Using Finite Elements in Mechanical Design
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Dr. J. Toby Mottram and Dr. Christopher T. Shaw
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Using Finite Elements in Mechanical Design
Dr. J. Toby Mottram and
Dr. Christopher T. Shaw

CONTENTS
Preface xi
Notation xiv
1 Relationship Between Design and Finite Elements 1
I . I The Design Process
1.2 Evaluating Designs
1.3 Using Computers for Design Evaluation
1.4 Typical Design Situations
1.5 Historical Review of Obtaining Structural Design Solutions
by Computer
1.6 Overview of the Modern Finite Element Method
1.7 Industrial Implementation and Research in Finite Elements
1.8 References and Further Reading
2 Structural Mechanics
2.1 The Use of Load-Bearing Structures
2.2 Engineering Stress and Strain
2.3 Some Simple Situations
2.4 Describing Problems in Structural Mechanics
2.5 References and Further Reading
3 Finite Element Solutions of the Equations
Developing the Finite Element Methodology
3.2 Satisfying Equilibrium and Compatibility
3.3 Use of Elements with Different Spatial Dimensions
3.4 Overview of Methods for Calculating Element Equations
3.5 The Role of Displacement Distributions
3.6 The Direct Approach to Element Formulation
3.7 Isoparametric Elements
3.8 Other Types of Elements
3.9 Producing a Solution from the Element Equations
3.10 References and Further Reading
4 Implementing a Computer-based Analysis Procedure
4.1 A Process for Analysing a Structure
CONTENTS
4.2 Generic Software Packages for Stress Analysis
4.3 Available Computer Hardware Systems
4.4 Matching Hardware Types to Analyses
4.5 Acquiring the Technologies
5 Building the Geometry Description and the Mesh 111
5.1 Think First—Compute Later
5.2 Overview of Computer Models of Geometry
5.3 Wireframe Models
5.4 Surface Models
5.5 Solid Models
5.6 Choice of Geometry Description
5.7 Overview of Mesh Generation
5.8 Advice on Choosing Element Types
5.9 Determining a Strategy for Distributing the Elements
5.10 Types of Mesh Structure
5.11 Building Meshes
5.12 Adapting a Mesh to Improve a Solution
5.13 Checking a Mesh
6 Obtaining and Analysing the Results 140
6.1 Specifying Material Properties
6.2 Finding the Boundaries of the Mesh
6.3 Imposing the Boundary Conditions on the Mesh
6.4 Controlling the Solution
6.5 Running the Solver
6.6 Troubleshooting
6.7 The Results Generated by the Solver
6.8 Using Computer Graphics to Evaluate Solutions
6.9 Checking a Solution for Accuracy
6.10 Making Refinements
6.11 Failure Criteria
7 Some Example Problems 161
7.1 The Brazilian or Split Tensile Test
7.2 A Pressure Vessel
161
174
8 More Advanced Problems 192
8.1 Composite Material Open-sectioned Beam
8.2 Design Example with Submodels
192
209
9 Industrial Case Studies 231
9.1 Analysing a Bicycle Frame
9.2 Design of a Connecting Rod
231
239CONTFNTS lx
9.3 A Composite Suspension Arm
9.4 Displacement Analysis of a Vehicle Body Shell
246
249
10 Solving More Complex Problems 252
10.1 Transient Problems
10.2 Vibration Problems
10.3 Optimization
10.4 Nonlinear Problems
10.5 Calculating Thermal Effects within Structures
10.6 References and Further Reading
References and Further Reading 262
Index 266INDEX
Accuracy. 12. 16, 36. 49, 54. 57. 71. 81. 108,
119-125, 142, 150
checking. 159
Backup of data. 98, 100 , 107
Bandwidth, 56, 86
algorithm, Culhill-McKee, 137
reduction, 136-137
Bar, 9-10, 15. 19, 30-32, 36 39
element, ( sec Element, bar)
Beam. 9-10, 15, 19. 37, 40, 182, 192-204.
20R-200, 254
element, ( see Element, beam)
Benchmarks, 16, 42, 71, 120
Bending, 15. 19. 40, 64, 66, 120, 124, 177,
184-185, 188, 192-210. 209-230, 238,
243, 249
element, Element, beam, plate, shell
moment. 40, 79. 194. 198. 214 217. 223. 225
Bezier surface, 114— 11 $, 129
Bicycle frame example, 231-239
Biharmonic equation, 41^32
Binary files, 107. 151 152
Body forces, 19, 24-25. 38^10, 48 49. 51 52, 77.
estimation of
, 16
of consistent element force vector. 76-77
of different elements, 56
of element stresses, 77-78
of examples. 16^169. 172-173. 180. 183. 185.

  1. 205.210, 213. 220
    of integration. 64. 69-75
    ol solution. 134-136. 152. 259
    Acquisition of technology. 4, 91, 101
    hardware, 106
    software, 42^13, 102
    Admissible functions, 12. 49. 50-51. 214, 223
    Advancing front method, 133
    Aircraft design, 5. 6, 10, 12, 192, 256
    Airy stress functions. 6. 14, 17, 41-42, 162
    Aluminium, 29 30. 35, 107, 240, 242. 244
    black. 201
    ASCII files. 107. 149, 151-153
    Analysis process:
    for structural problems. 43. 91-92. 94, 105,
    1. 130, 140, 149. 164- 165, 175
      goals, 108. 240
      initial thinking, 92, 140. 164, 175
      mesh generation, (sec Mesh, generation)
      numerical problem specification, 92, 142, 160
      numerical solution. 8, 47, 90, 9,7-94, 98, 140,
      92
      Bound:
      lower, 58
      upper, 57. 175
      Boundary conditions, (sec also Restraints)
      application to faces. 93, 143-146, 148, 180
      application to global equations, 84-85, 125
      essential, 14
      natural. 14
      symmetry. 118. 165, 182, 201-202. 243
      Boundary element (integral) method. 7
      B-Rcp solid model, 115-117
      Brazilian tensile test example, 42, 161-174
      Brick elements, (see Element, brick )
      Brittle material, (see Material, brittle)
      Duckling213 , 38 -214 , 44,—259 *5. 75, 81, 102, 109. 192.
