Dynamics of Structures, Global Edition

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 264 7.3 Equation of Motion and Controlling Parameters 265 7.4 Effects of Yielding 266 7.5 Response Spectrum for Yield Deformation and Yield Strength 273 7.6 Yield Strength and Deformation from the Response Spectrum 277 7.7 Yield Strength-Ductility Relation 277 7.8 Relative Effects of Yielding and Damping 279 7.9 Dissipated Energy 280 7.10 Supplemental Energy Dissipation Devices 283 7.11 Inelastic Design Spectrum 288 7.12 Applications of the Design Spectrum 295 7.13 Comparison of Design and Response Spectra 301 8 Generalized Single-Degree-of-Freedom Systems 305 8.1 Generalized SDF Systems 305 8.2 Rigid-Body Assemblages 307 8.3 Systems with Distributed Mass and Elasticity 309 8.4 Lumped-Mass System: Shear Building 321 8.5 Natural Vibration Frequency by Rayleigh's Method 328 8.6 Selection of Shape Function 332 Appendix 8: Inertia Forces for Rigid Bodies 336 PART II MULTI-DEGREE-OF-FREEDOM SYSTEMS 343 9 Equations of Motion, Problem Statement, and Solution Methods 345 9.1 Simple System: Two-Story Shear Building 345 9.2 General Approach for Linear Systems 350 9.3 Static Condensation 367 9.4 Planar or Symmetric-Plan Systems: Ground Motion 370 9.5 One-Story Unsymmetric-Plan Buildings 375 9.6 Multistory Unsymmetric-Plan Buildings 381 9.7 Multiple Support Excitation 385 9.8 Inelastic Systems 390 9.9 Problem Statement 390 9.10 Element Forces 391 9.11 Methods for Solving the Equations of Motion: Overview 391 10 Free Vibration 401 Part A: Natural Vibration Frequencies and Modes 402 10.1 Systems without Damping 402 10.2 Natural Vibration Frequencies and Modes 404 10.3 Modal and Spectral Matrices 406 10.4 Orthogonality of Modes 407 10.5 Interpretation of Modal Orthogonality 408 10.6 Normalization of Modes 408 10.7 Modal Expansion of Displacements 418 Part B: Free Vibration Response 419 10.8 Solution of Free Vibration Equations: Undamped Systems 419 10.9 Systems with Damping 422 10.10 Solution of Free Vibration Equations: Classically Damped Systems 423 Part C: Computation of Vibration Properties 426 10.11 Solution Methods for the Eigenvalue Problem 426 10.12 Rayleigh's Quotient 428 10.13 Inverse Vector Iteration Method 428 10.14 Vector Iteration with Shifts: Preferred Procedure 433 10.15 Transformation of kphi = omega2mphi to the Standard Form 438 11 Damping in Structures 445 Part A: Experimental Data and Recommended Modal Damping Ratios 445 11.1 Vibration Properties of Millikan Library Building 445 11.2 Estimating Modal Damping Ratios 450 Part B: Construction of Damping Matrix 452 11.3 Damping Matrix 452 11.4 Classical Damping Matrix 453 11.5 Nonclassical Damping Matrix 462 12 Dynamic Analysis and Response of Linear Systems 465 Part A: Two-Degree-of-Freedom Systems 465 12.1 Analysis of Two-DOF Systems without Damping 465 12.2 Vibration Absorber or Tuned Mass Damper 468 Part B: Modal Analysis 470 12.3 Modal Equations for Undamped Systems 470 12.4 Modal Equations for Damped Systems 473 12.5 Displacement Response 474 12.6 Element Forces 475 12.7 Modal Analysis: Summary 475 Part C: Modal Response Contributions 480 12.8 Modal Expansion of Excitation Vector p(t) = sp(t) 480 12.9 Modal Analysis for p(t) = sp(t) 484 12.10 Modal Contribution Factors 485 12.11 Modal Responses and Required Number of Modes 487 Part D: Special Analysis Procedures 494 12.12 Static Correction Method 494 12.13 Mode Acceleration Superposition Method 497 12.14 Mode Acceleration Superposition Method: Arbitrary Excitation 498 13 Earthquake Analysis of Linear Systems 511 Part A: Response History Analysis 512 13.1 Modal Analysis 512 13.2 Multistory Buildings with Symmetric Plan 518 13.3 Multistory Buildings with Unsymmetric Plan 537 13.4 Torsional Response of Symmetric-Plan Buildings 548 13.5 Response Analysis for Multiple Support Excitation 552 13.6 Structural Idealization and Earthquake Response 558 Part B: Response Spectrum Analysis 559 13.7 Peak Response from Earthquake Response Spectrum 559 13.8 Multistory Buildings with Symmetric Plan 564 13.9 Multistory Buildings with Unsymmetric Plan 576 13.10 A Response-Spectrum-Based Envelope for Simultaneous Responses 584 13.11 Response to