Optimization Methods in Structural Design (Solid Mechanics and Its Applications Book 242)

This book offers an introduction to numerical optimization methods in structural design. Employing a readily accessible and compact format, the book presents an overview of optimization methods, and equips readers to properly set up optimization problems and interpret the results. A {u2018}how-to-do-it{u2019} approach is followed throughout, with less emphasis at this stage on mathematical derivations. The book features spreadsheet programs provided in Microsoft Excel, which allow readers to experience optimization {u2018}hands-on.{u2019} Examples covered include truss structures, columns, beams, reinforced shell structures, stiffened panels and composite laminates. For the last three, a review of relevant analysis methods is included. Exercises, with solutions where appropriate, are also included with each chapter. The book offers a valuable resource for engineering students at the upper undergraduate and postgraduate level, as well as others in the industry and elsewhere who are new to these highly practical techniques.While the specific application is to structural design, the principles involved can be applied far more widely Preface 7 Contents 9 Principal Notation 13 1 The Conventional Design Process 17 Abstract 17 1.1 Fully Stressed Design 19 1.1.1 Structure Made of Different Materials 23 1.1.2 Structure Under Alternative Loads 25 1.2 Strength-to-weight Ratio 28 1.2.1 Feasibility 31 1.3 Comparison of Layouts 32 1.3.1 Classification of Optimization Problems 35 1.4 Spreadsheet Program 37 1.4.1 ‘Seven-Bar Truss’ 37 1.5 Summary 40 1.5 Summary 41 References 44 2 Optimality Criteria 45 Abstract 45 2.1 Circular Tube in Compression 46 2.1.1 Efficiency Formula 49 2.1.2 Material Limitation 54 2.2 Criterion for Maximum Stiffness 56 2.3 Spreadsheet Programs 60 2.3.1 ‘Circular and Square Tubes’ 60 2.3.2 ‘Truss with Tubular Members’ 64 2.4 Summary 66 Exercises 67 References 69 3 The General Optimization Problem 70 Abstract 70 3.1 Box Beam Structure 71 3.1.1 General Form of Design Space 73 3.2 The Lagrange Multiplier Method 76 3.2.1 Interpretation of Lagrange Multipliers 82 3.3 Inequality Constrained Problems 85 3.3.1 The Kuhn–Tucker Conditions 88 3.4 Spreadsheet Program 88 3.4.1 Eccentrically Loaded Column 89 3.5 Summary 93 Exercises 94 References 96 4 Numerical Methods for Unconstrained Optimization 97 Abstract 97 4.1 Unconstrained Optimization 98 4.1.1 Steepest Descent Method 99 4.1.2 Fletcher–Reeves Method 104 4.1.3 Quasi–Newton Methods 106 4.2 Line Search Methods 108 4.2.1 Region Elimination and the Golden Section Method 109 4.2.2 Polynomial Interpolation 111 4.3 Spreadsheet Program 114 4.3.1 ‘Hooke and Jeeves Method’ 114 4.4 Summary 117 Exercises 117 References 120 5 Numerical Methods for Constrained Optimization 121 Abstract 121 5.1 Constraint-Following Methods 122 5.1.1 Gradient Projection Method 123 5.1.2 Generalized Reduced Gradient Method 134 5.1.3 Other Methods for Constrained Optimization 140 5.1.4 Substitution of Variables 143 5.2 Penalty Function Methods 143 5.2.1 Interior Penalty Function 144 5.2.2 Exterior Penalty Function 147 5.2.3 Augmented Lagrangian Penalty Function 149 5.3 Spreadsheet Program 152 5.3.1 ‘Penalty Function Method’ 154 5.4 Summary 156 Exercises 157 References 159 6 Optimization of Beams 160 Abstract 160 6.1 Beam Cross Section 161 6.1.1 Thin-Walled Beams 163 6.1.2 Geometrically Similar Sections 166 6.2 Optimum Spanwise Distribution 167 6.2.1 Statically Determinate Beams 168 6.2.2 Statically Indeterminate Beams 171 6.3 Limit Design 173 6.3.1 Yield Moment 