Structural Optimization (eBook)

Preface 6
Contents 8
Contents of the CD 12
Readme 12
Computer Programs 12
Tools 14
1 Introduction 15
1.1 Problem Statement 15
1.2 An Optimization Problem 16
1.3 Elementary Calculus 19
1.4 Optimal Slope for Truss Bars 20
1.5 An Arch Problem 21
1.6 The Gradient of a Function 22
1.7 The Lagrange Multiplier Rule 25
1.8 Newton’s Method 26
1.9 Solving Linear Equations 28
1.10 Linear Systems Versus Optimization 28
1.11 Equations of Structures 29
1.12 Plastic Analysis 32
1.13 A Beam Problem 33
1.14 Quadratic Programming 34
1.15 Embedding 35
1.16 Geometric Programming 35
1.17 Plastic Design of Plane Frames 36
1.18 Problems 38
2 Some Tools of Optimization 42
2.1 The Lagrange Multiplier Rule 42
2.2 The Kuhn–Tucker Conditions 44
2.3 Calculus of Variations 47
2.4 Newton’s Method 49
2.5 Linear Programming 53
2.6 Sequential Linear Programming 57
2.7 Other Methods of Mathematical Programming 59
2.8 Genetic Algorithms 60
2.9 Problems 61
3 Sequential Linear Programming and the Incremental Equations of Structures 62
3.1 Introduction 62
3.2 The Incremental Equations of Structures 62
3.3 Application to Structural Optimization 64
3.4 An Example with a Displacement Constraint 65
3.5 Adding Stress Constraints 69
3.6 The 25-Bar Truss 70
3.7 A Frame Example 73
3.8 A Buckling Example 76
3.9 The Incremental Equations when Shape Change is Allowed 80
3.10 A Beam Example 85
3.11 A Plate Bending Problem 87
3.12 Problems 89
4 Optimality Criteria Methods 90
4.1 Introduction 90
4.2 The Most Simple Optimality Criteria Problem 91
4.3 Monotone Behavior 92
4.4 An Application 93
4.5 Sandwich Beams 97
4.6 A Generalization of the Truss Problem 97
4.7 Plastic Design of Frames 99
4.8 Sandwich Plate Design 101
4.9 Truss Design 102
4.10 A Plane Stress Problem 104
4.11 Prager and His Co-Workers 108
4.12 A Plate Problem 110
4.13 Problems 113
5 Some Basic Optimization Problems 115
5.1 Multiple Loading Conditions 115
5.2 Deflection Constraints 121
5.3 Optimal Shape 134
5.4 Generating New Designs Automatically 140
5.5 Problems 149
6 Beams and Plates: The Work of Rozvany 150
6.1 Introduction 150
6.2 Design of Plates 157
6.3 Problems 160
7 Some Problems of Dynamic Structural Optimization 161
7.1 Introduction 161
7.2 Optimization for Transient Vibrations 162
7.3 Steady-State Problems 172
8 Multicriteria Optimization 185
8.1 Introduction 185
8.2 Solving Multicriteria Optimization Problems 187
9 Practical Matters: The Work of Farkas and Jarmai 189
9.1 Introduction 189
9.2 Sizing Member Cross Sections 189
9.3 Tubular Trusses 192
9.4 Problems 196
10 On Going Work 197
10.1 Design of Tall Buildings 197
10.2 Heuristic Algorithms 202
10.3 Extending the Design Process 203
10.4 Design Theory 203
A Using the Computer 206
A.1 Using Computer Languages and Programs 206
A.2 Matlab 208
A.3 Microsoft Excel 209
A.4 Freeware 214
A.5 Graphical Interface Applications 214
B The Node Method for Trusses 215
B.1 Introduction 215
B.2 A Formal Description of the Truss Problem 216
B.3 A Decomposition 220
C Convex Sets and Functions: Homogeneous Functions 227
C.1 Convex Sets and Functions 227
C.2 Homogeneous Functions 228
D Structural Optimization Classics 231
D.1 Michell Trusses 231
D.2 Keller’s Optimal Column 241
D.3 The Paper of Venkayya, Khot, and Berke 253
References 297
Index 306