9.2 Piecewise Finite Element Approximation—Assembly Equations 258
9.3 Isoparametric Finite Elements for Three-Dimensional Elasticity 264
9.3.1 Eight-Node Hexahedral Element 264
9.3.2 Numerical (Gaussian) Quadrature for Hexahedral Isoparametric
Elements 272
9.3.3 Calculation of Boundary Integral Contributions to Nodal Forces 276
9.3.4 Higher-Order Hexahedral Isoparametric Elements 277
9.3.5 Tetrahedral Isoparametric Elements 277
9.3.6 Three-Dimensional Elements from Collapsed (Degenerated) Hexahedral
Elements 280
9.3.7 Concluding Remark: Continuity and Completeness Ensured by ThreeDimensional Isoparametric Elements and Use for Other Problems 281
Problems 287
Reference 288
10 Topics in Applied Finite Element Analysis 289
10.1 Concentrated Loads in Multidimensional Analysis 289
10.2 Effect of Autogenous (Self-Induced) Strains—The Special Case of Thermal
Strains 291
10.3 The Patch Test for Verification of Finite Element Analysis Software 294
10.4 Subparametric and Superparametric Elements 295
10.5 Field-Dependent Natural Boundary Conditions: Emission Conditions and
Compliant Supports 296
10.6 Treatment of Nodal Constraints 302
Contents ix10.7 Treatment of Compliant (Spring) Connections Between Nodal Points 309
10.8 Symmetry in Analysis 311
10.9 Axisymmetric Problems and Finite Element Analysis 316
10.10 A Brief Discussion on Efficient Mesh Refinement 319
Problems 321
References 323
11 Convergence of Multidimensional Finite Element Analysis, Locking
Phenomena in Multidimensional Solids and Reduced Integration 324
11.1 Convergence of Multidimensional Finite Elements 324
11.2 Effect of Element Shape in Multidimensional Analysis 327
11.3 Incompatible Modes for Quadrilateral Finite Elements 328
11.4 Volumetric Locking in Continuum Elements 332
11.5 Uniform Reduced Integration and Spurious Zero-Energy (Hourglass)
Modes 337
11.6 Resolving the Problem of Hourglass Modes: Hourglass Stiffness 339
11.7 Selective-Reduced Integration 346
11.8 The B-bar Method for Resolving Locking 348
Problems 351
References 352
12 Multifield (Mixed) Finite Elements 353
12.1 Multifield Weak Forms for Elasticity 354
12.2 Mixed (Multifield) Finite Element Formulations 359
12.3 Two-Field (Stress-Displacement) Formulations and the Pian-Sumihara
Quadrilateral Element 367
12.4 Displacement-Pressure (u-p) Formulations and Finite Element
Approximations 370
12.5 Stability of Mixed u-p Formulations—the inf-sup Condition 374
12.6 Assumed (Enhanced)-Strain Methods and the B-bar Method as a Special
Case 377
12.7 A Concluding Remark for Multifield Elements 381
References 382
13 Finite Element Analysis of Beams 383
13.1 Basic Definitions for Beams 383
13.2 Differential Equations and Boundary Conditions for 2D Beams 385
13.3 Euler-Bernoulli Beam Theory 388
13.4 Strong Form for Two-Dimensional Euler-Bernoulli Beams 392
13.5 Weak Form for Two-Dimensional Euler-Bernoulli Beams 394
13.6 Finite Element Formulation: Two-Node Euler-Bernoulli Beam Element 397
13.7 Coordinate Transformation Rules for Two-Dimensional Beam Elements 404
13.8 Timoshenko Beam Theory 408
13.9 Strong Form for Two-Dimensional Timoshenko Beam Theory 411
13.10 Weak Form for Two-Dimensional Timoshenko Beam Theory 411
