Static & Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward LWilson Professor Emeritus of Civil Engineering University of California, Berkeley Fourth Edition Revised June 2010 Computers and Structures, Inc Berkeley, California, USA
CONTENTS Preface Acknowledgments Personal Remarks Contents List of Tables List of Figures v vii ix xi xxi xxv 1 Material Properties 1 11 Introduction 1 12 Anisotropic Materials 1 13 Use of Material Properties within Computer Programs 3 14 Orthotopic Materials 4 15 Isotropic Materials 5 16 Plane Strain Isotropic Materials 5 17 Plane Stress Isotropic Materials 6 18 Properties of Fluid-Like Materials 6 19 Shear and Compression Wave Velocities 7 110 Axisymmetric Material Properties 8 111 Force-Deformation Relationships 8 112 Summary 9 2 Equilibrium and Compatibility 11 21 Introduction 11 22 Fundamental Equilibrium Equations 12 23 Stress Resultants Forces and Moments 12 24 Compatibility Requirements 13 25 Strain Displacement Equations 13 26 Definition of Rotation 14 27 Equations at Material Interfaces 14 28 Interface Equations in Finite Element Systems 15 29 Node Rotations in Finite Element Systems 16 210 Statically Determinate Structures 16 211 Displacement Transformation Matrix 18 212 Element Stiffness and Flexibility Matrices 19 213 Solution of Statically Determinate System 20 214 General Solution of Structural Systems 20 xi
xii Static And Dynamic Analysis Of Structures 215 Summary 21 3 Energy and Work 23 31 Introduction 23 32 Virtual and Real Work 24 33 Potential Energy and Kinetic Energy 25 34 Strain Energy 27 35 External Work 28 36 Stationary Energy Principle 29 37 The Force Method 31 38 Lagrange's Equation of Motion 32 39 Conservation of Momentum 33 310 Summary 35 4 One-Dimensional Elements 37 41 Introduction 37 42 Analysis of an Axial Element 37 43 Two-Dimensional Frame Element 40 44 Three-Dimensional Frame Element 43 45 Member End-Releases 46 46 Summary 47 5 Isoparametric Elements 49 51 Introduction 49 52 A Simple One-Dimensional Example 50 53 One-Dimensional Integration Formulas 52 54 Restriction on Locations of Mid-Side Nodes 53 55 Two-Dimensional Shape Functions 54 56 Numerical Integration in Two Dimensions 56 57 Three-Dimensional Shape Functions 59 58 Triangular and Tetrahedral Elements 60 59 Summary 61 6 Incompatible Elements 63 61 Introduction 63 62 Elements with Shear Locking 64 63 Addition of Incompatible Modes 65 64 Formation of Element Stiffness Matrix 66 65 Incompatible Two-Dimensional Elements 67 66 Example using Incompatible Displacements 67 67 Three-Dimensional Incompatible Elements 68 68 Summary 69 7 Boundary Conditions and General Constraints 71 71 Introduction 71
Contents xiii 72 Displacement Boundary Conditions 72 73 Numerical Problems in Structural Analysis 73 74 General Theory Associated with Constraints 74 75 Floor Diaphragm Constraints 75 76 Rigid Constraints 78 77 Use of Constraints in Beam-Shell Analysis 79 78 Use of Constraints in Shear Wall Analysis 81 79 Use of Constraints for Mesh Transitions 82 710 Lagrange Multipliers and Penalty Functions 83 711 Summary 84 8 Plate Bending Elements 87 81 Introduction 87 82 The Quadrilateral Element 88 83 Strain-Displacement Equations 93 84 The Quadrilateral Element Stiffness 93 85 Satisfying the Patch Test 94 86 Static Condensation 95 87 Triangular Plate Bending Element 95 88 Other Plate Bending Elements 95 89 Numerical Examples 96 891 One Element Beam 96 892 Point Load on Simply