Matrix analysis of structural dynamics : applications and earthquake engineering / Franklin Y. Cheng.

Format:

Book

Author/Creator:

Cheng, Franklin Y.

Contributor:

Rosengarten Family Fund.

Series:

Civil and environmental engineering ; 4.

Civil and environmental engineering ; 4

Language:

English

Subjects (All):

Structural dynamics.

Earthquake engineering.

Matrices.

Physical Description:

xv, 997 pages : illustrations ; 27 cm.

Place of Publication:

New York : Marcel Dekker, [2001]

Summary:

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

Contents:

1 Characteristics of Free and Forced Vibrations of Elementary Systems 1

1.2 Free Undamped Vibration 1

1.2.1 Motion Equation and Solution 1

1.2.2 Initial Conditions, Phase Angle and Natural Frequency 3

1.2.3 Periodic and Harmonic Motion 6

1.3 Free Damped Vibration 7

1.3.1 Motion Equation and Viscous Damping 7

1.3.2 Critical Damping, Overdamping and Underdamping 9

1.3.3 Logarithmic Decrement and Evaluation of Viscous Damping Coefficient 11

1.4 Forced Undamped Vibration 14

1.4.1 Harmonic Forces 14

1.4.2 Steady-State Vibration and Resonance 15

1.4.3 Impulses and Shock Spectra 19

1.4.4 General Loading

Step Forcing Function Method vs. Duhamel's Integral 24

1.5 Forced Damped Vibration 29

1.5.1 Harmonic Forces 29

1.5.2 Steady-State Vibration for Damped Vibration, Resonant and Peak Amplitude 30

1.5.3 General Loading

Step-Forcing Function Method vs. Duhamel's Integral 32

1.5.4 Transmissibility and Response to Foundation Motion 36

1.6 Evaluation of Damping 42

1.6.1 Equivalent Damping Coefficient Method 42

1.6.2 Amplitude Method and Bandwidth Method 43

2 Eigensolution Techniques and Undamped Response Analysis of Multiple-Degree-of-Freedom Systems 47

2.1.1 Characteristics of the Spring-Mass Model 47

2.1.2 Advantages of the Lumped Mass Model 48

2.2 Characteristics of Free Vibration of Two-Degree-of-Freedom Systems 49

2.2.1 Motion Equations, Natural and Normal Modes 49

2.2.2 Harmonic and Periodic Motion 52

2.3 Dynamic Matrix Equation 54

2.4 Orthogonality of Normal Modes 55

2.5 Modal Matrix for Undamped Vibration 56

2.5.1 Modal Matrices and Characteristics 56

2.5.2 Response to Initial Disturbances, Dynamic Forces and Seismic Excitation 58

2.5.3 Effect of Individual Modes on Response 64

2.5.4 Response to Foundation Movement 67

2.6 Eigensolution for Symmetric Matrix 72

2.6.1 Iteration Method for Fundamental and Higher Modes 72

2.6.2 Proof of Iterative Solution 77

2.6.3 Extraction Technique for Natural Frequencies 80

2.6.4 Choleski's Decomposition Method 81

2.6.5 Generalized Jacobi Method 87

2.6.6 Sturm Sequence Method 95

Part B Advanced Topics 98

2.7 Eigensolution Technique for Unsymmetric Matrix 98

2.7.1 Classification of Cases 99

2.7.2 Iteration Method 100

2.8 Response Analysis for Zero and Repeating Eigenvalues 105

2.8.1 Zero and Repeating Eigenvalue Cases 105

2.8.2 Orthogonality Properties 105

2.8.3 Response Analysis 109

3 Eigensolution Methods and Response Analysis for Proportional and Nonproportional Damping 117

3.2 Response Analysis for Proportional Damping 117

3.2.1 Based on a Modal Matrix 117

3.2.2 Proportional Damping 120

3.3 Evaluation of Damping Coefficients and Factors 121

3.3.1 Two Modes Required 121

3.3.2 All Modes Required 125

3.3.3 Damping Factors from Damping Coefficients 125

3.4 Determination of Proportional and Nonproportional Damping 126

