Structural Analysis with Finite Elements

Structural Analysis with Finite Elements
اسم المؤلف
Friedel Hartmann, Casimir Katz
التاريخ
20 فبراير 2019
المشاهدات
التقييم
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Structural Analysis with Finite Elements
Friedel Hartmann, Casimir Katz
Contents
1 What are finite elements? . 1
1.1 Introduction 1
1.2 Key points of the FE method . 1
1.3 Potential energy . 6
1.4 Projection 8
1.5 The error of an FE solution . 13
1.6 A beautiful idea that does not work 15
1.7 Set theory 16
1.8 Principle of virtual displacements 23
1.9 Taut rope 29
1.10 Least squares . 33
1.11 Distance inside = distance outside . 37
1.12 Scalar product and weak solution 40
1.13 Equivalent nodal forces . 42
1.14 Concentrated forces 44
1.15 Green’s functions 51
1.16 Practical consequences . 55
1.17 Why finite element results are wrong . 57
1.18 Proof 64
1.19 Influence functions . 69
1.20 Accuracy . 80
1.21 Why resultant stresses are more accurate . 86
1.22 Why stresses at midpoints are more accurate 88
1.23 Why stresses jump . 99
1.24 Why finite element support reactions are relatively accurate 99
1.25 Gauss points 104
1.26 The Dirac energy 110
1.27 How to predict changes 113
1.28 The influence of a single element 126
1.29 Retrofitting structures . 130
1.30 Local errors and pollution 136
1.31 Adaptive methods . 147
1.32 St. Venant’s principle 172
1.33 Singularities 175
1.34 Actio = reactio? . 177Contents
1.35 The output . 181
1.36 Support conditions 183
1.37 Equilibrium . 184
1.38 Temperature changes and displacement of supports 187
1.39 Stability problems . 193
1.40 Interpolation 197
1.41 Polynomials 199
1.42 Infinite energy 208
1.43 Conforming and nonconforming shape functions . 209
1.44 Partition of unity 211
1.45 Generalized finite element methods 213
1.46 Elements . 220
1.47 Stiffness matrices 221
1.48 Coupling degrees of freedom 224
1.49 Numerical details 226
1.50 Warning 235
2 What are boundary elements? . 239
2.1 Influence functions or Betti’s theorem 240
2.2 Structural analysis with boundary elements . 247
2.3 Comparison finite elements—boundary elements . 262
3 Frames 269
3.1 Introduction 269
3.2 The FE approach 270
3.3 Finite elements and the slope deflection method . 289
3.4 Stiffness matrices 292
3.5 Approximations for stiffness matrices 298
3.6 Cables . 305
3.7 Hierarchical elements 309
3.8 Sensitivity analysis 313
4 Plane problems 327
4.1 Simple example . 327
4.2 Strains and stresses 334
4.3 Shape functions . 337
4.4 Plane elements 338
4.5 The patch test 344
4.6 Volume forces . 346
4.7 Supports . 347
4.8 Nodal stresses and element stresses 357
4.9 Truss and frame models 363
4.10 Two-bay wall . 365
4.11 Multistory shear wall 365
4.12 Shear wall with suspended load . 370
XContents
4.13 Shear wall and horizontal load 375
4.14 Equilibrium of resultant forces 378
4.15 Adaptive mesh refinement 383
4.16 Plane problems in soil mechanics 386
4.17 Incompressible material 393
4.18 Mixed methods . 393
4.19 Influence functions for mixed formulations 399
4.20 Error analysis . 401
4.21 Nonlinear problems 401
5 Slabs 415
5.1 Kirchhoff plates . 416
5.2 The displacement model 421
5.3 Elements . 422
5.4 Hybrid elements . 425
5.5 Singularities of a Kirchhoff plate 429
5.6 Reissner–Mindlin plates 431
5.7 Singularities of a Reissner–Mindlin plate . 436
5.8 Reissner–Mindlin elements 439
5.9 Supports . 441
5.10 Columns . 443
5.11 Shear forces 451
5.12 Variable thickness . 452
5.13 Beam models . 459
5.14 Wheel loads 460
5.15 Circular slabs . 461
5.16 T beams . 462
5.17 Foundation slabs 469
5.18 Direct design method 476
5.19 Point supports 477
5.20 Study 480
5.21 Sensitivity analysis 480
6 Shells 485
6.1 Shell equations 485
6.2 Shells of revolution 488
6.3 Volume elements and degenerate shell elements 490
6.4 Circular arches 491
6.5 Flat elements . 493
6.6 Membranes . 498
XIContents
7 Theoretical details . 503
7.1 Scalar product 503
7.2 Green’s identities 508
7.3 Green’s functions 516
7.4 Generalized Green’s functions . 519
7.5 Nonlinear problems 526
7.6 The derivation of influence functions . 529
7.7 Weak form of influence functions 535
7.8 Influence functions for other quantities . 539
7.9 Shifted Green’s functions . 541
7.10 The dual space 552
7.11 Some concepts of error analysis 560
7.12 Important equations and inequalities . 568
References . 579
XII
Index
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