Stability of Structures Elastic, Inelastic, Fracture, and Damage Theories

by Bazant, Zdenek P.; Cedolin, Luigi

• ISBN13: 9780486425689
• ISBN10: 0486425681
• Format: Paperback
• Copyright: 2003-02-14
• Publisher: Dover Publications

Summary

Exploration of principles and applications of stability analysis emphasizes nonelastic stability. Topics include modern stability problems of fracture and damage, thermodynamic principles of stability in irreversible systems, viscoelastic and viscoplastic buckling, other key areas. Numerous examples; 700 exercise problems. 1991 edition.

Table of Contents

Preface	vii
Introduction	xxi
I ELASTIC THEORIES
Buckling of Elastic Columns by Equilibrium Analysis	3	(50)
Theory of Bending	4	(1)
Euler Load, Adjacent Equilibrium, and Bifurcation	5	(4)
Differential Equations of Beam-Columns	9	(3)
Critical Loads of Perfect Columns with Various End Restraints	12	(7)
Imperfect Columns and the Southwell Plot	19	(8)
Lateral Disturbing Load
Initial Curvature or Load Eccentricity
Behavior near the Critical Load
Southwell Plot
Code Specifications for Beam-Columns	27	(3)
Effect of Shear and Sandwich Beams	30	(5)
Pin-Ended Columns
Generalization
Sandwich Beams and Panels
Pressurized Pipes and Prestressed Columns	35	(3)
Pressurized Pipes
Prestressed Columns
Large Deflections	38	(8)
Solution of Rotations by Elliptic Integrals (Elastica)
Deflections and Shortening
Discussion of Results
Spatial Buckling of Beams under Torque and Axial Force	46	(7)
Buckling of Elastic Frames by Equilibrium Analysis	53	(91)
Stiffness and Flexibility Matrices of Beam-Columns	54	(7)
Stiffness Matrix for End Rotations
Stiffness Matrix for End Rotations and Relative Lateral Displacement
Critical Loads of Frames and Continuous Beams	61	(13)
Simple Structures
Difficulties with the Flexibility Method
General Approach for Computer Analysis
Buckling as a Matrix Eigenvalue Problem and Use of Finite Elements	74	(4)
Large Regular Frames	78	(8)
Postcritical Reserve in Redundant Trusses	86	(3)
Example of a Statically Indeterminate Truss
Generalization and Limit Analysis Method
Order of Approximation
Postcritical Behavior of Frames	89	(13)
L-Frame of Koiter and Roorda
Second-Order Solution of the L-Frame
Imperfection Sensitivity
Generalizations and Implications
Order of Approximation
Postcritical Reserve Due to Redundancy
Finite Element Computational Procedure
Built-Up Columns and Regular Frames as Columns with Shear	102	(6)
High Arches	108	(10)
Curvature Change
Approximate Theory for Perfect Arches with a Fixed Compression Line
Various Types of Arches and the Effect of Imperfections
General Linearized Theory for Uniformly Compressed Circular Arches
Long-Wave Buckling of Regular Frames	118	(11)
System of Difference Equations
Solution for Tall Building Frames
Numerical Results and Their Discussion
Continuum Approximation for Large Regular Frames	129	(15)
Transition from Difference to Differential Equations
Continuum Approximation Based on Potential Energy
Micropolar Continuum and Couple Stresses
Stresses and Boundary Conditions
Numerical Results and Discussion
Dynamic Analysis of Stability	144	(55)
Vibration of Columns or Frames and Divergence	144	(7)
Columns
Types of Motion and Dependence of Natural Frequency on Load
Effect of Damping
Frames and Other Generalizations
Nonconservative Loads and Flutter	151	(11)
Massless Column under Follower Load
Effect of Distributed Mass
Elastically Supported Rigid Plate under Aerodynamic Forces
Conservative and Nonconservative Forces
Equations Governing Flutter or Suspension Bridges
Pulsating Loads and Parametric Resonance	162	(8)
Axial Pulsating Load on a Column
