Intro to Structural Motion, j. Connor, Usa, 2002. 695p

Introduction to Structural Motion Control

ConTPCRv2.fm Page 1 Tuesday, July 9, 2002 9:48 AM

Jerome J. ConnorMassachusetts Institute of Technology

Introduction to Structural Motion Control

Pearson Education, Inc., Upper Saddle River, New Jersey 07458


Library of Congress Data Available

Vice President and Editorial Director, ECS: Marcia J. HortonAcquisitions Editor: Laura FischerEditorial Assistant: Erin KatchmarVice President and Director of Production and Manufacturing, ESM: David W. RiccardiExecutive Managing Editor: Vince OBrienManaging Editor: David A. GeorgeProduction Editor: Rose KernanDirector of Creative Services: Paul BelfantiCreative Director: Carole AnsonArt Director: Jayne CoateCover Designer: Bruce KenselaarArt Editor: Greg DullesManufacturing Manager: Trudy PisciottiManufacturing Buyer: Lisa McDowellSenior Marketing Manager: Holly Stark

All rights reserved. No part of this book may be reproduced, in any form or by any means,without permission in writing from the publisher.

The author and publisher of this book have used their best efforts in preparing this book.These efforts include the development, research, and testing of the theories and programs todetermine their effectiveness. The author and publisher make no warranty of any kind,expressed or implied, with regard to these programs or the documentation contained in thisbook. The author and publisher shall not be liable in any event for incidental or consequen-tial damages in connection with, or arising out of, the furnishing, performance, or use of theseprograms.

Printed in the United States of America10 9 8 7 6 5 4 3 2 1

ISBN 0-13-009138-3

Pearson Education Ltd., LondonPearson Education Australia Pty. Ltd., SydneyPearson Education Singapore, Pte. Ltd.Pearson Education North Asia Ltd., Hong KongPearson Education Canada, Inc., TorontoPearson Educacin de Mexico, S.A. de C.V.Pearson EducationJapan, TokyoPearson Education Malaysia, Pte. Ltd.Pearson Education, Inc., Upper Saddle River, New Jersey

2003 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458


vContents

Preface xiii

1 Introduction 1

1.1 Motivation for Structural Motion Control . . . . . . . . . . . . . . . . . .11.1.1 Limitations of Conventional Structural Design . . . . . . . . . .11.1.2 Motion-based Structural Design and Motion Control. . . . . . .2

1.2 Motion versus Strength Issues for Building-type Structures . . . . . . . .3Example 1.1: Cantilever shear beam . . . . . . . . . . . . . . . .5Example 1.2: Cantilever bending beam. . . . . . . . . . . . . . .6Example 1.3: Quasi-shear beam frame . . . . . . . . . . . . . . .8

1.3 Design of a Single-degree-of-freedom System for Dynamic Loading . . .91.3.1 Response for Periodic Excitation . . . . . . . . . . . . . . . . . .91.3.2 Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.3 Methodology for Acceleration-controlled Design . . . . . . . . 14

Example 1.4: An illustration of acceleration controlled design 161.3.4 Methodology for Displacement-controlled Design . . . . . . . 17

Example 1.5: An illustration of displacement-controlled design 171.3.5 Methodology for Force-controlled Design . . . . . . . . . . . . 18

Example 1.6: Force reduction . . . . . . . . . . . . . . . . . . . 201.4 Design of a Single-degree-of-freedom System for Support Motion . . . 20

1.4.1 Response for Periodic Support Motion. . . . . . . . . . . . . . 201.4.2 Design Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Example 1.7: Controlling acceleration due to ground motion 22Example 1.8: Controlling relative motion due to ground motion 22

1.5 Stiffness Distribution for a Two-degree-of-freedom System . . . . . . . 23Example 1.9: 2DOF system with equal masses. . . . . . . . . . 25

1.6 Control Strategies for Motion-based Design . . . . . . . . . . . . . . . 261.7 Scope of Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

ConnorTOC.fm Page v Tuesday, July 9, 2002 9:43 AM

vi Contents

I Passive Control 38

2 Optimal Stiffness Distribution 39

2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.2 Governing Equations: Transverse Bending of Planar Beams . . . . . . 40

2.2.1 Planar Deformation-Displacement Relations . . . . . . . . . . 412.2.2 Optimal Deformation and Displacement Profiles . . . . . . . . 422.2.3 Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . 432.2.4 Force-Deformation Relations . . . . . . . . . . . . . . . . . . . 44

Example 2.1: Composite sandwich beam . . . . . . . . . . . . . 45Example 2.2: Equivalent rigidities for a discrete truss beam . . 46

2.2.5 Governing Equations for Buildings Modeled as Pseudo Shear Beams . . . . . . . . . . . . . . . . . . . . . . 49

2.3 Stiffness Distribution for a Continuous Cantilever Beam under Static Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Example 2.3: Cantilever beam: quasi-static seismic loading . . 55Example 2.4: Truss beam revisited . . . . . . . . . . . . . . . . 55

2.4 Stiffness Distribution for a Discrete Cantilever Shear Beam: Static Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Example 2.5: 3DOF shear beam. . . . . . . . . . . . . . . . . . 592.5 Stiffness Distribution: Truss under Static Loading . . . . . . . . . . . . 59

Example 2.6: Application of least squares approaches . . . . . 63Example 2.7: Comparison of strength-versus

displacement-based design . . . . . . . . . . . . . . . . . . . 692.6 Stiffness Distribution for a Cantilever Beam: Dynamic Response . . . 73

2.6.1 Rigidity Distribution: Undamped Response . . . . . . . . . . . 732.7 Stiffness Distribution for a Discrete Shear Beam: Dynamic Response 77

Example 2.8: 3DOF shear beam. . . . . . . . . . . . . . . . . . 782.8 Stiffness Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.8.1 Governing Equations: Fundamental Mode Response of a Discrete Shear Beam . . . . . . . . . . . . . . . . . . . . 79

Example 2.9: 3DOF shear beam revisited . . . . . . . . . . . . 802.8.2 Governing Equations: Fundamental Mode Response of a

Continuous Beam83Example 2.10: One-dimensional parameters: continuous beam 86

2.8.3 Stiffness Calibration: Periodic Excitation . . . . . . . . . . . . 87Example 2.11: 3DOF shear beam revisited again . . . . . . . . 88Example 2.12: Stiffness calibration: continuous beam. . . . . . 90Example 2.13: Example 2.12 revisited . . . . . . . . . . . . . . 91

2.8.4 Stiffness Calibration: Seismic Excitation . . . . . . . . . . . . . 922.8.5 Seismic Response Spectra . . . . . . . . . . . . . . . . . . . . . 96

Example 2.14: Example 2.12 revisited for seismic excitation 102Example 2.15: 5DOF shear beam . . . . . . . . . . . . . . . . 104

ConnorTOC.fm Page vi Tuesday, July 9, 2002 9:43 AM

Contents vii

2.9 Examples of Stiffness Distribution for Buildings . . . . . . . . . . . . 1062.10 Stiffness Modification for Seismic Excitation . . . . . . . . . . . . . . 113

2.10.1 Iterative Procedure . . . . . . . . . . . . . . . . . . . . . . . . 1132.10.2 Multiple Mode Response . . . . . . . . . . . . . . . . . . . . 1142.10.3 Building Examples: Iterated Stiffness Distribution . . . . . . 116Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3 Optimal Passive Damping Distribution 136

3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1363.2 Viscous, Frictional, and Hysteretic Damping . . . . . . . . . . . . . . 139

3.2.1 Viscous Damping . . . . . . . . . . . . . . . . . . . . . . . . . 139Example 3.1: Viscous damper . . . . . . . . . . . . . . . . . . 140

3.2.2 Friction Damping . . . . . . . . . . . . . . . . . . . . . . . . . 1423.2.3 Hysteretic Damping . . . . . . . . . . . . . . . . . . . . . . . 144

Example 3.2: Stiffness of a rod hysteretic damper . . . . . . . 146Example 3.3: Stiffness of two hysteretic dampers in series . . 146

3.3 Viscoelastic Material Damping . . . . . . . . . . . . . . . . . . . . . . 147Example 3.4: Viscoelastic damper. . . . . . . . . . . . . . . . 150

3.4 Equivalent Viscous Damping . . . . . . . . . . . . . . . . . . . . . . . 151Example 3.5: Structural and hysteretic damping comparison:

seismic excitation . . . . . . . . . . . . . . . . . . . . . . . 153Example 3.6: Determining d for 3M ISD110 damping

material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1593.5 Damping Parameters: Discrete Shear Beam. . . . . . . . . . . . . . . 159

3.5.1 Damping Systems . . . . . . . . . . . . . . . . . . . . . . . . . 1593.5.2 Rigid Structural Members: Linear Viscous Behavior . . . . . 161

Example 3.7: Example 2.15 revisited . . . . . . . . . . . . . . 1633.5.3 Rigid Structural Members: Linear Viscoelastic Behavior. . . 164