Part B Advanced Topics 128

3.5 Characteristics of Complex Eigenvalues for Nonproportional Damping 128

3.6 Iteration Method for Fundamental and Higher Modes of Complex Eigenvalues 137

3.6.1 Fundamental Mode 137

3.6.2 Orthogonality Condition and Iteration for Higher Modes 139

3.6.3 Step-by-Step Procedures 139

3.7 Response Analysis with Complex Eigenvalues 149

3.8 Relationship Between Undamped, Proportional Damping, and Nonproportional Damping 156

4 Dynamic Stiffness and Energy Methods for Distributed Mass Systems 161

4.2 Derivation of Bernoulli-Euler Equation 161

4.3 Derivation of Dynamic Stiffness Coefficients 166

4.4 Characteristics of Dynamic Stiffness Coefficients 168

4.4.1 Numerals and Curves for Coefficients 168

4.4.2 Rayleigh's Dynamic Reciprocal Principle 171

4.4.3 Muller-Breslau's Principle 173

4.5 Dynamic Stiffness, Load, and Mass Matrices 175

4.5.1 Degree-of-Freedom of Plane Structural Systems 175

4.5.2 Equilibrium Matrices 176

4.5.3 Compatibility Matrices 177

4.5.4 Dynamic Stiffness Matrix 178

4.5.5 Dynamic Load Matrix 179

4.5.6 System Matrix Equation 180

4.6 Derivation of Dynamic Fixed-end Moments and Fixed-end Shears 180

4.6.1 Differential Equations 181

4.6.2 Uniform Load 182

4.6.3 Triangular Load 183

4.6.4 Concentrated Load between Nodes 185

4.6.5 Foundation Movement 186

4.7 Numerical Technique for Eigensolutions 186

4.8 Steady-State Response Analysis 198

4.9 Response for General Forcing Functions with and without Damping 203

4.9.1 Kinetic and Strain Energy 203

4.9.2 Orthogonality Condition 204

4.9.3 Dissipated Energy and Work 205

4.9.4 Response Equations 206

5 Dynamic Stiffness Method for Coupling Vibration, Elastic Media and P-[Delta] Effect 213

5.2 Longitudinal Vibration and Stiffness Coefficients 213

5.3 Longitudinal Vibration and Stiffness Coefficients with Elastic Media 214

5.4 Dynamic Analysis of Trusses and Elastic Frames 216

5.4.1 Dynamic Stiffness Coefficients of Pin-connected Member 216

5.4.2 Dynamic Stiffness Matrix of Trusses 218

5.4.3 Dynamic Stiffness Matrix of Elastic Frames 221

5.4.4 Coupling of Longitudinal and Flexural Vibration 224

5.5 Torsional Vibration and Stiffness Coefficients 229

5.6 Dynamic Stiffness Matrix of Grid Systems 230

5.7 Coupling of Torsional and Flexural Vibration 233

Part B Advanced Topics 237

5.8 Bernoulli-Euler Equation with Elastic Media 237

5.9 Bernoulli-Euler Equation with Elastic Media and P-[Delta] Effect 238

5.10 Timoshenko Equation (Bending and Shear Deformation and Rotatory Inertia) 240

5.10.1 Differential Equations 240

5.10.2 Stiffness Coefficients 243

5.10.3 Fixed-end Forces for Steady-State Vibration 246

5.10.4 Response Analysis for General Forcing Functions 247

5.10.5 Effect of Various Parameters on Frequencies 252

5.11 Timoshenko Equation with Elastic Media and P-[Delta] Effect 252

5.11.1 Differential Equations 253

5.11.2 Stiffness Coefficients 255

5.11.3 Fixed-end Forces 256

5.11.4 Case Studies of the Effect of Various Parameters on Frequencies 257

6 Consistent Mass Method for Frames and Finite Elements 261

6.2 Energy Method for Motion Equation 262

6.2.1 Rigid Frames 263

6.2.2 Elastic Frames 265

6.3 Stiffness, Mass and Generalized Force Matrices for Frame Members 265

6.3.1 Two-Force Member 265

6.3.2 Torsional Member 268

6.3.3 Flexural Member 270

6.4 Eigenvalue Comparisons Among Lumped Mass, Dynamic Stiffness and Consistent Mass Methods 283

