COURSE Dynamic of structure

Contents

PART I
SINGLE-DEGREE-OF-FREEDOM SYSTEMS
 1 Equations of Motion, Problem Statement, and Solution Methods
 1.1 Simple Structures
  1.2 Single-Degree-of-Freedom System
 1.3 Force–Displacement Relation
 1.4 Damping Force
 1.5 Equation of Motion: External Force
 1.6 Mass–Spring–Damper System
 1.7 Equation of Motion: Earthquake Excitation
 1.8 Problem Statement and Element Forces
1.9 Combining Static and Dynamic Responses
 1.10 Methods of Solution of the Differential Equation
 1.11 Study of SDF Systems: Organization
 Appendix 1: Stiffness Coefficients for a Flexural Element
 2 Free Vibration 39 2.1 Undamped Free Vibration
 2.2 Viscously Damped Free Vibration
 2.3 Energy in Free Vibration
 2.4 Coulomb-Damped Free Vibration
 3 Response to Harmonic and Periodic Excitations 65

Part A: Viscously Damped Systems: Basic Results
 3.1 Harmonic Vibration of Undamped Systems
 3.2 Harmonic Vibration with Viscous Damping 72

Part B: Viscously Damped Systems: Applications
 3.3 Response to Vibration Generator
  3.4 Natural Frequency and Damping from Harmonic Tests
 3.5 Force Transmission and Vibration Isolation
 3.6 Response to Ground Motion and Vibration Isolation
 3.7 Vibration-Measuring Instruments
 3.8 Energy Dissipated in Viscous Damping
 3.9 Equivalent Viscous Damping 103

Part C: Systems with Nonviscous Damping
 3.10 Harmonic Vibration with Rate-Independent Damping
 3.11 Harmonic Vibration with Coulomb Friction 109 Contents xi

Part D: Response to Periodic Excitation
 3.12 Fourier Series Representation
 3.13 Response to Periodic Force
 Appendix 3: Four-Way Logarithmic Graph Paper
 4 Response to Arbitrary, Step, and Pulse Excitations 125
Part A: Response to Arbitrarily Time-Varying Forces 125
4.1 Response to Unit Impulse 126 4.2 Response to Arbitrary Force 127
Part B: Response to Step and Ramp Forces 129 4.3 Step Force 129
4.4 Ramp or Linearly Increasing Force 131
4.5 Step Force with Finite Rise Time 132
 Part C: Response to Pulse Excitations 135
4.6 Solution Methods 135
4.7 Rectangular Pulse Force 137
4.8 Half-Cycle Sine Pulse Force 143
4.9 Symmetrical Triangular Pulse Force 148
 4.10 Effects of Pulse Shape and Approximate Analysis for Short Pulses 151
4.11 Effects of Viscous Damping 154
4.12 Response to Ground Motion 155
5 Numerical Evaluation of Dynamic Response 165
5.1 Time-Stepping Methods 165
5.2 Methods Based on Interpolation of Excitation 167
5.3 Central Difference Method 171
5.4 Newmark’s Method 174
5.5 Stability and Computational Error 180
5.6 Nonlinear Systems: Central Difference Method 183
5.7 Nonlinear Systems: Newmark’s Method 183
6 Earthquake Response of Linear Systems 197
6.1 Earthquake Excitation 197
6.2 Equation of Motion 203
6.3 Response Quantities 204
 6.4 Response History 205
6.5 Response Spectrum Concept 207
6.6 Deformation, Pseudo-velocity, and Pseudo-acceleration Response Spectra 208
6.7 Peak Structural Response from the Response Spectrum 217
6.8 Response Spectrum Characteristics 222
6.9 Elastic Design Spectrum 230
6.10 Comparison of Design and Response Spectra 239
6.11 Distinction between Design and Response Spectra 241
6.12 Velocity and Acceleration Response Spectra 242 Appendix 6: El Centro, 1940 Ground Motion 246
7 Earthquake Response of Inelastic Systems 257
7.1 Force–Deformation Relations 258
7.2 Normalized Yield Strength, Yield Strength Reduction Factor, and Ductility Factor 265
7.3 Equation of Motion and Controlling Parameters 266
7.4 Effects of Yielding 267
7.5 Response Spectrum for Yield Deformation and Yield Strength 274
7.6 Yield Strength and Deformation from the Response Spectrum 278
7.7 Yield Strength–Ductility Relation 278
7.8 Relative Effects of Yielding and Damping 280
7.9 Dissipated Energy 281
7.10 Supplemental Energy Dissipation Devices 284
7.11 Inelastic Design Spectrum 289
7.12 Applications of the Design Spectrum 296
7.13 Comparison of Design and Response Spectra 302
8 Generalized Single-Degree-of-Freedom Systems 307
8.1 Generalized SDF Systems 307
8.2 Rigid-Body Assemblages 309
8.3 Systems with Distributed Mass and Elasticity 311
8.4 Lumped-Mass System: Shear Building 323
8.5 Natural Vibration Frequency by Rayleigh’s Method 330
8.6 Selection of Shape Function 334 Appendix 8: Inertia Forces for Rigid Bodies 338

