Finite Element Analysis in Geotechnical Engineering Vol.1 - Theory

Finite Element Analysis in Geotechnical Engineering Vol.1 - Theory, 1ed, 1999, Scan, OCR......Page 1
2. Finite element theory for linear materials......Page 5
3. Geotechnical considerations......Page 6
5. Elastic constitutive models......Page 7
7. Simple elasto-plastic constitutive models......Page 8
8. Advanced constitutive models......Page 9
9. Finite element theory for nonlinear materials......Page 10
10. Seepage and consolidation......Page 11
12 Fourier series aided finite element method (FSAFEM)......Page 12
1.1 Synopsis......Page 19
1.3 Design objectives......Page 20
1.4 Design requirements......Page 21
1.5.2 Equilibrium......Page 22
1 .5.3 Compatibility......Page 23
1.5.4 Equilibrium and compatibility conditions......Page 24
1.5.5 Constitutive behaviour......Page 25
1.6.1 Plane strain......Page 26
1.6.2 Axi-symmetry......Page 27
1.7 Methods of analysis......Page 28
1.8 Closed form solutions......Page 29
1.9.1 Limit equilibrium......Page 30
1.9.2 Stress field solution......Page 32
1 .9.3 Limit analysis......Page 33
1.9.4 Comments......Page 36
1.10.1 Beam-spring approach......Page 37
1.10.2 Full numerical analysis......Page 38
1.11. Summary......Page 39
2.3 Overview......Page 41
2.4 Element discretisation......Page 42
2.5 Displacement approximation......Page 45
2.5.1 lsoparametric finite elements......Page 47
2.6 Element equations......Page 49
2.6.1 Numerical integration......Page 52
2.7.1 The direct stiffness assembly method......Page 54
2.9 Solution of global equations......Page 57
2.9.1 Storage of the global stiffness matrix......Page 58
2.9.2 Triangular decomposition of the global stiffness matrix......Page 59
2.9.3 Solution of the finite element equations......Page 61
2.9.4 Modification due to displacement boundary conditions......Page 63
2.11 Example......Page 65
2.12 Axi-symmetric finite element analysis......Page 67
2.13 Summary......Page 68
II. 1 .l Derivation of area coordinates......Page 69
11.1.2 lsoparametric formulation......Page 71
3.2 Introduction......Page 73
3.3 Total stress analysis......Page 74
3.4 Pore pressure calculation......Page 76
3.5.1 Introduction......Page 79
3.5.2 Strain definitions......Page 80
3.5.3 Constitutive equation......Page 81
3.5.4 Finite element formulation......Page 82
3.5.5 Membrane elements......Page 85
3.6.1 Introduction......Page 86
3.6.2 Basic theory......Page 87
3.6.3 Finite element formulation......Page 88
3.7.1 Introduction......Page 90
3.7.2 Local axes......Page 91
3.7.3 Prescribed displacements......Page 92
3.7.4 Tied degrees of freedom......Page 94
3.7.5 Springs......Page 96
3.7.6 Boundary stresses......Page 98
3.7.7 Point loads......Page 100
3.7.8 Body forces......Page 101
3.7.9 Construction......Page 102
3.7.10 Excavation......Page 104
3.7.1 1 Pore pressures......Page 105
3.8 Summary......Page 107
4.2 Introduction......Page 108
4.3.1 Behaviour under one dimensional compression......Page 109
4.3.2 Behaviour when sheared......Page 110
4.3.3 Effect of stress path direction......Page 112
4.3.4 Effect of the magnitude of the intermediate principal stress......Page 113
4.3.6 Behaviour at large strains......Page 115
4.4.1 Behaviour under one dimensional compression......Page 117
4.4.2 Behaviour when sheared......Page 118
4.4.3 Effect of the magnitude of the intermediate principal......Page 121
4.4.4 Anisotropy......Page 122
4.5.1 Comparison of sedimentary soils......Page 123
4.5.2 Residual soils......Page 128
4.5.3 Residual strength......Page 129
4.7 Summary......Page 130
5.3 invariants......Page 132
5.5 Linear isotropic elasticity......Page 136
5.6 Linear anisotropic elasticity......Page 138
5.7.1 Introduction......Page 140
5.7.3 K G model......Page 141
