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Table of Contents

I Structures Modeled as a Single-Degree-of-Freedom System.- 1 Undamped Single-Degree-Of-Freedom System.- 1.1 Degrees of Freedom.- 1.2 Undamped System.- 1.3 Springs in Parallel or in Series.- 1.4 Newton’s Law of Motion.- 1.5 Free Body Diagram.- 1.6 D’ Alembert’s Principle.- 1.7 Solution of the Differential Equation of Motion.- 1.8 Frequency and Period.- 1.9 Amplitude of Motion.- 1.10 Summary.- 1.11 Problems.- 2 Damped Single-Degree-of-Freedom System.- 2.1 Viscous Damping.- 2.2 Equation of Motion.- 2.3 Critically Damped System.- 2.4 Overdamped System.- 2.5 Underdamped System.- 2.6 Logarithmic Decrement.- 2.7 Summary.- 2.8 Problems.- 3 Response of One-Degree-of-Freedom System to Harmonic Loading.- 3.1 Harmonic Excitation: Undamped System.- 3.2 Harmonic Excitation: Damped System.- 3.3 Evaluation of Damping at Resonance.- 3.4 Bandwidth Method (Half-Power) to Evaluate Damping.- 3.5 Energy Dissipated by Viscous Damping.- 3.6 Equivalent Viscous Damping.- 3.7 Response to Support Motion.- 3.8 Force Transmitted to the Foundation.- 3.9 Seismic Instruments.- 3.10 Response of One-Degree-of-Freedom System to Harmonic Loading Using SAP2000.- 3.11 Summary.- 3.12 Analytical Problem.- 3.13 Problems.- 4 Response to General Dynamic Loading.- 4.1 Duhamel’s Integral-Undamped System.- 4.2 Duhamel’s Integral -Damped System.- 4.3 Response by Direct Integration.- 4.4 Solution of the Equation of Motion.- 4.5 Program 2-Response by Direct Integration.- 4.6 Program 3-Response to Impulsive Excitation.- 4.7 Response to General Dynamic Loading Using SAP2000.- 4.8 Summary.- 4.9 Analytical Problems.- 4.10 Problems.- 5 Response Spectra.- 5.1 Construction of Response Spectrum.- 5.2 Response Spectrum for Support Excitation.- 5.3 Tripartite Response Spectra.- 5.4 Response Spectra for Elastic Design.- 5.5 Influence of Local Soil Conditions.- 5.6 Response Spectra for Inelastic Systems.- 5.7 Response Spectra for Inelastic Design.- 5.8 Program 6-Seismic Response Spectra.- 5.9 Summary.- 5.10 Problems.- 6 Nonlinear Structural Response.- 6.1 Nonlinear Single Degree-of-Freedom Model.- 6.2 Integration of the Nonlinear Equation of Motion.- 6.3 Constant Acceleration Method.- 6.4 Linear Acceleration Step-by-Step Method.- 6.5 The Newmark Beta Method.- 6.6 Elastoplastic Behavior.- 6.7 Algorithm for the Step-by-Step Solution for Elastoplastic Single-Degree-of-Freedom System.- 6.8 Program 5-Response for Elastoplastic Behavior.- 6.9 Summary.- 6.10 Problems.- II Structures Modeled as Shear Buildings.- 7 Free Vibration of a Shear Building.- 7.1 Stiffness Equations for the Shear Building.- 7.2 Natural Frequencies and Normal Modes.- 7.3 Orthogonality Property of the Normal Modes.- 7.4 Rayleigh’s Quotient.- 7.5 Program 8-Natural Frequencies and Normal Modes.- 7.6 Free Vibration of a Shear Building Using SAP2000.- 7.7 Summary.- 7.8 Problems.- 8 Forced Motion of Shear Building.- 8.1 Modal Superposition Method.- 8.2 Response of a Shear Building to Base Motion.- 8.3 Program 9-Response by Modal Superposition.- 8.4 Harmonic Forced Excitation.- 8.5 Program 10-Harmonic Response.- 8.6 Forced Motion Using SAP2000.- 8.7 Combining Maximum Values of Modal Response.- 8.8 Summary.- 8.9 Problems.- 9 Reduction of Dynamic Matrices.- 9.1 Static Condensation.- 9.2 Static Condensation Applied to Dynamic Problems.- 9.3 Dynamic Condensation.- 9.4 Modified Dynamic Condensation.- 9.5 Program 12-Reduction of the Dynamic Problem.- 9.6 Summary.- 9.7 Problems.- III Framed Structures Modeled as Discrete Multi-Degree-of-Freedom Systems.- 10 Dynamic Analysis of Beams.- 10.1 Shape Functions for a Beam Segment.- 10.2 System Stiffness Matrix.- 10.3 Inertial Properties-Lumped Mass.- 10.4 Inertial Properties-Consistent Mass.- 10.5 Damping Properties.- 10.6 External Loads.- 10.7 Geometric Stiffness.- 10.8 Equations of Motion.- 10.9 Element Forces at Nodal Coordinates.- 10.10 Program 13-Modeling Structures as Beams.- 10.11 Dynamic Analysis of Beams Using SAP2000.