BUILDING-SPECIFIC LOSS ESTIMATION METHODS(chp1-5).pdf

Department of Civil and Environmental Engineering

Stanford University

Report No.

BUILDING-SPECIFIC LOSS ESTIMATION METHODS& TOOLS FOR SIMPLIFIED PERFORMANCE-BASED

EARTHQUAKE ENGINEERING

By

Carlos Marcelo Ramirez and

Eduardo Miranda

171

May 2009

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

©200 The John A. Blume Earthquake Engineering Center

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© Copyright by Carlos M. Ramirez 2009 All Rights Reserved

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ABSTRACT

The goal of current building codes is to protect life-safety and do not contain

provisions that aim to mitigate the amount of damage and economic loss suffered during an

earthquake. However, recent earthquakes in California and elsewhere have shown that

seismic events may incur large economic losses due to damage in buildings and other

structures, which in many cases were unexpected to owners and other stakeholders.

Performance-based earthquake engineering is aimed at designing structures that achieve a

performance that is acceptable to stakeholders. The approach developed the Pacific

Earthquake Engineering Research (PEER) center has showed promise by providing a fully

probabilistic framework that accounts for uncertainty from the ground motion hazard, the

structural response, and the damage and economic loss sustained. This framework uses

building-specific loss estimation methodologies to evaluate structural systems and help

stakeholders make better design decisions.

The objectives of this dissertation are to improve and simplify the current PEER

building-specific loss estimation methodology. A simplified version of PEER’s framework,

termed story-based loss estimation, was developed. The approach pre-computes damage to

generate functions (EDP-DV functions) that relate structural response directly to loss for

each story. As part of the development of these functions the effect of conditional losses of

spatially dependent components was investigated to see if it had a large influence on losses.

The EDP-DV functions were also developed using generic fragility functions generated

using empirical data to compute damage of components that do not currently have

component-specific fragilities. To improve the computation of the aleatoric variability of

economic loss, approximate analytical and simulation methods of incorporating building-

level construction cost dispersion and correlations, which are better suited to use

construction cost data appropriately, were developed. The overall loss methodology was

modified to incorporate the losses due to demolishing a building that has not collapsed but

cannot be repaired due to excessive residual drift. Most of these modifications to PEER’s

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methodology were implemented into computer tool that facilities the computation of

seismic-induced economic loss.

This tool was then used to compute and benchmark the economic losses of a set of

reinforced concrete moment-resisting frame office buildings available in literature that were

representative of both modern, ductile structures and older, non-ductile structures. The

average normalized economic loss of the ductile frames was determined to be 25% of the

building replacement value at the design basis earthquake (DBE) for this set of structures.

The non-ductile frames exhibited much larger normalized losses that averaged 61%. Of the

structural and architectural design parameters examined in this study, the height of the

building demonstrated the largest influence on the normalized economic loss. One of the 4-

story ductile structures was analyzed as a case-study to determine the variability of its

economic loss. Its mean loss at the DBE was estimated to be 31% of its replacement value

with a coefficient of variation of 0.67. To examine the effect of losses due to building

demolition, four example buildings (two ductile and two non-ductile frames) were

analyzed. It was found that this type of loss had the largest effect on the ductile structures,

increasing economic loss estimates by as much as 45%.

The economic losses computed in this investigation are large even for the code-conforming

buildings. The aleatoric variability of these losses is also large and heavily influenced by

construction cost uncertainty and correlations. The story-based loss estimation method

provides an alternative way of assessing structural performance that is efficient and less

computationally expensive than previous approaches. This allows engineers and analysts to

focus on the input – the seismic hazard analysis and the structural analysis – and the output

– the design decisions – of loss estimation rather than on the loss estimation procedure

itself. Limiting the amount of time and resources spent on the loss estimation process will

hopefully facilitate the acceptance of performance-based seismic design methods into the

practicing engineering community.

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ACKNOWLEDGEMENTS

This work was primarily funded by the Pacific Earthquake Engineering Research

(PEER) Center with support from the Earthquake Engineering Research Centers Program of

the National Science Foundation. Additional financial assistance provided by the John A.

Blume Fellowship and the by the John A. Blume Earthquake Engineering Center.

This report was initially published as the Ph.D. dissertation of the first author. The

authors would like to thank Professors Gregory Deierlein, Helmut Krawinkler and Jack

Baker for their valuable and insightful comments on this research. The authors would also

like to acknowledge Professors Abbie Liel and Curt Haselton for the use of their structural

simulation results and Professor Judith Mitrani-Reiser for the use of her MDLA toolbox.

This research was not possible without their collaboration and their contributions to this

work.

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TABLE OF CONTENTS

ABSTRACT .......................................................................................................................... IIACKNOWLEDGEMENTS .............................................................................................. IVTABLE OF CONTENTS .................................................................................................... VLIST OF TABLES ............................................................................................................. IXLIST OF FIGURES ........................................................................................................... XI1 INTRODUCTION ........................................................................................................ 1

1.1 MOTIVATION & BACKGROUND ................................................................................ 11.2 OBJECTIVES ............................................................................................................. 31.3 ORGANIZATION OF DISSERTATION ........................................................................... 4

2 PREVIOUS WORK ON LOSS ESTIMATION ........................................................ 82.1 LITERATURE REVIEW ............................................................................................... 82.2 REGIONAL LOSS ESTIMATION .................................................................................. 82.3 BUILDING-SPECIFIC LOSS ESTIMATION .................................................................. 102.4 LIMITATIONS OF PREVIOUS STUDIES ...................................................................... 14

3 STORY-BASED BUILDING-SPECIFIC LOSS ESTIMATION .......................... 173.1 INTRODUCTION ...................................................................................................... 173.2 STORY-BASED BUILDING-SPECIFC LOSS ESTIMATION ............................................ 20

3.2.1 Previous loss estimation methodology (component-based) ............................. 203.2.2 EDP-DV function formulation ......................................................................... 22

3.3 DATA FOR EDP-DV FUNCTIONS ........................................................................... 243.3.1 Building Components & Cost Distributions .................................................... 243.3.2 Fragility Functions Used .................................................................................. 28

3.4 EXAMPLE STORY EDP-DV FUNCTIONS ................................................................. 333.5 CONDITIONAL LOSS OF SPATIALLY INTERDEPENDENT COMPONENTS ..................... 403.6 DISCUSSION OF LIMITATIONS OF STORY-BASED APPROACH & EDP-DV FUNCTIONS 503.7 CONCLUSIONS ....................................................................................................... 51

4 DEVELOPMENT OF COMPONENT FRAGILTIY FUNCTIONS FROM EXPERIMENTAL DATA .................................................................................................. 53

4.1 AUTHORSHIP OF CHAPTER ..................................................................................... 53

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4.2 INTRODUCTION ...................................................................................................... 534.3 DAMAGE STATE DEFINITIONS ................................................................................ 564.4 EXPERIMENTAL RESULTS USED IN THIS STUDY ..................................................... 584.5 FRAGILITY FUNCTION FORMULATION .................................................................... 61

4.5.1 Fragility Functions for Yielding ....................................................................... 644.5.2 Fragility Functions for Fracture ....................................................................... 74

4.6 CONCLUSIONS ....................................................................................................... 77

5 DEVELOPMENT OF COMPONENT FRAGILITY FUNCTIONS FROM EMPIRICAL DATA ........................................................................................................... 79

5.1 AUTHORSHIP OF CHAPTER ..................................................................................... 795.2 INTRODUCTION ...................................................................................................... 795.3 SOURCES OF EMPIRICAL DATA .............................................................................. 82

5.3.1 Instrumented Buildings (CSMIP) .................................................................... 825.3.2 Buildings surveyed in the ATC-38 Report ....................................................... 84

5.4 DATA FROM INSTRUMENTED BUILDINGS ............................................................... 865.4.1 Structural response simulation ......................................................................... 865.4.2 Motion-damage pairs for each building ........................................................... 92

5.5 DATA FROM ATC-38 ............................................................................................. 955.5.1 Structural response simulation ......................................................................... 955.5.2 Motion-damage pairs for each building ........................................................... 98

5.6 FRAGILITY FUNCTIONS FORMULATION ................................................................ 1025.6.1 Procedures to compute fragility functions ..................................................... 1025.6.2 Limitations of fragility function procedures .................................................. 1075.6.3 Adjustments to fragility function parameters ................................................. 109

5.7 FRAGILITY FUNCTION RESULTS ........................................................................... 1125.7.1 Comparison with generic functions from HAZUS ........................................ 118

5.8 CONCLUSIONS ..................................................................................................... 119

6 DEVELOPMENT OF A STORY-BASED LOSS ESTIMATION TOOLBOX .. 1216.1 PROGRAM STRUCTURE ........................................................................................ 1216.2 GRAPHICAL USER INTER FACE ............................................................................. 124

6.2.1 Building Information & Characterization ...................................................... 1246.2.2 EDP-DV Function Editor ............................................................................... 1256.2.3 Main Window................................................................................................. 1296.2.4 Hazard Module ............................................................................................... 1306.2.5 Response simulation module .......................................................................... 1326.2.6 EDP-DV Module ............................................................................................ 1376.2.7 Loss Estimation Module ................................................................................ 1396.2.8 Loss Disaggregation and Visualization Module ............................................ 140

7 BENCHMARKING SEISMIC-INDUCED ECONOMIC LOSSES USING STORY-BASED LOSS ESTIMATION .......................................................................... 143

7.1 AUTHORSHIP OF CHAPTER ................................................................................... 1437.2 INTRODUCTION .................................................................................................... 1447.3 LOSS ESTIMATION PROCEDURE ........................................................................... 1467.4 DESCRIPTION OF BUILDINGS ................................................................................ 147

7.4.1 Architectural Layouts and Cost Estimates (developed by Spear and Steiner) 149

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7.4.2 Nonlinear Simulation Models and Structural Analysis (computed by Liel and Haselton) ..................................................................................................................... 152

7.5 ECONOMIC LOSSES .............................................................................................. 1567.5.1 Expected losses conditioned on seismic intensity .......................................... 1577.5.2 Expected Annual Losses ................................................................................ 1637.5.3 Present value of life-cycle costs ..................................................................... 1667.5.4 Comparison to Non-ductile Reinforced Concrete Frame Buildings .............. 1687.5.5 Loss Toolbox Comparison ............................................................................. 1707.5.6 Discussion of results relative to other loss estimation methodologies ........... 172

7.6 LIMITATIONS ....................................................................................................... 1747.7 CONCLUSIONS ..................................................................................................... 175

8 VARIABILITY OF ECONOMIC LOSSES ........................................................... 1788.1 AUTHORSHIP OF CHAPTER ................................................................................... 1788.2 INTRODUCTION .................................................................................................... 1788.3 TYPES OF LOSS VARIABILITY & CORRELATIONS ................................................. 180

8.3.1 Variability and Correlations in Construction Costs ....................................... 1818.3.2 Variability and Correlation in Response Parameters ..................................... 1898.3.3 Variability and Correlations in Damage Estimation ...................................... 198

8.4 VARIABILITY OF LOSS METHODOLOGY ............................................................... 2008.4.1 Mean annual frequency of loss & loss dispersion condition on seismic intensity ...................................................................................................................... 2008.4.2 Dispersion of loss conditioned on collapse .................................................... 2018.4.3 Dispersion of loss conditioned on non-collapse ............................................. 2028.4.4 Monte Carlo simulation method ..................................................................... 2118.4.5 Evaluation of quality of FOSM approximations ............................................ 212

8.5 DISPERSIONS OF ECONOMIC LOSS FOR EXAMPLE 4-STORY BUILDING .................. 2238.5.1 Variability of loss conditioned on non-collapse at the DBE .......................... 2248.5.2 Variability of loss conditioned on non-collapse as a function of IM ............. 2338.5.3 Variability of loss conditioned on collapse as a function of IM .................... 2378.5.4 Variability of loss as a function of IM & MAF of loss .................................. 240

8.6 CONCLUSIONS ..................................................................................................... 244

9 SIGNIFICANCE OF RESIDUAL DRIFTS IN BUILDING EARTHQUAKE LOSS ESTIMATION ....................................................................................................... 246

9.1 INTRODUCTION .................................................................................................... 2469.2 METHODOLOGY ................................................................................................... 2489.3 APPLICATIONS ..................................................................................................... 252

9.3.1 Description of Buildings Studied ................................................................... 2529.3.2 Results ............................................................................................................ 2559.3.3 Sensitivity of Loss to Changes in the Probability of Demolition ................... 2649.3.4 Limitations of results & discussion of residual drift estimations ................... 268

9.4 SUMMARY AND CONCLUSIONS ............................................................................ 269

10 SUMMARY AND CONCLUSIONS ....................................................................... 27110.1 SUMMARY ........................................................................................................... 27110.2 FINDINGS & CONCLUSIONS .................................................................................. 272

10.2.1 Story-based Loss Estimation ...................................................................... 27210.2.2 Improved Fragilities in support of EDP-DV Function Formulation .......... 273

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10.2.3 Implementing loss estimation methods into computer tool ....................... 27510.2.4 Benchmarking losses ................................................................................. 27510.2.5 Improved estimates on the uncertainty of loss ........................................... 27610.2.6 Accounting for Non-collapse losses due to building demolition ............... 278

10.3 FUTURE RESEARCH NEEDS .................................................................................. 27910.3.1 Data collection for fragility functions and repair costs .............................. 28010.3.2 Improvements to building-specific loss estimation methodology ............. 281

REFERENCES .................................................................................................................. 283APPENDIX A: COST DISTRIBUTIONS FOR EDP-DV FUNCTIONS .................. A-1

APPENDIX B: GENERIC STORY EDP-DV FUNCTIONS ....................................... B-1

APPENDIX C: SUBCONTRACTOR EDP-DV FUNCTIONS ................................... C-1

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LIST OF TABLES

TABLE 3.1 EXAMPLE BUILDING AND STORY COST DISTRIBUTIONS FOR MID-RISE OFFICE BUILDINGS ........... 26

TABLE 3.2 EXAMPLE COMPONENT COST DISTRIBUTION FOR A TYPICAL STORY IN A MID-RISE OFFICE

BUILDING ........................................................................................................................................... 27

TABLE 3.3 FRAGILITY FUNCTION & EXPECTED REPAIR COST (NORMALIZED BY COMPONENT REPLACEMENT

COST) PARAMETERS FOR DUCTILE STRUCTURAL COMPONENTS ......................................................... 28


COST) PARAMETERS FOR NON-DUCTILE STRUCTURAL COMPONENTS .................................................. 28


COST) PARAMETERS FOR NONSTRUCTURAL COMPONENTS ................................................................. 29

TABLE 4.1 PROPERTIES OF EXPERIMENTAL SPECIMENS CONSIDERED IN THIS STUDY ................................... 60

TABLE 4.2 INTERSTORY DRIFTS AT EACH DAMAGE STATE FOR EACH SPECIMEN .......................................... 61

TABLE 4.3 UNCORRECTED STATISTICAL PARAMETERS FOR IDRS CORRESPONDING TO THE DAMAGE STATES

FOR PRE-NORTHRIDGE BEAM-COLUMN JOINTS................................................................................... 64

TABLE 4.4 SUMMARY OF YOUSEF ET AL.’S BUILDING SURVEY RESULTS FOR TYPICAL GIRDER SIZES OF

EXISTING BUILDINGS ......................................................................................................................... 68

TABLE 4.5 REGRESSION COEFFICIENTS FOR RELATIONSHIP BETWEEN IDRY AND L/DB ................................. 69

TABLE 4.6 RECOMMENDED STATISTICAL PARAMETERS FOR FRAGILITY FUNCTIONS ................................... 69

TABLE 4.7 AVERAGE VALUES FOR PARAMETERS IN EQUATION (9), RELATING L/DB AND IDR ..................... 71

TABLE 5.1 CSMIP BUILDING PROPERTIES .................................................................................................. 83

TABLE 5.2 GENERAL DAMAGE CLASSIFICATIONS (ATC-13, 1985) ............................................................. 84

TABLE 5.3 ATC-13 DAMAGES STATES (ATC, 1985) .................................................................................. 84

TABLE 5.4 OCCUPANCY TYPES AND CODES (ATC-38) ............................................................................... 85

TABLE 5.5 MODEL BUILDING TYPES (ATC-38) .......................................................................................... 86

TABLE 5.6 FORMULAS USED FOR ESTIMATING STRUCTURAL BUILDING PARAMETERS ............................... 97

TABLE 5.7 PARAMETERS FOR SAMPLE FRAGILITY FUNCTIONS COMPUTED DIRECTLY AND WITH

ADJUSTMENTS FROM DATA FOR ACCLERATION NONSTRUCTRAL COMPONENTS (FROM CSMIP). ...... 111

TABLE 5.8 FRAGILITY FUNCTION PARAMETERS GENERATED FROM THE CSMIP DATA. ............................ 114

TABLE 5.9 FRAGILITY FUNCTION STATISTICAL PARAMETERS FOR SUBSETS OF ATC-38 DATA .................. 117

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TABLE 7.1 ARCHETYPE DESIGN PROPERTIES AND PARAMETERS ................................................................ 149

TABLE 7.2 COST ESTIMATES FOR STRUCTURES STUDIED ........................................................................... 152

TABLE 7.3 STRUCTURAL DESIGN INFORMATION AND COLLAPSE RESULTS (HASELTON AND DEIERLEIN 2007)

......................................................................................................................................................... 155

TABLE 7.4 EXPECTED LOSSES AND INTENSITY LEVELS .............................................................................. 156

TABLE 7.5 COMPARISON OF ASSUMED REPAIR COSTS FOR FINAL DAMAGE STATE OF GROUPS OF

COMPONENTS OF THE BASELINE 4-STORY BUILDINGS, NORMALIZED BY BUILDING REPLACEMENT

VALUE .............................................................................................................................................. 172

TABLE 8.1 STATISTICAL DATA OF CONSTRUCTION COSTS PER SUBCONTRACTOR (TOURAN & SUPHOT, 1997)

......................................................................................................................................................... 182

TABLE 8.2 CORRELATION COEFFICIENTS OF CONSTRUCTION COSTS BETWEEN DIFFERENT SUBCONTRACTORS

......................................................................................................................................................... 183

TABLE 8.3 EXAMPLE COST DISTRIBUTION BETWEEN CONSTRUCTION SUBCONTRACTORS OF EACH

COMPONENT IN A TYPICAL STORY OF AN OFFICE BUILDING .............................................................. 184

TABLE 8.4 AVERAGE OF EDP CORRELATION COEFFICIENTS FROM 1000 REALIZATIONS ........................... 194

TABLE 8.5 STANDARD DEVIATION OF EDP CORRELATION COEFFICIENTS FROM 1000 REALIZATIONS ....... 195

TABLE 8.6 COMPARISON OF STANDARD DEVIATIONS OF ECONOMIC LOSS DUE TO EDP VARIABILITY ONLY

USING FOSM (LOCAL DERIVATIVE) AND SIMULATION METHODS ..................................................... 218

TABLE 8.7 COMPARISON OF STANDARD DEVIATIONS OF ECONOMIC LOSS DUE TO EDP VARIABILITY ONLY

USING FOSM (AVERAGE SLOPE) AND SIMULATION METHODS .......................................................... 219

TABLE 8.8 COMPARISON OF INHERENT SUBCONTRACTOR LOSS CORRELATION COEFFICIENTS DUE TO EDP

VARIABILITY BETWEEN ANALYTICAL AND SIMULATION RESULTS .................................................... 230

TABLE 9.1COST ESTIMATES FOR BUILDINGS STUDIED ............................................................................... 254

TABLE 9.2 SUMMARY TABLE FOR EXPECTED ECONOMIC LOSS RESULTS AT DESIGN BASIS EARTHQUAKE

(DBE) AS A PERCENTAGE OF BUILDING REPLACEMENT VALUE ........................................................ 256

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LIST OF FIGURES

FIGURE 3.1 PEER METHODOLOGY .............................................................................................................. 18

FIGURE 3.2 STORY EDP-DV FUNCTIONS FOR TYPICAL FLOORS IN MID-RISE OFFICE BUILDINGS WITH

DUCTILE REINFORCED CONCRETE MOMENT RESISTING PERIMETER FRAMES. ...................................... 34

FIGURE 3.3 EDP-DV FUNCTIONS FOR LOW-RISE, MID-RISE AND HIGH RISE DUCTILE REINFORCED CONCRETE

MOMENT FRAME OFFICE BUILDINGS ................................................................................................... 36

FIGURE 3.4 COMPARISON BETWEEN DUCTILE AND NON-DUCTILE STRUCTURAL COMPONENT EDP-DV

FUNCTIONS OF TYPICAL FLOORS ......................................................................................................... 37

FIGURE 3.5 COMPARISON OF STRUCTURAL EDP-DV FUNCTIONS BETWEEN PERIMETER AND SPACE FRAME

TYPE STRUCTURES .............................................................................................................................. 38

FIGURE 3.6 INFLUENCE OF VARYING ASSUMED GRAVITY LOAD ON SLAB-COLUMN SUBASSEMBLIES ON

STRUCTURAL EDP-DV FUNCTIONS .................................................................................................... 39

FIGURE 3.7 HYPOTHETICAL FRAGILITY FUNCTIONS OF SPATIALLY INTERACTING COMPONENTS (SPRINKLERS

& SUSPENDED LIGHTING) (A) EXAMPLE WHERE LOSSES ARE UNAFFECTED (B) EXAMPLE WHEN LOSSES

ARE CONDITIONAL .............................................................................................................................. 42

FIGURE 3.8 PROBABILITY TREE FOR COMPONENTS CONSIDERED TO ACT INDEPENDENTLY .......................... 44

FIGURE 3.9 PROBABILITY TREE FOR INDEPENDENT COMPONENTS THAT USE DOUBLE-COUNTING TO

ACCOUNT FOR DEPENDENCY .............................................................................................................. 45

FIGURE 3.10 PROBABILITY TREE FOR PROPOSED APPROACH TO ACCOUNT FOR DEPENDENT COMPONENTS. . 46

FIGURE 3.11 EDP-DV FUNCTIONS FOR THREE DIFFERENT APPROACHES OF HANDLING COMPONENT

DEPENDENCY...................................................................................................................................... 47

FIGURE 3.12 FRAGILITY FUNCTIONS FOR PRE-NORTHRIDGE STEEL BEAMS AND PARTITIONS ...................... 48

FIGURE 3.13 PROBABILITY TREE FOR PROPOSED APPROACH, INCLUDING OTHER DS3 PARTITION-LIKE

COMPONENTS ..................................................................................................................................... 49

FIGURE 3.14 EDP-DV FUNCTIONS FOR PROPOSED APPROACH VS TREATING COMPONENTS INDEPENDENTLY,

WITH DS3 PARTITION-LIKE COMPONENTS INCLUDED. ........................................................................ 49

FIGURE 4.1 TYPICAL DETAIL OF PRE-NORTHRIDGE MOMENT RESISTING BEAM-TO-COLUMN JOINT ......... 54

FIGURE 4.2 TYPICAL TEST SETUPS (A) SINGLE SIDED (B) DOUBLE SIDED ................................................... 59

FIGURE 4.3 YIELDING WITHOUT CORRECTION FOR SPAN-TO-DEPTH RATIO (A) A36 (B) A572 GRADE 50 . 65

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FIGURE 4.4 SPAN-TO-DEPTH RATIO’S RELATIONSHIP TO INTERSTORY DRIFT (A) A36 (B) A572 GRADE 50 67

FIGURE 4.5 RECOMMENDED FRAGILITY FUNCTION CORRECTED FOR SPAN-TO-DEPTH RATIO WITH 90%

CONFIDENCE BANDS ........................................................................................................................... 69

FIGURE 4.6 FRAGILITY FUNCTIONS FOR TO BE USED IN CONJUNCTION WITH AN ANALYTICAL PREDICTION

OF IDRY (A) A36 (B) A572 GRADE 50 ................................................................................................. 73

FIGURE 4.7 EXAMPLE FRAGILITY FUNCTION FOR W36 BEAM GENERATED BY USING (A572 GRADE 50) .. 74

FIGURE 4.8 FRAGILITY FUNCTION FOR FRACTURE ...................................................................................... 75

FIGURE 4.9 RELATIONSHIP BETWEEN IDR AT FRACTURE AND BEAM DEPTH FOR ALL SPECIMENS. .............. 76

FIGURE 4.10 RECOMMENDED FRAGILITY FUNCTION CORRECTED FOR BEAM DEPTH WITH 90% CONFIDENCE

BANDS ................................................................................................................................................ 77

FIGURE 4.11 EXAMPLE CORRECTED FRAGILITY FOR W36 WHEN BEAM DEPTH IS KNOWN. .......................... 77

FIGURE 5.1 CONTINUOUS MODEL USED TO EVALUATE STRUCTURAL RESPONSE ......................................... 87

FIGURE 5.2 EXAMPLE OF SIMULATED STRUCTURAL RESPONSE COMPARED TO RECORDED RESPONSE ....... 88

FIGURE 5.3 CSMIP BUILDING RESPONSE COMPARISON SUMMARY SHEET LAYOUT .................................. 90

FIGURE 5.4 CSMIP BUILDING SUMMARY SHEET LAYOUT .......................................................................... 93

FIGURE 5.5 EXAMPLE OF RESULTS FROM SIMULATED STRUCTURAL RESPONSE. ........................................ 98

FIGURE 5.6 ATC-38 BUILDING SUMMARY SHEET LAYOUT ...................................................................... 100

FIGURE 5.7 DIFFERENCE BETWEEN OBSERVED VALUES AND VALUES PREDICTED BY A LOGNORMAL

DISTRIBUTION FOR DAMAGE STATE DS2 OF DRIFT-SENSITIVE NONSTRUCTURAL COMPONENTS BASED

ON DATA FROM CSMIP. ................................................................................................................... 104

FIGURE 5.8 DEVELOPING FRAGILITY FUNCTIONS USING THE BOUNDING EDPS METHOD. .......................... 106

FIGURE 5.9 LIMITATIONS OF FINDING UNIQUE SOLUTIONS FOR FRAGILITY FUNCTION PARAMETERS (A)

MULTIPLE SOLUTIONS FOR LEAST SQUARES AND MAXIMUM LIKELIKHOOD METHODS (B) MULTIPLE

SOLUTIONS FOR BOUNDED EDPS METHOD. ....................................................................................... 108

FIGURE 5.10 SAMPLE COMPARISONS OF DIFFERENT METHODS TO FORMULATE FRAGILITY FUNCTIONS (A)

EXAMPLE OF ALL THREE METHODS AGREEING (B) EXAMPLE OF 2 OUT OF 3 METHODS AGREEING. .... 109

FIGURE 5.11 (A) SAMPLE FRAGILITY FUNCTIONS COMPUTED FROM DATA FOR ACCLERATION

NONSTRUCTRAL COMPONENTS (FROM CSMIP) (B) SAMPLE FUNCTIONS AFTER ADJUSTMENTS. ....... 111

FIGURE 5.12 CSMIP FRAGILITY FUNCTIONS FOR (A) STRUCTURAL DAMAGE VS. IDR (B) NONSTRUCTURAL

DAMAGE VS. IDR AND (C) NONSTRUCTURAL VS. PBA. ................................................................... 113

FIGURE 5.13 EXAMPLE OF ATC-38 DATA SHOWING LIMITATIONS OF DATA .............................................. 116

FIGURE 5.14 FRAGILITY FUNCTIONS USING SUBSETS OF ATC-38 DATA BASED ON TYPE OF STRUCTURAL

SYSTEM (A) C-1: CONCRETE MOMENT FRAMES – DRIFT-SENSITIVE (B) S-1: STEEL MOMENT FRAMES –

DRIFT-SENSITIVE (C) C-1: CONCRETE MOMENT FRAMES – ACCELERATION-SENSITIVE (D) S-1: STEEL

MOMENT FRAMES – ACCELERATION-SENSITIVE ................................................................................ 117

FIGURE 5.15 COMPARISON TO HAZUS GENERIC FRAGILITY FUNCTIONS .................................................. 119

FIGURE 6.1 LOSS ESTIMATION TOOLBOX PROGRAM STRUCTURE ............................................................. 122

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FIGURE 6.2 BUILDING CHARACTERIZATION MODULE ................................................................................ 125

FIGURE 6.3 EDP-DV FUNCTION EDITOR MODULE ..................................................................................... 126

FIGURE 6.4 ADDING EDP-DV FUNCTIONS ................................................................................................ 127

FIGURE 6.5 VIEWING / EDITING / DELETING EDP-DV FUNCTIONS ............................................................ 128

FIGURE 6.6 MAIN WINDOW OF TOOLBOX ................................................................................................... 129

FIGURE 6.7 DEFINING THE SEISMIC HAZARD CURVE .................................................................................. 131

FIGURE 6.8 IMPORTING RESPONSE SIMULATION DATA. .............................................................................. 134

