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 |  موضوع: كتاب Concepts and Applications of Finite Element Analysis - Fourth Edition الجمعة 02 أغسطس 2024, 2:06 am | |
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أحضرت لكم كتاب
Concepts and Applications of Finite Element Analysis - Fourth Edition
Robert D. Cook
David S. Malkus
Michael E. Plesha
Robert J. Witt
University of Wisconsin - Madison
و المحتوى كما يلي :
CONTENTS
NOTATION
Chapter 1 INTRODUCTION 1
LI Finite Element Analysis 1
L2 Problem Classification, Modeling, and Discretization 3
1.3 Interpolation. Elements, Nodes, and D.O.F. 5
1.4 Example Applications. History of FEA 8
1.5 Solving a Problem by FEA 11
1.6 Learning and Using FEA 15
Analytical Problems 17
Chapter 2 ONE-DIMENSIONAL ELEMENTS AND COMPUTATIONAL
PROCEDURES 19
2.1 Introduction 19
2.2 Bar Element 20
2.3 Beam Element 24
2.4 Bar and Beam Elements of Arbitrary Orientation 29
2.5 Assembly of Elements 32
2.6 Properties of Stiffness Matrices 36
2.7 Boundary Conditions 40
2.8 Exploiting Sparsity. Solving Equations 42
2.9 Mechanical Loads. Stresses 46
2.10 Thermal Loads. Stresses 52
2.11 Structural Symmetry 54
2.12 Review. Remarks Regarding Modeling 57
2.13 An Application 59
Analytical Problems 62
Computational Problems 74
Chapter 3 BASIC ELEMENTS 78
3.1 Preliminaries 78
3.2 Interpolation and Shape Functions 83
3.3 Formulas for Element Matricies 88
3.4 Linear Triangle (CST) 91
3.5 Quadratic Triangle (LST) 95
3.6 Bilinear Rectangle (Q4) 96
3.7 Quadratic Rectangle (Q8, Q9) 100
3.8 Rectangular Solid Elements 102
3.9 Choice of Interpolation Functions 104
3.10 Improved Triangles and Quadrilaterals 106
ixX Contents
3.11 Nodal Loads 111
3.12 Stress Calculation 115
3.13 Nature of a Finite Element Solution 118
3.14 Example: A Simple Stress Concentration Problem 119
3.15 An Application with High Stress Gradient 121
Analytical Problems 124
Computational Problems 132
Chapter 4 FORMULATION TECHNIQUES: VARIATIONAL
METHODS 136
4.1 Introduction 136
4.2 Principle of Stationary Potential Energy 137
4.3 Problems Having Many D.O.F. 140
4.4 Potential Energy of an Elastic Body 142
4.5 The Rayleigh-Ritz Method 146
4.6 Comments Regarding the Rayleigh-Ritz Method 149
4.7 Strong Form and Weak Form 151
4.8 Finite Element Form of the Rayleigh-Ritz Method 156
4.9 Convergence of Finite Element Solutions 161
4.10 Additional Formulations. Hybrid Elements 165
Analytical Problems 171
Chapter 5 FORMULATION TECHNIQUES: GALERKIN
AND OTHER WEIGHTED RESIDUAL METHODS 179
5.1 Galerkin Method 179
5.2 Methods of Weighted Residuals (MWR) 182
5.3 Galerkin Finite Element Method in One Dimension 186
5.4 Integration by Parts 191
5.5 Galerkin Finite Element Method in Two Dimensions 193
5.6 A Mixed Formulation 195
Analytical Problems 198
Chapter 6 ISOPARAMETRIC ELEMENTS 202
6.1 Introduction 202
6.2 Bilinear Quadrilateral (Q4) 205
6.3 Quadrature: [k] Obtained by Numerical Integration 209
6.4 Quadratic Quadrilaterals (Q8, Q9) 213
6.5 Hexahedral Isoparametric Elements 217
6.6 Incompatible Modes. Nodeless D.O.F. 219
6.7 Static Condensation 221
6.8 Choices in Numerical Integration 223
6.9 Load Considerations 227
6.10 Stress Calculation 230
6.11 Effect of Element Geometry 234
6.12 Validity of Isoparametric Elements 237
6.13 Patch Test 238Contents xi
6.14 A 2D Application 240
6.15 A 3D Application 244
Analytical Problems 247
Computational Problems 255
Chapter 7 ISOPARAMETRIC TRIANGLES AND TETRAHEDRA 259
7.1 Reference Coordinates. Shape Functions 259
7.2 Element Characteristic Matrices 262
7.3 Analytical Integration. Area and Volume Coordinates 264
7.4 Numerical Integration 266
Analytical Problems 268
Chapter 8 COORDINATE TRANSFORMATION AND SELECTED
ANALYSIS OPTIONS 271
8.1 Transformation: Introduction and Vector Forms 271
8.2 Strain, Stress, and Material Property Transformation 273
8.3 Transformation of the Characteristic Matrix 275
8.4 Changing the Directions of Restraints 276
8.5 Connecting Dissimilar Elements. Rigid Elements 278
8.6 Higher Derivatives as Nodal D.O.F. 282
8.7 Fracture Mechanics. Singularity Elements 283
8.8 Elastic Foundations. Infinite Media 286
8.9 Structural Modification. Reanalysis 292
8.10 Tests of Element Quality 293
Analytical Problems 295
Computational Problems 299
Chapter 9 ERROR, ERROR ESTIMATION, AND CONVERGENCE 300
9.1 Sources of Error 300
9.2 IlFConditioning 302
9.3 The Condition Number 306
9.4 Diagonal Decay Test 308
9.5 Residuals 309
9.6 Discretization Error. Convergence Rate 310
9.7 Multimesh Extrapolation 315
9.8 Mesh Revision Methods 318
9.9 Gradient (Stress) Recovery and Smoothing 320
9.10 A-Posteriori Error Estimate 326
9.11 Adaptive Meshing 329
Analytical Problems 331
Computational Problems 335
Chapter 10 MODELING CONSIDERATIONS AND SOFTWARE USE 336
10.1 Introduction 336