      Bytes, 106-107, 151-152
      147
      results analysis. 43. 53. 78. 81. 93-94. 96.
      108-110, 122-124. 135, 141. 152-154,
      157-158. 160-251
      troubleshooting. 140. 150
      Analyst:
      team, 108. 191
      skills. 4. 17. 91. 108 110, 112
      Anisotropy. 30. 34, 44. 46. 81. 192-195. 197.
  2. 200
    Architecture:
    computer. 91. 97. 100
    novel. 98
    Assembly of equations. 7, 52, 58. 61. 82-84. 86
    Axial load. 31-32. 38- 39. 45, 61. 212. 214.
    216 217
    . 22.7. 243
    Axisymmetric elements, ( see Element,
    axisymmeiric)
    Computer-aided Design (CAD). 44. 74, 103,
    1. 247, 250
      model types:
      solid:
      B-Rep. 115-117
      Constructive Solid Geometry (CSG).
      115-116, 23.7
      surface:
      Bezier, 114-115. 129
      266I N D E X 267
      Coons patch. 114. 129
      Non-uniform rational B-spline ( NURBS ).
  3. 129
    wireframe, 112-115. 234, 247
    Computer-aided Engineering (CAE) svstems.
    96-97
    Conjugate gradient method. 89
    Calculus of variations. 9-10. 12
    post- processors. 16. 94. 96- 98, 103. 106,
    1. 153-154. 156. 158. 182.
      185-180. 204- 205. 207. 226. 238
      solvers. 85. 94-97, 103. 107-109. 111. 122.
    1. 149- 153. 159. 167. 182.
      195-196. 204. 252- 253. 258 259
      Ulililies. 94. 96-97
      specifying. 102-106
      speed:
      millions of instructions per second (mips).
  4. 106
    millions of floating point operations per
    second ( M FLOPS), 100. 106
    Carbon fibre composite material , (.see Material.
    Carbon fibre composite)
    Checking:
    data. 96, 144
    mesh. 134. 137
    coincident nodes. 138. 150-151
    exploded mesh. 139
    free-edge. 139
    free-faee, 138
    model , 94. 109. 117. 151, 255
    solution, 93-94, 150. 149
    Command files. 96. 109. 150-151. 153
    Commercial software, 7. 14- 16. 36. 54-57,
    70-7 ] , 74, 79. 92. 98. 101. 116. 129. 133,
    136-137. 167. 179. 184, 192. 209 210.
    212, 214, 231
    Communication of design solutions. 3, 100
    Comparison of Knite element and classical
    solutions, 169-173
    Compass point notation. 142
    Compatibility’. 6 8, 20. 28- 29, 37, 39 42. 49.
    53-54, 59.66. 79. 122. 125. 172. 206
    satisfying, 53- 54
    Completeness, 59, 71
    Complementary energy, Principle of minimum
    Concept design, 3
    Condensation technique. 77
    Conforming element, 59. 63
    non-, 79
    Connecting rod example. 239-245
    Connections. 9, 53. 113, 209 -210. 212
    semi-rigid. 9
    Connectivity list. 84. 97, 118, 138. 149. 152
    Consistent clement force vector, (.we Element,
    force vector)
    Consistent transformation, 51-52. 61 62.75 77.
    146, 180
    Contour plots. 78. 96, 154. 158-159. 167 173.
    182 191, 203- 204. 222. 226-230.
    239-240, 244-245.248
    Control ,
    of solution. 149-152, 253
    parameters, 92, 95, 235
    Convergence. 7, 59, 63, 66, 109, 119 -120. 122.
    1. 220, 253, 258- 259
      Constraints. 77. 81. 85, 108-109. 119, 124-125.
      149, 243
      on design variables, 3. 256-257
      multi-point, 108
      Coons patch. 114, 129
      Coordinates. 13-14, 21 -23, 33. 40. 48. 60.
      78- 79, 96. 107. H 3- II4. 118. 129.
      148-149. 152. 165, 177, 180, 182. 185.
  5. 221
    generalized. 49-50
    global. 57. 59. 61-62. 65. 76-77. 125. 128.
    156-158. 160
    local. 57. 63. 125. 128. 142. 154. 197, 200,
    57
    Compliances, 34
    Composite material, (see Material, composite)
    Composite material I-beam example. 192-209
    Computer, 97-101
    -aided design. 44. 74. 103, 113. 115, 129, 247.
    250
    hardware. 16, 43, 56, 91, 96-101. 103,
    106-108, 145, 152-154. 156. 191. 202
    graphics. 152-154
    minicomputers, 98. 101
    minisupercomputers. 98. 101. 152
    personal computers, 97. 100. 107
    parallel , 16. 97-8
    massively, 98
    supercomputers, 88, 98. 101, 107
    workstations, 97-8, 101, 107, 145. 152-153
    networks
    , 97-98, 109, 153
    peripherals, 97-100
    software:
    commercial, 7. 14-16, 36. 54, 56-57, 68.
    70 -71. 74. 79, 92. 98. 101, 116, 129, 133,
    1.36-137. 167. 179. 184. 192, 209 -210,
    212, 214, 231
    for finite element analysis:
    pre-processors, 16, 94-96, 106. 129-150.
