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

Contents......Page 12
Preface......Page 8
Introduction......Page 22
I ELASTIC THEORIES......Page 26
1 Buckling of Elastic Columns by Equilibrium Analysis......Page 28
1.1 Theory of Bending......Page 29
13.9 Size Effect......Page 0
1.3 Differential Equations of Beam-Columns......Page 34
1.4 Critical Loads of Perfect Columns with Various End Restraints......Page 37
1.5 Imperfect Columns and the Southwell Plot......Page 44
Lateral Disturbing Load......Page 45
Initial Curvature or Load Eccentricity......Page 46
Behavior Near the Critical Load......Page 47
Southwell Plot......Page 49
1.6 Code Specifications for Beam-Columns......Page 52
1.7 Effect of Shear and Sandwich Beams......Page 55
Pin-Ended Columns......Page 56
Generalization......Page 58
Sandwich Beams and Panels......Page 59
Pressurized Pipes......Page 60
Prestressed Columns......Page 62
1.9 Large Deflections......Page 63
Solution of Rotations by Elliptic Integrals (Elastica)......Page 64
Deflections and Shortening......Page 66
Discussion of Results......Page 68
1.10 Spatial Buckling of Beams under Torque and Axial Force......Page 71
References and Bibliography1......Page 74
2 Buckling of Elastic Frames by Equilibrium Analysis......Page 78
Stiffness Matrix for End Rotations......Page 79
Stiffness Matrix for End Rotations and Relative Lateral Displacement......Page 83
Simple Structures......Page 86
Difficulties with the Flexibility Method......Page 92
General Approach for Computer Analysis......Page 93
2.3 Buckling as a Matrix Eigenvalue Problem and Use of Finite Elements......Page 99
2.4 Large Regular Frames......Page 103
Example of a Statically Indeterminate Truss......Page 111
Generalization and Limit Analysis Method......Page 112
Order of Approximation......Page 113
L-Frame of Koiter and Roorda......Page 114
Second-Order Solution of the L-Frame......Page 117
Imperfection Sensitivity......Page 120
Generalizations and Implications......Page 121
Order of Approximation......Page 122
Postcritical Reserve Due to Redundancy......Page 123
Finite Element Computational Procedure......Page 124
2.7 Built-Up Columns and Regular Frames as Columns with Shear......Page 127
Curvature Change......Page 133
Approximate Theory for Perfect Arches with a Fixed Compression Line......Page 134
Various Types of Arches and the Effect of Imperfections......Page 136
General Linearized Theory for Uniformly Compressed Circular Arches......Page 138
System of Difference Equations......Page 143
Solution for Tall Building Frames......Page 145
Numerical Results and Their Discussion......Page 150
Transition from Difference to Differential Equations......Page 154
Continuum Approximation Based on Potential Energy......Page 156
Micropolar Continuum and Couple Stresses......Page 158
Stresses and Boundary Conditions......Page 159
Numerical Results and Discussion......Page 162
References and Bibliography......Page 163
3.1 Vibration of Columns or Frames and Divergence......Page 169
Columns......Page 170
Types of Motion and Dependence of Natural Frequency on Load......Page 171
Frames and Other Generalizations......Page 173
Massless Column under Follower Load......Page 176
Effect of Distributed Mass......Page 179
Elastically Supported Rigid Plate under Aerodynamic Forces......Page 181
Conservative and Nonconservative Forces......Page 183
Equations Governing Flutter of Suspension Bridges......Page 185
3.3 Pulsating Loads and Parametric Resonance......Page 187
Axial Pulsating Load on a Column......Page 188
Undamped Vibration......Page 189
Simple Energy Analysis of Parametric Resonance......Page 191
3.4 Other Types of Dynamic Loads......Page 195
3.5 Definition of Stability......Page 199
3.6 Theorems of Lagrange-Dirichlet and of Liapunov......Page 203
3.7 Stability Criteria for Dynamic Systems......Page 209
3.8 Stability of Continuous Elastic Systems......Page 212
3.9 Nonlinear Oscillations and Chaos......Page 214