10.2 Physical Behavior Versus Element Behavior 337
10.3 Element Shapes and Interconnection 340Contents
10.4
10.5
10.6
10.7
10.8
10.9
10.10
10.11
10.12
10.13
10.14
10.15
10.16
10.17
10.18
Test Cases and Pilot Studies 342
Material Properties 344
Loads and Reactions 347
Connections in Structures 348
Boundary Conditions 352
Repetitive Symmetry 354
Stress Concentrations. Submodels 356
Substructures 358
Planning an Analysis 360
Common Mistakes 363
Checking the Model 365
Critique of Computed Results 366
Design Optimization 369
Software 370
Concluding Remarks 371
Analytical Problems 371
Computational Problems 372
371
Chapter 11 FINITE ELEMENTS IN STRUCTURAL DYNAMICS
AND VIBRATIONS 373
11.1 Introduction 373
11.2 Dynamic Equations. Mass and Damping Matrices 374
11.3 Mass Matrices: Consistent, Diagonal, and Other 377
11.4 Natural Frequencies and Modes 383
11.5 Damping 388
11.6 Reduction of the Number of D.O.F. 390
11.7 Response History: Modal Methods 394
11.8 Response History: Ritz Vectors 398
11.9 Component Mode Synthesis (CMS) 400
11.10 Harmonic Response 405
11.11 Response History: Direct Integration Methods 407
11.12 Explicit Direct Integration 409
11.13 Implicit Direct Integration 416
11.14 Direct Integration: Stability and Accuracy Analysis 421
11.15 Analysis by Response Spectra 426
11.16 Remarks. Modeling Considerations 429
11.17 An Application: Vibration and Harmonic Response 436
11.18 An Application: Response History 439
Analytical Problems 444
Computational Problems 451
Chapter 12 HEAT TRANSFER AND SELECTED FLUID PROBLEMS 454
12.1 Heat Transfer: Introduction 454
12.2 Finite Element Formulation 459
12.3 Radiation. Nonlinear Heat Transfer Problems 462
12.4 Transient Thermal Analysis 464
12.5 Modeling Considerations. Remarks 467Contents xiii
12.6
12.7
12.8
12.9
An Application 469
Acoustic Frequencies and Modes 474
Fluid-Structure Interaction 477
Plane Incompressible Irrotational Flow 480
Analytical Problems 482
Computational Problems 486
Chapter 13 CONSTRAINTS: PENALTY FORMS, LOCKING, AND
CONSTRAINT COUNTING 489
13.1 Explicit Constraints. Transformation Equations 489
13.2 Lagrange Multipliers to Enforce Constraints 492
13.3 Penalty Functions to Enforce Constraints 493
13.4 Implicit Penalty Constraints and Locking 495
13.5 Constraint Counting 499
13.6 Remarks About Techniques for Incompressible Media 502
Analytical Problems 504
Chapter 14 SOLIDS OF REVOLUTION 508
14.1 Introduction. Elasticity Relations for Axial Symmetry 508
14.2 Axisymmetric Solid Elements 510
14.3 An Application 512
14.4 Loads Without Axial Symmetry: Introduction 516
14.5 Loads Without Axial Symmetry: Some Details of FEA 521
Analytical Problems 524
Computational Problems 527
Chapter 15 PLATE BENDING 530
15.1 Introduction. Plate Behavior 530
15.2 C1 (Kirchhoff) Plate Elements 536
15.3 (Mindlin) Plate Elements 542
15.4 Mindlin Beam. More Devices for C° Plate Elements 547
15.5 Boundary Conditions. Test Problems 551z
15.6 An Application 553
Analytical Problems 556
Computational Problems 559
Chapter 16 SHELLS 561
16.1 Introduction 561
16.2 Circular Arches and Arch Elements 563
16.3 Shells of Revolution 570
16.4 General Shells: Three- and Four-Node Elements 574
16.5 General Shells: Curved Isoparametric Elements 578
16.6 Test Cases. Remarks 583
16.7 An Axisymmetric Shell Application 586
Analytical Problems 588
Computational Problems 591xiv Contents
Chapter 17 NONLINEARITY: AN INTRODUCTION 595
17.1
17.2
17.3
17.4
17.5
17.6
17.7
17.8
17.9
17.10
Nonlinear Problems 595
Some Solution Methods 596
Plasticity: Introduction 602
Plasticity: General Formulation for Small Strains 606
Plasticity: Formulation for Von Mises Theory 609
Plasticity: Some Computational Procedures 612
Nonlinear Dynamic Problems 616
Problems of Gaps and Contact 619
Geometric Nonlinearity 621
Modeling Considerations. Remarks 626
Analytical Problems 630
Computational Problems 636
Chapter 18 STRESS STIFFNESS AND BUCKLING 639
18.1
18.2
18.3
18.4
18.5
18.6
18.7
Introduction. Energy Considerations 639
Bar and Beam Elements 642
Plate Elements 645
A General Formulation 646
Calculation of Buckling Loads 648
Remarks on Stress Stiffness and Its Uses 650
Remarks and Examples 653
Analytical Problems 656
Computational Problems 661
Appendix A MATRICES: SELECTED DEFINITIONS AND
MANIPULATIONS 663
Appendix B SIMULTANEOUS ALGEBRAIC EQUATIONS 668
Appendix C EIGENVALUES AND EIGENVECTORS 675
B.1
B.2
B.3
Overview 668
Direct Solvers 668
Iterative Solvers 671
REFERENCES 682
C.1
C.2
C.3
C.4
Overview 675
The Standard Eigenproblem 675
The General Eigenproblem 676
Solution Algorithms 679
INDEX 711NOTATION
Symbols used throughout most of the book are listed. Symbols less frequently used, or
that have different meanings in different contexts, are defined where they are used. Matrices and vectors are identified by boldface type.