    152-153, 160. 195, 20Q. 214, 241
    204
    natural. 64 78
    polar. 162
    Cost function. 256-257
    Coupon test, 31, 38, 175. 194. 197
    Crank-Nicholson method. 253
    Constructive Solid Geometry (CSG ) solid
    model. 115 116. 233
    Cuthill-McKee algorithm. 1.37
    Cutting plane, 154, 158
    Damping matrix, 254268 INDEX
    Data:
    checking, 96, 144
    files, 96-97, 107, 150-152, 155
    management, 93-94. 149’150
    storage:
    disk, 2, 97-100, 106-107, 149-150, 152,
    254, 259
    primary, 103, 106
    secondary, 98, 103, 106-107, 254
    Dead load, 19, 161, 189, 210, 227, 237-238
    deformed geometry. 96. 158, 167-168,
    182-184, 203-204, 216. 226, 238.
    243-244, 250-251
    vector plots, 96, 158
    Direct:
    method:
    solving simultaneous equations. 9, 16.
    85-89, 149, 152
    search method, 256-257
    stiffness, 9, 13-15, 85-86
    Divergence, 253, 258
    Domain creation, 129
    Drawings, engineering, 3, 5, 44, 177, 240
    Dynamic magnification, 210
    Dynamics of structure, 4, 43, 81, 101, 109, 127,
    210, 213, 227, 249, 254-255
    Decomposition:
    cell, 117
    LU, 86
    Deformation, 12-13, 18, 20, 26-35, 47-52. 75,
    82, 118, 120, 124
    of bar, 38
    of beam, 40, 56, 64, 78, 120
    of examples, 163, 167, 177, ISO, 182, 189-
    191, 195-206, 210, 214, 216, 225, 238,
    Eigenvalue, 149, 255, 259
    Eigenvector, 255
    259 Elastic:
    constants, 20, 29, 30, 33-34, 44, 81, 142, 165,
    175, 194-195, 200, 213, 215, 260
    range, 32, 34
    Elasticity, 8, 18-46, 53-54, 56, 202
    classical, 9, 12, 14-15, 21, 108, 161-163, 173
    Element, 54-56
    aspect ratio, 74, 121-122, 124, 156, 220, 222
    axisymmetric:
    formulation, 80
    use of, 80, 101, 119, 121, 176
    of shell, 80, 120
    large, 7, 19, 44, 191, 259
    Delaunay triangulation, 130, 133, 135
    Design:
    analysis, 4, 37
    communication, 3
    detailing, 3
    evaluation, 3-4
    generation of solutions, 3
    of frame, 209-230
    process, 1-6, 9-10, 12, 17, 25, 29, 32, 34-36,
    42, 44, 56, 58, 78, 91-93, 109. 175, 204,
    232, 240, 255-256
    situations, 5-6
    specification, 3, 92, 123, 141
    of examples, 160, 176, 179, 185, 189, 194,
    198, 204, 210, 244, 248-249
    synthesis, 3 ,
    variable, 175, 256-257
    Differential operator matrix, 28, 51, 67
    Dimensions, effect on model size, 37, 55-56,
  6. 121
    Discontinuities, 108-109, 120, 124-125, 159,
    174-175, 183, 185-186, 189, 191
    Discretization of space, (see Mesh, generation)
    effect on solution speed , 103
    Disk storage, 2, 97-100, 106-107, 149-150, 152.
    254, 259
    Displacement:
    distributions. 9-13. 15. 49-51, 53-54. 57-58.
    62, 66, 74-75, 80
    requirements. 58-59
    Display:
    of geometry, 154, 167, 179
    of results:
    contour plots, 78, 96, 154, 158-159,
    167-173, 182-191 , 203-204, 222,
    226-230, 239-240, 244-245.248
    bar:
    formulation, 54-55, 59-62, 66-68, 70, 72
    use of, 112-113, 118-119, 209, 225, 238
    beam:
    formulation, 78-79
    use of, 15, 112-113, 118, 120, 122,
    124-125, 209, 214-220, 228, 248
    brick, (sec Element, solid )
    characteristics, 11-15, 51-52, 57, 61-65, 75,
    77-78, 80
    compatibility, 53, 63, 79, 122
    contact, 122-123
    crack, 81
    distribution of, 158
    distortion of, 74
    equations, assembly, 7, 52, 61, 62, 82-85
    force vector, 11-14, 52, 56-57, 61. 82, 83
    consistent. 51-52, 57-58, 60, 61-63, 72,
    75-77, 146, 180
    gap, 81, 85, 119, 123
    hierarchical, 66
    interior angles of, 74
    lumped mass, 54, 253
    master, 64-68, 70, 74, 77, 166
    membrane, (See Element, plane stress/slrain
    or membrane)
    nodal degree of freedom vector, 11-15,
    50-54, 56-58, 60-63, 72, 75-79, 82-83,
    257INDEX 269
    plane stress/strain or membrane:
    formulation, 13-14, 54-56, 61-65, 68.
    70-75. 77-78. 80
    use of, 82-85, 119-120, 124, 126-127, 130,
    132-133, 136, 161-174
    Examples:
    bicycle frame, 231-239
    Brazilian tensile test. 161-174
    composite laminate beam, 192-209
    connecting rod, 239-245
    machine support frame, 209-230
    pressure vessel, 174—191
    suspension arm, 246-249
    vehicle body shell, 249-251
    Explicit methods. 253
    Eye position, 155-156
    plate bending:
    formulation, 55, 59, 77, 80
    use of, 115, 121-122
    requirements of, 53-54
    shape functions, li , 13, 51-52, 57-80, 177,
  7. 260
    shell:
    formulation, 55, 59, 64, 77-81, 177,
    195-197, 206
    Mindlin, 80, 177, 195, 206
    use of, 119-122, 124-125, 148. 174.
