building-specific loss estimation methods & tools for simplified ...

Department of Civil and Environmental Engineering

Stanford University

Report No.

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

©2009 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 .......................................................................................................................... II

ACKNOWLEDGEMENTS .............................................................................................. IV

TABLE OF CONTENTS .................................................................................................... V

LIST OF TABLES ............................................................................................................. IX

LIST OF FIGURES ........................................................................................................... XI

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

1.1 MOTIVATION & BACKGROUND ................................................................................ 1 1.2 OBJECTIVES ............................................................................................................. 3 1.3 ORGANIZATION OF DISSERTATION ........................................................................... 4

2 PREVIOUS WORK ON LOSS ESTIMATION ........................................................ 8

2.1 LITERATURE REVIEW ............................................................................................... 8 2.2 REGIONAL LOSS ESTIMATION .................................................................................. 8 2.3 BUILDING-SPECIFIC LOSS ESTIMATION .................................................................. 10 2.4 LIMITATIONS OF PREVIOUS STUDIES ...................................................................... 14

3 STORY-BASED BUILDING-SPECIFIC LOSS ESTIMATION .......................... 17

3.1 INTRODUCTION ...................................................................................................... 17 3.2 STORY-BASED BUILDING-SPECIFC LOSS ESTIMATION ............................................ 20

3.2.1 Previous loss estimation methodology (component-based) ............................. 20 3.2.2 EDP-DV function formulation ......................................................................... 22

3.3 DATA FOR EDP-DV FUNCTIONS ........................................................................... 24 3.3.1 Building Components & Cost Distributions .................................................... 24 3.3.2 Fragility Functions Used .................................................................................. 28

3.4 EXAMPLE STORY EDP-DV FUNCTIONS ................................................................. 33 3.5 CONDITIONAL LOSS OF SPATIALLY INTERDEPENDENT COMPONENTS ..................... 40 3.6 DISCUSSION OF LIMITATIONS OF STORY-BASED APPROACH & EDP-DV FUNCTIONS

50 3.7 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 ...................................................................................................... 53 4.3 DAMAGE STATE DEFINITIONS ................................................................................ 56 4.4 EXPERIMENTAL RESULTS USED IN THIS STUDY ..................................................... 58 4.5 FRAGILITY FUNCTION FORMULATION .................................................................... 61

4.5.1 Fragility Functions for Yielding ....................................................................... 64 4.5.2 Fragility Functions for Fracture ....................................................................... 74

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

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

5.1 AUTHORSHIP OF CHAPTER ..................................................................................... 79 5.2 INTRODUCTION ...................................................................................................... 79 5.3 SOURCES OF EMPIRICAL DATA .............................................................................. 82

5.3.1 Instrumented Buildings (CSMIP) .................................................................... 82 5.3.2 Buildings surveyed in the ATC-38 Report ....................................................... 84

5.4 DATA FROM INSTRUMENTED BUILDINGS ............................................................... 86 5.4.1 Structural response simulation ......................................................................... 86 5.4.2 Motion-damage pairs for each building ........................................................... 92

5.5 DATA FROM ATC-38 ............................................................................................. 95 5.5.1 Structural response simulation ......................................................................... 95 5.5.2 Motion-damage pairs for each building ........................................................... 98

5.6 FRAGILITY FUNCTIONS FORMULATION ................................................................ 102 5.6.1 Procedures to compute fragility functions ..................................................... 102 5.6.2 Limitations of fragility function procedures .................................................. 107 5.6.3 Adjustments to fragility function parameters ................................................. 109

5.7 FRAGILITY FUNCTION RESULTS ........................................................................... 112 5.7.1 Comparison with generic functions from HAZUS ........................................ 118

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

6 DEVELOPMENT OF A STORY-BASED LOSS ESTIMATION TOOLBOX .. 121

6.1 PROGRAM STRUCTURE ........................................................................................ 121 6.2 GRAPHICAL USER INTER FACE ............................................................................. 124

6.2.1 Building Information & Characterization ...................................................... 124 6.2.2 EDP-DV Function Editor ............................................................................... 125 6.2.3 Main Window................................................................................................. 129 6.2.4 Hazard Module ............................................................................................... 130 6.2.5 Response simulation module .......................................................................... 132 6.2.6 EDP-DV Module ............................................................................................ 137 6.2.7 Loss Estimation Module ................................................................................ 139 6.2.8 Loss Disaggregation and Visualization Module ............................................ 140

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

7.1 AUTHORSHIP OF CHAPTER ................................................................................... 143 7.2 INTRODUCTION .................................................................................................... 144 7.3 LOSS ESTIMATION PROCEDURE ........................................................................... 146 7.4 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 .............................................................................................. 156 7.5.1 Expected losses conditioned on seismic intensity .......................................... 157 7.5.2 Expected Annual Losses ................................................................................ 163 7.5.3 Present value of life-cycle costs ..................................................................... 166 7.5.4 Comparison to Non-ductile Reinforced Concrete Frame Buildings .............. 168 7.5.5 Loss Toolbox Comparison ............................................................................. 170 7.5.6 Discussion of results relative to other loss estimation methodologies ........... 172

7.6 LIMITATIONS ....................................................................................................... 174 7.7 CONCLUSIONS ..................................................................................................... 175

8 VARIABILITY OF ECONOMIC LOSSES ........................................................... 178

8.1 AUTHORSHIP OF CHAPTER ................................................................................... 178 8.2 INTRODUCTION .................................................................................................... 178 8.3 TYPES OF LOSS VARIABILITY & CORRELATIONS ................................................. 180

8.3.1 Variability and Correlations in Construction Costs ....................................... 181 8.3.2 Variability and Correlation in Response Parameters ..................................... 189 8.3.3 Variability and Correlations in Damage Estimation ...................................... 198

8.4 VARIABILITY OF LOSS METHODOLOGY ............................................................... 200 8.4.1 Mean annual frequency of loss & loss dispersion condition on seismic intensity ...................................................................................................................... 200 8.4.2 Dispersion of loss conditioned on collapse .................................................... 201 8.4.3 Dispersion of loss conditioned on non-collapse ............................................. 202 8.4.4 Monte Carlo simulation method ..................................................................... 211 8.4.5 Evaluation of quality of FOSM approximations ............................................ 212

8.5 DISPERSIONS OF ECONOMIC LOSS FOR EXAMPLE 4-STORY BUILDING .................. 223 8.5.1 Variability of loss conditioned on non-collapse at the DBE .......................... 224 8.5.2 Variability of loss conditioned on non-collapse as a function of IM ............. 233 8.5.3 Variability of loss conditioned on collapse as a function of IM .................... 237 8.5.4 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 .................................................................................................... 246 9.2 METHODOLOGY ................................................................................................... 248 9.3 APPLICATIONS ..................................................................................................... 252

9.3.1 Description of Buildings Studied ................................................................... 252 9.3.2 Results ............................................................................................................ 255 9.3.3 Sensitivity of Loss to Changes in the Probability of Demolition ................... 264 9.3.4 Limitations of results & discussion of residual drift estimations ................... 268

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

10 SUMMARY AND CONCLUSIONS ....................................................................... 271

10.1 SUMMARY ........................................................................................................... 271 10.2 FINDINGS & CONCLUSIONS .................................................................................. 272

10.2.1 Story-based Loss Estimation ...................................................................... 272 10.2.2 Improved Fragilities in support of EDP-DV Function Formulation .......... 273

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10.2.3 Implementing loss estimation methods into computer tool ....................... 275 10.2.4 Benchmarking losses ................................................................................. 275 10.2.5 Improved estimates on the uncertainty of loss ........................................... 276 10.2.6 Accounting for Non-collapse losses due to building demolition ............... 278

10.3 FUTURE RESEARCH NEEDS .................................................................................. 279 10.3.1 Data collection for fragility functions and repair costs .............................. 280 10.3.2 Improvements to building-specific loss estimation methodology ............. 281

REFERENCES .................................................................................................................. 283

APPENDIX 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

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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, λ[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 Floor

A. SUBSTRUCTURE2.3% 0.0% 0.0% 0.0%

B. SHELLB10 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)

Building Distribution (% of total bldg value)

Component GroupStory 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)

DispersionExpected

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)

DispersionExpected

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)

DispersionExpected

Value

DS1Visible damage and small cracks in gypsum board that can be repaired with taping, pasting and painting

0.21 0.61 0.10

DS2

Extensive crack in gypsum board that can be repaired with replacing the gypsum board, taping, pasting and painting

0.69 0.40 0.60

DS3

Damage 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

DS2

Occurrence 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

DS1

Hanging 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

DS3

Ceiling 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 Floor

Typ Floor

Top 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 Floor

Typ Floor

Top 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-rise

Mid-rise

High-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-rise

Mid-rise

High-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)

Ductile

Non-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)

Perimeter

Space

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 load

High 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 0.020

IDR

E[L | IDR]

Independent

Dependent, w/ Double-counting

Dependent 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 0.030

IDR

Beam - DM1

Partition - DM1

Partition - DM2

Partition - 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


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.

ReferencesBeam 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)i

Pn

(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:

1

2lowerpn

(4.3)

1

12upperp

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 StateMedian IDR

[%]Geometric Mean IDR LnIDR

Number of Specimens

(Outliers Removed)

DS1: YieldingA36: 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 2.0

Interstory 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

0.0 0.5 1.0 1.5 2.0


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 12

Span-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 12

Span-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 ratioWeight

Typical Girder

No 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

0.0 0.5 1.0 1.5 2.0


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

A36 Specimens

A572 Specimens

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 0.47DS2:

FractureType of connection and d b

A36 & 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 PIDR

EI L

(4.8)

3

212b cL

cc

h d PLIDR

EI h

(4.9)

1 1b

PZs b

h d PLIDR

h 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 LIDR

L d

(4.11)

Then equations (4.8)-(4.10) can be re-written in terms of L/db as follows:

2

11

3 2yc

b bcL

FdIDR L d

L E

(4.12)

3

2212 c

yb bc bd

ccL

Fh d IIDR L d

I Eh L

(4.13)

2

22

2.6c

yb bPZ bd

c cb cL

Fh d IIDR L d

t 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

A572 Specimens

Beam PropertiesCoupon Yielding Stress, F y [Mpa] 310 352

Beam Depth, d b [cm] 54 66

Moment 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 44

Web Thickness, t c [cm] 2.4 2.2

Moment 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

IDR

IDR (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 3.0

Correction 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 3.0

Correction 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 3.0

IDR [%]

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

0.0 1.0 2.0 3.0 4.0


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

0.0 1.0 2.0 3.0 4.0


P(DS| IDR)

Corrected Function


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

0.0 1.0 2.0 3.0 4.0


P(DS| IDR)

Corrected Function


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).


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 IDNo. of Stories

Lateral Resisting System

Occupancy TypeDist. To

Epicenter [km]

Interstory Ht. [cm]

EQ Direction

Period [s] AlphaDamping

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 Description

N None . No damage is visible, either structural or non-structural

I

Insignificant. 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.

M

Moderate. 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.

H

Heavy. 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 Damage

1 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 Code

Apartment 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 System

Steel 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

Flexuralbeam

Axially-rigid links

H

Shear beam

Flexuralbeam

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,

S2A Steel 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]

Flo

or

0 1 2 3

x 10-3

1

2

3

Peak IDRS

tory

0 500 10001

2

3

4

Peak Floor Acceleration [cm/s.2]

Flo

or

Building Response when Subjected to USGS080 - Comp 360

0 0.5 1 1.51

2

3

4

Peak Displacement [cm]

Flo

or

0 1 2

x 10-3

1

2

3

Peak IDR

Sto

ry

0 500 10001

2

3

4

Peak Floor Acceleration [cm/s.2]

Flo

or

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 1

i 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 iin

DSDS

LnEDP i ii

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 i

mP DS EDP edp

M (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 Data

Ratio of DS=1 per Bin

Fitted 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 Data

Ratio of DS=1 per Bin

Fitted 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 EDP

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]

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

MedianLN Standard

Deviation

Damage State

MedianLN 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]

MedianLN Standard

DeviationMedian


S-1: Steel Moment Frame

Damage State

Damage State

MedianLN Standard

Deviation

C-1: Conc. Moment Frame

MedianLN Standard

Deviation

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 0.020

IDR

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 2.0

PBA [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 0.020

IDR

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 2.0

PBA [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.

CHAPTER 6 121 Development of a Story-based Loss Estimation Toolbox

CHAPTER 6

6 DEVELOPMENT OF A STORY-BASED LOSS ESTIMATION TOOLBOX


Ramirez, C.M., and Miranda, E. (2008), Development of a Story-based Loss Estimation

Toolbox, Pacific Earthquake Engineering Research Center Report, (in preparation).

6.1 PROGRAM STRUCTURE

The Pacific Earthquake Engineering Research Center (PEER) has developed a

general framework to estimate the performance of structures in future earthquakes. PEER’s

approach is distinctively different from existing performance based approaches currently

being used by practicing structural engineers to evaluate the seismic safety of existing

buildings (e.g. ASCE 41). The PEER approach provides measures of seismic performance

that are directly relevant to stakeholders, specifically dollar losses, downtime and

casualties/fatalities. Furthermore, it provides continuous variables as measures of seismic

performance as opposed to discrete, and somewhat arbitrarily selected, performance levels

at discrete hazard levels. Another distinct feature is that it provides a fully probabilistic

framework which permits the incorporation and propagation of all relevant sources of

uncertainty involved in estimating ground motions at the site, in the structural response, and

in damage and losses.

Seismic damage and loss estimation in individual buildings requires a large amount

of data and numerical calculations such that it is practically impossible to do it manually.

Therefore it requires a computer to perform the numerical work involved. The main

objective of the project is the development of a computer tool to facilitate seismic loss

estimation for buildings. The Story-based loss estimation toolbox provides a framework


that: (a) facilitates the gathering, storage and visualization of information necessary to

conduct the damage and loss estimation; (b) conducts all required calculations; and (c)

facilitates the visualization and interpretation of loss estimation results.

The Story-based loss estimation toolbox has five main modules: (1) seismic hazard

module; (2) story loss functions module; (3) response simulation module; (4) loss

estimation module; and (5) disaggregation and visualization module. Figure 6.1 shows a

graphical representation of the program structure and the way the modules are organized.

Toolbox Main Window

Building Characterization

Story EDP-DV Function

Editor

Building Loss Estimation

Hazard Module

EDP-DV Module

SimulationResponse

Module

Loss Estimation

Module

Bldg Cost Distribution

Dissagregation& Visualization

Toolbox Main Window

Building Characterization

Story EDP-DV Function

Editor

Building Loss Estimation

Hazard Module

EDP-DV Module

SimulationResponse

Module

Loss Estimation

Module

Bldg Cost Distribution

Dissagregation& Visualization

Figure 6.1 Loss Estimation Toolbox Program Structure

The seismic hazard module permits the user to import and visualize a user-supplied

site-specific seismic hazard curve or to use a USGS seismic hazard curve which can then be

modified to account for local site conditions.


The story loss functions module calculates story loss functions that allow the

estimation of losses in individual stories as a function of scalar Engineering Demand

Parameters (EDPs) such as interstory drift ratio or peak floor acceleration. Users have the

option of using built-in (or previously computed) generic story loss functions for certain

building/use types by only knowing a small amount of information about the building such

as its structural lateral resisting system and its use. For a given building type (e.g., ductile

reinforced concrete office building) three generic loss functions are available in this module

of the toolbox: (i) structural components (assumed to be drift-sensitive); (ii) drift-sensitive

nonstructural components; and (iii) acceleration-sensitive nonstructural components. These

generic story loss functions are based on structural and nonstructural components that are

typically present in the selected building type and average distribution of construction costs

for the selected building type.

The response simulation module provides a user-friendly graphical user interface to

import and visualize response simulation results previously computed with PEER’s

OpenSees simulation platform or with other simulation software platform (e.g. SAP,

Perform, etc.). The module is capable of importing simulation results from either “cloud”

or “stripe” analyses and can import simulation results to compute the conditional

probability of collapse given (conditioned on) ground motion intensity.

The loss estimation module computes losses in the building conditioned on the

ground motion intensity and then computes several measures of seismic performance by

integrating these conditional building losses over all ground motion intensities. At the time

of this publication, the measures of seismic performance that can be computed by this

module are: (a) expected loss for a given level of ground motion intensity, and (b) expected

annual loss (EAL). Other types of results that will be implemented in the future will

include: (a) expected annual loss over a finite building life using a user-specified discount

rate; (b) net present value of the building considering earthquake losses; (c) probability of

exceeding a certain economic loss for a given ground motion intensity; (d) probability of

exceeding a certain dollar loss in the remaining life of the building; (e) economic loss

associated with a user-specified probability of exceedance.

The loss disaggregation module permits the disaggregation of economic loss to

identify the contribution of seismic events, response levels, stories, and performance groups

that primarily contribute to economic losses. Some examples of disaggregation computation

and visualization that will be included in this module include: contribution of collapse and


non-collapse building responses to economic losses, contribution of individual stories to

economic losses, disaggregation of losses due to damage in structural and nonstructural

components and identification of ground motion intensity levels primarily contributing to

expected annual losses or to specific dollar losses.

In addition to the five main modules, there are two other supplemental modules that

aid in inputting data into the toolbox. The first module, the building characterization

module, is where the user can input general building information such as the building

description, the building’s value and the building’s number of stories. This module is also

where the user will specify how the building’s value is distributed between its different

floors and substructure. The second module, the EDP-DV editor, allows the user to enter

and edit specific EDP-DV functions that relate structural response parameters directly to

economic loss (for more on EDP-DV Functions, see Chapter 3). Although toolbox has

several built-in functions, the user can to edit them, or input new functions as data and new

fragility functions become available.

6.2 GRAPHICAL USER INTER FACE

6.2.1 Building Information & Characterization

Purpose: To input general building information such as number of stories, total building

value, and building use and to define the way the total cost of the building is distributed in

each floor and substructure.

The use of this module is illustrated in Figure 6.2 in 5 steps:

Step 1: Enter in a general description of the building (e.g. name, address, owner, ..etc.). to

help identify the project for future reference.

Step2: Enter the number of stories of the building. This value is referenced multiple times

throughout the program for difference calculations.

Step3: Select the building use and type.

Step 4: Enter the total value of building. This is value can be estimated by standard cost

estimating procedures. This value is distributed to each story to calculate loss on a story-

by-story basis. Loss results are typically normalized by this value.


Step 5: Choose how the total value of the building will be distributed among the different

floors and substructure. There are four methods of distributing the value, which the user

can choose from depending on preference and how much cost information is available. The

user can specify: (a) the user can use a default distribution that was derived from typical

trends in R.S. Means cost data (Balboni, 2007), (b) the value of the foundation and use a

uniform distribution on all the stories (the value is equally distributed), (c) the value of the

foundation, the first floor, and the top floor, and the remaining value is uniformly

distributed among the remaining floors, or (d) the values for the foundation and all floors.

(1) Enter building description for reference

(2) Enter number of stories for building

(3) Select building use / type

(4) Enter value of building (replacement cost)

(5) Select method of distributing cost along height of building

Figure 6.2 Building characterization module

6.2.2 EDP-DV Function Editor

Purpose: To define, visualize, edit and delete EDP-DV functions used to by toolbox to

estimate loss.

The use of this module is illustrated in Figure 6.3.

The initial window has four option buttons to perform the following actions: (a) adding new

user-defined EDP-DV functions into the toolbox, (b) viewing existing EDP-DV functions


already entered into the toolbox (both function already built-in and user-defined), (c)

editing existing EDP-DV functions, and (d) deleting existing EDP-DV functions.

(1) Choose this button to add new EDP-DV functions

(2) Choose this button to view EDP-DV functions already inputted in the toolbox

(3) Choose this button to edit EDP-DV functions already inputted in the toolbox

(3) Choose this button to delete EDP-DV functions already inputted in the toolbox

(1) Choose this button to add new EDP-DV functions

(2) Choose this button to view EDP-DV functions already inputted in the toolbox

(3) Choose this button to edit EDP-DV functions already inputted in the toolbox

(3) Choose this button to delete EDP-DV functions already inputted in the toolbox

Figure 6.3 EDP-DV function editor module

Adding new EDP-DV functions is illustrated in Figure 6.4 in 5 steps:

Step 1: Select the type of EDP used to define the EDP-DV function. The toolbox only

considers two types of EDPs: peak interstory drifts (IDR) and peak floor accelerations

(PFA), for drift-sensitive and acceleration sensitive components respectively.

Step 2: Name the new EDP-DV function for future reference.

Step 3: Enter values of EDP-DV function using the spreadsheet-like interface. Values can

be copied and pasted in from other MS-Excel spreadsheets.

Step 4: Plot function to visualize the results and check for errors in data entry.

Step 5: Press “Add Function” button to finalize process of inputting the new function.


(1) Choose what type of EDP the function is dependent on

(2) Name the EDP-DV function for future reference

(3) Enter values of EDP-DV Function

(4) Press to plot function and check for errors

(5) Press to finalize and add function

Cancels adding EDP-DV function

(1) Choose what type of EDP the function is dependent on

(2) Name the EDP-DV function for future reference

(3) Enter values of EDP-DV Function

(4) Press to plot function and check for errors

(5) Press to finalize and add function

Cancels adding EDP-DV function

Figure 6.4 Adding EDP-DV functions

The process and interface for viewing, editing and deleting is similar and is illustrated in

Figure 6.5 in 3 steps:

Step 1: Select the EDP-DV function you want to view / edit / delete.

Step 2: Edit values as necessary (this step is not necessary if you are only viewing or

deleting functions).

Step 3: Press “Return / Accept Changes / Delete” button when finished.


(1) Choose EDP-DV function you want to view/edit/delete

(2) Edit values as if needed

(3) Press to finish viewing / editing / deleting

Press to plot function

Press to cancel

Press to close plot

(1) Choose EDP-DV function you want to view/edit/delete

(2) Edit values as if needed

(3) Press to finish viewing / editing / deleting

Press to plot function

Press to cancel

Press to close plot

Figure 6.5 Viewing / Editing / Deleting EDP-DV functions


6.2.3 Main Window

Purpose: This is window is the primary interface of the toolbox once the preliminary

building information and EDP-DV functions have been entered. The remaining modules of

the toolbox can be accessed from here. The use of this module is illustrated in Figure 6.7.

(1) Press to define seismic hazard using the Hazard Module

(3) Press to assign EDP-DV functions using the Fragility Module

(2) Press to import response simulation data using Response Simulation Module

(4) Press to perform loss estimation calculations using Loss Estimation Module

Press to exit Toolbox

Figure 6.6 Main window of toolbox


6.2.4 Hazard Module

Purpose: To define the seismic hazard curve for the site being considered. This is curve is

integrated with the vulnerability curves (expected losses given intensity) to obtain expected

annual losses (EAL).

The use of this module is illustrated in Figure 6.7. There are 3 steps to import a curve and 6

steps to look up a pre-defined curve.

Step 1: Select the desired method of defining the hazard curve. (A) The curve can be

imported from another program used to conduct a probabilistic seismic hazard analysis

(PSHA). (B) Alternatively, the curves for rock sites generated by USGS (USGS, 2005) can

be looked up by geographic coordinates and scaled to account for site-specific soil

conditions if desired.

To import a hazard curve:

Step 2: Enter the number of points needed to define the seismic hazard curve.

Step 3: Press ‘OK’ when finished.

To look up hazard curve using geographic coordinates:

Step 2: Input latitude coordinates.

Step 3: Input longitude coordinates.

Step 4: Enter the building’s fundamental period. (Seismic intensity is measured by spectral

acceleration at the fundamental period, Sa(T1))

Step 5: Select the desired method of scaling the hazard curve to account for site-specific

soil conditions. The values provided by USGS are for PSHA’s conducted assuming a rock

(or very stiff) site. Select this option to obtain values for a site located on rock (very stiff

soil).

Step 5B: Select this option to scale rock values using site amplification factors where the

type of soil defined by NEHRP soil classifications.


(1A) Select to import site-specific hazard curve from custom PSHA

(1B) Select to obtain hazard curve from USGS values

(2) Enter latitude coordinates for site

(3) Enter longitude coordinates for site

(4) Enter fundamental period of building

(5A) Select to use hazard curve for site ass

(5B) Select to scale rock values using NEHRP factors

(5C) Select to scale rock values using Surface Geology factors

(5D) Select to scale rock values using Geotechnical Data factors

(6) Press OK when finished

(2) Enter number of points needed to define hazard curve


(1A) Select to import site-specific hazard curve from custom PSHA

(1B) Select to obtain hazard curve from USGS values

(2) Enter latitude coordinates for site

(3) Enter longitude coordinates for site

(4) Enter fundamental period of building

(5A) Select to use hazard curve for site ass

(5B) Select to scale rock values using NEHRP factors

(5C) Select to scale rock values using Surface Geology factors

(5D) Select to scale rock values using Geotechnical Data factors


(2) Enter number of points needed to define hazard curve


Figure 6.7 Defining the seismic hazard curve


6.2.5 Response simulation module

Purpose: To input response simulation data generated by incremental dynamic structural

analysis. There are two primary types of simulation data required to conduct a loss

assessment: (1) collapse data, which is used to compute the probability of collapse for the

considered building, and (2) engineering demand parameter (EDP) data, which is used to

calculate losses due to repair given the structure has not collapsed.

The use of this module is illustrated in Figure 6.8. There is 1 step to import a data from a

pre-formatted EXCEL file and 10 steps to enter the data manually.

Step 1: Select the desired method of importing the response data. (A) The module allows

data to be entered manually such that it does not matter what computer program is used to

generate the simulation data. (B) The module also is capable of importing data from an MS

EXCEL file that has been preformatted such that it can be read in by the toolbox properly.

Step 2: Enter the number ground motion records that were used in the structural analysis.

Results produced from each ground motion are used to produce statistical parameters

(median and lognormal standard deviation) for the probability of collapse and EDP (e.g.

IDR and PFA) values at each story/floor.

Step 3: Enter the number of levels of ground motion intensity that each record was scaled

to during structural analysis.

Step 4: Enter collapse data into the spreadsheet form provided. Space is provided for the

user to enter the value of seismic ground motion intensity at which the structure collapses

for each ground motion record. In addition, there are cells where the user can enter the

statistical parameters for the collapse fragility directly (median and lognormal standard

deviation of the collapse fragility function).

Step 5: Once the collapse data has been entered, press the ‘Add’ button to write the data

into the program’s memory.

Step 6: Press the ‘EDP DATA’ tab to switch to the form to enter in EDP data.

Step 7: Select the value of seismic intensity (IM level) that is currently being entered using

the pull-down menu provided.

Step 8: Enter in the EDP data generated from structural analysis. The first few lines are for

entering in peak IDRs for each story. The next few lines are for entering peak residual


interstory drifts (RIDRs). The final lines are used for entering peak ground acceleration and

PFAs.

Step 9: Once the EDP data has been entered, press the ‘Add’ button to write the data into

the programs memory.

Step 10: Press the ‘OK’ button when you are finished to save the data that has been entered

in or Press the ‘Cancel’ button to exit the window without saving the data entered.


(1A) Select to enter in simulation data manually.

(1B) Select to import simulation data from pre-formatted file

Press to visualize data once data is imported (will become active only after data has been entered).

Press RETURN to return to main menu.

(2) Enter number of ground motions considered in simulation

(3) Enter number of intensity measure (IM) levels

(7) Select IM level for EDP data currently being entered.

(8A) Enter peak IDR’s

(8B) Enter residual IDR’s

(8C) Enter PGA & PFA’s

(9) Press to add data for current IM to file once data is entered

(10A) Press to OK when finished entering all simulation data

(10B) Press to CANCEL to return to previous window without saving data entered.

(6) Press to enter EDP data

(5) Press to add data for current IM to file once data is entered(4) Enter collapse

data.

(1A) Select to enter in simulation data manually.

(1B) Select to import simulation data from pre-formatted file

Press to visualize data once data is imported (will become active only after data has been entered).

Press RETURN to return to main menu.

(2) Enter number of ground motions considered in simulation

(3) Enter number of intensity measure (IM) levels

(7) Select IM level for EDP data currently being entered.

(8A) Enter peak IDR’s

(8B) Enter residual IDR’s

(8C) Enter PGA & PFA’s

(9) Press to add data for current IM to file once data is entered

(10A) Press to OK when finished entering all simulation data

(10B) Press to CANCEL to return to previous window without saving data entered.

(6) Press to enter EDP data

(5) Press to add data for current IM to file once data is entered(4) Enter collapse

data.

Figure 6.8 Importing response simulation data.

The following windows allow users to make adjustments to the response simulation

data after the raw values have been imported. The user will have the opportunity to make

adjustments to the parameters of the collapse fragility. Based on the final collapse fragility

parameters, the toolbox will determine if there is enough EDP data to calculate losses at

enough ground motion intensity levels to complete the integration for EAL. If there is not

enough data, the program gives users the option of extrapolating the EDP data for higher


ground motion intensity levels, or aborting the analysis. This process is illustrated in Figure

6.9 and is completed in the following 3 steps.

Step 11: After importing the raw data from response simulation, a window with a plot of the

collapse fragility will appear. Make any necessary adjustments to the collapse fragility

parameters using the textboxes provided. Users may need to modify the parameters of the

collapse fragility to account for phenomenon that influence the probability of collapse. For

instance, previous studies (Haselton and Deierlein 2007, Liel and Deierlein 2008) have

shown that spectral shape can significantly affect the collapse median, while considering

modeling uncertainty may influence the value of the lognormal standard deviation.

Step 12: Press the ‘OK’ button when the parameters of the collapse fragility have been

entered.

Step 13: Once the collapse fragility parameters have been established, the program will

check to see if there is sufficient amount of EDP data to complete the computation of EAL.

Select from the following options on how the EDP data will be extrapolated: (A)

extrapolate the EDP data for large values of ground motion intensity without smoothing,

(B) extrapolate the EDP data for large values of ground motion intensity with smoothing

using a polynomial regression, and (C) aborting the analysis and running more structural

response simulations.

The calculation of EAL involves integrating the expected losses as a function of

ground motion intensity with the seismic hazard curve (Aslani and Miranda, 2005) over the

full range of ground motion intensities up to that when the structure collapses. It was

decided that integrating up to intensity values associated with a 98% probability of collapse

would be sufficient to calculate EAL accurately. Ground motion intensities greater than

these values typically have a very low frequency of occurring and the economic loss

calculated at these intensities do not contribute a significant amount more to the total value

of EAL.

Calculation of the EAL requires expected losses for a given seismic intensity have

to be computed up to large intensities of ground motion associated with a 98% probability

of collapse. Since losses due to non-collapse are a function of peak EDP values, EDP data

for these large ground motion intensities have to be provided. This can be come difficult

when modeling collapse because many simulations at high levels of ground motion

intensity do not converge. Therefore, if not enough EDP data is provided then the user has

the option of running an extrapolation algorithm. The algorithm will extrapolate enough


EDP statistical parameters to compute enough expected losses at large intensity levels to

complete the integration.

The program also provides the user with the option of smoothing out the function

that relates EDP statistical parameters to ground motion intensity. Often at higher intensity

levels, estimates of the median and lognormal standard deviation for EDPs become less

reliable because the size of the sample of converged simulations becomes smaller. The

estimates become more erratic and may lead to less accurate estimates of the loss due to

non-collapse. The program offers the user the option to fit a 5th order polynomial to

represent the value of EDP as a function of ground motion intensity after extrapolation has

been performed.

(11A) Adjustments to the collapse median can be made here.

(11A) Adjustments to the lognormal standard deviation of the collapse probability distribution can be made here.

(12) Press to OK when finished editing the collapse parameters

(13A) Press to extrapolate EDPswithout smoothing using polynomial regression.

(13B) Press to extrapolate EDPs using polynomial regression to smooth the trends of the EDP parameters with IM level.

(13C) Press to abort analysis and run more structural analyses.

(11A) Adjustments to the collapse median can be made here.

(11A) Adjustments to the lognormal standard deviation of the collapse probability distribution can be made here.

(12) Press to OK when finished editing the collapse parameters

(13A) Press to extrapolate EDPswithout smoothing using polynomial regression.

(13B) Press to extrapolate EDPs using polynomial regression to smooth the trends of the EDP parameters with IM level.

(13C) Press to abort analysis and run more structural analyses.

Figure 6.9 Collapse fragility adjustments and EDP extrapolation options


Once the EDP data has been extrapolated, a visualization window will appear to

display the results of the response simulation data, as show in Figure 6.10. There are

several figures displaying the following information:

Graph A: This graph plots median EDP values for each floor/story as a function of ground

motion intensity.

Graph B: This graph plots the EDPs lognormal standard deviations for each floor/story as a

function of ground motion intensity.

Graph C: This graph plots median EDP values for each floor/story along the height of the

building for particular values of ground motion intensity.

Use pull-down menu to select type of EDP to view.

Use tabs to toggle between viewing EDP data and collapse data.

Graph A: This graph plots the median EDP as a function of ground motion intensity (Sa at T1, g)

Graph B: This graph plots the EDP’slognormal standard deviation as a function of ground motion intensity (Sa at T1, g)

Graph C: This graph plots the median EDP’sat each floor/story for particular ground motion intensity levels.

Press Close button to exit and return to main window.

Use pull-down menu to select type of EDP to view.

Use tabs to toggle between viewing EDP data and collapse data.

Graph A: This graph plots the median EDP as a function of ground motion intensity (Sa at T1, g)

Graph B: This graph plots the EDP’slognormal standard deviation as a function of ground motion intensity (Sa at T1, g)

Graph C: This graph plots the median EDP’sat each floor/story for particular ground motion intensity levels.

Press Close button to exit and return to main window.

Figure 6.10 Response simulation visualization.

6.2.6 EDP-DV Module

Purpose: To assign EDP-DV functions to drift-sensitive structural components, drift-

sensitive nonstructural components and acceleration-sensitive nonstructural components for

each floor.

The use of this module is illustrated in Figure 6.11. There are 4 steps required to complete

this process.

Step 1: Click on the pull-down menu to select the desired floor to make EDP-DV function

assignments.


Step 2: (A) Enter the number of the EDP-DV function that is to be assigned to drift-

sensitive structural components for the floor being considered. All the EDP-DV functions

are listed and enumerated in the list box above the textboxes. Use the numbering in this list

to reference the EDP-DV functions when making assignments. Repeat for the drift-

sensitive and acceleration-sensitive nonstructural components. (B) Alternatively, the

assignments can be made by typing in the number of the EDP-DV function directly into the

spreadsheet form provided.

Step 3: Press the ‘ASSIGN’ button to write the assignments into the spreadsheet form.

Step 4: Press the ‘OK’ button to save the EDP-DV assignments when finished. To return to

the Main window without saving the assignments press the ‘CANCEL’ button.

(1) Use pull-down menu to select the floor for which EDP-DV functions will be assigned.

This part of the form lists the EDP-DV functions currently stored in the toolbox. Each function is numbered to use as referenced.

(2A) Enter number of the EDP-DV function to be assigned for the drift-sensitive structural components. Repeat for nonstructural components.

(3) Press ASSIGN button to enter function assignments into spreadsheet form.

(2B) Alternatively, EDP-DV function numbers can be entered directly into the spreadsheet form.

(4A) Press OK button to save assignments and return to main window.

(4B) Press CANCEL button to return to main window without saving assignments.

(1) Use pull-down menu to select the floor for which EDP-DV functions will be assigned.

This part of the form lists the EDP-DV functions currently stored in the toolbox. Each function is numbered to use as referenced.

(2A) Enter number of the EDP-DV function to be assigned for the drift-sensitive structural components. Repeat for nonstructural components.

(3) Press ASSIGN button to enter function assignments into spreadsheet form.

(2B) Alternatively, EDP-DV function numbers can be entered directly into the spreadsheet form.

(4A) Press OK button to save assignments and return to main window.

(4B) Press CANCEL button to return to main window without saving assignments.

Figure 6.11 Assigning EDP-DV functions.


6.2.7 Loss Estimation Module

Purpose: To provide the user the option of considering building demolition losses given

that structure has not collapsed and to compute the loss estimation calculations.

The use of this module is illustrated in Figure 6.12. There are 5 steps to complete this

process.

Step 1: Click on the check box if monetary losses from building demolition conditioned on

non-collapsed are going to be considered in the analysis.

Step 2 & 3: If non-collapse losses due to building demolition are to be considered, enter the

statistical parameters for the probability of building demolition conditioned on residual IDR

in the textboxes provided.

Step 4: If non-collapse losses due to building demolition are to be considered, enter the

estimated percentage of the building’s replacement value that can be salvaged and reused.

Step 5: Press the ‘OK’ button when finished to start running the loss estimation

computations or press the ‘CANCEL’ button to return to the main window.

(1) Check box to include demolition losses given non-collapse.

(2) Enter median for probability of building demolition given RIDR

(3) Enter lognormal standard deviation for probability of building demolition given RIDR

(4) Enter estimated percent of building value that can be salvaged based on the building not collapsing.

(5A) Press OK to begin loss analysis.

(5B) Press CANCEL to return to main window.

Graph of cumulative probability distribution for probability of building demolition given RIDR.

(1) Check box to include demolition losses given non-collapse.

(2) Enter median for probability of building demolition given RIDR

(3) Enter lognormal standard deviation for probability of building demolition given RIDR

(4) Enter estimated percent of building value that can be salvaged based on the building not collapsing.

(5A) Press OK to begin loss analysis.

(5B) Press CANCEL to return to main window.

Graph of cumulative probability distribution for probability of building demolition given RIDR.

Figure 6.12 Loss Estimation Module - including building demolition losses given that the structure has not collapsed.


6.2.8 Loss Disaggregation and Visualization Module

Purpose: To display the results from the loss estimation computations. The module

displays total losses and disaggregated losses. Results for both expected losses as a

function of ground motion intensity and EALs are provided.

The types of results for expected loss as a function of ground motion intensity are illustrated

in Figure 6.13.

