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 |  موضوع: كتاب A First Course in the Finite Element Method الأحد 18 أغسطس 2024, 1:21 am | |
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A First Course in the Finite Element Method
Enhanced Sixth Edition, SI Version
Daryl L. Logan
University of Wisconsin–Platteville
و المحتوى كما يلي :
C O N T E N T S
Preface to the SI Edition ix
Preface x
Digital Resource xii
Acknowledgments xv
Notation xvi
1 Introduction 1
Chapter Objectives 1
Prologue 1
1.1 Brief History 3
1.2 Introduction to Matrix Notation 4
1.3 Role of the Computer 6
1.4 General Steps of the Finite Element Method 7
1.5 Applications of the Finite Element Method 15
1.6 Advantages of the Finite Element Method 21
1.7 Computer Programs for the Finite Element Method 25
References 27
Problems 30
2 Introduction to the Stiffness (Displacement) Method 31
Chapter Objectives 31
Introduction 31
2.1 Definition of the Stiffness Matrix 32
2.2 Derivation of the Stiffness Matrix
for a Spring Element 32
2.3 Example of a Spring Assemblage 36
2.4 Assembling the Total Stiffness Matrix by Superposition
(Direct Stiffness Method) 38
2.5 Boundary Conditions 40
2.6 Potential Energy Approach to Derive Spring Element Equations 55
Summary Equations 65
References 66
Problems 66
3 Development of Truss Equations 72
Chapter Objectives 72
Introduction 72
3.1 Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates 73vi Contents
3.2 Selecting a Displacement Function in Step 2 of the Derivation
of Stiffness Matrix for the One-Dimensional Bar Element 78
3.3 Transformation of Vectors in Two Dimensions 82
3.4 Global Stiffness Matrix for Bar Arbitrarily Oriented in the Plane 84
3.5 Computation of Stress for a Bar in the x – y Plane 89
3.6 Solution of a Plane Truss 91
3.7 Transformation Matrix and Stiffness Matrix for a Bar
in Three-Dimensional Space 100
3.8 Use of Symmetry in Structures 109
3.9 Inclined, or Skewed, Supports 112
3.10 Potential Energy Approach to Derive Bar Element Equations 121
3.11 Comparison of Finite Element Solution to Exact Solution for Bar 132
3.12 Galerkin’s Residual Method and Its Use to Derive
the One-Dimensional Bar Element Equations 136
3.13 Other Residual Methods and Their Application to a One-Dimensional Bar
Problem 139
3.14 Flowchart for Solution of Three-Dimensional Truss Problems 143
3.15 Computer Program Assisted Step-by-Step Solution for Truss Problem 144
Summary Equations 146
References 147
Problems 147
4 Development of Beam Equations 169
Chapter Objectives 169
Introduction 169
4.1 Beam Stiffness 170
4.2 Example of Assemblage of Beam Stiffness Matrices 180
4.3 Examples of Beam Analysis Using the Direct Stiffness Method 182
4.4 Distributed Loading 195
4.5 Comparison of the Finite Element Solution to the Exact Solution for a Beam 208
4.6 Beam Element with Nodal Hinge 214
4.7 Potential Energy Approach to Derive Beam Element Equations 222
4.8 Galerkin’s Method for Deriving Beam Element Equations 225
Summary Equations 227
References 228
Problems 229
5 Frame and Grid Equations 239
Chapter Objectives 239
Introduction 239
5.1 Two-Dimensional Arbitrarily Oriented Beam Element 239
5.2 Rigid Plane Frame Examples 243
5.3 Inclined or Skewed Supports—Frame Element 261
5.4 Grid Equations 262Contents vii
5.5 Beam Element Arbitrarily Oriented in Space 280
5.6 Concept of Substructure Analysis 295
Summary Equations 300
References 302
Problems 303
6 Development of the Plane Stress and Plane Strain
Stiffness Equations 337
Chapter Objectives 337
Introduction 337
6.1 Basic Concepts of Plane Stress and Plane Strain 338
6.2 Derivation of the Constant-Strain Triangular Element Stiffness
Matrix and Equations 342
6.3 Treatment of Body and Surface Forces 357
6.4 Explicit Expression for the Constant-Strain Triangle Stiffness Matrix 362
6.5 Finite Element Solution of a Plane Stress Problem 363
6.6 Rectangular Plane Element (Bilinear Rectangle, Q4) 374
Summary Equations 379
References 384
Problems 384
7 Practical Considerations in Modeling; Interpreting Results;
and Examples of Plane Stress/Strain Analysis 391
Chapter Objectives 391
Introduction 391
7.1 Finite Element Modeling 392
7.2 Equilibrium and Compatibility of Finite Element Results 405
7.3 Convergence of Solution and Mesh Refinement 408
7.4 Interpretation of Stresses 411
7.5 Flowchart for the Solution of Plane Stress/Strain Problems 413
7.6 Computer Program–Assisted Step-by-Step Solution, Other Models, and Results
for Plane Stress/Strain Problems 414
References 420
Problems 421
8 Development of the Linear-Strain Triangle Equations 437
Chapter Objectives 437
Introduction 437
8.1 Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations 437
8.2 Example LST Stiffness Determination 442
8.3 Comparison of Elements 444
Summary Equations 447
References 448
Problems 448viii Contents
9 Axisymmetric Elements 451
Chapter Objectives 451
Introduction 451
9.1 Derivation of the Stiffness Matrix 451
9.2 Solution of an Axisymmetric Pressure Vessel 462
9.3 Applications of Axisymmetric Elements 468
Summary Equations 473
References 475
Problems 475
10 Isoparametric Formulation 486
Chapter Objectives 486
Introduction 486
10.1 Isoparametric Formulation of the Bar Element Stiffness Matrix 487
10.2 Isoparametric Formulation of the Plane Quadrilateral (Q4) Element Stiffness
Matrix 492
10.3 Newton-Cotes and Gaussian Quadrature 503
10.4 Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature 509
10.5 Higher-Order Shape Functions (Including Q6, Q8, Q9, and Q12 Elements) 515
Summary Equations 526
References 530
Problems 530
11 Three-Dimensional Stress Analysis 536
Chapter Objectives 536
Introduction 536
11.1 Three-Dimensional Stress and Strain 537
11.2 Tetrahedral Element 539
11.3 Isoparametric Formulation and Hexahedral Element 547
Summary Equations 555
References 558
Problems 558
12 Plate Bending Element 572
Chapter Objectives 572
Introduction 572
12.1 Basic Concepts of Plate Bending 572
12.2 Derivation of a Plate Bending Element Stiffness Matrix and Equations 577
12.3 Some Plate Element Numerical Comparisons 582
12.4 Computer Solutions for Plate Bending Problems 584
Summary Equations 588
References 590
Problems 591Contents ix
13 Heat Transfer and Mass Transport 599
Chapter Objectives 599
Introduction 599
13.1 Derivation of the Basic Differential Equation 601
13.2 Heat Transfer with Convection 604
13.3 Typical Units; Thermal Conductivities, K; and Heat Transfer
Coefficients, h 605
13.4 One-Dimensional Finite Element Formulation Using
a Variational Method 607
13.5 Two-Dimensional Finite Element Formulation 626
13.6 Line or Point Sources 636
13.7 Three-Dimensional Heat Transfer by the Finite Element
Method 639