FIGURE 6.9 COLLAPSE FRAGILITY ADJUSTMENTS AND EDP EXTRAPOLATION OPTIONS ............................ 136

FIGURE 6.10 RESPONSE SIMULATION VISUALIZATION. .............................................................................. 137

FIGURE 6.11 ASSIGNING EDP-DV FUNCTIONS. ......................................................................................... 138

FIGURE 6.12 LOSS ESTIMATION MODULE - INCLUDING BUILDING DEMOLITION LOSSES GIVEN THAT THE

STRUCTURE HAS NOT COLLAPSED. .................................................................................................... 139

FIGURE 6.13 TOTAL AND DISAGGREGATION RESULTS FOR EXPECTED ECONOMIC LOSSES AS A FUNCTION OF

GROUND MOTION INTENSITY ............................................................................................................ 141

FIGURE 6.14 TOTAL AND DISAGGREGATION RESULTS FOR EXPECTED ANNUAL LOSSES. ............................ 142

FIGURE 7.1 GROUND MOTION PROBABILISTIC SEISMIC HAZARD CURVES (GOULET ET AL., 200&) ............ 148

FIGURE 7.2 EXAMPLE ARCHITECTURAL LAYOUT FOR HIGH-RISE BUILDINGS ............................................. 151

FIGURE 7.3 PEAK EDPS ALONG BUILDING HEIGHT FOR DESIGN 4-S-20-A-G (HAZELTON AND DEIERLEIN,

2007) ............................................................................................................................................... 153

FIGURE 7.4 COLLAPSE FRAGILITIES FOR 1, 2, 4, 8, 12 AND 20 STORY SPACE-FRAME BUILDINGS (HASELTON

AND DEIERLEIN, 2007) ..................................................................................................................... 154

FIGURE 7.5 EXPECTED LOSS GIVEN IM FOR 4-S-20-A-G (WITH COLLAPSE LOSS DISAGGREGATION) ......... 157

FIGURE 7.6 NORMALIZED EXPECTED ECONOMIC LOSS RESULTS AT DBE FOR 30 CODE-CONFORMING RC

FRAME STRUCTURES ......................................................................................................................... 158

FIGURE 7.7 EFFECT OF HEIGHT ON NORMALIZED EXPECTED LOSSES CONDITIONED ON GROUND MOTION

INTENSITY: (A) SPACE FRAMES AS A FUNCTION OF NORMALIZED GROUND MOTION INTENSITY (B)

PERIMETER FRAMES AS A FUNCTION OF NORMALIZED GROUND MOTION INTENSITY (C) NORMALIZED

LOSSES AT THE DBE AS A FUNCTION OF HEIGHT (D) COMPARISON OF PEAK IDRS BETWEEN 4 & 12-

STORY SPACE-FRAME BUILDINGS TO ILLUSTRATE CONCENTRATION OF LATERAL DEFORMATIONS. .. 160

FIGURE 7.8 EFFECT OF STRONG-COLUMN, WEAK-BEAM RATIO ON: (A) NORMALIZED EXPECTED LOSS AS A

FUNCTION OF NORMALIZED GROUND MOTION INTENSITY (B) NORMALIZED EXPECTED LOSS AT THE

DBE, DISAGGREGATED BY COLLAPSE & NON-COLLAPSE LOSSES. .................................................... 161

FIGURE 7.9 EFFECT OF DESIGN BASE SHEAR ON NORMALIZED EXPECTED LOSS AS A FUNCTION OF GROUND

MOTION INTENSITY ........................................................................................................................... 162

FIGURE 7.10 EAL RESULTS FOR 30 CODE-CONFORMING RC FRAME STRUCTURES .................................... 164

FIGURE 7.11 RESULTS OF MEAN ANNUAL FREQUENCY OF COLLAPSE FOR 30 CODE-CONFORMING RC FRAME

STRUCTURES (HASELTON AND DEIERLEIN, 2007). ........................................................................... 165

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FIGURE 7.12 SCATTER PLOTS AND CORRELATION COEFFICIENTS BETWEEN: (A) EAL & MAF OF COLLAPSE

(B) MAF OF COLLAPSE & YIELD BASE SHEAR COEFFICIENT (C) EAL & YIELD BASE SHEAR

COEFFICIENT .................................................................................................................................... 166

FIGURE 7.13 PRESENT VALUE OF NORMALIZED ECONOMIC LOSSES OVER 50 YEARS FOR 30 CODE-

CONFORMING RC FRAME STRUCTURES: (A) PRESENT VALUE OF LOSSES FOR EACH BUILDING AT A

DISCOUNT RATE OF 3% (B) RANGE OF PRESENT VALUE OF LOSSES AS A FUNCTION OF DISCOUNT RATE

(EXCLUDING DESIGN NUMBER 4). ..................................................................................................... 167

FIGURE 7.14 COMPARISON BETWEEN NORMALIZED ECONOMIC LOSS RESULTS BETWEEN MODERN, DUCTILE

(2003) AND OLDER, NON-DUCTILE REINFORCE CONCRETE FRAME STRUCTURES: (A) EXPECTED LOSS

AT DBE (B) EAL .............................................................................................................................. 169

FIGURE 7.15 COMPARISON OF EAL DISAGGREGATION OF COLLAPSE AND NON-COLLAPSE LOSSES FOR NON-

DUCTILE AND DUCTILE FRAMES ........................................................................................................ 170

FIGURE 7.16 COMPARISON OF VULNERABILITY CURVES FROM THIS STUDY AND FROM MDLA: (A)

PERIMETER FRAMES (B) SPACE FRAMES ............................................................................................ 171

1. FIGURE 8.1 CORRELATION BETWEEN SUBCONTRACTOR LOSSES DUE TO EDP VARIANCE (A) EDP-DV

FUNCTION FOR SUBCONTRACTOR K (B) EDP-DV FUNCTION FOR SUBCONTRACTOR K' ..................... 187

FIGURE 8.2 EDP DATA FROM INCREMENTAL DYNAMIC ANALYSIS AT INCREASING IM LEVELS ................. 190

FIGURE 8.3 EXAMPLE OF EDP RELATIONSHIPS WITH DIFFERENT LEVELS OF CORRELATION ...................... 191

FIGURE 8.4 CORRELATION TRENDS AT LOW AND HIGH SEISMIC INTENSITY LEVELS ................................... 192

FIGURE 8.5 VARIATION OF EDP CORRELATION WITH INTENSITY LEVEL.................................................... 193

FIGURE 8.6 RELATIONSHIP BETWEEN AVERAGE AND STANDARD ERROR OF CORRELATION COEFFICIENT

ESTIMATES ....................................................................................................................................... 195

FIGURE 8.7 DIFFERENCE BETWEEN 97.5TH AND 2.5TH PERCENTILES CONFIDENCE BANDS WITH MEDIAN

ESTIMATES OF CORRELATION COEFFICIENTS .................................................................................... 196

FIGURE 8.8 CONFIDENCE BANDS USING CLOSED FORM SOLUTION FOR DIFFERENT NUMBER OF GROUND

MOTIONS (A) BANDS FOR N = 10, 20, 40 AND 80 (B) COMPARISON WITH DATA FROM EXAMPLE

BUILDING. ........................................................................................................................................ 198

FIGURE 8.9 EDP-DV FUNCTIONS FOR ACCELERATION-SENSITIVE COMPONENTS IN A TYPICAL FLOOR FOR

THE EXAMPLE 4-STORY REINFORCED CONCRETE MOMENT-RESISTING FRAME OFFICE BUILDING ...... 206

FIGURE 8.10 EDP-DV FUNCTIONS FOR DRIFT-SENSITIVE COMPONENTS IN A TYPICAL FLOOR FOR THE

EXAMPLE 4-STORY REINFORCED CONCRETE MOMENT-RESISTING FRAME OFFICE BUILDING ............. 207

FIGURE 8.11 FOSM APPROXIMATIONS (A) LINEAR FUNCTION (B) NON-LINEAR FUNCTION ........................ 215

FIGURE 8.12 COMPUTING THE DERIVATIVE OF G(X) (A) LOCAL DERIVATIVE (B) AVERAGE SLOPE WITHIN

REGION THAT X WILL MOST LIKELY OCCUR IN. ................................................................................ 217

FIGURE 8.13 TYPICAL CASES OF EDP-DV FUNCTIONS FOR FOSM APPROXIMATIONS (A) UNDER-ESTIMATE

AT SMALL VALUES (B) OVER-ESTIMATE AT LARGE VALUES (C) GOOD APPROXIMATION AT MIDDLE

VALUES ............................................................................................................................................ 220

xv

FIGURE 8.14 QUANTITATIVE EXAMPLES OF FOSM APPROXIMATIONS USING THE DIFFERENT SLOPE

METHODS ......................................................................................................................................... 221

FIGURE 8.15 STANDARD DEVIATIONS FOR EACH SUBCONTRACTOR LOSS (A) DISPERSIONS DUE TO EDP

VARIANCE (B) DISPERSIONS DUE TO CONSTRUCTION COST VARIANCE ............................................... 225

FIGURE 8.16 MEAN VALUES OF ECONOMIC LOSS FOR EACH SUBCONTRACTOR AT THE DBE ..................... 227

FIGURE 8.17 COEFFICIENT OF VARIATIONS FOR EACH SUBCONTRACTOR LOSS (A) DISPERSIONS DUE TO EDP

VARIANCE (B) DISPERSIONS DUE TO CONSTRUCTION COST VARIANCE .............................................. 228

FIGURE 8.18 EFFECT OF SUBCONTRACTOR CORRELATION DUE TO EDP VARIABILITY .............................. 229

FIGURE 8.19 STANDARD DEVIATIONS OF LOSS CONDITIONED ON NON-COLLAPSE AT THE DBE CONSIDERING

DIFFERENT TYPES OF VARIABILITY AND CORRELATIONS .................................................................. 231

FIGURE 8.20 COEFFICIENT OF VARIATION OF LOSS CONDITIONED ON NON-COLLAPSE AT THE DBE

CONSIDERING DIFFERENT TYPES OF VARIABILITY AND CORRELATIONS ............................................ 231

FIGURE 8.21 STANDARD DEVIATION OF LOSS CONDITIONED ON NON-COLLAPSE AS A FUNCTION OF GROUND

MOTION INTENSITY (A) EDP VARIABILITY ONLY (B) CONSTRUCTION COST VARIABILITY ONLY (C)

EDP & COST VARIABILITY (D) EDP & COST VARIABILITY WITH EDP CORRELATIONS (E) EDP & COST

VARIABILITY WITH CONSTRUCTION COST CORRELATIONS (F) EDP & COST VARIABILITY WITH EDP &

COST CORRELATIONS. ....................................................................................................................... 234

FIGURE 8.22 ECONOMIC LOSS STANDARD DEVIATIONS CONDITIONED ON NON-COLLAPSE (NORMALIZED BY

THE BUILDING REPLACEMENT VALUE) AS A FUNCTION OF GROUND MOTION INTENSITY BASED ON THE

RESULTS FROM THE SIMULATION METHOD. ...................................................................................... 236

FIGURE 8.23 ECONOMIC LOSS STANDARD DEVIATIONS CONDITIONED ON NON-COLLAPSE (NORMALIZED BY

THE BUILDING REPLACEMENT VALUE) AS A FUNCTION OF GROUND MOTION INTENSITY FOR VALUES OF

SA(T1) 1.0G BASED ON THE RESULTS FROM THE SIMULATION METHOD. ........................................ 237

FIGURE 8.24 NORMALIZED STANDARD DEVIATION FOR OF LOSS (A) CONDITIONED ON NON-COLLAPSE (B)

CONDITIONED ON COLLAPSE. ............................................................................................................ 239

FIGURE 8.25 NORMALIZED EXPECTED LOSS AND DISPERSION GIVEN IM FOR EXAMPLE 4-STORY OFFICE

BUILDING ......................................................................................................................................... 241

FIGURE 8.26 COEFFICIENT OF VARIATION AS A FUNCTION OF INTENSITY LEVEL FOR EXAMPLE BUILDING. 242

FIGURE 8.27 MAF OF LOSS (A) EFFECT OF CORRELATIONS (B) COMPARISON BETWEEN ANALYTICAL AND

SIMULATION METHODS ..................................................................................................................... 243

FIGURE 9.1: PROBABILITY OF COLLAPSE FOR DUCTILE 4-STORY REINFORCED CONCRETE STRUCTURE

(HASELTON AND DEIERLEIN, 2007) ................................................................................................. 255

FIGURE 9.2: EDP DATA AS A FUNCTION OF BUILDING HEIGHT FOR DUCTILE 4-STORY REINFORCED

CONCRETE STRUCTURE (HASELTON AND DEIERLEIN, 2007)............................................................. 255

FIGURE 9.3 NORMALIZED EXPECTED ECONOMIC LOSS AS A FUNCTION OF GROUND MOTION INTENSITY. .. 257

xvi

FIGURE 9.4 EFFECT OF CONSIDERING LOSS DUE TO DEMOLITION CONDITIONED ON NON-COLLAPSE ON

NORMALIZED EXPECTED ECONOMIC LOSSES FOR A 4-STORY BUILDING AT THREE DIFFERENT LEVELS

OF SEISMIC INTENSITY. ..................................................................................................................... 258

FIGURE 9.5 COMPARISON OF THE PROBABILITY OF COLLAPSE WITH THE PROBABILITY OF BUILDING BEING

DEMOLISHED DUE TO RESIDUAL DEFORMATION AS A FUNCTION OF GROUND MOTION INTENSITY. ... 260

FIGURE 9.6 EFFECT OF CONSIDERING LOSS DUE TO DEMOLITION CONDITIONED ON NON-COLLAPSE ON

NORMALIZED EXPECTED ECONOMIC LOSSES FOR A 12-STORY BUILDING AT THREE DIFFERENT LEVELS

OF SEISMIC INTENSITY. ..................................................................................................................... 261

FIGURE 9.7 LOSS RESULTS FOR NON-DUCTILE BUILDINGS STUDIED (A) 4-STORY (B) 12-STORY ................ 262

FIGURE 9.8 COMPARISONS BETWEEN THE PROBABILITY OF COLLAPSE AND THE PROBABILITY OF

DEMOLITION FOR (A) A 4-STORY DUCTILE STRUCTURE (B) A 12-STORY DUCTILE STRUCTURE (C) A 4-

STORY NON-DUCTILE STRUCTURE AND (D) A 12 STORY NON-DUCTILE STRUCTURE. ......................... 264

FIGURE 9.9 DIFFERENT DISTRIBUTIONS FOR PROBABILITY OF DEMOLITION GIVEN RIDR (A) VARYING THE

MEDIAN (B) VARYING THE DISPERSION ............................................................................................ 266

FIGURE 9.10 RESULTS FOR SENSITIVITY ANALYSIS OF PROBABILITY OF DEMOLITION GIVEN RIDR FOR 4-

STORY DUCTILE REINFORCED CONCRETE MOMENT FRAME OFFICE BUILDING. .................................. 267

CHAPTER 1 1 Introduction

CHAPTER 1

1 INTRODUCTION

1.1 MOTIVATION & BACKGROUND

Despite significant improvements in seismic design codes (e.g. better detailing

requirements) that translate in better earthquake performance of modern buildings

compared to older structures, important deficiencies still exist. One of the inherent and

underlying problems with current structural design practice is that seismic performance is

not explicitly quantified. Instead, building codes rely on prescriptive criteria and overly

simplified methods of analysis and design that result in an inconsistent level of performance

(Haselton and Deierlein, 2005). One way of quantifying earthquake performance that has

been proposed by recent research (Krawinkler and Miranda 2004, Aslani and Miranda

2005, Mitrani-Reiser and Beck 2007) is using economic losses as a metric to gauge how

well structural systems respond when subjected to seismic ground motions.

While society and building owners’ main concern is the protection of life, there are

other risks that have traditionally been ignored in earthquake-resistant design. Namely,

current seismic design practice does not attempt to control economic loses or specify an

acceptable level of probability that a structure maintains its functionality after an

earthquake. During recent earthquakes in California, Loma Prieta in 1989 ($12 billion,

2008 US dollars) and Northridge in 1994 ($19-29 billion), substantial monetary losses were

incurred despite the relatively low loss in life (Insurance Information Institute, 2008). The

1989 Loma Prieta earthquake (Mw=6.9) resulted in 63 deaths, more than 3000 injuries and

produced between 8,000 and 12,000 homeless. The quake caused an estimated $6 billion to

$13 billion in property damage (Benuska, 1990). Similarly, the 1994 Northridge earthquake

resulted in 72 deaths and more than 9,000 injured including 1,600 that required

hospitalization. The direct economic loss has been estimated to be more than $25 billion


(Hall, 1995). Although the levels of ground motion intensity these seismic events

produced were considered relatively moderate, buildings experienced extensive structural

damage requiring substantial repairs.

A prominent example of how current design procedures fall short of building owners’

and users’ needs, was the nonstructural damage sustained by the Olive View Hospital

during the 1994 Northridge earthquake. Located in Sylmar, California, this six-story

structure was designed beyond minimum building code requirements in response to the

structural failure of the previous Olive View Hospital building during the 1971 San

Fernando earthquake. The replacement structure’s lateral force resisting systems consisted

of a combination of moment frames with concrete and steel plate shearwalls. Although the

building only experienced minor structural damage during the Northridge event, substantial

nonstructural damage was sustained. Particularly, sprinkler heads, rigidly constrained by

ceilings, ruptured when their connecting piping experienced large displacements. The

resulting water leakage caused the hospital to temporarily shut down. Not only was the

essential facility not able to treat injuries resulting form the earthquake, 377 patients being

treated at the time of the earthquake had to be evacuated (Hall, 1995). While the structure

conformed to building code standards for hospitals, the nonstructural damage resulted in the

loss of functionality of an essential facility directly after a seismic event. This damage

suffered by the Olive View Hospital illustrates how structural designs using prescriptive

codes may not be enough to achieve satisfactory seismic performance.

Damage, losses and loss of functionality sustained in these seismic events prompted

structural engineers to formulate preliminary documents (Vision 2000, FEMA 273 &

FEMA 356) that attempt to provide some guidance on how to achieve different levels of

performance that help stakeholders and design professionals make better and more

informed decisions that meet project-specific needs. Although these first generation

guidelines were a step towards making earthquake engineering adopt design approaches

that are more performance-based, the performance levels defined in these documents were

often qualitative, not well-defined and, consequently, open to subjectivity.

Recent advancements in performance-based earthquake engineering methods have

demonstrated the need for better quantitative measures of structural performance during

seismic ground motions and improved methodologies to estimate seismic performance. The

Pacific Earthquake Engineering Research (PEER) Center has conducted a significant

amount of research to address this need, by formulating a framework that quantifies


performance in metrics that are more relevant to stakeholders, namely, deaths (loss of life),

dollars (economic losses) and downtime (temporary loss of use of the facility). The PEER

methodology uses a probabilistic approach to estimate damage and the corresponding loss

based on the seismic hazard and the structural response. PEER’s work on performance-

based earthquake engineering is currently being implemented into seismic design standards

and guidelines by the Applied Technology Council through the ATC-58 project (ATC,

2007).

Building-specific economic loss estimation methods have advanced in recent years.

However, the process to calculate loss can become complicated because of the type and

amount of required computations. Practicing structural engineers are hard-pressed to

devote extra time towards detailed loss estimations in addition to delivering the structural

design. The successful adoption of performance-based design in the near future may hinge

on simplifying the loss estimation procedures and minimizing the computational effort

these procedures require.

1.2 OBJECTIVES

The goals of this is investigation are to improve areas of PEER’s economic loss

estimation framework by incorporating aspects that have been previously ignored, and, to

simplify it to decrease the amount of information required or time involved in performance

estimations. The resulting methods are then implemented into a computer tool that

estimates earthquake-induced economic losses as a quantitative metric of structural

performance. Specifically, the objectives of this study are as follows:

Introduce a new approach of estimating earthquake-induced monetary loss that

sums the losses by sub-contractor and by story, rather than by component, which is

more consistent with the way costs of construction projects are calculated and

requires less information to conduct the assessment.

Develop a simplified methodology of estimating mean economic losses by

consolidating fragility functions and normalized repair costs and collapsing out the

intermediate step of estimating damage to generate functions that relate response

simulation data directly to economic loss (EDP-DV functions).

Account for loss of a building’s entire inventory, given that the structure has not

collapsed, by developing generic fragility functions that estimate damage of


components that do not have specific fragilities. These fragilities will be derived by

establishing when damage initiates using empirical data, and then inferring the

probabilistic distribution parameters of more severe damage states.

Develop a computer toolbox that implements the new approach and to make

recommendations on how to address the computational challenges encountered.

Use the newly developed methods and tools to evaluate seismic-induced economic

losses of reinforced concrete moment frame buildings, including both ductile

concrete frames (that conform to current building seismic codes) and non-ductile

frames (that are representative of buildings built pre-1967 in California).

Propose a method of quantifying uncertainty on economic losses that incorporates

the correlations of construction costs at the building level. Cost correlations at the

component level have previously been considered at the building component-level,

however construction cost data is typically produced in terms of the entire building

or per subcontractor. A new procedure to integrate this type of data into the

computation of dispersion of economic losses is presented.

Evaluate the influence of the number of ground motions considered during

structural response analysis on the quality of estimates of response simulation

correlations.

Incorporate losses of a building that has not collapsed, but requires demolition due

to excessive residual drifts.

1.3 ORGANIZATION OF DISSERTATION

This dissertation is a collection of research papers on improving, simplifying and

implementing building-specific loss estimation methods. For chapters where co-authors

have contributed to the body of work, credit is documented at the beginning of the chapter

outlining the contributions of each author.

Chapter 2 presents a brief literature review of previous studies in building-specific

loss estimation methodologies and tools. The chapter chronologically outlines the most

relevant studies conducted by previous investigators for estimating seismic-induced

economic losses. Further, it summarizes the scope and limitations of the previous studies

and identifies gaps in research that have not yet been addressed. Addressing these gaps in

research provide the motivation for the objectives in this body of work.


Chapter 3 details the proposed method of simplifying PEER’s current building-

specific loss estimation methodology. It proposes collapsing out the intermediate step of

estimating damage by making assumptions on the building cost distribution among floors,

systems and components based on the building’s use, occupancy and structural system. The

formulation of generic EDP-DV functions is presented and example functions for

reinforced concrete moment frame office buildings are presented. The EDP-DV functions

are investigated to see which parameters have the greatest influence and how the issue of

conditional losses in spatially-interacting components affects the value of predicted loss.

Chapter 4 supplements the EDP-DV functions presented in Chapter 3 by

developing fragility functions for pre-Northridge beam-column joints. These functions can

be used to predict damage for pre-1994 steel moment frame buildings that have been found

to experience fracture at interstory drifts lower than previously expected. Results from

previous experimental studies are consolidated to formulate lognormal cumulative

distribution functions that predict yielding and fracture in these joints as a function of

interstory drift. Other parameters that significantly influence the functions were also

investigated.

Chapter 5 addresses the issue of estimating damage for components that do not

currently have fragility functions such that the entire building inventory is accounted for in

EDP-DV functions. Generic fragility functions are derived from empirical data gathered

during the 1994 Northridge earthquake. Two sources of data are considered in this study.

The first source generates motion-damage pairs from damage evaluations of instrumented

buildings documenting seismic performance (Naeim 1998). The second source relates

structural response to damage using damage data from the ATC-38 report (ATC 2000,

which documents damage for structures located close to ground motion stations) and

structural simulation to infer the response parameters. Functions are formulated for several

types of component groups, however, fragilities for drift-sensitive and acceleration sensitive

non-structural elements are of particular interest as these types of components typically lack

enough data to predict damage. The generic fragility functions for non-structural elements

presented here are used in Chapter 3 to supplement the formulation of the EDP-DV

functions. They are used for building components that do not have specific fragilities

generated from experimental data.

Chapter 6 documents the implementation of the simplified method presented in

Chapter 3, into an MS-EXCEL based computer tool. Despite the simplifications proposed


in this study, the performance-based framework still involves many variables and several

integrations that require a large amount of computation, necessitating a computer tool that

can facilitate these calculations. The tool also has the capability of computing economic

losses due to building demolition conditioned on non-collapse (as described in detail in

Chapter 9).

Chapter 7 presents economic seismic loss estimations for a set of archetypes of

reinforced concrete moment-resisting frame buildings, designed and analyzed by previous

investigators (Haselton and Deierlein, 2007, Liel and Deierlein, 2008), using the simplified

method presented in Chapter 3 and the computer tool illustrated in Chapter 6. The results

presented here are used to quantify loss results for both code-conforming structures, and

non-ductile concrete structures, representing buildings of an older vintage. The study

benchmarks performance in terms of economic loss for these types of structures, and

attempts to identify building parameters that have the strongest influence on seismic

performance.

Chapter 8 presents a modified approach of incorporating correlations into the

calculation of the uncertainty in predicting earthquake-induced economic losses. Aslani

and Miranda (2005) first introduced methods on how to incorporate repair cost correlations

at the component-level. However, estimates of these correlations at the component level

are not available, and collecting this type of data can be difficult. There is, however,

dispersion and correlation data available for construction costs between different

construction trades at the building level (Touran and Suphot, 1997). The approach

proposed in this investigation attempts to incorporate these correlations at the building

level, by first breaking down the costs associated with repair or replacement of each

component into different construction trades. The dispersions are then aggregated and

propagated for each trade until the uncertainty of the loss is calculated at the building level

where the construction cost correlations can be included. The influence of accounting for

these correlations on the loss dispersions is evaluated. The effect of correlations from

simulation data is also evaluated and the appropriate number of ground motions considered

in response simulation to accurately capture these correlations is investigated.

Chapter 9 proposes modifying the PEER loss estimation framework to incorporate

an intermediate building damage state in which demolition of a building becomes necessary

when excessive damage that cannot be repaired has occurred. The proposed approach uses

peak residual interstory drift as an engineering demand parameter to predict the likelihood


of having to demolish a building after an earthquake, given that the building has not

collapsed. The simplified method of Chapter 3 is used to evaluate losses of example

buildings taken from the study conducted in Chapter 6, to illustrate the effect of considering

these types of losses. It is shown that incorporating losses to due possible demolition has a

significant impact on predicted losses due to seismic ground motions.

Chapter 10 summarizes the results and contributions from this investigation.

Conclusions are drawn from these results and extended to identify what impact they have

on the field earthquake engineering. Finally, areas of future research are identified to lay

the groundwork for future investigators.

CHAPTER 2 8 Previous Work in Loss Estimation

CHAPTER 2

2 PREVIOUS WORK ON LOSS ESTIMATION

2.1 LITERATURE REVIEW

Current loss estimation methodologies can be categorized in two main types:

methodologies for regional loss estimation and methodologies for building-specific loss

estimation. Because regional methods do not provide the necessary level of detail required

by performance-based earthquake engineering (Aslani and Miranda, 2005), only a brief

review of these approaches is included here. This literature review primarily focuses on

previous studies in building-specific loss estimation. Although the review does not

document all previous research that has conducted on economic loss estimation, it attempts

to summarize the studies that directly influenced the direction of this dissertation and does

not discount the importance of other investigations that are not mentioned here,

2.2 REGIONAL LOSS ESTIMATION

Regional loss estimation attempts to quantify losses for a large number of buildings

within a specific geographic area. One of the first major studies that attempted to do this

was the study by Algermissen et al. (1972) which provided damage and loss estimates for

six scenario earthquakes in the San Francisco Bay Area (on the San Andreas & Hayward

Faults, with magnitudes 8.3. 7.0 and 6.0 on each fault). Although the study focused

primarily on injuries and casualties, economic losses were evaluated as well. Monetary

losses from repair costs were provided primarily for wood frame structures. This study was

the first of several similar studies to estimate seismic-induced losses in major metropolitan

areas (Los Angeles, Salt Lake City & Puget Sound).


One of the first investigations to explicitly consider the probabilistic nature of

seismic-induced monetary losses was the study by Whitman et al. (1973), which introduced

the concept of damage probability matrices into loss estimation methodology. These

damage probability matrices were developed for 5-story buildings with the following

structural systems: reinforced concrete moment frames, reinforced concrete shear walls and

steel moment frames. In this study, damage ratios were used to describe the amount of

estimated damage and seismic intensity was expressed as a function of Modified Mercalli

Intensity (MMI). Mean damage ratios were calculated for buildings in the San Francisco

Bay area and the Boston area to illustrate the use of this procedure.

The Applied Technology Council (ATC) conducted a study that provided data to

evaluate earthquake damage for California (ATC-13, 1985). The report developed a facility

classification scheme for 91 different types of facility classes (e.g. industrial, commercial,

residential…etc.). Damage probability matrices and the estimated amounts of time to repair

damaged facilities were constructed for the different classifications of structures. The

damage probability matrices, relating ground motion intensity to level of damage were

developed by expert opinion using a Delphi procedure. Damage estimation as a function of

MMI was then conducted using these matrices for different types of facilities in California.

