Department of Civil and Environmental Engineering Stanford...

Department of Civil and Environmental Engineering

Stanford University

PERFORMANCE EVALUATION OF VIBRATION CONTROLLED STEEL STRUCTURES UNDER SEISMIC LOADING

by

Luciana R. Barroso

and

H. Allison Smith

Report No. 133

June 1999

The John A. Blume Earthquake Engineering Center was established to promote research and education in earthquake engineering. Through its activities our understanding of earthquakes and their effects on mankind’s facilities and structures is improving. The Center conducts research, provides instruction, publishes reports and articles, conducts seminar and conferences, and provides financial support for students. The Center is named for Dr. John A. Blume, a well-known consulting engineer and Stanford alumnus. Address: The John A. Blume Earthquake Engineering Center Department of Civil and Environmental Engineering Stanford University Stanford CA 94305-4020 (650) 723-4150 (650) 725-9755 (fax) earthquake @ce. stanford.edu http://blume.stanford.edu

©1999 The John A. Blume Earthquake Engineering Center

PERFORMANCE EVALUATION OFVIBRATION CONTROLLED STEEL

STRUCTURES UNDER SEISMIC LOADING

by

Luciana R. Barrosoand

H. Allison Smith

The John A. Blume Earthquake Engineering Center

Department of Civil and Environmental Engineering

Stanford University

Stanford, CA 94305

Report No. 133

June 1999

c Copyright 1999 by Luciana R. Barroso

All Rights Reserved

ii

Abstract

The structural engineering community has been making great strides in recent years

to develop performance-based earthquake engineering methodologies for both new

and existing construction. For structural control to gain viability in the earth-

quake engineering community, understanding the role of controllers within the con-

text of performance-based engineering is of primary importance. Design of a struc-

ture/controller system should involve a thorough understanding of how various types

of controllers enhance structural performance, such that the most e�ective type of

controller is selected for the given structure and seismic hazard.

The goal of this research is to evaluate the role of structural control technol-

ogy in enhancing the overall structural performance under seismic excitations. This

study focuses on steel moment resisting frames and three types of possible controllers:

(1) friction pendulum base isolation system, FPS (passive); (2) linear viscous brace

dampers, VS (passive); (3) and active tendon braces, ATB. Two structures are se-

lected from the SAC Phase II project, the three story system and the nine story

system. Simulations of these systems, both controlled and uncontrolled, are prepared

using the three suites of earthquake records, also from the SAC Phase II project,

representing three di�erent return periods. Several controllers are developed for each

structure, and their performance is judged based on both roof and interstory drift,

normalized dissipated hysteretic energy, and peak oor acceleration demands.

This investigation has the following speci�c objectives: (1) To evaluate the ef-

fect of the various controller architectures on seismic demands as described through

performance-based design criteria; (2) To evaluate the sensitivity of the structure-

controller performance based on a variation of control parameters, load levels and

structural modeling; (3) To evaluate di�erent systems using a probabilistic format.

iii

The control parameters investigated for the FPS system include the isolation period

and the coeÆcient of friction. These parameters were varied to span a range of pos-

sible values. For the VS damper system, the e�ect of variation in e�ective damping

and its distribution over the height of the structure were evaluated. A representative

ATB control scheme was then designed with actuator saturation levels comparable

to the VS damper system for comparison.

Results indicate that structural control systems are e�ective solutions that can

improve structural performance. All three control strategies investigated can sig-

ni�cantly reduce the seismic demands on a structure, thereby reducing the expected

damage to the structure. However, no one system is consistently the best at all hazard

levels. The viscous system proves to be the most insensitive to modeling assumptions.

The isolation system can maintain the demands close to the structure's elastic limit.

However, the onset of nonlinear behavior decreases the system's e�ectiveness. The

active system is also sensitive to design assumptions, such as output parameters and

structural model parameters used in design. Peak responses alone do not describe the

possible damage incurred by the structure as cumulative damage results from several

incursions into the inelastic range. Accurate evaluations should involve consideration

of the dissipated hysteretic energy.

For isolation systems, selection of isolation period has the greatest impact in the

resulting seismic demands on the superstructure. Lowering the friction coeÆcient can

cause small reductions in drift demands, but the cost of this reduction in structural

demands is an increase in bearing displacements. This system of control proves to

be very e�ective system for both the 3-Story and 9-Story structures and all three

sets of ground motions. The median response of the superstructure remains close to

elastic even under severe ground motions. This system, however, is sensitive to the

sti�ness of the structure, and its e�ectiveness begins to deteriorate once noticeable

nonlinearities occur.

The viscous damper system is very sensitive to both the amount of e�ective damp-

ing provided and the distribution of dampers over the height of the structure. Di�er-

ent damper distributions have little impact on the roof drift. However, by distributing

dampers according to relative story sti�ness and expected peak plastic deformations,

the drift demands are more evenly distributed among the di�erent stories. If the

dampers are located in only a few stories for the same amount of e�ective damping,

iv

however, the system can be highly ine�ective and may increase story demands at

stories without dampers.

The capacity of the actuators for the ATB system contributes greatly to the

e�ectiveness of the control system. Higher actuator capacities provide the controller

a greater opportunity to reduce drift demands. The resulting systems may increase

story drift demands from the uncontrolled system, particularly in stories without

actuators. However, careful design of the control system for the 3-story structure

results in a system that consistently reduces the median story drift demands. The

impact on seismic demands of placing the actuators only at select story locations is

investigated in the 9-story. The result of this placement is that at high level excitations

the drift demands at stories without actuators are increased from the uncontrolled

case.

The use of a probabilistic format allows for a consideration of structural response

over a range of seismic hazards. Stable relationships can be developed between the

spectral acceleration and controlled structural demands. Similar relationships are also

possible for the demands on the control system, such as the peak bearing displacement

for the isolation system. As a result, fewer control analyses may be required to

estimate the expected structural behavior. The resulting annual hazard curves can

be used to evaluate the e�ect of di�erent control parameters as well as provide a basis

for comparison between di�erent control strategies.

v

Acknowledgements

The research presented in this report is based on the doctoral dissertation of Luciana

Barroso. The work presented here would not have been possible without the sup-

port from numerous individuals, a few of whom are presented here. Discussions with

Dr. Steven Winterstein into the extension of the research into the probabilistic realm

greatly in uenced the direction of the project. Dr. Helmut Krawinkler provided valu-

able technical advice and direction into the seismic performance of steel structures.

Special thanks are also due to Dr. Akshay Gupta the technical input and background

information for the case studies. The authors would also like to thank Scott Brene-

man for his collaboration in the development of the analysis software and research into

active control systems for the seismic resistance of steel moment-resisting frames.

vi

Contents

Abstract iii

Acknowledgements vi

List of Tables xii

List of Figures xx

Notation xxi

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objective and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Performance Evaluation of Structures 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Damage to Nonstructural Elements . . . . . . . . . . . . . . . . . . . 6

2.3 Damage to Structural Elements . . . . . . . . . . . . . . . . . . . . . 7

2.4 Damage Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 Maximum Deformation Damage Indices . . . . . . . . . . . . 9

2.4.2 Cumulative Damage Indices . . . . . . . . . . . . . . . . . . . 10

2.4.3 Combined Indices: MaximumDeformation and Cumulative Dam-

age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.4 Maximum Softening Damage Indices . . . . . . . . . . . . . . 13

2.4.5 Weighted Average of Damage Indices . . . . . . . . . . . . . . 13

vii

2.5 Recent Developments in Performance-Based Engineering . . . . . . . 14

2.5.1 Performance Levels . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5.2 Excitation Levels . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5.3 Structural Performance Parameters . . . . . . . . . . . . . . . 16

3 Structural Control in Civil Engineering Structures 18

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.1 Background and Recent Developments in Structural Control . 19

3.1.2 General Classi�cation of Control Systems . . . . . . . . . . . . 20

3.2 Isolation Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Elastomeric Bearings . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.2 Sliding Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Passive Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.1 Viscous and Viscoelastic Dampers . . . . . . . . . . . . . . . . 27

3.3.2 Friction-Slip Dampers . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Active Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4.1 Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4.2 Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5 Role of Structural Control in Performance-Based Engineering . . . . . 37

4 Description of Case Studies 40

4.1 Objective of Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Description of Structures . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Earthquakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 Control Systems Designed and Evaluated . . . . . . . . . . . . . . . . 46

4.4.1 Friction Pendulum Isolation System . . . . . . . . . . . . . . . 47

4.4.2 Fluid Viscous Damper . . . . . . . . . . . . . . . . . . . . . . 48

4.4.3 Active Tendon System . . . . . . . . . . . . . . . . . . . . . . 49

5 Description of Modeling and Analysis 51

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2 Structural Modeling Approach . . . . . . . . . . . . . . . . . . . . . . 52

5.2.1 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . 52

5.2.2 Modeling of P-delta E�ects . . . . . . . . . . . . . . . . . . . 54

viii

5.3 Evaluation Platform and Implementation . . . . . . . . . . . . . . . . 55

5.3.1 Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.3.2 Hysteresis Modeling . . . . . . . . . . . . . . . . . . . . . . . 57

5.3.3 P-M Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3.4 Geometric Nonlinearities: P-Delta . . . . . . . . . . . . . . . . 59

5.3.5 Viscous Damper . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3.6 Friction Pendulum Isolation (FPS) Element . . . . . . . . . . 62

5.3.7 Active Control . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.3.8 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . 65

6 Evaluation of Seismic Demands 68

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.2 Seismic Demands for Uncontrolled System . . . . . . . . . . . . . . . 69

6.3 E�ect of Controller Architecture Design . . . . . . . . . . . . . . . . . 74

6.3.1 FPS Isolation System . . . . . . . . . . . . . . . . . . . . . . . 74

6.3.2 Fluid Viscous-Brace Damper . . . . . . . . . . . . . . . . . . . 80

6.3.3 Active Tendon System . . . . . . . . . . . . . . . . . . . . . . 90

6.4 Comparison of Seismic Demands Across Control Systems . . . . . . . 94

6.4.1 Deformation Demands . . . . . . . . . . . . . . . . . . . . . . 95

6.4.2 Hysteretic Energy Demands . . . . . . . . . . . . . . . . . . . 109

6.4.3 Acceleration Demands . . . . . . . . . . . . . . . . . . . . . . 112

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7 E�ects of Modeling on Seismic Demands 120

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.2 E�ect of Nonlinearities on Controlled Structural Performance . . . . . 121

7.2.1 3-Story Structure . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.2.2 9-Story Structure . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.3 E�ect of Initial Sti�ness on Dynamic Response . . . . . . . . . . . . . 132

7.4 E�ect of Variations in Strain-Hardening Ratio . . . . . . . . . . . . . 137

7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

8 Probabilistic Seismic Control Analysis 141

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

ix

8.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

8.2.1 Probabilistic Seismic Hazard Analysis (PSHA) . . . . . . . . . 142

8.2.2 Probabilistic Seismic Demand Analysis (PSDA) . . . . . . . . 144

8.2.3 Probabilistic Seismic Control Analysis (PSCA) . . . . . . . . . 147

8.3 Spectral Acceleration Hazard . . . . . . . . . . . . . . . . . . . . . . 148

8.4 Relationship between Ground Motion and Demand Parameters . . . . 150

8.4.1 Estimate of Peak Story Drift . . . . . . . . . . . . . . . . . . . 150

8.4.2 Estimate of Control System Demand . . . . . . . . . . . . . . 157

8.4.3 Number of Analyses . . . . . . . . . . . . . . . . . . . . . . . 161

8.5 Drift Demand Hazard Curves . . . . . . . . . . . . . . . . . . . . . . 162

8.5.1 E�ect of Control Parameter Variation . . . . . . . . . . . . . . 162

8.5.2 Comparison Between Control Systems . . . . . . . . . . . . . 168

8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

9 Summary, Conclusions, and Future Work 178

9.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

9.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

9.2.1 Seismic Demands . . . . . . . . . . . . . . . . . . . . . . . . . 179

9.2.2 Modeling E�ects . . . . . . . . . . . . . . . . . . . . . . . . . 181

9.2.3 Probabilistic Seismic Control Analysis . . . . . . . . . . . . . 182

9.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

9.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Appendix A: Response Statistics 187

Bibliography 189

x

List of Tables

2.1 General Structural Performance Level De�nitions and Indicative Drifts

for Steel Moment Frames (FEMA 273). . . . . . . . . . . . . . . . . . 16

2.2 Probabilistic Hazard Levels and Corresponding Return Periods (FEMA

273). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1 Frictional Properties of PTFE in Contact with Polished Stainless Steel 25

3.2 High Capacity Fluid Viscous Dampers from Taylor Devices, Inc. . . . 30

4.1 Column Sections for 9-Story Structure - North-South Frame . . . . . 42

5.1 Modal Properties for Frames . . . . . . . . . . . . . . . . . . . . . . . 54

6.1 Statistics on Roof Drift Angle Demands . . . . . . . . . . . . . . . . 71

6.2 Frictional Properties for Isolator System . . . . . . . . . . . . . . . . 74

6.3 Peak Bearing Response for 3-Story Structure Isolation Bearing: 2 in

50 Set of Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . 76

6.4 Global Demand Parameters for 9-Story Structure with FPS Isolation,

2 in 50 Set of Ground Motions . . . . . . . . . . . . . . . . . . . . . . 80

6.5 Median Response Properties for Viscous Dampers, 2 in 50 Set of Ground

Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.6 Increases in Story Drift Demands for 3-Story Structure due to Addi-

tional Control, 2 in 50 Set of Ground Motions . . . . . . . . . . . . . 105

7.1 E�ect of Modeling on Percent Roof Drift Reduction . . . . . . . . . . 125

8.1 Parameters for Spectral Acceleration Hazard Curve Fit . . . . . . . . 149

xi

8.2 Parameters for Fit of Relationship Between Spectral Acceleration and

Story Drift, 3-Story Structure . . . . . . . . . . . . . . . . . . . . . . 152

8.3 Parameters for Fit of Relationship Between Spectral Value and Story

Drift, 3-Story Structure with FPS Isolation System . . . . . . . . . . 154


Drift, 3-Story Structure with FPS Isolation System - Ignoring Simu-

lated Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . 154


Drift for VS System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155


Drift for ATB System . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.7 Parameters for Fit of Relationship Between Spectral Value and Peak

Damper Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157


Bearing Displacement, 3-Story Structure with FPS Isolation System . 159


Bearing Displacement, 3-Story Structure with FPS Isolation System -

No Simulated Records . . . . . . . . . . . . . . . . . . . . . . . . . . 159

8.10 Parameters for Fit of Relationship Between Spectral Velocity and Story

Drift, Variation in Isolation Period of FPS Isolation System . . . . . . 163


Drift, VS Damping Systems . . . . . . . . . . . . . . . . . . . . . . . 165

8.12 Parameters for Fit of Relationship Between Spectral Value and Bearing

Displacements, Variation in Isolation Period of FPS Isolation System 167

8.13 Parameters Drift Hazard Calculation of Individual Stories, 3-Story

Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

8.14 Parameters Drift Hazard Calculation of Individual Stories, 9-Story

Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

xii

List of Figures

3.1 Free-body Diagram for FPS Isolation System . . . . . . . . . . . . . . 23

3.2 Basic Elements of a Closed-Loop Active Control . . . . . . . . . . . . 33

3.3 Design Process for Controlled Structural Systems . . . . . . . . . . . 39

4.1 3-Story Structure: North-South Moment-Resisting Frame . . . . . . . 42

4.2 9-Story Structure: North-South Moment-Resisting Frame . . . . . . . 43

4.3 Mean Elastic Spectral Acceleration for Ground Motion Sets . . . . . 45

4.4 Dispersion of the Elastic Spectral Acceleration for Ground Motion Sets 46

4.5 3-Story Structure with VS dampers . . . . . . . . . . . . . . . . . . . 49

5.1 Lumped Plasticity Model for Beam-Column Element. . . . . . . . . . 53

5.2 Bilinear diagram for P-M Interaction . . . . . . . . . . . . . . . . . . 59

5.3 P-� Forces Associated with a Gravity Column . . . . . . . . . . . . . 60

5.4 Schematic Diagram for Viscoelastic Damper . . . . . . . . . . . . . . 62

5.5 Flowchart of StructODE function . . . . . . . . . . . . . . . . . . . . 66

5.6 Comparison of Third Story Drift Response under la15 Ground Motion

with DRAIN-2DX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.1 Global Pushover Curves for LA 3- and 9-Story Structures . . . . . . . 70

6.2 Median Values for Peak Story Drift Angle for 3-Story Structure, All

Sets of Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.3 Median Values for Peak Story Drift Angle for 9-Story Structure, All

Sets of Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.4 Dispersion of Peak Story Drift Angle for 3-Story Structure, All Sets of

Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

xiii

6.5 Dispersion of Peak Story Drift Angle for 9-Story Structure, All Sets of

Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.6 Median Values for Peak Roof Drift Angle as Function of Isolation Pe-

riod, 3-Story Structure, 10 in 50 and 2 in 50 Set of Ground Motions . 75

6.7 Peak Bearing Displacements for 3-Story Frame with FPS Isolation, 2

in 50 Set of Ground Motions . . . . . . . . . . . . . . . . . . . . . . . 77

6.8 Median Values for Peak Story Drift Demands for 3-Story Frame with

FPS Isolation, 10 in 50 Set of Ground Motions . . . . . . . . . . . . . 78



6.10 84th Percentile Values for Peak Story Drift Demands for 3-Story Frame

with FPS Isolation, 2 in 50 Set of Ground Motions . . . . . . . . . . 79



6.12 Median Values for Peak Roof Drift Angle for 3-Story Frame as Function

of Percent of Critical Damping, 10 in 50 and 2 in 50 Set of Ground

Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82


Viscous-Brace Dampers D1, 10 in 50 Set of Ground Motions . . . . . 83


Viscous-Brace Dampers D1, 2 in 50 Set of Ground Motions . . . . . . 84

6.15 Dispersion of Peak Story Drift Angle for 3-Story Structure with Vary-

ing Added E�ective Damping Periods . . . . . . . . . . . . . . . . . . 84

6.16 Beam-Column Subassembly for an Interior Column . . . . . . . . . . 85

6.17 Median Values for Peak Roof Drift Demands for 3-Story Frame with

Viscous-Brace Dampers in Di�erent Distributions . . . . . . . . . . . 87

6.18 E�ect of Damping Distribution on Median Values for Peak Story Drift

Demands for 3-Story Frame, 2 in 50 Set of Ground Motions . . . . . 88


Viscous-Brace Dampers D1, 2 in 50 Set of Ground Motions . . . . . . 89

6.20 E�ect of Damping Distribution on Median Values for Peak Story Drift

Demands for 9-Story Frame, 2 in 50 Set of Ground Motions . . . . . 89

xiv


ATB Control, 10 in 50 Set . . . . . . . . . . . . . . . . . . . . . . . . 91

6.22 Median Values for Peak Story Drift Demands for 3-Story Frame - ATB

Control with Varying Saturation, 2 in 50 Set . . . . . . . . . . . . . . 92

6.23 84th Percentile Values for Peak Story Drift Demands for 3-Story Frame

- ATB Control with Varying Saturation, 2 in 50 Set . . . . . . . . . . 92


ATB Control, Variation in Design, 2 in 50 Set . . . . . . . . . . . . . 93

6.25 Dispersion of Peak Story Drift Angle for 3-Story Structure with ATB

Systems of Di�erent Controlled Outputs . . . . . . . . . . . . . . . . 93


ATB Control, 10 in 50 and 2 in 50 Set of Ground Motions . . . . . . 94

6.27 Maximum Values for Peak Story Drift Demands for 3-Story Frame, 50


6.28 Maximum Peak Story Drift Demands for 3-Story Frame, 10 in 50 Set

of Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.29 Maximum Peak Story Drift Demands for 3-Story Frame, 2 in 50 Set of

Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.30 Maximum Residual Story Drift Demands for 3-Story Frame, 50 in 50

Set of Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . 98





6.33 Median Peak Story Drift Demands for 3-Story Frame, 50 in 50 Set of

Earthquakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.34 84th Percentile Values for Peak Story Drift Demands for 3-Story Frame,

50 in 50 Set of Earthquakes . . . . . . . . . . . . . . . . . . . . . . . 102


Earthquakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103



xv


Earthquakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104


2 in 50 Set of Earthquakes . . . . . . . . . . . . . . . . . . . . . . . . 104


Earthquakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106




Earthquakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107







6.45 Comparison of Maximum Peak Story Drift Demands for VS and ATB

Control for 9-Story Structure . . . . . . . . . . . . . . . . . . . . . . 109

6.46 Median Values of Normalized Hysteretic Energy for 3-Story Frame, 50










6.51 Median Values of Floor Accelerations for 3-Story Frame, 50 in 50 Set




xvi

6.53 Median Values of Floor Accelerations for 3-Story Frame, 2 in 50 Set of

Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115



6.55 Median Values of Floor Accelerations for 9-Story Frame, 2 in 50 Set of

Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.1 E�ect of Modeling on Median Values for Peak Story Drift Demands, 2


7.2 Maximum Roof Drift Demands for 3-Story Frame L Model, 2 in 50 Set


7.3 Maximum Roof Drift Demands for 3-Story Frame NL2 Model, 2 in 50


7.4 Maximum Roof Drift Demands for 3-Story Frame NL3 Model, 2 in 50


7.5 Median Values for Peak Story Drift Demands for L Evaluation Models

- 3-Story, 2 in 50 Set . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.6 Median Values for Peak Story Drift Demands for NL3 Evaluation Mod-

els - 3-Story, 2 in 50 Set . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.7 E�ect of Modeling on Median Values for Peak Story Drift Demands

for FPS T3 - f1, 2 in 50 Set . . . . . . . . . . . . . . . . . . . . . . . 126


for VS - 30, D1, 2 in 50 Set . . . . . . . . . . . . . . . . . . . . . . . 127


for ATB - S1k, 2 in 50 Set . . . . . . . . . . . . . . . . . . . . . . . . 128


for 9-Story Structure, 2 in 50 Set . . . . . . . . . . . . . . . . . . . . 129


for 9-Story Structure with FPS T4 - f1, 2 in 50 Set . . . . . . . . . . 129


for 9-Story Structure with VS - 30, D1, 2 in 50 Set . . . . . . . . . . 130


for 9-Story Structure with ATB, 2 in 50 Set . . . . . . . . . . . . . . 131

xvii

7.14 Median Values for Peak Story Drift Demands for Linear Evaluation

Models of LA 9-Story Structure, 2 in 50 Set . . . . . . . . . . . . . . 131

7.15 Median Values for Peak Story Drift Demands for Linear Evaluation

Models of LA 9-Story Structure, 2 in 50 Set . . . . . . . . . . . . . . 132

7.16 Median Values for Peak Story Drift Demands for Uncontrolled System,

Varying Fundamental Period, 2 in 50 Set . . . . . . . . . . . . . . . . 133

7.17 Median Values for Peak Story Drift Demands for FPS Isolation System

T3, Varying Fundamental Period, 2 in 50 Set . . . . . . . . . . . . . . 134

7.18 Median Values for Peak Story Drift Demands for VS 30 System, Vary-

ing Fundamental Period, 2 in 50 Set . . . . . . . . . . . . . . . . . . . 134

7.19 Median Values for Peak Story Drift Demands for ATB - S1k, Varying

Fundamental Period, 2 in 50 Set . . . . . . . . . . . . . . . . . . . . . 135

7.20 Median Values for Peak Story Drift Demands for Half the Original


7.21 Median Values for Peak Story Drift Demands for Twice the Original


7.22 Median Values for Peak Story Drift Demands for Uncontrolled System,

Variation Strain-Hardening Ratio, 2 in 50 Set . . . . . . . . . . . . . 138

7.23 Median Values for Peak Story Drift Demands for FPS T3 System T3,

Variation Strain-Hardening Ratio, 2 in 50 Set . . . . . . . . . . . . . 138

7.24 Median Values for Peak Story Drift Demands for VS 30 System, Vari-

ation Strain-Hardening Ratio, 2 in 50 Set . . . . . . . . . . . . . . . . 139

7.25 Median Values for Peak Story Drift Demands for ATB System, Varia-

tion Strain-Hardening Ratio, 2 in 50 Set . . . . . . . . . . . . . . . . 139

8.1 Annual Hazard Curve for Spectral Acceleration, LA 3-Story Structure 149

8.2 Annual Hazard Curve for Spectral Acceleration, LA 9-Story Structure 150

8.3 Relationship between Spectral Acceleration and Maximum Peak Story

Drift for LA 3-Story Structure . . . . . . . . . . . . . . . . . . . . . . 152


Drift for LA 3-Story Structure with FPS Isolation . . . . . . . . . . . 155


Drift for LA 3-Story Structure with Viscous Brace System . . . . . . 156

xviii


Drift for LA 3-Story Structure with ATB System . . . . . . . . . . . 156

8.7 Relationship between Spectral Acceleration and Peak Damper Force

for LA 3-Story Structure VS Dampers . . . . . . . . . . . . . . . . . 158

8.8 Relationship between Spectral Acceleration and Peak Bearing Dis-

placement for LA 3-Story Structure FPS Isolation . . . . . . . . . . . 160

8.9 Relationship between Spectral Velocity and Peak Bearing Displace-

ment for LA 3-Story Structure FPS Isolation . . . . . . . . . . . . . . 160

8.10 Standard Error in Peak Drift Estimation due to Limited Sample Size

Using Full Data Set, 3-Story Structure . . . . . . . . . . . . . . . . . 161

8.11 Comparison of Drift Demand Hazard Curves of FPS Isolation System

for 3-Story Structure, Variation in Isolation Period . . . . . . . . . . 163

8.12 Comparison of Drift Demand Hazard Curves of VS Damping system

for 3-Story Structure, Variation in E�ective Damping . . . . . . . . . 164

8.13 Comparison of Drift Demand Hazard Curves of VS Damping system

for 3-Story Structure, Variation in Damping Distribution . . . . . . . 165

8.14 Comparison of Bearing Displacement Demand Hazard Curves for 3-

Story Structure, Variation in Isolation Period . . . . . . . . . . . . . 166

8.15 Comparison of Bearing Displacement Demand Hazard Curves for 3-

Story Structure, Variation in Isolation Period . . . . . . . . . . . . . 167

8.16 Comparison of Drift Demand Hazard Curves for LA 3-Story Structure 169

8.17 Comparison of Drift Demand Hazard Curves for LA 9-Story Structure 169

8.18 Comparison of Individual Story Drift Demand Hazard Curves for LA

3-Story Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171


3-Story Structure with FPS Isolation . . . . . . . . . . . . . . . . . . 171


3-Story Structure with VS Damping . . . . . . . . . . . . . . . . . . . 172


3-Story Structure with ATB System . . . . . . . . . . . . . . . . . . . 172


9-Story Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

xix


9-Story Structure with FPS Isolation . . . . . . . . . . . . . . . . . . 174


9-Story Structure with VS Damping . . . . . . . . . . . . . . . . . . . 175


9-Story Structure with ATB System . . . . . . . . . . . . . . . . . . . 176

xx

Notation

The following notation is used in this dissertation unless otherwise noted:

� strain hardening ratio;

�d ori�ce coeÆcient for uid damper;

a coe�. controlling dependency of friction on velocity;

A cross-sectional area;

Brx mapping matrix between nodal and relative displacements;

c equivalent viscous damping coeÆcient;

C numerical coeÆcient related to soil type and period;

C viscous damping matrix;

�t time step;

d nodal displacement vector;

E modulus of elasticity;

Ed work done by damping;

fmax coe�. of friction at high velocity;

fmin coe�. of friction at low velocity;

Es elastic-plastic work;

Fy yield strength;

Fd damping force vector;

Fs static resisting force vector;

�(�) gamma function;

g gravitational constant;

G shear modulus of elasticity;

h height;

I moment of inertia;

xxi

J Jacobian matrix;

J cost functional;

kE element elastic sti�ness;

kH element hysteretic sti�ness;

K00

d loss sti�ness for a viscous damper;

KISO sti�ness of isolation bearing;

K sti�ness matrix;

L(�) Lagrangian;

flg mapping vector to horizontal degrees of freedom;

L length;

�S sliding coeÆcient of friction;

M mass matrix;

R radius of curvarure for spherical sliding surface;

Sa spectral acceleration;

Sv spectral velocity;

Sd spectral displacement;

� equivalent viscous damping ratio;

� rotational displacement;

T fundamental period;

u horizontal displacement;

v vertical displacement;

Æ interstory drift ratio;

V base shear;

W seismically e�ective weight;

x displacement;

_x velocity;

�x acceleration;

shape function;

! circular natural frequency;

xxii

Chapter 1

Introduction

1.1 Motivation

In recent years, research in the development of control systems has made signi�cant

progress in the reduction of the overall response of civil structures subjected to seis-

mic excitations. However, much of this research has utilized highly simpli�ed linear

models of structural systems. To address the broader role of control technology in

improving the overall performance of structures, the control analyses presented here

consider more sophisticated structural models and include information about the non-

linear response of individual members.

In general, control studies in civil engineering can be divided into two categories:

those which address serviceability issues and those whose main concern is safety.

When serviceability is the main concern, control is used to reduce structural accel-

eration in order to increase occupant comfort during relatively mild wind or seismic

excitations. However, for those controllers developed for stronger excitations, where

occupant safety is the main concern, the goal is to improve structural response by

reducing peak interstory drift or by increasing energy dissipation. The majority of

these studies have dealt mainly with linear systems and analyses.

Improvement of structural performance under moderate to severe excitations re-

quires a reduction of damage under dynamic loading, and damage is an inherently

nonlinear process. Peak responses alone do not describe the possible damage incurred

by the structure as cumulative damage results from several incursions into the inelas-

tic range. As such, the reduction of peak interstory drifts alone is not suÆcient unless

1

Introduction Chapter 1

we also have information about the capacity of the structure. One cannot assume a

structure will remain linear even under moderate seismic loads.

The structural engineering community has been making great strides in recent

years to develop performance-based earthquake engineering methodologies for both

new and existing construction. Both SEAOC's Vision 2000 project (SEAOC 1995)

and BSSC's NEHRP Guidelines for Seismic Rehabilitation of Buildings (BSSC 1997)

present the �rst guidelines for multi-level performance objectives. One of the intents

of these provisions is to provide methods for designing and evaluating structures such

that they are capable of providing predictable performance during an earthquake.

For structural control to gain viability in the earthquake engineering community,

understanding the role of controllers within the context of performance-based engi-

neering is of primary importance. Design of a structure/controller system should

involve a thorough understanding of how various types of controllers enhance struc-

tural performance, such that the most e�ective type of controller is selected for the

given structure and seismic hazard. Controllers may be passive, requiring no ex-

ternal energy source, or active, requiring an external power source. Applications of

certain passive systems, including base isolation and viscous dampers, have become

more common, leading to a reasonable understanding of how such systems reduce the

dynamic behavior of structures. However, few full-scale applications of active con-

trollers exist and their enhancement of structural performance, particularly for larger

events, is less understood. Furthermore, neither passive or active systems have been

investigated with the objective of quantifying and comparing their ability to improve

structural performance under the parameters established by the recently developed

performance-based design criteria.

1.2 Objective and Scope

The objective of the research presented here is to evaluate the role of structural

control technology in enhancing the overall structural performance under seismic

excitations. This study focuses on steel moment-resisting frames, and three types of

possible controllers: (1) base isolation system (passive); (2) viscous brace dampers

(passive); (3) and active tendon braces. Two structures are selected from the SAC

Phase II project, the three story system and the nine story system. The lateral force

2

Chapter 1 Introduction

resisting system for both buildings is composed of perimeter steel moment-resisting

frames. These buildings are represented as two-dimensional nonlinear �nite element

models using centerline dimensions. Simulations of these systems, both controlled

and uncontrolled, are prepared using the three suites of earthquake records, also

from the SAC Phase II project, representing three return di�erent periods. Several

controllers are developed for each structure, and the resulting system performance is

judged based on drift, oor accelerations, and dissipated hysteretic energy demands.




controller performance to variation of control parameters, load intensities, and struc-

tural modeling techniques; and (3) To compare the bene�ts of the controllers in both

a deterministic and probabilistic format.

1.3 Overview

In Chapter 2 an overview of building damage and the available indices used for damage

assessment is presented. A discussion of the current performance-based guidelines and

their application to steel moment-resisting frames is included.

The basic ideas and concepts of structural control as applied to civil engineering

structures are discussed in Chapter 3. Previous work in the area is presented and re-

viewed. Current provisions for the use of supplemental control systems are discussed.

A description of how structural control methods �t within the goals of performance

engineering is then presented.

Chapter 4 provides a description of the two structures that are analyzed and the

ground motions utilized for seismic demand calculations. Three di�erent types of

control systems were then selected for implementation with these structures. The

reasoning behind the selection of these systems and the basic design philosophy of

each one is discussed.

The modeling of the structure and control systems is given in Chapter 5. Time

history analysis of these systems is performed using software written expressly for

this purpose. The representation of the element behavior in the analysis software

is discussed, including the modeling assumptions of element behavior. The analysis

3

Introduction Chapter 1

software was veri�ed by benchmarking the results against those from DRAIN-2DX.

Example results from this veri�cation process are given at the end of the chapter.

Global roof drift and story parameters for the di�erent systems investigated in

this study are presented in Chapter 6. The emphasis of the discussions are on roof

and story drift demands. The e�ect of variation in selected control design parameters

are presented. A representative system is chosen for each type of control system for

comparison on the basis of peak and residual drifts, dissipated hysteretic energy, and

peak oor accelerations.

Chapter 7 provides an analysis of the sensitivity of initial period and strain-

hardening ratio. The e�ect of performing di�erent types of analysis, for example

linear vs. nonlinear, are investigated for both structural systems.

The performance of the systems are developed in a probabilistic format in Chap-

ter 8. A procedure developed by Cornell (1996) is used in this process. The curves

generated from this procedure are used to assess the impact of di�erent control pa-

rameters and to perform a comparison between control systems for a given structure.

A summary of the research and its conclusions are presented in Chapter 8. Possible

directions of future research are then discussed.

4

Chapter 2

Performance Evaluation of

Structures

2.1 Introduction

The structural engineering profession su�ered signi�cant setbacks after the 1994

Northridge Earthquake in Los Angeles and the 1995 Great Hanshin Earthquake in

Kobe, Japan. Until that time, the general seismic design philosophy was to safeguard

against the collapse of structures and loss of lives. In these recent earthquakes, how-

ever, damage to structures and their contents lead to losses of billions of dollars. So

in addition to ensuring against collapse, structural engineers are being required to

design structures that are designed to minimize the damage based on the function of

the building and within the constraints of available resources.

The basic performance requirement of life-safety needs to be met for all structures.

However, depending on its function, the structure should conform to a variety of

performance requirements. For example, critical facilities such as hospitals, which

need to remain operational after a severe earthquake, should be designed for very

di�erent criteria than a warehouse.

New guidelines for building structures have been set forth by di�erent organiza-

tions to ful�ll these requirements. Two such set of guidelines are the Vision 2000

project by the Structural Engineers Association of California (SEAOC 1995) and

NEHRP Guidelines for Seismic Rehabilitation of Buildings (BSSC 1997) issued by

the Federal Emergency Management Agency (FEMA). These guidelines are the �rst to

5

Performance Evaluation of Structures Chapter 2

introduce a framework for performance-based design. In this framework, the seismic

demand of a structure needs to be calculated as accurately as possible and compared

with the allowable limits for the desired performance level.