Example 3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Example 3.9: Example 3.7 revisited . . . . . . . . . . . . . . . 166Example 3.10: Viscoelastic damper design . . . . . . . . . . . 167Example 3.11: Hysteretic damper design: diagonal element 169

3.5.4 Flexible Structural Members: Linear Viscoelastic Behavior 170Example 3.12: Coupled spring-damper model . . . . . . . . . 173

3.6 Damping Parameters: Truss Beam . . . . . . . . . . . . . . . . . . . . 1743.6.1 Linear Viscous Behavior . . . . . . . . . . . . . . . . . . . . . 1753.6.2 Linear Viscoelastic Behavior . . . . . . . . . . . . . . . . . . 177

3.7 Damping Distribution for MDOF Systems . . . . . . . . . . . . . . . 1773.7.1 Multimode Free Vibration Response . . . . . . . . . . . . . . 178

Example 3.13: Eigenvalue problem: 2DOF . . . . . . . . . . 181Example 3.14: Modal response for nonproportional damping 187

3.7.2 Stiffness Proportional Viscous Damping . . . . . . . . . . . . 190Example 3.15: Low-rise buildings . . . . . . . . . . . . . . . . 192

ConnorTOC.fm Page vii Tuesday, July 9, 2002 9:43 AM

viii Contents

Example 3.16: Building 4. . . . . . . . . . . . . . . . . . . . . 198Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

4 Tuned Mass Damper Systems 217

4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2174.2 An Introductory Example . . . . . . . . . . . . . . . . . . . . . . . . . 218

Example 4.1: Preliminary design of a TMD for a SDOF system . . . . . . . . . . . . . . . . . . . . . . . . . 220

4.3 Examples of Existing Tuned Mass Damper Systems . . . . . . . . . . 2224.3.1 Translational Tuned Mass Dampers . . . . . . . . . . . . . . 2224.3.2 Pendulum Tuned Mass Damper. . . . . . . . . . . . . . . . . 226

4.4 Tuned Mass Damper Theory for SDOF Systems . . . . . . . . . . . . 2294.4.1 Undamped Structure: Undamped TMD . . . . . . . . . . . . 2294.4.2 Undamped Structure: Damped TMD. . . . . . . . . . . . . . 233

Example 4.2: Design of a TMD for an undamped SDOF system . . . . . . . . . . . . . . . . . . . . . . . . . 245

4.4.3 Damped Structure: Damped TMD . . . . . . . . . . . . . . . 245Example 4.3: Design of a TMD for a damped SDOF system 251

4.5 Case Studies: SDOF Systems . . . . . . . . . . . . . . . . . . . . . . . 2514.6 Tuned Mass Damper Theory for MDOF Systems . . . . . . . . . . . 258

Example 4.4: Design of a TMD for a damped MDOF system 263Example 4.5: Design of TMDs for a simply supported beam 266

4.7 Case Studies: MDOF Systems. . . . . . . . . . . . . . . . . . . . . . . 271Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

5 Base Isolation Systems 286

5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2865.2 Isolation for SDOF Systems. . . . . . . . . . . . . . . . . . . . . . . . 287

5.2.1 SDOF Examples . . . . . . . . . . . . . . . . . . . . . . . . . 2875.2.2 Bearing Terminology . . . . . . . . . . . . . . . . . . . . . . . 2905.2.3 Modified SDOF Model. . . . . . . . . . . . . . . . . . . . . . 2915.2.4 Periodic Excitation: Modified SDOF Model . . . . . . . . . . 296

Example 5.1: Stiffness factors for prescribed structure andbase motion . . . . . . . . . . . . . . . . . . . . . . . . . . 298

5.2.5 Seismic Excitation: Modified SDOF Model . . . . . . . . . . 298Example 5.2: Stiffness parameters: modified SDOF

model of Building Example 2. . . . . . . . . . . . . . . . . 3025.3 Design Issues for Structural Isolation Systems . . . . . . . . . . . . . 303

5.3.1 Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3035.3.2 Rigidity Under Low-level Lateral Loads . . . . . . . . . . . . 3045.3.3 Energy Dissipation/Absorption . . . . . . . . . . . . . . . . . 3065.3.4 Modeling of a Natural Rubber Bearing (NRB) . . . . . . . . 3075.3.5 Modeling of a Lead Rubber Bearing (LRB) . . . . . . . . . . 309

ConnorTOC.fm Page viii Tuesday, July 9, 2002 9:43 AM

Contents ix

5.3.6 Applicability of Base Isolation Systems . . . . . . . . . . . . 3135.4 Examples of Existing Base Isolation Systems . . . . . . . . . . . . . . 313

5.4.1 USC University Hospital . . . . . . . . . . . . . . . . . . . 3145.4.2 Fire Department Command and Control Facility . . . . . . . 3145.4.3 Evans and Sutherland Manufacturing Facility. . . . . . . . . 3155.4.4 Salt Lake City Building . . . . . . . . . . . . . . . . . . . . 3155.4.5 The Toushin 24 Ohmori Building . . . . . . . . . . . . . . . 3165.4.6 Bridgestone Toranomon Building . . . . . . . . . . . . . . . 3165.4.7 San Francisco City Hall . . . . . . . . . . . . . . . . . . . . . 3195.4.8 Long Beach V.A. Hospital . . . . . . . . . . . . . . . . . . . . 320

5.5 Optimal Stiffness Distribution: Discrete Shear Beam . . . . . . . . . 3205.5.1 Scaled Stiffness Distribution. . . . . . . . . . . . . . . . . . . 320

Example 5.3: Scaled stiffness for a 4DOF beam with baseisolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

5.5.2 Fundamental Mode Response . . . . . . . . . . . . . . . . . . 324Example 5.4: Example 5.3 revisited . . . . . . . . . . . . . . . 324

5.5.3 Stiffness Calibration for Seismic Isolation . . . . . . . . . . . 325Example 5.5: Stiffness calibration for Example 5.4 . . . . . . 326

5.6 Optimal Stiffness Distribution: Continuous Cantilever Beam . . . . . 3285.6.1 Stiffness Distribution: Undamped Response. . . . . . . . . . 3285.6.2 Fundamental Mode Equilibrium Equation. . . . . . . . . . . 3365.6.3 Rigidity Calibration: Seismic Excitation . . . . . . . . . . . . 337

Example 5.6: Stiffness calibration: Example Building 2 . . . . 3385.7 Building Design Examples . . . . . . . . . . . . . . . . . . . . . . . . 339

5.7.1 Stiffness Distribution Based on Fundamental Mode Response . . . . . . . . . . . . . . . . . . . . . . . . 339

5.7.2 Stiffness Distribution Including the Contribution of the Higher Modes . . . . . . . . . . . . . . . . . . . . . 345

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

II Active Control 362

6 Introduction to Active Structural Motion Control 363

6.1 The Nature of Active Structural Control. . . . . . . . . . . . . . . . . 3636.6.1 Active versus Passive Control . . . . . . . . . . . . . . . . . . 3636.6.2 The Role of Feedback . . . . . . . . . . . . . . . . . . . . . . 3666.6.3 Computational Requirements and Models for Active Control 367

6.2 An Introductory Example of Quasi-static Feedback Control . . . . . 367Example 6.1: Shape control for uniform loading. . . . . . . . 370Example 6.2: Discrete displacement data. . . . . . . . . . . . 371

6.3 An Introductory Example of Dynamic Feedback Control . . . . . . . 372Example 6.3: Illustrative example: influence of

velocity feedback . . . . . . . . . . . . . . . . . . . . . . . 375

ConnorTOC.fm Page ix Tuesday, July 9, 2002 9:43 AM

x Contents

6.4 Actuator Technologies. . . . . . . . . . . . . . . . . . . . . . . . . . . 3796.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3796.4.2 Force Application Schemes . . . . . . . . . . . . . . . . . . . 3796.4.3 Large-scale Linear Actuators . . . . . . . . . . . . . . . . . . 3836.4.4 Large-scale Adaptive Configuration-based Actuators . . . . 3866.4.5 Small-scale Adaptive Material-based Actuators. . . . . . . . 388

6.5 Examples of Existing Large-scale Active Structural Control Systems 3966.5.1 AMD in Kyobashi Seiwa Building . . . . . . . . . . . . . . . 3976.5.2 AVS Control at Kajima Technical Research Institute . . . . 3976.5.3 DUOX Active-Passive TMD in Ando Nishikicho Building 4016.5.4 ABS: 600-Ton Full-Scale Test Structure . . . . . . . . . . . . 404Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

7 Quasi-Static Control Algorithms 410

7.1 Introduction To Control Algorithms . . . . . . . . . . . . . . . . . . . 4107.2 Active Prestressing of a Simply Supported Beam . . . . . . . . . . . . 411