Graph A: This graph plots the total expected losses as a function of increasing levels of

ground motion intensity. This type of information can be used for so called “scenario-

based” events where the monetary loss can be found for a ground motion intensity of

particular interest. For instance, this graph may be used to obtain expected monetary loss

given the ground motion intensity level associated with the design basis earthquake (DBE),

the event with an exceedance rate of occurrence of 10% in 50 years. Also plotted on this

graph are the losses disaggregated with respect to losses given that the building has

collapsed, losses given that the building has not collapsed due to building demolition, and

losses given that the building has not collapsed due to repair costs. For each level of ground

motion intensity, the user can identify how much each of these sources of economic losses

contribute to the overall losses.

Graph B: This graph plots the loss given that the building has not collapsed due to repair

costs as a function of ground motion intensity, disaggregated by floor. This information is

useful to identify which floor(s) are contributing the most to losses due to non-collapse

caused by repair costs.

Graph C: This graph plots the loss given that the building has not collapsed due to repair

costs as a function of ground motion intensity disaggregated by types of building

components grouped as follows: (1) drift-sensitive structural components (2) drift-sensitive

nonstructural components and (3) acceleration-sensitive nonstructural components.


Use tabs to toggle between results for expected loss as a function of ground motion intensity and expected annual results.

Use pull-down menu to select type of graph (line or bar) to display expected losses

Graph B: This graph plots expected loss results as a function of ground motion intensity. Also shown are the expected results for loss disaggregated for collapse losses, non-collapse losses due to demolition and non-collapse losses due to repair costs.

Graph A: This graph plots expected non-collapse losses due to repair disaggregated by floor/story.

Press RETURN button to return to main window.

Press SAVE & CLOSE button to save loss results in output file.

Graph C: This graph plots expected non-collapse losses due to repair disaggregated by component types.

Figure 6.13 Total and disaggregation results for expected economic losses as a function of ground motion intensity

The types of results for expected loss as a function of ground motion intensity are

illustrated in Figure 6.14. The total EAL and disaggregated EALs are tabulated in the upper

left part of the window.

Graph A: This pie graph plots the EAL disaggregated by collapse losses, non-collapse

losses due to building demolition and non-collapse losses due to repair costs.

Graph B: This pie graph plots the EAL of non-collapse losses due to repair costs

disaggregated by drift-sensitive structural components (ST | IDR), drift-sensitive


nonstructural components (NS | IDR) and acceleration-sensitive nonstructural components

(NS | PFA).

Graph C: This bar graph plots the total EAL disaggregated by ground motion intensity

level. Each bar represents the amount of EAL for intensities up to the value shown below

the bar.

Graph D: This bar graph plots non-collapse losses due to repair costs disaggregated by both

component type and intensity level.

Graph B: This graph plots EAL results for non-collapse losses due to repair disaggregated by component type

This part of the form summarizes key values for the EAL results

Graph C: This graph plots EAL results that have been disaggregated by ground motion intensity level.

Graph A: This graph plots EAL results disaggregated by collapse losses, non-collapse losses due to demolition and non-collapse losses due to repair. Press RETURN

button to return to main window.


Graph D: This graph plots EAL of non-collapse losses due to repair disaggregated by both component type and ground motion intensity level.

Graph B: This graph plots EAL results for non-collapse losses due to repair disaggregated by component type

This part of the form summarizes key values for the EAL results

Graph C: This graph plots EAL results that have been disaggregated by ground motion intensity level.

Graph A: This graph plots EAL results disaggregated by collapse losses, non-collapse losses due to demolition and non-collapse losses due to repair. Press RETURN

button to return to main window.


Graph D: This graph plots EAL of non-collapse losses due to repair disaggregated by both component type and ground motion intensity level.

Figure 6.14 Total and disaggregation results for expected annual losses.

CHAPTER 7 143 Benchmarking Seismic-induced Economic Losses Using Story-based Loss Estimation

CHAPTER 7

7 BENCHMARKING SEISMIC-INDUCED ECONOMIC LOSSES USING STORY-BASED

LOSS ESTIMATION

This chapter is based on the following publications:

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

Journal of Structural Engineering, (in preparation).

Ramirez, C.M., A.B. Liel, J. Mitrani-Reiser, C.B. Haselton, A.D. Spear, J. Steiner, G.G.

Deierlein, E. Miranda. (2009), “Performance-Based Predictions of Earthquake-Induced

Economic Losses in Reinforced Concrete Frame Structures,” Earthquake Spectra, (in

preparation).


Ramirez led this research effort that consisted of two primary phases. The first phase

involved evaluating economic losses using a component-based approach using a computer

tool (MDLA) developed by Mitrani-Reiser (2007). The results of this phase will be

published in the second paper listed above. The second phase repeated the loss analyses

using the story-based loss estimation approach presented in Chapter 3 and the computer tool

developed in Chapter 6.

Liel computed and supplied the structural analysis for the non-ductile ductile

reinforced concrete moment frame buildings in this study and wrote/edited portions of this

publication. Mitrani-Reiser provided the component-based loss estimation computer tool

(MDLA) used in the first phase of this investigation and wrote/edited portions of this

publication. Spear and Steiner developed the architectural layouts and cost estimations in


this study, and was responsible for completing the economic loss analyses for the results

produced by MDLA. Haselton computed and supplied the structural analysis for the ductile

reinforced concrete moment frame building data for used in this study, helped edit the

publication and served as an advisor for this study. Miranda and Deierlein served as

advisors to this investigation.

7.2 INTRODUCTION

Recent earthquakes in California, particularly Loma Prieta (1989) and Northridge

(1994), have demonstrated that while modern U.S. building codes have been relatively

successful in protecting human life in moderate magnitude events, significant economic

losses may still occur. These losses suggest that building owners and other stakeholders

may wish to evaluate other aspects of building seismic performance beyond protecting life-

safety. The Pacific Earthquake Engineering Research (PEER) Center has established a

framework for performance-based earthquake engineering (PBEE) that can be used to

assess several metrics of seismic performance, including economic losses, downtime and

number of fatalities (Krawinkler and Miranda, 2004). By providing quantitative measures

of structural performance, PBEE can be used to consider possible earthquake economic

losses and loss mitigation strategies when making design decisions.

Methodologies for predicting economic losses in earthquakes can be characterized

as either regional loss estimation or building-specific loss estimation. Regional loss

estimation predicts earthquake economic losses in a broad geographical area, by estimating

building inventory data and categorizing buildings into generic structural types to evaluate

earthquake performance 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 is calculated

using pre-established capacity and fragility functions, to determine repair costs.

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


estimating losses, but these methods cannot capture unique aspects of a specific building’s

structural and nonstructural design and are not reliable for individual structures.

Building-specific earthquake-induced economic losses can be computed by

evaluating damage in structural and non-structural components on the basis of structural

analysis of an individual building to estimate damage and determining the cost of repairing

this damage. Damage states are defined for each component in the building based on the

different repair actions needed to restore the components to its original state. Scholl (1979)

initially introduced a building-specific loss estimation methodology, which was further

developed by a variety of researchers over the past 20 years (Kustu et al. 1982; Gunturi and

Shah 1993; Singhal and Kiremidjian 1996). Porter and Kiremidjian 2001 established an

assembly-based approach to probabilistic performance evaluation that considers building

assemblies (groups of building components) at a greater level of detail than previous

techniques. Extending this work, PEER’s loss estimation methodology incorporates items

such as probabilistic seismic hazard analysis and the probability of structural collapse,

establishing a comprehensive framework for performance assessment. PEER’s framework

was documented by Miranda et al, (2004), Aslani and Miranda (2005) and Mitrani-Reiser

(2007). Mitrani-Reiser (2007) also implemented the framework into a computational tool,

the MATLAB Damage and Loss Analysis (MDLA) toolbox, which calculates seismic

economic losses for reinforced concrete (RC) frame structures. A simplified approach to

PEER’s loss methodology, referred to as story-based loss estimation, was developed and

documented in Chapter 3 of this dissertation and also implemented into a computer tool.

Mitrani-Reiser used the MLDA toolbox to benchmark the performance of different

design variations of a modern four-story RC office building for a site in Los Angeles,

California (Goulet et al. 2007). Their investigation studied the effects of different design

alternatives on predictions of collapse and direct economic losses, but because its scope was

limited to one type of building, other structural parameters – such as the building height,

bay width, and foundation design assumptions – were not examined. Haselton and

Deierlein (2007) took a broader approach in a separate investigation that analyzed a set of

archetypical RC frame buildings, to examine how structural design variations (height, bay

width, etc.) and other key code provisions (strong-column weak-beam requirements,

maximum interstory drift ratio, and R-factor) influenced performance; however, this

research focused on collapse performance, and did not consider economic losses.


In this study, the same set of archetypical modern RC buildings (Haselton 2007) is

investigated to benchmark the magnitude of the economic losses possible in modern code-

conforming RC frame structures in high seismic areas and to quantify the effect of

important design variables on estimated losses, using PEER’s story-based loss estimation

method (developed in Chapter 3), and the associated computer toolbox that implements this

approach (described in Chapter 6). Structural design parameters considered include

building height, type of framing system, strength and stiffness variability along the building

height, foundation design assumptions. We also investigate how selected code provisions,

primarily the design R-factor and the strong column-weak beam ratio, affect seismic

performance in terms of economic losses. Additionally, architectural variations are also

examined to see if they have a significant effect on predicted economic losses. This

investigation focuses on modern code-conforming, moment-resisting frame buildings

(ductile reinforced concrete), primarily used as office space; however, a brief comparison to

non-ductile reinforced concrete frame buildings will be conducted to evaluate the difference

in risk associated with older buildings. In this study, seismic performance is solely based

on economic losses (defined here as the cost of repairing or replacing structural and

nonstructural components damaged by earthquakes) and seismic performance metrics such

as downtime and fatalities are not considered.

7.3 LOSS ESTIMATION PROCEDURE

The decision variables considered in this study are all associated with the

repair/replacement effort needed to return a building back to its original (undamaged) state

after an earthquake and the repair/replacement effort is quantified in dollar terms. The

mean total repair cost for each hazard level, also known as the vulnerability function, is

calculated for the study RC frame buildings by summing the mean repair cost for all

damageable components in the building. The repair cost is disaggregated at each hazard

level to illustrate the contribution of each damageable building component type to the total

loss (Miranda et al. 2004, Aslani and Miranda 2005). Finally, the expected annual loss is

computed to estimate the amount an owner could expect to pay on average yearly to repair

earthquake damage, considering the frequency of occurrence and severity of all possible

earthquakes at the site.


The damage and loss analyses portions of the PEER methodology are evaluated in

this study using the computer toolbox described in Chapter 6 of this dissertation. In order to

estimate economic losses in a structure, the toolbox only needs the estimated replacement

value of the building and input from the hazard and structural analysis results. The loss is

calculated using story EDP-DV functions built into the toolbox, which relate response

demand parameters to expected loss directly. Interested readers are directed to Chapter 3 of

this dissertation for a more detailed explanation of the equations used in this study.

7.4 DESCRIPTION OF BUILDINGS

In this study, the performance-based loss estimation framework is applied to

evaluate earthquake-induced economic losses in modern, code-conforming RC frame

structures. All buildings in this study are assumed to be located at a site in the Los Angeles

basin, south of downtown Los Angeles. The location was selected as a typical urban

California site with high levels of seismicity in the transition zone of IBC design maps, but

not subject to near-fault directivity effects (Haselton et al. 2007). The site-specific hazard

curve (Figure 7.1), which predicts the likelihood of experiencing a ground motion

exceeding a specified intensity at the site of interest, is obtained from probabilistic seismic

hazard analysis by Goulet et al. (2007). Ground motion intensity in this study is quantified

using spectral acceleration at fundamental period, Sa(T1) (note that in the figures and tables

in this chapter will denote this value as Sa for brevity, however, it is implied that this value

is at the fundamental period).


1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

0.01 0.1 1 10IM = Sa(T1)

(IM)

T1 = 1 sec

T1 = 2 sec

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

0.01 0.1 1 10IM = Sa(T1)

(IM)

T1 = 1 sec

T1 = 2 sec

Figure 7.1 Ground motion probabilistic seismic hazard curves (Goulet et al., 200&)

The reinforced concrete frame structures were designed according to the 2003

International Building Code and related ACI and ASCE provisions by Haselton and

Deierlein (2007) (ACI 2002; ASCE 2002; ICC 2003,). Seismic design is based on the

mapped seismicity at the site of interest with Ss = 1.5g and S1 = 0.6g (these values define

corners of the design spectra, which is associated with a mean annual frequency of

approximately 10% in 50 years as specified in the International Building Code (ICC 2003)).

Each of the structures are designed to fully comply with the building code provisions for

special moment frames, including strong column-weak beam ratios, joint shear strength and

detailing provisions, and requirements for strength and stiffness. The buildings range in

height from 1 to 20 stories, and include both perimeter and space frame systems. Table 7.1

details the design parameters of the 30 buildings evaluated in this study. The shorter

buildings (1, 2 and 4-stories) have a plan area of 120 ft. x 180 ft. The taller buildings (8,

12, and 20 stories) have a smaller footprint, reflecting typical ratios of usable floor area to

lot sizes in high rise construction.


Table 7.1 Archetype design properties and parameters

Design Number

Design ID No. No. of storiesFrame Type

Bay Width

Strength/Stiffness

Distr. Over Ht.

Fdn Fixity Assumpti

on

1 1-S-20-A-G 1 Space 20 A GB2 1-S-20-A-P 1 Space 20 A P3 1-S-20-A-F 1 Space 20 A F4 1-P-20-A-G 1 Perimeter 20 A GB5 2-S-20-A-G 2 Space 20 A GB6 2-S-20-A-P 2 Space 20 A P7 2-S-20-A-F 2 Space 20 A F8 2-P-20-A-G 2 Perimeter 20 A GB9 4-P-20-A-G 4 Perimeter 20 A GB10 4-P-20-C-G 4 Perimeter 20 C GB11 4-S-20-A-G 4 Space 20 A GB12 4-P-30-A-G 4 Perimeter 30 A GB13 4-S-30-A-G 4 Space 30 A GB14 8-P-20-A-G 8 Perimeter 20 A GB15 8-S-20-A-G 8 Space 20 A GB16 8-S-20-C-G 8 Space 20 C GB17 8-S-20-B(1-65)-G 8 Space 20 B (65%)b GB18 8-S-20-B(1-80)-G 8 Space 20 B (80%)b GB19 8-S-20-B(2-65)-G 8 Space 20 B (65%)c GB20 8-S-20-B(2-80)-G 8 Space 20 B (80%)c GB21 12-P-20-A-G 12 Perimeter 20 A GB22 12-S-20-A-G 12 Space 20 A GB23 12-S-20-C-G 12 Space 20 C GB24 12-S-20-B(1-65)-G 12 Space 20 B (65%)b GB25 12-S-20-B(1-80)-G 12 Space 20 B (80%)b GB26 12-S-20-B(2-65)-G 12 Space 20 B (65%)c GB27 12-S-20-B(2-80)-G 12 Space 20 B (80%)c GB28 12-S-30-A-G 12 Space 30 A GB29 20-P-20-A-G 20 Perimeter 20 A GB30 20-S-20-A-G 20 Space 20 A GB

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

Architectural layouts for each building are used to create an inventory of

damageable non-structural components and to estimate the median total replacement cost.

Figure 7.2 shows the architectural design of a typical floor of the 8, 12 and 20 story

buildings, which is to generate a list (and quantities) of damageable components in the

building. This study uses the architectural layout developed by Mitrani-Reiser (2007) for

low/mid-rise buildings (the 1, 2, and 4-story buildings listed in Table 7.1).


This study also examined the effect of variations in two primary architectural design

parameters on the seismic loss estimations: the amount of partitions, and the quality of

interior and exterior finishes. Previous studies have shown that non-structural components

represent a substantial portion of the investment in office buildings (Taghavi and Miranda,

2002) and often also account for most of the economical losses (Miranda et al 2004, Aslani

and Miranda 2005). Damaged partitions, in particular, have been identified as a large

contributor to economic losses (Mitrani-Reiser 2007). The quantity of partitions is directly

related to the percent of open office space dictated by the architectural layout. Here, open

office space is defined as space that is occupied by multiple occupants and, in many cases,

segmented by removable, partial height partitions – a group of cubicles, for example,

represents open office space, unlike a single-occupant office, which is enclosed by non-

removable partitions. The original architectural layout (Figure 7.2) has 36% open office

space. Two alternate high-rise layouts with different percentages of open office space (14%

and 58%) to investigate how the quantity of partitions affected loss results. We also

investigated how finish quality affects losses estimations, by evaluating a building with

higher level of finishes, characteristic of a high-end tenant.


Figure 7.2 Example architectural layout for high-rise buildings

Building replacement costs were determined sing the RS Means Cost Estimating

Manuals (RS Means 2007). Table 7.2 reports the estimated replacements costs for all the

structures considered in this study, in terms of both total cost and costs per square foot.

Estimates developed using RS Means tend to be low compared to actual office building

construction costs in the geographical area being considered, but the method is used here as

a rational, systematic approach to estimate building replacement values. Since expected

losses are normalized by replacement costs to compare economic losses in different

structures underestimation of replacement costs is not a significant limitation, provided that

repair costs and replacement costs are consistent. Comparisons to other studies must be

made with caution as these normalized results are subjective to the method used to calculate

building replacement costs.


Table 7.2 Cost estimates for structures studied

Total Cost Cost per s.f. Total Cost Cost per s.f.

1-story 21,600 $3,310,000 $153 $3,180,000 $147

2-story 43,200 $6,070,000 $140 $6,510,000 $151

4-story 86,400 $12,500,000 $144 $12,000,000 $138

8-story 115,200 $19,900,000 $173 $19,400,000 $168

12-story 172,800 $29,100,000 $168 $28,100,000 $163

20-story 288,000 $49,500,000 $172 $47,900,000 $166

Space Frame Perimeter Frame

120'x180'

Number of Stories

Footprint

120'x120'

Floor Area (sf)

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

Each of the RC frame structures of interest in this study was modeled in OpenSees

(PEER 2006), using a two-dimensional, three-bay model of the lateral resisting system and

a leaning (P-Δ) column. The model does not incorporate strength or stiffness from

components designed to resist gravity loads only. Beams and columns are modeled with

concentrated plastic hinge (lumped plasticity) elements, which employs a material model

developed by Ibarra et al.(2005). This model is defined by a monotonic backbone and

associated hysteretic rules, such that it can capture critical aspects of strength and stiffness

deterioration for predicting structural behavior up to collapse. Model parameters for RC

elements are predicted from design and detailing parameters on the basis of empirical

relationships developed from calibrating the material model to experimental data from more

than 250 tests of RC columns (Haselton et al. 2007). All model parameters represent

expected (average) values. Haselton and Deierlein (2007) have shown that the lumped

plasticity modeling approach provides reasonable results, compared to fiber-element

models, at low levels of deformation and, in addition, captures material nonlinearities as the

structure approaches collapse.

Nonlinear simulation models for RC frames are analyzed using the incremental

dynamic analysis (IDA) technique (Vamvatsikos and Cornell, 2002). The analysis utilizes

a suite of ground motion records, in order to capture the effects of variation in ground


motion characteristics, such as frequency content, on the response. Selected ground

motions records are from large magnitude events, recorded at moderate fault-rupture

distances on stiff soil or rock sites (ATC, 2008). In IDA, the model is subjected to each

ground motion, recording the dynamic response (EDPs). The ground motion record is

subsequently scaled to increasing levels of intensity until collapse occurs from lateral

dynamic instability. This process is repeated for all ground motions in the set.

The outcome of IDA is the statistical prediction of critical engineering demand

parameters (EDP) at a number of ground motion intensities (stripes), and a collapse fragility

function, which describes the probability of collapse as a function of the ground motion

intensity. At lower ground motion intensities, losses are dominated by non-collapse results,

particularly damage to drift and acceleration-sensitive nonstructural components and

contents. Examples of EDPs used to calculate non-collapse losses are shown in Figure 7.3

for a 4-story space frame (4-S-20-A-G). Damage in individual components and associated

repair costs are computed from distributions in EDPs obtained at each intensity level.

1

2

3

4

0 0.02 0.04 0.06

PEAK IDR

0.05

0.25

0.50

1.00

2.00

STORY

Sa Values

2

3

4

5

0 1 2

PEAK FLR ACCEL [g]

FLOOR

1

2

3

4

0 0.02 0.04 0.06

PEAK IDR

0.05

0.25

0.50

1.00

2.00

STORY

Sa Values

2

3

4

5

0 1 2

PEAK FLR ACCEL [g]

FLOOR

Figure 7.3 Peak EDPs along building height for Design 4-S-20-A-G (Hazelton and Deierlein, 2007)


As indicated in Chapter 3, loss estimation requires the estimation of the probability

of collapse as a function of the ground motion intensity. The collapse fragility is adjusted

to account for uncertainties in the structural modeling process (Liel et al. 2008), and the

expected spectral shape of rare ground motions in California (Haselton and Deierlein 2007).

Figure 7.4(a) shows collapse fragilities for 2, 4, 8, 12, and 20 story space-frame buildings

included in this study. The geometric mean values of the probability of collapse (a measure

of central tendency for lognormal distributions, approximately equal to the median,

[ ] 0.5P C IM ) widely vary in intensities from 0.83g to 3.6g. However, comparing the

fragilities directly can be misleading, because buildings have different fundamental periods

and therefore respond to different regions of the earthquake spectra. For purposes of

comparison, it is more appropriate to normalize the fragilities by the intensity of the Design

Basis Earthquake (DBE), defined as an earthquake with a spectral-ordinate equal to two-

thirds of the Maximum Considered Earthquake (MCE)1, as shown in Figure 4(b). At the

DBE, the probability of collapse is fairly small for these modern code-conforming

structures (0.41% to 4.2%).

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0

IM = Sa [g]

1-S-20-A-G2-S-20-A-G4-S-20-A-G8-S-20-A-G12-S-20-A-G20-S-20-A-G

P(C | IM)

(a)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0

IM = Sa [g]


P(C | IM)

(a)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0

IM = Sa / Sa @ DBE


P(C | IM)

(b)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0

IM = Sa / Sa @ DBE


P(C | IM)

(b)

Figure 7.4 Collapse fragilities for 1, 2, 4, 8, 12 and 20 story space-frame buildings (Haselton and Deierlein, 2007)

1 The MCE is a seismic event that has a 2% in 50 years frequency of occurrence. The DBE is defined as two-thirds of the MCE, which is approximately equal to an event with a 10% in 50 years probability of occurring for regions in the Western US.


Table 7.3 summarizes structural analysis all collapse results of all 30 archetypical

RC frame buildings, including the building’s fundamental period, design base shear

coefficient, yield base shear coefficient (obtained from a nonlinear pushover analysis),

geometric mean and logarithmic standard deviation of the collapse fragility, spectral

acceleration at the fundamental period for the DBE (denoted as Sa @ DBE for brevity), the

probability of collapse at the DBE and the mean annual frequency (MAF) of collapse

(obtained by integrating the collapse fragility with the hazard curve). For more detail on the

formulation of collapse fragilities & the collapse results discussed in this study, readers are

referred to Haselton and Deierlein (2007).

Table 7.3 Structural design information and collapse results (Haselton and Deierlein 2007)

Design Number

Design ID No.1st Mode

Period (sec)

Design Base Shear Coeff.

Yield Base Shear Coeff.

Frm Pushover

Adjusted Geometric

Mean, Sa(T1) [g]

Adjusted Log Std. Dev., σLN

Sa @ DBEP(C | IM =

DBE)

MAF of collapse,

λcol [10-4]

1 1-S-20-A-G 0.42 0.125 0.47 3.35 0.68 1.00 3.8% 7.32 1-S-20-A-P 0.42 0.125 0.57 3.67 0.67 1.00 2.6% 4.73 1-S-20-A-F 0.42 0.125 0.47 3.36 0.68 1.00 3.8% 7.34 1-P-20-A-G 0.71 0.125 0.20 1.80 0.63 0.85 11.7% 17.35 2-S-20-A-G 0.63 0.125 0.39 3.55 0.65 0.95 2.2% 2.66 2-S-20-A-P 0.56 0.125 0.51 3.65 0.64 1.00 2.2% 2.27 2-S-20-A-F 0.63 0.125 0.37 2.94 0.66 0.95 4.4% 4.88 2-P-20-A-G 0.66 0.125 0.22 2.48 0.66 0.91 6.6% 8.79 4-P-20-A-G 1.12 0.092 0.14 1.56 0.62 0.54 4.3% 11.310 4-P-20-C-G 1.11 0.092 0.15 1.83 0.65 0.54 3.0% 8.111 4-S-20-A-G 0.94 0.092 0.24 2.22 0.63 0.64 2.4% 5.412 4-P-30-A-G 1.16 0.092 0.14 1.87 0.65 0.52 2.3% 8.213 4-S-30-A-G 0.86 0.092 0.27 3.17 0.65 0.70 1.0% 2.514 8-P-20-A-G 1.71 0.05 0.08 1.00 0.64 0.35 5.1% 25.515 8-S-20-A-G 1.80 0.05 0.11 1.23 0.62 0.33 1.9% 11.516 8-S-20-C-G 1.80 0.05 0.11 1.20 0.62 0.33 1.8% 9.217 8-S-20-B(1-65)-G 1.57 0.05 0.15 1.41 0.63 0.38 1.9% 10.018 8-S-20-B(1-80)-G 1.71 0.05 0.15 1.47 0.64 0.35 1.2% 8.719 8-S-20-B(2-65)-G 1.57 0.05 0.14 1.17 0.63 0.38 3.9% 16.820 8-S-20-B(2-80)-G 1.71 0.05 0.13 1.10 0.64 0.35 3.7% 16.821 12-P-20-A-G 2.01 0.044 0.08 0.85 0.62 0.30 4.6% 20.322 12-S-20-A-G 2.14 0.044 0.09 0.83 0.63 0.28 4.2% 15.523 12-S-20-C-G 2.13 0.044 0.09 0.96 0.63 0.31 3.9% 12.624 12-S-20-B(1-65)-G 1.92 0.044 0.14 1.06 0.63 0.31 2.6% 12.125 12-S-20-B(1-80)-G 2.09 0.044 0.10 0.95 0.62 0.29 2.7% 14.026 12-S-20-B(2-65)-G 1.92 0.044 0.11 0.95 0.62 0.31 3.7% 18.127 12-S-20-B(2-80)-G 2.09 0.044 0.10 0.84 0.63 0.29 4.3% 18.028 12-S-30-A-G 2.00 0.044 0.09 1.18 0.65 0.30 1.7% 8.529 20-P-20-A-G 2.63 0.044 0.07 0.71 0.62 0.23 3.3% 13.430 20-S-20-A-G 2.36 0.044 0.09 0.99 0.64 0.25 1.7% 9.0


7.5 ECONOMIC LOSSES

Economic losses were estimated for all 30 buildings in Table 1 in order to evaluate

seismic performance of modern RC frame buildings and to quantify the effects of important

design variables on economic losses. Metrics of economic loss of interest include the

expected value of the loss at the DBE, expected annual losses (EAL), and the present value

(PV) of life-cycle losses. The results of loss estimation are summarized in Table 7.4.

Table 7.4 Expected losses and intensity levels

Design Number

Design ID No. Sa @ DBELoss at DBE [%]

Normalized EAL [%]

1 1-S-20-A-G 1.00 32.3% 1.10%2 1-S-20-A-P 1.00 30.4% 1.10%3 1-S-20-A-F 1.00 32.4% 1.11%4 1-P-20-A-G 0.85 54.2% 3.31%5 2-S-20-A-G 0.95 30.9% 1.15%6 2-S-20-A-P 1.00 27.9% 0.97%7 2-S-20-A-F 0.95 31.9% 1.14%8 2-P-20-A-G 0.91 35.0% 1.22%9 4-P-20-A-G 0.54 30.7% 1.33%

10 4-P-20-C-G 0.54 29.2% 1.30%11 4-S-20-A-G 0.64 25.8% 1.03%12 4-P-30-A-G 0.52 31.4% 1.42%13 4-S-30-A-G 0.70 24.2% 0.91%14 8-P-20-A-G 0.35 25.1% 0.95%15 8-S-20-A-G 0.33 23.9% 1.11%16 8-S-20-C-G 0.33 24.3% 1.11%17 8-S-20-B(1-65)-G 0.38 19.7% 0.89%18 8-S-20-B(1-80)-G 0.35 21.8% 1.05%19 8-S-20-B(2-65)-G 0.38 20.1% 0.89%20 8-S-20-B(2-80)-G 0.35 22.1% 1.04%21 12-P-20-A-G 0.30 20.8% 0.65%22 12-S-20-A-G 0.28 20.7% 0.69%23 12-S-20-C-G 0.31 22.2% 0.87%24 12-S-20-B(1-65)-G 0.31 18.0% 0.68%25 12-S-20-B(1-80)-G 0.29 18.9% 0.67%26 12-S-20-B(2-65)-G 0.31 18.4% 0.68%27 12-S-20-B(2-80)-G 0.29 19.9% 0.69%28 12-S-30-A-G 0.30 18.5% 0.67%29 20-P-20-A-G 0.23 15.2% 0.44%30 20-S-20-A-G 0.25 13.2% 0.42%

MIN = 0.23 13.2% 0.42%MAX = 1.00 54.2% 3.31%AVG = 0.53 25.3% 1.02%

Loss ResultsDesign Information


7.5.1 Expected losses conditioned on seismic intensity

Loss vulnerability curves were computed for every building in the study, as shown

in Figure 7.5 for a 4-story space frame building (Design 4-S-20-A-GB). Vulnerability

curves have a characteristic shape like that in Figure 7.5; losses initially increase fairly

linearly with increasing levels of ground motion intensity. The relationship subsequently

becomes more nonlinear (for Design 4-S-20-A-GB at Sa(T1) = 1.0g) and saturates at the

total median replacement cost of the building. For the 4-story building in Figure 7.5,

expected losses of $3.2 million, or 25.8% of the median replacement cost, are predicted

under the design basis earthquake (Sa(T1) = 0.64g). Figure 7.5 also disaggregates the

expected losses into mean losses due to collapse, and mean losses due to non-collapse.

Design 4-S-20-A-G’s losses are primarily dominated by the non-collapse losses at the lower

intensity ground motions (Sa(T1) = 0.05g to 0.60g) which have greater rate of occurrence.

At the DBE (Sa(T1) = 0.64), 94% of the expected loss is due to non-collapse losses. For

this building, non-collapse losses peak at an approximate Sa(T1) of 1.60g and collapse

losses do not begin to dominate until an intensity of 2.20g.

0.00

0.20

0.40

0.60

0.80

1.00

0.0 1.0 2.0 3.0 4.0 5.0

IM = Sa [g]

E[L | IM]

Total

Non-Collapse

Collapse

Figure 7.5 Expected loss given IM for 4-S-20-A-G (with collapse loss disaggregation)


Expected losses were calculated at the design spectral acceleration at the

fundamental period (i.e. Sa(T1) at the DBE) for all buildings in this study. Expected losses

at the DBE, normalized by the replacement value of each building, are documented in Table

4 and illustrated in Figure 7.6. Given the DBE, predicted losses range from a minimum of

13% of building replacement value in a 20-story space frame (20-S-20-A-G), to a maximum

of 54% in a 1-story perimeter frame (1-P-20-A-G). The average expected loss of all the

buildings in this study is 25% of building replacement cost. This means that even though

these buildings are designed according to current codes and are successful in delivering

small probabilities of collapse in ground motions with intensity corresponding to the DBE,

they sustain very large economical losses that on average are one quarter of the initial

construction cost.

0%

10%

20%

30%

40%

50%

60%

1 2 3 4 5 6 7 8 9 10111213141516171819 202122232425 2627282930

Design Number

No

rma

lize

d L

oss

at D

BE

1 story 2 stories 4 stories 8 stories 12 stories 20 st

0%

10%

20%

30%

40%

50%

60%

1 2 3 4 5 6 7 8 9 10111213141516171819 202122232425 2627282930

Design Number

No

rma

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d L

oss

at D

BE


Figure 7.6 Normalized expected economic loss results at DBE for 30 code-conforming RC frame structures

The largest expected economic loss was experienced by the 1-story perimeter frame

(1-P-20-A-G) and is substantially greater than the other losses computed. According to

Curt Haselton (via verbal conversation), this building was designed with the specific intent

of making it as flexible as current US building codes would allow. The structure was

computed to have a median peak of IDR of 0.027 when subjected to ground motions equal

to the DBE (Sa(T1) = 0.85g). Although this value exceeds the code-specified limit of 0.025

for reinforced concrete moment frame structures (ICC, 2003) it is very close. Story drift


demands are often computed in design using equivalent static methods, and this building

does meet those requirements using this procedure according to Haselton and Deierlein

(2007). As a result, this building experiences large values of peak IDR at very small ground

motion intensity levels. For example, a median IDR of 0.011 is computed at an Sa(T1) =

0.35g, which is large enough to induce structural damage. These large drifts will result in

large estimations of damage and monetary loss as demonstrated by Figure 7.6. This design

may be unconventional and not necessarily representative of what is commonly constructed.

Therefore, it can be considered an outlier and removed when computing statistics on the

loss results. If this building is excluded, the mean expected loss for this set of buildings

reduces to 24%. Despite the fact that this building can be considered unconventional, it is

important to note that it still meets code requirements and was analyzed to demonstrate the

level of inconsistent performance that code may allow.

Haselton and Deierlein (2007) found that of all the design parameters, building

height had the largest influence on collapse safety. Similarly, results from this investigation

indicate that building height also has a significant influence on economic losses. Figure

7.7(a) shows the normalized loss vulnerability curves for 2, 4, 8, 12 and 20-story space

frames (2-S-20-A-G, 4-S-20-A-G, 8-S-20-A-G, 12-S-20-A-G and 20-S-20-A-G), and

Figure 7.7(b) plots the curves of perimeter frames for those corresponding heights (2-P-20-

A-G, 4-P-20-A-G, 8-P-20-A-G, 12-P-20-A-G and 20-P-20-A-G). It is shown that losses

normalized by the replacement cost of the building tend to decrease as the height of the

building increases (this trend is particularly apparent at higher ground motion intensities).

Expected loss at the DBE is plotted as a function of building height in Figure 7.7(c) for both

space and perimeter frame structures. Lower normalized losses are observed in taller

buildings because in moment frame buildings higher drift demands concentrate in a smaller

percentage of the height, while the shorter buildings’ drifts are more evenly distributed as

shown in Figure 7.7(d). Concentration of interstory drift demands typically increase as the

level of ground motion intensity increase and therefore the influence of height increases as

the ground motion increases. Consequently, damage and normalized loss will also be

concentrated in a smaller amount of floors in taller buildings, and produce a smaller

percentage of loss relative to the replacement cost of the building. This trend is more

pronounced in perimeter frames because soft story mechanisms are more likely to form in

these types of structures. While concentration of lateral deformations is considered

undesirable in terms of collapse because collapse may be triggered earlier, the concentration


may also significantly reduce owners’ repairs following moderate earthquakes in tall

buildings as damage concentrates in fewer stories.

0.00

0.20

0.40

0.60

0.80

1.00

0.0 1.0 2.0 3.0 4.0

IM = Sa / Sa @ DBE

E[L | IM]

2-P-20-A-G

4-P-20-A-G

8-P-20-A-G

12-P-20-A-G

20-P-20-A-G

0.00

0.20

0.40

0.60

0 5 10 15 20Num of Stories

E[L

oss]

@ D

BE

[%

of

repl

acem

ent

cost

] Space Frames

Perimeter Frames

1

2

3

4

0 0.02 0.04 0.06

PEAK IDR

0.05

0.25

0.50

1.00

2.00

STORY

Sa Values

0.00

0.20

0.40

0.60

0.80

1.00

0.0 1.0 2.0 3.0 4.0

IM = Sa / Sa @ DBE

E[L | IM]

2-S-20-A-G

4-S-20-A-G

8-S-20-A-G

12-S-20-A-G

20-S-20-A-G

1

2

3

4

5

6

7

8

9

10

11

12

0 0.05 0.1

PEAK IDR

0.05

0.25

0.50

1.00

2.04

STORY

Sa Values

(a) (b)

(c) (d)

0.00

0.20

0.40

0.60

0.80

1.00

0.0 1.0 2.0 3.0 4.0

IM = Sa / Sa @ DBE

E[L | IM]

2-P-20-A-G

4-P-20-A-G

8-P-20-A-G

12-P-20-A-G

20-P-20-A-G

0.00

0.20

0.40

0.60

0 5 10 15 20Num of Stories

E[L

oss]

@ D

BE

[%

of

repl

acem

ent

cost

] Space Frames

Perimeter Frames

1

2

3

4

0 0.02 0.04 0.06

PEAK IDR

0.05

0.25

0.50

1.00

2.00

STORY

Sa Values

0.00

0.20

0.40

0.60

0.80

1.00

0.0 1.0 2.0 3.0 4.0

IM = Sa / Sa @ DBE

E[L | IM]

2-S-20-A-G

4-S-20-A-G

8-S-20-A-G

12-S-20-A-G

20-S-20-A-G

1

2

3

4

5

6

7

8

9

10

11

12

0 0.05 0.1

PEAK IDR

0.05

0.25

0.50

1.00

2.04

STORY

Sa Values

(a) (b)

(c) (d)

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.

The results reported so far are for structures that conform to current US building

codes. Several code provisions that significantly change structural behavior were studied to

evaluate the effect of relaxing or escalating the restrictions on building performance,

particularly economic losses. ACI code provisions require columns to be 1.2 times stronger

than beams at each joint (ΣMcolumn/ ΣMbeam), referred to as the strong-column, weak-beam


ratio (SCWB), to diminish the possibility of plastic hinging in the columns. Haselton and

Deierlein (2007) demonstrated that this ratio has a significant influence on structural

collapse capacity. In this study the influence of the SCWB ratio on economic losses was

investigated. The variation of economic losses normalized by the replacement value as a

function of the level of ground motion intensity of four-story structures designed with

SCWB ratios of 0.4, 0.6, 0.8, 1.0, 1.2, 1.5, 2.0 and 3.0 are shown in Figure 7.8(a). Figure

7.8(b) shows the influence of SCWB ratio of the normalized loss in the DBE event, which

are also disaggregated to show the contributions from collapse and non-collapse losses.