Part B Advanced Topics 285

6.5 Stiffness, Mass and Generalized Force Matrices for Finite Elements 285

6.5.1 Finite Element Formulation

Generalized Coordinates 286

6.5.2 Finite Element Formulation

Natural Coordinates 291

6.6 Motion Equation with P-[Delta] Effect 303

6.6.1 Potential Energy and Motion Equation 303

6.6.2 Geometric Matrix with Rotation and Deflection 305

6.6.3 Geometric Matrix (String Stiffness) with Deflection 305

6.7 Timoshenko Prismatic Member with P-[Delta] Effect 306

6.7.1 Displacement and Shape Functions 306

6.7.2 Stiffness Matrix 308

6.7.3 Mass Matrix 309

6.7.4 Generalized Force Matrix 312

6.7.5 Geometric Matrix 312

6.8 Timoshenko Tapered Member with P-[Delta] Effect 314

6.8.1 Stiffness Matrix 314

6.8.2 Mass Matrix 315

6.8.3 Generalized Force Matrix 317

6.8.4 Geometric Matrix 317

6.9 Comments on Lumped Mass, Consistent Mass, and Dynamic Stiffness Models 318

7 Numerical Integration Methods and Seismic Response Spectra for Single-and Multi-Component Seismic Input 321

7.2 Earthquakes and Their Effects on Structures 321

7.2.1 Earthquake Characteristics 321

7.2.2 Intensity, Magnitude, and Acceleration of Earthquakes 322

7.2.3 Relationship Between Seismic Zone, Acceleration, Magnitude, and Intensity 326

7.2.4 Earthquake Principal Components 327

7.3 Numerical Integration and Stability 329

7.3.1 Newmark Integration Method 329

7.3.2 Wilson-[Theta]

Method 332

7.3.3 General Numerical Integration Related to Newmark and Wilson-[Theta] Methods 334

7.3.4 Runge-Kutta Fourth-Order Method 338

7.3.5 Numerical Stability and Error of Newmark and Wilson-[Theta] Methods 350

7.3.6 Numerical Stability of Runge-Kutta Fourth-Order Method 358

7.4 Seismic Response Spectra for Analysis and Design 361

7.4.1 Response Spectra, Pseudo-Spectra and Principal-Component Spectra 362

7.4.2 Housner's Average Design Spectra 369

7.4.3 Newmark Elastic Design Spectra 371

7.4.4 Newmark Inelastic Design Spectra 372

7.4.5 Site-Dependent Spectra and UBC-94 Design Spectra 378

7.4.6 Various Definitions of Ductility 380

Part B Advanced Topics 383

7.5 Torisonal Response Spectra 383

7.5.1 Ground Rotational Records Generation 383

7.5.2 Construction of Torsional Response Spectra 389

7.6 Response Spectra Analysis of a Multiple d.o.f. Systems 390

7.6.1 SRSS Modal Combination Method 393

7.6.2 CQC Modal Combination Method 394

7.6.3 Structural Response Due to Multiple-Component Seismic Input 397

7.7 Maximum (Worst-Case) Response Analysis for Six Seismic Components 399

7.7.1 Based on SRSS Method 400

7.7.2 Based on CQC Method 404

7.8 Composite Translational Spectrum and Torsional Spectrum 410

7.8.1 Construction of the Composite Response Spectrum 411

7.8.2 Composite Spectral Modal Analysis 412

8 Formulation and Response Analysis of Three-Dimensional Building Systems with Walls and Bracings 417

Part A Fundamentals 417

8.2 Joints, Members, Coordinate Systems, and Degree of Freedom (d.o.f.) 417

8.3 Coordinate Transformation Between JCS and GCS: Methods 1 and 2 418

8.4 Force Transformation Between Slave Joint and Master Joint 424

8.5 System d.o.f. as Related to Coordinate and Force Transformation 426

8.6 Beam-Columns 429

8.6.1 Coordinate Transformation Between ECS and JCS or GCS: Methods 1 and 2 429

8.6.2 Beam-Column Stiffness in the ECS 431

8.6.3 Beam-Column Stiffness in the JCS or GCS Based on Method 1 434