7.2.1 Passive Prestressing. . . . . . . . . . . . . . . . . . . . . . . . 4117.2.2 Active Prestressing . . . . . . . . . . . . . . . . . . . . . . . . 4137.2.3 Active Prestressing with Concentrated Forces . . . . . . . . . 418

Example 7.1: A single force actuator . . . . . . . . . . . . . . 418Example 7.2: Two force actuators . . . . . . . . . . . . . . . . 421

7.2.4 A General Active Prestressing Methodology . . . . . . . . . 428Example 7.3: Multiple actuators. . . . . . . . . . . . . . . . . 431

7.3 Quasi-Static Displacement Control of Beams . . . . . . . . . . . . . . 4347.3.1 Continuous Least Squares Formulation . . . . . . . . . . . . 4357.3.2 Discrete Least Squares Formulation . . . . . . . . . . . . . . 436

Example 7.4: Cantilever beamLeast squares algorithm . . 4377.3.3 Extended Least Squares Formulation . . . . . . . . . . . . . 443

Example 7.5: Example 7.4 revisited . . . . . . . . . . . . . . . 4457.4 Quasi-Static Control of MDOF Systems . . . . . . . . . . . . . . . . . 447

7.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4477.4.2 Selection of Measures . . . . . . . . . . . . . . . . . . . . . . 448

Example 7.6: Illustrative examples of observability . . . . . . 449Example 7.7: Illustrative examples of controllability . . . . . 451

7.4.3 Least Squares Control Algorithms . . . . . . . . . . . . . . . 452Example 7.8: Example 7.7 revisited . . . . . . . . . . . . . . . 453Example 7.9: Example 7.7 revisited with an extended least

squares algorithm . . . . . . . . . . . . . . . . . . . . . . . 456Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

8 Dynamic Control Algorithms: Time-Invariant Linear Systems 458

8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4678.2 State-Space Formulation: Time-Invariant Linear SDOF Systems . . . 468

ConnorTOC.fm Page x Tuesday, July 9, 2002 9:43 AM

Contents xi

8.2.1 Governing Equations. . . . . . . . . . . . . . . . . . . . . . . 4688.2.2 Free Vibration Uncontrolled Response . . . . . . . . . . . . 4708.2.3 General Solution: Linear Time-Invariant Systems . . . . . . 471

Example 8.1: Equivalence of Equations (8.18) and (8.24) . . 4738.2.4 Stability Criterion. . . . . . . . . . . . . . . . . . . . . . . . . 4748.2.5 Linear Negative Feedback . . . . . . . . . . . . . . . . . . . . 4758.2.6 Effect of Time Delay on Feedback Control . . . . . . . . . . 4778.2.7 Stability Analysis for Time Delay . . . . . . . . . . . . . . . . 481

8.3 Discrete Time Formulation: SDOF Systems. . . . . . . . . . . . . . . 4878.3.1 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . 4878.3.2 Linear Negative Feedback Control . . . . . . . . . . . . . . . 4898.3.3 Stability Analysis for Time-Invariant Linear

Feedback Control . . . . . . . . . . . . . . . . . . . . . . . 490Example 8.2: Stability analysisSDOF system with

no time delay. . . . . . . . . . . . . . . . . . . . . . . . . . 496Example 8.3: Stability analysisSDOF system with

time delay . . . . . . . . . . . . . . . . . . . . . . . . . . . 5038.4 Optimal Linear Feedback: Time-Invariant SDOF Systems . . . . . . 505

8.4.1 Quadratic Performance Index . . . . . . . . . . . . . . . . . . 5058.4.2 An Example: Linear Quadratic Regulator Control Algorithm 5068.4.3 The Continuous Time Algebraic Riccati Equation . . . . . . 5118.4.4 The Discrete Time Algebraic Riccati Equation . . . . . . . . 516

Example 8.4: Solution of the discrete time algebraic Riccatiequation for a SDOF system . . . . . . . . . . . . . . . . . 518

8.4.5 Finite Interval Discrete Time Algebraic Riccati Equation . . 525Example 8.5: Example 8.4 revisited . . . . . . . . . . . . . . . 526

8.4.6 Continuous Time Riccati Differential Equation . . . . . . . . 5278.4.7 Variational Formulation of the Continuous Time

Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . 528Example 8.6: Application to scalar case . . . . . . . . . . . . 533

8.5 State-Space Formulation for MDOF Systems . . . . . . . . . . . . . . 5388.5.1 Notation and Governing Equations. . . . . . . . . . . . . . . 5388.5.2 Free Vibration Response: Time-Invariant

Uncontrolled System . . . . . . . . . . . . . . . . . . . . . 540Example 8.7: Free vibration solution for proportional

damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543Example 8.8: General uncoupled damping . . . . . . . . . . . 545

8.5.3 Orthogonality Properties of the State Eigenvectors . . . . . . 546Example 8.9: Initial conditionsfree vibration response . . . 548

8.5.4 Determination of W and fj . . . . . . . . . . . . . . . . . . . . 5488.5.5 General Solution: Time-Invariant System . . . . . . . . . . . 5508.5.6 Modal State Space Formulation: Uncoupled Damping . . . . 5518.5.7 Modal State Space Formulation: Arbitrary Damping . . . . . 554

Example 8.10: Modal formulationundamped case . . . . . 557

ConnorTOC.fm Page xi Tuesday, July 9, 2002 9:43 AM

xii Contents

Example 8.11: Modal formulationuncoupled damping . . . 558Example 8.12: Modal parameters4DOF system. . . . . . . 559Example 8.13: Modal response for Example 8.12 . . . . . . . 566Example 8.14: Modal response with feedback

for Example 8.12. . . . . . . . . . . . . . . . . . . . . . . . 5698.5.8 Stability Analysis: Discrete Modal Formulation . . . . . . . . 574

Example 8.15: Stability analysis for Example 8.14 . . . . . . . 5778.5.9 Controllability of a Particular Modal Response . . . . . . . . 592

Example 8.16: Controllability analysis for a 20DOF Model 5938.5.10 Observability of a Particular Modal Response. . . . . . . . . 595

Example 8.17: Observability analysis for a 20DOF model . . 5978.6 LQR Control Algorithm: MDOF Time-Invariant Systems . . . . . . 598

8.6.1 Continuous Time Modal Formulation . . . . . . . . . . . . . 5988.6.2 Discrete Time Modal Formulation . . . . . . . . . . . . . . . 6008.6.3 Application Studies: LQR Control . . . . . . . . . . . . . . . 602

Example 8.18: Control force design studies for a 20DOFshear beam . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

Example 8.19: Alternate choice of response measures . . . . 6218.7 Advanced Control Theory Topics . . . . . . . . . . . . . . . . . . . . 623

8.7.1 State Controllability . . . . . . . . . . . . . . . . . . . . . . . 6248.7.2 State Observability . . . . . . . . . . . . . . . . . . . . . . . . 625

Example 8.20: 5DOF Model . . . . . . . . . . . . . . . . . . . 626Example 8.21: Example 8.20 revisited . . . . . . . . . . . . . 627

8.7.3 State Observer . . . . . . . . . . . . . . . . . . . . . . . . . . 6298.7.4 An Introduction to Input-Output Relationships. . . . . . . . 6368.7.5 SDOF Input-Output Relations . . . . . . . . . . . . . . . . . 637

Example 8.22: Frequency domain solution procedure . . . . 6408.7.6 Norm of Functions . . . . . . . . . . . . . . . . . . . . . . . . 6418.7.7 Input-Output Relationships Revisited . . . . . . . . . . . . . 643

Example 8.23: Periodic excitation . . . . . . . . . . . . . . . . 645Example 8.24: Impulsive loading . . . . . . . . . . . . . . . . 646Example 8.25: Seismic excitation . . . . . . . . . . . . . . . . 646Example 8.26: Estimate for , assuming . . . . . 648

8.7.8 MDOF Input-Output Relations . . . . . . . . . . . . . . . . . 650Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654

References 664

Electronic References 672

Bibliography 673

Index 675

h t( ) 1 2 1

ConnorTOC.fm Page xii Tuesday, July 9, 2002 9:43 AM

xiii

Preface

Conventional structural design procedures are generally based on two requirements,namely safety and serviceability. Safety relates to extreme loadings, which have avery low probability of occurring, on the order of 2%, during a structures life, and isconcerned with the collapse of the structure, major damage to the structure and itscontents, and loss of life. Serviceability pertains to medium to large loadings, whichmay occur during the structures lifetime. For service loadings, the structure shouldremain operational (i.e., the structure should suffer minimal damage, and further-more, the motion experienced by the structure should not exceed specified comfortlimits for humans and motion-sensitive equipment mounted on the structure). Typi-cal occurance probabilities for service loads range from 10% to 50%.

Safety concerns are satisfied by requiring the resistance (i.e., strength) of theindividual structural elements to be greater than the demand associated with theextreme loading. Once the structure is proportioned, the stiffness properties arederived and used to check the various serviceability constraints such as elasticbehavior. Iteration is usually necessary for convergence to an acceptable structuraldesign. This approach is referred to as strength-based design since the elements areproportioned initially according to strength requirements.