13.8 One-Dimensional Heat Transfer with Mass Transport 641
13.9 Finite Element Formulation of Heat Transfer with Mass Transport
by Galerkin’s Method 642
13.10 Flowchart and Examples of a Heat Transfer Program 646
Summary Equations 651
References 654
Problems 655
14 Fluid Flow in Porous Media and through
Hydraulic Networks; and Electrical Networks
and Electrostatics 673
Chapter Objectives 673
Introduction 673
14.1 Derivation of the Basic Differential Equations 674
14.2 One-Dimensional Finite Element Formulation 678
14.3 Two-Dimensional Finite Element Formulation 691
14.4 Flowchart and Example of a Fluid-Flow Program 696
14.5 Electrical Networks 697
14.6 Electrostatics 701
Summary Equations 715
References 719
Problems 720
15 Thermal Stress 727
Chapter Objectives 727
Introduction 727
15.1 Formulation of the Thermal Stress Problem and Examples 727
Summary Equations 752
Reference 753
Problems 754x Contents
16 Structural Dynamics and Time-Dependent
Heat Transfer 761
Chapter Objectives 761
Introduction 761
16.1 Dynamics of a Spring-Mass System 762
16.2 Direct Derivation of the Bar Element Equations 764
16.3 Numerical Integration in Time 768
16.4 Natural Frequencies of a One-Dimensional Bar 780
16.5 Time-Dependent One-Dimensional Bar Analysis 784
16.6 Beam Element Mass Matrices and Natural Frequencies 789
16.7 Truss, Plane Frame, Plane Stress, Plane Strain, Axisymmetric, and Solid
Element Mass Matrices 796
16.8 Time-Dependent Heat Transfer 801
16.9 Computer Program Example Solutions for Structural Dynamics 808
Summary Equations 817
References 821
Problems 822
Appendix A Matrix Algebra 827
Appendix B Methods for Solution of Simultaneous
Linear Equations 843
Appendix C Equations from Elasticity Theory 865
Appendix D Equivalent Nodal Forces 873
Appendix E Principle of Virtual Work 876
Appendix F Geometric Properties of Structural Steel Wide-Flange
Sections (W Shapes) 880
Answers to Selected Problems 908
Index 938938 938
A
Adjoint matrix, 837
Adjoint method, 835–837
Admissible variation, 58–59
Amplitude, 763
Approximation functions, 78–80, 213, 344–348, 438–
440, 455–457, 578–580, 607–608, 626–627
axisymmetric elements, 455–457
bar elements, 78–82
beam elements, 213
conforming (compatible) functions, 81
constant-strain triangular (CST) elements, 344–348
displacement functions, 79–82, 213, 344–348, 438–
440, 578–580
heat transfer, 607–608, 626–627
interpolation functions, 80
linear-strain triangle (LST) elements, 438–440
plate bending elements, 578–580
shape functions, 80, 347, 438–439, 456–457
temperature functions, 607–608, 626–627
Aspect ratios, 392–395
Axial (longitudinal) displacement, 73–76
Axial displacement function, 75
Axial symmetry, 109
Axis of symmetry (revolution) for, 452
Axisymmetric elements, 10,11, 451–485, 732–733,
799–800
applications of, 468–473
axis of symmetry (revolution) for, 452
body force distribution, 459–460
defined, 10, 11
discretization, 462–463
displacement functions, 455–457
element type selection, 455
mass matrix, 799–800
pressure vessels, 462–468
shape functions, 456–457, 799–800
stiffness matrix, 451–462, 463–467
strain/displacement relationships, 454, 457–458
stress/strain relationships, 452–454, 457–458
surface forces of, 460–461
thermal stress in, 732–733
von Mises stresses of, 470–472
B
Banded matrix, 857
Banded-symmetric matrices, 856–863
Bar elements, 72–168, 258–261, 487–492,
515–520, 731, 733, 764–768, 780–789
arbitrarily oriented in plane, 84–89
body forces and, 491
boundary conditions, 767–768
displacement functions, 78–82, 488–489, 764
dynamic analysis of, 764–768, 780–784
element type selection, 75, 487–488, 764
exact solution, 132–136
finite element comparison, 132–136
frames with beam elements and, 258–261
Galerkin’s residual method, 136–139
global equations, 767–768
gradient matrix, 124, 490, 498
isoparametric formulation, 487–492, 515–520
Jacobian matrix, 490
linear-strain (three-node), 515–520
local coordinates, 73–78
mass matrix, 765–767
natural frequency, 780–784
one-dimensional, 78–82, 136–143, 731, 733, 78–789
potential energy approach, 121–132
residual methods for, 136–143
shape functions, 80, 488–489
stiffness matrix, 73–78, 84–89, 487–492, 765–767
strain/displacement of, 75, 489–490, 764–765
stress, computation for in x–y plane, 89–91
stress/strain relationships, 75, 489–490, 764–765
surface forces and, 491–492
thermal stress in, 731, 733
three-dimensional, 100–109, 143
time-dependent (dynamic) analysis, 764–768,
780–789
transformation of vectors in two dimensions, 82–84
two-dimensional, 82–84
Beam elements, 169–238, 239–243, 280–294, 789–796
frames with bar elements and, 258–261
boundary conditions, 180–182
curvature (k ) of, 172
defined, 170
I N D E XIndex 939
direct stiffness method for, 182–195
displacement functions, 173–174
distributed loading, 195–208
element type selection, 172–173
Euler-Bernoulli beam theory, 171–177
exact solution, 208–214
finite element solution, 208–214
flexure formula for, 175
frame equations, 239–243, 280–294
Galerkin’s method for, 225–227
global equations, 180–182
load replacement, 197–198
mass matrix, 789–796
natural frequency, 789–796
nodal forces, 198–199
nodal hinges for, 214–221
plane, arbitrary orientation in, 239–243
potential energy approach to, 222–225
shape functions, 174
sign conventions, 170–171
space, arbitrary orientation in, 280–294
stiffness matrix for, 171–182
stiffness, 170–180
strain/displacement relationship, 174–175
strain energy, 225
stress/strain relationship, 174–175
time-dependent (dynamic) analysis,
789–796
Timoshenko beam theory, 177–180
transformation matrix, 239–243, 282–285
transverse shear deformation and, 177–180
work-equivalence method for, 196–198
Bending, 281–285, 378–379, 572–598
CST and Q4 displacement comparison, 378–379
frame equations, 281–285
geometry and deformation from, 573
Kirchhoff assumptions for displacement from,
573–575
plate elements, 572–598
pure, 378–379
Bilinear quadratic (Q6) elements, 521–522
Body forces, 357–359, 459–460, 491, 500, 543–544
axisymmetric elements, 459–460
bar elements, 491
centrifugal, 358
plane stress and plane strain from, 357–359
rectangular (Q4) plane elements, 500
tetrahedral elements, 543–544
Boundary conditions, 35, 40–55, 180–182,
353–355, 603–604, 612, 676–678, 682, 709–710,
767–768
bar elements, 767–768
beam elements, 180–182
constant-strain triangular elements, 353–355
differential equations and, 603–604, 676–678
electrostatics, 709–710
fluid flow, 676–678, 682, 709–710
heat transfer, 603–604, 612
homogeneous, 41–42
nodal displacements and, 40–44
nonhomogeneous, 42–44
spring elements, 35, 40–55
stiffness method and, 35, 40–44
time-dependent (dynamic) analysis, 767–768
Boundary elements for support evaluation, 120–121