8.6.4 Beam-Column Geometric Matrix (String Stiffness) in ECS and JCS or GCS Based on Method 1 438

8.7 Shear Walls 439

8.7.1 Shear-Wall ECS and GCS Relationship Based on Method 1 439

8.7.2 Shear-Wall Stiffness in the ECS 441

8.7.3 Shear-Wall Stiffness in the JCS or GCS Based on Method 1 447

8.7.4 Shear-Wall Geometric Matrix (String Stiffness) in the JCS or GCS Based on Method 1 455

8.8 Bracing Elements 455

8.8.1 Bracing-Element ECS and GCS Relationship Based on Method 1 455

8.8.2 Bracing-Element Stiffness in ECS 457

8.8.3 Bracing-Element Stiffness in the JCS or GCS Based on Method 1 457

8.9 Structural Characteristics of 3-D Building Systems 462

8.10 Rigid Zone Between Member End and Joint Center 462

8.11 Building-Structure-Element Stiffness with Rigid Zone 464

8.11.1 Beam-Column Stiffness in ECS Based on Method 2 464

8.11.2 Beam-Column Stiffness in GCS Based on Method 2 466

8.11.3 Beam-Column Geometric Matrix (String Stiffness) in JCS or GCS Based on Method 2 473

8.11.4 Beam Stiffness in the GCS Based on Method 2 475

8.11.5 Bracing-Element Stiffness in the JCS or GCS Based on Method 2 479

8.11.6 Shear-Wall Stiffness in the JCS or GCS Based on Method 2 482

8.11.7 Shear-Wall Geometric Matrix (String Stiffness) in the JCS or GCS Based on Method 2 487

Part B Advanced Topics 490

8.12 Assembly of Structural Global Stiffness Matrix 490

8.12.1 General System Assembly (GSA) 490

8.12.2 Floor-by-Floor Assembly (FFA) 498

8.13 Mass Matrix Assembly 504

8.14 Loading Matrix Assembly 508

8.14.1 Vertical Static or Harmonic Forces 509

8.14.2 Lateral Wind Forces 511

8.14.3 Lateral Dynamic Loads 513

8.14.4 Seismic Excitations 514

8.15 Analysis and Response Behavior of Sample Structural Systems 516

9 Various Hysteresis Models and Nonlinear Response Analysis 527

9.1.1 Material Nonlinearity and Stress-Strain Models 528

9.1.2 Bauschinger Effect on Moment-Curvature Relationship 528

9.2 Elasto-Plastic Hysteresis Model 529

9.2.1 Stiffness Matrix Formulation 532

9.3 Bilinear Hysteresis Model 534

9.3.1 Stiffness Matrix Formulation 535

9.4 Convergence Techniques at Overshooting Regions 538

9.4.1 State of Yield and Time-Increment Technique 538

9.4.2 Unbalanced Force Technique 539

9.4.3 Equilibrium and Compatibility Checks for Numerical Solutions 552

9.5 Curvilinear Hysteresis Model 555

9.5.1 Stiffness Matrix Formulation 556

9.5.2 Stiffness Comparison Between Bilinear and Curvilinear Models 560

9.6 Ramberg-Osgood Hysteresis Model 562

9.6.1 Parameter Evaluations of Ramberg-Osgood Stress-Strain Curve 562

9.6.2 Ramberg-Osgood Moment-Curvature Curves 563

9.6.3 Stiffness Matrix Formulation for Skeleton Curve 565

9.6.4 Stiffness Matrix Formulation for Branch Curve 570

Part B Advanced Topics 579

9.7 Geometric Nonlinearity 579

9.8 Interaction Effect on Beam Columns 589

9.9 Elasto-Plastic Analysis of Consistent Mass Systems 591

9.9.1 Stiffness Matrix Formulation 591

9.9.2 Moments, Shears and Plastic Hinge Rotations 595

9.10 Hysteresis Models of Steel Bracing, RC Beams, Columns and Shear Walls 604

10 Static and Dynamic Lateral-Force Procedures and Related Effects in Building Codes of UBC-94, UBC-97 and IBC-2000 607

10.2 Background of Lateral Force Procedures in Building Codes 608

10.2.1 Effective Earthquake Force and Effective Mass 608

10.2.2 Base Shear and Overturning Moment 610

10.3 UBC-94 and Design Parameters 612

