Advanced Mechanics of Materials and Applied Elasticity Book

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