Branch elements, 687–688
C
Cartesian coordinates, 537–538
Central difference method, 768–774
Centrifugal body forces, 358
Coarse-mesh generation, 343
Coefficient matrix inversion, 846–847
Coefficient of thermal expansion, 728
Cofactor matrix, 836
Cofactor method, 835–837
Collocation method, 140–141
Column matrix, 5, 827
Combining elements, 21, 23, 403–404
finite element method for, 21, 23
modeling, 403–404
Compatibility equations, 867–869
Compatibility of modeling results, 405–408
Complete pivoting, 854
Computational fluid dynamics (CFD), 21
Computer programs, 25–27, 144–145, 414–419, 584–
588, 808–816, 861–863
damping, 812–816
finite element method and, 25–27
plate bending element solutions, 584–588
simultaneous linear equations, 861–863
step-by-step modeling, 414–419
structural dynamics solutions, 808–816
time-dependent problem solutions, 808–816
truss problem solutions, 144–145
Computer role in finite element method 6–7
Concentrated (point) loads, 402
Conduction, 601–605, 608–612, 628–629
element conduction matrix, 608–612, 628–629
Fourier’s law of heat conduction, 602
one-dimensional heat transfer, 601–605, 608–612940 Index
Conduction (continued)
two-dimensional heat transfer, 603–604, 628–629
with convection, 604–605
without convection, 601–604
Conforming (compatible) functions, 81
Conservation of energy principle, 601–602, 642
Conservation of mass, 674–675
Consistent loads, 353
Consistent-mass matrix, 766–767, 790, 797–800
Constant-strain triangular (CST) elements,
342–357, 362–363, 377–379, 402, 444–447
boundary conditions, 353–355
displacement functions, 344–348
displacement results, 378–379
element forces (stresses), 355
element type selection, 343
explicit expression for, 362–363
global equations, 353–355
LST elements, comparison of, 444–447
nodal displacements, 355
plane stress and plane strain equations,
342–357, 362–363
potential energy approach for, 350–353
rectangular (Q4) plane elements comparison to,
377–379
shape functions, 347
stiffness matrix, 342–357
strain/displacement relationship, 348–350
stress results, 379
stress/strain relationship, 350
transition triangles for modeling, 402
Constitutive law, 12
Convection, heat transfer with, 604–606
Coordinates, 73–78, 487, 537–538
bar elements, 73–78
Cartesian, 537–538
local, 73–78
natural (intrinsic), 487
stiffness matrix, 73–78
stress analysis, 537–538
three-dimensional elements, 537–538
trusses, 73–78
Coulomb’s law, 701–702
Cramer’s rule, 845–846
Cross sections for grid equations, 265–267
CST elements, see Constant-strain triangular (CST)
elements
Cubic elements, 10, 11
Cubic rectangles (Q12), 525–526
Current flow, 698–699
Curvature, 170–171, 576–577, 580–581
beams, 170–171
moments of, 576–577, 580–581
plate bending elements, 576–577, 580–581
stress/strain relationships and, 576–577
strain/displacement relationships, 580–581
D
Damping, 812–816
Darcy’s law, 675
Deformation of plate bending elements, 573
Degrees of freedom, 15, 16, 32
Determinants, 835–837
Dielectric constants, 704–705
Differential equations, 601–605, 674–678
boundary conditions and, 603–604, 676–678
conservation of energy, 601–602
conservation of mass, 674–675
convection, 604–605
Darcy’s law, 675
fluid flow, 674–678
Fourier’s law of heat conduction, 602
heat transfer, 601–605
one-dimensional heat conduction, 601–603
pipes, flow through, 677–678
porous medium, flow through, 674–676
solid bodies, flow around, 677–678
two-dimensional heat conduction, 603–604
Differentiating a matrix, 833–834
Direct methods, 8, 14, 31–71
force (flexibility), 8
displacement (stiffness), 8, 31–71
Direct stiffness method, 3, 14
Direct stiffness method, 3, 14, 31, 38–40, 182–195, 612,
624–626. See also Superposition
beam analysis using, 182–195
global equations assembled using, 14, 612
heat transfer equations from, 612, 624–626
history of, 8
spring analysis using, 38–40
superposition methods as, 14, 31
total stiffness matrix from, 38–40
Discretization, 2, 9–10, 364–365, 395–397, 462–463
axisymmetric pressure vessels, 462–463
coarse-mesh generation, 343
finite method use of, 2, 9–10, 364–365
natural subdivisions for, 395–397
plane stress and, 364–365
Displacement, 32–33, 35, 40–41, 73–76, 78–82, 378–
379, 573–575. See also
Strain/displacement relationships
axial (longitudinal), 73–76
bar elements, 73–76Index 941
CST and Q4 result comparison, 378–379
Kirchhoff assumptions and, 573–575
nodal, 32–33, 35, 40–41, 73–76, 78–82
plate bending elements, 573–575
spring elements, 32–33, 35, 40–41
truss elements, 76, 78–82
Displacement functions, 10–12, 78–82, 173–174,
344–348, 375, 438–440, 455–457, 488–489, 495,
540–542, 578–580, 764
axisymmetric elements, 455–457, 488–489
bar elements, 78–82, 764
beam elements, 173–174
conforming (compatible) functions, 81
constant-strain triangular (CST) elements, 344–348
finite element method selection of, 10–12
interpolation functions, 80
isoparametric formulation and, 488–489, 495
linear-strain triangle (LST) elements, 438–440
one-dimensional bars, 78–82
plate bending elements, 578–580
rectangular (Q4) plane elements, 375, 495
shape functions and, 80, 174, 456–457, 488–489
tetrahedral elements, 540–542
time-dependent (dynamic) analysis, 764
Displacement (stiffness) method, 8, 31–71. See also
Stiffness method
Distributed loading, 195–208
beam elements, 195–208
fixed-end reactions and, 195–196
load replacement, 197–198
nodal forces, 198–199
work-equivalence method for, 196–198
Dynamic analysis, 761–826
axisymmetric elements, 799–800
bar elements, 764–768, 780–784
beam elements, 789–796
computer program solutions, 808–816
damping, 812–816
dynamic equations, 764–768
heat transfer, 801–807
mass matrix, 765–767, 789–800
natural frequency, 763, 780–784, 789–796, 808–812
numerical time integration, 768–780,
802–804
one-dimensional bar elements, 780–789
plane frame elements, 797–798
plane stress/strain elements, 798–799
spring-mass system, 762–764
tetrahedral (solid) elements, 800
time-dependent problems, 764–768, 784–789,
801–807
truss elements, 796–797
E
Effective nodal forces, 257
Elasticity theory, 865–872
compatibility equations, 867–869
equations of equilibrium, 865–867
strain/displacement equations, 867–869
stress/strain relationships, 867–872
Electric displacement fields, 707, 710
Electric field/potential gradient relationship,
706–707, 710
Electrical networks, 697–701
current flow, 698–699
fluid flow and, 697–701
Kirchhoff’s current law, 699
Ohm’s law, 697
sign convention, 698–699
voltage drop, 697–699
Electrostatics, 701–715
boundary conditions, 709–710
Coulomb’s law, 701–702
dielectric constants, 704–705
electric displacement fields, 707, 710
electric field/potential gradient relationship, 706–