PART II MULTI-DEGREE-OF-FREEDOM SYSTEMS 345
9 Equations of Motion, Problem Statement, and Solution Methods 347
9.1 Simple System: Two-Story Shear Building 347
9.2 General Approach for Linear Systems 352
9.3 Static Condensation 369
9.4 Planar or Symmetric-Plan Systems: Ground Motion 372
9.5 One-Story Unsymmetric-Plan Buildings 377
9.6 Multistory Unsymmetric-Plan Buildings 383
9.7 Multiple Support Excitation 387
 9.8 Inelastic Systems 392
9.9 Problem Statement 392
9.10 Element Forces 393
9.11 Methods for Solving the Equations of Motion: Overview 393
10 Free Vibration 403 Part A: Natural Vibration Frequencies and Modes 404
10.1 Systems without Damping 404
10.2 Natural Vibration Frequencies and Modes 406
10.3 Modal and Spectral Matrices 408
10.4 Orthogonality of Modes 409
10.5 Interpretation of Modal Orthogonality 410
10.6 Normalization of Modes 410
10.7 Modal Expansion of Displacements 420 Part B: Free Vibration Response 421
10.8 Solution of Free Vibration Equations: Undamped Systems 421
10.9 Systems with Damping 424
10.10 Solution of Free Vibration Equations: Classically Damped Systems 425 Part C: Computation of Vibration Properties 428
10.11 Solution Methods for the Eigenvalue Problem 428
10.12 Rayleigh’s Quotient 430
10.13 Inverse Vector Iteration Method 430
10.14 Vector Iteration with Shifts: Preferred Procedure 435
10.15 Transformation of kφ = ω2mφ to the Standard Form 440 11 Damping in Structures 447

Part A: Experimental Data and Recommended Modal Damping Ratios 447
11.1 Vibration Properties of Millikan Library Building 447
11.2 Estimating Modal Damping Ratios 452
Part B: Construction of Damping Matrix 454
11.3 Damping Matrix 454
11.4 Classical Damping Matrix 455
11.5 Nonclassical Damping Matrix 464

12 Dynamic Analysis and Response of Linear Systems 467
Part A: Two-Degree-of-Freedom Systems 467
12.1 Analysis of Two-DOF Systems Without Damping 467
12.2 Vibration Absorber or Tuned Mass Damper 470

Part B: Modal Analysis 472
12.3 Modal Equations for Undamped Systems 472
12.4 Modal Equations for Damped Systems 475
12.5 Displacement Response 476
12.6 Element Forces 477
12.7 Modal Analysis: Summary 477

Part C: Modal Response Contributions 482
12.8 Modal Expansion of Excitation Vector p(t) = sp(t) 482
12.9 Modal Analysis for p(t) = sp(t) 486
12.10 Modal Contribution Factors 487
12.11 Modal Responses and Required Number of Modes 489

Part D: Special Analysis Procedures 496
12.12 Static Correction Method 496
12.13 Mode Acceleration Superposition Method 499
12.14 Mode Acceleration Superposition Method: Arbitrary Excitation 500

13 Earthquake Analysis of Linear Systems 513
Part A: Response History Analysis 514
13.1 Modal Analysis 514 13.2 Multistory Buildings with Symmetric Plan 520
13.3 Multistory Buildings with Unsymmetric Plan 540
13.4 Torsional Response of Symmetric-Plan Buildings 551
13.5 Response Analysis for Multiple Support Excitation 555
13.6 Structural Idealization and Earthquake Response 561

