Static & Dynamic. Analysis of Structures. Edward L.Wilson. University of California, Berkeley. Fourth Edition. Professor Emeritus of Civil Engineering

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1 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

2 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 Axisymmetric Material Properties Force-Deformation Relationships Summary 9 2 Equilibrium and Compatibility Introduction Fundamental Equilibrium Equations Stress Resultants Forces and Moments Compatibility Requirements Strain Displacement Equations Definition of Rotation Equations at Material Interfaces Interface Equations in Finite Element Systems Node Rotations in Finite Element Systems Statically Determinate Structures Displacement Transformation Matrix Element Stiffness and Flexibility Matrices Solution of Statically Determinate System General Solution of Structural Systems 20 xi

3 xii Static And Dynamic Analysis Of Structures 215 Summary 21 3 Energy and Work Introduction Virtual and Real Work Potential Energy and Kinetic Energy Strain Energy External Work Stationary Energy Principle The Force Method Lagrange's Equation of Motion Conservation of Momentum Summary 35 4 One-Dimensional Elements Introduction Analysis of an Axial Element Two-Dimensional Frame Element Three-Dimensional Frame Element Member End-Releases Summary 47 5 Isoparametric Elements Introduction A Simple One-Dimensional Example One-Dimensional Integration Formulas Restriction on Locations of Mid-Side Nodes Two-Dimensional Shape Functions Numerical Integration in Two Dimensions Three-Dimensional Shape Functions Triangular and Tetrahedral Elements Summary 61 6 Incompatible Elements Introduction Elements with Shear Locking Addition of Incompatible Modes Formation of Element Stiffness Matrix Incompatible Two-Dimensional Elements Example using Incompatible Displacements Three-Dimensional Incompatible Elements Summary 69 7 Boundary Conditions and General Constraints Introduction 71

4 Contents xiii 72 Displacement Boundary Conditions Numerical Problems in Structural Analysis General Theory Associated with Constraints Floor Diaphragm Constraints Rigid Constraints Use of Constraints in Beam-Shell Analysis Use of Constraints in Shear Wall Analysis Use of Constraints for Mesh Transitions Lagrange Multipliers and Penalty Functions Summary 84 8 Plate Bending Elements Introduction The Quadrilateral Element Strain-Displacement Equations The Quadrilateral Element Stiffness Satisfying the Patch Test Static Condensation Triangular Plate Bending Element Other Plate Bending Elements Numerical Examples One Element Beam Point Load on Simply Supported Square Plate Uniform Load on Simply Supported Square Plate 894 Evaluation of Triangular Plate Bending Elements 895 Use of Plate Element to Model Torsion in Beams Summary Membrane Element with Normal Rotations Introduction Basic Assumptions Displacement Approximation Introduction of Node Rotation Strain-Displacement Equations Stress-Strain Relationship Transform Relative to Absolute Rotations Triangular Membrane Element Numerical Example Summary Shell Elements Introduction Ill 102 A Simple Quadrilateral Shell Element Modelling Curved Shells with Flat Elements Triangular Shell Elements 114

5 (K-Factors) xiv Static And Dynamic Analysis Of Structures 105 Use of Solid Elements for Shell Analysis Analysis of the Scordelis-Lo Barrel Vault Hemispherical Shell Example Summary Geometric Stiffness and P-Delta Effects Definition of Geometric Stiffness Approximate Buckling Analysis P-Delta Analysis of Buildings Equations for Three-Dimensional Buildings The Magnitude of P-Delta Effects P-Delta Analysis Without Computer Program Modification Effective Length Factors General Formulation of Geometric Stiffness Summary Dynamic Analysis Introduction Dynamic Equilibrium Step-By-Step Solution Method Mode Superposition Method Response Spectra Analysis Solution in the Frequency Domain Solution of Linear Equations Undamped Harmonic Response Undamped Free Vibrations Summary Dynamic Analysis using Mode Superposition Equations to be Solved Transformation to Modal Equations Response Due to Initial Conditions Only General Solution Due to Arbitrary Loading Solution for Periodic Loading Participating Mass Ratios Static Load Participation Ratios Dynamic Load Participation Ratios Summary Calculation of Stiffness and Mass Orthogonal Vectors Introduction Determinate Search Method Sturm Sequence Check Inverse Iteration 145 Gram-Schmidt Orthogonalization 156