ATC-13 also reviewed several inventory sources and introduced a method for estimating

large building inventories from economic data. The report provided a detailed description

of the inventory information, which is necessary when evaluating regional losses.

In 1992, the Federal Emergency Management Agency (FEMA) and the National

Institute of Building Sciences (NIBS) began funding the development of a geographic

information system (GIS)-based regional loss estimation methodology (Whitman et al.

1997), which eventually was implemented in the widely-used computer tool, HAZUS

(National Institute of Building Sciences, 1997). Based on a building’s lateral force resisting

system, height and occupancy, structural response and damage are calculated using pre-

established capacity and fragility functions to determine economic losses as a function of

the peak response of single-degree-of-freedom (SDOF) systems (i.e., spectral ordinates).

Generalizing buildings in this manner provides a simple and widely applicable way of

estimating loss; however, it does not capture unique and important aspects of a specific

building’s structural and nonstructural design.


2.3 BUILDING-SPECIFIC LOSS ESTIMATION

One of the first building-specific loss estimation methodologies was developed by

Scholl et al. (1982). The authors of this report developed and suggested improvements to

both empirical and theoretical loss estimation procedures. Part of the theoretical studies

included an in depth study of developing damage functions for a variety of building

components based on experimental test data. The report recommends a probabilistic,

component-based method of evaluating damage, and demonstrated applications of this

method. Three example buildings (the Bank of California Building and two hotel

buildings) damaged during the 1971 San Fernando earthquake were used to illustrate the

proposed damage-prediction methodology. To develop the theoretical motion-damage

relationships, only elastic analyses in combination with response spectrum analysis (using

spectral displacement to as the spectral ordinate) were used to estimate structural response

at each floor of each building being considered. The resulting relationships measured

damage using a damage factor, which is the ratio between the repair costs induced by

earthquake damage and the replacement value of the building.

The method proposed by Scholl et al. (1982) required component damage functions

(i.e. component fragility functions), to estimate damage on a component-by-component

basis. In conjunction with the Scholl et al. (1982) study, Kutsu et al. (1982) collected

laboratory test data to estimate damage in various high-rise building components to

implement the proposed component-based methodology. The investigators consolidated

experimental data for components commonly found in high-rise buildings and statistically

determined central tendency and variability values of exceeding particular levels of damage

in these components. The components evaluated included the following: reinforced

concrete structural members (beams, columns and shear walls), steel frames, masonry

walls, drywall partitions and glazing. Based on published building cost data, the study also

statistically determined proportions of construction costs for these components. This

information was then used in combination with the damage functions to calculate the

overall damage factor of the component (damage as percentage of the replacement values of

the component). Although no building damage results were produced by Kutsu et al.

(1982), these relationships were subsequently used by Scholl et al. (1982) to develop the

theoretical motion-damage relationships for the three example buildings mentioned

previously, using rudimentary elastic analyses to approximate the structural response


parameters. These relationships are limited because the analyses used do not capture

higher-mode effects and damage due to nonlinear behavior.

A scenario-based loss estimation methodology – assessing monetary losses of a

building from its structural response from a particular earthquake ground motion – was

introduced by Gunturi and Shah (1993). Damage to building components, categorized into

structural, nonstructural and contents elements, was calculated by obtaining structural

response parameters at each story from a nonlinear time history analysis, by scaling the

record to peak ground acceleration (PGA) levels of 0.4g, 0.5g and 0.6g. The response

parameters were related to damage levels for each component and loss was calculated per

story and summed to get the total building loss. An energy-based damage index developed

by Park and Ang (1985) was used to estimate damage in structural elements, while

interstory drift and peak floor accelerations were used to assess nonstructural damage.

Several strategies to map these damage indices to monetary losses, including a probabilistic

approach, but based on the available data at the time the study was published, a

deterministic mapping primarily based on expert opinion was used for the example

buildings presented. Losses were assessed for several reinforced concrete moment resisting

frame buildings as examples to illustrate their approach. Although their study examined

damage variation with different ground motions for one of the example buildings presented,

the frequency at which ground motions occur was not accounted for.

The variability in ground motions, as it relates to assessing economic losses for

buildings, was addressed in a study by Singhal and Kiremidjian (1996). A systematic

approach to developing motion-damage relationships was proposed by subjecting a

structure to a suite of simulated ground motions, and obtaining its probabilistic response

using Monte Carlo simulation. Methods for two types of motion-damage relationships,

building-level fragility curves and damage probability matrices (DPMs), were developed.

Each type of relationship predicted the probability of exceeding discrete damage states.

These damage states were defined using ranges of damage indices that quantified building-

level damage as the ratio between repair costs over the total replacement value of the

building. For the fragility curves, root mean square (RMS) acceleration and spectral

acceleration for a specified structural period range are used to characterize earthquake

ground motion. MMI was used as the ground motion parameter for the DPMs. Artificial

ground motions were generated using models that included the stationary Gaussian model

with modulating functions and the autoregressive moving-average (ARMA). Structural


response was computed using nonlinear dynamic analysis using DRAIN-2DX. Park and

Ang’s (1985) index was used to relate this response to damage level and to predict the

probability of damage occurring. Fragility curves and DPMs were generated for reinforced

concrete frame structures, classified into low-rise (defined in this study as 1-3 stories tall),

mid-rise (4-7 stories) and high-rise (8 stories or taller) categories. However, these curves

only account for structural damage do not consider damage due to nonstructural building

components.

Porter and Kiremidjian (2001) introduced an assembly-based framework that is

fully probabilistic. It also incorporates the uncertainty stemming from estimating building

damage and the associated repair costs, which previously had not been considered. Monte

Carlo simulation was used within this framework to predict building-specific relationships

between expected loss and seismic intensity (also known as vulnerability curves).

Techniques to develop fragility functions for common building assemblies were presented

and used to predict losses for an example office building. Ground motions used in the

examples presented in this study were simulated using the ARMA model to generate the

number of artificial time histories necessary to run structural analyses. Depending on the

structural response parameter of interest, the study used both linear and non-linear dynamic

analyses to compute peak structural responses. A simplified, deterministic sensitivity

analysis was also conducted to investigate which sources of uncertainty have the largest

effect on loss results; the uncertainty of the ground motion intensity was found to have the

largest influence. In the framework proposed by Porter and Kiremidjian (2001) no attempt

is made to explicitly compute the probability of collapse.

As part of the Pacific Earthquake Engineering Research (PEER) center’s effort to

establish performance-based assessment methods, Aslani and Miranda (2005) developed a

component-based methodology that incorporated the effects of collapse on monetary loss

by explicitly estimating the probability of collapse at increasing levels of ground motion

intensity. Both sidesway collapse and loss of vertical carrying capacity were integrated into

the calculation of seismic-induced expected losses, however, losses due to building

demolition resulting from large residual interstory drifts were not considered. This

investigation also proposed techniques to disaggregate building losses to identify the most

significant components that contribute to the overall loss. Additionally, the authors

presented a method for incorporating the effect of correlations into calculating the

dispersion associated with these losses at the component-level. Values of component cost


correlations were unavailable and so building-level cost data was used to approximate these

correlation coefficients. Component fragilities necessary to illustrate the use of these

techniques were developed and applied to an existing seven-story non-ductile reinforced

concrete moment frame building. Damage of components was primarily estimated with

minimal consideration of any dependent losses between spatially interacting components.

This study treated these component losses independently, assuming that they would not

have any affect on the overall losses due to non-collapse.

In coordination with the study by Aslani and Miranda (2005), PEER’s component-

based loss estimation methodologies were also developed and implemented by Mitrani-

Reiser and Beck (2007). This study developed a computer program, named the MATLAB

Damage and Loss Analysis (MDLA) toolbox, that implemented the PEER loss estimation

framework. This program was then used in an investigation to benchmark the performance

of a 4-story ductile reinforced concrete moment resisting frame office building, which

conformed to modern day seismic codes. Mean losses as a function of ground motion

intensity level and expected annual losses were calculated for multiple design variants to

examine how different structural and modeling parameters influenced monetary losses. The

design variants only consisted of 4-story structures, and consequently, losses for structures

of different heights were not examined. Losses due to non-collapse were calculated on a

component-by-component basis, however, much like previous studies, the estimations only

included losses from components with available fragility functions. The components

considered in this study included beams, columns, slab-column joints, partitions, glazing,

sprinklers and elevators. An attempt was made to account for dependent losses of spatially

interacting components by including the replacement cost of the dependent component in

the repair cost of the other component. However, this approach results in counting the loss

of the dependent component twice.

Zareian and Krawinkler (2006) developed a simplified version of PEER’s

performance-based design framework. This study uses a semi-graphical approach to

compute building-specific economic losses. Instead of computing monetary losses per

component, the approach computes losses by grouping components into subsystems (either

at the story-level or building-level) such that components that belong to the same subsystem

are well represented by a single structural response parameter. Although this study

provided a framework that was easier to work with and less complicated, the investigators

had to make assumptions about the relationships between structural response and economic


loss to evaluate performance due to the limited damage estimation and loss data available at

the time the research was published.

2.4 LIMITATIONS OF PREVIOUS STUDIES

Although building-specific loss estimation methods have advanced substantially in

recent years, there are many issues that have been left unaddressed. Some of the key

limitations that can be identified in previous studies described above are as follows:

One the one hand, regional loss estimation methods are typically based on single

degree of freedom (SDOF) systems and therefore are not able to adequately capture

many significant effects that building-specific approaches can. Effects that are not

captured by regional loss estimation methods include higher mode effects of multi-

degree of freedom (MDOF) systems, nonlinear behavior of structures and repair

cost variability. On the other hand, building-specific loss estimation methods can

become complicated and computationally intensive. These types of analyses are

more tedious and time-consuming than the regional loss estimation methods. A

simplified approach that combines the efficiency of regional methods while

maintaining the ability to capture an adequate level of detail that building-specific

techniques employ has yet to be developed.

The economic losses of certain building components are often dependent on the

damage state of another component. Losses due to this dependency have been

either ignored (Aslani and Miranda, 2005) or accounted for twice (i.e. “double

counting”) in the repair costs of both components (ATC, 2007). Methods to

account for this interaction such that the losses are not underestimated or

overestimated are not yet available.

Previous studies have made efforts to predict damage probabilistically by

developing fragility functions for various building components. Unfortunately,

many components found in a building’s inventory remain without established

fragilities because of a lack of available data. Previous studies (Porter and

Kiremidjian 2001, Mitrani-Reiser 2007) have either ignored components for which

there are no fragility functions or have treated them as rugged (i.e. components are

not damaged unless collapse occurs). Other investigators (Aslani and Miranda,


2005) have estimated the loss in some of these components by using generic

functions that were initially developed to be used in regional methods (HAZUS) for

some of these components. The data used to develop these generic functions,

however, are not well-documented and rely heavily on expert opinion that has yet to

be validated. Generic functions that estimate damage based on more reliable data

are required until data for component-specific fragilities become available, as these

components can contribute significantly to the building’s overall loss.

Modern structures are designed to be more ductile to protect life-safety by

preventing collapse. However, these structures have a higher probability of

experiencing residual interstory drifts that are large enough to warrant post-

earthquake building demolition. While previous investigations have been able to

account for losses due to non-collapse (Porter and Kiremidjian, 2001) and losses

due to collapse (Aslani and Miranda, 2005), there has been limited work conducted

to develop an approach that includes monetary losses from a building that has not

collapsed but requires demolishing the building. In particular, the probability that

the building will be demolished due to excessive permanent lateral drifts as a

function the probability of residual interstory drift exceeding a particular value for a

given ground motion intensity level has not been incorporated into the current

PEER loss estimation framework.

Aslani and Miranda (2005) derived methods to incorporate construction cost

correlations into quantifying the uncertainty of seismic-induced loss on a

component basis. Yet general contractors structure building construction costs by

incorporating estimates of various construction subcontractors. Therefore, much of

the cost data available to calculate cost dispersion and correlations is at the

building-level or trade/subcontractor-level and not the component-level. An

approach to incorporate cost correlations using the data available has not been

previously proposed to quantify its effect on uncertainty propagation.

Although previous studies have established the need of using multiple ground

motions to characterize the probabilistic nature of the structural response, the

number of ground motions to be considered in a loss analysis is much debated. One

issue that has not been considered when determining the number of necessary

ground motions, is how the number of ground motions considered influences the


quality of estimates of response parameter correlations coefficients computed from

response simulation.

Current building-specific loss estimation methods require a large amount of

computation (Aslani and Miranda, 2005), making hand predictions of losses tedious

and unpractical. Computer tools are needed to facilitate these computations such

that analysts can focus on the analytical data that is input into loss estimation and on

the evaluation the results, rather than on the process of predicting monetary losses.

Mitrani-Reiser and Beck (2007) created a MATLAB-based computer tool that

implements the current methods. However, there are a limited amount of tools that

exist for simplified building-specific loss estimation methods. Also, most computer

tools have not considered losses conditioned on non-collapse that are caused by to

building demolition.

Mitrani-Reiser and Beck (2007) collaborated with other PEER researchers (Goulet

et al., 2007) to evaluate and benchmark the seismic performance of a conventional

4-story reinforced concrete moment frame building in terms on monetary loss.

Losses for a range of design variations for this class of buildings have not been

evaluated. Benchmarking losses for an entire class of structures can help identify

trends and quantify how well these types of buildings perform when subjected to

seismic ground motions.

CHAPTER 3 17 Simplified Building Specific Loss Estimation

CHAPTER 3

3 STORY-BASED BUILDING-SPECIFIC LOSS ESTIMATION

This chapter is based on the following publication:

Ramirez, C.M., and Miranda, E. (2009), “Story-based Building-Specific Loss Estimation,”

Journal of Structural Engineering, (in preparation).

3.1 INTRODUCTION

Current seismic codes are aimed primarily at protecting life-safety by providing a

set of prescriptive provisions. Recently a few documents have been published which have

laid the ground work for performance-based design. In the United States, the two most

notable are Vision 2000 (SEAONC, 1995) and ASCE-41 (which was based the pre-standard

document FEMA-356 and the previous guidelines FEMA-273). Both documents define

discrete global performance goals. For instance, ASCE-41 (ASCE, 2007) describes four

structural performance levels as follows: operational, immediate occupancy, life safety and

collapse prevention. However measuring performance in this way is difficult because the

performance levels are not clearly defined or easy to work with. Recent research suggests

structural performance should be quantified in more useful terms on which stakeholders can

base their decisions. The Pacific Earthquake Engineering Research (PEER) Center suggests

that economic losses, down time and number of fatalities are better seismic performance

measures. Thus, there is a great need to develop procedures to estimate economic loss that

a structure is likely to experience in future seismic events.

PEER has established a fully probabilistic framework that uses the results from

seismic hazard analysis and response simulation to estimate damage and monetary losses

incurred during earthquakes. The methodology is divided into four basic stages that


account for the following: ground motion hazard of the site, structural response of the

building, damage of building components and repair costs. The first stage uses probabilistic

seismic hazard analysis to generate a seismic hazard curve, which quantifies the frequency

of exceeding a ground motion intensity measure (IM) for the site being considered. The

second stage involves using structural response analysis to compute engineering demand

parameters (EDPs), such as interstory drift and peak floor accelerations), and the collapse

capacity of the structure being considered. The third stage produces damage measures

(DMs) using fragility functions, which are cumulative distribution functions relating EDPs

to the probability of being or exceeding particular levels of damage. The fourth and final

stage establishes decision variables (DVs), in this case economic losses based on repair and

replacement costs of damaged building components, which stakeholders can use to help

them make more informed design decisions. The results of each stage serves as input to the

next stage as shown in schematically in Figure 3.1. Mathematically, if the metrics from

each stage are considered to be random variables, they can be aggregated using the theorem

of total probability as demonstrated by Cornell and Krawinkler (2000) using the following

equation:

DV G DV DM dG DM EDP dG EDP IM d IM (3.1)

where G[X|Y] denotes the complementary cumulative distribution function of X conditioned

on Y, [X|Y] denotes the mean annual occurrence rate of X given Y.

Intensity Measure (IM)

Engineering Demand Parameter (EDP)

Damage Measure (DM)

Decision Variable (DV)

PEER Methodology

EDP-DV Functions



Damage Measure (DM)


PEER Methodology

EDP-DV Functions



Damage Measure (DM)


Intensity Measure (IM)Intensity Measure (IM)



Damage Measure (DM)

Damage Measure (DM)



PEER Methodology

EDP-DV Functions



Damage Measure (DM)


PEER Methodology

EDP-DV Functions



Damage Measure (DM)


PEER Methodology

EDP-DV Functions



Damage Measure (DM)





Damage Measure (DM)



Damage Measure (DM)


PEER Methodology

EDP-DV Functions



Damage Measure (DM)


PEER Methodology

EDP-DV Functions



Damage Measure (DM)





Damage Measure (DM)

Damage Measure (DM)



PEER Methodology

EDP-DV Functions

Figure 3.1 PEER methodology


This framework involves several integrations of many random variables making it very

computational intensive. It also requires obtaining a complete inventory of the building

being evaluated which can be time consuming. The amount of data to keep track of (i.e. the

number of response parameters and their locations, the number of building components, the

number of damage states…etc) can become overwhelming. Consequently, the loss

estimation process can be very time consuming, making it prohibitively expensive to

conduct on a routine basis. Simplifying the economic loss estimation procedure would

allow decision makers to focus on the hazard and structural analysis that serve as input to

loss assessments, and the resulting output, structural performance results and design

decisions, rather than on the process of estimating losses.

A simplified version of PEER’s previous building-specific loss estimation

methodology is presented in this study. The proposed approach, hereon referred to as story-

based loss estimation, is predicated on conducting beforehand the third stage of PEER’s

framework, damage estimation (see Figure 3.1), thus reducing the amount of data and

computation that the design professionals would need to assess seismic structural

performance. This can be achieved by creating functions, termed EDP-DV functions,

which relate structural response parameters (EDPs) directly to economic losses (DVs).

These functions reduce the amount of computation by integrating fragility functions with

repair costs beforehand, and reduce the amount of data required to be tracked by making

assumptions regarding the building’s inventory based on its occupancy and structural

system. These functions are particularly useful when assessing seismic performance during

schematic design because many important design decisions, such as the type of lateral force

resisting system, are made during this stage, when much of the building’s inventory is

uncertain or unknown. Generic story EDP-DV functions are computed here for reinforced

concrete moment-resisting frame office buildings, as a demonstration of this approach.

Additionally, consolidating fragility function and repair costs in this manner provides the

opportunity to investigate the issue of conditional losses of spatially interdependent

components using EDP-DV functions to analyze how they can be accounted for using this

methodology.


3.2 STORY-BASED BUILDING-SPECIFC LOSS ESTIMATION

3.2.1 Previous loss estimation methodology (component-based)

The third and fourth stages of PEER’s methodology, as described in the previous

section, involve building-specific damage and loss estimation procedures that have been

developed at the component level. It is assumed that the total loss in a building, LT, is equal

the sum of repair and replacement costs of the individual components damaged during

seismic events. This loss can be computed as:

1 2 31

...n

T j j j j j nj

L L L L L L (3.2)

where Lj is the loss in the jth component and n is the total number of components in the

building (note that all the losses in this equations are random variables). Every damageable

component considered in the analysis is assigned fragility functions to estimate damage

based on the level of structural response. This is what will be herein referred to as

component-based loss estimation.

Previous studies (Krawinkler & Miranda 2006, Aslani 2005, Mitrani-Reiser, 2007)

have already derived the mathematical expressions used in PBEE. Calculating expected

losses conditioned on ground motion intensity, E[LT | IM], is the summation between losses

due to total collapse multiplied by the probability of collapse and the losses due to non-

collapse multiplied by the probability of non-collapse as shown by the following

expression,

| | , | | |T T TE L IM E L NC IM P NC IM E L C P C IM (3.3)

where E[LT | NC,IM] is the expected loss in the building provided that collapse has not

occurred for ground motions with an intensity level of IM, E[LT | C] is the expected loss in

the building when collapse has occurred in the building, P(NC | IM) is the probability that

the structure will not collapse conditioned on the occurrence of an earthquake with ground

motion intensity, IM, and P(C | IM) is the probability that the structure will collapse


conditioned on IM, which is complementary to P(NC | IM), that is, P(NC | IM) = 1 - P(C |

IM).

Determining the expected losses given collapse involve estimating the probability

of collapse from the structural response simulation of the building and estimating the

expected value of the loss given that collapse has occurred. The latter typically involves the

cost of removal of collapse debris from the site plus replacement value.

The simplifications proposed in this study will concentrate on the simplification of

calculating expected losses due to non-collapse. The expression for expected losses

conditioned on non-collapse is given as follows:

1 1

| , | , | ,N N

T j jj j

E L NC IM E L NC IM E L NC IM (3.4)

where E[Lj |NC , IM] is the expected loss in the jth component given that global collapse

has not occurred at the intensity level IM, and Lj is the loss in the jth component defined as

the cost of repair or replacement.

Using the total probability theorem, the expected loss given no collapse has

occurred can be calculated as follows:

0

| , | , | ,j j j j jE L NC IM E L NC EDP dP EDP edp NC IM (3.5)

where E[Lj | NC, EDPj] is the expected loss in the jth component when it is subjected to an

engineering demand parameter, EDPj, P(EDPj > EDPj | NC, IM ), is the probability of

exceeding EDPj, in component j given that collapse has not occurred in the building and the

level of ground motion intensity IM is im. Further detail on the estimation of the

conditional probability P(EDPj > EDPj | NC, IM ) and probabilistic seismic response

analysis, can be found in Aslani and Miranda (2005).


The expected loss in component j conditioned on EDP, E[Lj | NC, EDPj] is a

function of the component’s repair cost when it is in different damage states and the

probability of being in each damage state as illustrated in the following expression:

1

[ | , ] | , | ,m

j j j i i ji

E L NC EDP E L NC DS P DS ds NC EDP (3.6)

where m is the number of damage states in the jth component, E[Lj | NC, DSi] is the

expected value of the normalized loss in component j when it is in damage state i , DSi, and

P(DS = dsi | NC, EDPj) is the probability of the jth component being in damage state i, dsi ,

given that it is subjected to a demand of EDPj. The probability of being in each damage

state for component j can be obtained from component-specific fragility functions. The

reader is referred to Aslani and Miranda (2005) for further details on the development of

component-specific fragility functions.

3.2.2 EDP-DV function formulation

The first step in developing story EDP-DV functions is collapsing out the third

intermediate step of damage estimation by combining information from loss functions and

fragility function as shown in equation (3.6). This requires consolidating all the fragility

and expected repair costs for each component. However, if the repair costs are normalized

by the component’s replacement value, aj, this computation can be conducted without

having to provide these values for every damage state, which will save a substantial amount

of number keeping. Mathematically, aj can be factored out of equation (3.6), and canceled

on both sides equations such that:

1

[ | , ] | , | ,m

j j j j j i i ji

a E L NC EDP a E L NC DS P DS ds NC EDP (3.7)

where E’[Lj | NC, DSi] and E’[Lj | NC, EDPi] are now normalized by the component’s

replacement value, aj.


The second step involves summing the individual component losses for the entire

story of a building. Previously, this summation requires inventorying the number of

components and the value of each component type. However, generic EDP-DV functions

can be formulated if components of the same type are grouped together and assumed to

experience the same level of damage (i.e. all partitions in the same story experience the

same level of damage). The loss for each component type can be calculated by multiplying

the results of equation (3.7) by its value relative to entire value of the story, bj (that is, bj is

equal to the total value of components of the same type, j¸ divided by the total value of the

story). Component types can then by summed for the entire story using:

[ | , ] [ | , ]m

STORY k j j jj

E L NC EDP b E L NC EDP (3.8)

where [ | , ]STORY kE L NC EDP is the expected loss of the entire story normalized by the

replacement value of the story, conditioned on the kth EDP. This is how the generic EDP-

DV functions will be expressed. With the loss expressed in these terms, the analysts no

longer needs to specify j replacement values for each component, but rather only needs to

stipulate the total value of the story to determine the loss of the component. The monetary

value of the expected loss for the entire story can then be found with the following

equation:

[ | , ] [ | , ]STORY k l STORY kE L NC EDP c E L NC EDP (3.9)

where [ | , ]STORY kE L NC EDP is the economic loss of the story expressed in dollars and cl is

the replacement value of the story in dollars.

Note that because the results of equations (3.8) and (3.9) are conditioned on EDP,

separate functions need to be generated for each type of EDP sensitivity. EDP sensitivity

is defined by what type of EDP is used to determine building component damage.

Although there are many types of EDPs, the loss estimation process can be further

simplified if the choice of different EDPs is limited to a small number. The EDPs chosen in


this study are interstory drift ratio (IDR) and peak floor accelerations (PFA). Accordingly,

components can be categorized as either drift-sensitive or acceleration sensitive, depending

which type of parameter induces damage for each component. It is also useful for engineers

to differentiate between structural and nonstructural components. Assuming that structural

damage is primarily caused by IDR, it was determined that only the following seismic

sensitivities would be considered in this implementation: drift-sensitive structural

components, drift-sensitive nonstructural components, and acceleration sensitive

nonstructural components.

An important consideration when formulating EDP-DV functions is whether or not

the economic losses of components on the same story are dependent on one another due to

spatial and physical interactions between the components. This issue is described and

explored in greater detail in section 3.5 of this chapter. For the EDP-DV functions

presented in this study, it was found that these types of losses did not have a large influence

on the total economic losses for each story. However, this may not necessarily always be

the case for other types of structural systems and occupancies and a method of accounting

for these types of losses into EDP-DV functions is presented.

3.3 DATA FOR EDP-DV FUNCTIONS

3.3.1 Building Components & Cost Distributions

Generic story EDP-DV functions normalized by the story replacement value

requires knowing typical cost distributions for a given building occupancy and structural

system. The source chosen to establish the cost distribution for this investigation is the

2007 RS Means Square Foot Costs (Balboni, 2007). The publication gives cost

distributions of the entire building rather than the distributions at the story level.

Engineering judgment was used translate this data into story cost distributions, while

maintaining the overall building cost distribution.

Translating the building cost distribution to story distributions requires making

assumptions as to how the value varies along its height. This will be highly dependent on

how the building components are distributed amongst the different floors, which is typically

a function of the occupancy of the building. The sample functions generated by this study

are for typical commercial office buildings. Although different story cost distributions


could be generated for ever floor, the number of distributions used can be limited by

making the following assumptions:

The entire building will be used for office space (i.e. not a mixed-use facility)

The value of the first floor has significant differences from the other floors

because as the main entrance, the layout, facades and finishes are typically

different at this level.

The value of the top floor, typically the roof of the building, has significant

differences from the other floors because typically this is where most of the

buildings MEP equipment is located (this floor includes any equipment that

may be located in a mechanical penthouse).

The remaining intermediate floors are all dedicated to office use. These floors

will have the same story cost distribution.

Under these assumptions, it was decided that there would be three different types of story

cost distributions: one for the 1st floor, one for the top floor, and one for the intermediate

floors, which will be referred to as the typical floor.

The 2007 RS Means Square Foot Costs (Balboni, 2007) documents estimated cost

building distributions for many different types of common building occupancies (ex.

residential high-rise, commercial low-rise, hospitals…etc.). Table 3.1 displays an example

cost distribution for a 7-story commercial office building. The first column is the cost

distribution for the entire building taken directly from RS Means (Balboni, 2007). Based on

this information, the cost distributions for the 1st floor, the typical floors and the top floor

were approximated as shown in the second, third and fourth columns, respectively, in Table

3.1. Most of the story distributions are similar to the overall building distributions with the

exception of a couple of items that reflect the assumptions discussed in the previous

paragraph. For instance, the component group Exterior Enclosures has a higher

contribution to the story cost in the 1st floor because it is common to have more expensive

exterior elements around the building’s main entrances. Conversely, component groups

such as HVAC and Conveying have high cost contributions at the top floor because most of

the equipment associated with these groups is typically located on the building’s roof.


