# Advanced Mechanics of Materials and Applied Elasticity

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## Book Description

This book presents both differential equation and integral formulations of boundary value problems for computing the stress and displacement fields of solid bodies at two levels of approximation - isotropic linear theory of elasticity as well as theories of mechanics of materials. Moreover, the book applies these formulations to practical solutions in detailed, easy-to-follow examples.

Advanced Mechanics of Materials and Applied Elasticity presents modern and classical methods of analysis in current notation and in the context of current practices. The author's well-balanced choice of topics, clear and direct presentation, and emphasis on the integration of sophisticated mathematics with practical examples offer students in civil, mechanical, and aerospace engineering an unparalleled guide and reference for courses in advanced mechanics of materials, stress analysis, elasticity, and energy methods in structural analysis.

## Table of Contents

CARTESIAN TENSORS

Vectors

Dyads

Definition and Rules of Operation of Tensors of the Second Rank

Transformation of the Cartesian Components of a Tensor of the Second Rank upon Rotation of the System of Axes to Which They Are Referred

Definition of a Tensor of the Second Rank on the Basis of the Law of Transformation of Its Components

Symmetric Tensors of the Second Rank

Invariants of the Cartesian Components of a Symmetric Tensor of the Second Rank

Stationary Values of a Function Subject to a Constraining Relation

Stationary Values of the Diagonal Components of a Symmetric Tensor of the Second Rank

Quasi Plane Form of Symmetric Tensors of the Second Rank

Stationary Values of the Diagonal and the Non-Diagonal Components of the Quasi Plane, Symmetric Tensors of the Second Rank

Mohr's Circle for Quasi Plane, Symmetric Tensors of the Second Rank

Maximum Values of the Non-Diagonal Components of a Symmetric Tensor of the Second Rank

Problems

STRAIN AND STRESS TENSORS

The Continuum Model

External Loads

The Displacement Vector of a Particle of a Body

Components of Strain of a Particle of a Body

Implications of the Assumption of Small Deformation

Proof of the Tensorial Property of the Components of Strain

Traction and Components of Stress Acting on a Plane of a Particle of a Body

Proof of the Tensorial Property of the Components of Stress

Properties of the Strain and Stress Tensors

Components of Displacement for a General Rigid Body Motion of a Particle

The Compatibility Equations

Measurement of Strain

The Requirements for Equilibrium of the Particles of a Body

Cylindrical Coordinates

Strain-Displacement Relations in Cylindrical Coordinates

The Equations of Compatibility in Cylindrical Coordinates

The Equations of Equilibrium in Cylindrical Coordinates

Problems

STRESS-STRAIN RELATIONS

Introduction

The Uniaxial Tension or Compression Test Performed in an Environment of Constant Temperature

Strain Energy Density and Complementary Energy Density for Elastic Materials Subjected to Uniaxial Tension or Compression in an Environment of Constant Temperature

The Torsion Test

Effect of Pressure, Rate of Loading and Temperature on the Response of Materials Subjected to Uniaxial States of Stress

Models of Idealized Time-Independent Stress-Strain Relations for Uniaxial States of Stress

Stress-Strain Relations for Elastic Materials Subjected to Three-Dimensional States of Stress

Stress-Strain Relations of Linearly Elastic Materials Subjected to Three-Dimensional States of Stress

Stress-Strain Relations for Orthotropic, Linearly Elastic Materials

Stress-Strain Relations for Isotropic, Linearly Elastic Materials Subjected to Three-Dimensional States of Stress

Strain Energy Density and Complementary Energy Density of a Particle of a Body Subjected to External Forces in an Environment of Constant Temperature

Thermodynamic Considerations of Deformation Processes Involving Bodies Made from Elastic Materials

Linear Response of Bodies Made from Linearly Elastic Materials

Time-Dependent Stress-Strain Relations

The Creep and the Relaxation Tests

Problems

YIELD AND FAILURE CRITERIA

Yield Criteria for Materials Subjected to Triaxial States of Stress in an Environment of Constant Temperature

The Von Mises Yield Criterion

The Tresca Yield Criterion

Comparison of the Von Mises and the Tresca Yield Criteria

Failure of Structures - Factor of Safety for Design

The Maximum Normal Component of Stress Criterion for Fracture of Bodies Made from a Brittle, Isotropic, Linearly Elastic Material

The Mohr's Fracture Criterion for Brittle Materials Subjected to States of Plane Stress

Problems 179

FORMULATION AND SOLUTION OF BOUNDARY VALUE PROBLEMS USING THE LINEAR THEORY OF ELASTICITY

Introduction

Boundary Value Problems for Computing the Displacement and Stress Fields of Solid Bodies on the Basis of the Assumption of Small Deformation

