Statics and Influence Functions - from a Modern Perspective (eBook)

Friedel Hartmann has been Professor for Civil Engineering at the University of Kassel, Germany. He is author of Greens' Functions and Finite Elements and Structural Analysis with Finite Elements, both published by Springer.

Preface 6
Contents 8
1 Basics 14
1.1 Introduction 14
1.1.1 Principle of Virtual Displacements 15
1.1.2 Betti's Theorem 16
1.1.3 Influence Functions 17
1.1.4 Identities 19
1.2 Green's Identities 20
1.2.1 Longitudinal Displacement u(x) of a Bar 20
1.2.2 Shear Deformation ws(x) of a Beam 21
1.2.3 Deflection w of a Rope 22
1.2.4 Deflection w of a Beam 22
1.2.5 Deflection w of a Beam, Second-Order Theory 23
1.2.6 Beam on Elastic Support 23
1.2.7 Tensile Chord Bridge 23
1.3 Variational Principles of Structural Analysis 24
1.4 Zero Sums 25
1.5 Examples 27
1.5.1 The Principle of Virtual Displacements 27
1.5.2 Conservation of Energy 30
1.5.3 The Principle of Virtual Forces 30
1.6 Frames 32
1.7 Single Forces and Moments 34
1.8 Support Settlements 35
1.9 Springs 39
1.10 Temperature 40
1.11 Mohr's Equation 41
1.12 Duality 42
1.13 Principle of Virtual Forces Versus Betti 43
1.14 Weak and Strong Influence Functions 45
1.15 The Canonical Boundary Values 50
1.16 The Dimension of the fi 53
1.17 Reduction of the Dimension of a Problem 53
1.18 Boundary Element Method 55
1.19 Finite Elements and Boundary Elements 58
1.20 Must Virtual Displacements Be Small? 60
1.21 Only When in Equilibrium? 61
1.22 What is a Displacement and What is a Force? 62
1.23 The Number of Force and Displacement Terms 63
1.24 Why the Minus in -EAu'' = p? 63
1.25 The Virtual Interior Energy 64
1.26 Equilibrium 65
1.27 How a Mathematician Discovers the Equilibrium Conditions 67
1.28 The Mathematics Behind the Equilibrium Conditions 68
1.29 Sinks and Sources 69
1.30 The Principle of Minimum Potential Energy 69
1.30.1 Minimum or Maximum? 71
1.30.2 Cracks 73
1.30.3 The Size of the Trial Space mathcalV 75
1.31 Infinite Energy 78
1.32 Sobolev's Embedding Theorem 82
1.33 Reduction Principle 85
1.34 Nonlinear Problems or Symmetry Lost 87
References 88
2 Betti's Theorem 89
2.1 Basics 89
2.2 Influence Functions for Displacement Terms 92
2.2.1 Derivation of an Influence Function 93
2.3 Influence Functions for Force Terms 96
2.3.1 Influence Function for N(x) 98
2.3.2 Influence Function for M(x) 100
2.3.3 Influence Functions for Higher-Order Derivatives 101
2.3.4 Moments Differentiate Influence Functions 101
2.4 Statically Determinate Structures 104
2.4.1 Pole-Plans 107
2.4.2 Construction of Pole-Plans and the Shape of the Displaced Figure 108
2.4.3 How to Determine the Magnitude of Rotations 108
2.4.4 Influence Function for a Shear Force (Fig.2.20) 111
2.4.5 Influence Function for a Normal Force (Fig.2.21) 112
2.4.6 Influence Function for a Moment (Fig.2.22) 113
2.4.7 Influence Function for a Moment (Fig.2.23) 115
2.4.8 Influence Function for a Shear Force (Fig.2.24) 115
2.4.9 Influence Function for Two Support Reactions (Fig.2.25) 115
2.4.10 Abutment Reaction (Fig.2.26) 118
2.5 Statically Indeterminate Structures 119
2.6 Influence Functions for Support Reactions 121
2.7 The Zeros of the Shear Force 124
2.8 Dirac Deltas 125
2.9 Dirac Energy 127
2.10 Point Values in 2-D and 3-D 134
2.11 Duality 135
2.12 Monopoles and Dipoles 137
2.13 Influence Functions for Integral Values 143
2.14 Influence Functions Integrate 145
2.15 Second-Order Beam Theory 147
References 149
3 Finite Elements 150
3.1 The Idea of the FE-Method 150
3.2 Why the Nodal Values Are Exact 153
3.3 Adding the Local Solution 156
3.4 Projection 159
3.5 Equivalent Nodal Forces 161