MATHEMATICAL SYMBOLS
an
da
Rectangular matrix or square matrix, diagonal matrix
Column vector, row vector
Matrix transpose (thus { } = |_ J7)
Matrix inverse, transpose of inverse (= inverse of transpose)
Norm of a matrix or vector
Time differentiation; for example, u = du/dt. u = d2u/dr
Partial differentiation if the following subscript is a letter; for
example w,x = dw/dx, = d2w/dxdy
T
Represents an an an , where n = n(ab a2> • • •»
dtZj 3^2 ^an
1
LATIN SYMBOLS
A
{a}
B
[B]
cm
[C]
D
{D}, {d}
{D} -
d.o.f.
E
[E]
{F}
fGhI
rij
j
[j]
k
[K], [k]
[KJ, [kJ
Area or cross-sectional area
Generalized d.o.f. (also known as generalized coordinates)
Bulk modulus, B = E/(3 - 6p)
Spatial derivatives of field variables are [B] {d}
Field continuity of degree m (Section 3.2)
Damping matrix; constraint matrix
Displacement; flexural rigidity of a plate or a shell
Nodal d.o.f. of structure and element, respectively
Amplitudes of nodal d.o.f. (as in vibration or buckling)
Degree(s) of freedom
Modulus of elasticity
Matrix of elastic stiffnesses; [E] = £ in one dimension
Body forces per unit volume
Cyclic frequency of vibration,/ = a)/2ir, flux
Shear modulus
Characteristic length; convective heat transfer coefficient
Moment of inertia of cross-sectional area
Unit matrix, also called identity matrix
Determinant of [J]
Jacobian matrix (Section 6.2)
Spring stiffness, or bar stiffness AE/L, or thermal conductivity
Conventional stiffness matrix of structure, element
Stress stiffness matrix of structure, element
xvxvi Notation
L,Lp
I, m, n
[M], [m]
^els
[N]
O
[0], {0}
{P}
p
<7
{R}
{re}
5Tt
[T]
u,uQ
U, V, w
{u}
V
x,y,z
Length of element, length of structure
Direction cosines
Mass matrix of structure, element
Number of elements
Shape (or basis, or interpolation) functions
Order; for example O(h") = a term of order /z2
Null matrix, null vector
Externally applied concentrated loads on structure nodes
Pressure; degree of a complete polynomial
Distributed load, per unit length or per unit area
Total load on structure nodes; {R} = {P} + {re}
Loads applied to nodes by an element (Section 2.5)
Surface or surface area
Temperature
Thickness; time
Transformation matrix
Strain energy, strain energy per unit volume
Displacement components in coordinate directions
Vector of displacements, {u} = [w v
Volume
Cartesian coordinates
GREEK SYMBOLS
a
tn
{e}, {e0}
t?
0x, Oy,
[k], {k}
AV$ np
{«•}, {o-q}
°e
[ * ]
{$}
<^rCoefficient of thermal expansion; penalty number
Jacobian matrix inverse, [T] = [J]-1
Vector of strains, vector of initial strains
A global error measure, computed from the gradient field
Rotation components about coordinate axes
Matrix of thermal conductivities, vector of curvatures
Eigenvalue; Lagrange multiplier
Poisson’s ratio
Damping ratio (ratio of actual damping to critical damping)
Reference coordinates of isoparametric elements
A functional; for example fl^ = potential energy functional
Mass density
Vector of stresses, vector of initial stresses
von Mises stress, Eq. 3.12-2 (also called effective stress)
Modal matrix
Surface tractions
Circular frequency in radians per second, spectral matrix
INDEX
Accuracy, see Bounds; Convergence; Error
Acoustics, 474-477
Active column storage, 43
Adaptive meshing, 329-331
Added mass, 383, 431,479
Admissible displacement or field, 88, 138
Analogies, problem areas, 469
Anisotropy of element behavior, 105
Arches, theory and elements, 563-570 *
Arc-length method, 599
Area coordinates, 264-265
Assembly of elements
congruent transformation, 160-161
direct stiffness method, 23, 32-36
matching of d.o.f., 28
Attachment d.o.f. and modes, 359, 400
Average acceleration algorithm, 417-418,
420-426
Axial symmetry
strain-displacement relations, 510, 512
stress-strain relation, 509
see also Shells; Solids of revolution
Bandwidth, of matrix, 44
Bar element
heat conduction, 21-22
mass matrices, 377-378
stress stiffness matrix, 643
three nodes, 203-205 '
two nodes, 20-21, 29-31, 47-48, 89-90,
186-188
Basis, reduced, 390-391
Basis function, 84
Beam, curved, 244-247, 338-339, 341
Beam element
beam-column, 640-641
connection to solid, 279, 349
Euler-Bernoulli beam, defined, 24
limitations of, 28-29, 338
mass matrices, 378-379
Mindlin element, 495-496, 547-550
standard element, 24-29, 32, 49-52, 90-91,
145, 170-171
stress stiffness matrix, 643
Timoshenko beam, defined, 24
Bilinear element, 96-100, 205-209
Bimoment, 29
Biquadratic element, 101-102, 213-215
Blast loading, 408
Body force, see Loads
Boundary conditions
computational procedures, 404-2, 276-277,
305, 354
essential (principal), 137, 151—152, 155
in heat transfer, 457-4-63
inadequate support, 38-39, 364, 523-524