The Principle of Saint Venant

Methods for Finding Exact Solutions for Boundary Value Problems in the Linear Theory of Elasticity

Solution of Boundary Value Problems for Computing the Displacement and Stress Fields of Prismatic Bodies Made from Homogeneous, Isotropic, Linearly Elastic Materials

Problems

PRISMATIC BODIES SUBJECTED TO TORSIONAL MOMENTS AT THEIR ENDS

Description of the Boundary Value Problem for Computing the Displacement and Stress Fields in Prismatic Bodies Subjected to Torsional Moments at Their Ends

Relations among the Coordinates of a Point Located on a Curved Boundary of a Plane Surface

Formulation of the Torsion Problem for Prismatic of Arbitary Cross Section on the Basis of the Linear Theory of Elasticity

Interpretation of the Results of the Torsion Problem

Computation of the Stress and Displacement Fields of Bodies of Solid Elliptical and Circular Cross Section Subjected to Equal and Opposite Torsional Moments at Their Ends

Multiply Connected Prismatic Bodies Subjected to Equal and Opposite Torsional Moments at Their Ends

Available Results

Direction and Magnitude of the Shearing Stress Acting on the Cross Sections of a Prismatic Body of Arbitrary Cross Section Subjected to Torsional Moments at Its Ends

The Membrane Analogy to the Torsion Problem

Stress Distribution in Prismatic Bodies of Thin Rectangular Cross Section Subjected to Equal and Opposite Torsional Moments at Their Ends

Torsion of Prismatic Bodies of Composite Simply Connected Cross Sections

Numerical Solutions of Torsion Problems Using Finite Differences

Problems

PLANE STRAIN AND PLANE STRESS PROBLEMS IN ELASTICITY

Plane Strain

Formulation of the Boundary Value Problem for Computing the Stress and the Displacement Fields in a Prismatic Body in a State of Plane Strain Using the Airy Stress Function

Prismatic Bodies of Multiply Connected Cross Sections in a State of Plane Strain

The Plane Strain Equations in Cylindrical Coordinates

Plane Stress

Simply Connected Thin Prismatic Bodies (Plates) in a State of Plane Stress Subjected on Their Lateral Surface to Symmetric in x1 Components of Traction Tn2 and Tn3

Two-Dimensional or Generalized Plane Stress

Prismatic Members in a State of Axisymmetric Plane Strain or Plane Stress

Problems

THEORIES OF MECHANICS OF MATERIALS

Introduction

Fundamental Assumptions of the Theories of Mechanics of Materials for Line Members

Internal Actions Acting on a Cross Section of Line Members

Framed Structures

Types of Framed Structures

Internal Action Release Mechanisms

Statically Determinate and Indeterminate Framed Structures

Computation of the Internal Actions of the Members of Statically Determinate Framed Structures

Action Equations of Equilibrium for Line Members

Shear and Moment Diagrams for Beams by the Summation Method

Stress-Strain Relations for a Particle of a Line Member Made from an Isotropic Linearly Elastic Material

The Boundary Value Problems in the Theories of Mechanics of Materials for Line Members

The Boundary Value Problem for Computing the Axial Component of Translation and the Internal Force in a Member Made from an Isotropic, Linearly Elastic Material Subjected to Axial Centroidal Forces and to a Uniform Change in Temperature

The Boundary Value Problem for Computing the Angle of Twist and the Internal Torsional Moment in Members of Circular Cross Section Made from an Isotropic, Linearly Elastic Material Subjected to Torsional Moments

Problems

THEORIES OF MECHANICS OF MATERIALS FOR STRAIGHT BEAMS MADE FROM ISOTROPIC, LINEARLY ELASTIC MATERIALS

Formulation of the Boundary Value Problems for Computing the Components of Displacement and the Internal Actions in Prismatic Straight Beams Made from Isotropic, Linearly Elastic Materials

The Classical Theory of Beams

Solution of the Boundary Value Problem for Computing the Transverse Components of Translation and the Internal Actions in Prismatic Beams Made from Isotropic, Linearly Elastic Materials Using Functions of Discontinuity

The Timoshenko Theory of Beams

Computation of the Shearing Components of Stress in Prismatic Beams Subjected to Bending without Twisting

Build-Up Beams

Location of the Shear Center of Thin-Walled Open Sections

Members Whose Cross Sections Are Subjected to a Combination of Internal Actions

Composite Beams

Prismatic Beams on Elastic Foundation

Effect of Restraining the Warping of One Cross Section of a Prismatic Member Subjected to Torsional Moments at Its Ends