6 Contents XV 146 Block Subspace Iteration Solution of Singular Systems Generation of Load-Dependent Ritz Vectors A Physical Explanation of the LDR Algorithm Comparison of Solutions Using Eigen and Ritz Vectors Correction for Higher Mode Truncation Vertical Direction Seismic Response Summary Dynamic Analysis using Response Spectrum Seismic Loading Introduction Definition of a Response Spectrum Calculation of Modal Response Typical Response Spectrum Curves The CQC Method of Modal Combination Numerical Example of Modal Combination Design Spectra Orthogonal Effects in Spectral Analysis Basic Equations for Calculation of Spectral Forces, The General CQC3 Method Examples of Three-Dimensional Spectra Analyses Recommendations on Orthogonal Effects Limitations of the Response Spectrum Method Story Drift Calculations Estimation of Spectra Stresses in Beams Design Checks for Steel and Concrete Beams Calculation of Shear Force in Bolts Summary Soil-Structure Interaction Introduction Site Response Analysis Kinematic or Soil Structure Interaction Response Due to Multi-Support Input Motions Analysis of Gravity Dam and Foundation The Massless Foundation Approximation Approximate Radiation Boundary Conditions Use of Springs at the Base of a Stnicture Summary Seismic Analysis Modelling to Satisfy Building Codes Introduction Three-Dimensional Computer Model Three-Dimensional Mode Shapes and Frequencies 201

7 xvi Static and Dynamic Analysis Of structures 174 Three-Dimensional Dynamic Analysis Dynamic Design Base Shear Definition of Principal Directions Directional and Orthogonal Effects Basic Method of Seismic Analysis Scaling of Results Dynamic Displacements and Member Forces Torsional Effects Numerical Example Dynamic Analysis Method Summary Summary Fast Nonlinear Analysis Introduction Structures with a Limited Number of Nonlinear Elements Fundamental Equilibrium Equations Calculation of Nonlinear Forces Transformation to Modal Coordinates S olution of Nonlinear Modal Equations Static Nonlinear-Analysis of Frame Structure Dynamic Nonlinear Analysis offrame Structure Seismic Analysis of Elevated Water Tank Summary Linear Viscous Damping Introduction Energy Dissipation in Real Structures Physical Interpretation of Viscous Damping Modal Damping Violates Dynamic Equilibrium Numerical Example Stiffness and Mass Proportional Damping Calculation of Orthogonal Damping Matrices Structures with Non-Classical Damping Nonlinear Energy Dissipation Summary Dynamic Analysis using Numerical Integration Introduction Newmark Family of Methods Stability of Newmark's Method The Average Acceleration Method Wilson's ^-Factor The Use of Stiffness Proportional Damping The Hilber, Hughes and Taylor a-method Selection of a Direct Integration Method 241

8 Contents xvii 209 Nonlinear Analysis Summary Nonlinear Elements Introduction General Three-Dimensional Two-Node Element General Plasticity Element Different Positive and Negative Properties The Bilinear Tension-Gap-Yield Element Nonlinear Gap-Crush Element Viscous Damping Elements Three-Dimensional Friction-Gap Element Summary Seismic Analysis using Displacement Loading Introduction Equilibrium Equations for Displacement Input Use of Pseudo-Static Displacements Solution of Dynamic Equilibrium Equations Numerical Example Example Structure Earthquake Loading Effect of Time Step Size for Zero Damping Earthquake Analysis with Finite Damping The Effect ofmode Truncation Use of Load Dependent Ritz Vectors Solution Using Step-By-Step Integration Summary Fluid-Structure Interaction Introduction Fluid-Structure Interaction Finite Element Model ofdam-foundation Interface Loading Due to Uplift and Pore Water Pressure Pore Water Pressure Calculation Using SAP Selection of Gap Element Stiffness Value Fundamental Equations in Fluid Dynamics Relationship Between Pressure and Velocity Equilibrium at the Interface oftwo Materials Radiation Boundary Conditions Surface Sloshing Modes Vertical Wave Propagation The Westergaard Paper Dynamic Analysis of Rectangular Reservoir Energy Absorptive Reservoir Boundaries 289

9 xviii Static And Dynamic Analysis Of Structures 2316 Relative Versus Absolute Formulation The Effect of Gate Setback on Pressure Seismic Analysis of Radial Gates 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

10 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 General Formulation 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

11 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

Esben Byskov. Elementary Continuum. Mechanics for Everyone. With Applications to Structural Mechanics. Springer

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