كتاب A First Course in the Finite Element Method
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 كتاب A First Course in the Finite Element Method

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A First Course in the Finite Element Method
Fourth Edition
Daryl L. Logan
University of Wisconsin-Platteville  

كتاب A First Course in the Finite Element Method  A_f_c_10
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Contents
1 Introduction 1
Prologue
1.1 Brief History 2
1.2 Introduction to Matrix Notation 4
1.3 Role of the Computer 6
1.4 General Steps of the Finite Eleme~t Method 7
I.S Applications of the Finite Element Method 15
1.6 Advantages of the Finite Element Method 19
1.7 Computer Programs for the Finite Element ~ethod 23
References 24
Problems 27
2 Introduction to the Stiffness (Displacement) Method 28
Introduction 28
2.1 Definition of the Stiffness Matrix 28
2.2 Derivation of the Stiffness Matrix for a Spring Element 29
2.3 Example of a Spring Assemblage 34
2.4 Assembling the Total Stiffness Matrix by Superposition
'\ " (Direct Stiffness Method) 37 .
2.5 Boundary Conditions 39
2.6 ~otential Energy Approach to Derive Spring Element Equations 52iv A Contents
References 60
Problems 61
3 Development of Truss Equations 65
Introduction 65
3.1 Derivation of the Stiffness Matrix for a Bar Element
in Local Coordinates 66
3.2 Selecting Approximation Functions for Displacements 72
3.3 Transformation of Vectors in Two Dimensions 75
3.4 Global Stiffness Matrix 78
3.5 Computation of Stress for a Bar in the x-y Plane 82
3.6 Solution of a Plane Truss 84
3.7 Transformation Matrix and Stiffness Matrix for a Bar
in Three-Dimensional Space 92
3.8 Use of Symmetry in Structure 100
3.9 Inclined) or Skewed, Supports 103
3.10 Potential Energy Approach to Derive Bar Element Equations 109
3.11 Comparison of Finite Element Solution to Exact Solution for Bar 120
3.12 Galerkin's Residual Method and ItS Use to Derive the One-Dimensional
Bar Element Equations 124
3.13 Other Residual Methods and Their Application to a One-Dimensional
Bar Problem 127
References 132
Problems 132
4 Development of Beam Equations 151
Introduction 151
4.1 Beam Stiffness 152
4.2 Example of Assemblage of Beam Stiffness Matrices 161
4.3 Examples of Beam Analysis Using the Direct Stiffness Method 163
4.4 Distributed Loading 175
4.5 Comparison of the Finite Element Solution to the Exact Solution
for a Beam 188
4.6 Beam Element with Nodal Hinge 194
4.7 Potential Energy Approach to Derive Beam Element Equations 199. :
Contents .a. v
4.8 Galerkin's Method for Deriving Beam Element Equations 2U1
References 203
Problems 204
5 Frame and Grid Equations 214
Introduction 214
5.1 Two-Dimensional Arbitrarily Oriented Beam Element 214
5.2 Rigid Plane Frame Examples 218
5.3 Inclined or Skewed Supports-Frame Element 237
5.4 Grid Equations 238
5.5 Beam Element Arbitrarily Oriented in Space 255
5.6 Concept of Substructure Analysis 269
References 275
Problems 275
6 Development of the Plane Stress
and Plane Strain Stiffness Equations 304
Introduction 304
6.1 Basic Concepts of Plane Stress and Plane Strain 305
6.2 Derivation of the Constant-Strain Triangular Element
Stiffness Matrix and Equations 310
6.3 Treatment of Body and Surface Forces 324
'6.4 Explicit Expression for the Constant-Strain Triangle Stiffness Matrix 329
6.5 Finite Element Solution of a Plane Stress Problem 331
ReferenCes 342
Problems 343
7 Practical Considerations in Modeling;
Interpreting Results; and Examples
of Plane Stress/Strain Analysis
Introduction 350
7.1 Finite Element Modeling 350
7.2 Equilibrium and Compatibility of Finite Element Results 363
350vi A.. Contents
7.3 Convergence of Solution 367
7.4 Interpretation of Stresses 368
7.5 Static Condensation 369
7.6 Flowchart for the Solution of Plane Stress/Strain Problems 374
7.7 Computer Program Assisted Step-by-Step Solution, Other Models,
and Results for Plane Stress/Strain Problems 374
References 381
Problems 382
8 Development of the Linear-Strain Triangle Equations 398
Introduction 398
8.1 Derivation of the Linear-Strain Triangular Element
Stiffness Matrix and Eq"\lations 398
8.2 Example LST Stiffness Determination 403
8.3 Comparison of Elements 406
References 409
Problems 409
9 Axisymmetric Elements 412
Introduction 412
9.1 Derivation of the Stiffness Matrix 412
9.2 Solution of an Axisymmetric Pressure Vessel 422
9.3 Applications of Axisymmetric Elements 428
References 433
Problems 434
10 Isoparametric Formulation 443
Introduction 443
10.1 Isoparametric Formulation of the Bar Element Stiffness Matrix 444
10.2 Rectangular Plane Stress Element 449
10.3 lsoparametric Fonnulation of the Plane Element Stiffness Matrix 452
10.4 Gaussian and Newton-Cotes Quadratufe (Numerical Integration) 463
10.5 Evaluation' of the Stiffness Matrix and Stress Matrix
by Gaussian Qua~ture 469Contents • vU
10.6 Higher..Qrder Shape Functions 475
References 484
Problems 484
11' Three-Dimensional Stress Analysis 490
Introduction 490
11.1 Three-Dimensional Stress and Strain 490
11.2 Tetrahedral Element 493
11.3 Isoparametric Formulation 501
References 508
Problems 509
12 Plate Bending Element 514
Introduction 514
12.1 Basic ~ncep~ of Plate Bending 514
12.2 Derivation ofa Plate Bending Element Stiffness Matrix
and Equations 519
12.3 Some Plate EJemep.t Numerical Compa.tjsons 523
12.4 Computer Solution for a Plate Bending Problem 524
References 528
Problems 529·
13 Heat Transfer and Mass Transpor't S34
Introduction ·534
13.1 Derivation of the Basic Differenti3.l Equation 535
13.2 Heat Transfer'with Convection 538
13.3 Typical Units; Thermal Conductivities, K; and Heat-Transfer
Codficients,h 539
13.4 One-Dimensional Finite Element Formulation Using
a Variational Method S40
13.5 Two-Dimensional Finite Element FonnuJation 555
13.6 Line or Point sOurces . 564
13.7 Three-Dimensional Heat Transfer Finite Element FormUlation 566
13.8 One-Dimensional Heat Transfer with Mass Transport 569
,viii ... Contents
13.9 Finite Element Fonnulation of Reat Transferwith Mass Transport
by Galerkin's Method 569
13JO Flowchart and Examples ofa Heat-Transfer Program 574
References 577
Problems 577
;'
14 Fluid Flow 593
Introduction 593
14.f Derivation of the Basic Differential Equations 594
14.2 One-Dimensional Finite Element Fonnulation 598
14.3 Two-Dimensional Finite Element Formulation 606
14.4 Flowchart and Example of a Fluid-Flow Program 611
References 612
Problems 613
15 Thermal Stress 617
Introduction 617
15.1 Fonnulation of the Thermal Stress Problem'and Examples 617
Reference 640
Problems 641
16 Structural Dynamics and T'me-D~pen~ent Heat Transfer 647
Introduction 647
16.1 Dynamics of a Spring-Mass System 647
16.2 Direct Derivation of the Bar Element Equations 649
16'.3 Nwnerica1 Integratio~ in Time 653
16.4 Nat~l Frequencies of a One-Dimensional Bar 665
16.5 Time-Dependent One-Dimensional Bar Analysis 669
16.6 Beam Element Mass Matrices and Natural Frequ~cies 674
16.7 Truss, Plane Frame, Plane Stress/Strain, Axisymmetric,
and Solid 'Element Mass Matrices: 681
16.8 Time-Dependent Heat T~~f~ 686Contents A ix
16.9 Computer Program Example Solutions for Structural Dynamics 693
References 702
Problems 702
Appendix A Matrix Algebra 708
Introduction 708
A.l Definition of a Matrix 708
A.2 Matrix Operations 709 .