in modeling, 352-354
nonessential (natural), 137, 151-152, 155
for plate bending, 551-552
for solids of revolution, 512, 523-524
on stress, 82, 119
for symmetry, various types, 54-57, 354-356
Boundary elements, 290-291
Bounds
on bifurcation buckling load, 650
Gerschgorin, 413
with hybrid elements, 168
with incompatible elements, 110, 221
by modeling choices, 352, 436
on Rayleigh quotient, 678
on Rayleigh-Ritz solutions, 150-151
with reduced integration, 223
on vibration frequencies, 382-383
Box beams, 520
Brick element, 102-104, 217-219
Bubble function, 215
Buckling
bifurcation, 641-642, 648, 653-655
bounds, on bifurcation load, 650
collapse, 627, 642, 654, 655
imperfection sensitivity, 654
limit point, 599, 642, 653-654
nature of, 639, 642
nonlinearities, 650, 653-656
pressure load, 651
snap-through, 599, 642, 655
symmetry, use of, 57
thin-walled structures, 651, 654
see also Stress stiffening
Bulk modulus, 496
711712 Index
Cables and chains, 626, 651
Central difference methods, 40SM-16, 421-426
Centrifugal softening, 652-653
CFL condition, 413
Characteristic matrix, 19
Checkerboarding, 503
Checking for mistakes, 13-15,236-237,363-369
Choleski method, 670-671
Circulation modes, 480
Cm continuity, defined, 84
Collapse load, 642, 654
Collocation residual method, 183
Compatibility
in elasticity, 81
extent satisfied in FEA, 118-119
interelement, 105
see also Incompatible elements
Completeness
convergence and, 149-150, 313
of polynomial field, 105-106, 313
Component mode synthesis, 400-405
Condensation (reduction of order)
in buckling problems, 649-650
in dynamic problems, 390-394, 396, 399,
400-405
in modal methods, 396
by Ritz vectors, 399
in static problems, 221-222, 352, 359, 490
Condition number, 44, 306-308
see also Ill-conditioning
Conduction, see Heat conduction and transfer
Congruence transformation, 160-161, 677
Conjugate gradient solution, 672-674
Connections
of dissimilar elements, 279-282, 349, 358
interelement, 117-118, 342
partial, at nodes, 39
in structures, 348-352
Conservative system, defined, 137
Consistent penalty method, 503
Constant-strain triangle, 93-95, 102, 262-263
Constitutive matrix, see Stress-strain relations
Constraint modes, 400
Constraints
consistent penalty method, 503
counting of, 226, 500-502, 546-547,
568-569
incompressibility, 94, 496-497, 502-504
and Lagrange multipliers, 492-493, 620-621
multipoint, 281, 489
penalty, explicit, 493-495, 621
penalty, implicit, 495-499
and quadrature rule, 498-500
ratios, 501-502
shear, beams and plates, 495—496, 544-547
single-point, 489
transformation methods for, 276-282,
489—491
see also Locking
Contact, sliding, 353
Contact problems, 340, 492, 595, 619-621
Continuity, degree of, 84
Convected coordinates, 622
Convergence
equilibrium iterations for, 597-598
extrapolation for, 315-318
A-refinement, 318-320
monotonic, 162-163, 315-316
in nonlinear problems, 601-602, 626, 629
p-refinement, 318-320
rate of, dynamics, 388, 409, 411, 419^-20
rate of, statics, 164-165, 310-315, 325, 383
of Rayleigh-Ritz method, 149-151
requirements for, in FEA, 104, 161—163
see also Bounds; Error
Coordinate transformation, see Transformation
Corotational formulation, 622-625
Coupled field problems
defined, 3
fluid-structure, 477^480
Courant number, 414
Cracks, fracture and, 283-286
Craig-Bampton method, 400-405
Critical load, defined, 639
CST element, 93-95, 102, 262-263
Curved beam, 244-247, 338-339, 341
D.o.f., see Degrees of freedom
Damping
algorithmic, 389,419-421, 423^26
consistent matrix for, 376
modal, 390, 395
proportional, 389-390
ratio, 384
Rayleigh, 389-390
types of, 388-389
Degenerate (degraded) elements, 264, 285, 563
Degrees of freedom
defined, 8
generalized, 7-8, 146, 157
hierarchic, 305-306, 319
higher derivatives as, 282-283
nodeless, 109, 219
relative, 109, 305-306, 319Index 713
Design of experiments, 363
Determinant, calculation of, 671
Developable surface, 535
Diagonal decay test, 308-309
Direct integration in dynamics
accuracy of, 414—416, 419—420, 423^-26,
434
algorithmic damping and, 389, 419—421,
423^126
average acceleration algorithm, 417-418,
420-426
central difference methods, 409^416,
421-426
cost of, relative, 408 «
error, control of, 436, 619
error, order of, 409, 411, 419^420
explicit methods, 407-416, 616-618
in heat transfer, 465-466
implicit methods, 407^409, 416-421,
618-619
initial conditions, 412, 415-416, 418, 419
mass matrix for, 411, 413, 425-426, 434
mixed time integration, 421
Newmark methods, 416-421
nonlinear problems, 616-619, 629
operator splitting, 421
overview, 407^-09
spurious modes and, 227, 383, 407,412,431
stability of, 411, 419-420, 421^123,466,617
