Structural Theorems and Their Applications

Structural Theorems and Their Applications is an account of the various structural theorems and their applications. Topics covered range from the principles of superposition to virtual work and energy concepts, calculation of deflections, and analysis of indeterminate structures using the compatibility and equilibrium methods. Reciprocal theorems and theorems of plastic analysis for plane frames are also discussed. This book is comprised of eight chapters and begins with an overview of the problems of structural analysis and the importance of the principle of virtual work in this regard, followed by an analysis of the principles of superposition. The next chapter is devoted to virtual work and energy concepts such as strain energy and complementary energy. The principle of virtual work is used in the subsequent chapters as the basis for all of the indirect methods of structural analysis described in the text, including the analysis of indeterminate structures using the compatibility method and the equilibrium method. The principle of virtual work is also used to prove the reciprocal theorems and to establish the various theorems of plastic and incremental collapse for framed structures. This monograph will be of interest to mechanical and structural engineers.

Preface

Chapter 1. The Problems of Structural Analysis

1.1. Introduction

1.2. Basic Conditions

1.3. Exclusion of Gross Deformations

1.4. Statically Determinate and Indeterminate Structures

1.5. Choice of Variables

1.6. Direct and Indirect Approaches

Chapter 2. Principles of Superposition

2.1. Introduction

2.2. Superposition of Force Systems

2.3. Superposition of Displacements

2.4. Superposition for Linear Elastic Structures

2.5. Symmetry and Skew-Symmetry

Chapter 3. Virtual Work and Energy Concepts

3.1. Introduction

3.2. Principle of Virtual Work

3.3. Virtual Work Transformations

3.4. Strain Energy

3.5. Complementary Energy

3.6. Total Potential

Chapter 4. Indeterminate Structures by the Compatibility Method

4.1. Introduction

4.2. Truss with Single Redundancy

4.3. Truss with Several Redundancies

4.4. Proof of Engesser's Theorem of Compatibility

4.5. Lack of Fit

4.6. Castigliano's Theorem of Compatibility

4.7. Further Applications of Virtual Work

Chapter 5. Calculation of Deflections

5.1. Introduction

5.2. Statically Determinate Truss

5.3. Statically Indeterminate Truss

5.4. First Theorem of Complementary Energy

5.5. Castigliano's Theorem (Part II)

5.6. Further Examples

Chapter 6. Indeterminate Structures by the Equilibrium Method

6.1. Introduction

6.2. Truss with Two Deformation Variables

6.3. Lack of Fit

6.4. Theorem of Minimum Potential Energy

6.5. Castigliano's Theorem (Part I)

6.6. Beams and Frames

Chapter 7. Reciprocal Theorems

7.1. Introduction

7.2. Proofs of Reciprocal Theorems

7.3. Influence Lines for Redundant Structures

7.4. Influence Lines by Model Analysis

Chapter 8. Theorems of Plastic Analysis for Plane Frames

8.1. Introduction

8.2. Plastic and Incremental Collapse

8.3. Plastic Collapse Theorems

8.4. Upper and Lower Bounds on λc

8.5. Shake-Down and Incremental Collapse Theorems

8.6. Trial and Error Method for Determining λ8

Appendix A. Proof of Principle of Virtual Work for Plane Frames

Bibliography

Recommended Further Reading

Index