Structural Design Via Optimality Criteria : The Prager Approach to Structural Optimization

I.1 Why Optimality Criteria?.- I.2 Classes of Problems in Structural Optimization.- I.3 Case Studies Involving Simple Structures.- I.4 Case Studies Involving More Complex Structures.- I.5 Broader Implications of Optimality Criteria Methods.- 1. Static-Kinematic Optimality Criteria.- 1.1 Aims.- 1.2 An Introductory Example: What This Book Is All About.- 1.2.1 Solution via Static-Kinematic Optimality Criteria.- 1.2.2 Check by Differentiation.- 1.2.3 The Use of Optimality Criteria in Optimal Layout Problems.- 1.2.4 Check by Dual Formulation.- 1.2.5 Check through Numerical Examples.- 1.3 Plastic Design on the Basis of the Lower Bound Theorem.- 1.4 Basic Variables in Structural Mechanics.- 1.5 Fundamental Relations of Structural Mechanics.- 1.6 The Role of Static-Kinematic Optimality Criteria.- 1.7 The Prager-Shield Theory of Optimal Plastic Design.- 1.7.1 Proofs of the Prager-Shield Condition.- 1.8 The G-Gradient Operator.- 1.9 Extensions of the Prager-Shield Theory in Plastic Design.- 1.9.1 Reactions or Unspecified Forces of Non-Zero Cost.- 1.9.2 Optimal Plastic Design for External Load and Body Forces.- 1.9.3 Optimal Plastic Design for Alternate Loads.- 1.9.4 Optimal Plastic Design: Prescribed Cost Distribution.- 1.9.5 Allowance for the Cost of Connections.- 1.9.6 Optimization of Segmentation.- 1.9.7 Optimal Location of Supports.- 1.9.8 Special Features of Homogeneous Specific Cost Functions.- 1.9.9 Optimal Plastic Design with Bounded Spatial Gradients or "Taper" (Niordson-Constraints).- 1.9.10 Extended Duality Principles for Optimal Plastic Design.- 1.9.11 Generalized Specific Cost Functions in Optimal Plastic Design.- 1.9.12 Continuous and Segment-Wise Linear Cost Distribution.- 1.10 Optimal Elastic Design - Static Problems.- 1.10.1 Stress and Displacement Constraints - Continuously Varying Cross-Section.- Applications.- 1.10.2 Optimal Elastic Design with Prescribed Minimum and Maximum Values of the Cross-Sectional Parameters.- 1.10.3 Outline of Proof of Optimality Criteria for Elastic Structures.- 1.10.4 Prescribed Distribution of the Cross-Sectional Parameters over Given Segments.- 1.10.5 Allowance for Selfweight.- 1.10.6 Allowance for Cost of Reactions and Unspecified Actions.- 1.10.7 Elastic Design with Niordson-Constraints.- 1.10.8 Multicriterion Optimal Design and Pareto Optimality.- 1.11 Optimal Elastic Design - Buckling and Natural Frequency Constraints.- 1.12 Superposition Principles.- 1.13 Duality Principles in Elastic Design.- 1.14 Concluding Remarks.- 2. Optimal Plastic Design of Beams with Freely Variable Cross-Sectional Dimensions.- 2.1 General Concepts.- 2.2 Optimal Plastic Design of Beams Having a Moment-Dependent Specific Cost Function - Continuously Variable Cross-Section.- Problems and Solutions.- 2.3 Optimal Plastic Design of Beams Having a Moment and Shear Dependent Specific Cost Function - Continuously Variable Cross-Section.- Problems and Solutions.- 2.4 Dual Formulation for Plastically Designed Beams - Continuously Varying Cross-Section.- Problems and Solutions.- 2.5 Concluding Remarks.- 3. Optimal Plastic Design of Beams with Unspecified Actions or Reactions.- 3.1 Preliminary Remarks.- 3.2 External Actions (Reactions) at Prescribed Locations.- 3.3 External Actions or Reactions of Unspecified Location.- Problems and Solutions.- 3.4 Concluding Remarks.- 4. Optimal Plastic Design of Beams with Segmentation.- 4.1 Segmentation in Beam Design.