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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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CHOICE – Outstanding Academic Title – Award Winner
CHOICE – 2018 Outstanding Academic Title – Award Winner
Shingo Research and Professional Publication Award Winner
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