METHOD OF RELIABILITY-BASED CALIBRATION OF SEISMIC ... · Method of Reliability-Based Calibration...

UILU-ENG-94-2016

CIVIL ENGINEERING STUDIES STRUCTURAL RESEARCH SERIES NO. 595

ISSN: 0069-4274

METHOD OF RELIABILITY-BASED CALIBRATION OF SEISMIC STRUCTURAL DESIGN PARAMETERS

By SANG WHAN HAN AND Y. K. WEN

A Report on a Research Project Sponsored by the

". _" "- : ':-;"~:~}:~v:;-g A";.7enue G~':;.2..l"~? .C.i;:'I~QiB fJ18{)l

NATIONAL SCIENCE FOUNDATION (Under Grants BCS 91 -06390, NCEER 924001 C and 9341028)

DEPARTMENT OF CIVIL ENGINEERING UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, ILLINOIS NOVEMBER 1994

50272-101

REPORT DOCUMENTATION PAGE

4. Title and Subtitle

1. REPORT NO. 2. 3. Recipient's Accessio n No.

UILU-ENG-94-2016 5. Report Date

Method of Reliability- Based Calibration of Seismic Structural Design Parameters 6.

November 1994

7. Author(s)

S. W. Han and Y. K. Wen

9. Performing Organization Name and Address

University of Illinois at Urbana -Champaign Department of Civil Engineering 205 N. Mathews Avenue Urbana, Illinois 61801

12.Sponsoring Organization Name and Address

National Science Foundation 1800 "0" Street, NW Washigton, D.C. 20550

15. Supplementary Notes

16.Abstract (Limit: 200 words)

8. Performing Organization Report No.

SRS 595 10.Project(T"ask/Work Unit No.

11. Contract(C) or Grant(G) No.

/G)NSF BCS 91-06390 and 'NSF NCEER 924001C and 93410~1 13. Type of Report & Period Covered

14.

Structural design under seismic load requires proper consideration of uncertainties associated with loads and resistance as well as the limited ability of analytical models to describe the response of a structure. To account for these uncertainties, the design parameters in current seismic codes need to be calibrated based on required safety and satisfactory performance under future earthquakes in terms of reliability. The calibration is done by minimization of the difference between actual and target probabilities for both serviceability and ultimate limit states. The Response Surface Method (RSM) with a central composite design is used to expedite the calibration process. Since information is required on the actual probabilities of exceeding various limit state conditions for various structural configuratIOns which typically requires a large number of nonlinear time history structural response analyses, an Equivalent Nonlinear System (ENS) is used to replace the MDO F analytical model. The ENS retains the important properties of the original system, i.e., the dynamic characteristics of the first two modes, the global yield displacement and post-yielding behavior of the structure. Response scaling factors based on extensive regression analyses of structures of up to 12 stories under historical earthquakes are then applied to the responses calculated using the ENS in order to obtain responses comparable to the original structure. Numerical examples on the calibration are given, and parametric studies are carried out to show the dependence of the structural design parameters on the target reliabilities for both serviceability and ultimate limit states. The computational advantage and the accuracy of the proposed methods are also shown.

17. Document Analysis a. Descnptol"5

Equivalent Nonlinear Sy'Stem, Global Response Scaling Factor, Local Response Scaling Factor, Response Surface Method~ Central Composite Design, Reliability, Limit State Probability, Seismic Design, Simulation, Special Moment Resisting Steel Frame, Allowable. Story Drift, Load Factor

b. Identifiers/Open-Ended Terms

c. COSATI Field/Group

18.Availability Statement

Release Unlimited

(See ANSl-Z39.1B)

19.5ec1Jrity Class (This Report)

UNCLASSIFIED

20. Sec1Jrity Class (This Page)

UNCLASSIFIED

.. \ , .L

21. No. of Pages

170

22. Price

OPTIONAL FORM 2n (4-77) Department of Commerce

11

ACKNOWLEDGEMENTS

This report is based on the thesis of Dr. Sang Whan Han submitted in partial

fulfillment of the requirements for the Ph.D. degree in Civil Engineering at the

University of Illinois at Urbana-Champaign. The support of the National Science

Foundation under GrantsNSFBCS 910-06390, NSF - NCEER 924001C and 934102B

is gratefully acknowledged.

III

TABLE OF CONTENTS

CHAPTER

1 INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Objective and Scope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Organization................................................. 3

2 BACKGROUND. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

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

2.2 Reliability Evaluation of Structures .............................. 5

2.3 Calibration of Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Equivalent Systems for Evaluating the Response of Structures ....... 8

3 SEISMIC RISK ANALYSIS USING EQUIVALENT NONLINEAR SYSTEM 12

3.1 Introduction ................................................. 12

3.2 Equivalent Nonlinear System (ENS) ............................. 14

3.3 Designing Representative Structures for Calibrating Ro and RL ...... 16

3.4 Earthquakes Used for Calibrating Ra and RL . . . . . . . . . . . . . . . . . . . . . . 18

3.5 Response Scaling Factors (Ra and RL) ........................... 18

3.5.1 Global Response Scaling Factor (Ra) ....................... 19

3.5.2 Local Response Scaling Factor (RL) ........................ 20

3.6 Seismic Risk Analysis Using Equivalent System. . . . . . . . . . . . . . . . . . . . 22

3.6.1 Seismic Risk Analysis Using Equivalent Linear SDOF

System (ELSS) ......................................... 23

3.6.2 Seismic Risk Analysis Using Equivalent Nonlinear

System (ENS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.7 Evaluation of Limit State Probabilities and Verification of the

Proposed Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.8 Results and Conclusions ....................................... 31

4 SEIS~nC nESIST~~ DESIGN ACCOP~ING TO 1991 NEHPJ> PROVISIONS

4.1 Introduction ................................................ .

66

66

iv

4.2 Overview of Seismic Design Process Using the Equivalent Lateral Force

Procedure ................................................... 68

4.3 Equivalent Lateral Force Procedure (ELF)

4.3.1 Determination of Design Base Shear

69

69

4.3.2 Determination of Design Lateral Forces .................... 71

4.3.3 Ductility Requirements and P-Ll Effects . . . . . . . . . . . . . . . . . . . . 72

4.3.4 Combination of Load Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4 Design Example: Six Story Office Building ....................... 74

5 RESPONSE SURFACE METHOD FOR CALffiRATING DESIGN

P AAAMETERS .................................................... 84

5.1 Introduction ................................................. 84

5.2 Overview of Response Surface Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.3 Objective Function (Response) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4 Experimental Design Strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.5 Response Surface Fit ............. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.6 Determination of Design Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.6 Steps to Determine Design Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 NUMERI CAL EXAMPLES .. ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.1 Introduction ................................................. 95

6.2 Example I : Calibration of Load Factors for Live Load and Seismic Load

6.2.1 Objective Function ...................................... .

6.2.2 Selection of Load Factor Combinations Using Central Composite

Design ................................................ .

6.2.3 Representative Structures ................................ .

6.2.4 Evaluation of the Limit State Probabilities .................. .

96

97

97

98

98

6.2.5 Determination of Load Factors ............................ 99

6.2.6 Results................................................. 100

6.3 Example II : Calibration of Seismic Load Factor and Allowable Drift.. 102

6.3.1 Objective Function ..................................... " 102

v

6.3.2 Selection of Combinations for Seismic Load Factors and Allowable

Drifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 103

6.3.3 Representative Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 103

6.3.4 Brief Description of Latin Hypercube Sampling .............. 104

6.3.5 Evaluation of Limit State Probabilities .................... " 105

6.3.6 Determination of Seismic Load Factor and Allowable Drifts. . .. 105

6.3.7 Results................................................. 106

7 SUMMARY AND CONCLUSIONS .................................. 161

7.1 Summary.................................................... 161

7.2 Conclusions.................................................. 162

7.2 Future Research .............................................. 164

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 165

VITA .••...................•.•••.........................•...•.••....• 171

vi

LIST OF TABLES

TABLE

3.1 Properties of Representative Structures .............................. .

3.2 Ground Motion Records .......................................... .

3.3 Mean Value of Nonlinear Spectral Reduction Factor (FflG) ............. .

3.4 Nonlinear Spectral Reduction Factor (FJlG) .......................... .

3.5 Nonlinear Spectral Reduction Factor (FJlG)

3.6 Nonlinear Spectral Reduction Factor (FJlG) .......................... .

3.7 Global Limit State Probability (L.A. site) ............................ .

3.8 Local Limit State Probability (L.A. site) ............................. .

3.9 Global Limit State Probability (Imperial Valley site) ................... .

3.10Local Limit State Probability (Imperial Valley site) .................... .

4.1 Design Vertical Loads ............................................ .

4.2 Weight of the Building ............................................ .

35

35

39

40

41

42

43

43

44

45

77

77

4.3 Seismic Design Forces ............................................. 78

4.4 Member Sections of the North -South Frame. . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.5 Member Sections of the East-West Frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.6 Drift LiInit ...................................................... 79

6.1 Load Combinations Determined Using Central Composite Design ( Numerical

Example I) ...................................................... 110

6.2 Member Sizes and Properties of 1 Story SMRSF (Numerical Example I) .,. 110

6.3 Member Sizes and Properties of 2 Story SMRSF(Numerical Example I) ... 111

6.4 Limit State Probabilities of 1 Story SMRSF (Numerical Example I) ....... 111

6.5 Limit State Probabilities of 2 Story SMRSF (Numerical Example I) ....... 111

6.6 Live Load Factor (YL) and SeisIIl.ic Load Factor (YE) for (Numerical Example !)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 112

6.7 Live Load Factor (YL) and Seismic Load Factor (YE) for (Numerical Example I)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 112

vii

6.8 Central Composite Design Combinations for Load Case and Allowable Drift Limit

(Numerical Example II) .......................................... " 113

6.9 Design Variables (Numerical Example II) ...... . . . . . . . . . . . . . . . . . . . . . .. 113

6.10Combinations of Representative Values of Design Variables based on Latin

Hypercube Sampling (Numerical Example II) ......................... 114

6.11 Member Sizes and Dynamic Properties of 2 Story SMRSF (Numerical Example II)

................................ " .. . . .... . ...... .. ... . .... . . . . .. 115

6.12Member Sizes and Dynamic Properties of 3 Story SMRSF (Numerical Example II)

· ................................................................ 116


................................................................. 117


· ........................ " ..... " ... , ......................... " 118


................................................................. 119


· ............. , .......... '" .... . . . . .. . ... . ...... . ... . .. ... . . . . .. 120

6. 17Limit State Probabilities for 2 Story SMRSF (Numerical Example II) .... " 121

6. 18Lirnit State Probabilities for 3 Story SMRSF (Numerical Example II) 121

6.19Limit State Probabilities for 4 Story SMRSF (Numerical Example II) 121

6.20Limit State Probabilities for 5 Story SMRSF (Numerical Example II) 122

6.21 Limit State Probabilities for 6 Story SMRSF (Numerical Example II) 122

6.22Limit State Probabilities for 12 Story SMRSF (Numerical Example II) .... 122

6.23Weight Set Number 1 Used in Numerical Example II ................... 122

6.24Seismic Load Factor (YE) and Allowable Drift Limit (~a) for Various Target Limit

State Probabilities (Numerical Example II) . . . . . . . . . . . . . . . . . . . . . . . . . . .. 123

6.25 Seismic Load Factor (YE) and Allowable Drift Limit (~a) for Various Target Limit

State Probabilities (Numerical Example II) .......................... " 123





viii

6.28Weight Set Number 2 Used in Numerical Example II ................... 125





6.31 Seismic Load Factor (YE) and Allowable Drift Limit (~a) for Various Target Limit




ix

LIST OF FIGURES

FIGURE

3.1 Global Yield Displacement for Two Story SMRSF ..................... .

3.2 One Story SMRSF ............................................... .

3.3 Two Story SMRSF ................................................ .

3.4 Five Story SMRSF 1 .............................................. .

3.5 Five Story SMRSF 2 .............................................. .

3.6 Five Story SMRSF 3 ....... , ..................... , ................ .

3.7 Nine Story SMRSF ............................................... .

3.8 Twelve Story SMRSF ............................................. .

3.9 Global Response Scaling Factor RGFor One Story SMRSF ............. .

3.10Global Response Scaling Factor RG For Two Story SMRSF ............. .

3.11 Global Response Scaling Factor RG For Five Story SMRSF 1

3.12Global Response Scaling Factor RG For Five Story SMRSF 2

3.13Global Response Scaling Factor RG For Five Story SMRSF 3

46

46

46

47

47

47

48

48

49

49

50

50

51

3.14Global Response Scaling Factor RG For Nine Story SMRSF ............. 51

3.15 Global Response Scaling Factor RG For Twelve Story SMRSF . . . . . . . . . . . . 52

3.16Regression for Coefficient Co vs. RMP

3.17Regression for Coefficient Cl vs. RMP

52

53

3. 18Regression for Coefficient C2 vs. RMP ............................... 53

3.19Local Yield Displacement for Two Story SMRSF . . . . . . . . . . . . . . . . . . . . . . . 54

3.20Loca1 Yield Displacement for Two Story SMRSF ........... 0. . . . . . . . . . . . 54

3.21Loca1 Response Scaling Factor RL For Two Story SMRSF . . . . . . . . . . . . . . . 55

3.22Loca1 Response Scaling Factor RL For Five Story SMRSF . . . . . . . . . . . . . . . 55

3.23Local Response Scaling Factor RL For Five Story SMRSF 1

3.24Loca1 Response Scaling Factor RL For Five Story SMRSF 2

56

56

3.25Local Response Scaling Factor RL For Nine Story SMRSF 3 . . . . . . . . . . . . . 57

x

3.26Local Response Scaling Factor RL For Twelve Story SMRSF . . . . . . . . . . . . . 57

3.27Regression of RL w,t,t, RMP . . . . . . . .. . . . . . . . . . . . . .. . . . . .. . . . .. . . . . . . 58

3.28Nonlinear Spectral Reduction Factor (F) vs. Ductility for T=O.5 .......... 58

3.29Nonlinear Spectral Reduction Factor (F) vs. Ductility for T= 1.0 .......... 59

3.30Comparison of Global Limit State Probabilities from ENS and NMS (L.A. site)

· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.31 Comparison of Local Limit State Probabilities from ENS and NMS (L.A. site)

· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.32Comparison of Global Limit State Probabilities from ENS and NMS (Imperial

Valley Site) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.33 Comparison of Local Limit State Probabilities from ENS and NMS (Imperial Valley

Site) ............................................................ 61

3.34Histogram of Difference (Residue) of Global Limit State Probabilities (L.A. site)

· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.35Histogram of Difference (Residue) of Local Limit State Probabilities (L.A. site)

· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.36Histogram of Difference (Residue) of Global LLmit State Probabilities (Imperial

Valley site) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.37Histogram of Difference (Residue) of Local Limit State Probabilities (Imperial

Valley site) ...................................................... 63

3.38GlobaI Spectral Reduction Factor (F~G) vs. Ductility for 2 Story SMRSF . . . 63

3.39Loca.l Spectral Reduction Factor (FIlO) vs. Ductility for 2 Story SMRSF . . . . 64

3.40Histogram of Difference of Local Limit State Probabilities : (ELSS and NMS)

(2story SMRSF) .................................................. 64

3.41 Histogram of Difference of Local Limit State Probabilities: (ENS and NMS) (2story

SMRSF) . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . .. . . . .. . .. . . . . . . . . . . . . . . . . . 65

4.1 Seismic Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2 k vs. T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3 Force Distribution for k= 1, k=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4 Floor Plan ....................................................... 81

4.5 Evaluation and Member for the North-South SMRSF . .. . . .. .......... . 81

Xl

4.6 Evaluation and Member for the East-West SMRSF ......... . ..... ..... 82

4.7 Members for the North-South SMRSF .............................. 82

4.8 Members for the East-West SMRSF ................................ 83

5.1 Comparison of Penalty Terms Nomalized By Target Probability. . . . . . . . . . . 93

5.2 2k Factorial Design for k=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3 3k Factorial Design for k = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.4 Central Composite Design for k=3 .................................. 94

6.1 Configuration of Central Composite Design ... . . . . . . . . . . . . . . . . . . . . . . .. 127

6.2 Dimensions of Structures and Member Numbers of Structures ( 1 and 2 Story

SMRSFs) for Numerical Example I .................................. 127

6.3 Typical Simulated Earthquake Record far L.A. site. . . . . . . . . . . . . . . . . . . .. 128

6.4 Scatterness Diagram of Objective Function (Ratio of weights for service and ultimate

=1 :2) ............................................................ 129

6.5 Scatterness Diagram of Objective Function (Ratio of weights for service and ultimate

=1 :5) ............................................................ 130

6.6 Fitted 2nd Order Polynomial Representing Objective Function (Weights for

Serviceability and Ultimate Limit States = 1:5, Target Probabilities = 0.65 (service), 0.06

(ultimate) ) Numerical Example I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 131

6.7 Configuration of Central Composite Design for Seismic Load Factor (YE) and

Allowable Drift Ratio Limit (~a!hsx)(Numerical Example II) ............ 131

6.8 Dimensions and Member Numbers of 2 Story SMRSF (Numerical Example II)

· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 132

6.9 Dimensions and Member Numbers of 3 Story SMRSF (Numerical Example II)

· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 133

6.10Dimensions and Member Numbers of 4 Story SMRSF (Numerical Example II)

.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . . . .. .. . . . .. . . . . . . .. 134

6.11Dimensians and Member Numbers of 5 Story SMRSF (Numerical Example II)

· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 135


................................................................. 136


. .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. .. . . . . . . . . . . . . . . .. . . .. . . . . . . .. 137

xii

6.14Suggested Criteria for Acceptable Risk 138

6.15Scatterplots of Objective Function (Ratio of weights for service and ultimate =1:2 )

(Weights for Structures=8:8:1:1:1:1) ................................... 139











6.21 Scatterplots of Objective Function (Ratio of weights for service and ultimate =1: 15 )


6.22Scatterplots of Objective Function (Ratio of weights for service and ultimate =1: 15 )


6.23Fitted 2nd Order Polynomial Representing Objective Function (Ratio of weights for

service and ultimate =1:10, Weights for Structures=8:8:1:1:1:1, Target Probabilities:

0.30 I SCf\·1CC J. 0.060 (ultimate)) ...................................... 147

6.24Sensin\1r~ of 'rE to Ps * for Various Ratios of Weights for Service and

Cltirrl~te'\\·elfhts for Structures=8:8:1:1:1:1, Pu*= 0.06) .. , ............. '" 147

6.25 Senslt \ H! of ~.;hsx to P s * for Various Ratios of Weights for Service and Ultimate

('Velght~ L.l; Structures=8:8:1:1:1:1, Pu*= 0.06) .......................... 148

6.26Senslt\ It\ of ';'E to pu• for Various Ratios of Weights for Service and

CltirrwtC' \\C'l~hts for Structures=8:8:1:1:1:1, Ps*= 0.35) ................... 148

6.27Sensio\ l~ or ~.'hsx to Pu * for Various Ratios of Weights for Service and Ultimate

(Weight~ tor Structures=8:8:1:1:1:1, Ps*= 0.35) ........................... 149

6.28Derived \Veight Function ............................................ 149


(Weights for Structures=20:14:6:4:3:1) ................................. 150


xiii



(Weights for Structures=20: 14:6:4:3: 1) ................................. 152











6.37Fitted 2nd Order Polynomial Presenting Objective Function (Ratio of weights for service

and ultimate =1:1,Weights for Structures=20:14:6:4:3:1, Target Probabilities: 0.30

(service), 0.060 (ultimate)) .......................................... 158

6.38Sensitivity OfYE to Ps* for Various Ratios of Weights for Service and

Ultimate (Weights for Structures=20: 14:6:4:3: 1, Pu * = 0.06) ................ 158

6.39Sensitivity of ~afhsx to Ps* for Various Ratios of Weights for Service and Ultimate rUT"",; "'\..tC' ""yo C'+-r-.,,,h,yo.c.,,_'"){).' ;1.t;:.;1.'l.' D * - () ()t;:\ 1 "\Q "1'1'~15"'"" .lUI "'UU\;Llll~':)-""'U • .l"".u."",-".l, .I. S - v.VVJ •••••••••••••••••••••••• ......J./

6.40Sensitivity of YE to Pu * for Va.rious Ratios of Weights for Service and

Ultimate (Weights for Structures=20:14:6:4:3:1, Ps* = 0.35) ................ , 159

6.41Sensitivity of ~afhsx to Pu * for Various Ratios of Weights for Service and Ultimate

(Weights for Structures=20: 14:6:4:3: 1, P s * = 0.35) ........................ 160

1.1 Objective and Scope

CHAPTER 1

INTRODUCTION

The overall objective of earthquake-resistant design of building structures is to

ensure "satisfactory performance" during an earthquake. The meaning of "satisfactory

performance" can vary among different people or organizations. Most building codes

and seismic design provisions in the United States (for example, the Uniform Building

Code CUBe), the Standard Building Code, the BOCA National Building Code, the

ASCE 7-88 standards, and the NEHRP recommended provisions) have adopted

following set of performance objectives to describe satisfactory performance:

(1) The structure should resist minor earthquakes without structural or

non - structural damage,

(2) The structure should resist moderate earthquakes without structural

damage.

(3) The structure should resist severe earthquakes without collapse.

According to these performance objectives, structures designed based on the seismic

codes and standards may undergo inelastic deformation during a large severe earth

quake.

Designing a structure to achieve the above mentioned perfonnance objectives (or

any other performance objectives) is complicated by the uncertainties involved in the

design process. For example, there are large uncertainties in predicting the intensity as

well as the spatial and temporal characteristics of future earthquakes. Also, there are

2

uncertainties associated with the limited ability of analytical models to properly de

scribe the response of structures. In addition, structural capacity can not be determined

precisely because of the variation in material strength, workmanship, etc. Since earth

quake loading, structural response, and structural capacity are probabilistic in nature,

the performance of a structure is also probabilistic in nature and thus the performance

is somewhat uncertain. This uncertainty should be reflected in the design and code

procedures.

Probability-based ( reliability-based) methods can be used to account for the

uncertainties in the seismic design process. Direct use of probabilistic methods may not

be suitable for routine designs since it requires detailed reliability analyses and statisti

cal modeling of loads and structural resistance. This is too computationally intensive.

However, these methods can be used to calibrate code parameters and to develop

reliability- based design procedures. Therefore, use of reliability-based design pro

cedures and calibrated parameters should result in structural designs which have more

satisfactory performance (i.e., more consistent with the reliability levels desired by

profession ).

In this study, a method is presented by which code (design) parameters relevant

to seismic design of structures can be calibrated using probabilistic methods. In gener

al, the design parameters can be load factors, allowable drift limits, importance factor,

response modification factor, etc. The design parameters are calibrated according to

given target limit state probabilities. Limit states represent the various state of unde

sirable behavior of structures such as yielding, buckling, instability, severe displace-

ment, etc. The calibration is achieved by mininlizing an objective function which

measures the difference between "target" and "actual" probabilities of reaching vari

ous limit state conditions. The live load factor, seismic load factor, and allowable story

drift limits in the NEHRP recommended provisions are selected as the design parame-

3

ters to be calibrated.

In this study, the Equivalent Nonlinear System (ENS) is used to reduce the

computation involved in limit state probability evaluation for nonlinear multi - degree

of freedom (MDOF) structures. The use of an ENS provides an estimates of limit state

probabilities which are good approximations to those obtained from nonlinear dynam

ic analyses of MDOF structures. Also, the Response Surface Method (RSM) based on

a central composite design is used to expedite the minimization solution procedure.

1.2 Organization

Chapter 2 presents some background information on the evaluation of the reli

ability of a structure and existing procedures for calibrating design parameters based on

reliability.

Chapter 3 describes the Equivalent Nonlinear System (ENS) methodology for

evaluating the limit state probabilities of MDOF systems. To establish the validity of

the ENS methodology, the results using the ENS methodology are compared with the

results obtained from nonlinear dynamic analyses of seven MDOF structures.

Chapter 4 provides an outline of seismic design requirements presented in the

NEHRP recommended provisions (1991). One six-story Special Moment Resisting

Steel Frame (SMRSF) is designed as an example.

In Chapter 5, the Response Surface Method and the central composite design

method are discussed. These methods are used to minimize the objective function. The

rationale which leads to the selection of the proper form of the objective function is also

discussed.

Chapter 6 contains two numerical examples which demonstrates how the RSM

with central composite design can be used to calibrate design parameters. In the first

4

example, the load factors for live load and seismic load as the design parameters are

calibrated. In the second example, the seismic load factor and allowable drift limit are

considered as the design parameters since the drift limit is an dominant factor as far as

the reliability of the building is concerned. In these two examples, the "actual" limit

state probabilities of the structures are evaluated using ENS, and the design parame

ters are calibrated using RSM. In both examples, the design parameters are calibrated

for various values of the target limit state probabilities.

Chapter 7 provides a summary of the work done in this study and the significant

results and conclusions. Also, areas of future research are identified.

2.1 Introduction

5

CHAPTER 2

BACKGROUND

In this chapter, the previous research results on calibrating design parameters

and evaluating structural reliability are presented and discussed. Section 2.2 is devoted

to past research on evaluating the reliability of various types of structures. In Section

2.3, previous studies on calibrating the design parameters specified in current codes

and standards are reviewed. Section 2.4 describes various simplified analysis methods

for evaluating the response of nonlinear structural systems.

2.2 Reliability Evaluation of Structures

Given the many sources of uncertainty involved in the design of structures to

resist earthquakes, there is a need to quantify the safety (reliability) of structures

designed according to current procedures. Several researchers have evaluated the

reliability associate with current seismic design procedures as discussed below.

Shinozuka, et al. (1989) evaluated the seismic risk associated with bridges de

signed based on Allowable Stress Design (ASD) or Load and Resistance Factor Design

(LRFD). For the several bridge structures considered, they showed that the variability

of limit state probability (i.e., probability of unsatisfactory on undesirable perfor

mance) for bridges designed according to LRFD is smaller than that for bridges

designed according to ASD. Thus, the design procedure using the LRFD procedure is

more appropriate and accounts more explicitly for uncertainty. For the same reason,

many current design codes and standards (e.g., AISC LRFD manual, AISC 7-88

6

standards, etc.) have adopted the LRFD design procedure. Therefore, there is also a

need to calibrate current seismic codes in order to have consistent performance of a

structure.

Eliopoulos and Wen (1991) evaluated the seismic reliability of moment-resist

ing steel frames located near Los Angeles, California. A five story Special Moment

Resisting Steel Frame (SMRSF) was considered; this structure was designed in accor

dance with the Uniform Building Code (1988) and AISC ASD manual (1989). In their

study, seismic reliability was evaluated based on the probability of exceeding a story

drift ratio of 1.5 percent during the 50 year time window 1991 to 2041.

Wen, et al. (1992) and Foutch, et a1 (1992) considered several different types of

steel structures and evaluated the interstory drift levels corresponding to various ex

ceedence probabilities. A Special Moment Resisting Steel Frame (SMRSF), an

Ordinary Moment Resisting Steel Frame (OMRSF), a Concentrically Braced Steel

Frame (CBF), and a dual system with an Eccentrically Braced Frame (EBF) were

considered. The structures were assumed to be located at either Imperial Valley (5 km

from the Imperial fault) or Santa Monica Boulevard (60 Km from Mojave segment of

the Southern San Andreas Fault). They found that there is a variation of risks among

the different types of steel buildings designed according to UBC.

Saito and Wen (1994) performed a study similar to that by Wen etal (1992). They

eval uated the story drift ratios of structures corresponding to various exceedence prob

abilities. Seven - story and twelve - story reinforced concrete (RC) structures were

considered; the structures were assumed to be located at either Sendai or Tokyo in

Japan. The structures were designed according to the design guidelines of The Archi

tecture Institute of Japan (AIJ, 1990).

Loh et al. (1994) evaluated the story drift ratios of an eight - story steel structure

located in Taipei basin (Taiwan) corresponding to various exceedence probabilities.

7

The structure was designed based on the Taiwan code.