Applying a strength-based approach for preliminary design is appropriatewhen strength is the dominant design requirement. In the past, most structuraldesign problems have fallen in this category. However, the following developmentshave occurred recently that have limited the effectiveness of the strength-basedapproach. First, the trend toward more flexible structures such as tall buildings andlonger-span horizontal structures has resulted in more structural motion under ser-vice loading, thus shifting the emphasis from safety toward serviceability. Second,some of the new types of facilities such as space platforms and semiconductor manu-facturing centers have more severe design constraints on motion than the typicalcivil structure. For example, in the case of micro-device manufacturing, the environ-ment has to be essentially motion free. Third, recent advances in material science

Preface.fm Page xiii Tuesday, July 9, 2002 9:49 AM

xiv Preface

and engineering have resulted in significant increases in the strength of traditionalcivil engineering materials. However, the material stiffness has not increased at thesame rate. The lag in material stiffness versus material strength has led to a problemwith satisfying the requirements on the various motion parameters. Indeed, for veryhigh-strength materials, the motion requirements control the design. Fourth, experi-ence with recent earthquakes has shown that the cost of repairing structural damagedue to inelastic deformation was considerably greater than anticipated. This findinghas resulted in a trend toward decreasing the reliance on inelastic deformation todissipate energy and controlling the structural response with other types of energydissipation and absorption mechanisms.

Motion-based structural design is an alternate design paradigm that addressesthese issues. The approach takes as its primary objective the satisfaction of motion-related design requirements such as restrictions on displacement and accelerationand seeks the optimal deployment of material stiffness and motion control devicesto achieve these design targets as well as satisfy the constraints on strength. Struc-tural motion control is the enabling technology for motion-based design.

This book provides a systematic treatment of the basic concepts and computa-tional procedures for structural motion control. Examples illustrating the applica-tion of motion control to a wide spectrum of buildings are also presented. Topicscovered include optimal stiffness distributions for building-type structures, the roleof damping in controlling motion, tuned mass dampers, base isolation systems,quasi-static active control, and dynamic feedback control. The targeted audience ispracticing engineers and graduate students.

This work was motivated by the authors interest in the design of structuresfor dynamic excitation and by members of the Structural Engineering Communitywho have been enthusiastic supporters of this design paradigm. The author is par-ticularly indebted to Professor Akira Wada (Tokyo Institute of Technology) andDr. Mamoru Iwata (Nippon Steel Corporation) for their intellectual support. Theircontributions have significantly strengthened the discussion of the practical aspectsof passive control. Without the financial support provided by the School of Engi-neering, Massachusetts Institute of Technology, this work would not have been pos-sible. The author is most appreciative. Special thanks are also due to PetrosKomodromos, George Kokosalakis, Victor Pellon, and Paul Kassabian for theirpainstaking effort in producing the text files.

Lastly, the author is especially grateful for the continuous support of his wife,Barbara, who has enriched his life in many dimensions and provided the stable envi-ronment required for a concentrated effort such as this text.

JEROME J. CONNORMassachusetts Institute of Technology

Preface.fm Page xiv Tuesday, July 9, 2002 9:49 AM

C H A P T E R 1

1

Introduction

1.1 MOTIVATION FOR STRUCTURAL MOTION CONTROL

1.1.1 Limitations of Conventional Structural Design

The word design has two meanings. When used as a verb it is defined as the act of cre-ating a description of an artifact. It is also used as a noun, and in this case is definedas the output of the activity (i.e., the description). In this text, structural design is con-sidered to be the activity involved in defining the physical makeup of the structuralsystem. In general, the designed structure has to satisfy a set of requirements per-taining to safety and serviceability. Safety relates to extreme loadings that have a lowprobability of occuring during a structures life. The concerns here are the collapse ofthe structure, major damage to the structure and its contents, and loss of life. Service-ability pertains to moderate loadings that may occur several times during the struc-tures lifetime. For service loadings, the structure should remain fully operational(i.e., the structure should suffer negligible damage and, furthermore, the motionexperienced by the structure should not exceed specified comfort limits for humansand motion-sensitive equipment mounted on the structure). An example of a humancomfort limit is the restriction on the acceleration; humans begin to feel uncomfort-able when the acceleration reaches about . A comprehensive discussion ofhuman comfort criteria is given by Bachmann and Ammann (1987).

Safety concerns are satisfied by requiring the resistance (i.e., strength) of theindividual structural elements to be greater than the demand associated with theextreme loading. The conventional structural design process proportions the struc-ture based on strength requirements, establishes the corresponding stiffness proper-ties, and then checks the various serviceability constraints such as elastic behavior.Iteration is usually necessary for convergence to an acceptable structural design.This approach is referred to as strength-based design since the elements are propor-tioned according to strength requirements.

0.02g

ConCh01v2.fm Page 1 Thursday, July 11, 2002 3:29 PM

2 Chapter 1 Introduction

Applying a strength-based approach for preliminary design is appropriate whenstrength is the dominant design requirement. In the past, most structural design prob-lems have fallen in this category. However, a number of developments have occurredrecently that have limited the effectiveness of the strength-based approach.

First, the trend toward more flexible structures such as tall buildings andlonger span horizontal structures has resulted in more structural motion under ser-vice loading, thus shifting the emphasis from safety toward serviceability. Forinstance, the wind-induced lateral deflection of the Empire State Building in NewYork City, one of the earliest tall buildings in the United States, is several incheswhereas the wind-induced lateral deflection of the former World Trade Center tow-ers was several feet, an order of magnitude increase. This difference is due mainly tothe increased height and slenderness of the former World Trade Center towers incomparison with the Empire State tower. Furthermore, satisfying the limitation onacceleration is a difficult design problem for tall, slender buildings.

Second, some of the new types of facilities such as space platforms and micro-structure manufacturing centers have more severe design constraints on motionthan the typical civil structure. In the case of microdevice manufacturing, the envi-ronment has to be essentially motion free. Space platforms used to support mirrorshave to maintain a certain shape to a small tolerance in order for the mirror to prop-erly function. The design strategy for motion-sensitive structures is to proportion themembers based on the stiffness needed to satisfy the motion constraints, and thencheck if the strength requirements are satisfied.

Third, recent advances in material science and engineering have resulted insignificant increases in the strength of traditional civil engineering materials such assteel and concrete, as well as a new generation of composite materials. Although thestrength of structural steel has essentially doubled, its elastic modulus has remainedconstant. Also, there has been some increase in the elastic modulus for concrete, butthis improvement is still small in comparison with the increment in strength. The lagin material stiffness versus material strength has led to a problem with satisfying theserviceability requirements on the various motion parameters. Indeed, for very highstrength materials, it is possible for the serviceability requirements to be dominant.Examples presented in the following sections illustrate this point.

1.1.2 Motion-based Structural Design and Motion Control

Motion-based structural design is an alternate design process that is more effectivefor the structural design problems just described. This approach takes as its primaryobjective the satisfaction of motion-related design requirements, and views strengthas a constraint, not as a primary requirement. Motion-based structural designemploys structural motion control methods to deal with motion issues. Structuralmotion control is an emerging engineering discipline concerned with the broadrange of issues associated with the motion of structural systems, such as the specifi-cation of motion requirements governed by human and equipment comfort and theuse of energy storage, dissipation, and absorption devices to control the motiongenerated by design loadings. Structural motion control provides the conceptionalframework for the design of structural systems, where motion is the dominant


Section 1.2 Motion Versus Strength Issues for Building-type Structures 3

design consideration. Generally, one seeks the optimal deployment of material andmotion control mechanisms to achieve the design targets on motion as well as sat-isfy the constraints on strength.

In what follows, examples are presented that reinforce the need for an alter-nate design paradigm having motion rather than strength as its primary focus andillustrate the application of structural motion control methods to simple structures.The first three examples deal with the issue of strength versus serviceability from astatic perspective for building-type structures. The discussion then shifts to thedynamic regime. A single-degree-of-freedom (SDOF) system is used to introducethe strategy for handling motion constraints for dynamic excitation. The last exam-ple extends the discussion further to multi-degree-of-freedom (MDOF) systems andillustrates how to deal with one of the key issues of structural motion control, deter-mining the optimal stiffness distribution. Following the examples, an overview ofstructural motion control methodology is presented.

