Theory and Applications

1st Edition - January 1, 1974

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  • Author: Adel S. Saada
  • eBook ISBN: 9781483159539

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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


    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


    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


    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


    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


    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


    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


    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


    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


    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


    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


    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


    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


    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


    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


    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



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

James P. Hartnett

William F. Hughes

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