- 10.12 Summary.- 10.13 Problems.- 11 Dynamic Analysis of Plane Frames.- 11.1 Element Stiffness Matrix for Axial Effects.- 11.2 Element Mass Matrix for Axial Effects.- 11.3 Coordinate Transformation.- 11.4 Program 14-Modeling Structures as Plane Frames.- 11.5 Dynamic Analysis of Frames Using SAP2000.- 11.6 Summary.- 11.7 Problems.- 12 Dynamic Analysis of Grid Frames.- 12.1 Local and Global Coordinate Systems.- 12.2 Torsional Effects.- 12.3 Stiffness Matrix for a Grid Element.- 12.4 Consistent Mass Matrix for a Grid Element.- 12.5 Lumped Mass Matrix for a Grid Element.- 12.6 Transformation of Coordinates.- 12.7 Program 15-Modeling Structures as Grid Frames.- 12.8 Dynamic Analysis of Grid Frames Using SAP2000.- 12.9 Summary.- 12.10 Problems.- 13 Dynamic Analysis Ofthree-Dimensional Frames.- 13.1 Element Stiffness Matrix.- 13.2 Element Mass Matrix.- 13.3 Element Damping Matrix.- 13.4 Transformation of Coordinates.- 13.5 Differential Equation of Motion.- 13.6 Dynamic Response.- 13.7 Program 16-Modeling Structures as Space Frames.- 13.8 Dynamic Response of Three-Dimensional Frames Using SAP2000.- 13.9 Summary.- 13.10 Problems.- 14 Dynamic Analysis of Trusses.- 14.1 Stiffness and Mass Matrices for the Plane Truss.- 14.2 Transformation of Coordinates.- 14.3 Program 17-Modeling Structures as Plane Trusses.- 14.4 Stiffness and Mass Matrices for Space Trusses.- 14.5 Equation of Motion for Space Trusses.- 14.6 Program 18-Modeling Structures as Space Trusses.- 14.7 Dynamic Analysis of Trusses Using SAP2000.- 14.8 Summary.- 14.9 Problems.- 15 Dynamic Analysis of Structures Using the Finite Element Method.- 15.1 Plane Elasticity Problems.- 15.1.1 Triangular Plate Element for Plane Elasticity problems.- 15.1.2 SAP2000for Plane Elasticity Problem.- 15.2 Plate Bending.- 15.2.1 Rectangular Element for Plate Bending.- 15.2.2 SAP2000 for Plate Bending and Shell Problems.- 15.3 Summary.- 15.4 Problems.- 16 Time History Response of Multidegree-of-Freedom Systems.- 16.1 Incremental Equations of Motion.- 16.2 The Wilson-? Method.- 16.3 Algorithm for Step-by-Step Solution of a Linear System Using the Wilson-? Method.- 16.3.1 Initialization.- 16.3.2 for Each Time Step.- 16.4 Program 19-Response by Step Integration.- 16.5 The Newmark Beta Method.- 16.6 Elastoplastic Behavior of Framed Structures.- 16.7 Member Stiffness Matrix.- 16.8 Member Mass Matrix.- 16.9 Rotation of Plastic Hinges.- 16.10 Calculation of Member Ductility Ratio.- 16.11 Time-History Response of Multidegree-of-Freedom Systems Using SAP2000.- 16.12 Summary.- 16.13 Problems.- IV Structures Modeled with Distributed Properties.- 17 Dynamic Analysis of Systems with Distributed Properties.- 17.1 Flexural Vibration of Uniform Beams.- 17.2 Solution of the Equation of Motion in Free Vibration.- 17.3 Natural Frequencies and Mode Shapes for Uniform Beams.- 17.3.1 Both Ends Simply Supported.- 17.3.2 Both Ends Free (Free Beam).- 17.3.3 Both Ends Fixed.- 17.3.4 One End Fixed and the other End Free (Cantilever Beam).- 17.3.5 One End Fixed and the other End Simply Supported.- 17.4 Orthogonality Condition Between Normal Modes.- 17.5 Forced Vibration of Beams.- 17.6 Dynamic Stresses in Beams.- 17.7 Summary.- 17.8 Problems.- 18 Discretization of Continuous Systems.- 18.1 Dynamic Matrix for Flexural Effects.- 18.2 Dynamic Matrix for Axial Effects.- 18.3 Dynamic Matrix for Torsional Effects.- 18.4 Beam Flexure Including Axial-Force Effect.- 18.5 Power Series Expansion of the Dynamic Matrix for Flexural Effects.- 18.6 Power Series Expansion of the Dynamic Matrix for Axial and for Torsional Effects.- 18.7 Power Series Expansion of the Dynamic Matrix Including the Effects of Axial Forces.- 18.8 Summary.- V Special Topics: Fourier Analysis, Evaluation of Absolute Damping, Generalized Coordinates.- 19 Fourier Analysis and Response in the Frequency Domain.- 19.1 Fourier Analysis.- 19.2 Response to a Loading Represented by Fourier Series.- 19.3 Fourier Coefficients for Piecewise Linear Functions.- 19.4 Exponential Form of Fourier Series.