10.3.1 Criteria for Appropriate Lateral-Force Procedure 612

10.3.2 Base Shear of Static Lateral-Force Procedure and Related Parameters 612

10.3.3 Vertical Distribution of Lateral Force 620

10.3.4 Story Shear and Overturning Moment 620

10.3.5 Torsion and P-[Delta] Effect 621

10.3.6 Story Drift Limitations 623

10.3.7 3R[subscript w]/8 Factor 623

10.4 UBC-97 and Design Parameters 624

10.4.1 Criteria for Appropriate Lateral-Force Procedure 624

10.4.2 Base Shear of Static Lateral-Force Procedure and Related Parameters 624

10.4.3 R[subscript w] and R Relationship vs Load Combination 626

10.4.4 Load Combination for Strength Design and Allowable Stress Design 627

10.4.5 Story Shear, Overturning Moment and Restoring Moment 631

10.4.6 Story Drift, P-[Delta] Effect and Torsion 632

10.4.7 Relationships Among 3R[subscript w]/8, [Omega subscript 0] and 0.7R[Delta subscript s] 632

10.5 IBC-2000 and Design Parameters 633

10.5.1 Criteria for Appropriate Lateral-Force Procedure 633

10.5.2 Base Shear of Equivalent Lateral-Force Procedure and Related Parameters 633

10.5.3 Vertical Distribution of Lateral Force 638

10.5.4 Horizontal Shear Distribution and Overturning Moment 639

10.5.5 Deflection and Story Drift 639

10.5.6 P-[Delta] Effect 640

10.6 Summary Comparison of UBC-94, UBC-97 and IBC-2000 Lateral-Force Procedures 641

10.7 Numerical Illustrations of Lateral-Force Procedure for UBC-94, UBC-97 and IBC-2000 648

10.8 Techniques for Calculating Rigidity Center 672

10.8.1 Method A

Using Individual Member Stiffness for Rigid-floor Shear Buildings 672

10.8.2 Method B

Using Relative Rigidity of Individual Bays for General Buildings 673

Part B Advanced Topics 675

10.9 Dynamic Analysis Procedures of UBC-94, UBC-97 and IBC-2000 675

10.9.1 UBC-94 Dynamic Analysis Procedure 675

10.9.2 UBC-97 Dynamic Analysis Procedure 676

10.9.3 IBC-2000 Dynamic Analysis Procedure 678

10.9.4 Regionalized Seismic Zone Maps and Design Response Spectra in UBC-97 and IBC-2000 682

10.10 Summary Comparison of UBC-94, UBC-97 and IBC-2000 Dynamic Analysis Procedures 684

10.11 Numerical Illustrations of Dynamic Analysis Procedures for UBC-94, UBC-97 and IBC-2000 688

Appendix A Lagrange's Equation 817

Appendix B Derivation of Ground Rotational Records 823

Appendix C Vector Analysis Fundamentals 827

Appendix D Transformation Matrix Between JCS and GCS 831

Appendix E Transformation Matrix Between ECS and GCS for Beam Column 843

Appendix F Transformation Matrix [A'] and Stiffness Matrix [K superscript i subscript eg] of Beam Column with Rigid Zone 851

Appendix G Computer Program for Newmark Method 855

Appendix H Computer Program for Wilson-[Theta] Method 863

Appendix I Computer Program for CQC Method 865

Appendix J Jain-Goel-Hanson Steel-Bracing Hysteresis Model and Computer Program 875

Appendix K Takeda Model for RC Columns and Beams and Computer Program 895

Appendix L Cheng-Mertz Model for Bending Coupling with Shear and Axial Deformations of Low-Rise Shear Walls and Computer Program 913

Bending: Low-Rise Shear Wall Cheng-Mertz Hysteresis Model 913

Shear: Low-Rise Shear Wall Cheng-Mertz Hysteresis Model 932

Axial: Low-Rise Shear Wall Cheng-Mertz Hysteresis Model 952

Appendix M Cheng-Lou Axial Hysteresis Model for RC Columns and Walls and Computer Program 967.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Rosengarten Family Fund.

ISBN:

0824703871

OCLC:

43977713