707, 710
element type selection, 705
finite element method for, 705–715
fluid flow and, 701–715
Gauss’s law, 702–704
global equations, 709–710
Poisson’s equation, 704
potential function, 705–706
stiffness matrix, 707–709
two-dimensional triangular elements, 705–710
voltage (potential) gradient, 706–707
Elements, 9–13, 32–36, 38–40, 65–66, 72–168, 169–
238, 342–357, 362–363, 374–379, 392–395,
403–404, 437–438, 444–447, 455, 487–488,
493–495, 539–540, 578, 607–613, 626–629, 679,
681, 705, 764
axisymmetric, 455
bars, 72–168, 487–488, 764
beams, 169–238
conduction matrix and equations, 608–612,
628–629
connecting different types of, 403–404
constant-strain triangular (CST), 342–357, 362–363
discretization and, 9–11
electrostatics, 705
finite element steps for, 9–18
fluid flow, 679, 691, 705
forces, 36, 77, 355
heat transfer and, 607–613, 626–629942 Index
Elements (continued)
isoparametric formulation of, 487–488, 493–495
linear-strain triangle (LST), 437–438
LST and CST comparison of, 444–447
plane stress and plane strain in, 342–357,
362–363, 374–379
plate bending, 578
rectangular (Q4) plane, 374–379, 493–495
shapes for modeling, 392–395
springs, 32–36, 38–40, 65–66
stiffness matrix and equations, 12–13, 32–36, 38–40,
65–66
stresses, 355
temperature function, 607–608, 626–627
temperature gradients, 608, 613, 627–628
tetrahedral, 539–540
time-dependent (dynamic) analysis, 764
triangular, 691
trusses, 75, 77
Energy method, 12–13
Equations, 1–2, 6, 7, 12–14, 12–13, 32–36,
38–40, 55–66, 72–168, 169–238, 337–390, 437–
450, 601–605, 607–636, 647–678,
843–864, 865–872. See also Conduction matrix;
Stiffness matrix
bar elements, 72–168
beams,169–238
compatibility, 867–869
constant-strain triangular elements, 342–357,
362–363
differential, 601–605, 674–678
elasticity theory, 865–872
element stiffness matrix, 12–13, 32–36, 38–40, 65–66
equilibrium, 865–867
finite element method and, 1–2, 7
fluid flow, 674–678
global conduction, 612–613
global stiffness, 6,14, 35
heat transfer, 601–605, 607–636
homogeneous, 843
linear-strain triangle (LST) elements, 437–450
matrix forms, 12–14
nonhomogeneous, 843
plane stress and plane strain, 337–390
Poisson’s, 704
potential energy approach for, 55–65
simultaneous linear, 843–864
spring elements, 32–36, 38–40, 55–66
strain/displacement, 12, 34, 867–869
stress/strain, 12, 15, 34
trusses, 72–168
Equilibrium, equations of, 865–867
Equilibrium of modeling results, 405–408
Equivalent joint force replacement method, 253–258
Equivalent nodal forces, 198–199
Errors, checking models for, 404
Euler-Bernoulli beam theory, 171–177
Exact solution, 132–136, 208–214
bar elements, 132–136
beam elements, 208–214
finite element method compared to, 132–136,
208–214
External forces, potential energy and, 121–123
Explicit numerical integration method, 804
F
Field problems, 55
Finite element method, 1–30, 123–127, 132–136,
208–214, 363–374, 392–408, 607–636, 639–641,
642–646, 678–695, 705–715, 764–768
advantages of, 21, 24
applications of, 14–25
bar elements, 132–136, 764–768
beam elements, 208–214
computer programs for, 25–27
computer, role of, 6–7
defined, 1
degrees of freedom, 15
direct equilibrium method, 12
direct methods, 8
direct stiffness method and, 3, 624–626
discretization for, 2, 9–10, 364–365
displacement function selection, 10–12
electrostatics, 705–715
element conduction matrix and equations,
608–612, 628–629
element stiffness matrix and equations, 12
energy method, 12–13
exact solution compared to, 132–136, 208–214
fluid flow, 678–695, 705–715
Galerkin’s residual method for, 642–646
global stiffness matrix, 365–370
heat transfer equations, 607–636, 639–641, 642–646
history of, 3–4
hydraulic networks, 687–691
mass transport and, 642–646
matrix notation, 4–6
maximum distortion theory and, 373
modeling, 2, 392–408
nonstructural problems, 16
one-dimensional fluid flow, 678–691Index 943
one-dimensional heat conduction, 607–626
plane stress solutions, 363–374
potential energy approach for equations, 123–127
result interpretation, 15
steps of, 7–14
strain/displacement relationships, 12
stress/strain relationships, 12, 15
structural problems, 15
three-dimensional heat transfer, 639–641
time-dependent (domain) analysis, 764–768
two-dimensional fluid flow, 691–695
two-dimensional heat transfer, 626–636
variational methods, 8–9, 607–626
weight residual methods, 9–12
weighted residual method, 13–15
work method, 12–13
Fixed-end forces, 253–254
Fixed-end reactions, 195–196
Flexure formula, 175
Flowcharts, 143, 413, 646–650, 696–697,
771, 776
central difference method, 771
fluid flow, 696–697
heat transfer, 646–650
modeling, 413
Newmark’s method, 776
numerical integration, 771, 776
time-dependent (dynamic) analysis, 771, 776
truss analysis, 143
Fluid flow, 673–726
boundary conditions, 676–678, 682, 709–710
conservation of mass, 674–675
Coulomb’s law, 701–702
Darcy’s law, 675
differential equations, 674–678
electrical networks, 697–701
electrostatics, 701–715
finite element method for, 678–695
flowcharts, 696–697
Gauss’s law, 702–704
hydraulic networks, 687–691
Kirchhoff’s current law, 699
line or point sources, 695
Ohm’s law, 697
one-dimensional, 674–676, 678–691
pipes, 677–678
porous medium, 674–676
potential functions, 679, 691–692, 705–706
solid bodies, flow around, 677–678
two-dimensional, 676, 691–695
velocity potential function, 677–678
Force/displacement equations, 296–297
Force matrix, 377
Force (flexibility) method, 8
Forced convection, 604, 606
Forces, 36, 42, 77, 121–123, 198–199, 253–254, 355,
357–362, 459–461, 491–492, 500–501, 543–544,
764
axisymmetric elements and, 459–461
body, 357–359, 459–460, 491, 500, 543–544
centrifugal body, 358
effective nodal, 257
element solutions, 36, 77, 355
equivalent nodal, 198–199
external, 121–123
fixed-end, 253–254
friction, 764
nodal, 42, 198–199, 257
isoparametric formulation of, 491–492, 500–501
reactions (nodal force), 42
stresses as, 355
surface, 359–362, 460–461, 491–492,
500–501, 544
three-dimensional elements, 543–544
Forcing function, 808–809, 811
Foundation load, 403
Frame equations, 239–262, 280–336
bar elements, 258–261
beam elements, 239–243, 280–294
bending and, 281–285
effective nodal forces, 257
equivalent joint force replacement method, 253–258
fixed-end forces, 253–254
inclined supports, 261–262
plane, arbitrary beam orientation in, 239–243
rigid plane frames, 243–261
skewed supports, 261–262
space, arbitrary beam orientation in, 280–294
stiffness matrix, 293–243, 289–294
substructure analysis using, 295–299
three-dimensional orientation, 280–294
transformation matrix, 239–243, 282–285
two-dimensional orientation, 239–262
Free-end deflections, 377
Friction forces, 764
