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