A.3 Cofactor or Adjoint Method to Determine the Inver:se of a Matrix 716.
A.4 Inverse of a Matrix by Row Reduction 718
References 720
Problems 720
Appendix B Methods for Solution
of Simultaneous Linear Equations 722
Introduction 722
B.t General Form of tPe Equations 722
B.2 Uniqueness, Nonuniqueness, and Nonexistence of Solution 723
B.3 Methods for Solving Lin~r Algebraic Equat.ions 724
B.4 Banded-5ymmetric Matrices, Bandwidth, Skyline,
and Wavefront Methods 735'
References 741
Problems 742
Appendix C Eq~ations from Elasticity \'leory 744
IntfOduction 744
CJ Differential Equations ofEquilibrium 744
C.2 StrainfDisplacement and Compatibility Equations 746
C.3 Stress/S~in Relationships 748
Reference' 751x .A Contents
Appendix.D Equivalent Nodal Forces 752
Problems 152
Appendix E Principle of Virtual Work 755
References 758
Appendix F Properties of Structural Steel and Aluminum Shapes 759
Answers to Selected Problems 773
Index 79
A
Adaptive refinement, 355
Adjoint method, 718
Admissible variation, 55
Aluminum shapes, properties of,
759-772
Amplitude, defined, 649
Approximation'functions,
72-74
compatible, 73
complete, 73-74
conforming. 73
displacement, 72-74
interpolation, 74
Aspect ratio (AR), 35).
,352-353
Axial symmetIy, 100
Axis of revolution, 412
Axis of symmetry, 412
Axisymmetric element, 9, 412-442,
684-685
applications of, 428-433
body forces, 419-420
consistent-mass matrix, 6S4-685
defined, 9, 412
discretization, 423
displacement functions, 415-417
element type, selection of, 415
equations, 419-421
introduction to, 412
pressure vessel, solution of,
422-428
sti1fDess matrix, 412-422,
423-428
strainjdispl.acement relationships,
411-419
stress/strain relationships, 417-419
surface foras, 420-421
B
Banded-syrnm.etric method, 735-741
Bar elements, 67-72, 92-100,
109:-120, 120-124, 124-127,
127-131,444-449,665-669,
669-674. See also Truss equations
analysis of. 665-669, 669-674
collocation method, 129
consistent-mass matrix, 651-653
displacement function, 68, 446,
650
dynamic analysis of, 649-653,
665--669, 669-674
equations, 124-127,447-449,
649-653
exact'solution, 120-124
finite element solution, 12():"'124
Galerldn's residual method,
124-127, 131
isoparametric formulation,
444-449
least squares method, 130
lOCal coordin'ates for, 66-12
lumped·mass matrix, 651
mass matrix, 650-653
Datural frequencies, 665-669
one-127-131,665-669,609-674
, potential energy approach,
109-120
residual methods, 124-127,
127-131
selection of, 67, 444-446,650
stiffness matrix, 66-72, 92-100,
444-449, 6~653
strain/displacement relationships,
69,446-447,650
stress. computation of, 82-83
stress/strain relationships. 69,
446-447,650
subdomain method, 129-130
three-dimensional space, 92-100
time-dependent (dynamic) stress
analysis, 649-653
time..dependent problem,
669-674
transformation matrix, 92-100
Beam element, 152-161, 161-163,
194-199,214-218,218-236,
255-269, 674-681
arbitrarily oriented, 214-218,
255-269
bending, 153-158, 255-260
boundary conditions, 161-163
defined, 152
deformations, 153-158 '
displacement function, 155..:..156
equations, 157-158, 161-163
mass matrices, 674-681
natural frequencies, 674-681
nodal hinge, 194--199
rigid plane frames, 218-236
selection of, 154
shape functions, 155-156
sign conventions, 152, 256-257
space, arbitrarily oriented in,
255-269
stiffness, 152-161
stiffuess matrix, 153-158,
158-161
strain/displacement relatioDShips,
156-157
stress/strain relationships,
156--157
transformation matrix, 216,
259-260800 j, Index
Beam element (Continued)
transverse shear deformations,
158-161
twCHiimensionai, arbitJarily
oriented, 214-218
Beam equations, 151-213
bending deformations; 153-158
boundaryeonditions, 161-163
direct stiffness method, 163-175
displacement functions, 155-156
distributed loading, 175-188
EuIer-Bemouli theory, 153-158
exact solution, 188-194
finite element solution, 188-194
fixed-end reactions, 175
Galerkin's method, 201-203
introduction to, 151-152
load replacement, 177-178
nodal hinge, element with a,
194-199
potential energy approach, 199-201
sign conventions, 152
stiffness matrix, 153-158, 158-161,
161-163
stiffuess ofel.ement, 152-161
strain/dispJacement relationships,
156-157
stress/strain relationships, 156-157
TlIDosbcDko theory, 158-161
transverse shear deformations,
158-161
work-equiva1ence method, 176-177
Bending, 153-158,255-260,514-518
beam elements in arbitrary space,
255-260
defonna~ons in beam elements,
15~-158
plate element. 514-518
rigidity ofa plate, 517
Body forces; 324-326, 419-420, 448,
460; 497-498
axisymmetric elemcots, 419-420
bar element, 448
centrifugal, 325
natural coordinate system, 448
plane element, 460
teU'ahedral element, 497-498
treatment of, 324-326
Boundary conditions, 13-14,34,
39-52, 103-109, 161-163,
320-322, 601
beam elements. 161-163
constant-strain triangular (CST)
elemertt, 320-322
fluid flow, 601
homogeneous, 39-40
iDclincd supports, 103-109
introduction to, 13-14,34
DOnbomo.geneous, 39, 4O-4l
penalty method, 50-52
C
skewed supports, 103-109
stiffness method, 39-52
Castigli;lDO'S theorem, 12
Central difference method, 653,
654-659
Centrifugal body force, 325
Cireu1ar frequency, natural, 649
Coarse-mesh generation, 310
Coefficient matrix, inversion of, 726
Coefficient oftbennal expansion, 618
Cofactor method, 716-717
Collocation method, 129
Column matrices, 4, 708
Compatibility, 35, 363-367, 746-748
condition of, 748
equations, 746-748
finite element resultS, 363-367
requirement, 35
Compatible displacements, 155
Compatible functions, 73
Complete, approximation functions,
73-74
Computer programs, 6-7, 23-24,
314-380,524-528,693-701
finite element method, 23-24
plate bending element, solution for,
524-528
role of, 6-7
step-by-step solutions, 374-380
structuta1 dynamics, 693-701
Concentrated loads, 360-361
Condensation, see Static. condensation
Conduction,535-538,542-546,551-S58
element conduction matrix, .