13.11 Two-Node Timoshenko Beam Finite Element 415
13.12 Continuum-Based Beam Elements 417
13.13 Extension of Continuum-Based Beam Elements to General Curved
Beams 424
x Contents13.14 Shear Locking and Selective-Reduced Integration for Thin Timoshenko
Beam Elements 440
Problems 443
References 446
14 Finite Element Analysis of Shells 447
14.1 Introduction 447
14.2 Stress Resultants for Shells 451
14.3 Differential Equations of Equilibrium and Boundary Conditions
for Flat Shells 452
14.4 Constitutive Law for Linear Elasticity in Terms of Stress Resultants and
Generalized Strains 456
14.5 Weak Form of Shell Equations 464
14.6 Finite Element Formulation for Shell Structures 472
14.7 Four-Node Planar (Flat) Shell Finite Element 480
14.8 Coordinate Transformations for Shell Elements 485
14.9 A “Clever” Way to Approximately Satisfy C1 Continuity Requirements for
Thin Shells—The Discrete Kirchhoff Formulation 500
14.10 Continuum-Based Formulation for Nonplanar (Curved) Shells 510
Problems 521
References 522
15 Finite Elements for Elastodynamics, Structural Dynamics, and
Time-Dependent Scalar Field Problems 523
15.1 Introduction 523
15.2 Strong Form for One-Dimensional Elastodynamics 525
15.3 Strong Form in the Presence of Material Damping 527
15.4 Weak Form for One-Dimensional Elastodynamics 529
15.5 Finite Element Approximation and Semi-Discrete Equations of Motion 530
15.6 Three-Dimensional Elastodynamics 536
15.7 Semi-Discrete Equations of Motion for Three-Dimensional
Elastodynamics 539
15.8 Structural Dynamics Problems 539
15.8.1 Dynamic Beam Problems 540
15.8.2 Dynamic Shell Problems 543
15.9 Diagonal (Lumped) Mass Matrices and Mass Lumping Techniques 546
15.9.1 Mass Lumping for Continuum (Solid) Elements 546
15.9.2 Mass Lumping for Structural Elements (Beams and Shells) 548
15.10 Strong and Weak Form for Time-Dependent Scalar Field (Parabolic)
Problems 549
15.10.1 Time-Dependent Heat Conduction 549
15.10.2 Time-Dependent Fluid Flow in Porous Media 552
15.10.3 Time-Dependent Chemical Diffusion 554
15.11 Semi-Discrete Finite Element Equations for Scalar Field
(Parabolic) Problems 555
15.12 Solid and Structural Dynamics as a “Parabolic” Problem: The
State-Space Formulation 557
Problems 558
References 559
Contents xi16 Analysis of Time-Dependent Scalar Field (Parabolic) Problems 560
16.1 Introduction 560
16.2 Single-Step Algorithms 562
16.3 Linear Multistep Algorithms 568
16.3.1 Adams-Bashforth (AB) Methods 569
16.3.2 Adams-Moulton (AM) Methods 569
16.4 Predictor-Corrector Algorithms—Runge-Kutta (RK) Methods 569
16.5 Convergence of a Time-Stepping Algorithm 572
16.5.1 Stability of Time-Stepping Algorithms 572
16.5.2 Error, Order of Accuracy, Consistency, and Convergence 574
16.6 Modal Analysis and Its Use for Determining the Stability for Systems with Many
Degrees of Freedom 583
Problems 587
References 587
17 Solution Procedures for Elastodynamics and Structural Dynamics 588
17.1 Introduction 588
17.2 Modal Analysis: What Will NOT Be Presented in Detail 589
17.2.1 Proportional Damping Matrices—Rayleigh Damping Matrix 592
17.3 Step-by-Step Algorithms for Direct Integration of Equations
of Motion 594
17.3.1 Explicit Central Difference Method 595