Multi-Component Ground Motion 592 14 Analysis of Nonclassically Damped Linear Systems 613 Part A: Classically Damped Systems: Reformulation 614 14.1 Natural Vibration Frequencies and Modes 614 14.2 Free Vibration 615 14.3 Unit Impulse Response 616 14.4 Earthquake Response 617 Part B: Nonclassically Damped Systems 618 14.5 Natural Vibration Frequencies and Modes 618 14.6 Orthogonality of Modes 619 14.7 Free Vibration 623 14.8 Unit Impulse Response 628 14.9 Earthquake Response 632 14.10 Systems with Real-Valued Eigenvalues 634 14.11 Response Spectrum Analysis 642 14.12 Summary 643 Appendix 14: Derivations 644 15 Reduction of Degrees of Freedom 653 15.1 Kinematic Constraints 654 15.2 Mass Lumping in Selected DOFs 655 15.3 Rayleigh-Ritz Method 655 15.4 Selection of Ritz Vectors 659 15.5 Dynamic Analysis Using Ritz Vectors 664 16 Numerical Evaluation of Dynamic Response 669 16.1 Time-Stepping Methods 669 16.2 Linear Systems with Nonclassical Damping 671 16.3 Nonlinear Systems 677 17 Systems with Distributed Mass and Elasticity 693 17.1 Equation of Undamped Motion: Applied Forces 694 17.2 Equation of Undamped Motion: Support Excitation 695 17.3 Natural Vibration Frequencies and Modes 696 17.4 Modal Orthogonality 703 17.5 Modal Analysis of Forced Dynamic Response 705 17.6 Earthquake Response History Analysis 712 17.7 Earthquake Response Spectrum Analysis 717 17.8 Difficulty in Analyzing Practical Systems 720 18 Introduction to the Finite Element Method 725 Part A: Rayleigh-Ritz Method 725 18.1 Formulation Using Conservation of Energy 725 18.2 Formulation Using Virtual Work 729 18.3 Disadvantages of Rayleigh-Ritz Method 731 Part B: Finite Element Method 731 18.4 Finite Element Approximation 731 18.5 Analysis Procedure 733 18.6 Element Degrees of Freedom and Interpolation Functions 735 18.7 Element Stiffness Matrix 736 18.8 Element Mass Matrix 737 18.9 Element (Applied) Force Vector 739 18.10 Comparison of Finite Element and Exact Solutions 743 18.11 Dynamic Analysis of Structural Continua 744 PART III EARTHQUAKE RESPONSE, DESIGN, AND EVALUATION OF MULTISTORY BUILDINGS 751 19 Earthquake Response of Linearly Elastic Buildings 753 19.1 Systems Analyzed, Design Spectrum, and Response Quantities 753 19.2 Influence of T1 and on Response 758 19.3 Modal Contribution Factors 759 19.4 Influence of T1 on Higher-Mode Response 761 19.5 Influence of on Higher-Mode Response 764 19.6 Heightwise Variation of Higher-Mode Response 765 19.7 How Many Modes to Include 767 20 Earthquake Analysis and Response of Inelastic Buildings 771 Part A: Nonlinear Response History Analysis 772 20.1 Equations of Motion: Formulation and Solution 772 20.2 Computing Seismic Demands: Factors To Be Considered 773 20.3 Story Drift Demands 777 20.4 Strength Demands for SDF and MDF Systems 783 Part B: Approximate Analysis Procedures 784 20.5 Motivation and Basic Concept 784 20.6 Uncoupled Modal Response History Analysis 786 20.7 Modal Pushover Analysis 793 20.8 Evaluation of Modal Pushover Analysis 798 20.9 Simplified Modal Pushover Analysis for Practical Application 803 21 Earthquake Dynamics of Base-Isolated Buildings 805 21.1 Isolation Systems 805 21.2 Base-Isolated One-Story Buildings 808 21.3 Effectiveness of Base Isolation 814 21.4 Base-Isolated Multistory Buildings 818 21.5 Applications of Base Isolation 824 22 Structural Dynamics in Building Codes 831 Part A: Building Codes and Structural Dynamics 832 22.1 International Building Code (United States), 2009 832 22.2 National Building Code of Canada, 2010 835 22.3 Mexico Federal District Code, 2004 837 22.4 Eurocode 8, 2004 840 22.5 Structural Dynamics in Building Codes 842 Part B: Evaluation of Building Codes 848 22.6 Base Shear 848 22.7 Story Shears and Equivalent Static Forces 852 22.8 Overturning Moments 854 22.9 Concluding Remarks 857 23 Structural Dynamics in Building Evaluation Guidelines 859 23.1 Nonlinear Dynamic Procedure: Current Practice 860 23.2 SDF-System Estimate of Roof Displacement 861 23.3 Estimating Deformation of Inelastic SDF Systems 864 23.4 Nonlinear Static Procedures 870 23.5 Concluding Remarks 876 A Frequency-Domain Method of Response Analysis 879 B Notation 901 C Answers to Selected Problems 913 Index 929