542-546, 551-558
beat, one-dimensional, 535-537
beat, two-dimensiona~ 537-538
Conforming functions, 73
Gonnecting (mixing) different kinds
of elements, 361-362
Consistent-mass matrix, 651-653,
682-685
cOnstant-strain uianguJar (CST)
element, 304-305, 310-324,
324-329, 342, 406-408
body forces, 324-326
boundaly conditions. 320-322
eoarse--~ generation, 310
defects, 342
.displacement function, 311-315
equations, 310-324
forces (stresses), 322-324
global equations, 320-322
introduction to, 304-305
LST elements, comparison of,
406-408
matrix, 310-324, 329-331
nodal displacements, 322
penalty formulation, 331
selection of, 310-311
strain/displacement relationships,
315-320
stress/strain relationships, 315-320
surface forces, 326-329
Constitutive law, 11
Constitutive ma!rix, 309, 522
Continuity, 35, 73
requirement, 35
symbol,73
Convection, heat tJansfer with,
538-539, 540
Convergence of finite element
solution, 367-368
Coordinates, 66-72, 444-446
bar elements, 67-72,444-446
intrinsic system, 444
natura] system, 444
Coulomb-Mohr theory, 342
Cramer's rule, 724-725
CST, see Constant-strain triangular
(CST) element
Cubic elements, 9
Curvature matrix, 521-522
D
D'A!embert's principle, 755-756
Defects, CST elements, 342
Deformation, "33, 153-158, 158-161,
514-518
bending'in beams, 153-158
bending rigidity of a plate, 517
defined, 33
Kirchhoff assumptions, $]5-516
plate bending, 514-518
potential energy, 518
stress/strain relationships, 517-518
transVerse shear in beams, 158-161
Degrees offreedom, 14, IS, 29
defined,15
spring element, 29
unknown, 14
Determinant, defined, 716
Differential equations, 535-538,
594-596, 744-746
elasticity theory, 744-746
equilibrium, 744-746
fluid flow, 594-598 .
heat transfer, 535-538
Direct equilibrium method, 11
Direct integration, 653
Direct stiffness method, 2-4, 13-14,
28,37-39, 163-175.
See also Superposition
beam analysis using, 163-175
history of, 2-4, 28
total stiffness matrix, assembly by,
37-39
use of, 13-14Direction cosines, 85, 95-96
Directional stiffness bias, 371
Discontinuities, natural subdivisions
at, 354, 357
Discretization, 1,8-10,331-332,423
axisymmetric element, 423
finite element method, 1, 8-10.
331-332
plane stress, 331-332
Displacement function, 11,31-32,68,
15.5-156, 311-315, 399-401,446,
450-451, 455-456, 494-496,
519-521 -
bar element, 68, 446
beam element, 155-156
constant-strain triangular (CSlj
element, 11l-315
Hermite cubic interpolation,
155-156
interpolation, 32
isoparametric function, 446,
450-451,455-456
linear-strain triangle (LSl),
399-401
plane element" 455-456
plane stress element, 450-451
plate bending element, 519-521
selection o( 11
shape, 32, 155-156
spring element, 31-32
tetrahedral element, 494-496
DispIacement method, 7, 28-64. See
also Stiffness method
introduction to, 28-64
use of, 7
Displacements, 34, 70, 72-74,
755-758. See also Strain/"
djsp1acement relationships
appromnation functions for, 72-74
compatible, 755
nodal, 34, 10
virtual work, principles of, 755-758
Distributed loading, 175-188
beams, 175-188
eft'eetive global nodal forces,
- 181-182
fixed-end reactions,. 175 _
general formulation of, 178-179
load replacement, 171-178
work..equiva1ence method, 176-177
Dynamics, 647-700
axisymmetric element, analysis of,
684-685
bar element equations, 649-653
beam element DlaS.') mattia:s,
674-681
central difference method, 653,
654-659
computer program example
solutions, 693-701
E
introduction to, 647
mass matrices, 650-653, 674-681,
681-685
natural frequencies, 649, 665-669,
674-681
Newmark's method, 659-663
numerical integration in time,
653-665, 687-693
one-dimensional bar analysis,
665-669, 669-674
plane frame element, analysis of,
682-683
plane stress/strain element, analysis
of, 683--684
spring-mass system, 641-649
structural, 647-707
tetrahedral (solid) element mass
matria:s, analysiS of, 685
time, numerical integration in,
653-665, 687"'-693
time-dependent heat transfer,
686-693
t.ime-dependent stress analysis,
649-653, 669-674
truss element, analysis of, 681-682'
Wj]so~'s (Wdson·Theta) method,
664-665
EffectivC stress, 341
Elasticity theoIy, 744-751
compatibility equations, 746-748
condition of compatibility, 748
differential equations of
equilibrium, 744-746
equilibrium, differential equations
of. 744-746
introduction to, 744
modulus ofe1asticity, 748
strain/displacement, 746-748
stress/strain relationships, 748-751
Elements, 8-1(}, 11, 13-14,30-34,
65-150, 151-213, 304-30S,
310-324, 342,351-362. 398-403,
444--449,449-452, 480-482,
493-500,501-508, 514-533
aspect ratio (AR), 35]
axisymmetric; 9
bar, 65-150,444-449
beam, lS1-21~
eoarse--mesb. generation, 310
connecting (mixing), modeling,
361-362
constant...strain triangular (CST),
304-305, 310-324, 342
cubic,.9 __
defects, CST, 324
equations, 11, J3-J4, 34, 69-70,
402-403,451-452, 522-523
finite, 8
Index ... 84l
forces, 34, 70
heterosis, 523
isopara:metric, 446
laGrange, 482
linear, 9
linear hexahedral, 501-504-
linear-strain triangle (LST),
398-403
plane stress, 449-452
plate bending. 514-533
Q8,480
Q9,482
quadratic, 9
quadratic bexahedral, 504-508
refinement, methods of, 355-356,
358-359
selection of, 8-10, 30-31,310-311
399,444--446,449,519
serendipity,481
shapes. modeling, 351
sizing, 355-356, 358-359
spring, 30-34 -
stiffues:s matrix, It ~ 33-34, 66-72,
402-403,447-449,451-452,
522-523
tet:rah.edral. 493-500
transition triangles, 359-360
Energy method, 12
Equations, II, 13-14, 34, 52-60,
65-149, 151-213,214-237,
238-255,310-324,398-411,
419-422, 447-449, 451-452,
459-460, 497-498, 522-523,
535-538, 542-$46, 557-558,
594-596, 599-601. 608, 659-661,
664-665,722-743, 744-751.