17.3.2 Newmark Method 597
17.3.3 Hilber-Hughes Taylor (HHT or Alpha) Method 599
17.3.4 Stability and Accuracy of Transient Solution Algorithms 601
17.4 Application of Step-by-Step Algorithms for Discrete Systems with More than
One Degrees of Freedom 608
Problems 613
References 613
18 Verification and Validation for the Finite Element Method 615
18.1 Introduction 615
18.2 Code Verification 615
18.2.1 Order of Accuracy Testing 616
18.2.2 Systematic Mesh Refinement 617
18.2.3 Exact Solutions 618
18.3 Solution Verification 622
18.3.1 Iterative Error 623
18.3.2 Discretization Error 624
18.4 Numerical Uncertainty 627
18.5 Sources and Types of Uncertainty 629
18.6 Validation Experiments 630
18.7 Validation Metrics 631
18.8 Extrapolation of Model Prediction Uncertainty 633
18.9 Predictive Capability 634
References 634
xii Contents19 Numerical Solution of Linear Systems of Equations 637
19.1 Introduction 637
19.2 Direct Methods 638
19.2.1 Gaussian Elimination 638
19.2.2 The LU Decomposition 639
19.3 Iterative Methods 640
19.3.1 The Jacobi Method 642
19.3.2 The Conjugate Gradient Method 642
19.4 Parallel Computing and the Finite Element Method 644
19.4.1 Efficiency of Parallel Algorithms 645
19.4.2 Parallel Architectures 647
19.5 Parallel Conjugate Gradient Method 649
References 653
Appendix A: Concise Review of Vector and Matrix Algebra 654
A.1 Preliminary Definitions 654
A.1.1 Matrix Example 655
A.1.2 Vector Equality 655
A.2 Matrix Mathematical Operations 656
A.2.1 Exterior Product 657
A.2.2 Product of Two Matrices 657
A.2.3 Inverse of a Square Matrix 660
A.2.4 Orthogonal Matrix 660
A.3 Eigenvalues and Eigenvectors of a Matrix 660
A.4 Rank of a Matrix 662
Appendix B: Review of Matrix Analysis for Discrete Systems 664
B.1 Truss Elements 664
B.2 One-Dimensional Truss Analysis 666
B.3 Solving the Global Stiffness Equations of a Discrete System
and Postprocessing 671
B.4 The ID Array Concept (for Equation Assembly) 673
B.5 Fully Automated Assembly: The Connectivity (LM) Array Concept 680
B.6 Advanced Interlude—Programming of Assembly When the Restrained Degrees
of Freedom Have Nonzero Values 682
B.7 Advanced Interlude 2: Algorithms for Postprocessing 683
B.8 Two-Dimensional Truss Analysis—Coordinate Transformation
Equations 684
B.9 Extension to Three-Dimensional Truss Analysis 693
Problems 694
Appendix C: Minimum Potential Energy for Elasticity—Variational Principles 695
Appendix D: Calculation of Displacement and Force Transformations for
Rigid-Body Connections 700
Index 706
Contents xiii
Index
a
Adams-Bashforth (AB) methods 569,
578, 582
Adams-Moulton (AM) methods 569, 582
anisotropic material 194, 348
approximation
finite difference 5, 13
(piecewise) finite element 13, 26, 47, 51,
54, 59, 104, 121, 130, 141, 212, 226, 232,
258, 264, 289, 295, 317, 319, 326, 353,
359, 367, 377
qualitative definition of 3, 574, 629
of weak form 28, 39
Argyris, J. 6, 153
assembly 64, 146, 226, 258, 305, 373, 381,
397, 417, 485, 532, 555, 588, 611, 670,
673, 680, 682, 689
autogenous strains 291, 294, 444
axisymmetry 220, 316
b
B-bar method 348, 377
Belytschko, T. 92, 100, 104, 106, 155, 294,
339, 344, 603
bending 324, 328, 339, 343, 383, 391, 410,