Table 3.1 Example building and story cost distributions for mid-rise office buildings

Total1 1st Floor Typical Floor Top FloorA. SUBSTRUCTURE

2.3% 0.0% 0.0% 0.0%B. SHELL

B10 Superstructure 17.6% 17.9% 18.5% 15.4%B20 Exterior Enclosure 16.3% 18.8% 16.2% 16.9%B30 Roofing 0.6% 0.0% 0.0% 4.5%

C. INTERIORS19.4% 20.7% 21.4% 11.1%

D. SERVICESD10 Conveying 9.5% 9.1% 9.4% 11.8%D20 Plumbing 1.9% 1.9% 1.9% 2.0%D30 HVAC 13.0% 12.3% 12.7% 17.6%D40 Fire Protection 2.6% 2.6% 2.7% 2.8%D50 Electrical 16.8% 16.6% 17.2% 17.9%

100% 100% 100% 100%Notes: 1) Cost distribution of total bldg value take from RS Means Square Foot Costs (2007)

BuildingDistribution (% of total bldg value)Component Group

Story Distribution (% of story value)

Table 3.2 goes into greater detail of the story cost distribution for a typical story in

a 7-story office building by further dividing the cost of each component group into

individual components. The distribution of cost for each component group was primarily

based on engineering judgment. Also included in the table is information about each

component’s seismic sensitivity and assigned fragility group. Several of the components

were assumed to only be damaged if the entire structure collapsed. These components,

termed “rugged,” were assumed to not contribute to the loss due to non-collapse. The

fragilities assigned to the components that are deemed damageable are explained in greater

detail in Section 3.3.2. All the cost distributions for low-rise, mid-rise and high-rise

buildings used in this study can be found in Appendix A.


Table 3.2 Example component cost distribution for a typical story in a mid-rise office building

Building Height: Mid-riseFloor Type: Typical Floor

Component Seismic Sensitivity Fragility Group

B. SHELLB10 Superstructure

Slab Rugged 8.2%Beam-column Assembly IDR Structural 7.2%Slab-column Assembly IDR Structural 3.1%

B20 Exterior EnclosureExterior Walls IDR Partitions 9.1%Exterior Windows IDR Windows 6.2%Exterior Doors IDR Partitions 1.0%

B30 RoofingRoof Coverings Rugged 0.0%Roof Openings Rugged 0.0%

C. INTERIORSPartitions with finishes IDR Partitions 4.5%Interior Doors IDR Partitions 1.9%Fittings IDR Generic-Drift 0.6%Stair Construction IDR Generic-Drift 1.9%Floor Finishes - 60% carpet IDR DS3 Partition-like 4.4%

30% vinyl composite tile Rugged 2.2%10% ceramic tile Rugged 0.7%

Ceiling Finishes PFA Ceilings 5.1%

D. SERVICESD10 Conveying

Elevators & Lifts IDR Generic-Drift 0.9%PFA Generic-Accl 8.5%

D20 PlumbingPlumbing Fixtures IDR DS3 Partition-like 0.9%

Rugged 1.1%D30 HVAC

Terminal & Package Units PFA Generic-Accl 9.5%IDR Generic-Drift 3.2%

Other HVAC Sys. & Equipment –D40 Fire Protection

Sprinklers PFA Generic-Accl 2.0%Standpipes IDR Generic-Drift 0.7%

D50 ElectricalElectrical Service/Distribution PFA Generic-Accl 1.5%Lighting & Branch Wiring Rugged 1.1%Lighting & Branch Wiring PFA Generic-Accl 5.1%Lighting & Branch Wiring IDR DS3 Partition-like 4.5%Communications & Security Rugged 1.0%Communications & Security PFA Generic-Accl 1.5%Communications & Security IDR DS3 Partition-like 2.5%

= 100% 100%

18.5%

17.2%

Normalized costs

9.4%

16.2%

0.0%

12.7%

2.7%

1.9%

21.4%


3.3.2 Fragility Functions Used

Creating EDP-DV functions requires consolidating fragility and mean repair costs

for all the components being considered. Table 3.3, Table 3.4 and Table 3.5 display the

parameters for the fragility and normalized mean repair costs used in this study for ductile

concrete structural components, non-ductile concrete structural components and

nonstructural components, respectively. The first column identifies the type of component

and the second column lists the different damage states associated with each component.

The third and fourth columns list the medians and lognormal standard deviations of the

fragility functions used, respectively. The fifth column lists the expected value of the

corresponding cost of repair/replacement actions. The sixth and final column cites the

reference that developed the functions.

Table 3.3 Fragility function & expected repair cost (normalized by component replacement cost) parameters for ductile structural components

Repair CostMedian (% for IDR, g for PFA) Dispersion Expected

ValueDS1 Method of Repair 1 0.70 0.45 0.14DS2 Method of Repair 2 1.70 0.50 0.47DS3 Method of Repair 3 3.90 0.30 0.71DS4 Method of Repair 4 6.00 0.22 2.25DS1 Light Cracking 0.40 0.39 0.10DS2 Severe Cracking 1.00 0.25 0.40DS3 Punching Shear Failure 9.00 0.24 2.75

Reference

Brown & Lowes (2006)

Aslani & Miranda (2005), & Roberson

et al. (2002)

Slab-column Subassembly

Beam-column Subassembly

Fragility Function ParametersDamage StateComponent

Table 3.4 Fragility function & expected repair cost (normalized by component replacement cost) parameters for non-ductile structural components

Repair Cost

Median (% for IDR, g for PFA) Dispersion Expected

Value

DS1 Light Cracking 0.35 0.33 0.10DS2 Severe Cracking 1.00 0.44 0.50DS3 Shear Failure 2.60 0.55 2.00DS4 Loss of Vertical Carrying Capacity 6.80 0.38 3.00DS1 Method of Repair 1 0.65 0.35 0.14DS2 Method of Repair 2 1.20 0.45 0.47DS3 Method of Repair 3 2.20 0.33 0.71DS4 Method of Repair 4 3.00 0.30 1.41DS5 Method of Repair 5 3.60 0.26 2.31DS1 Light Cracking 0.40 0.39 0.10DS2 Severe Cracking 1.00 0.25 0.40DS3 Punching Shear Failure 4.40 0.24 1.00DS4 Loss of Vertical Carrying Capacity 5.40 0.16 2.75

Slab-column Subassembly

Columns

Fragility Function Parameters

Damage StateComponent

Beam-column Subassembly

Reference

Aslani & Miranda (2005)

Aslani & Miranda (2005), & Roberson

et al. (2002)

Pagni & Lowes (2006)


Table 3.5 Fragility function & expected repair cost (normalized by component replacement cost) parameters for nonstructural components

Repair CostMedian (% for IDR, g for PFA) Dispersion Expected

Value

DS1Visible damage and small cracks in gypsum board that can be repaired with taping, pasting and painting

0.21 0.61 0.10

DS2Extensive crack in gypsum board that can be repaired with replacing the gypsum board, taping, pasting and painting

0.69 0.40 0.60

DS3Damage to panel and also frame that can be repaired with replacing gypsum board and frame, taping, pasting and painting

1.27 0.45 1.20

DS3 Partition-like DS1 IDR 1.27 0.45 1.20 Aslani (2005)

DS1Some minor damages around the frame that can be repaired with realignment of the window

1.60 0.29 0.10

DS2Occurrence of cracking at glass panel without any fall-out of the glass that can be repaired with replacing of the glass panel

3.20 0.29 0.60

DS3Part of glass panel falls out of the frame. The damage state can be repaired with replacing of glass panel

3.60 0.27 1.20

DS1 Slight Damage 0.55 0.60 0.03DS2 Moderage Damage 1.00 0.50 0.10DS3 Extensive Damage 2.20 0.40 0.60DS4 Complete Damage 3.50 0.35 1.20

DS1Hanging wires are splayed and few panels fall down. The damage state can be repaired with fixing the hanging wires and replacing the fallen panel.

0.30 0.40 0.12

DS2

Damage to some of main runners and cross tee bars in addition to hanging wires. The damage state can be repaired with replacing the damaged parts of grid, fallen panels and damaged hanging wires.

0.65 0.50 0.36

DS3Ceiling grid tilts downward (near collapse). The damage state can be repaired with replacing the ceiling and panels.

1.28 0.55 1.20

DS1 Slight Damage 0.70 0.50 0.02DS2 Moderage Damage 1.00 0.50 0.12DS3 Extensive Damage 2.20 0.40 0.36DS4 Complete Damage 3.50 0.35 1.20

ATC (2007)

Partitions (including façade)

Windows

Generic-Drift

IDR

Ramirez & Miranda (2009)

Reference

Ceilings

Generic-Acceleration

ATC (2007)

Aslani & Miranda (2005)

Ramirez & Miranda (2009)

Fragility Function ParametersDamage StateComponent

PFA

Seismic Sensitivity

IDR

IDR

PFA

Most of the fragility functions were used directly from the reference cited in Table

3.5, without any additional modifications. However, several of the structural fragilities

required making assumptions to establish the functions’ parameters. The following section

will detail the assumptions and modifications to the made for this study.

3.3.2.1 Fragility functions for ductile reinforced concrete structural components

Fragility functions for beam-column subassemblies were based on Brown and

Lowes (2006), with slight modifications made to the parameters by the authors of this

study. The lognormal standard deviation of damage state 1 (DS1) was decreased from 0.89

to 0.45 because the original dispersion value published is substantially higher than other

values of dispersion for structural component fragility functions computed from

experimental data. A high value of dispersion in the first damage state of a component can


be problematic because it can estimate that the probability of damage occurring initiates at

very early levels of IDR. To demonstrate this, damage initiation in a building component

will be quantitatively defined as the value IDR that results in a 1% probability of the first

damage state occurring or being exceeded. Using this criterion, the original parameters of

the function for DS1 published by Brown and Lowes (2006) computes that damage initiates

at an IDR of 0.00085. This estimates that damage will first become probable when relative

floor displacements are equal to about and 1/8th of an inch (assuming a 13-foot story

height). At this level of relative lateral deformation, structural components will more than

likely still behave elastically and not require any repairs to be made.

There exists a range of initial EDP values, from zero to a threshold value (which

can be referred to as a “quiet zone”), where damage will not occur because the response

parameters below this threshold are not large enough to yield the building components. If

this elastic region is not considered when estimating damage using continuous probability

distributions, economic losses (particularly expected annual losses, which are very sensitive

to losses due to non-collapse at small levels of ground motion intensity, Miranda and

Aslani, 2005) may be overestimated. This is because large probabilities will be computed

at small response parameter values for lognormal distribution functions with large values of

dispersion as shown in this example. Thus the standard deviation for the first damage state

of this component was decreased.

The functions for the other damage states generated by Brown and Lowes (2006)

for this component and other functions computed form previous studies on structural

component fragility functions (Robertson et al. 2002, Aslani and Miranda 2005, Pagni and

Lowes 2006) have lognormal standard deviations that typically range from approximately

0.20 to 0.50. Therefore, this fragility was assigned a lognormal standard deviation of 0.45,

which is on the higher end of this range.

Adjustments made to the other damage states include rounding off the parameters of

damage states 2 and 3. The parameters of damage state 4 were adjusted such that it would

not cross the fragility for damage state 3 (this required more substantial adjustments than

the other modifications, however, this fragility is based on a smaller set of data, and may

not be as reliable as the other functions).

At the time of this publication, there were no fragilities available for ductile slab-

column subassemblies. To account for damage of these components, fragilities for non-

ductile slab-column subassemblies (Aslani and Miranda, 2005) were modified to represent


how these components would perform if they were ductile. The parameters of damage state

3, accounting for punching shear failure of the slab, needed be increased because more

recent codes have introduced shear reinforcing requirements into their provisions (ACI,

2002). An investigation conducted by Roberson et al. (2002) developed a relationship

between gravity load carried by the slab and the interstory drift at which punching shear

occurs for slab-column subassemblies with shear reinforcement. The median IDR for this

damage state was taken from this relationship by assuming a shear demand-capacity ratio of

0.15. The fourth damage state is defined as the loss of vertical carrying capacity which is

not considered for ductile joints because modern building codes require slab reinforcing

bars run continuous through the column joint, preventing this failure mode from occurring.

3.3.2.2 Fragility functions for non-ductile reinforced concrete structural components

Parameters for non-ductile concrete column fragility functions were taken directly

from Aslani and Miranda (2005). The value of IDR that column shear failure occurs, the

third damage state for this component, is dependent on the amount for axial (gravity) load it

is carrying. Consequently, the parameters of the fragility function for the third damage state

are also a function of the level of axial load creating a fragility surface. To determine the

parameters for this fragility, a relatively low level of axial load was assumed

( 50g cP A f , where P is the axial load, Ag is the gross cross-sectional area of the

column, fc is the compressive strength of the concrete and ’’ is the reinforcement ratio), for

low to mid-rise buildings and intermediate level of axial load ( 150g cP A f ) for high-

rise buildings.

Pagni and Lowes’ (2006) developed fragility functions for modern reinforced

concrete beam-column subassemblies. These functions were used in this study with minor

modifications made to some of their parameters. The dispersion of damage state 1 was

decreased from 0.47 to 0.35 to increase the range of IDR where these components behave

elastically and no damage occurs (i.e. increase the “quiet zone” as described in section

3.3.2.1. The other parameters for this function were adjusted to achieve a better fit with the

empirical data reported in the Pagni and Lowes (2006) paper.

Functions for non-ductile concrete slab-column subassemblies were taken directly

from the study conducted by Aslani and Miranda (2005). The level of deformation at which


punching shear failure occurs in slab-column subassemblies is a function of the level of

gravity load the slab is carrying, typically represented by the shear demand-capacity ratio

that occurs at a distance d/2 from the column face (where d is the average effective depth of

the slab). The median and dispersion of the fragility function for the third damage state of

these components varies as a function of the level of gravity load resulting in a fragility

surface. For the purposes of this study, a low level of gravity load was assumed, where a

shear demand-capacity ratio of 0.15 was used based on the fact that building’s occupancy is

defined as office use and large gravity loads are not expected for this type of use.

3.3.2.3 Fragility functions for drift-sensitive nonstructural components

Fragility functions derived by Aslani and Miranda (2005) for partitions and windows

and partition-like components were used in this study. Aslani and Miranda (2005)

introduced the concept of “partition-like” components – other components whose loss is

dependent on the damage state of the partition. Many of these components, such as

electrical wiring, plumbing…etc., are often contained within the partitions. If a partition is

damaged to an extent that it needs to be replaced, these other components have to be

replaced as well, regardless if they have been damaged independently. Consequently, these

components were assigned the same fragility as the function for the partition replacement,

the partitions’ third damage state, and were termed “DS3 partition-like components.” This

physical and spatial interaction between partitions and the components contained within the

partitions results in their losses being dependent. There are other building components that

exhibit this type of loss dependency and this phenomenon is discussed and investigated in

further detail in section 3.5. No modifications were made to the parameters of functions for

partitions, DS3 partition-like components, and window and were used as documented by

Aslani and Miranda (2005). For all other components, generic fragility functions derived

from empirical data, as described in Chapter 5 of this dissertation, were used to estimate

damage and loss.

3.3.2.4 Fragility functions for acceleration-sensitive nonstructural components

Ceiling fragility functions taken from preliminary data and documents from the ATC-

58 project (ATC, 2007) were used in this study with only one modification made to its first


damage state. Recent studies (Badillo-Almaraz et al., 2007) have shown that damage may

initiate at larger accelerations than previously thought. The median of the first damage state

was rounded up from 0.27 and 0.30 to take this into account. All other acceleration-

sensitive components were assigned generic fragility functions formulated from empirical

data as described in Chapter 5 of this dissertation.

3.4 EXAMPLE STORY EDP-DV FUNCTIONS

Example EDP-DV functions were created for several variations of reinforced

concrete moment-resisting frame office buildings. Functions for low-rise, mid-rise and

high-rise structures were calculated. Other structural variations were also considered, such

as frame type (perimeter or space frames) and the ductility of the concrete (ductile or non-

ductile), when generating EDP-DV functions for structural components. For all variations,

functions for the different floor types (1st floor, typical floor and top floor) were formulated.

The entire set of functions computed in this study are reported in Appendix B of this

document.

Figure 3.2 shows the EDP-DV story functions generated for mid-rise, ductile

perimeter frame buildings. Functions for drift-sensitive structural components, drift-

sensitive nonstructural components, and acceleration sensitive nonstructural components

are plotted in Figures (a), (b) and (c) respectively. On each graph, the functions for the 1st

floor, the typical floor and the top floor are plotted.


Structural components

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.05 0.10 0.15IDR

E(L | IDR)

1st FloorTyp FloorTop Floor

(a)


0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.05 0.10 0.15IDR

E(L | IDR)

1st FloorTyp FloorTop Floor

(a)

Nonstructural components

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.05 0.10 0.15IDR

E(L | IDR) Nonstructural components

0.00

0.20

0.40

0.60

0.80

1.00

0.00 2.00 4.00 6.00 8.00PFA [g]

E(L | PFA)

(b) (c)


0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.05 0.10 0.15IDR


0.00

0.20

0.40

0.60

0.80

1.00

0.00 2.00 4.00 6.00 8.00PFA [g]

E(L | PFA)Nonstructural components

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.05 0.10 0.15IDR


0.00

0.20

0.40

0.60

0.80

1.00

0.00 2.00 4.00 6.00 8.00PFA [g]

E(L | PFA)

(b) (c)

Figure 3.2 Story EDP-DV functions for typical floors in mid-rise office buildings with ductile reinforced concrete moment resisting perimeter frames.

Comparing plots in Figure 3.2(a) between the different floor types shows that losses

for drift-sensitive structural components are slightly higher for the 1st and typical floors than

the top floor. A similar trend is observed in Figure 3.2(b) for drift sensitive nonstructural

components. Conversely, acceleration-sensitive nonstructural components show the

opposite trend (Figure 3.2(c)), where the larger losses are observed in the top floor. These

trends can be explained by how the cost is distributed along the height of the building.

Drift-sensitive items, such as partitions and structural members, make more of the relative

story value in the lower stories, especially in the 1st floor where more expensive

components (ex. finishes) may be located. On the other hand, acceleration-sensitive

components may make up more of the story value at the top floor because mechanical items

(such as HVAC units) are typically located on the roof of these types of buildings.


The nonstructural, drift-sensitive functions indicate that these types of components

have the largest potential to contribute to the loss, especially for structural systems that are

designed to experience large IDRs, such as moment-frames. The functions saturate

between 0.46-0.53 of the total value of the story. Further, these functions estimate higher

economic losses at smaller IDR values than the losses estimated by the functions for

structural components. Beginning at an approximate interstory drift of 0.05, these

functions take a steep increase to a loss of about 0.32 of the story value as the IDR

approaches 0.02. By comparison, the structural components experience loss of 0.05 of the

total story value at an IDR of 0.025 (an IDR of 0.025 is a noteworthy value because modern

reinforced concrete moment frame buildings are designed to not to exceed this level of IDR

using equivalent static analyses when subjected to a ground motion intensity equal to the

design-basis earthquake as prescribed by US building codes, ICC 2006), which is about

600% smaller than the drift-sensitive nonstructural components. Previous studies (Aslani

and Miranda 2005, Taghavi and Miranda, 2006) have also suggested that nonstructural

components will make up the majority of seismic-induced losses as observed here. It

follows that if the value of the story is primarily comprised of nonstructural components,

the majority of associated losses will be made up of these elements.

The difference in EDP-DV functions for typical floors between low-rise, mid-rise

and high-rise buildings are shown in Figure 3.3 for each of the three different components

group categories. These figures show that the losses for structural components are lower

for stories in low-rise buildings than the losses in stories of high-rise buildings. The

opposite trend is true for nonstructural components, where the low-rise buildings exhibit the

largest normalized story losses. When Figure 3.3(b) is compared to Figure 3.3(c), it can be

observed that the differences in economic losses between the low-rise and high-rise

buildings are larger for drift-sensitive components than they are for acceleration-sensitive

components. These trends can also be attributed to the differences in cost distributions for

these types of elements between structures of different heights. For instance, the value of

structural components relative to the entire value building increases for taller buildings as

can be observed from the cost distributions in Appendix A (taken from Balboni, 2007).



0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.05 0.10 0.15IDR

E(L | IDR)

Low-riseMid-riseHigh-rise

(a)


0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.05 0.10 0.15IDR

E(L | IDR)

Low-riseMid-riseHigh-rise

(a)


0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.05 0.10 0.15IDR


0.00

0.20

0.40

0.60

0.80

1.00

0.00 2.00 4.00 6.00 8.00PFA [g]

E(L | PFA)

(b) (c)


0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.05 0.10 0.15IDR


0.00

0.20

0.40

0.60

0.80

1.00

0.00 2.00 4.00 6.00 8.00PFA [g]

E(L | PFA)

(b) (c)

Figure 3.3 EDP-DV Functions for low-rise, mid-rise and high rise ductile reinforced concrete moment frame office buildings

EDP-DV functions can be used to evaluate how varying different structural

parameters can influence loss at the story-level. Over the past 40 years, US seismic

building codes have introduced a variety of provisions to increase the ductility of structural

reinforced concrete. Based on observed performance during seismic events, more stringent

confinement requirements and other detailing provisions that delay or prevent certain

sudden, failure modes from occurring (ex. shear failure modes), were instituted to decrease

the probability of lives lost during an earthquake. How this improved performance

translates when using losses as a metric can be assessed using the EDP-DV functions

formulated in this study. Figure 3.4 compares structural story functions between ductile

and non-ductile concrete elements. Both functions initiate loss at approximately the same

IDR, but begin to deviate from each other at an IDR of about 0.015. The non-ductile


function indicates, as expected, that loss accumulates at faster rate than the ductile function

for increasing values of IDR. The largest deviation between the two curves occurs at the

IDR value of 0.052, where there is a maximum difference of about 0.13 of loss (relative to

the total story value). This represents a relative change of approximately 140% in

performance between non-ductile frames and ductile frames. The non-ductile function

saturates at an earlier drift of 0.075 whereas the ductile function levels off later at around

0.013.


0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.05 0.10 0.15IDR

E(L | IDR)

DuctileNon-ductile

Figure 3.4 Comparison between ductile and non-ductile structural component EDP-DV functions of typical floors

Different EDP-DV functions for perimeter frames and for space frames were

formulated to account for the different type of construction implemented for these systems.

The functions for perimeter frame buildings accounted for beam-column subassemblies and

slab-column subassemblies, whereas the functions for space frame buildings only

considered beam-column subassemblies. It was assumed that the value of the slab-columns

represented in the perimeter frame buildings would be replaced by an equivalent value of

beam-column components in the space frame buildings to keep the total percentage of story

value due to structural components consistent with the cost distributions taken from the RS

Means data (this is primarily because the data from RS Means did not make a distinction

between perimeter and space frame buildings). The value of beam-column subassemblies

was increased by the value of slab-column connections that was removed. Figure 3.5 plots


the comparison between different frame types for typical floors of mid-rise buildings. The

graph shows that there is very little difference in story loss between the two types of frames.

The maximum difference in loss – approximately 0.035 of the total story value – occurs

between the IDR range of 0.06 to 0.08. The relatively small difference in the functions may

suggest that it may not be important to differentiate between frame type when evaluating

non-collapse losses. Being able to make this assumption can further simplify the process in

assessing loss by not having to define and use separate EDP-DV functions for perimeter and

space frames separately.


0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.05 0.10 0.15IDR

E(L | IDR)

PerimeterSpace

Figure 3.5 Comparison of structural EDP-DV functions between perimeter and space frame type structures

As documented in section 3.3.2, several components have fragility functions whose

parameters are dependent on other variables. For instance, the probability of experiencing

or exceeding the damage state for shear punching failure of slab-column connections is

dependent on the amount of gravity load the slab is carrying (Aslani and Miranda, 2005).

Assumptions on the level of demand-capacity shear load ratio needed to be made to set the

median and dispersion to be used in calculating the EDP-DV function. This assumption

was evaluated by generating functions for both low and high levels of gravity loads to see

how sensitive this parameter affects the corresponding losses, and the results are shown in

Figure 3.6.


The primary differences between these two EDP-DV functions are the parameters

used for the fragility function of the third damage state of the slab-column connections

(punching shear failure). For the fragility assuming a low level of gravity load (shear

demand-capacity ratio = 0.15), the median IDR was computed to be 0.09 (based on

Robertson et al, 2002) with a lognormal standard deviation of 0.24 (based on Aslani and

Miranda, 2005). For the fragility assuming a high-level of gravity load (shear demand-

capacity ratio = 0.50), the median was IDR was computed to be 0.056 (based on Robertson

et al, 2002) with a lognormal standard deviation of 0.54 (based on Aslani and Miranda,

2005).

There appears to be some difference between the two functions, however, it does

not seem to be very substantial. Largest difference in losses occurs within the IDR range of

0.05 and 0.10, where the maximum difference in loss (0.05 of the story value) occurs at

around and IDR of 0.075. This represents a relative difference of 28% if the gravity load is

underestimated using the lower level assumption. This suggests that the assumed gravity

load on these components for perimeter frames may not have as strong an influence on

seismic-induced losses as other structural properties (i.e. ductility of concrete). Analysts

conducting loss assessments can take advantage of the fact that this assumption will not

affect their estimates significantly, by not having to deal with multiple functions that

account for different levels of gravity load.


0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.05 0.10 0.15IDR

E(L | IDR)

Low gravity loadHigh gravity load

Figure 3.6 Influence of varying assumed gravity load on slab-column subassemblies on structural EDP-DV functions


3.5 CONDITIONAL LOSS OF SPATIALLY INTERDEPENDENT COMPONENTS

Consolidating the loss components into story EDP-DV functions in this manner

provides the opportunity to investigate the issue of the interaction between the losses of

components that are spatially interdependent. These types of losses occurs when damage

from one component results in repair or replacement of another component because of their

physical relationship between the two elements. For instance, when sprinklers are

damaged, water may leak onto the components below, such as suspended lighting fixtures.

The fragility functions of lighting fixtures typically do not capture damage due to leaking

water, but still needs to be considered if losses are to be computed accurately.

The spatial interaction of components will influence economic loss estimates when

the fragility functions of the elements begin to significantly overlap. To illustrate this,

Figure 3.7 shows two example sets of hypothetical fragility functions for sprinklers and

suspending lighting fixtures, which may interact during a seismic event. Each figure, (a)

and (b), plots functions for two damage states, one in which repair is required (DM1) and

the other in which replacement is required (DM2), for both the suspended lighting (solid

lines) and the sprinklers (dashed lines). In this example, when the sprinklers’ first damage

state occurs, water leakage is assumed to occur as well damaging the lighting fixtures below

and initiating replacement of the lighting fixtures. If the components have fragilities as

those shown in Figure 3.7 (a), then this spatially interaction does not have that large of

influence on the economic losses. For instance, for a given PFA = 1.0g, the probability of

the light fixtures requiring replacement is 9%, whereas the probability of the sprinklers

leaking and needing repair is approximately 0%. This means that it is more likely that if the

lighting fixtures need replacement due to the accelerations induced by the applied ground

motion, rather than due to water leakage of damaged sprinklers. In this case the losses are

calculated correctly because for the given level of PFA there is almost no probability that

losses of lighting fixture replacement due to water damage will be incurred.

Conversely, if the components have fragilities that have greater overlap as shown in

Figure 3.7 (b), damage and loss estimation may be underestimated. In this case, at a PFA =

1.0 the probability of the sprinklers leaking and forcing replacement of the lighting fixtures

(24%) is higher than the probability of lighting fixture being replaced due to floor


accelerations directly (9%). There is a significant probability that the lighting fixture will

have to be replaced due to water damage, however, the only monetary losses that are

associated with this damage state (Sprinklers -DM1) are the repair costs of the sprinklers.

When economic losses are computed for the lighting fixtures, the methodology discussed

thus far will only account for losses due to damage caused by PFA directly and not water

damage. This approach ignores the interaction between the components (i.e. the

components’ losses are treated independent of one another). There is nothing in the current

framework that accounts for the conditional loss of replacing the lighting due to water

damage.

Previous studies (Beck et al., 2002), have attempted to account for these types of

conditional economic losses due to spatial interaction by including the replacement cost of

the dependent component into the other component’s repair cost. In the previous example

for a given PFA = 1.0g , this approach includes the cost of replacing the suspended lighting

fixtures into the repair cost associated with the first damage state of the sprinklers (DM1 –

Repair). Unfortunately, monetary losses from the replacement of the lighting fixtures due

to PFA directly are still computed because this event has a probability (9%) of occurring for

the given level of EDP. Consequently, the economic loss from the cost of replacing the

lighting fixtures is counted twice, or “double counted.”


0.00

0.20

0.40

0.60

0.80

1.00

0.0 1.0 2.0 3.0 4.0 5.0

PFA

Lighting - DM1:Repair

Lighting - DM2:Replacement

Sprinklers - DM1:Repair/Leakage

Sprinklers - DM2:Replacement

P(DM > dm | PFA = pfa)

0.00

0.20

0.40

0.60

0.80

1.00

0.0 1.0 2.0 3.0 4.0 5.0

PFA






(a)

(b)

0.00

0.20

0.40

0.60

0.80

1.00

0.0 1.0 2.0 3.0 4.0 5.0

PFA






0.00

0.20

0.40

0.60

0.80

1.00

0.0 1.0 2.0 3.0 4.0 5.0

PFA






(a)

(b)

Figure 3.7 Hypothetical fragility functions of spatially interacting components (sprinklers & suspended lighting) (a) example where losses are unaffected (b) example when losses are

conditional

Therefore, there are two possible errors in estimating loss that may arise if this

dependency is not accounted for properly: (1) underestimating the loss by ignoring this

dependency; or (2) overestimating the loss by counting the repair cost of a component twice

(double-counting). A proposed method that addresses these errors and accounts for these

losses correctly is presented here using a more detailed example of the dependent

relationship between steel beams and partitions. The failures of pre-Northridge steel-

column joints fracturing have been well-documented (FEMA-355E, 2000). Although these

types of structures avoided catastrophic collapse during this event, many stakeholders ended

up paying large sums of money to repair the steel structural members, which fractured at

smaller interstory drifts than expected. It has been demonstrated that steel beams,

particularly ones with large depths, are susceptible to fracture at very small amounts of


rotation. Consequently, there were many occurrences of fractured steel members behind

partitions that experienced little to no damage. To repair the damaged joint, contractors

must remove the partitions to access the structural members. The cost of replacing the

partition must be accounted for in loss estimates, despite the fact that very little damage to

the partition may have occurred.

