Structural Stability Theory and Practice

Foreword xvii Preface xix About the Companion Website xxiii 1 Structural Stability 1 1.1 Introduction 1 1.2 General Concepts 2 1.2.1 Bifurcation of Equilibrium 3 1.2.2 Limit Load Instability 4 1.2.3 Finite Disturbance Instability 4 1.3 Rigid Bar Columns 5 1.3.1 Rigid Bar Supported by a Translational Spring 5 1.3.1.1 The Displaced Shape Equilibrium Method 5 1.3.1.2 The Energy Method 6 1.3.2 Two Rigid Bars Connected by Rotational Springs 8 1.3.2.1 The Displaced Shape Equilibrium Method 8 1.3.2.2 The Energy Method 9 1.3.3 Three-Member Truss 11 1.3.3.1 The Energy Method 11 1.3.4 Three Rigid Bars with Two Linear Springs 13 1.3.4.1 The Displaced Shape Equilibrium Method 13 1.3.4.2 The Energy Method 15 1.4 Large Displacement Analysis 17 1.4.1 Rigid Bar Supported by a Translational Spring 17 1.4.1.1 The Displaced Shape Equilibrium Method 17 1.4.1.2 The Energy Method 17 1.4.2 Rigid Bar Supported by Translational and Rotational Springs 19 1.4.2.1 The Displaced Shape Equilibrium Method 19 1.4.2.2 The Energy Method 20 1.4.3 Two Rigid Bars Connected by Rotational Springs 22 1.4.3.1 The Energy Method 22 1.5 Imperfections 23 1.5.1 Rigid Bar Supported by a Rotational Spring at the Base 23 1.5.1.1 The Displaced Shape Equilibrium Method 23 1.5.1.2 The Energy Method 24 1.5.2 Two Rigid Bars Connected by Rotational Springs 26 1.5.2.1 The Displaced Shape Equilibrium Method 26 1.5.2.2 The Energy Method 26 Problems 28 References 29 2 Columns 31 2.1 General 31 2.2 The Critical Load According to Classical Column Theory 31 2.2.1 Pinned-Pinned Column 32 2.2.2 Fixed-Fixed Column 35 2.2.2.1 Symmetric Mode 35 2.2.2.2 Anti-Symmetric Mode 37 2.2.3 Cantilever Column 39 2.2.4 Fixed-Pinned Column 40 2.3 Effective Length of a Column 42 2.4 Special Cases 43 2.4.1 Pinned-Pinned Column with Intermediate Compressive Force 43 2.4.2 Cantilever Column with Intermediate Compressive Force 46 2.4.2.1 Case 1 49 2.4.2.2 Case 2 49 2.5 Higher-Order Governing Differential Equation 50 2.5.1 Boundary Conditions for Different Supports 52 2.5.1.1 Pinned Support 52 2.5.1.2 Fixed Support 52 2.5.1.3 Free End 52 2.5.1.4 Guided Support 52 2.5.2 Pinned-Pinned Column 52 2.5.3 Cantilever Column 54 2.5.4 Pinned-Guided Column 55 2.6 Continuous Columns 56 2.7 Columns on Elastic Supports 59 2.7.1 Column Pinned at One End and Elastic Support at the Other End 59 2.7.2 Column Fixed at One End and Elastic Support at the Other End 62 2.8 Eccentrically Loaded Columns 64 2.8.1 The Secant Formula 67 2.9 Geometrically Imperfect Columns 68 2.9.1 The Southwell Plot 71 2.10 Large Deflection Theory of Columns 73 2.10.1 Pinned-Pinned Column 73 2.10.2 Cantilever Column 79 2.10.3 Effective Length Approach 82 2.10.3.1 Pinned-Pinned Column 82 2.10.3.2 Cantilever Column 83 2.10.3.3 Fixed-Fixed Column 83 2.10.3.4 Fixed-Pinned Column 83 2.11 Energy Methods 84 2.11.1 Calculus of Variations 84 2.11.2 The Rayleigh-Ritz Method 87 2.11.3 The Galerkin Method 88 Problems 90 References 93 3 Inelastic and Metal Columns 95 3.1 Introduction 95 3.2 Double Modulus Theory 96 3.2.1 Rectangular Section 100 3.3 Tangent Modulus Theory 101 3.4 Shanley’s Theory for Inelastic Columns 103 3.5 Columns with Other End Conditions 108 3.6 Eccentrically Loaded Inelastic Columns 108 3.7 Aluminum Columns 116 3.7.1 North American and Australian Design Practice 117 3.8 Steel Columns 122 3.8.1 Buckling of Idealized Steel I-Section 123 3.8.1.1 Strong Axis Bending 125 3.8.1.2 Weak Axis Bending 126 3.8.2 Column Strength Curves for Steel Columns 126 3.8.3 Column Research Council Curve 128 3.8.4 Structural Stability Research Council Curves 129 3.8.5 European Multiple Column Curves 131 