Reliability studies have also been performed for structures other than buildings

and bridges. For example, Bazzurro and Cornell (1993) evaluated annual probabilities

of exceeding target ductility level for a jacket type offshore platform.

2.3 Calibration of Design Parameters

The design parameters in building code are generally determined by code com

mittees and largely based on judgement and experience. They may not guarantee

satisfactory performance of a structure under all design conditions because of the

inherent large uncertainties in the loadings. There have been several studies per

formed to calibrate design parameters taking the uncertainty into consideration by

probabilistic methods.

Ellingwood et al. (1980) calibrated load factors and resistance factors using

probabilistic methods corresponding to a target member level limit state probability

(e.g., a probability of reaching the fully plastic flexural capacity for a beam). These load

and resistance factors were adopted in current design codes such as AISC LRFD

manual and ANSI 7-88. However, load and resistance factors based on the member

level limit state probability may not necessarily guarantee satisfactory structural

system performance (Kim and Wen, 1987).

Shinozuka et al. (1989) presented a method for detennining load and resistance

factors based on system level target limit state probabilities for bridges. In this proce

dure, load and resistance factors were determined by minimizing the total difference

between the target limit state probability and the actual limit state probability of a

structure. For this purpose, they proposed an objective function which measures the

difference between target and actual limit state probabilities.

8

Hwang et al. (1991) evaluated the load factor for seismic load and importance

factor according to the system level target limit state probability for collapse ( 1/1000

per YT.) using a procedure sinlilar to that used by Shinozuka et al. (1989) in which the

same objective function was also used.

In the methods used by Shinozuka et aI. and Hwang et aI., non -linear program

ming solution procedures are generally necessary which may require a large number of

iterations to reach the minimum point of the objective function and therefore may be

computationally expensive. For practical applications, a more efficient way is needed

for calibrating design parameters. In the present study, the Response Surface Method

(RSM) and the central composite design method are used for calibrating the design

parameters. Additional background information of these methods is provided in Chap

ter 5.

2.4 Equivalent Systems for Evaluating the Response of Structures

Evaluation of structural reliability under earthquake loading often requires anal

ysis of nonlinear dynamic response of the structure which is generally time consuming

and expensive. Many researchers have proposed simplified methods to estimate the

nonlinear dynamic response in lieu of performing a nonlinear dynamic analysis of a

complex structure, mostly based on the response of a single-degree-of freedom

(SDOF) oscillator. Various scaling factors are also used to account for nonlinear

behavior and/or MDOF effects. Several of these methods are discussed below.

Newmark (1973) proposed a procedure to predict the inelastic response of a

nonlinear SDOF system using the response of linear SDOF system and a deamplifica

tion factor. The deamplification factor can be considered as a scaling factor which is

applied to the maximum force in a SDOF oscillator to obtain the yield force needed to

9

achieve a desired target ductility for the nonlinear SDOF system. (Ductility, as used

here, is defined as the maximum displacement divided by the yield displacement.) The

deamplification factor was obtained based on a statistical study of responses under

excitation of historical earthquakes. Riddel and Newmark (1979) later proposed an

improved deamplification factor based on a more comprehensive statistical study.

Kennedy et aL (1984) proposed a scaling factor, F~, which is conceptually similar

to the deamplification factor. First, a particular earthquake record is scaled up or down

so as to cause incipient yielding in a SDOF oscillator. Then, the scaling factor F~ is the

amount bywhich this earthquake record must be scaled up in order to attain a specified

damage level in a nonlinear SDOF oscillator.

Using a procedure similar to that of Kennedy, et aL(1984), Swell (1989) also

evaluated the force reduction factor based on a target ductility level, a target normal

ized hysteretic energy level, a target damage ratio, and a target cumulated damage

level. This study also focused on SDOF systems. His study showed that there is no

systematic dependence of the mean value of the force reduction factor on either earth

quake magnitude or site-to-source distance. He also observed that the record to

record vana~Llr\ of this factor is small, and the dependence of this factor on the

force-deformJ::,-lr. model and damage model is also smalL

Hwang anJ Jaw (1989) evaluated the response modification factor, R, for rein

forced concre~c structures using a statistical procedure. The response modification

factor is defmcc as the ratio of the absolute value of the linear elastic base shear to the

absolute value of the nonlinear base shear of a structure subjected to the same earth

quake.

Inoue and Cornell (1990) evaluated the annual probability of yielding for three

MDOF structures based on the linear response of a SDOF system. The first structure

was a three-story building having constant stiffness along its height. The second

10

structure was a three-story building with relatively large contributions from the second

and third modes. The third structure was ten - story building with a fundamental period

of 0.9 seconds. They proposed using two scaling factors for the response of a SDOF

system in order to predict the inelastic response of a MDOF structure based on that of a

linear SDOF system. The SDOF system has a natural period equal to the fundamental

period of the MDOF structure under consideration. One factor is a force reduction

factor similar to the one proposed in the study by Swell, et al (1989). The other factor is

a MDOF response factor which accounts for the difference between the linear re

sponses of a SDOF system and a MDOF system. Thus, this factor captures the effect of

the higher modes ofa MDOFstructure. Asuite of twelve strong ground motion records

and ten ground motion records are used for calculating the MDOF response factor and

the force reduction factor, respectively.

Nassar and Krawinkler (1991) proposed a modification parameter for predicting

the inelastic strength demand of a MDOF structure using the response of a linear

SDOF system. This modification factor is a function of the ductility and fundamental

period of a structure. The functional relationship was established by a regression

analysis using 48 earthquakes records.

Qi and Moehle (1991) proposed an equivalent SDOF model for representing the

overall behavior of multi -story building structures. This study focused on the roof

level displacement response of a building.

Bazzurro and Cornell (1993) performed a study similar to that by Inoue and

Cornell (1991). They evaluated the inelastic response of a jacket type offshore platform

using a linear SDOF system and scaling factors. A suite of 15 earthquakes were used to

calculate the scaling factors.

As discussed above, the studies by Inoue and Cornell (1990), and Bazzurro and

Cornell (1993) focused on MDOF structures. It is interesting to note that they arrived

11

at the same conclusions as did Swell (1990) for SDOF systems. They observed that the

mean values of scaling factors of a MDOF system has no systematic dependence on

either magnitude or site to source distance, the record to record variability of these

factors are small, and the dependence on the force-deformation model and damage

model is also small. Bazzuro and Cornell (1991) suggested that the scaling factors,

which are actually random variables, can be treated as deterministic quantities (repre

sented by their mean values) for the purpose of evaluating annual probabilities of

exceeding a given ductility. Inoue and Cornell also suggested that the mean value of the

scaling factors can be obtained by averaging the values calculated using 5 to 7 earth

quake records.

12

CHAPTER 3

SEISMIC RISK ANALYSIS USING EQUIVALENT NONLINEAR SYSTEM

3.1 Introduction

In seismic risk analysis of buildings and structures, the computational efforts

required can become excessive since the seismic risk is usually evaluated based on

repeated nonlinear response analyses of MDOF systems which are time consuming. In

order to reduce the computational cost, many researchers tried to develop methods of

estimating the nonlinear response of a MDOF system using a simple equivalent system.

Bazzuro and Cornell (1992) , and Inoue and Cornell (1990, 1991) proposed an

equivalent linear SDOF system (ELSS) which incorporated a nonlinear spectral re

duction factor (F) and a MuOF response factor (C). This equivalent system has the

same structural period as that of the first mode of the MDOF system being considered.

The nonlinear spectral reduction factor (F) was originally developed by Kennedy, et al

(1984). Given an earthquake which causes incipient yield in a structure, the factor F is

the amount by which this earthquake must be scaled up in order to attain a specified

damage level (e.g., a ductility (J1.) of 3.). Several other researchers have also developed

nonlinear spectral reduction factors using a slightly different procedure [ for example,

Riddell and Ne\\mark (1979), Bertero (1986), Hwang and Jaw (1989), Swell (1989) and

Nassar and KrawinkJer (1991)]. Using an equivalent linear system with the nonlinear

spectral reduction factor F, nonlinear response spectral values for a MDOF system at a

given damage level can be obtained. However, this response only includes the effect of

the 1st mode of a MDOF system. In order to capture the effects of all the modes of a

13

MDOF system, the use of a MDOF response factor (C) has been proposed by Inoue

and Cornell (1990, 1991). This factor accounts for the difference between the linear

response of MDOF system and that of the BLSS. The factor C is defined as the ratio of

the maximum elastic response of the ELSS to the maximum elastic response of the

MDOF system; the maximum MDOF and ELSS responses are normalized by their

respective story drift capacities. Cornell and others (1989,1992) also found that the

mean values of the factors F and C are only slightly dependent on earthquake magni

tude and source distance. Furthermore, they observed that the record to record vari

ability is also small compared to the variabilities in spectral response quantities such as

spectral acceleration. These findings imply that a moderate sample size of earthquake

records is sufficient to evaluate these factors for any given structure, and these factors

can be treated as deterministic values. Therefore, only the mean value or median value

is needed to evaluate the risk of a structural limit state.

The analysis of the ELSS with the use of factors C and F, however, may not be

satisfactory in predicting the seismic risk associated with a nonlinear multi -degree

of - freedom system ( NMS ) if good accuracy is required. To improve the accuracy,

more of the inelastic and dynamic response characteristics of the MDOF system need

to be considered; these include the yielding displacement, the modal mass participa

tion factors and mode shapes, etc. The use of the nonlinear spectral reduction factor

(F) to scale an earthquake to attain a target damage level of different intensity is

questionable since scaling a ground motion does not account for variations in ground

motion characteristics (e.g. frequency content) which change with intensity. Further

more, the use of the MDOF response factor (C) is also questionable for NMS since it is

derived based on a linear elastic MDOF system. In risk analysis of a large number of

structures, total computational effort for evaluating the factors F and C for each

individual structure can be significant (Detail procedure is described in section 3.6.1.).

14

In a reliability-based calibration of current seismic codes and provisions, such analy

ses are required; therefore, there is still a need for an equivalent system which is simple

to use, accurately represents a NMS, and does not require excessive computational

effort in response analysis.

In this study, the concept of an Equivalent Nonlinear System (ENS) is used in

conjunction with a global response scaling factor (%) and a local response scaling

factor (RL) for evaluating the seismic risk associated with a MDOF structure. The

response of the ENS is obtained by considering the first two modes and the yield

displacement of the MDOF structure. Response scaling factors can be defined as the

correction factors needed to obtain the response of a NMS from that of an Equivalent

Nonlinear System (ENS). To include the dependence of these factors on the response

level and structural system properties into consideration, the Rc and RL are functions

of the ductili ty ratio and ratio of sum of modal mass participation factors of the first two

modes (RMP) to that of all modes of a structure. Their relationships are established by

regression analyses of dynamic responses of MDOF system versus ENS under excita

tion of actual earthquake ground motions. To establish the functional relationships,

seven structures and eighty eight real earthquake records are used.

The seismic risks of MDOF structures are evaluated using the proposed equiva

lent system with response scaling factors. The validity of this procedure is then checked

by comparison of the results obtained from analyses using the ENS and actual nonlin

ear MDOF system.

3.2 Equivalent Nonlinear System (ENS)

An ENS is defined as the system which retains the first two modes of a MDOF

structure (i.e., having the same natural periods, mass participation factors, and mode

15

shapes). The ENS has a yield displacement equal to the global yield displacement

associated with the MDOF structure. Global yield displacement is determined based

on the results of a static nonlinear push -over analysis. The vertical distribution of

lateral force used in the analysis is that used in Uniform Building Code (UBe). In the

push - over analysis the lateral forces are proportionally increased, and the displace

ment at the top of the structure is monitored. A force - deflection diagram is then

constructed as shown in Figure 3.1. The resulting nonlinear force-deflection relation

ship is approximated by a bilinear one, and the yield displacement is the displacement

corresponding to the intersection point of the two lines. DRAIN - 2DX, developed by

R. Allahabadi and G.R. Powell (1988) for the analysis of inelastic two dimensional

structures under static and dynamic loadings, was used to perform this analysis. Figure

3.1 also shows how the global yield displacements were determined for a structure.

U sing the first two modes of a MDOF structure, two equations of motion, which

allow for inelastic responses, can be established. Both modes have the same global

yield displacement and restoring force model (e.g., elasto-plastic model). An analyti

cal restoring force model is determined based on the type of structural systems, the type

of material. etc. For example, an elasto-plastic or bi -linear model is often used to

represent the restoring force characteristics of steel frames (Wakabayashi, 1986).

Given an earthquake time history, the displacement is calculated from the equation of

motion of each mode at each time step. The displacements of two modes are then

combinec .J)' modal superposition. Although modal superposition is only valid for

linear system, it can be used to approximate the displacement ofNMS. The accuracy of

this procedure is discussed in section 3.8. To evaluate the maximum global displace

ment UE using ENS, the following procedure is used~

1. Design the MDOF structure based on a seismic code (e.g., UBC)

16

2. Calculate the natural periods ( TI , T2 ), mass participation

factors (rI , r2), and the normalized mode shapes ( <PI , ¢2) of

the first two modes. Note that the mode shapes are normalized

so that ¢iM<Pi = 1. (M is the mass matrix.)

3. Find the global yield displacement ( uy ) of the MDOF system

using static nonlinear push over analysis

4. Evaluate the displacement of the first two modes of the MDOF

system using the following nonlinear dynamic equation:

i = 1,2

where a is the strain hardening ratio, c; is the viscous damping

ratio, and ug is the ground acceleration. (note that each mode

has same global yield displacement.)

5. Evaluate the maximum global displacement by combining re

sponses obtained from solving equation (3.1) as follows:

2

U E = max(L <PinX qi(t) ) i= 1

where ¢in is the element of the mode shape vector correspond

ing to the horizontal displacement of the top floor in the ith

mode. [Note: Since axial deformation of the beams was not

considered, all nodes at the top floor have the same horizontal

displacement]

3.3 Designing Representative Structures for Calibrating R<; and IlL

(3.1)

(3.2)

As mentioned earlier, earthquake resistant design in current seismic codes at-

17

tempts to achieve three performance objectives; (1) resist minor earthquakes without

structural or non structural damage, (2) resist moderate earthquakes wi thout structural

damage and (3) resist severe earthquakes without collapse. To achieve those objec

tives, UBC (1988) defines a design earthquake with a ten percent probability of

exceedence in 50 years and uses an equivalent static lateral load design procedure to

design a structure for earthquakes. In UBC, the Rw factor is adopted to reduce the

design base shear in order to account for inelastic behavior of a structure.

For this study, seven typical office buildings are designed as representative struc

tures. These structures are assumed to be located near Imperial Valley or on Santa

Monica Boulevard in L.A. in Southern California. Both sites are classified as the

highest seismic zone in UBC (Zone 4 ). The ranges of the dimensions of a structure are

limited to one to four bays and one to twelve stories. The design seismic force is

determined in accordance with the provisions of UBC. This study concentrates on

buildings having lateral resistance provided by special moment resisting steel frames

(SMRSF) located on the perimeter of a structure. These SMRSFs are designed accord

ingto UBCand theAISCAllowable Stress Design manual using IGRESS-2 (comput

er software for analysis and design developed at University of Illinois at Urbana

Champaign). Figure 3.2 to 3.8 show the perimeter frames for the representative

structures. The properties of representative structures are shown in Table 3.1

In the dynamic analysis of each of these structures, two important parameters

which must be specified are the damping ratio and strain hardening ratio. Five percent

damping is used in all dynamic analysis. This appears to be consistent with the damping

levels referenced in current seismic codes and provisions for buildings (SEAOC, UBC,

NEHRP). In the dynamic analysis involving the modeling of individual members of the

frames, a strain hardening ratio of 5 % is assumed. However, in the ENS analyses, a

strain hardening ratio of 10 % is used to represent the strain hardening effects on a

18

global, or system level. Osteraas and Krawinkler (1990) note that a strain hardening

ratio of 10 % for a structural system may be on the conservative side for many struc

tures.

3.4 Earthquakes Used for Calibrating % and ~

A suite of eighty-eight real earthquake records are used in calibrating Rc and

RL. Eighty two of the records were recorded in North America, and six earthquakes

were recorded in Japan. Among the earthquake records from North America, 10

records are obtained from North Ridge earthquake ( California Department of Con

servation, 1994) and 66 earthquake records are obtained from the U.S. Geological

Survey digital data series, DDS-7, CD-Rom (1992). This USGS data base provides

uncorrected accelograms of earthquakes which occurred in North America and Hawaii

from 1933 to 1986. Basic Strong Motion Accelogram Processing Software (BAP) is

used for correcting the earthquake records extracted from the CD- Rom. The correc

tion procedures were similar to those outlined in the report by Naeim and Anderson

(1993). The magnitudes of the earthquakes range from 4.4to 8.1, and the peak ground

acceleration range from 0.03 g to 1.17g. The source distances range from 0 Km to 400

Km. Table 3.2 shows the pertinent information on the earthquakes.

3.5 Response Scaling Factors (% and ~ )

The response scaling factors can be considered as correction factors to be applied

to the response of the ENS to obtain the comparable response of a NMS. In a reliabil

ity- based code calibration, many prototype structures need to be considered when

evaluating the risk associated with current seismic code designs or calibrating the

coefficients in these codes based on reliability. In order to reduce this computational

19

burden, this study aims at establishing a functional form of the response scaling factors

based on regression analysis; the response scaling factors are assumed to be functions

of ductility and RMP. Seven (7) representative structures and eighty eight (88) real

earthquake records mentioned in sections 3.3 and 3.4 are used for this purpose. Once

the functional form of the factors RL and Rc are established, the response of a NMS

can be evaluated without performing a nonlinear dynamic response analysis of a

MDOF system.

3.5.1 Global Response Scaling Factor (Rc )

The global response scaling factor (Rc ) is defined as the ratio of the maximum

displacement of ENS ( UE ) to the maximum global displacement ( at the top) of a

MDOF structure ( UG) :

UE RG = U

G (3.3)

Figures 3.9-3.15 show Rc vs. ductility (PE=UEllly ) for ENS corresponding to

each representative structure. In each figure, there are eighty eight Rc versus fJ-E data

points for each representative structure since eighty eight different earthquakes are

used. Hence, 616 Rc versus fJ-E data points are used for establishing the functional

form of the global response scaling factor, Re. As shown in Figures 3.9-3.15, for a

given value of ductility (;lE ), there appears to be only a small amount of scatter.

However, considering the entire range of ductility values, there is significant scatter.

This suggests that Rc is a function of ductility (PE ). A two stage regression analysis is

carried out in 2D domain. In the first stage, the function for Rc vs. ductility (PE) is

regressed for discrete values of RMPs (Seven discrete points for seven representative

structures), and then the effect of the RMP is evaluated at the second stage. The

dependence of % on fJ-E is modeled by second order polynomial as follows:

20

(3.4)

in which coefficients Co, C1 and C2 are functions of the RMP to be determined by

linear regression analyses. The results of the two-stage regression analyses are:

Co = 0.9695 + 0.0178T (3.5)

C l = - 0.1664 + 0.2016T (3.6)

C2 - 0.1473 - O.1467T (3.7)

T = ITll + 1T21 #nodes

(3.8)

I lTil i= 1

where T represents the R?\1P. For the buildings considered in this study, RMP de-

creases as the number of stories of a structure increases since higher mode effects

become more significant for taller buildings. Ii is the mass participation factor of the ith

mode ofa structure. Figures 3.16-3.18 show the comparison of the data points with the

regression equations 3.4. 3.5, 3.6, and 3.7. In Figures 3.9-3.15, the solid line represents

the values from the refressed function of global response scaling factor (Eq.(3.4»). For

ductilities (u£) less than 0.5, the coefficient of variation (COV) of the factor Rc is

small; hence. the \,anability is neglected as an approximation. For ductility values

larger than 0.5. CO\' of ~ for 1, 2, 5, 9, and 12 story structures are evaluated, which

are 5%, 6C:;C. 10,(. 11 r-; . and 10% respectively. The variability in Rc will be accounted

for in evaiuatrnf the ilInl! state probabilities (Sections 3.7 and 3.8 ).

3.5.2 Locai Respon~f' Scaiing Factor (~ )

In addition to achIeve equivalence in global response, the local response which is

closely related to structural and non - structural damage is also important and needs to

be considered. The local limit state corresponding to the exceedence of an interstory

drift threshold of a structure is used for this purpose. Local limit states are more likely

21

to occur than global limit states, since damage does not spread throughout the entire

structure during an earthquake. Certain stories may be damaged more than others

even if a structure is well designed according to current seismic codes. This phenomena

is accounted for by the local response scaling factor,RL. The factor RL is the ratio of the

global ductility (PG) to maximum local ductility (PLd. Local ductility (;.iLi) is defined as

the ratio of the maximum inter story drift of itb floor to the story yield displacement.

The story yield displacement of ith floor (local yield displacement) is obtained by the

same procedure as used for the global yield displacement CUy) described in Section 3.2.

Figures 3.19 and 3.20 illustrate how the local yield displacement (Uyi) is determined for

a two story building. The factor RL can be written as follows:

(3.9)

where VLi is the maximum inter story drift and Uyi is the yield displacement of the ith

story (local yield displacement). VG is the maximum displacement at the top of the

structure and lly is global yield displacement.

Figures 3.21-3.26 show RL vs. global ductility (flG = VGllly ) for each of the 6

representative structures subjected to the 88 earthquakes. (For one story SMRSF, RL

factor is not needed.) Hence, for each representative structure, there are eighty-eight

data points. For establishing the functional form of the local response scaling factor,

RL by regression analysis, 616 data points were evaluated. As shown in Figures

3.21-3.26, RL does not appear to be strongly dependent on ductility (PG). However,

Figure. 3.27 indicates that RL does vary with respect to RMP The following regressed

function for the local response scaling factor is obtained bv a two step regression .... '-" <II.&. t.....I

analysis:

R L = 0.3627 + O.4774T (3.10)

22

where r represents the RMP In Figure 3.27, the solid line represents the values from

the regressed function of RMP (Eq.(3.10». The variability of RL is neglected.

3.6 Seismic Risk Analysis Using Equivalent System

The seismic performance of a structure can be measured in terms of a limit state

probability. Limit states represent the various states of undesirable behavior of struc-

tures such as yielding, excessive deflection, instability, buckling, severe damage, etc. In

this study, the limit states are defined in terms of interstory or total drift. It is reason

able to define the limit state using a drift limit, since SMRSFs designed according to

current seismic codes are usually governed by lateral stiffness requirements. In the

Uniform Building Code (UBC-1991), the maximum inters tory drift calculated

(3Rw/8) <\ must be less than 1.5 percent of the story height. (6x is the elastic drift

computed by the equivalent lateral force method in UBC) The elastic drift limit is also

limited to 0.04/Rw or 0.005 times the story height, whichever is smaller. The global limit

state probability can be represented as follows:

(3.11 )

where xUc

represents displacement threshold at the top floor. In Eq.(3.11), a global

displacement threshold, xUc

' can be expressed as the value of the global ductility

threshold (xJ1.c ) multiplied by global yield displacement (~). Thus, Eq.(3.11) can be

written as follows:

Uc = P(U

y > xJ1.) (3.12)

= P(;.1c > xJ1.c) (3.13)

The local limit state probability can be represented as follows:

23

PL = P(max/UL) > XU) (3.14)

= p(max{~~:) > x~) (3.15)

= P(max/Ji, Li) > xfi-) (3.16)

where PG is the globa1limit state probability and PL is the local limit state probability,

Xu L is the interstory drift threshold and Xfi-L is a local (interstory) ductility threshold.

UG is the maximum displacement at the top floor, and ULi is the maximum interstory

drift due to a given earthquake obtained by a nonlinear dynamic analysis of a MDOF

system. To avoid extensive computation, the limit state probabilities defined by

Eqs.(3.11) and Eq.(3.14) are calculated using either the Equivalent Linear System

(ELSS) with the factors F and C or the Equivalent Nonlinear System (ENS) with a

global response scaling factor (Re) and a local response scaling factor (RL) as de

scribed below.

3.6.1 Seismic Risk Analysis Using Equivalent Linear SDOF System (ELSS)

The procedures using the ELSS developed by Cornell and others (1990,1992) for

evaluating the sClsmic risk associated with a structure are as follows

(3.17)

s P L = PI rr...Ll, v~ L I > xfi-) = P(E(C)E(F fi-) > Sayld ) (3.18)

In these equations. 5,;; 1S the linear spectral acceleration of the first mode of a MDOF

structure. Slfild IS the yield level spectral acceleration of the MDOF structure. E(.)

denotes the mean value.

The factor F JiG is the global nonlinear spectral reduction factor which is defined

for a given global ductility (PG) as the scaling factor for ground motion necessary to

24

achieve a displacement at the top of the structure which is jiG times larger than the

displacement experienced by the top of the structure when incipient yield occurs

anywhere in the structure. Similarly, FJiL is the local nonlinear spectral reduction .

factor which is defined for a given local ductility threshold (;iL) as the scaling factor for

ground motion necessary to achieve the local ductility JiLi in at least one element of the

structure (Bazzurro and Cornell, 1992). Hence, the spectral acceleration at a certain

damage level (e.g., jiG =4), Sa DMO-'G =4), can be evaluated as follows:

(3.19)

The MDOF response factor (C) in Eg.(3.17) and (3.18) can be evaluated using

following equations:

DIsdoj C=-~

DI rruioj

(IVilmax)

DI mdof = maxi V. L,e

DI = IVIl max sdof V

I.e

(3.20)

(3.21)

(3.22)

where I Ch Imax is the maximum value oftheith story drift from linear analysis, and Ui,c is

the linear story drift capacity. In Eg. (3.21) and (3.22), the maximum story drift

(I Ch I max ) can be easily calculated by a response spectrum analysis as follows (Chopra,

1980) :

N

Ln = LmjPj,n j=l

(3.23)

(3.24)

25

N

Mn = Im/p},n j=l (3.25)

where </>~n is the normalized displacement of nth mode shape of i th floor, fni is the

lumped mass at the ith floor, and Sdn is the spectral displacement of the nth mode.

However, as mentioned earlier ( Chapter 3.1 ), using ELSS for evaluating the

seismic risk associated with a structure may not be satisfactory because of following

reasons:

1) The MDOF response factor, C, is derived based on a linear elastic

MDOF system, thus it may not be appropriate for a nonlinear

MDOF system (NMS).

2) Scaling a ground motion for estimating a nonlinear spectral reduction

factor to attain a certain target damage level does not reflect the

changes in true ground motions of different intensities. (e.g., fre-

quency content).

In order to test the accuracy ofELSS, a nonlinear SDOF system and the corresponding

ELSS are compared. For this purpose, Eq.(3.17) can be written as follows:

P G = P(j.lG > x/-l)

v = P(--li > x ) V /-lc y

V = P(-.!1 > V ) xf.J.c y

In the ELSS mentioned Eq.3.28 is replaced by

_ _. s;~ _. ? G = P(E(F f.J.~)E(C) > SaYid )

(3.26)

(3.27)

(3.28)

(3.29)

where UG is the maximum displacement of SDOF system under consideration. E( C) is

26

one for SDOF system. Divide both sides in Eq.(3.29) by square of the natural frequency

(()2 ) :

(3.30)

Sd = P(E(F

fLC) > Uy ) (3.31 )

From Eq.(3.28) and (3.31), it is seen that the following approximation is used:

UG Sd xfLc = E(F fLc)

(3.32)

Two nonlinear SDOF systems with fundamental periods of 0.5 and 1.0 are used to

examine the accuracy of Eg.(3.32). Global ductility threshold, xuc ,are assumed to be

2,3, and 4. The value of Ff-lG corresponding to each ductility are shown in Table 3.3. In

order to evaluate the mean value and standard deviation of Ffle shown in Figures 3.28

and 3.29, nine earthquake records are used (Numbers 4,8, 12,23,24,25,29, 72, and 73

in Table 3.2). The maximum response (Ue) is calculated by a nonlinear dynamic

response analysis. Elasto -plastic model is used to represent the behavior of restoring

forces. Tables 3.4 to 3.6 show the deviation of above two terms in Eg. (3.32) for 14

earthquakes ( no. 38 to 43 and 79 to 86 in Table 3.2) which is calculated as follows:

. . ( Sd/FfLc - Umax/IG(p.) ) devzanon( Cjc) = U / I . 100

max G(p.) (3.33)

Mean deviation for each SDOF system is calculated using the deviations for 14 earth

quakes. For the system with the fundamental period of 0.5, the mean deviations are

-32 %, -29 %,and -25% for ductility thresholds 2, 3, and 4. Similarly, for the system

with the fundamental period of 1.0, the mean deviations are -34.84 %, - 28 %, and

-14 % for ductility thresholds 2, 3, and 4. These deviations may not be negligible when

the ELSS is used to evaluate the seismic risk associated with a structure. In section 3.8,

27

the seismic risks associated with a 2 story SMRSF evaluated using the ELSS are also

compared with the risks using nonlinear MDOF system (NMS).