0.00

0.10

0.20

0.30

0.40

0.50

0.4 0.6 0.8 1.0 1.2 1.5 2.0 3.0

Strong-Col, Weak-Bm Ratio

Non-Collapse

Collapse

E[L | IM = Sa @ DBE]

0.00

0.20

0.40

0.60

0.80

1.00

0.0 1.0 2.0 3.0 4.0

IM = Sa / Sa @ DBE

E[L | IM]

4S30AG - SCWB = 0.44S30AG - SCWB = 0.64S30AG - SCWB = 0.84S30AG - SCWB = 1.04S30AG - SCWB = 1.24S30AG - SCWB = 1.54S30AG - SCWB = 2.04S30AG - SCWB = 3.0

(a) (b)

0.00

0.10

0.20

0.30

0.40

0.50

0.4 0.6 0.8 1.0 1.2 1.5 2.0 3.0

Strong-Col, Weak-Bm Ratio

Non-Collapse

Collapse

E[L | IM = Sa @ DBE]

0.00

0.20

0.40

0.60

0.80

1.00

0.0 1.0 2.0 3.0 4.0

IM = Sa / Sa @ DBE

E[L | IM]

4S30AG - SCWB = 0.44S30AG - SCWB = 0.64S30AG - SCWB = 0.84S30AG - SCWB = 1.04S30AG - SCWB = 1.24S30AG - SCWB = 1.54S30AG - SCWB = 2.04S30AG - SCWB = 3.0

(a) (b)

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.

Both figures demonstrate that SCWB ratios of 0.4 and 0.6 have larger economic

losses because of the higher probability of collapse in these cases. While the losses due to

collapse significantly decrease as the SCWB ratio increases, the total losses remain

approximately the same (~23-26%) for ratios greater than 0.8. In buildings with small

SCWB ratios column hinging occurs early and produces collapse mechanism in one story

and the probability of collapse increases. Since column hinging results in concentration of

lateral deformation demands in 1 or 2 stories, non-collapse losses are smaller and the total

losses are smaller and the total losses are dominated by collapse losses. Although

increasing the SCWB ratio reduces the probability of collapse by preventing hinging in the


columns, lateral deformation in these structures become more evenly distributed along the

height resulting in more nonstructural damage. While increasing the SCWB ratio to a value

greater than 1.2 could potentially have a substantial impact on collapse performance (and

perhaps life safety), it does not appear to help mitigate economic losses.

The strength reduction factor, R, (ASCE-05, 2005) which is used to calculate a

structure’s design base shear, demonstrated a larger influence on total expected losses than

the SCWB ratio. Loss results for structures with R = 4, R = 8 and R = 10 are shown in

Figure 7.9. These structures have design base shear coefficients (Cs = V/W) of 0.077, 0.044

and 0.028, respectively. When designed for a small base shear, structures become weaker

and more flexible and the losses incurred are higher. At the DBE, increasing the R-factor

from 8 to 10 increases the expected losses by 25%. On the other hand, decreasing the R-

factor from 8 to 4, leads to a 12% decrease in expected loss at the DBE. These results

suggests that there is a disincentive to relaxing the seismic design force by increasing R,

due to a predicted increase in economic losses. Decreasing R and increasing the design

base shear has the opposite effect, reducing economic losses, but the impact is less

significant.

0.00

0.20

0.40

0.60

0.80

1.00

0.0 1.0 2.0 3.0 4.0

IM = Sa / Sa @ DBE

E[L | IM]

R=4

R=8

R=10

Figure 7.9 Effect of design base shear on normalized expected loss as a function of ground motion intensity

The other design parameters considered in this study had minimal influence on

expected losses. Structures that maintained uniform strength and stiffness of structural


members over the height of the building, a conservative design, performed slightly better

than the typical designs which stepped down member strength and stiffness over height

(maximum difference in loss of 4%). Assuming the foundation is pinned in design results

in stronger structural members at lower stories (particularly for low-rise buildings) and the

economic losses calculated were somewhat lower (10% difference in loss). Since these

design parameters strengthen the structural system, these lower losses were not surprising.

Contrary to initial intuition, perimeter frames actually produced lower losses than space

frame structures, as shown in Figure 7.7(b) Haselton and Deierlein (2007) found that space

frames typically had superior collapse performance, because of the additional strength from

gravity-load design, and typically failed in full mechanisms. In contrast, we found that

perimeter frames experience lower losses because damage concentrates primarily in a few

stories (15% difference in loss). As with taller buildings, modest amounts of damage

concentration can be beneficial from a loss standpoint. Finally, the architectural design

parameters analyzed had a very small influence on estimated losses (≤2% difference in

loss). Different realistic office layouts, with varying open office space and amount of

partitions, had essentially the same predicted losses. Varying the level of finishes had a

significant effect on the magnitude of loss, but little effect when the loss was normalized by

the building replacement cost (because higher quality finishes are also more expensive to

repair and replace).

7.5.2 Expected Annual Losses

EAL results for all the modern, code-forming RC buildings analyzed in this study

are shown in Figure 7.10. The expected annual losses vary between 0.4% and 1.5% of the

replacement costs (with one extreme outlying value of 3.3%), depending on the structure,

with the mean EAL for this set of structures of approximately 1.0%. The 20-story space

frame (20-S-20-A-G) has the lowest predicted expected annual losses, primarily because

damage tends to concentrate in smaller fraction of stories in taller building. The highest

expected annual loss relative to its replacement cost (3.3%) was experienced by a 1-story

perimeter frame structure (1-P-20-A-G, Design #4). As was described previously, this

building may not be representative of common engineering practice and can be considered

an outlier. If this structure is excluded, the mean EAL for this structure reduces to 0.94%.


0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930

Design Number

No

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Exp

ecte

d A

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0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

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Design Number

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Exp

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d A

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0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

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Design Number

No

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Exp

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d A

nnua

l Lo

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Figure 7.10 EAL results for 30 code-conforming RC frame structures

Multiple measures of seismic performance are useful when comparing design

alternatives. Figure 7.11 shows the mean annual frequency of collapse for the set of RC

structures as calculated by Haselton and Deierlein (2007). The mean annual frequency of

collapse corresponds approximately to the probability of collapse in a given year, and is an

important metric for life safety. It is clear by comparing Figure 7.10 and Figure 7.11, and

referring to Figure 7.12, that understanding how likely a building is to collapse provides

little information on the value of EAL that building may experience. Design 8-P-20-A-G

(Design #14 in the figure), an 8-story perimeter frame building, has the worst collapse

performance, but its EAL is nearly average for the group of RC frame structures. Figure

7.12 confirms that there is not a strong relationship between EAL and the mean annual

frequency of collapse (correlation coefficient, = -0.40). These observations suggest that

increasing collapse capacity does not necessarily mitigate economic losses and that

consideration of economic losses, need to be addressed separately from collapse.


0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 1112 1314 1516 1718 1920 2122 232425 2627 2829 30

Design Number

MA

F C

olla

pse

x 1

0-4


0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 1112 1314 1516 1718 1920 2122 232425 2627 2829 30

Design Number

MA

F C

olla

pse

x 1

0-4


Figure 7.11 Results of mean annual frequency of collapse for 30 code-conforming RC frame structures (Haselton and Deierlein, 2007).

Performance-based approaches enable consideration of multiple seismic

performance metrics in design. Modern building codes aim to protect life-safety by

reducing the risk of collapse, but these goals do not necessarily translate into good

performance in terms of limiting economic loss. The weak relationship between critical

code design parameters and economic losses is further evident when collapse and loss

metrics are compared to the structural strength of the buildings. Figure 7.12(b) and (c) plots

MAF of collapse and EAL, respectively, as a function of the yield base shear coefficient

from a nonlinear pushover analysis. As expected, there is a relatively strong negative

correlation (ρ=-0.66) between MAF of collapse and the yield base shear coefficient because

collapse is less likely in stronger structures. The weak correlation (ρ=0.42) between EAL

and the yield base shear suggests that strength is not the only design parameter that needs to

be addressed by codes to mitigate large economic losses. This type of information is

important for stakeholders who are making design decisions to protect their investment.

The use of performance-based tools in this study illustrates how this framework is useful to

address both life-safety and economic loss in ways current code approaches cannot.


0.0%

0.4%

0.8%

1.2%

1.6%

0 10 20 30

MAF Collapse (x10-4)

EAL

=-0.41

(a)

0.0%

0.4%

0.8%

1.2%

1.6%

0 10 20 30

MAF Collapse (x10-4)

EAL

=-0.41

(a)

0

5

10

15

20

25

30

0.0 0.2 0.4 0.6

Yield Base Shear Coefficient

= -0.66

MAF Collapse (x10-4) (b)

0

5

10

15

20

25

30

0.0 0.2 0.4 0.6


= -0.66

MAF Collapse (x10-4) (b)

0.0%

0.4%

0.8%

1.2%

1.6%

0.0 0.2 0.4 0.6


EAL

= 0.42

(c)

0.0%

0.4%

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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

7.5.3 Present value of life-cycle costs

Another way of putting these results into a more understandable context, is to

evaluate the present value of the losses over a building’s typical lifespan. Expected Annual

Losses, like those shown in Figure 7.10, can be annualized over a period of time to give

insight on the life-cycle costs associated with earthquake damage. In this study, a

building’s life-span was taken to be 50 years. Each structure’s EAL was multiplied by this

expected lifetime and discounted to obtain the equivalent present value of 50 years of

losses, as shown in Figure 7.13(a) with a discount rate of 3%. For these code-conforming

structures present values of life cycle costs of earthquake damage range from 11% to 85%

of the total replacement value of the structure. Because the discount rate may vary, this


calculation was repeated for a variety of discount rates for the buildings with the minimum

and maximum present value of expected loss, as shown in Figure 7.13(b). Note that design

number 4 is considered as an extreme case and is not included in the calculation for Figure

7.13(b).

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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).

Figure 7.13(b) gives a likely range where present value of expected losses are likely

to be for buildings designed according to present code. This figure highlights two

important things: (1) the range of losses over buildings that are expected to have similar

performance, may end up varying by a factor as high as three, and (2) the losses are fairly

large (ranging from 11% to 36% at a discount of 3%). Assuming the discount rate of 3%,

stakeholders may lose up to 36% of their initial investment due to future earthquakes.


These losses are significant and may not be acceptable to stakeholders, even though these

buildings meet current code requirements. Considering that the structural system is

typically between 15 to 20% of the total initial construction cost (RS Means, 2007), the

additional cost of investing in a lateral resisting system could be worthwhile, compared to

the high life-cycle costs due to earthquakes. Performance-based design tools can be used to

communicate these types of concerns to building owners and other stakeholders.

7.5.4 Comparison to Non-ductile Reinforced Concrete Frame Buildings

Measuring performance in terms of economic losses can be used to quantify how

well changes in US building code provisions over the past 50 years have improved the

seismic design of reinforced concrete moment frame structures. Older reinforced concrete

buildings typically exhibit non-ductile behavior primarily because the reinforcing detailing

requirements were less stringent in previous editions of US building codes (Moehle, 1998).

Smaller amounts of shear reinforcement in beams, columns and joints in these frames

reduce the deformation capacity of these elements and make them more likely to experience

non-ductile failures (Scott et al., 1982). Discontinuous reinforcing through structural joints

and insufficient lap splices may also increase the likelihood of inadequate load transfer

between structural elements, which could lead to loss of vertical carrying capacity. These

types of structural deficiencies increase the probability of damage to structural components

and the probability of collapse that may lead to larger economic losses than those sustained

by modern buildings.

Liel and Deierlein (2008) investigated the seismic performance of non-ductile

reinforced concrete moment frame structures to benchmark these types of structures’

capacity to resist collapse. A set of 26 typical structures (archetypes) were designed, using

the 1967 Uniform Building Code, to be representative of older reinforced concrete moment

frame buildings in California built between 1950 and 1975. Similar modeling and structural

analysis techniques were use as described in section 7.4.2 to assess the collapse capacities

of each building (interested readers are directed to Liel and Deierlein, 2008, for further

details on the procedure and results of this investigation).

Economic loss results were computed for this set of non-ductile structures and use

to compare their performance with the modern, ductile structures created by Haselton and

Deierlein (2007). Figure 7.14(a) compares normalized economic losses conditioned on the


ground motion intensity corresponding to the DBE between non-ductile and ductile frames

for 2, 4, 8, and 12 space and perimeter frames. Only the baseline structures for each

building height are considered in this comparison. The economic losses for the non-ductile

structures range from 42% to 71% of the building’s replacement value with median value of

61%. Comparatively, the ductile structures’ normalized losses at the DBE range from 21%

to 35% with a median value of 25%. Normalized EAL results were also computed and

compared in Figure 7.14(b). The normalized EAL of the non-ductile structures range from

1.1% to 3.4%, with a median of 2.6%, whereas the ductile structures varied from 0.7% to

1.3% and has a median of 1.1%. In both types of loss metrics, the median economic losses

of the older, non-ductile structures are 1.4 times greater than those of the modern buildings.

This means that implementing better detailing requirements that increase structural

deformation capacity, not only reduces the probability of collapse, as demonstrated by Liel

and Deierlein (2008), but also significantly reduces economic losses.

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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


Figure 7.15 compares the disaggregation of EALs into collapse and non-collapse

losses for the 4-story perimeter frame structures. The non-ductile structure’s losses due to

collapse comprise 28% of the total EAL, whereas only 3% of the ductile structure’s EAL is

attributed to collapse. This suggests that collapse plays a larger role in economic loss for

older buildings, which have higher probabilities of collapse. It can be inferred that

introducing ductile detailing has been successful in reducing the likelihood of collapse and

the monetary losses associated with it. However, the results for modern buildings have

shown that the economic losses are still very large, and are primarily due to non-collapse

losses. Any attempt to mitigate losses further in these building would need to focus on

limiting interstory drifts and floor accelerations rather than other parameters that primarily

associated with preventing collapse .

Collpase28%

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Figure 7.15 Comparison of EAL disaggregation of collapse and non-collapse losses for non-ductile and ductile frames

7.5.5 Loss Toolbox Comparison

Previous loss estimation methods have relied heavily on expert opinion and

engineering judgment, making the results open to subjectivity (Aslani and Miranda, 2005).

One of the goals of PEER’s building-specific loss methodologies is to establish a

framework that is rational and generates vulnerability curves from data that is well-

documented. It is also important that the methodology produces results that are consistent

and robust given the same input data and parameters. Mitrani-Reiser and Beck (2007)

developed a computer tool, the MATLAB Damage and Loss Analysis (MLDA) toolbox,


that implements PEER’s component-based loss estimation framework. MLDA was used to

in a previous study (Ramirez et al., 2009) to estimate expected losses for the same set of 30

code-conforming reinforced concrete moment frame office buildings developed by

Haselton and Deierlein (2007), so that they could be compared to the results produced by

the simplified, story-based loss estimation methods used in this study. The results were

compared to identify any discrepancies and the sources of the inconsistencies.

Figure 7.16 compares vulnerability curves produced by this study (solid lines) to

those generated from MDLA (dashed lines). In Figure 7.16(a), the perimeter frame results

appear to be relatively consistent between the two approaches. The curves for space

frames, shown in Figure 7.16(b), demonstrate greater discrepancies between the results of

the two toolboxes. The differences can be attributed to the way the losses due to non-

collapse are formulated.

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Figure 7.16 Comparison of vulnerability curves from this study and from MDLA: (a) perimeter frames (b) space frames

The collapse losses in both approaches are identical because the main equation to

compute these losses (Chapter 3, equation 3.2) and the parameters that go into the equation,

the building replacement costs and collapse fragilities, are the same. Therefore, the way

non-collapse losses are computed can be the only source of discrepancy. The space frames

exhibit larger differences in estimations between the two approaches because the non-

collapse losses play a larger role in the overall losses. Space frames have lower


probabilities of collapse, which increases the percentage of losses that are due to non-

collapse, particularly at lower levels of ground motion intensity. The differences in the way

non-collapse losses are computed between the two approaches have a larger influence on

the space frame results.

Non-collapse losses are dependent on the fragility functions and the repair costs

assumed for each damage state. Although these items constitute many parameters that may

differ and these differences are difficult to track, summing the assumed repair costs of the

most extreme damage state for every component in the building can be used to help explain

some of the discrepancies between the two approaches. Table 7.5 shows the total building

repair costs for the final damage states of structural components and nonstructural

components, normalized by replacement cost of the building, for the 4 story buildings in

Figure 7.16. It can be observed that MDLA assumes that the beam-column components

represent a large portion of the repair costs, especially for space frames (space frames have

greater amount of beam-column joints). Therefore it can be assumed that the most of the

non-collapse losses are due to the beam-column joints in the MDLA losses for space

frames. Finally, the total repair costs value for the space frame is much larger than those

assumed in this study. This may be why MDLA’s estimates are larger than the results of

this study for the space frames as observed in Figure 7.16.

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

MDLA This study MDLA This study

Beam-Column 41.2% 11.4% 102.1% 16.4%Slab-Column 2.2% 6.0% 0.0% 0.0%Nonstructural 19.8% 91.1% 19.0% 91.1%TOTAL 63.1% 108.5% 121.1% 107.5%

Perimeter SpaceComponent

7.5.6 Discussion of results relative to other loss estimation methodologies

Although the expected economic loss results presented in this study are consistent

with results produced by other research efforts using PEER’s framework, they may differ,

in some cases, from those computed using other methodologies (e.g. regional loss


estimation methods, proprietary models in the insurance industry,…etc.). There are several

possible reasons for differences in estimating losses, which are described briefly here.

There may be differences in the way damage is estimated. The fragility functions

used in this study are updated functions that have been developed in the past few years.

These functions may differ from those used by previous loss estimation methods, as was

demonstrated in Chapter 5. The generic fragility functions used to estimate damage for a

large portion of drift-sensitive, nonstructural components in this study were compared to

those used by HAZUS. It was determined that the functions used in this study estimate that

damage will initiate at smaller values than those approximated by the HAZUS functions.

This may result in loss estimations that will be higher than those computed by HAZUS,

given that the displacement demands computed are the same.

The values of displacement demands used to estimate loss may also differ because

the methods of calculating structural response parameters may be different than other

approaches. Many regional loss estimation methods use first-order estimates of structural

response based on single degree of freedom (SDOF) systems such as spectral acceleration

and spectral displacement. The methodology established by PEER uses more complex

structural models to estimate response. The differences in computing structural response

may produce differences in response parameters, which are used to determine losses due to

repair of building components.

Estimation of inelastic displacement by other loss estimation methods is often

computed using equivalent linear methods. This may produce very different displacement

demands than those computed using detailed nonlinear simulation models. As was

observed by the disaggregated results in this study, economic losses in reinforced concrete

moment frame office buildings are largely due to drift-sensitive components. Differing

displacement demands may cause large differences in loss estimations.

Not only are the values of response parameters used in loss estimation dependent on

the way they are computed, they are also dependent on the building properties and

parameters that are estimated. Structural parameters that may have a strong influence on

economic loss estimations include the building’s assumed damping ratio, its estimated

ductility ratio, its height (as was demonstrated earlier in this chapter) and its estimated

fundamental period of vibration. Regional loss estimation methods often group structures

into generic building types and assume that these parameters are the same for structures that

fall under the same type. For instance, HAZUS categorizes modern reinforced concrete


moment frame buildings that are designed to current US building code standards (termed

“High-code seismic design level” by HAZUS, and based on code standards established after

1975) into three basic types: low-rise (1-3 stories), mid-rise (4-7 stories) and high-rise (≥8

stories) structures (according to HAZUS terminology, these building types are coded as

C1L, C1M and C1H, respectively). According to the HAZUS technical manual (NIBS,

2003), the fundamental periods estimated for these building types are 0.40s for the low-rise

structures, 0.75s for the mid-rise structures and 1.45s for the high-rise structures. These can

be compared to the periods computed by Haselton and Deierlein (2007) for the set of

buildings analyzed in this study. Table 7.3 reports that the periods for the 1 and 2-story

buildings vary from 0.42s to 0.71s, the periods for the 4-story buildings vary from 0.86s to

1.16s and the buildings for the 8, 12 and 20-story buildings have periods that vary from

1.57s to 2.63s. These values are all larger than the periods assumed by HAZUS for the low-

rise, mid-rise and high-rise building types, respectively. Larger periods (i.e. structures that

are more flexible) will produce larger displacement demands, which will result in larger

economic losses.

The differences between economic loss estimations computed by PEER’s

framework and those produced by other methods may be substantial, depending on the

approach, and need to be studied further. Future studies are required to reconcile these

differences and validate the results presented here.

7.6 LIMITATIONS

In conducting this study, we encountered a variety of challenges in implementing

the PEER methodology for predicting economic losses that warrant further research. These

items include:

In the current study, uncertainties in structural modeling were incorporated in

predictions of the structural collapse limit state, but not the non-collapse EDPs.

These modeling uncertainties represent the uncertainty in how well the model

reflects the true behavior of the structure. Modeling uncertainties have a particularly

important effect in highly nonlinear limit states such as collapse, because these

phenomena are more difficult to model. Previous studies have shown them to be

less significant for smaller levels of structural response that tend to dominate losses


(e.g. Porter 2002). A detailed examination of the effects of modeling uncertainties

on predictions of structural losses is a topic for future work.

The collapse fragility is appropriately adjusted to reflect the expected spectral shape

in large, rare ground motions (epsilon > 0) (Baker and Cornell 2005). Because the

smaller ground motions that dominate non-collapse losses are typically closer to

epsilon-neutral, this effect is not included for predictions of non-collapse

engineering response. A more complete approach would account appropriately for

the expected shape of ground motions at each stripe (intensity level). This change is

not expected to have significant effects on the overall results.

Economic losses in this study include only direct economic losses associating with

repairing damageable components. The costs associated with building closure and

the downtime needed to conduct repairs may be an even more significant source of

loss to building owners (Comerio 2006). In future, these studies should include as

many possible sources of economic loss related to seismic damage as possible. By

including all possible sources of economic losses, life cycle costs associated with

earthquakes can be more completely determined and used by building owners in

making important decisions about earthquake investment.

7.7 CONCLUSIONS

PEER’s performance-based earthquake engineering methodology was used to

evaluate economic losses for a set of 30 modern concrete moment frame buildings. All the

buildings were designed -- structurally and architecturally -- to provide a realistic inventory

of damageable components. The structural designs are representative of modern code-

conforming structures and were reviewed by a practicing engineer as part of the Applied

Technology Council Project 63 (ATC 2008). Results from probabilistic seismic hazard

analysis and incremental dynamic analysis performed by previous investigators (Goulet et

al. 2007, Haselton and Deierlein 2007) were used to evaluate the site’s hazard and structural

response respectively. Ramirez and Miranda’s toolbox (2009) was then used to calculate

damage and economic loss, defined as cost of repairing earthquake damage, of each

structure. Seismic performance metrics generated in this study include expected losses


conditioned on seismic intensity, expected annual losses, and present values of life cycle

costs associated with repairing earthquake damage.

Expected losses were calculated at several values of intensity, but results at the

DBE in particular were used to identify trends. Losses at this intensity were found to be a

significant percentage of the replacement value, ranging from 13% to 54%. This means that

although current code-conforming buildings have small probabilities of collapse at the DBE

intensity, the economic losses sustained are very large. The mean EAL for this set of

structures is approximately 1.0% of the replacement cost, but varied between 0.4% and

3.3%, suggesting that current US building codes provide an inconsistent level of seismic

performance when considering monetary loss. This set of structures also exhibited life-

cycle costs associated with earthquake damage that ranged from 8% to as much as 37%

over a 50 year period, despite conforming to modern building codes. Like the scenario-

based results for a given level of ground motion intensity, these losses are large and a

considerable amount of the buildings’ initial investment.

Different design parameters, both structure-related and architecture-related, were

varied within the set to examine how they affected performance. Of the code-conforming

structural parameters considered in this study, building height was found to have the largest

influence on expected losses. Taller buildings experienced less relative loss than shorter

buildings because damage concentrates in a small percentage of floors as opposed to

damage throughout all levels of the structure. Structural parameters that explored the effect

of particular code limitations were also investigated. Although increasing the design

SCWB ratio to values greater than one reduced the economic losses due to collapse, it did

not significantly decrease the overall expected losses. On the other hand, modifying the R-

factor had a larger effect on expected losses at the DBE. If code committees are interested

in adjusting code provisions to limit direct economic losses due to seismic ground motions,

this study suggests that they would be best served by making amendments to the R-factor.

Stiffness limitations and the Cd factor are also expected to have significant effects on

economic losses.

It was also demonstrated that designing for life-safety does not necessarily translate

to satisfactory economic loss performance. The parameters of EAL and the MAF of

collapse were found to be uncorrelated for these set of buildings. Performance-based

design tools provide separate quantifiable metrics to address collapse risks and economic

losses. The ability to quantify both metrics also provides decision-makers with information


to identify and consider possible tradeoffs between mitigating losses and reducing collapse

risk.

When the modern set of ductile reinforced concrete buildings were compared to a

set of non-ductile frame structures, representative of older buildings in California, it was

determined that these buildings pose an even greater risk with expected losses that were 1.4

times as great. The non-ductile detailing of these older structures make them more

susceptible to sudden, non-ductile structural collapses as reflected by their larger collapse

probabilities. Consequently, collapse losses increase the overall losses in these buildings

and constitute a larger percentage of the total loss. Although the introduction of improved

detailing requirements into US code provisions reduces the expected economic losses for

newer structures, a significant amount these older reinforced concrete frame buildings are

still in use (Liel and Deierlein, 2008) and vulnerable to losses much larger than their

modern counterparts.

While modern U.S. building codes seem to do a fairly good job at preventing

collapse and protecting life-safety in moderate magnitude earthquakes many owners may be

surprised to learn that these codes do not do as well in protecting their infrastructural

investment. As this study shows, structures designed according to seismic provisions may

incur significant earthquake damage and economic losses, and the losses are even more

significant in older structures. Advancements in code provisions over recent decades, such

as the SCWB ratio have not led to nearly as much reduction in losses as in collapse risk.

Performance-based design provides important metrics of losses and collapse risk for

building owners and engineers making design decisions. Life-cycle costs associated with

earthquake damage can be evaluated for different types of structural systems and the

benefits of investing in a better lateral-force resisting system can quantified to demonstrate

the value of different alternatives. Engineers will be able to provide stakeholders with the

right type of information that they can use to make wise business decisions that protect their

investment.

CHAPTER 8 178 Variability of Economic Losses

CHAPTER 8

8 VARIABILITY OF ECONOMIC LOSSES

This Chapter is based on the following publication:

Ramirez, C.M., Miranda, E., Baker, J.W. (2009), “Building-level Construction Cost

Correlations in Loss Estimation Uncertainty,” Journal of Earthquake Engineering and

Structural Dynamics, (in preparation).


Ramirez developed the proposed method, computed the results for the example

building and authored the publication. The structural analysis results for the example

building were taken from a previous study performed by Haselton and Deierlein (2007).

Baker and Miranda served as advisors for the project with Miranda being the principal

investigator.

8.2 INTRODUCTION

Estimating expected annual monetary losses in a building due to damage from

seismic ground motions is of great interest to stakeholders & decision makers.

Consequently, building-specific loss estimation is also currently being incorporated into

performance-based design methodologies. For instance, one approach taken by the Pacific

Earthquake Engineering Research (PEER) Center is to use expected losses as a metric for

structural performance during seismic events. In addition to knowing mean losses, there is

also interest in quantifying the probable range these economic losses can lie within and how

large these losses can become. Computing this type of information requires knowing the

variability associated with estimating these losses. Both the mean and variability of


seismic-induced economic losses are also needed to compute the mean annual frequency of

exceedance of particular value of loss. This metric provides information on the likelihood

of experiencing an economic loss greater than a particular value.

PEER’s probabilistic framework for assessing building seismic performance involves

segmenting the analysis into four basic modules that account for the following sources of

variability: ground motion hazard of the site, structural response of the building, damage of

building components and repair costs. Baker and Cornell (2003) presented methods of

characterizing and propagating variability from each of these stages of the framework.

Methods to incorporate correlations and models to approximate these correlations when

empirical data is lacking were presented. Aslani and Miranda (2005) expanded on these

techniques by formulating more specific closed-form equations that computed loss

dispersions and correlations per building component (component-level). This study also

was the first to consider correlations between construction costs of different subcontractors

and applied them to the component-level equations they developed. The equations

developed were used to calculate expected losses and dispersions for a test-bed structure to

illustrate and validate PEER’s framework. The findings of this study demonstrated that

correlations can significantly affect the value of the loss variability.

Unfortunately, incorporating correlations at the component level does not correspond

to the way construction costs of projects are typically formulated. Construction costs are

typically generated by accepting bids from contractors who in turn use bids from sub-

contractors to price projects. Consequently, much of the data available on dispersions and

correlations of construction costs are at a building-level, rather than at the component-level,

and divided by type of sub-contractors involved. This creates a discrepancy between the

framework that quantifies uncertainties of losses due to earthquakes and the data available

to generate dispersions and correlations of construction costs.

A new approach that calculates the variability of economic loss that is better suited to

use building-level construction cost data is proposed in this study. Instead of evaluating

dispersion of economic loss for each damaged building component, replacement and repair

costs for the entire building are grouped by construction subcontractor. The available cost

data can then be used to approximate the variability associated with the losses for each

subcontractor and the correlations between them. The approach is implemented using two

methods to propagate variability: an analytical method using FOSM approximations and a

method using Monte Carlo simulations. First, an overview of all the types of variability and


correlations that are being considered and the sources of data used to calculate them is

explained. Following this overview the proposed method is presented. Specific issues that

could significantly influence the computation of the variability of economic loss were

examined in detail, including: (1) the effect of the number of ground motions considered

during structural analysis on the quality of response parameter correlation coefficient

estimates, and (2) the effect of inherent correlation between subcontractor losses due to

response parameter variance. A sample 4-story reinforced concrete moment-resisting frame

office building (Design ID 4-P-30-A-G in Table 7.3) that was designed and analyzed by

Haselton & Deierlein (2007) was used to illustrate the new methodology and to examine the

effects of response parameter and cost correlations on loss variability. Dispersion results

are computed using both the analytical and Monte Carlo methods to assess how well the

FOSM approximations are able to reproduce the results from simulation.

8.3 TYPES OF LOSS VARIABILITY & CORRELATIONS

Current building-specific economic loss estimation methodologies (Krawinkler and

Miranda 2004, Aslani and Miranda 2005, Mitrani-Reiser 2007) compute expected losses for

a given level of seismic intensity using the total probability theorem as a weighted sum

between the expected losses conditioned on the building collapsing and the expected losses

conditioned on the building not collapsing (as described in greater detail in Chapter 3).

Correspondingly, the variability of loss for a given level of seismic intensity is a function of

the variability of losses conditioned on structural collapse and the variability of losses

conditioned on non-collapse and the correlation between the two. The variability of loss

conditioned on structural collapse is only due to the variability of construction costs

because the entire building needs to be replaced. This implies that variability of replacing

the building is the same as the variability in cost to construct a new building. The

variability of loss conditioned on non-collapse is more complex. Because estimating these

types of losses involves approximating structural response and estimating damage to

building components, the variability associated with these sources of variability must be

considered in addition to the variability in construction repair costs.

Consequently, there are three primary sources of variability to consider when

computing the variability of expected economic loss for a given seismic intensity: (1) the


variability in response parameters (which are also referred to as engineering demand

parameters, EDPs, using terminology established by PEER); (2) the variability in

construction costs; and (3) the variability in fragility functions (functions that are used to

estimate levels of damage in building components). There is additional variability caused

by three types of correlations that influence the value of economic dispersion: (1)

correlations between response parameters; (2) correlations between construction costs of

different subcontractors; and (3) correlations between fragility functions of different

damage states. In addition to these correlations, there exists an inherent correlation between

economic losses from the costs of different subcontractors that are conducting construction

work on the same floor/story. This type of correlation, which has not been considered

before, exists because building components and assemblies on the same floor/story are

affected by the same response parameters therefore even if the repair/replacement costs of

different subcontractors were uncorrelated the dispersion total cost in the building will be

affected because the damage to be repaired by different subcontractors affected by a given

EDP in one floor/story (e.g. IDR, FPA) is strongly correlated. The types of variability

presented here only consider aleatoric uncertainty, and the influence of epistemic/modeling

uncertainty is outside the scope of this study. The following sections present these sources

of variability in detail and correlations associated with each type of dispersion.

8.3.1 Variability and Correlations in Construction Costs

Data on construction costs are scarce because of the way contractors price

construction projects. Contractors keep internal cost databases and price projects based on

previous costs of past projects. These databases are usually kept private to maintain a

competitive edge when bidding for projects, and as a result, are not typically accessible to

researchers. Furthermore, although subcontractors may use unit costs for preparing their

bids, usually their budgets only include budgets and schedules for all of the work that they

will be conducting and not broken down component by component. Therefore, variability of

constructions costs and the correlation among them is not available because is typically not

provided to general contractors.

Building-specific loss estimation has typically computed expected earthquake losses

and loss dispersions on a component-by-component basis. Aslani and Miranda (2005)

developed closed-form solutions to compute the loss dispersion for the entire building by


summing the loss dispersions of individual building components. Their method also

introduced a way of incorporating component cost correlations into the calculation of loss

variability. However, the estimated values of component dispersions and correlations used

in their study were based on translating building-level cost data into component-level values

because component-level construction cost data was not available.

The construction cost data used by Aslani and Miranda (2005) were based on

research conducted by Touran and Wiser (1992) and Touran and Suphot (1997). These

investigations collected data on construction costs and determined dispersions and

correlation coefficients between different construction subcontractors. Data for these studies

was provided by R.S. Means, Inc., Kingston. Massachusetts. Values for variability were

extracted from unit cost data from 1,014 low-rise office buildings collected between the

years 1982 to 1992. Table 8.1 shows the different subcontractors considered in the cost

data. This table also shows the averages and standard deviations of the cost per square foot

of each of the construction items. As shown in this table, construction costs are

characterized by very large variabilities with logarithmic standard deviations ranging from

0.57 to 0.89 and coefficients of variations (COV) varying between 0.61 and 1.1. It should

be noted that these variabilities are higher than the typical variability in attenuation

relationships to estimate a spectral ordinate for a given magnitude and distance.

Table 8.1 Statistical data of construction costs per subcontractor (Touran & Suphot, 1997)

SubcontractorMean ($/sf)

StdDev ($/sf)

COV LN MeanLN

StdDevConcrete 5.8 5.52 0.95 1.44 0.8Masonry 3.86 3.28 0.85 1.08 0.74Metals 5.12 3.8 0.74 1.41 0.66Carpentry 3.58 3.96 1.11 0.88 0.89Moisture Protection 2.95 2.44 0.83 0.82 0.72Doors, windows, and glass 3.78 3.11 0.82 1.07 0.72Finishes 5.84 3.74 0.64 1.59 0.59Mechanical 8.94 5.84 0.65 2.01 0.6Electrical 5.61 3.45 0.61 1.56 0.57

Table 8.2 displays the corresponding correlation coefficients between the

construction subcontractor costs from the data gathered by Touran & Suphot (1997).

Correlations between subcontractor costs are typically caused by the similarities and

relationships between the type of work, the amount of work and the scope of work that each


subcontractor does. As shown in this table, with a few exceptions, costs from different

subcontractors are typically positively correlated with several of them strongly correlated.

For instance, the costs due to the mechanical and electrical subcontractors are highly

correlated (0.79). Typically the amount and complexity of work that the mechanical

subcontractor does is closely related to the work the electrical subcontractor conducts

because much the building’s mechanical equipment and infrastructure is dependent on the

building’s electrical system.

Table 8.2 Correlation coefficients of construction costs between different subcontractors

Subcontractor Concrete Masonry Metals CarpentryMoisture

Protection

Doors, Windows,

GlassFinishes Mech'l Electrical

Concrete 1.00 0.06 0.11 -0.04 0.19 0.12 0.21 0.38 0.40Masonry 0.06 1.00 -0.01 0.11 0.06 0.14 0.22 0.33 0.30

Metals 0.11 -0.01 1.00 -0.33 0.12 0.39 0.20 0.35 0.33Carpentry -0.04 0.11 -0.33 1.00 0.18 0.03 0.07 -0.02 0.07

Moisture Protection 0.19 0.06 0.12 0.18 1.00 0.16 0.18 0.18 0.15Doors, Windows, Glass 0.12 0.14 0.39 0.03 0.16 1.00 0.25 0.28 0.36

Finishes 0.21 0.22 0.20 0.07 0.18 0.25 1.00 0.42 0.44Mechanical 0.38 0.33 0.35 -0.02 0.18 0.28 0.42 1.00 0.79

Electrical 0.40 0.30 0.33 0.07 0.15 0.36 0.44 0.79 1.00

Unfortunately, the use of building-level cost dispersions and correlations to

approximate values at the component level is difficult to validate. An improved approach is

needed to make use of the cost data provided by Touran & Suphot (1997). The first step of

the approach proposed in this study is establishing a relationship between the cost of each

component to the costs of each subcontractor for the entire building. To accomplish this, the

cost of each component is divided into the contributions of each subcontractor. The cost of

each subcontractor can then be summed to get the total cost for the entire building.

In Chapter 3 of this dissertation, an inventory of building components was

developed to estimate losses for a reinforced concrete moment-frame office building, using

a simplified methodology of building-specific loss estimation. Table 8.3 shows the cost of

each component distributed into each of the subcontractors listed in Touran & Suphot

(1997). The assumed cost distributions were developed with the help of a former cost

estimator, Carlos Sempere, who now works as a design engineer for Forell/Elssesser

Structural Engineers, Inc. These distributions were used to compute loss dispersion for

each subcontractor for the entire building, so the correlations between the costs of different

sub-contractors can be accounted for.