1.2 MOTION VERSUS STRENGTH ISSUES FOR BUILDING-TYPE STRUCTURES

Building configurations must simultaneously satisfy the requirements of site (loca-tion and geometry), building functionality (occupancy needs), appearance, and eco-nomics. These requirements significantly influence the choice of the structuralsystem and the corresponding design loads. Buildings are subjected to two types ofloadings: gravity loads, consisting of the actual weight of the structural system andthe material, equipment, and people contained in the building; and lateral loads,consisting mainly of wind and earthquake loads. Both wind and earthquake loadingsare dynamic in nature and produce significant amplification over their static coun-terpart. The relative importance of wind versus earthquake depends on the site loca-tion, building height, and structural makeup. For steel buildings, the transition fromearthquake dominant to wind dominant loading for a seismically active region occurswhen the building height reaches approximately . Concrete buildings, becauseof their larger mass, are controlled by earthquake loading up to at least a height of

, since the additional gravity load increases the seismic forces. In regionswhere the earthquake action is low (e.g., Chicago, Illinois), the transition occurs at amuch lower height, and the design is governed primarily by wind loading.

When a low-rise building is designed for gravity loads, it is very likely that theunderlying structure can carry most of the lateral loads. As the building heightincreases, the overturning moment and lateral deflection resulting from the lateralloads increase rapidly, requiring additional material over and above that needed forthe gravity loads alone. Figure 1.1 illustrates how the unit weight of the structuralsteel required for the different loadings varies with the number of floors. There is asubstantial structural weight cost associated with lateral loading for tall buildings(Taranath, 1988).

To illustrate the dominance of motion over strength as the slenderness of thestructure increases, the uniform cantilever beam shown in Figure 1.2 is considered.The lateral load is taken as a concentrated force applied to the tip of the beam andis assumed to be static. The limiting cases of a pure shear beam and a pure bendingbeam are examined.

150 m

250 m

p



FIGURE 1.1: Structural steel quantities for gravity and wind systems.

FIGURE 1.2: Building modeled as a uniform cantilever beam.

0 1 2Relative weights of structural steel/unit floor area

3 4 50

20

40

60

80

100

120

140

Num

ber

of fl

oors

Gravity loads Lateral loads

FloorColumns

H

pu

d

d

aa

Section aa

w



Example 1.1: Cantilever shear beam

The shear stress is given by

(1.1)

where is the cross-sectional area over which the shear stress can be consideredto be constant. When the bending rigidity is very large, the displacement, , at thetip of the beam is due mainly to shear deformation and can be estimated as

(1.2)

where is the shear modulus and is the height of the beam. This model is calleda shear beam. The shear area needed to satisfy the strength requirement followsfrom Eq. (1.1):

(1.3)

where is the allowable stress. Noting Eq. (1.2), the shear area needed to satisfythe serviceability requirement on displacement is

(1.4)

where denotes the allowable displacement. The ratio of the area required to sat-isfy serviceability to the area required to satisfy strength provides an estimate of therelative importance of the motion design constraints versus the strength design con-straints

(1.5)

Figure E1.1 shows the variation of r with . Increasing places moreemphasis on the motion constraint since it corresponds to a decrease in the allow-able displacement, . Furthermore, an increase in the allowable shear stress, ,also increases the dominance of the displacement constraint.

p

As------=

Asu

upH

GAs-----------=

G H

As strengthp-----

As serviceabilitypG----- H

u------

u

rAs serviceability

As strength--------------------------------

G----- H

u------= =

H u H u

u



Example 1.2: Cantilever bending beam

When the shear rigidity is very large, shear deformation is negligible, and the beamis called a bending beam. The maximum bending moment in the structureoccurs at the base and equals

(1.6)

The resulting maximum stress is

(1.7)

where is the section modulus, is the moment of inertia of the cross section aboutthe bending axis, and is the depth of the cross section (see Figure 1.2). The corre-sponding displacement at the tip of the beam becomes

(1.8)

The moment of inertia needed to satisfy the strength requirement is given by

(1.9)

Using Eq. (1.8), the moment of inertia needed to satisfy the serviceability require-ment is

(1.10)

FIGURE E1.1: Plot of versus for a pure shear beam.

100 200 300 400

r

Hu*

1*

1*2

*

r H u

M

M pH=

MS----- Md

2I---------

pHd2I

------------= = =

S Id

u

upH3

3EI-----------=

IstrengthpHd2------------

IserviceabilitypH3

3Eu--------------



Here, and denote the allowable displacement and stress respectively. Theratio of the moment of inertia required to satisfy serviceability to the moment ofinertia required to satisfy strength has the form

(1.11)

Figure E1.2 shows the variation of with for a constant value of theaspect ratio ( for tall buildings). Similar to the case of the shear beam,an increase in places more emphasis on the displacement since it correspondsto a decrease in the allowable displacement, , for a constant H. Also, an increasein the allowable stress, , increases the importance of the displacement constraint.

For example, consider a standard strength steel beam with an allowable stressof , a modulus of elasticity of , and an aspect ratioof . The value of at which a transition from strength to serviceabilityoccurs is

(1.12)

For , and motion controls the design. On the other hand, if high-strength steel is utilized ( and ),

(1.13)

and motion essentially controls the design for the full range of allowable displacement.

FIGURE E1.2: Plot of versus for a pure bending beam.

u

rIserviceability

Istrength--------------------------

pH3

3Eu-------------- 2

pHd------------ 2H

3d--------

E------ H

u------ = = =

r H uH d H d 7

H uu

200 MPa= E 200,000 MPa=H d 7= H u

Hu------

r 1=

32--- E

------ d

H----- 200 =

H u 200> r 1> 400 MPa= E 200,000 MPa=

Hu------

r 1=

100

100 200 300 400

r

Hu*

1*

1*2

*

r H u



Example 1.3: Quasi-shear beam frame

This example compares strength vs. motion based design for a single bay frame ofheight and load (see Figure E1.3). For simplicity, a very stiff girder is assumed,resulting in a frame that displays quasi-shear beam behavior. Furthermore, the col-umns are considered to be identical, each characterized by a modulus of elasticity,

, and a moment of inertia about the bending axis, . The maximum moment, , in each column is equal to

(1.14)

The lateral displacement of the frame under the load is expressed as

(1.15)

where denotes the equivalent shear rigidity, which, for this structure, is given by

(1.16)

The strength constraint requires that the maximum stress in the column be lessthan the allowable stress :

(1.17)

where represents the depth of the column in the bending plane. Equation (1.17) iswritten as

(1.18)

FIGURE E1.3: Quasi-shear beam example.

H p

Ec IcM

MpH4

--------=

H Ec, Ic Ec, Ic

Ig u

p2

p2

upHDT--------=

DT

DT24Ec Ic

H2-----------------=

Md2Ic---------

pHd8Ic

------------ =

d

Ic strengthpHd8------------


Section 1.3 Design of a Single-degree-of-freedom System for Dynamic Loading 9

The serviceability requirement constrains the maximum displacement to beless than the allowable displacement :

(1.19)

The corresponding requirement for is

(1.20)

Forming the ratio of the moment of inertia required to satisfy the serviceabilityrequirement to the moment of inertia required to satisfy the strength requirement,

(1.21)

leads to the value of for which motion dominates the design:

(1.22)

1.3 DESIGN OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM FOR DYNAMIC LOADING

The previous examples dealt with motion based design for static loading. A similarapproach applies for dynamic loading once the relationship between the excitationand the response is established. The procedure is illustrated for the single-degree-of-freedom (SDOF) system shown in Figure 1.3.

1.3.1 Response for Periodic Excitation

The governing equation of motion of the system has the form(1.23)

FIGURE 1.3: Single-degree-of-freedom system.

u

pH3

24Ec Ic----------------- u

Ic

Ic serviceabilitypH3

24Ecu------------------

rIc serviceability

Ic strength------------------------------

3Ec--------- H

d----- H

u------ = =

H u

Hu------

3Ec--------- d

H-----

mu t( ) cu t( ) ku t( )+ + p t( )=

R p

u

m

k

c



where , , are the mass, stiffness, and viscous damping parameters of the system,respectively; is the applied loading; is the displacement; and is the independenttime variable. The dot operator denotes differentiation with respect to time. Ofinterest is the case where is a periodic function of time. Taking to be sinusoidalin time with frequency ,

(1.24)

the corresponding forced vibration response is given by

(1.25)

where and characterize the response. They are related to the system and loadingparameters as follows:

(1.26)

(1.27)

(1.28)

(1.29)

(1.30)

(1.31)

The term is the displacement response that would occur if the loadingwere applied statically; represents the effect of the time varying nature of theresponse. Figure 1.4 shows the variation of with the frequency ratio, , for vari-ous levels of damping. The maximum value of and corresponding frequencyratio are related to the damping ratio by

(1.32)

m k cp u t

p p

p t( ) p tsin=

u t( ) u t ( )sin=

u

upk--- H1=

H11

1 2[ ]2 2[ ]2+-------------------------------------------------=

km-----=

c2m------------ c

2 km----------------= =

---- m

k-----= =

tan 21 2--------------=

p kH1

H1 H1

H1 max1

2 1 2-------------------------=



(1.33)

When ,

(1.34)

(1.35)

Since is usually small, the maximum dynamic response is significantly greater thanthe static response and is close to 1. For example, for , which corre-sponds to a high level of damping, the peak response is

(1.36)

(1.37)

When the forcing frequency, , is close to the natural frequency, , theresponse is controlled by adjusting the damping. Outside of this region, dampinghas less influence and has essentially no effect for and .