3.6.2 Seismic Risk Analysis Using Equivalent Nonlinear System (ENS)

The equation for evaluating the global limit state probability using the ENS

proposed in this study is as follows:

UE Pc = P(- > xu) Rc G

(3.34)

For evaluating the local limit state probabilities using the ENS, Eq.(3.14) is divided

by Uyi as follows:

(3.35)

Using Eq.(3.9),

(3.36)

Using Eq.(3.3),

(3.37)

Then, multiply both sides by Uy,

(3.38)

Since,

xu=xu.(Uy)

G L U . yz (3.39)

the equation for evaluating the local limit state probabilities can be approximated as:

28

UE PL :=:::: peR R > Xu ) G L G

(3.40)

In the above equations, UE is the maximum displacement of the ENS. Rc is the global

response scaling factor to account for the difference between the maximum displace

ment of the equivalent system (ENS) and that of the MDOF structure. Similarly, RL is

the local response scaling factor applied to the global response of the MDOF structure

to obtain the maximum local (interstory) drift response.

3.7 Evaluation of Limit State Probabilities and Verification of the Proposed

Method

Limit state probabilities of the representative structures located at either Santa

Monica Boulevard site in L.A. or at the Imperial Valley site are evaluated for a time

window of 1994-2044 by the proposed method and compared with the results of

analyses of MDOF systems (NMS). The Santa Monica Boulevard site in Los Angeles is

60 Km from the Mojave segment of the Southern San Andreas fault, and Imperial

Valley site is 5 Km from the Imperial fault. For evaluating the limit state probabilities of

a structure at each site, ground motions are simulated. The method of simulation is

outlined in the following.

The potential earthquakes that present a threat to these sites can be modeled as

either characteristic earthquakes or non-characteristic earthquakes. Characteristic

earthquakes represent the earthquakes which occur along a fault. These earthquakes

have a relatively regular recurrence period and a narrow range of magnitudes. The

probability of occurrence of a characteristic earthquake depend on the elapsed time

since the last earthquake. Thus time dependence is typically modelled as a renewal

process with lognomally distributed recurrence time. Non -characteristic earthquakes

are minor local events which are randomly distributed in time ( i.e., oCCllrrences are

29

independent of the time since the last earthquake ). Hence, the occurrence of non

characteristic earthquakes is modeled as a Poission process [ Eliopoulos and Wen

(1991)]. Besides the recurrence time, the major parameters of the characteristic

earthquakes for seismic risk analysis are magnitude (M), epicentral distance to the site

(R), and attenuation function. For non -characteristic earthquakes, the major param

eters are local intensity (MMI) and duration (td). Using these parameters, future

earthquakes in these sites can be simulated. The parameters of ground motion model

used for the two sites are presented in the study by Eliopoulos and Wen (1991). The

synthetic accelograms are modeled as a sample function of a non stationary random

process exhibiting frequency modulation function, intensity function and power spec

tral density function. Details of the ground motion model are described in the study by

Yeh and Wen (1989) and Eliopoulos and Wen (1991).

For the Imperial Valley site, two sets of ground motions are generated. One set is

generated using parameters determined from the differential array seismic station

records obtained during the 1979 Imperial Valley Earthquake. This set contains the

effect of the fault rupture propagating toward the site. The other set is generated based

on parameters obtained from the 1940 El Centro earthquake record. This latter set has

the opposite directivity. For the L.A. site, seismic hazard due to both characteristic

earthquakes and non -characteristic earthquakes is investigated: hence, two sets of

ground motions are generated for this site. Since each site has two different sources of

ground motions, the contributions to the overall lifetime limit state probability from

these two different sources are treated separately. Hence, Eq.3.11 and 3.14 need to be

modified. Global limit state probability can be calculated as follows:

Pc = 1- (l-Pg1 )(1-Pg2)

For LA site:

(3.41 )

P gl = P(U G > xu)Ch)P(Ch)

P g2 = P(U G > xu)Nch)P(Nch)

For Imperial Valley site:

PI = P(U G > xucICht)P(Cht)

P 2 = P(UG > xUcICha)P(Cha)

30

(3.42)

(3.43)

(3.44)

(3.45)

where UG is the maximum displacement at the top of the structure and xUc

. This

displacement is evaluated using response analysis of the NMS or analysis of the ENS

usingtbefactorsRc andRL. The terms Ch, Nch, Chr andCha represent the occurrence

of the characteristic earthquake, non-characteristic earthquake, characteristic earth

quake toward the Imperial Valley site, and characteristic earthquake away from this

site, respectively. Local limit state probability can be calculated as follows:

For LA site:

Pll = P(maxi(UU ) > XUL1Ch)P(Ch)

P l2 = P(maxi(UL ) > xuLlNch)P(Nch)

For Imperial Valley site:

Pll = P(maxi(UU ) > XUL1Cht)P(Chc)

P l2 = P(maxi(U L) > Xu LICha)P(Cha)

(3.46)

(3.47)

(3.48)

(3.49)

(3.50)

where i is the story number in a structure. ULi can be evaluated using the analysis of a

NMS or an ENS with the factors RL and Re. Xu is story drift threshold. Additional L

detail of the above equations (Eq. 3.46 to 3.50 ) are given by Eliopoulos and Wen

(1991 ).

31

3.8 Results and Conclusions

The limit state probabilities of the structures located at both the L.A. site and the

Imperial Valley site are evaluated based on the response from analyses of NMS, ELSS

with the factors F and C, and ENS with the factors Ra and RL. To perform these

analyses, 50 non -characteristic and 20 characteristic earthquakes are generated for

the L.A. site, and 20 characteristic earthquakes away from the site and 50 characteris

tic earthquake toward the site are generated for the Imperial Valley site. Hence, there

are 70 maximum responses for each structure at each site. Based on these maximum

responses, the limit state probabilities are evaluated by Eq. (3.41) and Eq. (3.46).

Tables 3.7 and 3.8 show the global and local limit state probabilities of the structures for

the L.A. site using NMS and ENS. Tables 3.9 and 3.10 also show the global and local

limit state probabilities of the structures for the Imperial Valley site. Six different

global1imit states and six different local limit states are considered. The global limit

states for the displacement at the top of the structure are set at 0.5 %, 1.0 %, 1.5 %, 2.0

%, 2.5 % and 3.0 % of the height of the structure. Similarly, the local limit states for

interstory drift are set at 0.5 %, 1.0 %, 1.5 %,2.0 %,2.5 % and 3.0 % of the story height.

In Figures 3.30 and 3.31, limit state probabilities for the L.A. site obtained using NMS

and ENS are are compared for global and local limit states. Similarly, Figures 3.32 and

3.33 show the comparisons for the Imperial Valley site. The histograms in Figures 3.34

and 3.35 show the deviations of the limit state probabilities of ENS from those ofNMS.

Similarly, Figures 3.36 to 3.37 show the deviation of limit state probabilities of ENS

from !'-.J1v1S for the Imperial Valley site. The discrepancy berNeen the rNO sets of limit

state probabilities from ENS and NMS for the Imperial Valley site is lager than that for

the L.A. site. This result may be attributed to large inelastic behavior of the structure

at the Imperial Valley site, since the randomness in % and RL increase with the

32

inelastic deformation of the structure (Figures 3.9 to 3.15). The results are also shown

in Tables 3.7 to 3.10. In the view of the generally good comparison, it is concluded that

the limit state probabilities based on ENS with % and RL can be used as an approxi

mation to those of nonlinear multi-degree of freedom systems.

The global and local limit state probabilities of a 2 story SMRSF using ELSS with

the factors F and C are also calculated and compared to those using NMS. Using the

equation presented by Inoue and Cornell (1991), the C factor for the 2 story SMRSF is

0.9. Figures 3.38 and 3.39 show the global and local nonlinear spectral reduction

factors vs. ductility for the 2 story SMRSF. Nine different earthquake records (earth

quake records 4, 8, 12,23,24,25,29, 72, and 73 in Table 3.2) are used for evaluating the

mean value and the coefficient of variation of Ff-lG and FLi as shown in Figures 3.38 and

3.39. Figure 3.40 shows the histogram of the deviation of the limit state probabilities of

ELSS from the ~~1S. This figure also show that the limit state probabilities obtained

using ELSS is biased. The corresponding deviation of the ENS system limit state

probability from those of the NMS is shown in Figure 3.41. It is seen that the accuracy of

the ENS method lS ~atlsfactory.

It is noted that only regular SMRSF are considered in this study. The methodolo

gy may be extcndcJ ttl other structural systems, such as Ordinary Moment Resisting

Frame (O~fRFJ C uo('cntric Braced Frame (CBF), and Eccentric Braced Fame (EBF),

etc. Also. E~S mJ\ nc-cd to be modified when it is applied to irregular structures.

Notation

C

ELSS

ENS

F

FJ-lG

FJ-lL

MDOF

NMS

PG

PL

RMP(T)

Rc

RL

Sa

Sayld

Sdi

SaD/v!

SDOF

UE

33

MDOF response factor

Equivalent Linear System

Equivalent Nonlinear System

Nonlinear Spectral reduction factor

Global nonlinear spectral reduction factor

Local nonlinear spectral reduction factor

Global displacement threshold-

Interstory drift threshold

Global ductility threshold

Local ductility threshold

M uti - Degree Of Freedom

Nonlinear MDOF system

Global limit state probability

Local limit state probability

ratio of sum of modal mass participation factors of the first two

modes to that of all modes of a structure

Global response scaling factor

Local response scaling factor

Linear spectral acceleration

Yield level spectral acceleration

Spectral displacement of ith mode

Epectral acceleration at a certain damage level (e.g., Jl=4)

Single Degree of Freedom

Maximum displacement of ENS

Uc

ULi

U~c

I Vi Imax

Ii

flG

flLi

flE

(Pi

¢i,n

34

Maximum global displacement of a MDOF system

Maximum interstory drift of ith floor

Linear story drift capacity

Maximum linear value of ith story drift

Lumped mass of ith floor

Global yield displacement

Yield displacement of ith floor

Natural period of ith mode

Natural frequency of ith mode

Mass participation factor of ith mode

Global ductility ratio

Local ductility ratio

Ductility ratio of ENS

Normalized mode shape of ith mode

Nomalized displacement of nth mode shape of i th floor

35

Table 3.1 Properties of Representative Structures

n Structure T1 T2 <PIn <P2n f1 f2 RMP ~% Uyin hft 1 1 story SF 0.619 0.059 1.13 0.00 0.88 0.00 1.00 5 1.3 15

2 2 story SF 0.914 0.355 -0.74 0.45 -1.66 -0.51 0.99 5 2.7 28

3 5 story SF1 1.550 0.557 -0.42 -0.42 -3.26 1.31 0.71 5 6.4 67

4 5 story SF2 1.540 0.558 -0.43 0.42 -3.25 -1.32 0.69 5 6.6 67

5 5 story SF3 1.560 0.526 -0.43 0.42 -3.29 -1.31 0.68 5 6.8 67

6 9 story SF 2.330 0.831 0.46 0.46 -3.39 -1.29 0.61 5 9.1 119

7 12 story SF 2.700 0.962 0.39 0.39 -3.91 -1.53 0.58 5 12.4 158

Table 3.2 Ground Motion Records

Earth I rnmn I PGA (j I ni~t , I .... - ~ .. - .. ......, .. ~ "-'~ ••• .t" ~ - -0 _ ... -..... ..... - . ~

quake 1 1920 Hollister Ho llister Diff Array3 255 0.10 15 5.5 2 1935 Helena Caroll College 270 0.17 2 6.0

Montana 3 1938 N\V Ferndale City Hall 045 0.15 51 5.6 _ ,or

L.allIOrnla 4 1940 Elcentro Imperial Valley, Ir- SOOE 0.35 8 6.7

rigation District, CA

5 1941 Santa Bar- San ta Barbara Court 135 0.18 13 5.9 hara House

6 19~1 Santa Bar- San ta Barbara Court 045 0.24 13 5.9 bara House

7 1949 \\cstern Seattle Dist Engr. off. 182 0.07 36 7.1 \\:~shmg-

ton

8 1952 T.itt Taft,Lincoln School S69E 0.17 56 7.1 Kern County, CA

9 1952 Kerr; Santa Barbara Court 132 0.13 85 7.2 Count\· House

10 1954 Nortbern Ferndale City Hall 044 0.16 26 6.5 Califorinia

11 1961 Hollister Hollister public li- 271 0.09 21 5.5 brary

12 1966 Parkfield Cholame Array 2. N65E 0.49 0.1 5.6 Parkfld,CA

36

N yr. Earth- Station Comp. PGAg Dist. Mag. quake

13 1966 Mexico Infiemillo N-120 248 0.09 120 5.6

14 1971 San Fer- 3440 University Ave. 029 0.06 44 6.6 nando bsmt

15 1971 San Fer- 4867 Sunset blvd. 269 0.16 3'/ 6.6 nando bsmt

16 1971 San Fer- 3470 Wilshire Blvd. 000 0.14 39 6.6 nando subbsmt

17 1971 San Fer- Wilshire Blvd, bsmt 180 0.13 37 6.4 nando

18 1971 San Fer- 5260 Centuary Blvd. 000 0.06 52 6.4 nando 1st floor

19 1971 San Fer- 750 Garland Ave. 210 0.22 41 6.6 nand

20 1971 San Fer- 750 Garland Ave. 300 0.16 41 6.6 nand

21 1971 San Fer- Ventura Blvd.750 011 0.23 28 6.6 nand

22 1971 Andrean Adak, Alaska, U.S. 000 0.09 70 7.1 of islands Naval Base

23 1971 Castiac Castiac Old R - idge N21E 0.32 21 6.6 Route,CA

24 1971 Pacoima Pacoima Dam, San S16E 1.17 3 6.4 Fernando,CA

25 1971 Ventura 14724 Ventura Bl- N78W 0.2 28 6.6 vd, 1st fl., LA. CA

26 1972 Alaska SMO sitka Obsevato- 180 0.08 86 7.5 ry

27 1972 Managua, Managua, ESSO 090 0.37 37 6.2 Nicaragua

28 1972 Central Melendy Ranch 331 0.53 17 4.8 California

29 1972 Melendy Melendy Ranch N29W 0.52 Bam,BearVal.,CA

30 1973 Bear Bear Valley Station 4 130 0.05 23 4.5 Valley

31 1973 Managua, Managua, National 270 0.55 60 6.0 Nicaragua Univ.

37

N yr. Earth- Station Compo PGAg Dist. Mag. quake

32 1973 Mexico Pajaritos 000 0.06 240 6.8

33 1975 N. Califor- Petrolia, General 075 0.17 16 4.4 nia Store

34 1975 Ferndale City Hall 224 0.18 50 5.7

35 1975 N. Califor- Petrolia, General 336 0.03 16 4.4 nia Store

36 1975 Oroville Medical Center 336 0.27 0 4.7 Aftershock

37 1975 island of Hila University of Ha- 074 0.10 43 7.2 Hawaii waii

38 1978 Miyagiken Shiogama Habor, Ja- EW 0.28 110 7.4 Oki pan

39 1978 Miyagiken Shiogama Habor, Ja- NS 0.27 110 7.4 Oki pan

40 1978 Miyagiken Tarumizu Dam, Ja- N35E 0.24 120 7.4 Oki pan

41 1978 Miyagiken Tarumizu Dam, Ja- 0.19 120 7.4 Oki pan

42 1978 Miyagiken Touboku Univ. Ja- EW 0.21 110 7.4 Oki pan

43 1978 Miyagiken Touhoku Univ. Ja- NS 0.26 110 7.4 Oki pan

44 1979 Imperial Elcentro Array #2 230 OA2 31 6.6 Valley

45 1979 Imperial Elcentro Array #3 230 0.24 29 6.6 Valley


47 1979 Imperial Elcentro Array #5 230 OAO 27 6.6 Valley

48 1979 Imperial Holtville post office 310 0.23 20 6.6 Valley


50 1979 Southern Icy Bay 180 0.17 75 7.3 Alaska

38


51 1979 Mexico Villita Base Cortina 340 0.14 101 7.6

52 1979 Coyote SJB Overpass Bent 337 0.11 28 5.7 Lake 3.g.1

53 1980 Imperial Brawley Airport 315 0.23 42 6.6 Valley CA

54 1980 Mammoth Mammoth H.S. Gym. 254 0.33 10.8 6.5 Lake g.f. middle

55 1980 Mammoth Green Cburch(temp) 146 0.15 . 6.3 Lake

56 1980 Mammoth Long Valley Dam 000 0.10 13 5.5 Lake

57 1980 Victoria Cucapab 085 0.08 42 6.1

58 1981 Santa Bar- Salton sea 315 0.19 9 5.6 bara

59 1981 Westmore- Superstition Mt. 135 0.11 24 5.6 land

60 1983 Hawaii Kona Hospital 346 0.10 46 6.6

61 1983 Coalinga Slack Canyon 045 0.17 34 6.7

62 1983 Trinidad Pro Dell Overpass, 000 0.15 88 5.5 WGround

63 1983 Mexico Guacmayas, Mich 270 0.12 45 5.3

64 1983 Coalinga, Cantua Creek School 360 0.29 30 6.7 CA

65 1983 Alaska VDZ Valdez City Hall 360 0.05 64 6.2

66 1984 Morgan Gilroy Array #6 090 0.3 36 6.2 Hill

67 1984 Morgan Halls Valley 240 0.31 6.2 4.0 Hill

68 1985 Nahami site 1 010 0.92 7 6.9 69 1985 Nahami site 2 330 0.28 6.1 6.9 70 1985 Mexico N aranjito, Mich 155 0.06 67 5.4 71 1985 Guerrero, Paraiso 090 0.37 154 7.5

Mexico

72 1985 Chilean Vina Del Mar Chile S20W 0.36 7.8

73 1985 Mexico Secretary of Comm. N90W 0.17 400 8.1 &Trans,Mexico

39


74 1986 Alaska NAS Adak Hangar 197 0.19 135 7.9

75 1986 Santa Bar- Halls Valley 143 0.15 15 5.8 bara

76 1986 NortPann San J asinto - Soboba 90 0.25 34 5.6 Spring,CA

77 1986 Hollister Glorietta Wohs, g.f. N 000 0.14 12 5.5 wall

78 1986 N. Palm Stetson Ave Fore Sta- 360 0.14 46 5.6 Spring tion

79 1994 Northridge Newhall LA county 90 0.58 19 6.6 Fire Station

80 1994 Northridge Newhall LA county 360 0.59 19 6.6 Fire Station

81 1994 Northridge Pacoima Dam, 265 0.43 18 6.6 down stream

82 1994 Northridge Pacoima Dam, 175 0.42 18 6.6 down stream

83 1994 Northridge Santa Monica City 90 0.88 24 6.6 Hall

84 1994 Northridge Santa Monica City 360 0.37 24 6.6 Hall

85 1994 Northridge Sylmar County Hos- 90 0.60 15 6.6 pital Parking lot

86 1994 Northridge Sylmar County Hos- 360 0.84 15 6.6 pital Parking lot

87 1994 Northridge Arlerta - Nordhoff 90 0.34 9 6.6 Ave Fire Station

88 1994 Northridge Arlerta - Nordhoff 360 0.30 9 6.6 Ave Fire Station

Table 3.3 Mean Value of Nonlinear Spectral Reduction Factor (Ff.iG)

X!lG=2 x~=3 x~=4

T=0.5sec 2.15 3.12 3.98

T=1.0sec 2.14 2.95 3.42

40

Table 3.4 Nonlinear Spectral Reduction Factor (FDM)

T=O.5 x~=2.0 FIJ-G=2.15 T=1.0 x~=2.0 F!lG=2.14

N Sdl'FllG Umax/X~ Devation 0/0 ScVFllG Umax/X~ Devation %

62 0.7098 0.6756 5.06 2.6058 3.8080 -31.57

63 0.4372 1.8574 -76.46 1.6248 10.2339 -84.12

64 0.3373 0.6148 -45.14 1.1826 4.7802 -75.26

65 0.3740 0.4300 -12.82 0.8057 5.8119 -86.14

66 0.5713 0.9301 -38.57 0.9312 0.7934 17.37

67 0.3382 0.4514 -25.08 0.8465 0.8014 5.63

79 1.3759 2.2757 -39.53 3.3490 3.9454 -15.11

80 1.8615 6.2971 -70.43 5.3323 8.2909 -35.69

81 0.9113 0.5328 71.05 1.6982 2.2214 -23.55

82 1.1704 1.0706 9.31 1.0976 1.3014 -15.66

83 0.7930 1.6790 -52.77 1.4910 3.0385 -50.93

84 0.6185 0.5943 4.06 1.5632 0.9594 62.93

85 1.5309 3.4601 -55.76 2.8050 4.4210 -36.55

86 2.2595 4.7247 -52.18 3.9657 7.8339 -49.38

Average -31.61 Average -34.84

IlG denotes global ductility FflG denotes the global nonlinear spectral reduction factor for a given ductility ilG N denotes the Earthquake Number in Table 3.2

41

Table 3.5 Nonlinear Spectral Reduction Factor (F DM)

T=O.5 x!lG=3.0 FJlG=3.12 T=1.0 x!lG=3.0 F!lG=2.95

N Devation % Devation %

62 0.4891 0.4504 8.60 1.8903 2.5387 -25.54

63 0.3012 1.2382 -75.67 1.1786 6.8226 -82.72

64 0.2324 0.4099 -43.30 0.8579 3.1868 -73.08

65 0.2577 0.2860 -9.88 0.5845 3.8746 -84.92

66 0.3937 0.6201 -36.51 0.6755 0.5289 27.72

67 0.2331 0.3010 -22.56 0.6141 0.5343 14.94

79 0.9481 1.5171 -37.50 2.4295 2.6302 -7.63

80 1.2827 4.1981 -69.44 3.8682 5.5273 -30.01

81 0.6280 0.3552 76.81 1.2320 1.4810 -16.81

82 0.8065 0.7138 12.99 0.7962 0.8676 -8.22

83 0.5465 1.1194 -51.18 1.0816 2.0257 -46.60

184 1 0.4261 0.3962 7.56 " 1.1340 I 0.6396 77.39 I

85 1.0550 2.3068 -54.27 2.0348 2.9473 -30.96

86 1.5570 3.1498 -50.56 2.8769 5.2226 -44.92


f.!G denotes global ductility FI-lG denotes the global nonlinear spectral reduction factor for a given ductility JlG N denotes the Earthquake Number in Table 3.2

42

Table 3.6 Nonlinear Spectral Reduction Factor (F DM)

T=O.5 x~=3.0 F~G=3.12 T=I.0 x~=3.0 F!lG=2.95

N ScVF~G Umax/x~ Devation 0/0 SctIF~G Urnax/x1lG Devation %

62 0.3834 0.3378 13.51 1.6305 1.9040 -14.36

63 0.2362 0.9287 -74.57 1.0167 5.1170 -80.13

64 0.1821 0.3074 -40.73 0.7400 2.3901 --69.04

65 0.2020 0.2145 -5.81 0.5041 2.9060 -82.65

66 0.3086 0.4651 -33.64 0.5827 0.3967 46.89

67 0.1827 0.2257 -19.06 0.5297 0.4007 32.19

79 0.7433 1.1378 -34.68 2.0956 1.9726 6.23

80 1.0056 3.1485 -68.06 3.3366 4.1454 -19.51

81 0.4923 0.2664 84.80 1.0627 1.1107 -4.32

82 0.6322 0.5353 18.11 0.6868 0.6507 5.54

83 0.4284 0.8395 -48.97 0.9330 1.5193 -38.59

84 0.3341 0.2972 12.43 0.9781 0.4797 103.91

85 0.8270 1.7300 -52.20 1.7551 2.2105 -20.59

86 1.2206 2.3623 -48.33 2.4815 3.9170 -36.65


~G denotes global ductility F~G denotes the global nonlinear spectral reduction factor for a given ductility ~G N denotes the Earthquake Number in Table 3.2

43

Table 3.7 Global Limit State Probability (L.A. site)

Struc- Used O.5%of 1.0%of 1.5%of 2.0%of 2.5%of 3.0%of ture No. System height height height height height height

1 NMS 0.2415 0.0749 0.0387 0.0241 0.0167 0.0123

1 ENSS 0.2443 0.0741 0.0381 0.0237 0.0163 0.0120

2 NMS 0.3021 0.0906 0.0476 0.0300 0.0210 0.0156

2 ENSS 0.3050 0.0898 0.0468 0.0294 0.0205 0.0152

2 ELSS 0.3165 0.0804 0.0429 0.0259 0.0178 0.0140

3 ENSS 0.2459 0.0708 0.0368 0.0231 0.0161 0.0120

4 NMS 0.2477 0.0760 0.0403 0.0256 0.0180 0.0135

4 ENSS 0.2356 0.0692 0.0361 0.0227 0.0158 0.0112

5 NMS 0.2334 0.0695 0.0364 0.0230 0.0161 0.0120

5 ENSS 0.2513 0.0748 0.0392 0.0247 0.0172 0.0128

6 NMS 0.1650 0.0475 0.0233 0.0140 0.0094 0.0068

6 ENSS 0.1495 0.0422 0.0202 0.0119 0.0079 0.0056

7 NMS 0.1286 0.0375 0.0182 0.0108 0.0072 0.0052

7 ENSS 0.1212 0.0340 0.0161 0.0094 0.0062 0.0044

Table 3.8 Local Limit State Probability (L.A. site)

Struc- Used 0.5%of 1.0%of 1.5%of 2.0%of 2.5%of 3.0%of ture No. System height height height height height height

2 NMS 0.3588 0.1153 0.0621 0.0399 0.0282 0.0212

2 ENSS 0.4229 0.1235 0.0637 0.0401 0.0280 0.0208

2 ELSS 0.3658 0.0869 0.0515 0.0303 0.0243 0.0165

3 NMS 0.4282 0.1345 0.0717 0.0460 0.0324 0.0243

3 ENSS 0.4367 0.1278 0.0639 0.0402 0.0281 0.0209

4 NMS 0.4676 0.1536 0.0831 0.0538 0.0382 0.0289

4 ENSS 0.4335 0.1268 0.0641 0.0404 0.0282 0.0210

5 NMS 0.3986 0.1224 0.0646 0.0412 0.0289 0.0216

" EJ'.~SS f) 11t;;1" f) 1"2Q;1 f) f)'7f)"2 0.0444 f) f)'21 1 0.0232 -' V.-rVJ.-' V • .L-'U"T V.V I V-' V.VJ.L.1

6 NMS 0.2990 0.0917 0.0468 0.0289 0.0198 0.0145 c:. Et~SS II ,., A II,., II II()1 ,., ()()AA1 () ()""£""" () ()1""1A () ()1 1"\":-U V •• Y+V.J V.V711 V.V~l V.V~U~ V.V 1 1'+ V.Vl~.J

7 NMS 0.2401 0.0748 0.0377 0.0230 0.0156 0.0113

7 ENSS 0.2914 0.0789 0.0377 0.0222 0.0147 0.0105 I

44

Table 3.9 Global Limit State Probability (Imperial Valley site)


1 NMS 0.8264 0.7045 0.2956 0.1155 0.0383 0.0109

1 ENSS 0.8264 0.6902 0.2879 0.0916 0.0245 0.0062

2 NMS 0.8258 0.5519 0.1930 0.0589 0.0166 0.0045

2 ENSS 0.8230 0.5595 0.2154 0.0568 0.0140 0.0051

2 ELSS 0.8218 0.4835 0.1652 0.0421 0.0110 0.0029

3 NMS 0.8253 0.4241 0.0886 0.0147 0.0023 0.0003

3 ENSS 0.7942 0.4551 0.1020 0.0154 0.0021 0.0003

4 NMS 0.8252 0.4607 0.1209 0.0255 0.0051 0.0001

4 ENSS 0.7927 0.4495 0.1023 0.0160 0.0023 0.0003

5 NMS 0.8247 0.4282 0.0923 0.0157 0.0026 0.0004

5 ENSS 0.8026 0.4859 0.1192 0.0193 0.0028 0.0004

6 T

~!\fS 0.8178 0.1820 0.0152 0.0012 0.0001 0.7e-4

6 E~SS 0.7676 0.1506 0.0111 0.0008 0.0001 0.4e-4 "j ~~fS 0.7676 I 0.0821 0.0054 0.0004 I O.3e-4 I O.2e-5 I I I I I

7 E~SS 0.6702 0.0824 0.0065 0.0005 I OAe-4 I 0.3e-5

45

Table 3.10 Local Limit State Probability (Imperial Valley site)


2 NMS 0.8245 0.6632 0.3522 0.1718 0.0792 0.0350

2 ENSS 0.8257 0.7000 0.3798 0.1761 0.0759 0.0313

2 ELSS 0.8256 0.6083 0.3298 0.1009 0.0500 0.0179

3 NMS 0.8264 0.8237 0.4890 0.1695 0.0494 0.0133

3 ENSS 0.8213 0.6951 0.3931 0.1352 0.0367 0.0093 '*

4 NMS 0.8264 0.8246 0.6529 0.3364 0.1428 0.0549

4 ENSS 0.8212 0.6987 0.4034 0.1453 0.0415 0.0109

5 NMS 0.8264 0.8178 0.4769 0.1634 0.0441 0.0110

5 ENSS 0.8237 0.7235 0.4480 0.1738 0.0520 0.0142

6 NMS 0.8264 0.6832 0.2281 0.0416 0.0065 0.0010

6 ENSS 0.8254 0.5865 0.1688 0.0321 0.0056 0.0010

7 NMS 0.8264 0.5596 0.1381 0.0236 0.0038 0.0006

7 ENSS 0.8222 0.4473 0.1081 0.0219 0.0044 0.0009

500.00

400.00

--0..

