Structural Analysis with Finite Elements (eBook)

Preface 5
Preface to the second edition 7
Contents 8
1. What are finite elements? 12
1.1 Introduction 12
1.2 Key points of the FE method 12
1.3 Potential energy 17
1.4 Projection 19
1.5 The error of an FE solution 24
1.6 A beautiful idea that does not work 26
1.7 Set theory 27
1.8 Principle of virtual displacements 34
1.9 Taut rope 40
1.10 Least squares 44
1.11 Distance inside = distance outside 48
1.12 Scalar product and weak solution 51
1.13 Equivalent nodal forces 53
1.14 Concentrated forces 55
1.15 Green’s functions 62
1.16 Practical consequences 66
1.17 Why finite element results are wrong 68
1.18 Proof 75
1.19 Influence functions 80
1.20 Accuracy 91
1.21 Why resultant stresses are more accurate 97
1.22 Why stresses at midpoints are more accurate 99
1.23 Why stresses jump 110
1.24 Why finite element support reactions are relatively accurate 110
1.25 Gauss points 115
1.26 The Dirac energy 121
1.27 How to predict changes 124
1.28 The influence of a single element 137
1.29 Retrofitting structures 141
1.30 Local errors and pollution 147
1.31 Adaptive methods 158
1.32 St. Venant’s principle 183
1.33 Singularities 186
1.34 Actio = reactio? 188
1.35 The output 192
1.36 Support conditions 194
1.37 Equilibrium 195
1.38 Temperature changes and displacement of supports 198
1.39 Stability problems 204
1.40 Interpolation 208
1.41 Polynomials 210
1.42 Infinite energy 219
1.43 Conforming and nonconforming shape functions 220
1.44 Partition of unity 222
1.45 Generalized finite element methods 224
1.46 Elements 231
1.47 Stiffness matrices 232
1.48 Coupling degrees of freedom 235
1.49 Numerical details 237
1.50 Warning 246
2. What are boundary elements? 250
2.1 Influence functions or Betti’s theorem 251
2.2 Structural analysis with boundary elements 258
2.3 Comparison finite elements—boundary elements 273
3. Frames 280
3.1 Introduction 280
3.2 The FE approach 281
3.3 Finite elements and the slope deflection method 300
3.4 Stiffness matrices 303
3.5 Approximations for stiffness matrices 309
3.6 Cables 316
3.7 Hierarchical elements 320
3.8 Sensitivity analysis 324
4. Plane problems 338
4.1 Simple example 338
4.2 Strains and stresses 345
4.3 Shape functions 348
4.4 Plane elements 349
4.5 The patch test 355
4.6 Volume forces 357
4.7 Supports 358
4.8 Nodal stresses and element stresses 368
4.9 Truss and frame models 374
4.10 Two-bay wall 376
4.11 Multistory shear wall 376
4.12 Shear wall with suspended load 381
4.13 Shear wall and horizontal load 386
4.14 Equilibrium of resultant forces 389
4.15 Adaptive mesh refinement 394
4.16 Plane problems in soil mechanics 397
4.17 Incompressible material 404
4.18 Mixed methods 404
4.19 Influence functions for mixed formulations 410
4.20 Error analysis 412
4.21 Nonlinear problems 412
5. Slabs 426
5.1 Kirchhoff plates 427
5.2 The displacement model 432
5.3 Elements 433
5.4 Hybrid elements 436
5.5 Singularities of a Kirchhoff plate 440
5.6 Reissner–Mindlin plates 442
5.7 Singularities of a Reissner–Mindlin plate 447
5.8 Reissner–Mindlin elements 450
5.9 Supports 452
5.10 Columns 454
5.11 Shear forces 462
5.12 Variable thickness 463
5.13 Beam models 470
5.14 Wheel loads 471
5.15 Circular slabs 472
5.16 T beams 473
5.17 Foundation slabs 480
5.18 Direct design method 487
5.19 Point supports 488
5.20 Study 491
5.21 Sensitivity analysis 491
6. Shells 496
6.1 Shell equations 496
6.2 Shells of revolution 499
6.3 Volume elements and degenerate shell elements 501
6.4 Circular arches 502
6.5 Flat elements 504
6.6 Membranes 509
7. Theoretical details 514
7.1 Scalar product 514
7.2 Green’s identities 519
7.3 Green’s functions 527
7.4 Generalized Green’s functions 530
7.5 Nonlinear problems 537
7.6 The derivation of influence functions 540
7.7 Weak form of influence functions 546
7.8 Influence functions for other quantities 550
7.9 Shifted Green’s functions 552
7.10 The dual space 563
7.11 Some concepts of error analysis 571
7.12 Important equations and inequalities 579
References 590
Index 604