1st Edition

Advanced Mechanics of Materials and Applied Elasticity

By Anthony E. Armenàkas Copyright 2006
    996 Pages 732 B/W Illustrations
    by CRC Press

    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.

    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

    Biography

    Anthony E. Armenakas

    "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