Structural Dynamics: Theory and Computation – Mario Paz, William Leigh – 5th Edition

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.