Undamped Vibration
Damped Vibration
Simple Energy Analysis of Parametric Resonance
Other Types of Dynamic Loads	170	(4)
Definition of Stability	174	(4)
Theorems of Lagrange--Dirichlet and of Liapunov	178	(6)
Stability Criteria for Dynamic Systems	184	(3)
Stability of Continuous Elastic Systems	187	(2)
Nonlinear Oscillations and Chaos	189	(10)
Energy Methods	199	(106)
Positive-Definite Matrices, Eigenvalues, and Eigenvectors	199	(8)
Potential Energy for Discrete Elastic Systems	207	(9)
Structure-Load System
Second Variation of Potential Energy
Critical State
An Example
Effect of Higher-Order Derivatives of II
Difficulties with Complementary Energy
Overturning Instability of a Block: Discontinuous Π'
Bifurcation Buckling at Small Deflections	216	(11)
Calculation of Potential Energy of Beam-Columns
Equilibrium and Stability
Role of Axial Strain and Shortening due to Deflections
Calculation of II from the Work of Initial and Incremental Stresses
Example with Two Degrees of Freedom
Some Fundamental Aspects of Potential Energy
Snapthrough and Flat Arches	227	(11)
Von Mises Truss
Flat Arches
Effect of Imperfections
Other Examples of Snapthrough
Large-Deflection Postcritical Behavior and Types of Bifurcation	238	(23)
Symmetric Stable Bifurcation: Example
Symmetric Unstable Bifurcation: Example
Asymmetric Bifurcation: Example
L-Shaped Rigid-Bar Frame
Rigid-Bar Arch
Nonlinear Springs and Polynomial Approximation of Potential Energy
Two Degrees of Freedom: Example
Limit Points of Equilibrium Paths
Bifurcation Criterion in Terms of the Tangential Stiffness Matrix
Classification of Elementary Instabilities of Elastic Structures
Koiter's Theory, Imperfection Sensitivity, and Interaction of Modes	261	(9)
General Validity of Koiter's 1/2-Power and 2/3-Power Laws
Interaction of Buckling Modes
Catastrophe Theory and Breakdown of Symmetry	270	(8)
Snapdown at Displacement-Controlled Loading	278	(16)
Structures with Two Degrees of Freedom
Softening Specimen in a Testing Machine
Generalization of Snapdown Analysis
Equilibrium Paths with Bifurcations, Snapthrough, and Snapdown
Incremental Work Criterion at Equilibrium Displacements	294	(11)
Stability Criterion
Possibility of Generalization to Inelastic Systems
Energy Analysis of Continuous Structures and Approximate Methods	305	(65)
Indirect Variational Method and Euler Equation	306	(8)
Review of the Calculus of Variations
Application to Structures Possessing a Potential Energy
Review of Positive-Definite and Self-Adjoint Operators
Beam on Elastic Foundation	314	(9)
Potential Energy and Differential Equations
Solution for Different Boundary Conditions
Fiber on Elastic Foundation
Rayleigh Quotient	323	(8)
Upper-Bound Property of Rayleigh Quotient
Application to Beam-Columns
Relation to Differential Equation
Proof of Upper-Bound Property and Convergence
Extension to Free Vibration
Timoshenko Quotient and Relations between Various Bounds	331	(8)
Derivation
Examples
Relation to Differential Equation and Proof of Upper-Bound Property
Relation to Rayleigh Quotient and Inequalities
Inapplicability to Dynamics
The Question of Lower Bounds
Bound Approximation for Columns, Frames, and High Arches	339	(9)
Columns
Frames
Elastically Supported Beams
High Arches
Rayleigh--Ritz Variational Method	348	(8)
Galerkin Variational Method	356	(2)
Method of Successive Approximations and Lower Bounds	358	(3)
Formulation of the Method
Example
Lower Bound
Nonlinear Problems
Large Deflections of Columns	361	(9)
Thin-Walled Beams	370	(49)
Potential Energy and Differential Equations	371	(10)
Deformation of the Cross Section
Potential Energy
Differential Equations and Boundary Conditions
