Optimal Design of Flexural Systems

Preface

Introduction

Sign Conventions

List of Symbols


1. Review of Some Fundamental Concepts


1.1 Basic Variables in Structural Theories.


1.2 Fundamental Relations in Solid Mechanics.


1.3 Special Classes of Strain-Stress Relations.


1.4 Lower and Upper Bound Theorems of Plastic Limit Analysis.


1.5 Euler Equations.


1.6 Variational Problems with Equality Constraints.


1.7 Example - Equality Constraints (Kaliszky's problem).


1.8 Variational Problems with Movable Boundaries and Variable Boundary Values.


1.9 Example - Unspecified Boundary Values.


1.10 Variational Problems with Inequality Constraints.


1.11 Convexity, Local and Global Minima.


1.12 Uniqueness and Sufficiency; Minimum of Convex Function(al)s.


1.13 Example - Inequality Constraints, Convexity and Dual Formulation (Heyman's Problem).


1.14 Mixed Variational Problems.


1.15 Example - Mixed Variational Problem.


1.16 General Concepts in Optimal Design: Cost Functional, Structural Domain, Specific Cost, Behavioral and Geometric Constraints, Prescribed Cost Distribution.


1.17 Concluding Remarks.


2. Optimal Plastic Design - Single Load System and Unspecified Cost Distribution.


2.1 Introduction - Static-Kinematic Approach.


2.2 Historical Review.


2.3 Specific Cost Function, Stress Space, Stress Regimes, Constituent Cost Functions, Cost Gradient Vector and Cost Gradient Surface.


2.4 The Prager-Shield Optimality Condition.


2.5 Example - Optimal Plastic Design of a Circular Clamped Sandwich Plate.


2.6 Further Applications of the Prager-Shield Theory.


2.7 Proof of the Prager-Shield Optimality Criterion.


2.8 Upper and Lower Bounds on the Minimum Cost.


2.9 Dual Relation between Plastic Limit Analysis and Optimal Plastic Design.


2.10 Extension of the Prager-Shield Condition to Non-Convex Specific Cost Functions.


2.11 Optimal Plastic Design of Anisotropic Cylindrical Shells having a Variable Rib Depth.


2.12 Upper and Lower Limits on the Specific Cost.


2.13 Example - Optimal Design of Fiber-Reinforced Plate with Prescribed Minimum Specific Fiber Volume.


2.14 Discontinuous Specific Cost Functions.


3. Optimal Plastic Design - Alternate and Moving Loads, Multicomponent Systems, Partially Preassigned Cost Distribution and Generalized Cost Functions.


3.1 Introduction.


3.2 Optimal Plastic Design for Alternate Loads.


3.3 Optimal Plastic Design of Multi-Component Systems.


3.4 Example - Optimal Plastic Design of Fiber-Reinforced Plate for Alternate Loads.


3.5 Optimal Plastic Design for Moving Loads.


3.6 Optimal Plastic Design for Partially Prescribed Cost Distribution.


3.7 Example. Beam of Partially Prescribed Cost Distribution.


3.8 Proof of Optimality Conditions for Partially Prescribed Cost Distribution.


3.9 Derivation of Earlier Theories from the Optimality Criterion in Section 3.6.


3.10 Generalized Specific Cost Functions.


3.11 The Hemp-Prager-Nagtegaal Superposition Principle.


3.12 Optimal Multi-Criterion Design.


4. Plastic Design - Optimization of Segmentation, Reactions, Generalized Load, Joints, Discretization and Specified Topography.


4.1 Introduction.


4.2 Optimal Segmentation.


4.3 Examples. Optimal Segmentation.


4.4 Optimization of Unspecified Reactions or Loads.


4.5 Various Subclasses of Optimization Problems Involving Beams and Unspecified Reactions or Generalized Loads.


4.6 Optimal Design Taking the Cost of Joints Into Account.


4.7 Optimization of Preassigned Topography.


4.8 Optimal Design of Discrete Systems.


5. Optimal Elastic Design.


5.1 Introduction.


5.2 Literature Survey - Optimal Elastic Design by Analytical Methods.


5.3 Optimality Criteria for Elastic Beams - Strength Design.


5.4 Optimality Criteria for Elastic Beams - Deflection Design.


5.5 Proof of Optimality Conditions.


5.6 Generalizations of Optimality Conditions in Sections 5.3 and 5.4.


5.7 Prestressed Elastic Systems.


5.8 Concluding Remarks.


6. Optimal Flexure Fields I: Simple Boundary Conditions.


6.1 Introduction and Literature Survey.


6.2 Problem Formulation and Derivation of Optimality Criteria.


6.3 Geometrical Properties of Optimal Regions.


6.4 Solutions for Axially Symmetric Supports and Loading.


6.5 Optimal Design of Circular Footing Slabs of Constant Depth.


6.6 Solutions for Rectangular Domains with Various Support Conditions.


7. Optimal Flexure Fields II: General Theories.


7.1 Topographical Properties of Optimal Flexure Fields.


7.2 Properties of Solutions for Clamped Boundaries.


7.3 Examples - Optimal Flexure Fields for Clamped Boundaries.


7.4 Mixed Boundary Conditions - Properties of αα and αβ Branches.


7.5 Examples - Branches Containing α and β Fields only.


7.6 Mixed Boundary Conditions - Properties of γ and δ Fields.


7.7 Construction of Branches Containing γ and δ Fields.


7.8 γ- and δ-fields - Examples.


7.9 Solutions for Internal Supports.


7.10 Optimal Flexure Fields for Combined Upward and Downward Loads.


7.11 Solutions for Alternate Loads.


7.12 The Effect of Torsion and Shear on Optimal Flexure Fields.


7.13 The Problem of Free Edges.


8. Optimal Plastic Design of Grillages and Fiber-Reinforced Plates of Constrained Geometry.


8.1 Introduction.


8.2 Dense Grillages Having Prismatic Beams in Preassigned Directions - Theory.


8.3 Optimal Rectangular Prismatic Grillages and Balanced Prestressed Plates.


8.4 Rectangular Fiber-Reinforced Plates with Piece-wise Constant Reinforcement - Theory.


8.5 Minimum Volume of Reinforcement for Rectangular Slabs having a Single Step in the Strip Yield Moments.


8.6 Fiber-Reinforced Plates with Variable Straight Reinforcement in Given Directions - Piece-wise Linear Formulation with Non-Zero Torsion.


8.7 Optimization of Straight Variable Reinforcement in Preassigned Directions - Torsionless Solutions.


8.8 Partially Discretized Grillages with Beams in Prescribed Directions.


9. Implications of the Theoretical Results Presented.


9.1 Optimization of Reinforced Concrete Structures - Practical Aspects.


9.2 Optimization of Long Span Roof Structures.


9.3 Controversial Issues in the Literature.


9.4 Concluding Remarks.

References

Index