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Engineering Mechanics Textbooks

by Roman Solecki and R. Jay Conant

Cover type: HardbackEdition: 03

Copyright: 2003

Publisher: Oxford University Press

Published: 2003

International: No

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Advanced Mechanics of Materials bridges the gap between elementary mechanics of materials courses and more rigorous graduate courses in mechanics of deformable bodies (i.e., continuum mechanics, elasticity, plasticity) taken by graduate students. Covering both traditional and modern topics, the text is ideal for senior undergraduate and beginning graduate courses in advanced strength of materials, advanced mechanics of materials, or advanced mechanics of solids. Rather than exclusively emphasizing either fundamentals or applications, it provides a balance between the two, teaching fundamentals while using real-world applications to solidify student comprehension.

Advanced Mechanics of Materials features:

- applications to contemporary practice
- use of modern computer tools, including Mathcad
- an introduction to modern topics, such as piezoelectricity, fracture mechanics, and viscoelasticity

Chapters two through five cover theoretical and conceptual development and contain relatively simple examples aimed at enhancing student understanding. The remaining chapters apply the theory to specific classes of problems such as:

- beam bending, including the effects of piezoelectricity
- plate bending
- beam and plate vibration and buckling
- introductory concepts of fracture mechanics
- finite element analysis

The authors assume that students will have an understanding of elementary (statics, dynamics, strength of materials) and intermediate (aircraft structures, machine design) mechanics.

Each chapter starts with a Summary and ends with References and Problems.