The de�nition of limits are based on expected damage states for a given demand

level. This chapter presents an overview of building damage and the available indices

used for damage assessment. A discussion of the current performance-based guidelines

and their application to steel moment-resisting frames is presented in the last section.

2.2 Damage to Nonstructural Elements

The nonstructural system in a building is comprised of architectural components

(cladding, ceilings, partitions, windows, etc.), mechanical systems (ducts, HVAC,

elevators, etc.), electrical systems (security, communications, etc.), and contents (fur-

niture, computer equipment, etc.). Traditionally, building codes have emphasized

life safety as their primary goal. So, while structural integrity has been of primary

concern, little regard has been paid to nonstructural components. For example, a

survey conducted after the Loma Prieta Earthquake of 129 medium and large oÆce

buildings showed that only 9% of the buildings had structural damage, while 86% of

them had nonstructural damage, with a mean monetary value of $941,000/building

(LOMA 1990).

Three types of risk are associated with seismic damage to nonstructural compo-

nents (FEMA 74):

1. Life Safety: Damaged or falling components can injure or kill building oc-

cupants. Potentially life threatening hazards from past earthquakes include:

broken glass, overturned bookcases, and fallen ceiling panels and light �xtures.

2. Property Loss: For most commercial buildings, only 20-25% of the original

construction cost can be attributed to the foundation and superstructure. The

remaining cost is due to the mechanical, electrical, and architectural compo-

nents. Building contents introduced by the occupants are also at risk and can

often correspond to signi�cant additional expense.

3. Loss of Function: Damage incurred during an earthquake may also make it

diÆcult to carry out the normal activities performed at the location. This loss

6

Chapter 2 Performance Evaluation of Structures

of function can have signi�cant monetary consequences for businesses; however,

in critical facilities such as hospitals, a loss of function can also represent a life

safety risk.

Each of the nonstructural systems described above are governed by di�erent fac-

tors. One possible classi�cation, based on the governing mode of damage, for non-

structural components is:

� Acceleration-sensitive components: Components are sensitive to the inertial

forces experienced during an earthquake. Examples include �le cabinets, free

standing bookshelves, and oÆce equipment.

� Deformation-sensitive components: Components are sensitive to building dis-

tortion or separation joints between structures. Examples include glass panes,

partitions, and masonry in�ll or veneer.

2.3 Damage to Structural Elements

Damage of materials occurs through a progressive process in which they break. This

can be considered in three levels: the microscale level, the mesoscale level, and the

macroscale level. At the microscale level, damage is incurred by the accumulation of

microstresses at defects or interfaces and by bond breaking. At the mesoscale level,

damage is observed as the initiation and growth of cracks. At the macroscale level,

damage is related to the deterioration of parts of the entire structure.

In analyzing a structure, performing a damage evaluation in detail at every point

of the structure is impossible or not of primary interest (Williams and Sexmith 1995).

Several methods to determine an indicator of damage at the structure level have been

presented in literature. Generally, these methods can be divided into four categories

of structural demand parameters:

1. Strength demands, both elastic and inelastic

2. Ductility demands

3. Energy dissipation

4. Sti�ness degradation

7


Strength Demands

If strength demands remain below the yield capacity of the structure, the structural

damage will be small. However, if demands approach or exceed the ultimate strength

of the structure, the damage to structure may also be high. Once yield is exceeded,

strength capacity may become reduced in future cycles into the inelastic range.

Ductility Demands

Ductility is the ability of an element to deform inelastically without total fracture. It

is usually expressed in terms of a ratio between the maximum deformation incurred

during loading and the yield deformation. Any deformation quantity may be used to

determine the ductility demand.

Energy Dissipation

Energy dissipation is the capacity of member to dissipate energy through hysteretic

behavior. An element has a limited capacity to dissipate energy in this manner prior

to failure. As a result, the amount of energy dissipated serves as an indicator of how

much damage has occurred to structural members during loading.

Sti�ness Degradation

Damage su�ered during loading may result in a loss of sti�ness and, consequently,

longer natural periods for the structure. As the determination of the fundamental

period is easily accomplished, this parameter can also be used as a damage indicator.

2.4 Damage Indices

The major task in damage assessment is �nding clear quantitative measures to rep-

resent the amount of damage a structure has su�ered. During the past 20-30 years,

a considerable amount of research has been performed on the development of such

methods. Desirable characteristics of these procedures include:

1. General applicability - valid for a variety of structural systems under di�erent

load histories.

8


2. Simple to evaluate - indices are easily formulated and evaluated.

3. Physically interpretable - resulting value has a physical meaning.

In general, structural damage has been de�ned in terms of either economics or

safety/strength considerations. Economic damage indices are usually expressed as

some ratio of repair costs to replacement costs for a structure or structural element.

Though speci�c knowledge of this information is desired, an accurate determination

of repair costs can be diÆcult to determine and is usually taken to be related to a

physical response parameter. Safety/strength damage indices are normally related

to deterioration of structural resistance. The following sub-sections describe damage

indices based on safety/strength approach.

2.4.1 Maximum Deformation Damage Indices

Maximum deformation damage indices are based on the peak value of a speci�ed

deformation, such as element rotation or member displacement. Two of the earliest

and simplest forms of a damage index are the ductility and interstory drift. These

two indices as well as the exural damage ratio are described below.

Ductility Ratios

Ductility is de�ned as ability to deform inelastically without total fracture and sub-

stantial loss of strength. In literature, it is commonly expressed as a ductility ratio,

�R, as de�ned below:

�R =umuy

(2.1)

where um is the maximum deformation experienced and uy is the yield deformation.

The maximum deformation is determined from the load-deformation history of the

structure under a given load. The deformation quantity can be any one desired:

displacement, rotation, etc. At the structural level either displacements or drifts are

usually used. A problem with the ductility ratio is that it cannot account for both

duration and frequency content of the typical ground motion (Banon and Veneziano

1982). Also, determination of yielding can be diÆcult, especially at the structural

level.

9


Interstory Drift

Interstory drift is de�ned as the relative interstory displacement of a story. Cul-

ver (1975) proposed a damage index de�ned as the observed maximum story dis-

placement to the story displacement at failure. A problem with this index is that

determination of drift at failure is diÆcult.

Toussi and Yao (1983) proposed a damage index de�ned as the ratio between the

maximum interstory displacement, �i, and the story height, h, as given below, and

provided guidelines for interpretation of results. This drift ratio, Æi, has been widely

used in a variety of structural systems as an indicator of the deformation demands

on a structure.

Æi =�i

h(2.2)

As with ductility ratios, peak interstory drift measures cannot take into account the

e�ects of repeated cycling, which can be a signi�cant source of damage to structural

members.

Flexural Damage Ratio

To counteract the limitations of the above measures, a number of parameters related

to sti�ness degradation were proposed. Banon (1981) correlated damage to the ratio

of the initial structural sti�ness to the secant sti�ness at the maximum displacement

forming the Flexural Damage Ratio, (FDR). This index relies on sti�ness degradation

as an indicator of damage. Roufaiel and Mayer (1983) later suggested a modi�cation

of the exural damage ratio so that it was de�ned as the ratio of the secant sti�ness

at the onset of failure in a one-cycle test to the minimum reduced secant sti�ness.

A ratio of zero corresponds to no damage, while a ratio of 1 corresponds to failure.

However, the authors admitted that this index would be diÆcult to calculate for an

actual structure.

2.4.2 Cumulative Damage Indices

Capturing the accumulation of damage sustained during dynamic loading is of par-

ticular interest to structural engineers. This process is usually accomplished through

10


a low-cycle fatigue formulation or calculation of the energy absorbed by the system

during loading. In both those cases, inelastic behavior is assumed before any damage

is considered.

Normalized Cumulative Deformations

Early deformation-based indices tried to account for cumulative damage by extending

the concept of ductility for repeated loadings. Banon and Veneziano (1982) proposed

the normalized cumulative deformation (NCD) as a damage index. This index is

de�ned as the ratio of the sum over all half-cycles of all the maximum plastic defor-

mations to the deformation at yield as follow:

NCD =mXi=1

jupijuy

(2.3)

Normalized Cumulative Dissipated Energy

Additionally, Banon and Veneziano (1982) proposed the normalized cumulative dissi-

pated energy (NHE) as a damage index. The NHE is de�ned as the ratio of the total

energy dissipated in inelastic deformation to the elastic energy that would be stored

in a member. So for an element yielding in exure:

NHE =

Z tm

0

M(�)�(�)

Eed� (2.4)

where M(�) and �(�) are the moment and corresponding rotation at a given time,

Ee is the elastic energy capacity of the member, and tm is the time at the end of

the excitation. Though this index showed signi�cant variation at failure in reinforced

concrete components, Krawinkler (1991) has shown that the NHE provides a good

indication of damage in steel structure.

Low Cycle Fatigue

A low cycle fatigue model of damage uses accumulated plastic deformation as an

indicator of damage. One of the early indexes of that nature was proposed by Yao

and Munze (1968) where the index expressed as a function of the sum of a nonlinear

function of the inelastic deformation per response half-cycle.

11


Iemura (1980) proposed a similar index expressed in terms of the rotation ductility.

The index adequately represents damage to members, but the constants used in the

formulation are dependent on individual member properties and no general were rules

developed.

2.4.3 Combined Indices: Maximum Deformation and Cumu-

lative Damage

Park and Ang (1985) de�ned a local damage index which combines the in uence of

the normalized maximum deformation and absorbed hysteretic energy. The damage

index is expressed as the following linear combination:

DPA;i =umax

uu+

�

Qruu

XdE (2.5)

where umax is the maximum deformation, uu is the ultimate deformation under mono-

tonic loading, Qr is the yield strength, dE is the incremental absorbed energy, and �

is a non-negative strength deteriorating parameter.

The maximum and ultimate displacements are only well de�ned in the case of a

cantilever beam �xed at one end, so the deformation quantities for other cases, such as

beams and columns, is not clear. Therefore, Rodrigo-Gomez (1990) suggested the use

of curvatures instead of displacements. A further re�nement was introduced by Kun-

nath, Reinhorn, and Lobo (1992), in which the recoverable deformation is removed

from the �rst term of the equation. A drawback of the index is that the strength

deteriorating parameter � has to be found experimentally. Also, the damage scale is

nonlinear. Values of DPA;i in excess of 0.4 imply very severe damage, while di�eren-

tiating between levels of damage at the bottom end of the scale is diÆcult (Williams

and Sexmith 1995).

Banon and Veneziano (1982) proposed that the damage state of a structural mem-

ber can be described by two damage indices: exural damage ratio and the normalized

dissipated energy, as described in previous sections, and assembled into a damage vec-

tor. These two quantities were chosen from a correlation study of di�erent indices

proposed in literature. No attempt to combine them into a single index was made,

and interpretation of exactly what is meant by the resulting damage vector is diÆcult.

12


2.4.4 Maximum Softening Damage Indices

These indices are based on the variation of the vibrational periods of a structure

during a seismic event. In several papers (DiPasquale and Cakmak 1990; Nielsen,

Koyluoglu, and Cakmak 1992), a correlation was found between the damage state

of the structure and the maximum softening. DiPasquale and Cakmak (1990) de�ne

the maximum softening for the one-dimensional case, where only the fundamental

eigenfrequency is considered. The index is given by:

ÆM = 1� T0TMAX

(2.6)

where T0 is the initial fundamental eigenperiod for the undamaged structure and

TMAX is the maximum value of the fundamental eigenperiod during the earthquake.

The authors performed extensive study of this index and found consistent mapping

between the values of the index and the structure's damage state. A drawback is that

this index provides no information about the distribution of damage in the structure.

2.4.5 Weighted Average of Damage Indices

Several of the damage indices discussed previously are intended to be evaluated on

an element level. In order to determine an index for the entire structure, a method to

weigh these local values into a global parameter is necessary. Kunnath et al. (1991)

proposed an energy weighted average of the local damage indices as:

Dg =

Pni=1DiEiPni=1Ei

(2.7)

where Di is the local damage index at substructure i, Ei is the dissipated energy at

substructure i, and n is the number of substructures. Park, Ang, and Wen (1985)

suggested using the damage index selected as a weight leading to the following global

index:

Dg =

Pni=1D

2iPn

i=1D2i

(2.8)

13


The above weighing methods are only two possible methods as no unique solution

exists.

2.5 Recent Developments in Performance-Based

Engineering

Structural performance is a measure of the damage in a structure. The improvement

of structural response requires a reduction of damage under dynamic loading. In

general, this evaluation considers both structural and nonstructural components as

well as the contents of the structure. Performance evaluation consists of a structural

analysis with computed demands on structural elements compared against speci�c

acceptance criteria provided for each of the various performance levels. In order to

evaluate structural performance, the following information is required (Bertero 1996):

1. Sources of excitation during service life of structure

2. De�nition of performance levels

3. De�nition of excitation intensity

4. Types of failures (limit states) of components

5. Cost of losses and repairs

One of the �rst requirements of performance evaluation is the selection of one

or more performance objectives, i.e.: select desired performance level and associated

seismic hazard level. Since the evaluation relies on analysis rather than experimen-

tation, the criteria should be stated in terms of a response that can be calculated.

Depending on the intensity of the ground motion, a di�erent performance objec-

tive will be desired. According to the expected intensity, the designer must analyze

whether achieving the desired objective will be economically feasible. For frequent

events, the designer will probably desire that the structure remain fully operational.

For rarer events, ensuring prevention against collapse may be the only realistic goal.

Signi�cant work has been performed in the development of performance-based

design and evaluation. Discussions on the subject can be found in Bertero (1996),

Cornell (1996), and Krawinkler (1996). Recent guidelines, such as those in Vision

2000 (SEAOC 1995) and FEMA 273 (BSSC 1997), provide a framework for the

14


performance-based design and evaluation of structures under seismic loads. Both

qualitative and quantitative de�nitions for seismic hazard and structural performance

are provided. The following subsections brie y discuss the basic concepts outlined in

the guidelines.

2.5.1 Performance Levels

Performance may be concerned with structural and nonstructural systems as well

as contents, and behavior ranging from minor damage to failure. In general, di�er-

ent performance levels will require di�erent design criteria to be applied to di�erent

design parameters. At one end of the performance spectrum, content damage is of-

ten proportional to oor accelerations, which can be limited by reducing sti�ness.

At the other end of the spectrum, life safety and collapse prevention are controlled

by inelastic deformation capacity of ductile members and strength capacity of brit-

tle members. As a result, no single design parameter may satisfy all performance

requirements. Furthermore, con icting demands of strength and sti�ness may be

involved.

Both the NEHRP's FEMA 273 and SEAOC's Vision 2000 projects have qualita-

tively identi�ed similar performance level de�nitions with slightly di�erent naming

conventions. These performance levels for FEMA 273 are listed in Table 2.1.

2.5.2 Excitation Levels

In selecting a seismic hazard level, one �rst needs to de�ne what is meant by exci-

tation intensity. Earthquake shaking demands can be expressed in terms of ground

motion response spectra, parameters which de�ne these spectra, or suites of ground

motion histories depending on evaluation method utilized. These demands are a func-

tion of the location of the building and may be de�ned on either a probabilistic or

deterministic basis. In FEMA 273, probabilistic hazard levels are de�ned by their

corresponding mean return period, as shown in Table 2.2.

Theoretically, to evaluate a structure's performance, one needs to generate sample

ground motions that represent all future events in the region that may have an impact

on the building. This procedure, however, is computationally unmanageable. As a

result, suites of earthquake ground motions can be generated that, as a set, contribute

15


Table 2.1: General Structural Performance Level De�nitions and IndicativeDrifts for Steel Moment Frames (FEMA 273).

Level Description Drifta Limit (%)Transient Residual

CollapsePrevention

Little residual sti�ness and strength, butload bearing columns and walls function.Large permanent drifts. Building is nearcollapse.

5 5

Life Safety Some residual strength and sti�ness.Some permanent drift. Building may bebeyond economical repair.

2.5 1

ImmediateOccupancy

No permanent drift. Structure substan-tially retains original strength and sti�-ness.

0.7 negligible

aIndicative of a typical structure when responding within performance level.

Table 2.2: Probabilistic Hazard Levels and Corresponding Return Periods(FEMA 273).

Earthquake Probability of Exceedance Mean Return Period (years)50% in 50 years 7220% in 50 years 22510% in 50 years 4742% in 50 years 2,475

the most to the hazard of the site for a given probability of exceedance. Current

guidelines in FEMA 273 require at least 5 ground motions such that, on the average,

the spectra of the records are equal to or greater than the design spectrum across the

range of frequencies of interest.

2.5.3 Structural Performance Parameters

As the previous discussion on available damage measures indicate, no single damage

measure will provide all the information required to assess structural performance,

especially at all performance objectives. However, some indicator must be used to

provide quantitative limits. In FEMA 273, both the peak and residual interstory

drifts are utilized in de�ning performance levels as an indicator of damage. Tables

16


are provided that contain limiting drift values, for di�erent structural systems, for

the di�erent performance levels. The indicative drift values for steel moment-resisting

frames (SMRF) are shown in Table 2.1. The document emphasizes that these drift

values are indicative of the drifts a structure will experience at that performance

level. They should be used as indicators, and not as design or evaluation limits. For

use in the design and evaluation process, FEMA 273 further provides information on

component level peak deformation values relative to the analysis procedure used for

evaluation.

Peak transient drift serves as an indicator of damage to low strength rigid elements,

such as building cladding and partition walls, and the maximum deformation of the

structural elements. The permanent drift provides a rough indicator of damage to

structural members. Care must be taken in the interpretation of this value, however,

as it may be misleading. For example, a large cyclic loads may result in small residual

drifts, while a half-pulse load of signi�cant smaller amplitude may result in large

residual drifts.

The use of maximum values as an indicator of damage provides preliminary infor-

mation to be used in the evaluation of the structural system. However, the informa-

tion is incomplete, as it does not account for the cumulative damage incurred during

seismic loads. In the following discussions on the performance of the di�erent sys-

tems analyzed, the FEMA 273 values are used to provide an estimate of the expected

structural performance. However, discussions on cumulative damage, as indicated by

dissipated hysteretic energy, and nonstructural damage, as indicated by peak oor

acceleration, are also provided.

17

Chapter 3

Structural Control in Civil

Engineering Structures

3.1 Introduction

The protection of civil structures, including their material contents and human oc-

cupants, is of serious global importance. Such protection may range from reliable

operation and comfort to survivability. Examples of such structures include build-

ings, o�shore rigs, towers, roads, bridges, and pipelines. Events which cause the need

for such protective measures include earthquakes, winds, and waves. Research in this

�eld indicates that control methods will be able to make a genuine contribution to

this problem area.

The common feature of the di�erent proposed approaches is the modi�cation of

the dynamic interaction between the structure and the dynamic loads. The goal of the

modi�cation is to minimize the damage and vibrations throughout the structure. The

result is that well-designed controlled systems display enhanced safety and occupant

comfort.

This chapter provides an overview of the basic ideas and concepts of structural

control as applied to civil engineering structures. Previous work in the area is collected

and reviewed. A discussion of how structural control methods �t within the goals of

performance engineering is then presented.

18

Chapter 3 Structural Control in Civil Engineering Structures

3.1.1 Background and Recent Developments in Structural

Control

Various means of controlling structural vibrations produced by earthquake or wind

have been investigated by the structural engineering community. These means include

modifying rigidities, masses, damping, or shape through the provision of countering

forces. A structure that is designed solely on the basis of strength requirements

does not necessarily ensure that the building will respond dynamically in such a way

that the comfort and safety of the occupants is maintained. For example, a 47-

story building in San Francisco experienced peak accelerations of 0.45g on the top

oor (Housner et al. 1997), resulting in signi�cant nonstructural damage.

The notion of structural control in civil engineering can be traced back to John

Milne, over 100 years ago, who built a small wood house and placed it on bearings

to illustrate how it could be isolated from earthquake vibrations. During the �rst-

half of the twentieth century, development of system theory and its application to

vibration control was driven by the development of the internal combustion engine.

The engine, which was used in both automobiles and aircrafts, produced signi�cant

forces at connection points to the surrounding system. Structural control theory

was applied so as to counteract those forces. During the World War II the concepts

of vibration isolation and absorption were further developed and applied to aircraft

structures (Housner et al. 1997).

The structural engineering community began to seriously investigate the appli-

cation of control techniques in the 1960's. Knowledge has largely been adapted

from both the aerospace and automobile industries, where a signi�cant amount of

research and applications have occurred. In structural engineering, the means of vi-

bration control have taken several forms; for example, the use of base isolation for

low and medium height structures for seismic protection. For taller, more exible

structures, particularly those susceptible to high winds, the addition of supplemental

dampers have been successfully employed. One well-known example where viscoelas-

tic dampers are utilized to control wind-induced vibrations is the World Trade Center

in New York City.

Special considerations distinguish the application of control to civil engineering

structures as opposed to other engineering applications. One such di�erence lies in

19

Structural Control in Civil Engineering Structures Chapter 3

the fact that civil structures are anchored to the ground and so are statically stable.

In contrast, space structures when deployed require active control for stability. Also,

environmental disturbances, such as winds and earthquakes, are highly uncertain with

respect to both their occurrence and intensity. The loads found in most mechanical

applications, however, are fairly well documented.

Since the initial conceptual studies were conducted, the �eld of vibration control

has slowly matured and found application in the civil engineering arena. The U. S.

National Workshop on Structural Control, held at the University of Southern Califor-

nia in 1990, attracted nearly 100 participants from around the globe. By the Tenth

World Conference on Earthquake Engineering in Madrid, Spain in 1992, several tech-

nical sessions were dedicated to topics in structural control. At this conference, the

decision was made to form an international association and hold world conferences

dedicated to structural control. The resulting International Association of Struc-

tural Control (IASC) was formed the following year, with Professor G. Housner as

president. Their e�orts led to the First World Conference in Structural Control in

1994, several workshops and technical sessions at earthquake engineering conferences,

the �rst European Conference in Structural Control, and in 1998 the Second World

Conference in Structural Control in Tokyo.

3.1.2 General Classi�cation of Control Systems

The control of structural systems can be subdivided on the basis of the approach

taken to produce the desired response. The basic classi�cations used in structural

control are de�ned below.

Isolation systems introduce, at the foundation of the structure, a system that

is characterized by increased exibility and high energy absorption capacity. The

increased exibility provided shifts the fundamental period to a range of reduced

energy input from the ground motion. The isolation system's energy dissipation

capacity then further reduces the displacement demands on the superstructure.

Passive control systems do not require external forces. The forces imparted to

the structural system are a function of the response to the excitation. These systems

function by increasing the dissipated energy capacity of the structure and focusing

energy dissipation in specially designed devices.

20


Active controls systems contain external source powers actuators that apply forces

in speci�ed manner. They both add and dissipate energy in the structure. Closed-

loop control uses information about structural response only, measured with physical

sensors. Open-loop control results when control forces are determined only from mea-

sured excitations. Open-closed-loop control uses information from both the structural

response and the external excitation.

A semi-active control system can be considered as a passive device whose proper-

ties can be actively controlled. For example, a uid viscous damper whose damping

constant can be controlled. These systems require power inputs which are signi�cantly

reduced from a fully active system.

The term hybrid control implies the combination of both passive and active control

systems. Generally, a passive device is utilized to control the larger portion of the

response, while the active device is utilized to optimize the response to the given

excitation and maintain the passive system within desired parameters. For example,

a base isolation system could have excessive displacements in the isolation unit that

are then controlled through the use of an active actuator.

3.2 Isolation Systems

A seismic isolation system introduces, at the base of the structure, a system which is

characterized by high exibility and energy absorption capacity. Buckle and Mayes

(1990) identi�ed the basic elements in a practical isolation system as: (1) exibility

to lengthen the period and produce isolation e�ect, (2) energy dissipation capability

to reduce displacement demands to a practical design level, and (3) a means for

providing rigidity under service loads, such as winds

Examples of modern isolation systems include high damping elastomeric bearings,

lead-core rubber bearings, and friction sliding bearings. In hysteretic isolation systems

(high damping rubber bearings, lead-rubber bearings, sliding bearings), high levels

of energy dissipation can lead to out-of-phase, high accelerations in superstructure.

These accelerations, however, are still less than those observed in the uncontrolled

structure (Soong and Constantinou 1994). While high accelerations are undesirable

in the protection of acceleration-sensitive building contents, high levels of hysteretic

damping are bene�cial in protecting the structural system.

21


The �rst isolated building in the United States was the Foothill Communities Law

and Justice Center in San Bernadino, CA, which was constructed in 1986. Since that

time, a number of isolated structures, including buildings and bridges, have been

constructed. A survey of these structures is presented in Soong and Constantinou

(1994). The basic properties of elastomeric and sliding bearings are presented in the

following sections.

3.2.1 Elastomeric Bearings

Elastomeric bearings consist of thin layers of natural rubber which are vulcanized

and bonded to steel plates. Rubber has low shear modulus, G, ranging from 0.5-1

MPa at a shear strain of about 50% (Soong and Constantinou 1994). The insertion of

steel plates does not the a�ect shearing capacity, so the e�ective horizontal sti�ness

is given by:

Khz =ArGPtr

(3.1)

where Ar is the bonded rubber area andPtr is the total rubber thickness. The steel

plates are used to reduce the capacity of rubber to bulge in the vertical direction, so

the vertical sti�ness is large.

Elastomeric bearings with low-damping natural rubber have an equivalent viscous

damping in the order of 0.05 or less of critical. They are useful in the isolation of

structures with highly sensitive equipment, where reduction of accelerations is of

primary importance.

Lead-rubber bearings are constructed of low-damping natural rubber with a pre-

drilled center-hole in which a lead core is press-�tted. The lead core deforms in almost

pure shear and, at normal temperatures, yields at low levels of stress. The resulting

behavior is hysteretic curves which are stable over a number of cycles. Furthermore,

due to the recrystalization of the lead at normal temperatures, repeated yielding does

not cause fatigue.

High-damping rubber bearings utilize specially compounded rubber. The resulting

equivalent damping ratios range from 0.10 to 0.15 of critical. They were developed

originally in the United Kingdom and �rst found application in U.S. at the Foothill

22


Communities Center in California (Soong and Constantinou 1994).

3.2.2 Sliding Bearings

Flat isolation bearings limit the transmission of force across the isolation interface.

However, they also require recentering devices, otherwise large permanent displace-

ments result that can accumulate to unacceptable levels (Constantinou et al. 1991).

Constantinou et al. (1993) and Tsoupelas et al. (1994) have investigated various

means of providing the necessary restoring force to these systems. The most practi-

cal is the use of a spherical sliding surface as in the Friction Pendulum System (FPS)

bearings.

R

F

W

θ

u = R sin θ

N

Ff

Figure 3.1: Free-body Diagram for FPS Isolation System

The FPS bearing consists of an articulated slider on a spherical surface, which

is faced with a polished stainless-steel overlay. The restoring force is generated by

the rising of the structure along the spherical surface, while energy is dissipated

by friction. The force needed to produce a displacement in the bearing consists

of a restoring force, due to the rising of the structure along the spherical surface,

and a frictional force along the sliding interface. A free-body diagram of the FPS

23


isolation bearing is shown in Figure 3.1. The horizontal and vertical components of

displacement are respectively given by:

u = R sin(�)

v = R(1� cos(�))(3.2)

where R is the radius of curvature of the spherical sliding surface. From equilibrium,

the lateral force, F, is

F = W sin(�) +Ff

cos(�)(3.3)

whereW is the weight carried by the bearing and Ff is the friction force at the sliding

interface.

F =W

R cos(�)u+

Ffcos(�)

(3.4)

For small values of the angle �, Equation (3.4) can be simpli�ed to:

F =W

Ru+ �SW sgn( _u) (3.5)

where � is the friction coeÆcient, and _u is the velocity across the bearing. The FPS

bearings are typically designed for displacement u < 0:2R to minimize the lineariza-

tion error and reduce forces induced in structural columns. Previous analytical and

experimental investigations have demonstrated that these approximations are valid

in the calculation of the system responses for all practical purposes (Zayas et al.

1987; Mokha et al. 1991; Constantinou et al. 1993). If vertical ground motions are

expected to be signi�cant, recent research has shown that more complex models of

the isolation system are required (Llera and Almazan 1998).

The friction force which is generated at the sliding interface depends on the nor-

mal load, bearing pressure, direction and value of sliding velocity, and composition of

the sliding interface. The frictional properties of PTFE (te on) and stainless steel in-

terfaces have been studied by Mokha (1988) and Constantinou (1990). Constantinou

24


proposed that:

�S = fmax � (fmax � fmin) exp�aj _uj (3.6)

where �S is the sliding coeÆcient of friction, _u is the velocity of sliding, fmax is the

coeÆcient of sliding friction at high velocity, fmin is the coeÆcient of sliding friction

at low velocity, and a is a coeÆcient controlling the dependency of friction on sliding

velocity. In general, the parameters fmax, fmin, and a are determined experimentally

depending on bearing pressure, surface roughness, and the composition of PTFE.

Table 3.1 presents the values of these parameters from the tests of Mokha (1988),

where: UF = Un�lled Te on; 15GF = glass-�lled Te on at 15%; 25GF = glass-�lled

Te on at 25%; P = sliding parallel to the direction of the te on lay; T = sliding

perpendicular to direction of the te on lay.

Table 3.1: Frictional Properties of PTFE in Contact with Polished StainlessSteel

Type Pressure Sliding fmax fmin aTe on (MPa) Direction (%) (%) (sec/in)UF 6.9 P 11.93 2.66 2.2UF 13.8 P 8.70 1.75 2.3UF 20.7 P 7.03 1.51 3.7UF 44.9 P 5.72 0.87 4.315GF 6.9 P 14.61 4.01 2.115GF 13.8 P 10.08 4.28 1.415GF 20.7 P 8.40 4.32 1.315GF 44.9 P 5.27 2.15 2.225GF 6.9 P 13.20 5.54 1.425GF 13.8 P 11.20 4.87 1.425GF 20.7 P 9.60 4.40 1.525GF 44.9 P 5.89 3.19 1.8UF 6.9 T 14.20 2.39 3.0UF 13.8 T 10.50 1.72 4.4UF 20.7 T 8.20 2.90 1.5UF 44.9 T 5.50 1.11 3.2

Since the force is proportional to the weight carried by the bearings, the resultant

25


force of all bearings always develops at the center of mass of the structure, therefore

eliminating eccentricities even in the case of uneven mass distribution. Another useful

property of this system is that the period of isolation for the system is independent

of the mass of the structure. In Equation (3.5), the sti�ness of the bearing is given

by the quantity W=R. Therefore, the period of isolation is given by:

T = 2�

�W

Kg

�1=2

= 2�

�R

g

�1=2

(3.7)

Several structures have been built in recent years which utilize FPS isolators or

similar devices. The U.S. Court of Appeals in San Francisco is one recent and well

documented example (Mokha et al. 1996). This structure was damaged in the 1989

Loma Prieta earthquake and was retro�tted using 256 FPS isolators. The structure

has a oor area of 31500 square meters and a weight of 55000 metric tons.

3.3 Passive Control Systems

Passive Control Systems increase the hysteretic energy dissipation capacity of the

structure. By increasing the amount of energy dissipated through hysteretic behavior

of passive devices, the amount of energy needed to be dissipated through strain de-

formation of the structure is reduced. Energy dissipation is concentrated in specially

designed and detailed areas and away from critical load bearing members. Passive de-

vices have long been used to control response to wind loads. The towers in the World

Trade Center and Seattle's Columbia Sea�rst Center both utilize passive viscoelas-

tic dampers to control wind induced vibrations (Mahmoodi et al. 1987). Recently,

passive systems have also been found to be an eÆcient means of controlling seismic

excitations (Lin et al. 1991; Cherry and Filiatrault 1993).

Aiken et al. (1993) investigated passive energy dissipation systems for earthquake-

resistant design. They categorize passive energy dissipation devices into two broad

categories of rate-dependent or rate-independent on the basis of whether the hysteretic

characteristics of the device are dependent on the rate of loading. Examples of rate-

dependent devices include viscous and viscoelastic systems; rate-independent devices

include friction-slip, steel yielding and shape-memory systems.

26


The two most commonly used and researched devices are the viscoelastic and the

friction-slip systems. A major di�erence between these two devices is the maximum

force that each will develop during an earthquake (Hanson 1993). Viscoelastic devices

produce maximum forces which are proportional to the maximum displacements and

velocities across the the device. In contrast, friction-slip devices produce maximum

forces equal to the design friction force plus strain hardening. As a result, the maxi-

mum force experienced by the passive control device can be more readily controlled

in the friction devices.

3.3.1 Viscous and Viscoelastic Dampers

Though viscoelastic dampers have long been used in response control under wind

loads, their use in seismic response control has been investigated only in recent years.

Both analytical and experimental investigations have indicated that signi�cant reduc-

tions in structural response can be achieved by the use of viscoelastic dampers (Aiken

and Kelly 1990; Lin et al. 1991; Zhang et al. 1989). However, experimental stud-

ies have also shown that, while e�ective in controlling structural response, achieving

maximum eÆciency in their design requires careful consideration of several key fac-

tors, such as excitation frequencies and environmental temperature (Aiken and Kelly

1990).

Some viscoelastic damping is present in all building structures, and their e�ects

on the dynamic response of the structure has long been recognized and accepted.

Typical values of damping utilized in design range from 2% to 5%, which account

for the inelastic behavior in the structural members. By increasing the percentage

of critical damping in the system, the dynamic ampli�cation factor is correspond-

ingly reduced in the range of the system's natural frequency. Outside that frequency

range, an increase in the viscous damping will not signi�cantly impact the building

response (Hanson 1993). The behavior of the viscoelastic material itself is not con-

stant. Mahmoodi (1969) conducted experimental analysis on the viscoelastic material

and found that their material behavior is strongly in uenced by its shear modulus,

the shear storage modulus, and the material loss factor. These material properties,

in turn, are dependent on frequency, temperature, and deformation strain level.

27


Viscoelastic damping also has been utilized to develop energy dissipating connec-

tions. Hsu and Faftis (1992) have developed a viscoelastic-type connection isolator

for use with frame structures. The connection consists of a single tooth device which

transfers only shear forces into the device. A Kelvin-Voigt model was utilized to

develop an analytical model for the connection, which was then calibrated using

experimental results. The results of these analysis showed that the connections pro-

vided signi�cant improvement in response by reducing the lateral displacement of the

structure.

The uid viscous damper consists of a piston in a damper housing �lled with a

compound of silicon or other viscous uid. This type of damper dissipates energy

through the movement of the piston through the highly viscous uid. If the uid is

purely viscous, then the output force of the damper is directly proportional to the

velocity of the piston. Over a large frequency range, the damper exhibits viscoelastic

uid behavior.

