Matrix Analysis of Structural Dynamics: Applications and Earthquake Engineering

by Cheng; Franklin Y.

• ISBN13: 9780824703875
• ISBN10: 0824703871
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-10-19
• Publisher: CRC Press

Summary

Uses state-of-the-art computer technology to formulate displacement method with matrix algebra. Facilitates analysis of structural dynamics and applications to earthquke engineering and UBC and IBC seismic building codes.

Author Biography

Franklin Y. Cheng was appointed Curators' Professor of Civil Engineering in 1987, the highest professorial position in Missouri's university system, and is Senior Investigator, Intelligent Systems Center, University of Missouri, Rolla

Table of Contents

Preface	p. iii
Characteristics of Free and Forced Vibrations of Elementary Systems	p. 1
Introduction	p. 1
Free Undamped Vibration	p. 1
Motion Equation and Solution	p. 1
Initial Conditions, Phase Angle and Natural Frequency	p. 3
Periodic and Harmonic Motion	p. 6
Free Damped Vibration	p. 7
Motion Equation and Viscous Damping	p. 7
Critical Damping, Overdamping and Underdamping	p. 9
Logarithmic Decrement and Evaluation of Viscous Damping Coefficient	p. 11
Forced Undamped Vibration	p. 14
Harmonic Forces	p. 14
Steady-State Vibration and Resonance	p. 15
Impulses and Shock Spectra	p. 19
General Loading--Step Forcing Function Method vs. Duhamel's Integral	p. 24
Forced Damped Vibration	p. 29
Harmonic Forces	p. 29
Steady-State Vibration for Damped Vibration, Resonant and Peak Amplitude	p. 30
General Loading--Step-Forcing Function Method vs. Duhamel's Integral	p. 32
Transmissibility and Response to Foundation Motion	p. 36
Evaluation of Damping	p. 42
Equivalent Damping Coefficient Method	p. 42
Amplitude Method and Bandwidth Method	p. 43
Overview	p. 45
Bibliography	p. 45
Eigensolution Techniques and Undamped Response Analysis of Multiple-Degree-of-Freedom Systems	p. 47
Fundamentals	p. 47
Introduction	p. 47
Characteristics of the Spring-Mass Model	p. 47
Advantages of the Lumped Mass Model	p. 48
Characteristics of Free Vibration of Two-Degree-of-Freedom Systems	p. 49
Motion Equations, Natural and Normal Modes	p. 49
Harmonic and Periodic Motion	p. 52
Dynamic Matrix Equation	p. 54
Orthogonality of Normal Modes	p. 55
Modal Matrix for Undamped Vibration	p. 56
Modal Matrices and Characteristics	p. 56
Response to Initial Disturbances, Dynamic Forces and Seismic Excitation	p. 58
Effect of Individual Modes on Response	p. 64
Response to Foundation Movement	p. 67
Eigensolution for Symmetric Matrix	p. 72
Iteration Method for Fundamental and Higher Modes	p. 72
Proof of Iterative Solution	p. 77
Extraction Technique for Natural Frequencies	p. 80
Choleski's Decomposition Method	p. 81
Generalized Jacobi Method	p. 87
Sturm Sequence Method	p. 95
Advanced Topics	p. 98
Eigensolution Technique for Unsymmetric Matrix	p. 98
Classification of Cases	p. 99
Iteration Method	p. 100
Response Analysis for Zero and Repeating Eigenvalues	p. 105
Zero and Repeating Eigenvalue Cases	p. 105
Orthogonality Properties	p. 105
Response Analysis	p. 109
Bibliography	p. 114
Eigensolution Methods and Response Analysis for Proportional and Nonproportional Damping	p. 117
Fundamentals	p. 117
Introduction	p. 117
Response Analysis for Proportional Damping	p. 117
Based on a Modal Matrix	p. 117
Proportional Damping	p. 120
Evaluation of Damping Coefficients and Factors	p. 121