Preface

**1. Introduction**

Reference

**2. Stress and Equilibrium Equations**

2.1. Concept of Stress

2.2. Stress Components and Equilibrium Equations

2.2.1. Stress Components in Cartesian Coordinates--Matrix Representation

2.2.2. Symmetry of Shear Stresses

2.2.3. Stresses Acting on an Inclined Plane

2.2.4. Normal and Tangential Stresses--Stress Boundary Conditions

2.2.5. Transformation of Stress Components--Stress as a Tensor

2.2.7. Equilibrium Equations in Cartesian Coordinates

2.2.8. Equilibrium Equations in Polar Coordinates

2.2.9. Applicability of Equilibrium Equations

2.3. Principal Stresses and Invariants

2.3.1. Characteristic Equation

2.3.2. Principal Stresses and Principal Directions

2.3.3. Plane Stress--Principal Stresses and Principal Directions

2.3.4. Plane Stress--Mohr's Circle

2.3.5. Octahedral Stresses

2.3.6. Mean and Deviatoric Stresses

2.4. Three-dimensional Mohr's Circles

2.5. Stress Analysis and Symbolic Manipulation

**3. Displacement and Strain**

3.1. Introduction

3.2. Strain-Displacement Equations

3.3. Compatibility

3.4. Specification of the State of Strain at a Point

3.4.1. Strain Gages

3.5. Rotation

3.6. Principal Strains

3.7. Strain Invariants

3.8. Volume Changes and Dilatation

3.9. Strain Deviator

3.10. Strain-Displacement Equations in Polar Coordinates

**4. Relationships Between Stress and Strain**

4.1. Introduction

4.2. Isotropic Materials--A Physical Approach

4.2.1. Coincidence of Principal Stress and Principal Strain Axes

4.2.2. Relationship between G and E

4.2.3. Bulk Modulus

4.3. Two Dimensional Stress-Strain Laws--Plane Stress and Plane Strain

4.3.1. Plane Stress

4.3.2. Plane Strain

4.4. Restrictions on Elastic Constants for Isotropic Materials

4.5. Anisotropic Materials

4.6. Material Symmetries

4.7. Materials with a Single Plane of Elastic Symmetry

4.8. Orthotropic Materials

4.8.1. Engineering Material Constants for Orthotropic Materials

4.8.2. Orthotropic Materials under Conditions of Plane Stress

4.8.3. Stress-Strain Relations in Coordinates Other than the Principal Material Coordinates

4.9. Transversely Isotropic Materials

4.10. Isotropic Materials--A Mathematical Approach

4.11. Stress-Strain Relations for Viscoelastic Materials

4.12. Material Behavior beyond the Elastic Limit

4.12.1. Additional Experimental Observations

4.13. Criteria for Yielding

4.13.1. Maximum Shear Theory

4.13.2. Distortion Energy Theory

4.13.3. Comparison of the Two Theories

4.14. Stress-Strain Relations for Elastic-Perfectly Plastic Materials

4.15. Stress-Strain Relations when the Temperature Field is Nonuniform

4.16. Stress-Strain Relations for Piezoelectric Materials

**5. Energy Concepts**

5.1. Fundamental Concepts and Definitions

5.2. Work

5.2.1. Work Done by Stresses Acting on an Infinitesimal Element

5.3. First Law of Thermodynamics

5.4. Second Law of Thermodynamics

5.5. Some Simple Applications Involving the First Law

5.5.1. Maxwell's Reciprocity Theorem

5.6. Strain Energy

5.6.1. Complementary Strain Energy

5.6.2. Strain Energy in Beams

5.7. Castigliano's Theorem

5.8. Principle of Virtual Work

5.8.1. Principle of Virtual Work for Particles and Rigid Bodies

5.8.2. Principle of Virtual Work for Deformable Bodies

5.9. Theorem of Minimum Total Potential Energy

5.10. Applications of the Theorem of Minimum Total Potential Energy

5.11. Rayleigh-Ritz Method

5.12. Principle of Minimum Complementary Energy

5.13. Betti-Rayleigh Reciprocal Theorem

5.14. General Stress-Strain Relationships for Elastic Materials

**6. Numerical Methods I**

6.1. Method of Finite Differences

6.1.1. Application to Ordinary Differential Equations

6.1.2. Application to Partial Differential Equations

6.2. Method of Iteration

6.3. Method of Collocation

**7. Numerical Methods II: Finite Elements**

7.1. Introduction

7.2. Two-Dimensional Frames

7.3. Overall Approach

7.4. Member Force-Displacement Relationships

7.5. Assembling the Pieces

7.6. Solving the Problem

7.7. An Example

7.8. Notes Concerning the Structure Stiffness Matrix

7.10. Finite Element Analysis

7.11. Constant Strain Triangle

7.12. Element Assembly

7.13. Notes on Using Finite Element Programs

7.13.1. Interelement Compatibility

7.13.2. Inherent Overstiffness in a Finite Element

7.13.3. Bending and the Constant Strain Triangle

7.14. Closure

**8. Beams**

8.1. Bending of Continuous Beams

8.1.1. Introduction

8.1.2. Method of Initial Parameters

8.1.3. Application of Castigliano's Theorem

8.2. Unsymmetric Bending of Straight Beams

8.3. Curved Beams

8.3.1. Out-of-Plane Loaded Beams and Rings

8.3.2. A Transversely Loaded Circular Ring Supported by Three or More Supports (Biezeno's Theorem)

8.3.3. In-Plane Loaded Curved Beams (Arches) and Rings

8.3.4. Bending, Stretching, and Twisting of Springs

8.4. Beams on Elastic Foundations

8.4.1. Equilibrium Equation for a Straight Beam

8.4.2. Infinite Beams

8.4.3. Finite Beams

8.4.4. Stresses in Storage Tanks

8.5. Influence Functions (Green's Functions) for Beams