    177-184. 191, 195, 199-207, 210, 212,
    220-231, 233, 235, 238-239, 250-251
    Faces, of an element:
    applying boundary conditions, 93, 143-146,
    148, 180
    finding free faces. 138
    Failure criteria, 5, 23, 35-36, 108, 12.3, 141. 162
    Tresca or maximum shear stress, 36
    von Miscs’ or distortions!, 36, 78, 154. 160,
    168, 206, 222, 229-230, 239, 244
    Fatigue, 4. 6. 30. 35, 43, 209-210, 212, 218, 220,
    222, 227, 240, 243-234
    solid:
    formulation, 63, 74-75
    use of, 80, 119-121, 123, 133, 196, 220,
    241-245, 247-249
    special, 47, 79, 81
    stiffness matrix, (see Matrix, element
    stiffness)
    stress-hybrid, 57
    sub-parametric, 65
    super-parametric, 65
    transition, 81, 122
    types, choice of, 63, 112, 120, 123, 125, 177
    Embodiment design: 3
    Energy:
    method, 9-12, 15, 38. 47-52, 182, 253
    Principle of complementary, 57
    Principle of minimum potential, 10-12, 47-50
    Engineering drawings, 3, 3, 44, 177, 240
    Equations:
    algebraic, 9, 14-15, 50, 52. 61
    assembly, (see Assembly of equations)
    compatibility, 28-29, 39-41, 172
    equilibrium, 37, 40, 53, 172, 205
    stiffness, (see Stiffness, equations)
    stress-strain, (see Stress, -strain relationships)
    Equilibrium, satisfying, 7, 20-25, 37, 53-54,
    206, 212-213, 226
    Errors, 93, 107, 137, 150
    of discretization, 53-54, 70, 74. 79, 120
    of distorting element, 124, 130, 148
    estimation of, 15, 36
    handling of, 5
    of geometry modelling. 129-130
    of integration, 64
    of off-set element, 122
    residual, 88, 152
    round-off, 56, 74. 81, 108, 117, 120
    of solution, 88-89, 135
    in solving examples, 166, 171. 178-180, 182,
    184, 206-207, 209, 243, 249
    Essential boundary conditions, 14
    Evaluation of designs, 3 4
    Files:
    ASCII, 107, 149. 151-153
    binary, 107, 151-152
    command, 96, 109, 150-151, 153
    data, 96-97, 107, 150-152, 155
    EBCDIC, 151
    neutral
    , 97, 149
    results, 151-152
    Finite difference method, 7, 10, 42
    Finite element method,
    assembly of equations, 7. 52, 58, 61, 82. 84,
    86
    connectivity list, 84, 97, 118, 138, 149, 152
    Galerkin method, 10, 17
    research, 7, 14-15, 120, 122, 130
    shape functions, (.tee Element, shape
    function)
    stiffness matrix, ( see Stiffness, matrix)
    test (or trial) function, 11, 260
    variational form, 12, 47-50
    Flexural modulus, 197-198, 206, 209
    Force:
    body. 19, 24. 25, 38-tt), 48-49, 52. 76-77, 92
    matrix method. 10
    vector:
    element, ( see Element, force vector )
    global, (.tee Global, force vector)
    Fracture. 30. 35-36. 68. 81
    Free mesh, ( see Mesh, free)
    Free-face, 138-139, 144-145, 151, 154
    Functional, 12, 48-51, 91. 256
    Functions:
    admissible, 12. 49, 50-51, 214, 223
    cost. 256-257
    objective, 256-257
    shape, ( jee Shape, function)270 INDEX
    Galerkin method, 10. 17
    Gauss:
    Hooke’s Law, 29. 32, 194
    Homogeneity, 25. 29-30, 33-34. 195. 213, 141,
    Legendre quadrature rule, 69, 72 161. 163. 174. 192-193
    points, 69, 72. 75. 77-78, 124. 168. 184
    rule. 69
    -Seidel method, 87-88
    Generalized :
    coefficients, 59 60, 62 -63
    coordinates, 49- 50
    Generation of design solutions, 3
    Geometry:
    creation, 129
    description, choice of. 117
    models:
    wireframe, 112-115, 234, 247
    surface, 114-115, 129
    solid. 115-117, 233
    sources:
    analytical equations, 44
    Computer-aided Design (CAD), 44. 74,
    103, 113, 115, 129, 247, 250
    engineering drawings. 3, 5. 44, 177, 240
    measurement of models. 44
    specification, 44
    surface. 114-115, 129
    Global:
    stiffness matrix, (see Matrix, global stiffness)
    nodal degree of freedom vector, 14, 50.