Supported Square Plate 97 893 Uniform Load on Simply Supported Square Plate 894 Evaluation of Triangular Plate Bending Elements 895 Use of Plate Element to Model Torsion in Beams 99 99 100 810 Summary 101 9 Membrane Element with Normal Rotations 103 91 Introduction 103 92 Basic Assumptions 104 93 Displacement Approximation 105 94 Introduction of Node Rotation 106 95 Strain-Displacement Equations 107 96 Stress-Strain Relationship 107 97 Transform Relative to Absolute Rotations 107 98 Triangular Membrane Element 109 99 Numerical Example 109 910 Summary 110 10 Shell Elements 111 101 Introduction Ill 102 A Simple Quadrilateral Shell Element 112 103 Modelling Curved Shells with Flat Elements 113 104 Triangular Shell Elements 114
(K-Factors) xiv Static And Dynamic Analysis Of Structures 105 Use of Solid Elements for Shell Analysis 114 106 Analysis of the Scordelis-Lo Barrel Vault 114 107 Hemispherical Shell Example 116 108 Summary 117 11 Geometric Stiffness and P-Delta Effects 119 111 Definition of Geometric Stiffness 119 112 Approximate Buckling Analysis 121 113 P-Delta Analysis of Buildings 122 114 Equations for Three-Dimensional Buildings 124 115 The Magnitude of P-Delta Effects 125 116 P-Delta Analysis Without Computer Program Modification 126 117 Effective Length Factors 127 118 General Formulation of Geometric Stiffness 127 119 Summary 128 12 Dynamic Analysis 131 121 Introduction 131 122 Dynamic Equilibrium 132 123 Step-By-Step Solution Method 133 124 Mode Superposition Method 134 125 Response Spectra Analysis 134 126 Solution in the Frequency Domain 135 127 Solution of Linear Equations 135 128 Undamped Harmonic Response 136 129 Undamped Free Vibrations 137 1210 Summary 137 13 Dynamic Analysis using Mode Superposition 139 131 Equations to be Solved 139 132 Transformation to Modal Equations 140 133 Response Due to Initial Conditions Only 141 134 General Solution Due to Arbitrary Loading 141 135 Solution for Periodic Loading 146 136 Participating Mass Ratios 147 137 Static Load Participation Ratios 148 138 Dynamic Load Participation Ratios 149 139 Summary 150 14 Calculation of Stiffness and Mass Orthogonal Vectors 153 141 Introduction 153 142 Determinate Search Method 154 143 Sturm Sequence Check 154 154 144 Inverse Iteration 145 Gram-Schmidt Orthogonalization 156
Contents XV 146 Block Subspace Iteration 156 147 Solution of Singular Systems 156 148 Generation of Load-Dependent Ritz Vectors 157 149 A Physical Explanation of the LDR Algorithm 158 1410 Comparison of Solutions Using Eigen and Ritz Vectors 160 1411 Correction for Higher Mode Truncation 161 1412 Vertical Direction Seismic Response 163 1413 Summary 165 15 Dynamic Analysis using Response Spectrum Seismic Loading 167 151 Introduction 167 152 Definition of a Response Spectrum 168 153 Calculation of Modal Response 169 154 Typical Response Spectrum Curves 169 155 The CQC Method of Modal Combination 172 156 Numerical Example of Modal Combination 173 157 Design Spectra 176 158 Orthogonal Effects in Spectral Analysis 176 1581 Basic Equations for Calculation of Spectral Forces, 178 1582 The General CQC3 Method 179 1583 Examples of Three-Dimensional Spectra Analyses 180 1584 Recommendations on Orthogonal Effects 182 159 Limitations of the Response Spectrum Method 183 1591 Story Drift Calculations 183 1592 Estimation of Spectra Stresses in Beams 183 1593 Design Checks for Steel and Concrete Beams 184 1594 Calculation of Shear Force in