Axial-Torsional Buckling of Columms	381	(3)
Lateral Buckling of Beams and Arches	384	(8)
Axial-Torsional Buckling due to Eccentric Axial Force
Lateral Buckling due to Bending Moment
Approximate Solution for Variable M0z
Bimoment
Lateral Buckling of Arches
Beams of Arbitrary Open Cross Section	392	(9)
General Theory of Warping Torsion
Stresses and Bimoment in General Theory
Potential Energy and Differential Equations
Monosymmetric Cross Section
Large Deflections	401	(7)
Box Girders	408	(11)
Deformation Modes and Postcritical Energy
Examples
Finite Element Solution
Interaction with Local Buckling
Plates and Shells	419	(66)
Classical Plate Theory	420	(2)
Differential Equation and Strain Energy	422	(9)
Strains
Potential Energy
Differential Equations of Equilibrium
Boundary Conditions
Direct Derivation of Transverse Resultant of In-Plane Forces
Discussion and Summary
Buckling of Rectangular Plates	431	(9)
Buckling of Simply Supported Plates
Rectangular Plate with Arbitrary Boundary Conditions
Buckling of Plate Subjected to Shear
Nonuniform In-Plane Forces
Solutions by Other Variational Methods
Large Deflections and Postcritical Reserve of Plates	440	(9)
Von Karman--Foppl Differential Equations
Solution by Minimization of Potential Energy
Large Deflections and Ultimate Strength
Measurement of Critical Loads
Axisymmetric Buckling of Cylindrical Shells	449	(4)
Shallow or Quasi-Shallow Shells	453	(13)
Basic Relations for Cylindrical Shells
Donnell's Equation
Axially Compressed Cylindrical Shell
Effect of Lateral Pressure on Cylindrical Shells
Cylindrical Shell Subjected to Torsion
Variational Derivation from Potential Energy
Cylindrical Shell Panels
General Quasi-Shallow Shells
Nonlinear Analysis of Shell Buckling and Imperfections	466	(8)
Reduction Factors for Classical Critical Loads
Physical Source of Postcritical Load Drop and High Imperfection Sensitivity
Koiter's Laws of Imperfection Sensitivity
Buckling Modes and Their Interaction
Summary
Sandwich Plates and Shells	474	(11)
Basic Relations for a Sandwich Plate Element
Rectangular Sandwich Plate and Other Problems
II INELASTIC, DAMAGE, AND FRACTURE THEORIES
Elastoplastic Buckling	485	(99)
Perfect Columns or Structures and Shanley's Bifurcation	486	(20)
Reduced Modulus Load
Tangent Modulus Load
Column Strength Curve
Postbifurcation Load-Deflection Diagram
Bifurcation in Plastic Structures with Multiaxial Stress
Conclusion
Imperfect Columns and Structures	506	(7)
Shanley's Rigid-Bar Column: Exact Solution
Arbitrary Imperfect Columns: Approximate Solution
Effect of Cross-Section Nonsymmetry
Effect of Residual Stresses	513	(4)
Calculation of the Effect of Residual Stresses
Examples
Metal Columns and Structures: Design and Code Specifications	517	(15)
Centrically Loaded Columns
Load and Resistance Factor Design and Probabilistic Aspects
Beam-Columns
Plates, Shells, and Other Structures
Design Examples
Concrete Columns and Structures: Design and Code Specifications	532	(24)
Interaction Diagram (Failure Envelope)
Deflections and Interaction Diagram
Numerical Algorithm for Calculating Deflections and Interaction Diagram
Column Response for Unsmooth Stress-Strain Diagrams
Design Recommendations and the ACI Code
CEB Design Recommendations
Comparisons of Codes and Shortcomings
Prestressed Concrete Columns
Shells and Other Structures
Stress--Strain Relations for Strain Softening
Design Examples
Perfectly Plastic Large-Deflection Buckling, Impact, and Blast	556	(13)
Load-Deflection Curve or Perfectly Plastic Columns
Buckling of Perfectly Plastic Frames
Plastic Redistribution and Reserve Capacity of Structures
Dynamic Impact