The force in the uid damper may be expressed as:

Fv = c j _uj�d sgn( _u) (3.8)

where _u is the velocity of the piston rod, c is a damping constant and �d is a coeÆ-

cient in the range of approximately 0.5 to 2.0 depending on the device ori�ce. Fluid

dampers with a coeÆcient �d = 1, corresponding to a linear damper, were tested over

a wide temperature range and exhibited a much smaller variation in their damping

constant as compared to viscoelastic solid dampers (Constantinou and Symans 1992;

Constantinou et al. 1993).

The advantages of nonlinear viscous dampers with small values of the parameter

�d can be shown by looking at a single degree-of freedom (SDOF) system under

sinusoidal motion as given by:

u = u0 sin(!0t) (3.9)

where u0 is the amplitude of harmonic motion at the undamped natural frequency of

the system and !0 is the natural frequency.

28


The damping ratio for an SDOF system with a damper with characteristics de-

scribed in Equation (3.8) is de�ned by (Constantinou 1994):

� =Ed

2�Ku20(3.10)

where Ed is the dissipated energy in one cycle of loading, and K is the sti�ness of the

system. By substituting for the dissipated energy, the resulting expression becomes:

� =1

2�Ku20

Z T

0

Fv _udt =21+�dcu�d�10 !�d�20

�m� �2(1 + �d

2)

�(2 + �d)(3.11)

where T = 2�=!0, m is the inertial mass, and � is the gamma function.

For dampers with �d > 1, the damping ratio decreases with increasing amplitude

of motion, making them undesirable for vibration control problems. For dampers with

�d < 1, the damping ratio increases with increasing amplitude of motion, whereas for

linear dampers the damping ratio is independent of the amplitude of motion. One

advantage of dampers with �d < 1 is that the force tends to atten out at higher

velocities (Constantinou 1994), providing an upper-bound on the expected force

levels.

Taylor Devices, Inc. is one company that is widely involved in developing uid

damping devices for seismic protection. These devices are rated and proof tested

to a minimum burst pressure of 20,000 psi, per U. S. government standards. The

operating uid is inert silicon, with an operating temperature range from -40Æ to 160Æ

F. Table 3.2 lists the properties of a few of the high capacity linear uid viscous

dampers available from Taylor Devices, Inc.

Most early studies into the e�ect of viscous dampers on structural response treated

the structure as SDOF systems (Lin et al. 1991). Zhang and Soong (1992) have pro-

posed a sequential procedure for the optimal placement of the dampers. The location

of the dampers were determined based on the concept of degree of controllability.

Dampers were placed where the response of the uncontrolled structure is the largest.

The dampers were tested at several di�erent ambient temperatures. Though the per-

formance of the damper degraded at higher temperatures, it still contributed to the

overall damping of the structure for all temperature values tested. Recently, Shen and

29


Table 3.2: High Capacity Fluid Viscous Dampers from Taylor Devices, Inc.

Maximum Force Extended Length Bearing Width(kips) (inches) (inches)100 131 7.5200 132 9.0300 138 11.5600 155 16.01000 166 23.02000 180 26.0

Soong (1996) have proposed a simple design procedure based on the the minimiza-

tion of the Park and Ang damage index, described in Section 2.4.3. They propose

a transformation of an MDOF system to an equivalent SDOF model in the inelastic

range so that the damage function can be applied.

3.3.2 Friction-Slip Dampers

Friction-slip devices have also been studied as a means of controlling the seismic

response of buildings. These devices contain specially designed surfaces that are

designed to slip past one another at a predetermined load value, called the slip load.

During severe excitations, the device slips before any main structural members have

yielded. Until the load across the device reaches the slip load, no energy dissipation

occurs. Since the behavior of these devices rely on nonlinear behavior, a nonlinear

dynamic response calculation should be performed to verify that the desired building

response was achieved (Hanson 1993).

Friction devices have demonstrated excellent energy dissipating characteristics

(Aiken et al. 1993; Filiatrault 1990; Grigorian et al. 1992). Their hysteretic behavior

is regular and repeatable, with a nearly perfect rectangular shape to the hysteretic

curve. The force displacement response has been shown to be basically independent

of loading frequency, amplitude, number of loading cycles, and temperature (Aiken,

Nims, Whittaker, and Kelly 1993).

One type of friction device was developed by Pall in 1982, called a Pall Friction

Device (Filiatrault 1990; Cherry and Filiatrault 1993). This system consists of

30


a mechanism containing brake lining pads introduced at the intersection of frame

crossings. In order to be fully e�ective, the device must slip in both compression and

tension, otherwise energy will be dissipated only in the �rst few cycles. If the bracing

elements are designed not to buckle in compression, then a simple friction joint would

suÆce. However, designing braces to this requirement is often not economical. The

Pall device causes the friction design to slip in both tension and compression by

simultaneously connecting the friction mechanism at the intersection of the frame

cross-braces. The element sliding in tension then causes the other element to slip

back to its original position, so it can immediately absorb energy when the cycle

reverses. A. Filiatrault and S. Cherry (1987) have developed a simpli�ed analysis

technique for these devices.

Another type of friction device is a Slotted Bolted Connection (SBC's). SBC's

have recently been studied as simple and inexpensive friction-slip devices (Grigo-

rian, Yang, and Popov 1992; Grigorian and Popov 1994). SBC's are modi�ed bolted

connections designed to dissipate energy during rectilinear tension and compression

loading cycles. The devices were �rst tested on an MTS loading frame and tested

under displacement control (Grigorian, Yang, and Popov 1992). SBC's which include

a shim-like brass insert plate show almost no variation in the slip force magnitude.

These devices can be approximated as an elastic-perfectly-plastic connections. They

were also tested as part of a braced structural frame (Grigorian and Popov 1994).

Both analytical and experimental results were presented, showing that properly de-

signed SBC's incorporated into structural systems are highly e�ective at dissipating

seismic energy.

3.4 Active Control

In comparison with passive systems of the type described in previous sections, the

research and development of active structural control strategies is more recent. The

advantages that typically cited for active control systems are: (1) enhanced e�ec-

tiveness in motion control, limited only by capacity of the system, (2) applicability

to multi-hazard applications, and (3) selectivity of control objectives. Active control

systems are used to control the response of structures to internal or external exci-

tation, such as machinery, wind, or earthquakes. However, the addition of purely

31


active control carries a risk of possible destabilization of the structure. Given that

civil engineering structures are statically stable, as they are attached to the ground,

active control solutions have been slower to be accepted in civil engineering practice

as compared to passive control solutions.

3.4.1 Basic Principles

Active control is branch of engineering concerned with the design of systems that

are able to act so as to force a system to behave according to some prescribed spec-

i�cations. The behavior of the system is described by means of a dynamic system

characterized by state variables. The relationship between the system and the ex-

ternal environment is described in terms of input variables, which represent external

actions on the system, and output variables, which are measurable responses on the

system. As mentioned previously, two basic controller arrangements are: open-loop

and closed-loop. In open-loop control, the controller manipulates the system by try-

ing to drive output to a speci�ed behavior without using information regarding the

output. This type of control su�ers by not allowing it to compensate for errors due to

perturbations in either the input, from unknown or unmeasured disturbances, or sys-

tem parameters. In contrast, closed-loop control uses information about the system

response as feedback information. The basic elements found in a closed-loop control

system are shown in Figure 3.2.

The following naming conventions are used to describe the combination of the

structural system and active control components. An actuator architecture or actuator

system refers to the physical actuator system, with their corresponding location and

capacities. The sensor system is the set of sensors, their locations, and �ltering

qualities. Sensors can be used to measure both the input and output response. Closing

the loop is the controller, which is the system that processes the sensor measurements

and determines how the control system should respond by sending the appropriate

signals to the actuators. The control algorithm provides the mathematical theory

within which the controller is formulated.

Studies into active control of civil engineering have ourished since its introduction

to the �eld by Yao (1972). Topics of research conducted to date include H2 and H1

control (B. F. Spencer and Sain 1997; Suhardjo et al. 1992; B. F. Spencer et al. 1994),

32


Structure SensorsActuators

Controller

Excitation

Output

Control Signal

Figure 3.2: Basic Elements of a Closed-Loop Active Control

sliding mode control (Nonami et al. 1994; Yang et al. 1995), e�ects of actuator satura-

tion on control (Chase and Smith 1996), reliability-based control (B.F. Spencer et al.

1992; R.V. Field et al. 1996; R. V. Field and Bergman 1997), fuzzy control (Nagara-

jaiah 1994), neural control (Venini and Wen 1994; Ghaboussi and Joghataie 1995),

nonlinear control (Agrawal and Yang 1996), modelling issues (Smith and Schemmann

1996; Dyke et al. 1995), and benchmark studies (B.F. Spencer et al. 1997). Ref-

erences to the current state-of-the-art in active control can be found in Spencer et

al. (1997) and Housner et al. (1997).

Common actuator systems found in civil engineering applications include the ac-

tive mass driver (AMD) and the active tendon brace (ATB) systems. The AMD

actuation system is composed of a small, relative to the structure, additional mass,

that is activated by servo-hydraulic actuators which connects the mass to the struc-

ture. The system is typically located near the top of the structure so as to be most

e�ective at controlling the �rst mode of the structure. The ATB actuation system

uses a tendon/pulley system to transmit the force generated by the hydraulic actu-

ator to the structure. These actuators can be placed throughout the height of the

structure and can be useful in controlling various vibrational modes.

Consider a linear structure modeled by an n-dof shear building model, where the

matrix equation of motion can be written as:

M�x(t) +C_x(t) +Kx(t) = Gu(t) + Jf(t) (3.12)

33


whereM, C, andK are the system mass, damping and sti�ness matrices, respectively;

x(t) is the displacement vector; f(t) represents the vector of applied load or external

excitation; and the vector u(t) is the applied control force vector. The matrices G

and J de�ne the locations of the control force vector and the excitation, respectively.

Suppose that the control force is designed to be a linear function of the measured

displacement vector, x(t), and the velocity, _x(t). The control force vector takes the

form:

v(t) = Kux(t) +Cu _x(t) +Mu�x(t) (3.13)

where Ku and Cu are the control gains. By substituting, the resulting equation is of

the form:

(M�GMu) �x(t) + (C�GCu) _x(t) + (K�GKu)x(t) = J f(t) (3.14)

so that the e�ect of control is to modify the structural parameters so that it can

respond more favorably to the external excitation. The choice of the control gain

matrices in Equation (3.14) depends on the control algorithm selected (Rodellar

1994).

More typically, Equation (3.14) can be written as a system of �rst-order di�erential

equations, referred to as the state-space representation, as follows:

(_x(t)

�x(t)

)=

"0 I

�M�1K �M�1C

#(x(t)

_x(t)

)+

"0

M�1

#Fg(t) +"

0

M�1

#Fu(t)

(3.15)

where Fg is the vector of forces due to ground motion excitation and Fu is the vector

of applied control forces.

3.4.2 Control Algorithms

Several analytical theories are available to develop a control algorithm for active

structural control. Three di�erent theories discussed here are: 1) optimal control,

34


2) stochastic control, and 3) robust control. The optimal control design involves

minimizing or maximizing a performance measure, or cost function. One example for

linear systems is the minimization of a quadratic cost function, such as:

J =1

2x(tf)

TSx(tf) +

1

2

Z tf

t0

fx(t)T Qx(t) + u(t)T Ru(t)gdt (3.16)

where S, Q, R, are weighting matrices x(t) is the state of the system and u(t) is the

control vector. Though the cost function itself has no physical meaning, it is used

to weight two variables that do have a physical signi�cance. The matrix S penalizes

the state vector at the end of the analysis. By manipulating the relative values

of Q and R, a controller can be designed that compromises between maintaining

the state close to the origin while not allowing the control e�ort to be excessive.

The advantage of linear quadratic (LQ) control is its simplicity and the fact that it

results in a linear controller; however, these controllers have limited success in seismic

applications, largely due to their inability to account for uncertainties in the structural

system (Housner et al. 1997; Skelton and Shi 1996; Soong and Constantinou 1994).

The use of stochastic control encompasses a number of functions. Among these

are (Housner et al. 1997):

� The determination of control policy for a dynamic system subject to random

vibrations so as to achieve some speci�ed objective

� The use of incomplete or noisy information in the estimation of the states of

the dynamic system.

� Ability to assess the robustness of a control system with respect to the uncer-

tainty in the structure and control parameters and system inputs.

The time-optimal control problem considered requires the determination of a control

vector, u(t), that minimizes a scalar cost function of the form:

J = �(x(tf) ;w(tf) ;p(tf) ; tf) +

Z tf

t0

L(x(t) ;u(t) ;w(t) ;p(t) ; t) dt (3.17)

where x(t) is the n-dimensional state vector; u(t) is the control vector; w(t) is a

vector of disturbances; p(t) is a vector of system parameters; �(�) expresses the

terminal constraint; and L(�) is the Lagrangian. Though the above expression �nds

35


many applications in structural engineering, the solution of the control problem is not

easily determined. So as to simplify the problem, the cost function is generally given

as the expected value of the expression of Equation (3.17). Also, the disturbances

and sensor noise are assumed to be small and additive, and the parameters and

initial conditions of the system are assumed to be random variables. However, the

determination of the optimal control inputs requires knowledge of the state vectors,

implying a recursive nonlinear estimation and design procedure, limiting its general

application to civil engineering structures (Housner et al. 1997).

Robust control focuses on the issues of system performance in the presence of

uncertainty in both parameter and system inputs. LQ theory, which is discussed

brie y above, cannot explicitly accommodate uncertainties in the structure parame-

ters. This need led to the development of H1 problem, principally in the frequency

domain. The objective of H1 control is to design a controller such that the in�nity

norm of the transfer function from input excitations, w(t), to controlled responses,

y(t) is minimized. Mathematically, this expression is given by:

k Tyw k1= supw(t)

k y(t) k2k w(t) k2 � (3.18)

where Tyw is de�ned as the transfer function between w(t) and y(t), sup is the

supremum over all w(t), and is a positive, scalar attenuation constant, which is

speci�ed by the designer. k y(t) k2 indicates the 2-norm of y(t) and is given by

k y(t) k2= (y1(t)2 + y2(t)

2 + � � �+ yq(t)2)

1

2 , where q is the number of sensor measure-

ments. In general, this transfer function contains measures of nominal performance

and stability robustness. An important characteristic of this problem is that the H1

norm gives a measure of the worst case response of the system over an entire class

of input disturbances. Application of H1 control theory to civil engineering struc-

tures has been investigated by several researchers, and a good summary can be found

in Housner et al. (1997).

36


3.5 Role of Structural Control in Performance-Based

Engineering

Structural control provides an extra mechanism to improve structural performance.

For maximum e�ectiveness, minimal control e�ort would be required to achieve the

desired performance goals. In active control, this control e�ort is measured in terms of

the energy and forces required of the actuation system. Attempting to apply very large

control forces may not be physically realizable. Under severe seismic loads, meeting

the desired performance objective may be unattainable with a given controller design

and architecture. In such cases, a di�erent control solution should be investigated.

Extensive work in the �eld of passive energy dissipation have resulted in tenta-

tive requirements for the design and implementation of passive energy dissipation

devices in the United States. The Energy Dissipation Working Group of the Base

Isolation Subcommittee of the Structural Engineers Association of Northern Califor-

nia (SEAONC) has developed a document addressing these tentative requirements,

which provides design guidelines applicable to a wide variety of systems (Whittaker

1992; Whittaker 1993). The general philosophy is to con�ne inelastic deformation to

the energy dissipators, while the main structure remains elastic for the design basis

earthquake.

Recent guidelines (BSSC 1997; BSSC 1998) prepared by the Building Seismic

Safety Council (BSSC) for the Federal Emergency Management Agency (FEMA)

specify provisions which are similar to those developed by SEAONC in both scope

and philosophy. Simpli�ed design procedures are available for both the bearing system

and superstructure depending on the isolation approach. For new buildings, the un-

derlying design philosophy for base isolation systems has two performance objectives.

The primary performance objective is speci�ed for the �xed-base structure, which is

the protection of life-safety for major earthquake. The secondary performance objec-

tive provides damage reduction for the isolated system. For rehabilitation of existing

buildings, the performance objectives may be less stringent depending on the owner's

requirements. If the current structure does not adequately provide for protection of

life-safety, the primary goal of the application of an isolation system would be that

target.

37


Passive energy dissipation systems are also speci�cally addressed in in the design

guidelines. For applications in new buildings, the guidelines are only general in na-

ture. They state that the design of the system shall be based on \rational methods of

analysis, using the most appropriate analysis methods." For the rehabilitation of ex-

isting structures, speci�c guidelines are given concerning the modeling of the dampers

and structure, allowable analysis procedures, and the design and construction review.

Current guidelines allow response spectrum analysis for rate-dependent devices, such

as viscous dampers, as long as the remaining structure remains elastic. For all other

systems, such as friction dampers and inelastic systems, a nonlinear dynamic analysis

must be used. As the passive energy dissipation technology is still relatively new, a

conservative approach is taken. For example, an independent review panel must be

formed to conduct a review of any design involving additional passive dampers. Also,

all devices must be tested to shown that they meet design requirements. Though

speci�c performance-based guidelines have not been put forth, a reasonable assump-

tion is that they will be similar to those for base isolation systems. For example, the

primary objective would be the protection of life-safety while the secondary perfor-

mance objective would provide for damage reduction when compared to the initial

structural system.

Active control, which is the newest vibration control technology used in civil

engineering applications, is only brie y addressed in code provisions. The recent

guidelines indicate that these systems may be considered, though the design must be

reviewed by independent review panel, enpaneled by the owner prior to the prelimi-

nary design phase, and fully tested.

Incorporation of the control design process within the overall design process is

illustrated in Figure 3.3. The left side of these �gures represents the design process

for the structure based on the concept of performance-based design. Once the exci-

tation(s) are chosen and the initial structural design is developed based on collapse

prevention criteria, controllers can be used to help meet the performance criteria

associated with minimizing damage. An important consideration is that the uncon-

trolled structural response not compromise life safety; therefore, controllers are not

incorporated until minimal requirements are met.

Referring to Figure 3.3, the shaded region on the right represents the controller

design process. A controller can be designed for the original structure, which together

38


Structure

ResponseQuantities

LifeSafetyMet?

Analyze

DamageCriteria Met?

Controller

Excitation

ControllerDesign Model

Control Systemand Objectives

Control System

Control Design Process

yes

no

yes

no

END

(a) Passive-Control Systems

Structure

ResponseQuantities

LifeSafetyMet?

Analyze

DamageCriteria Met?

Controller

Excitation

DevelopUncertainty Model

Nominal ControllerDesign Model

Control Specifications:Objectives, Algorithm,

and Architecture

Control System

Control Design Process

yes

no

yes

no

END

(b) Active-Control Systems

Figure 3.3: Design Process for Controlled Structural Systems

with the original structure represents the controlled system. Per the iterative process

shown in Figures 3.3, control speci�cations can be updated until the performance

criteria for the system are met.

Speci�cations for active vibration controller include: (1) the control objectives,

which are dependent on the overall performance criteria for the structure; (2) the

control algorithm (i.e., software); and (3) the control architecture, which includes the

actuator and sensor requirements. Control analysis may be computationally intensive;

therefore, simulations are usually performed on a reduced-order, nominal model of

the structure. Furthermore, explicit incorporation of nonlinear e�ects is usually not

feasible when designing the controller.

39

Chapter 4

Description of Case Studies

4.1 Objective of Simulations

The objective of the research presented here is to evaluate the role of structural

control technology in enhancing the overall structural performance under seismic

excitations. This study focuses on steel moment resisting frames, and three types of

possible controllers: (1) base isolation system (passive); (2) viscous brace dampers

(passive); (3) and active tendon braces. Two structures are selected from the SAC

Phase II project: the three story system and the nine story system designed for the

Los Angeles region. Simulations of these systems, both controlled and uncontrolled,

are prepared using the three suites of earthquake records, also from the SAC Phase II

project, representing three return di�erent periods. Several controllers are developed

for each structure and the performance is judged based on the interstory demands

and dissipated hysteretic energy.




controller performance to variations of the control parameters, load levels and struc-

tural modeling techniques; (3) To compare the bene�ts of the controllers in both a

deterministic and probabilistic format.

This chapter provides a description of the two structures that are analyzed and

the ground motions utilized for seismic demand calculations. Three di�erent types

of control systems are then selected for implementation with these structures. The

40

Chapter 4 Description of Case Studies

reasoning behind the selection of these systems and the basic design philosophy of

each one is then presented.

4.2 Description of Structures

The structures analyzed are two steel moment-resisting frame buildings (SMRF), one

3-stories and the second 9-stories tall, designed as part of the SAC steel project for the

Los Angeles area. These buildings conform to local code requirements. All buildings

are oÆce buildings designed for gravity, wind, and seismic loads, with a basic live

load of 2.4 kPa (50 psf). The structural system for all buildings consists of steel

perimeter moment frames and interior gravity frames with shear connections. All

columns in the perimeter frame that are part of the lateral force-resisting system

bend about the strong axis. The nominal yield strength of beams is 36 ksi (248.0

MPa) and the nominal yield strength of the columns is 50 ksi (344.5 MPa). However,

the expected strength of the material is very di�erent from the nominal values used

in design (SPPC 1994). So the expected yield strength used for analysis is 49.2 ksi

for beams and 57.6 ksi for columns. The design of the moment frames in the two

orthogonal directions was found to be either identical or very similar, thus only one

direction was chosen for analysis. The typical di�erence in the two directions lies

in the orientation of the gravity beams and sub-beams, which are oriented in the

North-South direction. As a result, only the frames in the North-South direction are

analyzed.

The North-South frame of the 3-story structure has three fully moment-resisting

bays and one simply-connected bay, as shown in Figure 4.2. The columns are �xed at

the base and run the full height of the structure. The dimensions shown in Figure 4.2

are centerline dimensions, and the section sizes are listed next to each corresponding

member.

The North-South frame of the 9-story structure has four fully moment-resisting

bays and one partially moment-resisting bay, as shown in Figure 4.2. The simple

shear connection in the partial moment-resisting bay occurs as a result to avoid bi-

axial bending in the corner columns. One basement level is also present, which is

horizontally restrained at the ground level. The columns are pinned at the base of

the structure and spliced at every other story, indicated by the parallel horizontal lines

41

Description of Case Studies Chapter 4

3rd Floor

2nd Floor

1st Floor

Ground

W24x68 W24x68 W24x68 W21x44

W21x44

W21x44

W30x116 W30x116W30x116

W33x118 W33x118 W33x118

30' - 0'' 30' - 0'' 30' - 0'' 30' - 0''

Figure 4.1: 3-Story Structure: North-South Moment-Resisting Frame

Table 4.1: Column Sections for 9-Story Structure - North-South Frame

Limiting Floor Levels Moment Frame SectionsLower Upper Interior Exterior

-1 1/2 W 14x370 W 14x5001/2 3/4 W 14x370 W 14x4553/4 5/6 W 14x283 W 14x3705/6 7/8 W 14x257 W 14x2837/8 Roof W 14x233 W 14x257

across the columns, with the splice location six feet above the oor. The dimensions

shown in Figure 4.2 are centerline dimensions, and the section sizes for beams are

listed next to each corresponding member. Sections for column members are listed

in Table 4.1.

4.3 Earthquakes

Performance based design approaches are now being used to develop the next genera-

tion of buildings codes (e.g. SEAOC Vision 2000 (SEAOC 1995); SAC Steel Building

Project; FEMA-273 (BSSC 1997). The probabilistic ground motion response spectral

maps recently developed by the USGS provide a �rst-order, non site-speci�c estimate

42


W24x68 W24x68 W24x68 W24x68 W24x68

W27x84 W27x84 W27x84 W27x84 W27x84

W30x99 W30x99 W30x99 W30x99 W30x99

W36x135

W36x160

9th Flr

8th Flr

7th Flr

6th Flr

5th Flr

4th Flr

3rd Flr

2nd Flr

1st Flr

Ground

B-1

W36x135 W36x135 W36x135 W36x135

W36x135 W36x135 W36x135 W36x135 W36x135

W36x135 W36x135 W36x135 W36x135 W36x135

W36x135 W36x135 W36x135 W36x135 W36x135

W36x160 W36x160 W36x160 W36x160

W36x160 W36x160 W36x160 W36x160 W36x160

W36x160 W36x160 W36x160 W36x160 W36x160

30' - 0'' 30' - 0'' 30' - 0'' 30' - 0'' 30' - 0''

Figure 4.2: 9-Story Structure: North-South Moment-Resisting Frame

43


of the response spectra for use in performance-based design. However, the imple-

mentation of performance-based design requires more detailed speci�cation of input

ground motions than do conventional codes. In particular, many of the methods be-

ing used in the design and analysis of buildings for performance-based design require

ground motion time history inputs.

Suites of ten time histories were generated by Sommerville (1997) to represent

ground motions having probabilities of exceedance of 50% in 50 years, 10% in 50 years,

and 10% in 250 years in the Los Angeles region. These sets of ground motions are

referred to as the 50 in 50 Set, 10 in 50 Set, and 2 in 50 Set, respectively, throughout

this study. The time histories have magnitude-distance pairs that are compatible

with the deaggregation of the probabilistic seismic hazard. Individual time histories

were scaled so that their response spectra are compatible with the spectral ordinates

from the 1996 USGS probabilistic ground motion maps, adjusted for site conditions

from soft rock to sti� soil (from SB/SC boundary to SD), in the period range of 0.3

to 4 seconds. A single scaling factor was found for each time history that minimized

the squared error between the target spectrum and the average response spectrum of

the two horizontal components of the time history assuming lognormal distribution

of amplitudes. The weights used were 0.1, 0.3, 0.3, and 0.3 for periods of 0.3, 1, 2,

and 4 seconds respectively. The scale factor was then applied to all components of

the time history.

The time histories for the 50 in 50 Set are all derived from recordings of crustal

earthquakes on sti� soil (category SD). The time histories are derived from earth-

quakes in the magnitude range 5.7 to 7.7, and the distance range of about 5 to 100

km. With the exception of the Downey recording of the 1987 Whittier Narrows earth-

quake, which was scaled up by a factor of about 3.6, none of the recordings required

scaling by more than a factor of 3.

The time histories for the 10 in 50 Set are all derived from recordings of crustal

earthquakes on sti� soil (category SD). Most of the recorded time histories were

scaled up by factors between 1 and 3 to match the target values. Two recordings

of the magnitude 7.3 Landers earthquake of 1992 at distances of about 40 km are

included to represent large earthquakes on the San Andreas fault at a comparable

distance from Los Angeles. The other eight time histories are near-fault recordings

of strike-slip, oblique and thrust earthquakes in the magnitude range 6 to 7.

44


For the 2 in 50 Set, all of the time histories are from near-fault recordings or

simulations, and the scale factors required to match the response spectra to the

target spectra are relatively close to unity. The recorded time histories are from

the 1974 Tabas, 1989 Loma Prieta, 1994 Northridge and 1995 Kobe earthquakes.

The simulated time histories are for magnitude 7.1 earthquakes on the Palos Verdes

fault (a strike-slip fault), and on the Elysian Park fault (a blind thrust fault). The

simulated time histories were generated using di�erent methods for the short-period

and the long-period portions of the spectral acceleration curve. The results from

these two methods were then merged near a period of 1 second. As a result, results

for structures in this period range for these records may be questionable.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

Mean Elastic Spectral Acceleration LA Record Sets, ξ = 2%

Period (seconds)

Acc

eler

atio

n (g

)

50/50 Set10/50 Set2/50 Set

Figure 4.3: Mean Elastic Spectral Acceleration for Ground Motion Sets

The resulting mean spectral acceleration for all three ground motion sets are shown

in Figure 4.3. An important note regarding the earthquake sets is that they should be

used only as a set, and not individually or as small sub-sets as representative of the

probability levels speci�ed. At any particular period the median spectral acceleration

of the set may match the target value reasonably well; however, any individual record

may have a value quite di�erent than the expected target spectral acceleration. A

45


plot showing a measure of the dispersion for the spectral acceleration within each

ground motion set is given in Figure 4.4.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Measure of Dispersion: Elastic Spectral Acceleration LA Earthquake Sets: ξ = 2%

Period (seconds)

Sta

ndar

d D

evia

tion

of L

n(S a)

2 in 50 Set

10 in 50 Set

50 in 50 Set

Figure 4.4: Dispersion of the Elastic Spectral Acceleration for Ground Mo-tion Sets

4.4 Control Systems Designed and Evaluated

As discussed in Chapter 3, three basic types of control systems are used in vibration

control of civil engineering structures. These control types are: (1) isolation systems,

(2) passive damper systems, and (3) active control systems. A representative control

system was chosen from each type listed above for implementation with the two

structures to be analyzed. These systems are: (1) the friction pendulum isolation

system, (2) the linear uid viscous damper system, and (3) the active tendon brace

system. The basic design and implementation of these systems is described in the

following subsections.

46


4.4.1 Friction Pendulum Isolation System

The isolation system studied is the Friction Pendulum System described in Sec-

tion 3.2.2. The design of this system requires the selection of the isolation period,

the sliding surface of the bearings, and the number and location of the bearings. All

isolation systems designed are located at the ground level, have a sliding surface of

un�lled te on, and the direction of sliding is parallel to the lay of the te on. In the

9-Story structure, so as to place the bearings at ground level, the basement columns

are cut and the bearings are placed at their top. Investigations into locating the

bearings at the bottom of the basement resulted in an unrealistic system. The result-

ing bearing displacements were larger and the cost of excavating a gap around the

structure of that depth would not be cost-e�ective.

Isolation systems with varying isolation period and number of bearings are inves-

tigated to determine the impact of these parameters. The number of bearings used is

determined by matching the two target bearing pressures of 6.9 MPa (1 ksi) and 44.9

Mpa (6.5 ksi) for the individual bearings. These two bearing pressures were chosen as

the resulting frictional properties for the systems are representative of the high and

low frictional capacities for te on sliding surfaces (see Table 3.1).

The isolation period for the various systems is determined by selecting the radius

of curvature, R, for the sliding bearing as follows:

R = g

�T

2�

�2

(4.1)

The design requirements for friction isolation systems require that a time-history

analysis be performed with at least three pairs of horizontal time-history compo-

nents (BSSC 1998). The selection of the �nal isolation design for this structure is

based on the resulting demands from the nonlinear dynamic analyses of all three sets

of ground motions as discussed in Section 6.3.1. These designs were chosen on the full

suite of earthquakes, satisfying those design requirements. For the 3-Story structure,

the isolation period is varied from 1 to 4 seconds. Investigations into the resulting

seismic demands of these systems are presented and a �nal design chosen from those

systems. For the 9-Story structure, the 2 in 50 set of earthquakes were used to eval-

uate and design the isolation systems. As the original �xed base structure has a �rst

47


mode period of 2 seconds, the performance of isolation systems with isolation periods

of 4 and 5 seconds were evaluated as these systems would provide enough shift in the

fundamental period to act as true isolators.

4.4.2 Fluid Viscous Damper

Viscoelastic (VE) and viscous (VS) dampers can be classi�ed as one group of energy

dissipators since they dissipate energy depending on the relative velocity between the

two ends of the damper. VS dampers (such as uid cylinders) can be designed to

provide a purely viscous force to the surrounding structure. Although the damper

uid material may be temperature sensitive, this sensitivity can be reduced to achieve

stable damper behavior. When installing the dampers into a frame, support members

such as braces are required. In this study, the damper-brace component is collectively

referred to as the damper system.

The VS dampers studied are assumed to be linear in nature. The brace support for

the damper is assumed to be rigid compared to the damper, so that all deformation in

the damper system occurs through damper deformation. This assumption has been

shown to be valid when the brace sti�ness is at least 10 times the damper storage

loss sti�ness, given by K00

d = Cd! (Fu and Kasai 1998). The dampers are located

in or adjacent to the center of the moment-resisting frame and are arranged so that

one damper runs diagonally across each story. The placement of the dampers for the

3-Story structure are illustrated in Figure 4.4.2.

Damping systems are designed through the selection of the damping constant,

Cd;i, for each damper, i. The resulting systems have critical damping ranging from

10% to 40% of critical for the fundamental modes of vibration. The damping ratio

for the retro�tted frame was determined through the modi�ed modal strain energy

method (Fu and Kasai 1998), such that:

�n =1

2

f�ngT [K00

d ]f�ngf�ngT [K]f�ng (4.2)

where f�ng is the nth modal shape, [K] is the sti�ness matrix of the system, and [K00

d ]

is the matrix form of the viscous damper's loss sti�ness.

48


3rd Floor

2nd Floor

1st Floor

Ground

30' - 0'' 30' - 0'' 30' - 0'' 30' - 0''

damper

Cd,1

Cd,2

Cd,3

Figure 4.5: 3-Story Structure with VS dampers

4.4.3 Active Tendon System

An active tendon system was chosen for implementation in the two structures. This

particular actuator system was chosen because its force application is similar to that

of a passive viscous brace system, allowing for a more direct comparison between

the two systems. Speci�cations for active vibration controller include those for the

control algorithm and the control architecture, which includes the actuator and sensor

speci�cations. The actuators are located in or adjacent to the center of the moment-

resisting frame and are arranged so that one damper runs diagonally across each

story. For the 3-Story structure, an actuator is place in each story. In the 9-Story

structure, actuators are only placed in selected stories based on analysis results of

the uncontrolled structure. Accelerometers are utilized as sensors and are placed

to measure horizontal oor accelerations at all oors above ground level. A robust

H1 controller was developed for the above architecture using interstory drifts as

the regulator response quantity. The design model represents the real system to the

controller design optimization procedures, thus should be a realistic as needed to

characterize the behavior of the physical system that impact the e�ectiveness of the

controller.

During the controller design, the structural dynamics of all of the structural sys-

tems used in this research are modeled as linear dynamic systems. The control de-

sign utilized here augments the nominal dynamic system with additional frequency

49


weighted uncertainties. Ground motion excitation is modeled as an unknown external

excitation with characteristic frequency content modeled using Kanai-Tajimi �lters.

Sensors are assumed to be unbiased and each sensor has some small level of inde-

pendent white noise error that is modeled as an external excitation. Actuators in

the design model are assumed to be band-limited in capacity and have errors rep-

resented by an independent white noise excitation added to the command signal of

each actuator.

Control analyses may be computationally intensive; therefore, simulations are

usually performed on a reduced-order, nominal model of the structure. To reduce

the size of these systems, traditional structural dynamic reduction techniques such

as modal truncation or Guyan reduction are applied. The application of the Guyan

reduction is performed on the state-space form of so that the actuator and sensor

mapping matrices are transformed as well as the basic structural dynamics. Also, the

degrees-of-freedom (and states) of the reduced system are in terms of inter-story drift

values, not oor displacements relative to the ground. Designing active controllers

for these structures is discussed in detail in Breneman (1999).

50

Chapter 5

Description of Modeling and

Analysis

5.1 Introduction

During the design evaluation process, the engineer requires information about the

behavior of actual structure. The amount of detail required varies depending on the

application of the information; i.e., whether used for preliminary design or �nal design

veri�cation. Typically, evaluation of a structure is achieved through a mathematical

representation and analysis of the structure. The basic steps taken during this process

are: 1) the development of a mathematical model for both the structure and loading

conditions; 2) the performance of an analysis; and 3) the interpretation of results.