    52-53, 61, 77. 84
    force vector, 12, 14, 52, 77, 84
    Graphics:
    concepts:
    cutting plane, 154, 158
    eye position, 155-156
    hidden-line display, 145, 157—158
    magnification, 156, 182
    target position, 155-156
    up direction, 155-156
    view area, 156
    volume clipping, 157
    zooming in, 153, 156, 227
    hardware, 98, 100-101. 103. 107, 145,
    152 154, 202
    button box, 153, 156
    mouse, 153
    Visual Display Unit (VDU), 153
    software, 97 100 107, 141
    using, 152-155
    111-conditioning, 74, 121. 124, 213
    Implicit methods, 253
    Inconsistent:
    loading, 77
    Initial conditions. 13, 32. 44. 48-49. 52, 61,
    76-77. 79. 81, 91-93. 95. 109, 149. 254,
    2S7-258. 260
    Instability of mesh. 75, 119
    Integration, 47, 49- 50, 61 , 64, 79
    closed-form, 64
    full, 75, 78. 253
    numerical, 47. 64, 67, 73, 72-76, 253
    in one dimension, 68-70
    in two dimensions, 72
    points, (see Gauss, points)
    reduced, 75
    rule:
    Gauss quadrature, 69
    Gauss-Legendre quadrature, 69, 72
    Newton-Cotes, 69
    Simpson’s, 69
    Trapezoidal, 69
    Interfacing software, 95-96, 103-104, 109, 129
    Irregular mesh structure, (see Mesh, irregular)
    Isoparametric elements, 57, 62. 64-67, 70- 72,
    74-78, 80, 120. 128
    Isotropy, 63
    Iteration:
    controlling, 149, 258
    processes:
    in design, 3, 93-94
    in mesh generation. 130
    to resolve nonlinearity, 85, 254, 258 -259
    to solve simultaneous equations, 86-89,
    149
    , 152
    w’ith time variation, 254, 258-259
    solution methods:
    Crank-Nicholson, 253
    conjugate gradient, 89
    Gauss-Seidel, 87-88
    Jacobi, 87
    line relaxation, 88
    multigrid. 89
    point relaxation, 88
    preconditioning, 89
    Stone’s method, 88
    successive overelaxation. 88
    Half-bandwidth, 56. 86, 136 137
    Hardware specification, 43, 56, 96 101. 103,
    106, 108
    Heat transfer. 4. 6. 43. 82, 102
    links to structure, 252. 260
    Hidden-line display. 134 , 145, 157-158
    Hierarchical elements, 66
    Hierarchy of entities, 113, 120. 128-129. 154,
    Jacobi method, 87
    Jacobian. 67-68, 72, 124
    boundary. 76-77
    matrix, 72
    Joints. 109, 209-210, 246, 248
    analysis of, 123, 204, 210-222, 243
    fabrication of. 210, 233
    177
    Historical development, 1. 6-16, 47. 50INDEX 271
    pin-, 62. 148, 210. 212. 214-215
    rigid-. 210, 212-220, 232
    properties, 20, 29-32, 34-36, 38, 44-45, 81,
    92-93, 122-124, 141-142, 151, 193-195,
    197, 206, 209, 257, 260
    specification, 42-43, 141-142, 165, 175-176,
    200, 213, 229, 237, 242. 247-248, 250
    steel, 29-35, 45. 81, 167, 176, 213, 215, 229,
    237, 239, 250
    Keyboard, 153
    Kirchoffs assumption, 195, 197, 199
    Lagrajigian.
    elements, 67, 70-72. 76
    strain, 25
    Lamina, 193-195, 200
    Laminate:
    element , 195, 197, 201, 204-205
    material, 36. 81. 192-198, 200, 202-205
    Line relaxation method, 88
    Load , 18-20, 44-45, 47^19. 80-81, 108-109.
    119, 124
    applied to examples, 85, 161-163, 165-166,
    173-174. 176, 183, 189, 196, 201-202,
    209 212,
    214-217, 223-225, 231, 237-238, 241, 243.
  8. 248, 252, 254
    axial. 30- 32, 35 -36. 38- 39
    axisymmetric. 80
    boundary conditions. 36-37. 143, 146-148
    dead, (.tee Dead , load)
    distributed, 13, 52, 61, 75-77
    element nodal force vector, i^ec Element.
    nodal force vector)
    global nodal force vector. (see Global, nodal
    force vector)
    Matrix:
    damping, 254
    differential operator. 28. 51. 67
    element stiffness, 11-15, 28, 51-52. 56-58,
    60-68, 70, 72, 74-75. 78-79, 81-83, 124,
    195, 257. 259
    global stiffness, 14, 50-53, 56, 59, 61, 77-78.
    82-85, 149, 254-255, 257-258
    Jacobian, 72
    mass, 253-255
    material property, 33, 41, 48, 53, 57, 61-63,
    72, 77. 79, 81. 194-195, 258
    shape function. 51-52. 57-68, 75-76, 79, 253,
    260
    spatial derivative!s) of field variable(s), 51,
    33, 57, 60, 62-63, 67, 69-70. 72,
    77
    . 79
    system matrix, 255
    Vandermode, 60, 62, 64, 67, 79
    Membrane deformation, 13, 40-41, 62-63, 80.
    82, 119-120, 124, 127, 161-174, 177,
    184-185, 188, 194-195, 197, 199, 201,
    206, 208
    element, free plane stress/strain or membrane)
    Mesh, 7. 44, 64. 93, 96, 111-139
    adaption, 136
    enrichment, 134-136
    refinement. 53, 56, 109, 120-121, 123,
    134-136
    boundary, 92, 95, 130, 132
    finding, 137-138
    checking, 134, 137
    generation, 44, 71. 73, 81, 92, 95, 112-114,
    117-120. 123-125. 128-129, 159. 177.
    live, 19
    Locking of mesh, 79, 108, 119, 122
    LU decomposition, 86
    Lumping:
    load. 77
    mass, 54, 81. 253
    properties. 39. 80. 197. 204. 214
    Machine support frame example. 209-230
    Mapped mesh, ( see Mesh, mapped)
    Mapping. 65, 128
    from/to master element, 65-68, 72
    distortion, 70-71, 73-74. 77. 117. 122-124.
    178, 200. 221. 244
    Massively parallel computer. 98
    Master:
    degrees of freedom. 255
    element, (see Element , master )
    shape function, (see Shape function, master)
    Material, 29-36
    anisotropic. 30. 34. 44. 46. 81. 192-195. 197.
    206, 209
    brittle. 35-36.42, 161-163
    carbon fibre composite, 193-194, 196.
    200-201
    composite. 29 30. 32. 34, 36. 81, 141,
    192 -209, 231, 246-249
    orlhotropic. 193-194, 201
    plastic. 29-30, 32. 36. 174 -175, 189, 191
    250
    algorithms
    Delaunay triangulation, 130, 133, 135
    advancing front method, 133
    Octree methods, 131, 133
    Quadtree methods, 131, 133
    examples. 11, 56, 82-85, 131 -136, 166-167,
    177-180. 199-200. 215, 221-222, 234 ,
    237-237, 241-242, 247, 250-251
    need for. 118
    simple, 82-86, 114, 136, 145
    using:
    commercial software. 92
    local software, 92
    instability. 75, 119
    locking, 108. 119. 122
    parts:272 I N D E X
    mesh {continued)
    elements. 53, 56. 63, 77. 92. 94, 103, 112.
  9. 117-140, 142-144. 147-148.
    151-152, 154. 157. 160-161. 165-166.
    1. 177-179. 199-200. 214-215.