See also .Elasticity theory;
SimuJtaneous linear equations
axisymmetric eIeiner.tt, 419-422
bar element. 124-127, 447-449
beam, 151-213
beam-clement, 199-201, 201-203
compatibility, 746-748
constant--strain triangnlar (CST)
element, 310-324
diffcn:ntia1~ 535-538, 594-596,
744-745
element, 11, 13-14,69-70
element conduction, 542-546.
557-558
finite e1cmcnt, III
ftuid ftow, 599-601, 608
frame, 214-231
global, 13-14,34, 70, 161-163, S4E
601
grid. 214, 238-255
heat transfer, 535-518
i.soparametric formulation, 447-449
459-460
Jacobian function, 447802 ... ·Index
Equations (Continued)
linear-strain triangle (LST),
398-411
Newmark's, 659-661
one-dimensional, 124-127, 131,
542-546
plane element, 459-460
plane stress element., 451-452
plate bending element, 522-523
simultaneous linear, 722-743
spring element, 52-60
tetrahedral element, 497-498
total, 13-14, 70
truss, 65-149
two-dimensional,557-558
Wilson's, 664-665
Equilibrium, 363-367, 744-746
compatibility and, 363-367
differential equations 744-746
finite element results,.363-367
Equivalent stress., 341
Euler-Bemouli theory, 153-158
Exact solution, 120-124, 188-194
bar element, 120- t24
beams, 188-194
finite element solution, comparison
to, 120-124, 188-194
Explicit numerical integration method,
689
F
Field problems, 52
Finite element, defined, 8
Finite element method, 1-26,
120-124,350-363,540--;555,
555-564, 566-568, 569-574,
59&-606,606-610. See also
Madding
advantages of, 19-22
applications of, 15-19 .
boundary conditions, 13-14
computer, role of, 6-7
computer programs for, 23-24
constitutive law~ 11
defined, I, 8
degrees offreedom, 14, 15
direct equilibrium method, 11 \
direct stiffness method, 2-3,
13-14
discretization, 1,8-10
displacement function, selection of,
II
displacement method, 7
element conduction matrix,
542-546,557-558
element types, selection of, 8-10,
541, 555, 598
energy method, 12
exact solution, comparison to,
1~124
flexibility method, 7
fluid flow, 598-606, 606-610
force method, 7
functional, 12
generalized displacements, 14
global equations, 13-14
gradient/potential relationship, 599,
607
heat flux/temperature gradient
relationship, 542, 556-557
heat transfer, 540-555, 555-564,
566-568, 569-574
history of, 2-4
introduction to, 1-26
matrix notation, 4-6
modeling, 350-363
one-dimensional, 540-555, 569,
598-606
potential ftinction, 598-599, 607
primal}' unknowns, 14
results, interpretation of, 14
steps of, 7-14
stiffness method, 7
strain/displacement relationships, 11
stress/strain relationships, 11, 14
temperature fuJiction, 541, 556
temperature gradient/temperature
relationships, 542, 556-557
three-dimensional, 566-568
total equations, 13-14· •
truss equations, 120-124
two-dimensional, 555-564, 606-610
v;uiationai method, 540-555
velocity/gradient relationship, 599,
607
weighted residuals, methods of,
12-13
work method, 12
Finite element solution, 120-124,
188-194,331-342,363-367,
367-369
approximations in, 364-367
bar element, 120-124
beams, 188-194
compatibility of.results, 363-367
convergence of, ·361-368
CST defects, 342
discretization, 331-332
equifibriwn of results, 363-361
exact solution, comparison to,
120-124,188-194
plane stress, 305-309
stiffness matrix, assemblage of,
332-342
Fixed-end forces, 229-230
Fixed-end reactions, 115
Flexibility metbod, 7
Flowcharts, 374, 574,611,656,661
central difference method, 656
fluid dow; 611
heat transfer, 574
Newmark's equations, 661
nwnerical integration, 656
plane stress/strain, 374
Fluid flow, 593-616
boundary conditions, 601
differential equations, 594-598
equations, 599-601, 608
finite element formulation, 598-606,
606-610
flowchart for, 611
global equations, 601
gradient/potential relationship, 599,
607
introduction to, 593
nodal potentials, 601
one-dimensional, 598-601
pipes, 596-598
porous medium, 594-596
potential function, 589
program, example of. 611-61i
solid bodies, around, 596-598
stiffness matrix, 599-601, 608
two-dimensional, 606-6l0
velocities, 602
, velocity/gradient relationship, 599,
601 .