trapezoidal rule, 418
Direct stiffness method, 23, 32-36, 161
Discrete Kirchhoff elements, 538-541
Discretization, 4
Distorted elements, 234-236, 238, 340-342,
366, 570, 577-578
Divergence theorem, 191-192
Drilling d.o.f., 106-108
Dynamic problems
basic equations, 189, 374-376
classified, 373-374, 455
inverse problem (identification), 435
see also Damping; Direct integration in
dynamics; Eigenproblems; Harmonic
response; Mass and mass matrices; Modal
methods; Response history; Response
spectra; Vibration
Dynamic relaxation, 601
Dynamic stiffness matrix, 385
Effective stress, 117, 232-233, 609
Eigenproblems
buckling, 648-650
hand calculation, 385-387
orthogonality of modes, 395, 678
Rayleigh quotient, 387-388, 432, 678-679
theory and algorithms, 675-681
vibration, 385
Eigenvalue test of elements, 293-294
Elastica, 626
Elastic support, 286-288, 353
Energy
in buckling problems, 639-641
complementary, 167
conservation of, 139-140
error measure, use in, 294, 326-328
in nonlinear dynamics, 617, 619
stationary principle, 137-140
strain energy density, 142-143
in terms of d.o.f., 160
in vibration (Rayleigh quotient), 387
Equation solving, 42-46, 668-674
Equilibrium
differential equations of, 81-82,156
extent satisfied in FEA, 88, 119, 194
iteration to satisfy, 322, 597-598, 619, 625
nodal, 33-34
in patch stress recovery, 325
Error
a posteriori error estimate, 326-328
checking, in modeling, 363-364
discretization, 4, 165-166, 301, 310-315
extrapolation to reduce, 315-318
indicator, eigenproblems, 328,436, 650
indicator, flux or stress, 14, 326, 328, 468
indicator, modal methods, 396-397
indicator, nonlinear dynamics, 619
iterative reduction of, 310, 597-598, 619,
625
modeling, 3^4, 300-301
numerical, 4, 301-310
order of, dynamics, 383, 388, 409, 411, 419-
420
order of, statics, 165-166, 310-315, 325
of Rayleigh quotient, 388, 678
singularities, 314-315
sources, classified, 300-301
tests for, 301-302, 306-310, 326-328, 436,
601-602, 619
ZZ error estimate, 326-328
see also Bounds; Convergence; Illconditioning; Locking
Euler equations, 152-153
Examples, application of FEA
axisymmetric shell, 586-588714 Index
axisymmetric solid, 469-473, 512-516
beam element structure, 59-62
buckling, nonlinear, 655-656
elastic foundation, 553-556
harmonic response, 438^-39
heat transfer, 469-472
plane, 119-124, 240-244
plate bending, 553-556
response history, 439-442
response spectra, 442-443
shrink fit, 512-516
solid, 3D, 244-247
spinning disk, 512-516
thermal stress, 121-124, 472-473
vibration, 436-438
wave propagation, 414-416, 617-618
Excitation, 374
Experiment
comparison with, 367, 435
numerical, 363
Extrapolation, multimesh, 315-318
Fills, in equation solving, 43, 44, 46, 671
Finite element method
advantages of, 1-2, 9-10
analysis procedure, 13, 15, 20
dangers of, 15-16, 331, 371, 436
defined, 5, 7
history of, 10-11
nature of, 5-8, 118-119, 150, 161
see also Modeling
Finite prism method, 521
Finite strip method, 520
Flexural rigidity, 533
Fluid flow, 480-482
Fluid-structure interaction, 477—480
Follower force, 348, 626, 651
Forced vibration, 373, 405—407, 438—439
Forcing function, 374
Foundation, elastic, 286-288, 353
Fourier series, 516-524
Fracture mechanics, 283-286
Frame invariance, 105, 227
Framework analogy, 10
Frequency response analysis, 405-407, 438
439 . . '
Frontal solution, 44, 671
Fully stressed design, 370
Functionals, various, 136, 143, 152-153,197
198, 459
Galerkin method, 156, 179-197
Gaps and contact, 288, 340, 353, 595, 619-621
Gauss elimination, 45-46, 668-670
Gauss quadrature, 209-213
Generalized d.o.f., described, 7-8, 146
Geometric isotropy, 105, 227
Geometric nonlinearity, 340, 367-368, 535,
595, 621-626, 653-656
Geometry, element, 234-236, 238, 340-342,
366, 570, 577-578
Gerschgorin bound, 413
Global stiffness, defined, 22
Global-local model, 356-358
Green strain, 621-622
Guyan reduction, 390-394
Gyroscopic effects, 388
Harmonic function, 193
Harmonic response, 373, 405^107, 438^439
Heat conduction and transfer
bar element, 21-22, 190-191
boundary conditions, 457^463
error measure, 328
formulation, 153, 157-159, 456-462
modeling considerations, 467-468
nomenclature and units, 454-455
radiation, 462-463
transients, 464-466
Helmholtz equation, 475
Hilbert matrix, 302
Hinge, 39, 58, 279, 349, 351-352
History of FEA, 10-11
Hourglass mode, see Spurious modes
Hybrid formulation, 165-171, 233
Ill-conditioning
condition number, 306-308
in equation solving, 44-45, 302, 303, 672,
674
in least squares methods, 184
modeling situations, 281, 303-308, 494, 511
with slender ring elements, 511
Impact loading, 408
Imperfection sensitivity, 654