      221-223, 231, 235. 241. 247. 250. 259.
      National Agency for Finite Element Methods
      and Standards (NAFEMS). 15. 80
      benchmarks. 16. 71. 120
      Natural:
      boundary conditions, 14
      coordinates 64-78
      frequency. 255
      Need:
      recognition of. 2
      Network
      . 97-98. 109. 153
      Neutral files. 97, 149
      Nodal:
      260
      nodes. 92. 94. 103. 107, 114, 118, 122-123.
      125-127. 130-140. 142-144. 147-148.
    1. 166, 174, 178-179. 200,
      215 222 235 “’50 259
      refinement, 59. 108, 124. 134-136. 161-191
      h-, 109, 136, 169, 174. 179. 183. 189
      hp-. 136
      P-. 109, 136, 161, 166. 169
      structure, 125-129, 136
      comparison of, 56
      determining, 114, 118, 125-128
      regular, 10, 125-127, 135
      irregular, 13. 125-127
      topology, (see Mesh, structure)
      types:
      free (and unstructured), 74, 102. 125. 127.
      129, 133. 138-139. 154. 160-161. 166,
      degree of freedom vector, element, (see
      Eleinent. nodal degree of freedom
      vector)
      Node:
      interior. 68-71. 74. 77
      numbering, 136- 137
      Non-dimensional quantities, 162- 172
      Nonlinearity. 86, 109, 140. 149, 254
      of geometry, 7. 19. 44. 191. 259
      of material properties. 7. 15. 34-35, 44. 102.
      1. 259
        Number crunching, 96-98
        234 Numerical:
        control parameters, 92, 95. 149. 235, 252
        error. 56, 64. 70. 74. 81.90, 108, 117. 122.
      1. 178, 180. 185, 209
        integration , 47, 64. 68-70. 72 -76. 253
        solution process. 82-89. 119, 122. 135. 137.
        mapped (and structured ), 10, 125-127, 129,
        135, 142-146, 154. 157-160. 174.
        177-179, 221-222, 241, 247
        Millions of floating point operations per second
        (MFLOPS), 100, 106
        Mindlin theory. 79-80, 120, 177, 195, 197. 206
        Minicomputer, 98, 101
        Minimization, 9, 50-53, 256
        Millions of instructions per second (mips), 100,
        147
        Non-uniform rational B-spline ( NURBS), 115,
        129
        106 Objective function. 256-257
        Objectives, 246
        Octree methods, 131, 133
        Optimization. 13, 101-102, 140, 149, 171, 252.
        255-257
        Orthotropic material, (see Material, orthotropic)
        Mode:
        failure of, 36, 45. 194
        of instability, 36
        of vibration, 255
        rigid-body, 59
        shape, 252
        Model checking, (see Checking, model)
        Modulus:
        elastic constants, 20, 29, 30, 33-34, 44, 81,
        142, 165, 175, 194-195, 200, 213, 215,
        Parallel computer, massively, 16, 97-98
        Parent , (see Master)
        Partial differential equations, 6-14, 24-25,
        36-37, 54. 82. 91. 103. 109. 129. 136.
        260 142
        flexural, 197-198. 206. 209
        section, (see Section, modulus)
        shear, 34, 40, 194, 197-199, 208-209, 214
        Young’s, 29, 32-35, 40. 44. 92, 141, 175-176,
    1. 209, 213, 237, 248
      Mohr’s circle, 23, 28
      Moment, 47-52
      causing bending, 19-20, 40, 45, 79, 194, 198,
      212, 214-217, 223, 225
      -curvature expression, 79, 197-198, 204
      Mouse, 153
      Multigrid method, 89
      Multi-point constraint. 108
      Pascal’s triangle. 63. 71
      Patch test. 59. 80. 119-120. 177
      Pattern search method, 256
      Peripheral devices, 97- 100
      backup devices. 98. 100
      graphics displays, 97. 100. 134, 152 154
      hardcopy devices, 100
      data storage, 98. 106
      Personal computers. 97. 100. 107
      Pixel, 100, 153
      Plane:
      dement, (see plane stress/strain or membrane)
      strain. 10. 13.41^12INDEX 273
      stress, 13, 4fMl2,
      example of, 161-174
      Plastic material, (see Material, plastic)
      Plasticity, 7, 15, 34-35, 44, 102, 168, 251
      Plates, 13, 36-37
      element for bending, (see Element, plate
      bending)
      element for membrane action, (we Element,
      plane stress/strain or membrane)
      Plotting, ( see Display, contour plots)
      Point relaxation method , 88
      Poisson’s:
      equation, 260
      ratio, 29. 32, 44. 92, 175-176, 194, 213, 237,
      mesh, 110. 125-127. 129, 135, 142-146, 154,
      157-160, 177-179, 199-200, 222. 241,
      247
      Relaxation:
      method, 9
      successive over-, 88
      parameter, 148, 253, 258-259
      point, 88
      technique, 258
      Residual:
      error, 88, 152
      methods,
      weighted, 10, 15, 261
      stress, 48, 81
      248 Restraints, 6, 39, 43, 44, 84-85, 92, 146-148,
      Polymer, 174-175 150
      matrix, 193-194
      Polynomial, 11, 49-50, 54, 57-63, 69-75,
      113 115, 136
      incomplete, 63
      Legendre, 69
      Post-processing, 16, 94-98, 103, 106, 108, 109,
      X 37, ISO, 153 154, 156, 158, 182,
      185-186,
      204-207, 226, 238, 241
      results analysis, 151-159
      software tools, 94. 96, 103, 153—158
      Potential energy, 9-12, 47-51
      Principle of minimum, 10-12, 47-50
      Pre-processing, 6, 16, 94-96, 106, 152-153, 160,
      195, 200, 214, 241
      stages in process, 94-95, 129-150
      geometry generation, 94, 129-130
      mesh generation, (see Mesh, generation)
      setting boundary conditions, 93, 141-148
      setting initial conditions, 95
      setting numerical solution parameters, 95,
      of examples, 164-165, 175-177, 180-183,
      201-202, 215. 222-223, 238, 248, 250
      Results analysis, 43, 53, 78, 81, 93-94, 108-110,
      124-124, 135, 141, 152-154, 157-158,
      from ASCII files, 151-2
      of examples, 161-250
      using graphics, 16, 96-98, 103, 108, 109. 137,
      150, 153-154, 156, 158, 185-186,
      204-207, 226, 238
      Requirements:
      of the analysis, 42-43, 112, 161, 164.