Bolts 184 1510 Summary 185 16 Soil-Structure Interaction 187 161 Introduction 187 162 Site Response Analysis 188 163 Kinematic or Soil Structure Interaction 188 164 Response Due to Multi-Support Input Motions 191 165 Analysis of Gravity Dam and Foundation 193 166 The Massless Foundation Approximation 195 167 Approximate Radiation Boundary Conditions 195 168 Use of Springs at the Base of a Stnicture 197 169 Summary 197 17 Seismic Analysis Modelling to Satisfy Building Codes 199 171 Introduction 199 172 Three-Dimensional Computer Model 200 173 Three-Dimensional Mode Shapes and Frequencies 201
xvi Static and Dynamic Analysis Of structures 174 Three-Dimensional Dynamic Analysis 204 1741 Dynamic Design Base Shear 204 1742 Definition of Principal Directions 206 1743 Directional and Orthogonal Effects 206 1744 Basic Method of Seismic Analysis 206 1745 Scaling of Results 207 1746 Dynamic Displacements and Member Forces 207 1747 Torsional Effects 207 175 Numerical Example 208 176 Dynamic Analysis Method Summary 209 177 Summary 210 18 Fast Nonlinear Analysis 213 181 Introduction 213 182 Structures with a Limited Number of Nonlinear Elements 214 183 Fundamental Equilibrium Equations 214 184 Calculation of Nonlinear Forces 216 185 Transformation to Modal Coordinates 216 186 S olution of Nonlinear Modal Equations 218 187 Static Nonlinear-Analysis of Frame Structure 218 188 Dynamic Nonlinear Analysis offrame Structure 221 189 Seismic Analysis of Elevated Water Tank 224 1810 Summary 225 19 Linear Viscous Damping 227 191 Introduction 227 192 Energy Dissipation in Real Structures 228 193 Physical Interpretation of Viscous Damping 229 194 Modal Damping Violates Dynamic Equilibrium 229 195 Numerical Example 230 196 Stiffness and Mass Proportional Damping 231 197 Calculation of Orthogonal Damping Matrices 232 198 Structures with Non-Classical Damping 233 199 Nonlinear Energy Dissipation 233 1910 Summary 234 20 Dynamic Analysis using Numerical Integration 235 201 Introduction 235 202 Newmark Family of Methods 236 203 Stability of Newmark's Method 237 204 The Average Acceleration Method 239 205 Wilson's ^-Factor 239 206 The Use of Stiffness Proportional Damping 240 207 The Hilber, Hughes and Taylor a-method 241 208 Selection of a Direct Integration Method 241
Contents xvii 209 Nonlinear Analysis 242 2010 Summary 242 21 Nonlinear Elements 243 211 Introduction 243 212 General Three-Dimensional Two-Node Element 244 213 General Plasticity Element 245 214 Different Positive and Negative Properties 246 215 The Bilinear Tension-Gap-Yield Element 247 216 Nonlinear Gap-Crush Element 247 217 Viscous Damping Elements 249 218 Three-Dimensional Friction-Gap Element 251 219 Summary 252 22 Seismic Analysis using Displacement Loading 253 221 Introduction 253 222 Equilibrium Equations for Displacement Input 254 223 Use of Pseudo-Static Displacements 256 224 Solution of Dynamic Equilibrium Equations 257 225 Numerical Example 258 2251 Example Structure 258 2252 Earthquake Loading 260 2253 Effect of Time Step Size for Zero Damping 260 2254 Earthquake Analysis with Finite Damping 262 2255 The Effect ofmode Truncation 265 226 Use of Load Dependent Ritz Vectors 266 227 Solution Using Step-By-Step Integration 267 228 Summary 269 23 Fluid-Structure Interaction 271 231 Introduction 271 232 Fluid-Structure Interaction 272 233 Finite Element Model ofdam-foundation Interface 274 234 Loading Due to Uplift and Pore Water Pressure 275 235 Pore Water Pressure Calculation Using SAP2000 277 236 Selection of Gap Element Stiffness Value 279 237 Fundamental Equations in Fluid Dynamics 279 238 Relationship Between Pressure and Velocity 280 239 Equilibrium at the Interface oftwo Materials 281 2310 Radiation Boundary Conditions 283 2311 Surface Sloshing Modes 283 2312 Vertical Wave Propagation 284 2313 The Westergaard Paper 285 2314 Dynamic Analysis of Rectangular Reservoir 286 2315 Energy Absorptive Reservoir Boundaries 289