Two Modes Required	p. 121
All Modes Required	p. 125
Damping Factors from Damping Coefficients	p. 125
Determination of Proportional and Nonproportional Damping	p. 126
Advanced Topics	p. 128
Characteristics of Complex Eigenvalues for Nonproportional Damping	p. 128
Iteration Method for Fundamental and Higher Modes of Complex Eigenvalues	p. 137
Fundamental Mode	p. 137
Orthogonality Condition and Iteration for Higher Modes	p. 139
Step-by-Step Procedures	p. 139
Response Analysis with Complex Eigenvalues	p. 149
Relationship Between Undamped, Proportional Damping, and Nonproportional Damping	p. 156
Bibliography	p. 159
Dynamic Stiffness and Energy Methods for Distributed Mass Systems	p. 161
Introduction	p. 161
Derivation of Bernoulli-Euler Equation	p. 161
Derivation of Dynamic Stiffness Coefficients	p. 166
Characteristics of Dynamic Stiffness Coefficients	p. 168
Numerals and Curves for Coefficients	p. 168
Rayleigh's Dynamic Reciprocal Principle	p. 171
Muller-Breslau's Principle	p. 173
Dynamic Stiffness, Load, and Mass Matrices	p. 175
Degree-of-Freedom of Plane Structural Systems	p. 175
Equilibrium Matrices	p. 176
Compatibility Matrices	p. 177
Dynamic Stiffness Matrix	p. 178
Dynamic Load Matrix	p. 179
System Matrix Equation	p. 180
Derivation of Dynamic Fixed-end Moments and Fixed-end Shears	p. 180
Differential Equations	p. 181
Uniform Load	p. 182
Triangular Load	p. 183
Concentrated Load between Nodes	p. 185
Foundation Movement	p. 186
Numerical Technique for Eigensolutions	p. 186
Steady-State Response Analysis	p. 198
Response for General Forcing Functions with and without Damping	p. 203
Kinetic and Strain Energy	p. 203
Orthogonality Condition	p. 204
Dissipated Energy and Work	p. 205
Response Equations	p. 206
Bibliography	p. 212
Dynamic Stiffness Method for Coupling Vibration, Elastic Media and P-[Delta] Effect	p. 213
Fundamentals	p. 213
Introduction	p. 213
Longitudinal Vibration and Stiffness Coefficients	p. 213
Longitudinal Vibration and Stiffness Coefficients with Elastic Media	p. 214
Dynamic Analysis of Trusses and Elastic Frames	p. 216
Dynamic Stiffness Coefficients of Pin-connected Member	p. 216
Dynamic Stiffness Matrix of Trusses	p. 218
Dynamic Stiffness Matrix of Elastic Frames	p. 221
Coupling of Longitudinal and Flexural Vibration	p. 224
Torsional Vibration and Stiffness Coefficients	p. 229
Dynamic Stiffness Matrix of Grid Systems	p. 230
Coupling of Torsional and Flexural Vibration	p. 233
Advanced Topics	p. 237
Bernoulli-Euler Equation with Elastic Media	p. 237
Bernoulli-Euler Equation with Elastic Media and P-[Delta] Effect	p. 238
Timoshenko Equation (Bending and Shear Deformation and Rotatory Inertia)	p. 240
Differential Equations	p. 240
Stiffness Coefficients	p. 243
Fixed-end Forces for Steady-State Vibration	p. 246
Response Analysis for General Forcing Functions	p. 247
Effect of Various Parameters on Frequencies	p. 252
Timoshenko Equation with Elastic Media and P-[Delta] Effect	p. 252
Differential Equations	p. 253
Stiffness Coefficients	p. 255
Fixed-end Forces	p. 256
Case Studies of the Effect of Various Parameters on Frequencies	p. 257
Bibliography	p. 258
Consistent Mass Method for Frames and Finite Elements	p. 261
Fundamentals	p. 261
Introduction	p. 261
Energy Method for Motion Equation	p. 262
Rigid Frames	p. 263
Elastic Frames	p. 265
Stiffness, Mass and Generalized Force Matrices for Frame Members	p. 265
Two-Force Member	p. 265
Torsional Member	p. 268