Perfectly Plastic Buckling of Thick Plates
Transverse Impact or Blast on Plates or Columns with In-Plane or Axial Loads
Geometric Tensile Instability, Localization, and Necking	569	(15)
Role of Transverse Contraction and Finite Strain
Strain Localization
Necking
Creep Buckling	584	(49)
Viscoelastic Stress--Strain Relations	585	(5)
Compliance Function and Integral-Type Creep Law
Differential-Type Creep Law and Rheologic Models
Elastic--Viscoelastic Analogy
Viscoelastic Buckling	590	(7)
Deflection History and Long-Time Critical Load
The Concept of Stability for Viscoelastic Structures
Extensions and Ramifications
Viscoplastic Buckling	597	(6)
Rigid-Bar Model Column
Critical Time and Stability Concept
Real Columns
Buckling of Aging Viscoelastic Structures	603	(12)
Aging Maxwell Solid (Dischinger-Type Methods)
Deflections According to Aging Maxwell Model
Deflection According to More Realistic Rheologic Models
Deflection According to Effective Modulus
Deflection According to Age-Adjusted Effective Modulus
Deflection According to Integral-Type Stress--Strain Relation
Appendix I---Compliance Function and Relaxation Function of Concrete
Appendix II---Proof of Age-Adjusted Effective Modulus Method
Effect of Creep Deflection on Concrete Column Strength	615	(4)
Nonlinear Creep and Long-Time Strength of Concrete Structures	619	(4)
Creep Buckling at Finite Deflections	623	(10)
Example of Imperfection-Sensitive Rigid-Bar Column
Broader Implications and Ramifications
Variable Load
Stability of Inelastic Structures, Bifurcation and Thermodynamic Basis	633	(73)
Thermodynamic Criteria of Stable State	633	(17)
First and Second Laws of Thermodynamics
Tangentially Equivalent Elastic Structures
Total Energy U and Helmholtz Free Energy F
Second Variation of F or U
Path Dependence and Incremental Potentials
Second-Order Work of Stresses and Geometric Stiffness
Criterion of Stable State for the Case of Dead Loads
Extensions to Variable Loads
Stability at Critical State
Gibbs Free Energy and Enthalpy
Stability Criteria Based on Complementary Work
Structures with a Single Load or a Single Controlled Displacement
Summary
Thermodynamic Criteria of Stable Path	650	(8)
Path Stability for Basic Types of Control
Mixed Controls of Loads and Displacements
The Case of Equal (ΔS)in for Two Branches
Second-Order Work of Stresses along the Path
Structures with a Single Load or a Single Controlled Displacement
Stable States on Postbifurcation Branches
Further Comments and Conclusion
Application to Elastoplastic Columns and Broader Implications	658	(12)
Loading-Unloading Combinations and Equilibrium Paths
Second-Order Work
Stable Equilibrium States of Elastoplastic Column
Stable Equilibrium Path of an Elastoplastic Column
Breakdown of Symmetry
Hypothesis Implied in Present Thermodynamic Approach
Summary
Critical States of Stability and Bifurcation	670	(10)
Critical State for Structures with a Symmetric Stiffness Matrix
Critical States for Structures with a Nonsymmetric Stiffness Matrix
Example of a Nonsymmetric Stiffness Matrix
Symmetric and Asymmetric Bifurcations at the Critical State
Uniqueness
Bifurcation for Inelastic Structures and Hill's Linear Comparison Solid
Distribution of Bifurcation Points and Postbifurcation Branches
Numerical Finite Element Analysis
Summary
Stability at Infinitesimal Loading Cycles	680	(5)
Internal Entropy Changes for Cycles in Shanley's Column
Stability
Structures with a Single Cyclic Load or Displacement
Incremental Collapse
Drucker's and II'yushin's Postulates for Stable Materials	685	(8)
Drucker's Postulate
II'yushin's Postulate
Nonuniformly Strained Bodies
Normality Rule for Plasticity
