# Elasticity

### Theory and Applications

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Elasticity: Theory and Applications reviews the theory and applications of elasticity. The book is divided into three parts. The first part is concerned with the kinematics of continuous media; the second part focuses on the analysis of stress; and the third part considers the theory of elasticity and its applications to engineering problems. This book consists of 18 chapters; the first of which deals with the kinematics of continuous media. The basic definitions and the operations of matrix algebra are presented in the next chapter, followed by a discussion on the linear transformation of points. The study of finite and linear strains gradually introduces the reader to the tensor concept. Orthogonal curvilinear coordinates are examined in detail, along with the similarities between stress and strain. The chapters that follow cover torsion; the three-dimensional theory of linear elasticity and the requirements for the solution of elasticity problems; the method of potentials; and topics related to cylinders, disks, and spheres. This book also explores straight and curved beams; the semi-infinite elastic medium and some of its related problems; energy principles and variational methods; columns and beam-columns; and the bending of thin flat plates. The final chapter is devoted to the theory of thin shells, with emphasis on geometry and the relations between strain and displacement. This text is intended to give advanced undergraduate and graduate students sound foundations on which to build advanced courses such as mathematical elasticity, plasticity, plates and shells, and those branches of mechanics that require the analysis of strain and stress.

## Table of Contents

Preface

Part I Kinematics of Continuous Media (Displacement, Deformation, Strain)

Chapter 1 Introduction to the Kinematics of Continuous Media

1-1 Formulation of the Problem

1-2 Notation

Chapter 2 Review of Matrix Algebra

2-1 Introduction

2-2 Definition of a Matrix. Special Matrices

2-3 Index Notation and Summation Convention

2-4 Equality of Matrices. Addition and Subtraction

2-5 Multiplication of Matrices

2-6 Matrix Division. The Inverse Matrix

Problems

Chapter 3 Linear Transformation of Points

3-1 Introduction

3-2 Definitions and Elementary Operations

3-3 Conjugate and Principal Directions and Planes in a Linear Transformation

3-4 Orthogonal Transformations

3-5 Changes of Axes in a Linear Transformation

3-6 Characteristic Equations and Eigenvalues

3-7 Invariants of the Transformation Matrix in a Linear Transformation

3-8 Invariant Directions of a Linear Transformation

3-9 Antisymmetric Linear Transformations

3-10 Symmetric Transformations. Definitions and General Theorems

3-11 Principal Directions and Principal Unit Displacements of a Symmetric Transformation

3-12 Quadratic Forms

3-13 Normal and Tangential Displacements in a Symmetric Transformation. Mohr's Representation

3-14 Spherical Dilatation and Deviation in a Linear Symmetric Transformation

3-15 Geometrical Meaning of the aij'S in a Linear Symmetric Transformation

3-16 Linear Symmetric Transformation in Two Dimensions

Problems

Chapter 4 General Analysis of Strain in Cartesian Coordinates

4-1 Introduction

4-2 Changes in Length and Directions of Elements Initially Parallel to the Coordinate Axes

4-3 Components of the State of Strain at a Point

4-4 Geometrical Meaning of the Strain Components εij. Strain of a Line Element

4-5 Components of the State of Strain under a Change of Coordinate System

4-6 Principal Axes of Strain

4-7 Volumetric Strain

4-8 Small Strain

4-9 Linear Strain

4-10 Compatibility Relations for Linear Strains

Problems

Chapter 5 Cartesian Tensors

5-1 Introduction

5-2 Scalars and Vectors

5-3 Higher Rank Tensors

5-4 On Tensors and Matrices

5-5 The Kronecker Delta and the Alternating Symbol. Isotropic Tensors

5-6 Function of a Tensor. Invariants

5-7 Contraction

5-8 The Quotient Rule of Tensors

Problems

Chapter 6 Orthogonal Curvilinear Coordinates

6-1 Introduction

6-2 Curvilinear Coordinates

6-3 Metric Coefficients

6-4 Gradient, Divergence, Curl, and Laplacian in Orthogonal Curvilinear Coordinates

6-5 Rate of Change of the Vectors āi and of the Unit Vectors ēi in an Orthogonal Curvilinear Coordinate System

6-6 The Strain Tensor in Orthogonal Curvilinear Coordinates

6-7 Strain-Displacement Relations in Orthogonal Curvilinear Coordinates

6-8 Components of the Rotation in Orthogonal Curvilinear Coordinates

6-9 Equations of Compatibility for Linear Strains in Orthogonal Curvilinear Coordinates