References and Bibliography......Page 219
4 Energy Methods......Page 224
Structure-Load System......Page 232
Second Variation of Potential Energy......Page 233
Critical State......Page 234
An Example......Page 235
Difficulties with Complementary Energy......Page 236
Overturning Instability of a Block: Discontinuous  '......Page 239
Calculation of Potential Energy of Beam-Columns......Page 241
Equilibrium and Stability......Page 242
Role of Axial Strain and Shortening Due to Deflections......Page 243
Calculation of n from the Work of Initial and Incremental Stresses......Page 245
Example with Two Degrees of Freedom......Page 246
Some Fundamental Aspects of Potential Energy......Page 249
4.4 Snapthrough and Flat Arches......Page 252
Von Mises Truss......Page 253
Flat Arches......Page 256
Effect of Imperfections......Page 258
Other Examples of Snapthrough......Page 260
Symmetric Stable Bifurcation: Example......Page 263
Symmetric Unstable Bifurcation: Example......Page 266
Asymmetric Bifurcation: Example......Page 269
L-Shaped Rigid-Bar Frame......Page 272
Rigid-Bar Arch......Page 274
Nonlinear Springs and Polynomial Approximation of Potential Energy......Page 275
Two Degrees of Freedom: Example......Page 277
Limit Points of Equilibrium Paths......Page 279
Bifurcation Criterion in Terms of the Tangential Stiffness Matrix......Page 280
Classification of Elementary Instabilities of Elastic Structures......Page 281
General Validity of Koiter's ~-Power and i-Power Laws......Page 288
Interaction of Buckling Modes......Page 289
4.7 Catastrophe Theory and Breakdown of Symmetry......Page 295
Structures with Two Degrees of Freedom......Page 303
Softening Specimen in a Testing Machine......Page 309
Generalization of Snapdown Analysis......Page 310
4.9 Incremental Work Criterion at Equilibrium Displacements......Page 319
Stability Criterion......Page 320
References and Bibliography......Page 325
5 Energy Analysis of Continuous Structures and Approximate Methods......Page 330
Review of the Calculus of Variations......Page 331
Application to Structures Possessing a Potential Energy......Page 334
Review of Positive-Definite and Self-Adjoint Operators......Page 335
Potential Energy and Differential Equations......Page 339
Solution for Different Boundary Conditions......Page 341
Fiber on Elastic Foundation......Page 344
Upper-Bound Property of Rayleigh Quotient......Page 348
Application to Beam-Columns......Page 349
Relation to Differential Equation......Page 351
Proof of Upper-Bound Property and Convergence......Page 352
Extension to Free Vibration......Page 354
Derivation......Page 356
Examples......Page 357
Relation to Differential Equation and Proof of Upper-Bound Property......Page 358
Relation to Rayleigh Quotient and Inequalities......Page 359
Inapplicability to Dynamics......Page 362
The Question of Lower Bounds......Page 363
Columns......Page 364
Frames......Page 366
Elastically Supported Beams......Page 368
High Arches......Page 369
5.6 Rayleigh-Ritz Variational Method......Page 373
5.7 Galerkin Variational Method......Page 381
Formulation of the Method......Page 383
Example......Page 384
Lower Bound......Page 385
5.9 Nonlinear Problems; Large Deflections of Columns......Page 386
References and Bibliography......Page 391
6 Thin-Walled Beams......Page 395
6.1 Potential Energy and Differential Equations......Page 396
Deformation of the Cross Section......Page 397
Potential Energy......Page 399
Differential Equations and Boundary Conditions......Page 404
6.2 Axial-Torsional Buckling of Columms......Page 406
6.3 Lateral Buckling of Beams and Arches......Page 409
Axial-Torsional Buckling due to Eccentric Axial Force......Page 410
Lateral Buckling due to Bending Moment......Page 412
Approximate Solution for Variable M......Page 413
Bimoment......Page 415
Lateral Buckling of Arches......Page 416
General Theory of Warping Torsion......Page 417
Stresses and Bimoment in General Theory......Page 421
