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## 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

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

## Structure of atom

__Contents__Width and Shape of Spectral Lines

Lifetime Broadening

Collision or Pressure Broadening

Doppler Broadening

Atomic Orders of Magnitude

Other important Atomic quantities

The Central Field Approximation

The form of the Central Field

Finding the Central Field

The Central Field Approximation

The Physics of the Wave Functions

Energy

Angular Momentum

Radial wavefunctions

Parity

Multi-electron atoms

Electron Configurations

The Periodic Table

Gross Energy Level Structure of the Alkalis: Quantum Defect

Corrections to the Central Field: Spin-Orbit interaction

The Physics of Spin-Orbit Interaction

Finding the Spin-Orbit Correction to the Energy

The B-Field due to Orbital Motion

The Energy Operator

The Radial Integral

The Angular Integral: Degenerate Perturbation Theory

Degenerate Perturbation theory and the Vector Model

Evaluation of D sˆ · ˆl E using DPT and the Vector Model

Spin Orbit Interaction: Summary

Spin-Orbit Splitting: Alkali Atoms

Spectroscopic Notation

Two-electron Atoms: Residual Electrostatic Effects and LS-Coupling

Magnesium: Gross Structure

The Electrostatic Perturbation

Symmetry

Orbital effects on electrostatic interaction in LS-coupling

Spin-Orbit Effects in 2-electron Atoms

Nuclear Effects on Atomic Structure 37 6.1 Hyperfine Structure

The Magnetic Field of Electrons

Coupling of I and J

Finding the Nuclear Spin, I

Isotope Effects

Selection Rules 42 7.1 Parity

Configuration

Angular Momentum Rules

Atoms in Magnetic Fields 44 8.1 Weak field, no spin

8.2 Weak Field with Spin and Orbit

Anomalous Zeeman Pattern

Polarization of the radiation

Strong fields, spin and orbit

Intermediate fields

Magnetic field effects on hyperfine structure

Weak field

Strong field

X-Rays: transitions involving inner shell electrons 56 9.1 X-ray Spectra

X-ray series

Fine structure of X-ray spectra

X-ray absorption

Auger Effect

High Resolution Laser Spectroscopy 61 10.1 Absorption Spectroscopy

Laser Spectroscopy

Spectral resolution

“Doppler Free” spectroscopy

Crossed beam spectroscopy

Saturation Spectroscopy

Two-photon-spectroscopy

Calibration of Doppler-free Spectra

Comparison of “Doppler-free” Methods

## Structure of atom

After studying this unit you will be
able to

• know about the discovery of electron, proton and neutron and their characteristics;

• describe Thomson, Rutherford and Bohr atomic models;

• understand the important features of the quantum mechanical model of atom;

• understand nature of electromagnetic radiation and Planck’s quantum theory;

• explain the photoelectric effect and describe features of atomic spectra;

• state the de Broglie relation and Heisenberg uncertainty principle;

• define an atomic orbital in terms of quantum numbers;

• state aufbau principle, Pauli exclusion principle and Hund’s rule of maximum multiplicity;

• write the electronic configurations of atoms.

• know about the discovery of electron, proton and neutron and their characteristics;

• describe Thomson, Rutherford and Bohr atomic models;

• understand the important features of the quantum mechanical model of atom;

• understand nature of electromagnetic radiation and Planck’s quantum theory;

• explain the photoelectric effect and describe features of atomic spectra;

• state the de Broglie relation and Heisenberg uncertainty principle;

• define an atomic orbital in terms of quantum numbers;

• state aufbau principle, Pauli exclusion principle and Hund’s rule of maximum multiplicity;

• write the electronic configurations of atoms.

## ATOMIC STRUCTURE pdf file

At the end of this unit you will be able to:

• Calculate the electrostatic and gravitational forces between two bodies or
particles

• State the Heisenberg Uncertainty Principle and calculate the uncertainty
in position or velocity of a particle or body

• Define the de Broglie wavelength and calculate same for particles and
bodies

• Explain interference and diffraction in light and electrons

• Explain the terms wavefunction, Eigenfunction and Hamiltonian operator
as they appear in the Schrödinger Wave Equation

• Sketch the radial wavefunctions for the 1s, 2s and 2p orbitals

• Sketch the Radial Distribution Functions for 1s, 2s and 2p orbitals

• Define and depict radial and angular nodes on orbitals

• Define and give examples of principal, orbital angular momentum,
magnetic and spin quantum numbers
• Calculate the energy of the levels and the emission lines in the hydrogen
atom

• Explain the Orbital Approximation and apply it to the Helium atom

• State the Pauli Exclusion Principle, and rationalize it in terms of the
relative stability of different electronic configurations (e.g. Lithium).

• State Hund’s rule and explain it in terms of the relative stability of the
different electronic configurations of sub-shells (e.g. Carbon)
• Define Cartesian and Spherical Polar coordinates

• State advantages of expressing wavefunctions in Spherical Polar
coordinates

• Define radial wavefunction and angular wavefunction

• Calculate and plot the hydrogen 1s Radial wavefunction

• Define and calculate orbital angular momentum of an electron in different
orbitals

• Define and explain Space Quantization
• Define Ionization Enthalpy and explain its trend across the Li – Ne
period.

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