Structures designed for seismic resistance are expected to deform in the inelastic

range when subjected to design level earthquake ground motions. The elements of

the structure are thus called upon to dissipate the seismic energy in the form of

hysteretic energy. With the addition of control systems, however, the amount of

energy that must be dissipated within steel structural elements is reduced. Developing

a mathematical model for the system requires consideration of the basic element

behavior of both structural and control elements, including inelastic e�ects.

This chapter discusses the basic modeling approach taken for the structure and

control systems described in Chapter 4. Time history analysis of these systems is

performed using software written expressly for this purpose. The representation of

the element behavior in the analysis software is discussed, including the modeling

51

Description of Modeling and Analysis Chapter 5

assumptions of element behavior.

5.2 Structural Modeling Approach

The focus of this study is on developing a better understanding of the seismic demands

for a variety of control systems and structures when subjected to a multitude of ground

motions. A compromise between accuracy and eÆciency is usually necessary given

the scope of the problem being addressed. The critical modeling requirement is to

represent all major behavioral characteristics of the structure such that the behavior

represented is accurate with reasonable con�dence. This section describes the basic

modeling approach and assumptions taken in representing the structures described

in Chapter 4.

5.2.1 Finite Element Model

The structures are modeled as two-dimensional frames that represent half of the

structure in the north-south direction. The frame is given half of the seismic mass of

the structure at each oor level. The seismic weight and mass properties for the two

structure are given in Section 4.2.

A basic centerline model of the bare moment-resisting frame, MRF, is developed

for both structures. The beams and columns extend from centerline to centerline.

The strength, sti�ness, and shear distortions of panel zones is neglected. Moments in

the beams and columns are computed at the connection centerline as opposed to the

faces of columns and beams, which results in a high estimate of moments.

This type of centerline model is perhaps the most widely used in structural engi-

neering analyses. The basic argument in favor of this type of model is that the use of

centerline dimensions compensates for the disregard of panel zone shear deformations

and that sti�ness estimates based on bare frame properties ignore the contributions

from nonstructural elements and the gravity load-resisting system. One of the re-

sulting e�ects of this assumption is the increased importance given to beam versus

column sti�ness in drift control. Though this assumptions does not accurately cap-

ture the distribution of demands between beams and columns, the e�ect on global

demand estimates is minor for these structures (Gupta 1998).

52

Chapter 5 Description of Modeling and Analysis

Two basic modeling assumptions have been made throughout this study with re-

spect to the steel structural members: 1) designs use compact sections with adequate

lateral bracing provided, so that strength and sti�ness deterioration due to local and

lateral buckling can be neglected, and 2) that elements do not exhibit any undesir-

able characteristics under tension. Beams consist of elastic portions and partially or

completely plasti�ed regions whose location, length, and strain distribution depend

on geometric parameters, boundary conditions, the e�ect of gravity loading, and the

interstory drift demands imposed by the earthquake. For designs in which the e�ect

of the gravity load is signi�cantly smaller than the lateral load e�ect, the regions of

plasticity (if any) are usually located near the face of the column.

x1

θ1

y1

θ3 θ4

x2

θ2

y2

Figure 5.1: Lumped Plasticity Model for Beam-Column Element.

A �nite element model of the structure was developed where an assembly of in-

terconnected elements describes the hysteretic behavior of structural members. The

inelastic behavior of the members is taken to be concentrated at the end of girders

and beams. Thus each structural member is constructed using a lumped plasticity

model with nonlinear rotational springs at each end joined by a linear beam-column

element, as seen in Figure 5.1. The hinges are zero length elements with very high

initial sti�ness relative to the beam elements. The strength of the springs is equal

to the plastic moment capacity of the beam section, and the post-yield sti�ness is

calculated based on the elastic exural sti�ness of the beam.

The presence of signi�cant axial loads, as found in columns, leads to inelastic

strain distributions and moment-curvature relationships that are greatly a�ected by

the loading history. If the axial force is high, the plastic regions of the member may

extend over a signi�cant length of the column, and the point-hinge approximation

becomes extremely poor. The formation of regions of plasticity in columns can result

in detrimental behavior, especially if a story mechanisms forms.

For these reasons, current design practice limits the plasticity that occurs in

53


columns through the strong column-weak girder (SCWG) concept. If column hinging

is limited, then the point hinge model and the column interaction equations from the

1994 LRFD provide a reasonable approximation of column behavior. For the columns

in this research, the bending strength for the columns is de�ned by a bilinear P-M in-

teraction diagram. For columns bending about the strong-axis, the nominal bending

strength is equal to the plastic strength of the section for axial loads less than 15% of

the axial load. Beyond that point, a linear relationship exists between axial load and

nominal bending strength. For columns bending about the weak-axis, the transition

point is taken at 40% of the axial load-carrying capacity of the column.

Gravity loads are applied the columns as nodal loads. The basement of the 9-

Story frame is modeled as a typical story; however, the basement oor and ground

level are restrained against horizontal displacement. The seismic excitation is applied

equally at all laterally restrained degrees of freedom.

Unless speci�cally speci�ed, all models use 3% strain-hardening, strength proper-

ties are based on expected strength of the material, the oor slab is assumed to be

horizontally rigid, and 2% Rayleigh damping is enforced at the �rst mode period and

a period of 0.2 seconds. The resulting modal properties for both the 3- and 9-story

structures are given in Table 5.1.

Table 5.1: Modal Properties for Frames

First Mode Second Mode Third ModePeriod (sec) � (%) Period (sec) � (%) Period (sec) � (%)

3-Story 1.02 2.0 0.33 1.5 0.17 2.29-Story 2.27 2.0 0.85 1.1 0.49 1.1

5.2.2 Modeling of P-delta E�ects

Though the models used represent only the perimeter frames of the structure, the

P-� e�ect caused by vertical gravity loads on the interior gravity system cannot be

ignored. These loads are transferred to the perimeter MRF through the rigid oor

slab. As a result, for the structures considered, each MRF has half the structure

weight contributing the P-� e�ect for that frame.

As the loads are not directly carried by the MRF, the loads in the gravity columns

54


cannot simply be lumped with the loads applied directly to the columns of the lateral

system as that assumption generally results in an over-estimation of column demands.

Therefore, the e�ect is considered by attaching an elastic P-� column to the two-

dimensional model with frame elements. The P-� column is given a very high axial

sti�ness and negligible bending sti�ness, so that the column can take the de ected

shape of the MRF without contributing to the moment-resisting system. The analysis

program uses a geometric sti�ness matrix based on the axial load in the columns under

gravity loads only, as described in Section 5.3.4.

5.3 Evaluation Platform and Implementation

In order to evaluate the behavior of the structural model, an analysis tool capable

of analyzing inelastic structural behavior during seismic loads is required. This tool

also needs to include the basic behavior of the di�erent control elements. A �nite

element nonlinear dynamic analysis code was written utilizing MATLAB1 to evaluate

the systems for this study.

Currently, the code is limited to planar, two-dimensional, structures. Each node

can have up to three degrees-of-freedom, translations in the x- and y-direction and

rotation about the out-of-plane axis. The code has provisions for specifying nodal

displacement constraints, allowing for the slaving of one degree-of-freedom to another.

The mass is assumed to be lumped at the nodes, so that the structural mass matrix is

diagonal. Also, loads may only be applied at the nodes of the model. Viscous damping

inherent in the structure is assumed to be of the Rayleigh type. The damping matrix

is assumed to be constant, so the sti�ness proportional damping is based only on the

initial sti�ness of the system. However, di�erent amounts of damping can be speci�ed

to di�erent element types.

Most engineering analyses adopt a nonlinear beam model in which the plasti�ed

region is represented as a point while the remainder of the structural member remains

elastic (Chen and Powell 1982; Allahabadi 1987). Lumped models typically consist

1MATLAB is a registered trademark of The MathWorks Inc., 24 Prime Park Way, Natic, MA01760-1415

55


of several springs that are connected either in series or parallel. Each spring or \com-

ponent" has a predetermined force-deformation response. A structural member is

then constructed by connecting the components, and the resulting force-deformation

response for the structural member is a combination of the component responses.

For this study, structural beam members are constructed using a lumped plasticity

model with nonlinear rotational springs at each end joined by a linear beam-column

element, as seen in Figure 5.1.

The following sections describe how the basic element behavior is represented.

Once the system matrices are assembled, the resulting equation of equilibrium is

assembled as system of �rst-order di�erential equations. The solution process is

discussed in Section 5.3.8.

5.3.1 Beams

The sti�ness matrix for the linear beam element shown may be found in most basic

�nite element texts as is given in Equation (5.1).

Kb =EI

L3

266666666664

AL2

I0 0 �AL2

I0 0

0 12 6L 0 �12 6L

0 6L 4L2 0 �6L 2L2

�AL2

I0 0 AL2

I0 0

0 �12 �6L 0 12 �6L0 6L 2L2 0 �6L 4L2

377777777775

(5.1)

where Kb is beam element sti�ness, E is the modulus of elasticity, I is moment of

inertia about the bending-axis, A is the cross-sectional area, and L is the element

length. The beam element sti�ness matrix is based on the nodal displacements given

in Equation (5.2).

d = f u1 v1 �1 u2 v2 �2 gT (5.2)

56


5.3.2 Hysteresis Modeling

Various hysteretic models for the restoring force of an inelastic structure have been

developed in recent years (Baber and Noori 1985; Baber and Noori 1986). The

model chosen for the nonlinear rotational spring is the Bouc-Wen (1976) smooth-

varying hysteretic model. This model includes a number of parameters, allowing

a mathematically tractable state-space representation capable of expressing several

hysteretic properties.

The restoring force, (fR)i, for a single nonlinear element i may be decomposed

into two parts, (fE)i and (fH)i, representing the elastic and hysteretic components

respectively. The restoring force can then be written with the subscript i implied as:

fR = fE + fH = ke r(t) + kh z(t) = � kT (xa � xb) + (1� �) kT z(t) (5.3)

where � is ratio of the post-yielding to pre-yielding sti�ness and kT is the pre-yielding

sti�ness. The variable r(t) is the relative deformation, xa and xb are the absolute

displacements at nodes a and b respectively, and z(t) is the corresponding variable

introduced to describe the hysteretic component. The elastic component is used

to represent the strain-hardening in the element. The force-deformation curve is

described by:

_z = _rdz

dr= _r[A� (� + sgn( _rz))jzjn] (5.4)

where A, �, , n are shaping parameters (Wen 1976) and the term in square brackets,

dz=dr, describes the hysteretic curve. Since we require dz=dr to be unity at small

values of z, then A = 1 . The yield displacement Y = (� + )1=n; taking � and as

equal, Equation (5.4) can now be written as:

_z = _rh1� 0:5(1 + sgn( _rz))

�� zY

��ni (5.5)

The resulting hysteretic behavior described above is a stable force-deformation

curve. The use of constant strain-hardening with the stable hysteretic loop ignores

the presence of cyclic hardening and does not permit modeling of deterioration due to

local instabilities. These e�ects could be captured through modi�cation of the above

57


equations.

Now consider a structure idealized by an n degree-of-freedom system under a one-

dimensional earthquake ground motion. The equation of motion for the system can

be expressed as:

M �x(t) +C _x(t) +KE x(t) +KH z(t) = �M flg �ug(t) = Fg(t) (5.6)

in which x(t) is an vector containing the displacement of each degree of freedom

relative to the ground, and z(t) is a vector containing the corresponding hysteretic

information for each element. M is the mass matrix and C is the viscous damping

matrix. The ground motion, Fg, is found by mapping the horizontal ground accelera-

tion, �ug, to the horizontal degrees of freedom through the vector flg and multiplying

by M. As in the single element case, the elastic and hysteretic components of the

structural restoring force can be separated such that:

FR = FE + FH = KE x(t) +KH z(t) (5.7)

so that the restoring force is a function of both x(t) and z(t). The equation of motion

for the system can be written in a nonlinear state-space format, as follows:8>><>>:

_x(t)

�x(t)

z(t)

9>>=>>; =

2664

0 I 0

�M�1KE �M�1C �M�1KH

0 [dz=dx] 0

37758>><>>:

x(t)

_x(t)

z(t)

9>>=>>;+

2664

0

M�1

0

3775Fg(t)

(5.8)

where [dz=dx] is a non-square matrix function found by:

[dz=dx] = [dz=dr]Brx (5.9)

and [dz=dr] is a diagonal matrix function of _x(t) and z(t) with entries dzi=dri found

in Equation (5.5), and Brx is a non-square matrix mapping the displacements x(t) to

the relative deformations r(t). The system can now be solved using any of a number

58


of numerical algorithms, such as the Runge-Kutta method.

5.3.3 P-M Interaction

In order to account for the e�ect of axial force impact on the nominal bending strength

of the hysteretic elements, the bending strength for the point hinges is de�ned by a

bilinear P-M interaction diagram as shown in Figure 5.2.

P

Py

M

M y

Figure 5.2: Bilinear diagram for P-M Interaction

For each hinge where axial force e�ects are to be considered, the corresponding

axial member is assigned. The values needed to de�ne the bilinear interaction curve

are de�ned in terms of ratios to yield axial force and moment. During analysis,

the axial force present in the beam-column element is computed at each time step.

Utilizing the interaction equations for the two segments of the curve, the nominal

bending capacity is calculated and updated. This procedure neglects the e�ect of

bending moments on the axial sti�ness of the element.

5.3.4 Geometric Nonlinearities: P-Delta

The procedure utilized to account for the e�ects of P-� is derived directly from

the P-� e�ect on an individual cantilever column, as illustrated in Figure 5.3. The

cantilever column loaded with both a vertical load, P, and horizontal load, H. Due to

59


a horizontal displacement at the top of the column, �, the vertical load now generates

an additional bending moment at the bottom of the column.

P

H

Le

P

H

∆e

H + (P/Le) ∆e

Le

Figure 5.3: P-� Forces Associated with a Gravity Column

Alternatively, the problem can be reformulated so that the horizontal load in the

column is augmented to result in the same bending moment at the base of the column.

This additional horizontal load, or geometric shear, is given by P (�e=Le), where Le is

the height of the column. The equilibrium equation for the column in the horizontal

direction is now:

k�e = H+ P�e

Le

(5.10)

where k is the lateral sti�ness of the column. By rearranging the terms in Equa-

tion (5.10), the following expression results:

�k � P

L

�� = keff� = H (5.11)

were keff is the e�ective lateral sti�ness for the column. Thus, the introduction of

the P-� is equivalent to reducing the lateral sti�ness of the column. Though the

resulting variation of bending moment over the length of the column is not exactly

the same as in the original case, this method provides a reasonable approximation.

The above relationship between forces and the lateral translations at the top and

bottom of a column can be written in matrix form (White and Hajjar 1991) as:

(fxi

fxj

)=

P

Le

"1 �1�1 1

#(uxi

uxj

)(5.12)

60


where P is the axial force in the column (negative for compression), and Le is the col-

umn height. This matrix also corresponds to the geometric sti�ness of a truss element.

The full P-� contribution from a gravity column at any level can be represented in

a global matrix solution simply by assembling the sti�ness matrix of Equation (5.12)

into the global structure elastic sti�ness matrix,KE. If a rigid oor constraint is used,

these element sti�ness matrices can be grouped together onto one structure node per

oor.

The analysis program uses a geometric sti�ness matrix based on the axial load

in the columns under gravity loads only. As a result, since the geometric sti�ness

matrices are only calculated once, this method provides a �rst-order approximation

of a second-order elastic analysis.

5.3.5 Viscous Damper

The viscous dampers studied are assumed to be linear, and their constitutional force-

deformation relationship can be expressed as follows:

Fd(t) = Cd _ud(t) (5.13)

where Fd(t) and _ud(t) are the resistance force and deformation of the dampers re-

spectively. Cd is the damping constant of the VS damper. In the frequency domain,

assuming ud(t) = ud;max sin(!t), the above equation becomes:

Fd(t) = Cd ! ud;max cos(!t) = K00

d ud;max cos(!t) (5.14)

where K00

d is the loss sti�ness for the VS damper (Kasai, Munshi, Lai, and Maison

1993).

As mentioned in Section 4.4.2, the damper-brace component is collectively referred

to as the damper system. The mathematical model of the damper system is shown

in Figure 5.3.5, where KB is the sti�ness of the brace used to install the damper in

the frame structure. The brace element, if not assumed to be rigid, is modeled as a

linear truss element.

61


KB

KD

CD

Figure 5.4: Schematic Diagram for Viscoelastic Damper

The equations for the damper element, in matrix format, can be written as

(Fd;i

Fd;j

)= Cd

"1 �1�1 1

#(_ud;i

_ud;j

)+Kd

"1 �1�1 1

#(ud;i

ud;j

)(5.15)

where ud;i and ud;j are the displacements at the ith and jth end of the damper re-

spectively. The above equations are similar to those for linear truss elements, except

that it contains both velocity and displacement terms in the local coordinate system.

After transformation to the global coordinate system, the matrix corresponding to

the displacement proportional terms are assembled into the global elastic sti�ness

matrix, KE, while the matrix corresponding to the velocity proportional terms are

assembled into the global damping matrix, C.

5.3.6 Friction Pendulum Isolation (FPS) Element

The principles of operation of the FPS bearing are discussed in Section 3.2.2. The

bearing consists of a spherical sliding surface and an articulated slider that is lined

with a high capacity bearing material. The basic model implemented for the bearing

is similar to that of the nonlinear spring. The frictional force that is mobilized at

the sliding interface of sliding bearings depends on the normal load, bearing pressure,

direction and value of sliding velocity and composition of the sliding interface.

At an instance of time, the bearing carries a weight and is subjected to vertical

acceleration and additional seismic load due to overturning moments e�ects. The

force-displacement relationship that develops at the sliding interface are described

by:

Ff =W �

Ruf + �SW

�zf = ke;fuf + kh;f (5.16)

62


where:

W � = W (1 +�ufg+PSW

) (5.17)

is the normal load on the sliding interface, PS is the normal load on the bearing, W is

the seismic weight of the structure carried by the bearing, and �uf is the acceleration

across the bearing surface. The direct e�ects of variations in the normal force on the

behavior of the FPS bearing are to instantaneously change the sti�ness and friction

force of the bearing. However, approximating the normal load such that W � = W re-

sults in nearly the same global isolation system response and superstructure response.

This e�ect has been demonstrated by comparison of analytical results to shake table

results of a seven-story model in which the axial forces on individual bearings varied

from 0 to twice the gravity load (Al-Hussaini, Zayas, and Constantinou 1994).

Also, zf is a dimensionless variable which are governed by a system of di�erential

equations based on the Bouc-Wen model described above, such that

_zf = D�1y ( _uf � �ij _uf jjzf jn�1zf � _uf jzf jn) (5.18)

where Dy is the yield displacement; and � and govern the shape of the hysteresis

loop described in Section 5.3.2. When yielding commences, provided that

1

� + = 1 (5.19)

then zf is bounded by values �1 and account for the direction of sliding forces.

The coeÆcient of sliding friction, is described by

�S = fmax � (fmax � fmin) exp�aj _uj (5.20)

where �S is the sliding coeÆcient of friction, _uf is the velocity of sliding, fmax is

the coeÆcient of sliding friction at high velocity, and fmin is the coeÆcient of sliding

friction at low velocity, and a is a coeÆcient controlling the dependency of friction

on sliding velocity. In general, the parameters fmax, fmin, and a are determined ex-

perimentally depending on bearing pressure, surface roughness, and the composition

of PTFE.

63


As with the nonlinear spring element, the elastic and hysteretic components of

the structural restoring force can be separated such that:

Ff = FE;f + FH;f = KEf x(t) +KHf z(t) (5.21)

where KEf and KHf are the matrix formulation of the expressions given in Equa-

tion (5.16). The element also places the entry dzi=dri, as de�ned in Equation (5.18),

into [dz=dr], given in Equation (5.8).

5.3.7 Active Control

Speci�c elements are available to represent the actuators and sensors in the active

control system. Both require information regarding the noise bandwith, and a satu-

ration level for the actuator element. The controller algorithms integrated with the

above analytical model are linear dynamic output feedback controllers of the form:

_xc(t) = Acxc(t) +Bcy(t) (5.22)

w(t) = Ccxc(t) +Dcy(t) (5.23)

where xc(t) is the state vector of the dynamic controller, y(t) is the sensor reading

vector, w(t) is the controller command signal, and the constant matrices Ac, Bc, Cc

and Dc are the linear state description of the regulator.

The controller command signal w(t) is mapped to forces applied to the structure

by the equation:

Fu(t) = Buw(t) (5.24)

which is an additional force input to the state equations with the same mapping as

Fg in Equation (5.8). The sensor measurements are described by mapping the states

to the absolute accelerations at the sensor locations

y(t) = Cy(�M�1C_x(t) +�M�1KEx(t) +�M�1KHz(t) + Fu(t)) (5.25)

where Cy selects which accelerations are available. Equation (5.25) is combined with

64


the uncontrolled structure state space equation in Equation (5.8) to form the con-

trolled system with the augmented state vector xs(t) = fx _x z xcgT with the addi-

tional states of the controller added to the original state vector. The controller loop

from the sensors to the actuator forces is algebraically closed to form a state space

equation:

_xs(t) = A(xs(t)) xs(t) +BwFg (5.26)

which is used to perform the nonlinear time history analysis.

5.3.8 Solution Procedure

Based on the above discussions, the analysis platform is implemented in MATLAB

. Utilizing MATLAB as the platform for implementation o�ers the advantages of

simplicity of coding, a wide variety of inbuilt functions for matrix analysis, toolboxes

for active control linear analysis and design, and graphical functions for visualization

of the results. However, analyses performed using MATLAB are fairly ineÆcient

as the resulting code is not pre-compiled. Given the problem size, eÆciency and

memory management were special concerns. Care was taken to vectorize operations

as much as possible, as these operations are faster within MATLAB. In-built utilities

for sparse matrix operations and storage were used, and temporary variables cleared

from memory as soon as possible.

An input �le needs to be generated that de�nes the model, both its structural

and control elements. Sections that must be included in the input are: node def-

inition, boundary conditions, nodal constraints, point masses, material properties,

element de�nitions, static nodal loads, control elements, and the acceleration record

for seismic load. From this information the �nite element model is generated. The sys-

tem matrices are assembled, and the state-space model described in Equations (5.8)

and (5.26) is generated. The di�erent analyses available include a modal analysis

based on initial system properties, static linear analysis, and nonlinear time-history

analysis.

The time history analysis is performed using one of the built-in solver for or-

dinary di�erential equations. To use the MATLAB ordinary di�erential equation

65


{ x, z }, F

AnyElements with PM-Interaction Surfaces

Defined?

Calculate current axialforceUpdate moment capacity

PM_Interaction

AnyFPS Bearing Elements? Determine relative

velocityUpdate µs

Friction Coefficient

yes

no

Update [dz/dx]

Solve for {xd, zd}

{xd, zd}

Figure 5.5: Flowchart of StructODE function

(ODE) solvers, you must rewrite such equations as an equivalent system of �rst-order

di�erential equations as in Equation (5.26). Once the equation is represented in this

way, it can be coded as a function that a MATLAB ODE-solver can use. After test-

ing the di�erent solvers available, the sti�-system solver ode15s was chosen. The

ode15s is a variable-order, multistep solver based on the numerical di�erentiation

formulas. Optionally it uses the backward di�erentiation formulas (also known as

Gear's method) that are usually less eÆcient (Shampine and Reichelt 1998). The

analytical expression for the Jacobian matrix is provided for increased eÆciency. At

initial time-step, perform a linear static analysis under speci�ed static nodal loads.

The results are used to initialize the structure states for the dynamic analysis. At

every function evaluation, the process illustrated in Figure 5.5 are taken.

The output of the time-history analysis is a matrix containing the structure states

66


at every output time step speci�ed. Post-processing of these results include calcula-

tion of the following values for structural response evaluation: drift calculation, peak

and cumulative plastic deformation of nonlinear springs. Relevant response values

are also calculated for the control system being utilized. The quantities calculated

include: peak bearing displacement, peak damper force and displacement, and active

tendon peak force and energy requirements.

The results of the nonlinear analysis capabilities, without control, of the above

code were veri�ed by comparing it with DRAIN-2DX (Allahabadi 1987), a well-known

analysis software for planar nonlinear dynamic analysis of structures. Veri�cation was

done using a simple portal frame and the 3- and 9-Story structures used in this study.

A plot comparing the time-history response of the third story drift between the two

analyses is shown in Figure 5.6. The di�erences in the results shown are negligible.

0 5 10 15−2

−1

0

1

2

3

4

5

6

Time (seconds)

Drif

t (in

ches

)

Comparison of Drift Response of Third Story la15 Record: LA 3−Story Structure

DRAIN−2DX MATLAB Code: Bouc−Wen Model

Figure 5.6: Comparison of Third Story Drift Response under la15 GroundMotion with DRAIN-2DX

67

Chapter 6

Evaluation of Seismic Demands

6.1 Introduction

This chapter focuses on the quanti�cation of seismic demands at the structure level

for the 3- and 9-Story steel moment-resisting frames (SMRF) in Los Angeles. The

description for the structure design and basic control parameters is given in Chapter 4.

The behavior and response of the structures with and without control systems is

studied by subjecting nonlinear analytical models of the structures to sets of ground

motions representative of di�erent hazard levels.

The structural demand parameters used to evaluate and quantify the response

of the structures include roof and interstory drift angle, hysteretic energy demands

and peak oor accelerations. For each control system analyzed, the e�ect of varying

control parameters on the seismic demand of the structures is determined. From

these investigations, the performance of one design per control system is evaluated

based on characteristic values given in FEMA 273 (BSSC 1997), listed in Table 2.1.

A comparison between the di�erent systems is then presented.

As a representative values of the response, the primary statistic of interest is

a \best estimate" or central-estimate of the data referred to as the median1 value.

Also of interest are the maximum, 84th percentile, which is determined assuming

a logarithmic distribution to the data, and the standard deviation of the natural

log of the data, �ln, which is used as an indicator of the dispersion in the data.

1The median is more precisely de�ned as the geometric mean of the data that is found as theexponential of the average of the natural log of the observed values

68

Chapter 6 Evaluation of Seismic Demands

This information provides a good indication regarding the expected demands of the

structure as well as the scatter in the results. These statistical measures are discussed

in further detail in Appendix A.

A compromise between accuracy and eÆciency is usually necessary in the context

of the problem being addressed. The focus of this study is on developing a better

understanding of the seismic demands for a variety of control systems and structures

when subjected to a multitude of ground motions. The critical modeling require-

ment is to represent all major behavioral characteristics of the structure such that

the behavior represented is accurate with reasonable con�dence. Based on research

conducted on the uncontrolled frames (Gupta 1998), a bare-frame centerline model

that neglects the e�ects of P-� is utilized for all nonlinear dynamic analyses unless

otherwise indicated.

6.2 Seismic Demands for Uncontrolled System

In order to understand the impact of additional control systems, the seismic demands

for the uncontrolled system should be understood. Though extensive research has

been conducted on these structure (Gupta 1998), some of the basic results are dis-

cussed here, including the results of a nonlinear pushover analysis.

In a pushover analysis, static lateral loads are applied to the structure using

patterns that approximately represent the relative inertial forces in a structure. The

absolute magnitude of those forces is then increased until the desired deformation

levels are achieved, either at the structure or element level. A signi�cant contribution

of this procedure is the insight provided into the inelastic behavior of the structure. A

pushover analysis for the two structures was performed using the design load pattern

in the FEMA 222A guidelines, so as to agree with previous benchmarks for the

structures.

The resulting curves in Figure 6.1 show the relationship between the normalized

base shear, where the base shear is normalized by the structure seismic weight, versus

the roof drift angle. The curves for both the LA 3-Story and 9-Story structures

indicate that inelastic behavior occurs roughly at a global drift angle of about 1%.

The 3-Story structure indicates a higher elastic sti�ness, which corresponds to its

lower �rst mode period, and e�ective strength than the 9-Story structure.

69

Evaluation of Seismic Demands Chapter 6

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Roof Drift Angle

Nor

mal

ized

Bas

e S

hear

Roof Drift Angle vs. Normalized Base Shear Nonlinear Pushover Analysis: LA 3− and 9−Story

LA 3−StoryLA 9−Story

Figure 6.1: Global Pushover Curves for LA 3- and 9-Story Structures

A nonlinear dynamic analysis is necessary to more accurately capture the response

of the structure under seismic loads. The statistical values for the roof drift angle

demands for both structures are given in Table 6.1. Based on the results of the

pushover analyses, the median values of the roof drift angle indicate minor inelastic

behavior under the 50 in 50 set of ground motions. The roof drift angles under the 2

in 50 set of excitations are still below the collapse prevention limit of 5%. Signi�cant

dispersion is present in the data, however, as indicated by the high values of �ln.

While the dispersion remains fairly constant between the 50 in 50 set and 10 in 50

set, it is signi�cantly larger for the results of the simulations under the 2 in 50 set.

The dispersion is also higher in the results of the 3-Story frame as compared to those

for the 9-Story frame.

The drift demands over the height of the structures can be obtained from Fig-

ures 6.2 and 6.3. The values for story drift angle are indicated as points between

the bounding oors that are connected by straight lines. This plotting convention

is adopted throughout this report. Median values of story drift angle correlate well

with basic expected structural response (BSSC 1997): little inelasticity under fre-

quent events and collapse prevention under severe excitations. In the 50 in 50 set

70


Table 6.1: Statistics on Roof Drift Angle Demands

Structure Ground Motion Median 84th Perc. Maximum �ln

3-Story50 in 50 0.0112 0.0169 0.0300 0.4110 in 50 0.0159 0.0241 0.0371 0.412 in 50 0.0283 0.0519 0.0770 0.61

9-Story50 in 50 0.0067 0.0094 0.0110 0.3510 in 50 0.0114 0.0162 0.0197 0.352 in 50 0.0160 0.0261 0.0306 0.49

of ground motions, an average of 1% drift is observed for both the 3- and 9-Story

structures. Also, the distribution of drift demands over the height of the structures is

fairly constant in both cases. As the excitations become more severe, drift demands

are not evenly distributed.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

Median Values for Peak Story Drift Angles for 3 Earthquake Sets LA 3−Story Structure: α = 3%, no P−∆

Drift Angle

Flo

or L

evel

2 in 5010 in 50 50 in 50

Figure 6.2: Median Values for Peak Story Drift Angle for 3-Story Structure,All Sets of Ground Motions

For the 3-Story structure, the peak drift angle occurs in the third story for all

three sets of ground motions. The 10 in 50 set induce a peak drift angle of about

2.4%, which increases to about 4.7% under the 2 in 50 set. For the 9-Story structure,

the location of peak drift occurs in the eighth story for the 10 in 50 set; however, in

71


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

1

2

3

4

5

6

7

8

9

Median Values for Peak Story Drift Angles for 3 Earthquake Sets LA 9−Story Structure: α = 3%, no P−∆

Drift Angle

Flo

or L

evel

2 in 5010 in 50 50 in 50

Figure 6.3: Median Values for Peak Story Drift Angle for 9-Story Structure,All Sets of Ground Motions

the 2 in 50 set, the location of peak drifts has moved to the third story. As with the

roof drift angle, dispersion values in the data are quite high especially under the 2

in 50 set of excitations, as seen in Figures 6.4 and 6.5. In the 3-Story structure, the

increase in dispersion evenly increases in each story with increasing excitations. For

the 9-Story structure the dispersion is fairly similar under the 50 in 50 set and 10 in

50 set of excitations. The distribution under the 2 in 50 set, however, is signi�cantly

di�erent, with the highest values present around the second and third stories. The

standard deviation of the natural logarithm of the data, �ln(Æ), is used as an indicator

of the dispersion of the results. The dispersion of the peak drift values for each story

of the 3- and 9-story structures are shown in Figures 6.4 and 6.5. For both structures

the scatter in the data is signi�cant. The story dispersion for the 3-story structure

consistently increases with increasing hazard for all three stories. A di�erent behavior

is present in the 9-story structure, where the lines for di�erent hazard levels cross one

another. While the lower stories indicate a signi�cant increase in dispersion in the 2

in 50 Set, the dispersion in the upper three stories decreases below that of the 50 in

50 Set.

72


0 0.1 0.2 0.3 0.4 0.5 0.6

1

2

3

Dispersion of Peak Story Drift Angles LA 3−Story Structure: α = 3%, no P−∆

Std. Deviation of the Log. of Peak Drift Angle

Flo

or L

evel

2 in 50

10 in 50

50 in 50

Figure 6.4: Dispersion of Peak Story Drift Angle for 3-Story Structure, AllSets of Ground Motions

0 0.1 0.2 0.3 0.4 0.5 0.6

1

2

3

4

5

6

7

8

9

Dispersion of Peak Story Drift Angles LA 9−Story Structure: α = 3%, no P−∆

Flo

or L

evel


2 in 50

10 in 50

50 in 50

Figure 6.5: Dispersion of Peak Story Drift Angle for 9-Story Structure, AllSets of Ground Motions

73


6.3 E�ect of Controller Architecture Design

The design process of any control system is an iterative process, where trade-o�s are

made between improved structural response and the amount of control provided. In

order to identify the key parameters in the controller, the impact on seismic drift

demands due to variations in controller design is determined.

6.3.1 FPS Isolation System

In designing an isolation system, one must select: (1) isolation period, (2) sliding

surface, and (3) the number and location of isolators. The number of isolators will

determine the bearing pressure on a single isolator. As discussed in Section 3.2.2, the

bearing pressure and type of sliding surface together determine the frictional proper-

ties of the system. The isolation period determines the exibility to be introduced to

the system once sliding occurs. For all isolation systems considered, the bearings are

assumed to be evenly distributed so torsion is not signi�cant. The sensitivity to the

selection of an isolation period and frictional properties are investigated.

In this study, two di�erent frictional surfaces that are representative of the high

and low end of available frictional levels are evaluated and analyzed. The �rst slid-

ing surface, referred to as f1, has properties corresponding to un�lled te on with a

bearing pressure of 6.9 MPa (1.00 psi). The second sliding surface considered, f2, has

properties corresponding to un�lled te on with a bearing pressure of 44.9 MPa (6.51

psi). For both these systems, the sliding direction is parallel to the direction of the

lay of the te on, so that the only di�erence between the two systems is in the bearing

pressure. The resulting frictional properties for these bearings are given in Table 6.2.

Table 6.2: Frictional Properties for Isolator System

Surface Pressure (MPa) fmax fmin af1 6.9 11.93% 2.66% 0.6f2 44.9 5.72% 0.87% 0.5

Systems with periods of 1, 2, 3, and 4 seconds and both frictional surfaces were

evaluated using the 3-story structure to determine the impact of varying the isolation

74


period. As the �rst mode of this structure is 1.01 seconds, a system with an isolation

system of 1 second provides no additional exibility; as a result, no true isolation to

ground motion is provided. The only bene�t that may be derived from this system

lies in its energy dissipation ability. Also, as the behavior of the system can no longer

be approximated as that of a single degree of freedom system with a period of 1

second, the behavior of the isolation bearing and the superstructure are coupled.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Isolation Period (seconds)

Roo

f Drif

t Ang

le

Median Peak Roof Drift Angle for Varying Isolation Periods FPS System with Different Frictional Surfaces: 3−Story Frame, α = 3%, no P−∆

f1 − 2/50 f2 − 2/50 f1 − 10/50f2 − 10/50

Figure 6.6: Median Values for Peak Roof Drift Angle as Function of IsolationPeriod, 3-Story Structure, 10 in 50 and 2 in 50 Set of GroundMotions

The impact on roof drift angle for these systems for the 10 in 50 and 2 in 50 set of

ground motions is shown in Figure 6.6, where an isolation period of 0 seconds indicates

the original �xed base structure. While all isolation systems provide reductions in

the median values of peak roof drift for the 10 in 50 ground motions, the same does

not occur for the more severe ground motions in the 2 in 50 set where the systems

with isolation periods of 1 second can actually increase peak roof drift demands.

