Elasticity: Theory and Applications, now in a revised and updated second edition, has long been used as a textbook by seniors and graduate students in civil, mechanical, and biomedical engineering, since the first edition was published in 1974. The kinematics of continuous media and the analysis of stress are introduced through the concept of linear transformation of points and brought together to study in great detail the linear theory of elasticity as well as its application to a variety of practical problems. Elastic stability, the theory of thin plates, and the theory of thin shells are covered. Complex variables are introduced and used to solve two dimensional and fracture-related problems. Through theory, solved examples and problems, this authoritative book helps the student acquire the foundation needed to pursue advanced studies in all the branches of continuum mechanics. It also helps practitioners understand the source of many of the formulas they use in their designs. A solutions manual is available to instructors.

Key Features

Unifies the study of finite and linear strain, and that of stress under the notion of linear transformation of points

Uses matrices with an emphasis on geometry as the reader is introduced to the concept of tensor and its associated notations

Supports the theories and applications with a large number of solved examples throughout the book

Includes additional solved numerical examples and detailed explanations of key topics in the addendum

About the Author(s)

Adel S. Saada (Ing., E.C.P., Ph.D., Princeton University) is presently Professor of Civil Engineering at Case Western Reserve University, Cleveland, Ohio. Dr. Saada received his Ingénieur des Arts et Manufactures degree from École Centrale des Arts et Manufactures de Paris, France, and the equivalent of a Master of Science degree from the University of Grenoble, France. Before coming to Princeton University the author was a practicing structural engineer in France. Dr. Saada’s teaching activities are in two major areas: the first is that of the mechanics of solids and in particular elasticity; the second is that of mechanics applied to soils and foundations. His research activities are primarily in the area of stress-strain relations and failure of transversely isotropic geomaterials. Much of his research work has been supported by personal grants from the National Science Foundation, the U.S. Army, and the U.S. Air Force. Dr. Saada is a member of several professional societies, a consulting engineer, and the author of numerous papers published in both national and international journals.

Table of Contents

About the Author
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 e.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 a, and of the Unit Vectors Z, 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 1) Gaussian surface coordinates. First fundamental form. 2) Second fundamental form. 3) Curvature of a normal section. Meunier's theorem. 4) Principal directions and lines of curvature. 5) Principal curvatures, first and second curvatures. 6) Euler's theorem. 7) Rate of change of the vectors a, and the corresponding unit vectors along the parametric lines. 8) The Gauss-Codaizi conditions. 9) Application to surfaces of revolution. 10) Important remarks.
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

Chapter 19 Solutions of Elasticity Problems by Means of Complex Variables
19-1 Introduction
19-2 Complex Variables and Complex Functions: A Short Review
19-3 Line Integrals of Complex Functions. Cauchy's Integral Theorem
19-4 Cauchy's Integral Formula
19-5 Taylor Series
19-6 Laurent Series, Residues and Cauchy 's Residue Theorem
19-7 Singular Points of an Analytic Function
19-8 Evaluation of Residues
19-9 Conformal Representation or Conformal Mapping
19-10 Examples of Mapping by Elementary Functions
19-11 The Theorem of Harnack and the Formulas of Schwarz and Poisson
19-12 Torsion of Prismatic Bars Using Complex Variables
19-13 Torsion of Prismatic Bars with Various Shapes
19-14 The Plane Stress and Strain Problems and the Solution to the Biharmonic Equation 1) Displacements and stresses. 2) Boundary conditions. 3) The structure of the functions +(i)and ~(7.) in simply connected regions. 4) The structure of the functions +(z) and ~(z)in finite, multiply connected regions. 5) The structure of the functions +(z) and ~(z)in infinite, multiply connected regions. 6) The first and second boundary value problems in plane elasticity. 7) Displacements and stresses in curvilinear coordinates. 8) Conformal mapping for plane problems. 9) Solution by means of power series for simply connected regions. 10) Mapping of infinite regions.
19-15 Solutions Using Timoshenko's Equations. Westergaard's Stress Function
19-16 Simple Examples Using Complex Potentials
19-17 The Infinite Plate with a Circular Hole
19-18 The Infinite Plate under the Action of a Concentrated Force and Moment
19-19 The Infinite Plate with an Elliptic Hole Subjected to a Tensile Stress Normal to the Major Principal Axis of the Ellipse
19-20 Infinite Plate with an Elliptic Hole Subjected to a Uniform all around Tension S
19-21 Conformal Mapping Applied to the Problem of the Elliptic Hole
19-22 Infinite Plate with an Elliptic Hole Subjected to a Uniform Pressure P
19-23 Application to Fracture Mechanics

References
Appendices
Addendum: Comments and Detailed Explanations. Additional Solved Examples. Additional Problems.
Index