441, 451, 454, 458, 460, 463, 467, 475,
501, 508
boundary conditions
essential 21, 23, 26, 34, 38, 60, 67, 89,
105, 126, 136, 138, 208, 211, 220, 231,
257, 263, 306, 354, 359, 387, 393, 411,
414, 453, 471, 523, 526, 527, 529, 536,
538, 540, 543, 551, 552, 554, 555, 619,
621, 696, 697, 699
generalized (field-dependent)
natural 296, 298
natural 21, 23, 24, 29, 32, 34, 42, 63, 66,
85, 114, 126, 133, 136, 138, 139, 143,
151, 157, 170, 176, 208, 210, 218, 220,
236, 242, 245, 276, 285, 301, 311,
315, 387, 393, 401, 408, 411, 453,
456, 469, 479, 484, 526, 529, 551,
554, 619
bulk modulus 334, 347
c
calculus of variations 698
central difference method 595, 599,
605, 610
Chopra, A. 588, 589
Clough, R. 6, 7
coefficient matrix 25, 27, 40, 62, 64, 75, 91,
97, 130, 144, 162, 169, 178, 212, 230,
301, 363, 368, 374, 377, 379, 487, 555,
565, 585, 595
compatibility conditions 18, 191, 332
completeness 50, 91, 104, 147, 181, 232,
264, 281, 296, 325, 377, 396,
414, 622
compliant supports 296, 322
concentrated forces 106, 289
conductivity 34, 67, 127, 154, 164
conjugate gradient method 642, 649
connectivity array 673, 680
conservation laws 2, 33, 113, 124, 133, 137,
524, 549, 552, 624
consistency 474, 572, 574, 602
constraints 302, 311, 336, 464, 514, 704
continuity 6, 24, 26, 48, 91, 147, 159, 181,
189, 232, 264, 281, 325, 332, 365, 394,
399, 414, 448, 500
continuum-based structural elements
beams 417, 424
shells 510
convergence
conceptual definition of 3, 49, 91, 299
of mixed elements 374, 381
multi-dimensional analysis 147, 183,
277, 296, 324
one-dimensional analysis 100, 104
of time-stepping algorithms 572,
574, 577
coordinate transformation 199, 404, 485,
684, 693
curvature 390, 409, 451, 500
curved structural elements
beams 424
shells 510
cylindrical coordinates 220 see also polar
coordinates
d
damping (viscous) 524, 527, 537
damping matrix 534, 542, 588, 592
Darcy’s law 137, 552
density (mass) 133, 525, 536, 546
deviatoric stress/strain 332, 347, 371, 381
differential equations 1, 5, 12, 18, 22, 32,
34, 121, 125, 133, 136, 209, 218, 224,
336, 354, 371, 385, 411, 452, 523, 527,
536, 551, 557, 572, 618
diffusion 138, 554
Dirac delta 107, 289
direct methods (for systems of
equations) 638
Discrete Kirchhoff theory (DKT) 500
distributed memory 647
e
eigenvalue 200, 337, 579, 584, 586, 590,
603, 623, 660
elasticity 17, 189, 226, 257, 354, 456
embedded element constraints 306
emission boundary condition 296
equivalent right-hand-side vector 63, 145,
164, 178, 364, 561
error
conceptual definition of 3, 50, 100, 133,
216, 306, 621, 629
iterative 623
norm 100, 104, 325, 616
in transient integration 572, 574, 604
Euler-Bernoulli beam theory 388
explicit methods 339, 561, 568, 595, 599
f
fiber vector 417, 425, 511
Fick’s law 139
Fish, J. 92, 100, 104, 106, 155, 294
flow in porous media 137, 183, 552
Fourier’s law 34, 127, 550
functional 105, 356, 625, 696
g
Galerkin, B. 32, 45
gather-scatter array 60, 143, 171, 228, 244,
300, 408, 531, 668, 669, 674
Gauss elimination 41, 638
Gaussian quadrature
one-dimensional 91
two-dimensional 165
three-dimensional 272
generalized midpoint rule 562
generalized strains 392, 398, 410, 450,
456, 500