      174-175, 185, 192, 196-197, 209-212,
      220, 231, 240, 246, 249
      of displacement distributions, 58-S9
      Rigid-body modes, 59
      Saint-Venant’s principle, 38-39, 186, 223
      Search method:
      direct. 256
      pattern, 256-257
      Section:
      modulus. 197-199, 208-209
      properties, 215
      Semi-discrete form, 253
      Sensitivity analysis, 257
      Shape function, 11, 13, 51-52, 57-80, 253, 260
      Hermitian, 66
      Lagrangian, 67-68, 70-72, 77
      master, 64-67, 70-71, 74-75, 77, 80, 177
      matrix, (see Matrix, shape function)
      serendipity. 70-71. 74
      Shear:
      deformation, 78-80, 195-200, 208-209
      force, 198, 206, 212, 216, 223
      modulus, 34. 39^10. 194. 197-198. 208-209.
      149
      software tools, y4~9b
      Preconditioning methods, 89
      Pressure vessel example, 174-191
      Properties of a material, (see Material,
      Modulus, Poisson’s ratio, strength)
      Quadrature, (see Integration, numerical)
      Quadtree methods, 131, 133
      Quality assurance (QA), 103-104
      Quasi:
      -isotropic, 201, 204
      -static load. 210-211
      214
      Random Access Memory ( RAM ), 97-100, 106
      Rayleigh-Ritz method, 9, 49-50
      Refinement:
      mesh, 59. 108-109, 124, 134-136, 161-191
      model, 94, 159-160, 161-191. 209-231
      Regular structure:
      geometry, Ml
      strain, 25-29, 32-37, 3?M2, 206
      stress, 21-25, 32-37, 39-42, 77, 164, 168-171,
  10. 205-206. 216.225
    Shell:
    element, (see Element, shell)
    theory, 79-80, 197
    Mindlin. 80. 120. 177. 195. 206274 INDEX
    Simultaneous equations. 9-10, 52-53. 58, 77
    solution:
    direct method, 85-86
    iterative methods:
    Crank- Nicholson. 253
    conjugate gradient, 89
    Gauss Seidel. 87-88
    Jacobi. 87
    line relaxation, 88-89
    multigrid relaxation, 89
    point relaxation, 88
    preconditioning, 89
    Stone’s, 88-89
    successive overrelaxation. 88
    Singularity, 68. 161-163
    Slender:
    members, 9. 17, 45, 56
    Software:
    commercial, (see Commercial, software)
    perpetual licences, 104-105
    specification, 94-97, 102-106
    suppliers, 103-110
    Solid element, (eee Element, solid)
    Solid model, 112, 115-117
    B-rep, 115-117
    Constructive Solid Geometry (CSG),
    1 IS—117, 233
    of bicycle frame, 231-234
    of connecting rod, 240-241
    Solution:
    accuracy, 36, 77-78. 81, 119-125, 134-130,
    142, 152
    of examples, (we Accuracy, of examples)
    checking, 93-94, 150. 159
    control, 92, 95, 149-152, 253
    of linear simultaneous equations, 85-89
    direct methods, 85-86
    iterative methods, 85
    Crank-Nicholson, 253
    conjugate gradient, 89
    Gauss-Scidel, 87-88
    Jacobi, 87
    line relaxation, 88-89
    multigrid. 89
    point relaxation, 88
    preconditioning, 89
    Stone’s, 88-89
    successive overrelaxation, 88
    singular, 82, 151, 260
    stability. 254
    Solver. 85, 94-97. 103, 107-109, 111. 122. 136,
    140, 149-153, 159, 167. 182. 195-196,
  11. 252-253, 258-259
    running, 107-108. 150-151. 167, 182
    re-running, 159
    Specification:
    design, ( tee Design, specification)
    hardware. 43. 56, 96 -101. 103. 106. 108
    geometry of examples, 162, 164-165,
    174-175, 177, 199, 211, 214-215,
    220-222, 225. 231, 233. 240-241. 247
    geometry of structure. 44
    material properties. 42-43. 141-142
    for examples, (.we Material, specification)
    of problem. 2-3, 42-45, 92. 111-112, 162,
    171-177, 196-197, 210-212, 232.
    240 -241. 246, 249
    software, 102-106
    Speed:
    of solution. 55-56, 96-103
    Split tensile test example, ( see Brazilian test
    example)
    Steel:
    frame. 9, 209-230
    material, (see Material, steel )
    members, 45, 198, 209-230
    Stiffness:
    equations, 13-15, 50-53. 61, 82-85, 98
    matrix, 11-15, 28. 50-53, 56-58. 60-68, 70,
    72, 74-75, 77-79, 81-85, 124, 149. 195,
    254- 255, 257-259
    method:
    direct, 9, 13-15, 85-86
    Stone’s method, 88-89
    Strain:
    axial, 25-29, 32-36. 39^41, 56, 67, 78
    -displacement relationship, 27-28, 51, 64
    direct, (we Strain, axial)
    energy, 12, 48-51, 57-78, 75
    engineering, 25-27
    initial, 13, 48-51, 61, 76-77, 81, 92, 260
    Lagrangian, 25
    normal, (see Strain , axial)
    plane, 10, 13, 41, 56, 62, 64
    principal values, 28
    shear, (we Shear, strain )
    true, 27
    Strength:
    yield, 32, 35 -36, 45. 160, 213, 222. 228-230
    ultimate, 35-36, 45, 123
    Stress:
    at a point, 20-21. 25
    axial, 21- 25. 30^12. 56. 79-80. 216 248
    concentration, 108, 135 136, 160, 161-174,
    223
    distribution, 10, 57, 123. 260
    direct, (s-ee Stress, axial)
    engineering, 20
    experimental, 30-36, 246. 248
    from elements. 14, 53, 61-63, 75. 77 78, 119.
