Structural Optimization with Uncertainties (eBook)

Structural Optimization with Uncertainties 1
Part I Prototype Problems 12
1 Guaranteed Approaches 13
1.1 Prototype Problem 1 13
1.2 Prototype Problem 2 Illustrating Nonexistenceof a Worst Load 18
2 Probabilistic Uncertainties 23
2.1 Prototype Problem 3: Moment Constraints 23
2.2 Prototype Problem 4: Optimization with Direct Probabilistic Constraints 27
Part II Optimization in Frame of Guaranteed Approach 30
3 Uncertainties and Worst Case Scenarios 31
3.1 Transformation Schemes 31
3.2 Uncertainties in Loading Conditions 33
3.3 Some Optimal Solutions 35
3.4 Optimal Design with Uncertainties Versus Optimal Multipurpose Design 37
4 Optimal Design of Beams and Plates with Uncertainties 44
4.1 Beams Having the Smallest Weight with Constraints on Strength 44
4.2 Some Rigidity Optimization Problems for Elastic Beams and Plates 53
4.2.1 Minimization of Elastic Deflection 54
4.2.2 Rigidity Estimation for Thin-Walled Elastic Structures and Worst Case Loading 58
5 Uncertainties in Fracture Mechanics and Optimal Design Formulations 61
5.1 Basic Relations of Fracture Mechanics 62
5.2 Model Assumptions and Optimization Problems 66
6 Beams and Plates with Brittle-Fracture Constraints 71
6.1 Optimization of Beams 71
6.2 Optimum Shapes of Holes in Elastic Plates 73
6.3 Optimal Design of Bimaterial Layered Beams 74
7 Optimization of Axisymmetric Shells Against Brittle Fracture 80
7.1 Basic Relations of the Membrane Shell Model 80
7.2 Some Problems of Optimal Thickness Distribution 85
7.2.1 Thickness Distribution for Toroidal Shell 86
7.2.2 Thickness Distribution for Conical Shell 86
7.2.3 Thickness Distribution for Spherical Shell 88
7.3 Some Examples with Prescribed Middle Surfaces 90
8 Shape and Thickness Distribution of Pressure Vessels 95
8.1 Optimizing the Shape of a Shell of Revolution 95
8.2 The Search for the Shape and Thickness Distribution of an Optimal Shell 97
8.3 Some Properties of the Optimal Solution 105
9 Brittle and Quasi-Brittle Materials 107
9.1 Mass Effectiveness and Its Maximization 107
9.2 Analytical Solution and Convexity Properties 110
10 Gravity Forces and Snow Loading 118
10.1 Equilibrium of Shells Under Gravity Forces 118
10.2 Weight Minimization 119
10.3 Shape and Thickness Optimization by Genetic Algorithm 122
11 Damage Characteristics and Longevity Constraints 127
11.1 Transformations of the Longevity Constraint 127
11.2 Investigation of Optimal Design 130
11.3 Loader Crane Optimization with Longevity Constraints 132
11.3.1 First Example 133
11.3.2 Second Example 134
11.3.3 Third Example 135
12 Optimization of Shells Under Cyclic Crack Growth 137
12.1 Basic Relations and Optimization Modeling 137
12.2 Optimal Thickness Distribution for Shells of Given Geometry 143
12.3 Shape Optimization of Axisymmetric Shells 145
12.4 Simultaneous Optimization of the Meridian Shape and the Thickness Distribution of the Shell 149
12.4.1 Optimum Shells of Positive Gaussian Curvature 150
12.4.2 Optimum Shells of Negative Gaussian Curvature 154
12.4.3 Some Properties of Optimal Solution 157
13 Uncertainties in Material Characteristics 158
13.1 Discrete Sets of Materials with Uncertain Properties 158
13.1.1 Basic Representations 158
13.1.2 Minimization of Reaction Force 162
13.2 Uncertainties in Elastic Moduli 169
Part III Probabilistic and Mixed Probabilistic – GuaranteedApproaches 172
14 Some Basic Notions of Probability Theory 173
14.1 Random Variables 173
14.2 Some Continuous Distributions 176
14.3 Functions of Random Variables 179
15 Probabilistic Approaches for Incomplete Information 185
15.1 Probabilistic Problems with Fracture Mechanics Constraints 186
15.2 Beams with Random Crack Length 188
15.3 Beams with Randomly Placed Cracks 190
15.4 The Probabilistic Approach with Constrainton the Stress Intensity Factor 193
15.5 Conclusions 195
16 Optimization Under Longevity Constraint 196
16.1 Basic Assumptions and Relations 196
16.2 Probabilistic Optimization 198
16.3 Examples of Probabilistic Problems 201
16.3.1 Simply Supported Beam 201
16.3.2 Cantilever Beam 203
16.3.3 Optimal Design of a Frame 204
16.3.4 Clamped-Simply Supported Beam 204
16.4 Moment Constraints 206
17 Mixed Probabilistic-Guaranteed Optimal Design 211
17.1 Probabilistic-Guaranteed Optimization 211
17.2 Analytic Solution 213
17.3 A Toroidal Shell 215
17.4 A Shell Loaded by Forces at the Free Ends 217
References 219
Index 226