Table 8.3 Example cost distribution between construction subcontractors of each component in a typical story of an office building

A. SUBSTRUCTUREFoundation 25% 75% 0% 0% 0% 0% 0% 0% 0% 0%


Columns 0% 100% 0% 0% 0% 0% 0% 0% 0% 0%Slab 0% 100% 0% 0% 0% 0% 0% 0% 0% 0%Beam-column joint 0% 100% 0% 0% 0% 0% 0% 0% 0% 0%Slab-column joint 0% 100% 0% 0% 0% 0% 0% 0% 0% 0%

B20 Exterior EnclosureExterior Walls 0% 0% 0% 0% 0% 0% 0% 100% 0% 0%Exterior Windows 0% 0% 0% 0% 0% 0% 100% 0% 0% 0%Exterior Doors 0% 0% 0% 0% 0% 0% 100% 0% 0% 0%

B30 RoofingRoof Coverings 0% 0% 0% 0% 0% 100% 0% 0% 0% 0%Roof Openings 0% 0% 0% 0% 0% 100% 0% 0% 0% 0%

C. INTERIORSPartitions with finishes 0% 0% 0% 40% 0% 0% 0% 60% 0% 0%Interior Doors 0% 0% 0% 0% 0% 0% 100% 0% 0% 0%Fittings 0% 0% 0% 0% 0% 0% 50% 50% 0% 0%Stair Construction 0% 5% 0% 90% 0% 0% 0% 5% 0% 0%Floor Finishes 0% 0% 0% 0% 0% 0% 0% 100% 0% 0%

0% 0% 0% 0% 0% 0% 0% 100% 0% 0%0% 0% 0% 0% 0% 0% 0% 100% 0% 0%

Ceiling Finishes 0% 0% 0% 0% 0% 0% 0% 100% 0% 0%


Elevators & Lifts 0% 0% 0% 0% 0% 0% 0% 0% 100% 0%0% 0% 0% 0% 0% 0% 0% 0% 100% 0%

Escalators & Moving WalksD20 Plumbing

Plumbing Fixtures 0% 0% 0% 0% 0% 0% 0% 0% 100% 0%0% 0% 0% 0% 0% 0% 0% 0% 100% 0%

Domestic Water Distribution 0% 0% 0% 0% 0% 0% 0% 0% 100% 0%Rain Water Drainage 0% 0% 0% 0% 0% 0% 0% 0% 100% 0%

D30 HVACEnergy Supply 0% 0% 0% 0% 0% 0% 0% 0% 100% 0%Heat Generating Systems 0% 0% 0% 0% 0% 0% 0% 0% 100% 0%Cooling Generating Systems 0% 0% 0% 0% 0% 0% 0% 0% 100% 0%Terminal & Package Units 0% 0% 0% 0% 0% 0% 0% 0% 100% 0%

0% 0% 0% 0% 0% 0% 0% 0% 100% 0%Other HVAC Sys. & Equipment 0% 0% 0% 0% 0% 0% 0% 0% 100% 0%

D40 Fire ProtectionSprinklers 0% 0% 0% 0% 0% 0% 0% 0% 100% 0%Standpipes 0% 0% 0% 0% 0% 0% 0% 0% 100% 0%

D50 ElectricalElectrical Service/Distribution 0% 0% 0% 0% 0% 0% 0% 0% 0% 100%Lighting & Branch Wiring 0% 0% 0% 0% 0% 0% 0% 0% 0% 100%Lighting & Branch Wiring 0% 0% 0% 0% 0% 0% 0% 0% 0% 100%Lighting & Branch Wiring 0% 0% 0% 0% 0% 3% 0% 0% 0% 100%Communications & Security 0% 0% 0% 0% 0% 0% 0% 0% 0% 100%Communications & Security 0% 0% 0% 0% 0% 0% 0% 0% 0% 100%Communications & Security 0% 0% 0% 0% 0% 0% 0% 0% 0% 100%Other Electrical Systems 0% 0% 0% 0% 0% 0% 0% 0% 0% 100%

Performance Group CarpentryDoors,

Windows, Glass

FinishesSitework Concrete Masonry Metals Mechanical ElectricalMoisture

Protection

8.3.1.1 Inherent subcontractor cost correlation due to work performed on the same floor/story

In addition to the correlations between subcontractor costs that are obtained from

construction cost data, there is another type of correlation between subcontractor losses that

needs to be accounted for. Monetary losses due to work done by different subcontractors on

drift-sensitive components on the same floor/story will be fully correlated because the

damage on that floor/story will depend on the same EDP. Similarly, economic losses due to

repairs done by different subcontractors on acceleration-sensitive components on the same

floor/story will be fully correlated because the damage on that floor/story will depend on

the same EDP. As a result, when the losses due to each subcontractor are summed for the

entire building, the total losses due to each subcontractor will be correlated because their

losses per floor are fully correlated. That is, a significant amount of work done by different

subcontractors will be correlated because portions of the work are triggered by the same


EDPs. The covariance associated with this type of “inherent” correlation is computed from

EDP variability and occurs even when EDPs are uncorrelated. This type of correlation is

different from the correlations arising from correlated subcontractor cost or from correlated

EDPs and has not been considered in previous economic loss estimation research.

This phenomenon is best understood by considering a simple, hypothetical example.

Consider a 2-story building with losses resulting only from repair work conducted by two

subcontractors, subcontractor k (e.g. concrete subcontractor) and subcontractor k’ (e.g.

metals subcontractor). For this example, assume that economic losses are caused by only

two EDPs located in the 1st and 2nd floors (e.g. IDR in the 1st and 2nd stories) which are

denoted by EDP1 and EDP2, respectively.. Economic losses can be calculated using

relationships that relate EDP to monetary loss, termed EDP-DV functions, as described in

Chapter 3 and later in section 8.4.3.1.

Figure 8.1 plots EDP-DV functions for each subcontractor loss for this example. If

the EDP-DV function for subcontractor k is denoted as gk(EDPj) for and EDP at the jth

story, then economic losses for the 1st and 2nd stories are computed as follows:

,1 1k kL g EDP (8.1)

,2 2k kL g EDP (8.2)

where Lk,1 and Lk,2 denotes the value of loss due to subcontractor k due to EDPs in the 1st

and 2nd floors, respectively. Similarly, Lk’,1 and Lk’,2 denote the losses due to subcontractor

k’ in the 1st and 2nd floors and can also be computed using the EDP-DV function gk’(EDPj).

Note that losses computed from both subcontractors are dependent on the same EDPs. To

compute the total economic losses per subcontractor for the entire building, these losses can

be summed using the following equations:

,1 ,2k k kL L L (8.3)

,1 ,2k k kL L L (8.4)


where Lk and Lk’ are the total losses for all stories of subcontractor k and subcontractor k’,

respectively. To facilitate the understanding of the source of this additional correlation on

the cost, equations (8.1) and (8.2) can be substituted into (8.3), and the analogous equations

for subcontractor k’ can be substituted into (8.4), to obtain:

1 2k k kL g EDP g EDP (8.5)

1 2k k kL g EDP g EDP (8.6)

These equations demonstrate that because 1kg EDP and 1kg EDP are both dependent

on EDP1 they are highly correlated. Similarly, 2kg EDP and 2kg EDP are also highly

correlated because both are dependent on EDP2. This implies that correlation will also

exists between the losses due to work performed by subcontractor k and subcontractor k’, Lk

and Lk’ even if constructions costs of subcontractors k and k’ are uncorrelated.

Lk,1

Lk,2

EDP2

EDP

E[Lk | EDP]

EDP1

11 22

(a) Subcontractor k

k,1k,1

k,2

Lk,1

Lk,2

EDP2

EDP

E[Lk | EDP]

EDP1

11 22

(a) Subcontractor k

k,1k,1

k,2


Lk’,1

Lk’,2

EDP2

EDP

E[Lk’ | EDP]

EDP1

11 22

(b) Subcontractor k'

k’,1

k’,1

k’,2

k’,2

Lk’,1

Lk’,2

EDP2

EDP

E[Lk’ | EDP]

EDP1

11 22

(b) Subcontractor k'

k’,1

k’,1

k’,2

k’,2

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'

This type of correlation will influence the variability of Lk and Lk’ even when the

EDPs are uncorrelated. EDP1 and EDP2 can be modeled as random variables using

probability distributions as shown in the figure with standard deviations, 1 and 2,

respectively. The EDP-DV functions can be used to propagate the dispersion of the EDPs

to loss dispersions for each subcontractor at each floor using either FOSM or simulation

methods (as will be demonstrated in greater detail later in section 8.4.3.2). For the purposes

of this illustrative example, FOSM can be used to compute the dispersion of the losses due

to EDP variability as follows:

2,1k

1

2

2 2,1 1

1EDP

kk

m

g

EDP

(8.7)

1

2

2 2,2 2

2EDP

kk

m

g

EDP

(8.8)


where k,1 and k,2 are the resulting standard deviations of the loss of subcontractor k in the

1st and 2nd floors, respectively, and 1

1EDP

k mg EDP and

22

EDPk m

g EDP are the

derivatives of k jg EDP evaluated at the mean values of EDP1 (mEDP1) and EDP2 (mEDP1),

respectively. Likewise, the standard deviations of the loss of subcontractor k’, denoted k’,1

and k’,2 for dispersions at the 1st and 2nd floor respectively, can be found using analogous

expressions using the derivatives of k jg EDP . Assuming that the EDPs at different

stories are uncorrelated, the total dispersion for each subcontractor can be found using the

sum of squares as shown using these equations:

2 2 2,1 ,2k k k (8.9)

2 2 2,1 ,2k k k (8.10)

where k and k’ and are the standard deviations of economic losses for work performed by

subcontractor k and subcontractor k’ in all floors, respectively. To obtain the dispersion of

economic losses of the entire building, the results from equations (8.9) and (8.10) can be

summed together. However, because this inherent correlation between losses due to work

performed by different subcontractors exists, the covariance produced by this correlation

needs to be accounted for as follows:

2 2 2,2BLDG k k k k (8.11)

where BLDG is the standard deviation of economic loss for the entire building and k,k’ is the

covariance between losses due work performed by subcontractor k and subcontractor k’ that

results from this inherent correlation. This covariance can be estimated as follows:

1 1 2 2

2 2, 1 2

1 1 2 2EDP EDP EDP EDP

k k k kk k

m m m m

g g g g

EDP EDP EDP EDP

(8.12)


Note that equations (8.11) and (8.12) do not involve EDP correlations or construction cost

correlations described in the previous section. This means that this inherent correlation may

affect the overall loss dispersion of the building even if the EDPs and constructions costs

between different subcontractors are uncorrelated. A more generalized version of

computing the variances that account for all these sources of correlation is described in

greater detail in section 8.4.3.

8.3.2 Variability and Correlation in Response Parameters

Capturing the response of a structure to estimate economic loss can be done by

using response parameters such as interstory drift ratio (IDR) and peak floor accelerations

(PFA). Values of peak response parameters can widely vary depending on the ground

motion the structure is subjected to. Figure 8.2 shows the interstory drift in the second story

for a 4-story reinforced concrete moment-resisting frame (ID1009 Haselton & Deierlein,

2007) when subjected to 80 different ground motions scaled to the same intensity level

(measured as spectral acceleration at the fundamental period, Sa(T1) = 0.05g. Note that in

the figures and tables in this chapter will denote this value as Sa for brevity, however, it is

implied that this value is at the fundamental period). The structural analysis was performed

using incremental dynamic analysis (IDA). Record-to-record variability due to frequency

content and other ground motion characteristics introduce this type of dispersion into

predicting structural response and consequently into our estimation of economic loss.


IDR @ 2nd Floor

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0% 2% 4% 6% 8% 10%

IDR [%]

IM = Sa [g] IDR @ 2nd Floor

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0% 2% 4% 6% 8% 10%

IDR [%]

IM = Sa [g]

Figure 8.2 EDP data from incremental dynamic analysis at increasing IM levels

In addition to the record-to-record variability in EDPs, additional variability can

arise from the correlations between EDPs. EDP correlations typically occur because

physical relationships between the parameters based on several factors which include: the

type of response parameter (e.g. IDR vs. PFA), the spatial proximity of the response

parameters, the degree of participation of higher modes and the level and location of

nonlinearities (i.e. whether the elements behave elastically or inelastically). The type of

parameter and spatial proximity of the response parameters may produce highly correlated

EDPs when the parameters are of the same type and located close to each other. However,

this may not be the only situation in which correlations exist depending on the other two

factors. If the structure’s higher modes do not participate in its response and its behavior is

first mode dominated, then EDPs at floors/stories may have high correlation even if they are

not located close to each other. Even different types of response parameters may be highly

correlated if the structure behaves elastically because linear relationships between floor

displacements and accelerations exist before the structure yields.

Figure 8.3 illustrates examples of different levels of correlation between EDPs for

the example 4-story building at a ground motion intensity level that has a spectral

acceleration value of 0.55g at the fundamental period. Figure 8.3(a) shows the relationship

between interstory drift ratios in the first and second stories as an example EDPs that are

highly correlated. The calculated correlation coefficient is 0.97 for these two parameters.


An example of partial correlation is plotted in Figure 8.3(b). The relationship between PFA

in the second floor and IDR in the second story has a correlation coefficient of 0.58.

Finally, EDPs can also be uncorrelated as shown in Figure 8.3(c); this figure shows the

relationship between IDR in the second floor and PFA in the fourth floor, which has a

correlation coefficient of 0.041.

0.00

0.02

0.04

0.06

0 0.02 0.04 0.06 0.08

IDR @ 1st Story

IDR @ 2nd Story

= 0.97

(a) Sa = 0.55g

0.0

0.5

1.0

1.5

0 0.01 0.02 0.03

IDR @ 4th Story

PFA @ 4th Floor [g]

= 0.58

(b) Sa = 0.55g

0.0

0.5

1.0

1.5

0 0.02 0.04 0.06 0.08

IDR @ 1st Story

PFA @ 4th Floor [g]

= 0.041

(c) Sa = 0.55g

Figure 8.3 Example of EDP relationships with different levels of correlation

Correlations between response parameters can also vary as the intensity level of the

ground motions is increased. Figure 8.4 shows scatter plots between different EDPs at two

different levels of ground motion intensity. It can be observed from these plots that

parameters that are highly correlated at lower intensity levels can either remain highly

correlated, or decrease their level of correlation at higher levels of intensity (Figure 8.4 (a)

to (d)). On the other hand, parameters that are initially poorly correlated, can increase their

level of correlation as the earthquake intensity is increased (Figure 8.4 (e) to (f)). To better

illustrate this, Figure 8.5 plots correlations coefficients for a range of seismic intensities

(measured in spectral acceleration at the fundamental period, Sa(T1)). The figure shows the

same three trends of variation in correlation coefficients in Figure 8.4. Most changes in

correlation occur because of the change from elastic to inelastic behavior when damage

occurs to the structure. For instance, IDR in the fourth story and PFA in the fifth floor are

highly correlated at lower ground motion intensities because they are close in proximity to

each other, the structure is likely first-mode dominated (low-rises structure) and because

linear relationships exist between displacements and accelerations before the structure

yields. However, the EDP correlation decreases at higher seismic intensities because at


these ground motions the structure has yielded and lateral deformation concentrates in

members where plastic hinges have developed. Once these hinges form, the linear

relationships between displacements and accelerations do not necessarily hold.

0.000

0.002

0.004

0.006

0.008

0.010

0 0.002 0.004 0.006 0.008

IDR @ 1st Story

IDR @ 2nd Story

= 0.97

(a) Sa = 0.25g

0.00

0.02

0.04

0.06

0.08

0.10

0 0.02 0.04 0.06 0.08 0.1

IDR @ 1st Story

IDR @ 2nd Story

= 0.95

(b) Sa = 1.6g

0.0

0.1

0.2

0.3

0.4

0.5

0 0.001 0.002 0.003 0.004

IDR @ 4th Story

PFA @ 5th Floor [g]

= 0.87

(c) Sa = 0.25g

0.0

0.5

1.0

1.5

2.0

2.5

0 0.02 0.04 0.06 0.08 0.1

IDR @ 4th Story

PFA @ 5th Floor [g]

= 0.18

(d) Sa = 1.6g

0.00000

0.00005

0.00010

0.00015

0.00020

0.00025

0 0.001 0.002 0.003 0.004

IDR @ 4th Story

RIDR @ 4th Story

= 0.22

(e) Sa = 0.25g

0.00

0.02

0.04

0.06

0.08

0.10

0 0.02 0.04 0.06 0.08 0.1

IDR @ 4th Story

RIDR @ 4th Story

= 0.79

(f) Sa = 1.6g

Figure 8.4 Correlation trends at low and high seismic intensity levels


-1.00

-0.50

0.00

0.50

1.00

0.0 0.5 1.0 1.5 2.0

IM = Sa [g]

IDR2 & IDR1

-1.00

-0.50

0.00

0.50

1.00

0.0 0.5 1.0 1.5 2.0

IM = Sa [g]

PFA5 & IDR4

-1.00

-0.50

0.00

0.50

1.00

0.0 0.5 1.0 1.5 2.0

IM = Sa [g]

RIDR4 & IDR4

(a) (b) (c)

-1.00

-0.50

0.00

0.50

1.00

0.0 0.5 1.0 1.5 2.0

IM = Sa [g]

IDR2 & IDR1

-1.00

-0.50

0.00

0.50

1.00

0.0 0.5 1.0 1.5 2.0

IM = Sa [g]

PFA5 & IDR4

-1.00

-0.50

0.00

0.50

1.00

0.0 0.5 1.0 1.5 2.0

IM = Sa [g]

RIDR4 & IDR4

(a) (b) (c)

Figure 8.5 Variation of EDP correlation with intensity level

8.3.2.1 EDP correlation coefficient estimates

Although structural analysis data is readily available to determine EDP correlations,

it must be recognized that the values calculated are only estimations of the true correlations.

Obtaining the true correlations requires subjecting the considered building to all ground

motions possible at the structure’s site. This is impossible because all possible ground

motions are not known. Instead, the building is subjected to a sample of ground motions

from which only approximate what the EDP correlations can be obtained. The accuracy of

these approximate correlations depends on the number of ground motions being sampled in

the structural analysis. Unfortunately, using a large number of ground motions can be

computational expensive and time consuming. The issue of how many ground motions

should be considered has been much debated. However, previous work has been primarily

aimed at determining the number of ground motions required to obtain estimates of median

response/loss (e.g. expected loss, probability of collapse…etc.). There has been little work

on how many ground motions are necessary to obtain good estimates of EDP correlation

coefficients.

Since the actual distribution of correlation coefficients is unknown, an approximate

distribution can be obtained by using bootstrap sampling. Bootstrap sampling is a

resampling method (with replacement) that uses data collected for a single sample to

simulate results for multiple samples, without repeating the procedure required to obtain the

original data (Efron 1979, 1982). It is often used in studies where data collection is limited,

such as in medical studies. Bootstrap sampling will used here to demonstrate how this

procedure can be used to obtain reliable estimates of response parameter correlation


coefficients. Further, it will also be shown how bootstrap sampling can be used to evaluate

how reliable the correlation coefficient estimates are depending on the sample size of the

ground motions being considered.

The same 4-story concrete moment frame building taken from Haselton & Deierlein

(2007) previously discussed in the preceding paragraphs can be used to illustrate the use of

the bootstrap method to predict EDP correlations coefficients. Haselton & Deierlein used a

sample of 80 ground motions to obtain EDPs at different values of ground motion intensity.

For a given intensity level, only one set of EDP correlations coefficients can computed from

this sample. However, there is no way of knowing how variable the estimates of the

correlation coefficients are. When using bootstrapping, a new sample of 80 ground motions

are randomly extracted out of the existing 80 ground motions, where each ground motion

can be selected at most 80 times (i.e. a ground motion can be considered more than once

during resampling). The EDPs computed from the new sample of ground motions can be

used to generate another realization of correlation coefficients. Repeating this process for a

large number of realizations, probability distributions of the correlation coefficients can be

estimated by obtaining the mean and standard deviation of each coefficient.

Table 8.4 shows the mean correlation coefficient matrix for the Haselton &

Deierlein (2007) 4-story concrete moment frame building using 1000 bootstrap realizations.

Correlations were computed for IDRs, PFAs and residual interstory drifts (RIDRs). Also

documented are the standard deviations of these coefficients in Table 8.5. In these tables

the type of EDP is listed followed by the location where it is located along the height of the

building (e.g. IDR1 corresponds to IDR in the 1st story and PFA2 corresponds to PFA in the

2nd floor).

Table 8.4 Average of EDP correlation coefficients from 1000 realizations

IDR1 IDR2 IDR3 IDR4 RIDR1 RIDR2 RIDR3 RIDR4 PGA PFA2 PFA3 PFA4 PFA5IDR1: 1.00 0.97 0.55 0.05 0.95 0.93 0.68 0.16 0.12 0.06 0.01 0.04 0.03IDR2: 0.97 1.00 0.70 0.18 0.93 0.94 0.79 0.30 0.16 0.09 0.05 0.08 0.07IDR3: 0.55 0.70 1.00 0.69 0.54 0.63 0.89 0.73 0.33 0.28 0.30 0.34 0.35IDR4: 0.05 0.18 0.69 1.00 0.07 0.14 0.47 0.86 0.56 0.55 0.58 0.58 0.60

RIDR1: 0.95 0.93 0.54 0.07 1.00 0.99 0.74 0.19 0.12 0.06 0.01 0.03 0.01RIDR2: 0.93 0.94 0.63 0.14 0.99 1.00 0.82 0.28 0.15 0.09 0.05 0.06 0.05RIDR3: 0.68 0.79 0.89 0.47 0.74 0.82 1.00 0.63 0.20 0.16 0.16 0.19 0.19RIDR4: 0.16 0.30 0.73 0.86 0.19 0.28 0.63 1.00 0.36 0.38 0.40 0.40 0.41

PGA [g]: 0.12 0.16 0.33 0.56 0.12 0.15 0.20 0.36 1.00 0.89 0.84 0.79 0.78PFA2 [g]: 0.06 0.09 0.28 0.55 0.06 0.09 0.16 0.38 0.89 1.00 0.93 0.84 0.85PFA3 [g]: 0.01 0.05 0.30 0.58 0.01 0.05 0.16 0.40 0.84 0.93 1.00 0.87 0.88PFA4 [g]: 0.04 0.08 0.34 0.58 0.03 0.06 0.19 0.40 0.79 0.84 0.87 1.00 0.85PFA5 [g]: 0.03 0.07 0.35 0.60 0.01 0.05 0.19 0.41 0.78 0.85 0.88 0.85 1.00


Table 8.5 Standard deviation of EDP correlation coefficients from 1000 realizations

IDR1 IDR2 IDR3 IDR4 RIDR1 RIDR2 RIDR3 RIDR4 PGA PFA2 PFA3 PFA4 PFA5IDR1: 0.000 0.009 0.101 0.103 0.020 0.033 0.089 0.119 0.087 0.098 0.098 0.106 0.098IDR2: 0.009 0.000 0.086 0.115 0.019 0.021 0.068 0.130 0.091 0.106 0.106 0.124 0.108IDR3: 0.101 0.086 0.000 0.063 0.116 0.105 0.028 0.068 0.085 0.106 0.108 0.124 0.103IDR4: 0.103 0.115 0.063 0.000 0.110 0.120 0.103 0.028 0.075 0.076 0.088 0.083 0.073

RIDR1: 0.020 0.019 0.116 0.110 0.000 0.006 0.091 0.135 0.081 0.095 0.088 0.117 0.099RIDR2: 0.033 0.021 0.105 0.120 0.006 0.000 0.073 0.144 0.087 0.104 0.099 0.133 0.112RIDR3: 0.089 0.068 0.028 0.103 0.091 0.073 0.000 0.109 0.099 0.123 0.128 0.156 0.134RIDR4: 0.119 0.130 0.068 0.028 0.135 0.144 0.109 0.000 0.085 0.097 0.109 0.127 0.117

PGA [g]: 0.087 0.091 0.085 0.075 0.081 0.087 0.099 0.085 0.000 0.026 0.030 0.042 0.058PFA2 [g]: 0.098 0.106 0.106 0.076 0.095 0.104 0.123 0.097 0.026 0.000 0.015 0.033 0.053PFA3 [g]: 0.098 0.106 0.108 0.088 0.088 0.099 0.128 0.109 0.030 0.015 0.000 0.026 0.035PFA4 [g]: 0.106 0.124 0.124 0.083 0.117 0.133 0.156 0.127 0.042 0.033 0.026 0.000 0.043PFA5 [g]: 0.098 0.108 0.103 0.073 0.099 0.112 0.134 0.117 0.058 0.053 0.035 0.043 0.000

Figure 8.6 shows the standard deviations as a function of the average values for the

correlation coefficients listed in Table 8.4. EDPs that are well correlated have smaller

variability than those that are not well correlated. Figure 8.6 also includes results for

correlation coefficients when only 10 ground motions are sampled for each realization of

the bootstrapping process. The standard deviations are much larger when compared to

results using 80 ground motions, suggesting that the estimates are much less reliable.

0.00

0.10

0.20

0.30

0.40

0.50

0.00 0.20 0.40 0.60 0.80 1.00

Avg. of Correlation Coeff Estimations

Std

. D

ev.o

f C

orr

elat

ion

Co

eff

Est

imat

ion

s

80 Ground Motions

10 Ground Motions

Figure 8.6 Relationship between average and standard error of correlation coefficient estimates

Perhaps a better way of capturing how well correlation coefficients are estimated is

to calculate 95% confidence bands of the bootstrap samples. The upper confidence band in

this context is defined by the difference between the 97.5th percentile and the median value

of the correlation coefficient estimates, whereas the lower confidence band is defined by the


difference between the 2.5th percentile and the medina value. Figure 8.7 shows the 95%

confidence bands for the correlation coefficient estimates of the example building for two

cases: when 80 ground motions are considered and when 10 records are considered. Also

plotted are quadratic regressions of each set of datum. As shown in this figure, the

difference between the confidence bands is much larger when only considering 10 ground

motions.

-1.0

-0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Median of Correlation Coeff. Estimates

95%

Con

fiden

ce B

ands

80 Ground Motions

10 Ground Motions

Figure 8.7 Difference between 97.5th and 2.5th percentiles confidence bands with median estimates of correlation coefficients

This trend – correlation coefficients are more difficult to estimate as their value

approaches zero – is not something that is specific to response parameter correlations and is

a well established statistical phenomenon. A closed-form solution exists to compute the

bands shown in Figure 8.7 (Neter et al, 1996). Estimating these intervals is usually

conducted by using an approximate procedure based on a transformation. This

transformation, known as the Fisher z transformation, is as follows:

1 1

ln2 1

z (8.13)


where is the value of correlation coefficient for which confidence bands are being

computed. The distribution of z is assumed to be approximately normal such that its

variance is equal to:

2 1

3z N

(8.14)

where N is the number of ground motions being considered. Correspondingly, if z is

assumed normal, the approximate 95% confidence limits for the expected value of z is:

1.96 zz (8.15)

where 1.96 is the value of the 1 2 100 percentile of the standard normal distribution

for confidence value of 95% ( = 5). The 95% confidence limits for can then be

computed by transforming the limits obtained from equation (8.15) back in terms of using

the inverse solution of equation (8.13). Figure 8.8(b) compares the results from this

procedure to the bands obtained from the example building’s EDP data for N = 10 and 80.

The figure shows that there is fairly good agreement between the EDP data generated from

the bootstrap sampling and the analytical procedure. Figure 8.8(a) uses the analytical

procedure to approximate the 95% confidence bands for N = 10, 20, 40 and 80 ground

motions. As the number of ground motions considered are decreased, the difference

between confidence bands and median increases. This procedure can be used to help

engineers decide how many ground motions to use during structural analysis based on how

well analysts would like to approximate EDP correlation coefficients.


-1.0

-0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0


95%

Con

fiden

ce B

ands

N = 80: AnalyticalN = 10: AnalyticalN = 10: DataN = 80: Data

-1.0

-0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0


95%

Con

fiden

ce B

ands

N = 10N = 20N = 40N = 80

(a) (b)

-1.0

-0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0


95%

Con

fiden

ce B

ands

N = 80: AnalyticalN = 10: AnalyticalN = 10: DataN = 80: Data

-1.0

-0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0


95%

Con

fiden

ce B

ands

N = 10N = 20N = 40N = 80

(a) (b)

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.

The confidence bands in Figure 8.8 illustrate that the number of ground motions

considered can have a large effect on the EDP correlation coefficients estimated and,

consequently, can have a large effect on the value of economic loss dispersion computed.

For instance, the mean correlation coefficient between the 3rd story IDR and the 3rd floor

PFA is equal to 0.30 for the example building (given that the spectral acceleration at the

fundamental period equal to 0.55g’s) as shown in Table 8.4. According to the results in

Figure 8.8(a), the estimated value of this coefficient could vary between approximately -

0.40 and 0.75 if only 10 ground motions are considered. This means that if only 10 ground

motions are considered, the correlation between these two EDPs can be estimated to be

either negatively correlated or strongly positively correlated. This will yield opposite

effects on the value of the overall economic loss dispersion, resulting in a poor estimate of

the loss variability.

8.3.3 Variability and Correlations in Damage Estimation

Damage to building components due to seismic ground motions is typically

conducted by defining discrete damage states, DS, for each component i. The probability

of experiencing or exceeding each damage state is quantified by defining cumulative


probability distributions that are based on the level of EDP that the component is subjected

to. These probabilistic relationships between EDP and DS are termed fragility functions.

The variability in damage estimation is quantified by the dispersion of these fragility

functions. These functions can be obtained from experimental data gathered from

laboratory experiments conducted on building components (Porter et al. 2007), from

empirical data observed from earthquake damage experienced in previous seismic events

(Chapter 5) or from using analytical models.

The correlations between building component damage states are difficult to quantify

because it requires knowing the joint probability distribution of components i and i’

conditioned on EDPj and EDPj’ respectively (Aslani and Miranda, 2005). Unfortunately,

data to estimate these joint probably distributions are unavailable and difficult to obtain. In

the absence of data, simplifying assumptions need to be used to account for correlations

between building component damage states. For the purposes of this study, the following is

assumed:

For a given EDP, the levels of damage in the same type of components

located in the same story are fully correlated.

For a given EDP, the damage levels of different types of components have

no correlation.

The assumption that the damage levels are fully correlated for the same type of components

located in the same story is based on the notion that the same types of components are

typically constructed by the same subcontractor. If this is true, the quality of construction

will be the same for these types of components and will most likely experience the same

damage when subjected to the same level of EDP. Conversely, assuming that the levels of

damage are not correlated between different building components is based on the notion

that the quality of construction of one component is not related to another. This is primarily

based on the fact that different sub-contractors tend to operate fairly independently of one

another. The vulnerability of one type of component based on the quality of its construction

is most likely not related to the vulnerability of a different type of component because it

was constructed by a different subcontractor. It may be argued that the quality of

construction between sub-contractors may be related because they are chosen by a common

general contractor; however, there are many factors that go into selecting sub-contractors,

such that any correlation is likely to be weak.


8.4 VARIABILITY OF LOSS METHODOLOGY

8.4.1 Mean annual frequency of loss & loss dispersion condition on seismic intensity

Previous studies (Aslani and Miranda, 2005) have established that a building’s

mean annual frequency of exceeding a particular value of monetary loss, lT, can be

calculated using:

0

T T T T

dv IMv L l P L l IM dIM

dIM

(8.16)

where T TP L l IM is the probability of exceeding a certain value of monetary loss,

given non-collapse at a seismic intensity level IM=im, and v IM is the mean annual

frequency of exceeding a seismic intensity IM (taken from the seismic hazard curve,

generated from probabilistic seismic hazard analysis). Aslani and Miranda (2005) have

suggested that T TP L l IM can be assumed to follow a cumulated lognormal

distribution. To define this probability distribution, the mean and dispersion of the

estimated economic loss need to be computed. Computing the mean of the loss is described

in greater detail in Chapter 3. The following paragraphs will present an approach to

compute the dispersion of loss conditioned on the level of ground motion intensity, 2 TL IM.

Baker and Cornell (2003) and Aslani and Miranda (2005) have proposed using the

following equation to compute the variance of losses conditioned on IM:

2 2 2,

2

2

,

T T TL IM L NC IM L C

T T

T T

P NC IM P C IM

E L NC IM E L IM P NC IM

E L C E L IM P C IM

(8.17)


where P NC IM and P C IM are the probabilities of non-collapse and collapse

conditioned on IM, respectively (note that these probabilities are complimentary such that

P NC IM = 1 P C IM . The expected loss conditioned the building not collapsing is

denoted by ,TE L NC IM , the expected loss given that the building collapsed is denoted

by TE L C (taken to be the mean replacement value of the building) and the expected

loss conditioned on the level of ground motion intensity is denoted by TE L IM . The

variance of loss conditioned on collapse and the variance of loss conditioned on non-

collapse are represented by 2

TL C

and 2,

TL NC IM, respectively.

8.4.2 Dispersion of loss conditioned on collapse

The variance of loss conditioned on collapse, 2 TL C, can be computed from

construction cost data similar to those shown in Table 8.2. The expected loss conditioned

on collapse, TE L C , is equal to the replacement value of the building because it is

defined as the cost of constructing a new structure to replace the one that has collapsed

(including the costs for removing the collapsed building). The dispersion associated with

this loss is assumed to be similar to the variability of the cost of constructing a new building

of the same occupancy that can be obtained from construction cost data.

The variance of economic loss conditioned on collapse, 2 TL C, can be obtained

form the standard deviations of the loss conditioned on collapse due to each subcontractor

k, kL C, as follows:

2,

1 1

2,

1 1 1

T k k

k k k

l l

k kL C L C L Ck k

l l l

k kL C L C L Ck k k

k k

(8.18)


where k,k’ are the cost correlation coefficients that can be obtained from construction cost

data. Note that the equation in (8.18) represents the sum of the covariance matrix for the

economic loss conditioned on collapse. The first line of equation (8.18) can be expanded to

separate the variance (the diagonal terms in the covariance matrix) and covariance (the off-

diagonal terms) terms of the covariance matrix as shown on the second line. The standard

deviation of the loss conditioned on collapse due to each subcontractor k, kL C, is

computed using:

k

k kL CE L C (8.19)

where k is the coefficient of variation of the construction costs for the kth subcontractor

and kE L C is expected economic loss conditioned on collapse for subcontractor k. k

is extracted from construction cost data and kE L C can be found computed from the

building’s replacement value with the following equation:

k k TE L C b E L C (8.20)

where b’k is the normalized proportion of the building replacement cost that is due to the kth

subcontractor. This value can also be determined from construction cost data by dividing

the mean cost of each subcontractor by the total mean cost of the entire building.

8.4.3 Dispersion of loss conditioned on non-collapse

Computing the variance of loss conditioned on non-collapse, 2,

TL NC IM, is more

complicated. Not only does estimating this type of loss dispersion involve the variability of

construction costs, it must account for variabilities from structural response and fragility

functions. The variability due to fragility functions is accounted for indirectly by collapsing


out the estimation of damage to create functions that relate economic loss and EDPs directly

(EDP-DV functions). These functions, which group building components by subcontractor

and EDP, can be used to propagate the variability and correlations of the building’s

structural response parameters. The variances can be summed for the entire building for

each subcontractor loss. The variability and correlations due to construction costs obtained

from building-level cost data can then be incorporated by adding them to the variances per

subcontractor computed thus far. The variance of each subcontractor can then be summed

together to obtain the dispersion of the economic loss conditioned on non-collapse for the

entire building at a given level of ground motion intensity.

Two implementations of this part of the methodology were developed. The first

implementation, presented in this section, is an approximate analytical approach that relies

on FOSM to compute dispersions. The second implementation, presented in section 8.4.4,

follows the same basic principles as the first but uses Monte Carlo simulation to obtain

dispersion results. Both implementations have advantages and disadvantages. The

analytical approach is not as computationally-intensive but the results are approximate and

may yield poor results depending on the type of building being considered. The approach

using simulation propagates variability accurately but can be computationally expensive

and may take longer to compute.

The nomenclature for the variables presented in the following equations can become

lengthy and confusing if carried out thoroughly and explicitly. Therefore, to simplify

presentation of this approach, it will be implied that from this point forward, all the

variables contained in this section will be conditioned on the building not collapsing, NC,

whether it is explicitly expressed or not. For example, when the expected loss conditioned

on EDP is expressed as E L EDP , it is implied that this value is also conditioned on non-

collapse as well (i.e. it is implied that the expected loss is conditioned on EDP and on the

building not collapsing as if it were expressed using , E L NC EDP ).

8.4.3.1 Collapse out damage estimation and group by subcontractor cost

Estimating damage is collapsed out because the damage states for each component

are discrete random variables. The next step in this approach will involve using FOSM

methods to propagate response parameter dispersions. Using FOSM methods can become


problematic when the random variables are not continuous. Further, collapsing out damage

estimation can simplify the process of computing the variability of economic loss if this

step is conducted beforehand. This step is also the basis of the story-based loss estimation

framework introduced in chapter 3 that relates response parameters directly to loss.

The expected loss for the ith component conditioned on a particular value of the jth

EDP has been experienced, i jE L EDP , can be calculated using the following equation

(Miranda and Aslani, 2002; Krawinkler and Miranda, 2004):

1

p

i j i q q jq

E L EDP E L DS P DS ds EDP (8.21)

where i qE L DS is the expected repair cost of the ith component given it has experienced

damage state, DSq=dsq, and q jP DS ds EDP is the probability of the component being

in that damage state.

In previous studies (Baker and Cornell, 2003, Aslani and Miranda, 2005), the

variance of each component, 2 i jL EDP

, was also computed from the variance of the repair

costs for each damage state. However, since data on the dispersion of component costs

were unavailable, these investigations used building-level cost data to estimate the variance

of the repair costs for each damage state. Instead, the approach here will apply the

variability associated with construction costs when the losses are summed for the entire

building as will be described later in this chapter. This process more closely represents the

way costs are assembled in construction projects. Although the variance of loss conditioned

on EDP, 2 i jL EDP

, is not explicitly calculated, the dispersions of the fragility functions are

embedded in the values of i jE L EDP . This is because i jE L EDP is a function of

y jP DS ds EDP , which is defined by its mean and dispersion parameters.