Problems

Part II Theory of Stress

Chapter 7 Analysis of Stress

7-1 Introduction

7-2 Stress on a Plane at a Point. Notation and Sign Convention

7-3 State of Stress at a Point. The Stress Tensor

7-4 Equations of Equilibrium. Symmetry of the Stress Tensor. Boundary Conditions

7-5 Application of the Properties of Linear Symmetric Transformations to the Analysis of Stress

7-6 Stress Quadric

7-7 Further Graphical Representations of the State of Stress at a Point. Stress Ellipsoid. Stress Director Surface

7-8 The Octahedral Normal and Octahedral Shearing Stresses

7-9 The Haigh-Westergaard Stress Space

7-10 Components of the State of Stress at a Point in a Change of Coordinates

7-11 Stress Analysis in Two Dimensions

7-12 Equations of Equilibrium in Orthogonal Curvilinear Coordinates

Problems

Part III The Theory of Elasticity Applications to Engineering Problems

Chapter 8 Elastic Stress-Strain Relations and Formulation of Elasticity Problems

8-1 Introduction

8-2 Work, Energy, and the Existence of a Strain Energy Function

8-3 The Generalized Hooke's Law

8-4 Elastic Symmetry

8-5 Elastic Stress-Strain Relations for Isotropic Media

8-6 Thermoelastic Stress-Strain Relations for Isotropic Media

8-7 Strain Energy Density

8-8 Formulation of Elasticity Problems. Boundary-Value Problems of Elasticity

8-9 Elasticity Equations in Terms of Displacements

8-10 Elasticity Equations in Terms of Stresses

8-11 The Principle of Superposition

8-12 Existence and Uniqueness of the Solution of an Elasticity Problem

8-13 Saint-Venant's Principle

8-14 One Dimensional Elasticity

8-15 Plane Elasticity

8-16 State of Plane Strain

8-17 State of Plane Stress

8-18 State of Generalized Plane Stress

8-19 State of Generalized Plane Strain

8-20 Solution of Elasticity Problems

Problems

Chapter 9 Solution of Elasticity Problems by Potentials

9-1 Introduction

9-2 Some Results of Field Theory

9-3 The Homogeneous Equations of Elasticity and the Search for Particular Solutions

9-4 Scalar and Vector Potentials. Lame's Strain Potential

9-5 The Galerkin Vector. Love's Strain Function. Kelvin's and Cerruti's Problems

9-6 The Neuber-Papkovich Representation. Boussinesq's Problem

9-7 Summary of Displacement Functions

9-8 Stress Functions

9-9 Airy's Stress Function for Plane Strain Problems

9-10 Airy's Stress Function for Plane Stress Problems

9-11 Forms of Airy's Stress Function

Problems

Chapter 10 The Torsion Problem

10-1 Introduction

10-2 Torsion of Circular Prismatic Bars

10-3 Torsion of Non-Circular Prismatic Bars

10-4 Torsion of an Elliptic Bar

10-5 Prandtl's Stress Function

10-6 Two Simple Solutions Using Prandtl's Stress Function

10-7 Torsion of Rectangular Bars

10-8 Prandtl's Membrane Analogy

10-9 Application of the Membrane Analogy to Solid Sections

10-10 Application of the Membrane Analogy to Thin Tubular Members

10-11 Application of the Membrane Analogy to Multicellular Thin Sections

10-12 Torsion of Circular Shafts of Varying Cross Section

10-13 Torsion of Thin-Walled Members of Open Section in which some Cross Section is Prevented from Warping

A-10-1 The Green-Riemann Formula

Problems

Chapter 11 Thick Cylinders, Disks, and Spheres

11-1 Introduction

11-2 Hollow Cylinder with Internal and External Pressures and Free Ends

11-3 Hollow Cylinder with Internal and External Pressures and Fixed Ends

11-4 Hollow Sphere Subjected to Internal and External Pressures

11-5 Rotating Disks of Uniform Thickness

11-6 Rotating Long Circular Cylinder

11-7 Disks of Variable Thickness

11-8 Thermal Stresses in Thin Disks

11-9 Thermal Stresses in Long Circular Cylinders

11-10 Thermal Stresses in Spheres

Problems

Chapter 12 Straight Simple Beams

12-1 Introduction

12-2 The Elementary Theory of Beams

12-3 Pure Bending of Prismatical Bars

12-4 Bending of a Narrow Rectangular Cantilever by an End Load

12-5 Bending of a Narrow Rectangular Beam by a Uniform Load

12-6 Cantilever Prismatic Bar of Irregular Cross Section Subjected to a Transverse End Force

12-7 Shear Center

Problems

Chapter 13 Curved Beams

13-1 Introduction

13-2 The Simplified Theory of Curved Beams

13-3 Pure Bending of Circular Arc Beams

13-4 Circular Arc Cantilever Beam Bent by a Force at the End

Problems

Chapter 14 The Semi-Infinite Elastic Medium and Related Problems

14-1 Introduction

14-2 Uniform Pressure Distributed over a Circular Area on the Surface of a Semi-Infinite Solid