Functional, defined, 13
G
Galerkin’s residual method, 136–139, 142–143,
225–227, 642–646
bar elements (one-dimensional), 137–139, 142–143944 Index
Galerkin’s residual (continued)
beam elements, 225–227
finite element formulation and, 642–646
general formulation of, 136–137
heat transfer with mass transport equations by,
642–646
weighting functions and, 142–143
Gauss points, 550
Gaussian elimination, 847–854
complete pivoting, 854
partial pivoting, 854
sequential pivot elements, 853–854
simultaneous linear equations and, 847–854
zero pivot element, 852–853,
Gaussian quadrature, 506–515
integration using, 506–509
stiffness matrix evaluation by, 509–513
stress evaluation by, 513–515
two-point formula for, 507–508
Gauss-Jordan method, 837–839
Gauss’s law, 702–704
Gauss-Seidel iteration, 854–856
Global conduction equations, 612–613
Global mass matrix, 796–797, 799
Global stiffness equations, 6, 14, 35, 76, 180–182, 353–
355, 682, 709–710, 767–768
beam elements, 180–182
bar elements, 767–768
boundary conditions and, 14
constant-strain triangular elements, 353–355
defined, 6
electrostatics, 709–710
finite element assembly of, 14
fluid flow, 682
spring elements, 35
time-dependent (dynamic) analysis, 767–768
truss elements, 76
Global stiffness matrix, 32, 73–78, 365–370
bars arbitrarily oriented in the plane, 73–78
defined, 32
plane stress, 365–370
superposition for assembly of, 365–370
Gradient matrix, 124, 211, 350, 490, 498, 608, 627–628,
679–680, 706–707
beam equations, 211
hydraulic, 679–680
temperature, 608, 627–628
strain-displacement, 350
truss equations, 124
voltage (potential), 706–707
Grid, defined, 262
Grid equations, 262–280
cross sections for, 265–267
polar moment of inertia (J) for, 265–266
shear center (SC) for, 267
stiffness matrix for, 263–269
torsional constant for, 265–267
transformation matrix for, 268
H
Heat flux/temperature gradient relationships, 608,
627–628
Heat transfer, 599–672, 801–807
boundary conditions for, 603–604, 612
conduction (with convection), 604–605
conduction (without convection), 601–604
conservation of energy principle, 601–602, 642
convection and, 604–606
differential equations for, 601–605
finite element method for, 607–613, 626–636, 639–
641, 642–646
flowcharts for, 646–650
Fourier’s law of heat conduction, 602
Galerkin’s residual method for, 642–646
line sources for, 636–638
mass matrix, 802
mass transport and, 601, 641–646
numerical time integration, 802–804
one-dimensional, 601–605, 607–626, 641–642
point sources for, 636–638
temperature distribution and, 600
temperature function, 607–608, 626–627
thermal conductivities, 605–606
three-dimensional, 639–641
time-dependent (dynamic) analysis, 801–807
two-dimensional, 603–604, 626–636
units of, 605–606
variational method for, 607–626
Heat-transfer coefficients, 605–606
Hexahedral elements, 547–555
Gauss points for, 550
isoparametric formulation and, 547–555
linear, 547–550
quadratic, 550–555
shape functions for, 548, 551
stress analysis for, 547–555
Hinge nodes in beam elements, 214–221
Homogeneous boundary conditions, 41–42
Homogeneous equations, 843Index 945
Hydraulic gradient, 679–680
Hydraulic networks, fluid flow through, 687–691
I
Identity matrix, 831
Inclined supports, 112–121, 261–262
frame equations for, 261–262
truss equations for, 112–121
Infinite medium, 403
Infinite stress, 402
Integration, 503–509, 834–835
Gaussian quadrature, 506–509
isoparametric formulation and, 503–509
matrix, 834–835
Newton-Cotes method for, 503–506
two-point formula for, 507–508
Interpolation functions, 80
Inverse of a matrix, 832, 835–839
adjoint method, 835–837
cofactor method, 835–837
defined, 832
determinants, 835–837
Gauss-Jordan method, 837–839
row reduction, 837–839
Isoparametric, defined, 487
Isoparametric formulation, 486–535, 547–555
bar elements, 487–492
bilinear quadratic (Q6) elements, 521–522
cubic rectangles (Q12), 525–526
Gaussian quadrature, 506–515
hexahedral elements, 547–555
higher-order functions, 515–526
linear-strain bars, 515–520
natural (intrinsic) coordinates, 487
Newton-Cotes numerical method, 503–506
quadratic rectangles (Q8 and Q9), 522–525
rectangular (Q4) plane elements, 492–503
stiffness matrix from, 486–503, 509–513
transformation mapping and, 487
Iteration, Gauss-Seidel method for, 854–856
J
Jacobian matrix, 490
K
Kirchhoff assumptions for plate bending elements,
573–575
Kirchhoff’s current law, 699
L
Least squares method, 141–142
Line sources, 636–638, 695
Linear elements, 10, 11
Linear equations, see Simultaneous linear equations
Linear hexahedral elements, 547–550
Linear-strain bars, isoparametric formulation for, 515–520
Linear-strain triangle (LST) elements, 437–450
CST elements, comparison of, 444–447
defined, 440
displacement functions for, 438–440
element type selection, 437–438
Pascal triangle for, 438–439
shape functions for, 438–439, 443
stiffness, determination of, 442–444
stiffness matrix for, 437–442
strain/displacement relationships, 440–441
stress/strain relationships, 440–441
Load discontinuities, natural subdivisions at, 395–397
Load replacement, 197–198
Loading, 195–208, 402–403
concentrated (point), 402
distributed, 195–208
foundation, infinite medium for, 403
Local stiffness matrix, 35
LST elements, see Linear-strain triangle (LST) elements
Lumped-mass matrix, 765–766, 789–790, 797–799
M
Mass flow rate, 641
Mass matrix, 765–767, 789–800
axisymmetric elements, 799–800
bar elements, 765–767
beam elements, 789–796
consistent, 766–767, 790, 797–800
global, 796–797, 799
lumped, 765–766, 789–790, 797–799
plane frame elements, 797–798
plane stress/strain elements, 798–799
shape functions and, 797–799
tetrahedral (solid) elements, 800
time-dependent (dynamic) analysis, 765–767, 789–800
truss elements, 796–797
Mass transport, 601, 641–646
conservation of energy principle, 642
finite element method for, 642–646
Galerkin’s residual method for, 642–646
heat transfer with, 601, 641–646
one-dimensional heat transfer with, 641–642946 Index
Matrices, 4–6, 32–36, 38–40, 73–78, 82–89, 100–109,
124, 342, 350, 377, 490, 498, 608–612, 627–629,
679–680, 692, 731–733, 765–767, 789–800, 827,
831–833, 836–837, 839–840, 856–863. See also
Matrix algebra; Stiffness matrix
adjoint, 837
cofactor, 836
column, 5, 827
banded, 857
banded-symmetric, 856–863
defined, 4, 827
element conduction, 608–612, 628–629
force, 377
global stiffness, 35, 73–78
gradient, 124, 490, 498
hydraulic gradient, 679–680, 692
identity, 831
Jacobian, 490
local stiffness, 35
mass, 765–767, 789–800
notation for, 4–6
orthogonal, 832–833
positive definite, 839
positive semidefinite, 840
rectangular, 5, 827
row, 827
singular, 837
square, 827
stiffness, 32–36, 38–40, 73–78, 84–89,