Incompatible elements, 7, 105, 109-111, 117—
118,219-221,537
Incompressible materials, 94, 308, 496^497,
502-504
Inextensibility condition, 565
Infinite elements and media, 286-291
Initial stiffness method, 599
Initial stress and strain
bar elements, 52-54
calculation procedure, 52, 115
element load formula, 89Index 715
energy expression, 143
stress calculation, 53, 115
stress-strain relation, 78-79
Instability, see Buckling; Mechanisms;
Spurious modes
Integration
analytical, triangles and tetrahedra, 264-266
by parts, 154, 156, 181, 191-192
see also Numerical integration
Interaction
fluid-structure, 477^80
in general, 3
Internal forces, 374,412, 617
Interpolation
choice of functions, 104-106
defined, 83
Hermitian (C1), 86-87
Lagrange’s formula (C°), 85-86
Intrinsic coordinates, 203,205-206,259,264-265
Inverse iteration, 680
Isoparametric elements
bar example, 203-205
basics, plane, 205-209
basics, solid, 217-218
defined, 202
shells, 578-583
triangles and tetrahedra, 259-264
validity of, 237-240
see also Numerical integration
Iterative improvement, 310, 322, 329-331,
369-370, 597-598, 625
Jacobian, 204, 207, 218, 262, 582
Joints, see Connections
Kinematic mode, see Spurious modes
Kirchhoff plate elements, 536-541
Lagrange elements, 97, 101, 215, 219, 545-
547, 583
Lagrange multipliers, 492-493
Lagrange’s interpolation formula, 85-86
Lanczos method, 390, 674, 680
Lap joint, 351
Laplace’s equation, 193
Large displacement and strain, 38, 57,163-
164, 535, 570, 621-623, 628
Least squares residual methods, 183
Least squares solution, 184, 185, 233, 324
Limit point, 599, 653-654
Loads
acceleration, 82
axisymmetric, 347, 512
body force, 81, 113-114
concentrated, 90, 112-114, 347
consistent, 47,49, 51, 89,111-115
in contact problems, 619-621
corrective (imbalance), 397-398, 597-601,
612, 615, 625
distributed, and pressure, 47-49, 344, 348,
626, 651
by element, to element, 21, 33
follower forces, 348, 626, 651
Fourier series for, 517-518
gravity, 46, 113-114
on incompatible elements, 111-112, 220,
221
inertia, 378
initial strain and stress, 52, 89,145-146, 160
mesh layout, effect of, 344, 541
moment (couple), 114, 279, 347
moving, 435
multiple load cases, 45-46, 668, 674
on plates and shells, 536, 541
prestress, 350
reduced (lumped), 46, 49, 51-52
spinning, 512
symmetry and, 55-57, 355
thermal, 52-53, 79, 89-90, 145-146, 348
tractions (surface), 82, 112-114, 227-230
work-equivalent, 111-115
Locking
constraint counting and, 500-502, 546-547,
568-569
dilatational, 94, 497, 612
discussed, 93-95, 498-500
incompressibility and, 94,497
membrane, 567-570, 582-583
penalty constraints, implicit, 495-500
quadrature rule and, 498-500
selective integration and, 227, 500-501,
543-549, 568-570
shear, 98, 99, 496, 543, 546, 548-549
volumetric, 94, 497, 612
LST element, 95-96, 102, 107, 263
Marguerre shell theory, 566
Mass and mass matrices
added mass, 287, 383, 431, 479
choice of, modeling, 382-383, 411, 413,
426, 430^-31, 434, 436
condensation of, 390-394
consistent, 376, 378-379
HRZ lumping, 380-381
nonstructural, 287, 383, 431, 479716 Index
optimal lumping, 381-382
particle-lumped, 377-378
Master d.o.f., 359, 391
Material nonlinearity, see Plasticity
Material properties, see Stress-strain relations
Mathematical model, defined, 3
Matrices, definitions and manipulations, 663
667
Mechanisms, see Spurious modes
Membrane-bending coupling, 535, 563, 566
567
Membranes, 585-586, 651
Mesh generation and revision, 242, 318-320,
329-331, 341-342
Mindlin elements
arches, 568-570
beams, 495^196, 547-550
plates, 534—535, 542-550
shells, 573-574, 578-583
Mistakes, common, 363-365
Mixed formulation, 166, 195-197, 538
Modal methods
error correction for, 396-398
in harmonic response, 406^107
in heat transfer, 464—465
modal synthesis, 400-405
mode acceleration method, 397
in nonlinear problems, 398
number of modes needed, 397, 433, 442
orthogonality of modes, 388, 395, 678
problem type for, 408, 433
spurious modes and, 227, 412, 431
static correction, 397-398
theory of, 395-397
versus Ritz vectors, 398
Modeling
dynamic problems, 429-436
element selection, 337-339
error of, 3-4, 300-301
general procedure, 11-13, 336-337, 360-363
heat transfer, 467-468
mathematical model, 3
nonlinear problems, 596, 626-629, 651-654
see also specific problem areas
Modification, of structures, 292-293, 363
Multigrid methods, 320, 673-674
Natural coordinates, 203, 205-206, 259,
264-265
Natural frequencies, see Vibration
Newmark methods, 416-421, 425-426
Newton-Raphson methods, 597-598
Nodeless d.o.f., 109, 219
Nonconforming elements, see Incompatible
elements
Nonlinearity
convergence, and criteria for, 601-602, 629
