1st Edition

Limit Analysis of Solids and Structures



ISBN 9780849328732
Published July 24, 1996 by CRC Press
464 Pages

USD $250.00

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Book Description

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:

Table of Contents

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

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