- 19.5 Discrete Fourier Analysis.- 19.6 Fast Fourier Transform.- 19.7 Program 4-Response in the Frequency Domain.- 19.8 Summary.- 19.9 Problems.- 20 Evaluation of Absolute Damping from Modal Damping Ratios.- 20.1 Equations for Damped Shear Building.- 20.2 Uncoupled Damped Equations.- 20.3 Conditions for Damping Uncoupling.- 20.4 Program 11-Absolute Damping From Modal Damping Ratios.- 20.5 Summary.- 20.6 Problems.- 21 Generalized Coordinates and Rayleigh’s Method.- 21.1 Principle of Virtual Work.- 21.2 Generalized Single-Degree-of-Freedom System-Rigid Body.- 21.3 Generalized Single-Degree-of-Freedom System-Distributed Elasticity.- 21.4 Shear Forces and Bending Moments.- 21.5 Generalized Equation of Motion for a Multistory Building.- 21.6 Shape Function.- 21.7 Rayleigh’s Method.- 21.8 Improved Rayleigh ’ s Method.- 21.9 Shear Walls.- 21.10 Summary.- 21.11 Problems.- VI Random Vibration.- 22 Random Vibration.- 22.1 Statistical Description of Random Functions.- 22.2 Probability Density Function.- 22.3 The Normal Distribution.- 22.4 The Rayleigh Distribution.- 22.5 Correlation.- 22.6 The Fourier Transform.- 22.7 Spectral Analysis.- 22.8 Spectral Density Function.- 22.9 Narrow-Band and Wide-Band Random processes.- 22.10 Response to Random Excitation: Single-Degree-of-Freedom System.- 22.11 Response to Random Excitation: Multiple-Degree-of-Freedom System.- 22.11.1 Relationship Between Complex Frequency Response and Unit Impulse Response.- 22.11.2 Response to Random Excitation: Two-degree-of-freedom System.- 22.11.3 Response to Random Excitation: N Degree of Freedom System.- 22.12 Summary.- 22.13 Problems.- VII Earthquake Engineering.- 23 Uniform Building Code 1997: Equivalent Lateral Force Method.- 23.1 Earthquake Ground Motion.- 23.2 Equivalent Lateral Force Method.- 23.3 Earthquake-Resistant Design Methods.- 23.4 Seismic Zone Factor.- 23.5 Base Shear Force.- 23.6 Distribution of Lateral Seismic Forces.- 23.7 Story Shear Force.- 23.8 Horizontal Torsional Moment.- 23.9 Overturning Moment.- 23.10 P-Delta Effect (P-?).- 23.11 Redundancy/Reliability Factor p.- 23.12 Story Drift Limitation.- 23.13 Diaphragm Design Forces.- 23.14 Earthquake Load Effect.- 23.15 Irregular Structures.- 23.16 Summary.- 23.17 Problems.- 24 Uniform Building Code 1997: Dynamic Method.- 24.1 Modal Seismic Response of Buildings.- 24.1.1 Modal Equation and Participation Factor.- 24.1.2 Modal Shear Force.- 24.1.3 Effective Modal Weight.- 24.1.4 Modal Lateral Forces.- 24.1.5 Modal Displacements.- 24.1.6 Modal Drift.- 24.1.7 Modal Overturning Moment.- 24.1.8 Modal Torsional Moment.- 24.2 Total Design Values.- 24.3 Provisions of UBC-97: Dynamic Method.- 24.4 Scaling of Results.- 24.5 Program 24-UBC 1997 Dynamic Lateral Force Method.- 24.6 Summary.- 24.7 Problems.- 25 International Building Code IBC-2000.- 25.1 Response Spectral Acceleration: SS, S1.- 25.2 Soil Modification Response Spectral Acceleration: SMS, SM1.- 25.3 Design Response Spectral Acceleration: SDS, SD1.- 25.4 Site Class Definition: A, B,.....F.- 25.5 Seismic Use Group (SUG) and Occupancy Importance Factor (IE).- 25.6 Seismic Design Category (A, B, C, D, E and F).- 25.7 Design Response Spectral Curve: Sa v.s. T.- 25.8 Determination of the Fundamental Period.- 25.9 Minimum lateral Force Procedure [IBC-2000: Section 1616.4.1].- 25.10 Simplified Analysis Procedure [IBC-2000: Section 1617.5].- 25.10.1 Seismic Base Shear.- 25.10.2 Response Modification Factor R.- 25.10.3 Vertical Distribution of Lateral Forces.- 25.11 Equivalent Seismic Lateral Force Method: [IBC-2000: Section 1617.4].- 25.11.1 Distribution of Lateral Forces.- 25.11.2 Overturning Moments.- 25.11.3 Horizontal Torsional Moment.- 25.11.4 P-Delta Effect (P-?).- 25.11.5 Story Drift.- 25.12 Redundancy/Reliability Factor.- 25.13 Earthquake Load Effect.- 25.14 Building Irregularities.- 25.15 Summary.- Appendices.- Appendix I: Answers to Problems in Selected Chapters.- Appendix II: Computer Programs.- Appendix III: Glossary.- Selected Bibliography.