Basic Structural Dynamics

PREFACE xi 1 BASIC CONCEPTS OF STRUCTURAL DYNAMICS 1 1.1 The Dynamic Environment 1 1.2 Types of Dynamic Loading 2 1.3 Basic Principles 3 1.4 Dynamic Equilibrium 9 2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS 13 2.1 Reduction of Degrees of Freedom 13 2.2 Time-Dependent Force 15 2.3 Gravitational Forces 17 2.4 Earthquake Ground Motion 18 2.5 Formulation of Equation of Motion 19 2.5.1 d’Alembert’s Principle 19 2.5.2 Virtual Work (Virtual Displacements) 20 2.6 Generalized Coordinates 22 2.6.1 Discrete Parameters 23 2.6.2 Continuous Parameters 31 2.6.3 Transformation Factors 38 2.6.4 Axial Load Effect 42 2.6.5 Linear Approximation 44 3 FREE-VIBRATION RESPONSE OF SINGLE-DEGREE-OF-FREEDOM SYSTEMS 51 3.1 Undamped Free Vibration 51 3.1.1 Alternate Solution 53 3.1.2 Rayleigh’s Method 58 3.1.3 Selection of Deflected Shape 60 3.2 Damped Free Vibration 61 3.2.1 Rotating Vector Form 63 3.2.2 Logarithmic Decrement 66 3.2.3 Radical Positive 67 4 RESPONSE TO HARMONIC LOADING 77 4.1 Undamped Dynamic System 77 4.2 Damped Dynamic System 84 4.3 Tripartite Logarithmic Plot 91 4.4 Evaluation of Damping 91 4.5 Seismic Accelerometers and Displacement Meters (Seismographs) 95 5 RESPONSE TO IMPULSE LOADS 101 5.1 Rectangular Pulse 104 5.2 Damped Rectangular Pulse 108 5.3 Triangular Pulse 109 5.4 Approximate Analysis for Short-Duration Impulse Load 112 6 RESPONSE TO ARBITRARY DYNAMIC LOADING 121 6.1 Duhamel Integral 121 6.2 Numerical Formulation of the Equation of Motion 123 6.3 Numerical Integration Methods 124 6.4 Newmark’s Numerical Method 127 7 MULTIPLE-DEGREE-OF-FREEDOM SYSTEMS 137 7.1 Elastic Properties 137 7.1.1 Flexibility 137 7.1.2 Stiffness 138 7.1.3 Inertia 139 7.1.4 Viscous Damping 139 7.2 Undamped Free Vibration 141 7.3 Free Vibration 149 7.4 Betti’s Law 153 7.5 Orthogonality Properties of Mode Shapes 155 7.6 Changing Coordinates (Inverse Transformation) 156 7.7 Holzer Method for Shear Buildings 159 7.8 Axial Load Effects (Linear Approximation) 162 7.9 Modal Equations for Undamped Time-Dependent Force Analysis 165 7.10 Modal Equations of Damped Forced Vibration 174 7.11 Modal Equations for Seismic Response Analysis 179 8 NONLINEAR RESPONSE OF MULTIPLE-DEGREE-OF-FREEDOM SYSTEMS 183 8.1 Static Nonlinear Analysis 184 8.2 Dynamic Nonlinear Analysis 185 8.3 Gauss Reduction 189 8.4 MATLAB Applications 190 9 SEISMIC RESPONSE OF STRUCTURES 201 9.1 Introduction 201 9.2 Linear Elastic Response Spectra 203 9.3 Elastic Design Response Spectrum 208 9.4 Earthquake Response of SDOF Systems 215 9.5 Earthquake Response Analysis of MDOF Systems 219 9.5.1 Time History Modal Analysis 221 9.5.2 Modal Combinations for Spectral Analyses 222 9.6 Structural Dynamics in the Building Code 232 9.6.1 Equivalent Lateral Force Procedure 234 9.6.2 Modal Response Spectrum Procedure 237 APPENDIX—HISTORICAL DEVELOPMENT OF BUILDING CODE SEISMIC PROVISIONS 249 SELECTED REFERENCES 263 INDEX 265