Potential Energy and Differential Equations......Page 422
Monosymmetric Cross Section......Page 425
6.5 Large Deflections......Page 426
Elements of matrix K......Page 428
Element of matrix......Page 429
Elements of matrix......Page 430
6.6 Box Girders......Page 433
Deformation Modes and Postcritical Energy......Page 434
Examples......Page 436
Finite Element Solution......Page 437
Interaction with Local Buckling......Page 438
References and Bibliography......Page 439
7 Plates and Shells......Page 444
7.1 Classical Plate Theory......Page 445
7.2 Differential Equation and Strain Energy......Page 447
Strains......Page 448
Potential Energy......Page 449
Differential Equations of Equilibrium......Page 450
Boundary Conditions......Page 452
Direct Derivation of Transverse Resultant of In-Plane Forces......Page 454
Discussion and Summary......Page 455
7.3 Buckling of Rectangular Plates......Page 456
Buckling of Simply Supported Plates......Page 457
Rectangular Plate with Arbitrary Boundary Conditions......Page 459
Buckling of Plate Subjected to Shear......Page 460
Nonuniform In-Plane Forces......Page 461
Solutions by Other Variational Methods......Page 462
7.4 Large Deflections and Postcritical Reserve of Plates......Page 465
Von Karman-Foppl Differential Equations......Page 466
Solution by Minimization of Potential Energy......Page 467
Large Deflections and Ultimate Strength......Page 471
Measurement of Critical Loads......Page 473
7.5 Axisymmetric Buckling of Cylindrical Shells......Page 474
7.6 Shallow or Quasi-Shallow Shells......Page 478
Basic Relations for Cylindrical Shells......Page 479
Donnell's Equation......Page 481
Axially Compressed Cylindrical Shell......Page 482
Effect of Lateral Pressure on Cylindrical Shells......Page 484
Cylindrical Shell Subjected to Torsion......Page 487
Variational Derivation from Potential Energy......Page 488
General Quasi-Shallow Shells......Page 489
7.7 Nonlinear Analysis of Shell Buckling and Imperfections......Page 491
Reduction Factors for Classical Critical Loads......Page 492
Physical Source of Postcritical Load Drop and High Imperfection Sensitivity......Page 493
Koiter's Laws of Imperfection Sensitivity......Page 495
Buckling Modes and Their Interaction......Page 496
Basic Relations for a Sandwich Plate Element......Page 499
A Redangular Sandwich Plate and Other Problems......Page 500
References and Bibliography......Page 502
II 
INELASTIC, DAMAGE, AND
FRACTURE THEORIES......Page 510
8 Elastoplastic Buckling......Page 512
8.1 Perfect Columns or Structures and Shanley's Bifurcation......Page 513
Reduced Modulus Load......Page 515
Tangent Modulus Load......Page 518
Column Strength Curve......Page 520
Postbifurcation Load-Deflection Diagram......Page 521
Bifurcation in Plastic Structures with Multiaxial Stress......Page 527
Conclusion......Page 529
Shanley's Rigid-Bar Column: Exact Solution......Page 533
Arbitrary Imperfect Columns: Approximate Solution......Page 535
Effect of Cross-Section Nonsymmetry......Page 538
8.3 Effect of Residual Stresses......Page 540
Calculation of the Effect of Residual Stresses......Page 541
Examples......Page 542
8.4 Metal Columns and Structures: Design and Code Specifications......Page 544
Centrically Loaded Columns......Page 545
Load and Resistance Factor Design and Probabilistic Aspects......Page 549
Beam-columns......Page 551
Design Examples......Page 555
8.5 Concrete Columns and Structures: Design and Code Specifications......Page 559
Interaction Diagram (Failure Envelope)......Page 560
Deflections and Interaction Diagram......Page 563
Numerical Algorithm for Calculating Deflections and Interaction Diagram......Page 565
Column Response for Unsmooth Stress-Strain Diagrams......Page 568
Design Recommendations and the ACI Code......Page 569
CEB Design Recommendations......Page 572
Comparisons of Codes and Shortcomings......Page 576
Prestressed Concrete Columns......Page 578