volumetric dow rates, 602
Force, 7, 34, 36, 70, 178-182,
229-230,232-233,322-324,
324-329,419-421, 44&--449, 460,
497-498, 752-754
axisymmetric elements, 419-421
bar element, 70, 448--449
body, 324-326,419-420,448,460,
497-498
centrifugal body, 325
constant-strain triangular (CST)
element, 322-324, 324-329
equivalent nodal, 178-180,752-154
fixed-end, 22~-230
global nodal matrix, 36
method,7
nodal, 178-182,232-233
plane element, 460
rigid plane frames, 229-230,
232-233
spring element, 34
stresses, 322-324
sulface, 326-329,420-421,
448-449,460,498
tetrahedral element, 497-498
Forced convection, 538, 540
Frame equations, 214-237
effective nodal forces, 232-233
fixed-end forces, 229-230
inclined supports, 137
introduction to, 214
rigid plane frames. 218-236
skewed supports, 237Free convection. 538,540
Fringe carpet., 369
Functional, defined, 12
G
Galerkin's method, 12-13, 124-127,
131,201-203
bar element formulation, 125-127
beam element equations, 201-203
general fonnuIation, 124-125
one-dimensionaJ bar element
equations, 124-127, 131
residual method, 124-127, 131
lISe of, 12-13
Gauss·Jordan method, 718-720
Gauss-Seidel iteration, 733-735
Gaussian elimination, 726-733
Gaussian quadrature, 463-466,
469-475
element stresses, evaluation of,
473-475
one-point, 463-464
stiffness matrix, evaluation of,
.469-413
three-point, 465-466
two-point fonnula. 464-465
Global equations, 13-]4, 34,70,
161-163,320-322,601
assemblage of, 13-14
bar clement, 70
beam element, 161-163
constant-strain triangular (CST)
element, 320-322
Huid flow, 601
spring element, 34
Global stiffness matrix, 36, 78-81. See
also Total stifihess matrix
bar element, 78-81
inverse. 80
spring assembly, 36
. transverse, 80
Gradient/potential relationsbip, 599,
607
Grid, defined, 238 .
Grid equations, 214, 238-255
dete:nnination of, 238-255
introduction to, 214
H
open sections, 241
polar moment of inertia, 240
torsional constant, 240-241,242
h method of refinement, 355-356
Hannonic motion, simple, 649
Hea~ fiux, 542, 546
Heat flux/temperature gradient
relationship, 542, 556-557
Heat transfer, 534-593, 686-6!a
coefficients, 539-540
convection, 538-539, S40,
differential equations, 535-538
element conduction matrix,
542-546, 557-S5S
finite element fonntdation, 540-555,
555-564, 566-568, 569-574
flowchart for, 574
Galerkin's method, 569-574
heat conduction, one-dimensional,
535-537
beat conduction, two-dimensional,
537-538
heat flux/temperature gradient
relationship, 542, 556-557
heat·transfercoefficients, 539-540
introduction to, 534-535
line sources, 564-566
mass uansport, 569-574
nodal temperature, 546
nwneric:al time integration, 687-683
one..point sources, 564-566
program, examples of, 574-576
temperature function, 541, 556
temperature gradient/tenlf)erature
relationships, 542, 556-557
thermal conductivities, 539-540
three-dimensional, 566-568
time-dependent, 686-693
two-dimensional, 555-564, 574-567
units of, 539-540
variational method, 540-555
Hermite cubic interpolation function,
155-156
Heterosis element, 523
Hooke's Jaw, 11,67
I
Identity matrix, 712
Inclined supports,_ 103-109, 237
frame equations, 237
truss equations, 103-109
Infinite medium, 361
Infinite stress, 360-361
Integration, !fee Numerical Integration .
Interpolation functions, 32, 74. See
also Approximation functions
Intrinsic coordinate system, 444-
Inverse, defined, 80
Inverse ofa matrix, 7l2, 716-718,
71&-720
adjoint method, 718
cofactor method, 716-717
defined, 712
Gauss-Jordan method, 71&-720
row reduction, 718-720
lsoparametric formulati9ll, 443~89,
501-508
bar element stiffness matrix.,
444--449
defined, 444, 483
J
Index A. 80.
element stresses, evaluation of,
473-475
Gaussian quadrature, 463-466,
469-475
intrinsic coordinate system, 444
introduction to, 443
linear hexahedral element, 501-50.:
natural coordinate system, 444
Newton-Cotes quadrature, 467-49.
numerical integration, 463-469
plane element stiffness matrix,
452-462
plane stress element, 449-452
quadratic hexabedral e~ment,
504-508
shape functions, higher-order,
475-484
stiffness matrix, evaluation of,
469-473
stress analysis. 501-508
transformation mapping, 444
Jacobian function, 447
Joint force, see Nodal force
K
Kirchhoff assumptions, 515-517
L \
L:aGrange interpolation, 482
Least squares method, 130
Line elements, defined, 3M
Line sources, 564-566
Linear elements, 9 .
Linear..elastic bar element, see Bar
elements; Truss equations
Linear hexahedral element, SOI-5M
Linear-strain triangle (LSlj equation:
398-411
CSTelements,comparisonof,406-4{
defined, 398, 401
derivation of, 389-403
displacement function, 399-401
element type, selection of, 399
introduction to, 398
Pa.sc:al triangle, 400
quadratic-strain triangle (QST)
element., 400
stiffness, determination of, 4Ol-4Ot
st:i.f6less matrix. 398-403
strain/displacement relationships,
401-402
stress/strain relationships, 401-402
Load rep1acc:ment, 177-178
Local stiffness matrix, 34
Longitudinal wave velocity, 670
LST.'see Linear-strain triangle (I..ST.
equations
Lumped-mass matrix, 651, 682804 .. Index
M
Mass matrix, 650-653,674-681,
681-685
axisymmetric element, 684--(;85
bar eJemeDt, 650-653
beam element, 674-681
consistent-mass, 651-653, 682-985
lumped-mass, 651, 682
natural frequencies and. 674-681
plane frame element, 682-683
plane stress/strain element,
683-684
tetrahedral (solid) element, 685
truss element., 681-682
Mass transport, 569-574
Galerkin's method, 569-574
heat transfer and, 569-574
mass flow rate, 569
Matrix, 4-6, 11, 28-29, 29-34, 36,
37-39,06-72,78-81,92-100,
216,259-260,304-305,309,
310-324,329-331,519-523,
_.542-546,557-558,620-622,
650-653,647-681.681-68S,
708-721. See also Matrix algebra;
Mass matrix; Stiffness matrix
algebra, 708-721
column, 4, 708
consistent-mass., 651-653
constant-strain,triangular (CST) .