Stability of Frictional Materials and Structures	693	(13)
Frictional Block Preloaded by a Spring
Generalization to Frictional Continuum
Stability Condition of Frictional Materials
Plastic Strain Increment for Frictional Materials
Inverse Material Friction
Frictional Phenomena in Other Constitutive Theories
Three-Dimensional Continuum Instabilities and Effects of Finite Strain Tensor	706	(54)
Finite Strain	707	(8)
Notations and Basic Requirements
Lagrangian (Green's) Finite Strain Tensor
Biot's Finite Strain Tensor
Second-Order Approximations of Other Finite Strain Tensors
Further Measures of Finite Strain
The Special Case of Thin Bodies
Decomposition of Strain into Elastic and Inelastic Parts
Stresses, Work, and Equilibrium at Finite Strain	715	(5)
Virtual Work Relations and Equilibrium
True (Cauchy) Stress
Stress Referred to Initial Configuration and Working on Displacement Gradient
Stress Referred to Initial Configuration and Working on Finite Strain
Incremental Equilibrium and Objective Stress Rates	720	(6)
Incremental Equilibrium Conditions
Increments of Cauchy (True) Stresses
Objective Stress Increments Conjugate to Strain Increments
Objective Stress Rates
Tangential Moduli at Large Initial Stress	726	(6)
Stable States and Paths for Multidimensional Continuous Bodies	732	(2)
Column or Plate with Shear: Finite-Strain Effect	734	(7)
Differential Equations
Theories of Engesser and Haringx
Correlation to Built-Up Columns
Summary
Surface Buckling and Internal Buckling of Anisotropic Solids	741	(7)
Basic Relations for Incompressible Orthotropic Solids
Surface Buckling of an Orthotropic Half-Space
Internal Buckling and Other Instabilities
General Solution
Consistent Geometric Stiffness Matrix of Finite Elements	748	(2)
Buckling of Curved Fibers in Composites	750	(10)
Macroscopic Elastic Stress--Strain Relations
Decrease of Elastic Moduli Due to Fiber Undulation
Generalization to Three Dimensions
Stresses Due to Fiber Undulation
Fracture as a Stability Problem	760	(69)
Linear Elastic Fracture Mechanics	760	(12)
Stress Singularity and Fracture Energy
Energy Release Rate
Determination of G and Gf from Compliance Changes
Some Simple Elastic Solutions
Approximation by Stress Relief Zone
Examples Solvable by Bending Theory
Herrmann's Method to Obtain Approximate KI by Beam Theory
Nonlinear Fracture Mechanics and Size Effect	772	(10)
Inelastic Zone and Equivalent Elastic Crack
Fracture Models with a Nonlinear Zone
Size Effect
Crack Stability Criterion and R-Curve	782	(6)
R-Curve and Fracture Equilibrium Condition
Fracture Stability Criterion and Critical State
Determination of Geometry-Dependent R-Curve from Size Effect Law
Crack Propagation Direction
Kinking of Cracks and Three-Dimensional Instability of Front Edge
Snapback Instability of a Crack and Ligament Tearing	788	(10)
General Procedure for Load-Displacement Relation at Growing Crack
Snapback Instability at Crack Coalescence in Two Dimensions
Snapback Instability at Tearing of Circular Ligament
General Condition for Snapback at Ligament Tearing
Alternative Calculation of Displacement from Compliance Variation
Stable States and Stable Paths of Interacting Cracks	798	(18)
Conditions of Equilibrium and Stability in Terms of Crack Length
Stability of Parallel Cooling or Shrinkage Cracks
Stable Path and Bifurcation at Advancing Cooling Front
Three-Dimensional Pattern of Cooling or Shrinkage Cracks
Stability of Parallel Cracks in Reinforced Concrete
Stability Analysis in Terms of Displacements
Crack Spacing	816	(13)
Spacing of Parallel Initial Drying or Cooling Cracks: Energy Balance
Size of Initial Hexagonal Cracking Cells from Energy Balance