    123, 152, 166, 168, 183-185. 220. 258
    from isoparametric elements. 77-78
    functions (Airy), 6. 14, 17, 41-42, 162
    gradient, 127. 161. 165-166, 168-174
    -hybrid element. 57
    infinite. 163
    initial, 48- 49. 52. 61, 76-77, 81, 92, 260
    intensity, 96I N D E X 275
    from laminate. 196-197. 204- 208
    local. 209
    mean, 6. 244
    normal, (see Stress, axial)
    peak, 43. 117, 212, 227
    peel, 205
    plane. 13, 40-41. 61 64, 80. 161 174
    plots, 78, 96. 154, 158-159. 167-173.
    184-191, 226-230, 239-240. 244-245.
    Taylor series. 24, 26-27
    Thermal. 4. 6, 43. 81-82, 102, 127. 160. 260
    conductivity. 29
    effects on structure, 4. 81. 109, 252. 260
    elements, 55. 81
    Poisson’s equation. 260
    Three dimensions:
    modification from two dimensions, 55-56,
    120, 129
    248 numerical integration, 72
    principal values, 23, 28. 36, 78. 160. 185-189.
  12. 227-229
    Time:
    dependence, 18. 30. 43, 81. 101-102. 140,
    210, 252-255
    Explicit methods. 253
    Implicit methods, 89. 253
    variation. 56, 85, 254-255
    Topology, of mesh. 125-126, 136
    Torsion. 9. 19, 39. 249
    dement, 214
    Tractions, surface, .37, 48-49. 52. 77
    Transformation:
    axis, 23. 28. 61, 78, 125, 157, 194
    consistent, 51-52, 61-62, 75-77, 146, 180
    Tresca criterion, 36
    Troubleshooting, 140, 150-151
    proof , 35
    range, 6, 183
    residual
    . 48, 81
    resultants. 194
    shear, (see Strain, shear)
  • strain relationships, 6, 8. 30-34, 48. 50, 61,
    64
    , 79. 81, 168. 172, 194, 260
    superaccurate, 78
    superconvergent. 78
    tensor, 21
    total, 21
    von Mises’, 154, 159, 160, 168. 206. 222,
    229-230. 239. 244
    yield, (see Strength, yield)
    Structured mesh, (see Mesh, mapped)
    Structures:
    load-bearing, 18-20
    Sub-:
    region model. 9-13, 69, 92
    parametric element, 65
    structuring, 108, 121, 191-192, 210, 215, 218.
    220-230
    Universities, I. 7, 14, 71, 110
    Unstructured mesh, (see Mesh, free)
    Up-direction, 155-156
    User, 96, 98, 100, 107, 137, 145, 150, 200
    current, 105
    end-, 110
    experience, 93-95, 104, 111, 127, 135
    friendliness, 95, 104-105
    or graphics, 152-159
    interface, 96, 104
    multi-
    , 97-98
    single-, 97
    programmed-, 152
    successful-, 108-111
    support, 16, 34, 71, 79, 104-106, 110
    Utility programs, 94, 96-97
    Successive overtaxation method. 88
    Supercomputers, 88. 98, 101, 107
    mini-, 98. 101, 152
    Super-parametric element, 65
    Superposition, principle of 32
    Surface:
    descriptions, 44, 73-74, 112-117, 128-129,
    177, 234-235, 247
    Bezier
    . 114-115, 129
    Coons, 114, 129
    Non-uniform rational B-spline (NURBS),
    115, 129
    tractions, 37. 48-49, 51-52, 76-77
    Suspension arm example, 246-249
    Symmetry. 108. 117-118. 161. 164-166.
    173-174, 176-177, 180, 182-183. 194,
    199, 211, 225, 241
    Vandermode matrix, (see Matrix, Vandermode)
    Variational formulation method, 9-10, 12, 15
    Variational, Calculus of, 9-10, 12
    Visual Display Unit (VDU), 95, 153
    Vector plots, 96, 154, 158-159, 167
    Vehicle body shell example, 249-251
    Vibration, 4. 43. 81, 101, 109, 127, 210, 213.
    227, 249, 254-255
    mode of, 255
    View area, 156
    Visual display system , (see Visual Display Unit)
    Visualization, 108, 117, 134, 139
    of results, 23, 152-153
    Volume clipping, 157
    Volume fraction. 141, 195
    axial, 118
    boundary condition , 1 18, 165, 182, 201-202,
    243
    cvclic, 118
    mirror, 118, 162, 174. 176, 199, 243
    repetitive, 118
    Target position. 155-156276 INDEX
    Yield, 35-36, 160, 163, 168, 229
    criteria, 36, 78, 154, 160, 168, 229
    point, 35-36
    lower, 35
    upper. 35
    strength, (JW Strength, yield)
    stress, ( see Strength, yield)
    Young’s modulus, (see Modulus, Young’s)
    von Mises’:
    criterion. 36, 78, 154, 160, 168, 229
    stress, 154, 159, 160. 168, 206, 222, 229-230,
    239, 244
    Weight, (see Load, dead)
    Weighted :
    residual method, 10, 15
    Welded joints, 175, 209-231
    Windows. Icons. Menus, Pull-Down Screens
    (WIMPS). 95
    Workstations. 97-98, 101. 107. 145. 152-153
    Zooming-in, 153, 156, 227

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