Advanced Mechanics of Materials and Applied Elasticity  book cover
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Advanced Mechanics of Materials and Applied Elasticity




ISBN 9780849398995
Published August 19, 2005 by CRC Press
992 Pages 732 B/W Illustrations

 
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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

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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