In order to make use of the dispersions and correlations obtained from available

construction cost data, the losses need to be grouped by the different subcontractors being

considered. Up to this point the expected losses have been expressed per component. The


value of each component must be divided in proportion to how much the cost of each

subcontractor contributes to the total value of the repair costs, such that:

, k i j k i jE L EDP b E L EDP (8.22)

where bk is the fraction of the component’s replacement value that is attributed to

subcontractor k. In this study, bk is assumed to be deterministic. Although bk may be

considered a random variable, the relative size of dispersion is deemed to be much smaller

compared to the other sources variability considered in this methodology.

In order to limit the amount of computation required to propagate the variance of

loss given EDP to loss for a given intensity level, IM=im, it is convenient to group

components’ expected values and dispersion by the number of EDPs being considered. The

number of EDPs, m, is a function of the number of stories and the types of EDPs

considered. In this chapter, only two types of EDPs, peak interstory drift (IDR) and peak

floor acceleration (PFA), are considered for each story (the total number EDPs is equal to 2

x the number of stories) for the purposes of computing economic loss dispersion (RIDR is

used as an EDP that is used to estimate the probability of building demolition conditioned

on non-collapse in Chapter 9). Expected values of losses ( , k j jE L EDP ) for the jth EDP

can be summed using the following equation:

, ,1

n

k j j k i ji

E L EDP E L EDP (8.23)

where n is the number of components whose damage depends on the same EDP. This

procedure is the same the one used to formulate the EDP-DV functions described Chapter 3

except that these functions in addition to being grouped by type of EDP, these functions are

also grouped by subcontractor cost using the assumed distributions in Table 8.3. Figure 8.9

and Figure 8.10 shows the subcontract floor used to calculate the dispersions for the

example 4-story reinforced concrete moment-resisting frame building. The expected

economic losses shown in these figures are normalized by the entire value of the story.


More detail on the fragility functions and cost distributions used to generate the EDP-DV

functions can be found in Chapter 3 of this dissertation.

Drift-sensitive

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.000 0.020 0.040 0.060 0.080 0.100

IDR

E( L | IDR)

ConcreteMetalsDoors, Window s, GlassFinishesMechanicalElectrical

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


Acceleration-sensitive

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.0 1.0 2.0 3.0 4.0 5.0

PFA [g]

E( L | PFA )

Finishes

Mechanical

Electrical

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

8.4.3.2 Propagate variance from response parameters

The subcontractor EDP-DV functions shown in Figure 8.9 and Figure 8.10 can be

combined with the variability of EDPs conditioned on a given intensity level, IM=im, to

compute the dispersion of economic loss as a function of ground motion intensity. The

variability of EDPs for a given ground motion intensity level can be computed from

structural analysis. The EDP-DV functions, which express loss as function of EDP, can

then be used to translate the structural response dispersion into the variability of economic

loss due to the variability of EDP.

It has been suggested by previous investigators to use FOSM approximations to

accomplish this step in variability propagation (Benjamin and Cornell 1970, Ang and Tang

1984, Baker and Cornell 2008). Using the FOSM approach, the economic loss due to work

performed by subcontractor k to repair damage caused by EDP j conditioned on ground

motion intensity, ,k jL IM , can be calculated using a first-order Taylor expansion of the

economic loss due to work performed by subcontractor k to repair damage caused by EDP j,


conditioned on EDP, ,k jL EDP . If jEDP IM is denoted as Xj , and ,k jL IM is denoted as

Yk,,j, then the EDP-DV functions that relate the two variables can be denoted as

, ,k j k j jY g X . Using this notation, the FOSM approximation of the expected value of

,k jL IM is calculated by:

, , k j k j jE Y g E X (8.24)

The variance of this expected economic loss, ,

2

k jY , can be calculated using FOSM as

follows:

,

2

,2 2

k j j

x j

k jY x

jm

g

x (8.25)

where 2

,x j

k j jm

g x is the derivative of gk.j(X) evaluated at the mean of Xj, mxj and 2jx is

the variance of Xj for the jth response parameter. The dispersion computed in equation

(8.25) only accounts for EDP variability and the variability from the fragility functions that

is embedded in the EDP-DV functions. The variability due to construction cost will be

incorporated in the following section. The correlations that exist between EDPs will also

produce variability and its influence needs to be considered by quantifying its associated

covariance. The covariance of economic losses between component groups whose damage

depends on response parameters j and j’, due to work performed by subcontractor k, ,kj kjY

,

is accounted for by using the following equation:

,

, ,,

kj kj j j j j

x xj j

k j k jY x x x x

j jm m

g g

x x (8.26)


where 2

,x j

k j jm

g x

is the derivative of gk.j’(X) evaluated at the mean of Xj’, mxj’, 2

jx is

the variance of Xj’ for the j’th response parameter and ,j jx x is the correlation coefficient

between Xj and Xj’ for the jth and j’th response parameter, respectively.

8.4.3.3 Sum losses and dispersions for entire building

The total expected economic loss for each subcontractor k, kE Y , can be

determined by summing the repair/replacement costs done by the subcontractor at each

individual story computed in equation (8.24) such that:

,1

m

k k jj

E Y E Y

(8.27)

The total variance of the economic loss for each subcontractor k, 2

kY , is computed using the

results from equations (8.25) and (8.26), and adding the variability due to construction costs

as illustrated in the following expression:

,

, ,

2 2

1 1

2 2

1 1 1

k kj kj k

k j kj kj k

m m

Y Y Yj j

m m m

Y Y Yj j j

j j

(8.28)

where 2

kY is the variance due construction costs for subcontractor k, which can be obtained

from cost databases. The dispersion results for this study will use cost data gathered by

Touran and Suphot (1997) as described in section 8.3.1. The second line of equation (8.28)

separates the variance terms (computed from equation (8.25)) and covariance terms

(computed from equation (8.26)) for clarity.


The covariance between economic losses due to work performed subcontractor k

and subcontractor k’, ,k kY

, is a function of both response parameter correlations and

correlations from construction costs. This type of variability can be estimated using the

FOSM method with the following equation:

,

, ,, ,

1 1k k j j j j k k k k

x xj j

m mk j k j

Y x x x x Y Y Y Yj j j j

m m

g g

x x

(8.29)

where kY

is the standard deviation due construction costs for subcontractor k’ and ,k kY Y is

the correlation coefficient between the construction costs of subcontractors k and k’. Both

kY and ,

k kY Y can be determined from construction cost databases similar to the

information described in section 8.3.1. Note that the first term in equation (8.29) is the term

that accounts for the inherent subcontractor loss correlations due EDP variability previously

described in section 8.3.1.1. The second term represents the covariance that results from the

correlations between construction costs of different subcontractors.

The expected economic loss for the entire building conditioned on non-collapse at a

given level of ground motion intensity, E Y , is computed by summing the losses due to

the work performed by each subcontractor such that:

1

l

kk

E Y E Y

(8.30)

where l is the total amount of subcontractors involved in repairing damage due to

earthquakes. Correspondingly, the variance of the loss for the entire building conditioned

on non-collapse at a given level of ground motion intensity, 2Y , can be found by:


,

,

2

1 1

2

1 1 1

k k

k k k

l l

Y Yk k

l l l

Y Yk k k

k k

(8.31)

where the second line of equation (8.31) separates the variance and covariance terms. The

first term of the second line is obtained from the results of equation (8.28) and the second

term is obtained from the results of (8.29). The final dispersion result in (8.31) accounts for

all three types of variability – variability from fragility functions, from response parameters

and from construction costs as well as their correlations.

8.4.4 Monte Carlo simulation method

Alternatively, the variability of economic loss can be computed by using Monte Carlo

simulation. Unlike the analytical method presented in the previous sections, Monte Carlo

simulations do not rely on approximations in variability propagation that the FOSM

methods do. However, Monte Carlo simulations are computationally intensive and may be

more difficult to implement.

Monte Carlo simulation can be used to predict the variability of economic loss using

the following steps:

1. Formulate EDP-DV functions for each subcontractor as described in section 8.4.3.1.

2. For each realization, simulate values of EDPs based on the statistical parameters at

given level of ground motion intensity computed from structural analysis.

3. Use EDP-DV functions and simulated EDP values to compute the expected

economic losses for each realization of EDP at each floor.

4. Sum the losses at each floor to obtain the total expected losses per subcontractor for

all floors.

5. Simulate the economic losses for each subcontractor as random variables by using

the total expected losses per subcontractor computed in step 4 as the mean and the

variability information extracted from construction cost data as the dispersion.

6. Sum the subcontractor losses simulated in step 5 to obtain the loss for the entire

building.


7. Repeat steps 2 to 7 for a large number of realizations.

8. Compute the standard deviation of the loss for all of the realizations

9. Repeat steps 2 to 8 for different levels of ground motion intensity levels.

In this study, dispersion results for the example building using the Monte Carlo

simulation method described above was implemented by assuming lognormal probability

distributions for both the EDP simulations and the subcontractor loss simulations. To

obtain reliable results, 3000 realizations were used to obtain dispersion results at each level

of ground motion intensity.

8.4.5 Evaluation of quality of FOSM approximations

Understanding how well FOSM methods approximate the direct solutions requires

examining the mean-valued Taylor series expansion of ,k jg X evaluated around a point

x , where the random variable X = EDPj | IM. If Y = ,k jL IM , then

22

2

( )( )

2!

!

x x

nn

n x

X xg gY g x X x

x x

X xgx

x n

(8.32)

when n . Expanding the second term, obtains:

2 ( )2 ( )

2 2! !

x x

nn

nx x

g gY g x X x

x x

X x X xg gx

x x n

(8.33)

A first-order approximation truncates higher order terms such that equation (8.33) becomes:


x x

g gY g x X x

x x

(8.34)

Recognizing that only the second term in equation (8.34) is random and the other terms are

treated as constants, the mean value of Y is computed by applying the expectation operator

as follows:

x x

g gE Y g x E X x

x x

(8.35)

A “mean-centered” FOSM approximation is one that evaluates equation (8.35) by choosing

xx m , where xm is the mean of X. Equation (8.35) can be reduced, recognizing that

xE X m , such that:

xE Y g m (8.36)

The corresponding approximate variance of Y can be obtained by:

22Y yE Y m

(8.37)

where ym is the mean value of Y. Recognizing that y xm E Y g m from equation

(8.36), equation (8.35) can be substituted into equation (8.37), which becomes:

2

2Y y

x x

g gE g x X x g m

x x

(8.38)


Once again choosing xx m , this equation reduces to:

2

2

x x

Y xm m

g gE X m

x x

(8.39)

Rearranging and recognizing that xm

g x is a constant, equation (8.39) becomes:

2

22

x

Y xm

gE X m

x

(8.40)

2

22

2

2

x

x

Y xm

Xm

gE X m

x

g

x

(8.41)

The resulting equation in (8.41) shows the calculation of the derivative of g(X), xm

g x ,

becomes very important in the ability of FOSM to approximate the exact solution of 2Y

well.

FOSM methods rely on approximating nonlinear functions using linear relationships

at the point of interest (which in this case is at the mean of X, xm ). Therefore, the quality

of FOSM approximations is dependent on how nonlinear g(X) is around this point and on

the level of dispersion of X. Figure 8.11 shows examples of two extremes that illustrate this

concept. In Figure 8.11(a), g(X) is linear and the solution the FOSM method yields is exact.

This is because the higher order terms in equation (8.33) are equal to zero for this case.

Conversely, if g(X) is highly nonlinear, as shown in Figure 8.11(b), then the FOSM

approximation deviates from the actual solution. As the nonlinearity of the function


increases, the values of the higher-order terms in equation (8.33) also increase, resulting in

these deviations.

Y

X

Y = g(X)

X

xx

y

y

g(x)

(a)

Y

X

Y = g(X)

X

xx

y

y

g(x)

Y

X

Y = g(X)

X

xx

y

y

g(x)

(a)

Y

X

Y = g(X)

X

xx

yy

g(x)xm

g

x

(b)

Y

X

Y = g(X)

X

xx

yy

g(x)xm

g

x

Y

X

Y = g(X)

X

xx

yy

g(x)xm

g

x

(b)

Figure 8.11 FOSM approximations (a) linear function (b) non-linear function

The error of FOSM methods can be quantified by applying the expectation operator

to equation (8.33), substituting xx m and rearranging such that:

2 ( )

2 ( )

2 2! !x x

nn

xnm m

E X x E X xg gE Y g m x

x x n

(8.42)

If it is assumed that the first term of this equation is much larger than the other terms and

recognize that 22x E X x , the error becomes:

22

2 2!x

xx

m

gE Y g m

x

(8.43)


This quantity represents the error in the FOSM approximation of the mean value of Y, E[Y],

however, this error term would also have to be carried through equations (8.38) to (8.41)

when computing the variance of Y. Therefore, this error term also contributes to the

difference between the economic loss variances computed by FOSM and the exact results.

This means that the size of the possible errors using FOSM is not only dependent on the

nonlinearity of g(X) (as represented by the second derivative 22

xmg x ) but also on the

size of the variance of X, 2x .

In order to improve the approximations made by the FOSM methods, an alternative

way of computing the slope of g(X) was investigated. Instead of taking the derivative

locally at the mean of X, the slope was calculated over a larger range of X. A rational range

of X to choose would be one that captures the most probable values of X. One such range

would be the values between the 16th percentile, x0.16, and the 84th percentile, x0.84, or

equivalently, one standard deviation below and above the mean of X (i.e. xm +/- x ). This

alternative method of computing the slope of g(X), x x

x x

m

mg x

, can then be determined as

follows:

0.84 0.16

0.84 0.16

x x

x x

m

m

g x g xg

x x x

(8.44)

where,

0.16

0.84

x x

x x

x m

x m

(8.45)

Using this method of computing the slope of g(X) improved dispersion results because

computing the derivative locally does not capture everything that occurs in g(X) within the

probable region of X (the most likely region of X being between xm +/- x ).


Figure 8.12 shows an example where computing the derivative locally will not

approximate the y as well as using this alternative slope method. In this example, the

mean value of X occurs in a region where g(X) plateaus. Figure 8.12(a) shows that if the

derivative is computed locally, it would produce very small value of y . The figure also

shows that there is a large deviation between g(X) and the line produced by xm

g x

within the likely region of X, suggesting that errors between the exact solution and the

linear approximation are significant. Conversely, Figure 8.12(b) plots the variability

propagation using the alternative slope calculation ( x x

x x

m

mg x

). Using this method

produces a larger value of y and a line that does a better job of following g(X). Rather

than obtain a slope based on one value of X ( xm ), x x

x x

m

mg x

computes an average slope

within the range of xm +/- x which captures more of the function g(X) in this region.

Y

X

Y = g(X)

Xxx

y

g(x)

xm

g

x

(a)

Y

X

Y = g(X)

Xxx

y

g(x)

xm

g

x

Y

X

Y = g(X)

Xxx

y

g(x)

xm

g

x

(a)

Y

X

Y = g(X)

Xxx

y

g(x)

x x

x x

m

m

g

x

(b)

Y

X

Y = g(X)

Xxx

y

g(x)

x x

x x

m

m

g

x

Y

X

Y = g(X)

Xxx

y

g(x)

x x

x x

m

m

g

x

(b)

Figure 8.12 Computing the derivative of g(X) (a) local derivative (b) average slope within region that X will most likely occur in.

The two methods of computing the slope of g(X) were used on the example building

to see which approach produced better results for this case-study. Standard deviations

(,

k jY ) were calculated using both types of slope computations for all the possible

combinations of types of subcontractors, k, and number of EDPs, m. These values were


then compared to the corresponding standard deviations obtained from Monte Carlo

simulations, which are considered to be more representative of the exact dispersions.

Table 8.6 shows the standard deviation results of economic loss for each type of

subcontractor and each type of EDP considered at an intensity level equal to the design

basis earthquake using the local derivative to compute the slope of g(X). The top third of

the table shows results using FOSM, the middle third reports the results using simulation

and the bottom third shows the percent difference between the FOSM and simulation

results. Table 8.7 reports the same type of results as Table 8.6, except that the FOSM

standard deviation values were computed using the average slope in equation (8.44). The

results of these two tables show that the slope of g(X) computed using the local derivative

produced larger differences from the simulation results (max = 73%, min = -46%) than

those calculated using the average slope (max = 36%, min = -34%) for this particular

building. Therefore, it was determined that the average slope method would be used to

generate the dispersion results for this building.

Table 8.6 Comparison of standard deviations of economic loss due to EDP variability only using FOSM (local derivative) and simulation methods

IDR1 IDR2 IDR3 IDR4 PFA2 PFA3 PFA4 PFA5 TOTAL

Concrete $21,625 $17,277 $26,964 $39,519 $0 $0 $0 $0 $55,271Masonry $0 $0 $0 $0 $0 $0 $0 $0 $0

Metals $22,625 $21,007 $23,100 $16,294 $0 $0 $0 $0 $41,860Carpentry $0 $0 $0 $0 $0 $0 $0 $0 $0

Moisture Protection $0 $0 $0 $0 $0 $0 $0 $0 $0Doors, Windows, Glass $39,620 $28,181 $25,352 $20,101 $0 $0 $0 $0 $58,401

Finishes $164,433 $122,845 $154,071 $189,000 $45,820 $28,796 $21,177 $9,721 $324,129Mechanical $29,495 $27,006 $23,645 $36,838 $40,483 $13,661 $8,384 $12,769 $74,661

Electrical $84,308 $71,037 $81,100 $119,202 $18,983 $6,406 $3,932 $4,718 $182,701






Electrical $63,028 $56,544 $60,783 $72,106 $26,237 $11,105 $7,324 $8,083 $131,037


Concrete 0.00 -0.04 0.26 0.73 N/A N/A N/A N/A 0.31Masonry N/A N/A N/A N/A N/A N/A N/A N/A N/A

Metals 0.13 0.12 0.18 0.41 N/A N/A N/A N/A 0.17Carpentry N/A N/A N/A N/A N/A N/A N/A N/A N/A

Moisture Protection N/A N/A N/A N/A N/A N/A N/A N/A N/ADoors, Windows, Glass -0.18 -0.18 0.00 -0.20 N/A N/A N/A N/A -0.15

Finishes 0.22 0.16 0.25 0.56 0.18 0.01 -0.05 -0.04 0.29Mechanical 0.17 0.15 0.18 0.29 -0.28 -0.42 -0.46 -0.42 -0.10

Electrical 0.34 0.26 0.33 0.65 -0.28 -0.42 -0.46 -0.42 0.39

STANDARD DEVIATTION OF LOSS (FOSM - LOCAL DERIVATIVE)

STANDARD DEVIATTION OF LOSS (SIMULATION)

% DIFFERENCE BETWEEN FOSM & SIMULATION

Subcontractor

Subcontractor

Subcontractor


Table 8.7 Comparison of standard deviations of economic loss due to EDP variability only using FOSM (average slope) and simulation methods






Electrical $76,325 $66,580 $72,637 $97,815 $21,258 $7,921 $4,823 $5,674 $160,226






Electrical $63,028 $56,544 $60,783 $72,106 $26,237 $11,105 $7,324 $8,083 $131,037


Concrete 0.04 0.02 0.14 0.31 N/A N/A N/A N/A 0.14Masonry N/A N/A N/A N/A N/A N/A N/A N/A N/A

Metals 0.11 0.11 0.11 0.25 N/A N/A N/A N/A 0.12Carpentry N/A N/A N/A N/A N/A N/A N/A N/A N/A

Moisture Protection N/A N/A N/A N/A N/A N/A N/A N/A N/ADoors, Windows, Glass -0.13 -0.10 -0.03 -0.20 N/A N/A N/A N/A -0.11

Finishes 0.14 0.12 0.15 0.35 0.14 0.03 -0.04 -0.02 0.18Mechanical 0.12 0.12 0.10 0.13 -0.19 -0.29 -0.34 -0.30 -0.09

Electrical 0.21 0.18 0.20 0.36 -0.19 -0.29 -0.34 -0.30 0.22

STANDARD DEVIATTION OF LOSS (FOSM - AVERAGE SLOPE)

STANDARD DEVIATTION OF LOSS (SIMULATION)

% DIFFERENCE BETWEEN FOSM & SIMULATION

Subcontractor

Subcontractor

Subcontractor

As has been shown in the previous paragraphs, the accuracy of the FOSM method is

highly dependent on the level of dispersion of EDP and the shape and characteristics of

g(X). Since g(X) represents EDP-DV functions in the context of this dissertation, the

characteristics of typical EDP-DV functions can be identified to examine what common

situations will produce good FOSM approximations and what common situations produce

poor approximations. It can be observed from this dissertation (Chapter 3 and section

8.4.3.1) that many of these EDP-DV functions have typically an “S-shaped” curve. For

these types of functions, there are two primary regions where the curves are highly

nonlinear. Consequently, the FOSM method does not do as well in computing loss

dispersions in these ranges. The first nonlinear region, shown in Figure 8.13(a), is located

at the beginning of the curve at small values of X. In this range, the FOSM method tends to

underestimate the dispersion of economic loss in this range. The second nonlinear region

occurs when the function begins to level off (i.e. the top of the “S”) at large values of X as

shown in Figure 8.13(b). For the example building at the DBE, the loss dispersions


computed in this range using FOSM tend to be overestimated. On the other hand, the

portion of the function located between these two regions is typically much more linear.

Figure 8.13(c) shows this region where the FOSM method will compute a fairly good

approximation of the economic loss dispersion.

Y

X

Y = g(X)

X

xx

y

y

g(x)

(a)

Y

X

Y = g(X)

X

xx

y

y

g(x)

Y

X

Y = g(X)

X

xx

y

y

g(x)

(a)

Y

X

Y = g(X)

X

xx

y

y

g(x)

(b)

Y

X

Y = g(X)

X

xx

y

y

g(x)

Y

X

Y = g(X)

X

xx

y

y

g(x)

(b)

Y

X

Y = g(X)

X

xx

y

y

g(x)

(c)

Y

X

Y = g(X)

X

xx

y

y

g(x)

Y

X

Y = g(X)

X

xx

y

y

g(x)

(c)

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

Figure 8.14 illustrates specific quantitative examples of the different cases discussed

in the previous paragraph using data from the example 4-story building and the EDP-DV

function for repair work done by the electrical subcontractor for drift-sensitive components.

Each graph plots economic loss normalized by the replacement value of the story as a

function of IDR and the slope using both types of methods described previously. The first


example, shown in Figure 8.14(a), demonstrates what happens for the value of the interstory

drift at the 4th story for a ground motion intensity level equal to an spectral acceleration of

0.25g at the fundamental period (Sa(T1)). The mean value of IDR at this intensity level (mx

= 0.056) occurs in the nonlinear region at the beginning of this EDP-DV function. The

normalized economic loss dispersion computed from Monte Carlo simulation is equal to

0.013 of the replacement value of the story. Using local derivative to compute the slope

( xm

g x = 2.6), also shown in the figure, the FOSM method results in a normalized

economic loss dispersion of 0.0075, which is 42% less than the value calculated using

simulation. If instead the average slope is used ( x x

x x

m

mg x

= 2.9), the normalized

economic loss dispersion is equal to 0.0083 using FOSM and is 35% less than the value

computed using simulation. Both methods of computing the slope for the FOSM approach

significantly underestimates the standard deviation because this region of the EDP-DV

function is highly nonlinear; however, using the average slope method produces results that

are closer to the value computed from simulation.

0.00

0.02

0.04

0.000 0.005 0.010

X = IDR

E[Y] = E[L | IDR]

xm

g

x

x x

x x

m

m

g

x

xmx xm

x xm

0.00

0.02

0.04

0.000 0.005 0.010

X = IDR

E[Y] = E[L | IDR]

xm

g

x

x x

x x

m

m

g

x

xmx xm

x xm

0.00

0.02

0.04

0.06

0.08

0.10

0.000 0.010 0.020 0.030

X = IDR

E[Y] = E[L | IDR]

xm

g

x

x x

x x

m

m

g

x

xm

x xm

x xm

0.00

0.02

0.04

0.06

0.08

0.10

0.000 0.010 0.020 0.030

X = IDR

E[Y] = E[L | IDR]

xm

g

x

x x

x x

m

m

g

x

xm

x xm

x xm

(a) IM = 0.25g (b) IM = 0.80g

Figure 8.14 Quantitative examples of FOSM approximations using the different slope methods


0.00

0.02

0.04

0.06

0.08

0.10

0.000 0.010 0.020 0.030

X = IDR

E[Y] = E[L | IDR]

xm

g

x

x x

x x

m

m

g

x

xm

x xm

x xm

0.00

0.02

0.04

0.06

0.08

0.10

0.000 0.010 0.020 0.030

X = IDR

E[Y] = E[L | IDR]

xm

g

x

x x

x x

m

m

g

x

xm

x xm

x xm

(c) IM = 0.35g

Figure 8.14 Quantitative examples of FOSM approximations using the different slope methods

(cont.)

Figure 8.14(b) shows the same EDP-DV function when the example building is

subjected to a higher ground motion intensity (Sa(T1) = 0.80g). At this intensity, a mean

IDR of 0.016 is computed, which occurs later on in the EDP-DV function where the curve

begins to level off (i.e. at the top of the “S-shaped” curve). Monte Carlo simulation

computes a normalized standard deviation equal to 0.028 of the story replacement value.

Also shown in the figure is the slope of g(X) computed using xm

g x , which is equal to

4.7, and the slope computed using x x

x x

m

mg x

, which is equal to 4.5. These values result

in normalized standard deviations of 0.042 and 0.039 of the story replacement value,

respectively, using the FOSM method. Both these values overestimate the dispersion

computed using simulation because of the nonlinearity of the function for these values of

IDR. The difference between the value using simulation and the value computed using the

average slope method (41%), however, is smaller than the value calculated using the local

derivative.

The final example, shown in Figure 8.14(c), illustrates an instance where the mean

IDR occurs within a linear region of the EDP-DV function. This figure shows the interstory

drift at a ground motion intensity level of Sa(T1) = 0.30 which has a mean of 0.0096. The

normalized standard deviation computed using simulation is equal to 0.19. Both types of


slope computational methods approximate values of dispersion closer to the one computed

using simulation. Using the local derivative, the FOSM method calculates the normalized

dispersion to be 20% greater and equal to 0.23. The FOSM method using the average slope

approach computes the normalized standard deviation to be 0.21, which represents a smaller

difference of 8%.

8.5 DISPERSIONS OF ECONOMIC LOSS FOR EXAMPLE 4-STORY BUILDING

The proposed methodology described above was applied to the sample 4-story

reinforced concrete moment resisting frame office building designed by Haselton and

Deierlein (2007). For greater detail on the structural analysis, readers are directed to that

investigation and to Chapter 9 of this dissertation. The building’s mean losses as a function

of seismic intensity (quantified here as spectral acceleration at the fundamental period,

Sa(T1)) is calculated by the loss methodology and assumptions documented in Chapter 3,

and the computer tool described in Chapter 6, was used to generate the loss results for this

study.

Several types of dispersion results were computed and are presented in the next few

sections. Results for dispersions of loss conditioned on non-collapse at a ground motion

intensity level equal to that of the design basis earthquake (DBE, see Chapter 7 for

definition) were computed for each subcontractor. These values were then summed

together to obtain the dispersion of loss due to non-collapse for the entire building. This

computation is repeated for a range of ground motion intensities to obtain dispersion

conditioned on non-collapse as a function of ground motion intensity. The dispersion of the

loss conditioned on collapse was then computed and combined with the results from the

non-collapse values to obtain the total dispersion as a function of ground motion intensity.

The mean and the variability of loss conditioned on ground motion intensity were then

integrated with the site’s hazard curve to obtain the mean annual frequency of exceeding a

particular value of loss. Results from both the analytical and simulation approaches are

presented. The simulation results are considered to be more accurate because of the FOSM

approximations used in the analytical approach. Most of the standard deviation results are

normalized by the building’s replacement value which was calculated to be $12 million as

reported in Chapter 7.


8.5.1 Variability of loss conditioned on non-collapse at the DBE

The first set of results presented here are dispersions that only consider EDP

variability only. The results of this part of the overall dispersion are particularly important

because it is in this part of the analytical method that FOSM approximations are

implemented. Therefore, any differences between the analytical and simulation approaches

will occur in these results.

As was reported earlier, Table 8.7 shows the standard deviation results of economic

loss for each type of subcontractor and each type of EDP considered. The largest

underestimation of the standard deviation of economic loss occurs for the dispersion of the

electrical subcontractor losses at the 4th floor. The FOSM method computes a standard

deviation of $4,823, whereas simulation calculates a standard deviation of $7,324,

representing an underestimation of 34%. The underestimation occurs because the probable

range of this EDP is within a nonlinear region of the EDP-DV function. Specifically, the

probable values of this EDP are located in the bottom part of the “S”(as plotted in Figure

8.13(a) for illustrative purposes).

The largest overestimation occurred for the standard deviation of the economic

losses due to the drift-sensitive components within the finishes subcontractor at the 4th

story. Using the FOSM method, a standard deviation of $91,815 is computed, whereas the

standard deviation calculated from simulation is $72,106, which represents an

overestimation of 36%. This overestimation occurs because the probable values of IDR in

the 4th story are within a nonlinear region of the EDP-DV function for the losses due to the

finishes subcontractor. In this case, the values are overestimated because the probable

values of this EDP are located in the top part of “S” curve as illustrated in Figure 8.13(b).

These trends of underestimations and overestimations also occur for the remaining standard

deviations in Table 8.7. The magnitude of the differences from the simulation values

depend on how nonlinear the region of the EDP-DV function is within the region being

considered. As demonstrated by Table 8.7, these differences can be very large and may

lead to large errors in overall dispersions for the entire building.

The individual dispersions due to EDP variability in Table 8.7 can be summed

together to obtain dispersions for the entire building for each subcontractor. The standard

deviations due to construction cost variability can be also computed for each subcontractor


at the building level. Figure 8.15 shows the normalized standard deviations of the example

building’s earthquake-induced economic loss conditioned on non-collapse and on a ground

motion intensity level associated with the design basis earthquake (DBE, see Chapter 7 for

definition of DBE). The dispersions shown in this figure are categorized by subcontractor

and normalized by the replacement value of the building. The standard deviations

associated with masonry, carpentry and moisture protection are equal to zero because losses

from these subcontractors are not expected based on the assumed cost distribution for

reinforced concrete moment-frame office buildings. Figure 8.15(a) shows the dispersion

due to EDP variability while (b) shows the dispersion due to construction cost variability.

The standard deviations due to EDP variability are smaller than the corresponding standard

deviations due to construction cost variability. The values of standard deviation of loss

conditioned on non-collapse due to EDP variability range from 0.0029 and 0.021 of the

building’s replacement value The dispersions due to construction cost variability vary

between 0.014 to 0.10 of the building replacement value.

0.000

0.020

0.040

0.060

0.080

0.100

Concrete Masonry Metals Carpentry MoistureProtection

Doors,Window s,

Glass

Finishes Mechanical Electrical

FOSM

Simulation

[Lk | NC, IM] (a) Due to EDP Variability

Figure 8.15 Standard deviations for each subcontractor loss (a) dispersions due to EDP variance (b) dispersions due to construction cost variance


0.000

0.020

0.040

0.060

0.080

0.100


Doors,Window s,

Glass


FOSM

Simulation

[Lk | NC, IM] (b) Due to Cost Variability

Figure 8.15 Standard deviations for each subcontractor loss (a) dispersions due to EDP variance (b) dispersions due to construction cost variance (coont.)

In both figures, the finishes subcontractor has the largest standard deviation of all

the other construction trades. It has standard deviation of 0.021 for EDP variability and

0.99 for construction cost variability. The finishes standard deviations are the largest

primarily because the repairs associated with this subcontractor make up large portion of

the building’s mean losses. This can be inferred from Figure 8.10 where the EDP-DV

function for finishes subcontractor is larger than the other functions for the entire range of

IDR (the EDP-DV functions for acceleration-sensitive components can also be considered,

however, in Chapter 7, it was shown that for this type of structure, reinforced concrete

moment frame office building, the majority of the economic loss is due to repairs of drift-

sensitive components). Although it is useful to be able to disaggregate the overall standard

deviation to see which subcontractors have the largest contributions, these values are highly

dependent on their corresponding mean economic losses. The largest standard deviations

do not necessarily correspond to the largest variations relative to the mean values.

Standard deviations must be understood in the context of means such that variation

between random variables with wildly different means can be compared. The standard

deviations in Figure 8.15 can be normalized by their corresponding mean values of loss for

each subcontractor, k, (shown in Figure 8.16) to obtain coefficient of variation values, k,

which are shown in Figure 8.17. As was observed in Figure 8.15, these graphs also show

that the dispersions due to construction cost variability (Figure 8.17(b)) are greater than the

dispersions due to EDP variability (Figure 8.17(a)). However, unlike the standard


deviations in Figure 8.15, the finishes subcontractor does not have the largest quantification

of dispersion in either the EDP variability values or construction cost variability values.

The mechanical subcontractor has the largest value of [Lk | NC, IM] for EDP variability

(0.30) and the electrical subcontractor has the largest value for construction cost variability

(0.98). The mechanical subcontractor has the largest variation due to EDP variability

because it has variability from both drift and acceleration-sensitive components. The

concrete subcontractor has the largest variation due to construction cost variability because

that is what the data from the Touran and Suphot (1997) study produced as shown in Table

8.1.

0.00

0.05

0.10

0.15

0.20

0.25


Doors,Window s,

Glass


FOSM

Simulation

E [Lk | NC, IM]

Figure 8.16 Mean values of economic loss for each subcontractor at the DBE


0.00

0.20

0.40

0.60

0.80

1.00


Doors,Window s,

Glass


FOSM

Simulation

[Lk | NC, IM] (a) Due to EDP Variability

0.00

0.20

0.40

0.60

0.80

1.00


Doors,Window s,

Glass


FOSM

Simulation

[Lk | NC, IM] (b) Due to Cost Variability

Figure 8.17 Coefficient of variations for each subcontractor loss (a) dispersions due to EDP variance (b) dispersions due to construction cost variance

Figure 8.15 and Figure 8.17 also compare dispersion results that were computed

using both the analytical and simulation approach. The dispersion results due to

construction costs show good agreement between the two approaches because the Monte

Carlo simulations are simulating the value of dispersions in Table 8.1 directly. That is, the

variances are not being propagated from one type of dispersion to another. The dispersion

results due to EDP variability do not match up as well as the results due to construction cost

variability because FOSM approximations are used to propagate the EDP variability.

Particularly, the standard deviations of the finishes and electrical subcontractors show the

largest discrepancies between the two methods, with differences of 18% and 22%,

respectively. The analytical results of these subcontractor dispersions do not agree as well


with the simulation results because the nonlinearity of the corresponding EDP-DV

functions at the values of EDPs produced at this level of ground motion intensity.

The variance conditioned on non-collapse for the entire building due to EDP

variability can be computed analytically by summing the squares of the standard deviations

for each subcontractor shown in Figure 8.15(a) using equation (8.25). However, when

these results are compared to the results from simulation, it was determined that using

equation (8.25) only was not enough to capture the dispersion correctly. This is due to the

inherent correlations between subcontractor losses described section 8.3.1.1. Figure 8.18

compares the analytical results to those generated by Monte Carlo simulation for cases

where this effect is not considered (left bar graphs) and when it is accounted for (right bar

graphs). When only equation (8.25) is used, the analytical standard deviations (0.030)

underestimate the simulation results (0.048) by 38%. In order for the analytical

computations to better match the results from simulation, the first term in equation (8.29)

that computes the covariance associated with this type of correlation needs to added to the

results of equation (8.25) using equation (8.31). The analytical approximations that

consider the inherent correlations (0.055) show better agreement and reduce the difference

between the analytical method and the simulation method to 15%.

0.000

0.025

0.050

0.075

0.100

Equation (8.25) only Equation (8.25) w ith inherentcorrelation

Analytical

Simulation

[L | NC, IM]

Figure 8.18 Effect of subcontractor correlation due to EDP Variability

How well the analytical results can reproduce the results from simulation depend on

the quality of the variance approximations in equation (8.28) and covariance

approximations computed from equation (8.29). Results from these equations were used to


compute correlation coefficients and compared to coefficients generated from simulation to

examine how well the analytical approach captures this effect in the example building.

Table 8.8 compares the correlation coefficients from both implementations and reports the

percent differences between the two approaches. Although most of the coefficients show

relatively good agreement, the mechanical subcontractor coefficients have substantial

differences between the two approaches (9% to 26% difference). The discrepancies

between FOSM and simulation in the covariances and variances associated with these

coefficients are likely causing the 15% difference observed in Figure 8.18.

Table 8.8 Comparison of inherent subcontractor loss correlation coefficients due to EDP variability between analytical and simulation results

Concrete MetalsDoors,

Windows, Glass


Concrete 1.00 0.95 0.92 0.98 0.72 0.99Metals 0.95 1.00 0.98 0.95 0.70 0.95

Doors, Windows, Glass 0.92 0.98 1.00 0.93 0.69 0.92Finishes 0.98 0.95 0.93 1.00 0.84 0.99

Mechanical 0.72 0.70 0.69 0.84 1.00 0.82Electrical 0.99 0.95 0.92 0.99 0.82 1.00


Windows, Glass


Concrete 1.00 0.96 0.87 0.96 0.57 0.95Metals 0.96 1.00 0.91 0.94 0.57 0.93




Windows, Glass


Concrete 0.00 0.00 0.07 0.02 0.26 0.04Metals 0.00 0.00 0.07 0.01 0.23 0.02



INHERENT CORRELATIONS (ANALYTICAL)

INHERENT CORRELATIONS (SIMULATION)

% DIFFERENCE BETWEEN ANALYTICAL & SIMULATION

Subcontractor

Subcontractor

Subcontractor

Figure 8.19 shows the total standard deviation of economic loss conditioned on non-

collapse at the DBE for the example building, normalized by the replacement value of the

building, when considering different types of variability. There are six sets of bar graphs


representing six different cases (case 1 is denoted as (1), case 2 as (2)…etc.) that consider

different combinations of types of variability. Cases (1) to (3) do not consider any

variability from correlations, while cases (4) to (6) account for different types of

correlations in addition to EDP and construction cost variability. The corresponding

coefficients of variation associated with these dispersions are shown in Figure 8.20.

