1st Edition

Limit Analysis of Solids and Structures

By Jacov A. Kamenjarzh Copyright 1996
    464 Pages
    by CRC Press

    Solids subjected to sufficiently large loads undergo plastic strain that does not vanish after unloading. Limit analysis is used to find out whether a given loading is safe against capacity loss due to intensive plastic deformation. Over the past 25 years, the theory and methods of limit analysis have undergone substantial development. This book gives a clear and complete presentation of the state of the art of limit analysis, including:

    Rigid Perfectly Plastic Body
    Plastic Deformation
    Elastic and Residual Strain
    Yield Stress
    Elasticity Domain
    Rate Insensitivity
    Basic Properties of Plastic Materials
    Admissible Stresses. Yield Surface
    Stress Space
    Admissible Stresses
    Yield Surface
    Constitutive Relations
    Normality Flow Rule
    Constitutive Maximum Principle
    Rigid Perfectly Plastic Body
    Dissipation
    Constitutive Maximum Principle and Dissipation
    Properties of Dissipation
    Yield Surface Determined by Dissipation
    Rigid-Plastic Problem
    Quasistatic Problem
    First Example of the Limit Load
    Rigid-Plastic Problem
    On the Admissibility and Equilibrium Conditions
    Beams and Trusses
    Beam. Kirchhoff Hypothesis
    Internal Forces in a Beam
    Constitutive Relations for Rigid Perfectly Plastic Beam
    Rigid-Plastic Problem for a Beam
    Truss
    Internal Efforts in a Truss
    Rigid-Plastic Problem for a Truss
    Appendix A. Stress and Strain
    Stress
    Deformation
    Elasticity Law
    Appendix B. Convex Sets and Convex Functions
    Definitions and Examples
    Separation Theorem
    Appendix C. Extremums
    Minimum and Maximum
    Extremum Problems
    Conditions for the Minimum of Convex Function
    Comments
    Virtual Work Principle
    Bodies under Standard Loading
    Virtual Velocity Fields and Power
    Virtual Work Principle
    Generalized Equilibrium Conditions
    Bodies under Mixed Boundary Conditions
    Tangent Load and Surface Slip
    Rigid Punch Loading
    Beams and Trusses
    Virtual Velocity Fields and Power for a Beam
    Virtual Work Principle for a Beam
    Generalized Conditions for Equilibrium of a Beam
    Virtual Work Principle for a Truss
    Comments
    Fundamentals of Limit Analysis
    Rigid-Plastic Problem
    Stress Fields and Equilibrium Conditions
    Admissible Stress Fields
    Velocity Fields
    Formulation of the Problem
    Safety Factor
    Admissible and Inadmissible Loads
    Safety Factor
    Safe and Limit Loads
    Safe Loads
    Safety Criterion
    Limit Load and Failure Mechanism
    On the Failure Mechanism's Existence
    Problems of Limit Analysis
    Limit Stress State Principle
    Safety of aLoading
    Limit Surface
    Limit Analysis in Presence of Permanent Load
    Basic Statements and Methods of Limit Analysis
    Lower Boundary for Safety Factor: Static Multiplier
    Upper Boundary for Safety Factor: Kinematic Multiplier
    Criterion for Static and Kinematic Multipliers Equality
    Rigid-Plastic Solution Method
    On the Kinematic Method
    Examples of Limit Analysis
    Formulation of the Axially Symmetric Plain Strain Problem
    A Pipe under Internal Pressure
    A Pipe under Internal Torsion
    Comments
    Limit Analysis: General Theory
    Rigid-Plastic Problem: General Formulation
    Equilibrium Conditions
    Kinematic Relations
    Constitutive Relations
    General Formulation of the Problem
    Local Description of Material Properties
    Examples
    Rigid-Plastic Problem for a Beam
    Rigid-Plastic Problem for a Discrete System
    Safe, Limit, and Inadmissible Loads
    Safety Factor
    Safe Stress Fields
    Safe Loads
    Safety Criterion
    Limit Analysis
    Static and Kinematic Multipliers
    Static Multiplier
    Kinematic Multiplier
    Criterion for Static and Kinematic Multipliers Equality
    On Methods for Limit Analysis
    Integral Formulation of Constitutive Maximum Principle
    Integral Formulation
    Extremum Property and Calculation of Dissipation
    Admissible Stresses: A Set-Valued Mapping
    Conditions for Equivalence of Constitutive Principle Formulations
    Extremum of Integral Functional
    Integral Functional
    Evaluating the Extremum
    Appendix A. Linear Spaces
    Definitions and Examples
    Subspace
    Linear Operator
    Pairing between Linear Spaces
    Appendix B. Measurable Sets and Measurable Functions
    Measure
    Measurable Functions
    Comments
    Extremum Problems of Limit Analysis
    Static and Kinematic Extremum Problems
    Limit Static and Kinematic Multipliers
    Main Results
    Static and Kinematic Extremum Problems: Standard Formulation
    Minkowski Function
    Static Extremum Problem Standard Form
    Kinematic Extremum Problem Standard Form
    Dual Extremum Problem
    Fenhel Transformation
    Constructing the Dual Problem
    Applying the Dual Problem
    Conditions for Extremums Equality
    Conditions for Equality of Limit Multipliers - I
    Static Extremum Problem Dual of the Kinematic Problem
    Repeated Fenhel Transformation
    Limit Multipliers Equality
    Bodies with Bounded Yield Surfaces