Furthermore, no one frictional surface consistently produces greater reductions in the

roof drift demands at all values of isolation period for both sets of ground motions. At

low values for the isolation period, the systems with lower frictional values have less

75


of an impact on roof drift demands, while the reverse is true at higher isolation levels.

As sliding surface f2 has a smaller coeÆcient of friction than f1, systems with those

surfaces reach their sliding force more quickly. This behavior is desirable at high

values of isolation period, where the system period is shifted to values of reduced

spectral accelerations.

Because provisions must be made to allow for the lateral motion of the structure at

the base, generally through a gap with the surrounding area, a key issue in isolation

system design is maintaining bearing displacements as low as possible while still

achieving the desired structural response. Systems with the sliding surface f2 isolator

generally result in signi�cantly higher bearing displacements then those associated

with the corresponding f1 system (see Table 6.3). As the systems are identical except

for the pressure on the bearings, correct estimation of the building weight is critical in

determining the system's performance. For the systems with isolation periods of 1 and

2 seconds, bearing displacements are greater than 20% of their radius of curvature. As

a result, the small angle approximations utilized in describing the bearing behavior

no longer hold, and signi�cant vertical motion occurs in the bearing, which is not

captured in the model analyzed. This vertical motion leads to a signi�cant increase

in axial forces in the columns (Llera and Almazan 1998). This e�ect reduces the

columns load carrying capacity and may result in higher drift demands than indicated

by these results. For the same isolation period, f2 systems result in higher median

bearing displacements, as their sliding force is lower.

Table 6.3: Peak Bearing Response for 3-Story Structure Isolation Bearing: 2in 50 Set of Ground Motions

IsolationPeriod

Median (inches) % Curvaturef1 f2 f1 f2

1 sec 4.78 5.45 49 562 sec 12.4 17.2 31 433 sec 15.7 24.7 18 284 sec 15.2 21.3 10 14

The scatter in the response is also greater for bearings with surface f2 than their

counterparts with sliding surface f1, as shown in Figure 6.7. The scatter also in-

creases with increasing isolation period. In a few cases, the bearing displacement

76


exceeds three feet, which would may not be feasible for implementation depending

on the structure's actual surroundings. At the high isolation periods typically used

for isolation systems, the improvement gained by using sliding surfaces with lower

frictional levels is minor compared to the increase in bearing displacement; therefore

systems with higher values of friction are more desirable desirable.

T1, f1 T1, f2 T2, f1 T2, f2 T3, f1 T3, f2 T4, f1 T4, f20

10

20

30

40

50

60

Drif

t Ang

le

Maximum Bearing Displacement for FPS System 2 in 50 Set of Earthquakes: 3−Story Structure, α = 3%, no P−∆

Figure 6.7: Peak Bearing Displacements for 3-Story Frame with FPS Isola-tion, 2 in 50 Set of Ground Motions

The distribution of story drift over the height of the structure is fairly constant

under the 10 in 50 ground motions as the superstructure remains essentially linear,

as seen in Figure 6.8. However, the story drift demands under the 2 in 50 Set for

the system with an isolation period of 2 seconds are beyond the desirable range, as

shown in Figure 6.9. Little additional reduction in drift demands is gained by using

an isolation period of 4 seconds. Hence, once an isolation system is designed such that

the superstructure is kept essentially linear and undergoes reasonable displacements,

only minor additional reductions in drift demands can be achieved by increasing the

isolation period.

Though increasing the isolation period increases the scatter in the bearing re-

sponse, the scatter in the drift demands of the superstructure are reduced. The 84th

77


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

Median Values for Peak Story Drift Angles 10 in 50 Set of Earthquakes: 3−Story, α = 3%, no P−delta

FPS − T2, f1FPS − T3, f1FPS − T4, f1Uncontrolled

Figure 6.8: Median Values for Peak Story Drift Demands for 3-Story Framewith FPS Isolation, 10 in 50 Set of Ground Motions

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3


Drift Angle

Flo

or L

evel



78


percentile of the peak story drift demands are plotted in Figure 6.10. The demands

for the 2 second isolation system are beyond 5% and comparable to those of the un-

controlled system. The demands at the �rst story are actually slightly greater than

those of the uncontrolled system. In contrast, the response of systems with isola-

tion periods of 3 and 4 seconds are greatly reduced from those of the uncontrolled

structure. As a result, of the systems analyzed, the FPS with an isolation period of

3 seconds and frictional surface f1 produces the desired structural response with the

least control.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

Drift Angle

Flo

or L

evel

84th Percentile Values for Peak Story Drift Angles 2 in 50 Set of Ground Motions: 3−Story Structure with FPS Isolation, α = 3%, no P−∆


Figure 6.10: 84th Percentile Values for Peak Story Drift Demands for 3-StoryFrame with FPS Isolation, 2 in 50 Set of Ground Motions

Based on the previous results, two FPS isolation systems were designed for the

9-Story structure. As surfaces with a higher frictional surface reduced the amount

of bearing displacement, both sliding surfaces are of type f1. This structure has a

fundamental period of about 2 seconds; therefore, to ensure enough separation with

the isolation period and minimize any coupling e�ects, the systems designed have

isolation periods of 4 and 5 seconds, labeled T4 and T5 respectively. Table 6.4 gives

the values of peak roof drift angle and peak bearing displacement under the 2 in 50

set of excitations. Median values for both parameters are fairly similar under both

79


systems. Increasing the isolation period has only minor impact on the demands.

Referring back to the pushover results, the roof drift angle demands indicate only

minor inelastic behavior.

Table 6.4: Global Demand Parameters for 9-Story Structure with FPS Iso-lation, 2 in 50 Set of Ground Motions

Median 84th Perc. MaximumPeak Roof Drift

AngleT4 0.0078 0.0117 0.0181T5 0.0075 0.0101 0.0134

Peak BearingDisplacement

T4 16.63 in. 27.09 in. 38.25 in.T5 17.43 in. 34.94 in. 48.64 in.

The story drift angle demands under the 2 in 50 set are shown in Figure 6.11,

and again the reduction in demands is only minor with increased isolation period.

The demands are fairly evenly distributed over the height of the structure, though

the system is slightly less e�ective at reducing the peak drift demands of the upper

stories. The main impact of isolation systems occurs in the demand contributions from

the fundamental mode of the structure (Soong and Constantinou 1994). The small

increase demands observed at the top stories are mostly due to higher mode e�ects,

as they have a greater impact on the upper story demands of tall structures (Gupta

1998). The story drift demands are between 1% and 1.5%, which fall well below the

life-safety limit of 2.5%. As the bene�t of a higher isolation period is minor and

results in higher bearing displacements, an isolation period of 4 seconds is chosen as

the �nal design for this structure.

6.3.2 Fluid Viscous-Brace Damper

Two important design parameters for a uid viscous damper (VS) are: (1) the percent

damping to be provided, and (2) the distribution of damping along the height of the

structure. The �rst parameter that needs to be determined is the amount of damping

required to meet the desired performance goals. An initial determination can be

made by determining the reduction of roof drift as a function of increasing levels of

damping.

In this study, the addition of damping to the structure is modeled in two ways.

The �rst model includes the viscous dampers explicitly, using the element described

80


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

4

5

6

7

8

9

Median Values for Peak Story Drift Angles 2 in 50 Set of Earthquakes: 9−Story, α = 3%, no P−∆

Drift Angle

Flo

or L

evel

FPS − T4, f1

FPS − T5, f1

Uncontrolled


in Section 5.3.5. The second model represents the additional damping as Rayleigh

damping. In the latter case, the modal damping values for the �rst two modes of

the system are designed to match the corresponding system with explicitly modeled

dampers. As Rayleigh damping implies damping that is proportional to mass and

sti�ness distribution, the distribution of damping over the structure is di�erent than

that of placing equal size dampers in each story.

The response of the systems was then simulated with the 10 in 50 set and the

2 in 50 set of ground motions. The variation of roof drift angle versus the percent

equivalent damping in the �rst mode are shown in Figure 6.12. The model using

Rayleigh damping does not satisfactorily capture the e�ects of the viscous dampers.

This model indicates a smooth reduction of roof drift with increased damping. When

the dampers are explicitly modeled, however, the trend is not as clear, with no sig-

ni�cant di�erence in roof drift between the 30% and 40% damping systems. Another

important factor is that the relationship between the two models is not consistent

and can provide non-conservative estimates of the peak roof drift demand. Under

the 10 in 50 set of ground motions, using Rayleigh damping underestimates the peak

81


0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03

Percent Critical Damping

Roo

f Drif

t Ang

le

Effect of Damping on Median Peak Roof Drift Angle 3−Story Frame, α = 3%, no P−∆

VS, D1 − 2/50

Rayleigh − 2/50

VS, D1 − 10/50

Rayleigh − 10/50

Figure 6.12: Median Values for Peak Roof Drift Angle for 3-Story Frame asFunction of Percent of Critical Damping, 10 in 50 and 2 in 50Set of Ground Motions

roof drift demands. However, under the 2 in 50 set of ground motions, the Rayleigh

damping model overestimates the roof drift demand.

All three VS systems produce very similar story drift demands under the 10 in

50 ground motions (Figure 6.13). However, under the 2 in 50 set of ground motions

(Figure 6.14), the di�erences between the systems become more apparent. In both

cases, the resulting decrease in story drift demands resulting in the increase in damp-

ing from 30% to 40% is slightly less than that of increasing the e�ective damping from

20% to 30%. This behavior re ects the e�ect observed on peak roof drift demands,

shown in Figure 6.12. Also, the VS systems demonstrate a di�erent distribution of

drift demands over the height of the structure in comparison with results from the 10

in 50 set of ground motions. This di�erence results from the signi�cant yielding in

the �rst oor of the structure at this higher demand level. Though the distribution

of drift demands for all three levels of e�ective damping are fairly equivalent, slight

di�erences are apparent with increasing damping. In the system with 20% e�ective

damping, the story with the greatest drift demand is the second story, while the third

82


story has the greatest peak drift demands for the other two VS damping systems.

Unlike the FPS isolation systems, varying the e�ective damping had little impact on

the dispersion of the analysis results. As seen in Figure 6.15, dispersion values for

all 3 stories remain fairly similar for both ground motion sets and the three damping

values. Since the scatter in all data sets for these systems are so similar, the di�er-

ences in peak drift demands at any hazard level or e�ective damping in the range

considered will be dominated by the e�ect of the dampers on the median values.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3


Drift Angle

Flo

or L

evel

VS − 20, D1 VS − 30, D1 VS − 40, D1 Uncontrolled

Figure 6.13: Median Values for Peak Story Drift Demands for 3-Story Framewith Viscous-Brace Dampers D1, 10 in 50 Set of Ground Mo-tions

For the same value of added damping, several di�erent distributions of damper

sizes are possible. Three di�erent damper distributions were designed for the 3-

story structure. The �rst damper distribution, labeled D1, has equal size dampers

located in each story. As seen in Figure 6.14, this system results in uneven story drift

demands in the presence of nonlinearities, with the drift being smaller in the upper

stories. Story sti�ness typically decreases as a function of the height of the structure.

As a result, the same damper force will result in greater drift reductions. To account

for this e�ect, a second damper distribution, referred to as D2, has damper sizes

83


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3


Drift Angle

Flo

or L

evel

VS − 20, D1 VS − 30, D1 VS − 40, D1 Uncontrolled

Figure 6.14: Median Values for Peak Story Drift Demands for 3-Story Framewith Viscous-Brace Dampers D1, 2 in 50 Set of Ground Motions

0 0.1 0.2 0.3 0.4 0.5

1

2

3

Dispersion of Peak Story Drift Angles − Variation of Effective Damping LA 3−Story Structure with Evenly Distributed VS Dampers: α = 3%, no P−∆

Flo

or L

evel


2/50 − V20

2/50 − V30

2/50 − V40

10/50 − V20

10/50 − V30

10/50 − V40

Figure 6.15: Dispersion of Peak Story Drift Angle for 3-Story Structure withVarying Added E�ective Damping Periods

84


distributed proportional to an estimate of story sti�ness.

In order to estimate relative story sti�nesses, an estimate of story yield drifts is

required. An estimate for the story drift can be readily obtained if the following

simplifying assumptions are made (Krawinkler 1978; Gupta 1998):

1. The in ection points are assumed to be at the mid-heights of the columns and

midspans of the beams for structures that have no apparent strength or sti�ness

irregularities.

2. Yielding can be assumed to occur in beam elements �rst. For structures de-

signed according to strong-column/weak-beam building code criterion, this as-

sumption is reasonable.

3. Lateral de ections due to axial deformation of the columns as well as second-

order e�ects can be neglected.

In the situations where the above conditions hold, the story yield drift is not expected

to be signi�cantly di�erent between adjacent stories. Consider a typical subassembly

as shown in Figure 6.16, where in ection points are assumed to be at the mid-heights

of the columns. Assuming that the beams are the �rst elements to yield at the con-

δ

h

l/2 l/2

Ic

Ib

Figure 6.16: Beam-Column Subassembly for an Interior Column

nection, then the shear force in the columns, Vcol, can be estimated by the following:

Vcol =�M

h

0@1� dc

l

1A

=2Mpb

h

0@1� dc

l

1A

(6.1)

85


where Mpb is the plastic moment capacity of the beams, dc is the depth of the column

cross-section, h is the height between the two in ection points in the columns, and

l is the length between in ection points in the beams framing into the cross-section.

Once the shear force in the columns is estimated, this information is combined with

element material and geometric properties to compute the associated drift in the

elements. The lateral drift contribution from the beams, Æb, and the columns, Æc, are

computed as follows:

Æb =h2

6

0@IEb

l

1AVcol (6.2)

Æc =h3

12EIcVcol (6.3)

The rigidity, R, of the subassembly can then be found by:

R =Vcol

Æb + Æc(6.4)

Once the rigidity for each story is computed, the relative values between stories can

be determined and used to weigh the distribution of damping over the height of

the structure. Also, since inelastic demands are not equal for all stories, damping

distribution can further be weighted by relative inelastic drift demands. Using the

median values under the 2 in 50 set, a third damping distribution, D3, was designed.

While damping distribution as a function of story sti�ness results in small di�er-

ences in roof drift demands (see Figure 6.17), signi�cant di�erences in the distribu-

tion of the story drift demands occur in the presence of nonlinear structural behavior.

Not much di�erence between damping distributions occurs under the medium ground

motions when the structure remains essentially linear. Under the 2 in 50 ground mo-

tions, however, the distribution of damping becomes much more signi�cant for story

drift demands. As expected, the VS system with distribution D2, which has larger

dampers on the �rst story, has much greater impact on drift demands at the bottom

of the structure. The VS system D3 results in drift demands that are roughly evenly

86


0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Percent Critical Damping

Roo

f Drif

t Ang

le

Effect of Damping Distribution on Median Peak Roof Drift Angle Variation over Damping: 3−Story Frame, α = 3%, no P−∆

D1 − 2/50

D2 − 2/50

D3 − 2/50

D1 − 10/50

D2 − 10/50

D3 − 10/50

Figure 6.17: Median Values for Peak Roof Drift Demands for 3-Story Framewith Viscous-Brace Dampers in Di�erent Distributions

distributed over the three stories. Both of these trends are also re ected in the peak

forces through the dampers, as seen in Table 6.5. These forces are on the high end

of those achievable through one damper per story.

A similar design process was adopted for the 9-story structure. The e�ect of three

di�erent damping levels, 10%, 20% and 30%, and two di�erent damping distributions,

D1 and D3 as described above, were investigated. The e�ect of variation on the

amount of damping provided is similar to that observed in the 3-Story structure, as

illustrated in Figure 6.19. Even a small amount of additional damping has a signi�cant

impact on the amount and distribution drift demands. The largest impacts are found

at the top stories, where story sti�ness and inertial mass are smaller. Redistribution

of damping, D3, results in a more even distribution of drift demands over the height

of the structure, as well as reducing the maximum peak demand over all stories, as

seen in Figure 6.20. Though not a perfectly even distribution, this approach can

provide a better starting point for an iterative design process where the goal is to

evenly distribute drift demands over the height of the structure.

87


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3


Drift Angle

Flo

or L

evel

VS − 30, D1VS − 30, D2VS − 30, D3Rayleigh 30

Figure 6.18: E�ect of Damping Distribution on Median Values for Peak StoryDrift Demands for 3-Story Frame, 2 in 50 Set of Ground Motions

Table 6.5: Median Response Properties for Viscous Dampers, 2 in 50 Set ofGround Motions

Percent ofDistribution

Peak Damper Force (kips)Critical Damping Story 1 Story 2 Story 3

20%D1 587 657 590D3 811 698 305

30%

D1 872 907 696D2 1,278 921 342D3 1,127 1,012 399

40%D1 1,103 1,085 764D3 1,425 1,151 466

88


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

4

5

6

7

8

9


Drift Angle

Flo

or L

evel

VS − 10, D1

VS − 20, D1

VS − 30, D1

Uncontrolled

Figure 6.19: Median Values for Peak Story Drift Demands for 9-Story Framewith Viscous-Brace Dampers D1, 2 in 50 Set of Ground Motions

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

4

5

6

7

8

9

Median Values for Peak Story Drift Angles for Various Damping Distributions 2 in 50 Set of Earthquakes: 9−Story, α = 3%, no P−∆

Drift Angle

Flo

or L

evel

VS − 30, D1VS − 30, D3

Figure 6.20: E�ect of Damping Distribution on Median Values for Peak StoryDrift Demands for 9-Story Frame, 2 in 50 Set of Ground Motions

89


6.3.3 Active Tendon System

Several di�erent decisions are made in the design of an active control system which

directly impact the system's ultimate performance. A detailed analysis of these con-

cerns can be found in Breneman (1999). In this investigation, only the e�ect of control

e�ort, represented by the actuator saturation level, for an active-tendon brace control

system (ATB) is investigated. For this structure, the active tendon-braces are located

in each story of the structure and designed as described in Section 4.4.3, with only a

single output variable used to represent the regulated output of story drifts. One of

the consequences of this design decision is that, since a linear combination of drifts

is used, drift values of opposite signs cancel one another during the design process.

This e�ect is discussed in detail in Breneman (1999). Two di�erent saturation levels

were speci�ed for the same control architecture and design process. The �rst system

saturates at a force of 325 kips and is referred to as the S325 system. This satura-

tion level of about 5% of the frame's seismic weight seems a realistic value for this

structure. However, in order to evaluate a system with control forces of comparable

magnitude as those of the viscous-brace system, the second system with a saturation

level of 1000 kips, labeled S1k, was also designed.

For the 10 in 50 set of ground motions, both systems reduce the story drifts, as

seen in Figure 6.21. As expected, the S1k system produces signi�cantly higher drift

reductions than the S325 system. A moderate amount of nonlinear behavior still

occurs with the S325 system, but the system is robust enough to deal with those

variations in system properties.

However, under the 2 in 50 ground motions, both ATB systems perform unsatis-

factorily, as seen in Figure 6.22. The resulting nonlinearity from these ground motions

leads to a system that is not adequately controlled. If the only concern is the reduc-

tion of the absolute maximum story drift value, then these two systems do provide a

small measure of improvement in the mean peak story drift value. The reduction in

drift of the upper story is made at the cost of increasing the drift at the lower stories.

As mentioned previously, the story sti�ness and seismic mass at the upper stories

is less than those values for the lower stories. So for the same amount of actuator

force a greater impact will be felt at the top stories. Since the controller weights

all story drifts equally, the �rst story drift is sacri�ced to achieve improvements in

90


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3


Drift Angle

Flo

or L

evel

ATB − S0 − S325k ATB − S0 − S1k Uncontrolled

Figure 6.21: Median Values for Peak Story Drift Demands for 3-Story Framewith ATB Control, 10 in 50 Set

the other stories. The poor performance of these two ATB control systems becomes

more apparent when looking at the 84th percentile values, shown in Figure 6.23. For

these controllers, increasing the the actuator capacity results in a system with higher

seismic demands than the uncontrolled system.

Due to the problems with the above control systems, a second design for the ATB

system was developed with an actuator saturation level of 1000 kips. In the design of

this system, multiple output variables were used to represent the regulated output of

story drifts. The drifts were combined, with equal weights, such that they would not

negate one another. The original design is labeled S0, while the new design is labeled

S1. The impact on drift demands for the S1 controller is signi�cantly improved over

those from the S0 controller, as seen in Figure 6.24 for the 2 in 50 set of ground

motions. Story drift demands are virtually equal in all three stories at just under 3%.

Even in the presence of nonlinear behavior in the structural members, this active

control system reduces the peak drift demands.

For the 9-Story structure, placing tendon braces on each story was deemed unreal-

istic and impractical, so a system utilizing only three active braces was designed. The

91


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3


Drift Angle

Flo

or L

evel

ATB − S0 − S325k ATB − S0 − S1k Uncontrolled

Figure 6.22: Median Values for Peak Story Drift Demands for 3-Story Frame- ATB Control with Varying Saturation, 2 in 50 Set

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

Drift Angle

Flo

or L

evel

84th Percentile Values for Peak Story Drift Angles 2 in 50 Set of Earthquakes: 3−Story, α = 3%, no P−∆

ATB − S0 − S325

ATB − S0 − S1k

Uncontrolled

Figure 6.23: 84th Percentile Values for Peak Story Drift Demands for 3-StoryFrame - ATB Control with Varying Saturation, 2 in 50 Set

92


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

Median Values for Peak Story Drift Angles 2 in 50 Set of Earthquakes: 3−Story with ATB, α = 3%, no P−∆

Drift Angle

Flo

or L

evel

ATB − S0 − A1k

ATB − S1 − S1k

Uncontrolled

Figure 6.24: Median Values for Peak Story Drift Demands for 3-Story Framewith ATB Control, Variation in Design, 2 in 50 Set

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1

2

3

Dispersion of Peak Story Drift Angles − Different Control Design LA 3−Story Structure with ATB System, Saturation 1k: α = 3%, no P−∆

Flo

or L

evel


2/50 − S0

2/50 − S1

10/50 − S0

10/50 − S1

Figure 6.25: Dispersion of Peak Story Drift Angle for 3-Story Structure withATB Systems of Di�erent Controlled Outputs

93


tendons were placed based on the location of peak drifts for the uncontrolled struc-

ture and highest force requirement of the purely passive viscous system. As a result,

braces are only located on the �rst, second, and eighth stories for this structure. The

resulting story drift demands are seen in Figure 6.26. Due to the limited number of

actuators, the impact on drift demands is small, with the controller performing worse

on the stories lacking an actuator. In the 2 in 50 set, the demands in the stories

in the middle of the structure, stories four and �ve, are actually greater than in the

uncontrolled case.

0 0.01 0.02 0.03 0.04 0.05 0.060

1

2

3

4

5

6

7

8

9

Median Values for Peak Story Drift Angles 9−Story Structure with ATB System: α = 3%, no P−∆

Drift Angle

Flo

or L

evel

ATB − 2/50 Set ATB − 10/50 Set Uncontrolled − 2/50 SetUncontrolled − 10/50 Set

Figure 6.26: Median Values for Peak Story Drift Demands for 9-Story Framewith ATB Control, 10 in 50 and 2 in 50 Set of Ground Motions

6.4 Comparison of Seismic Demands Across Con-

trol Systems

Several possible control approaches were discussed in the previous section. In order to

select a control strategy, the relative performance of these systems is compared. The

comparison is based on story drift, hysteretic energy, and oor acceleration demands.

94


As discussed earlier, story drift angles and hysteretic energy demands provide an

indication of structural damage in steel moment resisting frames, while oor acceler-

ations are used as an indication of the content and acceleration-sensitive nonstructural

damage.

6.4.1 Deformation Demands

A wide consensus exists in the earthquake engineering community that for moment-

resisting frames the interstory drift demand, expressed in terms of the interstory drift

angle, is the best indicator of expected damage. As a global parameter, interstory drift

is much more appropriate than the roof drift angle because in individual stories it may

exceed the latter by a factor of two or more (Krawinkler and Gupta 1998). The use

of story substructures permits also the estimation of element force and deformation

demands from the story drift angle.

As a representative value, the median (exponent of the average value of the log of

the data) is selected. The control systems selected for comparison with the 3-Story

structure are: 1) VS with 30% damping and distribution D1, 2) FPS isolation with

isolation period of 3 seconds and frictional surface f1, and 3) ATB design S1 with

saturation level of 1000 kips. The values suggested for peak and residual story drift

angles in FEMA 273, which are listed in Table 2.1, can be used as one set of guidelines

by which performance of a SMRF structure may be judged.

The maximum story drifts over the height of the structure (maximum drift in

any story for a given ground motion) for each system are shown in Figures 6.27

through 6.29. The �gures show the 20 data points corresponding to the 20 records

in each set; the median value for the responses are shown by the solid dash mark2.

Results from the 50 in 50 set of ground motions, shown in Figure 6.27 indicate that

for a short return period the control systems can reduce the seismic demands by

nearly 50%. For this structure, the median response moves to below the immediate-

occupancy level for all 3 types of control strategies. Only a few records drive the

structure above the immediate-occupancy level, and only one beyond the life-safety

2Note that the median values indicated are the median values of the maximum peak drifts overthe height of the structure over the 20 individual ground motions. These results are not the sameas the maximum of the median individual story peak drift values, which are plotted in other �gures

95


Uncontrolled VS − 30, D1 FPS − T3, f1 ATB − S1k0

0.01

0.02

0.03

0.04

0.05

Drif

t Ang

le

Maximum Story Drift Angles 50 in 50 Set of Earthquakes: 3−Story, α = 3%, no P−delta

Figure 6.27: Maximum Values for Peak Story Drift Demands for 3-StoryFrame, 50 in 50 Set of Ground Motions


0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Drif

t Ang

le


Figure 6.28: Maximum Peak Story Drift Demands for 3-Story Frame, 10 in50 Set of Ground Motions

96


limit in the uncontrolled system. Though the lowest median value for peak story drift

occurs with the viscous system, the scatter in the response is lowest in the presence of

the isolation system. In the 10 in 50 set, while the median uncontrolled response lies

just below the life-safety limit, several records push the uncontrolled response between

the life-safety and collapse-prevention limits, as seen in Figure 6.28. However, the ap-

plication of control reduces both the median structural response as well as its scatter.

The median response of the passive systems, while above the immediate-occupancy

limit, falls just at the linear limit of this structure, with the median response of the

ATB system just above those values. All records have responses below the life-safety

limit after the application of control.


0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

Drif

t Ang

le


Figure 6.29: Maximum Peak Story Drift Demands for 3-Story Frame, 2 in50 Set of Ground Motions

The results from the 2 in 50 set of ground motions, shown in Figure 6.29, indicate

a median response for the uncontrolled systems just under the collapse prevention

limit and great scattering in the response. The original system response contains

clear outliers with story drift demands exceeding 10% drift, which are unsustainable

by the physical frame system of the SMRF and are not present in any of the controlled

systems. The ATB control system performs better under these excitations, both in

97



0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Drif

t Ang

le

Maximum Residual Story Drift Angles 50 in 50 Set of Earthquakes: 3−Story, α = 3%, no P−∆

Figure 6.30: Maximum Residual Story Drift Demands for 3-Story Frame, 50in 50 Set of Ground Motions

terms of the median value and the scatter of the data, with only one record resulting

in a structural response about the collapse-prevention limit. Both passive systems

reduce both the median and the scatter of the response, though the scatter is still

pronounced. However, the increase in scatter for the FPS system is greater than

that for viscous system, possibly due to the near-�eld type ground motions which

are present in that set. The only system with all data points below the collapse

prevention limit is the viscous system.

The residual drift is the permanent drift of the story, with respect to its original

position, at the end of the excitation. Residual drift is often used as an indicator

of inelastic damage to the structure. While this parameter can be useful in certain

situations, it can also be misleading. For example, a small displacement pulse in

one direction, which takes the structure into the inelastic range, will result in a

larger residual drift value than two much larger and equal magnitude pulses that

occur in opposite directions, which result in zero drift. So care must be taken in the

interpretation of these results.

The values of the maximum residual story drift over all the stories are presented in

98



0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Drif

t Ang

le




0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Drif

t Ang

le



99


Figures 6.30 through 6.32, with the median value for each system indicated by a solid

horizontal line. Results from the 50 in 50 set of ground motion have median values

that are fairly close to zero for all systems, though the median values for the controlled

systems are lower than the uncontrolled system. Only one case corresponding to the

uncontrolled system is beyond the life-safety limit. The addition of control reduces the

scatter, especially for the isolation system. Under the 10 in 50 set of ground motions,

the scatter in the results increases signi�cantly for all systems. Outlier values are

apparent for all systems. Three cases in the uncontrolled system and one with the

ATB control are beyond the life-safety limit. Median values in the 2 in 50 set of ground

motions are below the life-safety limit for the uncontrolled system and the two passive

systems. However, the median value for the ATB controlled system is greater than

that of the uncontrolled structure. This example illustrates the incomplete nature

of the information given by the residual drift. In several ground motions, the use

of the ATB control is e�ective at reducing the repeated excursions into the inelastic

range relative to the uncontrolled case. As a results, in these cases the addition of

control appears to increase damage, when in reality the control system is reducing

the inelastic damage to the structural members. A comparison of the dissipated

hysteretic energy, which is discussed in Section 6.4.2, con�rms this fact.

Peak story drift demand over the height of the structure is not suÆcient to fully

understand the demands on the structural system. One must also evaluate the dis-

tribution of drift demands over the height of the structure. In the 50 in 50 set of

ground motions, shown in Figure 6.33, the median values of drift demands remain

fairly constant over the height of the structure. The largest drift ratio occurs at the

top story except for the viscous system, where the largest drift ratio occurs in the

second story. These same trends can be observed for the median values of the 10 in

50 set of ground motions, shown in Figure 6.35. However, the relative placement of

the demand curves for each system changes. In both cases, all three control strate-

gies reduce the story drift demands over the entire height of the structure. In the

50 in 50 set of ground motions, the viscous control system results in the lowest drift

demands. Under the stronger ground motions of the 10 in 50 set, the FPS isolation

system becomes more e�ective at reducing story drift demands. The isolation system

only starts to become e�ective once the base shear reaches the sliding force of the

bearing. As a result, this system is not as e�ective at low level ground motions. The

100


ATB system has comparable drift demands with the passive system. The di�erences

between systems becomes more pronounced at the 84th percentile values, though the

same trends occur.

Under the 2 in 50 set of ground motions, signi�cant nonlinearities occur in the

controlled systems. Furthermore, the di�erences between control systems becomes

more pronounced, as seem in Figure 6.37. While the FPS isolation system results

in the lowest median drift demands, the e�ect of higher scatter can be seen when

comparing the median and the 84th percentile values. In comparison, the results for

the viscous system display less scatter. The ATB system has drift demands which are

higher than both passive systems, though signi�cantly reduced from the uncontrolled

case. Again the 84th percentile values indicate drift demands which increase more

than those for the viscous system. However, for all controlled systems these values

are well below the collapse prevention limit. Since the placement of the actuators

and their saturation level were designed to provide a comparable system between the

ATB and VS controller, the only di�erence lies in the control strategy and not the

physical limitations of the systems.

Care must be taken not to assume that control strategies which reduce statistical

values for the ground motion sets will reduce demands for all individual excitations. In

a motion-by-motion comparison, occasionally both active and passive control system

do make performance worse. For this structure and sets of ground motions, the FPS

isolation system always reduced roof and story drift angles. The same is not true for

the VS and ATB system, as seen in Table 6.6. In the VS controlled system, three

ground motions within the 2 in 50 set led to slightly higher drifts in the �rst story.

Similarly, two ground motions in the 2 in 50 set increase peak drift angles for both

�rst and second stories with the ATB control system.

Results for the 9-story frame story drift demands are given in Figures 6.41 and 6.44.

When comparing the e�ect of the ATB system, consideration must be given to the

fact that this system only has three actuators located on the �rst, second, and eighth

stories. The viscous system, in contrast, has dampers located on each story. In the

10 in 50 set of excitations, all control systems produce reductions in the story drift

demands. The passive systems have the most signi�cant reduction in drifts, with

comparable performance. The FPS isolation system proves to be more e�ective in

the lower structures, while the VS system is more e�ective in the upper stories. The

101


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3


Drift Angle

Flo

or L

evel

Uncontrolled

Viscous − 30, D1

FPS − T3, f1

ATB − 1000k

Figure 6.33: Median Peak Story Drift Demands for 3-Story Frame, 50 in 50Set of Earthquakes

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

Drift Angle

Flo

or L

evel

84th Percentile Values for Peak Story Drift Angles 50 in 50 Set of Earthquakes: 3−Story, α = 3%, no P−delta

Uncontrolled

Viscous − 30, D1

FPS − T3, f1

ATB − 1000k

Figure 6.34: 84th Percentile Values for Peak Story Drift Demands for 3-StoryFrame, 50 in 50 Set of Earthquakes

102


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3


Drift Angle

Flo

or L

evel

Uncontrolled

Viscous − 30, D1

FPS − T3, f1

ATB − S1 − S1k


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

Drift Angle

Flo

or L

evel


Uncontrolled

Viscous − 30, D1

FPS − T3, f1

ATB − S1 − S1k


103


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3


Drift Angle

Flo

or L

evel

Uncontrolled

Viscous − 30, D1

FPS − T3, f1

ATB − 1000k


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

Drift Angle

Flo

or L

evel


Uncontrolled

Viscous − 30, D1

FPS − T3, f1

ATB − 1000k


104


Table 6.6: Increases in Story Drift Demands for 3-Story Structure due toAdditional Control, 2 in 50 Set of Ground Motions

Ground Motion Story Percent Increase

VS 30

la41 1 9.1la42 1 12.0la52 1 6.6

ATB

la41 1 9.5la41 2 16.8la52 1 9.8la52 2 2.5

impact of the active system is minor, with some stories showing almost no improve-

ment. In the 2 in 50 set, the two passive systems again show signi�cant reductions

in drift demands, with all story drifts below the life-safety limit of 2.5%. Again, the

FPS isolation has the lowest drift demands for the lower stories, while the VS system

has the lowest values for the upper stories. On average, however, the FPS isolation

performs better, with its worst story drift demand below that for the VS system. In

contrast, the ATB is only e�ective in reducing the demands near the location of its

actuators. The drift demands in the fourth and �fth stories actually increases when

compared to the original uncontrolled structure.