Aslani and Miranda (2005) treated the components separately by defining their loss

functions that excluded any loss from related components. Figure 3.8 shows the probability

trees for a pre-Northridge steel beam and a partition when the components are treated

separately. Each branch of the trees represents possible damage states for each component.

The probability for each outcome is computed by the fragility function associated with each

damage state. Also shown in the figure is the expected loss due to repair actions for each

damage state, E[Lk | DM=k]. A numerical example, for the expected loss when the

interstory drift is equal to 0.01, E[Li | IDR], is given in the figure. For each component, i¸

the probability of being in damage state, k, is calculated, P(DM = k | IDR), and using the

theorem of total probability, the expected loss is calculated using equation (3.7).

The resulting losses in the beams and partitions are 0.01 and 0.024 (normalized by

the total value of the story), respectively, for a total of 0.034 for both components. In this

approach the two components are not dependent as illustrated by the fact that the branches

do not intersect. The losses are calculated entirely independently, and does not account for

any loss due to the partition being removed to access the steel beam for inspection and

repair.


DM = 0, No Damage

DM =1, Fracture

E[Lbms | IDR] = 0.01

P(DM = 0 | IDR) = 74%

P(DM = 1 | IDR) = 26%

E[L | DM = 0] = 0.0

E[L | DM = 1] = 2.0

DM = 0, No Damage

DM = 3, Replacement req’d

E[Lparts | IDR]=0.024

DM = 1, Small cracking

DM = 2, Extensive cracking

P(DM = 0 | IDR) = 5%

P(DM = 1 | IDR) = 13%

P(DM = 2 | IDR) = 47%

P(DM = 3 | IDR) = 35%

E[L | DM = 0] = 0.0

E[L | DM = 1] = 0.1

E[L | DM = 2] = 0.6

E[L | DM = 3] = 1.2

STEEL BEAMSabms = 2.0%

PARTITIONSaparts = 3.3%

TOTALE[LTOTAL | IDR] = 0.034

Figure 3.8 Probability tree for components considered to act independently

Another approach to account for this loss, is to include the cost of replacing the

partitions, as part of the repair cost of the steel beams. An example of this approach is

documented by Beck et al. (2002), where they list the replacement cost of partitions and

other nonstructural components as part of total estimate of the repair cost function. This

creates the second issue mentioned above of double-counting, where the partitions are being

assessed a loss twice. It is being counted in the repair cost function of the beam, as well as

a separate component that experiences damage. This approach is illustrated through

probability trees shown in Figure 3.9. This figure is the same as Figure 3.8, with the

exception that the expected loss of repairing the steel beams is increased from 2.0 to 5.0

(these losses are normalized by the value of a new component) to account for the cost of

replacing nonstructural components demolished to access the structural member. If the

same numerical example for IDR = 0.01 is carried out, the total loss from both components

results in 0.05 of the total value of the story, this is a 47% increase over the approach that

treats the components independently. Although this approach accounts for the partitions’

dependency on the steel beams, a significant portion of this increase in loss may be

attributed to the fact that the repair cost of the partitions are counted twice.


DM = 0, No Damage

DM =1, Fracture

E[Lbms | IDR] = 0.026

P(DM = 0 | IDR) = 74%

P(DM = 1 | IDR) = 26%

E[L | DM = 0] = 0.0

E[L | DM = 1] = 5.0

DM = 0, No Damage





P(DM = 0 | IDR) = 5%

P(DM = 1 | IDR) = 13%

P(DM = 2 | IDR) = 47%

P(DM = 3 | IDR) = 35%

E[L | DM = 0] = 0.0

E[L | DM = 1] = 0.1

E[L | DM = 2] = 0.6

E[L | DM = 3] = 1.2




DM = 0, No Damage

DM =1, Fracture

E[Lbms | IDR] = 0.026

P(DM = 0 | IDR) = 74%

P(DM = 1 | IDR) = 26%

E[L | DM = 0] = 0.0

E[L | DM = 1] = 5.0

DM = 0, No Damage





P(DM = 0 | IDR) = 5%

P(DM = 1 | IDR) = 13%

P(DM = 2 | IDR) = 47%

P(DM = 3 | IDR) = 35%

E[L | DM = 0] = 0.0

E[L | DM = 1] = 0.1

E[L | DM = 2] = 0.6

E[L | DM = 3] = 1.2




Figure 3.9 Probability tree for independent components that use double-counting to account for dependency

Thus far we have demonstrated that the first approach ignores the loss produced by

repair actions that affect more than one component, therefore, may be underestimating the

combined loss of both components. Although the second method accounts for this

dependency, it may be overestimating the loss because it double counts the repair of the

dependent component. The actual loss will be somewhere in between the two methods.

Therefore an approach that captures this dependency without double-counting is required.

The proposed approach computes the loss such that the estimation of the dependent

components’ damage is conditional on the damage state of the other component. Figure

3.10 shows the probability tree that illustrates this method. The partitions’ damage is now

conditional on what damage state the steel beams are in, as represented by the branches of

the partitions being stacked behind those of the beams. If the steel beam does not

experience damage, the partitions’ damage is estimated by using the same fragility

functions as before. If the beam has been damaged, then only two possible damage states

are considered: no damage and replacement required. Note that the replacement damage

state is assigned a conditional probability of 100% to ensure that the partition will be

replaced if we know that the beam has fractured.


DM = 0, No Damage

DM =1, Fracture

E[LTOTAL | IDR] = 0.038

P(DM = 0 | IDR) = 74%

P(DM = 1 | IDR) = 26%

DM = 0, No Damage




P(DM = 0 | IDR) = 5%

P(DM = 1 | IDR) = 13%

P(DM = 2 | IDR) = 47%

P(DM = 3 | IDR) = 35%

DM = 0, No Damage


P(DM = 0 | IDR) = 0%

P(DM = 3 | IDR) = 100%

E[L | DM = 0] = 0.0

E[L | DM = 1] = 0.1

E[L | DM = 2] = 0.6

E[L | DM = 3] = 1.2

E[L | DM = 0] = 0.0

E[L | DM = 1] = 2.0



TOTAL

DM = 0, No Damage

DM =1, Fracture


P(DM = 0 | IDR) = 74%

P(DM = 1 | IDR) = 26%

DM = 0, No Damage




P(DM = 0 | IDR) = 5%

P(DM = 1 | IDR) = 13%

P(DM = 2 | IDR) = 47%

P(DM = 3 | IDR) = 35%

DM = 0, No Damage


P(DM = 0 | IDR) = 0%

P(DM = 3 | IDR) = 100%

E[L | DM = 0] = 0.0

E[L | DM = 1] = 0.1

E[L | DM = 2] = 0.6

E[L | DM = 3] = 1.2

E[L | DM = 0] = 0.0

E[L | DM = 1] = 2.0



TOTAL

Figure 3.10 Probability tree for proposed approach to account for dependent components.

Using the previous numerical example in the descriptions of the previous

approaches, the total probability theorem can be carried though both branches to calculate

that the expected loss from both components is 0.038 given an IDR of 0.01. As expected,

this value is between the values generated by the previous approaches. It 12% greater than

the first approach, where the components’ losses are calculated independently; however, it

is 32% less than if the partition losses are double counted. This infers that double-counting

creates a greater deviation in loss than assuming the components act independently.

Instead at looking at a single value of IDR, a better comparison of the three

approaches can be made by contrasting the resulting EDP-DV functions when both

components are integrated using equation (3.7) over a range of IDRs. Figure 3.11 shows

the EDP-DV functions of all three approaches. It appears that the trends from the numerical

example of the loss at a given IDR = 0.01 can also be observed when comparing EDP-DV

functions. The proposed approach is slightly larger than the approach that treats each

component independently. The approach that double counts loss due to partition repairs,

however, is significantly higher than the proposed approach. Again, this suggests that

double-counting may introduce more error into loss estimates than treating each component

independently


0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

0.000 0.004 0.008 0.012 0.016 0.020IDR

E[L | IDR]

IndependentDependent, w/ Double-countingDependent w/o Double-counting

Figure 3.11 EDP-DV functions for three different approaches of handling component dependency

For this particular example, using conditional fragility functions did not have a

significant difference in loss than if the components were treated independently. This can

be explained examining the fragility functions of both components as shown in Figure 3.12.

How much loss increases due to component dependence, depends on how much the fragility

of the steel beam overlaps with the first two damage states of the partitions. If the steel

beams’ fragility overlaps with these damage states, it means that there is a probability that

the beams may fracture, initiating replacement, before small or extensive cracking is

experienced in the partitions. The greater the overlap, the higher this probability is and the

greater expected loss is. Although, these particular components have small levels of

overlap, other dependent components may have greater overlap, and have a more significant

difference in expected losses than if the components were considered independent.


0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.010 0.020 0.030IDR

Beam - DM1Partition - DM1Partition - DM2Partition - DM3

P(DM | IDR)

Figure 3.12 Fragility functions for Pre-Northridge steel beams and partitions

The effect of component dependency will also be more significant if there are more

than one component whose loss is conditioned on the damage of other components. Aslani

and Miranda (2005) introduced the concept of “partition-like” components – other

components that needed to be replaced when a partition was replaced (ex. electrical wiring,

plumbing…etc.). They were assigned the same fragility as the one for the partitions’ third

damage state for replacement, and therefore termed “DS3 partition-like components.” If the

value of these DS3 partition-like components is incorporated into the previous example, and

the calculations for expected loss at an IDR of 0.01 are repeated, the resulting loss is equal

to 0.114. This is 33% greater than if the components were treated independently (which has

an expected loss of 0.085 when DS 3 partition-like components are included). Figure 3.14

plots the corresponding EDP-DV functions to illustrate the effect of including more

dependent components. Note that the losses become larger at smaller values of IDR


DM = 0, No Damage

DM =1, Fracture


P(DM = 0 | IDR) = 74%

P(DM = 1 | IDR) = 26%

DM = 0, No Damage




P(DM = 0 | IDR) = 5%

P(DM = 1 | IDR) = 13%

P(DM = 2 | IDR) = 47%

P(DM = 3 | IDR) = 35%

DM = 0, No Damage


P(DM = 0 | IDR) = 0%

P(DM = 3 | IDR) = 100%

E[L | DM = 0] = 0.0

E[L | DM = 1] = 0.1

E[L | DM = 2] = 0.6

E[L | DM = 3] = 1.2

E[L | DM = 0] = 0.0

E[L | DM = 1] = 2.0


PARTITIONS & DS3 COMPSaparts = 3.3% & aDS3 = 12.2%

TOTAL

DM = 0, No Damage

DM =1, Fracture


P(DM = 0 | IDR) = 74%

P(DM = 1 | IDR) = 26%

DM = 0, No Damage




P(DM = 0 | IDR) = 5%

P(DM = 1 | IDR) = 13%

P(DM = 2 | IDR) = 47%

P(DM = 3 | IDR) = 35%

DM = 0, No Damage


P(DM = 0 | IDR) = 0%

P(DM = 3 | IDR) = 100%

E[L | DM = 0] = 0.0

E[L | DM = 1] = 0.1

E[L | DM = 2] = 0.6

E[L | DM = 3] = 1.2

E[L | DM = 0] = 0.0

E[L | DM = 1] = 2.0


PARTITIONS & DS3 COMPSaparts = 3.3% & aDS3 = 12.2%

TOTAL

Figure 3.13 Probability tree for proposed approach, including other DS3 Partition-like components

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

0.000 0.004 0.008 0.012 0.016 0.020IDR []

E[L | IDR]

IndependentDependent, w/o Double-counting

Figure 3.14 EDP-DV functions for proposed approach vs treating components independently, with DS3 Partition-like components included.


3.6 DISCUSSION OF LIMITATIONS OF STORY-BASED APPROACH & EDP-DV FUNCTIONS

The simplifications presented here offer numerous advantages in terms of

computational efficiency and ease of use. Using a story-based approach in combination

with EDP-DV functions to evaluate seismic-induced economic loss is not as complicated or

computationally intensive as component-based methods, while still being able to capture

building-specific behavior – notably higher mode effects of multi-degree of freedom

systems, nonlinear behavior of structures and repair cost variability – that regional loss

estimation methods can not. However, making the simplifications discussed in this chapter

results in limitations on the level of detail of the loss analysis.

The EDP-DV functions formulated in this study compute economic loss two

dimensionally. This means that it is assumed that all components in a story are subjected to

response parameters that act in only one direction and all components experience damage in

same direction (i.e. the approach does not consider structural response that may result from

both directional components of a ground motion nor does it account for the fact that

components may be oriented in different directions). For example, the damage computed

for all partitions in a story is only dependent on one value of IDR even though there may be

structural displacements occurring in both primary planar directions of a structure. Further,

it is assumed that all partitions are oriented in the same direction even though many of these

walls may be perpendicular to each other. The EDP-DV functions can be modified to

account for this by making assumptions on how the value of building components are

distributed based on their orientation, such that there are functions available for both

primary planar directions of a story. However, other three-dimensional effects, such as

building torsion and vertical displacements/accelerations, are not accounted for.

The loss of building components that are dependent between stories due to spatial or

physical interaction is not captured by evaluating losses in each individual story. The

expected economic losses of each story are assumed to be independent of each other.

However, there may be instances where the loss of a component in one story is dependent

on another story. For instance, a building’s sprinkler and/or piping system may span across

several stories. If the system is centralized and is damaged on one floor, it may cause

leakage and damage to other floors that were not necessarily subjected to large EDPs. This


dependency can be accounted for in a similar way that dependent losses between

components on the same story were considered in section 3.5.

The functions developed in this study were limited on only three primary types of

subsystems of building components: drift-sensitive structural components, drift-sensitive

nonstructural components and acceleration-sensitive nonstructural components.

Considering only three types of subsystems reduces the amount of number tracking and

computations required but may not always produce the best estimates for certain types of

components. For instance, damage of structural components, such as reinforced concrete

shear walls, may be better estimated by floor accelerations rather than interstory drifts. It

has also been suggested that the damage of other building components is more correlated to

floor velocities (i.e. velocity-sensitive components) rather than PFA or IDR.

Further research is required to investigate how sensitive economic loss estimations

are to these limitations that result from the simplifying assumptions made when using

story-based methods.

3.7 CONCLUSIONS

A simplified approach to implementing the PEER loss estimation methodology,

referred to here as story-based loss estimation, has been presented. The new approach

collapses out the intermediate step of estimating damage to create generic relationships at

the story-level between structural response parameters and loss (EDP-DV functions). These

relationships can be established without knowing the building’s exact inventory ahead of

time by using an assumed cost distribution based on knowing the building’s structural

system and occupancy, and normalizing all repair costs by the entire value of the story.

Functions for reinforced concrete moment resisting frames were developed and documented

in this study to for use in loss assessments for these types of structures. The functions

aggregate building components into 3 primary groups: drift-sensitive structural components,

drift-sensitive nonstructural components, and acceleration-sensitive nonstructural

components

Assessment of the story EDP-DV functions yielded several significant findings.

Comparing the functions of the 3 different component groups shows that losses due to drift-

sensitive, non-structural components will comprise the majority of the repair costs in a story


of a standard office building. Functions for different locations along the height of the

building (1st floor, typical floors, top floor), did not show significant difference in resulting

losses, however, functions for buildings of varying heights (low-rise, mid-rise and high-rise

buildings), were found to differ substantially, suggesting that separate EDP-DV functions

are required when analyzing buildings of different heights. As expected, the ductility of the

structural concrete elements also had a substantial in affect on losses. The largest

difference was observed at an approximate IDR of 6%, where losses decreased by 11% of

the story value (which represents a percent difference of 45%). Conversely, other structural

variables examined, namely frame type (space vs. perimeter frame) and the level of gravity

load on slab-column connections, did not have a significant impact on expected losses

conditioned on EDP. Finally, the issue of loss estimation on spatially-interdependent

components was evaluated and the approach of conditional fragilities was introduced. It

was found that treating components independently does not underestimate the losses

substantially, and not as significantly as double counting the losses of dependant

components overestimates the loss.

The story-based loss estimation approach presented in this study, makes assessing

earthquake-induced losses more efficient by not having to inventory every component in the

considered building. Collapsing out the intermediate step of estimating damage also allows

loss analysts to predict losses without having to deal directly with every fragility function

associated with the inventoried components. As demonstrated in this study, the generic

story EDP-DV functions developed can be used to identify what variables and fragilities

significantly influence non-collapse loss results. The functions developed in this study,

however, are limited by the data available at the time of publication. Assumptions using

expert opinion were made where fragility function or cost data was unavailable. As

relevant data is collected, these story EDP-DV functions need to be updated accordingly.

Further, there are many components of the nonstructural components that did not have

specific fragility functions and generic fragilities were used in their place. Although using

these generic functions is an improvement from previous studies that have ignored their

contribution to the loss, they must be eventually replaced with fragilities developed from

experimental data.

CHAPTER 4 53 Development of Component Fragility Functions from Experimental data

CHAPTER 4

4 DEVELOPMENT OF COMPONENT FRAGILTIY FUNCTIONS FROM EXPERIMENTAL DATA


Ramirez, C.M., Kolios, D., and Miranda, E. (2008), “Fragility Assessment of Pre-

Northridge Welded Steel Moment-Resisting Beam-Column Connection,” Journal of

Structural Engineering, (in press).

4.1 AUTHORSHIP OF CHAPTER

Ramirez headed up this research effort by computing the fragility functions and

confidence bands, developing methods to account for other parameters that influence the

probability distribution parameters and authoring the publication. Kolios consolidated the

experimental data used to develop the response-damage relationships and formulated some

of the preliminary functions for this investigation. Miranda served as advisor and principal

investigator for this project.

4.2 INTRODUCTION

Prior to the 1994 Northridge earthquake, steel moment resisting frame buildings

were widely regarded as one of the best structural systems to resist lateral loads generated

by seismic events. In particular, moment resisting beam-to-column connections in welded

steel moment frames (WSMF) were considered to be able to withstand large inelastic

deformations without developing significant strength degradation or instabilities. Should

damage occur in these frames, it would be limited to ductile yielding of beams and beam-

column connections (FEMA, 2000a). The moment connection detail most commonly used


in seismic regions in the U.S. between 1970 and 1994 (prior to the Northridge earthquake)

was the welded flange-bolted web connection shown in Figure 4.1. In this type of

connection the beam flanges are connected to the column using complete joint penetration

(CJP) single-bevel groove welds while the beam web is bolted to a single shear plate tab

which is welded to the column.

Complete joint penetrationTop & Bottom Flange

W Steel Beam

Fillet WeldEach side

Shear tabw/ bolts

Complete joint penetrationTop & Bottom Flange

W Steel Beam

Fillet WeldEach side

Shear tabw/ bolts

Figure 4.1 Typical Detail of Pre-Northridge Moment Resisting Beam-to-Column Joint

While investigating the effects of supplement welds placed between the beam web

and the shear tab, Engelhardt and Husain (1993) observed that welded flange-bolted web

steel moment connections could experience fractures at relatively low levels of

deformation. Of the eight specimens they tested, only one was able to reach a plastic

rotation of 0.015. Analysis of their results, together with a re-examination of the plastic

rotation capacity attained in five previous experimental investigations, led them to conclude

that this type of connection had highly variable performance with a significant number of

specimens having poor or marginal performance when subjected to cyclic loading. They

expressed concern that the performance of this widely used connection was not as reliable

as once thought. Soon after the publication of their study, the January 17th, 1994

Northridge, California confirmed their concerns.

Consolidated damage reports from the Northridge earthquake and found that of 155

steel moment resisting frame buildings inspected, 90 of them experienced some connection

damage (FEMA-355E, 2000). Close inspection of buildings following the earthquake


showed that many steel moment-frame buildings experienced fractures in their beam-to-

column connections. Damaged buildings were between one to 26 stories and with ages

ranging from as old as 30 years to brand new buildings. (FEMA 350) Although several

different types of fractures were observed, observations from damaged buildings as well as

experimental results conducted after the earthquake as part of the SAC joint venture

indicated that fracture of the bottom flange is more likely to occur in this type of connection

and that it is typically initiated at the center of the beam flange. The occurrence of the initial

fracture produces a large and sudden loss in moment-resisting capacity which in many cases

leads to a subsequent fracture of the other flange and/or fracture in the web shear

connection either by fracturing the shear single plate tab by shearing off one of more bolts

connecting the tab to the beam web. Even more disconcerting was that, in certain cases,

several studies indicate that these fractures occurred in buildings that experienced ground

motions less intense than the code specified design level earthquake (FEMA, 2000c).

Consequently, building owners, insurance companies and other stakeholders suffered

significant economic losses associated with repairing these connections.

There is a growing trend in earthquake engineering to move towards a performance-

based design where, in addition to having an adequate safety against collapse, a structure is

designed to reduce the risk of economic losses and temporary loss of use (downtime) to

levels that are acceptable to owners and other stakeholders. Whether one is interested in

assessing the probability of collapse or in assessing possible economic losses or downtime

of WSMF buildings built prior to 1994 a necessary component in this assessment is a

procedure to predict the occurrence of different damage states in the beam-to-column

connections at different levels of ground motion intensity. One way of estimating damage

is by using fragility functions. Fragility functions are cumulative probability distributions

that estimate the probability that a building component will reach or exceed a level of

damage when subjected to a particular value of a structural response parameter. These

functions are used as part of the Pacific Earthquake Engineering Research (PEER) Center’s

performance-based design methodology to estimate damage and corresponding economic

losses as a measure of seismic performance.(Krawinkler and Miranda, 2004; Miranda

2006).

There have been previous attempts to consolidate experimental data on pre-

Northridge steel moment frame beam-to-column connections, however, none these studies

have successfully related damage limit states to drift or other demand parameters.


Engelhardt and Husain’s (1993) study did include a review of five previous experimental

investigations, but they did not conduct any statistical analyses the data they collected.

Roeder and Foutch (1996) conducted an extensive study of past experiments to investigate

possible causes of fracture in these types of connections. They compared different test

programs and performed statistical analyses on the data, and determined that panel zone

yielding and beam depth have significant influences on the flexural ductility of pre-

Northridge connections. Unfortunately, Roeder and Foutch (1996) did not develop fragility

functions for this type of connections. There have been some studies that have developed

fragility functions. For example, Song and Ellingwood (1999) developed fragility functions

for steel moment resisting frame buildings; however, the fragility functions in that study

were only concerned with the reliability of the structure as whole, rather than the estimation

of damage in individual beam-to-column connections. The study provided estimates of the

probability of being or exceeding qualitative measures of performance, similar to those

defined in FEMA-356 (2000), as a function of a ground motion intensity measure, namely

spectral acceleration. Measuring performance in this manner makes loss estimation

difficult because the limit states are not well-defined and can not be easily translated into

quantifiable metrics of loss (i.e. dollars, downtime…etc.).

The objective of this study is to consolidate existing experimental test data of Pre-

Northridge moment resisting connections and use it to develop fragility functions to

estimate damage in pre-Northridge welded flange-bolted web beam-to-column connections

as a function of interstory drift ratio, IDR.

4.3 DAMAGE STATE DEFINITIONS

Pre-Northridge steel moment-resisting beam-column connections may typically

experience different types of damage, such as yielding, local buckling and fracture, and this

damage may occur at various locations (e.g. at the column flanges, the column web, the

beam flanges, the beam web…etc.). Two distinct damage states, yielding and fracture,

were adopted in this study. These damage states can be related to specific repair actions

that will help estimate the economic loss, and eventually downtime and casualties. Local

buckling of the beam and column flanges may have important consequences related to

repair/replacement actions and therefore was also considered as another possible damage


state, however, this failure mode did not occur very often in the experiments included in

this study or the drift at which it was first observed was often not reported. Therefore, there

was not enough data reported on local buckling to generate reliable fragility functions for

this type of damage.

DS1 Yielding: In a beam-column subassembly, yielding may first occur at different

locations such as flanges or webs of beams or columns. The experiments reviewed during

this study did not always clearly document how the occurrence of yielding was identified.

In some cases yielding may be identified from strain gages or displacement transducers at

locations where displacements are imposed in the subassembly. Most of the studies

reported the drift at which yielding was initiated, and cases where it was not reported, it was

inferred in our study from the force-displacement plots presented in the reports. Yielding in

pre-Northridge connections primarily takes the form of flange beam yielding or column

panel zone yielding. However, for the purposes of this study, the first reported occurrence

of yielding anywhere on the specimen was used to define the IDR at which this damage

state is induced. Note that this damage state is not as important when estimating economic

losses because typically no repair actions are required when a structural steel member yields

(assuming that any residual displacement is small). However, the information provided by

a fragility function that estimates yielding may be used to help identify the threshold at

which nonlinear behavior initiates in the steel member. This type of information can be

useful when trying to predict structural parameters such as residual story drifts.

DS2 Fracture: Fracture is a failure mode occurring when molecular bonds in the metal

matrix begin to physically separate, resulting in a sudden loss in the joint’s strength.

Fracture often occurs in the complete joint penetration welds that connected the beam

flanges to the column face; however, fracture was also observed in the beam flanges and the

column flanges. It is particularly important to be able to predict this damage state because

it leads to expensive repairs and downtime. Further, if it occurs in sufficient number of

connections, it may lead to a local or global collapse of the structure. As with the damage

state for yielding, the first reported occurrence of fracture anywhere on the specimen by the

experimental study was used to define the IDR this damage state initiates.


4.4 EXPERIMENTAL RESULTS USED IN THIS STUDY

Previous experimental research conducted on Pre-Northridge steel welded flange,

bolted web moment-resisting beam-to-column connections were reviewed and included as

part of this study. Data was drawn from the SAC Phase 1 (SAC, 1996) project and from

other studies that have been conducted over the past 26 years (Popov and Stephen, 1970;

Popov et al., 1985; Tsai and Popov, 1986; Anderson and Linderman, 1991; Engelhardt and

Husain, 1992; Whittaker et al.,1998; Uang and Bondad, 1996; Shuey et al., 1995; Popov et

al., 1995; Kim et al., 2003). Most of the data was taken from single-sided tests, where there

was only one beam attached to a column (Figure 4.2(a)), but one of the investigations,

Popov et al. (1985), used a setup that conducted double-sided tests that had beams on either

side of the column (Figure 4.2(b)). Only specimens that used complete-joint penetration

single bevel groove welds to connect the beam flanges to the column and bolted shear tabs

that connected the beam web to the column were considered in this study. Overall, data was

taken from 10 experimental studies, five conducted before the Northridge earthquake and

five conducted after the Northridge earthquake for a total of 51 test specimens. Table

4.1summarizes all the experimental results considered to formulate our fragility functions.

Both yielding and fracture occurred in all the specimens.


Location of Applied Load

W Steel Beam

W Steel Column

LcL

h/2

h/2


W Steel Beam

W Steel Column

LcL

h/2

h/2

(a)

LcL

W Steel Beam

W Steel Column


LcL

W Steel Beam

W Steel Column


(b)

Figure 4.2 Typical Test Setups (a) Single Sided (b) Double Sided

With the exception of the tests conducted by Popov et al. (1985), all of the

specimens were set up in a single-sided configuration and loaded by displacing the free-end

of the beam as shown in Figure 4.2a. This displacement of the beam’s free end, can be

used to calculate the joint rotation, and thus the equivalent interstory drift, by dividing it by

the length between the beam end and the column centerline, LcL. Popov et al.’s (1985)

investigation used a two-sided configuration, as shown in Figure 4.2b, and loaded these

specimens by displacing the free ends of the upper and lower columns. Interstory drift was

calculated by taking this displacement, and dividing it by the half the total height of the


column. The interstory drifts at which each damage state occurs for each specimen is

reported in Table 4.2.