xviii Static And Dynamic Analysis Of Structures 2316 Relative Versus Absolute Formulation 290 2317 The Effect of Gate Setback on Pressure 291 2318 Seismic Analysis of Radial Gates 293 2319 Final Remarks 296 APPENDICES 299 A Vector Notation 301 Al Introduction 301 A2 Vector Cross Product 302 A3 Vectors to Define a Local Reference System 304 A 4 Fortran Subroutines for Vector Operations 305 B Matrix Notation 309 B l Introduction 309 B2 Definition of Matrix Notation 309 B3 Matrix Transpose and Scalar Multiplication 312 B4 Definition ofa Numerical Operation 313 B5 Programming Matrix Multiplication 313 B 6 Order of Matrix Multiplication 314 B 7 Summary 314 C Solution or Inversion of Linear Equations 317 C l Introduction 317 C2 Numerical Example 318 C3 The Gauss Elimination Algorithm 319 C4 Solution of a General Set of Linear Equations 321 C5 Alternative to Pivoting 321 C6 Matrix Inversion 323 C7 Physical Interpretation of Matrix Inversion 325 C8 Partial Gauss Elimination, Static Condensation and Substructure Anal ysis 326 C9 Equations Stored in Banded or Profile Form 328 CIO LDLFactorization 329 C101 Triangularization or Factorization of the A Matrix 329 C102 Forward Reduction of the b Matrix 330 C103 Calculation of x by Back substitution 331 C 11 Diagonal Cancellation and Numerical Accuracy 332 C 12 Summary 332 D The Eigenvalue Problem 335 D l Introduction 335 D2 The Jacobi Method 336 D3 Calculation of 3D Principal Stresses 338 D4 Solution of the General Eigenvalue Problem 339
Contents xix D 5 Summary 340 E Transformation of Material Properties 341 E l Introduction 341 E2 Summary 343 F A Displacement-Based Beam Element with Shear Deformations 345 F l Introduction 345 F2 Basic Assumptions 345 F 3 Effective Shear Area 348 G Numerical Integration 351 G l Introduction 351 G2 One-Dimensional Gauss Quadrature 352 G3 Numerical Integration in Two Dimensions 354 G4 An Eight-Point Two-Dimensional Rule 354 G5 An Eight-Point Lower-Order Rule 356 G6 A Five-Point Integration Rule 356 G7 Three-Dimensional Integration Rules 357 G8 Selective Integration 359 G 9 Summary 360 H Speed of Computer Systems 363 H l Introduction 363 H2 Definition of One Numerical Operation 363 H3 Speed of Different Computer Systems 364 H4 Speed of Personal Computer Systems 364 H5 Paging Operating Systems 365 H 6 Summary 366 I Method of Least Square 367 I 1 Simple Example 367 12 General Formulation 368 13 Calculation of Stresses within Finite Elements 370 J Consistent Earthquake Acceleration and Displacement Records 373 Jl Introduction 373 J2 Ground Acceleration Records 374 J3 Calculation of Acceleration Record from Displacement Record 375 J4 Creating Consistent Acceleration Record 377 J5 Summary 378 K Example of a SAP2000 Model for a Slab/Wall System 381 Kl Introduction 381 K2 The Problem 382
XX Static And Dynamic Analysis Of Structures K3 A Shell/Beam Model of the Slab/Wall System 382 K4 Shell Element Modification 384 Bibliography 385 Index 389