8.5.1. Straight Beams

8.5.2. Straight Beams on Elastic Foundations

8.6. Thermal Effects

8.7. Composite Beams

8.7.1. Stresses, Bending Moments, and Bending Stiffness of a Laminated Beam

8.7.2. Differential Equation for Deflection of a Laminated Beam

8.8. Limit Analysis

8.9. Fourier Series and Applications

8.10. Approximate Methods in the Analysis of Beams

8.10.1. Finite Differences--Examples

8.10.2. Rayleigh-Ritz Method--Examples

8.11. Piezoelectric Beams

8.11.1. Piezoelectric Bimorph

8.11.2. Piezoelectric Multimorph

8.11.3. Castigliano's Theorem for Piezoelectric Beams

8.11.4. Thin Curved Piezoelectric Beams

8.11.5. Castigliano's Theorem for Thin Curved Piezoelectric Beams

**9. Elementary Problems in Two- and Three-Dimensional Solid Mechanics**

9.1. Problem Formulation--Boundary Conditions

9.2. Compatibility of Elastic Stress Components

9.3. Thick-Walled Cylinders and Circular Disks

9.3.1. Equilibrium Equation and Strains

9.3.2. Elastic, Homogeneous Disks and Cylinders

9.3.3. Thermal Effects

9.3.4. Plastic Cylinder

9.3.5. Composite Disks and Cylinders

9.3.6. Rotating Disks of Variable Thickness

9.4. Airy's Stress Function

9.5. Torsion

9.5.1. Circular Cross Section

9.5.2. Noncircular Prisms--Saint-Venant's Theory

9.5.3. Membrane Analogy

9.5.4. Rectangular and Related Cross Sections

9.5.5. Torsion of Hollow, Single-Cell and Multiple-Cell Members

9.5.6. Pure Plastic Torsion

9.6. Application of Numerical Methods to Solution of Two-Dimensional Elastic Problems Elastic Problems

**10. Plates**

10.1. Introduction

10.2. Axisymmetric Bending of Circular Plates

10.2.1. General Expressions

10.2.2. Particular Solutions for Selected Types of Axisymmetric Loads

10.2.3. Solid Plate: Boundary Conditions, Examples

10.2.4. Solid Plate: Influence Functions (Green's Functions)

10.2.5. Solid Plate with Additional Support

10.2.6. Annular Plate: Boundary Conditions and Examples

10.2.7. Annular Plate: Influence Functions (Green's Functions)

10.3. Bending of Rectangular Plates

10.3.1. Boundary Conditions

10.3.2. Bending of a Simply Supported Rectangular Plate

10.4. Plates on Elastic Foundation

10.5. Strain Energy of an Elastic Plate

10.6. Membranes

10.7. Composite Plates

10.7.1. Laminated Plates with Isotropic Layers

10.7.2. Laminated Plates with Orthotropic Layers

10.8. Approximate Methods in the Analysis of Plates and Membranes

10.8.1. Application of Finite Differences

10.8.2. Examples of Application of the Rayleigh-Ritz Method

**11. Buckling and Vibration**

11.1. Buckling and Vibration of Beams and Columns

1.1. Equation of Motion and Its Solution

11.1.2. Frequencies and Critical Loads for Various Boundary Conditions

11.1.3. Applications of Rayleigh-Ritz Method

11.2. Buckling and Vibration of Rings, Arches, and Thin-Walled Tubes

11.2.1. Equations of Motion and Their Solution

11.3. Buckling and Vibration of Thin Rectangular Plates

**12. Introduction to Fracture Mechanics**

12.1. Introductory Concepts

12.2. Linear Cracks in Two-Dimensional Elastic Solids--Williams' Solution, Stress Singularity

12.3. Stress Intensity Factor

12.4. Crack Driving Force as an Energy Rate

12.5. Relation Between G and the Stress Intensity Factors

12.6. Some Simple Cases of Calculation of Stress Intensity Factors

12.7. The J-Integral

Appendix A. Matrices

Appendix B. Coordinate Transformations

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Summary

Advanced Mechanics of Materials bridges the gap between elementary mechanics of materials courses and more rigorous graduate courses in mechanics of deformable bodies (i.e., continuum mechanics, elasticity, plasticity) taken by graduate students. Covering both traditional and modern topics, the text is ideal for senior undergraduate and beginning graduate courses in advanced strength of materials, advanced mechanics of materials, or advanced mechanics of solids. Rather than exclusively emphasizing either fundamentals or applications, it provides a balance between the two, teaching fundamentals while using real-world applications to solidify student comprehension.

Advanced Mechanics of Materials features:

- applications to contemporary practice
- use of modern computer tools, including Mathcad
- an introduction to modern topics, such as piezoelectricity, fracture mechanics, and viscoelasticity

Chapters two through five cover theoretical and conceptual development and contain relatively simple examples aimed at enhancing student understanding. The remaining chapters apply the theory to specific classes of problems such as:

- beam bending, including the effects of piezoelectricity
- plate bending
- beam and plate vibration and buckling
- introductory concepts of fracture mechanics
- finite element analysis

The authors assume that students will have an understanding of elementary (statics, dynamics, strength of materials) and intermediate (aircraft structures, machine design) mechanics.

Table of Contents

Each chapter starts with a Summary and ends with References and Problems.