0.000

0.050

0.100

0.150

0.200

0.250

(1) EDPvariability only

(2) Costvariability only

(3) EDP & costvariability

(4) EDP & costvariability w /

EDP correlationsonly


costcorrelations

(6) EDP & costvariability w /EDP & costcorrelations

Analytical

Simulation

[L | NC, IM]

Figure 8.19 Standard deviations of loss conditioned on non-collapse at the DBE considering different types of variability and correlations

0.00

0.20

0.40

0.60

0.80

1.00

(1) EDPvariability only

(2) Costvariability only

(3) EDP & costvariability


EDP correlationsonly


costcorrelations

(6) EDP & costvariability w /EDP & costcorrelations

Analytical

Simulation

[L | NC, IM]

Figure 8.20 Coefficient of variation of loss conditioned on non-collapse at the DBE considering different types of variability and correlations


Case (1) shows the dispersion when only considering EDP variability, case (2)

shows the dispersion when only construction cost is considered and case (3) shows the

dispersion when both types of variability is considered. The standard deviation due to EDP

variability (0.049) is twice as small as the standard deviation due to construction cost

(0.11). This suggests that the construction cost dispersion has a larger contribution to the

total standard deviation due to both types of variability (case (3)) for this building. It is

important that the variability of construction costs is captured properly, using the type of

cost data available, because this type of variability can be very large and may play a

significant role in the overall variability of economic loss.

The effect of correlations on the dispersions at the DBE is illustrated by cases (4)

to (6) in Figure 8.19. Case (4) considers EDP and construction cost variability with only

the correlations between EDPs, case (5) considers EDP and construction cost variability

with only correlations between construction costs and case (6) considers EDP and

construction cost variability with both EDP and construction cost correlations. All three

cases show that correlations significantly increase the dispersion. EDP correlations increase

the loss dispersion by 20%, whereas construction cost correlations increase the loss

dispersion by 25% when these standard deviations are compared to case (3). When both

types of correlations are accounted for, the standard deviation is 36% larger than the

dispersion value computed without correlations in case (3). This means that the variability

of economic loss conditioned on non-collapse may be substantially underestimated if

correlations due to response parameters and construction costs are ignored.

Each set of bar graphs compares the results generated from the analytical method

with the results from the simulation method. The differences between the two methods

range from as small as 2% (case (2)) to as large as 13% (case (1)). The difference in

standard deviations due to EDP variability between the two methods is fairly significant

because of the FOSM approximations. These differences appear to be decrease when

combined with the variability of construction costs. The construction cost standard

deviations, which show better agreement between the analytical and simulation results,

contribute a larger portion of the total dispersion and, as a result, there is a smaller

difference between the two methods for standard deviations when both sources of

variability are combined.


8.5.2 Variability of loss conditioned on non-collapse as a function of IM

The analyses to generate results shown in section 8.5.1 were repeated for a range of

seismic intensities to obtain dispersion results as function ground motion intensity. The

standard deviations of economic loss conditioned on non-collapse as function of ground

motion intensity, measured here as spectral acceleration at the fundamental period, Sa(T1),

are shown in Figure 8.21 for six cases that were reported in Figure 8.19. The standard

deviations reported here are normalized by the replacement value of the building. Figure

8.21(a) shows loss dispersions due to EDP variability only, Figure 8.21(b) shows loss

dispersions due to construction cost variability only and Figure 8.21(c) shows loss

dispersions for both EDP and construction cost variability. The remaining three graphs,

Figure 8.21(d), Figure 8.21(e) and Figure 8.21(f), plot the dispersion results for EDP and

construction cost variability when combined with EDP correlations only, with construction

cost correlations only, and with both EDP and cost correlations combined, respectively.


0.00

0.10

0.20

0.30

0.40

0.0 0.5 1.0 1.5 2.0 2.5 3.0

IM = Sa [g]

[L | IM, NC]

Analytical

Simulation

(b) Cost Variability Only

0.00

0.10

0.20

0.30

0.40

0.0 0.5 1.0 1.5 2.0 2.5 3.0

IM = Sa [g]

[L | IM, NC]

Analytical

Analytical without Inherent Correlation

Simulation

(a) EDP Variability Only

0.00

0.10

0.20

0.30

0.40

0.0 0.5 1.0 1.5 2.0 2.5 3.0

IM = Sa [g]

[L | IM, NC]

Analytical

Simulation

(c) EDP & Cost Variability

0.00

0.10

0.20

0.30

0.40

0.0 0.5 1.0 1.5 2.0 2.5 3.0

IM = Sa [g]

[L | IM, NC]

Analytical

Simulation

(d) EDP & Cost Variability w/ EDP Correlations

0.00

0.10

0.20

0.30

0.40

0.0 0.5 1.0 1.5 2.0 2.5 3.0

IM = Sa [g]

[L | IM, NC]

Analytical

Simulation

(e) EDP & Cost Variability w/ Cost Correlations

0.00

0.10

0.20

0.30

0.40

0.0 0.5 1.0 1.5 2.0 2.5 3.0

IM = Sa [g]

[L | IM, NC]

Analytical

Simulation

(f) EDP & Cost Variability w/ All Correlations

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.


All six graphs in Figure 8.21 compare the results from the analytical method and the

Monte Carlo simulations. As was observed with the results for the DBE, all six cases show

fairly good agreement between the two methods. The differences between the two methods

are largest for the standard deviations results for EDP variability only. These differences

are slightly larger because of the FOSM approximations that are used to propagate the EDP

variability. The differences occur at ground motion intensity levels that induce values of

EDPs at which the EDP-DV functions that are used to compute loss behave nonlinearly.

The differences between the results of the two methods for the construction cost standard

deviations are entirely due to the variability of the simulations which is dependent on the

number of realizations that are conducted.

Similar trends observed from the results at the DBE can also be observed for the

dispersion results at different levels of ground motion intensity. The standard deviations of

the total loss due to variability in construction costs are substantially larger than those due

to variability in EDP. At higher intensity levels, the standard deviations due to cost

variability can be as much as 3 times the standard deviations due EDP variability. Also

shown in Figure 8.21(a) are economic loss standard deviations computed analytically

without considering the inherent subcontractor loss correlations due work performed on the

same floor/story (section 8.3.1.1). As was the case when examining results at the DBE, the

dispersions of total loss conditioned on non-collapse as a function of ground motion are

significantly underestimated by an average of 30% if these correlations are not accounted

for.

The influence of correlations on the dispersion of economic losses as a function of

ground motion intensity also show similar trends to those observed at the DBE. Figure 8.22

plots and compares the standard deviations of economic loss, normalized by the

replacement value of the building, computed from the simulation results for all six cases.

As was the case for the dispersions at the DBE, the standard deviations that account for

correlations are either equal to or greater than those that do not. This trend holds true for all

values of ground motion intensity for this building. This re-emphasizes the importance of

incorporating correlations into the computation of economic loss uncertainty.


0.00

0.10

0.20

0.30

0.40

0.0 0.5 1.0 1.5 2.0 2.5 3.0

IM = Sa [g]

[L | IM, NC]

EDP Variability OnlyCost Variability OnlyEDP & Cost Var., No Corr.EDP & Cost Var. w/ EDP Corr.EDP & Cost Var. w/ Cost Corr.EDP & Cost Var.w/ Cost & EDP Corr.

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.

As will be observed in the next section, economic loss dispersions conditioned on

non-collapse have a larger influence on the overall loss dispersion at smaller ground motion

intensities. Therefore it is important to understand how correlations influence economic

loss dispersions conditioned on non-collapse at these ground motion intensities. Figure

8.23 shows the same results as in Figure 8.22, but only for ground motion intensity levels

that are smaller than Sa(T1) = 1.0g. From this graph, it can be observed that EDP

correlations have a stronger influence at small values of ground motion intensity, while cost

correlations have a larger effect at higher intensities. EDP correlations have a stronger

influence at small ground motion intensities because EDP correlations tend to be higher in

this range of spectral accelerations. EDPs are highly correlated at small intensities because

in this range, a large portion of the building and its components will behave elastically. For

example, displacements and floor accelerations will be strong correlated for initial values of

spectral acceleration as shown in Figure 8.5(b). As the ground motion intensity increases

and the structure begins to exhibit inelastic behavior, the correlation decreases. As the EDP


correlations decreases, the construction cost correlations start to play a larger role in

increasing the overall dispersion as shown in Figure 8.23.

0.00

0.10

0.20

0.00 0.25 0.50 0.75 1.00

IM = Sa [g]

[L | IM, NC]

EDP Variability OnlyCost Variability OnlyEDP & Cost Var., No Corr.EDP & Cost Var. w/ EDP Corr.EDP & Cost Var. w/ Cost Corr.EDP & Cost Var.w/ Cost & EDP Corr.

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.

8.5.3 Variability of loss conditioned on collapse as a function of IM

Up to this point, the variability of loss conditioned on collapse, 2 TL C, has not been

considered. Using the procedure in section 8.4.2 and the data in Table 8.2, this value was

computed to be 0.43 when considering construction cost correlations and 0.27 when cost

correlations were not considered. These values are only due to the variability of

construction costs and do not vary with the level of ground motion intensity. Figure 8.24(a)

show the results for the variability of loss conditioned on non-collapse computed in the

previous section and Figure 8.24 (b) plots the variability of loss conditioned on collapse as

function of ground motion intensity. Each figure plots standard deviations that are

normalized by the replacement value of the building and reports results for two cases:


dispersions that consider correlations and dispersions that do not consider correlations. The

economic loss dispersions conditioned on collapse are large and are greater than the

dispersions that are conditioned on non-collapse. As was the case with the dispersions

conditioned non-collapse, the dispersions of losses associated with collapse are driven by

the large values of variability due to construction costs. Because both collapse and non-

collapse loss dispersions primarily depend on construction cost variability, it can be inferred

that when these types of dispersions are combined, the total economic loss uncertainty will

also depend largely on the variability of construction costs.


0.00

0.10

0.20

0.30

0.40

0.50

0.0 0.5 1.0 1.5 2.0 2.5 3.0

IM = Sa [g]

[L | IM, C]

w/ No Correlations

w/ Cost & EDP Correlations

0.00

0.10

0.20

0.30

0.40

0.50

0.0 0.5 1.0 1.5 2.0 2.5 3.0

IM = Sa [g]

[L | IM, NC]

w/ No Correlations


(a) (b)

0.00

0.10

0.20

0.30

0.40

0.50

0.0 0.5 1.0 1.5 2.0 2.5 3.0

IM = Sa [g]

[L | IM, C]

w/ No Correlations


0.00

0.10

0.20

0.30

0.40

0.50

0.0 0.5 1.0 1.5 2.0 2.5 3.0

IM = Sa [g]

[L | IM, NC]

w/ No Correlations


(a) (b)

0.00

0.10

0.20

0.30

0.40

0.50

0.00 0.50 1.00 1.50 2.00 2.50 3.00

IM = Sa [g]

[L | IM]

w/o Correlationsw/ Non-collapse Correlationsw/ Collapse Correlationsw/ All Correlations

Figure 8.24 Normalized standard deviation for of loss (a) conditioned on non-collapse (b) conditioned on collapse.

Equation (8.17) was used to combine the dispersion conditioned on non-collapse

with the dispersion conditioned on collapse to obtain the total dispersion as a function of

ground motion intensity level. Different combinations of the plots in Figure 8.24(a) and

Figure 8.24(b) were used to investigate the effect of correlations from the collapse and non-

collapse conditions. Four different cases are shown in Figure 8.24(c): dispersions with no

correlations, dispersions with correlations from loss conditioned on non-collapse,


dispersions with correlations from loss condition on collapse and dispersions with

correlations from both the collapse and non-collapse conditions.

At smaller intensity levels, the non-collapse correlations have a larger influence on

the overall dispersion, whereas the collapse correlations have a more substantial effect at

higher intensities. This is an expected result because equation (8.17) is a sum that is

weighted by the probability of collapse, P C IM . At small ground motion intensities,

where the probability of collapse is small, the terms involving the dispersion conditioned on

non-collapse in equation (8.17) are very large compared to the terms involving the

dispersion conditioned on collapse. As P C IM increases with increasing values of

ground motion intensity, collapse plays a larger role in economic losses and its

corresponding dispersion has a larger contribution to the overall variability. Consequently,

the non-collapse correlations are more influential at small ground motion intensities and

collapse correlations are more important at higher ground motion intensities.

8.5.4 Variability of loss as a function of IM & MAF of loss

Figure 8.25 shows the expected losses conditioned on ground motion intensity and

the associated standard deviations for particular values of intensity. The results are

normalized by the replacement value of the building. The seismic intensity associated with

the DBE, is equal to 0.52g. At this intensity, the expected loss is 0.31, with a standard

deviation of 0.21. This means that expected loss could be as small as 0.10 or as large as

0.52 of the building’s replacement value based on a 68% confidence interval (i.e. within

one standard deviation of the mean). This represents a coefficient of variation ([L | IM]) of

0.67.


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 1.0 2.0 3.0

IM = Sa [g]

E[L | IM]

Mean Losses

Median Losses

+/- Std Dev

Figure 8.25 Normalized expected loss and dispersion given IM for example 4-story office building

The coefficient of variation, [L | IM], as a function of ground motion intensity in this

building is plotted in Figure 8.26. At very small ground motion intensities, [L | IM] is very

large and is as high as 0.97 at an Sa(T1) = 0.10g. Previous economic loss estimation studies

(ATC 1997, National Institute of Building Sciences 1997) have typically assumed

dispersion values that vary between 0.6 and 0.8, which is, in general, in the same range as

those shown in Figure 8.26. However, for very small intensities, these studies may

underestimate the dispersion. The values in Figure 8.26 decrease as ground motion

intensity increases until it levels off at 0.43. This is equal to the coefficient of variation that

is computed for the economic losses conditioned on collapse computed from the data

gathered by Touran and Suphot (1997). The probability of collapse increases as the ground

motion intensity level increases, such that the variability of loss will approach the values of

dispersion conditioned on collapse shown in Figure 8.24(b).


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.50 1.00 1.50 2.00 2.50 3.00

IM = Sa [g]

[L | IM]

Figure 8.26 Coefficient of variation as a function of intensity level for example building

The results in Figure 8.25 can be integrated with the building site’s hazard curve

using equation (8.16) to obtain values for the mean annual frequency (MAF) of loss. Figure

8.27(a) compares the MAF of loss between economic loss dispersions that consider

correlations and economic loss dispersions that do not consider correlations. The figure

shows that for high probability events that induce losses smaller than $2M, the effect of

correlations is negligible. For losses greater than this value the effect of correlations

becomes much more significant. At a loss of $12M, the assumed mean replacement value

of the building, the MAF of loss is equal to 6.6x10-5 without considering correlations. This

increases by 110% to 1.4x10-4 when correlations are considered. This trend is similar to the

results obtained by Aslani and Miranda (2005) and confirms the importance of accounting

for correlations when computing the variability of seismically-induced economic loss.


1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

$0 $5 $10 $15 $20

Loss [$M]

(L>l)

Simulation

Analytical

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

$0 $5 $10 $15 $20

Loss [$M]

(L>l)

Losses w ithoutCorrelations

Losses w ithCorrelations

(a) Effect of correlations (b) Analytical vs. Simulation (With Correlations)

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

$0 $5 $10 $15 $20

Loss [$M]

(L>l)

Simulation

Analytical

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

$0 $5 $10 $15 $20

Loss [$M]

(L>l)

Losses w ithoutCorrelations

Losses w ithCorrelations

(a) Effect of correlations (b) Analytical vs. Simulation (With Correlations)

Figure 8.27 MAF of loss (a) effect of correlations (b) comparison between analytical and simulation methods

Figure 8.27(b) compares the MAF of loss computed using the analytical approach

with MAF of loss computed using simulation. For this building, there is very little

difference between the loss curves computed from the different methods. This is because

most of the economic loss dispersion calculated in this study was due to the large values of

construction cost variability. This type of variability does not require the use of any FOSM

approximations to propagate uncertainty using the analytical method because they are

already expressed in terms of monetary value and taken directly from cost data. The

contributions of the EDP variability, which rely on FOSM approximations using the

analytical method, are small for this building. Further, the FOSM method, in this particular

case, does a relatively good job of estimating the EDP variability generated from simulation

(as shown in Figure 8.21) because the errors in approximating loss dispersions at each floor

(as reported in Table 8.7) seem to offset each other when they are summed together. There

may be cases in which these errors in dispersion at each floor amplify each other and result

in large errors in total loss dispersion due to EDP variability. This means that the

agreement between the analytical and simulation methods shown in Figure 8.27(b) may not

necessarily always hold true if the errors in EDP variability are large and the contributions

of EDP variability are also large.


8.6 CONCLUSIONS

A method that computes the variability of seismic-induced economic loss by grouping

the losses by subcontractor and incorporating building-level construction cost dispersions

and correlations was presented. The method is more compatible to the way construction

projects are priced and is better suited to use the type of construction cost data that is

currently available. Two types of implementations of the method, an analytical approach

using FOSM approximations and an approach that used Monte Carlo simulation, were

developed to provide analysts different alternatives on how to propagate variability.

Other issues related to the variability of economic loss that were addressed included

accounting for inherent correlations between subcontractor losses due work performed on

the same floor/story. It was shown that ignoring this type of correlation could significantly

underestimate the variance due to structural response parameters (15% at the DBE for the

example building). The use of the bootstrap sampling method was suggested as a way of

obtaining reliable EDP correlation estimates. It was also demonstrated how the bootstrap

method could be used to evaluate how many number of ground motions are needed in

structural analysis to compute robust EDP correlations. An attempt was also made to

improve the FOSM approximations by proposing an alternative way of computing the slope

of the nonlinear functions (EDP-DV functions) used to propagate response parameter

dispersion. It was determined that computing the slope over the range of values within a

standard deviation of the mean EDP, instead of computing the slope locally at the mean

EDP improved the agreement between results computed by the FOSM approximations and

those generated by simulation for the example building being considered.

Variability results from an example 4-story reinforced concrete moment-frame office

building, taken from previous studies (Haselton and Deierlein, 2007), were computed to

illustrate the use of the new economic loss dispersion methodology. At the DBE, the

coefficient of variation for the example building was computed to be 0.67. This value is

fairly large despite the fact that it does not account for epistemic uncertainty. As was

demonstrated in previous investigations (Aslani and Miranda, 2005) correlations due to

EDP variability and construction cost variability significantly increases the overall value of

dispersion conditioned on ground motion intensity (36% at the DBE). Correlations also

significantly increased the MAF of loss by 110% at a loss greater or equal to the

replacement value of the building ($12 million). When the economic loss dispersion was


disaggregated, it was determined that the variance due to construction costs had larger

contributions to the overall variability than the variance due to EDPs for this building.

Comparing the results generated from the analytical and the simulation

implementations showed that there was fairly good agreement between the two approaches

for the total dispersion in this particular building. However, when the dispersion is

disaggregated, the standard deviations due to EDP variability for each subcontractor

demonstrated larger deviations between the two implementations. This means that the good

agreement between the approaches of the overall dispersions may be misleading because

the errors at the subcontractor level could be offsetting each other in this particular building.

The differences are also mitigated by the fact that dispersions due to construction costs,

which have better agreement between the two approaches, comprised a larger portion of the

total variability. Although this was true for the example building examined in this study, it

may not necessarily be a general trend for all types of buildings. Further research is needed

to see whether these trends hold for other buildings. Analysts must be aware that despite

the good agreement demonstrated in this particular example building, fairly significant

inaccuracies are possible when using FOSM approximations.

CHAPTER 9 246 Losses due to Demolition given Non-collapse

CHAPTER 9

9 SIGNIFICANCE OF RESIDUAL DRIFTS IN BUILDING EARTHQUAKE LOSS ESTIMATION


Ramirez, C.M., and Miranda, E. (2009), “Significance of Residual Drifts in Building

Earthquake Loss Estimation,” Earthquake Spectra, (in preparation).

9.1 INTRODUCTION

Observation of the performance of buildings in previous earthquakes indicates that

significant residual displacements may occur. Recent analytical and experimental studies

(Mahin and Bertero 1981, MacRae and Kawashima 1997, Pampanin et al. 2002, Ruiz-

Garcia and Miranda 2005) have shown that the likelihood of experiencing residual

deformations increases as the level of inelastic deformation increases. This suggests that

lateral force resisting systems that are capable of sustaining large lateral displacements are

more likely to experience residual deformations when subjected to seismic ground motions.

Although it has been demonstrated that structural systems with large deformation capacity

do a relatively good job in reducing the probability of collapse in buildings (Haselton and

Deierlein, 2007), these structures are more likely sustain residual drifts that may result in

poor performance when considering other metrics such as economic losses and the loss of

facility use after and earthquake has occurred.

Residual displacements and its consequences play an important role in structural

performance (Ruiz-Garcia & Miranda 2005). In the 1985 Michoacan earthquake, for

instance, numerous reinforced concrete buildings were demolished in Mexico City because

of the complications associated with repairing and straightening the permanent

deformations the structures experienced (Rosenblueth and Meli, 1986). In another example,


scores of Kobe’s bridge reinforced concrete bridge piers sustained excessive residual

displacements during the Hyogo-Ken-Nambu earthquake, necessitating demolition and

replacement (Kawashima, 2000). In both examples, the structures performed well in terms

of preventing collapse, as they were able to remain standing. However, from a loss

standpoint, these structures exhibited poor structural performance as they had to be

demolished leading not only to large economic losses associated with demolition and

reconstruction, but also with the loss of use of the facility for long periods of time until

reconstruction was completed.

Recent advances in earthquake engineering – particularly the advent of the Pacific

Earthquake Engineering Research Center’s (PEER) performance-based design framework

– have attempted to use economic loss as a performance metric for design. PEER’s

evaluation framework can be traced back to building-specific loss estimation methodologies

first proposed by Scholl et al. (1982). Initially, PEER’s framework was based on methods

that only considered economic losses due to repairable damage given the building has not

collapsed. Monetary losses induced by seismic ground motions are calculated by using

structural response parameters (peak interstory drift, peak floor accelerations…etc.) to

estimate damage in different building components. The cost to repair each damaged

component is then summed for the entire building to obtain the total loss of the building.

This methodology was eventually improved to include economic losses due to building

replacement given that the building has collapsed (Miranda et al. 2004, Aslani and Miranda

2005). However, the current framework does not account for the economic loss due to

demolishing a building that has not collapsed but cannot be repaired because of excessive

residual (permanent) drifts.

The objective of this study is to present an improved loss estimation methodology

that explicitly incorporates losses resulting from having to demolish buildings that have

experienced large residual drifts. The improved methodology is illustrated by evaluating

economic losses in four buildings. The case-study buildings are reinforced concrete

moment-frames structures, whose modeling and response simulation was conducted by

other investigators (Haselton and Deierlein 2007, Liel and Deierlein 2008). In each case

economic losses are estimated with the existing and improved methodologies.


9.2 METHODOLOGY

PEER’s building specific loss estimation methodology uses the total probability

theorem to compute the expected value of the total economic loss in a building conditioned

on a ground motion intensity IM=im, E[LT | IM], as the weighted sum of expected losses in

two mutually exclusive, collectively exhaustive events. Namely: (1) collapse does not

occur (non-collapse, NC) and damage in the building is repaired, R, (i.e., NC ∩ R); and (2)

collapse occurs and the building is rebuilt, C (Miranda et al. 2004, Aslani and Miranda

2005, Mitrani-Rieser 2007). The expected value of the loss in the building for a given

ground motion intensity IM=im is computed as:

| | , | | |T T TE L IM E L NC R IM P NC R IM E L C P C IM (9.1)

where E[LT | NC ∩ R, IM] is the expected value of the total loss in the building provided

that collapse does not occur (non-collapse) and the building is repaired given that it has

been subjected to earthquakes with a ground motion intensity IM=im, and E[LT | C] is the

expected loss in the building when collapse occurs in the building. The weights on these

two expected losses are P(NC ∩ R | IM) which is the probability that the building will not

collapse and that it will be repaired given that it has been subjected to earthquakes with a

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

collapse under a ground motion with a level of intensity, im.

Previous investigations have not included the consequence of earthquake damage

when collapse does not occur but the building has to be demolished, D (i.e., NC ∩ D). To

include this effect, this study proposes that a third intermediate term be added to equation

(9.1), to account for economic losses that may result from this outcome. With this term, the

equation becomes:

| | , |

| | | |

T T

T T

E L IM E L NC R IM P NC R IM

E L NC D P NC D IM E L C P C IM

(9.2)


where E[LT | NC ∩ D] is the expected loss in the building when there is no collapse but the

building needs to be demolished. The weight on this expected loss is P(NC ∩ D | IM),

which is the probability that the building will not collapse but that it has to be demolished,

given that it has been subjected to earthquakes with a ground motion intensity IM=im. As

will be shown later, in many cases, ignoring this term can lead to significant

underestimations of earthquake losses.

The probability that the building will not collapse and that it will be repaired given

that it has been subjected to earthquakes with a ground motion with a level of intensity, im

is given by:

| | , |P NC R IM P R NC IM P NC IM (9.3)

Similarly, the probability that the building will not collapse but that it will be

demolished given that it has been subjected to earthquakes with a ground motion intensity

IM=im can be computed as:

| | , |P NC D IM P D NC IM P NC IM (9.4)

Since the repair and demolition events are mutually exclusive events, given that no

collapse has occurred and collapse and non-collapse are also mutually exclusive events,

then:

| , 1 | ,P R NC IM P D NC IM (9.5)

| 1 |P NC IM P C IM (9.6)

Substituting (9.3)-(9.6) into (9.2) we obtain:


| | , 1 | , 1 |

| | , 1 | | |

T T

T T

E L IM E L NC R IM P D NC IM P C IM

E L NC D P D NC IM P C IM E L C P C IM

(9.7)

Previous studies (Aslani and Miranda 2005, Mitrani-Rieser 2007) have used a

component-based methodology to calculate E[LT | NC ∩ R, IM]. The damage and

corresponding loss for every component in the building is first calculated, and then summed

up using the following equation:

1

, ,N

T j jj

E L NC IM a E L NC IM

(9.8)

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

collapse has not occurred at the intensity level IM, aj is the cost of a new jth component,

and Lj is the normalized loss in the jth component defined as the cost of repair or cost to

replace the component normalized by aj.

A method to simplify the calculation of this term using a story-based approach was

presented in Chapter 3. This approach relies on assumptions about the building’s cost

distribution to calculate loss at each story instead of at every component. If the repair costs

are normalized in the same manner show in equation (9.8), only the value of the story needs

to be specified, rather than the values and quantities of each component. Further,

relationships that relate structural response directly to loss can be developed, such that the

intermediate step of estimating component damage become unnecessary. This approach is

used in this study and the reader is referred to Chapter 3 of this dissertation for further

details of the methodology.

In Equation (9.7) E[LT | C] corresponds to cost of removing the collapse structure

from the site plus the replacement cost of the building. Similarly, E[LT | NC ∩ D] is equal

to the cost of demolishing the existing structure, plus cost of removing debris from the site

plus the replacement cost minus the cost of building components that may be salvageable

because the structure did not collapse.

As shown in Equation (9.7) all three terms are influenced by the probability of

collapse, P(C | IM). Previous studies (e.g., Krawinkler et al. 2005, Haselton and Deierlein


2007, Liel and Deierlein 2008) have used nonlinear structural simulation models analyzed

using incremental dynamic analysis (Vamvatsikos and Cornell, 2002) to estimate the

probability of collapse as a function of the ground motion intensity. Using a suite of ground

motion records, incremental dynamic analysis captures the effects of variation in frequency

content and other ground motion characteristics, on structural response. The outcome of

incremental dynamic analysis is a set of statistical results that permit the estimation of the

probability of exceedance of critical engineering demand parameters (EDPs) at a number of

ground motion intensities and a collapse fragility function, which describes the probability

of collapse as a function of the ground motion intensity.

In this study, the probability that a structure is forced to be demolished is computed

as a function of the maximum residual interstory drift in the building. The probability that a

building will need to be demolished after an earthquake, given that it has not collapse in an

earthquake with intensity IM =im, can be computed applying the theorem of total

probability as follows:

|0| , | , | ,D RIDRP D NC IM im G D RIDR ridr NC dP RIDR ridr NC IM im

(9.9)

where, GD|RIDR(D | RIDR = ridr, NC) is the cumulative distribution function (CDF) for the

probability that the structure will be demolished, given that the building has not collapsed

but that it has experienced a maximum residual drift is equal to ridr, when subjected to an

earthquake with intensity level, im, and P(RIDR > ridr | NC, IM=im) is the probability of

exceeding ridr given the that the building has not collapsed at the intensity, im. Note that

RIDR is the maximum residual drift in any story in the building. P(RIDR > ridr | NC,

IM=im) can be determined from the structural response simulation similarly to other types

of engineering demand parameters. In this study, GD|RIDR(D | RIDR = ridr, NC) was

assumed to be lognormally distributed. This probability may be interpreted as the

percentage of engineer that would recommend demolition of the building with increasing

levels of residual interstory drift. There is very little data available to obtain an accurate

estimate of the parameters for this probability distribution. For the purposes of this study,

GD|RIDR(D | RIDR = ridr, NC) was assumed to have a median of 1.5% and a lognormal

standard deviation of 0.30. These values were deemed reasonable using engineering


judgment and a sensitivity analysis was conducted to see how large an influence it had on

estimating losses. The details of this analysis and the reasoning behind the assumed values

are described in section 9.3.3.

9.3 APPLICATIONS

9.3.1 Description of Buildings Studied

To evaluate the influence of the explicit consideration of having to demolish a

building due to excessive residual drifts, four reinforced concrete frame buildings whose

seismic response was previously studied by other investigators were considered. The four

case study buildings are: a 4-story building with ductile detailing, a 12-story building with

ductile detailing, a 4-story building with non-ductile detailing, and a 12-story building with

non-ductile detailing. These buildings are taken from the same sets of structures reported in

Chapter 7. As was described in that chapter, all four structures were assumed to be located

at a site in Los Angeles, CA, south of the city’s downtown area, and is representative of a

typical urban California site with high levels of seismicity, but not subject to near-fault

directivity effects (Haselton et al., 2007). Probabilistic seismic hazard analysis was

conducted by Goulet et al. (2007) to calculate this site’s seismic hazard curve (Figure 7.1),

which provides information on the likelihood of experiencing a ground motion exceeding a

specified intensity. In this study, the spectral acceleration at the first mode period of the

structure, Sa(T1) [g], was used to quantify characterize the ground motion intensity (note

that in the figures and tables in this chapter will denote this value as Sa for brevity,

however, it is implied that this value is at the fundamental period).

The two structures with ductile detailing were modern buildings designed by

Haselton and Deierlein (2007) according to the 2003 International Building Code and

related ACI and ASCE provisions (ACI 2002, ASCE 2002, ICC, 2003). Design spectra for

the site was based on adjusted mapped values of Ss = 1.5g and S1 = 0.6g. Both concrete

frame structures are perimeter, special moment frame structures that comply with capacity

design provisions, strong column-weak beam ratios, joint shear strength and detailing

regulations, in addition to requirements for strength and stiffness. As modeled, the 4-story

building and the 12-story building have fundamental periods of 0.94 and 2.14 seconds,

respectively. Interested readers can refer to Haselton and Deierlein (2007) and Haselton et


al. (2007) for more details about the designs and modeling parameters of the modern

example structures.

For the structures with non-ductile detailing, the 1967 Uniform Building Code

(UBC 1967) was used to design buildings that were more representative of older concrete

frame structures erected in California from approximately 1950 to 1975. Both designs

complied in accordance with Zone 3 requirements, the highest seismic design provisions of

this era. Although these designs met all requirements of the 1967 UBC (ICBO, 1967) (such

as maximum and minimum reinforcement ratios, maximum stirrup spacing, bar spacing and

anchorage…etc.), these provisions did not require as much transverse reinforcing as modern

day building codes, which result in members that are non-ductile. Further, at the time, no

special provisions for design or reinforcement of beam-column joints were required, and the

strong-column weak beam ratio requirement had yet to be introduced. Thus, these

structures are susceptible to joint-shear failure and column hinging. The buildings are also

much more flexible because of lower reinforcing restrictions and lower seismic design

forces, as demonstrated by their fundamental periods of 2.0 and 2.8 seconds for the 4-story

and 12-story buildings, respectively. Detailed information on the designs and modeling

parameters of these structures can be found in Liel and Deierlein (2008).

Obtaining realistic loss estimations requires developing architectural layouts for the

buildings being considered, to inventory the amount of damageable non-structural

components and to determine the median total replacement cost. This study uses

architectural layouts documented in Mitrani-Reiser (2007) and in Chapter 7 of this

dissertation. A rectangular footprint that is 120-ft wide and by 180-ft long, was used for the

4-story buildings. A smaller layout, with a 120-ft square footprint, was developed for the

12-story buildings. The architectural layouts were used to develop estimates of the total

replacement costs of the buildings. Cost estimates were developed using the RS Means

Cost Estimating Manuals (Balboni, 2007). Table 9.1 lists the estimates for all different

building design variants. Also listed in Table 9.1 are the costs per square foot for each

building design. Although the authors recognize that estimates developed using RS Means

are relatively low compared to typical office building construction costs in the geographical

area being considered, this method follows a rational, systematic approach to estimate

building replacement values. All of the expected loss results that will be presented were

normalized by these values to make valid comparisons within this study.


Table 9.1Cost estimates for buildings studied

Building Type Footprint Area (sf)Replacement

Cost ($)Cost per

sf ($)4-story 180'x120' 86,400 $12,000,000 $138.89

12-story 120'x120' 172,800 $47,900,000 $277.20

Each of the reinforced concrete frame structures of interest in this study was

modeled in OpenSees (PEER, 2006) using a two-dimensional, three-bay model of the

lateral resisting system and a leaning (P-Δ) column. The model does not incorporate

strength or stiffness from components designed to resist gravity loads. Beams and columns

were modeled with concentrated hinge (lumped plasticity) elements and employ a material

model developed by Ibarra et al. (2005). Haselton and Deierlein (2007) have shown that

the lumped plasticity modeling approach provides reasonable results (compared to fiber-

element models) at low levels of deformation and in addition, captures material

nonlinearities as the structure collapses.

The nonlinear simulation models of reinforced concrete frames were analyzed using

the incremental dynamic analysis technique. The selected ground motions records are from

large magnitude events and recorded at moderate fault-rupture distances on stiff soil or rock

sites (ATC, 2008). The collapse fragility, typically represented by cumulative probability

distribution, was adjusted to account for uncertainties in the structural modeling process and

the expected spectral shape of rare ground motions in California. As an example of the type

simulation data used, the collapse fragility and the EDP data for the ductile, 4-story building

considered in this study are shown in Figure 9.1 and Figure 9.2, respectively. Interested

readers are referred to (Haselton and Deierlein, 2007) for the remaining simulation results

of the example buildings used in this study and for more details on how the structural

simulation was conducted.


PROBABILITY OF COLLAPSE

0.0

0.2

0.4

0.6

0.8

1.0

0.0 2.0 4.0 6.0

IM = Sa [g]

P(Collapse | IM)

Figure 9.1: Probability of collapse for ductile 4-story reinforced concrete structure (Haselton and Deierlein, 2007)

1

2

3

4

0 0.02 0.04 0.06E[EDP = IDR | IM=Sa]

STORY

2

3

4

5

0 1 2E[EDP = PFA | IM=Sa]

FLOOR

1

2

3

4

0 0.02 0.04E[EDP = RIDR | IM=Sa]

STORY

Sa = 0.05g

Sa = 0.25g

Sa = 0.50g

Sa = 1.00g

Sa = 2.00g

1

2

3

4

0 0.02 0.04 0.06E[EDP = IDR | IM=Sa]

STORY

2

3

4

5

0 1 2E[EDP = PFA | IM=Sa]

FLOOR

1

2

3

4

0 0.02 0.04E[EDP = RIDR | IM=Sa]

STORY

Sa = 0.05g

Sa = 0.25g

Sa = 0.50g

Sa = 1.00g

Sa = 2.00g

1

2

3

4

0 0.02 0.04E[EDP = RIDR | IM=Sa]

STORY

Sa = 0.05g

Sa = 0.25g

Sa = 0.50g

Sa = 1.00g

Sa = 2.00g

Sa = 0.05g

Sa = 0.25g

Sa = 0.50g

Sa = 1.00g

Sa = 2.00g

Figure 9.2: EDP data as a function of building height for ductile 4-story reinforced concrete structure (Haselton and Deierlein, 2007)

9.3.2 Results

The economic loss results at the DBE normalized by the building replacement value

for all four structures considered in this study are summarized in Table 9.2. The normalized

values of each type of loss – non-collapse losses due to repair, non-collapse losses due to

demolition and collapse losses – are reported in columns (1) through (3). Each of the


values in columns (1), (2) and (3) are computed by the first, second and third terms in

equation (9.2), respectively. Column (4) reports the overall economic loss when the three

types of losses are summed together. The last three columns in the table report percent

contributions of each type of loss (i.e. these values represent the losses computed in

columns (1) to (3) but are normalized by the total loss in column (4)).

Table 9.2 Summary table for expected economic loss results at design basis earthquake (DBE) as a percentage of building replacement value

Eq. (9.2) Term 1

Eq. (9.2) Term 2

Eq. (9.2) Term 3

TotalEq. (9.2) Term 1

Eq. (9.2) Term 2

Eq. (9.2) Term 3

(1) (2) (3) (4) (1) / (4) (2) / (4) (3) / (4)

4-story Ductile 25% 15% 3% 42% 58% 36% 6%

12-story Ductile 15% 13% 6% 34% 44% 38% 18%

4-story Non-ductile 12% 12% 51% 74% 16% 16% 68%

12-story Non-ductile 4% 12% 65% 81% 5% 15% 80%

DisaggregationExpected Loss at design level EQ

Building

The first building analyzed in this study was a 4-story ductile, reinforced concrete

structure. Figure 9.3 shows the expected losses as a function of the intensity of ground

motion (measured here in terms of spectral acceleration at the fundamental period, Sa(T1)).

Economic losses here are normalized by the replacement value of the building. Losses

appear to rise very quickly as the earthquake intensity level increases. Structures in the US

are designed to a level of ground motion intensity typically referred to as the Design Basis

Earthquake (DBE), which is defined in Chapter 7. For this site, the DBE has a spectral

acceleration of 0.52g, and is identified on Figure 9.3. At this intensity level, the total

expected losses are approximately 59% of the mean replacement cost of the structure.


