Stability of Structures
Contents of the book
Buckling of Elastic Columns by Equilibrium Analysis 3
1.1 Theory of Bending 4
1.2 Euler Load, Adjacent Equilibrium, and Bifurcation 5
1.3 Differential Equations of Beam-Columns 9
4.8 Snapdown at Displacement-Controlled Loading 278
4.9 Incremental Work Criterion at Equilibrium Displacements 294
1.1 Theory of Bending 4
1.2 Euler Load, Adjacent Equilibrium, and Bifurcation 5
1.3 Differential Equations of Beam-Columns 9
1.4 Critical Loads of Perfect Columns with Various End Restraints 12
1.5 Imperfect Columns and the Southwell Plot 19
1.6 Code Specifications for Beam-Columns 27
1.7 Effect of Shear and Sandwich Beams 30
1.8 Pressurized Pipes and Prestressed Columns 35
1.9 Large Deflections 38
1.10 Spatial Buckling of Beams under Torque and Axial Force 46
2 Buckling of Elastic Frames by Equilibrium Analysis 53
2.1 Stiffness and Flexibility Matrices of Beam-Columns 54
2.2 Critical Loads of Frames and Continuous Beams 61
2.3 Buckling as a Matrix Eigenvalue Problem and Use of Finite Elements 74
2.4 Large Regular Frames 78
2.5 Postcritical Reserve in Redundant Trusses 86
2.6 Postcritical Behavior of Frames 89
2.7 Built-Up Columns and Regular Frames as Columns with Shear 102
2.8 High Arches 108
2.9 Long-Wave Buckling of Regular Frames 118
2.10 Continuum Approximation for Large Regular Frames 129
3 Dynamic Analysis of Stability 144
3.1 Vibration of Columns or Frames and Divergence 144
3.2 Nonconservative Loads and Flutter 151
3.3 Pulsating Loads and Parametric Resonance 162
3.4 Other Types of Dynamic Loads 170
3.5 Definition of Stability 174
3.6 Theorems of Lagrange-Dirichlet and of Liapunov 178
3.7 Stability Criteria for Dynamic Systems 184
3.8 Stability of Continuous Elastic Systems 187
3.9 Nonlinear Oscillations and Chaos 189
4 Energy Methods 199
4.1 Positive-Definite Matrices, Eigenvalues, and Eigenvectors 199
4.2 Potential Energy for Discrete Elastic Systems 207
4.3 Bifurcation Buckling at Small Deflections 216
4.4 Snapthrough and Flat Arches 227
4.5 Large-Deflection Postcritical Behavior and Types of Bifurcation 238
4.6 Koiter's Theory, Imperfection Sensitivity, and Interaction of Modes 261
4.7 Catastrophe Theory and Breakdown of Symmetry 270
1.5 Imperfect Columns and the Southwell Plot 19
1.6 Code Specifications for Beam-Columns 27
1.7 Effect of Shear and Sandwich Beams 30
1.8 Pressurized Pipes and Prestressed Columns 35
1.9 Large Deflections 38
1.10 Spatial Buckling of Beams under Torque and Axial Force 46
2 Buckling of Elastic Frames by Equilibrium Analysis 53
2.1 Stiffness and Flexibility Matrices of Beam-Columns 54
2.2 Critical Loads of Frames and Continuous Beams 61
2.3 Buckling as a Matrix Eigenvalue Problem and Use of Finite Elements 74
2.4 Large Regular Frames 78
2.5 Postcritical Reserve in Redundant Trusses 86
2.6 Postcritical Behavior of Frames 89
2.7 Built-Up Columns and Regular Frames as Columns with Shear 102
2.8 High Arches 108
2.9 Long-Wave Buckling of Regular Frames 118
2.10 Continuum Approximation for Large Regular Frames 129
3 Dynamic Analysis of Stability 144
3.1 Vibration of Columns or Frames and Divergence 144
3.2 Nonconservative Loads and Flutter 151
3.3 Pulsating Loads and Parametric Resonance 162
3.4 Other Types of Dynamic Loads 170
3.5 Definition of Stability 174
3.6 Theorems of Lagrange-Dirichlet and of Liapunov 178
3.7 Stability Criteria for Dynamic Systems 184
3.8 Stability of Continuous Elastic Systems 187
3.9 Nonlinear Oscillations and Chaos 189
4 Energy Methods 199
4.1 Positive-Definite Matrices, Eigenvalues, and Eigenvectors 199
4.2 Potential Energy for Discrete Elastic Systems 207
4.3 Bifurcation Buckling at Small Deflections 216
4.4 Snapthrough and Flat Arches 227
4.5 Large-Deflection Postcritical Behavior and Types of Bifurcation 238
4.6 Koiter's Theory, Imperfection Sensitivity, and Interaction of Modes 261
4.7 Catastrophe Theory and Breakdown of Symmetry 270
4.8 Snapdown at Displacement-Controlled Loading 278
4.9 Incremental Work Criterion at Equilibrium Displacements 294