Problems

NON-PRISMATIC MEMBERS - STRESS CONCENTRATIONS

Computation of the Components of Displacement and Stress of Non-Prismatic Members

Stresses in Symmetrically Tapered Beams

Stress Concentrations

Problems

PLANAR CURVED BEAMS

Introduction

Derivation of the Equations of Equilibrium for a Segment of Infinitesimal Length of a Planar Curved Beam

Computation of the Circumferential Component of Stress Acting on the Cross Sections of Planar Curved Beams Subjected to Bending without Twisting

Computation of the Radial and Shearing Components of Stress in Curved Beams

Problems

THIN-WALLED, TUBULAR MEMBERS

Introduction

Computation of the Shearing Stress Acting on the Cross Sections of Thin-Walled, Single-Cell, Tubular Members Subjected to Equal and Opposite Torsional Moments at Their Ends

Computation of the Angle of Twist per Unit Length of Thin-Walled, Single-Cell, Tubular Members Subjected to Equal and Opposite Torsional Moment at Their Ends

Prismatic Thin-Walled, Single-Cell, Tubular Members with Thin Fins Subjected to Torsional Moments

Thin-Walled, Multi-Cell, Tubular Members Subjected to Torsional Moments

Thin-Walled, Single-Cell, Tubular Beams Subjected to Bending without

Thin-Walled, Multi-Cell, Tubular Beams Subjected to Bending without Twisting

Single-Cell, Tubular Beams with Longitudinal Stringers subjected to Bending without Twisting

Problems

INTEGRAL THEOREMS OF STRUCTURAL MECHANICS

A Statically Admissible Stress Field and an Admissible Displacement Field of a Body

Derivation of the Principle of Virtual Work for Deformable Bodies

Statically Admissible Reactions and Internal Actions of Framed Structures

The Principle of Virtual Work for Framed Structures

The Unit Load Method

The Principle of Virtual Work for Framed Structures, Including the Effect of Shear Deformation

The Strong Form of One-Dimensional, Linear Boundary Value Problems

Approximation of the Solution of One-Dimensional, Linear Boundary Value Problems Using Trial Functions

The Classical Weighted Residual Form for Second Order, One-Dimensional, Linear Boundary Value Problems

The Classical Weighted Residual Form for Fourth Order, One-Dimensional, Linear Boundary Value Problems

Discretization of Boundary Value Problems Using the Classical Weighted Residual Methods

The Modified Weighted Residual (Weak) Form of One-Dimensional, Linear Boundary Value Problems

Total Strain Energy of Framed Structures

Castigliano's Second Theorem

Betti-Maxwell Reciprocal Theorem

Proof That the Center of Twist of a Cross Section Coincides with Its Shear Center

The Variational Form of the Boundary Value Problem for Computing the Components of Displacement of a Deformable Body - Theorem of Stationary Total Potential Energy

Comments on the Modified Gallerkin Form and the Theorem of Stationary Total Potential Energy

Problems

ANALYSIS OF STATICALLY INDETERMINATE FRAMED STRUCTURES

The Basic Force or Flexibility Method

Computation of Components of Displacement of Points of Statically Indeterminate Structures

Problems

THE FINITE ELEMENT METHOD

Introduction

The Finite Element Method for One-Dimensional, Second Order, Linear Boundary Value Problems as a Modified Galerkin Method

Element Shape Functions

Assembly of the Stiffness Matrix for the Domain of One-Dimensional, Second Order, Linear Boundary Value Problems from the Stiffness Matrices of Their Elements

Construction of the Force Vector for the Domain of One-Dimensional, Second Order, Linear Boundary Value Problems

Direct Computation of the Contribution of an Element to the Stiffness Matrix and the Load Vector of the Domain of One-Dimensional, Second Order, Linear Boundary Value Problems

Approximate Solution of Linear Boundary Value Problems Using the Finite Element Method

Application of the Finite Element Method to the Analysis of Framed Structures

Approximate Solution of Scalar Two-Dimensional, Second Order, Linear Boundary Value Problems Using the Finite Element Method

Problems

PLASTIC ANALYSIS AND DESIGN OF STRUCTURES

Strain-Curvature Relation of Prismatic Beams Subjected to Bending without Twisting

Initiation of Yielding Moment and Fully Plastic Moment of Beams Made from Isotropic, Linearly Elastic, Ideally Plastic Materials

Distribution of the Shearing Component of Stress

Acting on the Cross Sections of Beams Where M2Y

## Reviews

"The author successfully presents the transition of applied elasticity from the eighteenth century to the twenty first century; from Mohr circle to finite elements. The material is well organized, well written, and well presented."

-J. Genin in Zentralblatt MATH 1089