100–109, 839–840
strain-displacement gradient, 350
stress–strain (constitutive), 342
symmetric, 831
system stiffness, 38
temperature gradient, 608, 627–628
thermal force, 731–733
thermal strain, 731–732
total (global) 38–40, 76
transformation (rotation), 82–84, 100–109, 832
unit, 831
Matrix algebra, 827–842
addition, 829
adjoint method, 835–837
cofactor method, 835–837
definitions, 827–828
determinants, 835–837
differentiation, 833–834
integration, 834–835
inverse of a matrix, 832, 835–839
multiplication by a scalar, 828
multiplication of matrices, 829–830
operations, 827–835
orthogonal uses, 832–833
row reduction, 837–839
stiffness matrix properties, 839–840
symmetry, 831, 839
transpose of a matrix, 830–831
types of matrices, 827
Maximum distortion energy theory, 373. See also von
Mises stress
Mechanical event simulation (MES), 18
Mesh revision and convergence, 397–401, 408–411
modeling accuracy from, 397–401
patch test for, 408–411
refinement methods for, 397–401
Mindlin plate theory, 583, 585
Modeling, 2, 391–436
aspect ratios, 392–395
compatibility of results, 405–408
computer program-assisted step-by-step solution,
414–419
concentrated (point) loads, 402
connecting (mixing) different elements, 403–404
element shapes for, 392–395
equilibrium of results, 405–408
errors, 404
finite element method, 2, 392–408
flowcharts, 413
foundation load and, 403
infinite medium and, 403
infinite stress and, 402
load discontinuities and, 395–397
mesh revision and convergence, 397–401, 408–411
methods of refinement, 397–401
natural subdivisions, 395–398
patch test for, 408–411
plane stress and plane strain problems, 413–419
postprocessor results, 404–405
results, 404–408
stresses interpreted for, 411–413
symmetry and, 395
transition triangles for, 402
von Mises (equivalent) stress and, 417–419
Moments of curvature, 576–577, 580–581
N
Natural (free) convection, 604, 606
Natural (intrinsic) coordinates, 487
Natural frequency, 763, 780–784, 789–796, 808–816
circular, 763
computer analysis and, 808–816
beam elements, 789–796
damping, 812–816Index 947
one-dimensional bar elements, 780–784
time-dependent (dynamic) analysis, 763,
780–784, 789–796, 808–812
Natural subdivisions at load discontinuities, 395–398
Natural subdivisions, 395–398
Newmark’s method, 774–778
Newton-Cotes numerical integration, 503–506
Newton’s second law of motion, 762
Nodal degrees of freedom, 15, 16, 32, 173–174
Nodal displacements, 10–12, 32–33, 35, 40–41, 73–76,
78–82, 355
bar elements, 73–76
boundary conditions and, 40–41
constant-strain triangular elements, 355
degrees of freedom, 32
determination of, 35
displacement functions for, 10–12, 78–82
spring elements, 32–33, 35, 40–41
stiffness method, 32–33, 36
truss elements, 76, 78–82
Nodal forces, 198–199
Nodal hinges for beam elements, 214–221
Nodal potentials, 682
Nodal temperatures, 613
Nodes, 10–11, 32, 395–398, 623–624
defined, 32
heat balance equations at, 623–624
natural subdivisions at, 395–398
placement of in elements, 10–11
Nonexistence of solution, 845
Nonhomogeneous boundary conditions, 42–44
Nonhomogeneous equations, 843
Nonstructural problems, finite element method for, 16
Nonuniqueness of solution, 844
Normal (longitudinal) strains, 341
Numerical comparisons, plate bending elements,
582–584
Numerical time integration, 768–780, 802–804
central difference method, 768–774
explicit, 804
heat transfer solution by, 802–804
Newmark’s method, 774–778
time-dependent (dynamic) analysis, 768–780,
802–804
trapezoid rule for, 802
Wilson’s method, 779–780
O
Ohm’s law, 697
One-dimensional bar elements, 78–82, 136–143, 731,
733, 780–789
displacement functions, 78–82
Galerkin’s residual method for, 136–139, 142–143
natural frequency, 780–784
residual methods for, 136–142
thermal strain matrix, 731, 733
time-dependent (dynamic) analysis,
780–789
truss bars, 78–82, 136–143
One-dimensional fluid flow, 674–676, 678–691
boundary conditions, 676–678, 682
branch elements, 687–688
conservation of mass, 674–675
Darcy’s law, 675
differential equations, 674–676
element type selection, 679
finite element method for, 678–691
global equations for, 682
hydraulic gradient, 679–680
hydraulic networks, 687–691
nodal potentials, 682
permeability of materials, 680
Poiseuille’s law, 687
potential function, 679
stiffness matrix for, 680–681
velocity/gradient relationship, 679–680, 682
volumetric flow rate, 682
One-dimensional heat transfer, 601–605, 607–626,
641–642
boundary conditions for, 603, 612
conduction with convection, 604–605
conduction without convection, 601–604
differential equations for, 601–605
direct stiffness method for, 612, 624–626
element conduction matrix, 608–612
element type selection, 607
finite element method for, 607–626
global conduction equations, 612–613
heat balance equations at nodes, 623–624
heat flux/temperature gradient relationships, 608
mass transport and, 641–642
nodal temperatures, 613
temperature function, 607–608
temperature gradients, 608, 613
variational methods for, 607–626
Open sections, 266
Orthogonal matrix, 832–833
Orthogonality condition, 284
P
Parasitic shear, 378
Partial pivoting, 854
Pascal triangle, 438–439948 Index
Patch test, 408–411
Period of vibration, 763
Permeability of materials, 680
Pinned boundary condition, 145
Pipes, fluid flow through, 677–678
Pivot elements, 852–854
Plane stress/strain, 337–390, 413–419, 731–732,
798–799
body forces and, 357–359
computer program-assisted step-by-step solution,
414–419
constant-strain triangular (CST) elements,
342–357, 362–363, 377–379
defined, 338–339
discretization for, 364–365
finite element solution for, 363–374
flowcharts for, 413
global stiffness matrix for, 365–370
mass matrix, 798–799
maximum distortion theory for, 373
modeling problems, 413–419
principal angle, 340
principal stresses, 340
rectangular (Q4) plane elements, 374–379
shape functions for, 347, 798
surface forces and, 359–362
thermal strain matrix, 731–732
two-dimensional state of, 339–342
Plane trusses, solutions for, 91–100
Planes, 239–261, 374–379, 492–503, 797–798. See also
Two-dimensional elements
arbitrary beam orientation in, 239–243
equations for elements in, 243–261
frames, 239–261, 797–798
isoparametric formulation for, 492–503
mass matrix, 797–798
rectangular (Q4) elements, 374–379, 492–503
rigid frames, 243–261
Plate bending elements, 572–598
computer solutions for, 584–588
concept of, 572
curvature relationships, 580–581
deformation of, 573
displacement and, 573–575
displacement functions, 578–580
element type selection, 578
geometry of, 573
Kirchhoff assumptions for, 573–575
Mindlin plate theory, 583, 585
numerical comparisons, 582–584
potential energy of, 577