Table 4.1 Properties of experimental specimens considered in this study

Shape db [cm] Lb [cm] Lcl [cm] Coupon [Mpa] Shape dc [cm] Hcol [cm] Coupon

[Mpa]1 Whittaker et al. (1998) W30x99 75.4 340 360 347 W14x176 38.6 345 3412 Whittaker et al. (1998) W30x99 75.4 340 360 335 W14x176 38.6 345 3693 Whittaker et al. (1998) W30x99 75.4 340 360 325 W14x176 38.6 345 3864 Uang & Bondad (1996) W30x99 75.4 361 361 321 W14x176 38.6 345 3535 Uang & Bondad (1996) W30x99 75.4 361 361 321 W14x176 38.6 345 3536 Uang & Bondad (1996) W30x99 75.4 361 310 321 W14x176 38.6 345 3537 Shuey et al. (1996) W36x150 91.2 340 361 292 W14x257 41.7 345 3368 Shuey et al. (1996) W36x150 91.2 340 361 292 W14x257 41.7 345 3369 Shuey et al. (1996) W36x150 91.2 340 361 292 W14x257 41.7 345 33610 Popov et al. (1995) W36x150 91.2 342 362 418 W14x257 41.7 351 33311 Popov et al. (1995) W36x150 91.2 342 362 418 W14x257 41.7 351 37212 Popov et al. (1995) W36x150 91.2 342 362 280 W14x257 41.7 351 33313 Popov & Stephen (1970) W18x50 45.7 213 227 310 W12x106 32.8 229 24814 Popov & Stephen (1970) W18x50 45.7 213 227 310 W12x106 32.8 229 24815 Popov & Stephen (1970) W18x50 45.7 213 227 310 W12x106 32.8 229 24816 Popov & Stephen (1970) W24x76 60.7 213 227 248 W12x106 32.8 229 24817 Popov & Stephen (1970) W24x76 60.7 213 227 248 W12x106 32.8 229 24818 Engelhardt & Husain (1992) W24x55 59.9 244 261 287 W12x136 34.0 366 37919 Engelhardt & Husain (1992) W24x55 59.9 244 261 287 W12x136 34.0 366 37920 Engelhardt & Husain (1992) W24x55 59.9 244 261 287 W12x136 34.0 366 37921 Engelhardt & Husain (1992) W18x60 46.3 244 261 282 W12x136 34.0 366 37922 Engelhardt & Husain (1992) W18x60 46.3 244 261 282 W12x136 34.0 366 37923 Engelhardt & Husain (1992) W21x57 53.5 244 261 265 W12x136 34.0 366 37924 Engelhardt & Husain (1992) W21x57 53.5 244 261 265 W12x136 34.0 366 37925 Engelhardt & Husain (1992) W21x57 53.5 244 261 265 W12x136 34.0 366 379

Beam Properties Column PropertiesSpecimen No. References

Shape db [cm] Lb [cm] Lcl [cm] Coupon [Mpa] Shape dc [cm] Hcol [cm] Coupon

[Mpa]26 Anderson & Linderman (1991) W16X26 39.9 132 146 322 BOX 11-1.25-0.75 27.9 112 33527 Anderson & Linderman (1991) W16X40 40.6 132 146 290 BOX 11-0.75-0.75 27.9 112 35628 Anderson & Linderman (1991) W16X40 40.6 132 146 380 BOX 11-1.25-0.75 27.9 112 33529 Anderson & Linderman (1991) W16X26 39.9 132 146 417 BOX 11-1-0.75 27.9 112 32130 Anderson & Linderman (1991) W16X26 39.9 132 146 341 BOX 11-1-0.75 27.9 112 32131 Anderson & Linderman (1991) W16X40 40.6 132 146 322 BOX 11-0.75-0.75 27.9 112 28632 Anderson & Linderman (1991) W16X40 40.6 132 146 324 BOX 11-0.75-0.75 27.9 112 26133 Anderson & Linderman (1991) W16X40 40.6 132 149 322 W12X136 34.0 112 28534 Popov et al. (1985) W18x50 45.7 142 164 320 Built-up 45.7 145 33835 Popov et al. (1985) W18x50 45.7 142 164 320 Built-up 45.7 145 33836 Popov et al. (1985) Built-up: 47.6 140 164 262 Built-up 48.6 145 33837 Popov et al. (1985) Built-up: 47.6 140 164 262 Built-up 48.6 145 33838 Popov et al. (1985) Built-up: 47.6 140 164 262 Built-up 48.6 145 33839 Popov et al. (1985) W18x71 47.0 137 164 300 W21x93 54.9 145 41440 Popov et al. (1985) W18x71 47.0 137 164 300 W21x93 54.9 145 41441 Tsai & Popov (1986) W18x35 45.0 165 182 356 W12x133 34.0 156 38942 Tsai & Popov (1986) W21x44 52.6 160 179 335 W14x176 38.6 156 38643 Tsai & Popov (1986) W18x35 45.0 161 180 353 W14x159 38.1 156 43844 Tsai & Popov (1986) W21x44 52.6 161 180 308 W14x159 38.1 156 31645 Tsai & Popov (1986) W18x35 45.0 161 180 N/A W14x159 38.1 156 N/A46 Tsai & Popov (1986) W21x44 52.6 161 180 308 W14x159 38.1 156 31647 Tsai & Popov (1986) W18x35 45.0 161 180 319 W14x159 38.1 156 38448 Tsai & Popov (1986) W21x44 52.6 161 180 290 W14x159 38.1 156 29049 Kim et al. (2003) W33x118 83.6 206 229 419 BC18x18x257 45.7 417 41950 Kim et al. (2003) W36x232 94.2 371 411 393 BC31.5x13x464 80.0 417 39351 Kim et al. (2003) W36x210 93.2 374 411 390 W27x281 74.4 417 376

Specimen No. References

Beam Properties Column Properties


Table 4.2 Interstory drifts at each damage state for each specimen

1 0.74 1.98 27 0.52 1.672 0.74 3.95 28 0.74 1.833 0.74 2.97 29 0.78 1.834 0.65 0.94 30 0.43 1.575 0.95 0.94 31 0.52 1.446 0.82 1.87 32 0.42 1.677 0.70 0.70 33 0.60 1.538 0.53 1.41 34 0.91 1.859 0.53 0.70 35 0.93 3.7110 0.55 2.10 36 0.56 2.1611 0.60 1.40 37 0.55 2.4712 0.63 2.10 38 0.57 2.0113 0.62 2.76 39 0.77 3.4914 0.61 3.65 40 0.74 3.8615 0.50 2.65 41 0.41 1.5016 0.56 4.90 42 0.51 1.2117 0.56 1.77 43 0.52 1.8418 0.73 1.17 44 0.43 2.5419 0.73 1.17 45 ** 0.4120 0.97 1.70 46 0.34 0.7721 0.97 1.07 47 0.33 1.9122 0.88 2.19 48 0.39 1.6523 0.97 1.95 49 0.75 0.7624 0.97 2.43 50 0.38 0.5925 0.88 1.95 51 0.38 0.5826 0.41 1.30

IDRDS1 [%] IDRDS2 [%]Specimen No.

IDRDS2 [%]IDRDS1 [%] Specimen No.

4.5 FRAGILITY FUNCTION FORMULATION

The data consolidated from the experimental studies listed in Table 4.2 was used to

develop drift-based fragility functions for yielding and fracture. As observed from Table

4.2, the interstory drift ratio at which the beam-column specimens reach these damage

states varies significantly from test to test. When estimating damage in existing WSMF

buildings built prior to 1994, it is important to account for this variability. Drift-based

fragility functions capture this specimen-to-specimen variability, providing a way of

estimating the probability that the joint will experience or exceed a particular damage state

given an imposed interstory drift demand. Fragility functions were created for each damage

state as described in the preceding sections.


Fragility functions are cumulative frequency distribution functions that provide the

variation of increasing probability of reaching or exceeding a damage state as interstory

drift increases. These functions are generated by first sorting the data, in ascending order,

by the interstory drift ratio at which the damage was reported for each damage state. These

values are then plotted against their cumulative probability of occurrence. In this study the

cumulative probability of occurrence was computed using the following equation:

( 0.5)iPn

(4.1)

where i is the position of the peak interstory drift ratio within the sorted data and n is the

number of specimens. This equation is also known as Hazen’s Model, and is one of several

commonly-used equations used to compute quantiles. This particular definition was

selected because previous research has shown this definition limits the amount of bias

introduced into the plotting position (Cunnane 1978). It also prevents the first data point in

a sample to be assigned a probability of 0 (i.e. the damage state will never occur at this

drift), and the last data point with a probability of 1 (i.e. the damage state is guaranteed to

occur at this drift), which is unrealistic. However, the differences between the different

definitions of the quantile are subtle and for large sample sizes, all of them converge to the

same value.

After the data is plotted, a lognormal cumulative distribution function was fitted to

the data points, by using its logarithmic statistical parameters to define the function. It has

been well-established that the lognormal distribution provides relatively good fit to

empirical cumulative distributions computed from experimental data (Aslani 2005; Aslani

and Miranda 2005; Pagni and Lowes 2006; Brown and Lowes 2007). The equation of this

fitted function is given by:

( ) ( )( )iLnIDR

Ln idr Ln IDRP DS ds IDR idr (4.2)


where P(DS dsi IDR = idr) is the probability of experiencing or exceeding damage state

i, Ln IDR is the natural logarithm of the counted median of the interstory drift ratios

(IDRs) at which damage state i was observed, LnIDR is the standard deviation of the natural

logarithm of the IDRs, and is the cumulative standard normal distribution. Alternatively,

Ln IDR can be replaced by the geometric mean, which is the mean of the natural

logarithm of the data.

To ensure that the fitted functions are not skewed by outlying data points, outliers

were identified and removed from our fragilities. Chauvenet’s outlier criterion (Barnett

1978, Hawkins 1980, and Barnett and Lewis 1995), given by the following equations, was

used to determine whether or not the data points would be included:

12lowerp

n (4.3)

112upperp

n (4.4)

If a data point’s cumulative probability was smaller than the probability calculated with

Equation (4.3), or was greater than the probability calculated with Equation (4.4), then it

was excluded from the data set.

Kolmogorov-Smirnov goodness-of-fit tests (Benjamin and Cornell 1970) were

conducted to verify that the cumulative distribution function could be assumed to be

lognormally distributed. This was done by plotting graphical representations of this test for

10% significance levels on the same graph as the data and its fitted cumulative distribution

function. If all the data points lie within the bounded significance levels, the assumed

cumulative distribution function fits the empirical data adequately.

In addition to the specimen to specimen variability, statistical uncertainty was

considered to account for inherent uncertainty in the proposed fragility functions because

their parameters have been established using data with a limited amount of specimens

(finite-sample uncertainty). This uncertainty is quantified by bounding our fitted lognormal

cumulative distribution function with confidence intervals of the median and dispersion


parameters of IDR for each damage state. Conventional statistical methods can be used to

establish the confidence intervals because our underlying probability distribution is

lognormal. Crow et al. (1960) proposed the following equation to approximate the

confidence intervals of a lognormally distributed sample:

/ 2exp LnIDRIDR zn

(4.5)

where z /2 is the value in the standard normal distribution such that the probability of a

random deviation numerically greater than z /2 is , and n is the number of data points.

90% confidence intervals can were obtained and plotted for each fragility function.

4.5.1 Fragility Functions for Yielding

The experimental data used included specimens that were fabricated from both A36

and A572 grade 50 steel (Fy = 36 ksi and 50 ksi, respectively). The data was divided into

these two categories because the A36 specimens are expected to yield at lower drifts. It was

found that the difference in IDRs in the two groups was statistically significant. Therefore,

two separate fragility functions were created based on the yielding stress of the specimen.

When the test specimen consisted of members fabricated from differing types of steel (e.g.

the beam made from A36 and the column made from A572 grade 50), the specimen was

categorized based on the member where yielding first occurred. Figure 4.3a displays the

fragility function for test specimens whose yielded members were fabricated from A36

steel, while Figure 4.3b displays the fragility function for A572 grade 50 steel. Statistical

parameters for both functions are listed in Table 4.3.

Table 4.3 Uncorrected statistical parameters for IDRs corresponding to the damage states for Pre-Northridge beam-column joints


Damage State Median IDR [%]

Geometric Mean IDR LnIDR

Number of Specimens

(Outliers Removed)DS1: Yielding

A36: Uncorrected Raw Data 0.56 0.59 0.32 32A572: Uncorrected Raw Data 0.74 0.71 0.19 16

DS2: Fracture (Uncorrected) 1.85 1.79 0.47 50

DS = Yielding (A36)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0Interstory Drift, IDR [%]

P(DS|IDR)

Data Fitted Curve K-S Test, 10% Signif.

(a)

DS = Yielding (A572)

0.0

0.2

0.4

0.6

0.8

1.0


P(DS|IDR)

Data Fitted Curve K-S Test, 10% Signif.

(b)

Figure 4.3 Yielding without Correction for Span-to-Depth Ratio(a) A36 (b) A572 grade 50


Given an IDR, the fragility functions shown in Figure 4.3a and 3b can be used to

estimate the probability of experiencing yielding in pre-Northridge beam-column

connections. However, the accuracy of these predictions are dependent on whether the test

setups and member sizes of the experimental data used to create the fragility functions are

representative of the beam-to-column connections used in practice. Parameters that

strongly influence the IDR at which yielding occurs must be identified, to determine if their

values in the experimental data are representative of those used in practice. The influence

of various parameters was identified by plotting them against their corresponding IDR and

linear regression was conducted to develop relationships. A T-test (Benjamin and Cornell

1970) was used to determine which parameters statistically influenced the yielding IDR.

Test specimen parameters tested included beam depth, flange thickness and beam span-to-

depth ratio. Of these parameters, beam span-to-depth ratio was the only that exhibited

statistical significance based on the T-test criterion.

The relationship between span-to-depth ratio (L/db, where L, the centerline span of

the beam , is equal to 2*LcL) and the joint’s yielding IDR, was further investigated by

conducting linear regression on the two random variables. Figure 4.4a plots the natural

logarithm of the test specimens’ yielding IDR as a function the beam’s L/db for the A36

specimen. As shown in this figure, there is a clear trend that beams with small span-to-

depth ratios require less deflection/drift to initiate yielding in the beam-column joint. 95%

confidence intervals on the regressed linear trend computed from the data are also plotted in

the figure. The regression yields a significant correlation coefficient of 0.64, confirming the

statistical significance indicated by the T-test. The mean beam’s L/db for the A36 specimens

is 8.25, that corresponds to a median IDR of 0.56%. The trend between L/db and ln(IDR)

for the A572 grade 50 specimens is not as strong. The correlation coefficient in this case is

only 0.19. The mean L/db for the A572 grade 50 specimens is 8.0 at a corresponding

median IDR of 0.74%. The lower correlation of the A572 grade 50 specimens may be

largely due to the smaller sample size of experimental joints made from this type of steel,

but also due to the fact that in most A572 grade 50 specimens yielding was not initiated in

the beams but in the panel zones and therefore the beam’s L/db has a smaller influence.


-1.2

-0.8

-0.4

0.0

0.4

6 8 10 12Span-to-Depth Ratio, SDR

ln(IDR) [%]

Data Fitted Data95% Confid. on Mean Theoretical

(a)

-1.2

-0.8

-0.4

0.0

0.4

4 6 8 10 12Span-to-Depth Ratio, SDR

ln(IDR) [%]

Data Fitted Line 95% Confid. on Mean Theoretical

(b)

Figure 4.4 Span-to-Depth Ratio’s relationship to Interstory drift (a) A36 (b) A572 grade 50

Shortly after the 1994 Northridge earthquake, Youssef et al. (1995) conducted a

survey of steel moment-resisting frame buildings that were affected by the earthquake.

According of their report, the buildings surveyed had a mean L/db of 10. Assuming that this

can serve as a fairly accurate representation of typical span-to-depth ratio’s used in practice,

we can conclude that the functions displayed in Figure 4.3, if used directly to estimate

damage in existing steel moment-resisting frame buildings built prior to 1994, may result in


underestimations of yielding IDR (which would lead to over predicting yielding), because

the test specimens used to develop these functions have a smaller span-to-depth ratios.

Table 4.4 Summary of Yousef et al.’s Building Survey Results for Typical Girder Sizes of Existing Buildings

d b Weighted Avg. L/d b Weighted Avg.(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

W14/16 6 48 4.6 5.8 8.5 0.04 38 1.5 15.2 0.6W18 9 46 3.7 6.1 12.2 0.04 46 1.7 13.3 0.5W21 12 112 3.4 5.5 12.2 0.09 53 4.9 10.3 1.0W24 23 135 4 7 10.4 0.11 61 6.8 11.5 1.3W27 19 56 4.9 7.9 12.2 0.05 69 3.2 11.5 0.5W30 20 106 4 7.6 12.8 0.09 76 6.7 10.0 0.9W33 20 174 4.9 8.5 12.8 0.14 84 12.1 10.1 1.5W36 30 533 4.6 7.9 14 0.44 91 40.3 8.6 3.8

1210 77.2 10.0

Beam Depth [cm] Span-to-depth ratioWeightTypical

GirderNo of Bldgs

Floor-Frames

Min Bay [m]

Avg Bay [m]

Max Bay [m]

Notes:- Column (10) is the calculated average span-to-depth ratio calculated by dividing (5) by (8) accounting for unit conversion- A weighted average was used based on the number of floor frames included in Yousef et al.'s study. The weight (7) is found by taking the value of (3) and dividing it by the sum of column (3). For example, the weighted average for span-to-depth ratio is obtained by multiplying (7) by (10) to get (11), and then summing up column (11).

There are different alternatives approaches that one may use to modify the fragility

functions shown in Figure 4.3. A first approach is to compute the median IDR as a function

of the L/db ratio as follows:

exp[ ]bIDR a b L d (4.6)

where IDRy is the geometric mean IDR at yielding, and a and b are dimensionless

coefficients the y-intercept and the slope of the linear regression relationship respectively.

Table 4.5 documents the values of the regression coefficients, a and b, for the data

considered in this study for both A36 and A572 grade 50 specimens. In cases in which the

L/db ratio is not known, one could use an L/db of 10 (based on Yousef et al.’s survey), in

equation (4.6) and then the median IDRy become 0.77and 0.76 for A36 and A572 gr 50

specimens, respectively (see also Table 4.6). In cases where L/db is known, small

reductions in dispersion are achieved to 0.24 for A36 specimens and to 0.18 for A572 grade

50 specimens (Table 4.6).


DS = Yielding (A36)

0.0

0.2

0.4

0.6

0.8

1.0


P(DS|IDR)

Original Function Corrected Function 90% Confid. Intervals

Figure 4.5 Recommended Fragility Function corrected for Span-to-Depth Ratio with 90% confidence bands

Table 4.5 Regression coefficients for relationship between IDRy and L/db

Regression Parameters

A36Specimens

A572Specimens

Y-intercept, a -1.83 -0.56Slope, b 0.16 0.028

Table 4.6 Recommended statistical parameters for fragility functions

Damage State Information Available Type of Steel Median IDR [%] LnIDR

A36 0.77 0.32

A572 Gr. 50 0.76 0.19

A36 exp [-1.83 + 0.16*(L/d b )] 0.24

A572 Gr. 50 exp [-0.56 + 0.028*(L/d b )] 0.18

A36 1.42(IDR y ) 0.21

A572 Gr. 50 1.25(IDR y ) 0.32

DS1: Yielding

Only type of connection

Type of connection and L/d b

Type of connection and member data

A36 & A572 Gr. 50 1.85 0.47

DS2: Fracture

Type of connection and d bA36 & A572

Gr. 50 0.44exp [-0.99 + 0.0074*(d b )]

Only type of connection

Notes:L = centerline span = 2LcL

IDRy analytically computed insterstory drift at yieldingFor Fracture, there is no statistical difference with the type of steel.


Correction to account for different L/db ratios can also be done by deriving an

analytical expression of the IDR at yield as a function of L/db. Krawinkler et al. (2000)

derived equations that calculated IDRs for typical steel moment-resisting frame beam-

column connection test setups. In their approach, the IDR was computed as the sum of the

terms corresponding to three separate equations that represent the contributions of beam

flexure (IDRb), column flexure (IDRc), and panel zone shear (IDRPZ) as follows:

y b c PZIDR IDR IDR IDR (4.7)

where,

3

2

3

cdcL

bb cL

L PIDREI L

(4.8)

3

212b cL

cc

h d PLIDREI h

(4.9)

1 1bPZ

s b

h d PLIDRh A G d h

(4.10)

where P is the load imposed on the specimen, dc is the depth of the column, h is the height

of the column, Ic is the column’s moment of inertia, db is the depth of the beam, Ib is the

beam’s moment of inertia, L is the distance from the beam-end to the face of the column, E

is Young’s modulus and G is the corresponding shear modulus. Equations (4.7) to (4.10)

are only true under the following simplifying assumptions (Krawinkler et al. 2000): (i)

inflection points are assumed to occur at mid-height and at mid-span in columns and beams,

respectively; (ii) the is no vertical deflection in the point of inflection in the beam; (iii)

localized deformations in welds or slippage in bolted connections are ignored.


Assuming yielding will occur first in the beam we can express IDR due to beam

flexure in terms of the beam’s strain, y, and replace Equation (4.8) with:

2 13 2

c cLb y

cL b

d LIDRL d

(4.11)

Then equations (4.8)-(4.10) can be re-written in terms of L/db as follows:

21 13 2

ycb b

cL

FdIDR L dL E

(4.12)

3

2212 c

yb bc bd

ccL

Fh d IIDR L dI Eh L

(4.13)

2

22

2.6c

yb bPZ bd

c cb cL

Fh d IIDR L dt d Eh d L

(4.14)

where Fy is the material yield stress and tc is the column’s web thickness. By using

equations (4.12)-(4.14) in equation (4.7) it is then possible to compute IDR at yield as a

function of L/db Table 4.7 displays mean values taken from our test specimens for the

parameters that are used in the derived analytical equations above. Figure 4.4a and b show

the analytical expression computed with these equation using the mean values indicated in

Table 4.7. It can be seen that, in both cases, the analytical expression falls within the 95%

confidence intervals suggesting that the linear regression of our data set follows the

analytical prediction.

Table 4.7 Average values for parameters in Equation (9), relating L/db and IDR


A36 Specimens

A572Specimens

Beam PropertiesCoupon Yielding Stress, F y [Mpa] 310 352Beam Depth, d b [cm] 54 66Moment of Inertia, I b [cm 4 ] 79,084 145,681Span-to-depth ratio, SDR 8.25 7.91

Column PropertiesHeight of Column, h [cm] 234 264Column Depth, d c [cm] 34 44Web Thickness, t c [cm] 2.4 2.2Moment of Inertia, I c [cm] 59,937 90,322

Mean Values Parameters

In cases in which there is enough information (e.g., section geometry and nominal

material properties) to analytically compute the interstory drift at which yielding will be

initiated, then experimental information shown in Table 4.2 can be used to obtain a fragility

function specifically for each connection by considering a random variable , defined as the

ratio of the IDR in which yielding was observed in the test to the analytical yielding IDR as

follows:

Observed

y

IDRIDR

(4.15)

Figure 4.6 shows the cumulative distribution functions of for both the A36

specimens and the A572 grade 50 specimens. Figure 6 also shows fitted lognormal

distributions computed with parameters listed in Table 4.6 and 10% significance curves

corresponding to the Kolmogorov-Smirnov test, suggesting that the random variable can

also be assumed to be lognormally distributed.


for A36

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0Correction Factor,

P(DS | )

Data

Fitted Curve

K-S Test, 10%Significance

(a)

for A572

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0Correction Factor,

P(DS | )

Data

Fitted Curve


(b)

Figure 4.6 Fragility Functions for to be used in conjunction with an analytical prediction of IDRy(a) A36 (b) A572 grade 50

For a given beam-to-column connection one would first compute analytically the

IDR corresponding to the onset of yielding and the median of the fragility function is then

computed as the product of IDRy and the median of Meanwhile the dispersion in the

fragility is set equal to the dispersion of shown in Table 4.6. It should be noted that using

this procedure a further reduction in dispersion was obtained for A36 specimens, however

for A572 grade 50 specimens the dispersion increased due to the fact that in several


specimens the location of yielding observed in the test was different to that predicted

analytically.

As an example of the latter approach, consider a pre-Northridge beam-to-column

connection between a W36x150 beam with a L/db ratio of 10 and W14x257 column both

made from A572 grade 50 steel. Using equations(4.7), (4.12), (4.13) and (4.14) one

computes IDRAnalytical =0.84 which considering the median and dispersion of shown in

table 6 results in the fragility function shown in Figure 4.7, which has a median of 1.05%

and logarithmic standard deviation of 0.32. Ninety percent confidence levels that account

for the statistical uncertainty are also shown in the figure.

A572

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0IDR [%]

P(DS|IDR)

Fitted Function

90% Confid. Intervals

Figure 4.7 Example Fragility Function for W36 beam generated by using (A572 grade 50)

4.5.2 Fragility Functions for Fracture

Fragility functions were also generated to estimate the probability of fracture as a

function of IDR. Unlike the yielding limit state in which an analytical prediction is possible

and therefore fragility functions were derived by using either a purely empirical approach

(i.e., entirely based on the experimental results) or by using a hybrid analytical-

experimental approach, for fracture there is not a reliable way to analytically estimate the

drift at which fracture is likely to occur in these connections, therefore in this case fragilities

were only based on experimental results. In the case of fracture the drifts at which A36


specimens fractured were not statistically different from the drifts at which A572 grade 50

specimens fractured, therefore all specimens were analyzed in the same group. Figure 4.8

shows the fragility function for fracture and its K-S goodness-of-fit test for 90%

significance levels corresponding to all specimens. The counted median IDR is 1.85%, the

geometric mean is 1.79% and the logarithmic standard deviation is 0.47. It should be noted

that the variability in fracture is significantly larger than that observed for yielding.

DS = Fracture

0.0

0.2

0.4

0.6

0.8

1.0


P(DS| IDR)

Data

Fitted Curve


Figure 4.8 Fragility Function for Fracture

Roeder and Foutch (1996) showed that beams of pre-Northridge joints with larger

depths have significantly smaller flexural ductility than shallower beams. Furthermore,

their work was the basis for creating relationships between beam depth plastic rotation

capacity as described in FEMA-355D (FEMA, 2000b). This study also investigated the

effect of beam depth on the drift at which fracture is likely to occur in this type of

connections by using the data in Table 4.2. Figure 4.9 shows the plot of the natural

logarithm of IDR as a function of beam depth. Consistent with Roeder and Foutch

observations, it can be seen that the IDR at which fracture occurs decreases as beam depth

increases. Using linear regression on this sample, the median IDR at which fracture occurs

can be estimated as a function of the beam depth by using the following equation:


exp[0.99 0.0074 ]bIDR d (4.16)

where db is the depth of the beam (in cm). Equation (4.16) is also plotted in Figure 4.9

along with 95% confidence intervals on the mean. It should noted that this equation is not

directly comparable to the equations developed by Roeder and Foutch in FEMA-355D

(2000) because those equations were based on beam ductility and beam plastic rotation

capacity, rather than on the IDR of the connection. The fragility function corresponding to

fracture can then constructed by first estimating the median parameter using equation (4.16)

and using a somewhat smaller logarithmic standard deviation of 0.44. Figure 4.10

illustrates an example of a fragility created with this procedure, using the calculated

weighted average of beam depth from Yousef et al.’s survey (1995, see Table 4.4), and

enveloped by 90% confidence intervals associated with the statistical uncertainty produced

by computing the parameters of the fragility function using a small sample size (i.e., a small

number of experimental tests). Figure 4.11 also implements this procedure using the beam

depth of a specific steel shape (W36x150, the same shape used above in the yielding

fragility example) as an example for users that have this information.

-1.0

0.0

1.0

2.0

0 30 60 90 120

Beam Depth, db [cm]

ln(IDR) [% ]

Data Fitted Data 95% Confid. Intervals

Figure 4.9 Relationship between IDR at fracture and beam depth for all specimens.


DS = Fracture

0.0

0.2

0.4

0.6

0.8

1.0


P(DS| IDR)

Corrected Function 90% Confid. Intervals

Original Function

Figure 4.10 Recommended fragility function corrected for beam depth with 90% confidence bands

DS = Fracture

0.0

0.2

0.4

0.6

0.8

1.0


P(DS| IDR)

Corrected Function

90% Confid. Intervals

Figure 4.11 Example corrected fragility for W36 when beam depth is known.