in dynamic problems, 398,409,412,616-619,
629
gaps and contact, 288, 340,353, 595, 619-621
geometric, 340, 367-368, 535, 595, 621
626, 653-656
material, 340, 595, 606, 627
modeling, 596, 626-629, 602
radiation heat transfer, 463—464
solution methods, general, 464, 596-602
sources of, 288, 340, 463, 535, 595
substructures, value of, 360
see also Buckling; Plasticity
Norms, matrix, 666-667
Numerical dissipation, see Damping,
algorithmic
Numerical experiments, 363
Numerical integration
and accuracy, 213, 235-236
full, 223, 499
Gauss quadrature, 209-213
reduced and selective, 221, 223-227, 499
500, 543-547, 569-570
thickness direction, 213, 582, 585, 629
shell elements, 582
triangles and tetrahedra, 266-268
see also Locking; Spurious modes
Offsets, 280-281
Optimization, design, 369-370
Ovalization, 338-339, 585
Overlays, 320
Parasitic shear, 98-100, 227, 496
see also Locking
Patch recovery for gradients, 323-326
Patch test, 238-240, 552
Penalty function, 493
Perforated plate, 345-346
Petrov-Galerkin method, 182
Pilot studies, 344
Pipe bend, 338-339, 585
Plane strain, 94-95, 501-502
Plane stress, 79, 533, 581
Plasticity
calculations, general, 612-616
flow rule, 606-608, 611
formulation, general, 606-609
hardening rule, 604, 606-607, 610Index 717
uniaxial, 603-605, 608-609
von Mises theory, 609-612
yield criterion, 604, 606-607, 609
Plate bending and plate elements
boundary conditions, 551-552
discrete shear elements, 550
FE surface definition, 368, 536
finite strip method, 520
folded plates, 520, 576, 583
Kirchhoff (C1) elements, 536-541
layered, 535
limitations of, 530, 535
membrane-bending coupling, 535
Mindlin (C°) elements, 542-550 3
stress calculation, 552
test cases, 552-553
theory of plates, 531-535
various formulations, 550
Poisson equation, 193
Postprocessing, 13, 365-369
Potential energy principle, 138-140
Potential function (fluids), 480
Prandtl-Reuss relations, 611
Preliminary analysis, 13, 337, 361
Preprocessing, 13
Pressure, see Loads
Pressure calculation, 503-504
Prestress, 350, 513-514
see also Initial stress and strain
Profile, of matrix, 43, 44
Programs and programming, 370-371
Q4 element, 96-100, 102,168-170, 205-209
Q6 and QM6 elements, 109-111, 219-221
Q8 and Q9 elements, 100-102, 213-217
Quadrature, see Numerical integration
Quarter-point elements, 284-286
Quasiharmonic equation, 193-194
Quasi-Newton methods, 600-601
Raasch problem, 584-585
Radiation (acoustics), 476-477
Radiation, see Heat conduction and transfer
Rank deficiency, 213, 223, 226, 665
Rayleigh quotient, 387-388, 678-679
Rayleigh-Ritz method, 136, 146-150, 156-161
Reanalysis, after modifications, 292-293
Reciprocal theorem, 37, 344
Recordkeeping, 360, 364—365
Reduction of order, see Condensation
Refinement methods, mesh, 318-320, 329-331
Reflection, of waves, 287, 290, 476-477
Release of d.o.f., 351-352
Repetition of form, 354-356, 359
Residual bending flexibility, 549
Residuals
in dynamics, 396-398,619
as error measure, 309-310
in nonlinear problems, 597-598, 601-602,
625
weighted residual methods, 155-156,179-197
Resonant frequency, 405
Response history
choice of method, 373-374, 408—4-09, 432
434
defined, 373-374
response spectra, 426—4-29, 442-443
Ritz vectors for, 398—4-00
see also Direct integration; Modal methods
Response spectra, 426-429, 442-443
Restart capability, 366
Richardson extrapolation, 315-316
Rigid body motion, 38, 104, 163-164, 293,
364, 385, 396,407, 523-524
Rigid links and elements, 278-282
Ritz vectors, 398^400
Rotational periodicity, 354-356
Sampling points, see Numerical Integration
Secant stiffness methods, 600-601
Serendipity elements, 100, 215, 219, 545-546,
583
Shape function, meaning of, 84
Shear center, 29, 338
Shear deformation, see Transverse shear
deformation
Shear lag, 338
Shells
arches, as special case, 563-570
axisymmetric, 339, 561-562, 570-574
behavior of, 561-562
C° elements, 573-574, 578-583
C1 elements, 571-573,574-578
FE surface definition, 368, 536
isoparametric elements for, 578-583
layered, 585
Marguerre theory, 566
mechanisms in elements, 575-576
membrane locking, 567-570
membrane-bending coupling, 566-567, 576
modeling suggestions, 562-563, 576,
585-586
test cases, 583-585
warped quadrilateral elements, 577-578
Shock loading, 408718 Index
Shock spectrum, 427
Shrink fit, 350, 468, 513-514
Single element test, 294
Singularities
and convergence, 314-315
elements for, 284-286
of field quantity, 124, 283-284, 290, 291,
329, 330, 341, 347
stiffness matrix, causes of, 38-39, 108, 364,
523-524, 599, 651
Skyline, of matrix, 42-43
Slave d.o.f., 280, 359, 391
Software, remarks about, 370-371
Solids of revolution