EXPECTED LOSS GIVEN IM

0%

20%

40%

60%

80%

100%

0.0 0.5 1.0 1.5 2.0

IM = Sa [g]

E[L | IM]

Mean Repair CostsCollapseNoncollapse - DemolishNoncollapse - Repair

Sa at DBE

EXPECTED LOSS GIVEN IM

0%

20%

40%

60%

80%

100%

0.0 0.5 1.0 1.5 2.0

IM = Sa [g]

E[L | IM]

Mean Repair CostsCollapseNoncollapse - DemolishNoncollapse - Repair

Sa at DBE

Figure 9.3 Normalized expected economic loss as a function of ground motion intensity.

Also illustrated in Figure 9.3 is the value of normalized economic losses

disaggregated by collapse losses, non-collapse losses due to demolition and non-collapse

losses due to repair as a function of intensity level. Non-collapse losses dominate the

performance at low intensities (Sa(T1) < 0.5g), with repair losses initially making up the

large contributions to the overall loss. Between spectral accelerations of 0.6g and ~ 1.7g,

the repair losses begin to decrease, and the demolition losses alone comprise the largest

portion of the total expected losses. Beyond Sa(T1) = 1.7g, collapse starts to govern the

losses as expected for large magnitudes of ground shaking.

Results from the 4-story, modern structure with ductile detailing reveal a substantial

increase in total expected losses when the effects of excessive residual drift are included in

economic performance. Figure 9.4 compares the expected economic losses with and

without considering losses due to demolition for three different levels of seismic hazard.

The middle pair of bars corresponds to the expected economic losses that are incurred at the

DBE. The pair to the left corresponds to the losses for a seismic due to a seismic event with

a seismic intensity that has a probability of exceedance of 50% in 50 years (this hazard level


occurs more frequently and is often referred to as the service-level earthquake). The pair to

the right corresponds to the losses due to seismic event that has a probability of exceedance

of 2% in 50 years (this hazard level occurs less frequently and is often referred to as the

Maximum Credible Earthquake, MCE). The values of the seismic ground motion intensity

that correspond to all three hazard levels are listed at the bottom of the figure. For each

hazard level, the left bar corresponds to losses that do not consider losses due to demolition

and the right bar corresponds to the losses that consider losses due to demolition.

0%

20%

40%

60%

80%

100%NC - Repair

NC - Demolish

Collapse

E[L | IM] DUCTILE 4-STORY

ServiceSa = 0.31

DBESa = 0.52

MCESa = 0.78

0%

20%

40%

60%

80%

100%NC - Repair

NC - Demolish

Collapse


ServiceSa = 0.31

DBESa = 0.52

MCESa = 0.78

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.

At the service-level earthquake, the effect of losses due to building demolition does

not have an influence on the overall normalized loss. On the other hand, the normalized

economic losses increase from 31% to 42% at the DBE. This represents a relative increase

of 35% (the relative increase is the difference between the two values of expected loss, with

and without considering losses due to demolition, divided by the expected loss with

considering losses due to demolition, multiplied by 100). At the MCE, the normalized

economic losses increase from 48% to 73% representing a relative increase of 52%. This

means that considering the losses due to demolition has a large influence on the overall loss

estimate, particularly for seismic events that have smaller ground motion intensities but


occur more frequently. At this level of ground motion intensity, the effect of economic

losses due to demolition is even larger than it was at the DBE.

Loss results at these levels were disaggregated to observe how much each term in

equation (9.2) contributes to the overall performance as shown in Figure 9.4 (Loss

disaggregation was performed in a similar manner as documented by Aslani and Miranda,

2005). Each bar in the figure is divided up into collapse losses, non-collapse (NC) losses

due to building demolition and non-collapse losses due to repair costs. The proportions of

each bar are equal to how much each type of loss contributes to the overall loss.

Demolition losses have the largest contributions to the overall loss at the MCE. At this

intensity level, losses conditioned on non-collapse due to demolition represent 60% of the

total loss. This is more than twice as large as the contributions from losses conditioned on

collapse, which comprise 27% of the overall loss at the MCE. The high contributions of

non-collapse losses due to demolition can be explained by comparing the probability of

collapse with the probability of demolition as a function of ground motion intensity, which

is illustrated in Figure 9.5. At the MCE, the probability of demolition is much higher (45%)

than the probability of collapse (8%). This means that the considered structure is more

likely to experience large residual deformations that will lead to demolition, as compared to

collapse due to an earthquake for the given level of ground motion intensity. Consequently,

the non-collapse losses due to demolition are much larger than the losses due to collapse

because the total expected loss is the sum of these two types of losses weighted by the

probability that these events will occur as demonstrated by equation (9.7).


0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

IM = Sa / Sa10/50

P(DM = Demolish | IM)

P(DM = Collapse | IM)

P(DM | IM) DUCTILE 4-STORY

IM = Sa / Sa @ DBE

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

IM = Sa / Sa10/50




IM = Sa / Sa @ DBE

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.

These results suggest that ignoring the financial consequences of excessive residual

drift can severely underestimate economic loss predictions. Previous loss estimation

methods that do not include these effects may be misleading, predicting better performance

than actually experienced. Performance can be underestimated by as much as 35% at the

DBE and 52% at the MCE as demonstrated by this case-study.

The results in Chapter 7 demonstrated that the effect of building height can have a

substantial influence on predicted economic losses. The effect of building height was also

investigated in this study by comparing the resulting economic losses of the 4-story

structure to the results of a 12-story structure. Figure 9.6 shows the loss results for the 12-

story building at the service-level earthquake, the DBE and the MCE. For each hazard

level, the total normalized economic losses are smaller than the losses of the 4-story

structure shown in Figure 9.4 (Note that the losses are smaller when normalized by the

replacement cost. The replacement costs and losses in the 12-story are larger in

magnitude.). This trend holds whether or not non-collapse losses due to demolition are

considered.


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ServiceSa = 0.31

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ServiceSa = 0.31

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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.

Figure 9.6 also compares the losses with and without considering loss due

demolition, for each hazard level. Despite demonstrating lower total losses, the relative

increase in losses due to considering the effect of demolition losses is larger in the 12-story

structure than it is in the 4-story building. When demolition losses are considered at the

DBE, normalized losses are 45% greater than when the demolition losses are ignored. This

is larger than the 35% increase in normalized loss observed in the loss analysis of the 4-

story structure. This is also true for the MCE, where the relative increase due to demolition

losses is 63% for the 12-story building and only 52% for the 4-story building. This

suggests that large residual drifts may play a more significant role in estimating loss for

high-rise buildings, than it does for low-rise buildings.

The loss results in Figure 9.6 were disaggregated as described previously to

determine the value of contributions from each of the terms in equation (9.2). As was the

case for the 4-story building, demolition losses comprise the largest portion of the overall

loss at the MCE. At this level, losses due to demolition make up 51% of the overall loss.

These case-studies suggest that accounting for excessive residual drift in loss estimations is

important in both low-rise and high-rise structures. Although the proportion of demolition

losses relative to the overall loss is approximately the same in both cases, it was


demonstrated in the previous paragraph that demolition losses may have a much larger

influence in high-rise buildings than they do in low-rise buildings.

Despite exhibiting larger overall losses, the effects of residual drift on performance

did not have as significant an influence on the older structures (non-ductile detailing) as it

did with the modern structures (ductile detailing). Figure 9.6(a) shows that the total

economic loss at the DBE for a 4-story non-ductile building (74%) is greater than the

corresponding total loss for the ductile 4-story structure reported previously (42%).

However, when comparing the effect of demolition losses, the influence of loss due to

demolition is not as great in the non-ductile structure as it is in the ductile structure. At the

DBE, the increase in loss experienced by the non-ductile building is only 12% while the

increase in the ductile structure is 35%. Similar results can be observed in the non-ductile,

12-story structure as illustrated by Figure 9.6(b). The probability of collapse plays a much

larger role in building performance than the probability of demolition for non-ductile

structures. This can be better demonstrated by examining the deaggregation of these losses.

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E[L | IM] (b) NON-DUCTILE 12-STORY

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Figure 9.7 Loss results for non-ductile buildings studied (a) 4-story (b) 12-story

In addition to showing total losses, Figure 9.7(a) and Figure 9.7(b) also

disaggregates the loss of non-ductile buildings. Comparing these disaggregations with

those shown in Figure 9.4 and Figure 9.6, it can be determined that collapse plays a much

larger role in loss for the non-ductile buildings than it does for the ductile structures.

According to these figures, the contributions of demolition losses to the overall losses are

greatest at the DBE (unlike the ductile structures, where the demolition loss contributions


were largest at the MCE). However, the demolition loss contributions are much smaller in

these building than they are in the ductile structures (16% for the non-ductile 4-story and

15% for the non-ductile 12-story). Note that the collapse loss contributions are much larger

than the other two types of losses. The 4-story non-ductile structure attributes 86% of its

total loss to collapse, while the 12-story non-ductile structure accredits 89%.

This trend can be explained by observing that the collapse probabilities are much

higher than the probability of demolition. Figure 9.8(a), Figure 9.8(b), Figure 9.8(c) and

Figure 9.8(d) compare the probability of collapse to the probability of building demolition

as a function of ground motion intensity for all four structures. In each plot, ground motion

intensity is normalized by the intensity level of the DBE such that buildings of different

heights can be compared (buildings of different heights typically have different periods and

are designed for different levels of spectral acceleration based on the code-specified design

spectrum). In the ductile structures, the probability of demolition is greater than the

probability of collapse up until 3 times the ground motion intensity of the DBE for the 4-

story building and until 2 times the ground motion intensity of the DBE for the 12-story

building. Conversely, the probability of collapse is significantly larger than the probability

of demolition throughout most of the range of the ground motion intensity levels for both

the 4-story and the 12 story building. Collapse is more likely to occur in the older, non-

ductile buildings for the following reasons: 1) these buildings were designed to withstand

lower seismic forces 2) these buildings are more susceptible to beam-column joint shear

failure, which may result in loss of vertical carrying capacity 3) column hinging is more

likely to occur in these structures because no minimum strong-column, weak-beam

(SCWB) ratio was considered in the design of these buildings 4) column shear failure is

more likely to occur and the plastic hinges developed in these structures will experience fast

degradation due to less confinement and tie reinforcement in non-ductile concrete members.

Because the effect of collapse dominates the monetary losses, the economic consequences

of residual drift had little to no effect on performance for these non-ductile structures.


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(a) (b)

(c) (d)



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.

9.3.3 Sensitivity of Loss to Changes in the Probability of Demolition

Incorporating non-collapse losses due to demolition into the building-specific loss

framework previously presented required estimating the statistical parameters of the

probability distribution GD|RIDR(D | RIDR = ridr, NC) in equation (9.9). The statistical

parameters of this variable were difficult to calculate because there was very little data

available to make adequate estimates. A sensitivity analysis was conducted to examine how

different estimates of these statistical parameters would affect estimations of loss.


The results of the case-study buildings in this study were primarily conducted

assuming that the median and lognormal standard deviation of GD|RIDR(D | RIDR = ridr,

NC) were equal to 1.5% and 0.30, respectively. The criteria used to select these parameters

was based on what value RIDR would have to exceed to create physical reasons that would

most likely lead to demolishing a building. Using the assumed parameters, an RIDR of

2.2% has a 90% probability of triggering building demolition. In stories that have

experienced an RIDR of 2%, doorways may start to become unusable. For a 7 foot

doorway, a 2% RIDR corresponds to a lateral displacement of approximately 1.7 inches,

which would make it impossible to close the door. Further, 2% drift approximately

corresponds to a 3 inch lateral displacement for a 13 foot story height. At this value of

RIDR, the lateral deformation will be visibly noticeable even to the untrained eye, which

may make building owners and occupants uncomfortable, and may affect the building’s

monetary value. Therefore, it seems reasonable that RIDRs in the range of 2% will have a

high probability of initiating building demolition.

The sensitivity of the assumed median and lognormal standard deviation of

GD|RIDR(D | RIDR = ridr, NC) on the resulting loss estimations was analyzed. To examine

the sensitivity of the median of GD|RIDR(D | RIDR = ridr, NC), loss analyses were conducted

using medians of 0.5%, 1.0%, 1.5%, 2.0%, 2.5%, 3.0%, 5.0% and 10% (Figure 9.9(a))

while holding the dispersion constant at 0.30. To examine the sensitivity of the dispersion

of GD|RIDR(D | RIDR = ridr, NC), its median was held constant at 1.5% and loss estimations

were performed for varying dispersion values of GD|RIDR(D | RIDR = ridr, NC) equal to 0.1,

0.2, 0.3, 0.4, 0.5, 0.6, 0.7, and 0.8 (Figure 9.9(b)).


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Figure 9.9 Different distributions for probability of demolition given RIDR (a) Varying the median (b) Varying the dispersion

The economic loss results at the DBE for different values of the median RIDR of

the cumulative probability distribution GD|RIDR(D | RIDR = ridr, NC) are displayed in

Figure 9.10(a). The plot shows that the total loss decreases as the median increases. This

decrease is larger at smaller values of the median than it is at larger values of the median

GD|RIDR(D | RIDR = ridr, NC). Note that as the median is increased and approaches very

large values, the performance is equivalent to the case where demolition losses are not

considered. In this figure, only the value of the repair losses and the demolition losses vary,

whereas the collapse losses remain constant. This is because the statistical parameters of

GD|RIDR(D | RIDR = ridr, NC) only affect the probability of demolition.

A similar, but much weaker trend is observed when using expected annual loss

(EAL) as a metric for performance. Expected annual loss is the average economic loss due

to seismic ground motions that will be incurred each year. It is obtained by integrating

expected losses as a function of ground motion intensity with the site’s seismic hazard

curve (Figure 7.1). Figure 9.10(b) shows the EAL results for the different median values

for the probability of building demolition given RIDR. As was observed in Figure 9.10(a),

the values of loss decrease as the median of GD|RIDR(D | RIDR = ridr, NC) increases. The

decrease in EAL between RIDRs of 0.5% to 2.0%, however, is much smaller than it is for

the expected losses at the DBE. Median values of greater than 2.0% do not produce

differences that are as substantial.


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EAL DUCTILE 4-STORY

Figure 9.10 Results for sensitivity analysis of probability of demolition given RIDR for 4-story ductile reinforced concrete moment frame office building.

Changing the dispersion of GD|RIDR(D | RIDR = ridr, NC) does not seem to have a

significant effect on both the expected economic losses for the DBE and EAL, as shown in

Figure 9.10(c) and (d), respectively. The expected loss at the DBE does show some

increase as the dispersion increases, but the difference is very small. There is virtually no

change in EAL as the dispersion increases.

Based on this sensitivity analysis, it appears that loss estimations are much more

sensitive to changes in the median than the dispersion of the probability of demolition given

RIDR. Although the estimates of the statistical parameters of the probability of demolition

given RIDR used in this study (median = 1.5%, dispersion = 0.3) seem reasonable based on

the criteria described previously, it is recommended that further study be conducted to

obtain data that will validate the use of these values in this loss estimation method to

eliminate any subjectivity used to estimate these parameters.


9.3.4 Limitations of results & discussion of residual drift estimations

It is recognized that the resulting economic loss estimates computed in this study

and the conclusions on the influence of residual deformations on seismic-induced economic

loss are dependent on how the values of the RIDR are computed. The simulation results

used in this study were taken from previous research that was conducted on the

performance of reinforced concrete moment frame structures. Although the simulation

techniques were state-of-the-art, the focus of these studies was on collapse performance and

not on capturing residual deformations. It is the opinion of the authors that the RIDR

values computed may be large and possibly overestimated. This is primarily because of the

type of hysteretic model used by Haselton and Deierlein (2007) and Liel and Deierlein

(2008) in their nonlinear simulations. These simulations implemented a material model

developed by Ibarra et al. (2005), which unloads using a stiffness equal to the initial

stiffness. For reinforced concrete members, this may lead to overestimations in residual

displacements as was demonstrated by Ruiz-Garcia and Miranda (2006).

Ruiz-Garcia and Miranda (2006) determined that hysteretic models that capture

stiffness-degradation will compute smaller residual displacements than models that do not

equate the unloading stiffness to the initial stiffness. The decrease in residual displacement

is primarily because stiffness-degradation models have a tendency to unload toward the

origin when subjected to cyclic loading. Systems that do not degrade have larger unloading

stiffness, which resists displacements that would restore the structural member being

modeled back to its original position. Although degrading stiffness systems are more

representative of actual behavior observed by reinforced concrete elements, the model

developed by Ibarra et al. (2005) does not model behavior. This suggests that residual

displacements computed by Haselton and Deierlein (2007) and Liel and Deierlein (2008)

may be overestimated.

If the residual displacements used in this study are overestimated, then the

economic loss results that consider replacement costs due to building demolition will also

be overestimated and may explain why the losses are fairly large. Although the actual

economic loss results may not be as high as those reported in this study, it is still important

to note that considering residual displacements will increase seismic-induced monetary

losses and that this increase may be large.


9.4 SUMMARY AND CONCLUSIONS

An approach to expand and improve the PEER performance-based design

methodology has been presented. The proposed approach now explicitly takes into account

the fact that a building may have to be demolished after an earthquake. The economic loss

conditioned on the level of ground motion intensity is computed as the sum of three terms:

two terms previously identified corresponding to losses resulting if the building collapses

and to losses associated with repairs given that the structure has not collapsed plus a third

term which accounts for losses resulting from having to demolish buildings that have

experienced excessive residual drifts.

Four case-study buildings were examined to investigate the impact of incorporating

losses due to forced demolition on total economic losses. Results indicate that losses due to

demolition can had a large influence in both the 4-story and the 12-story ductile reinforced

concrete moment-resisting frame buildings that were considered in this study. Including

losses due to demolition increased loss estimates for the DBE event by as much as 45%

larger than the loss estimates that ignored its contribution. This may suggest at that current

methods of loss estimation may be severely underestimating the building performance of

these types of structures by not accounting for the effects of permanent displacement in

structural damage. The influence of demolition losses was not as substantial in the 4-story

and 12-story non-ductile reinforced concrete moment-resisting frame structures. These

building exhibited a much larger probability of collapse at the ground motion intensities of

interest, and consequently yielded economic loss results that were primarily due to collapse.

The proposed approach provides practicing engineers a way of quantifying

differences in building performance between structures with systems that are susceptible to

large permanent deformations and those that use lateral force resisting systems that do not

rely on structure member damage to dissipate energy. The higher economic losses

associated with residual deformation in structures may promote the implementation of self-

centering structural systems that provide alternative means of dealing with excessive

permanent displacement. Although the initial construction cost of these systems may be

larger than that of conventional systems, the ability of these structures to minimize residual

drifts, avoids the potential of demolishing the structure and associate economic costs need

to replace the building.


The importance of permanent displacement in loss estimation, as illustrated here,

signifies a need for obtaining improved estimates on residual drift. The effect of vertical

permanent displacements should also be examined as they may also lead to building

demolition. Finally, this study limited its evaluation to using expected values of loss,

whereas, there is much value and interest into investigating these issues in terms of other

loss metrics, such as the mean annual frequency of loss.

CHAPTER 10 271 Summary & Conclusions

CHAPTER 10

10 SUMMARY AND CONCLUSIONS

10.1 SUMMARY

Performance-based earthquake engineering (PBEE) has taken tremendous strides in

the past decade. The collaborative effort of PEER researchers has resulted in a methodology

that can quantify structural performance in terms that stakeholders can more readily

understand and use in making decisions regarding seismic risk (e.g. deaths, dollars and

downtime). Unfortunately, evaluating seismic-induced economic loss is no easy task. In

particular, it requires a significant amount of information and is computationally intensive.

Successful adoption of PBEE by practicing engineers may hinge on providing a version of

PEER’s framework that is easier to use and more efficient than previously presented.

This study has presented a new story-based loss estimation methodology, a simplified

implementation of PEER’s previous approaches, as a more efficient way of quantifying

structural seismic performance (Chapter 3). Relationships between structural response and

monetary loss in the form of EDP-DV functions were developed based on the assumptions

on the building’s cost distribution and structural system. Engineers using this method do not

need to conduct the intermediate step of estimating damage, because it has been already

been computed based on assumptions on the building’s cost distribution. Fragility functions

for structural components were developed from experimental data to supplement the

creation of EDP-DV functions (Chapter 4). Generic fragility functions for non-structural

components were generated from empirical data, making it possible to account for the entire

inventory of components in a story when generating EDP-DV functions, yielding more

complete predictions of non-collapse losses (Chapter 5).

These simplifications were implemented into a user-friendly computer tool that can

be used to estimate economic losses as a metric of structural performance (Chapter 6). The


tool was then used to estimate the performance of a set of reinforced concrete moment

resisting frame office buildings to benchmark how much expected losses may be incurred at

a given seismic intensity level, and to evaluate the life-cycle costs due to earthquake

damage (Chapter 7).

This investigation has also proposed a number of improvements to PEER’s loss

estimation methodology to predict better earthquake-induced economic losses. An

alternative method of calculating the uncertainty of predicted losses that incorporates

correlations in construction cost at the building level, rather than at the component-level,

was presented, because dispersion and correlation data for losses on individual components

is not available (Chapter 8). Additionally, an approach was introduced (Chapter 9) that

accounts for losses due to building demolition triggered by excessive permanent

deformations (i.e. residual interstory drift).

10.2 FINDINGS & CONCLUSIONS

10.2.1 Story-based Loss Estimation

Story-based loss estimation, a simplified version of PEER’s previous loss estimation

methodology, is based on creating direct relationships between structural response and loss

by collapsing out the intermediate step of estimating building component damage. These

relationships, termed story EDP-DV functions, were developed for reinforced concrete

moment resisting frame office buildings, by consolidating fragility functions for every

building component and integrating over every damage state for each component. This

integration allows losses to be estimated without knowing the exact cost of repairs if these

values are normalized by the replacement cost of the component. The components can be

summed together at each story without a detailed building inventory, provided that

assumptions can be made about the cost distribution of story based on the building’s

occupancy and use.

Several types of EDP-DV functions for structural components were developed in this

study including functions for ductile and non-ductile reinforced concrete elements. In

general, the functions for ductile components estimated lower losses for a given IDR than

the non-ductile components. The largest change in losses was observed at an IDR of 0.052,

where the losses of ductile concrete components were 140% less than the non-ductile


components. This improved performance is a result of modern building codes requiring

better detailing, such as increasing minimum confinement requirements in order to increase

the member ductility (ACI, 2005). The increase in ductility delays or prevents particular

failure modes (e.g. shear failure of columns, punching shear of slab-column connections,

shear failure of beam-column joints). This reduces the amount economic loss between

ductile and non-ductile members computed by the EDP-DV function between values of

IDR between 0.03 and 0.10.

Consolidating the fragilities and repair costs of building components into story EDP-

DV functions in this manner provides the opportunity to investigate the issue of conditional

damage in spatially-interacting components. In previous studies, when the loss due to

damage of one component is dependent on the damage state of another component, it has

either been ignored leading to an underestimation of losses or double-counted leading to an

overestimation of economic losses. EDP-DV functions were used to examine the

dependency of partitions on pre-Northridge steel beam-column joints by structuring the

functions such that the partitions’ fragilities are conditional on the fragilities of the steel

joints. It was determined that double counting may overestimate the loss significantly while

treating the components as independent underestimates the loss. It was also demonstrated

that as more and more conditional components are accounted for, the underestimation of

independent components becomes more substantial.

Engineers using story-based loss estimation need only provide the economic

investment per story – rather than a detail inventory of all components and their

replacement for the entire story – to calculate loss, making the process much more efficient

and less computationally intensive. Furthermore, performance evaluations are often most

useful during preliminary design – when building inventory is not well defined – because

many important design decisions, such as the structural system, are being determined.

Reducing the amount of computation, time and resources that go into the loss estimation

process using simplified methods allows decision makers to focus on considering the

benefits of the different design alternatives and making better design decisions.

10.2.2 Improved Fragilities in support of EDP-DV Function Formulation

Fragility functions were generated from both experimental and empirical data to

supplement the formulation of story EDP-DV functions. Experimental data from testing of


Pre-Northridge beam-column joints was consolidated and used to create fragilities that can

be used to assess loss of buildings supported by steel moment resisting frames constructed

prior to 1994. Fragilities for two damage states, yielding and fracture, were developed based

on the data available. The fragility for fracture is more relevant to loss estimation because

yielding of a steel joint does not typically result in any required repair actions being taken.

The median IDR and corresponding lognormal standard deviation for fracture were 1.85%

and 0.47, respectively. It was also observed that the yielding parameters were dependant on

span-to-depth ratio, and the parameters for fracture were dependent on beam depth.

Relationships to determine median IDRs based on these dependent section properties were

also developed.

Generic fragility functions were created to account for losses from components that

previously did not have fragilities established from experimental data. Empirical data from

damage reports taken during the 1994 Northridge earthquake was combined with response

values from instrumentation and simulation to create motion-damage pairs, to which

lognormal cumulative distribution functions were fitted. Although many types of functions

were generated, the most valuable relationships derived from this investigation were the

results for drift-sensitive and acceleration-sensitive non-structural components, because it

has been shown that they can comprise a large part of overall loss (Aslani and Miranda

2005, Taghavi and Miranda 2003) and there are only a few fragility functions available to

estimate damage for these components. Although most of the data available were for early

stages of damage, identifying when damage initiates is particularly valuable because losses

at low intensities contribute the most when evaluating expected annual losses (EALs). It

was determined that damage initiates at a median IDR of 0.30% and a median PBA of

0.39g’s based on the data from the CSMIP data. When these fragilities were compared to

previous generic fragilities functions created for HAZUS (NIBS, 1999), the empirical

fragilities estimate a median IDR that is 28% less than those approximated by HAZUS.

Conversely, damage initiation for acceleration-sensitive components occurred at a median

value 58% greater than the median value of the HAZUS fragility functions, suggesting that

previous fragilities may overestimate losses.

Generic fragility functions can be used to improve estimates of seismic-induced

losses given that the considered building has not collapsed. Fragility functions have not

been generated for every type of component that may be included in a building’s inventory.

In the absence of functions formulated from experimental data, these generic functions can


be used as an adequate substitute to account for losses due to these components. Although

these functions may not capture some damage behavior that may be component specific,

they are relationships that are based on data taken from earthquake reconnaissance for these

types of items. They are also likely to yield better loss estimations than using functions

based on expert opinion or ignoring the damage due from components that do not have

specific fragilities.

10.2.3 Implementing loss estimation methods into computer tool

The improved and simplified methods described were implemented into an MS

EXCEL based computer tool. Step-by-step instructions on how to use the program were

documented in this study. During the process of developing this computer tool,

implementation challenges were encountered that needed to be addressed such that the

program estimated economic losses as intended by the framework.

10.2.4 Benchmarking losses

The methods and tools developed in this study were used to predict losses for a set

of typical reinforced concrete moment frame office buildings. The group of 30 modern

code-conforming buildings was designed with ductile detailing (e.g. well confined concrete)

by a previous investigation (Haselton and Deierlein, 2007). A brief comparison was made

to a second set of 25 buildings that were designed with non-ductile concrete moment

frames, representative of structures designed in California prior to 1967 (Liel and Deierlein,

2008). The findings and conclusions from these analyses are as follows:

For the ductile structures, expected losses results at the DBE were found

to be a significant percentage of the replacement value, ranging from 13%

to 54%. When a hazard curve from a site in L.A. was considered, the

EAL computed for this set of buildings varied from 0.4% to 3.3%. Both

metrics indicate that the economic losses can be very large considering

that they conform to current US codes. The wide range of results also

demonstrates that US building codes do not provide a consistent level of

monetary loss performance.

Of the code-conforming structural parameters considered in this study,

building height was found to have the largest influence on expected losses


normalized by the replacement value of the building. Taller buildings

experienced less relative monetary loss than shorter buildings because

lateral deformations tend to concentrate in a smaller percentage of stories.

Consequently, high levels of damage are confined to only a few stories as

opposed to being spread throughout the height of the structure.

Although increasing the design SCWB ratio reduced the losses due to

collapse, increasing the ratio beyond a value of 1.2 did not significantly

affect the total expected losses of the ductile structures. On the other

hand, modifying the R-factor had a larger effect on expected losses at the

DBE. If code committees are interested in adjusting code provisions to

limit direct economic losses due to seismic ground motions, this study

suggests that they would be best served by modifying the R-factor.

Designing for life-safety does not necessarily translate to satisfactory

economic loss performance. The parameters of EAL and the MAF of

collapse were found to be uncorrelated for these sets of buildings.

Performance-based design tools provide separate quantifiable metrics to

address collapse risks and economic losses. The ability to quantify both

metrics also provides decision-makers with information to identify and

consider possible tradeoffs between mitigating losses and reducing

collapse risk.

The older, non-ductile reinforced concrete frame buildings performed

significantly worse in terms of economic losses. The losses experienced

by these structures were on average 1.4 times greater than the ductile

buildings. The larger losses are primarily due to larger losses due to

collapse. These frames are more likely to experience sudden, brittle

failures that are associated with non-ductile detailing.

10.2.5 Improved estimates on the uncertainty of loss

An approach that incorporates building-level construction cost dispersions and

correlations, making it more compatible with the way construction cost data is produced

and recorded, was presented. One of the 4-story buildings from the set of structures


evaluated in Chapter 7 was used as an example to illustrate the use of this approach. It was

determined that for this building the coefficient of variation was computed to be 0.67 at the

DBE and a mean annual frequency (MAF) of loss equal 1.4x10-4 for losses equal to the

replacement value of the building. This MAF is 5.9 times larger than the MAF of collapse

(8.2 x10-4). This means that for this building, it would be almost 6 times more likely to

experience an economic loss equal to the replacement value than to experience collapse.

The largest source of variability was attributed to the variability of construction costs for

this building. It was also demonstrated that the correlations from building-level construction

costs and EDPs had a substantial influence on the economic loss dispersion, increasing the

MAF of loss by 110% in this building for losses equal or greater to the replacement value of

the example building.

The approach was implemented using both Monte Carlo simulation and first-order,

second-moment methods of estimating uncertainty to evaluate the accuracy as well as

advantages and disadvantages of each method. It was demonstrated that there was fairly

good agreement between the two approaches for the total dispersion for the example

building. However, this trend would not necessarily be true for other buildings because

there were fairly significant discrepancies between the two methods when the economic

loss dispersion due to EDP variability was disaggregated and compared at the story-level

for each subcontractor. The differences between the two methods were primarily due to

errors of the FOSM approximations occurring within the nonlinear regions of EDP-DV

functions used to evaluate the economic loss dispersion. For this particular building, the

errors happen to compensate for each other when the loss per story were summed together,

which resulted in smaller differences between the two methods when the total dispersions

for the entire building were compared. This may not necessarily be the case for other

buildings where these errors may amplify each other (i.e. the errors are additive).

Evaluating the variability of economic loss by dividing the losses into subcontractor

costs resulted in an inherent correlation between losses due to work performed by different

subcontractors that need to be accounted when evaluating dispersions analytically. This

correlation exists because the amount of work performed by each subcontractor on the same

floor/story is dependent on the same response parameters. Consequently, this correlation

occurs even when the EDPs and construction costs are uncorrelated. It was determined that

if this type of correlation is not accounted for, economic loss dispersions computed


analytically could significantly underestimate the variability due EDP variability (by as

much as 15% for the example building).

Finally, the bootstrap method of sampling was presented as a way of obtaining

reliable estimates of EDP correlations. It was also demonstrated how bootstrap confidence

intervals could be employed to evaluate the number of ground motions required to be used

in structural analysis to obtain reliable EDP correlation coefficients. A closed-form solution

taken from statistical theory was presented as a way of computing these confidence

intervals efficiently.

10.2.6 Accounting for Non-collapse losses due to building demolition

To incorporate this new building damage state into PEER’s loss estimation

framework it is necessary to calculate the probability of building demolition given that it

has not collapsed at a ground motion intensity IM=im. It is proposed that this random

variable be computed as a function of two other probabilities: 1) the probability of

demolition given the building has not collapsed and has experienced a certain peak residual

interstory drift, RIDR = ridr, and 2) the probability of experiencing various levels of RIDR

with increasing levels of seismic intensity IM=im. The first probability accounts for the

variability associated with the amount of permanent deformation that will trigger building

demolition. The second probability accounts for the record-to-record variability of residual

drifts which can be determined from structural analysis.

The effect of including this source of economic loss into PEER’s loss estimation

framework was examined, and it was observed that a substantial increase in loss was

experienced in the ductile reinforced concrete moment resisting frame office buildings

analyzed in this study. The 4-story building analyzed in this part of the investigation

showed a relative increase of 35% in expected loss at the design-basis earthquake (DBE)

and an increase of 45% was observed in the 12-story building when compare to equivalent

buildings that do not consider these losses.

The effect of incorporating these types of monetary losses has a greater influence in

taller buildings because permanent lateral deformations are more likely to be higher in these

structures. The higher gravity loads in the lower stories of tall buildings tend to localize

most of the lateral deformation in these lower stories because of P-delta effects, whereas the

deformation is more evenly spread throughout the height of the building for short structures.


This results in a higher probability that larger residual interstory drifts will occur in the

lower stories of tall buildings. It was also determined that the influence of demolition given

non-collapse is much less significant in older structures built with non-ductile reinforced

concrete. The relative increase in loss at the DBE of the 4-story non-ductile structure was

only 12%. This is because at large lateral deformations, non-ductile frames are more likely

to collapse rather than remain standing with significant permanent drifts.

The increase in losses is also found to be strongly dependent on the statistical

parameters (median and dispersion) estimated for the probability of demolition given RIDR.

A sensitivity analysis was conducted to examine how the variation in estimates of these

parameters affects earthquake-induced losses. It was determined that the estimation of the

median is more sensitive than the dispersion. Due to a lack of data, a median RIDR = 1.5%

and a lognormal standard deviation of 0.30 was recommended. These parameters are based

on values of permanent story drift that would likely cause practical problems if the damaged

building continued to be used was not demolished. Using these parameters implies that

approximately 9% of the buildings with a RIDR of 1% would be demolished while

approximately 85% of buildings with a RIDR of 2% would be demolished. The lower value

corresponds to values that make door, windows and elevators stop functioning while the

higher level corresponds to levels that are clearly visible produce significant P-delta effects.

Accounting for losses due to demolition given non-collapse can be used to evaluate

the difference in performance between new innovative structural systems and conventional

systems. Particularly, this new part of the methodology can be used to highlight improved

performance of structures that implement self-centering systems, which prevent permanent

deformations, over systems that use structural damage to dissipate energy.

10.3 FUTURE RESEARCH NEEDS

Although much progress has been made in performance-based earthquake

engineering and building-specific loss estimation, there are many areas that require future

research. Many simplifying assumptions were made during this investigation that

necessitate further development. Many of these future research needs involve data

collection to establish more fragility functions and repair cost relationships, on which EDP-

DV functions are based. There are also aspects of the current loss estimation methodology


that can be further developed and many other aspects of performance loss results that can be

extended.

10.3.1 Data collection for fragility functions and repair costs

The introduction of PEER’s performance-based earthquake engineering framework

has established the types of data that need to be collected from experiments, construction

cost databases or post-earthquake reconnaissance. Data of particular interest as it relates to

this study include the following:

Experimental data for fragility function development is ongoing such that

damage can be estimated for common types of components that may be found

in a building’s inventory. These fragilities can then be used to create EDP-DV

functions for story-based building-specific loss estimation. Having a complete

set of fragility functions will improve loss estimations by reducing the reliance

on generic fragilities and functions based on expert opinion.

Correlations between fragilities were assumed to be uncorrelated; however, this

needs to be validated by further investigation. Incorporating correlations

quantitatively is complicated because fragility function damage states are

discrete rather than continuous. A method of capturing these fragility function

correlations despite this complication needs to be developed. Once a

methodology has been established, a way of obtaining the value of these

correlations needs to be developed.

Non-collapse losses depend heavily on the values of component repair costs

and how they are formulated. One of the principles that the story-based loss

estimation method relies on is the relative costs of components’ repair costs to

the replacement value of said component. Although there have been some

attempts by previous studies to establish these values, many components still

lack this information and assumptions on their values needed to be made to

complete this study.

Computing the uncertainty of the estimated expected economic losses as

presented in this dissertation necessitates establishing how the replacement

value of each building component is distributed between the different


subcontractors that participate in its construction. Developing relationships

between components costs and subcontractor costs will bridge the gap between

damage estimation (which is typically conducted per component) and the

available construction cost data (which is typically expressed at the building

level).

The dispersion and correlation data presented in Chapter 8 was based on data

that only considered commercial office buildings. Similar studies that gather

this type data for other structural classification types are necessary to compute

the variability of economic loss for other building occupancies.

Statistical data on the values residual interstory drift that trigger building

demolition. This data can then be used to establish the probability

distribution parameters used in Chapter 9 to account for economic losses

due to building demolition.

EDP-DV functions for other building types need to be established to

evaluate the economic losses for structures of all types of classifications.

Functions that consider different structural systems and different

occupancies need to be developed such that computer tools, like the one

presented in Chapter 6, can offer a broad library of functions for users to

choose from.

10.3.2 Improvements to building-specific loss estimation methodology

There are several areas in building-specific loss estimation that require further

development. These areas include the following:

The economic losses presented in this investigation and other recent building-

specific losses estimation studies are typically larger than those computed using

regional loss estimation methods. Validation studies using data collected from

previous and future earthquakes can be used to identify parts of the

methodology, if any, that may not capture the computation of earthquake-

induced economic losses well.

The concepts and framework presented in this dissertation for the computation

of economic losses can be extended to the calculation of fatalities and facility


downtime caused by seismic events. The same basic framework and

probabilistic concepts used the compute of economic losses can be applied to

these other metrics and used to gage performance. Although there have been

studies that have addresses these types of losses (Liel and Deierlein 2008,

Mitrani-Reiser 2007), there has not been as much work conducted in these areas

as there has been in economic losses.