4.6 CONCLUSIONS

Fragility functions for pre-Northridge steel beam-column joint connections have been

developed in this study based on experimental results of 51 specimens. Fragility functions

for two damage states, yielding and fracture, were generated to establish relationships

between building response parameters, namely interstory drift ratio, and the level of damage

experienced in the beam-column connection. The fragility functions presented in this study


allow the incorporation of variability of deformation demands at which the two damage

states may occur. For the case of yielding three different sets of fragility functions were

developed. The first two fragility function employ a purely empirical approach in which the

parameters of the fragility function are computed with a span-to-depth ratio that

approximately represent those used in practice, or by using connection-specific span-to-

depth ratios. In the third fragility function uses a hybrid analytical-empirical approach in

which an analytical estimate of the drift at yielding is first computed and is then modified to

account for the bias and variability found using statistical information obtained between

observed and analytical drifts at yield for each specimen. Both empirical and hybrid

approaches indicate that the drift at which yielding is likely to occur increases with

increasing span-to-depth ratios, therefore consideration of the span-to-depth ratio when

estimating the likelihood of yielding does not only results in an improved estimate of the

median drift but also in a smaller dispersion. For the specimens considered in this study

analytical obtaining analytical predictions of the interstory drift at yield resulted in

insignificant further reduction in dispersion for A36 specimens and in an increment in

dispersion for A572 grade 50 specimens.

For estimating the probability of fracture only the empirical approach was used. It

was found that the drift at which fracture occurs decreases with increasing beam depths. For

beam depths between 76 and 91 cm (30 and 36 inches), which are commonly used in

practice in moment connections, median drifts that produce fracture in pre-Northridge

welded beam-to-column connections are between 1.47% and 1.31%, respectively.

Furthermore, the fragility functions developed in this study indicate that there is a

probability between 70% and 80% that WSMF buildings with pre-Northridge connections

experience fractures in their beam-to-column connections if they are subjected to interstory

drift demands of 2%, which is the maximum allowed in current U.S. codes in the design

level earthquake.

CHAPTER 5 79 Development of Component Fragility Functions from Empirical Data

CHAPTER 5

5 DEVELOPMENT OF COMPONENT FRAGILITY FUNCTIONS FROM EMPIRICAL DATA


Ramirez, C.M., Cheong, K.F., Schrotenboer, T., and Miranda, E. (2008), Development of

Empirical Fragility Functions in Support of the Story-based loss estimation toolbox, Pacific

Earthquake Engineering Research Center Report, (in preparation).

5.1 AUTHORSHIP OF CHAPTER

Ramirez aided with developing the models used for structural analysis, computed the

fragility functions, and authored the publication. Cheong and Schrotenboer were

responsible for completing the analyses required to obtain the structural response

parameters. Miranda served as advisor and principal investigator for this project.

5.2 INTRODUCTION

Quantifying structural performance in terms of economic loss induced by seismic

ground motions requires estimating damage as a function of structural response. Fragility

functions compute the probability of being in to or exceeding a given damage state or

performance level as a function of a structural response parameter. Fragility functions need

to be assigned to all components in a building’s inventory to estimate the damage and

associated monetary loss that is representative of the entire value of the building. However,

functions for every type of building component are currently not available. Most studies

(see Chapter 2) on seismic-induced economic losses have ignored the loss due to

components without fragilities or accounted for their loss only when the structure collapsed


(Goulet et al. 2007). Other investigators (Aslani and Miranda, 2005) have estimated the loss

in some of these components by using generic functions that were initially developed to be

used in regional methods (HAZUS) for some of these components. The data used to

develop these generic functions, however, are not well-documented and rely heavily on

expert opinion that has yet to be validated. Failing to account for economic losses due to

these types of components may lead to significant errors in the total estimated economic

loss and inaccurate projections of the loss’ composition. Until fragilities are available for all

building components, economic losses due to components without fragility functions need

to be accounted for in a reliable manner.

One way of accounting for these losses due to components without fragilities is using

generic fragility functions. Generic fragility functions are fragilities that are not component-

specific, but rather estimate the damage of groups of building components that are of the

same type. Components can be grouped into structural components and nonstructural

components for loss assessment purposes. Building components that do not have fragility

functions available can be grouped into these general categories and assigned these generic

fragilities such that their damage can be estimated collectively. Although damage estimated

from these functions may not be as precise as estimates generated from component-specific

functions, it yields better economic loss assessments than if the damage due to these

components is ignored.

Many generic fragility functions that are currently available are based on motion

damage relationships developed from expert opinion, such as those used in ATC-13 (ATC,

1985). Other functions, such as those used in HAZUS99 (FEMA, 1999), merge expert

opinion with analytical results. Empirical data collected after earthquakes have been used

to update these motion-damage relationships, however, the improvements made to these

models have been limited by the lack of relevant building performance data collected

(Anagnos et al., 1995, FEMA, 2000, Lizundia and Holmes, 1997). These empirical datasets

have been hampered by a variety of deficiencies, which include: small sample sizes that do

not provide enough data points, datasets that have a bias towards damaged or noteworthy

buildings, datasets that are limited by the amount of building types that are included,

datasets that are not collected in a consistent and complete manner, datasets that are

collected by private companies and are not available to the public and datasets that include

buildings that are not located close to free-field recording instruments (King et al., 2005).


Recently, there have been efforts to improve the quality of empirical building

performance data. The ATC-38 project (ATC, 2000) conducted after the 1994 Northridge

earthquake attempted to systematically document the damage of buildings located near

locations of strong ground motion recording stations. Engineers inspected more than 500

buildings located within 1000 feet of 30 strong motion recording stations. As a result, a

thorough non-proprietary database now exists that includes the building properties and

damage performance, photos, and strong motion recordings. Degenkolb Engineers (Heintz

and Poland, 2001) also developed a similar database from an investigation conducted in

Taiwan after the 1999 Chi-Chi earthquake.

King et al. (2005) developed motion-damage relationships using the ATC-38 project

and other similar datasets to create lognormal fragility curves and damage probability

matrices. Spectral acceleration and interstory drift ratio, the latter estimated by using

spectral displacement and using a method proposed by Miranda (Miranda, 2000), were used

as structural response parameters to develop these fragility functions. Unfortunately, these

response parameters are based on single-degree of freedom systems that neglect the

contribution of higher mode effects and assume the structure is first mode dominated. This

may lead to inaccuracies in economic loss estimates for components that are damaged by

floor accelerations (acceleration-sensitive components) because floor accelerations are

strongly dependent on higher mode effects even in buildings with moderate heights. Even in

a first mode dominated structure, the building’s roof acceleration could be 20% to 60%

higher than the spectral acceleration because spectral acceleration is not equivalent to its

peak floor accelerations. Additionally, King et al. (2005) computed both spectral

accelerations and spectral displacements using approximations of the structural fundamental

period used in US building codes (ICC, 2003) with parameters proposed by Goel and

Chopra (1997). The equations used to approximate the fundamental period are not building-

specific and may lead to inaccurate estimations of interstory drift ratios.

This study is primarily aimed at obtaining generic fragility functions for nonstructural

components where there is very scarce information. Of particular interest is to obtain

information about the levels of structural response which trigger nonstructural damage since

this information is particularly important when computing expected annualized losses

(EALs). This study also attempts to improve the way empirical generic fragility functions

are developed by capturing structural response using better response parameters and more

advanced methods of structural simulation to estimate these parameters. Instead of using


spectral acceleration and spectral displacement as response parameters, floor accelerations

and interstory drift ratios computed from analyses that model multi-degree of freedom

systems are used. Two sets of buildings that had damage data collected and documented

after the 1994 Northridge earthquake were modeled. The first set of data was extracted from

19 instrumented buildings established the California Strong Motion Instrumentation

Program (CSMIP). The responses of these buildings were computed using a continuous

model developed by Miranda (1999) that has been shown to approximate floor accelerations

and interstory drift ratios relatively well (Miranda, 1999; Miranda and Taghavi, 2005). The

response parameters computed from the continuous model were validated using the

response data recorded by the accelerographs contained in these buildings. The second set

of data was taken from the ATC-38 report (ATC, 2000). These buildings were not

instrumented but because they were located close to ground motion recording stations, the

ground motions recorded at these stations can be used in structural simulation to estimate

what the peak structural response parameters were during this earthquake. A probabilistic

approach using Monte Carlo simulation was used to obtain the most probable values of the

response parameters. Once the response parameters were established, they were paired with

the reported damage states for different groups of building components to create motion-

damage pairs. The motion-damage pairs were used to create fragility functions. These

fragility functions were then used to estimate damage for building components without

component-specific fragilities in Chapter 3 of this dissertation to develop EDP-DV

functions that relate structural response directly to monetary loss and used as part of the

Pacific Earthquake Engineering Research (PEER) center’s loss estimation toolbox, detailed

in Chapter 6.

5.3 SOURCES OF EMPIRICAL DATA

5.3.1 Instrumented Buildings (CSMIP)

The California Strong Motion Instrumentation Program (CSMIP) was established in

1972 to gather seismic activity data by instituting a network of accelerographs throughout

the state. The network includes 170 instrumented buildings, 19 of which have damage

documented during the 1994 Northridge Earthquake. For every building, a brief description

of the structure was provided which included several structural characteristics. A summary


of these building properties is listed in Table 5.1. These characteristics included the

building’s number of stories, occupancy type and type of lateral force resisting system. The

building’s location in terms of latitude and longitude is reported and in particular its

distance to the earthquake’s epicenter. Sensors are located throughout each building on the

ground floor and selected floors above ground. Each sensor recorded accelerations during

the Northridge earthquake.

Table 5.1 CSMIP Building Properties

Station ID No. of Stories

Lateral Resisting System

Occupancy TypeDist. To

Epicenter [km]

Interstory Ht. [cm]

EQ Direction Period [s] Alpha Damping

Ratio

24231 7 Steel MRF School 18 411 EW 1.10 4.1 0.05424231 7 Steel MRF School 18 411 NS 0.62 7.0 0.10024236 14 Rconcrete MRF Warehouse 23 320 EW 0.81 8.5 0.08824236 14 Rconcrete MRF Warehouse 23 320 NS 2.30 5.0 0.07524322 13 Rconcrete MRF Commercial 9 358 EW 2.92 29.5 0.05024322 13 Rconcrete MRF Commercial 9 358 NS 2.60 19.1 0.09024332 3 Shear Walls Commercial 20 508 EW 0.48 6.0 0.03524332 3 Shear Walls Commercial 20 508 NS 0.59 30.0 0.05024370 6 SMRF Commercial 22 396 EW 1.39 30.0 0.04024370 6 SMRF Commercial 22 396 NS 1.38 30.0 0.02924385 10 Shear Walls Residential 21 207 EW 0.59 2.0 0.05524385 10 Shear Walls Residential 21 207 NS 0.60 2.0 0.05924386 7 Rconcrete MRF Hotel 7 265 EW 1.98 8.9 0.13024386 7 Rconcrete MRF Hotel 7 265 NS 1.60 5.0 0.13024463 5 Rconcrete MRF Warehouse 36 725 EW 1.45 10.2 0.03724463 5 Rconcrete MRF Warehouse 36 725 NS 1.62 15.5 0.03524464 20 Rconcrete MRF Hotel 19 257 EW 2.60 9.1 0.05024464 20 Rconcrete MRF Hotel 19 257 NS 2.79 29.0 0.03024514 6 Shear Walls Hospital 16 472 EW 0.30 3.5 0.18024514 6 Shear Walls Hospital 16 472 NS 0.37 3.0 0.18024579 9 Rconcrete MRF Office Building 32 396 EW 1.29 4.1 0.05324579 9 Rconcrete MRF Office Building 32 396 NS 1.04 29.5 0.05024580 2 Base Isolation Office Building 38 NA NA NA NA NA24580 2 Base Isolation Office Building 38 NA NA NA NA NA24601 17 Shear Walls Residential 32 264 EW 1.08 1.8 0.03324601 17 Shear Walls Residential 32 264 NS 1.14 1.5 0.03624602 52 Steel MRF Office Building 31 396 EW 6.20 6.9 0.01524602 52 Steel MRF Office Building 31 396 NS 5.90 9.8 0.01024605 7 Base Isolation Hospital 36 NA NA NA NA NA24605 7 Base Isolation Hospital 36 NA NA NA NA NA24629 54 Steel MRF Office Building 32 396 EW 5.60 30.0 0.00524629 54 Steel MRF Office Building 32 396 NS 6.20 27.5 0.00924643 19 Steel MRF Office Building 20 406 EW 3.90 30.0 0.02124643 19 Steel MRF Office Building 20 406 NS 3.47 4.0 0.02624652 6 Steel MRF Office Building 31 427 EW 0.91 30.0 0.04924652 6 Steel MRF Office Building 31 427 NS 0.86 13.0 0.03224655 6 Steel MRF Parking Structure 31 305 EW 0.40 1.4 0.06724655 6 Steel MRF Parking Structure 31 305 NS 0.51 2.0 0.102

The damage collected for each building used the criteria established by ATC-13

(ATC, 1985). Overall damage for the entire building was reported using the following

damage states: none, insignificant, moderate and heavy. These damage states are defined in

Table 5.2. In addition to the reporting the building’s overall damage, damage to specific


components of the structure was evaluated. Damage experienced by the buildings’

structural components, its nonstructural components, its equipment and its contents were

also documented. Seven damage states, defined in Table 5.3, for each sub-category was

used to measure the performance of each component group. The economic loss associated

with each damage state was expressed as a percentage of the replacement cost of the group

of components.

Table 5.2 General Damage Classifications (ATC-13, 1985)

Code DescriptionN None . No damage is visible, either structural or non-structural

IInsignificant. Damage requires no more than cosmetic repair. No structural repairs are necessary. For non-structural elements this would include spackling partition cracks, picking up spilled contents, putting back fallen ceiling tiles, and righting equipement.

MModerate. Repairable structural damage has occurred. The existing elements can be repaired in place, without substantial demolition or replacement or elements. For non-structural elements this would include minor replacement of damaged partitions, ceilings, contents or equipment.

HHeavy. Damage is so extensive that repair of elements is either not feasible or requires major demolition or replacement. For non-structural elements this would include major or complete replacement of damaged partitions, ceilings, contents or equipment.

Table 5.3 ATC-13 Damages States (ATC, 1985)

State Description Percent Damage1 None 0%2 Slight 0% - 1%3 Light 1% - 10%4 Moderate 10% - 30%5 Heavy 30% - 60%6 Major 60% - 100%7 Destroyed 100%

5.3.2 Buildings surveyed in the ATC-38 Report

ATC-38 was conducted to collect data that would improve motion-damage

relationships for earthquake damage and loss modeling. During the days after the 1994

Northridge earthquake the Applied Technology Council (ATC), the United States

Geological Survey (USGS) and other Northern California organizations concerned with


earthquake engineering systematically documented building performance of structures

located within 300 meters (~1000 feet) of strong ground motion recording stations. 530

buildings near 31 recording stations were surveyed during the study. Eighteen of the

stations are operated by the California Division of Mines and Geology (CDMG), 7 are

operated by the University of Southern California (USC), and 6 are operated by USGS.

Digitized strong motion recordings were collected.

Standardized survey forms were used to evaluate the buildings and collect key

information. The buildings were categorized by their occupancy type as reported in Table

5.4. Photographs were taken to document the size, shape and visible damage. The survey

documented important structural characteristics for each building such as its design date,

predominant structural framing type (as defined by ATC, see Table 5.5), occupancy type

and number stories. The building’s nonstructural characteristics, equipment and contents

were also recorded. Building performance was evaluated by recording the degree of damage

experienced by the structural system, nonstructural components, equipment and contents.

Damage was measured using the same criteria established by ATC-13 and used for the

instrumented buildings described in Section 5.3.1.

Table 5.4 Occupancy Types and Codes (ATC-38)

Occupancy Type Refence CodeApartment AAuto Repair ARChurch CDwelling DData Center DCGarage GGas Station GSGovernment GVHospital HHotel HLManufacturing MOffice ORestaurant RRetail RSSchool STheater TUtility UWarehouse WOther OTHUnknown UNK


Table 5.5 Model Building Types (ATC-38)

Framing SystemSteel Moment Frame S1 - Stiff Diaphragms S1A - Flexible DiaphragmsSteel Braced Frame S2 - Stiff Diaphragms S2A - Flexible DiaphragmsSteel Light FrameSteel Frame w/ Concrete Shear Walls S4 - Stiff Diaphragms S4A - Flexible DiaphragmsSteel Frame w/ Infill Masonry Shear Walls S5 - Stiff Diaphragms S5A - Flexible DiaphragmsConcrete Moment Frame C1 - Stiff Diaphragms C1A - Flexible DiaphragmsConcrete Shear Wall Building C2 - Stiff Diaphragms C2A - Flexible DiaphragmsConcrete Frame w/ Infill Masonry Shear Walls C3 - Stiff Diaphragms C3A - Flexible DiaphragmsReinforced Masonry Bearing Wall RM1 - Flexible Diaphragms RM2 - Stiff DiaphragmsUnreinforced Masonry Bearing Wall URM - Flexible Diaphragms URMA - Stiff DiaphragmsPrecast/Tiltup Concrete Shear Walls PC1 - Flexible Diaphragms PC1A - Stiff DiaphragmsPrecast Concrete Frame w/ Concrete Shear WallsWood Light FrameCommercial or Long-Span Wood Frame W2

Reference Codes and Diaphragm Types

PC2

S3

W1

5.4 DATA FROM INSTRUMENTED BUILDINGS

5.4.1 Structural response simulation

The response parameters being considered in this study to create fragility functions

are listed and defined as the following:

Peak Building Acceleration (PBA): The maximum acceleration

experienced by the building at any floor during the earthquake.

Peak Interstory Drift (IDR): The maximum interstory drift experienced

by the building at any story during the earthquake

These response parameters are also often referred to as engineering demand

parameters (EDPs) under the terminology established by PEER for their performance-based

earthquake engineering framework (Krawinkler and Miranda, 2004). Calculating these

parameters requires knowing what the maximum displacements and accelerations are for

every floor of each building. The sensors for each CSMIP building were not located at

every floor. Therefore, approximate methods of structural analysis were used to evaluate the

building’s response at the intermediate floors that did not have any motion recordings.

Taghavi and Miranda (2005) have shown that in many cases it is possible to obtain

a relatively good approximation of the response of buildings subjected to earthquake ground

motions. In their model, the building is replaced by a continuous system consisting of a

shear beam laterally connected to a flexural beam by axially ridge struts (Figure 5.1). The

continuous system’s primary advantage is that it requires only three parameters to calculate


the dynamic properties of the structure: the building’s fundamental period of vibration, its

damping ratio and a non-dimensional parameter, 0, which controls the degree of flexural

or shear deformation. Closed-form solutions solving the dynamic equation of motion for the

continuous system were derived that compute the considered building’s mode shapes, its

corresponding modal participation factors and period ratio. Once the considered structure’s

dynamic characteristics have been determined, they can be used in combination with

traditional time-integration schemes to evaluate the structural response when subjected to

seismic ground accelerations.

H

Shear beam

Flexural beam

Axially-rigid links

H

Shear beam

Flexural beam

Axially-rigid links

Figure 5.1 Continuous Model used to evaluate structural response

The distribution of the building mass and stiffness was assumed to be uniform along

the height of the structure. Although making this assumption may seem restrictive, Miranda

and Taghavi (2005) have shown that, provided that there are no large sudden changes in

mass or stiffness along the height, this model leads to reasonable approximations of the

dynamic characteristics of many types of buildings. Any deviation in response that was

produced by nonuniform mass or stiffness, was small enough to neglect or could be

accounted for by using approximate equations.

The continuous model has been shown to have produced similar structural

responses for structures responding elastically that were predicted by more rigorous models

that required greater computational effort. Furthermore, by using the building’s


fundamental period, damping ratio and nondimensional parameter 0, inferred using system

identification techniques, the model was able to produce results that showed good

agreement with the structural response data recorded by the instrumented floors. A

representative example is shown in Figure 5.2. The last three columns of Table 5.1 list the

inferred parameters for each building.

Figure 5.2 Example of Simulated Structural Response compared to Recorded Response


Detailed summary sheets comparing how well each of the 19 instrumented

buildings’ simulated response matched its recorded response were complied. The computed

peak floor acceleration, the floor acceleration spectra, the floor acceleration and

displacement response histories are compared to those recorded by the buildings’ sensors.

An example of the type of data reported is shown in Figure 5.3.


Figure 5.3 CSMIP Building Response Comparison Summary Sheet Layout


Figure 5.3 CSMIP Building Response Comparison Summary Sheet Layout (cont.)


5.4.2 Motion-damage pairs for each building

Summary sheets were assembled to consolidate all the structural response and

damage information gathered for each of the 19 instrumented CSMIP buildings included in

this study. Figure 5.4 illustrates an example of the type of data generated for each building.

The example building shown is a 17 story shear wall residential building. General building

characteristics (e.g. number of stories, type of lateral resisting system, occupancy

type…etc) and a summary of the reported damage are documented. A map showing the

locations of the sensors in each building and plots of peak response parameters, similar to

those shown in Figure 5.2, are also reported in the summary sheets.


Figure 5.4 CSMIP Building Summary Sheet Layout


Figure 5.4 CSMIP Building Summary Sheet Layout (cont.)


5.5 DATA FROM ATC-38

5.5.1 Structural response simulation

Unlike the CSMIP buildings in the previous section, the actual structural response

was not recorded in buildings surveyed by ATC-38 and therefore is not known. The ground

motion accelerations for each specific building is also not known. However, because the

buildings in the ATC-38 report are located in close proximity to ground motion recording

stations, the ground motion accelerations for these structures are assumed to be the same as

those recorded at the nearby stations. By making this assumption, the ground motion

recordings were used to estimate the probable structural response during the 1994

Northridge earthquake by using simplified structural analyses of all 500 buildings.

Most of the buildings surveyed in ATC-38 were 5 stories or less. For low-rise

buildings the continuous model, used to model the CSMIP building, does not do as well in

simulating response parameters. This is because the distribution of mass along the height of

a building is much more discrete in low-rise buildings, than it is in taller structures. Instead

of using the continuous model described in Section 5.4.1, a more traditional discrete, linear

model was used to simulate the structural response. The discrete model consisted of a two

dimensional linear one-bay frame with lumped mass at the floor heights. Assuming a

linearly elastic model to estimate the response parameters of these buildings in this study

was deemed reasonable because the ground motion intensities observed during this

earthquake were, for the most part, not large enough to induce inelastic behavior in the

majority of these structures. The model was assumed to have uniform mass and uniform

stiffness throughout the height of the structure. Like the continuous model, this approach

uses the same three parameters (the building’s fundamental period, its damping ratio and

the nondimensional parameter, 0) to calculate the discrete model’s dynamic properties.

The dynamic properties were calculated by assembling a stiffness and mass matrix for a

uniform, one bay frame and solving the resulting eigenvalue problem. Once the dynamic

properties were established, the model was subjected to the recorded ground motion from

the nearby recording station. The response was calculated by using Newmark’s time-

integration algorithm in combination with modal superposition to solve the equations of

motion for the multi-degree-of-freedom problem.


The fundamental period, damping ratio and nondimensional parameter 0 for each

building reported in ATC-38 is not known because the buildings were not instrumented.

Therefore, instead of computing only one solution through a deterministic approach, a

probabilistic method of estimating each building’s structure properties and corresponding

response was used to account for modeling uncertainty. Each building’s fundamental

period, damping ratio and parameter 0 were treated as independent random variables that

were lognormally distributed. A lognormal distribution was assumed because realizations of

this distribution can not be less than or equal to zero and some studies have shown to be

appropriate for the fundamental period and damping ratios.

The median and dispersion of each random variable were estimated using formulas

determined for twelve general model building types, shown in Table 5.6. After defining

these probability distributions, Monte Carlo methods were used to generate 200 different

realizations with combinations of the random variables. Each combination of building

parameters was used together with an assumed interstory height (also found in Table 5.6) as

input to define the discrete model. A time history analysis was conducted using the nearby

recorded ground motion to simulate the building’s response in terms of the parameters

defined in Section 5.4.1. Statistical analysis was then conducted on the results of the 200

simulations to establish the median (50th percentile) and dispersion (15th and 85th

percentiles) of the simulations. The response was computed for each directional component

of the recorded ground acceleration producing response results in both component

directions. In order to find a numerical average for the predicted motions, the geometric

mean was calculated from the two component directions. Figure 5.5 displays an example of

the results for the simulated structural response of one of the ATC-38 buildings. Each graph

plots the three response parameters along the height of the building and displays the results

of all 200 simulations, with the 15th, 50th, and 85th percentiles highlighted


Table 5.6 Formulas used for Estimating Structural Building Parameters

Predominant MBT Typical

Interstory

Height

[ft]

Period Alpha Damping Ratio

Code Description

Median,

1T

Dispersion

Parameter,

1T

Median, Dispersion

Parameter, Median, Dispersion

Parameter,

S1,

S1A

Steel Moment Resisting

Frame 13.75 0.035H 0.805 0.3 25 0.2 0.1057NS -0.565 0.40

S2,

S2ASteel Brace Frame 13.75 0.017H 0.9 0.3 6 0.2 0.03 0.35

S3 Steel Light Frame 13.0 0.038H 0.8 0.3 20 0.2 0.1057NS -0.565 0.35

S4,

S4A

Steel Frame w/

Concrete Shear Walls 13.75 0.017H 0.9 0.3 10 0.2 0.03 0.35

S5,

S5A

Steel Frame w/ Infill

Masonry Shear Wall 13.75 0.023H 0.85 0.3 18 0.2 0.04 0.35

RM1,

RM2,

URM,

URMA

Masonry Buildings 12.0 0.017H 0.3 5 0.2 0.278NS -0.701 0.20

C1,

C1A

RC Moment Resisting

Frame

9.0 – res., hotel

13.8 – other 0.017H 0.92 0.3 25 0.2 0.0889NS -0.235 0.20

C2,

C2A

Concrete Shear Wall 12.45 0.0069H 0.3 3 0.2 0.0889NS -0.235 0.30

C3,

C3A

Concrete Frame w/

Infill Masonry Shear

Wall

9.0 – res., hotel

13.8 – other 0.015H 0.9 0.3 18 0.2 0.09NS -0.24 0.25

PC1,

PC1A

Precast/Tiltup Concrete

Shear Wall 16.0 0.007H 0.3 3 0.2 0.06 0.30

W1 Wood Light Frame

Buildings 10.0 0.032H 0.55 0.3 7 0.2 0.077 0.40

W2

Commercial Wood

Frame Buildings

(Longspan)

13.4 0.032H 0.55 0.3 15 0.2 0.077 0.40

NOTE: H = height of building [ft] ( = typical interstory height [ft] * number of stories )

NS = number of stories above ground


Building Response when Subjected to USGS080 - Comp 270

0 1 21

2

3

4

Peak Displacement [cm]0 1 2 3

x 10-3

1

2

3

Peak IDR0 500 1000

1

2

3

4

Peak Floor Acceleration [cm/s.2]

Building Response when Subjected to USGS080 - Comp 360

0 0.5 1 1.51

2

3

4

Peak Displacement [cm]0 1 2

x 10-3

1

2

3

Peak IDR0 500 1000

1

2

3

4

Peak Floor Acceleration [cm/s.2]

Floo

r

Figure 5.5 Example of Results from Simulated Structural Response.

5.5.2 Motion-damage pairs for each building

The following summary sheets were assembled to consolidate all the structural

response and damage information gathered for each of the buildings from the ATC-38

report included in this study. Figure 5.6 illustrates how the information on each building

summary is laid out. General building characteristics and the assumed median and

dispersions of the building’s structural properties were reported. Figures showing peak

response parameters along the height of the building and tables summarizing the peak


response values, for each direction of ground motion component and the geometric mean of

the two components, are also documented. Lastly, a summary of the reported damage is

tabulated for both general damage and nonstructural damage of specific components.


Figure 5.6 ATC-38 Building Summary Sheet Layout


Figure 5.6 ATC-38 Building Summary Sheet Layout (cont.)


5.6 FRAGILITY FUNCTIONS FORMULATION

5.6.1 Procedures to compute fragility functions

The values of engineering demand parameters (EDPs) at which the structures

exceed particular damage states can significantly vary from building to building. This

variability can be accounted for using cumulative distribution functions (cdf) to

approximate the likelihood of each damage state occurring. These functions, termed

fragility functions, approximate the probability that building components will experience or

exceed a particular damage state given its structural response (expressed as one of the two

EDPs defined in Section 5.4.1). The motion-damage pairs were separated into the different

types of damage reported from the CSMIP and ATC-38 reports. For each type of damage

(e.g. general damage, structural damage, nonstructural damage…etc.), cumulative

frequency distribution functions were developed for each damage state that was observed in

the dataset.

The probability of experiencing or exceeding a particular damage state conditioned

on a particular value of EDP, ( )jP DS ds EDP edp , is modeled using a lognormal

probability distribution, ( )F edp , given by the following equation:

( ) ( )( ) ( )jLnEDP

Ln edp Ln EDPF edp P DS ds EDP edp (5.1)

where P(DS dsj EDP = EDP) is the probability of experiencing or exceeding damage

state j, EDP is the median of the EDPs at which damage state j was observed, LnEDP is the

standard deviation of the natural logarithm of the EDPs, and is the cumulative standard

normal distribution (Gaussian distribution).