finite elements for, 510-512
nonaxisymmetric conditions, 516-524
strain-displacement relations, 510-512, 519
stress-strain relation, 509, 519
Sparsity, matrix, 37, 42^44, 671
Spectral matrix, 395
Spectral stability, 421-423
Spin softening, 652-653
Spurious modes
communicable, 224—227, 546
definition and terminology, 39, 223
drilling d.o.f. and, 108, 575, 583
in dynamics, 227, 383, 407, 412, 431
plates and shells, 544-546, 575-576, 583
pressure (checkerboarding), 503
stabilization of, 227, 383, 412, 431, 545,
575, 583
tests for, 240, 293, 364
see also Hinge
Stationary principles, see Variational methods
Stiff region, 281-282, 302-306, 494
Stiffener, eccentric, 280-281
Stiffness matrix
assembly of, 23, 32-36, 51, 160-161
formula for, derived, 88-89, 159-160, 194
195
formula for, from eigensolution, 307, 676
numerically integrated, 209-213
physical meaning of, 21
properties of, 36-39, 141-142
see also Stress stiffening
Strain energy density, 142-143
Strain-displacement relations
arches, 564
Cartesian coordinates, 80
large strains, 621-622
plate bending, 532, 537, 542
shell of revolution, 572, 573
solid of revolution, 510, 512, 519
Stream function, 480
Stress and stress calculation
accuracy of, 6, 115, 151, 314, 325
averaging and smoothing, 114, 116, 320-326
bars and beams, 47-54
best locations for, 6, 225, 230-231, 266, 323
concentrations, 119-121, 356-357
contours, as error indicator, 14
deviatoric, 502, 609
discontinuous, 116-118
effective stress, 117, 232-233, 609
element geometry, effect of, 234-236, 238,
340-342, 366, 577-578
extrapolation from Gauss points, 231-232
with incompatible elements, 221
intensity factor, 283-284, 286
invariants of, 116-117
iteration for improvement, 322
from nodal displacements, 115
from nodal forces, 233-234
patch recovery, 323-326
residual, 627
singularities, 124, 283-286, 290, 329, 330,
341, 347
superconvergence, 314, 325
surface definition and, 368, 536
thermal stress, 52-54, 115, 234, 467
trajectories, 368
von Mises stress, 117, 232-233, 609
Stress stiffening
bar, beam, plate elements, 643, 645-646
discussed, 625, 639
forms of matrix, 650-651
general formulation, 646-648
spin softening, 652-653
Stress-strain relations
anisotropic, 78, 344-345, 509, 519, 535
isotropic, 79, 94, 509, 519
plane strain, 94
plane stress, 79, 533, 581
plates and shells, 533-535, 571, 581
substitute, 345-347, 549
temperature-dependent, 79
Strong form, 136, 151-156
Structure, defined, 1
Sturm sequence, 680
Sub- and super-parametric elements, 202, 238
Subdomain residual method, 183
Submodels, 356-358
Subspace iteration, 390, 680
SubstructuresIndex 719
discussed, 358-360
dynamic, 400-405
repeating, 355, 359-360
Superelement, 359
Supports, see Boundary conditions
Surface tractions, 82
Surface waves, 479
Symmetry conditions
antisymmetry, 55-57
buckling problems and, 57
cyclic, 354-356
heat transfer, 468
nonlinear problems and, 57, 628
reflective, 54-56 *
repetitive, 354-356
skew, 56-57
vibration problems and, 57, 434
System, defined, 137
Test cases, in general, 342-344
Thermal stress calculation, 52-54, 79,115,
234, 467
Thermal transients, 464-466
Thin-walled construction, 337-339
see also Shells
Transformation
isoparametric, 204, 206-208, 218, 262
of material properties, 275
modeling applications, 276-282
of stiffness matrix, 29-32, 276
of stress and strain, 273-274
of support directions, 276-277
of vectors, 271-273
see also Constraints
Transition elements, 215-216, 358, 585
Transverse shear deformation
beams, 26-27, 170-171, 495-496, 549
plates, 533-535
Trapezoidal rule, 418, 605
Trefftz elements, 166
Trilinear solid element, 102-103
Underrelaxation, 600
Unsupported structure, 38, 364, 385, 396, 407
Variational methods, 136-171
Vibration
acoustical, 474-477
bars, 380, 387
beams and frames, 380, 385-386,436-437
bound on frequencies, 382-383
computational considerations, 431^132
forced vibration, 405-407, 438-439
frequency estimates, 432, 437
harmonic response, 373, 405-407, 438-439
multiple d.o.f., 384-385
order of error, 383
plates, 381, 393
single d.o.f., 383-384
spurious modes and, 227, 383, 412,431
symmetry, use of, 57, 434
see also Eigenproblems
Virtual work principle, 88, 156
Viscous relaxation, 601
Von Mises stress, 117, 232-233, 609
Warped elements, 341, 574, 577-578, 584
Wave equation, 474, 477
Wavefront solution method, 44, 671
Wave propagation, 408, 414-416, 433-434,
617-618
see also Response history
Waves, fluid surface, 479
Weak form, 136, 151-156
Weight, see Loads
Weighted residual methods, 179-198
Winkler foundation, 287-288, 353
Work, see Energy
Zero energy mode, see Spurious modes
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