In order to allow for a better comparison between the e�ects of the ATB and VS

systems, two additional VS systems were designed for this structure. The �rst system,

VS 30 - 3 brace, utilizes the same size dampers as in the viscous system with 30%

damping evenly distributed (VS 30, D1). This design results in a controlled system

with approximately 10% damping in the �rst mode of the structure. The second

damping system uses that 10% damping evenly over the height of the structure, VS

10, D1. The resulting drift demands are shown in Figure 6.45. When only the 3

braces are utilized, the VS system also increases the drift demands of the middle

stories. The additional dampers cause those stories to be more diÆcult to deform.

As a result, the seismic demands are redistributed to locations which are easier to

deform. However, the VS system still performs better than the ATB system when

comparing the maximum value of story drift over the all stories. In comparison, using

the damping evenly throughout the structure results in reduction of drift demands

for all stories.

105


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

1

2

3

4

5

6

7

8

9


Drift Angle

Flo

or L

evel

Uncontrolled

Viscous − 30, D1

FPS − T4, f1

ATB − S1k


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

4

5

6

7

8

9

Drift Angle

Flo

or L

evel


Uncontrolled

Viscous − 30, D1

FPS − T3, f1

ATB − S1k


106


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

1

2

3

4

5

6

7

8

9


Drift Angle

Flo

or L

evel

Uncontrolled

Viscous − 30, D1

FPS − T4, f1

ATB − S1k


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

4

5

6

7

8

9

Drift Angle

Flo

or L

evel


Uncontrolled

Viscous − 30, D1

FPS − T3, f1

ATB − S1k


107


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

1

2

3

4

5

6

7

8

9


Drift Angle

Flo

or L

evel

Uncontrolled

Viscous − 30, D1

FPS − T4, f1

ATB − S1k


0 0.01 0.02 0.03 0.04 0.05 0.060

1

2

3

4

5

6

7

8

9

Drift Angle

Flo

or L

evel


Uncontrolled

Viscous − 30, D1

FPS − T3, f1

ATB − S1k


108


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

1

2

3

4

5

6

7

8

9


Drift Angle

Flo

or L

evel

Uncontrolled VS 30: 3−brace VS 10 − D1: all storiesATB − S1k

Figure 6.45: Comparison of Maximum Peak Story Drift Demands for VS andATB Control for 9-Story Structure

6.4.2 Hysteretic Energy Demands

While peak interstory drift provides a good indication of performance, the resulting

information is incomplete as it does not take into account the cumulative damage to

the structure. Experimental investigations have demonstrated that structural dam-

age is a function of both peak as well as cumulative values. As normalized hysteretic

energy (NHE) provides a good indication of cumulative damage in steel structures,

median values of NHE are compared for control systems for each set of ground mo-

tions. Normalization of the hysteretic energy values of each element is performed by

normalizing by that elements elastic energy capacity, Ee, given by:

Ee = Mp �p =M2

p l

6EI(6.5)

where Mp is the plastic moment capacity, l is the member length, E is the modulus

of elasticity, and I is the moment of inertia.

The plots give values of the cumulative NHE dissipated by the frame members

109


at each oor of the structure. Results for the 3-Story structure are given in Fig-

ures 6.46 through 6.48. The results from the 50 in 50 set of ground motions, shown

in Figure 6.46 indicate that only the original uncontrolled system undergoes notice-

able nonlinearities. Since the dissipated energy values for the control systems are

essentially zero, their curves in Figure 6.46 lie along the y-axis. As expected from in-

creasing story drift demands, as the severity of ground motions increases, the amount

of hysteretic energy dissipated by the structure members increases. Under the 2 in 50

set of ground motions, all systems undergo some amount of nonlinearity, as indicated

by Figure 6.48. As expected from the peak drift demands, both the FPS and viscous-

brace damper systems signi�cantly decrease the NHE. Results from the active-tendon

system (ATB) also show signi�cant reductions in NHE.

0 5 10 15 20 25

0

1

2

3

Median Normalized Hysteretic Energy 50 in 50 Set of Earthquakes: 3−Story, α = 3%, no P−delta

NHE

Flo

or L

evel

Uncontrolled

VS − 30, D1

FPT − T3, f1

ATB − S1k

Figure 6.46: Median Values of Normalized Hysteretic Energy for 3-StoryFrame, 50 in 50 Set of Ground Motions

Similar results are observed in the 9-Story structure, as seen in Figures 6.49

and 6.50. In the 10 in 50 set of excitations, both passive systems reduce cumulative

energy dissipation to nearly zero, as they are successful in maintaining the structure

within the limits of elastic behavior. The ATB system also reduces the energy dissi-

pation throughout the height of the structure, and its impact is more noticeable than

110


0 5 10 15 20 25

0

1

2

3


NHE

Flo

or L

evel

Uncontrolled

VS − 30, D1

FPT − T3, f1

ATB − S1k


0 5 10 15 20 25

0

1

2

3


NHE

Flo

or L

evel

Uncontrolled

VS − 30, D1

FPT − T3, f1

ATB − S1k


111


in the reduction of peak drift demands. In the 2 in 50 set, only the FPS isolation

system is successful at keeping the system response essentially linear, as indicated

by very low values of NHE. The VS system reduces NHE dissipation by nearly half

in the lower stories and signi�cantly more in the upper stories, where the system is

most e�ective. Again, due to the limited capacity of the ATB system with only 3

actuators, the bene�ts provided are limited, though they occur evenly over the height

of the structure.

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

6

7

8

9

Median Normalized Hysteretic Energy 2 in 50 Set of Earthquakes: 9−Story, α = 3%, no P−∆

NHE

Flo

or L

evel

Uncontrolled

VS − 30, D1

FPT − T3, f1

ATB − S1k


6.4.3 Acceleration Demands

Acceleration demands are of concern for the nonstructural components of the building.

Added seismic control systems have the bene�t of being capable of reducing the

acceleration demands on the structure while reducing drift demands, which more

traditional methods such as increasing the building sti�ness cannot. The value of

ground level, oor level zero, acceleration indicated in the plots is the peak ground

acceleration (PGA) of the ground motion, except in the case of the FPS isolation

112


0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

6

7

8

9

Median Normalized Hysteretic Energy 2 in 50 Set of Earthquakes: 9−Story, α = 3%, no P−∆

NHE

Flo

or L

evel

Uncontrolled

VS − 30, D1

FPT − T3, f1

ATB − S1k


system. In that system, the acceleration reported is that observed immediately above

the isolation bearing, which will di�er from the PGA due to deformation in the

bearing.

Under the 50 in 50 set of ground motions shown in Figure 6.51, all three structural

control systems reduce oor accelerations above the ground level. For the FPS isola-

tion system, the acceleration at the ground oor level is slightly higher than the peak

ground acceleration. The accelerations at the higher oors is virtually identical to

that observed at the base. This behavior is consistent with the assumption that the

super-structure remains fairly rigid in comparison with the bearing. The VS brace

reduces the oor accelerations of the upper oors below that of the ground level. The

ATB system is not as e�ective at reducing the oor accelerations, though they are

still signi�cantly less than for the uncontrolled structure. As the controlled output of

interest in the design was story drift values only, this active system is not designed

to explicitly control oor accelerations.

Similar trends are observed under the 10 in 50 set of ground motions, as seen

in Figure 6.52. All three control systems reduce oor accelerations; however, the

113


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0

1

2

3

Median Values for Peak Floor Acceleration 50 in 50 Set of Earthquakes: 3−Story, α = 3%, no P−delta

Acceleration (g)

Flo

or L

evel

Uncontrolled

VS − 30, D1

FPT − T3, f1

ATB − S1k

Figure 6.51: Median Values of Floor Accelerations for 3-Story Frame, 50 in50 Set of Ground Motions

e�ect of the ATB system on the 3rd oor acceleration is to increase it beyond the

uncontrolled system. The accelerations for both passive systems are fairly constant

at about the value of the peak ground acceleration. Under the 2 in 50 set of ground

motions, as shown in Figure 6.53, the ATB system increases oor accelerations at all

three oors, while the passive systems still perform well. To achieve the reductions

in drift desired, the active system \sacri�ces" oor accelerations, which is not one of

its controlled outputs.

Impacts on the oor accelerations of the 9-Story structure follow a similar pattern

as that for the 3-Story. In the 10 in 50 set, shown in Figure 6.54, both the VS

and FPS controlled systems reduce oor accelerations, with the greatest reductions

provided by the viscous system. The acceleration at ground level for the isolation

system is just slightly higher than that of the ground motion. The ATB system has

mixed impact on oor accelerations, reducing accelerations in the upper oors while

increasing them from the �rst through third oors. A similar pattern is observed

in the 2 in 50 set, seen in Figure 6.55. While both passive system show signi�cant

reductions in oor accelerations, the impact of the ATB system is minor, though all

114


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0

1

2

3


Acceleration (g)

Flo

or L

evel

Uncontrolled

VS − 30, D1

FPT − T3, f1

ATB − S1k


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0

1

2

3


Acceleration (g)

Flo

or L

evel

Uncontrolled

VS − 30, D1

FPT − T3, f1

ATB − S1k


115


but the seventh and ninth oors display reduced peak accelerations.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

1

2

3

4

5

6

7

8

9

Median Values for Peak Floor Acceleration 10 in 50 Set of Earthquakes: 9−Story, α = 3%, no P−∆

Acceleration (g)

Flo

or L

evel

UncontrolledVS − 30, D1 FPT − T4, f1ATB − S1k


6.5 Conclusions

For isolation systems, selection of isolation period has the greatest impact in the re-

sulting seismic demands on the superstructure. Lowering the friction coeÆcient can


demands is an increase in bearing displacements. Large values of bearing displace-

ments should be avoided if at all possible as they must be accommodated through

a \moat" around the base of the structure. This system of control proves to be

very e�ective system for both the 3-Story and 9-Story structures and all three sets

of ground motions. The superstructure remains close to elastic behavior even under

severe ground motions, represented by the 2 in 50 set.




116


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

1

2

3

4

5

6

7

8

9

Median Values for Peak Floor Acceleration 2 in 50 Set of Earthquakes: 9−Story, α = 3%, no P−∆

Acceleration (g)

Flo

or L

evel

UncontrolledVS − 30, D1 FPT − T4, f1ATB − S1k



the drift demands are more evenly distributed among the di�erent stories. For the

same amount of e�ective damping, if the dampers are located in only a few stories the

system can by highly ine�ective. In the 9-Story structure, a system with dampers in

only three stories and 10% e�ective damping was designed. The resulting peak story

drift demands were higher than those of the uncontrolled structure in those stories

lacking dampers. In contrast, if smaller dampers are added at every story to produce

the same 10% e�ective damping, signi�cant reductions in drift result for all stories.

The capacity of the actuators for the ATB system contribute greatly to the ef-

fectiveness of the control system. Higher actuator capacities provide the controller

a greater opportunity to reduce drift demands. However, this same increased capac-

ity can result in systems which increase the demands from those of the uncontrolled

system.

Comparisons of the performance of three di�erent control strategies are presented

for all three sets of ground motions. The comparison is based on story drift, hys-

teretic energy, and oor acceleration demands. All three controllers were successful

117


in reducing the seismic demands, though no one system is consistently better than

the others at all three hazard levels.

For story drift demands, the values suggested for peak story drift angles in FEMA

273 are used as guidelines for judging structural performance. For short return peri-

ods, the addition of structural control reduced the median story drift by nearly 50%.

The most signi�cant impact of control occurs with the 2 in 50 ground motions. The

uncontrolled structure has a median response just under the collapse prevention limit.

Clear outliers are present in the data, with some response values exceeding 10% drift,

a response that is unsustainable by the physical system. The ATB and VS control

systems bring the median response to about 3%, with the VS system resulting in less

scatter in the data. The FPS isolation has the lowest median response, with values

of about 1.5%. Results for the 9-story structure follow a similar trend as those of the

3-story structure. Both the VS and FPS isolation are extremely e�ective at reducing

the peak story drift demands. The VS system proves to be most e�ective at higher

stories, while the FPS system is slightly less e�ective at the upper stories. For the

ATB control system, the actuators are placed on only the 1st, 2nd, and 8th stories.

The e�ect of this placement is that at high level excitations the drift demands at

stories without actuators are increased from the uncontrolled case. For comparison,

a VS system with dampers in only the same three stories and comparable peak forces

was designed. The resulting peak story drift demands were higher than those of

the uncontrolled structure in those stories lacking dampers. In contrast, if smaller

dampers are added at every story to produce the same amount of e�ective damping,

signi�cant reductions in drift result for all stories.

Residual drift values are used by FEMA 273 as an indicator of the inelastic damage

to the structural system. However, care must be taken in interpreting these values as

they can be misleading. In both the 50 in 50 set and 10 in 50 set of ground motions,

the median values for residual drifts of the control systems are close to negligible.

However, in the 2 in 50 set of ground motions, median values for all systems are

signi�cantly higher. The addition of the two passive controls decreases the median

drift demand value. However, the median value for the ATB system is greater than

that for the uncontrolled case, resulting in the appearance that the addition of this

control system appears to increase structural damage. However, closer analysis of the

response indicates that the ATB control keeps the structure from repeated inelastic

118


deformations, as indicated by dissipated hysteretic energy results.

Normalized hysteretic energy (NHE) is used to provide information regarding the

cumulative damage to the structure. Normalization of the hysteretic energy values of

each element is performed by normalizing by the element's elastic energy capacity. In

the 50 in 50 set and 10 in 50 set, the addition of structural control reduces the amount

of energy that must be dissipated by the structural system to negligible values. The

only exception occurs with the active control for the 9-story structure under the 10

in 50 set, as its capacity is signi�cantly less than the two passive control strategies.

The ATB control consistently reduces the NHE demands at every story. In the 2 in

50 set, some dissipation of hysteretic energy occurs for all systems. The FPS system

proves to result in the least amount of energy demands for the structure, which agrees

with the results from the drift analyses.

Floor accelerations are of concern for damage to several nonstructural compo-

nents. Structural control systems have the bene�t of being capable of reducing both

acceleration and drift demands on the structure, which traditional methods such as

increasing building sti�ness cannot accomplish. The two passive control systems in-

vestigated are particularly e�ective at reducing oor accelerations as compared with

the uncontrolled structure. In the 3-story structure, the peak oor accelerations at

all oors were about that of the peak ground acceleration. In the 9-story structure,

accelerations at the middle oor levels were reduced below that of the ground. The

active control system had no signi�cant reduction in oor accelerations, as the system

was not speci�cally designed to control those responses. In the 2 in 50 set for the

3-story structure, oor accelerations were increased from the uncontrolled case with

the addition of the ATB controller.

119

Chapter 7

E�ects of Modeling on Seismic

Demands

7.1 Introduction

The seismic demands presented in Chapter 6 were based on several assumptions con-

cerning structural parameters and modeling. The response of any structure depends

on careful selection of those parameters so as to capture the signi�cant e�ects of

the structure. The focus of this chapter is to evaluate the e�ect of: 1) the level of

nonlinear modeling and analyses of the structure, 2) the initial sti�ness of the struc-

ture, and 3) the strain-hardening assumptions in force-deformation relationships of

the elements.

These investigations are performed using the 2 in 50 set of ground motions, for

which the e�ects of parameter variation is expected to be the greatest. The control

systems, as described in Section 6.4 are used in these studies. These systems are:

1. VS 30, D1: evenly distributed VS dampers with e�ective damping of 30%

2. FPS T3, f1: FPS isolation system with an isolation period of 3 seconds and a

sliding surface f1

3. FPS T4, f1: FPS isolation system with an isolation period of 4 seconds and a

sliding surface f1

4. ATB - S1k: ATB active system with actuator saturation of 1,000 kips

120

Chapter 7 E�ects of Modeling on Seismic Demands

The results are primarily analyzed considering global structural demands, specif-

ically the peak values of roof and story drift demands. Sensitivity investigations of

the e�ect of the initial period and strain-hardening ratio are performed only on the 3-

Story structure. The range of values chosen for those parameters endeavor to contain

the expected range in variation. Investigations into the e�ect of nonlinear analyses

also include investigations with the 9-Story structure, where the increased height of

the structure might impact secondary nonlinear e�ects, such as P-�.

7.2 E�ect of Nonlinearities on Controlled Struc-

tural Performance

One of the basic requirements for performance-based evaluation is con�dence in the

analyses results. A compromise between accuracy and eÆciency is usually necessary,

requiring a determination of the level of modeling and type of analysis required for this

purpose. An evaluation of the e�ect of nonlinear behavior on the demands resulting

from time history analyses is required. For this purpose, four di�erent evaluation

models containing di�erent levels of increasing nonlinearity were created for the 3-

story structure. They are:

� L - a centerline linear model with no second-order e�ect, such as P-�, being

considered.

� NL1 - �rst nonlinear model including only material yielding

� NL2 - builds upon NL1 and includes the e�ect of axial loads on the moment

capacity

� NL3 - same as NL2 with the e�ects of P-� introduced

Each control system as well as the original structure were then analyzed using the

above models using the 2 in 50 set of ground motions. A comparison of the rela-

tive demand reductions from each control strategy to the original structure is made

between the di�erent models. For a speci�c control system, the di�erences in drift

demands resulting from each evaluation model are compared.

121

E�ects of Modeling on Seismic Demands Chapter 7

7.2.1 3-Story Structure

The median values for peak story drifts of the uncontrolled structure are shown in

Figure 7.1. The results of model NL1 are essentially identical to those of model NL2,

indicating that the axial loads on the columns are not high enough to signi�cantly

a�ect the moment capacity of the columns. As a result, those values are not repeated

on the plot. A di�erence in the distribution of drift of the height can be seen in story

drift demands between the linear and nonlinear models NL2 and NL3. Though the

median value for drift demands are all below the level where any negative sti�ness

to lateral loads is observed, as indicated by the global pushover analysis shown in

Figure 6.1, P-� e�ects impact the demands of some of the extreme ground motions,

resulting in a increase in the median peak story drifts. The scatter in the peak

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

Median Values for Peak Story Drift Angles for Uncontrolled System 2 in 50 Set of Earthquakes: 3−Story, α = 3%, no P−∆

Flo

or L

evel

Drift Angle

L Model

NL2 Model

NL3 Model

Figure 7.1: E�ect of Modeling on Median Values for Peak Story Drift De-mands, 2 in 50 Set of Ground Motions

roof drift demands of each system for these models can be observed in Figures 7.2

through 7.4, with the median indicated by a solid horizontal line. The presence of

outliers occurs in all models, especially for the uncontrolled structure. The linear

model, L, displays a fairly even scatter of the response about the median values. The

122


FPS isolation system displays the least scatter, with only 3 records having higher

demands than the tight clustering about the median. The demands from the NL2

evaluation model have slightly less scatter for most records. However, the severe

outlier values are still present. For the isolation system, the response of the outlier

points is signi�cantly higher. The consideration of P-� e�ects with the NL3 model

only has a signi�cant impact on the uncontrolled response, with more outlier points

than were present in the NL2 model


0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Drif

t Ang

le

Maximum Roof Drift Angles for L Model 2 in 50 Set of Earthquakes: 3−Story Structure

Figure 7.2: Maximum Roof Drift Demands for 3-Story Frame L Model, 2 in50 Set of Ground Motions

The amount of peak roof drift reduction for each control system from the original

structure, based on each type of nonlinear model, is shown on Table 7.1. For this

structure, little di�erence exists between estimated roof drift reductions among the

di�erent nonlinear models. However, linear models can severely overestimate the

e�ectiveness of the systems when comparing controlled performance with uncontrolled

response. Note that the impact of P-� is apparent in the percent reduction factors

in the 2 in 50 set, especially for the VS and ATB control systems, whose demands

are only minutely impacted by P-� e�ects.

Di�erences are also apparent in the distribution of peak story drift demands over

123



0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Drif

t Ang

le

Maximum Roof Drift Angles for NL2 Model 2 in 50 Set of Earthquakes: 3−Story, α = 3%, no P−∆

Figure 7.3: Maximum Roof Drift Demands for 3-Story Frame NL2 Model, 2in 50 Set of Ground Motions


0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Drif

t Ang

le

Maximum Roof Drift Angles for NL3 Model 2 in 50 Set of Earthquakes: 3−Story, α = 3%, P−∆

Figure 7.4: Maximum Roof Drift Demands for 3-Story Frame NL3 Model, 2in 50 Set of Ground Motions

124


Table 7.1: E�ect of Modeling on Percent Roof Drift Reduction

10 in 50 2 in 50Linear NL1 NL2 NL3 Linear NL1 NL2 NL3

VS 30, D1 61.8 46.2 46.0 44.8 47.8 47.8 47.3 40.8FPS T3, f1 69.5 50.1 49.8 47.4 72.5 65.9 65.7 62.2ATB S1k 67.4 46.7 46.5 39.7 49.3 43.6 43.7 30.4

the height of the structure within individual modeling cases, and seen in Figures 7.5

and 7.6 for the L and NL3 evaluation models. The distribution of peak drift demands

over the height of the structure for the various models is very di�erent, especially for

the VS and ATB systems. The di�erences in drift demands between the systems, and

therefore their relative e�ectiveness, increases with the use of the NL3 model.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

Median Values for Peak Story Drift Angles 2 in 50 Set of Earthquakes: 3−Story, Linear Model

Drift Angle

Flo

or L

evel

Uncontrolled

Viscous − 30, D1

FPS − T3, f1

ATB − 1000k

Figure 7.5: Median Values for Peak Story Drift Demands for L EvaluationModels - 3-Story, 2 in 50 Set

Though a linear model overestimates the percent of drift reduction for the FPS, it

does provide good estimates of the actual story drift demands based on the nonlinear

models (see Figure 7.7). This result is reasonable as the FPS allows only minor

nonlinearities to occur in the superstructure.

125


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

Median Values for Peak Story Drift Angles 2 in 50 Set of Earthquakes: 3−Story, α = 3%, P−∆

Drift Angle

Flo

or L

evel

Uncontrolled

Viscous − 30, D1

FPS − T3, f1

ATB − 1000k

Figure 7.6: Median Values for Peak Story Drift Demands for NL3 EvaluationModels - 3-Story, 2 in 50 Set

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

Median Values for Peak Story Drift Angles for T3, f1 System 2 in 50 Set of Earthquakes: 3−Story, α = 3%, no P−∆

Flo

or L

evel

Drift Angle

L Model

NL2 Model

NL3 Model

Figure 7.7: E�ect of Modeling on Median Values for Peak Story Drift De-mands for FPS T3 - f1, 2 in 50 Set

126


The same is not the case for the viscous system. The story drift demands are well

within the range where nonlinearities occur. As a result, the linear model provides

very poor estimates of the story drifts resulting from the nonlinear models, especially

in the �rst story, where most of the nonlinearity occurs. However, the di�erence

between the two nonlinear models is minor. A small di�erence can be observed

between peak drift demands in the �rst story, where P-� e�ects are greatest.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

Median Values for Peak Story Drift Angles for VS 30, D1 System 2 in 50 Set of Earthquakes: 3−Story, α = 3%, no P−∆

Flo

or L

evel

Drift Angle

L Model

NL2 Model

NL3 Model

Figure 7.8: E�ect of Modeling on Median Values for Peak Story Drift De-mands for VS - 30, D1, 2 in 50 Set

For the ATB system, a linear model provides inaccurate estimates of peak story

drift demands, as seen in Figure 7.9. The distribution of the drift demands is also

di�erent between the linear and nonlinear systems, as nonlinear demands are virtually

identical for all 3 stories. The seismic demands for both nonlinear systems are also

virtually identical. The active system is robust enough to compensate for the e�ect

of P-�. As full actuator e�ort is used in both cases, the resulting behavior is the

same.

127


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

Median Values for Peak Story Drift Angles for T3, f1 System 2 in 50 Set of Earthquakes: 3−Story, α = 3%, no P−∆

Flo

or L

evel

Drift Angle

L Model

NL2 Model

NL3 Model

Figure 7.9: E�ect of Modeling on Median Values for Peak Story Drift De-mands for ATB - S1k, 2 in 50 Set

7.2.2 9-Story Structure

Similar trends are observed with the 9-Story structure as discussed in the previous

section. A linear model analysis results in signi�cantly higher story drift demands to

those of the nonlinear analyses as seen in Figure 7.10. Also, the distribution of drift

demands over the stories for the linear model does not re ect the increased story drift

demands in the lower stories. Again, little is gained through the consideration of P-�

e�ects even with this taller structure. The drift demands are all below the level where

any negative sti�ness to lateral loads is observed, as indicated by the stable plateau

in the global pushover analysis shown in Figure 6.1

As the FPS system maintains the superstructure close to its elastic limit, a linear

dynamic analysis does provide good estimates of the actual drift demands as com-

pared to those from the nonlinear models, as shown in Figure 7.11. Only very minor

di�erences can be observed by including the e�ects of P-� in the analyses.

Though the story drift demands for the structure with VS dampers, principally

for the lower 6 stories, are well within the range where nonlinearities occur, The linear

model provides reasonable estimates of the story drifts resulting from the nonlinear

128


0 0.01 0.02 0.03 0.04 0.05 0.060

1

2

3

4

5

6

7

8

9

Median Values for Peak Story Drift Angles for Uncontrolled System 2 in 50 Set of Earthquakes: 9−Story, α = 3%, no P−∆

Drift Angle

Flo

or L

evel

NL2

Linear

NL3

Figure 7.10: E�ect of Modeling on Median Values for Peak Story Drift De-mands for 9-Story Structure, 2 in 50 Set

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

4

5

6

7

8

9

Median Values for Peak Story Drift Angles for System FPS T4 − f1 2 in 50 Set of Earthquakes: 9−Story, α = 3%, no P−∆

Drift Angle

Flo

or L

evel

NL2

Linear

NL3

Figure 7.11: E�ect of Modeling on Median Values for Peak Story Drift De-mands for 9-Story Structure with FPS T4 - f1, 2 in 50 Set

129


models, with only small di�erence when including P-� e�ects, as shown in Figure 7.12.

For the structure with the ATB system, a linear analysis provides a very poor estimate

of the nonlinear analysis results. This statement is true with respect to both the

absolute value of the story drift demands and to the distribution of the drifts over

the height of the structure. As the story drift demands indicated are signi�cantly

nonlinear, linear analyses of the system cannot adequately capture the structures

response to these excitations.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

4

5

6

7

8

9

Median Values for Peak Story Drift Angles for System VS 30 − D1 2 in 50 Set of Earthquakes: 9−Story, α = 3%, no P−∆

Drift Angle

Flo

or L

evel

NL2

Linear

NL3

Figure 7.12: E�ect of Modeling on Median Values for Peak Story Drift De-mands for 9-Story Structure with VS - 30, D1, 2 in 50 Set

When determining the impact of additional control, utilizing linear models can

produce misleading results. The linear model analysis results in higher drift demands

for the uncontrolled structure as compared to those from nonlinear analyses. However,

for both passive systems a linear analysis results in comparable drift demands as the

nonlinear analyses. As a result, if looking at the percent drift reduction, those system

would misleadingly indicate greater improvements. The case with the ATB controller

is even more serious. Though according to nonlinear analyses, the ATB controller

provides minimal improvements in performance (see Figure 7.15), performing only a

linear analysis indicates noticeable reduction in story drift demands for all stories.

130


0 0.01 0.02 0.03 0.04 0.05 0.060

1

2

3

4

5

6

7

8

9

Median Values for Peak Story Drift Angles for System ATB 2 in 50 Set of Earthquakes: 9−Story, α = 3%, no P−∆

Drift Angle

Flo

or L

evel

NL2

Linear

NL3

Figure 7.13: E�ect of Modeling on Median Values for Peak Story Drift De-mands for 9-Story Structure with ATB, 2 in 50 Set

0 0.01 0.02 0.03 0.04 0.05 0.060

1

2

3

4

5

6

7

8

9

Median Values for Peak Story Drift Angles 2 in 50 Set of Earthquakes: 9−Story, Linear Model

Drift Angle

Flo

or L

evel

Uncontrolled

Viscous − 30, D1

FPS − T3, f1

ATB − S1k

Figure 7.14: Median Values for Peak Story Drift Demands for Linear Evalu-ation Models of LA 9-Story Structure, 2 in 50 Set

131


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

4

5

6

7

8

9

Median Values for Peak Story Drift Angles 2 in 50 Set of Earthquakes: 9−Story, α = 3%, P−∆

Drift Angle

Flo

or L

evel

Uncontrolled

Viscous − 30, D1

FPS − T4, f1

Figure 7.15: Median Values for Peak Story Drift Demands for Linear Evalu-ation Models of LA 9-Story Structure, 2 in 50 Set

7.3 E�ect of Initial Sti�ness on Dynamic Response

The response of the structure is a�ected by the presence of cladding, partition walls,

and other nonstructural elements. These elements are expected to increase the elastic

sti�ness of the structure, though the exact amount is unknown. On the other hand,

moment-resisting connections may have some additional exibility associated with

them that is not originally considered. Furthermore, uncertainty in material prop-

erties may result in a more exible structure. To assess the range of e�ect changes

in sti�ness may have on the seismic response, the elastic sti�ness of all columns and

beams is reduced by constant amounts. The change is equivalent to reducing the �rst

mode period by factors of 0.5, 0.75, 1.25, and 2.00.

The median story drift demands for the uncontrolled 3-story structure under the 2

in 50 set of ground motions are shown in Figure 7.16. The changes in sti�ness a�ects

the drift demands in the upper stories more than in the �rst story. In general, the

impact of initial period variation can be estimated from the spectrum plots.

The median story drift demands for the 3-story structure with a FPS system

designed with an isolation period of 3 seconds under the 2 in 50 set of ground motions

132


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.5

1

1.5

2

2.5

3

Drift Angle

Flo

or L

evel

Effect of Fundamental Period Variation 2 in 50 Set of Earthquakes: LA 3−Story Structure

T = T0

T = T0/0.5

T = T0/0.75

T = T0/1.25

T = T0/2.00

Figure 7.16: Median Values for Peak Story Drift Demands for UncontrolledSystem, Varying Fundamental Period, 2 in 50 Set

are shown in Figure 7.17. As the relationship between the structural and isolation

period can be critical as that ratio approaches 1, the initial sti�ness was reduced by

factors of 0.33 and 0.25 in addition to those previously mentioned. The changes in

sti�ness a�ects the drift demands in the upper stories more than in the �rst story for

sti�er systems. At a reduction factor of 0.33 the fundamental period of the structure

equals the isolation period and large drift demands occur in the third story. Up to that

point the variation in drift demands is fairly \constant" and still reduces demands

from the uncontrolled case.

The median story drift demands for the 3-story structure with a viscous-brace

damping system designed with 30% critical damping under the 2 in 50 set of ground

motions are shown in Figure 7.18. As in the uncontrolled case, the changes in sti�ness

a�ects the drift demands in the upper stories more than in the �rst story. Much less

variation in the drift demand curves occurs than has been observed in the two previous

systems. The small variations can be estimated from spectrum plots developed for

30% damping.

The results of variations in the initial period for ATB control system are shown

133


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

1

2

3

Drift Angle

Flo

or L

evel

Effect of Fundamental Period Variation on System with FPS 2 in 50 Set of Earthquakes: LA 3−Story Structure

T = To

T = To/0.25

T = To/0.33

T = To/0.5

T = To/0.625

T = To/0.75

T = To/1.25

T = To/2.00

Figure 7.17: Median Values for Peak Story Drift Demands for FPS IsolationSystem T3, Varying Fundamental Period, 2 in 50 Set

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

1

2

3

Drift Angle

Flo

or L

evel

Effect of Fundamental Period Variation on System with Viscous Dampers 2 in 50 Set of Earthquakes: LA 3−Story Structure

T = To

T = To/0.5

T = To/0.625

T = To/0.75

T = To/1.25

T = To/2.00

Figure 7.18: Median Values for Peak Story Drift Demands for VS 30 System,Varying Fundamental Period, 2 in 50 Set

134


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

Drift Angle

Flo

or L

evel

Effect of Fundamental Period Variation on System with ATB Control 2 in 50 Set of Earthquakes: LA 3−Story Structure

T = To

T = To/0.5

T = To/0.625

T = To/0.75

T = To/1.25

T = To/2.00

Figure 7.19: Median Values for Peak Story Drift Demands for ATB - S1k,Varying Fundamental Period, 2 in 50 Set

in Figure 7.19. This system proves to be very sensitive to variations in the initial

period of the structure. The demands on the third story are particularly sensitive

to the variations. Though the design of the system is robust to small variations in

system parameters, signi�cant changes in the parameters can greatly a�ect system

performance.

Comparisons between the four systems for the cases of half and twice the period

of the original system are shown in Figures 7.20 and 7.21, respectively. Even in those

extreme cases, all three control systems result in reductions of seismic demands from

the uncontrolled case. However, the relative performance of the controlled systems

is di�erent. The demands for the viscous system remains roughly the same. The

isolation system has the lowest demands with the sti�er system. The ATB also

performs better with the sti�er system, and has the highest demands of the controlled

systems for the longer period case.

135


0 0.01 0.02 0.03 0.04 0.050

1

2

3

Drift Angle

Flo

or L

evel

Comparison of Median Values of Peak Story Drift Demands: T = To/2 2 in 50 Set of Earthquakes: 3−Story Structure

Uncontrolled

Viscous Dampers

FPS Isolation

ATB

Figure 7.20: Median Values for Peak Story Drift Demands for Half the Orig-inal Fundamental Period, 2 in 50 Set

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.120

1

2

3

Drift Angle

Flo

or L

evel

Comparison of Median Values of Peak Story Drift Demands: T = To/0.5 2 in 50 Set of Earthquakes: 3−Story Structure

Uncontrolled

Viscous Dampers

FPS Isolation

ATB

Figure 7.21: Median Values for Peak Story Drift Demands for Twice theOriginal Fundamental Period, 2 in 50 Set

136


7.4 E�ect of Variations in Strain-Hardening Ratio

The modeling of the post-yield sti�ness of frame element involve two major assump-

tions. The �rst assumption concerns the use of a stable force-deformation relationship,

which may not account for strength deterioration or sti�ness degradation. The second

assumption is in the determination of the value assigned to the post-yield sti�ness

of the element, �, which in this study is given as a fraction of the elastic sti�ness

of the frame member. This section investigates the e�ect of the second assumption

on the resulting drift demands of the structure. The 3-Story structure is analyzed

using the basic nonlinear model utilized in Chapter 6, NL2, under the 2 in 50 set of

ground motions. The values of the strain-hardening ratio of all frame members are

kept the same and taken to be either 0%, 3%, 5%, or 10%. These values are expected

to contain the expected range of possible strain-hardening for steel frame members.

The impact of varying the strain-hardening ratio is minor for the original un-

controlled structure, as seen in Figure 7.22. The trend in increasing the post-yield

sti�ness follows expectations, with increasing sti�ness resulting in a decrease in the

peak story drift demands.

The impact on strain-hardening variation is very minor with the FPS isolation

system, as seen in Figure 7.23. As the structure remains very close to the elastic limit,

the impact of changing post-yield behavior is small. However, in the VS system which

undergoes signi�cantly greater inelasticity, changing strain-hardening ratios has an

even smaller impact, shown in Figure 7.24. The ATB system proves to be the most

sensitive of the three control systems to variations in the strain-hardening ratio, as

shown in Figure 7.25. The impact is still small, however, and noticeably less than the

impact on the uncontrolled structure.