A lognormal distribution is chosen because it has been shown that it fits damage

data well for both structural components and nonstructural components (Porter and

Kiremidjian 2001, Aslani and Miranda 2005, Pagni and Lowes 2006). Theoretically, the

lognormal distribution is ideal because it equals zero probability for values of EDP that are


less than or equal to zero. The lognormal distribution also can be completely defined by two

parameters: the median EDP ,and the lognormal standard deviation, LnEDP. Three different

methods were used to determine the statistical parameters of the lognormal distribution for

the fragility functions produced in this investigation: (1) the least squares method, (2) the

maximum likelihood method, and (3) the second method (“Method B”) for bounding EDPs

as proposed by Porter et al. (2007).

5.6.1.1 Least squares method

The least squares method is a common statistical approach that attempts to fit

observed data to the values produced by a predicting function. This is accomplished by

minimizing the sum of the square of the differences between the observed data and the

values, g edp , predicted by the proposed function, F edp . Mathematically, this can be

expressed as:

2

11

,..., minN

N i j ii

g edp edp F edp DS edp (5.2)

where N is the number of data points, EDPi is the peak EDP observed for data point i, and

DSj(EDPi) indicates whether damage state j has been exceeded by taking on a binary value

of 1 when the damage state has been exceeded and 0 when the damage state has not

occurred. Figure 5.7 illustrates this procedure for damage state DS2 of drift-sensitive

nonstructural components based on the CSMIP data. The parameters EDP and LnEDP are

varied until the sum of the distances i j iF edp DS edp is minimized. The “Solver”

function in MS EXCEL was used to vary EDP and LnEDP, until a minimum value of

g edp was found.


0.00

0.20

0.40

0.60

0.80

1.00

0 0.002 0.004 0.006 0.008 0.01

IDR

Predicted Value

Observed value

P(DS2 | IDR)

i j iF edp DS edp

0.00

0.20

0.40

0.60

0.80

1.00

0 0.002 0.004 0.006 0.008 0.01

IDR

Predicted Value

Observed value

P(DS2 | IDR)

i j iF edp DS edp

Figure 5.7 Difference between observed values and values predicted by a lognormal distribution for damage state DS2 of drift-sensitive nonstructural components based on data from CSMIP.

5.6.1.2 Maximum likelihood method

In the method of maximum likelihood (Rice, 2007), it is assumed that each

realization dsi of the random variable DS is a sampled outcome of separate random

variables DSi (i.e. instead of regarding ds1, ds2,…, dsN as N realizations of the random

variable DS, each outcome dsi is a realization of DSi). The joint probability density function

(PDF) conditioned on the parameters of the lognormal distribution, g(ds1, ds2,…, dsN |

EDP , LnEDP), is defined as the likelihood function such that:

1 2, , ,..., ,LnEDP n LnEDPL EDP g ds ds ds EDP (5.3)

where L( ) is the likelihood operator, the joint density is a function of EDP and LnEDP

rather than a function of dsi. If DSi are assumed to be identically distributed and

independent random variables, then their PDF is the product of the marginal densities such

that equation (5.3) becomes


, ,n

LnEDP i LnEDPi

L EDP g ds EDP (5.4)

The values of EDP and LnEDP that maximizes the likelihood function – that is, the values

that make the observing the damage state DS “most probable” – are the values that are

selected to define the parameters of fragility functions.

Since our damage states are discrete and binary (i.e. either the damage state has

occurred or not occurred), it is assumed that each DSi observation is an ordinary Bernoulli

random variable, where:

1

0 1i i i

i i i

P DS EDP edp F edp

P DS EDP edp F edp

such that its probability distribution can be represented as:

1, 1 ii DSDS

i LnEDP i ig ds EDP F edp F edp (5.5)

Substituting equation (5.5) into (5.4), our likelihood function becomes:

1, 1 ii

n DSDSLnEDP i i

i

L EDP F edp F edp (5.6)

where F(EDPi) is the lognormal cdf as defined in equation (5.1) defined by the parameters

EDP and LnEDP. As was the case when using the least squares method, the MS EXCEL

solver tool was used to find the maximum value of equation (5.6) by varying the two

parameters of the lognormal distribution.


5.6.1.3 Bounding EDPs method (Porter et al. 2001, Method B)

The bounding EDPs method determines the probability of a damage state occurring

from observed data, by dividing the data set into discrete bins based on equal increments of

EDPs as shown in Figure 5.8. For each subset of data in every bin, the probability of

damage state DS occurring is calculated in each bin, according to:

1 imP DS EDP edpM

(5.7)

where m is number of data points that experienced this level of damage and M is the total

number of data points within the bin being considered. These probabilities are then plotted

at the midpoints of the EDP ranges in each bin as shown by the hollow points in Figure 5.8.

A lognormal distribution is then fitted to these points by varying the parameters EDP and

LnEDP. The interested reader is directed to Porter et al. (2007) for more information

regarding this procedure for developing fragility functions.

0.00

0.20

0.40

0.60

0.80

1.00

0 500 1000 1500 2000

PBA [cm/s2]

Observed DataRatio of DS=1 per BinFitted function

P(DS2 | PBA)

Bin Size

0.00

0.20

0.40

0.60

0.80

1.00

0 500 1000 1500 2000

PBA [cm/s2]

Observed DataRatio of DS=1 per BinFitted function

P(DS2 | PBA)

Bin Size

Figure 5.8 Developing fragility functions using the bounding EDPs method.


5.6.2 Limitations of fragility function procedures

How robust the methods were in finding reliable parameters was dependent on the

data available. The three different methods were used to formulate fragility functions

because each of the methods had limitations in their ability to find reliable solutions for the

parameters of the lognormal distribution given the available data. In cases where one

method was not able to find a solution that was reliable, the other methods were used to

determine the values of EDP and LnEDP. Having several methods was necessary to

confirm that estimates of EDP and LnEDP obtained were fairly accurate based on the

available data extracted from the CSMIP and ATC-38 buildings.

There were situations where the available data made finding unique solutions for

EDP and LnEDP using the least squares and the maximum likelihood methods impossible.

For instance, finding unique solutions was impossible when the range of EDPs for the

buildings that experienced damage state DSj did not overlap with the range of EDPs for the

buildings that did not experience this level of damage. This situation is illustrated Figure

5.9(a) which shows the fragility function for DS5 of acceleration-sensitive nonstructural

components derived from the CSMIP data. The range of EDPs that do not experience

damage ends at a PBA value of 1080 cm/s2, while the range of EDPs that experience DS5

begins at a PBA of 1550 cm/s2. For a given value of EDP , there can be multiple values of

LnEDP that will yield a fitted lognormal distribution that passes through all the data points.

This is shown in Figure 5.9(a) for an assumed EDP value of 1,315 cm/s2 (the midpoint

between the bounding data points with PBA values of 1,080 and 1,550 cm/s2) where the

range of possible solutions for the fitted functions is represented in the bounded area with

the diagonally striped hatching. Similarly, for a very small value of LnEDP, such that the

fitted function is almost vertical, EDP can take on any value between 1,080 and 1,550

cm/s2 and still yield a lognormal distribution that pass through all the data points as shown

in Figure 5.9(a). Under these circumstances, it was decided that EDP would be taken to be

the midpoint of the bounding EDP data points. The associate dispersion, LnEDP, would be

chosen as the largest value that would still produce a fitted function that passes through all

the data points.


0.00

0.20

0.40

0.60

0.80

1.00

0 500 1000 1500 2000

PBA [cm/s2]

P(DS5 | PBA)

Area in which possible fitted

functions can vary within

Range of possible

values of EDP

(a)

0.00

0.20

0.40

0.60

0.80

1.00

0 500 1000 1500 2000

PBA [cm/s2]

P(DS5 | PBA)

Area in which possible fitted

functions can vary within

Range of possible

values of EDPRange of possible

values of EDP

(a)

0.00

0.20

0.40

0.60

0.80

1.00

0 500 1000 1500 2000

PBA [cm/s2]

Observed Data

4 bins

3 bins

P(DS4 | PBA) (b)

0.00

0.20

0.40

0.60

0.80

1.00

0 500 1000 1500 2000

PBA [cm/s2]

Observed Data

4 bins

3 bins

P(DS4 | PBA) (b)

Figure 5.9 Limitations of finding unique solutions for fragility function parameters (a) multiple solutions for least squares and maximum likelikhood methods (b) multiple solutions for bounded

EDPs method.

The bounded EDPs method can also produce multiple solutions for estimated

fragility function parameters. The plotting position of the points that are used to fit the

lognormal distribution are highly dependent on number of data points and the distribution

of those points along the range of EDPs. Porter et al. (2007) suggests that this method

works best for data sets containing greater than 25 data points. The number of data points

and their distribution are important because the size and number of bins can be chosen

subjectively and consequently can change the resulting fitted functions. Figure 5.9(b) shows

the damage state DS4 for acceleration-sensitive nonstructural components that plots data

from the CSMIP buildings. The same data was used to derive fragility functions using 4

bins (solid line) and 3 bins (dashed line), yielding two very different probability

distributions. Typically, when confronted with different functions produced by selecting

different bin sizes, the function chosen was the one that was most similar to those produced

by the other methods.

For the most part, good agreement was shown between the three methods as shown in

Figure 5.10(a) for the DS2 damage state of drift-sensitive nonstructural components. In

instances where two of the methods showed good agreement and one did not, the

parameters from the methods that displayed closer results were used to define the fragility

function. An example of this is shown in Figure 5.10(b) for the DS 2 damage state of drift-


sensitive structural components. The maximum likelihood and least squares methods

typically produced similar results because their solution algorithms are very similar,

whereas the results from the bounded EDPs method were dependent on the number bins

used. In cases where none of the methods showed any agreement, the one of functions

produced from either the least squares or the maximum likelihood was chosen because

these methods do not introduce the same level of subjectivity as the bounded EDP method

(which was shown to be highly depended on the bin size). The choice between the fragility

produced by least squares and the fragility derived from maximum likelihood was based on

which method yielded a definite unique solution or which function made more engineering

sense based on previous data from other fragility functions previously derived from

experimental data.

0.00

0.20

0.40

0.60

0.80

1.00

0 0.005 0.01 0.015 0.02

IDR

Least Squares

Max Likelihood

Porter Method B

P(DS2 | IDR)

0.00

0.20

0.40

0.60

0.80

1.00

0 0.005 0.01 0.015 0.02

IDR

Least Squares

Max Likelihood

Porter Method B

P(DS2 | IDR)(a) Nonstructural components

(b) Structural components

0.00

0.20

0.40

0.60

0.80

1.00

0 0.005 0.01 0.015 0.02

IDR

Least Squares

Max Likelihood

Porter Method B

P(DS2 | IDR)

0.00

0.20

0.40

0.60

0.80

1.00

0 0.005 0.01 0.015 0.02

IDR

Least Squares

Max Likelihood

Porter Method B

P(DS2 | IDR)(a) Nonstructural components

(b) Structural components

Figure 5.10 Sample comparisons of different methods to formulate fragility functions (a) example of all three methods agreeing (b) example of 2 out of 3 methods agreeing.

5.6.3 Adjustments to fragility function parameters

Once the parameters for the fragility functions were established using the

procedures presented in the previous sections the results were examined to see if the

resulting distributions were reasonable. Although the fragilities were based on actual

empirical data from earthquake reconnaissance, the motion-damage pairs have limitations


that may produce results that become problematic when estimating damage. These

limitations include the following:

The motion-damage pairs generated from the ATC-38 buildings are based

on probabilistic response simulation results and not based on the actual

structural response.

Although the motion-damage pairs from the CSMIP buildings are derived

from recorded response data, the sample size of this set is relatively small

(19 data points)

The both sets of data have limited information on the more severe damage

states because only a very limited number of buildings suffered these high

levels of damage.

Both sets of data rely on subjective interpretations of damage states, by

the engineers who assessed the damage to each building.

Given these limitations, some of the resulting fragility functions, particularly for

certain damages states where data is scarce, needed to be adjusted after their

parameters have been computed. Functions were adjusted based on the level

confidence in how well the resulting probability distribution represented the actual

behavior. The level of confidence in the distributions computed was highly dependent

on both the total number of data points that were used to generate the functions and

the number of points that exceeded a particular damage state.

Figure 5.11 shows an example set of fragility functions that illustrate the

types of adjustments that were made to the parameters EDP and LnEDP. This set of

functions is for acceleration-sensitive nonstructural components based on data from

the CSMIP buildings. The example functions computed directly from the data using

the procedures described in the preceding sections are shown in Figure 5.11(a) and

their corresponding parameters are listed in Table 5.7. It can be observed that for

large accelerations (>1,300 cm/s2) that these functions estimate that the probability of

the damage state DS5 (Heavy damage) occurring, is higher than the probability of

DS3 (Light damage) or DS4 (Moderate damage), which, by definition of the damage

states, is impossible and problematic when estimating economic losses. Examining

the data used to compute the fragility for DS5 closer (as illustrated in Figure 5.9)

reveals that this function was based only one building experiencing damage that


exceeded this damage state. Consequently, no unique solution exists to define this

function using the least squares or maximum likelihood method based on this dataset

(as described in detail previously in 5.6.2). Since this function was formulated based

on only one data point, the level of confidence in this probability distribution

representing the actual behavior is not high and adjustments to its parameters is

required to obtain more realistic loss estimation results. Similar observations can be

made about the functions for DS3 and DS4 which are only based on four data points

and two data points experiencing or exceeding this damage state, respectively.

0.00

0.20

0.40

0.60

0.80

1.00

0 500 1000 1500 2000

PBA [cm/s2]

P(DS | PBA)

0.00

0.20

0.40

0.60

0.80

1.00

0 500 1000 1500 2000

PBA [cm/s2]

DS 2: Slight

DS 3: Light

DS 4: Moderate

DS 5: Heavy

P(DS | PBA)(a) (b)

0.00

0.20

0.40

0.60

0.80

1.00

0 500 1000 1500 2000

PBA [cm/s2]

P(DS | PBA)

0.00

0.20

0.40

0.60

0.80

1.00

0 500 1000 1500 2000

PBA [cm/s2]

DS 2: Slight

DS 3: Light

DS 4: Moderate

DS 5: Heavy

P(DS | PBA)(a) (b)

Figure 5.11 (a) Sample fragility functions computed from data for accleration nonstructral components (from CSMIP) (b) Sample functions after adjustments.

Table 5.7 Parameters for sample fragility functions computed directly and with adjustments from

data for accleration nonstructral components (from CSMIP).

DS2 387 0.52 387 0.52DS3 995 0.80 995 0.50DS4 1202 0.42 1202 0.36DS5 1300 0.07 1300 0.30

Damage State

Nonstructural | PBA (CSMIP)Unadjusted

Geometric Mean

LN Standard Deviation

Geometric Mean


Adjusted


The adjusted fragility functions are shown in Figure 5.11(b) and their corresponding

parameters are shown in the third and fourth columns of Table 5.7. Only minor adjustments

were made to the lognormal standard deviations of DS3 and DS5 because the confidence in

these results was not that high based on the limited number of data points that experienced

or exceeded these damage states. The lognormal standard deviation of DS3 was decreased

from 0.80 to 0.50. It has been observed from previous studies (Aslani and Miranda, 2005)

on fragility functions derived from experimental data that lognormal standard deviations for

more severe damage states tend to be less than or equal to the dispersion values of the

damage states that preceded them. In this example, the lognormal value of DS3 was

adjusted to 0.50 because the lognormal standard deviation of the preceding damage state,

DS2, is 0.52. Increasing the lognormal standard deviation of DS5 from 0.07 to 0.3 was

rationalized by noting that the value of LnEDP for this function was dictated by only one

data point as shown in Figure 5.9(a) at a PBA of 1550 cm/s2. This building was the only one

of the CSMIP structures that exceeded DS5, and for the fitted function to pass through this

point, a very small value of LnEDP was estimated for this distribution. Because the

parameters of this function are not based on several data points, it is not as reliable as other

damage states which have more observations that indicate damage was sustained. Therefore

the dispersion was increased to a more realistic value. This type of rationale was used to

make similar adjustments to the other fragility functions computed in this study.

5.7 FRAGILITY FUNCTION RESULTS

Three types of fragility functions were developed using the methods described in

the previous sections. Functions that estimate the probability of experiencing structural

damage conditioned on peak IDR, the probability of experiencing nonstructural damage

conditioned on peak IDR, and the probability of experiencing nonstructural damage

conditioned on PBA were produced. Figure 5.12 shows the functions for the three types of

fragilities that were computed from the CSMIP data. The corresponding statistical

parameters for these functions are reported in Table 5.8.


0.00

0.20

0.40

0.60

0.80

1.00

0 0.005 0.01 0.015 0.02

IDR

P(DS | IDR) (a)

0.00

0.20

0.40

0.60

0.80

1.00

0 0.005 0.01 0.015 0.02

IDR

P(DS | IDR) (a)

0.00

0.20

0.40

0.60

0.80

1.00

0 0.005 0.01 0.015 0.02

IDR

P(DS | IDR)

0.00

0.20

0.40

0.60

0.80

1.00

0 500 1000 1500 2000

PBA [cm/s2]

DS 2: Slight

DS 3: Light

DS 4: Moderate

DS 5: Heavy

P(DS | PBA)(b) (c)

0.00

0.20

0.40

0.60

0.80

1.00

0 0.005 0.01 0.015 0.02

IDR

P(DS | IDR)

0.00

0.20

0.40

0.60

0.80

1.00

0 500 1000 1500 2000

PBA [cm/s2]

DS 2: Slight

DS 3: Light

DS 4: Moderate

DS 5: Heavy

P(DS | PBA)(b) (c)

Figure 5.12 CSMIP Fragility Functions for (a) Structural Damage vs. IDR (b) Nonstructural Damage vs. IDR and (c) Nonstructural vs. PBA.

The value of EDPs at which damage initiates is of particular interest because it can

play a large role in computing the value expected annual loss (EAL). Expected annual loss

is the average economic loss that is expected to accrue every year in the building being

considered. It is a function of the expected economic losses as a function of seismic

intensity and the mean annual frequency of seismic ground motion intensity. The frequency

of occurrence for small ground motion intensities (intensity levels at which damage

initiates) is very high and has been shown to significantly contribute to value of EAL

(Aslani and Miranda, 2005). Therefore, to estimate EAL accurately, it is important that the

function of the first damage state does a relatively good job in capturing when damage

initiates.


Table 5.8 Fragility Function Parameters generated from the CSMIP data.

Nonstructural | IDR Nonstructural | PBA [cm/s2]

DS2 0.003 0.32 387 0.52DS3 0.012 0.30 0.011 0.30 995 0.50DS4 0.015 0.30 0.016 0.30 1202 0.36DS5 1300 0.30


Structural | IDRCSMIP

Median LN Standard Deviation

DamageState

Median LN Standard DeviationMedian

The CSMIP functions for drift-sensitive structural components are shown in Figure

5.12(a). This figure does not include the function for the DS2 damage state for slight

damage because the level of damage associated with this damage state is very small.

Structural damage at this level is typically too small to warrant any repair actions and

therefore is excluded from the results presented here. The first damage state of consequence

is DS3 (light damage), which represents a damage associated with 1-10% of the

replacement cost, has a median of IDR of 0.012 and a lognormal standard deviation of 0.30.

This value is in the same range of other fragility functions for structural components that

were computed using experimental data (see Chapter 4, Pagni and Lowes 2006). To

compare values of structural response that initiates damage, damage initiation is assumed to

occur at the level of EDP that results in a probability of experiencing or exceeding the first

level of damage of consequence equal to 1%. Using this criteria, the resulting value of IDR

at which damage initiates of structural components occurs at 0.006.

Figure 5.12(a) and Figure 5.12(b) show the resulting fragility functions for drift-

sensitive and acceleration-sensitive nonstructural components, respectively. These functions

are of particular interest because nonstructural components make up a large portion of a

building’s value and consequently can play a large role in economic losses due to

earthquake ground motions (Taghavi and Miranda, 2003). The median IDR and lognormal

standard deviation for the first damage state (DS2: Slight) of drift-sensitive components are

0.0030 and 0.30, respectively. According the criteria assumed in the previous paragraph,

this function estimates that damage initiates at an IDR of approximately 0.0014. These

parameters are in the same range of as other component-specific fragility functions that

have been computed from experimental data. For instance, the median IDR for the first

damage state of partitions has been previously computed to be 0.0021 by the ATC-58

project (ATC, 2007).


For acceleration-sensitive components, the median PBA and lognormal standard

deviation of the first damage state are 387 cm/s2 (0.39g) and 0.52, respectively. For this

function, damage initiates at an acceleration equal to 116 cm/s2 (0.12g). The functions for

the first damage state for both drift and accelerations sensitive occur at much earlier EDP

values than the other damage states. This may be primarily due the fact that minor damage

due to cracking (e.g. cracking in partitions and facades) can occur very early while more

severe damage that requires more substantial repair actions of nonstructural elements will

tend to occur at much larger values of EDP.

The initial resulting fragility functions from the ATC-38 data were less realistic.

Many of the functions computed using this data produced probability distributions that had

very large lognormal standard deviations, ranging from 2.3 to 6.4. This produced functions

that did not clearly define where damage initiated or a distinct range of EDPs where the

damage state is exceeded. Even for the first damage states, where the sample size of data

points that experienced or exceeded the initial damage state was large enough to be

considered reliable, the functions were problematic because of the way the data points were

distributed. The data points were distributed such that there was no clear transition of EDP

values between buildings that experienced damage and the buildings that did not exceed

this level of damage. An example of a fragility function computed from data that produced

a large dispersion due to this type of limitation is illustrated in Figure 5.13

Figure 5.13 shows the fragility function of DS2 (Slight Damage) for acceleration-

sensitive nonstructural components. The data points that experienced or exceeded this

damage state are plotted on the top axis of the graph and the data points that did not

experience this damage state are plotted on the bottom axis. These data points do not show

a clear transition in EDP values between these two groups of data because of the way they

overlap. The range of values for data points that did not experience damage falls entirely

within the range of values for data points that did experience or exceed this damage state.

This makes it impossible to determine what range of EDPs where there is little to no

probability that damage will be observed, what range of EDPs where there is a very high

probability that damage will be observed and the range of EDPs that transitions between

these two extremes. The nature of this type of data distribution can most likely be attributed

to the subjective interpretations of the damage states definitions by the engineers that

collected the damage data. This results in damage being reported in an inconsistent manner.


Although the ATC-38 dataset did not yield useful results as a whole, subsets of this data

offer improved results.

0.00

0.20

0.40

0.60

0.80

1.00

0 500 1000 1500 2000

PBA [cm/s2]

P(DS2 | EDP)

Figure 5.13 Example of ATC-38 data showing limitations of data

The ATC-38 data is comprised of different types of structures as described in

section 5.3.2. The data was categorized by structural type (see Table 5.5) and these subsets

were used to create fragility functions to see if there were better relationships between

structural response and nonstructural damage. Fragilities for drift-sensitive and

acceleration-sensitive nonstructural components were developed using data from concrete

(C-1) and steel moment frame buildings (S-1). Figure 5.14 shows the fragility functions for

these types of structures for both types of components as follows: (a) drift-sensitive

components for concrete moment frames, (b) drift-sensitive components for steel moment

frames, (c) acceleration-sensitive components for concrete moment frames, (d)

acceleration-sensitive components for steel moment frames. The corresponding statistical

parameters for these functions are reported in Table 5.9.


0.00

0.20

0.40

0.60

0.80

1.00

0 0.005 0.01 0.015 0.02

IDR

P(DS | IDR) (a) C-1

0.00

0.20

0.40

0.60

0.80

1.00

0 0.005 0.01 0.015 0.02

IDR

DS2: Slight

DS3: Light

DS4: Moderate

P(DS | IDR) (b) S-1

0.00

0.20

0.40

0.60

0.80

1.00

0 500 1000 1500 2000

PBA [cm/s2]

P(DS | PBA) (c) C-1

0.00

0.20

0.40

0.60

0.80

1.00

0 500 1000 1500 2000

PBA [cm/s2]

P(DS | PBA) (d) S-1

Figure 5.14 Fragility functions using subsets of ATC-38 Data based on type of structural system (a) C-1: concrete moment frames – drift-sensitive (b) S-1: steel moment frames – drift-sensitive (c) C-1: concrete moment frames – acceleration-sensitive (d) S-1: steel moment frames – acceleration-

sensitive

Table 5.9 Fragility function statistical parameters for subsets of ATC-38 data

DS2 0.002 0.25 0.0026 0.38DS3 0.004 0.60 0.0050 0.40DS4 0.0080 0.40

DS2 200 0.40 200 0.70DS3 569 0.81 1000 0.73

Nonstructural | PBA [cm/s2]

Median LN Standard Deviation Median LN Standard

Deviation

S-1: Steel Moment Frame

DamageState

DamageState


C-1: Conc. Moment Frame


C-1: Conc. Moment FrameNonstructural | IDR

S-1: Steel Moment Frame


5.7.1 Comparison with generic functions from HAZUS

Generic fragility functions for nonstructural components have been used in

HAZUS, a US regional loss estimation methodology and computer program, to estimate

losses due to earthquake ground motions (NIBS, 1999). The data used to create these

functions has not been well documented. Where data is lacking, these functions are

sometimes generated by engineering judgment. The generic functions for nonstructural

components generated from the data documented in this study, can be used as a point of

comparison to either validate or update functions from previously implemented by HAZUS.

Figure 5.15 plots comparisons of generic HAZUS functions for drift and acceleration-

sensitive components with the first damage state of the functions calculated in this

investigation. Only the first damage states are plotted because these functions have the most

data to support their validity and therefore the most reliable.

Figure 5.15(a) compares HAZUS functions for drift-sensitive components to the

functions calculated using the CSMIP data. The CSMIP function indicates that damage

initiates at smaller values of IDR than the HAZUS functions predict. The median for the

first damage state of the HAZUS function (IDR = 0.004) is 28% larger than the one from

the CSMIP function. These results suggest that the HAZUS function for drift-sensitive

nonstructural components may lead to underestimations for drift-induced damage in

commercial buildings (which were primarily used to generate the CSMIP buildings). When

comparing the acceleration-sensitive functions in Figure 5.15(b), it can be observed that

there is a substantial difference between the HAZUS functions and the one produced by

empirical data. The median for the function developed from the CSMIP data is 58% greater

than the median of the first damage state of the HAZUS fragilities. These results suggest

that HAZUS functions may significantly overestimate earthquake-induced damage and

corresponding economic losses in acceleration-sensitive components.


0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.005 0.010 0.015 0.020IDR

P(DS | IDR)

HAZUS DS1:Slight

CSMIP DS2:Slight

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0PBA [g]

P(DS|PBA)

HAZUSDS1: Slight

CSMIPDS2:Slight

(b)(a)

0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.005 0.010 0.015 0.020IDR

P(DS | IDR)

HAZUS DS1:Slight

CSMIP DS2:Slight

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0PBA [g]

P(DS|PBA)

HAZUSDS1: Slight

CSMIPDS2:Slight

(b)(a)

Figure 5.15 Comparison to HAZUS generic fragility functions

5.8 CONCLUSIONS

The preceding study consolidated data from instrumented CSMIP buildings and

buildings documented in the ATC-38 report to create data points that related structural

response parameters to damage states, or motion-damage pairs. Each building in the study

had detailed information and consistent measurements of the amount of damage these

building experienced. Approximate structural analyses using multi-degree of freedom

models were used to simulate structural response and estimate the EDPs associated with the

observed damage. Although these models are approximate, they yield more accurate

estimates of response parameters, as compared to those previously computed using spectral

single-degree of freedom systems.

Summary sheets detailing each building’s structural characteristics, response

parameter results and a summary of damage experienced were created to form a motion-

damage database. The sheets report two primary engineering demand parameters: peak

building acceleration and peak interstory drift ratio. The building summaries also report

each structures general damage, structural damage, nonstructural damage, equipment

damage and contents damage. The ATC-38 buildings also include more detailed

nonstructural damage information on structures’ partitions, lighting and ceiling. Once these

motion-damage pairs were generated, they were used to create fragility functions that

estimate the probability of experiencing or exceeding discrete levels of damage conditioned

on EDP.


The relationships derived from the CSMIP were used to compute the EDP-DV

functions described in Chapter 3 of this dissertation. They were used as generic fragilities

that account for losses from components that previously did not have specific functions.

The EDP-DV functions were then included into the Story-based loss estimation toolbox

described in Chapter 6. Information that these generic fragility functions provide, give a

more complete picture of losses due to non-collapse and improves the accuracy of overall

loss predictions. Furthermore, they can be used to validate and update other generic

functions currently being used as demonstrated by the comparison with the fragilities from

HAZUS. Elevating the ability to accurately predict the amount of loss a building can expect

to experience during an earthquake will help stakeholders realize the value of investing in

more innovative performance-based structural systems.

Documents

BUILDING-SPECIFIC LOSS ESTIMATION METHODS(chp1-5).pdf