FIGURE 1.4: Plot of versus and .

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

H1

0.0

0.2

0.4

mk

H1

max 1 22=

2



Differentiating twice with respect to time leads to the acceleration, ,

(1.38)

Noting Eq. (1.26), the magnitude of can be written as

(1.39)

where

(1.40)

The variation of with for different damping ratios is shown in Figure 1.5. Notethat the behavior of for small and large is opposite to . The maximum valueof is the same as the maximum value for , but the location (i.e., the corre-sponding value of ) is different. They are related to by

(1.41)

(1.42)

The ratio is the acceleration the mass would experience if it were unrestrainedand subjected to a constant force of magnitude . We can interpret as a modifi-cation factor that takes into account the time-varying nature of the loading and thesystem restraints associated with stiffness and damping.

Once the system and loading are defined (i.e., , , , , and are specified),we determine and compute the peak amplitudes using the following relations:

(1.43)

(1.44)

(1.45)

Note that for periodic response, the acceleration is related to the displace-ment by the square of the forcing frequency. We can also work with insteadof .

u a

a t( ) u t( ) 2u t ( )sin a t ( )sin= = =

a

apk--- 2H1

pm----- H2= =

H2 2H1

4

1 22

22

+

----------------------------------------------= =

H2 H2 H1

H2 H1

max1

1 22---------------------=

H2 max1

2 1 2-------------------------=

p mp H2

m k c p H1

upk--- H1=

a 2u=

H1 H1 m k c, , ,( ) H1 ,( )= =

H2H1



1.3.2 Design Criteria

The design problem differs from analysis in that we start with the mass of the sys-tem, , and the loading characteristics, and , and determine and such thatthe motion parameters, and , satisfy the specified criteria. In general, we havelimits on both displacement and acceleration:

(1.46)

(1.47)

where and are the target design values. In this case, since and are relatedby

(1.48)

we need to determine which constraint controls. If , the acceleration limitcontrols and the optimal solution will be

(1.49)

(1.50)

If, on the other hand, , the displacement limit controls. For this case, theoptimal solution satisfies

FIGURE 1.5: Plot of versus and .

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

H2

0.0

0.2

12

mk

H2

m p k cu a

u u

a a

u a u a

a 2u=

a 2u

ua

2------ u


(1.51)

(1.52)

In what follows, both cases are illustrated.

1.3.3 Methodology for Acceleration-controlled Design

We work with Eq. (1.39). Expressing the target design acceleration as a function ofthe gravitational acceleration,

(1.53)

and defining as

(1.54)

the design constraint takes the form

(1.55)

where is the weight of the system.The totality of possible solutions is contained in the region below .

Figure 1.6 illustrates the region for . For low damping, the intersection of and the curve for a particular value of , , establishes two limiting val-

ues, and . Permissible values of for the damping ratio are

FIGURE 1.6: Possible values of .

u u=

a 2u a


(1.56)

The second region does not exist when .Noting Eq. (1.40), the expressions for and are

(1.57)

These functions are plotted in Figure 1.7 for representative values of . The limiting values of for reduce to

(1.58)

FIGURE 1.7: Plot of and versus and .

0 1 H2 ,[ ] 2 H2 ,[ ]

k

AX P=

X An m A

S 1 Li2

X

f12---XTX=

nm n

f n

A1 A2X1X2

P=

A1 n n X1

X1 X1 BX2+=

X2 XX2

X X1

0

B

IX2+ X CX2+= =


68 Chapter 2 Optimal Stiffness Distribution

Substituting for , the objective function expands to

(2.83)

Requiring to be stationary with respect to leads to the following set of equations:

(2.84)

We solve Eq. (2.84) for and then determines with Eq. (2.81). This solutioncorresponds to an absolute minimum value of (see Strang, 1993). The MATLABstatement, , generates this least squares solution.

The mean value least squares approach works with the deviation from the mean,

(2.85)

For this case, there are m members, and is given by

(2.86)

Using matrix notation, the deviation vector, , can be expressed as

(2.87)

where the entries in row i of D are

(2.88)

The objective function is a quadratic form in .

(2.89)

Substituting for using Eq. (2.87) and enforcing stationarity with respect to leads to the equation for .

(2.90)

X

f12--- X CX2+( )

TX CX2+( )=

f X2 m n

CTCX2 CTX=

X2 X1f

X pinv A( )*P=

ei xi xmean=

xmean

xmean1m----- x1 x2 xm+ + +( )=

e

e DX D X CX2+( )= =

D i i,( ) 1 1m----- =

D i j,( ) 1m----- for j i=

e

f12---eTe=

e X2X2

DTCTCD( )X2 DTCTX=


Section 2.5 Stiffness Distribution: Truss under Static Loading 69

An alternative way of generating the solution is based on using Lagrangianmultipliers to incorporate the constraint equations in the objective function(Strang, 1993). The generalized function is defined as

(2.91)

where is a vector containing the n Lagrangian multipliers. Requiring J to be sta-tionary with respect to both X and leads to the following set of equations:

(2.92)

These equations can be combined,

(2.93)

and solved in a single step using a linear equation solver such as one of the MATLABfunctions.

Example 2.7: Comparison of strength-versus displacement-based design

Consider the three-member truss shown in Figure E2.7a. Suppose members 1 and 3to have the same properties. It follows that and . The member elonga-tions are

(1)

FIGURE E2.7a

J12---eTe T AX P( )+=

12---XT DTD( )X T AX P( )+=

DTDX AT+ 0=AX P =

DTD AT

A 0

X

0

P

=

u1 0= e3 e1

e1 u2 cos=e2 u2=

132

P2, u2

u1



Noting Eq. (1), the elongations are constrained by

(2)

The member flexibility factor, , is defined as

(3)

Then the member force-deformation relation can be expressed as

(4)

Using (4), the geometric compatibility equation expressed in terms of memberforces has the form

(5)

The solutions for and are

(6)

Strength-Based Design

Ideally, we would want the member stresses to be equal to an allowable stress, .However, a state of uniform stress is not possible because of the indeterminatenature of the structure. Noting that

(7)

for a member, Eq. (1) can be written as

(8)

Then assuming the same material for members 1 and 2 and substituting results in the constraint equation for the member stresses:

(9)

e1 e2 cos=

f

f 1k--- L

AE---------= =

e fF=

f1F1 f2F2cos=

F1 F2

F1 cosf1f2----F2=

F2P2

1 2 f1f2----cos2+

-----------------------------------=

a

e L=

L11 cos L22=

L2 L1 cos=

1 cos2( )2=


Section 2.5 Stiffness Distribution: Truss under Static Loading 71

According to Eq. (9), .Let

(10)

represent the design values for the stresses. The member cross-sectional areas are

(11)

Substituting for and in the equilibrium equation,

(12)

leads to a relationship between the cross-sectional areas:

(13)

The allowable solutions lie outside the triangle shaded in Figure E2.7b.The constraint equation is expressed as

(14)

FIGURE E2.7b

1 2

2 a=

1 a cos2=

A2F2a-----=

A1F11-----

F1a cos2--------------------= =

F1 F2

F2 2F1 cos+ P2=

A2 2 cos3[ ]A1+P2a------

A2 A1+ =

A2

Acceptable zone

A10

P2a

(2cos3 )1P2

a



There is no unique solution for the areas. One approach is to apply a least squareapproach to the member areas. Minimizing the function

(15)

subject to the constraint, Eq. (14), results in the following design:

(16)

Motion-Based Design

The member forces are expressed in terms of the displacement, , by using Eqs. (1)and (4).

(17)

Substituting for the forces in Eq. (12) leads to

(18)

This equation is similar to the strength-based constraint equation. The equationsdiffer only with respect to the right-hand sides. Comparing these terms, it followsthat motion-based design controls when

(19)

which translates to

(20)

J A12 A22 A32+ +( )=

A1

2 2+---------------=

A22

2 2+---------------=

u2

F1A1EL1

-----------e1 A1Eu2L1------ cos= =

F2A2EL2

-----------e2 A2Eu2L2------= =

A2 2 cos3[ ]A1+P2L2Eu2-------------

P2L2Eu2-------------

P2a------>

u2* L2aE----- Sv

T 0.1=

Sv 1.2 m s( )= 0.02=

Sv


Section 2.8 Stiffness Calibration 99

FIGURE 2.19: Normalized pseudo spectral velocity plots.

FIGURE 2.20: Pseudo spectral velocity plots for and .