element, 304-305, 310-324,
329-331
constitutive, 309,522
curvature, 521-522
defined, 4,708-709
element conduction, 542-546,
557-558
element stiffness., 11
global nodal displacement, 36
global nodal force, 36
global stiffiless, 36, 78-81
identity, 712
local stiffness, 34
lumped-m~ 651
mass, 650-653, 647-681. 681-685
moment, 521-522
notation for, 4-6
orthogonal,713-714
quadra~c form, 716
rectangular, 4, 708
row, 708
singular, 718
square, 708
stiffneSs, 28-29, 29-34, 66-7~
92-100, 519-523, ~50-653
stiffness influence coefficients, 5
stft'SS/strain, 309
symmcttic, 712
system stif1hess, 36
thermal strain, 620-622
three dimensions, for bars in,
92-100
total stiffness, 36, 37-39
transfonnatiOD (rotation), 92-100,
216, 259-260
unit, 712
Matrix algebra, 708-72t
addition of matrices, 710
adjoint meth~ 718
cofactor method,.716-717
definitions of, 708-709
differentiation's, 71+-.715
Gauss-Jordan method, 7]8-720
identity matrix, 721
integrating, 715-716
inverse of, 712,716-718,718-720
multiplication by a scalar, 709
mUltiplication of matrices, 710-711
operations, 709-716
orthogonal matrix, 713-714
fOW reduction, 718-720
symmetric matrices, 712
tIanspose, 711-712
unit matrix, 712
Maximum distortiQD energy theory,
341=-342
Mindlin plate theory, 523, 526
Minimum potential energy, principle
of, 52-53, 57-59, III
finite element equations, III
spring element equations, 52-53,
57-59
Modeling,350-397
adaptive refinement, 355
aspect ratio (AR). 351, 352-353
checking, 362
compatibility of results, 363-367
computer program assisted step-bystep solutions, 374-380
concentrated loads, 360-361
connecting (mixing) elements,
361-362 '
convergence of solution, 367-368
discontinuities, natural subdivisions
at, 354,357
equilibrium of resu1ts, 363-367
finite element, 350-363
flowcharts, 374
general considerations, 351
h method of refinement, 355-356
infinite medium, 361
infinite stress, 360-361
introduction to, 350
natural subdivisions, 354, 357
p method of refinement, 358-359
point loa~ 360-361
postprocessor results. 362-363
refinement, 355-356, 35&-359
static:: condensation, 369-373
stresses, interpretation of, 368-369
symmetry, 351-354, 355-356
tIansition triangles, 359-360
Modes, natural, 666, 668
Modulus of elasticity, 748
Moment matrix, 521-522
N
Natural convection, 538, 540
Natural coordinate system, 444, 447
Jacobian function. 447
use of, 444
Natural frequencies, 649, 665-669,
674-681
amplitude. 649
bar element, one-dimensional,
665-669
beam element, 674-681
circular, 649
mass matrices, 674-681
modes., 666, 668
role of thumb for, 668
Natura1 subdivisions at
discontinuities, 354, 357
Newmark's method of numerical
integration, 659-663
Newton-Co~es quadrature. 467-469
intervals, 467
numerical integration, 467-469
Nodal displacements, 34, 36, 70, 322
bar element, 70
constant-strain triangular (CST)
element, 322
global matrix, 36
spring element, 34
Nodal forces, 178-182,232-233,
752-754
effective, 232-233
effec:'tive global, 181-182
equivalent., 178-180,752-754
load displacement, beams, 178-182
rigid plane frames, 232-233
Nodal hinge, beam elements, 194--199
Nodal potentials, 601
Nodal temperature, 546
Nodes, 29, 152, 370
actual. 370
condensed out, 370
defined, 29
sign conventions for beams, 152
Nonexistence of solution, 724
Nonuniqueness of solution, 723-724
.Numerical comparisons. plate bending
element, 523-524
Numerical integration, 463-469,
653-665,687-693
central difference method, 653,
654-659
direct integration, 653
dynamic systems, 653-665
explicit, 689o
flowcharts for, 656, 661
Gaussian quadrature, 463-466,
469-475
heat-transfer, 687-693
Newmark's method, 659-663
Newton-Cotes quadrature, 467-469
Simpson one-third rule, 463, 467
time, 653-665, ~87-693
trapezoid rule, 463, 467-468, 687
Wilson's method, 664-665
One-dimcnsional elements, 124-127,
127-13!, 540-555, 569, 598-601,
665-·669, 669-674
bar analysis. 665-669, 669-674
bar element equations, 124-127
bar element problems, 127-131
fluid flow. 598-601
heat-tran~fer problems, 540-555,
569
mass transport, 569
natural frequencies, 665-669
time-dependent, 669-674
Open sections, 241
Orthogonal matrix, 713-714
p
p method of refinement, 358-359
Parasitic shear. 342
Pa."Cal triangle, 400
Penalty formulation, 331
Penalty method, 50-52
Period of vibration. 649
Pipes, fluid flow in, 5%-598
Plane element, 452-463, 682-684
body forces, 460
consistent-mass matrix, 683-684
displacement functions, 455-456
equations, 459-460
isoparametric formulation,
452-463
mass matrices, 682-684
quadrilateral element, 684
selection of, 453-455
stiffness matrix, 452-463
strain/displacement relationships,
456-459
stress/strain relationships. 456-459,
683-684
surface forces, 460
Plane {rames, 218-236,682-683
element, 682-683
mass matrices, 682-683
rigid, 218-236 .
Plane strain, 305-309, 374-380,
683-684
concept of, 305-309
consistent-mass matrix, 683-684
defined, 305
flowchart fOf, 374
program assisted step-by-step
solutions, 374-380
Plane stress, 305-309, 331-342,
374-380,449-452, 683-684
concept of, 305-309
consistent-mass mall'ix, 683-684
defined,305
discretization, 331-332
displacement functions, 450-451
element, 449-452
finite element solution of, 331-342
flowchart for, 374
isoparametrlc formulation. 449-452
maximum distortion energy theory,
34!-342
principal angle, 307
program assisted step-by-step
solutions, 374-380
rectangular element, 449-452
stiffness matrix assemblage for,
332-341
von Mises (von Mises-Hencky)
theory, 341-342
Plane truss, solution of, 84-92
Plate bending element, 514-533
computer solution for, 524--528
concept of, 514-518
deformation of, 514-515
displacement function, 519-521
equations, 519-523
geometry of, 514-515
heterosis element., 523
introduction to, 514
Kirchhoff assumptions, 515-517
Mindlin plate theory, 523, 526
numerical comparisons, 523-524
potential energy, 518
rigidity of, 517
selection of, 519
stiffness matrix, 519-523
strain/displacement relationships,
521-522
stress/strain relationships, 517-518,
521-522
Point loads, 360-361
Point sources, 564-566
Polar moment of inertia, 240
Porous medium, fluid flow in,
594-596
Potential energy approach, 52-60,
109--120, 199-201,518
admissible variation, 55
bar element equations, 109-120
beam element equations. 199--201
mi.nimwn potential energy,
principle of, 52-53, 57-59, 111
plate bending element, 518
spring element equations, 52-60
stationary value, 54
Index ... 80S
total potential energy, 53, 518
truss equations, 1'09-) 20
variation, 55
Potential function, 589
Pressure vessel, axisymmetric,
solution of, 422-428
Primary unknowns, defined, 14
Principal angle, 307
Principal stresses, 307
Q
Q8 element, 480
Q9 element, 482
Quadratic elements, 9
Quadratic form, 716
Quadratic hexahedral element,
504-508
Quadratic-strain triangle (QST)
element, 400
Quadrilateral element consistent-mass
matrix, 684
.R
Refinement, 355-356, 358-359
adaptive, 355
h method, 355-356
p method. 358-359
Reflective (mirror) symmetry, 100-103
Rigid plane frames, 218-236
defined, 218
examples of, 218-236
Row reduction, 718-720
S
Serendipity element, 431
Shape functions, 32, 155-156,
475-484
beam element, 155-156
defined,32
higher-order, 475--484
isoparametric formulation, 475-484
laGrange element, 482
Q8 element, 480
Q9 element, 482
serendipity element, 481
Shear locking, 342
Sign conventions, beams, H2,
256-257 .