~ ':' 300.00 ctS ~

..s:: CI'J ~

~ 200.00 CO

100.00

0.00 0.00

46

Yield Displacement

1.00 2.00 3.00 4.00

Global Displacement (in) Figure 3.1 Global Yield Displacement

for Two Story SMRSF

25'

Weight of each node: 150 kip

Figure 3.2 1 Story SMRSF

r,,<-,' ,~' ,~' ,~' :~~~':::;.-:~~~~~ ~-:~~~~~;;::-~:-'::~" ~7 ,7r~7 ,7 ':,.7' ,7r~7

25' 25' 25'

::-~J

Weight of each node: 163,12 kip (1 st floor) 128.25kip (2nd floor)


5.00

: 13' I 15'

6.00

47

t:>~>~:>~:p»:>5>5~ »5:>~:»:>:>:»:P:»:»:>~~~i>:>:»:>:>5:>~:»:»>:»>:»:»:>~:»:>~>~:>~:» >~:>~>~>~>~:»>~:»:>t

4@30'

Weight of each node: lS6kip(1st f1.), IS4kip(2nd-4th fl.), 133kip(Sth fl.)

Figure 3.4 S Story SMRSF 1

1<)')';") ~ )5»:>:>:>:»:»:>=>:P:>=>:»:»:»:»:»:»:»:»:»:r>:»:»:r>:»:»:>=>:>~

4@30'

Weight of c~ .. :h node: 273kip(1 st fl.), 271kip(2nd-4th fl.),233kip(Sth fl.)

Figure 3.5 5 Story SMRSF 2

40' 30' 40'

Weight of each node: 23 Skip (1 st-4th fl.), 198kip(Sth fl.)

Figure 3.6 5 Story SMRSF 3

4@13'

IS'

4@13'

IS'

4@13'

IS'

48

8@13'

15' p~>,>~>~>~>~~~>~~~5~>~>~~~~~~~>~5,>~~~>~>~~~>~~~5~~1

3@25' Weight of each node: 163kip(1st fl.), 162kip(2nd-8th fi.),128kip(9th fl.)


11@13'

15' t~~~~~~~~~~~~~~~.~~~~~~~~~~~~~~~~~~~~~~~~~~~>~>~~~~~~4

3@25' Weight of each node: 163kip( 1 st fl.), 162kip(2nd-ll th fl.), 128kip( 12th fl.)


RG

Rc

2.50

2.00

1.50

1.00

0.50

0.00 0.00

2.50

2.00

1.50

1.00

0.50

0.00 0.00

49

a 0 0

1.00 2.00 3.00

UEJUy

Figure 3.9 Global Response Scaling Factor Rc For One Story SMRSF

0 8 rP

0°0 aOo

1.00

° 0 0

0 0

2.00

UEJUy

° ° °

3.00

Figure 3.10 Global Response Scaling Factor Rc For Two Story SMRSF

o

4.00

4.00

RG

Rc;

2.50

2.00

1.50

1.00

0.50

0.00 0.00

2.50

2.00

1.50

1.00

0.50

0.00 0.00

o o

o

1.00

50

o

2.00

UEIUy

3.00

Figure 3.11 Global Response Scaling Factor Ro For Five Story SMRSF 1

Co 0

1.00

0

2.00

UEIUy

3.00

Figure 3.12 Global Response Scaling Factor Rc For Five Story SMRSF 2

o

4.00

4.00

RG

Rc

2.50 ~

2.00

1.50

1.00

0.50

0.00 0.00

2.50

2.00

1.50

1.00

0.50

0

0 o 0

i " 0

1.00

S 0

51

0

2.00

UEIUy

3.00

Figure 3.13 Global Response Scaling Factor Rc For Five Story SMRSF 3

o

o

•

o

4.00

0.00 ~t.,~~~~~~~~~~~~, .'~~' .'~~~~'~'~~~~~~'~' , 0.00 1.00 2.00

UEIUy

.3.00

Figure 3.14 Global Response Scaling Factor % For Nine Story SMRSF

4.00

2.50

2.00

1.50

Rc

0 U ...... t: .~ u E (l)

0 U

1.00

0.50

0.00 0.00

1.20

1.10

1.00

0.90

0.80 r., "

0.50

0

• 0

0

1.00

0

52

2.00

UEIUy

3.00

Figure 3.15 Global Response Scaling Factor Rc For Twelve Story SMRSF

DO 0

0

0

4.00

0

I , j , , I , , , , , , i ' , I , , , , , , i ' , , , , , , , , I , , , , , , , , , I

0.60 0.70 0.80 0.90 1.00

RMP

Figure 3.16 Regression for Coefficient 4 vs. RMP

0.10

0.06

~

U ....... 0.02 t: .~ U

S g -0.02 U

C"-I U -c: .~ u

-0.06

-0.10 0.50

0.20

0.12

0.04

tE -0.04 0

U

-0.12

0.60

53

0.70

RMP 0.80 0.90

Figure 3.17 Regression for Coefficient Cl VS. RMP

0 0

o

1.00

-0.20 I""",!! I , " , , ! !, ,I"" !, ! "I" , , , , , "I", '" " ,I 0.50 0.60 0.70 0.80 0.90 1.00

RMP Figure 3.18 Regression for Coefficient C2 VS. RMP

500.00

400.00 ".-..,

0.. ~ 'i::' 300.00 :'\l Q)

...c C/')

>-. E 200.00 C/')

c... ~ '-' ~ ... v ~

C/.: >-. ~

0 Uj

100.00

0.00 0.00

500.00

400.00

300.00

200.00

100.00

f"I f"If"I U.UU

0.00

54

Yield Displacement

1.00 2.00 3.00 4.00

1 st Story driftCin)

Figure 3.19 Local Yield Displacement for Two Story SMRSF

Yield Displacement

1.00 2.00 3.00 4.00

2nd Story ririft(in)

Figure 3.20 Local Yield Displacement for !wo Story SMRSF

5.00 6.00

5.00 6.00

3.00

2.00

1.00

0.00 0.00

3.00

2.00

1.00

1.00

55

o 00 0 0 ° 0

VGIVy

• o

2.00

Figure 3021 Local Response Scaling Factor RL For Two Story SMRSF

o ° 0°0 0

00

- 0 c 0 _0 ° 0 0

°

I , I ! ! I ,

1.00 2.00

Figure 3.22 Local Response Scaling Factor RL For Five Story SMRSF 1

o

3.00

0

! I 3.00

3.00 ~

2.00

1.00

0.00 0.00

3.00

2.00

1.00

0.00 0.00

56

00 o~

o

1.00 2.00 VGIVy


o 0 o 0

o OQ 0 o

1.00 2.00

VGIVy


o

3.00

o

3.00

57

3.00 ~ ~ ~

l-I-~ ~ I-

2.00 ~ ~

~

0 0 00 0

otB°o 0 0 0

~ ~cP 0 \I. 0-I- ~ t9 0 ~ 0

0 - 101

0 0 0 0

0

~

0.00 I- • I I • t . . . . I • . . . . 0.00 1.00 2.00

3.00 ! 2.00

1.00

Vc/Vy

Figure 3.25 Local Response Modification Factor RL For Nine Story SMRSF

1.00 2.00

Figure 3.26 Local Response Modification Factor RL For Twelve Story SMRSF

0 0

3.00

3.00

1.00

0.90

0.80

0.70

0.60

0.50 0.50

6.00

5.00

~.OO

l;..

3.00

2.00

1.00

I

~

1.00

0.60

58

0.70

RMP

0.80 0.90

Figure 3.27 Regression of RL W.r.t. Rf'v1P

a- --

E(F) E(F)+OF E(F)-OF

---.---_ .. -_------.0-- . .c::t-------

., .. -.. ' .#"-.. -

2.00

-.-_a-----

3.00

VGIVy

4.00 5.00

---.---

6.00

Figure 3.28 Nonlinear Spectral Reduction Factor (F) vs. ductility for T=O.S

1.00

......

6.00

5.00

4.00

3.00

2.00

1.00 1.00

0---

6- --

0·-------

2.00


/'

/' /'

59

-¥ - --

---------_-----8-------_ ..

... -----

3.00

VGIVy

4.00

---

/ / .....

-------~--------------

5.00 6.00

Figure 3.29 Nonlinear Spectral Reduction Factor (F) vs. ductility for T=1.0

·e 10 -2

:J ";3 .D o G

10 -3 ~~ ______________ '_' __ '~'I~ __ ~ __ ~~'~'~'~'_'~'~I ____ ~ __ ~ __ ~~,.,~,I 10 -3 10 -2 10 -1 10 0

Global Limit State Probability (NMS)

Figure 3.30 Comparison of Global Limit State Probabilities of ENS and NMS (L.A. site)

~

CZl Z ~

---->. ..... . -:0

co:: .D 0

0: ~

~ ..... CZl ..... E :5

co:: u 0

..J

..... . § :5

10 0

10 -1

10 -2

10 -3

10-3

60

10 -2 10 -1 10 0

Local Limit State Probability (NMS )

Figure 3.31 Comparison of Local Limit State Probabilities of ENS and NMS (l.....,.A. site)

o

c;10 -3 .D o 5 o

10 -4

10 -4 1 0 -3 1 0 -2 10 -1 10 0

Global Limit State Probability (NMS) .

Figure 3.32 Comparison of Global Limit State Probabilities of ENS and NMS (Imperial Valley site)

,,-.... en Z ~ '-" >. ...... -:.0 ~ .0 0

Q: d) ...... ~

r/5 ...... . § :J c; u 0

....J

61

10 0

10 -1

0

10 -2 0

10-3

10 -4

10 -4 10 -3 10 -2 10 -1 10 0

Local Limit State Probability (NMS)

Figure 3.33 Comparison of Local Limit State Probabilities of ENS and NMS (Imperial Valley site)

~ c.)

.0 E z

35~------------------------------~

30~----------------------------~

25r-------------~

20r-------------~

15r-----~------~ I---

10r-------------~

5~------------~

o~------------~--~~------------100-80 -60 -40 -20 0 20 40 60 80 100

Deviation (%) Figure 3.34 Histogram of Difference (Residue) of Global Limit State Probabilities

(L.A. site)

Imperial

.... Q)

.D E :::l Z

62

25

20

15

t---

10

5

o -1 00-80 ~O -40 -20 0 20 40 60 80 100

Deviation (%) Figure 3.35 Histogram of Difference (Residue) of Local Limit State Probabilities

(L.A. site)

!

I --I I ,

12 ~ <:,;

..c ,.... :: -- i

Z 8

I r----

i r----

I I n I I I I I o -100-80 -60 -40 -20 0 20 40 60 80 100

Deviation (%)

Figure 3.36 Histogram of Difference (Residue) of Global Limit State Probabilities

(Imperial Valley site)

63

20

16

.-----~

12

8

4 -

r- n-n o -100-80 -60 -40 -20 a 20 40 60 80 100

Deviation (%)

Figure 3.37 Histogram of Difference (Residue) of Local Limit State Probabilities

(Imperial Valley site)

I.;...

7.00

6.00

5.00

4.00

3.00

2.00

1.00 1.00

0--0 __ _

.6.-------


_ .. ---2.00

----_ ..... ------..... -- - -

3.00 4.00

UGIUy

Figure 3.38 Global Spectral Reduction Factor (FfJ,G) vs. ductility for 2 story SMRSF

5.00

f.;t;.,

7.00 0--

6.00 0 __ _

6·-------5.00

4.00

3.00

2.00

1.00 1.00

E(F) E(F)+OF E(F)-oF

2.00

64

_ ..... --

3.00

VG/Vy

---- -4------

4.00

Figure 3.39 Local Spectral Reduction Factor (FflG) vs. ductility for 2 story SMRSF

.... e)

20

15

E 10 ::l Z

5

o

r---

r---

r---

-100-80 -60 --40 -20 0 20 40 60 80 100

Deviation (%)

5.00

Figure 3.40 Histogram of Difference of Local Limit State Probabilities (ELSS and NMS) (2 story SMRSF )

.... Q.)

20

15

E 10 :::

Z

5

o

65

.----

r---

I -100-80 -60 -40 -20 0 20 40 60 80 100

Deviation (%)

Figure 3.41 Histogram of Difference of Local Limit State Probabilities (ENS and NMS ) ( 2 story SMRSF )

66

CHAPTER 4

SEISMIC RESISTANT DESIGN ACCORDING TO 1991 NEHRP PROVISIONS

4.1 Introduction

In the United States, the first mandatory requirements for seismic resistant de

sign came after the 1925 Santa Babara and the 1933 Long Beach earthquakes. Modifi-

cations in seismic design codes have been made as new research findings have become

available and new experience has been gained from past earthquake investigations. In

1985, 1988, and most recently in 1991, the Federal Emergency Management Agency

(FEMA) and the Building Seismic Safety Council (BSSC) developed recommended

seismic design provisions as part of the National Earthquake Hazard Reduction Pro

gram (NEHRP). These provisions (NEHRP, 1992) are based on the seismic design

provisions were intended to be a source document for use by any interested member of

the building community.

The purpose of seismic resistant design is well stated in the NEHRP recom

mended provisions (1991) as follows:

The purpose of the provisions is to minimize the hazard to life for all

buildings, to increase the performance of higher occupancy structure,

and to improve the capability of essential facilities to function during and

after an earthquake.

However, it is not economical to design or construct buildings and facilities to sustain

no damage during the rare event of a large earthquake. Hence, the NEHRP provisions

67

allow for some structural and non structural damage during a large earthquake. It

should be noted that the provisions only provide the minimum requirements for the

seismic design of buildings.

The NEHRP provisions use a design earthquake to define the seismicity of a site

at which the building under consideration is located. The design earthquake is defined

as an earthquake with a 10 percent probability of being exceeded in 50 year. To

characterize the intensity of a design earthquake, two parameters, Aa andAv, are used

which represent effective peak ground acceleration and effective peak velocity- re

lated acceleration, respectively. These parameters can be obtained using the seismic

zoning maps included with the NEHRP provisions.

A necessary aspect of the seismic design procedure is the determination of the

design values of seismic-induced forces and deformations in buildings. There are five

standard procedures which can be used to determine the design forces and deforma

tions [NEHHP Commentary, 1991 ] : (1) The equivalent lateral force procedure

(ELF); (2) The modal analysis procedure with one degree of freedom per floor in the

direction considered; (3) The modal analysis procedure with several degrees of free

dom per floor; (4) The inelastic response history analysis involving a step by step

integration of the coupled equation of motion with one degree of freedom per floor in

the direction being considered; and (5) The inelastic response history analysis involv

ing a step by step integration of the coupled equations of motion with several degrees of

freedom per floor.

The type of analysis which can be used to obtain seismic design forces and its

distribution is determined based on the seismic performance category to which the

building belongs. Buildings are assigned to one of these performance categories based

on the building'S function and the required degree of protection needed. In the

NEHRP provisions, the seismic performance category is determined according to the

68

coefficient,Av, and seismic hazard exposure group. For a building assigned to seismic

performance category A, no seismic analysis is required. For regular or irregular

buildings assigned to categories B or C, the equivalent lateral force procedure or a

more rigorous analysis should be used. For the buildings assigned to categories D and

E, the type of required analyses are given in Table 3.5.3 in the NEHRP provisions. The

seismic analyses performed in this study will be based on the ELF procedure.

4.2 Overview of Seismic Design Process Using the Equivalent Lateral Force

Procedure

The overall design process based on the equivalent lateral force procedure (ELF)

can be broken down into several basic steps. The steps outlined below are based on the

steps described by Schiff (1988), and Elliopoulos and Wen (1991).

1. Determine the overall building configuration to be designed at a

given site.

2. Determine the seismic base shear using formulas provided in the

code.

3. Determine the vertical distribution of the seismic base shear along the

height of the building.

4. Perform a static linear analysis of the structure for the prescribed load

combinations, and calculate interstory drift, member forces and

overturning moments.

5. Check the results of the analysis against story drift and strength

requirements of the provisions, and design the members (if

necessary) so that theses provisions are satisfied.

6. Calculate torsional effect.

69

7. Recalculate the equivalent lateral force considering torsional effects.

10. Repeat steps 4 and 5.

11. Recalculate the natural period of a structure (T) using an appropriate

method. (e.g., the formula based on Rayleigh's method ).

12. Repeat the steps 2 to 10 as necessary to obtain design which satisfy

code requirements.

Some of the details associated with the above steps described in the following sections.

4.3 Equivalent Lateral Force Procedure (ELF)

4.3.1 Detennination of Design Base Shear

The design seismic base shear ell) in a given direction is calculated by the follow-

ing equation:

v = Cs W (4.1)

where Cs is the seismic design coefficient and W is the "seismic" dead load (weight).

The weight W includes partitions, permanent equipment and the effective snow load.

In an area used for storage, 25 percent of floor live load is also included. The seismic

coefficient, Cs , is calculated using the following equation:

c - 1.2Av S s - RT2/3

< 2.5A a R (4.2)

where T is the fundamental period of the structure, S is the site coefficient, R is the

response modification factor, andAv andAa are the effective acceleration quantities

discussed earlier. Figure 4.1 shows the seismic coefficient (Cs) versus the fundamental

period of a structure (T). ( To draw this figure, 0.4 is used for 64V and..AAtl , 8 is used for R,

and 1.2 is used for S. )

The site coefficient S represents the soil profile characteristics of a site. Soil

70

profiles can be classified into four types: (1) rock or shallow soil over rock (5= 1.0); (2)

deep stiff soil over rock (5 = 1.2); (3) soft soil (5 = 1.5); and (4) soft soil which has a

shear wave velocity less than 500 feet per second (S=2.0).

The response modification factor, R, represents the ratio of the maximum forces

that would develop in a linear elastic response of a structure subjected to a given

earthquake to the static forces that would cause significant yield (Berg, 1982). As

shown in Eq. (4.1) and (4.2), R factor reduces the estimated "elastic" base shear in

order to account for the inherent system ductility, the energy dissipation capacity

beyond the elastic strain energy, and the unexpected additional strength of the non

structural components. The NEHRP provisions assign differentR factors according to

the type of structural system. ( However, the R factors listed in the provisions are

independent of the structural period.) The R factors in the NEHRP provisions range

from 1 1/4 to 8.

At the beginnIng of the design process, the period, T, cannot be calculated by

conventional structuraJ d)llamic procedures since member sizes are unknoWD. Howev-

er, the code imposes an upper bound on T. The fundamental period of a structure

should not exceed the product of the coefficient for upper limit (Ca) and approximate

fundamental PC:IOC {Ta i. Different values of Ca are assigned according to the

magnitude of .... h to account for the fact that buildings located in lower seismic zones

may be more f1cuhlc An approximate fundamental period can be determined by the

following equatIon

(4.3)

where CT is assigned according to the type of structural system ( e.g., CT is 0.035 for

moment resisting frame systems of steel) and hn is the total height of the structure in

feet. Alternatively, for concrete and steel frames not exceeding 12 stories in height and

71

having a maximum story height of 10 feet, the following equation can be used:

Ta = O.IN (4.4)

where N is the number of stories.

Once member sizes are known, T can be calculated using structural dynamics

procedures.

4.3.2 Determination of Design Lateral Forces

The design base shear calculated using Eq. (4.1) is distributed along the height of

the building in accordance with the following equation:

(4.5)

(4.6)

where Cv.t is the vertical distribution factor, Vis the design base shear, Wi and Wx are the

portion of the weight (W) assigned to level i or x of a building, ~ and hx represent the

height from the base to level i or X, and k is the exponent related to the fundamental

period of a building. For a building having a period ofD.5 second or less, k is equal to 1.

For a building having a period 2.5 seconds or more, k is equal to 2. For the buildings

with periods between 0.5 seccnds and 2.5 seconds, k is found using interpolation. The

coefficient k in Eq. (4.6) accounts for the effects of higher modes for buildings with a

longer fundamental period. Figure 4.2 shows the variation in k for different values of

th e fundamental period of a building. Due to the coefficient k, the vertical distribution

of the design base shear changes from a linear distribution for a short period structure

to a parabolic distribution with the vertex at the base for a long period structure. Figure

4.3 shows the vertical distributions of the design shear forces when k is equal to 1 and 2.

72

4.3.3 Ductility Requirements and P-A Effects

The drift limits specified in the NEHRP provisions are different from the drift

limits of the Uniform Building Code (UBe, 1991) because the NEHRP procedures are

based on limit state design instead of working stress design. The design story drift (L1 )

should be smaller than the allowable story drift (L1a). The allowable story drifts are

given in Table 3.8 in the NEHRP provisions. The design story drift is computed as the

difference of the deflections at the top and the bottom of the story under seismic forces.

The deflection (Ox) of level x at the center of mass is calculated by the following

equation:

(4.7)

where Cd is the deflection amplification factor and bxe is the deflection calculated by

elastic analysis. For the purpose of the drift analysis, the seismic forces can be calcu

lated using the computed fundamental period of the structure without considering an

upper limit such as the one discussed earlier.

P-LJ effects result from the moment induced by the lateral displacement of

gravity loads. This causes additional moments in the column. These effects lead to an

increase in member forces and story drifts, and the increase may be significant under

some circumstances (Berg, 1982). The NEHRP provisions evaluate the importance of

p - ~ effects by calculating a stability coefficient ( e) which is defined as :

(4.8)

wherePx is the total vertical design load at and above level x, L1 is the design story drift,

Vx is the seismic shear force acting between levelx and x-I , and hsx is the story height.

Cd is the deflection amplification factor. If e is less than 0.1 for every story, P-LJ effects

can be neglected according to the NEHRP provisions. Otherwise, the story drift

73

determined by Eq. (4.7) needs to be multiplied by 1/(1-8). The stability coefficient

should not exceed a maximum value, 8max , defined as :

8 - 0.5 max - f3C

d

(4.9)

where f3 is the ratio of shear demand to shear capacity for the story between levels x and

x-I. The value of f3 can be conservatively taken as 1.0. When 8l is greater than 8max ,

the structure is potentially unstable and needs to be redesigned.

4.3.4 Combination of Load Effects

When designing a structure, the following load combinations involving earth

quake loads need to be considered:

0.1 + 0.5A v)QD.±.1.0QE + (l.OQL + O.7Qs)

(0.9 - 0.5A v)QD.±.1.0QE

( 4.10)

(4.11 )

where QD is the effect of dead load, QL is the effect of live load, Qs is the effect of snow

load, (2£ is the effect of seismic load, and Av is the effective acceleration quantity

discussed earlIer In Eqs. (4.10) and (4.11), a factor ±Av accounts for the effects of

vertical ac.celeratlon \Vhen estimating QE, orthogonal effects may need to be consid

ered. This may ~c done by combining 100 percent of the forces due to ground motion in

one direction \li,lth 30 percent of the forces due to the ground motion in the perpendicu

lar directioD. TIus requirements may influence the corner columns which resist the

earthquake forces from both directions.

For the columns supporting a discontinuous lateral force resisting element, the

axial force in the column shall be calculated as follows:

( 4.12)

The axial forces in the column need not exceed the capacity of other elements which

74

transfer such loads to that column. (Examples are discussed in VBC (1991) and

SEAOC tentativ commentary (1988).) The design forces are increased for critical

elements "whose yield or failure could result in local or general col

lapse."(SEAOC,1988) Such critical elements include columns under discontinuous

shear walls or bracing systems and other elements in discontinuous force transfer force.

For brittle materials, systems and connections the following combination also

needs to be considered:

(4.13)

In Eq. (4.12) and (4.13), the factor 2R/5 shall be greater or equal to 1.0. In Eq. (4.10)

through (4.13), the term O.5Av can be neglected whereAy is equal or less than 0.05.

4.4 Design Example: Six Story Office Building

As an example, a typical 6 story office building (Seismic Hazard Exposure Group

I) is designed according to the NEHRP provisions and the AISC LRFD manual. The

computer program STAAD-III ( STructural Analysis And Design, 1993) is used to

facilitate the design process. The building is assumed to be located in the highest

seismic zone (Aa =0.4 andAy =0.4). The soil profile of the site is assumed to be a deep

stiff soil overlying rocks, and this is classified as soil type 2 (S2 = 1.2). According to

Table 1.4.3 in the NEHRP provisions, the seismic performance category of this building

is detennined asD. Since this building is assigned to the seismic hazard exposure group

I, the allowable story drift is 0.015 times the story height based on Table 3.8 in the

NEHRP provisions.

The building has 3 bays in the north-south direction and 6 bays in the east-west

direction. In each floor, the center of mass is assumed to coincide with the center of

75

rigidity. The story height is 14 feet for all stories except for the 1st story which is 17 feet

in height. The span length of all bays in both directions is 25 feet. In order to provide

the lateral resistance and stability against an earthquake, special moment resisting

steel flames (SMRSF) are placed on the perimeter of the building. For SMRSF, the

response modification factor (R) is 8 and deflection amplification factor (Cd) is 5 1/2.

The connections of interior frames are pinned, so that they only resist vertical load.

Pinned connections are used for the outer bays of the east-west perimeter frames.

Therefore, there is no comer column to resist the lateral seismic forces from both

orthogonal directions. The orthogonal SMRSFs are assumed to behave independent

ly; thus, each frame can be treated as a plane frame. Figure 4.4 shows the plan view of

the building. An Elevation view for each direction is shown in Figures 4.5 and 4.6. In

Figures 4.4, 4.5, and 4.6, the thicker lines represent the SMRSFs.

Dead loads on the floor and the roof are 95 psf and 82 psf respectively. Live loads

on the floor and the roof are 45 psf and 16 psf, respectively_ Table 4.1 shows the vertical

loads used in this study. The weight-of each story is given in Table 4.2. The design base

shear is 233 kips for each SMRSF. Design seismic lateral forces applied to each story

are shown in Table 4.3.