Part B: Response Spectrum Analysis 562
13.7 Peak Response from Earthquake Response Spectrum 562
13.8 Multistory Buildings with Symmetric Plan 567
13.9 Multistory Buildings with Unsymmetric Plan 579
13.10 A Response-Spectrum-Based Envelope for Simultaneous Responses 587
13.11 Response to Multicomponent Ground Motion 595
14 Analysis of Nonclassically Damped Linear Systems 617
Part A: Classically Damped Systems: Reformulation 618
14.1 Natural Vibration Frequencies and Modes 618
14.2 Free Vibration 619
14.3 Unit Impulse Response 620
14.4 Earthquake Response 621
Part B: Nonclassically Damped Systems 622
14.5 Natural Vibration Frequencies and Modes 622
14.6 Orthogonality of Modes 623
14.7 Free Vibration 627
14.8 Unit Impulse Response 632
14.9 Earthquake Response 636
14.10 Systems with Real-Valued Eigenvalues 638
14.11 Response Spectrum Analysis 646
14.12 Summary 647 Appendix 14: Derivations 648
15 Reduction of Degrees of Freedom 657
15.1 Kinematic Constraints 658
15.2 Mass Lumping in Selected DOFs 659
 15.3 Rayleigh–Ritz Method 659
15.4 Selection of Ritz Vectors 663
15.5 Dynamic Analysis Using Ritz Vectors 668
16 Numerical Evaluation of Dynamic Response 673
16.1 Time-Stepping Methods 673
16.2 Linear Systems with Nonclassical Damping 675
16.3 Nonlinear Systems 681
17 Systems with Distributed Mass and Elasticity 697
17.1 Equation of Undamped Motion: Applied Forces 698
17.2 Equation of Undamped Motion: Support Excitation 699
17.3 Natural Vibration Frequencies and Modes 700
17.4 Modal Orthogonality 707
17.5 Modal Analysis of Forced Dynamic Response 709
 17.6 Earthquake Response History Analysis 716
17.7 Earthquake Response Spectrum Analysis 721
17.8 Difficulty in Analyzing Practical Systems 724

18 Introduction to the Finite Element Method 729
Part A: Rayleigh–Ritz Method 729
18.1 Formulation Using Conservation of Energy 729
18.2 Formulation Using Virtual Work 733
18.3 Disadvantages of Rayleigh–Ritz Method 735
Part B: Finite Element Method 735 18.4 Finite Element Approximation 735
18.5 Analysis Procedure 737
18.6 Element Degrees of Freedom and Interpolation Functions 739
18.7 Element Stiffness Matrix 740 18.8 Element Mass Matrix 741
18.9 Element (Applied) Force Vector 743
18.10 Comparison of Finite Element and Exact Solutions 747
18.11 Dynamic Analysis of Structural Continua 748
PART III EARTHQUAKE RESPONSE, DESIGN, AND EVALUATION OF MULTISTORY BUILDINGS 755
19 Earthquake Response of Linearly Elastic Buildings 757
19.1 Systems Analyzed, Design Spectrum, and Response Quantities 757
19.2 Influence of T1 and ρ on Response 762
19.3 Modal Contribution Factors 763
19.4 Influence of T1 on Higher-Mode Response 765
19.5 Influence of ρ on Higher-Mode Response 768
19.6 Heightwise Variation of Higher-Mode Response 769
19.7 How Many Modes to Include 771
20 Earthquake Analysis and Response of Inelastic Buildings 775
Part A: Nonlinear Response History Analysis 776
20.1 Equations of Motion: Formulation and Solution 776
20.2 Computing Seismic Demands: Factors To Be Considered 777
 20.3 Story Drift Demands 781
20.4 Strength Demands for SDF and MDF Systems 787
Part B: Approximate Analysis Procedures 788
20.5 Motivation and Basic Concept 788
20.6 Uncoupled Modal Response History Analysis 790
20.7 Modal Pushover Analysis 797
20.8 Evaluation of Modal Pushover Analysis 802
20.9 Simplified Modal Pushover Analysis for Practical Application 807
21 Earthquake Dynamics of Base-Isolated Buildings 809
21.1 Isolation Systems 809
21.2 Base-Isolated One-Story Buildings 812
21.3 Effectiveness of Base Isolation 818
21.4 Base-Isolated Multistory Buildings 822
21.5 Applications of Base Isolation 828

22 Structural Dynamics in Building Codes 835
Part A: Building Codes and Structural Dynamics 836
22.1 International Building Code (United States), 2009 836
22.2 National Building Code of Canada, 2010 839
22.3 Mexico Federal District Code, 2004 841
22.4 Eurocode 8, 2004 844
22.5 Structural Dynamics in Building Codes 846
Part B: Evaluation of Building Codes 852
22.6 Base Shear 852
22.7 Story Shears and Equivalent Static Forces 856
22.8 Overturning Moments 858
22.9 Concluding Remarks 861
23 Structural Dynamics in Building Evaluation Guidelines 863
23.1 Nonlinear Dynamic Procedure: Current Practice 864
23.2 SDF-System Estimate of Roof Displacement 865
23.3 Estimating Deformation of Inelastic SDF Systems 868
23.4 Nonlinear Static Procedures 874
23.5 Concluding Remarks 880
A Frequency-Domain Method of Response Analysis 883
B Notation 905 C Answers to Selected Problems 917 Index




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