5.7.4 Hyperbolic model......Page 142
5.7.5 Small strain stiffness model......Page 143
5.7.6 Puzrin and Burland model......Page 145
5.8 Summary......Page 149
6.2 introduction......Page 150
6.3 Uniaxial behaviour of a linear elastic perfectly plastic material......Page 151
6.5 Uniaxial behaviour of a linear elastic strain softening plastic material......Page 152
6.7 Extension to general stress and strain space......Page 153
6.8.2 A yield function......Page 154
6.8.3 A plastic potential function......Page 155
6.8.4 The hardeninglsoftening rules......Page 156
6.9 Two dimensional behaviour of a linear elastic perfectly plastic material......Page 157
6.10 Two dimensional behaviour of a linear elastic hardening plastic material......Page 158
6.11 Two dimensional behaviour of a linear elastic softening plastic material......Page 159
6.12 Comparison with real soil behaviour......Page 160
6.13 Formulation of the elasto-plastic constitutive matrix......Page 161
6.14 Summary......Page 164
7.2 Introduction......Page 165
7.3 Tresca model......Page 166
7.4 Von Nlises model......Page 168
7.5 Mohr-Coulomb model......Page 169
7.6 Drucker-Prager model......Page 173
7.7 Comments on simple elastic perfectly plastic models......Page 175
7.8 An elastic strain hardeninglsoftening Mohr-Coulomb model......Page 176
7.9 Development of the critical state models......Page 178
7.9.1 Basic formulation in triaxial stress space......Page 179
7.9.2 Extension to general stress space......Page 184
7.9.3 Undrained strength......Page 186
7.10.1 Yield surface on the supercritical side......Page 187
7.10.2 Yield surface for K, consolidated soils......Page 189
7.10.3 Elastic component of the model......Page 190
7.1 0.4 Plastic behaviour inside the main yield surface......Page 191
7.1 1.1 Introduction......Page 193
7. 'l 1.2 Development of a new expression in triaxial stress space......Page 194
7.12 The effect of the plastic potential in plane strain deformation......Page 199
7.13 Summary......Page 203
Appendix VII. 1 : Derivatives of stress invariants......Page 204
Appendix V11.2: Analytical solutions for triaxial test onmodified Cam clay......Page 205
Appendix V11.3: Derivatives for modified Cam clay model......Page 213
Appendix V11.4: Undrained strength for critical statemodels......Page 215
8.2 Introduction......Page 218
8.3.1 Introduction......Page 219
8.3.2. 1 Yield surface......Page 220
8.3.2.2 Plastic potential......Page 221
8.3.2.3 Finite element implementa tion......Page 222
8.4 Formulation of the elasto-plastic constitutive matrix when two yield surfaces are simultaneously active......Page 223
8.5.2 Overview of model......Page 226
8.5.4 Failure criterion......Page 227
8.5.7 Conical hardening law......Page 228
8.5.1 1 Comments......Page 229
8.6.1 Introduction......Page 230
8.6.2 Bounding surface plasticity......Page 231
8.7.2 Transformed variables......Page 233
8.7.3 Hysteretic elasticity......Page 234
8.7.4 Behaviour on the bounding surface......Page 236
8.7.5 Behaviour within the bounding surface......Page 241
8.7.6 Comments......Page 244
8.8.2 Behaviour of a kinematic yield surface......Page 245
8.9.1 Bounding surface and bubble......Page 247
8.9.2 Movement of bubble......Page 248
8.9.3 Elasto-plastic behaviour......Page 249
8.10 Summary......Page 250
Appendix VIII. 1 Derivatives for Lade's double hardening model......Page 251
9.2 Introduction......Page 255
9.4.1 introduction......Page 256
9.4.2 Finite element implementation......Page 257
9.4.3 Uniform compression of a Mohr-Coulomb soil......Page 258
9.4.4 Uniform compression of modified Cam clay soil......Page 263
9.5.1 Introduction......Page 264
9.5.2 Finite element application......Page 265
9.5.3 Choice of time step......Page 268
9.5.5 Uniform compression of a Mohr-Coulomb soil......Page 269