To be consistent with the current state of practice, the following assumptions are

made in the design of the SMRSFs :

1. Cross sections of columns and girders only change in every two stories.

2. Lateral supports are provided to the flanges of the girders by bracing if

they are necessary.

3. The girders of the SMRSF are not connected to the slab. Hence, the

slab provides no contribution to the resistance of seismic forces.

The member sizes of each frame are listed in Tables 4.4 and 45, and shown in

76

Figures 4.11 and 4.12. Wide flange sections are used for all members. The design is

controlled by drift requirements rather than strength requirements. Table 4.6 presents

the story drift of each SMRSF calculated from Eq. 4.7 along with the allowable drift

limit. The calculated fundamental periods of the north -south and east-west

SMRSFs are 1.57 and 1.59 seconds, respectively.

77

Table 4.1 Design Vertical Loads

Load Floor (psf) Roof (psf)

Concrete Slab with Decking 42 42

D Insulation and Membrane 0 11 E A D

Ceiling 10 10

Mechnical and Electrical 4 4 L 0

Structural Steel 20 15 A D

Partition 20 0

Total 95 82

Exterior Wall and Facade 30 30

Live Load 45 16

Table 4.2 Weight of the building Components

~ Story No. Weight for floor

(1) Weight for Wall

(2) Weifht for story

1)+(2)

6 923 95 1018

5 1068 189 1257

4 1068 189 1257

3 1068 189 1257

2 1068 189 1257

1 1068 209 1277

I 6263 1060 7323

78

Table 4.3 Forces Used for Seismic Design

story Wx(kip) hx(ft) WxhxK (WxhxK)j Fx(kip) Fx/2 (for ~WxhxK) each

frame)

1 1277 17 58519 0.038 18 9

2 1257 31 129621 0.084 39 20

3 1257 45 214374 0.139 65 33

4 1257 59 309019 0.200 93 47

5 1257 73 411928 0.266 124 62

6 1018 87 422765 0.273 127 64

z: 7323 1546226 1.000 466 233

Table 4.4 Member Sections of the North-Sourth Frame

me Section me Section me Section me Section me Section m m m m m

1 W33x141 10 W33x118 19 W24X84 28 W30xl08 37 W24X76

2 W33X141 11 W33xl18 20 W24X94 29 W30x108 38 W24X76

3 W33X141 12 W30X130 21 W24X94 30 W30x108 39 W24X76

4 W33X141 13 W30Xl30 22 W24X84 31 W30x108 40 W24X76

5 W33X141 14 W33xl18 23 W24X84 32 W30xlO8 41 W24X76

6 W33X141 15 W33xl18 24 W24X94 33 W30xlO8 42 W24X76

7 W33X141 16 W30X130 25 W30x108 34 W30x108

8 W33X141 17 W27x94 26 W30x108 35 W30xl08

9 W30X130 18 W24X84 27 W30x108 36 W30x108

79

Table 4.5 Member Sections of the East-West Frame

me Section me Section me Section me Section me Section m m m m m

1 W33x130 12 W30x116 23 W24X84 34 W30x90 45 W27x84

2 W30x116 13 W30x116 24 W24X84 35 W30x90 46 W27x84

3 W30x116 14 W30X116 25 W24X84 36 W30x90 47 W24x62

4 W30x116 15 W30X124 26 W24X84 37 W30x90 48 W24x62

5 W33x130 16 W30X124 27 W24X84 38 W30x90 49 W24x62

6 W33x130 17 W30X116 28 W24X84 39 W27x84 50 W24x62

7 W30xl16 18 W30X116 29 W24X84 40 W27x84 51 W24x62

8 W30xl16 19 W30X116 30 W24X84 41 W27x84 52 W24x62

9 W30x116 20 W30X124 31 W30x90 42 W27x84 53 W24x62

10 W33x130 21 W24X84 32 W30x90 43 W27x84 54 W24x62

11 W30X124 22 W24X84 33 W30x90 44 W27x84

Table 4.6 Drift Limit

story 1 2 3 4 5 6 drift limit, in 3.06 2.52 2.52 2.52 2.52 2.52

N-S frame 2.14 2.50 2.51 2.26 2.52 1.80 E-W frame 2.14 2.46 2.52 2.37 2.50 1.70

0.20

0.15

~ >- 0.10 II I:r.I

U

0.05

0.00 0.00

3.0

2.0

1.0

0.0

80

0.50 1.00 1.50 2.00 2.50 3.00

T (second)

Figure 4.1 Seismic Coefficient

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 T (second) k=l k=2

Figure 4.2 k vs. T Figure 4.3 Force Distribution for k=l and k=2

25'

25'

25'

81

@ @ © @ ® ® @ :: - - - - - - ~ ~ - - - - - - ! ~ - - - - - - : ~ - - - - - - ~ -: - - - - - - ~ -:- - - - - -:: CD I I

, .. ~ -~

-~ ,

.. ~ .. ~ -~

, -~ t I I t I I I I I I -~ .!..L- ______ l.J ______ I .J ______ I -1 ______ J _1 ______ J _1_ - - - - - ~

25' 25'

40 2 I

37 I 7

34 ':l ~ 1

31 9

28 5

25

25' 25'

6 @ 25' = 150'

Figure 4.4 Floor Plan

41 42 22 2~

38 39 1~ IS

35 36 1<4 IS

32 33 lC 11

29 30 6 7

26 27

3@2S'

2A

2C

It

1~

8

25'

1

I~ 25' \l)

5@14'

17'

Figure 4.5 Elevation and Member Number for the North-South SMRSF

82

@ @ © @ ® ® @ -- 51 52 2~ 53 2< 54 3( 2t 2--

-,...

-..... 47 48 49 50 2 2~ 2' 2, 2:

..... 43 44 45 46 '"" l( l'" U 1~ 2( - I-

5@14' 39 40 41 42

L 1: 1~ lL 1: - 35 36 37 38 '"" t 7 8 S l( - I-.....

31 32 33 34 -~

1 2 1 -' ~ S 17'

Figure 4.6 Elevation and Member Number for the East-West SMRSF

Column Girder

ext. tnt

I w24x76

I w30xl08

I w30xlO8

Figure 4.7 Members for the North-South SMRSF

83

Column

ext. int. ® © @ ® Girder

w24x84 I w24x62

w30x124 w30x116 w27x84

r----t----+---1--......J I w30x90

Figure 4.8 Members for the East-West SMRSF

84

CHAPTER 5

RESPONSE SURFACE METHOD FOR CALIBRATING DESIGN PARAMETERS

5.1 Introduction

In this study, the design parameters ( such as load factors and allowable story

drift) are calibrated by minimizing an objective function which measures the total

difference between target limit state probabilities and actual limit state probabilities of

structures. Limit states represent various states of undesirable behavior of structures

such as yielding, excessive interstory drift, instability, buckling or severe damage.

Calibrating design parauleters based on reIiabilit'j as foullulated above requires the

use of nonlinear programming methods (Shinozuka, 1989 ; Hwang, 1991) which are, in

general, computationally intensive. These methods typically require redesigning the

structure ( s) under consideration and recalculating the limit state probabilities in each

iteration. Furthermore, if the target limit state probability is changed, then the entire

nonlinear programming analysis must be repeated. The calibration process, therefore,

could become very time consuming and costly. In view of this, a more efficient method

is needed for practical applications. The Response Surface Method (RSM) is an

alternative method which can be used for this purpose. The RSM will be used in this

study.

5.2 Overview of Response Surface Method

Response Surface Method (RSM) is a statistical analysis method which esta

blishes a functional relationship between system parameters (input variables) and

system response. For example, the RSM is often used to

85

(1) Find a suitable approximate function in order to predict future

responses ; or

(2) Optimize the response (determine the values of input variables

which result in maximum or minimum response).

Conceptually, the functional relation between input variables and system response can

be expressed as

(5.1)

where y is the value of the response, Xi (i= 1, .. ,k) are the input variables. The functional

fonn, Q, is usually unknown and could be extremely complicated. The fundamental

premise of the RSM is that the function Q can be approximated by a low order

polynomiaL For example, if the approximate function is assumed to be linear, the

response function can be written as follows;

k.

Y = f3cI - I;6,x1 + c (5.2) I" 1

where Po a.nd;J, are constant coefficients. The above first order model is useful when

the cun:aturc m n is believed to be negligible. Othery.rise, higher order polynomials

should be used However, polynomials of order higher than 2 are not often used in the

Response Surface ~1ethod. The response modeled by a 2nd order polynomial can be

expressed as follov.-s :

t k k i-I

Y = fie; - Ip,xi - LfiiiXT + L Lf30iXj + C (5.3) 1"= 1 i=1 i=lj=l

The constant coefficients (/3, f3o, A, and Aj ) are estimated based on observed data. The

estimated response function is often called a response surface or fitted surface. The

rationale for the polynomial approximation of the response function, Q, is based on the

86

Taylor series expansion of Q. For example, if the expansion is taken around the point

Xl = X2 = X3 = .... =Xk = 0, the Eq.(5.2) is developed by the first order expansion as

shown below.

y = Qx=o + (~f) xl + (~~) x2 + ...... + (~~) Xk + C I x=o 2 x=o k x=O

(5.4)

/30 = Yx=O (5.5)

_ (aQ) /3i - ax. Xi

l x=O

(5.6)

In order to use RSM, the following assumptions need to be considered (Myers, 1971) :

1. The response, y, exists.

2. The input variables are continuous and quantifiable parameters.

3. The response function, Q, is approximated in the region of interest

by a low order polynomial, such as Eq.(5.2) or (5.3).

4. The input variables can be controlled in the observational process and

measured with negligible error.

In this study, the RSM will be used to determine "optimum" values of the input parame

ters. The objective function used in the optimization procedure will be the response

function, y. As explained in section 5.3, the response function will be formulated in

terms of some measure of the consequences of various limit states.

5.3 Objective Function (Response)

As mentioned earlier, the objective function measures the difference between

actual and target probabilities for all limit states and all types of structures under

consideration for a certain combination of design parameters (input variables). The

following objective functions have been proposed by Siu (1975), and Shinozuka (1989) :

87

m n 2

Q(X) - I .2:WsI~sfX) - .B 1*] s= 1/= 1

(5.7)

2

_ ~ ~ [lnpS!X) - lnp z*] Q(x) - L L CVsz *

s=1/=1 InPz (5.8)

where Q(x) is the objective function depending on design parameters (x) , 1 refers to a

limit state, s refers to the type of structure, Pz * andfiz * are the target limit state probabili

ty and target safety index, respectively, of each limit state (I), Psz and f3sz are the actual

probability and actual safety index, respectively, of the structure(s) for limit state (I).

CVsI is a weighting factor assigned according to the limit state and the type of the

structure, however, the basis for assigning CVsI has not been clearly indicated in these

procedures.

The squared terms in Eq.(5.7) and (5.8) are the penalties for the deviation from a

target value. It is more reasonable to express the deviation directly in terms of limit

state probability. Since if the penalties are expressed proportional to limit state proba

bility, and if one can assign the weighting factors in proportion to the consequence of

exceeding a limit state, then the objective function ( summation of the product of the

penalties and weights over all possible limit states (I) and structure types (s) ) can be

interpreted as the total deviation of the "expected consequence." Therefore, in this

study, the following objective function is used (Han, 1992 and Wen, 1993) :

(){YI .... \,,(1 J

2 ~ ~ frl _ (PsAX) - pn L L"'" SL---p-*--s=11=1 /

in which the penalty term is directly proportional to limit state probabilities. Figure 5.1

shows the comparison of the penalty terms of these three objective functions divided by

88

target limit state probability (Pz j. In each case,Psl is equal to 0.4 Pz *. This figure shows

the penalty term ofEq.(S.7) (when divided by (~l *)2) and the penalty term ofEq.(S.8)

have distorted values depending on the level of target limit state probability. This

distortion is more serious when the level of target limit state probability is low or high,

whereas Eq.(S.9) by definition has constant values irrespective of the level of the target

limit state probability. Therefore, Eq.(S.9) is more appropriate for problems in which

limit states of different target probability levels are combined.

5.4 Experimental Design Strategy

Once pertinent input variables have been selected, it is necessary to decide which

combinations of input variables must be considered in the "experiments" used to obtain

observed data. Since each "experiment" or observation involves a certain amount of

cost and effort, the decision on the appropriate combinations of input variables affects

the efficiency of the procedure. The selection of these combinations can be interpreted

as a form of experimental design. The experimental design should be formulated based

on the relative precision needed in estimating the coefficients and the amount of

experimental (observational) effort involved.

A conceptually simple experimental design is the factorial design (Myers, 1971).

The most elementary class of factorial design is 2k factorial design. The configuration

for combinations of three input variables (k=3) is shown in Figure S.2. This 2k factorial

design is useful for cases in which the response can be approximated by a 1st order

polynomial since two values of each input variable are considered (see Figure S.2).

A factorial experimental design associated with three values of each input vari

ables taken as three levels is referred to as a 3k factorial design. The 3k factorial design

is useful when the response is well represented by a 2nd order polynomial. Figure S.3

89

shows the 3k factorial design configuration for three input variables (k=3). However,

when the number of input variables is large, the number of experiments (observations)

can become excessive. For example, when k=5, 243 observations are required to fit

only 21 coefficients; this is unreasonable from a practical point of view.

Box and Wilson (1951) have developed an alternative to the 3k factorial design

which is called central composite design. The central composite design consist of 2k

factorial points, nc center points, and 2k axial points. The center point is normally

chosen based on the experience of the experimenter with the problem at hand. In this

investigation, it corresponds the values of the parameters recommended in current

codes. Factorial points are located at equal distance from the center point. If d

represents the distance from the center point to one of the factorial points, then the

axial points are located on the axes of the parameter space at a distance a times the

component of d along that axis. For example, if k=5 and nc= 1, 43 observations are

required. The center points make it possible to measure the pure error in the system

response. Unlike usual experimental design problems, there is no pure error in this

study, therefore, only one center point is necessary. The axial points allow the design to

fit higher order surfaces without the efforts required in 3kfactorial design. The configu

ration of input variables for k=3 is shown in Figure 5.4.

The three design properties to consider for the proper selection of the number of

the center points and the distance of the axial points from the center points are ortho

gonality, rotatability, and uniformity of precision (Petersen, 1985). Orthogonality

implies that x'x matrix becomes diagonal, and constant coefficients in Eq.(5.3) are

uncorrelated. Rotatable design ensure that the variance of the estimated response at a

point is a function only of the distance of that point from the center of the design and

does not depend on the orientation of the design to the true response surface. For

central composite design, a should be 2k!4 to satisfy a rotatable design. If the variance of

90

the response is constant for a distance of 1 (in the normalized parameter space) from

the center of the design, the design is considered to have unifonn precision. By proper

selection of the number of center points, a rotatable central composite design may gain

uniform precision and orthogonality (Petersen, 1985).

5.5 Response Surface Fit

As mentioned earlier, the coefficients (/30, /3i, /3ij ), in Eq.(5.2) and (5.3) can be

determined by minimizing the error between the real observational responses and the

calculated response from the second order polynomial associated with a certain com

bination of input variables ( design parameters) defined by the central composite

design rule. A least squares method can be employed to carry out the minimization

process. The general fonn of Eqs.(5.2) and (5.3) can be expressed as follows (Myers,

1971) :

y=xfJ+c (5.10)

wherey istheresponsevector. x is the matrix consisting of l's andx/s forEq.(5.1), or

1 's, XiS, and XiXj'S for Eq.(5.3), fJ is the vector of the constant coefficients, and e is the

vector of random errors. Let vector E (scalar) be defined as follows;

E = e'e (5.11)

where e' is the transpose vector of e. Eq.(5.5) can be re-written using Eq.(5.4) as

follows;

E = y'y - 2fJ'x'y + p'x'xP

~ can be obtained by minimizing E.

aE = - 2x'y + 2(x'x)P = 0 ap

Solving Eq.(5.13) for the coefficient vector fJ gives

(5.12)

(5.13)

91

(5.14)

5.6 Detennination of Design Parameters

Once the 2nd order polynomial representing the response surface (objective

function) is established, the optimum design parameters are those points correspond

ing to the minimum value of the response surface function. At the minimum pointxi,

the following relations must hold:

i = l, ... ,k

i = l, ... ,k

For example, when k=2, Eq.(S.3) can be written as follows;

y = f3 0 + f3 1x 1 + f3~2 + f311xi + f32iX~ + f3 1iXlx2

Based on Eq.(S.lS) and (S.16),

a2y - = 2fJ.." > 0 ax2 --

2

(5.1S)

(5.16)

(5.17)

(5.18)

(5.19)

(5.20)

(5.21 )

Form Eqs.(5.18) and (S.19), design parameters, xl and x2, can be easily calculated as

follows;

(5.22)

92

Then;

(5.23)

5.7 Steps to Determine Design Parameters

The complete process of determining design parameters using RSM can be bro

ken into several steps as follows:

1. Select target limit state probabilities.

2. Select representative structures.

3. Determine the configuration of input variables (combinations of

design parameters) according to central composite design.

4. For each combination of design parameters, design the representative

structures according to appropriate code procedures and provisions.

5. Evaluate the limit state probabilities of representative structures.

6. Calculate the value of the objective function (Eq.(5.9)) for each com

bination of design parameters.

7. Based on the calculated values of the objective function, establish the

2nd order polynomial representing the objective function. The poly

nomial should be a function of the design parameters.

8. Find the design parameters which minimize the fitted 2nd order pol

ynomial ( fitted 2nd order objective function ).

Numerical examples for calibration of design parameters based on reliability

using RSM will be shown in Chapter 6.

10 3

10 2

5.. 10 1

---~ c; C Il)

0.. 10 0

10 -1

93

-

p-p p ~ ~2/ * (f3-f3 )/~.)2

((Iog(p)-Iog(p *)/Iog(p ))2

........... ........ ,', ......-...... ,'~ ,

- .. _ ' ,I _, : "

- .. _ " 'J -_ A

-__ ---_.. .,,' ,I \ ....... _ ......... _.......... r / '

.... _---- ... _- ................ " , ' -, ------------:..---:..---:---~:.=:.----'"

-.. -.. -

10 -2f ~--~~~~I ~I~II~I~I __ ~~I~I~I ~I~II~,~I __ ~ __ ~'~'~I~,~,~,,~I __ ~ __ ~~,.,.,~,.I 1 0 4 1 0 -.3 1 0 -2 1 0 -1 1 0 0

Target failure probability, p*

Figure 501 Comparison of Penalty Terms Nomalized By Target Probability

X3

~Xl

Figure 5.2 2k Factorial Design for k= 3

94

X3

~Xl

Figure 5.3 3k Factorial Design for k=3

I I ,

I I

I I

I

I

Axial Point (2k)

Factorial Point (2k)

__ ----~.--2-----~----- ___ ~ ____ __ ~-----~-----

, ,

I

Center Point (nc)

Figure 5.4 Central Composite Design for k=3

6.1 Introduction

95

CHAPTER 6

NUMERICAL EXAMPLES

In this chapter, two numerical examples are presented. These examples demon

strate how the methods described in Chapter 3, 4, and 5 can be used to calibrate the

design parameters based on required target reliabilities of the system (e.g., target

probabilities of exceedence for the serviceability limit state and the ultimate limit

state). As mentioned earlier (Chapter 5 ), calibrated values of the design parameters

are determined by minimizing the objective function which is expressed as a 2nd order

polynomial. The objective function measures the difference between actual (com

puted) and target limit state probabilities. The Response Surface Method (RSM) and

the method of a central composite design are used for this purpose (see Chapter 5).

In the first example (Section 6.2), the factors for live load and seismic load are

calibrated based on required reliabilities. One and two story special moment resisting

steel frames (SMRSF) are considered. The second example (Section 6.3) demon

strates the calibration of the load factor for seismic load and the allowable drift limit

based on required reliability. Multi-story SMRSFs of various configurations (e.g.,

different span lengths, story heights, etc.) are considered.

For both examples, all structures are assumed to be office buildings located at

Santa Monica Boulevard in Los Angeles. This site is located approximately 60 km from

the Mojave segment of the Southern San Andreas fault. Two limit states, serviceability

and ultimate, are used. The serviceability limit state is defined as an interstory drift

ratio of 0.5 % ; similarly, the ultimate limit state is defined as an interstory drift ratio of

96

1.5 %. In order to evaluate the limit state probabilities of exceeding these limit states

for each of the structures under consideration, the Equivalent Nonlinear System

(ENS) technique (described in Chapter 3) is used.

6.2 EXAMPLE I : Calibration of Load Factors for Live Load and Seismic Load

According to the Load and Resistance Factor Design (LRFD) philosophy, the

capacity of members should be larger than or equal to the demand determined for

appropriate critical combination(s) of factored loads as follows:

n

¢R 2: I YiSi (6.1) i == 1

where ¢ is a resistance factor, R is the nominal capacity, Yi is a load factor, and Si is a

load or a load effect. According to Eq.(3.1) in the NEHRP recommended provisions

(BSSC, 1991). the LRFD design equation is :

(6.2)

where YD. YL. :'1· and YS represent the load factors for dead load, live load, seismic

load, and snow load, respectively.

In thIS example. the load factor for dead load, YD, is defined as (1.1 +0.5 Av)

according to Eq ( 3.1 ) Ul the NEHRP provisions. The coefficient Av represents the

effective peal vcloclty-related acceleration. The term 0.5Av is intended to account

for the effects of vc rtlcal acceleration. Since the representative structures are assumed

Thus, the value of YD is 13 ( = 1.1 +0.5xO.4). According to the NEHRP recommended

provisions, snow load can be neglected if the design snow load is less than 30 pounds per

square foot. Since the structures are assumed to be located in L.A., snow load is not

considered. Therefore, for this example, the load combination in Eq.(6.2) reduces to:

97

(6.3)

6.2.1 Objective Function

The purpose of this example is to determine the appropriate combination of

earthquake and live load factors. Load factors YL and YE are the input variables of the

response (objective function) to be used in the Response Surface Method (Chapter 5).

Therefore, the objective function ( Eq.(5.9) ) can be written as follows:

(6.4)

where Q(YL, YE) is the objective function which depends on the input variables YL and

YE, I refers to a limit state, s refers to the type of structure, Pz * is the target limit state

probability for each limit state "[", PsI is the actual probability of the structure type "s"

for limit state" I," and OJ sf is a weighting factor assigned to each combination of limit

state and type of structure.

6.2.2 Selection of Load Factor Combinations Using Central Composite Design

To begin the analysis, several pairs of the live load factor, YL, and seismic load

factor, YE, must be selected. This selection is made using a central composite design

discussed in Chapter 5. The number of the input variables (k) is 2. Thus, according to a

central composite design, 9 different pairs are required. Of the 9 combinations (pairs),

four are factorial points, four are axial points and 1 is a central point. The 9 pairs of load

factors (YL, YE) are shown in Figure 6.1. The values given by Eq.(3.1) in the NEHRP

provisions are used as the center point (YL, YE)=(1.0, 1.0). The distances of the

factorial points and axial points from the center point are determined based on consid

eration of the range within which the values of load factors, YL and YE, that minimize

98

the objective function are expected to occur. For this study, the distances aE and aL

shown in Figure 6.1 are assumed to be 0.5. Table 6.1 shows the load combination

formats corresponding to the load factor pairs shown in Figure 6.1.

6.2.3 Representative Structures

For each load format shown in Table 6.1, one story and two story SMRSFs are

designed according to the NEHRP provisions and the AISC LRFD manual. The load

format shown in Eq.(6.3) typically governs the selection of member sizes; thus, the load

format shown in Eq.( 6.3) is considered in this study. The dimensions of the structures

are shown in Figure 6.2. Dead load and live load are assumed to be 100 psf and 50 psf,

respectively. For this example problem, nine one-story SMRSFs are designed; each

structure corresponds to one of the pairs of (YL, YE) shown in Figure 6.1. Similarly, nine

two- story SMRSFs are designed. Member sizes of each of the SMRSFs are shown in

Tables 6.2 and 6.3.

It should be noted that the 18 SMRSFs used in this example were designed to

satisfy strength criteria only; no attempt was made to satisfy the drift criteria. This

example is included only to demonstrate how load factors can be calibrated for given

target probabilities and load combination format. If the drift criteria are considered in

designing the representative structures, then the member sizes of structures are almost

the same for each of the nine different load combination fonnats shown in Table 6.1

since drift controls the design. Therefore, the response surface will be relatively flat,

and the minimum point of the objective function (response surface) may be difficult to

find. Numerical Example II in Section 6.3 will consider drift explicitly.

6.2.4 Evaiuation of the Limit State Probabiiities

The limit state probabilities, PsI(YL, YE), are calculated considering the uncer-

99

tainties in both the earthquake excitation and the live load. Monte- Carlo simulation is

used to determine the conditional probabilities of exceeding drift limit thresholds given

the occurrence of an earthquake (See Chapter 3, Eq.(3.47), and Eq.(3.48)). The

conditional probabilities are detennined based on the results of dynamic response

analyses which are carried out using the Equivalent Nonlinear System (ENS) proce

dure described in Chapter 3. Twenty simulated characteristic earthquakes and fifty

simulated non -characteristic earthquakes are considered. Some simulated earth

quake samples are shown in Figure 6.3. For each dynamic analysis, a simulated value of

the random live load is generated using the statistical information given in the refer

ence by Wen (1990). Once the conditional probabilities are determined, the limit state

probabilities, Ps!(YL, YE), are determined using Eq.(3.46). These limit state probabili

ties represent exceedence probabilities over an assumed design life of the structure.

6.2.5 Detennination of Load Factors

For each pair of load factors (YL YE) shown in Figure 6.1, the "actual" value of the

objective function can be calculated using Eq.(6.4). Using these calculated values of

the objective function for each pair of load factors (YL, YE) for given target limit state

probabilities~ a "best fit" 2nd order polynomial (Eq.(S.17)) is determined by the least

squares method. The details of the procedures are described in Chapter 5. The result

ing 2nd order polynomial is a function of load factors YL and YE. Figures 6.4 and 6.5

provide comparisons of the actual and predicted values of objective function. Figure

6.6 shows the "best fit" 2nd order polynomial representation of response surface (ob

jective function). The load factors are detennined by minimizing the 2nd order

polynomial.

100

6.2.6 Results

The lifetime limit state probabilities of one story SMRSFs and two story SMRSFs

designed according to all nine load formats are shown in Tables 6.4 and 6.5. In this

example, the design lifetime of each structure is assumed to be 50 years.

Based on the results in Tables 6.4 and 6.5, it appears that load formats 2 and 6 for

the one story SMRSF and load format 6 for the two story SMRSF correspond to the

most conservative design; load case 4 for the one story SMRSF and load case 8 for the

!\Va story SMRSF correspond to the least conservative design.

The target limit state probabilities for the 50 year time window (1994-2044) for

the serviceability limit state are selected as 0.5, 0.55, 0.60, and 0.65, and the probabili

ties for the ultimate limit state are selected as 0.06, 0.07, 0.08, and 0.09. Two different

sets of weights, Wsb are considered. In the first set, the ratio of weights (serviceability:

ultimate) is 1 :2. In the second set, the ratio is 1 :5. One and !\Va story SMRSFs are

assigned equal weights.

Target probabilities for serviceability and ultimate limit states implied in the

NEHRP recommended provisions without consideration of allowable drift are 0.55

and 0.083, respectively, which is obtained as follows:

1) Multiply the limit state probability of each representative structure

for set number 1 in Tables 6.4 and 6.5 by the corresponding weights

for each structure. (Note that set number one represents the load

format used in the NEHRP provisions.)

2) Add all values obtained in step l.

3) Divide the results obtained from step 2 by the sum of the weights for

the structures. In this example, the weight for each structure type can

be considered as one since equal weights are assigned for the struc-

101

tures.

Tables 6.6 and 6.7 show the calibrated load factors corresponding to each of the

target limit state probabilities. A comparison of Tables 6.6 and 6.7 shows that putting

more weight on the ultimate limit state generally results in higher load factors. Also,

putting more weight on the serviceability limit state causes a wider range of variation in

the live load factor. These tables also show that the lower load factors correspond to

the higher target probabilities.