generalized stresses 392, 410, 460, 544
generalized trapezoidal rule 562, 573
gradient 116
Green’s formula 118
Green’s theorem 118
h
heat conduction 32, 37, 66, 123, 141, 159,
181, 291, 296, 301, 314, 319, 549,
556, 566
Hellinger-Reissner principle 353,
358, 699
hexahedral element 264, 339
hourglass modes 337, 339
hourglass stiffness 339
Index 707Hughes, T. 177, 325, 336, 348, 381, 389,
426, 484, 510, 563, 575, 595, 599, 625
hybrid elements 508
i
ID array 673, 680
implicit methods 561, 568, 569, 582, 595,
599, 625
incompatible modes 328, 332, 353
inf-sup condition 374
initial conditions 526, 535, 551, 558, 561,
565, 597, 610
integration by parts 25, 37, 112, 118, 120,
129, 394, 412
isoparametric element
hexahedral 264, 280
quadrilateral 158, 237
tetrahedral 277
triangle 178, 235
isotropic material 127, 194, 198, 204, 212,
315, 321, 334, 342, 346, 348, 371, 381,
422, 458, 510, 544
iterative methods (for systems of
equations) 640
j
Jacobian determinant 165, 169, 180, 243,
275, 327, 483
Jacobian matrix (or Jacobian array) 161,
165, 169, 221, 235, 236, 240, 243, 269,
275, 368, 429, 482
Jacobi method 642
k
Kirchhoff-Love theory 448, 500
Kronecker delta 52, 59, 148, 157, 161, 279,
291, 333
l
Lagrange polynomials 56
Lamé’s constants 198, 334, 347
lamina 424, 430, 510, 513
LEFM, 328
linear elasticity 19, 189, 194, 231, 257, 297,
321, 456, 528, 698
linearization 14, 298
linearly elastic fracture mechanics see LEFM
linear multistep methods (LMS) 568
link see rigid link
LM array see connectivity array
LU decomposition 639
m
mapping 94, 151, 159, 179, 221, 235, 264,
279, 290, 295, 327, 368, 424, 481,
502, 511
mass matrix 532, 542, 546, 588
master node 302, 700
material stiffness matrix 194, 199, 236, 271,
316, 346, 358, 423, 456, 517
matrix algebra 654
membrane 187, 203, 447, 462, 467, 474,
501, 508
method of manufactured solutions
(MMS) 618
mixed elements 353, 508
modulus of elasticity 206, 321, 334, 342,
392, 410, 458, 665, 693
monoclinic material 195
multidimensional problems 12, 141, 189,
226, 257, 289, 324, 536
multifield weak form 354, 699
n
Newmark method 597, 604, 609
nullspace of matrix 663
numerical integration see Gaussian
quadrature
numerical stability 374, 572, 583, 601
numerical uncertainty 627
o
order of accuracy 574, 604, 607, 616
orthogonality of modal vectors 585,
591, 661
orthogonal matrix 200, 660
orthotropic material 195, 204
p
parallel computing
efficiency of 645
architecture 647
parasitic shear stiffness 329, 353, 441
partition of unity property 53, 59, 148, 157,
160, 279
Pian-Sumihara element 367
708 Indexplane strain 203, 209, 291
plane stress 203, 209, 291, 320, 332, 417,
456, 516
Poisson’s ratio 196, 198
polar coordinates 220, 316
potential energy for elasticity 356, 695
pressure modes 376
principal stresses/strains
in three dimensions 201
in two dimensions 205
proportional damping matrix 592
pseudo-code 681
q
quadrature
Gaussian see Gauss quadrature
on tetrahedral 280
on triangles 180
quadrilateral element 158, 165, 176, 181,
237, 245, 296, 326, 328, 337, 348, 367
r
rank (of a matrix) 642, 662
rank-deficiency 337, 348, 662
Rayleigh damping matrix 592
reduced integration
uniform 337
selective 346, 381, 440, 521
