# Limit Analysis of Solids and Structures

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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