Comparisons of the actual and predicted values of the objective function are

shown in Figures 6.4 and 6.5. The differences between actual and predicted values can

be estimated using average residual. In this study, the average residual is defined as

follows;

ns

I (Q pij- Qai)2

j=l

nc ns-l

E(e-) = 1-I------~-- . 100 (%) l nc E(Qa)

i= 1

(6.5)

where ei denotes the residual for each pairs of target probabilities shown in Tables 6.6

and 6.7. i is the number of the pairs of target probabilities in Tables 6.6 and 6.7

(i = 1 , ... ,l1c) and j is the number of the sets of design parameters shown in Figure 6.1

G = 1, ... ,lls where 11:s is 9). Qaij and Qpij represent the actual and predicted values of the

objective function at design parameter setj for a given target probability set i. E(.)

denotes the average value. E(Qai) denotes the mean of the actual values of the objec

tive function for nine sets of design parameters given target probability set i. The

average residual is 4.75 % ; therefore, it can be concluded that the 2nd order polynomial

approximation of the objective function is valid.

Figure 6.6 presents a plot of the response surface (objective function). This

figure clearly illustrates how the response (objective function) is strongly dependent

102

on seismic load factor (YE). However, the dependence of the response on live load

factor (YL) is negligible. Because of the small dependency of the objective function on

the live load factor, it may be difficult to find the point minimizing the objective

function. In fact, for this example, it is difficult to minimize the objective function when

the target limit state probabilities are outside the ranges 0.5 -0.65 and 0.06-0.09 for

the serviceability and ultimate limit states, respectively. This clearly indicates that the

objective function should only be a function of those input variables which strongly

affect the response and reliability.

6.3 EXAM:PLE II : Calibration of Seismic Load Factor And Allowable Drift

This example will focus on calibrating the seismic load factor and the allowable

story drift since these parameters are expected to significantly influence the reliability

of the building. The seismic load factor and allowable story drift are calibrated based

on required (target) limit state probabilities using the RSM. Since the live load factor

(YL) is equal to 1.0 in the NcHRP provisions, the load format shown in Eq.(6.3) is can

be expressed as :

(6.6)

The allowable drift limit (Lla ) is especially important in the design of moment

resisting frames in a higb seismic zone. According to the NEHRP recommended

provisions (Table 3.8), the allowable drift limit is expressed as a fraction of the story

height (hsx). The fraction varies depending on the seismic hazard exposure group and

type of structure.

6.3.1 Objective Function

The input variables are seismic load factor (YE) and allowable drift limit (Lla ).

Therefore, the response (objective function) can be ,vritten as :

103

(6.7)

6.3.2 Selection of Combinations for Seismic Load Factors and Allowable Drifts

A central composite design is used to determine the combinations of YE andL1a to

be used in the analysis. Since there are 2 input variables, 9 pairs of (YE, Lla) are

required according to a central composite design (22 + 2x2 + 1). The values given in

the NEHRP recommended provisions forYE and L1a are used as the center point. The

seismic load factor in the NEHF~ recommended provisions is 1.0. A . ..s mentioned

earlier, the allowable drift depends on the seismic hazard exposure group and type of

structure. In this study, the SMRSF office buildings are classified into seismic hazard

exposure group 1. The corresponding allowable drift limit is O.015h:sx. As discussed in

Section 6.2.2, the factorial points and axial points are determined based on consider

ation of the range of values of YE and L1a within which the minimum of the objective

function is expected to occur. Figure 6.7 shows the configuration of the pairs of YE and

Lla values considered in this example. The load formats corresponding to each set in

Figure 6.7 are sho'WIl in Table 6.8.

6.3.3 Representative Structures

SMRSF structures ranging from 1 to 12 stories are considered. The design

variables (e.g., story height, number of bays and stories, etc.) considered in the design

of the structures are shown in Table 6.9. The ranges of the design variables are selected

based on current practice. In order to design the representative structures, representa

tive values of the design variables need to be selected. The selected values are also

shown in Table 6.9. For example, the representative values of the number of stories are

chosen as 2,3,4,5, 6, and 12. According to an ATe report (Rojahn and Sharpe, 1985 ),

104

structures with 1 to 3 stories, 4 to 7 stories, and 8 or more stories are classified as

low-rise, medium-rise, and high-rise structures, respectively. This report also

points out that the population of high -rise structures is very small compared with the

populations of low and medium-rise structures. For this reason, only one high-rise

structure (12 stories) is considered.

Considering all possible combinations of the representative values of design

variables in Table 6.9 is not appropriate for practical purposes due to the significant cost

and computational effort involved. Hence, a judicious selection procedure of input

variables is required. The latin hypercube sampling technique ( Iman and Conover,

1980) is used for this purpose. A brief description of this sampling technique is provided

in Section 6.3.4.

In this example, 6 different combination vectors of design variables (one vector

per representative structure) are established by the latin hypercube technique. These

combinations are shown in Table 6.10. The dimensions of the 6 representative struc

tures are shown in Figure 6.8 through Figure 6.13. Since there are 9 different pairs of

(YE, L1a) to be considered for each representative structure, fifty-four structures ( 6

structures x 9 load factor pairs per structure) are designed according to the NEHRP

recommended provisions and AISC LRFD manual. A detailed description of the

design procedure is shown in Chapter 4. As in Example I, only one load combination

(Eq.(6.6» is considered in the design. The member sizes and properties of each

structure are shown in Tables 6.11 to 6.16.

6.3.4 Brief Description of Latin Hypercube Sampling

The latin hypercube sampling technique can be used to the particular values from

a vector of input variables X=(X}, .... , XK). This procedure is as follows (Iman and

Conover, 1980) :

105

The range of each input variable is divided into N intervals , and one

observation on the input variable is made in each interval using random

sampling within each interval. Thus there are N observations (by strati

fied sampling) on each of the K input variables. One of the observations

on Xl is randomly selected (each observation is equally likely to be

selected) ,matched with a randomly selected observation X2, and so on

through XK. These collectively constitute &. . One of the remaining

observations on &is then matched at random with one of the remaining

observations on X2, and so on, to get Xl. A similar procedures is fol

lowed for Xl, X4, ... , XN, which exhausts all of the observations and

results in a Latin hypercube sample.

Hwang (1987) demonstrated that the latin hypercube sampling technique is a

systematic and efficient technique for random sampling. This technique is also used by

Hsu and Hwang (1991) for combining representative values of design variables.

6.3.5 Evaluation of Limit State Probabilities

As mentioned earlier (Section 6.2.4), limit state probabilities of each MDOF

structure are evaluated based on dynamic analyses using the Equivalent Nonlinear

System (ENS) methodology. The relevant dynamic properties of the MDOF structures

to be used for the ENS are shown in Tables 6.11 through 6.16. The procedure for

evaluating the limit state probabilities is the same as that described in Chapter 3 and

Section 6.2.4.

6.3.6 Determination of Seismic Load Factor and Allowable Drifts

The value of the objective function ( Eq.(6.7) ) is calculated for each pair of

seismic load factor (YE) and allowable drift limit (Lla ) and for each target limit state

106

probability. Then, a second order polynomial is established to represent the objective

function. The details of the procedure are described in Chapter 5. For a given target

limit state probability, YE and Lta are calibrated by minimizing the 2nd order polyno

mial representation of the objective function.

6.3.7 Results

The lifetime (50 years) limit state probabilities are shown in Tables 6.17 through

6.22. The target limit state probabilities for the time window 1994-2044 for the

serviceability limit state are selected as 0.25, 0.30, 0.35, 0.40, and 0.45. The target

probabilities for the ultimate limit state are chosen at 0.035, 0.040, 0.045, 0.050, 0.055,

0.060, 0.065, and 0.070.

Four different sets of weights for serviceability and ultimate limit state are consid

ered; the weight ratios (serviceability: ultimate) are 1:2, 1:5, 1:10, and 1:15. Two

different sets of weights for representative structure type are considered. The first set

of weights for 2,3,4,5 ,6, and 12 story SMRSFs is assumed to be 8, 8, 1, 1, 1, and 1,

respectively. (see Table 6.23) The second set of weights is assumed to be 20, 14,6,4,3,

and 1 for 2, 3, 4, 5, 6, and 12 story SMRSFs, respectively.

The rationale for the first set of weights for representative structures is based on

the ATe report by Rojahn and Sharpe (1985). It is based on the total floor area

associated with low-rise, medium-rise, and high-rise buildings as a measure of

importance. (The definition of low-rise, medium-rise, and high-rise is described in

Section 6.3.3) They report that 80 % of the total floor area of buildings in California is

associated with low-rise buildings and the remaining 20 % is associated with me

dium -rise buildings. (The floor area associated with high-rise building is negligible.)

For this study, to assign a weight to each structure type, 2 and 3 story SMRSFs are

classified as low rise buildings and 4, 5, 6, and 12 story SMRSFs are classified as

107

medium rise buildings. Therefore, the weight for each of the two types of low rise

buildings is 40 % (=80%/2), and the weight for each of the four types of medium rise

buildings is 5 % (=20%/4). If the weights are nonnalized by the weight assigned to

medium rise buildings, the weights of low and medium rise buildings become 8 and l.

The weights in set 1 drop suddenly at the transition from low-rise to medium

rise buildings. Intuitively, weights should increase gradually as the number of stories

decreases. For this reason, a weight function is proposed which is dependent on the

number of stories. As mentioned earlier, the ratio of the floor area of low-rise

buildings to that of medium rise buildings is 4 to 1. Since a two-story building is in the

middle of the low-rise building category, assume its weight is 4. Similarly, a weight of 1

is assumed for a building with 5.5 stories. (Even though 5.5 is not a realistic value for

the number of stories, this value is in the middle of the range of stories in the medium

rise building category ( (4+ 5 + 6+ 7)/4). Using these two data points and assuming that

the weight varies exponentially with the number of stories, the weight function be

comes

Ws = 8.83 exp( - 0.3959 * Ns) (6.8)

where Ws is weight assigned to a building with Ns stories. Figure 6.28 shows how the

weight function varies exponentially with the number of stories, and Table 6.28 shows

the corresponding normalized weights for the representative structures considered

herein.

Using the results presented From Tables 6.17 through 6.22, the probabilities for

serviceability and ultimate limit states implied in the NEHRP recommended provi

sions can be calculated using a procedure similar to that described in Section 6.2.6. For

the first set of weights for representative structures (i.e., 8:8:1:1 :1:1), the implied proba

bilities for serviceability and ultimate limit states are 0.39 and 0.060, respectively. For

108

the second set of weights (20:14:6:4:3: 1), the probabilities are 0.38 and 0.059, respec

tively, slightly lower than those of the first set. Acceptable risk have been proposed by

many researchers (Hays (1985), Galambos (1990), Hwang (1991), Kennedy (1992».

Figure 6.14 shows the acceptable levels of risk proposed by Hays for various classifica

tion of structures. According to this figure, the above implied risks (when converted to

0.0076 and 0.0012 annual exceedence probability) correspond to the suggested accept

able risk levels associated with conventional, residential, and commercial buildings and

facilities.

Figures 6.15 through 6.22 and Figures 29 through 36 show the differences be

tween actual values and predicted values of the objective function. The average residu

al can be estimated using Eq.(6.S), which is 2.92 %. The residual of this example

(Example II) is smaller than that of Example 1. Therefore, the second order polynomial

approximation of the objective function using the RSM is valid. Tables 6.24 through

6.27 and Tables 6.29 through 6.32 show the "calibrated" values of YE and L1a which

minimize thc ohjective function for a given set of target limit state probabilities.

Figures 6.23 and 6.37 present plots of the response surfaces (objective functions ).

These figures InJlwte that the objective function is strongly dependent on L1a.

The rcsu! ts fenerally show that the higher target limit state probabilities lead to a

higher allowa~k dnft limit ratio and a lower seismic load factor as expected. Figures

6.24 and 6.3~ prcse-n! plots of the seismic load factor \::. target serviceability limit state

probability (P.,·) fur pu • =0.06. These figures show the trend of dependence OfYE on

the target p~. They also show the sensitivity of the seismic load factor to a change in

target serviceabilIty limit state probability (Ps *) as the relative weight on the service

ability limi t sta te increases. Figures 6.25 and 6.39 are corresponding plots for ~a/hsx vs.

Ps *. It is seen that the allowable drift ratio (~a/hsx ) generally increases with target

exceedence probability; also, the allowable drift ratio becomes more sensitive to

109

chan ge in P s * as the weigh t of the serviceabili ty limi t state is in creased. Figures 6.26 an d

6.40 show seismic load factor vs. target ultimate limit state probability (Pu *) for Ps *

=0.35. Figures 6.27 and 6.41 are the plots for ~a/hsx vs. Pu *. These figures show the

trend of dependence of seismic load factor and allowable drift ratio on the target

ultimate limit state probability. It is seen that these two design parameters also become

more sensitive to change in Pu * as the weight of the ultimate limit state is increased.

It is important to note that this example considered only one site. If another site

within the same seismic zone (as defined in the NEHRP provisions) were considered,

the calibrated design parameters corresponding to given target probabilities may be

different froln those presented herein due to variations in local seismicity which are not

reflected in current seismic zoning maps. Micro-zoning (i.e., breaking down current

seismic zones into smaller zones of uniform seismicity) would reduce the site-to-site

variability in design parameters.

110

Table 6.1 Load Combinations Based on Central Composite Design

(Numerical Example I )

Set No. Load Fonnat

1 1.3D + 1.0L + 1.0E

2 1.3D + 1.5L + 1.5E

3 1.3D + 1.5L + 0.5E

4 1.3D + 0.5L + 0.5E

5 1.3D + 0.5L + 1.5E

6 1.3D + 1.0L + 1.7E

7 1.3D + 1. 7L + 1.0E

8 1.3D + 1.0L + 0.3E

9 1.3D + 0.3L + 1.0E

Table6.2MemberSizesandPropertiesofl StorySMRS(NumericalExampleI)

Mem Set Set Set Set Set Set Set Set no. no.l no.2 no.3 no.4 no.5 no.6 no.7 no.8

1, 2 2Ix44 24x55 16x40 18x35 24x55 24x55 21x50 18x35

3 18x40 21x50 18x35 16x31 21x44 21x50 21x44 16x31

R(1,2) 0.987 0.938 0.985 0.814 0.868 0.976 0.939 0.844

R(3) 0.969 0.894 0.962 0.972 0.894 0.899 0.883 0.984

T 0.68 0.54 0.83 0.87 0.55 0.54 0.61 0.87

Uy 1.25 1.10 1.50 1.40 1.08 1.10 1.20 1.38

Note: set numbers refer to number shown in Table 6.1 R(i) : Ratio of Factored Demand to Factored Capacity T : Fundamental period (sec) Uy : Global yield displacement (in) All members are W sections and member numbers are shown in Figure 6.1

Set no.9

18x40

18x35

0.963

0.931

0.78

1.32

111

Table 6.3 MemberSizesandPropertiesof2StorySMRSF(NumericalExam pleI)

Mem Set Set Set Set Set Set Set Set Set no. no.1 no.2 no.3 no.4 no.5 no.6 no.7 no.8 no.9

1,2 2Ix42 24x76 16x50 16x45 21x76 24x84 2Ix62 18x35 18x60

3,4 24x55 24x62 16x45 18x35 24x55 24x62 24x55 18x35 21x50

5 24x62 24x76 18x46 21x44 24x76 24x76 24x62 18x40 24x55

6 2Ix50 24x55 18x40 18x35 24x55 24x55 24x55 18x35 21x44

R(1,2) 0.946 0.971 0.950 0.925 0.919 0.949 0.974 0.915 0.910

R(3,4) 0.848 0.948 0.980 0.918 0.957 0.956 0.942 0.915 0.910

R(5) 0.888 0.932 0.990 0.927 0.870 0.974 0.963 0.947 0.982

R(6) 0.880 0.952 0.960 0.944 0.851 0.979 0.920 0.894 0.918

T 1.01 0.86 1.44 1.47 0.87 0.84 0.99 1.59 1.15

Uy 2.60 2.55 3.25 3.15 2.60 2.60 2.60 3.07 2.92

Note: set numbers refer to number shown in Table 6.1 R(i) : Ratio of Factored Demand to Factored Capacity T : Fundamental period (sec) Uy : Global yield displacement (in) All members are W sections and member numbers are shown in Figure 6.2

Table 6.4 Limit State Probabilities of 1 Story SMRSF (Numerical Example I)

Limit Set Set Set Set Set Set Set Set Set State nO.l no.2 nO.3 no.4 no.5 no.6 no.7 no.8 no.9

O.5%hsx 0.5456 0.4155 0.7156 0.7598 0.4350 0.4155 0.4645 0.7593 0.6555

1.5%hsx 0.0766 0.0693 0.1355 0.1517 0.0745 0.0693 0.0609 0.1512 0.1194

hsx : Story height (Note: Set number refers to the number shown in Table 6.1)

Table 6.5 Limit State Probabilities of2 Story SMRSF (Numerical Example I)

I Limit I Set I ~t~tf> I nil 1

...... ----- ----O.5%hsx 0.5586

l.5%hsx 0.0883

Set Set Set Set Set Set Set Set n () ? n () ~ nil .1 nn ~ TIn h nn 7 nn Q TIn 0 ---- -_.- .... _ ..

.&,.,A"".-" ~'-'."" ....... ""'., ..1..&",.\..1 .LAv./

0.3707 0.7385 0.7504 0.3886 0.3304 0.5256 0.8108 0.6327

0.0581 0.1536 0.1540 0.0610 0.0480 0.0851 0.2251 0.1056

hsx : Story height (Note: Set number refers to the number shown in Table 6.1)

Ps'"

Pu'"

YL YE

112

Table 6.6 Live Load Factor (YL) and Seismic Load Factor (YE) for

(Numerical Example I)

0.50 0.55 0.60 0.65 0.60 0.60

0.08 0.08 0.08 0.08 0.06 0.07

1.24 1.15 0.99 0.66 1.24 1.16

1.24 1.19 1.15 1.12 1.22 1.19

Note: ps• = target limit state for serviceability (0.5%hsx) pu• = target limit state for ultimate failure (1.5%hsx)

ps•

pu •

YL

YE

Weights for serviceability and ultimate limit states = 1:2 Weights for 1 and 2 story SMRSFs= 1:1

Table 6.7 Live Load Factor (YL) and Seismic Load Factor (YE) for

(Numerical Example I)

0.50 0.55 0.60 0.65 0.60 0.60

0.08 0.08 0.08 0.08 0.06 0.07

1.50 1.49 1.47 1.45 1.43 1.42

1.23 1.20 1.18 1.16 1.26 1.22

Note: ps• = target limit state for serviceability (0.5%hsx) Pu " = target limit state for ultimate failure (1.5%hsx) Weights for serviceability and ultimate limit states = 1:5 Weights for 1 and 2 story SMRSFs= 1:1 * Denot that load factors can not be determined.

0.60

0.09

0.67

1.13

0.60

0.09

* *

113

Table 6.8 Central Composite Design Combinations for Load Case and

Allowable Drift Limit (Numerical Example II)

Set No. Load Case Allowable Drift Limit (L1a)

1 1.3D + 1.0L + 1.0E 0.015 hsx 2 1.3D + 1.0L + 1.5E 0.018 hsx 3 1.3D + 1.0L + 0.5E 0.018 hsx 4 I.3D + 1.0L + 0.5E 0.012 hsx 5 1.3D + 1.0L + loSE 0.012 hsx 6 1.3D + l.OL + 1.7E 0.015 hsx 7 1.3D + 1.0L + 1.0E 0.020 hsx 8 I.3D + 1.0L + 0.3E 0.015 hsx 9 1.3D + 1.0L + 1.0 E 0.010 hsx

Table 6.9 Design Variables (Numerical Example II)

No. Design Variables Range Repesentative Values

1 No. of Stories 1-15 2,3,4,5,6,12

2 Story Height (ft.) 14-18 for 1st story 15,17 for 1st story 11-15 for all others 12,14 for all others

3 No. of Bays (N -S) 2-5 3,4

4 No. of Bays (E-W) 2-5 5

5 Span Length (ft.) 20-35 25,30

6 Dead Load (floor,psf) 90-100 95,100

7 Dead Load (top, psf) 75-85 80,85

8 Live Load (floor, psf) 40-55 45,50

9 Live Load (top, psf) 12-20 16

114

Table 6.10 Combinations of Representative Values of Design

Variables based on Latin Hypercube Sampling

(Numerical Example II)

Design Van- Frame 1 Frame 2 Frame 3 Frame 4 Frame 5 able Number

1 12 6 3 4 2

2 12(15) 14(17) 12(15) 14(17) 14(17)

3 4 3 3 4 3

4 5 5 5 5 5

5 25 25 30 30 30

6 100 95 100 95 95

7 80 85 85 80 85

8 50 45 45 50 45

9 16 16 16 16 16

Note : for design variable 2, a(b) denotes story height b for 1 st story and story height a for other stories

Frame 6

5

12(15)

4

5

25

100

80

50

16

115

Table 6.11 Member Sizes and Dynamic Properties of 2 Story SMRSF


Mem Set Set Set Set Set Set Set Set no. nO.l no.2 no.3 no.4 no.5 no.6 nO.7 nO.8

1,4 24x76 3000 24x68 27x84 3Ox90 3009 24x76 24x68

2,3 24x76 3000 24x68 27x94 30x90 3009 24x76 24x68

5,8 24x68 24x55 24x68 27x84 27x84 24x55 21x44 24x62

6, 7 24x68 24x55 24x68 24x94 24x94 24x62 21x50 24x68

9tol1 24x76 24x76 21x57 24x76 24x84 24x84 24x68 24x68

12to14 24x55 24x55 21x50 24x76 24x68 24x55 21x44 21x57

Tl 0.94 0.95 1.08 0.83 0.80 0.89 1.11 1.02

T2 0.29 0.26 0.30 0.23 0.23 0.25 0.33 0.30

<PI 0.71 0.72 0.71 0.71 0.72 0.72 0.71 0.71

¢2 -0.41 -0.40 -0.40 -0.41 -0.40 -0.40 -0.42 -0.42

fl 1.70 1.69 1.69 1.70 1.69 1.69 1.70 1.71

f2 0.51 0.54 0.53 0.50 0.53 0.54 0.50 0.48

RMP 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

Uy 2.6 3.1 2.9 2.6 2.6 3.1 3.2 2.8

Notes : Ti = ~eriod of ith mode (sec) <Pi = it modal displacement at the top fj = mass participation factor of ith mode RMP = ratio of Ifll + If21 to sum of Ifil of all modes Uy = global yield displacement (in) Member numbers are shown in Figure 6.13

Set nO.9

30x99

27xl02

30x99

27x102

24x94

24x94

0.72

0.20

0.71

-0.41

1.70

0.51

1.0

2.4

116



Mem Set Set Set Set Set Set Set Set no. no.1 no.2 nO.3 no.4 no.5 no.6 no.7 no.S

1,5,4,8 3Ox108 3OX10S 27x94 33x118 33x118 3Ox108 24x84 30x99

2,6,3,7 3Ox108 36x10S 27x94 33xl18 33x130 4Ox149 3Ox108 30x99

9,12 24x68 21x50 21x62 27x102 27xl02 24x55 21x44 24x62

10,11 24x68 24x55 21x62 27x84 27x94 24x55 21x44 24x62

13t015 24x84 27x84 24x68 27x102 27xl02 30x90 24x76 24x84

16t018 24x84 27x84 24x68 27x102 27xl02 30x90 24x76 24x84

19t021 21x50 21x50 18x46 21x57 21x57 24x55 21x44 21x50

T1 1.05 1.07 1.21 0.89 0.87 0.87 1.17 1.08

T2 0.34 0.35 0.40 0.29 0.28 0.32 0.42 0.36

<1>1 -0.59 -0.61 -0.59 -0.59 -0.59 -0.61 -0.60 -0.59

<1>2 0.44 0.42 0.44 0.44 0.44 0.42 0.43 0.44

r 1 -2.23 -2.19 -2.24 -2.24 -2.24 -2.19 -2.22 -2.24

r2 -0.85 -0.92 -0.85 -0.86 -0.86 -0.92 -0.88 -0.84

RMP 0.87 0.86 0.85 0.88 0.88 0.86 0.87 0.84

Uy 3.4 3.5 3.6 3.1 3.1 3.0 3.6 3.4

Notes: Ti = reriod of ith mode (sec) <Pi = it modal displacement at the top ri = mass participation factor of ith mode RMP = ratio of If 1 I + I r 21 to sum of 1 fi I of all modes Uy = global yield displacement (in) Member numbers are shown in Figure 6.14

Set nO.9

36x160

36x170

30x90

30x90

30x99

30x99

24x76

0.76

0.24

-0.59

0.43

-2.22

-0.S7

0.87

2.8

117



Mem Set Set Set Set Set Set Set Set no. no.1 no.2 nO.3 no.4 no.5 no.6 no.7 nO.8

1,6,5,10 3Ox124 33x141 30x99 33x130 33x141 36x150 30x99 3Ox116 2,7,3,8,4,9 3Ox124 3Ox124 3Ox116 36x135 36x135 36x135 33x118 3Ox116 11,16,15, 27x114 21x68 27x94

20 3Ox116 3Ox116 27x102 24x84 27x102

12,17,13, 27x102 24x84 27x102 18,14,19

3Ox124 3Ox124 30x99 24x84 27x102

21t028 30x99 30x99 27x84 3Ox132 3Ox132 30x99 24x84 30x99

29t032 24x94 24x84 24x68 3Ox108 3Ox108 24x76 24x76 24x94

33t036 24x55 24x55 14x48 16x57 16x57 16x57 21x50 16x50

T1 1.22 1.26 1.50 1.09 1.07 1.14 1.53 1.30

T2 0.40 0.49 0.51 0.38 0.35 0.40 0.48 0.45

<P1 -0.49 -0.49 -0.48 -0.49 -0.49 -0.49 -0.48 -0.49

<P2 -0.45 -0.45 0.44 -0.45 -0.45 -0.45 -0.44 -0.45

f1 -2.74 -2.72 -2.73 -2.75 -2.73 -2.76 -2.76 -2.73

f2 1.02 1.07 1.06 1.01 1.06 0.98 0.99 1.03

RMP 0.82 0.82 0.82 0.82 0.82 0.81 0.80 0.80

Uy 4.8 4.8 5.3 4.8 4.7 4.8 5.3 5.2

Notes: Tj = period of it:! mode (sec) <Pi = it modal displacement at the top fj = mass participation factor of ith mode RMP = ratio of If11 + If21 to sum of Ifil of all modes Uy = global yield displacement (in)

Member numbers are shown in Figure 6.15

Set no.9

4Ox167

4Ox167

4Ox167

36x170

33x141

3Ox108

21x62

0.92

0.31

-0.50

-0.44

-2.71

1.09

0.83

4.0

I

118




1,6,5, 10 27xl02 33x118 27x84 3Ox116 33x118 33x130 27x94 27x94 2,7,3,8,4,9 27xl02 33x118 24x94 3Ox124 33x118 36x135 27x114 27xl02 11,16,15,

20 27x84 24x76 24x68 3Ox99 3Ox99 24x84 24x68 24x84

12,17,13, 27x84 24x76 24x76 18,14,19

3Ox90 3Ox99 24x84 24x68 24x84

21,25 16x50 18x35 18x40 21x57 21x68 21x44 14x43 14x48 22,23,24 16x57 18x40 18x40 21x50 21x62 21x44 14x43 16x57

26t033 24x94 27x84 24x76 3Ox99 3Ox99 30x90 24x68 24x94

34t041 24x76 24x76 24x68 3Ox90 3Ox99 24x84 24x62 24x76

42t045 14x48 14x48 14x34 18x46 18x40 21x44 14x43 14x48

Tl 1.41 1.42 1.58 1.16 1.12 1.18 1.60 1.44

T2 0.48 0.42 0.54 0.39 0.37 0.40 0.53 0.49

<PI -0.47 -0.48 -0.47 -0.46 -0.46 -0.49 -0.47 -0.47

<P2 -0.49 -0.48 -0.50 -0.50 -0.51 -0.48 -0.49 -0.49

f1 -2.88 -2.85 -2.89 -2.90 -2.89 -2.84 -2.87 -2.89

f2 1.03 1.09 1.00 0.98 1.01 1.11 1.03 1.01

RMP 0.78 0.77 0.78 0.79 0.78 0.77 0.78 0.78

Dy 5.0 4.5 5.0 4.1 3.8 4.2 5.0 4.9

Notes : Ti = feriod of it] mode (sec) <Pi = it modal displacement at the top fi = mass participation factor of ith mode RMP = ratio of If11 + If21 to sum of Ifd of all modes Uy = global yield displacement (in) Member numbers are shown in Figure 6.16