stiffness matrix for, 577–582
strain/displacement relationships, 580–581
stress (moment)-curvature relationships, 580–581
stress/strain relationships, 575–577
Veubeke “subdomain” formulation, 583
Point sources, 636–638, 695
Poiseuille’s law, 687
Poisson’s equation, 704
Polar moment of inertia (J), 265–266
Porous medium, fluid flow through, 674–676
Positive definite matrix, 839
Positive semidefinite matrix, 840
Postprocessor modeling results, 404–405
Potential energy approach, 55–65, 121–132,
222–225, 350–353
admissible variation, 58–59
approximate values from, 123
bar elements, 121–132
beam elements, 222–225
constant-strain triangular elements, 350–353
external forces, 121–123
finite element equations from, 123–127
matrix differentiation, 127
principle of minimum potential energy, 56, 123
spring element equations by, 55–65
stationary value of a function, 58
strain energy and, 57, 225
total potential energy, 56–59, 61, 123–127,
222–224
truss equations, 121–132
variation of, 58–59
Potential energy of plate bending elements, 577
Potential functions, 679, 691–692, 705–706
electrostatics, 705–706
fluid flow, 679, 691–692
one-dimensional flow, 679
two-dimensional flow, 691–692
Pressure vessels, 462–468
discretization, 462–463
stiffness matrix, 463–467
Primary unknowns, 15
Principal angle, 340
Principal stresses, 340
Pure bending, 378–379
Q
Q4 symbol, 374
Quadratic elements, 10, 11, 515–526, 550–555. See also
Rectangular (Q4) plane elements
bilinear quadratic (Q6), 521–522Index 949
cubic rectangles (Q12), 525–526
defined, 10, 11
hexahedral, 550–555
isoparametric formulation for, 515–526, 550–555
linear-strain bars, 515–520
quadratic rectangles (Q8 and Q9), 522–525
Quadratic rectangles (Q8 and Q9), 522–525
Quadrilateral consistent-mass matrix, 799
R
Reactions (nodal force), 42
Rectangular (Q4) plane elements, 374–379, 492–503
body forces and, 500
constant-strain triangular (CST) comparison to,
377–379
displacement functions, 375, 495
displacement results, 378–379
element type selection, 374, 493–495
force matrix for, 377
free-end deflections, 377
isoparametric formulation for, 492–503
plane stress and plane strain in, 374–379
shape functions for, 375, 494–495
stiffness matrix for, 374–377, 492–503
strain/displacement relationship, 376,
496–499
stress results, 379
stress/strain relationship, 376, 496–499
surface forces and, 500–501
Rectangular matrix, 5, 827
Refinement, 397–401
h method of refinement, 399
mesh revision and convergence, 397–401
modeling accuracy from, 397–401
p method of refinement, 399–401
r method of refinement, 401
Reflective symmetry, 109–112
Residual methods, 136–143
bar element formulation for, 137–139
collocation method, 140–141
Galerkin’s residual method, 136–139, 142–143
least squares method, 141–142
one-dimensional bar elements, 136–142
subdomain method, 141
truss equations, 136–143
Resistors, see Electrical networks
Rigid plane frames, 243–261
Rotation matrix, see Transformation (rotation) matrix
Row matrix, 827
Row reduction, 837–839
S
Shape functions, 80, 174, 211, 347, 375, 438–439, 443,
456–457, 488–489, 494–495, 541–542, 548, 551,
797–800
axisymmetric elements, 456–457, 799–800
bar elements, 80, 488–489
beam elements, 174, 211
constant-strain triangular (CST) elements, 347
hexahedral elements, 548, 551
isoparametric formulation and, 488–489, 494–495
linear-strain triangle (LST) elements, 438–439, 443
mass matrix and, 797–800
plane stress/strain elements, 347, 798–799
rectangular (Q4) plane elements, 375, 494–495
tetrahedral elements, 541–542, 800
truss elements, 797
Shear center (SC) for grid equations, 267
Shear locking, 378
Shear strain, 341
Simple harmonic motion, 763
Simultaneous linear equations, 843–864
banded-symmetric matrices, 856–863
coefficient matrix inversion, 846–847
computer programs for, 861–863
Cramer’s rule, 845–846
Gaussian elimination, 847–854
Gauss-Seidel iteration, 854–856
general form of, 843
nonexistence of solution, 845
nonuniqueness of solution, 844
pivot elements, 852–854
skyline, 856–857
uniqueness of solution, 844
wavefront method, 858–861
Singular matrix, 837
Skewed supports, 112–121, 261–262
frame equations for, 261–262
truss equations for, 112–121
Skyline, 856–857
Solid bodies, fluid flow around, 677–678
Space, arbitrary beam orientation in, 280–294
Spring constant, 32–33
Spring elements, 31–71
assemblage of, 36–38
boundary conditions for, 35, 40–55
compatibility (continuity) requirement for, 36
degrees of freedom, 32
element forces of, 36
element type selection, 33–34
global equation for, 35
nodal displacements for, 32–33, 35950 Index
Spring (continued)
potential energy approach, 55–65
stiffness matrix for, 32–36, 38–40
stiffness method for, 32–38, 55–65
strain/displacement relationship, 34
stress/strain relationship, 34
superposition for, 38–40
system (global) stiffness matrix, 38
total stiffness matrix for, 38–40
strain energy (internal) in, 57
Spring-mass system, 762–764
amplitude, 763
dynamic analysis, 762–764
period of vibration, 763
simple harmonic motion, 763
vibration, 763–764
Square matrix, 827
Stationary value of a function, 58
Stiffness, 170–180, 442–444
beams, 170–180
LST determination of, 442–444
Stiffness equations, see Stiffness matrix
Stiffness influence coefficients, 5
Stiffness matrix, 32–36, 38–40, 73–78, 84–89, 100–
109, 171–182, 239–243, 289–294, 337–390,
437–442, 451–462, 463–467, 486–503, 509–513,
543–547, 577–582, 680–681, 692–693, 707–709,
765–767, 839–840
axisymmetric elements, 451–462, 463–467
bar elements, 73–78, 84–89, 487–492, 765–767
beam elements, 171–182
boundary conditions, 35, 180–182
constant-strain triangular (CST) elements, 342–357
defined, 32
electrostatics, 707–709
element types, 33–34, 75, 172–173
Euler-Bernoulli beam theory, 171–177
fluid flow, 680–681, 692–693
frame elements, 293–243, 289–294
Gaussian quadrature evaluation of, 509–513
global, 32, 73–78, 180–182
integration for, 441–442
isoparametric formulation of, 486–503, 509–513
linear-strain triangle (LST) elements, 437–442
local coordinates, 73–78
local, 35
plane stress and plane strain, 337–390
plate bending elements, 577–582
potential energy approach to, 350–352
pressure vessel equations, 463–467
properties, 839–840
rectangular (Q4) plane elements, 374–377, 492–503
spring elements, 32–36, 38–40
stress analysis using, 543–547
superposition, assemblage by, 38–40
system (global), 38
tetrahedral elements, 543–547
time-dependent (dynamic) analysis, 765–767
Timoshenko beam theory, 177–180
total (global) 38–40, 76, 181–182
transformational matrix for, 100–109, 239–243,
282–285
truss equations, 73–78, 84–89, 100–109
Stiffness (displacement) method, 8, 31–71
boundary conditions for, 35, 40–55