refinement 47, 86, 103, 319, 327, 617, 625
Reissner-Mindlin shell theory 447,
460, 543
restraints 67, 314
Richardson extrapolation 626
rigid bar 304, 420, 514, 700
rigid link 302
Runge-Kutta (RK) methods 569
s
Saint Venant’s principle 321
scatter array see gather-scatter array
self-induced strains see autogenous strains
serendipity element 176, 277, 326, 337,
348, 503, 547
shear modulus 196, 198, 389, 410
slave node 302, 700
shape functions 52, 95, 106, 130, 141, 150,
155, 160, 176, 181, 233, 237, 259, 265,
279, 290, 295, 307, 319, 325, 329, 359,
399, 415, 424, 472, 482, 503, 507, 530,
539, 625
shear locking 329, 440, 521
shells 337, 447
singularity 67, 328, 666
spurious zero-energy modes 337, 376
stability see numerical stability
state-space formulation 557
static condensation 331, 591
stationary value of functional 698
stiffness matrix (for finite element
analysis) 62, 66, 98, 230, 242, 262, 271,
293, 298, 304, 318, 330, 337, 342, 347,
366, 397, 404, 418, 422, 431, 441, 478,
485, 510, 519, 534, 592, 594, 611,
666, 670
strain 19, 27, 34, 50, 61, 103, 189, 194, 199,
224, 234, 257, 291, 313, 316, 319, 325,
332, 338, 346, 353, 359, 372, 377, 389,
396, 409, 422, 428, 440, 449, 456, 463,
474, 500, 516, 524, 665
strain energy 194, 338, 356, 695
stress 17, 27, 100, 191, 196, 200, 203, 212,
219, 233, 257, 291, 313, 316, 319, 325,
331, 348, 353, 358, 366, 371, 376, 381,
388, 396, 409, 417, 427, 447, 451, 456,
510, 516, 524, 528, 537, 625, 664, 695
stress resultants see generalized stresses
strong form 17, 28, 32, 123, 131, 208, 215,
218, 392, 411, 452, 525, 527
superconvergent patch recovery (SPR) 626
symmetry in analysis 311
systematic mesh refinement 617
system response quantity (SRQ) 616
t
temperature 1, 32, 36, 47, 66, 100, 123, 138,
141, 181, 291, 353, 523, 549, 555, 618
temperature-induced strains 291, 444
tensor 189, 191, 198, 205, 224, 332, 354,
368, 450, 537
thermal strains see temperature-induced
strains
tie constraints 307
time-dependent 523, 560, 588
time-stepping algorithm (step-by-step
algorithm) 560, 566, 573, 583, 589,
594, 602, 608
Index 709Timoshenko beam theory 408, 411, 415,
440, 447
traction 20, 27, 208, 218, 233, 244, 270, 294,
298, 313, 318, 330, 354, 387, 396, 414,
456, 469, 520, 525, 619
transversely isotropic material 197
Turner, J. 6
u
uncertainty 627, 629
uniform reduced integration see reduced
integration
v
validation 295, 615, 630, 631
validation hierarchy 630
variational principles 28, 353, 358, 695
verification 294, 615, 622
virtual work 27, 214, 317, 356, 361, 372,
396, 412, 465, 471, 478, 502, 544
volumetric locking 332, 336, 346, 348, 351,
370, 442
volumetric strain 324, 332, 335, 346, 350,
371, 381
w
weak form 24, 28, 37, 129, 212,
215, 257, 354, 394, 411, 464, 529,
549, 695
weighted residuals 32
y
Young’s modulus 18, 34, 97, 194, 198, 402,
525, 664, 683 see also modulus of
elasticity
z
Zero-energy modes
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رابط من موقع عالم الكتب لتنزيل كتاب Fundamentals of Finite Element Analysis - Linear Finite Element Analysis
رابط مباشر لتنزيل كتاب Fundamentals of Finite Element Analysis - Linear Finite Element Analysis 
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