100

Period (s)101

Spec

tral

vel

ocit

y (m

/s)

101

100

101

102

0.02Sv 1.2 m/s

Spectral velocity plots scaled such that Sv 1.2 for 0.02

101

100

Period (s)101

Spec

tral

vel

ocit

y (m

/s)

102

Spectral velocity plots for 0.30

100

101

101

101

0.30Sv 0.47 m/s

100

Period (s)101

Spec

tral

vel

ocit

y (m

/s)

102

Spectral velocity plots for 0.15

100

101

101

101

0.15Sv 0.60 m/s

0.15= 0.30=



The pseudo spectral velocity function based on the bilinear log-log relation-ship can be used to solve Eq. (2.160). Since there is no unique solution, a family ofsolutions for is generated by specifying different values of . For each value of ,the computation reduces to iterating on the following equations:

(2.178)

(2.179)

(2.180)

The analytical expressions developed for the pseudo spectral velocity tend tobe conservative, since the piecewise linear log-log approximation for the averagedesign spectrum is oversimplified. An improvement in the process of establishing can be obtained if an average spectral velocity function is generated by taking theensemble average of the spectral velocity functions, instead of attempting to createan analytical expression. Equation (2.160) can be solved by iterating through theaverage spectral velocity function and finding the values of and that satisfy theequation for a given value of . Figure 2.22 shows average spectral velocity func-tions for various values of . For a given value of , numerical functions similar tothat of Figure 2.21 can be used to determine for a given value of .

FIGURE 2.21: versus .

0 0.05 0.1

S vm

ax (m

/s)

0.15

0.2 0.250

0.2

0.4

0.6

0.8

1

1.2

Svmax

T

T 0.6

T 2q*

Svmax---------------=

T 0.6< Tmax=

T a 1+2q*

Svmax---------------

Tmax( )a=

aSvmax Svmin( )logTmax Tmin( )log

-----------------------------------------=

Sv

TSv



Example 2.14: Example 2.12 revisited for seismic excitation

The properties of the beam considered in Example 2.12 are

FIGURE 2.22: Average pseudo spectral velocity functions.

100

Period T (s)101

S v (m

/s)

100

101

101

0.1

100

Period T (s)101

S v (

m/s

) 100

101

101

0.02, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3

H 100 m=

m 20 000 kg m=

s 0.4=

m 1.082 106 kg=



Using Example 2.10 Eq. (6), the corresponding value of is . The design valuefor is established by specifying the desired peak transverse shear deformation.Taking and noting that [see Eq. (2.129)] results in

An estimate for is determined by first evaluating the right-hand side of Eq.(2.178). If the value is greater than , that estimate is the actual value. If less than

, Eq. (2.179) applies. For this example, the right-hand term is

Taking and ,

The modal damping parameter is determined with

Increasing damping to and taking the corresponding maximum value of according to Figure 2.21,

leads to

We evaluate the rigidity parameters and by substituting for inEqs. (2.102) and (2.103).

1.12q

* 1 200= q* H=

q* *H 0.5 m= =

T0.6

0.6

2q*

Svmax--------------- 2.80

Svmax-----------=

0.02= Svmax 1.2=

T 2.801.2---------- 2.34 s = =

2T------ 2.68 rad/s= =

c 2m 2( ) 0.02( ) 2.68( ) 1.082( ) 106 116 (kN) s m( )= = =

0.1 Sv

0.1 Svmax 0.7= =

T 2.800.7---------- 4.0 s = =

1.57 rad/s=

c 340 (kN) s m( )=

DT DB 2



Example 2.15: 5DOF shear beam

Consider a 5DOF shear beam having equal nodal masses. The fundamental modevector is specified such that the relative nodal displacements for the five elementsare equal. Its form is

The equivalent modal mass follows from Eq. (2.121):

where m is the constant nodal mass. Substituting for and in Eq. (2.114)leads to the following set of stiffness coefficients:

The actual stiffness values depend on (i.e., ).Specializing Eq. (2.124) for seismic excitation and equal nodal masses, we

obtain

Then, noting the definition of ,

The maximum displacement at node 5 (the top level) is considered to be the con-trolling motion measure.

U q*=

* 15---

12345

=

m *( )TM* 2.2 m= =

* M

k'1 15 m= k'2 14 m= k'3 12 m= k'4 9 m= k'5 5 m=

k 2k'=

p mag *i 3mag= =

pmag---------- 3m

2.2m------------ 1.36= = =

u5 maxqmax q*=



Assuming the beam is a model of a building, is related to the height of thebuilding and the story shear deformation.

To illustrate the remaining steps, suppose H and have the following values:

Then, (meters) and the right-hand term in Eq. (2.178) becomes

Case 1 = 0.02Taking and , the value of the preceding term is less than and Eq. (2.179) applies. Noting Figure 2.19, , and . Usingthese values leads to

(N-s/m)

Case 2

= 0.1

Taking for ,

and Eq. (2.178) applies.

Increasing the damping reduces and the required stiffness. In this case, thereduction factor is , a % decrease in stiffness.

u5 max

u5 max*H=

*

H 20 (meters)=

* 1 200=

q* 0.1=

2q*

Svmax--------------- 0.462

Svmax-------------=

0.02= Svmax 1.2= 0.6Tmin 0.1= Svmin 0.085=

T 0.49 s=

12.82 rad/s=

c 2m 1.128 m= =

Svmax 0.7= 0.1=0.462Svmax------------- 0.66=

T 0.66=

9.52 rad/s=

c 4.19 m=

Svmax9.52 12.82( )2 0.55= 45


Section 2.9 Examples of Stiffness Distribution for Buildings 105

The modal damping parameter is related to the system damping matrix by Eq.(2.122):

Given , we need to establish . When linear viscous dampers are connected toadjacent nodes, is similar in form to the stiffness matrix . Using defined pre-viously, the triple matrix product evaluates to

where is the damping coefficient for element . Since there is only one equationrelating the five coefficients, there is no unique solution for the s. Additionalequations can be derived by imposing conditions on the damping ratios for thehigher modes and optimizing cost and performance. Establishing the optimaldamping distribution is addressed in the next chapter.

2.9 EXAMPLES OF STIFFNESS DISTRIBUTION FOR BUILDINGS

In this section, a series of simulation studies for a representative range of build-ings are presented. The primary focus here is on assessing the adequacy of therigidity distribution derived with the single mode expression to control the defor-mation profiles. How the rigidity distribution can be corrected to account for theinfluence of the higher modes is discussed in the following section. Only shearrigidity distributions and shear deformation profiles are presented here. The cor-responding bending rigidity and deformation plots can be found in Abboud-Klink(1995).

The motion-based design data are the uniformly distributed mass , thebuilding height , the s factor relating the shear and bending deformations, andthe values of the maximum allowable shear deformation corresponding to aspecified level of ground excitation as measured by the pseudo spectral velocity

. Table 2-3 lists the levels of excitation considered here and their correspond-ing allowable deformations. For preliminary design, the extreme earthquakelevel is used. Equation (2.178) gives the fundamental period corresponding tothe rigidity distributions based on the fundamental mode response. These distri-butions are generated using Eqs. (2.102) and (2.103). The relevant designparameters for the buildings modelled as a continuous beam are listed inTable 2-4.

c *( )TC *( )=

c CC K *

c1

25------ c1 c2 c3 c4 c5+ + + +( )=

ci ic

mH

Sv



The continuous building models are discretized and the correspondinglumped parameter MDOF models are then subjected to earthquake excitation.The time history response due to seismic excitation is generated by numericallyintegrating the MDOF matrix equilibrium equations. Two accelerograms, El Cen-tro (component S00E) and Taft (component N21E), both scaled to a maximumpseudo spectral velocity of corresponding to the extreme loading condi-tion, are used. Table 2-5 contains data for the first three modes of the MDOFmodels. Stiffness proportional damping is assumed to establish the damping ratiosfor the higher modes. These data are included here to provide background infor-mation needed to interpret the response. An alternative solution procedure basedon working with the equilibrium equations expressed in terms of modal coordi-nates is described in the next section.

TABLE 2-3: Design loads and allowable deformation values

Earthquake level Sv(m/s) *

Service load 0.6 1/400

Extreme load 1.2 1/200

Ultimate load 2.4 1/100

TABLE 2-4: Design parameters for the continuous models

Building #

m (kg/m)

H(m) H/B f* s *

Sv(m/s)

1(%)

T1 (s)

1 20,000 25 2 7 0.15 1/200 0.9 2 0.49

2 20,000 50 3 6 0.25 1/200 1.2 2 1.06

3 20,000 100 4 5 0.40 1/200 1.2 2 2.34

4 20,000 200 5 4 0.63 1/200 1.2 2 5.35

TABLE 2-5: Modal parameters for the discrete models

T1 (s) T2(s) T3(s) 1(%%%%) 2(%%%%) 3(%%%%) 2/1 2/1Bldg #1 0.52 0.21 0.13 2.00 5.04 8.17 0.40 0.23

Bldg #2 1.09 0.43 0.26 2.00 5.09 8.36 0.43 0.25

Bldg #3 2.39 0.92 0.55 2.00 5.18 8.64 0.45 0.26

Bldg #4 5.48 2.06 1.21 2.00 5.32 9.06 0.48 0.27

1.2 m/s



Figures 2.23 through 2.30 show the generated shear rigidity distributions andthe resulting maximum shear deformations for the different cases. The verticaldashed line in the deformation plots indicates the target maximum deformation; thesolid line shows the maximum deformation due to El Centro excitation; and thedashed line shows the maximum deformation due to Taft excitation. The mean (and ) and standard deviation ( and ) of a deformation profile providemeasures of the adequacy of the rigidity distribution. These values are summarizedin Table 2-6.