Flexural Member	p. 270
Eigenvalue Comparisons Among Lumped Mass, Dynamic Stiffness and Consistent Mass Methods	p. 283
Advanced Topics	p. 285
Stiffness, Mass and Generalized Force Matrices for Finite Elements	p. 285
Finite Element Formulation--Generalized Coordinates	p. 286
Finite Element Formulation--Natural Coordinates	p. 291
Motion Equation with P-[Delta] Effect	p. 303
Potential Energy and Motion Equation	p. 303
Geometric Matrix with Rotation and Deflection	p. 305
Geometric Matrix (String Stiffness) with Deflection	p. 305
Timoshenko Prismatic Member with P-[Delta] Effect	p. 306
Displacement and Shape Functions	p. 306
Stiffness Matrix	p. 308
Mass Matrix	p. 309
Generalized Force Matrix	p. 312
Geometric Matrix	p. 312
Timoshenko Tapered Member with P-[Delta] Effect	p. 314
Stiffness Matrix	p. 314
Mass Matrix	p. 315
Generalized Force Matrix	p. 317
Geometric Matrix	p. 317
Comments on Lumped Mass, Consistent Mass, and Dynamic Stiffness Models	p. 318
Bibliography	p. 319
Numerical Integration Methods and Seismic Response Spectra for Single-and Multi-Component Seismic Input	p. 321
Fundamentals	p. 321
Introduction	p. 321
Earthquakes and Their Effects on Structures	p. 321
Earthquake Characteristics	p. 321
Intensity, Magnitude, and Acceleration of Earthquakes	p. 322
Relationship Between Seismic Zone, Acceleration, Magnitude, and Intensity	p. 326
Earthquake Principal Components	p. 327
Numerical Integration and Stability	p. 329
Newmark Integration Method	p. 329
Wilson-[Theta] Method	p. 332
General Numerical Integration Related to Newmark and Wilson-[Theta] Methods	p. 334
Runge-Kutta Fourth-Order Method	p. 338
Numerical Stability and Error of Newmark and Wilson-[Theta] Methods	p. 350
Numerical Stability of Runge-Kutta Fourth-Order Method	p. 358
Seismic Response Spectra for Analysis and Design	p. 361
Response Spectra, Pseudo-Spectra and Principal-Component Spectra	p. 362
Housner's Average Design Spectra	p. 369
Newmark Elastic Design Spectra	p. 371
Newmark Inelastic Design Spectra	p. 372
Site-Dependent Spectra and UBC-94 Design Spectra	p. 378
Various Definitions of Ductility	p. 380
Advanced Topics	p. 383
Torisonal Response Spectra	p. 383
Ground Rotational Records Generation	p. 383
Construction of Torsional Response Spectra	p. 389
Response Spectra Analysis of a Multiple d.o.f. Systems	p. 390
SRSS Modal Combination Method	p. 393
CQC Modal Combination Method	p. 394
Structural Response Due to Multiple-Component Seismic Input	p. 397
Maximum (Worst-Case) Response Analysis for Six Seismic Components	p. 399
Based on SRSS Method	p. 400
Based on CQC Method	p. 404
Composite Translational Spectrum and Torsional Spectrum	p. 410
Construction of the Composite Response Spectrum	p. 411
Composite Spectral Modal Analysis	p. 412
Overview	p. 414
Bibliography	p. 414
Formulation and Response Analysis of Three-Dimensional Building Systems with Walls and Bracings	p. 417
Fundamentals	p. 417
Introduction	p. 417
Joints, Members, Coordinate Systems, and Degree of Freedom (d.o.f.)	p. 417
Coordinate Transformation Between JCS and GCS: Methods 1 and 2	p. 418
Force Transformation Between Slave Joint and Master Joint	p. 424
System d.o.f. as Related to Coordinate and Force Transformation	p. 426
Beam-Columns	p. 429
Coordinate Transformation Between ECS and JCS or GCS: Methods 1 and 2	p. 429
Beam-Column Stiffness in the ECS	p. 431
Beam-Column Stiffness in the JCS or GCS Based on Method 1	p. 434