3.8.6 AISC Design Criteria for Steel Columns 132 Problems 135 References 136 4 Beamcolumns139 4.1 Introduction 139 4.2 Basic Differential Equations of Beam Columns 139 4.3 Beam Column with a Lateral Concentrated Load 141 4.3.1 Concentrated Lateral Load at the Mid-span 144 4.3.2 Beam Columns with Several Concentrated Loads 146 4.3.3 Beam Column with Lateral Uniformly Distributed Load 148 4.3.4 Beam Columns with Uniformly Distributed Load Over a Portion of Their Span 152 4.3.5 Beam Columns with Uniformly Increasing Load Over a Portion of Their Span 153 4.4 Beam Columns Subjected to Moments 153 4.4.1 Span Moment on Beam Column 154 4.4.2 End Moment on a Beam Column 156 4.4.3 Moments at Both Ends of Beam Column 159 4.4.3.1 Two Equal Moments 161 4.4.3.2 Moments at Both Ends of the Beam Column: Alternate Method 162 4.4.3.3 End Moments of the Same Sign Giving Double Curvature 163 4.5 Columns with Elastic Restraints 165 4.6 Beam Columns with Different End Conditions and Loads 167 4.6.1 Pinned-fixed Beam Columns with a Concentrated Load 167 4.6.2 Pinned-fixed Beam Columns Subjected to Uniformly Distributed Load 169 4.6.3 Fixed-fixed Beam Column with Concentrated Force 171 4.6.4 Fixed-fixed Beam Column with Uniformly Distributed Load 172 4.7 Alternate Method Using Basic Differential Equations 174 4.7.1 Fixed-fixed Beam Column with Uniformly Distributed Load 174 4.7.2 Pinned-fixed Beam Column with Uniformly Distributed Load 176 4.8 Continuous Beam Columns 177 4.9 Slope Deflection Equations for Beam Columns 182 4.9.1 Matrix Inversion 183 4.9.2 Beam Columns Subjected to Rotations and Relative Displacement at the Ends 184 4.9.3 Beam Columns Having One End Hinged 185 4.9.4 Beam Columns with Transverse Loading 186 4.9.5 Beam Columns in Single Curvature 188 4.10 Inelastic Beam Columns 188 4.10.1 Case 1: Yielding on the Compression Side Only 190 4.10.2 Case 2: Yielding on Both the Compression and Tension Sides 194 4.11 Design of Beam Columns 202 4.11.1 Concept of Equivalent Moment and Factor Cm 204 4.11.2 AISC Design Criteria for Steel Beam Columns 206 4.11.2.1 Doubly and Singly Symmetric Members Subjected to Flexure and Compression 206 4.11.2.2 Unsymmetric and Other Members Subject to Flexure and Axial Force 207 4.11.3 Eurocode 3 (ECS, 1993) Design Criteria 210 4.11.4 Canadian Standards Association (CSA 1994 – CSA-S16.1) 210 4.11.5 Australian Standard AS4100-1990 211 Problems 215 References 218 5 Frames 221 5.1 Introduction 221 5.2 Critical Loads by the Equilibrium Method 221 5.2.1 Portal Frame Without Sidesway 221 5.2.1.1 Portal Frame Without Sidesway with Rigid or Extremely Flexible Beam 225 5.2.2 Portal Frame with Sidesway 225 5.2.2.1 Portal Frame Having Sidesway with a Rigid or Extremely Flexible Beam 229 5.2.3 Frame with Prime Bending and Without Sidesway 230 5.3 Critical Loads by Slope Deflection Equations 232 5.3.1 Portal Frame Without Sidesway 232 5.3.2 Portal Frame with Sidesway 234 5.3.3 Two-Story Frame Without Sidesway 237 5.3.4 Two-Bay Frame Without Sidesway 239 5.3.5 Frames with Prime Bending and Without Sidesway 242 5.3.5.1 Frame with Hinged Supports 242 5.3.5.2 Frame with Fixed Supports 245 5.3.6 Frames with Prime Bending and Sidesway 247 5.3.7 Box Frame Without Sidesway 251 5.3.8 Multistory-Multibay Frames Without Sidesway 253 5.4 Critical Loads by Matrix and Finite Element Methods 257 5.4.1 Formation of the Element Stiffness Matrix 258 5.4.2 Formation of the Structure Stiffness Matrix 262 5.4.3 In Span Loading 264 5.4.4 Buckling of a Frame Pinned at the Base and with Sidesway Permitted 266 5.4.5 Nonlinear Geometric or Large Deflection Analysis (Second-Order Elastic Analysis) 273 5.5 Design of Frame Members 