7.5 Conclusions

The use of linear evaluation models generally result in inaccurate estimation of the

seismic demands for the di�erent systems analyzed, both in terms of relative and

absoluted demand values. The relative impact of the addition of the di�erent control

systems depends on how far the sesimic demands drive the structure into the nonlinear

range. The use of a linear evaluation model for the FPS isolation system represents

137


0 0.01 0.02 0.03 0.04 0.05

1

2

3

Drift Angle

Flo

or L

evel

Effect of Variation in Percent Strain Hardening 2 in 50 Set of Earthquakes: LA 3−Story Structure

α = 0% α = 3% α = 5% α = 10%

Figure 7.22: Median Values for Peak Story Drift Demands for UncontrolledSystem, Variation Strain-Hardening Ratio, 2 in 50 Set

0 0.01 0.02 0.03 0.04 0.05

1

2

3

Drift Angle

Flo

or L

evel

Effect of Variation in Percent Strain Hardening on System with FPS 2 in 50 Set of Earthquakes: LA 3−Story Structure

α = 0% α = 3% α = 5% α = 10%

Figure 7.23: Median Values for Peak Story Drift Demands for FPS T3 Sys-tem T3, Variation Strain-Hardening Ratio, 2 in 50 Set

138


0 0.01 0.02 0.03 0.04 0.05

1

2

3

Drift Angle

Flo

or L

evel

Effect of Variation in Percent Strain Hardening on System with ATB 2 in 50 Set of Earthquakes: LA 3−Story Structure

α = 0% α = 3% α = 5% α = 10%

Figure 7.24: Median Values for Peak Story Drift Demands for VS 30 System,Variation Strain-Hardening Ratio, 2 in 50 Set

0 0.01 0.02 0.03 0.04 0.05

1

2

3

Drift Angle

Flo

or L

evel

Effect of Variation in Percent Strain Hardening on System with ATB 2 in 50 Set of Earthquakes: LA 3−Story Structure

α = 0% α = 3% α = 5% α = 10%

Figure 7.25: Median Values for Peak Story Drift Demands for ATB System,Variation Strain-Hardening Ratio, 2 in 50 Set

139


the only case where a reasonable estimate of the seismic demands is achieved. This

behavior occurs as the isolation system maintains the behavior of the superstructure

close to its elastic range. However, even in the isolated system, the linear evaluation

models underestimate the amount of scatter in the results. Furthermore, though the

superstructure can be modeled elastically, the behavior of the bearing is inherently

nonlinear, so a nonlinear time history analysis of the system needs to be performed.

Nonlinear evaluation models are required for the other systems in order to achieve

reasonable demand estimates. The distribution of drifts over the height of the struc-

ture are particularly a�ected by the choice of linear vs. nonlinear analyses. In the

ATB controlled system, the seismic demands resulting from a linear analysis indicate

that peak drift demands are reduced in all stories of the structure. Drift demands

resulting from a nonlinear analysis, however, indicate that peak drift demands in sto-

ries without actuators are actually increased from the uncontrolled response. In the

structures considered, the consideration of P-� e�ects in the calculation of demands

has little e�ect on the results, especially for the control systems. As these systems

reduce the demands on the structure, the chance of a system entering the negative-

sti�ness response range is reduced. This assumption should be checked with systems

where P-� e�ects are more pronounced.

A sensitivity analysis of these control strategies to the initial period and strain-

hardening ratio was also presented. As these systems reduce drift demands, the

excursion into the inelastic range is reduced. As a result, sensitivity to the strain-

hardening ration is minor. The systems are more sensitive to variations in the initial

period of the structure. The isolation and ATB systems are particularly sensitive

to increases in period. In contrast, the viscous damper system is fairly insensitive

to variations in initial period. In all cases, however, the addition of control reduced

seismic demands from the uncontrolled structure.

140

Chapter 8

Probabilistic Seismic Control

Analysis

8.1 Introduction

Current performance-based codes provide guidelines for calculation of seismic demand

for a scenario event; the resulting seismic demand is then compared with the allowable

limit given, such as those provided in Section 2.5. This procedure generally accounts

for the uncertainty present in ground motion excitation; however, the uncertainty

and variability in the demand estimation generally is not considered. One approach

to addressing this issue is the development of probabilistic seismic demand curves,

so that the probability that any damage measure exceeds a pre-determined allowable

limit can be determined.

Procedures for developing probabilistic seismic demand curves can be found in

literature (Cornell 1996; Wen 1995). These methodologies generally relate a record-

speci�c quantity, typically the linear elastic spectral acceleration at a characteristic

building frequency, to a structural response parameter, such as interstory drift, so as

to estimate the probability of exceeding the selected structural response parameter.

Di�erences between methods arise, at least in part, due to di�ering objectives by the

researchers. A comparison between the methods proposed by Cornell and Wen and

the current guidelines in FEMA 273 was performed by Shome, Cornell, Bazzurro, and

Carballo (1997).

In this study, the procedure proposed by Cornell, and explicitly described in Luco

141

Probabilistic Seismic Control Analysis Chapter 8

and Cornell (1998) with respect to its application in the SAC project, is used to

assess the annual exceedance probabilities for interstory drift demands. This chapter

summarizes this procedure and utilizes it to generate interstory drift hazard curves

for the di�erent control strategies previously discussed in Chapter 6.

8.2 Background

The �nal objective of the methodology devised by Cornell is the estimation of the

annual probability of exceedance of a given level of inelastic response in a speci�c

MDOF structure. The structure is located at a speci�c site with an associated poten-

tial seismic hazard. The response of interest can be any structural response param-

eter, representing either local or global structural demands. The procedure couples

conventional Probabilistic Seismic Hazard Analysis (PSHA) with nonlinear dynamic

structural analyses in order to determine the annual probability of exceedance of a

given response parameter. This procedure is referred to as the Probabilistic Seis-

mic Demand Analysis (PSDA). These concepts are then extended to the analysis of

controlled response, referred to as Probabilistic Seismic Control Analysis (PSCA).

8.2.1 Probabilistic Seismic Hazard Analysis (PSHA)

In view of the rareness of seismic events and the variability of their characteristics, the

description of the seismic threat in probabilistic terms has become common practice.

The earliest work in this area has de�ned this threat in terms of some measure of

the ground motion intensity, such as peak ground acceleration (PGA). More recent

work has commonly sought to characterize the ground motion by its response spectral

acceleration Sa|i.e., the peak acceleration the earthquake will induce in a 1-DOF

system with a speci�ed period T and damping ratio �. Commonly, T and � are chosen

to describe the properties of the �rst mode of the building of interest. Alternatively,

one may instead consider the peak spectral velocity Sv or displacement Sd. In any

case, the aim is to describe the probability that the ground motion characteristic of

interest (PGA, Sa, Sv, or Sd) is exceeded over a reference period of interest.

For example, an existing site hazard curve for spectral acceleration provides the

probability of exceeding any particular spectral acceleration, Sa(T; �), for a given

142

Chapter 8 Probabilistic Seismic Control Analysis

period, T , and damping ratio, �. The elastic spectral acceleration at the fundamental

period of the model structure is typically used since it is usually an e�ective structure-

speci�c measure of ground motion intensity for predicting the nonlinear response of

buildings (Shome et al. 1997). Research has also been conducted that evaluates the

e�ect of performing the analysis using a mixture or vectorized Sa values (Shome and

Cornell 1998). An \e�ective" intensity measure for earthquake records is one for

which the record-to-record dispersion of the drift response given the intensity level is

relatively small, and for which a hazard analysis is available. The particular choice

of ground motion intensity measure, as indicated previously, is not critical to the

procedure.

The probability that Sa exceeds a critical value scr in an arbitrary earthquake is

formally written as follows:

P [Sa > scr] = GSa(scr) =

ZZm;r

P [Sa > scr jM = m;R = r] fM;R(m; r) dmdr (8.1)

in which fM;R(m; r) is the joint probability density function that describes the mag-

nitude M and site-to-source distance R of an arbitrary event. Equation (8.1) is com-

monly evaluated numerically, and has become known as Probabilistic Seismic Hazard

Analysis (Cornell 1968) or \PSHA". Final results are generally rescaled by a rate � of

all events being modeled, leading to a mean hazard rate H(s) associated with those

seismic events that produce Sa > scr:

HSa(scr) = �GSa(scr) = �

ZZm;r

P [Sa > scrjM = m;R = r] fM;R(m; r) dmdr (8.2)

The quantity HSa, expressed as a function of scr, is the typical output of a PSHA

computation. As a simple parametric representation, Cornell (1996) has suggested

that the net result is often well-approximated by a power-law relation of the form:

H(Sa) = k0S�k1a (8.3)

where H(Sa) is the annual probability of exceeding a given Sa, and k0 and k1 are

constants obtained by �tting the function to results from a PSHA near the return

period of interest. The coeÆcients k0 and k1 thus serve to characterize the seismic

143


threat at a given site of interest. The above approximation has the advantage of being

linear in log-log space and has been shown to be satisfactory over a range of spectral

accelerations (Shome 1999).

8.2.2 Probabilistic Seismic Demand Analysis (PSDA)

Unfortunately, Sa does not provide suÆcient information to determine the precise

response of actual buildings, which generally show both nonlinear and multi-degree-

of-freedom (MDOF) behavior. Indeed, a growing trend toward \performance-based"

seismic design exists, in which a range of increasingly rare hazards are speci�ed, to-

gether with a correspondingly increasing level of permissible \damage" (i.e., nonlinear

behavior). These requirements imply the need for explicit recognition, and the statis-

tical quanti�cation, of the degree of nonlinear behavior|e.g., the drift demand Æ|as

a function of the desired exceedance rate, such as the return period.

Extension of conventional PSHA, to directly describe the seismic demand Æ of

a complex, nonlinear structure, has become known as probabilistic seismic demand

analysis (\PSDA"). In general, the calculation of the probability of exceedance of

a level y of any damage parameter Y involves the summation of the probability

of exceeding a level y for the parameter Y over all possible seismic sources. For

a given seismic source, the determination of the probability of exceeding a level y

for the parameter Y must consider 1) the probability of exceeding a level y for a

given spectral acceleration, earthquake magnitude, and distance to source, 2) the

probability distribution of the spectral acceleration given a magnitude and distance,

and 3) the probability distribution of magnitude and distance for the source. This

relationship can be expressed mathematically as follows:

P (Y > y) =NXi=1

�i

ZZZP [Y > y j sa; m; r]i � fSajM;R(sa j m; r)i

� fM;R(m; r)i dsa dm dr

(8.4)

where N is the number of seismic sources at a site, �i is the mean annual rate of oc-

currence of earthquakes from source i, fM;R(m; r) is the joint probability distribution

of magnitude M and distance R of the site, P [Y > y j sa; m; r] is the conditional

probability that the response or damage parameter Y exceeds a level y due to an

144


earthquake of magnitude m at distance r that generates spectral acceleration sa at

the site, and fSajM;R(sajm; r) is the conditional probability distribution of Sa for an

event of magnitude m at a distance r. However, given knowledge of the ground mo-

tion's intensity, as measured by Sa at the building's �rst mode, the nonlinear behavior

is often found to be not substantially in uenced by additional ground motion param-

eters (e.g., M , R, duration). Though somewhat counterintuitive, Shome (1999) has

demonstrated this e�ect through detailed comparisons. This result permits Sa to be

used as a powerful, scalar \pinch-point" to summarize the ground motion threat.

In practice, considerable variability exists in Sa, which re ects the elastic demand

of a simpli�ed, 1-DOF structure. A positive consequence of this behavior is that

one need not have a very precise description of the true nonlinear (MDOF) demand

as a function of the elastic demand. In particular, it generally suÆces to have a

reasonably accurate estimate of the median demand for a given level of Sa. Taking

drift, Æ, as the structural response parameter, then the median demand is represented

by Æ. Formally, this idea is expressed by the following:

Æ = Æ(Sa) � � (8.5)

in which the random variable � has unit median, and variability that is small compared

with that of Æ(Sa). In view of Equation (8.5), one may construct a result analogous

to Equation (8.1):

P [Æ > Æcr] = GÆ(Æcr) =

Z�

P [Æ(Sa) � � > Æcr] f�(�) d�

=

Z�

GSa[s(Æcr=�)] f�(�) d� (8.6)

in which s(Æ) denotes the value of Sa for which the median drift Æ(Sa) is given by Æ.

Multiplying each side of Equation (8.6) by the rate � of all modeled seismic events,

an analogous result is found for the drift hazard rate HÆ:

HÆ(Æcr) =

Z�

HSa [s(Æcr=�)] f�(�) d� (8.7)

The median relationship between spectral acceleration and drift is established by

performing nonlinear dynamic analyses of the model structure for numerous ground

145


motions at di�erent levels of intensity (as measured by spectral acceleration). The

spectral acceleration at the fundamental period of the model structure for each ground

motion is simply obtained from its elastic response spectrum. The response of the

model structure subjected to each earthquake record provides the corresponding drift.

The functional dependence between drift, Æ and Sa is taken to be:

Æ = aSba (8.8)

The exponent b in Equation (8.8) is included to capture potential \softening" of the

nonlinear relationship between spectral acceleration and median drift. Note that a

regression of the form given in Equation (8.8) is equivalent to a linear regression of

the log of drift on the log of spectral acceleration. Utilizing this relationship, then:

s(Æcr=�) =

�Æcra�

�1=b

(8.9)

Substituting this expression into the spectral acceleration hazard approximation,HSa,

which given in Equation (8.3), and then substituting into the drift hazard expression

from Equation (8.7) gives the following expression:

HÆ(Æcr) =

Z�

k0

�Æcra�

�(�k1=b)

f�(�) d�

= k0

�Æcra

�(�k1=b)

| {z }H

Z�

�(k1=b) f�(�) d�| {z }Cf

(8.10)

The �rst term in Equation (8.10) is equivalent to direct substitution of Equation (8.8)

into Equation (8.3). The second term, Cf , in Equation (8.10) is the �rst moment of

a lognormally distributed random variable, given by:

Cf = exp

�1

2k21

��ln(Æ)jSab

�2�(8.11)

where �ln(Æ)jSa , is calculated as the mean squared deviation of the spectral acceleration

versus drift data points from the regression �t. Note that a value of 1 for Cf would

ignore the e�ect of the dispersion of structural response in the drift demand hazard.

146


8.2.3 Probabilistic Seismic Control Analysis (PSCA)

We believe it is critical, in the practical evaluation of the performance of buildings|

whether controlled or not|under seismic threats to re ect (1) the realistic potential

for nonlinear behavior and (2) the realistic characteristics of ground motion excita-

tions (i.e., by imposing a suite of recorded ground motion records, as opposed to

an idealized probabilistic, random vibration description). We seek here to evaluate,

rather than to assume apriori, the assumption that the presence of structural control

will ensure that the building behavior remains linear.

This research proposes an extension of PSDA, which statistically describes the

drift demand Æ of a complex nonlinear structure, to analyze the modi�ed behavior of

the same structure in the presence of supplemental control (PSCA). Computationally,

the only practical di�erence that arises is the need to perform suÆcient analyses of

the controlled structure to predict the median demand Æ as a modi�ed power-law

function as follows:

Æ = acSbca (8.12)

The subscripts \c" here re ects that the parameters are to be �t to the controlled,

rather than the original, structure. One expects that the bene�t of control is re ected,

statistically, through reduced values of ac (and/or bc) with respect to the correspond-

ing values a and b that describe the original, uncontrolled system in Equation (8.8).

In the presence of control, structural response analysis under any single, speci�ed

earthquake becomes increasingly costly. For example, the analysis of a practical model

of controlled 3-story structure, with n = 49 degrees of freedom, has been found to

take 20 CPU minutes on a Sun Ultra 60 machine, with a 359 MHz UltraSPARC-

II processor and 640 Mb of memory to analyze a 60-second earthquake. For the

9-story structure, with n = 216 degrees of freedom, the analysis takes 720 CPU

minutes on the same machine. In other words, the ratio between analysis time and

ground motion duration is signi�cantly high so as to be an obstacle to real-time

earthquake simulation. In view of this fact, a major bene�t would be gained if

only the response to a relatively small number of records are required to adequately

estimate the coeÆcients ac, bc that characterize the modi�ed response trend in the

presence of control.

147


The presence of control, whether active or passive, often changes the e�ective

�rst-mode period and/or damping. This suggests that the de�nition of Sa, on which

the response regression in Equation (8.12) is based, should change as a function of the

e�ective period, Tc, and damping ratio, �c, of the controlled system. The following

sections show that our de�nitions of Tc and �c are relatively easily determined from

our choice of control strategy.

As mentioned previously, an\e�ective" intensity measure for earthquake records is

one for which the record-to-record dispersion of the drift response given the intensity

level is relatively small, and for which a hazard analysis is available. While the above

discussions have focused on Sa, other ground motion parameters may prove to result

in smaller record-to-record dispersion of the drift response. While the methodology

discussed remains applicable, a probabilistic seismic hazard analysis for that param-

eter may not be available. Similarly, PSHA is generally not available for the higher

damping values, �c, that result from the additional control.

8.3 Spectral Acceleration Hazard

The spectral acceleration hazard curve is obtained by �tting a curve of the form given

in Equation (8.3) to the points de�ned by the annual probabilities of exceedance and

corresponding median spectral accelerations for the three sets of ground motions.

Median values were utilized as the ground motions were scaled so that, on \average",

the entire set would correspond to the speci�ed hazard level. The elastic spectral

acceleration at the fundamental period is a measure of ground motion intensity that

is structure speci�c. However, the addition of supplemental control devices changes

the fundamental properties of the structural system. As a result, a di�erent spectral

acceleration hazard curve needs to be developed for each system. Figures 8.1 and

8.2 shows the hazard curve for the uncontrolled structure as well as the three points,

representing the three ground motion sets, that were used to obtain it for the 3- and

9-Story structures respectively. The values of the parameters k0 and k1 for all systems

are given in Table 8.1.

148


Table 8.1: Parameters for Spectral Acceleration Hazard Curve Fit

Structure Control System T1 (sec) � (%) k0 k1

3-Story

Uncontrolled 1.01 2.0 1.42e-3 3.25FPS T3, f1 3.00 2.0 1.10e-4 2.14VS 30 - D1 1.01 30.0 2.84e-4 2.33

ATB 1.00 36.7 2.59e-4 2.42

9-Story

Uncontrolled 2.27 2.0 2.62e-4 2.08FPS T4, f1 4.00 2.0 3.90e-5 1.89VS 30 - D1 2.27 30.0 6.67e-5 2.22

ATB 2.27 36.7 2.62e-4 2.08

0 0.5 1 1.5 2 2.510

−4

10−3

10−2

10−1

Annual Hazard Curce for Spectral Acceleration LA 3−Story Structure

Spectral Acceleration, Sa(T

1 = 1.01s, ξ

1 = 2%)

Ann

ual P

roba

bilit

y of

Exc

eeda

nce,

H(S

a)

Figure 8.1: Annual Hazard Curve for Spectral Acceleration, LA 3-StoryStructure

149


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

−4

10−3

10−2

10−1

Annual Hazard Curce for Spectral Acceleration LA 9−Story Structure

Spectral Acceleration, Sa(T

1 = 2.3s, ξ

1 = 2%)

Ann

ual P

roba

bilit

y of

Exc

eeda

nce,

H(S

a)

Figure 8.2: Annual Hazard Curve for Spectral Acceleration, LA 9-StoryStructure

8.4 Relationship between Ground Motion and De-

mand Parameters

8.4.1 Estimate of Peak Story Drift

A functional relationship of the form given in Equation (8.8) can be �t to the data

by: 1) using the median values for Sa and Æmax for each hazard set, or 2) utilizing the

full data set of Sa and Æmax points. A linear regression of the log of the data selected

was performed to establish the parameters a and b. Regardless of whether the full

data set or only the median values were used in the regression the dispersion of Æmax

given Sa must be calculated as the square root of the mean squared deviation of the

full data set from the �tted curve, as shown below:

�ln(Æ)jSa = sqrt1

n� 2

nXi=1

�ln(Æ)� ln(�Æ)

�2(8.13)

150


where �Æ is the value of peak drift predicted from the �tted curve. The data was �tted

using both the full data set and the median values of the data. Plots of spectral

acceleration, Sa, versus the maximum story drift over the height of the structure,

Æmax, from the nonlinear dynamic analyses are presented in Figures 8.3 through 8.6.

The �t of the regression curve for the uncontrolled structure, shown in Figure 8.3,

displays a signi�cant amount of scatter to the �tted curve. Most of the contribution

to the scatter comes from the results of the 2 in 50 set of ground motions. The

resulting parameters for the regression �t to all the records are listed in Table 8.2.

When �tting through the median values, the resulting exponent coeÆcient, b, is only

slightly larger than 1, indicating a slight softening of the system. However, when the

�t is performed using the entire data set, the exponent b is signi�cantly smaller. This

trend proves to be consistent for all cases analyzed. One possible explanation lies in

the scaling procedure used. As discussed in Section 4.3, the earthquake records were

scaled to match speci�ed spectral acceleration valuea at four di�erent periods. As a

result, a record with a higher than average spectral acceleration value at the period on

interest to the structure is associated, by design, with spectral acceleration values that

are systematically lower than average at some of the other periods. As nonlinearities

occur, the period of the structure moves away from from the fundamental period and

into the range where the spectral acceleration is lower than average. So this record will

produce lower drifts than the \typical" records with the same spectral acceleration.

A reverse e�ect occurs with records with spectral accelerations lower than average

at the period of interest. The net result is a \ attening" of the �tted curve and

lower b values than might be expected from a random sample of record. Another

concern with this structure are the results due to the simmulated ground motions.

As discussed in Section 4.3, results due to the simulated, or synthetic, time histories

for structures with a period of 1 second can be misleading. Previous researchers have

found that to be true for this structure when determining the relationship between

spectral acceleration and drift (Luco and Cornell 1998). Results for the regression

analysis with no simulated records are also listed in Table 8.2 for the uncontrolled

structure. The dispersion given is now signi�cantly lower. Note, however that the

resulting �t is close to the original result. Both curve �ts are shown in Figure 8.3.

The relationship between ground motion and structural demand was analyzed

for the FPS isolation system with a 3 second isolation period and maximum fricion

151


Table 8.2: Parameters for Fit of Relationship Between Spectral Accelerationand Story Drift, 3-Story Structure

Records Considered Fitted Data a b �ln(Æ)jSa

All Reccordsmedian 0.0291 1.099 0.383all 0.0288 0.842 0.346

No Simulatedmedian 0.0283 0.984 0.251all 0.0267 0.789 0.219

0 0.5 1 1.5 2 2.5 30

0.02

0.04

0.06

0.08

0.1

0.12

Drif

t Ang

le

Spectral Acceleration, Sa(T = 1.0s, ξ = 2%) [g]

Relationship between Spectral Acceleration and Peak Story Drift LA 3−Story Structure: α = 3%, no P−∆

50 in 50 Set

10 in 50 Set

2 in 50 Set

Fit: All

Fit: No Simulated

Figure 8.3: Relationship between Spectral Acceleration and Maximum PeakStory Drift for LA 3-Story Structure

152


coeÆcient of about 12% (FPS - T3, f1). The results of the regression analysis with the

response due to all records is listed in Table 8.3. As this system alternates between a

�xed base response, with a period of 1 second, and a sliding system, with a period of

3 seconds, analyses were performed with spectral acceleration values at both periods.

The resulting regression of the form in Equation (8.8) utilizing the median values

of each analysis set is poor, especially at high Sa values. Performing the regression

analysis utilizing all data points, rather than median values, provides only a small

bene�t, as indicated by the parameters in Table 8.3. Attempts to improve the �t

by determining Sa at the exact post-sliding period of 3.14 seconds or correlating to

another ground-motion parameter, such as the spectral velocity, had only a minimal

e�ect. The results also do not clearly indicate which parameters result in the better

�t.

The analyses were then performed ignoring the response of the simmulated, or

synthetic, time histories. The results, listed in Table 8.4, indicate signi�cantly re-

duced dispersion. Also, these results indicate that structural demand is more closely

associated with the spectral response at the �xed base period of 1 second. Utilizing

spectral velocity results in slightly less dispersion to the �tted curve. At low values

of Sa, the isolation system is not activated, as the horizontal forces observed by the

bearing are not greater than the frictional slip force of the bearing system. As a

result, the structure still has a fundamental period of 1 second and no reduction of

peak story drift is observed. Once the bearings begin to slide, the regression results

indicate a\hardening" relationship, where increases in Sa produce correspondingly

smaller changes in Æmax. This response results from the fact that the isolation bear-

ing limits the amount of force being transmitted to the superstructure. A comparison

of the regression �ts with and without the synthetic records is shown in Figure 8.4.

In contrast, the relationship between Sa and Æmax for the VS system is very strong,

as seen in Figure 8.5. The curve resulting from the regression �t is nearly linear, as

seen from the parameter b in Table 8.5 being nearly unity, and the scatter about the

curve is also low. One of the advantages of this result is that only a few number of

nonlinear dynamic analyses should be required to determine the functional relation-

ship between Sa and Æmax for this system. Also, this system is relatively una�ected

by the inclusion of the results from the simulated time histories.

153


Table 8.3: Parameters for Fit of Relationship Between Spectral Value andStory Drift, 3-Story Structure with FPS Isolation System

Spectral Parameter Period Fitted Data a b �ln(Æ)jSa

Sa

1.0 secmedian 1.3 e-2 0.754 0.377all 1.2 e-2 0.525 0.348

3.0 secmedian 2.2 e-2 0.484 0.377all 2.0 e-2 0.380 0.291

3.14 secmedian 2.4 e-2 0.492 0.383all 2.0 e-2 0.381 0.372

Sv

1.0 secmedian 5.2 e-4 0.770 0.367all 1.3 e-3 0.549 0.341

3.0 secmedian 1.4 e-3 0.536 0.328all 1.7 e-3 0.483 0.326

Table 8.4: Parameters for Fit of Relationship Between Spectral Value andStory Drift, 3-Story Structure with FPS Isolation System - Ignor-ing Simulated Ground Motions

Spectral Parameter Period Fitted Data a b �ln(Æ)jSa

Sa

1.0 secmedian 1.2 e-2 0.567 0.216all 1.1 e-2 0.432 0.199

3.0 secmedian 1.8 e-2 0.378 0.324all 1.4 e-2 0.205 0.296

Sv

1.0 secmedian 1.1 e-3 0.567 0.201all 1.7 e-3 0.454 0.188

3.0 secmedian 1.9 e-3 0.434 0.281all 3.1 e-3 0.305 0.268

154


0 0.5 1 1.5 2 2.5 30

0.01

0.02

0.03

0.04

0.05

0.06

Drif

t Ang

le


Relationship between Spectral Acceleration and Peak Story Drift LA 3−Story Structure with FPS Isolation: α = 3%, no P−∆

50 in 50 Set

10 in 50 Set

2 in 50 Set

Fit: All

Fit: No Simulated

Figure 8.4: Relationship between Spectral Acceleration and Maximum PeakStory Drift for LA 3-Story Structure with FPS Isolation

A similar result is seen for the ATB system, shown in Figure 8.6, with little scat-

ter about the resulting curve, especially when compared to the uncontrolled results.

Though the scatter is a little higher than that of the viscous case, a much smaller

number of nonlinear dynamic analyses should be required to determine the functional

relationship between Sa and Æmax for this system. As these analyses can be extremely

computationally intensive, reducing the number of analyses required becomes a great

advantage. The resulting curve is essentially the same independent of whether the

simulated time histories are included, as indicated in Table 8.6.

Table 8.5: Parameters for Fit of Relationship Between Spectral Value andStory Drift for VS System

Records Considered �tted data a b �ln(Æ)jSa

All Reccordsmedian 2.97 e-2 1.10 0.132all 2.84 e-2 1.04 0.128

No Simulatedmedian 2.74 e-2 1.04 0.0978all 2.63 e-2 0.996 0.0935

155


0 0.5 1 1.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Drif

t Ang

le


Relationship between Spectral Acceleration and Peak Story Drift LA 3−Story Structure with VS Dampers: α = 3%, no P−∆

50 in 50 Set

10 in 50 Set

2 in 50 Set

Fit: All

Fit: No Simulated

Figure 8.5: Relationship between Spectral Acceleration and Maximum PeakStory Drift for LA 3-Story Structure with Viscous Brace System

0 0.5 1 1.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Drif

t Ang

le

Spectral Acceleration, Sa(T = 1.0s, ξ = 36.7%) [g]

Relationship between Spectral Acceleration and Peak Story Drift LA 3−Story Structure with ATB System: α = 3%, no P−∆

50 in 50 Set

10 in 50 Set

2 in 50 Set

Fit: All

Fit: No Simulated

Figure 8.6: Relationship between Spectral Acceleration and Maximum PeakStory Drift for LA 3-Story Structure with ATB System

156


Table 8.6: Parameters for Fit of Relationship Between Spectral Value andStory Drift for ATB System

Records Considered �tted data a b �ln(Æ)jSa

All Reccordsmedian 3.31 e-2 1.02 0.287all 3.30 e-2 1.01 0.287

No Simulatedmedian 2.99 e-2 0.944 0.280all 2.98 e-2 0.943 0.280

8.4.2 Estimate of Control System Demand

An estimate of the demand on the control system can also be determined using a

similar process. For the viscous dampers, the dependence between the peak force in

the damper and the spectral acceleration is taken to be:

FD = aSba (8.14)

where FD is the peak damper force. This relationship is essentially the same as that

for peak drift, given in Equation (8.8). The regression results are listed in Table 8.7

for the VS damper systems for 20%, 30%, and 40% e�ective damping using equal

sized dampers in every story.

Table 8.7: Parameters for Fit of Relationship Between Spectral Value andPeak Damper Force

E�ective Damping �tted data a b �ln(Æ)jSx20% all 825 0.761 0.1930% all 1,122 0.786 0.2040% all 1,326 0.793 0.21

For the FPS isolation system, the displacement of the bearings can be consid-

ered as a drift-type quantity. As a result, a similar relationship as that given in

Equation (8.8) can be developed for the beak bearing displacement as shown below:

ÆB = aSbx (8.15)

where ÆB is the peak bearing displacement and Sx represents any spectral quantity

of interest. Both spectral acceleration, Sa, and spectral velocity, Sv, at the isolation

157


0 0.5 1 1.50

200

400

600

800

1000

1200

1400

1600

Pea

k D

ampe

r F

orce

[kip

s]


Relationship between Spectral Acceleration and Peak Damper Force LA 3−Story Structure with VS Dampers: α = 3%, no P−∆

50 in 50

10 in 50

2 in 50

Figure 8.7: Relationship between Spectral Acceleration and Peak DamperForce for LA 3-Story Structure VS Dampers

period of 3 seconds and the �xed base period of 1 second were used to determine

the expected peak bearing displacement curve. The parameters from the regression

analysis using the response to all time histories are listed in Table 8.8. For the bearing

displacement, using spectral velocity results in less dispersion to the �tted curve than

using spectral accelerations. Also, a better �t is achieved using the isolation period

of 3 seconds. As the bearing only displaces once the isolation system is activated, the

period of interest is that of the isolated structure. One possible reason for a better

�t using spectral velocity is that the coeÆcient of friction of the bearing is velocity

dependent. Also, at high periods, the system is entering the velocity sensitive section

of the response spectra. However, the resulting dispersion is still relatively high.

Ignoring the synthetic time histories does not reduce the dispersion, as indicated in

Table 8.9. Those records only present a problem for systems with a period near 1

second, where the two simmulated segments are merged. As the bearing displacement

is related to the spectral response at 3 seconds, the simmulated records do not present

a problem.

158


Table 8.8: Parameters for Fit of Relationship Between Spectral Value andPeak Bearing Displacement, 3-Story Structure with FPS IsolationSystem

Spectral Parameter Period Fitted Data a b �ln(Æ)jSx

Sa

1.0 secmedian 7.08 2.252 0.809all 6.24 1.529 0.668

3.0 secmedian 41.44 1.472 0.755all 25.68 1.146 0.706

Sv

1.0 secmedian 5.7 e-4 2.268 0.809all 0.0098 1.553 0.678

3.0 secmedian 0.0080 1.622 0.534all 0.0161 1.441 0.517

Table 8.9: Parameters for Fit of Relationship Between Spectral Value andPeak Bearing Displacement, 3-Story Structure with FPS IsolationSystem - No Simulated Records

Spectral Parameter Period Fitted Data a b �ln(Æ)jSx

Sa

1.0 secmedian 6.43 2.076 0.707all 5.45 1.433 0.582

3.0 secmedian 38.17 1.432 0.806all 19.19 1.016 0.743

Sv

1.0 secmedian 0.0012 2.055 0.704all 0.0132 1.449 0.598

3.0 secmedian 0.0077 1.629 0.578all 0.0192 1.389 0.556

159


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

Pea

k B

earin

g D

ispl

acem

ent [

inch

es]


Relationship between Spectral Acceleration and Peak Story Dr if t LA 3−Story Structure with FPS Isolation: α = 3%, no P−∆

50 in 50

10 in 50

2 in 50

Figure 8.8: Relationship between Spectral Acceleration and Peak BearingDisplacement for LA 3-Story Structure FPS Isolation

0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

30

35

40

Pea

k B

earin

g D

ispl

acem

ent [

inch

es]

Spectral Velocity, Sv(T = 3.0s, ξ = 2%) [in/sec]

Relationship between Spectral Velocity and Maximum Bearing Displacement LA 3−Story Structure with FPS Isolation: α = 3%, no P−∆

50 in 50

10 in 50

2 in 50

Figure 8.9: Relationship between Spectral Velocity and Peak Bearing Dis-placement for LA 3-Story Structure FPS Isolation

160


8.4.3 Number of Analyses

The required minimum number of simulations, n, can be determined based on the

estimation error of the median demand, which is expressed as a fraction of the median.

The standard error, �ln Æ, due to the use of a limited sample size is approximately equal

to:

�ln Æ =�ln(Æ)jSap

n

1 +

ln(Sa)� ln( �Sa)

�2ln(Sa)

!

� �ln(Æ)jSapn

(8.16)

for Sa � Sa. This relationship is illustrated in Figure 8.10 for all four systems. In all

cases, 20 analyses ensure that the one-sigma con�dence band on the estimation of the

median peak drift demand lying within 10% of the median value. The VS and ATB

systems require signi�cantly fewer analyses, with the VS system having a one-sigma

con�dence band within 10% of the median with only 3 analyses.

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Standard Error in Estimation due to Limited Sample Size LA 3−Story Structure: Regression on All Data Points

Number of Analyses

Sta

ndar

d E

rror

Uncontrolled

FPS − T3, f1

VS − 30, D1

ATB − S1k

Figure 8.10: Standard Error in Peak Drift Estimation due to Limited SampleSize Using Full Data Set, 3-Story Structure

161


8.5 Drift Demand Hazard Curves

Once the median relationship between spectral acceleration and drift (i.e., the median

drift given spectral acceleration), and the dispersion of drift given spectral acceleration

are known, the spectral acceleration hazard curve can be used to create a drift demand

hazard curve. Median values are used as the ground motion sets were developed to

be used as a unit, so that the seismic demand for a structure for a given hazard is

best approximated by the median response. The resulting drift demand curves can be

used to determine the impact of control parameter design as well as provide a basis

for comparison between systems.

