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