Preface

**1. Introduction**

Reference

**2. Stress and Equilibrium Equations**

2.1. Concept of Stress

2.2. Stress Components and Equilibrium Equations

2.2.1. Stress Components in Cartesian Coordinates--Matrix Representation

2.2.2. Symmetry of Shear Stresses

2.2.3. Stresses Acting on an Inclined Plane

2.2.4. Normal and Tangential Stresses--Stress Boundary Conditions

2.2.5. Transformation of Stress Components--Stress as a Tensor

2.2.7. Equilibrium Equations in Cartesian Coordinates

2.2.8. Equilibrium Equations in Polar Coordinates

2.2.9. Applicability of Equilibrium Equations

2.3. Principal Stresses and Invariants

2.3.1. Characteristic Equation

2.3.2. Principal Stresses and Principal Directions

2.3.3. Plane Stress--Principal Stresses and Principal Directions

2.3.4. Plane Stress--Mohr's Circle

2.3.5. Octahedral Stresses

2.3.6. Mean and Deviatoric Stresses

2.4. Three-dimensional Mohr's Circles

2.5. Stress Analysis and Symbolic Manipulation

**3. Displacement and Strain**

3.1. Introduction

3.2. Strain-Displacement Equations

3.3. Compatibility

3.4. Specification of the State of Strain at a Point

3.4.1. Strain Gages

3.5. Rotation

3.6. Principal Strains

3.7. Strain Invariants

3.8. Volume Changes and Dilatation

3.9. Strain Deviator

3.10. Strain-Displacement Equations in Polar Coordinates

**4. Relationships Between Stress and Strain**

4.1. Introduction

4.2. Isotropic Materials--A Physical Approach

4.2.1. Coincidence of Principal Stress and Principal Strain Axes

4.2.2. Relationship between G and E

4.2.3. Bulk Modulus

4.3. Two Dimensional Stress-Strain Laws--Plane Stress and Plane Strain

4.3.1. Plane Stress

4.3.2. Plane Strain

4.4. Restrictions on Elastic Constants for Isotropic Materials

4.5. Anisotropic Materials

4.6. Material Symmetries

4.7. Materials with a Single Plane of Elastic Symmetry

4.8. Orthotropic Materials

4.8.1. Engineering Material Constants for Orthotropic Materials

4.8.2. Orthotropic Materials under Conditions of Plane Stress

4.8.3. Stress-Strain Relations in Coordinates Other than the Principal Material Coordinates

4.9. Transversely Isotropic Materials

4.10. Isotropic Materials--A Mathematical Approach

4.11. Stress-Strain Relations for Viscoelastic Materials

4.12. Material Behavior beyond the Elastic Limit

4.12.1. Additional Experimental Observations

4.13. Criteria for Yielding

4.13.1. Maximum Shear Theory

4.13.2. Distortion Energy Theory

4.13.3. Comparison of the Two Theories

4.14. Stress-Strain Relations for Elastic-Perfectly Plastic Materials

4.15. Stress-Strain Relations when the Temperature Field is Nonuniform

4.16. Stress-Strain Relations for Piezoelectric Materials

**5. Energy Concepts**

5.1. Fundamental Concepts and Definitions

5.2. Work

5.2.1. Work Done by Stresses Acting on an Infinitesimal Element

5.3. First Law of Thermodynamics

5.4. Second Law of Thermodynamics

5.5. Some Simple Applications Involving the First Law

5.5.1. Maxwell's Reciprocity Theorem

5.6. Strain Energy

5.6.1. Complementary Strain Energy

5.6.2. Strain Energy in Beams

5.7. Castigliano's Theorem

5.8. Principle of Virtual Work

5.8.1. Principle of Virtual Work for Particles and Rigid Bodies

5.8.2. Principle of Virtual Work for Deformable Bodies

5.9. Theorem of Minimum Total Potential Energy

5.10. Applications of the Theorem of Minimum Total Potential Energy