Snapthrough Formation of Cracks According to LEFM
Crack Spacing in Loaded Reinforced Concrete Beams
Snapthrough Crack Formation in a Drying Tube
Damage and Localization Instabilities	829	(124)
Wave in Strain-Softening Materials	830	(15)
Exact Solution of Strain-Softening Bar
Stability Aspects and Unrealistic Properties of Solution of a Bar
Bar with Rehardening or Residual Yield Strength
Cylindrically or Spherically Converging Waves
General Three-Dimensional Condition for Waves to Exist
Summary
Series-Coupling Model for Localization Due to Softening	845	(12)
Stable States
Surface of Second-Order Work
Application to Uniaxially Stressed Bars or Specimens
Effects of Size and Support Stiffness
Specimen Ductility as a Function of Length and Loading Frame Stiffness
Inadmissibility of Arbitrarily Small Size of Localization Region
Bifurcation and Stable Path
Alternative: Imperfection Approach to Bifurcation
Identification of Softening Stress-Strain Relations from Tests
Relation of Strain Softening to Fracture Energy
Summary
Localization of Softening Damage into Planar Bands	857	(12)
Stability Condition for the Softening Band Within a Layer or Infinite Solid
Discussion of Various Cases
Numerical Examples
Generalization for Geometrically Nonlinear Effects
Bifurcation and Stable Path
Localization into Shear Bands Due to Nonassociatedness in Frictional Materials
Sand Liquefaction as a Localization Instability
Summary
Localization of Softening Damage into Ellipsoidal Regions	869	(11)
Eshelby's Theorem
Stability of Uniform Strain against Ellipsoidal Localization
Numerical Examples of Stability Limits and Discussion
Bifurcation and Stable Path of Ellipsoidal Localization
Simpler Derivation of Bifurcation Condition
Summary
Localization of Softening Damage into Spherical or Circular Regions	880	(7)
Localization Instability for Spherical Geometry
Localization Instability for Circular or Cylindrical Geometry
Numerical Examples
Bifurcation and Stable Path
Summary
Localization in Beams and Softening Hinges	887	(15)
Stability Limit and Snapback
Rotation Capacity or Ductility of Hinges in Concrete Beams
Length of the Softening Region
Bifurcation Due to Interaction of Softening Hinges
Imperfection Approach
Bifurcation and Localization in Redundant Structures
Bifurcation at Simultaneous Formation of Several Softening Hinges
Softening Frames and Trusses
Softening in Metallic Structures
Summary
Friction: Static and Dynamic	902	(5)
Paradox in Sudden Friction Drop
Bifurcation, Stable Path, and Localization of Frictional Slip
Frictional Supports in Columns
Structures with Stiffness Matrix Asymmetry
Bifurcations Due to Interaction of Softening Damage Zones	907	(7)
Interaction of Damage (Cracking) Fronts and Stable Paths
Convergence of Iterations Depends on Stability of State, not Path
Multiple Interacting Crack Band Fronts
Interaction of Multiple Shear Bands
Example: Buckling in Direct Tensile Test
Size Effect, Mesh Sensitivity, and Energy Criterion for Crack Bands	914	(7)
Localization as a Cause of Size Effect
Inobjectivity or Spurious Mesh Sensitivity
Energy Criterion for Crack Band and Stability
Nonlocal Continuum and Its Stability	921	(16)
Crack Band Model
Nonlocal Continuum Concept
Periodic Instabilities Due to Nonlocal Concept
Nonlocal Continuum with Local Strain
One-Dimensional Localization Instability
Measurement of Characteristic Length of Nonlocal Continuum
Example: Stability of a Tunnel
Gradient Approximation to Nonlocal Continuum
Summary
Constitutive Equations for Strain Softening	937	(16)
Glossary of Symbols	953	(10)
Author Index	963	(10)
Subject Index	973	(12)
Appendix to the Dover Edition	985