278 5.5.1 Braced Frames (Sidesway Inhibited) 279 5.5.2 Unbraced Frames (Sidesway Not Inhibited) 284 5.5.3 Inelastic Buckling of Frames 290 Problems 310 References 311 6 Torsional Buckling and Lateral Buckling of Beams 313 6.1 Introduction 313 6.2 Pure Torsion of Thin-Walled Cross-Sections 313 6.3 Non-uniform Torsion of Thin-Walled Open Cross-Sections 315 6.3.1 I-section 315 6.3.2 General Thin-Walled Open Cross-Sections 316 6.3.3 Warping Constant Cw of a Channel Section 320 6.4 Torsional Buckling of Columns 322 6.5 Torsional Buckling Load 326 6.5.1 Thin-Walled Open Sections with Rectangular Elements Intersecting at a Point 326 6.5.2 Thin-Walled Open Doubly Symmetric Sections 327 6.5.2.1 Pinned-pinned Columns 327 6.5.2.2 Fixed-fixed Columns 329 6.6 Torsional Flexural Buckling 332 6.6.1 Pinned-pinned Columns 335 6.6.2 Fixed-fixed Columns 337 6.6.3 Singly Symmetric Sections 338 6.6.3.1 Pinned-pinned Columns 339 6.6.3.2 Fixed-fixed Columns 342 6.7 Torsional Flexural Buckling: The Energy Approach 345 6.7.1 Strain Energy of Torsional Flexural Buckling 345 6.7.2 Potential Energy of External Loads in Torsional Flexural Buckling 348 6.8 Lateral Buckling of Beams 352 6.8.1 Lateral Buckling of Simply Supported, Narrow Rectangular Beams in Pure Bending 352 6.8.2 Lateral Buckling of Simply Supported I Beams in Pure Bending 356 6.8.3 Lateral Buckling of Simply Supported I Beams: Concentrated Load at the Mid-Span 359 6.8.4 Lateral Buckling of Cantilever I Beams: Concentrated Load at the Free End 364 6.8.4.1 Lateral Buckling of Cantilever Narrow Rectangular Beams: Concentrated Load at the Free End 367 6.8.5 Lateral Buckling of Narrow Rectangular Cantilever Beams Acted on by Uniform Moment 367 6.9 The Energy Method 369 6.9.1 Lateral Buckling of Simply Supported I Beams: Concentrated Load at the Mid-Span 369 6.9.1.1 Lateral Buckling of Simply Supported, Narrow Rectangular Beams: Concentrated Load at the Mid-Span 372 6.9.2 Lateral Buckling of Simply Supported I Beams: Uniformly Distributed Load 373 6.9.2.1 Lateral Buckling of Simply Supported, Narrow Rectangular Beams: Uniformly Distributed Load along the Centroidal Axis 378 6.9.3 Lateral Buckling of Cantilever Rectangular Beams: Concentrated Load at the Free End 378 6.10 Beams with Different Support and Loading Conditions 380 6.10.1 Different Support Conditions 381 6.10.2 Different Loading Conditions 382 6.10.2.1 Beams with Unequal Moments 383 6.11 Design for Torsional and Lateral Buckling 383 6.11.1 AISC Design Criteria for Steel Beams 383 6.11.1.1 Local Buckling 384 6.11.1.2 Lateral Torsional Buckling 385 Problems 395 References 397 7 Buckling of Plates 399 7.1 Introduction 399 7.2 Theory of Plate Bending 399 7.3 Buckling of Thin Plates 405 7.3.1 In-plane Forces 405 7.4 Boundary Conditions 408 7.4.1 Simply Supported Edge 408 7.4.2 Built-in Edge 408 7.4.3 Free Edge 408 7.4.4 Elastically Supported and Elastically Built-in Edge 409 7.5 Buckling of Rectangular Plates Uniformly Compressed in One Direction 410 7.5.1 Buckling of Rectangular Plates with Simply Supported Edges 410 7.5.2 Buckling of Rectangular Plates with Other Boundary Conditions 415 7.5.3 Loading Edges Simply Supported, the Side y = 0 Is Clamped, and the Side y = b Is Free 416 7.5.4 Loading Edges Simply Supported and the Sides y = 0 and y = b Are Clamped 421 7.5.5 Loading Edges Simply Supported, the Side y = 0 Is Simply Supported, and the Side y = b Is Free 423 7.5.6 Loading Edges Simply Supported, the Side y = 0 Is Elastically Built-in and the Side y = b Is Free 427 7.5.7 Loading Edges Simply Supported, the Sides y= ±b/2 Are Elastically Built-in 431 7.5.7.1 Loading Edges Simply Supported, the Sides y= ±b/2 