5.11. Rayleigh-Ritz Method

5.12. Principle of Minimum Complementary Energy

5.13. Betti-Rayleigh Reciprocal Theorem

5.14. General Stress-Strain Relationships for Elastic Materials

**6. Numerical Methods I**

6.1. Method of Finite Differences

6.1.1. Application to Ordinary Differential Equations

6.1.2. Application to Partial Differential Equations

6.2. Method of Iteration

6.3. Method of Collocation

**7. Numerical Methods II: Finite Elements**

7.1. Introduction

7.2. Two-Dimensional Frames

7.3. Overall Approach

7.4. Member Force-Displacement Relationships

7.5. Assembling the Pieces

7.6. Solving the Problem

7.7. An Example

7.8. Notes Concerning the Structure Stiffness Matrix

7.10. Finite Element Analysis

7.11. Constant Strain Triangle

7.12. Element Assembly

7.13. Notes on Using Finite Element Programs

7.13.1. Interelement Compatibility

7.13.2. Inherent Overstiffness in a Finite Element

7.13.3. Bending and the Constant Strain Triangle

7.14. Closure

**8. Beams**

8.1. Bending of Continuous Beams

8.1.1. Introduction

8.1.2. Method of Initial Parameters

8.1.3. Application of Castigliano's Theorem

8.2. Unsymmetric Bending of Straight Beams

8.3. Curved Beams

8.3.1. Out-of-Plane Loaded Beams and Rings

8.3.2. A Transversely Loaded Circular Ring Supported by Three or More Supports (Biezeno's Theorem)

8.3.3. In-Plane Loaded Curved Beams (Arches) and Rings

8.3.4. Bending, Stretching, and Twisting of Springs

8.4. Beams on Elastic Foundations

8.4.1. Equilibrium Equation for a Straight Beam

8.4.2. Infinite Beams

8.4.3. Finite Beams

8.4.4. Stresses in Storage Tanks

8.5. Influence Functions (Green's Functions) for Beams

8.5.1. Straight Beams

8.5.2. Straight Beams on Elastic Foundations

8.6. Thermal Effects

8.7. Composite Beams

8.7.1. Stresses, Bending Moments, and Bending Stiffness of a Laminated Beam

8.7.2. Differential Equation for Deflection of a Laminated Beam

8.8. Limit Analysis

8.9. Fourier Series and Applications

8.10. Approximate Methods in the Analysis of Beams

8.10.1. Finite Differences--Examples

8.10.2. Rayleigh-Ritz Method--Examples

8.11. Piezoelectric Beams

8.11.1. Piezoelectric Bimorph

8.11.2. Piezoelectric Multimorph

8.11.3. Castigliano's Theorem for Piezoelectric Beams

8.11.4. Thin Curved Piezoelectric Beams

8.11.5. Castigliano's Theorem for Thin Curved Piezoelectric Beams

**9. Elementary Problems in Two- and Three-Dimensional Solid Mechanics**

9.1. Problem Formulation--Boundary Conditions

9.2. Compatibility of Elastic Stress Components

9.3. Thick-Walled Cylinders and Circular Disks

9.3.1. Equilibrium Equation and Strains

9.3.2. Elastic, Homogeneous Disks and Cylinders

9.3.3. Thermal Effects

9.3.4. Plastic Cylinder

9.3.5. Composite Disks and Cylinders

9.3.6. Rotating Disks of Variable Thickness

9.4. Airy's Stress Function

9.5. Torsion

9.5.1. Circular Cross Section

9.5.2. Noncircular Prisms--Saint-Venant's Theory

9.5.3. Membrane Analogy

9.5.4. Rectangular and Related Cross Sections

9.5.5. Torsion of Hollow, Single-Cell and Multiple-Cell Members

9.5.6. Pure Plastic Torsion

9.6. Application of Numerical Methods to Solution of Two-Dimensional Elastic Problems Elastic Problems

**10. Plates**

10.1. Introduction

10.2. Axisymmetric Bending of Circular Plates

10.2.1. General Expressions

10.2.2. Particular Solutions for Selected Types of Axisymmetric Loads