- 4.2 Optimality Conditions for Segmented Beams with Prescribed Segment Boundaries.- 4.3 Optimization of Segmentation.- 4.4 Segmented Beams with Multiple Load Conditions.- Problems and Solutions.- 4.5 Concluding Remarks.- 5. Optimal Plastic Design of Beams: Allowance for Selfweight, Bounded Spatial Gradients (Niordson-Constraints) and Linear Segments.- 5.1 Introductory Remarks.- 5.2 Allowance for the Effect of Selfweight - Continuously Variable Cross-Section.- Problems and Solutions.- 5.3 Bounded Spatial Gradients of the Specific Cost (Niordson-Constraints).- 5.3.1 Beams without Selfweight.- Problems and Solutions.- 5.3.2 Allowance for Selfweight.- Problems and Solutions.- 5.3.3 Dual Formulation for Beams with Niordson-Constraints.- Problems and Solutions.- 5.4 Beams with Segmentation and Selfweight.- Problems and Solutions.- 5.5 Beams with Linear Segmentation.- Problems and Solutions.- 5.6 Concluding Remarks.- 6. Optimal Elastic Design of Beams - Stress and Deflection Constraints.- 6.1 Optimal Elastic versus Optimal Plastic Design.- 6.2 Linearly Elastic Beams with Stress and Displacement Constraints - Freely Variable Cross-Sectional Dimensions.- 6.2.1 Bernoulli-Beams with Flexural Stress Constraints and a Single Displacement Constraint.- Problems and Solutions.- 6.2.2 Timoshenko Beams with Normal and Shear Stress Constraints and a Single Displacement Constraint.- 6.3 Prescribed Distribution of the Cross-Sectional Parameters over Given Beam Segments.- Problems and Solutions.- 6.4 Concluding Remarks.- 7. Optimal Elastic Design of Beams - Optimization of Segmentation, Constraints on Spatial Gradients (Niordson-Constraints) and Multicriteria Design.- 7.1 Introductory Remarks.- 7.2 Optimization of Beam Segmentation and Location of Hinges and Supports.- Problems and Solutions.- 7.3 Optimization of Elastic Beams with Stress, Deflection and Niordson-Constraints.- Problems and Solutions.- 7.4 Multicriteria Optimization of Elastic Beams.- Problems and Solutions.- 7.5 Concluding Remarks.- 8. The Theory of Optimal Layouts and a Brief Review of Its Applications.- 8.1 Introductory Remarks.- 8.2 The Concept of Structural Universe.- 8.3 Introductory Examples.- 8.4 Classical and Advanced Layout Theories.- 8.5 Applications of the Classical Layout Theory.- (a) Micheli-Frames or Least-Weight Trusses.- (b) Least-Weight Grillages or Beam Layouts of Given Depth.- (c) Archgrids and Cable Nets of Optimal Layout (Prager Structures).- 8.6 Applications of the Advanced Layout Theory.- (a) Optimal Plastic Design of Solid Plates.- (b) Optimal Plastic Design of Perforated Plates.- (c) Optimal Elastic Design of Perforated Plates with a Compliance Constraint.- 9. A Short History of Optimality Criteria Methods.- 9.1 The Origins of Optimality Criteria in Structural Mechanics.- 9.2 Later Developments.- 9.3 Historical Notes on Optimal Layout Theory.- Closing Remarks.- Appendix. A Brief Review of Variational Methods.- A.1 Aims.- A.2 Necessary Conditions (Euler Equations) for the Minimum of a Functional - Given Boundary Conditions and No Constraints.- Problems and Solutions.- A.3 Variational Problems with Equality Constraints.- Problems and Solutions.- A.4 Transversality Conditions (Variational Problems with Variable Boundary Conditions).- Problems and Solutions.- A.5 Inequality Constraints.- Problems and Solutions.- A.6 Mixed Variational Problems.- Problems and Solutions.- A.7 Discontinuous Extremals.- Problems and Solutions.- A.8 The Rocket Problem (Variational Solution).- Selected Bibliography.- R.1 Books.- R.2 Review Papers.- R.3 Research Papers.- Name Index.