5 Energy Analysis of Continuous Structures and Approximate Methods 305
5.1 Indirect Variational Method and Euler Equation 306
5.2 Beam on Elastic Foundation 314
10.2 Thermodynamic Criteria of Stable Path 650
10.3 Application to Elastoplastic Columns and Broader Implications 658
10.4 Critical States of Stability and Bifurcation 670
10.5 Stability at Infinitesimal Loading Cycles 680
10.6 Drucker's and Il'yushin's Postulates for Stable Materials 685
5.1 Indirect Variational Method and Euler Equation 306
5.2 Beam on Elastic Foundation 314
5.3 Rayleigh Quotient 323
5.4 Timoshenko Quotient and Relations between Various Bounds 331
5.5 Bound Approximation for Columns, Frames, and High Arches 339
5.6 Rayleigh-Ritz Variational Method 348
5.7 Galerkin Variational Method 356
5.8 Method of Successive Approximations and Lower Bounds 358
5.9 Nonlinear Problems; Large Deflections of Columns 361
6 Thin-Walled Beams 370
6.1 Potential Energy and Differential Equations 371
6.2 Axial-Torsional Buckling of Columms 381
6.3 Lateral Buckling of Beams and Arches 384
6.4 Beams of Arbitrary Open Cross Section 392
6.5 Large Deflections 401
6.6 Box Girders 408
7 Plates and Shells 419
7.1 Classical Plate Theory 420
7.2 Differential Equation and Strain Energy 422
7.3 Buckling of Rectangular Plates 431
7.4 Large Deflections and Postcritical Reserve of Plates 440
7.5 Axisymmetric Buckling of Cylindrical Shells 449
7.6 Shallow or Quasi-Shallow Shells 453
7.7 Nonlinear Analysis of Shell Buckling and Imperfections 466
7.8 Sandwich Plates and Shells 474
II Inelastic, Damage, and Fracture Theories
8 Elastoplastic Buckling 485
8.1 Perfect Columns or Structures and Shanley's Bifurcation 486
8.2 Imperfect Columns and Structures 506
8.3 Effect of Residual Stresses 513
8.4 Metal Columns and Structures: Design and Code Specifications 517
8.5 Concrete Columns and Structures: Design and Code Specifications 532
8.6 Perfectly Plastic Large-Deflection Buckling, Impact, and Blast 556
8.7 Geometric Tensile Instability, Localization, and Necking 569
9 Creep Buckling 584
9.1 Viscoelastic Stress-Strain Relations 585
9.2 Viscoelastic Buckling 590
9.3 Viscoplastic Buckling 597
9.4 Buckling of Aging Viscoelastic Structures 603
9.5 Effect of Creep Deflection on Concrete Column Strength 615
9.6 Nonlinear Creep and Long-Time Strength of Concrete Structures 619
9.7 Creep Buckling at Finite Deflections 623
10 Stability of Inelastic Structures, Bifurcation and Thermodynamic Basis 633
10.1 Thermodynamic Criteria of Stable State 633
5.4 Timoshenko Quotient and Relations between Various Bounds 331
5.5 Bound Approximation for Columns, Frames, and High Arches 339
5.6 Rayleigh-Ritz Variational Method 348
5.7 Galerkin Variational Method 356
5.8 Method of Successive Approximations and Lower Bounds 358
5.9 Nonlinear Problems; Large Deflections of Columns 361
6 Thin-Walled Beams 370
6.1 Potential Energy and Differential Equations 371
6.2 Axial-Torsional Buckling of Columms 381
6.3 Lateral Buckling of Beams and Arches 384
6.4 Beams of Arbitrary Open Cross Section 392
6.5 Large Deflections 401
6.6 Box Girders 408
7 Plates and Shells 419
7.1 Classical Plate Theory 420
7.2 Differential Equation and Strain Energy 422
7.3 Buckling of Rectangular Plates 431
7.4 Large Deflections and Postcritical Reserve of Plates 440
7.5 Axisymmetric Buckling of Cylindrical Shells 449
7.6 Shallow or Quasi-Shallow Shells 453
7.7 Nonlinear Analysis of Shell Buckling and Imperfections 466
7.8 Sandwich Plates and Shells 474
II Inelastic, Damage, and Fracture Theories
8 Elastoplastic Buckling 485
8.1 Perfect Columns or Structures and Shanley's Bifurcation 486
8.2 Imperfect Columns and Structures 506
8.3 Effect of Residual Stresses 513
8.4 Metal Columns and Structures: Design and Code Specifications 517
8.5 Concrete Columns and Structures: Design and Code Specifications 532
8.6 Perfectly Plastic Large-Deflection Buckling, Impact, and Blast 556
8.7 Geometric Tensile Instability, Localization, and Necking 569
9 Creep Buckling 584
9.1 Viscoelastic Stress-Strain Relations 585