174 6.3.2 Limit Load 176 6.4 Spreadsheet Programs 181 6.4.1 ‘I-Section Beam’ 182 6.4.1.1 Modelling 182 6.4.1.2 Optimization 184 6.4.2 ‘Beam Under Lateral Load’ 186 6.4.2.1 Modelling 186 6.4.2.2 Optimization 189 6.5 Summary 191 Exercises 192 References 194 7 Reinforced Shell Structures 195 Abstract 195 7.1 Bending Stress 197 7.1.1 Effect of Yielding 199 7.1.2 Modelling of Discrete Stiffeners 201 7.2 Shear Stress 204 7.2.1 Torsional Stiffness 210 7.2.2 von Mises Criterion 212 7.3 Buckling Formulae 213 7.3.1 Buckling in Compression 213 7.3.1.1 Local Buckling 215 7.3.1.2 Flexural Buckling 218 7.3.2 Buckling in Shear 220 7.3.3 Efficiency Formula for a Compression Panel 222 7.3.4 Shear Web Efficiency 226 7.3.5 Post-buckled Shear Webs 230 7.4 Spreadsheet Programs 233 7.4.1 ‘Stiffened Panel’ 233 7.4.1.1 Modelling 235 7.4.1.2 Optimization 237 7.4.2 ‘Rectangular Box Beam’ 239 7.4.2.1 Modelling 239 7.4.2.2 Optimization 241 7.4.3 ‘Circular Fuselage Section’ 242 7.4.3.1 Modelling 243 7.4.3.2 Optimization 245 7.5 Summary 247 Exercises 248 References 251 8 Composite Laminates 252 Abstract 252 8.1 Lamination Theory 254 8.1.1 Transformed Stiffness Matrix 256 8.1.2 Laminate Stiffness Coefficients 258 8.1.3 Failure Criteria 263 8.1.4 Change in Temperature 267 8.1.5 Practical Restrictions on Lay-up 269 8.2 Laminate Optimization 270 8.2.1 Netting Analysis 271 8.2.2 Iterative Redesign 275 8.2.3 Numerical Optimization 278 8.2.4 Genetic Algorithm 281 8.3 Spreadsheet Program 286 8.3.1 ‘Composite Laminate’ 286 8.3.1.1 Modelling 286 8.3.1.2 Optimization 288 8.4 Summary 289 Exercises 291 References 293 9 Optimization With Finite Element Analysis 294 Abstract 294 9.1 Sensitivity Analysis 295 9.2 Reduction in Design Variables 301 9.3 Spreadsheet Program 302 9.3.1 ‘Design Variable Linking’ 302 9.4 Summary 305 Exercises 306 References 307 Appendix 308 A.1ƒUse of Solver in Microsoft Excel 308 A.2ƒMatrix Calculation and Custom Functions 316 Recommended Further Reading 318 Solutions to Selected Exercises 319 Index 321 This book offers an introduction to numerical optimization methods in structural design. Employing a readily accessible and compact format, the book presents an overview of optimization methods, and equips readers to properly set up optimization problems and interpret the results. A 'how-to-do-it' approach is followed throughout, with less emphasis at this stage on mathematical derivations. The book features spreadsheet programs provided in Microsoft Excel, which allow readers to experience optimization 'hands-on.' Examples covered include truss structures, columns, beams, reinforced shell structures, stiffened panels and composite laminates. For the last three, a review of relevant analysis methods is included. Exercises, with solutions where appropriate, are also included with each chapter. The book offers a valuable resource for engineering students at the upper undergraduate and postgraduate level, as well as others in the industry and elsewhere who are new to these highly practical techniques. While the specific application is to structural design, the principles involved can be applied far more widely Front Matter....Pages i-xvi The Conventional Design Process....Pages 1-28 Optimality Criteria....Pages 29-53 The General Optimization Problem....Pages 55-81 Numerical Methods for Unconstrained Optimization....Pages 83-106 Numerical Methods for Constrained Optimization....Pages 107-145 Optimization of Beams....Pages 147-181 Reinforced Shell Structures....Pages 183-239 Composite Laminates....Pages 241-282 Optimization With Finite Element Analysis....Pages 283-296 Back Matter....Pages 297-314