8.5.1 E�ect of Control Parameter Variation

Interstory drift demand hazard curves were developed for the FPS isolation system

with sliding surface f1 and isolation periods of 2, 3, and 4 seconds. The parameters

utilized in generating these curves are listed in Table 8.10. In Section 6.3.1, increasing

the isolation period is shown to cause a decrease in the median values of peak inter-

story drift demands, especially at the high hazard level. Consequently, increasing

the isolation period results in a decrease in the probability of exceeding a given drift

demand. However, the slopes of the 3 curves are signi�cantly di�erent, so that the dif-

ferences between the systems increases for higher target drifts. One major reason for

this result lies in the impact on the response scatter associated with each system. As

the isolation period increases, the scatter in the seismic demand response decreases,

resulting in a decrease in the uncertainty associated with that parameter. Also, the

e�ect of increasing isolation period was demonstrated to be more signi�cant for the 2

in 50 set, or high hazard level, than for lower seismic hazards, since a minimal force

must be generated across the sliding surface before the isolation system is triggered.

Similarly, maximum peak story drift hazard curves were developed for VS damp-

ing systems with di�erent e�ective damping values and are shown in Figure 8.12.

Increasing the damping in the system corresponds the expectation that higher damp-

ing values reduce drift demands and, therefore, the probability of exceeding a target

drift level. Increasing the e�ective damping in the system produces a fairly even shift

downwards of the curve over the range of return periods considered. Maximum peak

162


0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0510

−5

10−4

10−3

10−2

10−1

Maximum Story Drift Ratio, δ

Ann

ual P

roba

bilit

y[ P

(δ >

δcr

) ]

Annual Hazard Curves for Maximum Peak Story Drift Angle LA 3−Story Structure with FPS Isolation (α = 3%, no P−∆)

T2, f1 − All Records



T2, f1 − No Simulated Records



Figure 8.11: Comparison of Drift Demand Hazard Curves of FPS IsolationSystem for 3-Story Structure, Variation in Isolation Period

Table 8.10: Parameters for Fit of Relationship Between Spectral Velocityand Story Drift, Variation in Isolation Period of FPS IsolationSystem

Records Isolation Period a b �ln(Æ)jSv

All

2.0 sec 1.56 e-2 0.781 0.5733.0 sec 1.24 e-2 0.525 0.3484.0 sec 1.13 e-2 0.450 0.197

NoSimulated

2.0 sec 1.29 e-2 0.610 0.2973.0 sec 1.11 e-2 0.432 0.1994.0 sec 1.08 e-2 0.412 0.174

163


story drift hazard curves were also generated for the three damper distributions of

the VS damper system with 30% equivalent damping, seen in Figure 8.13. Little

variation is seen between the resulting hazard curves for the three distributions. This

result corresponds well with the behavior discussed in Chapter 6. The redistribution

of dampers over the height of the structure only had a noticeable impact under the 2

in 50 set of ground motions. In that case, though the distribution of drift demands

were signi�cantly di�erent, the values for peak drift were reduced by small amounts.

The parameters for the relationship between drift and spectral acceleration for these

viscous damping systems are listed in Table 8.11.

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0510

−4

10−3

10−2

10−1


Ann

ual P

roba

bilit

y[ P

(δ >

δcr

) ]

Annual Hazard Curves for Maximum Peak Story Drift Angle LA 3−Story Structure with VS Dampers (α = 3%, no P−∆)

VS 20 − D1

VS 30 − D1

VS 40 − D1

Figure 8.12: Comparison of Drift Demand Hazard Curves of VS Dampingsystem for 3-Story Structure, Variation in E�ective Damping

Hazard curves can also be developed for parameters of the control system. The

demand curves for the isolation bearing displacement for di�erent isolation periods is

shown in Figure 8.14. Once the sliding force for the bearings is reached, the systems

with higher isolation periods result in higher bearing displacements. The trend in the

hazard curves re ects that fact, with the hazard associated with an isolation period of

4 seconds being higher than that for the system with an isolation system of 2 seconds.

The speci�c parameters used in developing these hazard curves are listed in Table 8.12.

164


0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0510

−4

10−3

10−2

10−1


Ann

ual P

roba

bilit

y[ P

(δ >

δcr

) ]

Annual Hazard Curves for Maximum Peak Story Drift Angle LA 3−Story Structure with Varying Distribution of VS Dampers (α = 3%, no P−∆)

VS 30 − D1

VS 30 − D2

VS 30 − D3

Figure 8.13: Comparison of Drift Demand Hazard Curves of VS Dampingsystem for 3-Story Structure, Variation in Damping Distribution

Table 8.11: Parameters for Fit of Relationship Between Spectral Value andStory Drift, VS Damping Systems

E�ective Damping Distribution a b �ln(Æ)jSa20% D1 2.97 e-2 1.0113 0.154

30%

D1 2.84 e-2 1.0449 0.128D2 2.70 e-2 1.0208 0.103D3 2.66 e-2 1.0192 0.109

40% D1 2.63 e-2 1.0698 0.149

165


A second set of hazard curves were developed by developing a relationship between

bearing displacement and spectral velocity, Sv, evaluated at the isolation period. This

�t of the data results in less scatter from the �tted curve, as seen from the values of

�ln(Æ)jSv . The hazard curves are much atter than those developed using Sa, so that

the hazard at low bearing displacements is noticeably lower. A word of caution must

be made regarding the hazard curves developed using Sv. The assumption is made

that the hazard for the median spectral velocity of each ground motion set is the

same as the hazard for spectral acceleration of the corresponding ground motion set.

Since the sets were developed for spectral acceleration hazard, this assumption may

not be true and should be veri�ed.

5 10 15 20 25 30 35 4010

−4

10−3

10−2

Maximum Bearing Displacement

Ann

ual P

roba

bilit

y [ P

( θ

> θ

cr )

]

Annual Hazard Curves for Maximum Bearing Displacement 3−Story Structure

FPS − T2, f1

FPS − T3, f1

FPS − T4, f1

Figure 8.14: Comparison of Bearing Displacement Demand Hazard Curvesfor 3-Story Structure, Variation in Isolation Period

These drift hazard curves can be used in the design process of the control system.

With this information, optimization of the control design for the structural system

can be accomplished over a range of seismic hazards.

166


Table 8.12: Parameters for Fit of Relationship Between Spectral Value andBearing Displacements, Variation in Isolation Period of FPS Iso-lation System

Isolation Period �tted data Spectral Parameter a b �ln(Æ)jS

T = 2.0 sec allSa 13.0 1.168 0.517Sv 0.020 1.345 0.405

T = 3.0 sec allSa 25.7 1.142 0.706Sv 0.016 1.445 0.403

T = 4.0 sec allSa 35.5 0.961 0.840Sv 0.027 1.368 0.617

5 10 15 20 25 30 35 4010

−4

10−3

10−2

Maximum Bearing Displacement

Ann

ual P

roba

bilit

y [ P

( θ

> θ

cr )

]

Annual Hazard Curves for Maximum Bearing Displacement 3−Story Structure

FPS − T2, f1

FPS − T3, f1

FPS − T4, f1

Figure 8.15: Comparison of Bearing Displacement Demand Hazard Curvesfor 3-Story Structure, Variation in Isolation Period

167


8.5.2 Comparison Between Control Systems

This probabilistic performance assessment can also be utilized to compare the per-

formance of di�erent control strategies for the same structural system. The curves

showing the maximum interstory drift hazard for each of the control strategies dis-

cussed in Section 6.4 for the 3-Story structure are plotted in Figure 8.16. All three

control strategies result in signi�cant reductions in drift hazard. However, no single

strategy proves to be most e�ective at all target drift limits. The drift hazard for the

VS damper system and ATB system are very similar over the drift range considered.

The VS system has a smaller drift hazard over this range; however, the two curves

begin to converge at the high end. The FPS isolation drift hazard curve crosses both

other controlled drift hazard curves, exhibiting higher drift hazard at low return peri-

ods and lower drift hazard at high return periods. This condition re ects the fact that

frictional isolation systems only impact the structural response if the forces through

the bearing are large enough to overcome the frictional slip force for the bearing. The

other factor a�ecting the performance of the FPS system with respect to the two

other control systems is the scatter in the structural response for a given spectral

acceleration.

Similar curves were developed for the control strategies designed for the 9-Story

structure, shown in Figure 8.17. Again, though all three control strategies reduce

the drift hazard, no one strategy is the most e�ective at all return periods or target

drift levels. The ATB system is consistently less e�ective than the two passive control

system. However, recall that this system only utilizes 3 braces located at the �rst,

second, and eighth stories, and as such has a much smaller capacity than the VS

system plotted here. The drift hazard curves for the FPS isolation VS systems cross

at a drift limit of about 3%. The VS system has a lower drift hazard at the low return

periods, while the FPS isolation has a lower drift hazard at higher return periods.

One of the disadvantages of utilizing the maximum peak story drift over the height

of the structure is the loss of the location of the critical points in the structure. Also,

the location of the maximum peak story drift is not the same for all analyses of a

given system, much less across systems. For di�erent systems, peak drift locations

may be completely di�erent. However, drift hazard curves can be developed for

individual stories. Then, for a given story, a probabilistic performance assessment

168


0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0510

−4

10−3

10−2

10−1

Ann

ual P

roba

bilit

y[ P

(δ >

δcr

) ]

Annual Hazard Curves for Maximum Peak Story Drift Angle All Records: LA 3−Story Structure (α = 3%, no P−∆)

Uncontrolled FPS − T3, f1 Viscous − 30%, D1ATB − S1k

Figure 8.16: Comparison of Drift Demand Hazard Curves for LA 3-StoryStructure

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0510

−5

10−4

10−3

10−2

10−1

Maximum Peak Story Drift Ratio

Ann

ual P

roba

bilit

y[ P

(θ >

θcr

) ]

Annual Hazard Curves for Maximum Peak Story Drift Angle LA 9−Story Structure

Uncontrolled

FPS − T4, f1

Viscous − 30%, D1

ATB − S1k

Figure 8.17: Comparison of Drift Demand Hazard Curves for LA 9-StoryStructure

169


can be performed. This information also provides insight of which story contributes

most to the risk at di�erent input levels.

This analysis was performed for all three stories of the 3-Story structure. The

resulting curves are shown in Figures 8.18 through 8.21, and the speci�c parameter

values are listed in Table 8.13. For the uncontrolled system the risk is concentrated

in the two upper stories, particularly the third. Similarly, the risk associated with

the third story dominates the system. However, for the VS damper system, the risk

associated with the third story is now the lowest of the three stories, with the second

story having the highest risk over all return periods. The ATB system does not

provide a clearly dominant story as all three hazard curves are very close in value and

cross one another with an increasing return period. At low values of return period,

the risk is highest in the third story. However, at high values for the return period,

the second story has the highest drift hazard.

Table 8.13: Parameters Drift Hazard Calculation of Individual Stories, 3-Story Structure

System Story a b �ln(Æ)jSa Cf1

Uncontrolled

1 2.2 e-2 1.09 0.405 2.072 2.7 e-2 1.09 0.382 1.903 2.9 e-2 1.13 0.397 1.91

FPS - T3, f11 1.7 e-2 0.57 0.383 2.782 2.1 e-2 0.51 0.359 3.133 2.1 e-2 0.46 0.381 4.71

VS - 30, D11 2.6 e-2 1.09 0.155 1.062 2.9 e-2 1.07 0.122 1.043 2.0 e-2 0.98 0.120 1.04

ATB

1 2,8 e-2 1.12 0.287 1.212 3.3 e-2 1.27 0.164 1.053 3.1 e-2 0.98 0.266 1.24

Individual story drift hazard curves for the 9-Story structure were also developed.

The results for story numbers 1, 2, 3, 7,and 8 are shown in Figures 8.22 through 8.25,

and the parameters are listed in Table 8.14. For the uncontrolled system, drift hazard

is highest for the eighth story. Note that the hazard associated with the lower stories

begins to increase with respect to the upper stories for higher drift levels. Recall from

Chapter 6 that drift demand for those stories increased most with respect to other

170


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0510

−5

10−4

10−3

10−2

10−1

Maximum Story Drift Ratio

Ann

ual P

roba

bilit

y[ P

(θ >

θcr

) ]

Annual Hazard Curves for Peak Story Drift Angle LA 3−Story Structure: α = 3%, no P−∆

First Story

Second Story

Third Story

Figure 8.18: Comparison of Individual Story Drift Demand Hazard Curvesfor LA 3-Story Structure

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0510

−5

10−4

10−3

10−2

10−1


Ann

ual P

roba

bilit

y[ P

(θ >

θcr

) ]


First Story

Second Story

Third Story

Figure 8.19: Comparison of Individual Story Drift Demand Hazard Curvesfor LA 3-Story Structure with FPS Isolation

171


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0510

−5

10−4

10−3

10−2

10−1


Ann

ual P

roba

bilit

y[ P

(θ >

θcr

) ]


First Story

Second Story

Third Story

Figure 8.20: Comparison of Individual Story Drift Demand Hazard Curvesfor LA 3-Story Structure with VS Damping

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0510

−5

10−4

10−3

10−2

10−1


Ann

ual P

roba

bilit

y[ P

(θ >

θcr

) ]


First Story

Second Story

Third Story

Figure 8.21: Comparison of Individual Story Drift Demand Hazard Curvesfor LA 3-Story Structure with ATB System

172


stories when comparing the drift demands from the 10 in 50 set and the 50 in 50 set.

This same phenomenon is illustrated here.

Table 8.14: Parameters Drift Hazard Calculation of Individual Stories, 9-Story Structure

System Story a b �ln(Æ)jSa Cf1

Uncontrolled

1 4.5 e-2 0.63 0.242 2.782 4.3 e-2 0.84 0.240 3.133 4.6 e-2 0.89 0.253 3.137 3.3 e-2 0.57 0.293 3.138 3.9 e-2 0.56 0.357 4.71

FPS - T4, f1

1 4.2 e-2 0.61 0.379 1.982 3.5 e-2 0.76 0.412 1.683 3.6 e-2 0.77 0.399 1.617 2.4 e-2 0.47 0.371 3.038 2.7 e-2 0.44 0.385 3.83

VS 30, D1

1 6.6 e-2 0.63 0.242 1.372 5.5 e-2 0.84 0.239 1.193 5.1 e-2 0.89 0.253 1.197 2.4 e-2 0.57 0.293 1.758 1.9 e-2 0.56 0.357 2.36

ATB

1 3.8 e-2 1.01 0.139 1.052 3.8 e-2 1.11 0.132 1.043 4.3 e-2 1.09 0.129 1.047 3.1 e-2 0.99 0.126 1.058 3.8 e-2 0.99 0.128 1.05

The drift hazard curves for the 9-Story structure with FPS isolation system is

shown in Figure 8.23. At low return periods the drift hazard is highest for the upper

stories. As the return periods increase, the drift hazard becomes dominated by the

lower stories.

The 9-Story structure with VS dampers maintains relationship between stories

constant for the hazard range of interest, as seen in Figure 8.24. The highest drift

hazard is consistently associated with the �rst story, and the lowest drift hazard is

consistently associated with the uppermost story. This phenomenon was also observed

in the seismic story demand plots for individual ground motion sets, where a constant

decrease in drift demands was observed as one went up the structure.

The individual story drift hazard curves for the 9-Story structure with ATB are

173


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0510

−5

10−4

10−3

10−2

10−1


Ann

ual P

roba

bilit

y[ P

(θ >

θcr

) ]

Annual Hazard Curves for Peak Story Drift Angle LA 9Story Structure: α = 3%, no P−∆

First Story

Second Story

Third Story

Seventh Story

Eigth Story

Figure 8.22: Comparison of Individual Story Drift Demand Hazard Curvesfor LA 9-Story Structure

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0510

−5

10−4

10−3

10−2

10−1


Ann

ual P

roba

bilit

y[ P

(θ >

θcr

) ]

Annual Hazard Curves for Peak Story Drift Angle LA 9Story Structure with FPS Isolation − T4, f1: α = 3%, no P−∆

First Story

Second Story

Third Story

Seventh Story

Eigth Story

Figure 8.23: Comparison of Individual Story Drift Demand Hazard Curvesfor LA 9-Story Structure with FPS Isolation

174


0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0510

−5

10−4

10−3

10−2

10−1


Ann

ual P

roba

bilit

y[ P

(θ >

θcr

) ]


First Story

Second Story

Third Story

Seventh Story

Eigth Story

Figure 8.24: Comparison of Individual Story Drift Demand Hazard Curvesfor LA 9-Story Structure with VS Damping

plotted in Figure 8.25. The curves for all the stories lie very close to one another and

no clear trend exists as to which stories have the highest drift hazard. The controller

was designed with equal weighting placed on all story drift values, resulting in an

\evening out" of the drift hazard over the height.

8.6 Conclusions

Current performance guidelines provide for the determination of seismic demands

for a given scenario event, which is then compared with an allowable limit. These

procedures do not account for the variability in seismic demand estimation. One

approach to address this issue is the development of probabilistic seismic demand

curves. This study uses the methodology developed by (Cornell 1996) that combines

information regarding the seismic hazard and nonlinear structural response. The

procedure relates a record speci�c quantity, such as the elastic spectral acceleration, to

a structural response parameter, such as drift, to estimate the probability of exceeding

the selected structural parameter.

175


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0510

−5

10−4

10−3

10−2

10−1


Ann

ual P

roba

bilit

y[ P

(θ >

θcr

) ]


First Story

Second Story

Third Story

Seventh Story

Eigth Story

Figure 8.25: Comparison of Individual Story Drift Demand Hazard Curvesfor LA 9-Story Structure with ATB System

The elastic spectral acceleration at the fundamental period of the structure is a

measure of ground motion intensity that is structure speci�c. However, the addition

of supplemental control devices changes the fundamental properties of the structural

system. Therefore, the determination of spectral acceleration hazard curves and re-

lationships between record quantities and structure response need to be determined

with respect to the new system. Relationships between spectral acceleration and

interstory drift are developed for all systems. For the uncontrolled structure, a signif-

icant amount of scatter is present, especially due to responses from the 2 in 50 set of

ground motions. The addition of control reduces the scatter of the response in all 3

cases. The relationship for the FPS system contains signi�cant outliers at high values

of spectral acceleration and has essentially the same dispersion as the uncontrolled

case. The period of the structure with FPS isolation alternates from 1 to 3 seconds,

with most of the response to the 50 in 50 set being dominated by the 1 second system

while the 2 in 50 set being dominated by the isolated 3 second system. As a result

determining the relationship over all hazards based on a spectral accelerations deter-

mined at a single period proves ine�ective. For the VS and ATB control systems,

176


the scatter is greatly reduced. This fact implies that a smaller number of analyses

could be used to determine the relationship between drift and spectral accelerations.

Estimates on the error associated with the estimate of the system demand indicate

that signi�cantly fewer analyses could have been used. As control analyses can be

computationally intensive, reducing the number of required analyses becomes a great

advantage to the designer.

Once this information is combined with the spectral hazard, the resulting demand

hazard curves can be used to compare the e�ects of di�erent parameters for a single

control type as well as the performance between the di�erent control strategies. Re-

sults from these curves indicate that no single control strategy is the most e�ective

at all hazard levels. For example, at low return periods the VS system has the lowest

drift demands. However, at higher return periods, the FPS isolation system becomes

the most e�ective strategy.

177

Chapter 9

Summary, Conclusions, and

Future Work

9.1 Summary

The structural engineering community has been making great strides in recent years

to develop performance-based earthquake engineering methodologies for both new

and existing construction. Both SEAOC's Vision 2000 projects and BSSC's NEHRP

Guidelines for Seismic Rehabilitation of Buildings present the �rst guidelines for

multi-level performance objectives. One of the intents of these provisions is to pro-

vide methods for designing and evaluating structures such that they are capable of

providing predictable performance during an earthquake.

Structural control can provide an additional method to meet desired performance

objectives. Design of a structure/controller system should involve a thorough un-

derstanding of how various types of controllers enhance structural performance, such

that the most e�ective type of controller is selected for the given structure and seismic

hazard. Controllers may be passive, requiring no external energy source, or active,

requiring an external power source.

The goal of the research presented here is to evaluate the role of structural con-

trol technology in enhancing the overall structural performance under seismic ex-

citations. This investigation has the following speci�c objectives: (1) To evaluate

the e�ect of the various controller architectures on seismic demands as described

through performance-based design criteria; (2) To evaluate the sensitivity of the

178

Chapter 9 Summary, Conclusions, and Future Work

structure/controller performance based variation of control parameters, load levels

and structural modeling techniques; and (3) To compare the bene�ts of the con-

trollers in both a deterministic and probabilistic format. This study focuses on steel

moment resisting frames and three types of possible controllers: (1) base isolation

system (passive); (2) linear viscous brace dampers (passive); and (3) active tendon

braces. Two structures are selected from the SAC Phase II project, the three story

system and the nine story system. Simulations of these systems, both controlled and

uncontrolled, are prepared using the three suites of earthquake records, also from the

SAC Phase II project, representing three return di�erent periods. Several controllers

are developed for each structure, and their performance is judged based on the drift,

dissipated hysteretic energy, and oor acceleration demands.

9.2 Results

9.2.1 Seismic Demands

Comparisons of the performance of three di�erent control strategies are presented for

all three sets of ground motions. The comparison is based on story drift, hysteretic

energy, and oor acceleration demands. All three controllers were successful in reduc-

ing the seismic demands, though no one system is consistently better than the others

at all hazard levels. For story drift demands, the values suggested for peak story drift

angles in FEMA 273 are used as guidelines for judging structural performance.

For short return periods, the addition of structural control reduced the median

story drift by nearly 50%. In the 10 in 50 ground motion set, the median response

of the uncontrolled system lies just below the life-safety limit of 2.5% drift. Several

individual records push the uncontrolled response beyond the collapse prevention limit

of 5% The application of control is e�ective in reducing both the median response and

its scatter, with the median response falling at about 1% drift for all three control

systems. The most signi�cant impact of control occurs with the 2 in 50 ground

motions. The uncontrolled structure has a median response just under the collapse

prevention limit. Clear outliers are present in the data, with some response values

exceeding 10% drift that are unsustainable by the physical system. The ATB and

VS control systems bring the median drift response to about 3%, with the VS system

179

Summary, Conclusions, and Future Work Chapter 9

resulting in less scatter in the data. The FPS isolation has the lowest median response,

with values of about 1.5%. However, noticeable outliers are present in the data, with

a couple of records having demands close to the collapse limit.

Results for the 9-story structure follow a similar trend as those of the 3-story

structure. Both the VS and FPS isolation are extremely e�ective at reducing the

peak story drift demands. The VS system proves to be most e�ective at higher

stories, while the FPS system is slightly less e�ective at the upper stories. For the

ATB control system, the actuators are placed on only the 1st, 2nd, and 8th stories.

The e�ect of this placement is that at high level excitations the drift demands at

stories without actuators are increased from the uncontrolled case. For comparison,

a VS system with dampers in only the same three stories and comparable peak forces

was designed. The resulting peak story drift demands were higher than those of

the uncontrolled structure in those stories lacking dampers. In contrast, if smaller

dampers are added at every story to produce the same amount of e�ective damping,

signi�cant reductions in drift result for all stories.

Residual drift values are used by FEMA 273 as an indicator of the inelastic damage

to the structural system. However, care must be taken in interpreting these values as

they can be misleading. In both the 50 in 50 set and 10 in 50 set of ground motions,

the median values for residual drifts of the control systems are close to negligible.

However, in the 2 in 50 set of ground motions, median values for all systems are

signi�cantly higher. The addition of the two passive controls decreases the median

drift demand value. However, the median value for the ATB system is greater than

that for the uncontrolled case, such that the addition of this control system appears

to increase structural damage. However, closer analysis of the response indicates

that the ATB control keeps the structure from repeated inelastic deformations, as

indicated by dissipated hysteretic energy results.

Normalized hysteretic energy (NHE) is used to provide information regarding the

cumulative damage to the structure. Normalization of the hysteretic energy values of

each element is performed by normalizing by the element's elastic energy capacity. In

the 50 in 50 set and 10 in 50 set, the addition of structural control reduces the amount

of energy that must be dissipated by the structural system to negligible values. The

only exception occurs with the active control for the 9-story structure under the 10

in 50 set, as its capacity is signi�cantly less than the two passive control strategies.

180


The ATB control consistently reduces the NHE demands at every story. In the 2 in

50 set, some dissipation of hysteretic energy occurs for all systems. The FPS system

proves to result in the least amount of energy demands for the structure, which agrees

with the results from the drift analyses.

Floor accelerations are of concern for damage to several types of nonstructural

components. Structural control systems have the bene�t of being capable of reducing

both acceleration and drift demands on the structure, which traditional methods such

as increasing building sti�ness cannot accomplish. The two passive control systems

investigated are particularly e�ective at reducing oor accelerations as compared with

the uncontrolled structure. In the 3-story structure, the peak oor accelerations at

all oors were about that of the peak ground acceleration. In the 9-story structure,

accelerations at the middle oor levels were reduced below that of the ground. The

active control system had no signi�cant reduction in oor accelerations, as the system

was not speci�cally designed to control those responses. In the 2 in 50 set for the

3-story structure, oor accelerations were increased from the uncontrolled case with

the addition of the ATB controller.

9.2.2 Modeling E�ects

The seismic demands discussed previously were based on several assumptions con-

cerning structural parameters and modeling. The response of any structure depends

on careful selection of those parameters to capture the signi�cant e�ects of the struc-

ture. Investigations were conducted to evaluate the e�ect of: 1) the level of nonlinear

modeling and analyses of the structure, 2) the initial sti�ness of the structure, and 3)

the strain-hardening assumptions in force-deformation relationships of the elements.

The use of linear models consistently results in inaccurate seismic demands, with the

exception of the FPS isolation system. As the isolation system reduces the demands

to close to the elastic limit of the structure, using a linear model can provide very

good estimates of the seismic demands from nonlinear analyses.

To assess the range of response due to changes in sti�ness, the elastic sti�ness of all

columns and beams is reduced by equal factors ranging from 0.5 to 2.00. At all values

evaluated, the addition of control reduces the median story drift demands. However,

the sensitivity of individual systems to changes in sti�ness varies, with the VS system

181


showing little change in drift demands over the entire range of sti�ness variation. The

FPS isolation system is more sensitive to period variations, especially as that period

approaches the isolation period. The active control system is the most sensitive of

the di�erent control strategies to initial period variations. As this control is designed

based on the nominal structure parameters, once the system varies signi�cantly from

the properties used in design then the performance deteriorates signi�cantly.

One of the basic assumptions regarding the post-yield behavior of frame elements

involves the determination of the value assigned to the post-yield sti�ness. This

parameter is de�ned as a fraction of the initial elastic sti�ness for this study and

is varied from a value of 0 to 10%. The impact of strain-hardening variations is

minor, both for the uncontrolled and controlled systems. Controlled systems are

less sensitive than the uncontrolled structure, as the addition of control reduces the

amount of inelastic behavior.

9.2.3 Probabilistic Seismic Control Analysis

Current performance guidelines provide for the determination of seismic demands

for a given scenario event, which is then compared with an allowable limit. These

procedures do not account for the variability in seismic demand estimation. One

approach to address this issue is the development of probabilistic seismic demand

curves. This study uses the methodology developed by Cornell (1996) that combines

information regarding the seismic hazard and nonlinear structural response. The

procedure relates a record speci�c quantity, such as the elastic spectral acceleration, to

a structural response parameter, such as drift, to estimate the probability of exceeding

the selected structural parameter.

The elastic spectral acceleration at the fundamental period of the structure is a

measure of ground motion intensity that is structure speci�c. However, the addition

of supplemental control devices changes the fundamental properties of the structural

system. Therefore, the determination of spectral acceleration hazard curves and re-

lationships between record quantities and structure response need to be determined

with respect to the new system. Relationships between spectral acceleration and

interstory drift are developed for all systems. For the uncontrolled structure, a sig-

ni�cant amount of scatter is present, especially due to responses from the 2 in 50 set

182


of ground motions. The addition of control reduces the scatter of the response in

all 3 cases. The relationship for the FPS system contains signi�cant outliers at high

values of spectral acceleration. For the VS and ATB control systems, the scatter is

greatly reduced. This fact implies that a smaller number of analyses could be used

to determine the relationship between drift and spectral accelerations. As control

analyses can be computationally intensive, reducing the number of required analyses

becomes a great advantage to the designer. Once this information is combined with

the spectral hazard, the resulting demand hazard curves can be used to compare the

e�ects of di�erent parameters for a single control type as well as the performance

between the di�erent control strategies. Results from these curves indicate that no

single control strategy is the most e�ective at all hazard levels. For example, at low

return periods the VS system has the lowest drift demands. However, at higher return

periods, the FPS isolation system becomes the most e�ective strategy.

9.3 Conclusions

For isolation systems, selection of isolation period has the greatest impact in the

resulting seismic demands on the superstructure. Lowering the friction coeÆcient can


demands is an increase in bearing displacements. This system of control proves to

be very e�ective system for both the 3-Story and 9-Story structures and all three

sets of ground motions. The median response of the superstructure remains close to

elastic even under severe ground motions, represented by the 2 in 50 set. This system,

however, is sensitive to the sti�ness of the structure, and its e�ectiveness begins to

deteriorate once noticeable nonlinearities occur.





the drift demands are more evenly distributed among the di�erent stories. If the

dampers are located in only a few stories for the same amount of e�ective damping,

however, the system can be highly ine�ective and may increase story demands at

stories without dampers.

183


The capacity of the actuators for the ATB system contributes greatly to the

e�ectiveness of the control system. Higher actuator capacities provide the controller

a greater opportunity to reduce drift demands. However, this same increased capacity

can result in systems that increase the demands from those of the uncontrolled system.

However, careful design of the control system for the 3-story structure results in a

system that consistently reduces the median story drift demands. The impact on

seismic demands of placing the actuators only at select story locations is investigated

in the 9-story. The result of this placement is that at high level excitations the drift

demands at stories without actuators are increased from the uncontrolled case for the

active control system presented. Further research by Breneman (1999) into active

control systems for this structure suggest that active controllers can be designed so

that peak drift demands are reduced over all stories.

Structural control systems are e�ective solutions that can improve structural per-

formance. All three control strategies investigated can signi�cantly reduce the seismic

demands on a structure, therefore reducing the expected damage to the structure. No

one system is consistently the best at all hazard levels. However, the viscous system

proves to be the most insensitive to modeling assumptions. The isolation system

can maintain the demands close to the structure's elastic limit. However, the onset

of nonlinear behavior decreases the system's e�ectiveness. The active system is also

sensitive to design assumptions, such as output parameters and structural model pa-

rameters used in design. Peak responses alone do not describe the possible damage

incurred by the structure as cumulative damage results from several incursions into

the inelastic range. Thus accurate evaluations should involve consideration of the

dissipated hysteretic energy. The use of a probabilistic format allows for a consider-

ation of structural response over a range of seismic hazards. Stable relationships can

be developed between the spectral acceleration and controlled structural demands.

Similar relationships are also possible for the demands on the control system, such

as the peak bearing displacement for the isolation system. As a result, fewer control

analyses may be required to estimate the expected structural behavior.

184


9.4 Future Work

Based on the results of the current research, several future research directions may

be identi�ed. Possible research areas include:

� Evaluate the performance of structural control on other structures, such as

taller steel moment-resisting frames, or other structural systems, such as braced

frames or concrete systems,

� Evaluate the performance of structural control on degrading structural systems,

� Evaluate the performance of other seismic control strategies, such as nonlinear

dampers and semi-active systems, and

� Developing methods to be used in the design of structural control systems using

performance-based approaches

The investigations conducted in the course of this research focused on two speci�c

steel moment-resisting frames located in the the Los Angeles region. One character-

istic of these systems is that they were both fairly exible. In order to more fully

understand the impacts of additional control systems, investigations into other struc-

tural systems need to be performed, such as braced frames and wall systems that are

signi�cantly sti�er in nature. Also, the e�ectiveness of these systems in other areas of

the country, with di�erent loading characteristics, should be addressed. For example,

the ground motions in the Boston area are characterized by their high frequency con-

tent. As a result, structural response is dominated by higher mode e�ects, in contrast

to the �rst mode response observed in the Los Angeles region.

One important issue addressed in this research was the consideration of the hys-

teretic behavior of the structures under seismic loads. Previous research into the

e�ects of structural control on seismic demands, particularly active systems, had

utilized very simpli�ed and often linear evaluation models of the structural system.

However, the structures evaluated in this research were characterized by a stable non-

linear behavior. The e�ects of degrading structural properties, whether due to local

buckling of members or connection fractures, were not considered. The performance

of control strategies under those situations need to be considered.

185


While the performance of 3 di�erent control strategies were presented, several

other structural control systems are available. These systems include nonlinear vis-

cous dampers and semi-active controllers. Current research into these systems indi-

cate that they can be highly e�ective at reducing seismic demands on an structure.

Semi-active systems are becoming increasingly attractive as they o�er the adaptabil-

ity of active control systems while requiring minimal power input.

Ultimately, one wants to be able to utilize this knowledge during the design process

of the structure and be able to account for the e�ects of the control systems during

this process. Simpli�ed design guidelines need to be developed for individual control

systems that can be veri�ed later with the more detailed nonlinear analyses.

186

Appendix A

Response Statistics

In this study, the best estimate or \central value" for a data set if referred to as the

median. This estimator is more accurately the geometric mean of the data and is the

exponential of the average of the natural logarithms of the observed values, xi, of the

sample. Mathematically this parameter can be written as follows:

x = exp

�Pni=1 lnxin

�(A.1)

where n is the number of observations.

The geometric mean is a logical estimator of the median, especially if the data are

sampled from a lognormal distribution (see Benjamin and Cornell, 1970). The seismic

response data is generally observed to have asymmetry in their histogram, displaying

a longer right-hand tail. For ground-motion estimations, a lognormal distribution is

nearly a universal choice, and Shome (1999) has argued that the nonlinear response

of a structure is also lognormally distributed. The other advantage of considering

the above estimator of the median is that the estimate is less tail-sensitive than the

estimator of the mean in the presence of \outlier values", which are data points that

are much higher or lower than the others. This characteristic of response data is very

common in nonlinear seismic response analysis.

In this study, the dispersion measure denoted by � is the standard deviation of

187

Appendix A Chapter A

the natural logarithms of the data.

� =

�Pni=1(lnxi � ln x)2

n� 1

� 1

2

(A.2)

Under the assumption of lognormally distributed data, the standard deviation of

the natural logs of the data, �lnx, and the coeÆcient of variation, �x=�x, are related

to one another such that:

�2ln x = ln

1 +

��x�x

�2!

(A.3)

For relatively small values of �ln x, e.g., 0.3 or less, �ln x is approximately equal to

the coeÆcient of variation for the data. Under the lognormal assumptions, it is

the natural dispersion measure for the data (Benjamin and Cornell 1970). The

above estimator of dispersion is almost universally used in ground-motion estimation.

Because of the advantages described, the above dispersion measure was utilized. The

term \dispersion" is loosely utilized when referring to this parameter in the text.

188

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