9.5.6 Uniform compression of modified Cam clay soil......Page 270
9.6.1 Introduction......Page 274
9.6.2. 1 Introduction......Page 275
9.6.2.3 Re turn algorithm......Page 276
9.6.2.4 Fundamental comparison......Page 277
9.6.4 Uniform compression of Mohr-Coulomb and......Page 278
9.7.1 Introduction......Page 279
9.7.2 Idealised triaxial test......Page 281
9.7.3 Footing problem......Page 285
9.7.4 Excavation problem......Page 288
9.7.5 Pile problem......Page 291
9.7.6 Comments......Page 293
9.8 Summary......Page 294
IX. 1 .l Introduction......Page 295
IX. 1.2 Overview......Page 296
IX.l .3 Modified Euler integration scheme with error control......Page 298
IX.l .5 Correcting for yield surface drift in elasto-plastic finiteelement analysis......Page 301
Appendix IX.2: Return stress point algorithm......Page 304
Appendix IX.3: Comparison of substepping and returnalgorithms......Page 314
10.2 Introduction......Page 323
10.3 Finite element formulation for coupled problems......Page 324
10.4 Finite element implementation......Page 329
10.5 Steady state seepage......Page 330
10.6.2 Prescribed pore fluid pressures......Page 331
10.6.3 Tied degrees of freedom......Page 332
10.6.4 Infiltration......Page 333
10.6.6 Precipitation......Page 334
10.7.2 Linear isotropic permeability......Page 336
10.7.4 Nonlinear permeability related to void ratio......Page 337
10.8 Unconfined seepage flow......Page 338
10.9 Vaiidation example......Page 339
10.10 Summary......Page 341
11.2 Introduction......Page 343
11.3 Conventional 3D finite element analysis......Page 344
11.4.2 General iterative solution......Page 350
1 1.4.3 The gradient method......Page 352
11.4.4 The conjugate gradient method......Page 354
1 1.4.5 Comparison of the conjugate gradient and banded......Page 355
11.4.6 Normalisation of the stiffness matrix......Page 359
11.5 Summary......Page 360
12.2 Introduction......Page 362
1 2.3.1 Formulation for linear behaviour......Page 363
12.3.2 Symmetrical loading conditions......Page 370
12.3.3 Existing formulations for nonlinear behaviour......Page 372
12.3.4 New formulation for nonlinear behaviour......Page 373
12.3.5 Formulation for interface elements......Page 377
12.3.6 Bulk pore fluid compressibility......Page 379
12.3.7 Formulation for coupled consolidation......Page 382
1 2.4.1 Introduction......Page 388
1 2.4.2 Evaluating Fourier series harmonic coefficients......Page 389
12.4.2.1 The step wise linear method......Page 390
12.4.2.2 The fitted method......Page 391
12.4.3. 1 Introduction......Page 392
12.4.3.2 Right hand side correction......Page 393
12.4.4 Data storage......Page 394
12.4.6 Stiffness matrices......Page 395
12.4.7. 1 Introduction......Page 396
12.4.7.2 Examples of problems associated with parallel and orthogonal analysis......Page 398
1 2.5.1 Introduction......Page 403
12.5.2 Description of the discrete FSAFEM method......Page 404
12.6 Comparison between the discrete and the......Page 409
12.7 Comparison of CFSAFEM and full 3D analysis......Page 414
12.8 Summary......Page 416
Appendix XII.1: Harmonic coefficients of force fromharmonic point loads......Page 417
Appendix Xl1.2: Obtaining the harmonics of force fromharmonic boundary stresses......Page 418
Appendix Xi1.3: Obtaining the harmonics of force fromelement stresses......Page 419
Appendix X11.4: Resolving harmonic coefficients of nodalforce......Page 421
Appendix X11.5: Fourier series solutions for integratingthe product of three Fourier series......Page 422
Appendix X11.6: Obtaining coefficients for a stepwiselinear distribution......Page 423
Appendix X11.7: Obtaining harmonic coefficients usingthe fitted method......Page 425
References......Page 429
List of symbols......Page 443
Index......Page 453