10.2.3. Solid Plate: Boundary Conditions, Examples

10.2.4. Solid Plate: Influence Functions (Green's Functions)

10.2.5. Solid Plate with Additional Support

10.2.6. Annular Plate: Boundary Conditions and Examples

10.2.7. Annular Plate: Influence Functions (Green's Functions)

10.3. Bending of Rectangular Plates

10.3.1. Boundary Conditions

10.3.2. Bending of a Simply Supported Rectangular Plate

10.4. Plates on Elastic Foundation

10.5. Strain Energy of an Elastic Plate

10.6. Membranes

10.7. Composite Plates

10.7.1. Laminated Plates with Isotropic Layers

10.7.2. Laminated Plates with Orthotropic Layers

10.8. Approximate Methods in the Analysis of Plates and Membranes

10.8.1. Application of Finite Differences

10.8.2. Examples of Application of the Rayleigh-Ritz Method

**11. Buckling and Vibration**

11.1. Buckling and Vibration of Beams and Columns

1.1. Equation of Motion and Its Solution

11.1.2. Frequencies and Critical Loads for Various Boundary Conditions

11.1.3. Applications of Rayleigh-Ritz Method

11.2. Buckling and Vibration of Rings, Arches, and Thin-Walled Tubes

11.2.1. Equations of Motion and Their Solution

11.3. Buckling and Vibration of Thin Rectangular Plates

**12. Introduction to Fracture Mechanics**

12.1. Introductory Concepts

12.2. Linear Cracks in Two-Dimensional Elastic Solids--Williams' Solution, Stress Singularity

12.3. Stress Intensity Factor

12.4. Crack Driving Force as an Energy Rate

12.5. Relation Between G and the Stress Intensity Factors

12.6. Some Simple Cases of Calculation of Stress Intensity Factors

12.7. The J-Integral

Appendix A. Matrices

Appendix B. Coordinate Transformations

Publisher Info

Publisher: Oxford University Press

Published: 2003

International: No

Published: 2003

International: No

Advanced Mechanics of Materials bridges the gap between elementary mechanics of materials courses and more rigorous graduate courses in mechanics of deformable bodies (i.e., continuum mechanics, elasticity, plasticity) taken by graduate students. Covering both traditional and modern topics, the text is ideal for senior undergraduate and beginning graduate courses in advanced strength of materials, advanced mechanics of materials, or advanced mechanics of solids. Rather than exclusively emphasizing either fundamentals or applications, it provides a balance between the two, teaching fundamentals while using real-world applications to solidify student comprehension.

Advanced Mechanics of Materials features:

- applications to contemporary practice
- use of modern computer tools, including Mathcad
- an introduction to modern topics, such as piezoelectricity, fracture mechanics, and viscoelasticity

Chapters two through five cover theoretical and conceptual development and contain relatively simple examples aimed at enhancing student understanding. The remaining chapters apply the theory to specific classes of problems such as:

- beam bending, including the effects of piezoelectricity
- plate bending
- beam and plate vibration and buckling
- introductory concepts of fracture mechanics
- finite element analysis

The authors assume that students will have an understanding of elementary (statics, dynamics, strength of materials) and intermediate (aircraft structures, machine design) mechanics.

Each chapter starts with a Summary and ends with References and Problems.