Beam-Column Geometric Matrix (String Stiffness) in ECS and JCS or GCS Based on Method 1	p. 438
Shear Walls	p. 439
Shear-Wall ECS and GCS Relationship Based on Method 1	p. 439
Shear-Wall Stiffness in the ECS	p. 441
Shear-Wall Stiffness in the JCS or GCS Based on Method 1	p. 447
Shear-Wall Geometric Matrix (String Stiffness) in the JCS or GCS Based on Method 1	p. 455
Bracing Elements	p. 455
Bracing-Element ECS and GCS Relationship Based on Method 1	p. 455
Bracing-Element Stiffness in ECS	p. 457
Bracing-Element Stiffness in the JCS or GCS Based on Method 1	p. 457
Structural Characteristics of 3-D Building Systems	p. 462
Rigid Zone Between Member End and Joint Center	p. 462
Building-Structure-Element Stiffness with Rigid Zone	p. 464
Beam-Column Stiffness in ECS Based on Method 2	p. 464
Beam-Column Stiffness in GCS Based on Method 2	p. 466
Beam-Column Geometric Matrix (String Stiffness) in JCS or GCS Based on Method 2	p. 473
Beam Stiffness in the GCS Based on Method 2	p. 475
Bracing-Element Stiffness in the JCS or GCS Based on Method 2	p. 479
Shear-Wall Stiffness in the JCS or GCS Based on Method 2	p. 482
Shear-Wall Geometric Matrix (String Stiffness) in the JCS or GCS Based on Method 2	p. 487
Advanced Topics	p. 490
Assembly of Structural Global Stiffness Matrix	p. 490
General System Assembly (GSA)	p. 490
Floor-by-Floor Assembly (FFA)	p. 498
Mass Matrix Assembly	p. 504
Loading Matrix Assembly	p. 508
Vertical Static or Harmonic Forces	p. 509
Lateral Wind Forces	p. 511
Lateral Dynamic Loads	p. 513
Seismic Excitations	p. 514
Analysis and Response Behavior of Sample Structural Systems	p. 516
Overview	p. 523
Bibliography	p. 525
Various Hysteresis Models and Nonlinear Response Analysis	p. 527
Fundamentals	p. 527
Introduction	p. 527
Material Nonlinearity and Stress-Strain Models	p. 528
Bauschinger Effect on Moment-Curvature Relationship	p. 528
Elasto-Plastic Hysteresis Model	p. 529
Stiffness Matrix Formulation	p. 532
Bilinear Hysteresis Model	p. 534
Stiffness Matrix Formulation	p. 535
Convergence Techniques at Overshooting Regions	p. 538
State of Yield and Time-Increment Technique	p. 538
Unbalanced Force Technique	p. 539
Equilibrium and Compatibility Checks for Numerical Solutions	p. 552
Curvilinear Hysteresis Model	p. 555
Stiffness Matrix Formulation	p. 556
Stiffness Comparison Between Bilinear and Curvilinear Models	p. 560
Ramberg-Osgood Hysteresis Model	p. 562
Parameter Evaluations of Ramberg-Osgood Stress-Strain Curve	p. 562
Ramberg-Osgood Moment-Curvature Curves	p. 563
Stiffness Matrix Formulation for Skeleton Curve	p. 565
Stiffness Matrix Formulation for Branch Curve	p. 570
Advanced Topics	p. 579
Geometric Nonlinearity	p. 579
Interaction Effect on Beam Columns	p. 589
Elasto-Plastic Analysis of Consistent Mass Systems	p. 591
Stiffness Matrix Formulation	p. 591
Moments, Shears and Plastic Hinge Rotations	p. 595
Hysteresis Models of Steel Bracing, RC Beams, Columns and Shear Walls	p. 604
Overview	p. 604
Bibliography	p. 605
Static and Dynamic Lateral-Force Procedures and Related Effects in Building Codes of UBC-94, UBC-97 and IBC-2000	p. 607
Fundamentals	p. 607
Introduction	p. 607
Background of Lateral Force Procedures in Building Codes	p. 608
Effective Earthquake Force and Effective Mass	p. 608
Base Shear and Overturning Moment	p. 610
UBC-94 and Design Parameters	p. 612
Criteria for Appropriate Lateral-Force Procedure	p. 612