Set no.9

33x141

36x150

33x118

33x130

24x76

24x68

3Ox116

3Oxl08

18x50

1.00

0.34

-0.47

-0.50

-2.89

1.01

0.80

3.8

119



Mem Set Set Set Set Set Set Set Set no. nO.l no.2 no.3 nO.4 no.5 no.6 no.7 no.8

1,5,4,8 33xl18 33x130 3Oxl16 33x118 36x135 33x130 3Oxl16 3Oxl16 2,6,3,7 3Ox108 36x135 3Ox116 33xl18 33x118 4Ox149 33x118 3Ox108

9,13,12,16 3Oxl16 30x99 3Ox108 3Ox108 3Ox124 30x99 3Ox116 30x90 10,14,11,

15 30x99 3Ox116 3Ox108 3Ox116 3Oxl16 30x90 33x118 30x99

17,21,20, 24

24x76 24x62 24x62 24x84 24x94 24x68 24x68 24x62

18,22,19, 24x68 23

24x68 24x62 24x94 24x84 24x68 24x68 24x68

25t030 27x94 24x84 24x84 3Ox116 3Ox116 30x90 24x68 27x94

31 t036 27x94 24x84 24x84 3Oxl16 3Ox116 30x90 24x68 27x94

37t042 24x62 24x55 24x55 24x76 24x84 24x68 21x50 24x62

T} 1.58 1.68 1.75 1.36 1.34 1.46 1.87 1.61

T2 0.57 0.59 0.62 0.49 0.48 0.53 0.65 0.58

cp} -0.58 -0.59 -0.57 -0.58 -0.58 -0.59 -0.58 -0.58

CP2 0.59 0.58 0.58 0.58 0.59 0.58 0.59 0.57

rl -2.35 -2.32 -2.35 -2.36 -2.35 -2.31 -2.33 -2.36

r2 -O~~ -0.90 -0.88 -0.87 -0.88 -0.92 -0.89 -0.87

RMP o -'1 068 0.71 0.70 0.70 0.68 0.70 0.71

Uy bh 7.1 7.0 6.2 6.1 6.6 6.9 6.6

Notes: T i ::; period of ith mode (sec) ¢l. ::: l~:' modal displacement at the top r 1 = mas.s panicipation factor of ith mode R~{P :::. ratlO of I r 11 + I f21 to sum of I fd of all modes tJ, ;; glohal ~leId displacement (in) Memhcr numhers are shown in Figure 6.17

Set no.9

36x150

36x150

33x130

36x150

30x90

30x90

33x118

33x118

27x84

1.18

0.42

-0.59

0.58

-2.36

-0.88

0.72

5.7

120

Table 6.16 Member Sizes and Dynamic Properties of 12 Story SMRSF (Numerical Ex~'TIple II)


1,6,5,10 36x160 40x244 3Ox132 36x170 40x244 40x268 36x160 33x141 2,7,3,8,4,9 33x152 33xl18 3Ox132 36x170 36x170 33x130 33x118 33x152 11,16,15, 36x160 3Ox124 3Ox124 36x170 36x170 40x215 3Ox124 33x141

20

12,17,13, 33x152 3Ox116 3Ox132 36x170 36x170 33x130 3Ox124 33x152 18,14,19

21,26,25, 33x141 3Ox108 3Oxl16 36x170 36x170 36x210 3Ox116 33x118 30

22,27,23, 28,24,29

33x141 3Oxl16 3Oxl16 36x170 36x170 33x130 3Ox116 33x130

31,36,35, 40

3Ox132 30x90 30x99 36x170 36x170 3Ox124 30x99 3Ox116

32,37,33, 38,34,39

33x141 30x99 30x99 36x170 36x160 3Ox124 30x99 33x130

41,46,45, 3Ox99 50

27x84 27x102 3Ox173 3Ox132 30x90 27x84 30x99 42,47,43, 48,44,49

3Ox124 27xl02 27x102 3Ox173 3Ox173 30x90 27xl02 3Ox108

51,56,55, 60

21x83 21x62 21x73 24x84 24x84 24x68 21x68 21x73

52,57,53, 21x93 21x83 21x83 24x84 24x84 24x68 21x68 21x83 58,54,59

61to68 3Ox124 3Ox108 3Oxl08 33x141 33x141 3Ox124 3Oxl08 3Ox124 69to76 3Ox124 3Ox108 3Oxl08 33x141 33x141 3Ox124 3Oxl08 3Ox124 77to84 3Ox124 3Oxl08 3Ox108 3Ox418 3Ox148 3Ox124 3Oxl08 3Ox124 85to92 3Ox99 30x90 30x90 30x124 3Ox124 3Ox108 30x90 30x99 93to100 27x84 24x76 24x76 3Ox90 3Ox90 27x94 24x76 27x84 101to 108 21x62 21x50 21x50 24x76 24x76 21x57 21x57 21x62

Tl 2.13 2.24 2.42 1.89 1.80 1.99 2.33 2.15 T2 0.77 0.84 0.89 0.671 0.65 0.71 0.85 0.77 cJ>1 -0.36 -0.38 -0.36 -0.36 -0.37 -0.37 -0.37 -0.36 cJ>2 0.41 0.43 0.42 0.39 0.40 0.42 0.41 0.41 rl -3.87 -3.77 -3.92 -3.88 -3.82 -3.83 -3.84 -3.87 rz -1.49 -1.53 -1.46 -1.51 -1.54 -1.47 -1.50 -1.49

RMP 0.61 0.55 0.62 0.62 0.58 0.57 0.59 0.62 Uy 10.5 10.5 10.5 9.5 9.5 10.0 10.3 10.2

Notes: Ti = ~eriod of it:l mode (sec) cJ>i = it modal displacement at the top ri = mass participation factor of ith mode RMP = ratio of I rll + Ir 21 to sum of Iril of all modes Uy = global yield displacement (in) Member numbers are shown in Figure 6.18

Set no.9

4Ox183 36x182 4Ox183

36x182

4Ox183

36x182

36x182

36x182

33x141

33x141

24xl03

24xl03

4Ox167 40x167 36x170 33x152 33x130 24x94 1.59 0.55

-0.36 0.38

-3.89 -1.52 0.63 8.5

121

Table 6.17 Limit State Probabilitiesfor2StorySMRSF (Numerical Example II)

Limit Set Set Set Set Set Set Set Set Set State no.1 no.2 nO.3 nO.4 no.5 no.6 no.7 no.8 no.9

O.5%hsx 0.3996 0.4146 0.4732 0.2546 0.2322 0.3237 0.5229 0.4392 0.2323

l.O%hsx 0.1231 0.1285 0.1391 0.0817 0.0765 0.1020 0.1525 0.1338 0.0791

l.5%hsx 0.0655 0.0682 0.0717 0.0431 0.0405 0.0545 0.0764 0.0704 0.0388

2.0%hsx 0.0418 0.0431 0.0451 0.0272 0.0256 0.0347 0.0478 0.0449 0.0213

2.5%hsx 0.0295 0.0316 0.0314 0.0190 0.0179 0.0244 0.0332 0.0316 0.0185

3.0%hsx 0.0221 0.0223 0.0233 0.0142 0.0134 0.0183 0.0246 0.0237 0.0140

Table 6.18 LimitStateProbabilitiesfor3StorySMRSF(NumericaIExampleII)

Limit Set Set Set Set Set Set Set Set Set State no.1 no.2 no.3 no.4 no.5 no.6 no.7 no.8 no.9

O.5%hsx 0.4186 0.4578 0.5015 0.2893 0.2661 0.3695 0.4865 0.4760 0.2031

1.0%hsx 0.1216 0.1287 0.1452 0.0885 0.0845 0.1169 0.1397 0.1304 0.0577

l.5%hsx 0.0638 0.0688 0.0716 0.0500 0.0448 0.0560 0.0764 0.0730 0.0339

2.0%hsx 0.0405 0.0422 0.0446 0.0294 0.0283 0.0348 0.0469 0.0433 0.0244

2.5%hsx 0.0283 0.0300 0.0309 0.0205 0.0198 0.0269 0.0343 0.0312 0.0155

3.0%hsx 0.0211 0.0229 0.0228 0.0153 0.0148 0.0187 0.0254 0.0232 0.0119

Table 6.19 Limit State Probabilities for4 Story SMRSF (Numerical Example II)


O.5%hs.x 0.3012 0.3128 0.4140 0.2323 0.2247 0.2585 0.4332 0.3315 0.1666 1.0 % hsx 0.0842 0.0870 0.1170 0.0728 0.0686 0.0753 0.1249 0.0911 0.0524 1.5%hsx 0.0432 0.0442 0.0584 0.0382 0.0357 0.0392 0.0621 0.0463 0.0273 2.0%hsx 0.0270 0.0283 0.0366 0.0240 0.0223 0.0245 0.0388 0.0289 0.0171 2S%hsx 0.0187 0.0191 0.0254 0.0168 0.0155 0.0170 0.0270 0.0200 0.0119 3.0%hsx 0.0139 0.0141 0.0189 0.0125 0.0115 0.0126 0.0200 0.0148 0.0089

122

Table 6.20 Limit State Probabilitiesfor5 StorySMRSF(Numerical Exam pleII)


O.5%hsx 0.3527 0.3612 0.4263 0.2337 0.2044 0.2718 0.4346 0.3712 0.1864

1.0%hsx 0.0938 0.0987 0.1196 0.0699 0.0644 0.0782 0.1234 0.1005 0.0625

l.5%hsx 0.0456 0.0470 0.0593 0.0364 0.0334 0.0406 0.0612 0.0495 0.0332

2.0%hsx 0.0282 0.0288 0.0369 0.0229 0.0208 0.0254 0.0381 0.0307 0.0211

2.5%hsx 0.0195 0.0207 0.0255 0.0159 0.0144 0.0177 0.0264 0.0212 0.0149

3.0%hsx 0.0144 0.0148 0.0189 0.0118 0.0107 0.0131 0.0195 0.0156 0.0112

Table 6.21 LimitStateProbabilitiesfor6StorySMRSF(NumericalExampleII)

Limit Set Set Set Set Set Set Set Set Set State no.1 no.2 no.3 nO.4 no.5 no.6 no.7 no.8 no.9

0.5o/chsx 0.2967 0.3449 0.3665 0.2288 0.2158 0.2890 0.4120 0.3106 0.1651

1.0%hsx 0.0891 0.1042 0.1085 0.0624 0.0605 0.0804 0.1243 0.0945 0.0529

1.5%hsx 0.0470 0.0548 0.0567 0.0319 0.0313 0.0413 0.0646 0.0500 0.0278

2.0%hsx 0.0299 0.0348 0.0359 0.0200 0.0196 0.0261 0.0409 0.0318 0.0176

2.5%hsx 0.0210 0.0244 0.0252 0.0139 0.0136 0.0183 0.0286 0.0224 0.0123

3.0%hsx 0.0157 0.0183 0.0188 0.0103 0.0101 0.0137 0.0213 0.0168 0.0092

Table 6.22 LimIt Stdte Probabilitiesfor12StorySMRSF(NumericaIExampleII)

Limit Set Sct Set Set Set Set Set Set Set State nO.l no.2 no.3 no.4 Do.5 no.6 no.7 no.8 no.9

0.5 % hsx O.~lb 0.lb26 0.3001 0.1939 0.1917 0.1991 0.2848 0.2217 0.1407

1.0%hsx 0.06""-; 00731 0.0785 0.0623 0.0605 0.0634 0.0762 0.0677 0.0466

l.5%hsx O.O:;"t) 00356 0.0375 0.0318 0.0334 0.0367 0.0367 0.0346 0.0245 2.0%hsx 0.0:: 14 U.0213 0.0222 0.0207 0.0201 0.0211 0.0217 0.0214 0.0155 2.5%hsx 0.01";- 0.0142 0.0148 0.0144 0.0140 0.0148 0.0145 0.0147 0.0109 3.0%hsx 0.0108 0.0102 0.0106 0.0107 0.0104 0.0110 0.0103 0.0108 0.0081

Table 6.23 Weight Set Number 1 Used in Numerical Example II

I No. of Stories 2 3 4 5 6 12

Weight 8 8 1 1 1 1

I

123

Table 6.24 SeismicLoad Factor (YE) and Allowable Drift Limit (~a) for Various

Target Limit State Probabilities (Numerical Example II)

I"'.aliU Ul yvel~lll~ lUI '::>C;1 Yl\,.;-c;dUlillY dllU UlLllUaLC; l.JilUlL \.:Ha.L~;' = .1.":;"

Ratio of Weights for 2, 3, 4; 5,6, and 12 Story SMRSFs= 8:8:1:1:1:1

Ps • 0.25 0.30 0.35 0.40 0.45 0.50 0.35

pu • 0.060 0.060 0.060 0.060 0.060 0.060 0.040

YE * 1.41 1.30 1.11 0.92 0.73 1.41

~a!hsx * 0.013 0.015 0.016 0.017 0.018 0.014

Ps • 0.35 0.35 0.35 0.35 0.35 0.35 0.35

Pu'" 0.045 0.050 0.055 0.060 0.065 0.070 0.075

YE 1.38 1.35 1.33 1.3 1.28 1.26 1.25

~a/hsx 0.014 0.015 0.015 0.015 0.015 0.015 0.016

Note: Ps'" = target limit state probability for serviceability (0.5%hsx) p u • = target limit state probability for ultimate failure (1.5%hsx) * denote that design parameters can not be determined.

Table 6.25 SeismicLoadFactor(YE)andAllowableDriftLimit(~a)forVarious


Ratio of Weights for Serviceability and Ultimate Limit States = 1:5 T"l~.:~ ~&'"lT~:_l..~&~_'" 'J ..1 t: C. ~_....l 1.., C'.~_. C'l.A"TlC'1:'~_ 0.0.1.1.1.1 I".a.L1U Ul YVC::l~Ul~ lUI L.., .:J, '+, ..J, U, dUU .lL.. .::>lUlY '::>lV.lL"..::>r~'-' 0.0.1.1.1.1

P * s 0.25 0.30 0.35 0.40 0.45 0.50 0.35 P ,.

u 0.060 0.060 0.060 0.060 0.060 0.060 0.040

YE 1.30 1.41 1.28 1.12 0.96 0.82 1.47

~a/hsx 0.012 0.014 0.015 0.016 0.017 0.017 0.012

Ps • 0.35 0.35 0.35 0.35 0.35 0.35 0.35 P . u 0.045 0.050 0.055 0.060 0.065 0.070 0.075

YE 1.45 1.39 1.34 1.28 1.23 1.19 1.15

~a/hsx 0.013 0.014 0.015 0.015 0.016 0.016 0.017

Note: Ps"' = target limit state probability for serviceability (0.5%hsx) p "' ~ t':lT"Opt lirnit C't-:ltp T"\T"nh-:lh;1;n7 fr'\'r ll1t;rn-:ltp f'::l;ll1'rp (1 'OJ;..'J.... \ .&. u ........ 6 ...... ~ • .u ....... ~ "'~u. .. ~ t'.I.,-,..,u."'~.I."J ,&,'-'.1. L.£ ... I..J..I. ....... ~ .I.'-U.I.L.£I."-' \.1. • ..,1 /UI.I.SX)

* denote that design parameters can not be determined.

124

Table 6.26 Seismic Load Factor (YE) and Allowable Drift Limit (L\a) for Various


Ratio of Weights for Serviceability and Ultimate Limit States = 1: 1 0 Ratio of Weights for 2, 3, 4, 5, 6, and 12 Story SMRSFs= 8:8:1:1:1:1

ps • 0.25 0.30 0.35 0.40 0.45 0.50 0.35

pu • 0.060 0.060 0.060 0.060 0.060 0.060 0.040

YE 1.45 1.38 1.25 1.13 1.00 0.89 * L\alhsx 0.013 0.015 0.015 0.016 0.017 0.017 *

ps • 0.35 0.35 0.35 0.35 0.35 0.35 0.35

Pu* 0.045 0.050 0.055 0.060 0.065 0.070 0.075

YE 1.52 1.44 1.34 1.25 1.17 1.10 1.03

L\alhsx 0.013 0.014 0.015 0.015 0.016 0.017 0.018

Note: Ps = target limit state probability for serviceability (0.5%hsx) Pu lit = target limit state probability for ultimate failure (1.5%hsx)

* denote that design parameters can not be determined.

Table 6.27 Seismic Load Factor (YE) and Allowable Drift Limit (L\a) forVariou s


Ratio of Weights for Serviceability and Ultimate Limit States = 1:15 Ratio of Weights for 2, 3, 4, 5, 6, and 12 Story SMRSFs= 8:8:1:1:1:1

Ps * 0.25 0.30 0.35 0.40 0.45 0.50 0.35

Pu • 0.060 0.060 0.060 0.060 0.060 0.060 0.040

YE 1.45 1.35 1.23 1.12 1.02 0.94 * L\alhsx 0.014 0.015 0.016 0.016 0.017 0.017 *

Ps" 0.35 0.35 0.35 0.35 0.35 0.35 0.35 ..

0.045 0.050 0.055 0.060 0.065 0.070 0.075 Yu

YE 1.56 1.47 1.36 1.23 1.13 1.04 0.96

L\alhsx 0.012 0.014 0.015 0.015 0.016 0.017 0.018

Note: P s = target limit state probability for serviceability (0.5%hsx) Pu lit = target limit state probability for ultimate failure (1.5%hsx) * denote that design parameters can not be determined.

125

Table 6.28 Weight Set Number 2 Used in Numerical Example II

I No. of ~tories I 2 3 4 5 6 12

20 14 6 4 3 1

Table 6.29 Seismic Load Factor (YE) and Allowable Drift Limit (~a) for Various Target Limit State Probabilities (Numerical Example II)


Ps • 0.25 0.30 0.35 0.40 0.45 0.50 0.35

pu • 0.060 0.060 0.060 0.060 0.060 0.060 0.040

YE * 1.40 1.21 1.02 0.85 0.75 1.34

~a!hsx * 0.014 0.015 0.016 0.017 0.018 0.014

P * s 0.35 0.35 0.35 0.35 0.35 0.35 0.35

P '" u 0.045 0.050 0.055 0.060 0.065 0.070 0.075

YE 1.30 1.26 1.23 1.21 1.19 1.17 1.15

~a!hsx 0.014 0.015 0.015 0.015 0.016 0.016 0.016

Note: Ps = target limit state probability for serviceability (0.5%hsx) Pu * = target limit state probability for ultimate failure (1.5%hsx) * denote that design parameters can not be determined.

Table 6.30 Seismic Load Factor (YE)andP.JlowableDriftLimit(~a)forVarious Target Limit State Probabilities (Numerical Example II)


Ps * 0.25 0.30 0.35 0.40 0.45 0.50 0.35

Pu'" 0.060 0.060 0.060 0.060 0.060 0.060 0.040

YE 1.48 1.36 1.18 1.02 0.88 0.76 1.42

~a!hsx 0.013 0.014 0.015 0.016 0.017 0.018 0.013 p.

s 0.35 0.35 0.35 0.35 0.35 0.35 0.35 p '" 0.045 0.050 0.055 0.060 0.065 0.070 0.075 .... u

YE 1.36 1.30 1.24 1.18 1.13 1.09 1.05

~a!hsx 0.014 0.015 0.015 0.015 0.016 0.016 0.017

Note: Ps = target limit state probability for serviceability (0.5%hsx) P u * = target limit state probability for ultimate failure (1.5%hsx) * denote that design parameters can not be determined.

126

Table6.31 SeismicLoadFactor(YE)andA1lowableDriftLimit(~a)forVarious


Ratio of Weights for Serviceability and Ultimate Limit States = 1:10 Ratio of Weights for 2, 3, 4, 5,6, and 12 Story SMRSFs= 20:14:6:4:3:1

Ps* 0.25 0.30 0.35 0.40 0.45 0.50 0.35

pu· 0.060 0.060 0.060 0.060 0.060 0.060 0.040

YE 1.46 1.30 1.15 1.01 0.91 0.82 1.44

~a!hsx 0.014 0.015 0.016 0.017 0.017 0.018 0.011

Ps"' 0.35 0.35 0.35 0.35 0.35 0.35 0.35

Pu* 0.045 0.050 0.055 0.060 0.065 0.070 0.075

YE 1.43 1.33 1.23 1.15 1.07 0.99 0.93

~a/hsx 0.013 0.014 0.015 0.015 0.016 0.016 0.017

Note: P s = target limit state probability for serviceability (0.5 %hsx) p u• = target limit state probability for ultimate failure (1.5%hsx)

Table 6.32 SeismicLoadFactor(YE) andA1lowableDriftLimit(~a)forVarious



ps· 0"'-.,D 0.30 0.35 0.40 0.45 0.50 0.35 p

u• 0.060 0.060 0.060 0.060 0.060 0.060 0.040

YE 1.41 1.25 1.12 1.01 0.92 0.85 * ~a!hsx 0.014 0.015 0.016 0.017 0.017 0.017 *

ps· 0.35 0.35 0.35 0.35 0.35 0.35 0.35 pu· 0.045 0.050 0.055 0.060 0.065 0.070 0.075

""T" 1.47 1 ~h 1 'J'\ 1.12 1.02 0.93 0.86 It:, ..L • ..JV .&. • ....,~

~a!hsx 0.013 0.014 0.015 0.015 0.016 0.017 0.018

Note: Ps = target limit state probability for serviceability (0.5%hsx) Pu * = target limit state probability for ultimate failure (1.5%hsx) * denote that design parameters can not be detennined.

6@30'

127

YE 64 t (1.0,1.7)

(05,1.5) (15,1.5)

5 2

(0.3,1.0 ) 1 (i.0,1.0)

(1.7,1.0)

--9 7

YL

4 3 (05,05) (15,05)

&. (i.0,0.3 )

j2aL aL Figure 6.1 Configuration of Central Composite Design

for Load Factors (Numerical Example I)

6

15' 3

3 5

25' 25'

Section 1-1 Section 1-1

Figure 6.2 Dimensions of Structures and Member Numbers of Structures ( 1 and 2 Story SMRSFs ) for Numerical Example I

50 Q) ~ 30 ~ ~ 10 5 ~ -10 on c: ~-30

-50

50 ~ C'I 30 ~ 1s 10 5 ~-10 on 3 -30

-50

50 d) C'I 30 t$ ~ 10 ~ ~ ~ -> -10 on 2- -30

-50

o 10

o 10

o 10

, I , I

o 10

128

Ch-record 02

20 Ch-record 09

20 Ch-record 12

20 Non-Ch record 05

, : , ' , I , , , ,

20

30 40

30 40

30 40

, I , , I 30 40

200 Non-Ch record 08

ON120~ I ~ ~ 40

~ !j~g~~=~:~-; : ,-,: -, I ,------I,

50 - C'I 30 § ~ 10 5 ~-10 on § =30

-50

o 10

t ! , I ,

o 10

20 Non-Ch record 41

, I ! I I J !

20

time(sec)

30

, I I

30

Ch: characteristic EQ. Non Ch : non characteristic EQ.

Figure 6.3 Typical Simulated Earthquake Records for L.A. Site

40

, I

40

129

1: ! • • :s :I

"6 600 a 600 > >

'tJ 'tJ

• ! ... c 0 ii 400 1; 400 ! ! Cl

11.

200 200

200 ~ 600 SX) 1(01 1(0)

kWaI a,e Acblalvaiue

1 (XX) 1CXXl

SX)

/-!IX)

• • :s :I

'0 600 a 600 > > v r ] ! / 0 c lj 400 ii 400 • ! ~ Cl 11.

200 200

0 0 200 -400 600 SX) 1(01 200 400 600 IDl 1(0)

kWaI wail! AcbIaI waIue

1: ~ • • 11D) :s :I

"6 600 a > > 'tJ 'tJ

• • ... ... 0 c 1; 400 ii • ! ~ 500 Cl 11.

200

0 200 400 600 ID) 1(0) 0 SiX) 1(0) 1500

IdJIai WIM iduaiwaiue

Figure 6.4 Scatterplots of Objective Function (Ratio of Weights for Serviceabilty to Ultimate Limit States =1:2)

130

1600

• • :I :I 'ij 1200 'ij 1200 > >

" " • • 0 1) ~ !Xl ~ IX) • ! L a. a.

.G) 400

0 0 400 !Xl 1200 11Xl 2CXX)

ktuaI vakJe kbD_ue

2(XX) ml

1600 1600

• • .2 :l

" 1200 'ij 1200 > >

" " ! • 0 0 ~ &Xl ~ IDl • ! L a. a.

400 400

ActuaI_ue

2500 ml

2(XXi 0 2500

• • ml :I :I C 1500 'ij > >

" " 1SXl • • ., .. 0 0 ~ HDJ ~ • ! L 1(0) a. a.

Target Probabilities: 50) 0.60 (service)

I 0.06 (ultimate)

I I I I I I I I I I I I I I I I

0 1(0) 15CX1 m 2SOO 5CXI 1COl 15CX1 2CDl 2SOO ml ktuaI vakJe /dual waIue

Figure 6.5 Scatterplots of Objective Function ( Ratio of Weights for Serviceabilty to Ultimate Limit States = 1 :5)

131

Figure 6.6 Fined 2nd Order Polynomial Representing Objective Function (Ratio of Weights for Serviceability to Ultimate Limit States = 1 :5)

(Target Probabilities = 0.65 (service), 0.06 (ultimate) ) Example I

L1a1hsx

7 • (i.0,0.020)

I f f) , ~<' (15,0.018) _' '~' J l .I

3 2

tl ~ , . , (1.7,0.015 ' - 1 (1.0,0.015)

~ 6

YE

J 5

,t ( , ...... ,.., f1~ n7?1

(i.O,O.OiO)

Figure 6.7 Configuration of Central Composite Design for Seismic Load Factor (YE) and Allowable Drift Ratio Limit (~a!hsx)


14I 171

T

5@30'

CD t I

12

5 9 6

1 2

132

13

10 "'lJ

3

Section 1-1

3@30'

14

11 8

4

CD t I

Figure 6.8 Dimensions and Member Numbers of 2 Story SMRSF

(Numerical Example 2)

12'

15'

5@30'

133

19 20

9 16 10 17 1]

5 13 6 14 7

1 2 3

Section 1-1

I I I I 11 • I I I

-~ -I-

----

----

-I---I I I I I I I I I I • I

3@30'

21

1~ 18 ...

15 8

4

I I

.~

.~

.-

--

--I I I I



33

1"', 29

25 12

134

34

30 18

1~ 26 .-

35 36

31 IS 32 2C

27 lL 28 13

14'

Ie

11

6 21 7 22 8 23 9 24 1C

17' 1 2 3 4 5

5@30'

L}}~{:~:~:}}~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~t:~:r~:~:~:~:~:~:~:~:~:~:~:}~:::~:~:~:~:~:r~t:~:~:~:~:~:~:~:~:~:~:~:~:}~:~:~:~:~:~:}~:}~{:~:r~:~:~:~:~:~:~:~:~:~:~:~:~:}~:~:~:~:~:~:~:~:~:~:~:~:~:~:f~:~:f!:~{:!:!:~:!:!:fri:!t:!:~

CD Section 1-1 CD I • I I I I I I

I I I I I I I I I

I I I I

.~ ---I- --

.-.-

.~ -l-.- -I-J

1 I ,I, ,I, '"

,I, ,I f

4@30'



135

42 43 44 45

12' 21 38 22 39 2: 40 2~ 41 23

12' Ie 34 I~ 35 I~ 36 IS 37 2C

11 30 1~ 31 1~ 32 lL 33 1:

6 26 7 27 8 28 9 29 lC

1 2 3 -4 5

Section 1-1 CD

j • • • •• I • I • I I j

I I I I I I I I I 1

-~ ---~ --

-- -~ -- -~

5@25'

-~ ---- --

-.. -~ -~ -..