Preface

**1. Introduction**

Reference

**2. Stress and Equilibrium Equations**

2.1. Concept of Stress

2.2. Stress Components and Equilibrium Equations

2.2.1. Stress Components in Cartesian Coordinates--Matrix Representation

2.2.2. Symmetry of Shear Stresses

2.2.3. Stresses Acting on an Inclined Plane

2.2.4. Normal and Tangential Stresses--Stress Boundary Conditions

2.2.5. Transformation of Stress Components--Stress as a Tensor

2.2.7. Equilibrium Equations in Cartesian Coordinates

2.2.8. Equilibrium Equations in Polar Coordinates

2.2.9. Applicability of Equilibrium Equations

2.3. Principal Stresses and Invariants

2.3.1. Characteristic Equation

2.3.2. Principal Stresses and Principal Directions

2.3.3. Plane Stress--Principal Stresses and Principal Directions

2.3.4. Plane Stress--Mohr's Circle

2.3.5. Octahedral Stresses

2.3.6. Mean and Deviatoric Stresses

2.4. Three-dimensional Mohr's Circles

2.5. Stress Analysis and Symbolic Manipulation

**3. Displacement and Strain**

3.1. Introduction

3.2. Strain-Displacement Equations

3.3. Compatibility

3.4. Specification of the State of Strain at a Point

3.4.1. Strain Gages

3.5. Rotation

3.6. Principal Strains

3.7. Strain Invariants

3.8. Volume Changes and Dilatation

3.9. Strain Deviator

3.10. Strain-Displacement Equations in Polar Coordinates

**4. Relationships Between Stress and Strain**

4.1. Introduction

4.2. Isotropic Materials--A Physical Approach

4.2.1. Coincidence of Principal Stress and Principal Strain Axes

4.2.2. Relationship between G and E

4.2.3. Bulk Modulus

4.3. Two Dimensional Stress-Strain Laws--Plane Stress and Plane Strain

4.3.1. Plane Stress

4.3.2. Plane Strain

4.4. Restrictions on Elastic Constants for Isotropic Materials

4.5. Anisotropic Materials

4.6. Material Symmetries

4.7. Materials with a Single Plane of Elastic Symmetry

4.8. Orthotropic Materials

4.8.1. Engineering Material Constants for Orthotropic Materials

4.8.2. Orthotropic Materials under Conditions of Plane Stress

4.8.3. Stress-Strain Relations in Coordinates Other than the Principal Material Coordinates

4.9. Transversely Isotropic Materials

4.10. Isotropic Materials--A Mathematical Approach

4.11. Stress-Strain Relations for Viscoelastic Materials

4.12. Material Behavior beyond the Elastic Limit

4.12.1. Additional Experimental Observations

4.13. Criteria for Yielding

4.13.1. Maximum Shear Theory

4.13.2. Distortion Energy Theory

4.13.3. Comparison of the Two Theories

4.14. Stress-Strain Relations for Elastic-Perfectly Plastic Materials

4.15. Stress-Strain Relations when the Temperature Field is Nonuniform

4.16. Stress-Strain Relations for Piezoelectric Materials

**5. Energy Concepts**

5.1. Fundamental Concepts and Definitions

5.2. Work

5.2.1. Work Done by Stresses Acting on an Infinitesimal Element

5.3. First Law of Thermodynamics

5.4. Second Law of Thermodynamics

5.5. Some Simple Applications Involving the First Law

5.5.1. Maxwell's Reciprocity Theorem

5.6. Strain Energy

5.6.1. Complementary Strain Energy

5.6.2. Strain Energy in Beams

5.7. Castigliano's Theorem

5.8. Principle of Virtual Work

5.8.1. Principle of Virtual Work for Particles and Rigid Bodies

5.8.2. Principle of Virtual Work for Deformable Bodies

5.9. Theorem of Minimum Total Potential Energy

5.10. Applications of the Theorem of Minimum Total Potential Energy

5.11. Rayleigh-Ritz Method

5.12. Principle of Minimum Complementary Energy

5.13. Betti-Rayleigh Reciprocal Theorem

5.14. General Stress-Strain Relationships for Elastic Materials

**6. Numerical Methods I**

6.1. Method of Finite Differences

6.1.1. Application to Ordinary Differential Equations

6.1.2. Application to Partial Differential Equations

6.2. Method of Iteration

6.3. Method of Collocation

**7. Numerical Methods II: Finite Elements**

7.1. Introduction

7.2. Two-Dimensional Frames

7.3. Overall Approach

7.4. Member Force-Displacement Relationships

7.5. Assembling the Pieces

7.6. Solving the Problem

7.7. An Example

7.8. Notes Concerning the Structure Stiffness Matrix

7.10. Finite Element Analysis

7.11. Constant Strain Triangle

7.12. Element Assembly

7.13. Notes on Using Finite Element Programs