Shells and Other Structures......Page 579
Design Examples......Page 580
Load-Deflection Curve of Perfectly Plastic Columns......Page 584
Buckling of Perfectly Plastic Frames......Page 587
Plastic Redistribution and Reserve Capacity of Structures......Page 589
Perfectly Plastic Buckling of Thick Plates......Page 591
Transverse Impact or Blast on Plates or Columns with In-Plane or Axial Loads......Page 592
Role of Transverse Contraction and Finite Strain......Page 596
Strain Localization......Page 599
Necking......Page 600
References and Bibliography......Page 602
9 Creep Buckling......Page 611
Compliance Function and Integral-Type Creep Law......Page 612
Differential-Type Creep Law and Rheologic Models......Page 613
Elastic-Viscoelastic Analogy......Page 616
9.2 Viscoelastic Buckling......Page 617
Deflection History and Long-Time Critical Load......Page 619
The Concept of Stability for Viscoelastic Structures......Page 620
Extensions and Ramifications......Page 622
9.3 Viscoplastic Buckling......Page 624
Rigid-Bar Model Column......Page 625
Critical Time and Stability Concept......Page 626
Real Columns......Page 627
Aging Maxwell Solid (Dischinger-Type Methods)......Page 630
Deflections According to Aging Maxwell Model......Page 632
Deflection According to Effective Modulus......Page 635
Deflection According to Age-Adjusted Effective Modulus......Page 636
Deflection According to Integral-Type Stress-Strain Relation......Page 639
Appendix I-Compliance Function and Relaxation Function of Concrete......Page 640
Appendix II-Proof of Age-Adjusted Effective Modulus Method......Page 641
9.5 Effect of Creep Deflection on Concrete Column Strength......Page 642
9.6 Nonlinear Creep and Long-Time Strength of Concrete Structures......Page 646
Example of Imperfection-Sensitive Rigid-Bar Column......Page 650
Broader Implications and Ramifications......Page 652
References and Bibliography......Page 654
10.1 Thermodynamic Criteria of Stable State......Page 660
First and Second Laws of Thermodynamics......Page 661
Tangentially Equivalent Elastic Structure......Page 662
Total Energy U and Helmholtz Free Energy F......Page 663
Second Variation of   or......Page 665
Path Dependence and Incremental Potentials......Page 668
Second-Order Work of Stresses and Geometric Stiffness......Page 669
Criterion of Stable State for the Case of Dead Loads......Page 670
Extensions to Variable Loads......Page 671
Gibbs Free Energy and Enthalpy......Page 672
Stability Criteria Based on Complementary Work......Page 673
Structures with a Single Load or a Single Controlled Displacement......Page 674
Summary......Page 676
10.2 Thermodynamic Criteria of Stable Path......Page 677
Path Stability for Basic Types of Control......Page 678
Mixed Controls of Loads and Displacements......Page 680
Second-Order Work of Stresses along the Path......Page 682
Stable States on Postbifurcation Branches......Page 683
10.3 Application to Elastoplastic Columns and Broader Implications......Page 685
Loading-Unloading Combinations and Equilibrium Paths......Page 686
Stable Equilibrium States of Elastoplastic Column......Page 689
Second-Order Work......Page 687
Stable Equilibrium Path of an Elastoplastic Column......Page 691
Breakdown of Symmetry......Page 693
Hypothesis Implied in Present Thermodynamic Approach......Page 694
Summary......Page 695
Critical State for Structures with a Symmetric Stiffness Matrix......Page 697
Critical States for Structures with a Nonsymmetric Stiffness Matrix......Page 698
Example of a Nonsymmetric Stiffness Matrix......Page 699
Uniqueness......Page 700
Bifurcation for Inelastic Structures and Hill's Linear Comparison Solid......Page 701
Distribution of Bifurcation Points and Postbifurcation Branches......Page 703
Numerical Finite Element Analysis......Page 704
Summary......Page 705
Internal Entropy Changes for Cycles in Shanley's Column......Page 707
Stability......Page 709
Incremental Collapse......Page 710