Base Shear of Static Lateral-Force Procedure and Related Parameters	p. 612
Vertical Distribution of Lateral Force	p. 620
Story Shear and Overturning Moment	p. 620
Torsion and P-[Delta] Effect	p. 621
Story Drift Limitations	p. 623
3R[subscript w]/8 Factor	p. 623
UBC-97 and Design Parameters	p. 624
Criteria for Appropriate Lateral-Force Procedure	p. 624
Base Shear of Static Lateral-Force Procedure and Related Parameters	p. 624
R[subscript w] and R Relationship vs Load Combination	p. 626
Load Combination for Strength Design and Allowable Stress Design	p. 627
Story Shear, Overturning Moment and Restoring Moment	p. 631
Story Drift, P-[Delta] Effect and Torsion	p. 632
Relationships Among 3R[subscript w]/8, [Omega subscript 0] and 0.7R[Delta subscript s]	p. 632
IBC-2000 and Design Parameters	p. 633
Criteria for Appropriate Lateral-Force Procedure	p. 633
Base Shear of Equivalent Lateral-Force Procedure and Related Parameters	p. 633
Vertical Distribution of Lateral Force	p. 638
Horizontal Shear Distribution and Overturning Moment	p. 639
Deflection and Story Drift	p. 639
P-[Delta] Effect	p. 640
Summary Comparison of UBC-94, UBC-97 and IBC-2000 Lateral-Force Procedures	p. 641
Numerical Illustrations of Lateral-Force Procedure for UBC-94, UBC-97 and IBC-2000	p. 648
Techniques for Calculating Rigidity Center	p. 672
Method A--Using Individual Member Stiffness for Rigid-floor Shear Buildings	p. 672
Method B--Using Relative Rigidity of Individual Bays for General Buildings	p. 673
Advanced Topics	p. 675
Dynamic Analysis Procedures of UBC-94, UBC-97 and IBC-2000	p. 675
UBC-94 Dynamic Analysis Procedure	p. 675
UBC-97 Dynamic Analysis Procedure	p. 676
IBC-2000 Dynamic Analysis Procedure	p. 678
Regionalized Seismic Zone Maps and Design Response Spectra in UBC-97 and IBC-2000	p. 682
Summary Comparison of UBC-94, UBC-97 and IBC-2000 Dynamic Analysis Procedures	p. 684
Numerical Illustrations of Dynamic Analysis Procedures for UBC-94, UBC-97 and IBC-2000	p. 688
Overview	p. 705
Bibliography	p. 705
Problems	p. 707
Problems	p. 715
Problems	p. 721
Problems	p. 723
Problems	p. 727
Problems	p. 733
Problems	p. 741
Problems	p. 745
Problems	p. 753
Problems	p. 757
Solutions	p. 763
Solutions	p. 767
Solutions	p. 773
Solutions	p. 777
Solutions	p. 783
Solutions	p. 791
Solutions	p. 797
Solutions	p. 799
Solutions	p. 807
Solutions	p. 811
Lagrange's Equation	p. 817
Derivation of Ground Rotational Records	p. 823
Vector Analysis Fundamentals	p. 827
Transformation Matrix Between JCS and GCS	p. 831
Transformation Matrix Between ECS and GCS for Beam Column	p. 843
Transformation Matrix [A'] and Stiffness Matrix [K superscript i subscript eg] of Beam Column with Rigid Zone	p. 851
Computer Program for Newmark Method	p. 855
Computer Program for Wilson-[Theta] Method	p. 863
Computer Program for CQC Method	p. 865
Jain-Goel-Hanson Steel-Bracing Hysteresis Model and Computer Program	p. 875
Takeda Model for RC Columns and Beams and Computer Program	p. 895
Cheng-Mertz Model for Bending Coupling with Shear and Axial Deformations of Low-Rise Shear Walls and Computer Program	p. 913
Bending: Low-Rise Shear Wall Cheng-Mertz Hysteresis Model	p. 913
Shear: Low-Rise Shear Wall Cheng-Mertz Hysteresis Model	p. 932
Axial: Low-Rise Shear Wall Cheng-Mertz Hysteresis Model	p. 952
Cheng-Lou Axial Hysteresis Model for RC Columns and Walls and Computer Program	p. 967
Notation	p. 979
Index	p. 989
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