Stability and Optimization of Structures (eBook)

Series Preface 7
Preface 8
Contents 11
Part I: Generalized Sensitivity of Nonlinear Elastic Systems 17
Introduction to Design Sensitivity Analysis 18
1.1 Introduction 18
1.2 General Framework of Elastic Stability 19
1.3 Design Parameterization 22
1.4 Design Sensitivity Analysis for Linear Response 23
1.5 Design Sensitivity Analyses for Nonlinear Responses 24
1.6 Historical Development 28
1.7 Summary 29
Methods of Design Sensitivity Analysis 30
2.1 Introduction 30
2.2 Sensitivity of Bifurcation Load: Pedagogic Example 31
2.3 Minor and Major Design Modifications 35
2.4 Linear Eigenvalue Analysis Approach 38
2.5 Interpolation Approach 39
2.6 Explicit Diagonalization Approach 41
2.7 Numerical Examples for Design Sensitivity 44
2.8 Summary 49
Imperfection Sensitivity Analysis 50
3.1 Introduction 50
3.2 Mathematical Preliminaries 51
3.3 Classification of Critical Points 56
3.4 Derivation of Imperfection Sensitivity Laws 60
3.5 Imperfection Sensitivity for Simple Critical Points 62
3.6 Imperfection Sensitivity for Coincident Critical Points 65
3.7 Imperfection Sensitivity of Four-Bar Truss Tent 69
3.8 Historical Development 71
3.9 Summary 72
Part II: Optimization Methods for Stability Design 73
Optimization Under Stability Constraints 74
4.1 Introduction 74
4.2 Introduction to Nonlinear Programming Problem 75
4.3 Structural Optimization Problem and Gradient- Based Optimization Algorithm 76
4.4 Optimization Under Stability Constraints 80
4.5 Optimization of a Symmetric Shallow Truss Dome 82
4.6 Bar–Spring Model 85
4.7 Historical Development 88
4.8 Summary 89
Optimal Structures Under Snapthrough Constraint 90
5.1 Introduction 90
5.2 Optimization Problems for Structures Undergoing Snapthrough 91
5.3 Two-Bar Truss 91
5.4 Symmetric Shallow Truss Dome 95
5.5 Summary 98
Shape Optimization of Compliant Mechanisms 100
6.1 Introduction 100
6.2 Illustrative Examples of Bistable Compliant Mechanisms 102
6.3 Shape Optimization Problem for Multistable Compliant Mechanism 105
6.4 Examples of Multistable Compliant Mechanisms 108
6.5 Summary 112
Optimal Braced Frames with Coincident Buckling Loads 113
7.1 Introduction 113
7.2 Optimization Problem of a Braced Frame 115
7.3 Imperfection Sensitivity of Semi-Symmetric Bifurcation Point 117
7.4 Non-Optimal and Optimal Frames 119
7.5 Summary 126
Hilltop Branching Point I: Simple Bifurcation 127
8.1 Introduction 127
8.2 Imperfection Sensitivity Laws 128
8.3 Bar–Spring Model: Hilltop with Asymmetric Bifurcation 135
8.4 Summary 138
Hilltop Branching Point II: Multiple Bifurcations 139
9.1 Introduction 139
9.2 Imperfection Sensitivity 140
9.3 Arch-Type Truss: Hilltop with Multiple Symmetric Bifurcations 141
9.4 Regular-Polygonal Truss Tents: Hilltop with Group- Theoretic Double Point 145
9.5 Summary 147
Hilltop Branching Point III: Degenerate 149
10.1 Introduction 149
10.2 Degenerate Behaviors 150
10.3 Four-Bar Truss Tent 151
10.4 Symmetric Shallow Truss Dome 155
10.5 Spherical Truss 159
10.6 Summary 163
Part III: Worst and Random Imperfections 164
Worst Imperfection: Asymptotic Theory 165
11.1 Introduction 165
11.2 Asymptotic Theory of Worst Imperfection 166
11.3 Optimization Incorporating Worst Imperfection 170
11.4 Worst Imperfection for Four-Bar Truss: Hilltop Branching Point 171
11.5 Optimum Designs of Trusses with Worst Imperfection 173
11.6 Summary 177
Worst Imperfection: Anti-optimization by LP and QP 178
12.1 Introduction 178
12.2 Numerical Procedure to Obtain Worst Imperfection Mode 179
12.3 Dominant Worst Imperfection of Braced Column Structures 182
12.4 Summary 189
Worst Imperfection for Stable Bifurcation 190
13.1 Introduction 190
13.2 Maximum Load Factor for Stable Bifurcation 191
13.3 Anti-Optimization Problem 192
13.4 Worst Imperfection of Column-Type Trusses 194
13.5 Summary 200
Random Imperfections: Theory 201
14.1 Introduction 201
14.2 Probability Density Functions of Critical Loads 202
14.3 Numerical Procedure 203
14.4 Probabilistic Variation of Strength of Truss Domes 204
14.5 Historical Development 208
14.6 Summary 209
Random Imperfections of Elasto-Plastic Solids 211
15.1 Introduction 211
15.2 Probability Density Function of Critical Loads 212
15.3 Probabilistic Strength Variation of Steel Blocks 214
15.4 Summary 219
Random Imperfections: Higher-Order Analysis 220
16.1 Introduction 220
16.2 Higher-Order Asymptotic Theory 221
16.3 Numerical Procedure 222
16.4 Four-Bar Truss Tent 223
16.5 Truss Tower Structure 227
16.6 Summary 231
Appendix 232
A.1 Introduction 232
A.2 Interpolation Approach for Coincident Critical Points 233
A.3 Derivation of Explicit Diagonalization Approach 235
A.4 Block Diagonalization Approach for Symmetric System 239
A.5 Details of Quadratic Estimation of Critical Loads 241
A.6 Differential Coefficients of Bar–Spring Model 242
A.7 Imperfection Sensitivity Law of a Semi-Symmetric Bifurcation Point 243
A.8 Imperfection Sensitivity Laws of Degenerate Hilltop Point I: Asymmetric Bifurcation 246
A.9 Imperfection Sensitivity Laws of Degenerate Hilltop Point II: Unstable- Symmetric Bifurcation 250
A.10 Summary 254
Bibliography 255
Index 272