14-3 Uniform Pressure Distributed over a Rectangular Area

14-4 Rigid Die in the Form of a Circular Cylinder

14-5 Vertical Line Load on a Semi-Infinite Elastic Medium

14-6 Vertical Line Load on a Semi-Infinite Elastic Plate

14-7 Tangential Line Load at the Surface of a Semi-Infinite Elastic Medium

14-8 Tangential Line Load on a Semi-Infinite Elastic Plate

14-9 Uniformly Distributed Vertical Pressure on Part of the Boundary of a Semi-Infinite Elastic Medium

14-10 Uniformly Distributed Vertical Pressure on Part of the Boundary of a Semi-Infinite Elastic Plate

14-11 Rigid Strip at the Surface of a Semi-Infinite Elastic Medium

14-12 Rigid Die at the Surface of a Semi-Infinite Elastic Plate

14-13 Radial Stresses in Wedges

14-14 M. Levy's Problems of the Triangular and Rectangular Retaining Walls

Chapter 15 Energy Principles and Introduction To Variational Methods

15-1 Introduction

15-2 Work, Strain and Complementary Energies. Clapeyron's Law

15-3 Principle of Virtual Work

15-4 Variational Problems and Euler's Equations

15-5 The Reciprocal Laws of Betti and Maxwell

15-6 Principle of Minimum Potential Energy

15-7 Castigliano's First Theorem

15-8 Principle of Virtual Complementary Work

15-9 Principle of Minimum Complementary Energy

15-10 Castigliano's Second Theorem

15-11 Theorem of Least Work

15-12 Summary of Energy Theorems

15-13 Working Form of the Strain Energy for Linearly Elastic Slender Members

15-14 Strain Energy of a Linearly Elastic Slender Member in Terms of the Unit Displacements of the Centroid G and of the Unit Rotations

15-15 A Working Form of the Principles of Virtual Work and of Virtual Complementary Work for a Linearly Elastic Slender Member

15-16 Examples of Application of the Theorems of Virtual Work and Virtual Complementary Work

15-17 Examples of Application of Castigliano's First and Second Theorems

15-18 Examples of Application of the Principles of Minimum Potential Energy and Minimum Complementary Energy

15-19 Example of Application of the Theorem of Least Work

15-20 The Rayleigh-Ritz Method

Problems

Chapter 16 Elastic Stability: Columns and Beam-Columns

16-1 Introduction

16-2 Differential Equations of Columns and Beam-Columns

16-3 Simple Columns

16-4 Energy Solution of the Buckling Problem

16-5 Examples of Calculation of Buckling Loads by the Energy Method

16-6 Combined Compression and Bending

16-7 Lateral Buckling of Thin Rectangular Beams

Problems

Chapter 17 Bending of Thin Flat Plates

17-1 Introduction and Basic Assumptions. Strains and Stresses

17-2 Geometry of Surfaces with Small Curvatures

17-3 Stress Resultants and Stress Couples

17-4 Equations of Equilibrium of Laterally Loaded Thin Plates

17-5 Boundary Conditions

17-6 Some Simple Solutions of Lagrange's Equation

17-7 Simply Supported Rectangular Plate. Navier's Solution

17-8 Elliptic Plate with Clamped Edges under Uniform Load

17-9 Bending of Circular Plates

17-10 Strain Energy and Potential Energy of a Thin Plate in Bending

17-11 Application of the Principle of Minimum Potential Energy to Simply Supported Rectangular Plates

Problems

Chapter 18 Introduction to the Theory of Thin Shells

18-1 Introduction

18-2 Space Curves

18-3 Elements of the Theory of Surfaces

18-4 Basic Assumptions and Reference System of Coordinates

18-5 Strain-Displacement Relations

18-6 Stress Resultants and Stress Couples

18-7 Equations of Equilibrium of Loaded Thin Shells

18-8 Boundary Conditions

18-9 Membrane Theory of Shells

18-10 Membrane Shells of Revolution

18-11 Membrane Theory of Cylindrical Shells

18-12 General Theory of Circular Cylindrical Shells

18-13 Circular Cylindrical Shell Loaded Symmetrically with Respect to Its Axis

Problems

Index

## Product details

- No. of pages: 660
- Language: English
- Copyright: © Pergamon 1974
- Published: January 1, 1974
- Imprint: Pergamon
- eBook ISBN: 9781483159539

## About the Author

### Adel S. Saada

## About the Editors

### Thomas F. Irvine

#### Affiliations and Expertise

Department of Mechanical Engineering
State University of New York at Stony Brook
Stony Brook, New York