9.2 Viscoelastic Buckling 590
9.3 Viscoplastic Buckling 597
9.4 Buckling of Aging Viscoelastic Structures 603
9.5 Effect of Creep Deflection on Concrete Column Strength 615
9.6 Nonlinear Creep and Long-Time Strength of Concrete Structures 619
9.7 Creep Buckling at Finite Deflections 623
10 Stability of Inelastic Structures, Bifurcation and Thermodynamic Basis 633
10.1 Thermodynamic Criteria of Stable State 633
10.2 Thermodynamic Criteria of Stable Path 650
10.3 Application to Elastoplastic Columns and Broader Implications 658
10.4 Critical States of Stability and Bifurcation 670
10.5 Stability at Infinitesimal Loading Cycles 680
10.6 Drucker's and Il'yushin's Postulates for Stable Materials 685
10.7 Stability of Frictional Materials and Structures 693
11 Three-Dimensional Continuum Instabilities and Effects of Finite Strain Tensor 706
11.1 Finite Strain 707
11.2 Stresses, Work, and Equilibrium at Finite Strain 715
11.3 Incremental Equilibrium and Objective Stress Rates 720
11.4 Tangential Moduli at Large Initial Stress 726
11.5 Stable States and Paths for Multidimensional Continuous Bodies 732
11.6 Column or Plate with Shear: Finite-Strain Effect 734
11.7 Surface Buckling and Internal Buckling of Anisotropic Solids 741
11.8 Consistent Geometric Stiffness Matrix of Finite Elements 748
11.9 Buckling of Curved Fibers in Composites 750
12 Fracture as a Stability Problem 760
12.1 Linear Elastic Fracture Mechanics 760
12.2 Nonlinear Fracture Mechanics and Size Effect 772
12.3 Crack Stability Criterion and R-Curve 782
12.4 Snapback Instability of a Crack and Ligament Tearing 788
12.5 Stable States and Stable Paths of Interacting Cracks 798
12.6 Crack Spacing 816
13 Damage and Localization Instabilities 829
13.1 Wave in Strain-Softening Materials 830
13.2 Series-Coupling Model for Localization Due to Softening 845
13.3 Localization of Softening Damage into Planar Bands 857
13.4 Localization of Softening Damage into Ellipsoidal Regions 869
13.5 Localization of Softening Damage into Spherical or Circular Regions 880
13.6 Localization in Beams and Softening Hinges 887
13.7 Friction: Static and Dynamic 902
13.8 Bifurcations Due to Interaction of Softening Damage Zones 907
13.9 Size Effect, Mesh Sensitivity, and Energy Criterion for Crack Bands 914
13.10 Nonlocal Continuum and Its Stability 921
13.11 Constitutive Equations for Strain Softening 937
11 Three-Dimensional Continuum Instabilities and Effects of Finite Strain Tensor 706
11.1 Finite Strain 707
11.2 Stresses, Work, and Equilibrium at Finite Strain 715
11.3 Incremental Equilibrium and Objective Stress Rates 720
11.4 Tangential Moduli at Large Initial Stress 726
11.5 Stable States and Paths for Multidimensional Continuous Bodies 732
11.6 Column or Plate with Shear: Finite-Strain Effect 734
11.7 Surface Buckling and Internal Buckling of Anisotropic Solids 741
11.8 Consistent Geometric Stiffness Matrix of Finite Elements 748
11.9 Buckling of Curved Fibers in Composites 750
12 Fracture as a Stability Problem 760
12.1 Linear Elastic Fracture Mechanics 760
12.2 Nonlinear Fracture Mechanics and Size Effect 772
12.3 Crack Stability Criterion and R-Curve 782
12.4 Snapback Instability of a Crack and Ligament Tearing 788
12.5 Stable States and Stable Paths of Interacting Cracks 798
12.6 Crack Spacing 816
13 Damage and Localization Instabilities 829
13.1 Wave in Strain-Softening Materials 830
13.2 Series-Coupling Model for Localization Due to Softening 845
13.3 Localization of Softening Damage into Planar Bands 857
13.4 Localization of Softening Damage into Ellipsoidal Regions 869
13.5 Localization of Softening Damage into Spherical or Circular Regions 880
13.6 Localization in Beams and Softening Hinges 887
13.7 Friction: Static and Dynamic 902
13.8 Bifurcations Due to Interaction of Softening Damage Zones 907
13.9 Size Effect, Mesh Sensitivity, and Energy Criterion for Crack Bands 914
13.10 Nonlocal Continuum and Its Stability 921
13.11 Constitutive Equations for Strain Softening 937
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