1 I • I • •• I I • • I I I I •• I I I I

4@25' Figure 6.11 Dimensions and Member Numbers of 5 Story SMRSF


14'

14'

T

5@25'

1

CD t I

21

1~

1~

9

5

1

I I I I

-~ --

-----t--t-

-~ -t-

• I I I

40

37

34

31

28

25

136

41

2~ 38 2:

1~ 35 IS

lL 32 1:

1( 29 1]

6 26 7

2 3

Section 1-1

I I I I I I I I

_J I I I I I I I

3@25'

42

39

36

33

30

27

2~

2(

1t

1~

8

4

I I T -.

-~

-~

-~ -~

----

----I I 1 -I

CD t I



11@12'

5@25'

137

56

51

46

41

36 31

26

21

16

4@25' (Section 1-1) CD

I I I I I , , I I I I I I I I I I • I I

-~ -~

-~

-- -~ -- -~

-- -~ -- -'-

I I I I I I I I I • I I I I I I I I

4@25' Figure 6.13 Dimensions and Member Numbers of 12 Story SMRSF


138

Common Perception of Risk Proposed Land Uses in Terms of Annual

For Significant Probability of Risk

For Fatality and Occurrence Economic loss

High Open Space, Recreational and Essential Uses Only

--- 10-1 C) High Non-occupancy is co recommende. u

CIJ c.o Moderate 0

-J -- 10-2 C) :..;, -u5nven tlonal, ResIdentIal c C) and Commercial Build-~ ~

::s Low ings and Facilities (Build-u u 0 ... 10-3 0

C

ing according to building code)

Important Building and .-.c ~ .c 0 ~

Moderate Facilities having High Oc-cupancy (Site investiga-tions are required.)

Q.. 10-4 ~ ::l C .... ~

Critial and Emergency Fa-Negligible cilities (Site investigations

are required.)

10-5 Nuclear Power Plants and

Low Large Dams required to Remain Functional (Site investigations are re-auired.1

Figure 6.14 Suggested Criteria for Acceptable Risk (Hays 1985)

~ ~----------------------------~

~ ml.O '0 > "0 • .. o :s ~ 1cmD

0.0 HXnD

xaw ~----------------------------~

~ zmo '0 >

" ! o ii ~ lImO

~~ .. ~~~~~~ .. ~~~~~~~ 0.0

139

~ ~-----------------------------.

~~~ .... ~~~~~~~~~ .. ~~ ~ 11mO

~ ~----------------------------~

~ m.o i >

" • ii ii ~ 11mO

~

Figure 6.15 Scatterplots of Objective Function (Ratio of Weights for Serviceability to Ultimate Limit States=l :2)

(Weights for Structures=8:8: 1: 1: 1: 1)

~ r-----------------------------~

~ 2IDl.O o >

" • '0 ;:; • ~ lcm.o

0.0 Hmo

Target Probabilities: 0.35 (service) 0.04 (ultimate)

~ r-----------------------------~

~ zmo '0 >

" • ... o ;:;

~ lcmD

~ ~~ .. ----~~~~~----~----~ 0.0 lImO

140

~~-------------------------------

~ m.o o > v • '0 ;:;

~ umo

~~~ .. ~~~~~--~--~~~--~ 0.0 Hmo

~ r-----------------------------~

~ ml.O '0 >

" • ... o ;; ~ l(XXl.O

~~~--~~~--~~ .. --~~~~~ 0.0 1CXXl.O


(Weights for Structures=8:8: 1: 1:1: 1)

141

4CmO

• :J Ci ml.O >

" • .. 0 ;; !mO • ~

Q.

lema

0.0 0.0 Hmo 2£mO :ml.O 4CDl.O m.o 1CDlO zmo :ml.O m.o smo

/duaI'*'e kbIaI"-

scmo

4(XXl.O «mO

• • ::I :J

Ci ml.O j ml.O >

" " • .! 0 0 ;; mtO ;; m.o • ! .. Q. Q.

lema lImO

0.0 0.0 0.0 1(DlQ zmo ml.O 4CDl.O scmo 0.0 lema 2IXXUl mLO m.o sma /dual_ kbIaI_

scmo m.o

4CmO «mO

• • ::I ::I

Ci 3lmD j ml.O >

" " ! : 0 0

;; ml.O ii m.o ! ! Q. Q.

lema 1(01.0

a.o 0.0 Q.O lrmG zmo ml.O m.o m.o a.o lema 2IXXUl mLO m.o smo /dual_ Acbd"*'e

Figure 6.17 Scatterplots of Objective Function (Ratio of Weights for Serviceability to Ultimate Limit States=1 :5)

(Weights for Structures=8: 8: 1 : 1 : 1 : 1)

142

smot 71 ~t 71 4Cm.O 4CmO

• • :J :::I

~ JmO 1 ml.D ~ " • • ~

., 0

~ mlO ii ml.O ~ ! Q. Q.

lcm.o limo

0.0 0.0 0.0 Hmo zmo 3CmO 4CDl.O m.o 0.0 lema 2lmO m.o GnO sma

ktuala.e ktuala.e 5(Dl.O ~

4CmO «mO

• • 2 :::I a ml.O l ml.D > ~ " • ! 0 0 ~ 2OOl.O ~ m.o ~ ! Q. Q.

1(XXl.Q lImO

0.0 0.0 0.0 HmO 2IDlO mul 4CDl.O m.o 0.0 lCDl.O m.o m.o m.o sma

ktuala.e Actual_

5(Dl.O m.o

4CXXl..O mLO

• • :J :J 0 lXXl.O l ml.D > ~ " • • 0

., 0

~ 2OCQ..O ~

~ m.o !

Q. D.

11XXl.O lImO

Q.D 0.0 0.0 lImO 2IXnO ml.O 4CmO m.o 0.0 lCDl.O m.o m.o m.o sma

Acbda.e Acbd_


(Weights for Structures=8:8:1:1:1:1)

143

ImO m.o

scmo scmo

• 4CXW) ~ «mO :1 '0 "0 > > 'tI JmO 1 mtO ! 0 '0 ii ii • • ~ zmo i. 2ImO

lcmD lcmD

0.0 Q.O

0.0 Hmo zmo :mlO «mD sma m.o Q.O nmo zmo lDlO «m.O sma fDXl.O ktuIi wakIe ActuIi •

71 &mOt 71 9Wr

• ~ «mO :1 '0 a > >

1 JmO :; mtO 0 0

ii ii

lml.O lamo

lCXXl.O lcmD

0.0 Q.O

0.0 1cmD am.c JIIl.O «mO sma m.o 0.0 lcmD zmo m.o 4CDlO m.o 6lmO ......... Actual ..

mtO mtO

5IDl.O sma

• ano ~ «mO :1 '0 a > > 'tI ml.O 1 m.o • 0 1i ii ii

~ ml.D ~amo

lcmD lImO

0.0 0.0 0.0 Hmo 2DlU mJI cm.o m.o m.o Q.O 1(D).O zmo JmO 4CDlO m..o 6lmO

~ ... ActuIi ,.

Figure 6.19 Scatterplots of Objective Function (Ratio of Weights for Serviceability to Ultimate Limit States= 1: 1 0)


144

m.o eano

mtO mtO

• 4<mO • m.o :t :t 'ij 'ij > >

" ml.O " m.o ! ! 0 0 ;; ii • [mJ,O ~ mJ,O

lema lC1Xl.O

0.0 0.0 0.0 limo m.o m.o «mo m.o m.o 0.0 1CDl.O zmo mtO m.o m.o amo AcIuaI_

ktlIai wile

mul ml.O

sma mtO

• 4(D).O • m.o :l ~

'ij 'ij > > 'II

.unO "II

miD ! i 0

0 '0

;; ii

~ml.O ~2lDl.O

limo Target Probabilities: lC1Xl.O Target Probabilities: 0.35 (service) 0.35 (service) 0.045 (ultim.ate) 0.06~ (ultimate)

0.0 0.0 0.0 HmO m.o ml.O «mD sana m.o 0.0 1rDJ.O m.o lDl.O 4CDl.O m.o amo 1cluaI_ AcIuaI_

ml.O amo

5(D1() sma

• 4(DlO ~ 4CmO :l 'ij 1 >

" mlO 1 m.o • .. .. 0 0 ;; ;;

~ml.O ~ zmo

limo limo

Q.D 0.0 Q.O 1(0).0 mtO 3lmO «mo m.o m.o 0.0 1(0).0 2lDl.D m.o 4CmO m.o amo

IcluaIwabI kbdwile

Figure 6.20 Scatterplot of Objective Function (Ratio of Weights for Serviceability to Ultimate Limit States=1:10)


146

7CD) 7EXXl

1m) laX)

!m) SO)

• • :l :l

"ii "ii > GXl > 4lDl

" " • • .. .. 0 ;m) 0 ~ :0 :0 • ! ... a. a.

2CD1 ZDl

1(01 1CXXl

0 0 0 lCXXl m> m .4(XX) !mJ IXXl 1m) 0 1 COl m> m 4(D) 5(Q) 1m) 1m)

ktlIaI value ActuaIlIiue 7CD) 7EXXl

fm) laX)

5(XX) SO) • • :l :l "ii 0 > 4(nj > 4lDl

" " • .! 0 lDJ 0 3D) :0 :0 • ! ... a. a.

2C01 2m)

1(01 1(01

1(11) - JIX) 4(0) 5(XX) 6C01 7CXX) 1 COl m> lXX) 4(D) 5000 1m) 100J AdIG value

Actual_ 7CD) 7EXXl

IDXl am

5(XX) 5lXX) • • :l :l

'0 0 > 4CQj > 4CXXl

" " .! • 0 lXIJ 0 JIXX) ~ :0 ~ ! a. a.

2CD1 2m)

1CXKl 100)

1 CD) m .JD) .4(XX) 5(D) fm1 7CD) 1 COl m> nI) .mJ 5(XX) 6C01 7aX)

ktJIi value Actual_

Figure 6.22 Scatterplots of Objective Function (Ratio of Weights for Serviceability to Ultimate Limit States=l: 15)


147

Figure 6.23 Fitted 2nd Order Polynomial Representing Objective Function Ratio of weights for service and ultimate =1: 10

1.50

1.40

1.30

1.20

YE 1.10

1.00

0.90

0.80

0.70 0.30

(Weights for Structures=8:8: 1: 1: 1: 1) (Target Probabilities: 0.30 (service), 0.060 (ultimate) )

service:ultimcte= 1:2 service:ultimcte= 1:5 service:ultimcte= 1: 10 service:ultimote= 1: 15

0.35 0.50

Figure 6.24 Sensitivity of YE to Ps * for Various Ratios of Weights for Service and Ultimate (Weights for Structures=8:8: 1: 1: 1: 1, Pu * = 0.06)

0.018

0.017

0.016

~a!hsx 0.015 ~----~.....-

0.014

148

0--0-/::;.--

x--

service:ultimate= 1 :2 service:ultimate= 1 :5 service:ultimate=1:10 service:ultimate= 1: 15

0.013 ~_-'--_-'&' __ ~_---iL--_""""' __ ..L-_---,-_---,1 0.30 0.35 0.40

Ps* 0.45 0.50

Figure 6.25 Sensitivity of ~a/hsx to Ps * for Various Ratios of Weights for Service and

YE

Ultimate (Weights for Structures=8:8:1:1:1:1, Pu*= 0.06)

1.50

1.40

1.30

1.20

1.10

1.00

0.90 0.05

service:ultimate= 1 :2 service:ultimate= 1 :5 service:ultimate=1: 10 service:ultimate=1 :15

0.06 0.06

Pu* 0.07 0.07 0.08

Figure 6.26 Sensitivity of YE to Pu * for Various Ratios of Weights for Service and Ultimate (Weights for Structures=8:8:1:1:1:1, Ps*= 0.35)

149

0.018

0.017

0.016

0.015 [A-----~IF_---_F__---_A_----o£_j"

0.014

0.013 0.05 0.06 0.06

0-- service:ultimote=1:2 o -- service:ultimote= 1 :5 I:J. -- service:ultimote=1:10 X -- service:ultimote=1:15

0.07 0.07 0.08

Pu* Figure 6.27 Sensitivity of ~a/hsx to Pu * for Various Ratios of Weights for Service and

Ultimate (Weights for Structures=8:8:1:1:1:1, Ps*= 0.35)

10

9

8

7

... 6

...c C'I 5 Q)

~ 4

3

2

0

no. of story

Figure 6.28 Derived Weight Function

150

IDl.O ano

5(XXl.O scmo

• 4CmO ~ «mO :1 "0 j >

" ml.O 1 m10 • .. .. 0 0 li i • • a zmo i zmo

1ema 11mO

Q.O 0.0 0.0 limo m.o m.o ano m.o mLO 0.0 nOlO 2ImO m.o 4CXXlO 5ImO sma

IduaIm IduaIm ano mLO

S&lXI.O

m..o • 4(D)..O • ::I :l

"0 "0 > > "0

.JmO -: «mO • 0 .. 0

li :;

~zmo ! 11.

zmo 1ema

a.o Q.O 0.0 lImO m.o m.o m.o sma mLO 0.0 m.o m.o amo lImO

1duaI_ Actual_

ml.O amo

scm.o scmo

• 4(D)..O • «mD :I ::J 0 "0 > >

" .JmO 1lmD • .. .. 0 0 li ii

~ ml.O [ zmo

lanD lImO

Q.O 0.0 0.0 lImO m.o m.o «mO sma m.o a.o u01.o m.o m.o 4CD1O m.o sma 1duaI_ Actual_


(Weights for Structures=20: 14:6:4:3: 1)

ana

scmo

• 4CmO :J ii >

" ml.O ! 0 ii • a ml.O

Hmo

~ ~~ .. ~~~~~~~ .. ~~~~~ ~

ll1mO ,..----------------

smo

• ::I

-0 amo > "0

! 0 ii 4(Dl.O ! Q.

~~~~~~~~~~~~~~~~~

~ 1 lImO

~ .------------------------------~

~ 4CmO ii >

~ ml.O ... o ii

~ ml.O

ltm.O Target Probabilities: V. 0.35 (senrice) I 0.0 ' I 1 1 , 1 I 1 1 I 1 I 1 ~.~; I ~~t,~it~) I I 1

Q.O

151

ImO

scmo

• «mO :J

j

" m.o ! 0 ii [zmo ~2t:

1CDl.O

~~~~~~~~~~~~~~~uu

~

~~--------------------------~

~

• ::I

'0 >

" «mO ! 0 ii ! Q.

mtO

~~~~~~~~~~~~~~~~~

0.0

~ ~----------------------------~

llDJ.O larget Probabilities : 035 (ser.nce)

Q.O V, I , , , , I , 1 1 , I ,~.~7? I ~~l~~it~) I , I I Q.O



152

m..o • • :::J :::J "0 U > > 1I " «mO ! • .. 0 0 li li • • L. L.

~ ~

zmo

0.0 2mO .ml.O m.o m.o 0.0 m.o 4CDl.O ana sma

Actual_ Actual_ fml.O lrm1.O

m.o m.o

• 4(D)j) • :::J :J U 'ii amo > > II

ml.O II

! • 0 ii li ;; 4CDl.O ~ 2mO

! ~

lCDl.O m.o

0.0 0.0 0.0 lCXXl.O 2IDl.O m.o m.o m.o &anD 0.0 m.o m.o amo lImO lama

kba_ Adual1aiIe

tml.O m..o

scmo ml.D

• • «mO :::J ::J

a u > > 1I 4ClXl.O " mlD ! • .. 0 0 li ;; ~ ~mLO ~

2OOl.O

UDl.O

0.0 0.0 0.0 2IDl.O .ml.O amo m.o 0.0 1(Dl.l) zmo m.o 4CDl.O sana amo 1duaI_ ActuaI_

Figure 6.31 Scatterplots of Objective Function (Ratio of Weights for Serviceability to Ultimate Limit States= 1: 5)


m.o

scm.o

• 4CDlO ::J a >

'0 ;mw ! 0 :0 • ~ 2CmO

lcmD

0.0 0.0 Hmo

ltm1.O

IIDl.O

• .2 a sma > 'tJ

! 0 :0 «me ! 0.

zmo

0.0 Q.D zmo

ml.O

5CXXl.O

• 4CXXl.O ::J 'ii >

" DXl.O • .. 0 :0 ~ 2ml.O

0.0 ._ .. I~

153

121DlO

1cxmD

• m.o ::J

1 " ImO • b :v ! 0. m.o

2IDl.O

0.0 zmo DXl.O GWI sma m.o 0.0 2IDl.D 4CmO m.o ano lcxmD l2OOJ.O

leba_ ktuaI_ ano

ImO

• ::J D >

" «mO ! 0 :0 ! 0.

2IDl.D

a.o 4aXl.O amo m.o 1txmo 0.0 mtD 4CmO &emO sma khIaI_ Actual_

m.o

amo • ::I

a >

" «mO • .. 0 :v ! 0.

2fDU)

Target Probabilities:

CD tG':, 0.35 (service) 0.070 (ultimate) , , , , , , , , , , , , ,

=" mn m.o mn am" .... mn _ .. _ ..

"'" ~ QUUIMI

Achaa. /dual ..

Figure 6.32 Scatterplots of Objective Function (Ratio of Weights for Serviceability to Ultimate Limit States= l: 5)


lImO

• ::I C >

" • ... 0 :0 ~ a.

154

ml.O m.o • :J a >

smo -: «mil 0

... 0 .!!

~ Il.

lDl.O m.o

~~~~~~--~----~----------- ~.-~----~----~--~~~--~~~ 0.0 a.o



ana • ::I a :> ~ 4CXIl.O ! a 1; • L. CL

mlO

0.0 a.o

12!Dl.O

m.o • ::I a :> '0 m10 ! 0 1; • L. CL

ml.O

0.0 0.0

ml.O

Q.O

155

1sma

12lDlO

• :l

1 ImO

" • 1i i ImO ~ A.

JmO

Q.O

2OOlO .m1O m.o m.o a.o m.o uno m.o 1:ml.C lmlO ~.

AckD_ lml.O

amo

• :l 'ii ml.O >

" • ii i 4CmO ~ 0..

2lDlO

0.0 m..c IIItD m.o lmlO 0.0 m.o «mO m.o mtO ltml.O ...., ... AcIuaI_

m.o

ml.O

• ::J

a > ~

«mO • ... 0 1; ~ A.

m.o lJJ'~t Probabilities: Target Probabilities: 035 (service) 0.35 (service)

, I , , t I 0065 (ultimate) " ,'," , , , , , I

0.070 (ultimate) " ,I" ,

Q.O



• ::s 'ij >

" • 0 ;; ~ Cl.

0.0

llXmO

ImO

• ::s 'ij amo >

" ! 0 ;; 4CDl.O ~ Q.

mJ.O

Q.!)

0.0

ml.O

amo • ::s 'ij >

" 4CmO ! 0 ;; ! Q.

m.o

0.0

156

11Dl.O r---------------_

• 1ml.O ::s

1 " • .. 0 ;; ~ amo Il.

a.o m.o lmlO lsma mno Q.D amo 121Dl.O

Acbdwakle ActuaIwakie

1sana

lml.O

• ::s 'ij m.o > 11

! 0 ;; amo ! Q.

ml.O

M 2CmO 4(Dl.O m.o m.o ltxmO Q.D mJ.O m.o m.o lmJ.O

ActuaId.e Actual_ IDXl.O

IDl.O

• :J 'ij >

11 anD • .. 0 0

;; ! Il.

2IDlO Target Probabilities:

, I ,

0.45 (service )1 , , I ,O.~~ f4ti~a~e), , , , ,

Q.D m.o



11mO

1sana

157

l!mO 15(Q1O

l2CmO

• l2CmO • ::J ::J '0 0 m.o > >

" " • • ... b 0 ;; ;; ann • ~ L sma C. c.

JmO

0.0 0.0 0.0 m.o l2CDlO lImO a.o m.o ann m.o l2CmO lsmD khIaI_

kblm

lZDlO ~ )Y) 12CmO ~

-J m.o • • ::I ~ "0 0 > >

" m.o " ImO ! • 0 g ;; ;; • ~ L C. C.

m.o JmO

0.0 a.o 0.0 ml.O Im1.O m.o 121Dl.O 0.0 ml.O m.o 9IDlO 12OXl.O khIaI_

AcIId1GiJe

ltm1.O 15(Q1O

m.o l2CmO

• • ::I ::I

"0 fml.O j m.o >

" " • • ... b 0 ;; «mD ;; mLO ~ ~ 0 c. c.

ml.O JmO

Q.O 0.0 0.0 2IX110 -4OlI..O a:mo sma 10Dl.0 Q.O m.o amD m.o 121X110 151mO khIaI_

AcUI_

Figure 6.36 Scatterplots of Objective Function (Ratio of Weights for Serviceability to Ultimate Limit States= 1: 15)


YE

158

Figure 6.37 Fitted 2nd Order Polynomial Presenting Objective Function Ratio of Weights for Serviceability to Ultimate Limit States= 1: 10

(weights for Structures=20: 14:6:4:3: 1) (Target Probabilities: 0.30 (service), 0.060 (ultimate) )

1.40 ~--------------------------------------------~

1.30

1.20

1.'0

1.00

0.90

0.80

0.70 0.300

o -- service:ultimote= 1 :2 :J -- service:ultimate= 1 :5 6 -- service:ultimote= 1: 10 x -- service:ultimate=1 :15

0.350 0.400

Ps*

0.450 0.500

Figure 6.38 Sensitivity of YE to Ps * for Various Ratios of Weights for Serviceability to Ultimate Limit States (Weights for Structures=20:14:6:4:3:1, Pu* = 0.06)

159

0.018

0.017

~a!hsx 0.0'6

0.015 o -- service:ultimate=1:2 o -- service:ultimate= 1 :5 6 -- service:ultimate= 1 :1 0 X -- service:ultimate= 1: 15

0.014 ~ __ ~ __ ..a.-__ ""--__ ",,,-___________________ ~

0.300 0.350 0.400

Ps* 0.450 0.500

Figure 6.39 Sensitivity of ~afhsx to Ps'* for Various Ratios of Weights for Serviceability to Ultimate Limit States (Weights for Structures=20: 14:6:4:3: 1, Pu * = 0.06)

YE

1.50

1.40

1.30

1.20

1.10

1.00

f\ ~f\ \,1.;;;1 V

0.80

service:ultimate= 1 :2 o -- service:ultimate= 1 :5 L 6 -- service:ultimate=1:1 0

[ X -- service:uitimate= 1: 15

I 0.045 0.050 0.055 0.060

Pu * 0.065 0.070 0.075

Figure 6.40 Sensitivity of YE to P u * for Various Ratios of Weights for Serviceability to Ultimate Limit States (Weights for Structures=20:14:6:4:3:1, Ps* = 0.35)

0.019

0.018

0.017

0.016

~a!hsx 0.015

0.014

0.013

0.012 0.045 0.050 0.055

160

o -- service:ultimate= 1:2 o -- service:ultimate=1:5 6 -- service:ultimate=1:10 X -- service:ultimate=1 :15

0.075

Figure 6.41 Sensitivity of ~afhsx to P u * for Various Ratios of Weights for Serviceability to Ultimate Limit States (Weights for Structures=20: 14:6:4:3: 1, Ps * = 0.35)

161

CHAPTER 7

SUMMARY AND CONCLUSIONS

7.1 Summary

This study presents a method to calibrate (determine) design values of selected

parameters based on required target reliabilities of the structures. The design parame

ters are calibrated by minimizing an objective function which measures the total differ

ence between target and actual limit state probabilities of the structural system under

seismic and other loads.

A procedure using the Equivalent Nonlinear System (ENS) with response scaling

factors has been developed in order to evaluate the limit state probabilities of a nonlin

ear MDOF system efficiently. The ENS system retains the dynamic properties of the

first two modes of the MDOF structure under consideration and uses a global yield

displacement which is determined based on the results of a nonlinear static pushover

analysis. A global response scaling factor and a local response scaling factor are then

developed to modify the response of the ENS in order to obtain the comparable

response of a nonlinear tviDOF system.

The Response Surface Method (RSM) is used to expedite the minimization

solution procedure for determination of the design parameters. An appropriate num

ber of combinations of design parameters are selected to perfonn the RSM. For each

combination of the design parameters, representative structures (moment resisting

steel frames from 1 to 12 stories) are designed according to the NEHRP recommended

provisions and AISC LRFD manual, and their lifetime limit state probabilities are

evaluated.

162

In this study, the live load factor, seismic load factor, and allowable drift limit are

chosen as the design parameters to be calibrated. Two examples are considered. In the

first example, the load factors for live load and seismic load are calibrated according to

several sets of target probabilities for serviceability and ultimate limit states. In the

second example, the seismic load factor and allowable story drift r~:~io are calibrated

according to various sets of required target probabilities.

7.2 Conclusions

The following conclusions can be drawn based on the results of this study:

(1) The global response (i.e., maximum displacement at the top of a structure) and local

responses ( i.e., maximum interstory drift) evaluated using the ENS are comparable

to the global and local responses obtained from the dynamic analysis of correspon

ding nonlinear MDOF systems. Moreover, the probabilities of exceeding a global

limit state (e.g., a probability of exceeding a certain displacement threshold at the

top of the structure) and a local limit state (e.g., a probability of exceeding a certain

interstory drift ratio threshold) calculated using the ENS model agree well with

those calculated from nonlinear dynamic analyses of the corresponding MDOF

system.

(2) The Response Surface Method (RSM) with central composite design is an efficient

method in minimizing the objective function. Furthermore, it is not required to

repeat the cycle of design, response and reliability analysis (as in a nonlinear pro

gramming solution procedure) when the target limit state probability is changed.

Therefore, ENS and RSM in combination provides an efficient and accurate method

for calibrating design parameters

(3) Limit state probabilities for high-rise structures (>7 stories, ATC-13) which are

163

designed according to the NEHRP recommend provisions and AISC LRFD manual

are lower than those for low - rise (1 to 3 stories) and medium - rise structures (4 to 7

stories) designed according to the same provisions and manuaL This implies that

the design procedures in the NEHRP recommended provisions and AISC LRFD

manual are more conservative when applied to high - rise structures. This observa

tion is reasonable since the failure of a high -rise structure can lead to more serious

consquences than that of a low-rise or medium-rise structure.

(5) It is found that the allowable story drift ratio of the structures have a more significan t

effect than the seismic load factor on the reliability of serviceability limit state as well

as ultimate state. For the seismic load factor of 1.0 and the allowable drift ratio of

0.015 recommended 'in the NEHRP provisions, the implied target limit state proba

bilities for serviceability and ultimate limit states are repeatively 0.38 and 0.060 for

50 years, which correspond to annual risks of 0.0076 and 0.0012. These implied

risks are in agreement with suggested acceptable risk levels associated with build

ings designed according to building code (Hays, 1985). The two different sets of

relative weights assigned to the buildings of2 to 12 stories have only a small effect on

these probabilities.

(6) Load factors for seismic load and live load, and allowable story drift ratio can be

calibrated based on target probabilities for serviceability and ultimate limit states.

Therefore, the procedure provides a rational method for selecting these design pa

rameters. As expected, the relative weights assigned to the limit states have an

influence on the design parameters. Therefore, the weights need to be assigned in a

rational manner, e.g., according to the seriousness (expected cost) of the conse

quence of the limit states.

164

7.3 Future Research

The proposed research has focused on calibration of the allowable drift limit and

load factors for live load and seismic load. Other design parameters, such as the snow

load factor, wind load factor, etc. can be also included in the calibration. Also, only

SMRSFs have been considered in developing the Equivalent Nonlinear System

(ENS), in deriving the response scaling factors, and in calibrating design parameters.

In future studies, other types of buildings, e.g., Ordinary Moment Resisting Frames

(OMRFs), Concentrically Braced Frames (CBFs), Eccentrically Braced Frames

(EBFs), and dual systems should be also considered.

165

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168

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