## Description

The fundamental concepts of Theory of Elasticity have been widely used at various stages in the study of solid mechanics, fluid mechanics, geomechanics and material engineering. The basic concepts are used in their simpler form for initial analysis before preliminary design of engineering structures and the advanced principles are used for complex analysis before final design. Many of the undergraduate and postgraduate courses employ the concepts of elasticity as the basis for further development of engineering principles. Therefore, it is necessary as a first step to understand clearly the basic principles and corresponding mathematical expressions involved in theory of elasticity. Since the solutions to most of the engineering problems are obtained by solving the governing equations, it is imperative that one should learn the origin of such equations and the basis on which they have been derived. For this reason, attention has been given in this book to present the derivations of various fundamental equations in an extensive manner.

## Table of Content

**Chapter 1 INTRODUCTION TO THEORY OF ELASTICITY**

1.1 Overview of Theory of Elasticity

1.2 Differences between Elementary Theory and Theory of Elasticity

1.3 Procedure to be followed in Theory of Elasticity

1.4 Conditions Applied in Theory of Elasticity

1.5 Assumptions made in Elementary Theory and in Theory of Linear Elasticity

1.6 Classifications of Materials

1.7 Applications of Elasticity

**Chapter 2 THE STRESS FIELD**

2.1 Introduction

2.2 Types of Forces

2.3 Concept of Three-dimensional Stress

2.4 General State of Stress on an Element

2.4.1 Sign Conventions

2.4.2 Index Notation

2.4.3 Two-dimensional State of Stress

2.5 Differential Equation of Equilibrium in a General Three-dimensional Stress System

2.6 Stress on a General Plane

2.6.1 Direction Cosines

2.6.2 Axis Transformation

2.6.3 Stress on Oblique Plane through a Point (Cauchy’s Formula)

2.6.4 Stress Transformation

2.7 Principal Stresses and Planes

2.8 Boundary Conditions

**Chapter 3 THE DISPLACEMENT FIELD AND STRAIN FIELD**

3.1 Introduction

3.2 Elementary Concept of Strain

3.3 Strain Displacement Relation

3.4 Strain at a Point

3.5 Strain Components at a given Point in any Direction

3.6 Principal Strains and their Directions

3.7 Strain Rosettes

3.7.1 Rectangular Strain Rosette

3.7.2 Delta Rosette

3.8 Mohr’s Circle of Strain

**Chapter 4 CONSTITUTIVE RELATIONS**

4.1 Introduction

4.2 Response Model

4.3 1-D Hooke’s Law

4.4 Generalized Hooke’s Law (Anisotropic Form)

4.5 Non-isotropic Linear Elastic Behaviour

4.6 Stress-strain Relation for Isotropic Material

4.7 Stress-strain Relation for Orthotropic Material

4.8 Stress-strain Relation for Transverse Isotropic Material

**Chapter 5 TWO-DIMENSIONAL PROBLEMS OF ELASTICITY**

5.1 Introduction

5.1.1 Two-dimensional State of Stress

5.1.2 Two-dimensional State of Strain

5.2 Plane Stress Problems

5.3 Plane Strain Problems

5.4 Equation of Compatibility

5.5 Mathematical Conditions of Compatibility

**Chapter 6 TWO-DIMENSIONAL PROBLEMS IN CARTESIAN**

COORDINATE SYSTEM

6.1 Introduction

6.2 Airy’s Stress Functions

6.3 Saint-Venant’s Principle

6.4 Two-dimensional Problems in Cartesian coordinate

6.4.1 Airy’s Stress Function Method

6.4.2 Solution by Polynomials

6.5 Solution for Bending of a Cantilever Loaded at the Free End using Stress Function as

a Polynomial

6.6 Bending of a Beam by Uniform Load using the Stress Function as a Polynomial

**Chapter 7 TWO-DIMENSIONAL PROBLEMS IN POLAR COORDINATES**

SYSTEM