Simultaneous linear equations,
722-743
banded...syrtl.metric method, 735-741
Cramer's rule, 724-725
Gauss-Seidel iteration, 733-735
Gaussian elimination, 726-733
general fonn of, 722-723
introduction to, 722
inversion ofcoefficient matrix, 726
methods for solving, 724-735
nonexistence ofsolution, 724
nonuniqu.eness of solution, 723-724806 • Index
Simultaneous linear equations
(Continued)
skyline method, 735-741
uniqueness of solution, 723
wavefront method, 735-741
Sizing ofclements, 355-356, 35&-359
Skew, defined, 370-371
Skewed supports, 103-109,237
frame equations, 237
truss equations, 103-109
Skyline method, 735-741
Smoothing process, 369
Solid bodies, fluid flow around,
596-598
Solid element, see Tetrahedral element
Spring clements, 29-34,34-37,52-60
assemblage of, 34-37
compatibility requirement, 35
continuity requirement, 35
degrees of freedom, 29
displacement function, 31-32
element type, 30-31
equations, 52-60
global equation for, 34
nodal displacements, 34
nodes, 29
potential energy approach, 52-60
spring constant, 29
stiffness matrix for, 29-34
Spring-mass system, 647-649
amplitude, 649
dynamics of, 647-649
hannonic motion, simple, 649
natural circular frequency, 649
period of vibration, 649
Static condensation, 369-373
concept of, 369-373
condensed load vector, 370
condensed out nodes, 370
condensed stiffness matrix, 370
directional stiffness bias, 371
skew, 370-371
Stationary value, 54
Stiffness equations, 304-349
constant-strain triangular (CS11
element, 304;-305, 310-324,
324-329, 329-331
explicit expression, 329-331
finite element solution, 331-341
introduction to, 304-305
maximum distortion energy theory,
341-342
plane strain, 305-309
plane stress, 305-309, 331-342
von Mises (von Mises-Hencky)
theory, 341-342
Stiffuess inftuence eoefficients, 5
Stiffness matrix, 28-29, 29-34, 36,
66-72,92.:..100,153-158,
'158-161, 161-163,304-305,
310-324,332-341,369-313,
402-403,403-406, 419-422,
423-428,444-449,451-452,
452-463, 469-473, 497-500,
519-523,599-601,608,735-741
axisymmetric element, 419-422,
423-428
banded-symmetric method, 735-741
bar element, 66-72, 444-449
beam equations, 153-158, 158-161,
161-163
beams, examples of assembJage of,
161-163
bending deformations, 153-158
body forces, 419-420, 448
condensed, 370
constant-strain triangular (CST)
element, 304-305, 310-324
defined, 28-29
Euler-Bemouli theory, based on,
153-158
evaluation of, 469.473
fluid flow, 599-601,608
Gaussian quadrature, 469-473
isoparametric formulation,
444-449, 469-473
linear-strain triangle (l.S1) element,
402-403, 403-406
local, 34
plane element, 452-463
plane sl.reSS element, 451-452
plane stress problem, assemblage
offor, 332-341
plate bending element, 519-523
skyline method, 735-741
spring element, 29-34
static condensation, 369-373
superposition, assemblage by,
332-341,423-428
suIface forces, 420-421,448-449
tetrahedral element, 497-500
threedimensions,forbarsin,92-100
Timoshenko theory, base(! on,
158-161 .
total (global), 36,37-39, 332-341
transition matrix and, 92-100
transverse shear deformations,
158-161
wavefront method, 735-741
Stiffness method, 7, 28-64
boundary conditions, 34, 39-52
direct, 37-39
introduction to, 28-64
minimum potential energy,
principle of, 52-53, 57-59
penalty method, 50-52
potential energy approach., 52-60
spring constant, 29
spring clements, 29-34, 34-37,
52-60
stiffness matrix, 28-29, 29-34, 36
superposition, 37-39
total potential energy, 53
total stiffness matrix, 37-39
use of, 7
Strain, 306-309. See also Plane strain
normal,308
shear, 308
two-dimensional state of, 306-309
Strain/displacement relationships, 11,
33,69,156-157,315-320,
401-402,417-419,446-447,451,
456-459,490-493,496-497,
521-522,146-748
axisymmetric element, 417-419
bar element, 69
beam element, 156-157
condition ofcompatibility, 748
constant*strain triangular (CS1)
element, 315-320
deformation, 33
elasticity theory, 746-748
Hooke's law, I I, 67
isoparametric formulatioll,
446-:447,456-459
linear-strain triangle (LST)
elements, 401-402
plane clement, linear, 456-459
plane stress element, 451
plate bending element, 521-522
spring element, 33
stress analysis, 490-493
tetrahedral element, 496-497
Stress, 82-83, 306-309, 341-342,
360-361, 368-369, 473-475. See
also Plane stress; Thenna1 stress
computation of for a bar element,
82-83
Coulomb-Mohr theory, 342
effective, 341
equivalent, 341
evaluation of, 473-475
fringe carpet, 369
Gaussian quadrature, 473-475
infinite, 360-361
interpretation of, 368-369
maximum distortion energy theory,
341-342
principal, 307
smoothing process, 369
two-dimensional state of, 306-309
von Mises (von Mises-Hendcy)
theory, 341-342
Stress analysis, 490-513
isoparametric formulation, SOl-50S
linear hexahedral element, 501-504
quadratic hexahedral element,
504-508
strain/displacement relationships,
490-493stress/strain relationships, 490-493
tetrahedral element, 493-500
three-dimensional, 490-513
Stress/suain relationships, II, 14,33,
69, 156-157,315-320,401-402,
417-419,446-447,451,456-459,
490-493,496-497,517-518,
521-522,748-751
axisymmetric element, 417-419
bar dement, 69
beam element, 156-157
constant-strain triangular (CST)
element, 315-320
constitutive law, 11
defonnation, 33
elasticity theory, 748-751
isoparametric formulation,
446-447, 456-459
linear-strain triangle (LST)
elementS, 40],..402
modulus of elasticity, 748
plane element, linear, 456-459
plane stress element, 451
plate bending element, 517-518,
521-522
solving for, 14
spring element, 33 .