    Set of Admissible Stress Fields
    Spaces of Stress and Strain Rate Fields
    Equality of Limit Multipliers
    Conditions for Equality of Limit Multipliers - II
    Kinematic Extremum ProblemDual of the Static Problem
    Continuity of Convex Functions
    Equality of Limit Multipliers
    Bodies with Cylindrical Yield Surfaces
    Cylindrical Yield Surfaces
    Spaces of Stress and Strain Rate Fields
    Equality of Limit Multipliers
    Another Case of Limit Multipliers Equality
    Counterexamples
    Unequality of Limit Multipliers
    Unattainability of Extremums Over Smooth Fields
    Appendix A. Normed Spaces
    Definitions and Examples
    Space of Essentially Bounded Functions
    Convergency. Closure. Continuity.
    Conjugate Space
    Appendix B. Duality Theorum
    Comments
    Reduction of Limit Analysis Extremum Problems
    Reduction of Static and Kinematic Extremum Problems
    Static Problem Reduction
    Kinematic Problem Reduction
    Reduced Extremum Problems: Main Results
    Safety Factor as Extremum in the Reduced Problems
    Pressure Field Restoration
    Regularity of Body Boundary
    Distribution Restoration
    Pressure Field in a Body with Fixed Boundary
    Pressure Field in a Body with Fixed Part of Boundary
    Approximations to Vector Fields
    Regularity of the Free Part of the Body Boundary
    Conditions for Approximation
    Approximations to a Solenoidal Vector Fields
    Approximation in theCase of Fixed Boundary
    Vector Fields with a Given Divergence
    Approximation Conditions - I
    Approximation Conditions - II
    Appendix A. Distributions
    Appendix B. Sobolev Spaces
    Definition and Main Properties
    Spaces of Traces
    Comments
    Limit State
    Stress Field
    Limit State Problem
    Stress Field
    Failure Mechanism
    Strain Rate Field
    Extension Scheme
    Rigid-Plastic Problem Weak Formulation
    Limit State
    Comments
    Discontinuous Fields in Limit Analysis
    Kinematic Multiplier for Discontinuous Velocity Field
    On Definition of Kinematic Multiplier
    Dissipation at Discontinuity Surface
    Surface Slip
    Main Property of Kinematic Multiplier
    Methods for Limit Analysis
    Kinematic Method
    Criterion for Static and Kinematic Multipliers Equality
    Rigid-Plastic Solutions Method
    Discontinuity Relations
    Normality Law
    Maximum Principle
    Normality Law for Velocity Jump
    On the Possibility of Velocity Jump
    Bodies with Jump Inhomogeneity
    Jump Inhomogeneity
    Dissipation at the Discontinuity Surface
    Kinematic Multiplier and Kinematic Method
    Rigid-Plastic Solutions Method
    Discontinuity Relations
    Examples of Limit Analysis
    Lateral Stretching of Strip
    Stretching of a Strip with a Hole
    Limit Surface for Biaxial Stretching of the Plane with Holes
    A Pipe under Internal Torsion
    Shear of a Parallelepiped with Jump Inhomogeneity
    Derivation of the Formula for Kinematic Multiplier
    Formula for Kinematic Multiplier
    Smoothing the Jump
    Smoothing the Jump on a Standard Domain Boundary
    Smoothing with a Given Trace
    Derivation of the Formula for Kinematic Multiplier
    Comments
    Numerical Methods for Limit Analysis
    Approximations for the Kinematic Extremum Problem
    Formulation of the Problem
    Approximations
    Discretization: Finite Element Method
    Idea of the Method
    Approximation for Velocity Field Space
    Approximation for Solenoidal Velocity Field Space
    Discretized Problem of Limit Analysis
    Minimization: Separating Plane Method
    Subgradients
    Infimum and e-Subdifferentials
    Separating Plane Method
    Algorithm and Convergence of Iterations
    Finding a Subgradient
    Comments
    Shakedown Theory
    Elastic-Plastic Problem
    Elastic Perfectly Plastic Body
    Elastic-Plastic Problem: Strong Formulation
    A Way to Generalize Formulation: Examples
    General Formulation of the Problem
    Formulation in Stresses
    Elastic--Plastic Body under Variable Loading: Examples
    Residual Stresses and Shakedown
    Nonshakedown at Bounded Plastic Strain
    Nonshakedown at Unbounded Plastic Strain
    Shakedown at Nonstop Plastic Flow
    Conditions for Shakedown. Safety Factor
    Definitions of Shakedown and Nonshakedown
    Elastic Reference Body
    Shakedown Conditions and Safety Factor: Main Results
    Shakedown and Nonshakedown Theorems
    Shakedown Theorem
    Lower Boundary for Plastic Work
    Damaging Cyclical Loading
    Nonshakedown Theorem
    Reduction of Nonshakedown Theorum Assumptions
    Problems of Shakedown Analysis
    Shakedown to a Set of Loads
    One-Parametric Problems of Shakedown Analysis
    Shakedown under Mechanical and Thermal Loading
    Extremum Problems of Shakedown Analysis
    Static Extremum Problem
    Kinematic Extremum Problem
    Conditions for Equality of the Extremums - I
    Conditions for Equality of the Extremums - II
    Kinematic Method for Safety Factor Evaluation
    Formulation of the Method
    Modified Kinematic Problem
    Formula for the Safety Factor Upper Boundary
    Possibility of Safety Factor Evaluation
    Finite Element Method
    Comments
    Bibliography
    Index

    Biography

    Jacov A. Kamenjarzh