Drucker's Postulate......Page 712
lI'yushin's Postulate......Page 713
Nonuniformly Strained Bodies......Page 714
Normality Rule for Plasticity......Page 716
10.7 Stability of Frictional Materials and Structures......Page 720
Frictional Block Preloaded by a Spring......Page 721
Generalization to Frictional Continuum......Page 722
Stability Condition of Frictional Materials......Page 724
Plastic Strain Increment for Frictional Materials......Page 725
Inverse Material Friction......Page 726
References and Bibliography......Page 727
11 Three-Dimensional Continuum Instabilities and Effects of Finite Strain Tensor......Page 733
11.1 Finite Strain......Page 734
Notations and Basic Requirements......Page 735
Lagrangian (Green's) Finite Strain Tensor......Page 736
Biot's Finite Strain Tensor......Page 737
Further Measures of Finite Strain......Page 738
The Special Case of Thin Bodies......Page 740
Decomposition of Strain into Elastic and Inelastic Parts......Page 741
Virtual Work Relations and Equilibrium......Page 742
True (Cauchy) Stress......Page 743
Stress Referred to Initial Configuration and Working on Displacement Gradient......Page 744
Stress Referred to Initial Configuration and Working on Finite Strain......Page 745
Incremental Equilibrium Conditions......Page 747
Objective Stress Increments Conjugate to Strain Increments......Page 748
Objective Stress Rates......Page 750
11.4 Tangential Moduli at Large Initial Stress......Page 753
11.5 Stable States and Paths for Multidimensional Continuous Bodies......Page 759
Differential Equations......Page 761
Theories of Engesser and Haringx......Page 763
Correlation to Built-Up Columns......Page 765
Summary......Page 767
11.7 Surface Buckling and Internal Buckling of Anisotropic Solids......Page 768
Surface Buckling of an Orthotropic Half-Space......Page 769
Internal Buckling and Other Instabilities......Page 771
General Solution......Page 773
11.8 Consistent Geometric Stiffness Matrix of Finite Elements......Page 775
11.9 Buckling of Curved Fibers in Composites......Page 777
Decrease of Elastic Moduli Due to Fiber Undulation......Page 778
Generalization to Three Dimensions......Page 780
Stresses Due to Fiber Undulation......Page 781
References and Bibliography......Page 782
Stress Singularity and Fracture Energy......Page 787
Energy Release Rate......Page 790
Determination of  and Gf from Compliance Changes......Page 792
Some Simple Elastic Solutions 1......Page 793
Approximation by Stress Relief Zone......Page 795
Examples Solvable by Bending Theory......Page 796
Herrmann's Method to Obtain Approximate K1 by Beam Theory......Page 797
12.2 Nonlinear Fracture Mechanics and Size Effect......Page 799
Inelastic Zone and Equivalent Elastic Crack......Page 800
Fracture Models with a Nonlinear Zone......Page 802
Size Effect......Page 803
R Curve and Fracture Equilibrium Condition......Page 809
Fracture Stability Criterion and Critical State......Page 811
Determination of Geometry-Dependent R Curve from Size Effect Law......Page 812
Crack Propagation Direction......Page 814
12.4 Snapback Instability of a Crack and Ligament Tearing......Page 815
General Procedure for Load-Displacement Relation at Growing Crack......Page 816
Snapback Instability at Crack Coalescence in Two Dimensions......Page 818
Snapback Instability at Tearing of Circular Ligament......Page 820
General Condition for Snapback at Ligament Tearing......Page 821
Alternative Calculation of Displacement from Compliance Variation......Page 824
Conditions of Equilibrium and Stability in Terms of Crack Length......Page 825
Stability of Parallel Cooling or Shrinkage Cracks......Page 828
Stable Path and Bifurcation at Advancing Cooling Front......Page 831
Three-Dimensional Pattern of Cooling or Shrinkage Cracks......Page 833
Stability of Parallel Cracks in Reinforced Concrete......Page 834
Stability Analysis in Terms of Displacements......Page 836