7.13.1. Interelement Compatibility

7.13.2. Inherent Overstiffness in a Finite Element

7.13.3. Bending and the Constant Strain Triangle

7.14. Closure

**8. Beams**

8.1. Bending of Continuous Beams

8.1.1. Introduction

8.1.2. Method of Initial Parameters

8.1.3. Application of Castigliano's Theorem

8.2. Unsymmetric Bending of Straight Beams

8.3. Curved Beams

8.3.1. Out-of-Plane Loaded Beams and Rings

8.3.2. A Transversely Loaded Circular Ring Supported by Three or More Supports (Biezeno's Theorem)

8.3.3. In-Plane Loaded Curved Beams (Arches) and Rings

8.3.4. Bending, Stretching, and Twisting of Springs

8.4. Beams on Elastic Foundations

8.4.1. Equilibrium Equation for a Straight Beam

8.4.2. Infinite Beams

8.4.3. Finite Beams

8.4.4. Stresses in Storage Tanks

8.5. Influence Functions (Green's Functions) for Beams

8.5.1. Straight Beams

8.5.2. Straight Beams on Elastic Foundations

8.6. Thermal Effects

8.7. Composite Beams

8.7.1. Stresses, Bending Moments, and Bending Stiffness of a Laminated Beam

8.7.2. Differential Equation for Deflection of a Laminated Beam

8.8. Limit Analysis

8.9. Fourier Series and Applications

8.10. Approximate Methods in the Analysis of Beams

8.10.1. Finite Differences--Examples

8.10.2. Rayleigh-Ritz Method--Examples

8.11. Piezoelectric Beams

8.11.1. Piezoelectric Bimorph

8.11.2. Piezoelectric Multimorph

8.11.3. Castigliano's Theorem for Piezoelectric Beams

8.11.4. Thin Curved Piezoelectric Beams

8.11.5. Castigliano's Theorem for Thin Curved Piezoelectric Beams

**9. Elementary Problems in Two- and Three-Dimensional Solid Mechanics**

9.1. Problem Formulation--Boundary Conditions

9.2. Compatibility of Elastic Stress Components

9.3. Thick-Walled Cylinders and Circular Disks

9.3.1. Equilibrium Equation and Strains

9.3.2. Elastic, Homogeneous Disks and Cylinders

9.3.3. Thermal Effects

9.3.4. Plastic Cylinder

9.3.5. Composite Disks and Cylinders

9.3.6. Rotating Disks of Variable Thickness

9.4. Airy's Stress Function

9.5. Torsion

9.5.1. Circular Cross Section

9.5.2. Noncircular Prisms--Saint-Venant's Theory

9.5.3. Membrane Analogy

9.5.4. Rectangular and Related Cross Sections

9.5.5. Torsion of Hollow, Single-Cell and Multiple-Cell Members

9.5.6. Pure Plastic Torsion

9.6. Application of Numerical Methods to Solution of Two-Dimensional Elastic Problems Elastic Problems

**10. Plates**

10.1. Introduction

10.2. Axisymmetric Bending of Circular Plates

10.2.1. General Expressions

10.2.2. Particular Solutions for Selected Types of Axisymmetric Loads

10.2.3. Solid Plate: Boundary Conditions, Examples

10.2.4. Solid Plate: Influence Functions (Green's Functions)

10.2.5. Solid Plate with Additional Support

10.2.6. Annular Plate: Boundary Conditions and Examples

10.2.7. Annular Plate: Influence Functions (Green's Functions)

10.3. Bending of Rectangular Plates

10.3.1. Boundary Conditions

10.3.2. Bending of a Simply Supported Rectangular Plate

10.4. Plates on Elastic Foundation

10.5. Strain Energy of an Elastic Plate

10.6. Membranes

10.7. Composite Plates

10.7.1. Laminated Plates with Isotropic Layers

10.7.2. Laminated Plates with Orthotropic Layers

10.8. Approximate Methods in the Analysis of Plates and Membranes

10.8.1. Application of Finite Differences

10.8.2. Examples of Application of the Rayleigh-Ritz Method

**11. Buckling and Vibration**

11.1. Buckling and Vibration of Beams and Columns

1.1. Equation of Motion and Its Solution

11.1.2. Frequencies and Critical Loads for Various Boundary Conditions

11.1.3. Applications of Rayleigh-Ritz Method

11.2. Buckling and Vibration of Rings, Arches, and Thin-Walled Tubes

11.2.1. Equations of Motion and Their Solution

11.3. Buckling and Vibration of Thin Rectangular Plates

**12. Introduction to Fracture Mechanics**

12.1. Introductory Concepts

12.2. Linear Cracks in Two-Dimensional Elastic Solids--Williams' Solution, Stress Singularity

12.3. Stress Intensity Factor

12.4. Crack Driving Force as an Energy Rate

12.5. Relation Between G and the Stress Intensity Factors

12.6. Some Simple Cases of Calculation of Stress Intensity Factors

12.7. The J-Integral

Appendix A. Matrices

Appendix B. Coordinate Transformations