3.6 Fixed-End Forces 163
3.7 Shape Forces and the FE-Load Case 164
3.8 Slabs and the FE-Load Case 169
3.9 Computing Influence Functions with Finite Elements 171
3.10 Functionals 173
3.11 Weak and Strong Influence Functions 176
3.12 Examples 176
3.13 The Local Solution 184
3.14 The Central Equation 186
3.15 State Vectors and Measurements 188
3.16 Maxwell's Theorem 190
3.17 The Inverse Stiffness Matrix 193
3.18 Examples 193
3.19 General Form of an FE-Influence Function 197
3.20 Finite Differences and Green's Functions 198
3.21 Stresses Jump, Displacements Don't 200
3.22 The Path from the Source Point to the Load 201
3.23 The Inverse Stiffness Matrix as an Analysis Tool 204
3.24 Mohr and the Flexibility Matrix F = K-1 207
3.25 Non-uniform Plates 209
3.26 Sensitivity Plots 211
3.27 Support Reactions 212
3.28 Influence Function for a Rigid Support 215
3.29 Influence Function for an Elastic Support 219
3.30 Accuracy of Support Reactions 222
3.31 Point Loads and Point Supports in Plates 222
3.32 Point Supports are Hot Spots 226
3.33 The Amputated Dipole 227
3.34 Single Forces as Nodal Forces 232
3.35 The Limits of FE-Influence Functions 233
References 233
4 Betti Extended 234
4.1 Proof 235
4.2 At Which Points is the FE-Solution Exact? 237
4.3 Exact Values 242
4.4 Error at the Nodes 242
4.5 One-Dimensional Problems 244
4.6 Plates and Slabs 246
4.7 Point Supports of Plates and Slabs 248
4.8 If the Solution Lies in mathcalVh 250
4.9 Adaptive Refinement 253
5 Stiffness Changes and Reanalysis 256
5.1 A First Try 257
5.2 Second Example 258
5.3 Strategy 259
5.4 Variations in the Stiffness 260
5.5 Dipoles and Monopoles 261
5.6 Displacement Terms and Force Terms 263
5.7 The Decay of the Effects 264
5.8 The Relevance of These Results 265
5.9 Frames 269
5.10 Rigid Support 269
5.11 The Force Method 272
5.12 Replacement as Alternative 275
5.13 Engineering Approach 276
5.14 Local Analysis 278
5.15 Observables 281
5.16 Springs 283
5.17 How a Weak Influence Function Operates 284
5.18 Close by and Far Away 284
5.19 Summary 286
5.19.1 Loss of a Hinged Support 286
5.19.2 Loss of a Clamped Support 286
5.19.3 Change in a Spring 287
5.19.4 Change of a Torsional Spring 287
5.19.5 Change in the Longitudinal Stiffness of a Bar 287
5.19.6 Change in the Bending Stiffness of a Beam 287
5.19.7 Calculating the Deflection wc of a Spring 288
5.20 Optimal Shape of a Structural Member 290
5.21 One-Click Reanalysis 293
5.21.1 Modifications on the Diagonal 293
5.21.2 Plastic Hinges 294
References 295
6 Singularites 296
6.1 Singular Stresses 296
6.2 A Paradox? 297
6.3 Single Forces 298
6.4 The Decay of the Stresses 301
6.5 Cantilever Beam 305
6.6 Infinitely Large Stresses 305
6.7 Symmetry of Adjoint Effects 310
6.8 Cantilever Plate 310
6.9 Standard Situations 314
6.10 Singularities in Influence Functions 315
Reference 321
7 Energy Principles of Plates and Slabs and Supplements 322
7.1 Sign Conventions 324
7.2 Basic Operations 325
7.3 Gateaux Derivative 327
7.4 Potential Energy 327
7.5 Galerkin 328
7.6 Timoshenko Beam 330
7.7 Laplace Operator 331
7.8 Linear Elasticity 332
7.9 Kirchhoff Plate 334
7.10 Reissner--Mindlin Plate 336
7.11 Geometrically Nonlinear Beam 337
7.12 Geometrically Nonlinear Kirchhoff Plate 339
7.13 Nonlinear Theory of Elasticity 340
7.14 Green's First Identity and Finite Elements 342
7.14.1 Potential Energy 343
7.14.2 Galerkin 343
7.14.3 Stiffness Matrices 344
7.15 Supplements 344
7.15.1 Single Force Acting on a Plate 345
7.15.2 Multipoles 346
Reference 347
8 Postscript 348
Reference 351
9 Erratum to: Statics and Influence Functions—From a Modern Perspective 352
Erratum to:& #6
Bibliography 353
Index 355