12.6 Crack Spacing......Page 843
Spacing of Parallel Initial Drying or Cooling Cracks: Energy Balance......Page 844
Snapthrough Formation of Cracks According to LEFM......Page 845
Crack Spacing in Loaded Reinforced Concrete Beams......Page 846
Snapthrough Crack Formation in a Drying Tube......Page 848
References and Bibliography......Page 849
13 Damage and Localization Instabilities......Page 856
13.1 Wave in Strain-Softening Materials......Page 857
Exact Solution of Strain-Softening Bar......Page 858
Stability Aspects and Unrealistic Properties of Solution of a Bar......Page 862
Bar with Rehardening or Residual Yield Strength......Page 863
Cylindrically or Spherically Converging Waves......Page 866
General Three-Dimensional Condition for Waves to Exist......Page 867
Summary......Page 871
Stable States......Page 872
Surface of Second-Order Work......Page 874
Application to Uniaxially Stressed Bars or Specimens......Page 875
Effects of Size and Support Stiffness......Page 876
Specimen Ductility as a Function of Length and Loading Frame Stiffness......Page 877
Bifurcation and Stable Path......Page 878
Identification of Softening Stress-Strain Relations from Tests......Page 881
Relation of Strain Softening to Fracture Energy......Page 882
Summary......Page 883
13.3 Localization of Softening Damage into Planar Bands......Page 884
Stability Condition for the Softening Band within a Layer or Infinite Solid......Page 885
Discussion of Various Cases......Page 887
Numerical Examples......Page 889
Generalization for Geometrically Nonlinear Effects......Page 890
Bifurcation and Stable Path......Page 893
Sand Liquefaction as a Localization Instability......Page 894
Summary......Page 895
Eshelby's Theorem......Page 896
Stability of Uniform Strain against Ellipsoidal Localization......Page 898
Numerical Examples of Stability Limits and Discussion......Page 900
Bifurcation and Stable Path of Ellipsoidal Localization......Page 904
Simpler Derivation of Bifurcation Condition......Page 906
Localization Instability for Spherical Geometry......Page 907
Localization Instability for Circular or Cylindrical Geometry......Page 910
Numerical Examples......Page 912
Bifurcation and Stable Path......Page 913
13.6 Localization in Beams and Softening Hinges......Page 914
Stability Limit and Snapback......Page 915
Rotation Capacity or Ductility of Hinges in Concrete Beams......Page 919
Bifurcation Due to Interaction of Softening Hinges......Page 920
Bifurcation and Localization in aedundant Structures......Page 922
Bifurcation at Simultaneous Formation of Several Softening Hinges......Page 925
Softening Frames and Trusses......Page 927
Summary......Page 928
13.7 Friction: Static and Dynamic......Page 929
Paradox in Sudden Friction Drop......Page 930
Frictional Supports in Columns......Page 932
Structures with Stiffness Matrix Asymmetry......Page 933
Interaction of Damage (Cracking) Fronts and Stable Paths......Page 934
Multiple Interacting Crack Band Fronts......Page 936
Interaction of Multiple Shear Bands......Page 937
Example: Buckling in Direct Tensile Test......Page 939
Localization as a Cause of Size Effect......Page 941
In objectivity or Spurious Mesh Sensitivity......Page 943
Energy Criterion for Crack Band and Stability......Page 945
Crack Band Model......Page 948
Nonlocal Continuum Concept......Page 950
Periodic Instabilities Due to Nonlocal Concept......Page 952
Nonlocal Continuum with Local Strain......Page 954
One-Dimensional Localization Instability......Page 956
Measurement of Characteristic Length of Nonlocal Continuum......Page 960
Example: Stability of a Tunnel......Page 961
Gradient Approximation to Nonlocal Continuum......Page 962
13.11 Constitutive Equations for Strain Softening......Page 964
References and Bibliography......Page 966
Glossary of Symbols......Page 980
Other Symbols......Page 989
Author Index......Page 990
Subject Index......Page 1004
Appendix to the WSP Edition1......Page 1012