The results show that the deviation from the state of uniform deformationoccurs in the upper portion of the building and is far more pronounced in tallbuildings. This trend is attributed to the contributions of the higher modes, whichhave the greatest deformation near the top (see Figure 2.14). For low-rise build-ings, the deformation pattern is essentially dominated by the fundamental moderesponse. A procedure for incorporating the effect of the higher modes on therigidity distribution is presented in the next section. An alternate strategy is toincrease the damping for the fundamental mode. These results are for a low levelof damping ( ). Damping is discussed in the next chapter.

The results also show that the magnitude of the maximum deformation issensitive to the excitation. For example, the difference in the behavior of Build-ing 4 under the two excitations is mainly due to the difference in the responsespectra. For El Centro, the actual value of at a period of about , shown inFigure 2.31, is much less than the value used for the initial rigidity estimate,

, and as a result the actual contribution of the first mode is overesti-mated. The Taft spectrum, on the other hand, shows a closer agreement with thedesign value of and, consequently, the actual and design deformations arecloser together.

TABLE 2-6: Deformation results for the example buildings

El Centro Taft

(103) (103) (105) (106) (103) (104) (105) (106)

Bldg #1 3.42 1.25 3.84 1.14 1.87 1.27 2.18 0.80

Bldg #2 4.40 4.42 4.44 4.10 3.42 6.37 3.61 6.06

Bldg #3 6.13 9.82 4.61 10.06 4.42 16.52 3.61 15.88

Bldg #4 3.50 19.42 2.20 14.22 4.54 7.66 2.74 5.55

mm sd sd

m sd m sd m sd m sd

1 2%=

Sv

5 s

Sv

1.2 m/s=

Sv



FIGURE 2.23: Shear rigidity distribution for Building 1.

FIGURE 2.24: Maximum shear deformation for Building 1.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 109

20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Shear rigidity distribution DT (N)

Nor

mal

ized

hei

ght

x H

Building #1Quadratic basedInitialH 25 m m 20000 kg/ms 0.15Sv 1.2 m/s 1 2%

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

El Centro

0.009 0.010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Taft

Maximum shear deformation (m/m)

Nor

mal

ized

hei

ght

x H

*






0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 109

20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Nor

mal

ized

hei

ght

x H


0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

El Centro

0.009 0.010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Taft


Nor

mal

ized

hei

ght

x H

*




2.10 STIFFNESS MODIFICATION FOR SEISMIC EXCITATION

2.10.1 Iterative Procedure

The numerical studies presented in the previous section show that the method forestablishing the shear and bending rigidities based on the fundamental mode is ade-quate for low-rise buildings but needs to be modified for moderate-rise buildings. Thissection presents a procedure for adjusting the stiffness that is based on updating theshear and bending deformation profiles by including the contribution of the highermodes and then determining improved estimates for the rigidity measures with

(2.181)

(2.182)

The iteration is continued until the change in the rigidity is within the acceptablerange. In what follows, we present details of the procedure for seismic excitationapplied to beam type structures.

FIGURE 2.31: Modified response spectrum for Building 4.

101102

101

100

101

100

Period T (s)

Pse

udo

spec

tral

vel

ocit

y S v

(m

/s)

101

Taft N21E (Scale 3.00)

El Centro S00E (Scale 1.66)

2%

DTi 1+( ) x( ) DTi( )

i( ) x( )[ ]max

----------------------------=

DBi 1+( ) x( ) DBi( )

i( ) x( )[ ]max

-----------------------------=


Section 2.10 Stiffness Modification for Seismic Excitation 113

2.10.2 Multiple Mode Response

Given the mass distribution , we specify the following: , the desired deforma-tion; , the level of earthquake to be designed for; and the parameter relating the shear and bending deformations. An initial estimate of and

is generated using the fundamental mode response approach presented inthe previous sections. The structure is then discretized as an nth-order MDOF sys-tem, leading to the following governing equations:

(2.183)

where , , and represent the mass, damping, and stiffness matrices of the dis-crete system; is the displacement vector relative to the base; and is a vector rep-resenting the rigid body motion of the system due to a unit translation at the base.

Equation (2.183) can be transformed into a set of uncoupled equations byexpressing in terms of a special set of vectors, , and assuming a particularform for C.

(2.184)

These vectors are solutions of the following eigenvalue problem:

(2.185)

and satisfy the orthogonality relation:

(2.186)

where is the dirac delta function. Background material on the eigenvalue prob-lem is presented in Strang, 1993. Assuming to be proportional to (the basis forthis assumption is discussed in Section 3.7),

(2.187)

and noting Eq. (2.186) leads to the following set of uncoupled scalar equations:

(2.188)

m S

v1 1,( ) s

DT0( ) x( )

DB0( ) x( )

MU t( ) CU t( ) KU t( )+ + P t( ) MEag t( )= =

M C KU E

U j

U t( ) j qj t( )j 1=

n

q t( )= =

Kj j2 Mj= j 1, 2, 3, . . . , n=

iTKj j2iTMji j=

i jC K

C K=

qj 2jjq j j2qj+ + jag= j 1, 2, 3, . . . , n=



The damping ratio and modal participation factor for the jth mode are

(2.189)

(2.190)

The individual modal responses are determined following the procedureestablished for the SDOF system. Noting Eq. (2.161), the solution for the jth modalresponse is

(2.191)

(2.192)

(2.193)

Finally, the relative displacement vector due to all modal responses is obtained bysuperposition:

(2.194)

To determine an estimate of the earthquake response of a lumped MDOFsystem, we can make use of the response spectra method. The maximum responsefor a mode is obtained from the response spectrum for the SDOF system using thecorresponding values of frequency, damping, and participation factor. Theexpressions for the jth mode are

(2.195)

(2.196)

where is the pseudo spectral velocity corresponding to the damping andfrequency of the jth mode. The shear and bending deformations in the ith elementdue to the maximum response of the jth mode are determined with the followingdifferential approximations:

jj

2---------=

jjTMEjTMj--------------------=

qj t( )j'j------j t( )=

j t( ) ag ( )e jj t ( ) 'j t ( )[ ]sin d0

t=Uj t( ) jqj t( )=

U t( ) q t( ) jjj t( )'j

------------------------j 1=

n

= =

qj maxj'j------S

vj j,( )=

Uj max jqj max=

Sv

j j,( )


Section 2.10 Stiffness Modification for Seismic Excitation 115

(2.197)

(2.198)

where the displacement measures are for the nodes at the base and upper end ofelement i.

The individual modal time histories are generally not in phase, so we need tointroduce an assumption as to the relative phasing. Taking the algebraic sumassumes they are all in phase. Using a square root sum of squares (SRSS) procedureassumes the phase lag is a uniformly distributed random variable and tends to placemore emphasis on the dominant terms. The latter procedure is discussed here.

The maximum value of a variable, say v, is determined with

(2.199)

where is the number of modes retained for the stiffness modification. This com-putational algorithm is used to determine and for each segment.

We start with , compute the peak modal responses using Eq. (2.195), andthen evaluate the peak deformations with Eqs. (2.197) and (2.198). The rigiditycoefficients are then updated with Eqs. (2.181) and (2.182).

The process is now repeated. Given and , the system stiffnessmatrix is generated, and updated mode shapes, frequencies, and participationfactors are evaluated, leading to revised peak deformation distributions andultimately to new estimates for rigidities.

2.10.3 Building Examples: Iterated Stiffness Distribution

The iterative procedure is applied to the four building examples defined in Table 2-4.Table 2-7 lists the periods for the first three modes of the discretized systemcorresponding to the iterated rigidity distributions. The mean and standard devia-tion of the shear and bending deformations resulting from the El Centro and Taftexcitations are summarized for different iterations in Table 2-8. Results in the formof converged shear rigidity distributions and maximum shear deformation profilesfor El Centro and Taft excitations are provided in Figures 2.32 through 2.37. Theseprofiles illustrate the convergence of the iterative procedure over the spectrum ofbuilding examples starting with the quasi-quadratic distribution as an initial esti-mate for the rigidity. Significant improvement in the maximum deformation profilesis obtained after just one iteration. The contribution o

Documents

Intro to Structural Motion, j. Connor, Usa, 2002. 695p