stress analysis, 490-493
tetrahedral clement, 496-497
Structural dynamics, see Dynamics
Structural Sled, properties of,
759-712
Structures, 100-103,214-303
frame equations, 214-237
grid equations. 238-255
rigid plane frames, 218-236
substructure analysis, 269-275
symmetry in, 100-103
Subdivisions, natura.!, 354, 357
Subdomain method, 129-130
,Subparametric formulation,
'483-484
Substructure analysis, 269-275
Superposition, 37-39, 332-341,
423-428. See alst) Direct stiffness
method
axisymmetric element, assemblage
for by, 423-428
plane stress problem, assemblage
for by, 332-341
total (global) stiffness matrix,
assemblage by, 37-39, 332-341
Surface forces, 326-329, 420-421,
448-449, 460, 498 ,
axisymmetric elements, 420-421
bar element, 448-449
natural coordinate system, 448-449
plane element, 46(}
tetrahedral element. 498
treatment of, 326-329
Symmetry, 100-103,351-354,
355-356
axial,IOO
finite element modeling, 351-354,
355-356
reflective (mirror), 100-103, 351
structures, use of in, 100-103
Symmetric matrix. 712
System stiffness matrix, see Total
stiffness matrix
T
Temperature, 541-542, 546, 556,
574-576
distribution, examples of, 574-576
function, 541, 556
gradients, 542, 546
nodal, 546
Temperature gradient/temperature
relationships, 542, 556-557
Tetrahedral element, 493-500, 685
body forces, 497-498
consistent~mass matrix, 685
displacement functions, 494-496
equations, 497-498
selection of, 493-494
stiffness matrix, 497-500
strain/displacement relationships,
496--497
stress/strain relationships, 496-497
surface forces, 498
Thermal conductivities, 539-540
Thermal strain matrix, 620-622
Thermal stress, 617-646
coefficient of thermal expansion,
618
formulation of,617-640
introduction to, 617
thermal strain matrix, 620-622
Three-dimensional elements, 490-513,
566-568
heat·transfer problems, 566-568
space, 92-100
stiffness matrix for a bar, 94-100
stress analysis, 490-513
tetrahedral element, 493-500
transformation matrix for a bar,
92-94
Time, numerical integration in,
653-665,687-689
Time~dependent, 649-653, 669-674,
686-693
bar analysis, one-dimensional,
669-674
heat transfer, 686-693
longitudinal wave velocity, 670
numerical time integration, 681-693
stress analysis., 649-653
structural dynamics, 649-653,
669-674
Index .. 8
Timoshenko theory, 158-161
Torsional conscant, 240-241, 242
Total equations, see Global equatio
Total potential energy, defined, 53
Total stiffness matrix, 36, 37-39, It:
See also Global stiffness matrix
beam elemen[, 162
direct stiffness' method, assembly
by, 37-39
spring assembly, 36
superposition, assembly by,
37-39
Transfonna(ion mapping, 444
Transformation (rotation) matrix,
92-100,216,259-260, 713
Transition triangles, 359-360
Transpose of a matrix, 711
Transverse, defined, SO
Transverse shear deformations,
158-161
Trapezoid rule, 467-468, 687
Truss equations, 65-149, 681-682.
See a/st) Bar elements
approximation functions, 72-74
bar elements, 67-72, 92-100,
109-120, 12()-124, 124-127,
I27-13l
boundary conditions, 103-109
collocation method, 129
consistent-mass matrix, 682
displacements, 72-74
exact solution, ]20-124
finite element solution, 120-124
Galerkin's residual method,
124-127, 13l
global stiffness matrix, 78-8}
inclined supports, 103-109
introduction to, 65
~st squares method, 130
local coordinates for, 66-72
lumped-mass matrix, 682
mass matrices, 681-682
plane truss, solution of, 84-92
potential energy approach,
109-120
residual methods, 124-127,
127-131
skewed suppons, 103-109
stiffness matrix, 66-72, 92-100
strain/displacement relationships,
stress, computation of for a bar
element, 82-83
stress/strain relationships, 69
subdomain method, 129-t3O
symmetry, use of in structures,
100-103
transfonnation (rotation) matrix,
92-100
vectors, transformation ofin two
dimensions, 75-77808 .i. Index
Two dimensional elements, 75-77,
214-218, 304-349, 555-564,
574-576. 606-610
U
beam clements, arbitrarily oriented,
214-218
flowchan for heat-transfer process
fluid flow, 606-610
heat-transfer problems, 555-564
plane stress and strain equations,
304-349
temperature distribution, 574-576
vectors, transformation of in, 75-77
Uniqueness of solution, 723
Unit matrix, 712
V
Variation, defined, 55
Variational methods, 52, 540-555
Vectors, 75-77, 370
condensed load, 370
transfonnation of in two
dimensions, 75-77
Velocity, 602, 670
fluid flow 602
longitudinal wave, 670
Velocity/gradient relationship, 599,
607
Virtual work, principle or, 755-758
compatible displacements, 755
D'Alembert's principle,
155-756
Volumetric flow rates, 602
Von Mises (von Mises·Hencky)
theory, 341-342
W
Wavefront method. 735-741
Weighted residuals, methods of.
12-13, 124-127, 127-131,
201-203
bar element equations, 124-127,
127-l3I
beam element equations, 201-203
collocation method, 129
Galerlcin's method, 12--13,
124-l27, 131,201-203
introduction to, 12-13
least squares meth.od, 130
one-dimensional problems, 127-131
subdomain method, 129-130
Wilson's (Wilson-Theta) method of
numerical integration, 664-665
Work methods, l2, 52-53, 57-59,
176-177,755-758
Castigliano's theorem, 12
introduction to, 12
minimum potential energy,
principJe of, 52-53, 57-59
virtual work, principle of,
755-758
work-equivalence, 176-177


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