Are Elastically Restrained by Rotational Springs 434 7.5.7.2 Loading Edges Simply Supported, the Sides y = 0 and y = b Are Elastically Built with Different Flange Sizes 435 7.5.8 Loading Edges Simply Supported, the Sides y=0 and y = b Are Supported by Elastic Beams 438 7.6 The Energy Method 442 7.6.1 Strain Energy Due to Bending in Plates 442 7.6.2 Potential Energy of the External Forces in Plates 443 7.6.2.1 Potential Energy Due to Nx and Ny 443 7.6.2.2 Potential Energy Due to Nxy 444 7.6.3 Rectangular Plate Subjected to Uniaxial Compressive Force and Fixed on All Edges 445 7.6.4 A Rectangular Plate with Clamped Edges under Compressive Pressure in Two Perpendicular Directions 447 7.6.5 Buckling of Simply Supported Rectangular Plates Under the Action of Shear Forces 449 7.6.6 Buckling of Simply Supported Rectangular Plates Under Combined Bending and Compression 453 7.6.7 Buckling of Plates with Stiffeners 458 7.6.8 Simply Supported Rectangular Plates with Longitudinal Stiffeners 459 7.6.8.1 Plates with Two Longitudinal Stiffeners Dividing the Width of the Plate 463 7.6.9 Simply Supported Rectangular Compressed Plate with Transverse Stiffeners 464 7.6.10 Simply Supported Rectangular Plate with Stiffeners in Both the Longitudinal and Transverse Directions 468 7.7 Buckling of Circular Plates 470 7.7.1 Clamped Plate 472 7.7.2 Simply Supported Plate 473 7.8 The Finite Difference Method 474 7.8.1 Critical Load for a Simply Supported Plate Subjected to Biaxial Loading 477 7.9 The Finite Element Method 481 7.10 Large Deflection Theory of Plates 487 7.10.1 Post-buckling Behavior of Plates 491 7.11 Inelastic Buckling of Plates 495 7.11.1 Rectangular Plates with Simply Supported Edges 497 7.11.2 Plate with Loading Edges Simply Supported and the Sides y = 0 and y = b Are Clamped 499 7.12 Ultimate Strength of Plates in Compression 500 7.13 Local Buckling of Compression Elements: Design 504 Problems 506 References 508 8 Buckling of Shells 511 8.1 Introduction 511 8.2 The Large Deflection Theory of Cylindrical Shells 512 8.3 The Linear Theory of Cylindrical Shells 517 8.3.1 Linear Membrane Equations for Cylindrical Shells 518 8.4 Donnell’s Linear Equations of Stability of Cylindrical Shells 519 8.5 The Energy Method 524 8.6 Application of the Linear Stability Equations 526 8.6.1 Circular Cylinders Under Axial Compression 526 8.6.2 Circular Cylinders Under Uniform Lateral Pressure 531 8.6.2.1 Critical Pressures for Cylinders Subjected to External Pressure 533 8.6.3 Cylinders Subjected to Torsion 534 8.6.4 Cylinders Subjected to Combined Axial Compression and Uniform External Lateral Pressure 537 8.6.5 Cylindrical Shells with Different End Conditions 539 8.7 Failure and Post-buckling Behavior of Cylindrical Shells 540 8.7.1 Post-Buckling Behavior of Cylindrical Shells 541 8.7.2 Post-buckling Behavior of Cylindrical Panels 543 8.8 General Shells 548 8.8.1 Nonlinear Equations of Equilibrium 548 8.8.2 Linear Equations of Stability (the Donnell Type) 553 8.9 Shells of Revolution 558 8.9.1 Stability Equations Where Pre-buckling Rotations Are Retained 559 8.9.2 Stability Equations with Pre-buckling Rotations Neglected 561 8.9.3 Circular Flat Plates 563 8.9.3.1 Clamped Plate 566 8.9.3.2 Simply Supported Plate 566 8.9.4 Shallow Spherical Caps 567 8.9.5 Conical Shells 573 8.9.6 Toroidal Shells 578 Problems 583 References 584 Answers to the Problems 587 Appendix A Slope Deflection Coefficients for Beam Column Buckling 593 Appendix B Torsion Properties of Thin-Walled Open Cross-Sections 597 Appendix C Calculus of Variations 599 Appendix D Euler Equations 603 Appendix E Differential Geometry in Curvilinear Coordinates 607 Index 619