direct method of, 8, 31
equations for, 32, 34–35, 65–66
examples for solution of assemblages, 44–55
nodal displacements, 32–33, 35, 40–41
potential energy approach, 55–65
spring elements, 32–38, 55–65
stiffness matrix for, 32–36, 38–40
superposition for, 38–40
Strain, 339–342, 537–539, 731–733. See also Plane strain
normal (longitudinal), 341
shear, 341
state of, 339–342
thermal strain matrix, 731–733
three-dimensional elements, 537–539
two-dimensional (plane) elements, 339–342
Strain/displacement relationships, 12, 34, 75, 174–175,
348–350, 376, 440–441, 454, 457–458, 489–490,
496–499, 542–543, 580–581, 764–765, 867–869
axisymmetric elements, 454, 457–458
bar elements, 75, 489–490, 764–765
beam elements, 174–175
constant-strain triangular (CST) elements, 348–350
curvature and, 580–581
elasticity theory, 867–869
equations, 867–869
finite element step, 12
isoparametric formulation and, 489–490, 496–499
linear-strain triangle (LST) elements, 440–441
plate bending elements, 580–581
rectangular (Q4) plane elements, 376, 496–499
spring elements, 34
tetrahedral elements, 542–543
time-dependent (dynamic) analysis, 764–765
Strain energy, 57, 225, 729–730
beam elements, 225
potential energy approach, 57, 225
thermal stress and, 729–730Index 951
Stress, 89–91, 337–390, 402, 411–419, 470–472, 513–
515, 537–539, 580–581
bar in x–y plane, 89–91
CST and Q4 result comparison, 379
element forces as, 355
finite element model interpretation of, 411–413
Gaussian quadrature evaluation of, 513–515
infinite, 402
moment-curvature relationships, 580–581
plane, 337–390, 414–419
plate bending elements, 580–581
three-dimensional elements, 537–539
von Mises (equivalent), 373, 417–419, 470–472
Stress analysis, 536–571
Cartesian coordinates, 537–538
hexahedral elements, 547–555
isoparametric formulation and, 547–555
strain representation, 537–539
strain/displacement relationships, 542–543
stress representation, 537–539
stress/strain relationships, 538, 542–543
tetrahedral elements, 539–547
three-dimensional elements, 536–571
Stress–strain (constitutive) matrix, 342
Stress/strain relationships, 12, 15, 34, 75, 174–175,
350, 376, 440–441, 452–454, 457–458, 489–490,
496–499, 538, 542–543, 575–577, 764–765,
867–872
axisymmetric elements, 452–454, 457–458
bar elements, 75, 489–490, 764–765
beam elements, 174–175
constant-strain triangular (CST) elements, 350
elasticity theory, 867–872
finite element, 12, 15
isoparametric formulation and, 489–490, 496–499
linear-strain triangle (LST) elements, 440–441
moments of curvature and, 576–577
plate bending elements, 575–577
rectangular (Q4) plane elements, 376, 496–499
spring elements, 34
tetrahedral elements, 542–543
three-dimensional elements, 538, 542–543
time-dependent (dynamic) analysis, 764–765
Structural dynamics, see Dynamic analysis
Structural problems, finite element method for, 15
Subdomain method, 141
Substructure analysis, 295–299
Superposition, 38–40, 365–370. See also Direct stiffness
method
global stiffness matrix assembled by, 365–370
total stiffness matrix assembled by, 38–40
Supports, 112–121
boundary elements for, 120–121
inclined, 112–121
skewed, 112–121
truss equations for, 112–121
Surface forces, 359–362, 460–461, 491–492, 500–501, 544
axisymmetric elements, 460–461
bar elements, 491–492
plane stress and plane strain from, 359–362
rectangular (Q4) plane elements, 500–501
tetrahedral elements, 544
Symmetric matrix, 831
Symmetry, 109–112, 395, 451–485, 831
axis of revolution, 452
axisymmetrical elements, 451–485
matrix operations, 831
modeling problems and, 395
structures, 109–112
T
Temperature, 600, 607–608, 613, 626–628, 728–729
distribution, 600
function, 607–608, 626–627
gradients, 608, 613, 627–628
heat transfer and, 600, 607–608, 613, 626–628
nodal, 613
thermal stress and, 728–729
uniform change, 728–729
Tetrahedral elements, 539–547, 800
body forces, 543–544
displacement functions for, 540–542
element type selection, 539–540
mass matrix, 800
shape functions for, 541–542
stiffness matrix for, 543–547
strain/displacement relationships, 542–543
stress analysis of, 539–547
stress/strain relationships, 542–543
surface forces, 544
Thermal conductivities, 605–606
Thermal force matrix, 731–733
Thermal strain matrix, 731–732
Thermal stress, 727–760
axisymmetric elements, 732–733
coefficient of thermal expansion, 728
one-dimensional bars, 731, 733
solution procedure, 733
strain energy and, 729–730
two-dimensional (plane) elements, 731–733
uniform temperature change and, 728–729952 Index
Three-dimensional elements, 10–11, 100–109, 143,
280–294, 536–571
bar elements, 100–109, 143
beams, arbitrary orientation of, 280–294
bending in, 281–285
Cartesian coordinates, 537–538
flowcharts for, 143
frame equations for, 280–294
hexahedral, 547–555
mass matrix, 800
node placement, 10–11
solid, 539–547, 800
space, 10–11, 100–109, 143, 280–294
stiffness matrix, 100–109
stress analysis for, 536–571
tetrahedral, 539–547, 800
transformation matrix, 100–109
Three-dimensional heat transfer, 639–641, 646–650
Time-dependent problems, 764–768, 784–789, 801–816
bar elements, 764–768, 780–784
boundary conditions, 767–768
computer program solutions, 808–816
displacement functions, 764
dynamic equations, 764–768
element type selection, 764
finite element equations, 764–768
forcing function, 808–809, 811
global equations, 767–768
heat transfer, 801–807
mass matrix, 765–767
numerical time integration, 768–780, 802–804
one-dimensional bar analysis, 784–789
stiffness matrix, 765–767
strain/displacement relationships, 764–765
stress/strain relationships, 764–765
Timoshenko beam theory, 177–180
Torsional constant (J), 265–267
Total potential energy, 56, 61
Transformation (rotation) matrix, 82–84, 100–109,
239–243, 268, 282–285, 832–833
beam elements, 239–243, 282–285
displacement vectors in two dimensions, 82–84
frame equations and, 239–243, 282–285
grid equations and, 268
orthogonal matrix uses, 832–833
stiffness matrix from, 100–109, 239–243, 282–285
three-dimensional (space) orientation, 100–109,
282–285
truss equations and, 82–84, 100–109
two-dimensional (plane) orientation, 239–243
Transformation mapping, 487
Transition triangles, 402
Transverse shear deformation, 177–180
Trapezoid rule, 802
Triangular elements, 342–357, 362–363, 377–379, 402,
437–450, 451–485, 691, 705–710
axisymmetric elements, 451–485
constant-strain (CST), 342–357, 362–363,
377–379, 402, 444–447
electrostatic flow, 705–710
linear-strain (LST), 437–450
two-dimensional fluid flow, 691, 705–710
Truss equations, 72–168, 796–797. See also Bar
elements
bar elements, 72–168
boundary elements for, 120–121
collocation method for, 140–141
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