7.1 Introduction

7.2 Two-dimensional Differential Equation of Equilibrium in Polar Coordinates.

7.3 Derivations of Airy’s Stress Function in Polar Coordinates

7.4 Stress-strain Relationship in Polar Coordinates

7.5 Strain−displacement Relations

7.6 Compatibility Equation

7.7 Stresses due to Concentrated Loads

7.8 Bending of a Curved Bar by a Force at the End

7.9 Semi-infinite Medium Loaded with a Concentrated Force at the Boundary

**Chapter 8 AXI-SYMMETRIC STRESS DISTRIBUTION**

8.1 Introduction

8.2 Plane Stress and Plane Strain

8.3 Compatibility Equation for Axi-symmetric Case

8.4 Rotating Circular Disc

8.5 Thick Cylinder Subjected to Internal and External Radial Pressure or Lame’s Problem 199

8.6 Pure Bending of Curved Bars

**Chapter 9 TORSIONON PRISMATIC BARS**

9.1 Introduction

9.2 Saint-Venant’s Theory

9.3 Torsion of Elliptical Cross Section

9.4 Torsion of Equilateral Triangle Cross Section Bar

**Chapter 10 THEOREMS OF ELASTICITY**

10.1 Introduction

10.2 Uniqueness Theorem

10.3 Principle of Superposition

10.4 Method of Virtual Work and Minimum Potential Energy Principle of Elasticity

10.5 Complimentary Strain Energy

10.6 The Crotti-Engesser Theorem

10.7 Castigliano’s Theorem

10.8 Maxwell Reciprocal Theorem

10.9 Clapeyron’s Theorem in Linear Elastic Theory

**Chapter 11 STRESS CONCENTRATION**

11.1 Introduction

11.2 Stresses Concentration around Circular Hole

**Chapter 12 STRESSES DUE TO ROTATION**

12.1 Introduction

12.2 Rotational Stresses in Thin Cylinder or Rotating Ring

12.3 Expression for Stresses in a Rotating Thin Disc

12.3.1 Expression for Radial and Circumferential Stresses in a Solid Disc 12.3.2 Expression for Radial and Circumferential Stresses for Disc with Central Hole

12.4 Disc of Uniform Strength

12.5 Long Cylinders

**Chapter 13 CURVED BEAMS**

13.1 Introduction

13.2 Assumptions made in the Derivation of Stresses in a Curved Bar

13.3 Expression for Stresses in a Curved Bar

13.3.1 To Find the Position of Neutral Axis

13.4 Determination of Factor K for Various Sections

13.4.1 Rectangular Section

13.4.2 Triangular Section

13.4.3 Trapezoidal Section

13.4.4 Circular Section

13.4.5 T-section

13.4.6 I-section

13.5 Resultant Stress in a Curved Bar Subjected to Direct Stresses and Bending Stresses

13.5.1 Resultant Stress in a Hook

13.5.2 Stresses in Circular Ring

13.5.3 Stresses in a Chain Link

**Chapter 14 SHEAR CENTER**

14.1 Introduction

14.2 Shear Flow

14.3 Principle Involved in Finding the Shear Centre

14.3.1 Shear Centre for a Channel Section

14.3.2 Shear Centre for Unequal I-section

14.3.3 Shear Centre for an Angle Section

14.3.4 T-section

14.3.5 I-section

14.3.6 Half Thin Walled Cylindrical Section

**Chapter 15 UNSYMMETRICAL BENDING**

15.1 Introduction

15.2 Product of Inertia for an Area

15.3 Parallel-axis Theorem

15.4 Moment of Inertia of an Area about Inclined Axes

15.5 Principal Moments of Inertia

15.6 Shear Centre (Unsymmetrical Sections)

15.7 Unsymmetrical Bending

15.8 Determination of Bending Stress through Product of Inertia

## About The Author

**P N Chandramouli** is Professor, Department of Civil Engineering, The National Institute of Engineering, Mysore. He received his B.E in Civil Engineering from University of Mysore, M.E from Indian Institute of Science, Bangalore and Ph.D from Indian Institute of Technology, Roorkee. He has over 30 years of teaching experience at The National Institute of Engineering. He is a life member of ISTE and ACCE.

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