The methodology to compute the variability of economic losses in this

dissertation was limited to aleatoric uncertainty. Modeling uncertainty

associated with the variability of EDPs was not considered in this study. The

incorporation of epistemic uncertainty and its propagation into the building-

specific loss estimation as it relates to the variability associated with structural

response has been previously investigated, however, has not been combined

with the aleatoric findings in this dissertation. This type of uncertainty needs to

be accounted for to obtain a better value of dispersion and to see which type of

uncertainty has the larger contribution to the overall variability.

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APPENDIX A A-1 Cost Distributions for EDP-DV Functions

APPENDIX A

COST DISTRIBUTIONS FOR EDP-DV FUNCTIONS

This appendix contains the assumed building and story cost distributions used to

compute the generic story EDP-DV functions. The cost distributions grouped by three

different categories of building height (low-rise, mid-rise and high rise structures). For

each category of building height, there are three types of story distributions based on floor

type (1st floor, typical floor and top floor). The building cost distributions are normalized

by the replacement value of the building and the story cost distributions are normalized by

the replacement value of the story.


Building Height: Low-rise (1 to 5 stories)


A. SUBSTRUCTURE4.7% 0.0% 0.0% 0.0%


C. INTERIORS22.7% 26.4% 27.3% 14.0%






Building Height: Low-riseFloor Type: 1st Floor

Performance Group Seismic Sensitivity Fragility Group











Rugged 1.1%D30 HVAC





= 100% 100%

12.3%

Normalized costs

17.7%

0.6%

13.6%

2.0%

26.4%

8.0%

19.3%

0.0%


Building Height: Low-riseFloor Type: Typical Floor












Rugged 1.2%D30 HVAC





= 100% 100%

Normalized costs

18.3%

0.6%

14.1%

2.1%

12.7%

27.3%

8.3%

16.6%

0.0%


Building Height: Low-riseFloor Type: Top Floor












Rugged 1.2%D30 HVAC


Other HVAC Sys. & Equipment PFA Generic-Accl 0.0%D40 Fire Protection



= 100% 100%

Normalized costs

14.0%

2.2%

10.2%

17.1%

7.3%

18.8%

0.6%

19.2%

10.5%


Building Height: Mid-rise (6 to 10 stories)


A. SUBSTRUCTURE2.3% 0.0% 0.0% 0.0%


C. INTERIORS19.4% 20.7% 21.4% 11.1%






Building Height: Mid-riseFloor Type: 1st Floor












Rugged 1.1%D30 HVAC





= 100% 100%

17.9%

Normalized costs

12.3%

16.6%

2.6%

1.9%

20.7%

9.1%

18.8%

0.0%


Building Height: Mid-riseFloor Type: Typical Floor












Rugged 1.1%D30 HVAC





= 100% 100%

18.5%

17.2%

Normalized costs

9.4%

16.2%

0.0%

12.7%

2.7%

1.9%

21.4%


Building Height: Mid-riseFloor Type: Top Floor












Rugged 1.1%D30 HVAC





= 100% 100%

15.4%

17.9%

2.8%

Normalized costs

17.6%

2.0%

11.1%

11.8%

16.9%

4.5%


Building Height: High-rise (>10 stories)


A. SUBSTRUCTURE1.5% 0.0% 0.0% 0.0%


C. INTERIORS17.1% 17.7% 18.2% 9.5%






Building Height: High-riseFloor Type: 1st Floor












Rugged 0.7%D30 HVAC





= 100% 100%

Normalized costs

16.6%

4.1%

1.3%

15.9%

17.7%

4.7%

16.6%

0.0%

23.1%


Building Height: High-riseFloor Type: Typical Floor




B20 Exterior EnclosureExterior Walls IDR Partitions 8.0%Exterior Windows IDR Windows 5.4%Exterior Doors 0.9%








Rugged 0.7%D30 HVAC





= 100% 100%

Normalized costs

23.8%

1.3%

17.0%

4.3%

16.3%

18.2%

4.9%

14.2%

0.0%


Building Height: High-riseFloor Type: Top Floor












Rugged 0.8%D30 HVAC





= 100% 100%

17.8%

4.4%

22.7%

Normalized costs

9.5%

19.8%

1.4%

6.1%

14.9%

3.5%

APPENDIX B B-1 Generic Story EDP-DV Functions

APPENDIX B

GENERIC STORY EDP-DV FUNCTIONS

This appendix contains the graphs and data points for the generic story EDP-DV

functions developed in Chapter 3. There are 36 functions for reinforced concrete moment-

resisting frame office buildings. On each sheet, a set of three functions for drift-sensitive

structural components, drift-sensitive nonstructural components and acceleration-sensitive

nonstructural components are reported. The sets of functions are categorized by the type of

frame four different building properties: the building height (low-rise, mid-rise or high-

rise), the type of material behavior (ductile or non-ductile reinforced concrete), the type of

structural frame (perimeter or space) and the location/type of floor (1st floor, typical floor or

top floor). The values of economic loss shown here are normalized by the replacement

value of the story. Interstory drift ratio (IDR) is unit-less and peak floor acceleration is

expressed in terms of the acceleration of gravity (g).


EDP-DV Function Data

Building Height: Low-rise (1 to 5 stories) IDR E(LS | IDR) IDR E(LNS | IDR) PFA E(LNS | PFA) IDR E(LS | IDR) IDR E(LNS | IDR) PFA E(LNS | PFA)

0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.165 0.101 0.572 5.05 0.356

Strucutral Material: Ductile reinforced-concrete 0.002 0.000 0.002 0.009 0.10 0.000 0.102 0.166 0.102 0.572 5.10 0.3570.003 0.000 0.003 0.016 0.15 0.000 0.103 0.167 0.103 0.572 5.15 0.358

Structural System: Perimeter moment-frame 0.004 0.001 0.004 0.027 0.20 0.002 0.104 0.167 0.104 0.572 5.20 0.3580.005 0.002 0.005 0.044 0.25 0.004 0.105 0.168 0.105 0.573 5.25 0.359

Occupancy: Office 0.006 0.003 0.006 0.067 0.30 0.006 0.106 0.169 0.106 0.573 5.30 0.3600.007 0.006 0.007 0.094 0.35 0.009 0.107 0.170 0.107 0.573 5.35 0.361

Floor Type: 1st Floor 0.008 0.008 0.008 0.124 0.40 0.013 0.108 0.170 0.108 0.573 5.40 0.3610.009 0.011 0.009 0.154 0.45 0.016 0.109 0.171 0.109 0.573 5.45 0.3620.010 0.014 0.010 0.185 0.50 0.020 0.110 0.172 0.110 0.573 5.50 0.3630.011 0.017 0.011 0.213 0.55 0.024 0.111 0.172 0.111 0.573 5.55 0.3630.012 0.019 0.012 0.240 0.60 0.029 0.112 0.173 0.112 0.574 5.60 0.3640.013 0.021 0.013 0.265 0.65 0.034 0.113 0.173 0.113 0.574 5.65 0.3640.014 0.023 0.014 0.288 0.70 0.040 0.114 0.174 0.114 0.574 5.70 0.3650.015 0.024 0.015 0.308 0.75 0.045 0.115 0.174 0.115 0.574 5.75 0.3650.016 0.025 0.016 0.326 0.80 0.051 0.116 0.175 0.116 0.574 5.80 0.3660.017 0.026 0.017 0.342 0.85 0.057 0.117 0.175 0.117 0.574 5.85 0.3670.018 0.027 0.018 0.356 0.90 0.064 0.118 0.176 0.118 0.574 5.90 0.3670.019 0.028 0.019 0.369 0.95 0.070 0.119 0.176 0.119 0.574 5.95 0.3680.020 0.028 0.020 0.381 1.00 0.077 0.120 0.176 0.120 0.575 6.00 0.3680.021 0.029 0.021 0.391 1.05 0.084 0.121 0.177 0.121 0.575 6.05 0.3680.022 0.029 0.022 0.400 1.10 0.091 0.122 0.177 0.122 0.575 6.10 0.3690.023 0.030 0.023 0.409 1.15 0.098 0.123 0.178 0.123 0.575 6.15 0.3690.024 0.030 0.024 0.417 1.20 0.104 0.124 0.178 0.124 0.575 6.20 0.3700.025 0.030 0.025 0.425 1.25 0.111 0.125 0.178 0.125 0.575 6.25 0.3700.026 0.031 0.026 0.433 1.30 0.118 0.126 0.178 0.126 0.575 6.30 0.3710.027 0.031 0.027 0.440 1.35 0.125 0.127 0.179 0.127 0.575 6.35 0.3710.028 0.032 0.028 0.447 1.40 0.132 0.128 0.179 0.128 0.575 6.40 0.3710.029 0.032 0.029 0.453 1.45 0.138 0.129 0.179 0.129 0.575 6.45 0.3720.030 0.033 0.030 0.460 1.50 0.145 0.130 0.179 0.130 0.575 6.50 0.3720.031 0.033 0.031 0.466 1.55 0.151 0.131 0.180 0.131 0.575 6.55 0.3720.032 0.034 0.032 0.472 1.60 0.158 0.132 0.180 0.132 0.576 6.60 0.3730.033 0.034 0.033 0.478 1.65 0.164 0.133 0.180 0.133 0.576 6.65 0.3730.034 0.035 0.034 0.484 1.70 0.170 0.134 0.180 0.134 0.576 6.70 0.3730.035 0.035 0.035 0.489 1.75 0.176 0.135 0.180 0.135 0.576 6.75 0.3740.036 0.036 0.036 0.494 1.80 0.182 0.136 0.181 0.136 0.576 6.80 0.3740.037 0.037 0.037 0.499 1.85 0.188 0.137 0.181 0.137 0.576 6.85 0.3740.038 0.038 0.038 0.504 1.90 0.193 0.138 0.181 0.138 0.576 6.90 0.3750.039 0.039 0.039 0.508 1.95 0.199 0.139 0.181 0.139 0.576 6.95 0.3750.040 0.040 0.040 0.512 2.00 0.204 0.140 0.181 0.140 0.576 7.00 0.3750.041 0.041 0.041 0.516 2.05 0.209 0.141 0.181 0.141 0.576 7.05 0.3750.042 0.042 0.042 0.520 2.10 0.215 0.142 0.181 0.142 0.576 7.10 0.3760.043 0.044 0.043 0.523 2.15 0.220 0.143 0.181 0.143 0.576 7.15 0.3760.044 0.045 0.044 0.526 2.20 0.224 0.144 0.181 0.144 0.576 7.20 0.3760.045 0.047 0.045 0.529 2.25 0.229 0.145 0.182 0.145 0.576 7.25 0.3760.046 0.049 0.046 0.532 2.30 0.234 0.146 0.182 0.146 0.576 7.30 0.3770.047 0.051 0.047 0.534 2.35 0.238 0.147 0.182 0.147 0.576 7.35 0.3770.048 0.053 0.048 0.537 2.40 0.243 0.148 0.182 0.148 0.576 7.40 0.3770.049 0.056 0.049 0.539 2.45 0.247 0.149 0.182 0.149 0.577 7.45 0.3770.050 0.058 0.050 0.541 2.50 0.251 0.150 0.182 0.150 0.577 7.50 0.3770.051 0.061 0.051 0.543 2.55 0.255 0.151 0.182 0.151 0.577 7.55 0.3780.052 0.063 0.052 0.544 2.60 0.259 0.152 0.182 0.152 0.577 7.60 0.3780.053 0.066 0.053 0.546 2.65 0.263 0.153 0.182 0.153 0.577 7.65 0.3780.054 0.069 0.054 0.547 2.70 0.266 0.154 0.182 0.154 0.577 7.70 0.3780.055 0.072 0.055 0.549 2.75 0.270 0.155 0.182 0.155 0.577 7.75 0.3780.056 0.075 0.056 0.550 2.80 0.273 0.156 0.182 0.156 0.577 7.80 0.3790.057 0.078 0.057 0.551 2.85 0.277 0.157 0.182 0.157 0.577 7.85 0.3790.058 0.081 0.058 0.552 2.90 0.280 0.158 0.182 0.158 0.577 7.90 0.3790.059 0.084 0.059 0.553 2.95 0.283 0.159 0.182 0.159 0.577 7.95 0.3790.060 0.087 0.060 0.554 3.00 0.286 0.160 0.182 0.160 0.577 8.00 0.3790.061 0.090 0.061 0.555 3.05 0.289 0.161 0.182 0.161 0.577 8.05 0.3790.062 0.093 0.062 0.556 3.10 0.292 0.162 0.182 0.162 0.577 8.10 0.3790.063 0.096 0.063 0.557 3.15 0.295 0.163 0.182 0.163 0.577 8.15 0.3800.064 0.099 0.064 0.558 3.20 0.298 0.164 0.182 0.164 0.577 8.20 0.3800.065 0.102 0.065 0.559 3.25 0.300 0.165 0.183 0.165 0.577 8.25 0.3800.066 0.105 0.066 0.559 3.30 0.303 0.166 0.183 0.166 0.577 8.30 0.3800.067 0.107 0.067 0.560 3.35 0.305 0.167 0.183 0.167 0.577 8.35 0.3800.068 0.110 0.068 0.561 3.40 0.308 0.168 0.183 0.168 0.577 8.40 0.3800.069 0.113 0.069 0.561 3.45 0.310 0.169 0.183 0.169 0.577 8.45 0.3800.070 0.115 0.070 0.562 3.50 0.312 0.170 0.183 0.170 0.577 8.50 0.3810.071 0.118 0.071 0.562 3.55 0.314 0.171 0.183 0.171 0.577 8.55 0.3810.072 0.120 0.072 0.563 3.60 0.316 0.172 0.183 0.172 0.577 8.60 0.3810.073 0.123 0.073 0.563 3.65 0.319 0.173 0.183 0.173 0.577 8.65 0.3810.074 0.125 0.074 0.564 3.70 0.321 0.174 0.183 0.174 0.577 8.70 0.3810.075 0.127 0.075 0.564 3.75 0.322 0.175 0.183 0.175 0.577 8.75 0.3810.076 0.129 0.076 0.565 3.80 0.324 0.176 0.183 0.176 0.577 8.80 0.3810.077 0.131 0.077 0.565 3.85 0.326 0.177 0.183 0.177 0.577 8.85 0.3810.078 0.133 0.078 0.566 3.90 0.328 0.178 0.183 0.178 0.577 8.90 0.3810.079 0.135 0.079 0.566 3.95 0.330 0.179 0.183 0.179 0.577 8.95 0.3810.080 0.137 0.080 0.566 4.00 0.331 0.180 0.183 0.180 0.577 9.00 0.3820.081 0.139 0.081 0.567 4.05 0.333 0.181 0.183 0.181 0.577 9.05 0.3820.082 0.141 0.082 0.567 4.10 0.334 0.182 0.183 0.182 0.578 9.10 0.3820.083 0.142 0.083 0.567 4.15 0.336 0.183 0.183 0.183 0.578 9.15 0.3820.084 0.144 0.084 0.568 4.20 0.337 0.184 0.183 0.184 0.578 9.20 0.3820.085 0.146 0.085 0.568 4.25 0.339 0.185 0.183 0.185 0.578 9.25 0.3820.086 0.147 0.086 0.568 4.30 0.340 0.186 0.183 0.186 0.578 9.30 0.3820.087 0.149 0.087 0.569 4.35 0.341 0.187 0.183 0.187 0.578 9.35 0.3820.088 0.150 0.088 0.569 4.40 0.343 0.188 0.183 0.188 0.578 9.40 0.3820.089 0.151 0.089 0.569 4.45 0.344 0.189 0.183 0.189 0.578 9.45 0.3820.090 0.153 0.090 0.569 4.50 0.345 0.190 0.183 0.190 0.578 9.50 0.3820.091 0.154 0.091 0.570 4.55 0.346 0.191 0.183 0.191 0.578 9.55 0.3820.092 0.155 0.092 0.570 4.60 0.347 0.192 0.183 0.192 0.578 9.60 0.3820.093 0.157 0.093 0.570 4.65 0.348 0.193 0.183 0.193 0.578 9.65 0.3830.094 0.158 0.094 0.570 4.70 0.349 0.194 0.183 0.194 0.578 9.70 0.3830.095 0.159 0.095 0.571 4.75 0.350 0.195 0.183 0.195 0.578 9.75 0.3830.096 0.160 0.096 0.571 4.80 0.351 0.196 0.183 0.196 0.578 9.80 0.3830.097 0.161 0.097 0.571 4.85 0.352 0.197 0.183 0.197 0.578 9.85 0.3830.098 0.162 0.098 0.571 4.90 0.353 0.198 0.183 0.198 0.578 9.90 0.3830.099 0.163 0.099 0.571 4.95 0.354 0.199 0.183 0.199 0.578 9.95 0.3830.100 0.164 0.100 0.572 5.00 0.355 0.200 0.183 0.200 0.578 10.00 0.383

Loss Functions Loss Functions (cont'd)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.170 0.101 0.540 5.05 0.374




Floor Type: Typical Floor 0.008 0.008 0.008 0.119 0.40 0.014 0.108 0.176 0.108 0.541 5.40 0.3800.009 0.011 0.009 0.148 0.45 0.018 0.109 0.177 0.109 0.541 5.45 0.3800.010 0.015 0.010 0.177 0.50 0.022 0.110 0.177 0.110 0.541 5.50 0.3810.011 0.017 0.011 0.205 0.55 0.027 0.111 0.178 0.111 0.541 5.55 0.3820.012 0.020 0.012 0.231 0.60 0.032 0.112 0.179 0.112 0.541 5.60 0.3820.013 0.022 0.013 0.255 0.65 0.037 0.113 0.179 0.113 0.541 5.65 0.3830.014 0.024 0.014 0.276 0.70 0.043 0.114 0.180 0.114 0.542 5.70 0.3840.015 0.025 0.015 0.296 0.75 0.049 0.115 0.180 0.115 0.542 5.75 0.3840.016 0.026 0.016 0.313 0.80 0.055 0.116 0.181 0.116 0.542 5.80 0.3850.017 0.027 0.017 0.329 0.85 0.062 0.117 0.181 0.117 0.542 5.85 0.3850.018 0.028 0.018 0.342 0.90 0.069 0.118 0.182 0.118 0.542 5.90 0.3860.019 0.029 0.019 0.354 0.95 0.076 0.119 0.182 0.119 0.542 5.95 0.3860.020 0.029 0.020 0.365 1.00 0.083 0.120 0.182 0.120 0.542 6.00 0.3870.021 0.030 0.021 0.375 1.05 0.090 0.121 0.183 0.121 0.542 6.05 0.3870.022 0.030 0.022 0.384 1.10 0.097 0.122 0.183 0.122 0.542 6.10 0.3880.023 0.031 0.023 0.392 1.15 0.105 0.123 0.183 0.123 0.542 6.15 0.3880.024 0.031 0.024 0.400 1.20 0.112 0.124 0.184 0.124 0.543 6.20 0.3890.025 0.031 0.025 0.407 1.25 0.119 0.125 0.184 0.125 0.543 6.25 0.3890.026 0.032 0.026 0.414 1.30 0.126 0.126 0.184 0.126 0.543 6.30 0.3890.027 0.032 0.027 0.420 1.35 0.133 0.127 0.185 0.127 0.543 6.35 0.3900.028 0.033 0.028 0.427 1.40 0.140 0.128 0.185 0.128 0.543 6.40 0.3900.029 0.033 0.029 0.433 1.45 0.147 0.129 0.185 0.129 0.543 6.45 0.3910.030 0.034 0.030 0.439 1.50 0.154 0.130 0.185 0.130 0.543 6.50 0.3910.031 0.034 0.031 0.444 1.55 0.161 0.131 0.186 0.131 0.543 6.55 0.3910.032 0.035 0.032 0.450 1.60 0.168 0.132 0.186 0.132 0.543 6.60 0.3920.033 0.035 0.033 0.455 1.65 0.174 0.133 0.186 0.133 0.543 6.65 0.3920.034 0.036 0.034 0.460 1.70 0.181 0.134 0.186 0.134 0.543 6.70 0.3920.035 0.037 0.035 0.465 1.75 0.187 0.135 0.186 0.135 0.543 6.75 0.3930.036 0.037 0.036 0.470 1.80 0.193 0.136 0.187 0.136 0.543 6.80 0.3930.037 0.038 0.037 0.474 1.85 0.199 0.137 0.187 0.137 0.543 6.85 0.3930.038 0.039 0.038 0.478 1.90 0.205 0.138 0.187 0.138 0.543 6.90 0.3940.039 0.040 0.039 0.482 1.95 0.211 0.139 0.187 0.139 0.544 6.95 0.3940.040 0.041 0.040 0.486 2.00 0.217 0.140 0.187 0.140 0.544 7.00 0.3940.041 0.042 0.041 0.489 2.05 0.222 0.141 0.187 0.141 0.544 7.05 0.3940.042 0.044 0.042 0.492 2.10 0.227 0.142 0.187 0.142 0.544 7.10 0.3950.043 0.045 0.043 0.495 2.15 0.233 0.143 0.187 0.143 0.544 7.15 0.3950.044 0.047 0.044 0.498 2.20 0.238 0.144 0.187 0.144 0.544 7.20 0.3950.045 0.049 0.045 0.501 2.25 0.243 0.145 0.188 0.145 0.544 7.25 0.3950.046 0.051 0.046 0.503 2.30 0.247 0.146 0.188 0.146 0.544 7.30 0.3960.047 0.053 0.047 0.505 2.35 0.252 0.147 0.188 0.147 0.544 7.35 0.3960.048 0.055 0.048 0.507 2.40 0.257 0.148 0.188 0.148 0.544 7.40 0.3960.049 0.057 0.049 0.509 2.45 0.261 0.149 0.188 0.149 0.544 7.45 0.3960.050 0.060 0.050 0.511 2.50 0.265 0.150 0.188 0.150 0.544 7.50 0.3970.051 0.063 0.051 0.513 2.55 0.270 0.151 0.188 0.151 0.544 7.55 0.3970.052 0.065 0.052 0.514 2.60 0.274 0.152 0.188 0.152 0.544 7.60 0.3970.053 0.068 0.053 0.516 2.65 0.278 0.153 0.188 0.153 0.544 7.65 0.3970.054 0.071 0.054 0.517 2.70 0.281 0.154 0.188 0.154 0.544 7.70 0.3970.055 0.074 0.055 0.518 2.75 0.285 0.155 0.188 0.155 0.544 7.75 0.3980.056 0.077 0.056 0.519 2.80 0.289 0.156 0.188 0.156 0.544 7.80 0.3980.057 0.080 0.057 0.521 2.85 0.292 0.157 0.188 0.157 0.544 7.85 0.3980.058 0.084 0.058 0.522 2.90 0.296 0.158 0.188 0.158 0.544 7.90 0.3980.059 0.087 0.059 0.523 2.95 0.299 0.159 0.188 0.159 0.544 7.95 0.3980.060 0.090 0.060 0.523 3.00 0.302 0.160 0.188 0.160 0.544 8.00 0.3980.061 0.093 0.061 0.524 3.05 0.305 0.161 0.188 0.161 0.544 8.05 0.3980.062 0.096 0.062 0.525 3.10 0.308 0.162 0.189 0.162 0.544 8.10 0.3990.063 0.099 0.063 0.526 3.15 0.311 0.163 0.189 0.163 0.545 8.15 0.3990.064 0.102 0.064 0.527 3.20 0.314 0.164 0.189 0.164 0.545 8.20 0.3990.065 0.105 0.065 0.527 3.25 0.317 0.165 0.189 0.165 0.545 8.25 0.3990.066 0.108 0.066 0.528 3.30 0.319 0.166 0.189 0.166 0.545 8.30 0.3990.067 0.111 0.067 0.529 3.35 0.322 0.167 0.189 0.167 0.545 8.35 0.3990.068 0.114 0.068 0.529 3.40 0.324 0.168 0.189 0.168 0.545 8.40 0.3990.069 0.117 0.069 0.530 3.45 0.327 0.169 0.189 0.169 0.545 8.45 0.4000.070 0.119 0.070 0.530 3.50 0.329 0.170 0.189 0.170 0.545 8.50 0.4000.071 0.122 0.071 0.531 3.55 0.331 0.171 0.189 0.171 0.545 8.55 0.4000.072 0.124 0.072 0.531 3.60 0.333 0.172 0.189 0.172 0.545 8.60 0.4000.073 0.127 0.073 0.532 3.65 0.336 0.173 0.189 0.173 0.545 8.65 0.4000.074 0.129 0.074 0.532 3.70 0.338 0.174 0.189 0.174 0.545 8.70 0.4000.075 0.131 0.075 0.533 3.75 0.340 0.175 0.189 0.175 0.545 8.75 0.4000.076 0.134 0.076 0.533 3.80 0.341 0.176 0.189 0.176 0.545 8.80 0.4000.077 0.136 0.077 0.533 3.85 0.343 0.177 0.189 0.177 0.545 8.85 0.4000.078 0.138 0.078 0.534 3.90 0.345 0.178 0.189 0.178 0.545 8.90 0.4010.079 0.140 0.079 0.534 3.95 0.347 0.179 0.189 0.179 0.545 8.95 0.4010.080 0.142 0.080 0.534 4.00 0.349 0.180 0.189 0.180 0.545 9.00 0.4010.081 0.144 0.081 0.535 4.05 0.350 0.181 0.189 0.181 0.545 9.05 0.4010.082 0.145 0.082 0.535 4.10 0.352 0.182 0.189 0.182 0.545 9.10 0.4010.083 0.147 0.083 0.535 4.15 0.353 0.183 0.189 0.183 0.545 9.15 0.4010.084 0.149 0.084 0.536 4.20 0.355 0.184 0.189 0.184 0.545 9.20 0.4010.085 0.150 0.085 0.536 4.25 0.356 0.185 0.189 0.185 0.545 9.25 0.4010.086 0.152 0.086 0.536 4.30 0.358 0.186 0.189 0.186 0.545 9.30 0.4010.087 0.154 0.087 0.537 4.35 0.359 0.187 0.189 0.187 0.545 9.35 0.4010.088 0.155 0.088 0.537 4.40 0.360 0.188 0.189 0.188 0.545 9.40 0.4010.089 0.156 0.089 0.537 4.45 0.362 0.189 0.189 0.189 0.545 9.45 0.4020.090 0.158 0.090 0.537 4.50 0.363 0.190 0.189 0.190 0.545 9.50 0.4020.091 0.159 0.091 0.538 4.55 0.364 0.191 0.189 0.191 0.545 9.55 0.4020.092 0.161 0.092 0.538 4.60 0.365 0.192 0.189 0.192 0.545 9.60 0.4020.093 0.162 0.093 0.538 4.65 0.366 0.193 0.189 0.193 0.545 9.65 0.4020.094 0.163 0.094 0.538 4.70 0.368 0.194 0.189 0.194 0.545 9.70 0.4020.095 0.164 0.095 0.539 4.75 0.369 0.195 0.189 0.195 0.545 9.75 0.4020.096 0.165 0.096 0.539 4.80 0.370 0.196 0.189 0.196 0.545 9.80 0.4020.097 0.166 0.097 0.539 4.85 0.371 0.197 0.189 0.197 0.545 9.85 0.4020.098 0.167 0.098 0.539 4.90 0.372 0.198 0.189 0.198 0.545 9.90 0.4020.099 0.168 0.099 0.539 4.95 0.373 0.199 0.189 0.199 0.545 9.95 0.4020.100 0.169 0.100 0.539 5.00 0.373 0.200 0.189 0.200 0.545 10.00 0.402



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0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.140 0.101 0.478 5.05 0.397




Floor Type: Top Floor 0.008 0.007 0.008 0.097 0.40 0.010 0.108 0.145 0.108 0.480 5.40 0.4040.009 0.009 0.009 0.122 0.45 0.013 0.109 0.145 0.109 0.480 5.45 0.4040.010 0.012 0.010 0.146 0.50 0.016 0.110 0.146 0.110 0.480 5.50 0.4050.011 0.014 0.011 0.169 0.55 0.020 0.111 0.146 0.111 0.480 5.55 0.4060.012 0.016 0.012 0.191 0.60 0.024 0.112 0.147 0.112 0.480 5.60 0.4070.013 0.018 0.013 0.211 0.65 0.029 0.113 0.147 0.113 0.480 5.65 0.4070.014 0.019 0.014 0.230 0.70 0.034 0.114 0.148 0.114 0.480 5.70 0.4080.015 0.021 0.015 0.246 0.75 0.040 0.115 0.148 0.115 0.480 5.75 0.4090.016 0.021 0.016 0.261 0.80 0.046 0.116 0.149 0.116 0.481 5.80 0.4100.017 0.022 0.017 0.275 0.85 0.052 0.117 0.149 0.117 0.481 5.85 0.4100.018 0.023 0.018 0.287 0.90 0.059 0.118 0.149 0.118 0.481 5.90 0.4110.019 0.023 0.019 0.297 0.95 0.065 0.119 0.150 0.119 0.481 5.95 0.4110.020 0.024 0.020 0.307 1.00 0.072 0.120 0.150 0.120 0.481 6.00 0.4120.021 0.024 0.021 0.315 1.05 0.079 0.121 0.150 0.121 0.481 6.05 0.4130.022 0.025 0.022 0.324 1.10 0.086 0.122 0.151 0.122 0.481 6.10 0.4130.023 0.025 0.023 0.331 1.15 0.094 0.123 0.151 0.123 0.481 6.15 0.4140.024 0.026 0.024 0.338 1.20 0.101 0.124 0.151 0.124 0.481 6.20 0.4140.025 0.026 0.025 0.345 1.25 0.109 0.125 0.151 0.125 0.481 6.25 0.4150.026 0.026 0.026 0.351 1.30 0.116 0.126 0.152 0.126 0.482 6.30 0.4150.027 0.027 0.027 0.358 1.35 0.123 0.127 0.152 0.127 0.482 6.35 0.4160.028 0.027 0.028 0.364 1.40 0.131 0.128 0.152 0.128 0.482 6.40 0.4160.029 0.027 0.029 0.370 1.45 0.138 0.129 0.152 0.129 0.482 6.45 0.4170.030 0.028 0.030 0.376 1.50 0.146 0.130 0.152 0.130 0.482 6.50 0.4170.031 0.028 0.031 0.381 1.55 0.153 0.131 0.153 0.131 0.482 6.55 0.4180.032 0.029 0.032 0.387 1.60 0.160 0.132 0.153 0.132 0.482 6.60 0.4180.033 0.029 0.033 0.392 1.65 0.167 0.133 0.153 0.133 0.482 6.65 0.4180.034 0.030 0.034 0.397 1.70 0.174 0.134 0.153 0.134 0.482 6.70 0.4190.035 0.030 0.035 0.402 1.75 0.181 0.135 0.153 0.135 0.482 6.75 0.4190.036 0.031 0.036 0.406 1.80 0.188 0.136 0.153 0.136 0.482 6.80 0.4200.037 0.031 0.037 0.411 1.85 0.194 0.137 0.154 0.137 0.482 6.85 0.4200.038 0.032 0.038 0.415 1.90 0.201 0.138 0.154 0.138 0.482 6.90 0.4200.039 0.033 0.039 0.419 1.95 0.207 0.139 0.154 0.139 0.482 6.95 0.4210.040 0.034 0.040 0.423 2.00 0.214 0.140 0.154 0.140 0.483 7.00 0.4210.041 0.035 0.041 0.426 2.05 0.220 0.141 0.154 0.141 0.483 7.05 0.4210.042 0.036 0.042 0.429 2.10 0.226 0.142 0.154 0.142 0.483 7.10 0.4220.043 0.037 0.043 0.432 2.15 0.232 0.143 0.154 0.143 0.483 7.15 0.4220.044 0.039 0.044 0.435 2.20 0.237 0.144 0.154 0.144 0.483 7.20 0.4220.045 0.040 0.045 0.438 2.25 0.243 0.145 0.154 0.145 0.483 7.25 0.4220.046 0.042 0.046 0.440 2.30 0.248 0.146 0.154 0.146 0.483 7.30 0.4230.047 0.043 0.047 0.443 2.35 0.254 0.147 0.154 0.147 0.483 7.35 0.4230.048 0.045 0.048 0.445 2.40 0.259 0.148 0.154 0.148 0.483 7.40 0.4230.049 0.047 0.049 0.447 2.45 0.264 0.149 0.155 0.149 0.483 7.45 0.4240.050 0.049 0.050 0.449 2.50 0.269 0.150 0.155 0.150 0.483 7.50 0.4240.051 0.051 0.051 0.450 2.55 0.274 0.151 0.155 0.151 0.483 7.55 0.4240.052 0.054 0.052 0.452 2.60 0.278 0.152 0.155 0.152 0.483 7.60 0.4240.053 0.056 0.053 0.454 2.65 0.283 0.153 0.155 0.153 0.483 7.65 0.4250.054 0.059 0.054 0.455 2.70 0.287 0.154 0.155 0.154 0.483 7.70 0.4250.055 0.061 0.055 0.456 2.75 0.292 0.155 0.155 0.155 0.483 7.75 0.4250.056 0.064 0.056 0.457 2.80 0.296 0.156 0.155 0.156 0.483 7.80 0.4250.057 0.066 0.057 0.459 2.85 0.300 0.157 0.155 0.157 0.483 7.85 0.4250.058 0.069 0.058 0.460 2.90 0.304 0.158 0.155 0.158 0.483 7.90 0.4260.059 0.071 0.059 0.461 2.95 0.308 0.159 0.155 0.159 0.483 7.95 0.4260.060 0.074 0.060 0.462 3.00 0.311 0.160 0.155 0.160 0.483 8.00 0.4260.061 0.077 0.061 0.462 3.05 0.315 0.161 0.155 0.161 0.483 8.05 0.4260.062 0.079 0.062 0.463 3.10 0.318 0.162 0.155 0.162 0.483 8.10 0.4260.063 0.082 0.063 0.464 3.15 0.322 0.163 0.155 0.163 0.483 8.15 0.4270.064 0.084 0.064 0.465 3.20 0.325 0.164 0.155 0.164 0.484 8.20 0.4270.065 0.087 0.065 0.466 3.25 0.328 0.165 0.155 0.165 0.484 8.25 0.4270.066 0.089 0.066 0.466 3.30 0.331 0.166 0.155 0.166 0.484 8.30 0.4270.067 0.091 0.067 0.467 3.35 0.334 0.167 0.155 0.167 0.484 8.35 0.4270.068 0.094 0.068 0.467 3.40 0.337 0.168 0.155 0.168 0.484 8.40 0.4270.069 0.096 0.069 0.468 3.45 0.340 0.169 0.155 0.169 0.484 8.45 0.4280.070 0.098 0.070 0.469 3.50 0.343 0.170 0.155 0.170 0.484 8.50 0.4280.071 0.100 0.071 0.469 3.55 0.346 0.171 0.155 0.171 0.484 8.55 0.4280.072 0.102 0.072 0.470 3.60 0.348 0.172 0.155 0.172 0.484 8.60 0.4280.073 0.104 0.073 0.470 3.65 0.351 0.173 0.155 0.173 0.484 8.65 0.4280.074 0.106 0.074 0.471 3.70 0.353 0.174 0.155 0.174 0.484 8.70 0.4280.075 0.108 0.075 0.471 3.75 0.356 0.175 0.155 0.175 0.484 8.75 0.4280.076 0.110 0.076 0.471 3.80 0.358 0.176 0.155 0.176 0.484 8.80 0.4280.077 0.112 0.077 0.472 3.85 0.360 0.177 0.155 0.177 0.484 8.85 0.4290.078 0.113 0.078 0.472 3.90 0.362 0.178 0.155 0.178 0.484 8.90 0.4290.079 0.115 0.079 0.473 3.95 0.364 0.179 0.155 0.179 0.484 8.95 0.4290.080 0.116 0.080 0.473 4.00 0.366 0.180 0.155 0.180 0.484 9.00 0.4290.081 0.118 0.081 0.473 4.05 0.368 0.181 0.155 0.181 0.484 9.05 0.4290.082 0.120 0.082 0.474 4.10 0.370 0.182 0.155 0.182 0.484 9.10 0.4290.083 0.121 0.083 0.474 4.15 0.372 0.183 0.155 0.183 0.484 9.15 0.4290.084 0.122 0.084 0.474 4.20 0.374 0.184 0.155 0.184 0.484 9.20 0.4290.085 0.124 0.085 0.475 4.25 0.376 0.185 0.155 0.185 0.484 9.25 0.4290.086 0.125 0.086 0.475 4.30 0.377 0.186 0.155 0.186 0.484 9.30 0.4300.087 0.126 0.087 0.475 4.35 0.379 0.187 0.155 0.187 0.484 9.35 0.4300.088 0.127 0.088 0.476 4.40 0.380 0.188 0.155 0.188 0.484 9.40 0.4300.089 0.129 0.089 0.476 4.45 0.382 0.189 0.155 0.189 0.484 9.45 0.4300.090 0.130 0.090 0.476 4.50 0.383 0.190 0.155 0.190 0.484 9.50 0.4300.091 0.131 0.091 0.476 4.55 0.385 0.191 0.155 0.191 0.484 9.55 0.4300.092 0.132 0.092 0.477 4.60 0.386 0.192 0.155 0.192 0.484 9.60 0.4300.093 0.133 0.093 0.477 4.65 0.388 0.193 0.155 0.193 0.484 9.65 0.4300.094 0.134 0.094 0.477 4.70 0.389 0.194 0.155 0.194 0.484 9.70 0.4300.095 0.135 0.095 0.477 4.75 0.390 0.195 0.155 0.195 0.484 9.75 0.4300.096 0.136 0.096 0.477 4.80 0.391 0.196 0.155 0.196 0.484 9.80 0.4300.097 0.137 0.097 0.478 4.85 0.393 0.197 0.155 0.197 0.484 9.85 0.4310.098 0.138 0.098 0.478 4.90 0.394 0.198 0.155 0.198 0.484 9.90 0.4310.099 0.139 0.099 0.478 4.95 0.395 0.199 0.155 0.199 0.484 9.95 0.4310.100 0.139 0.100 0.478 5.00 0.396 0.200 0.155 0.200 0.484 10.00 0.431



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0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

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0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)



Building Height: Mid-rise (5 to 10 stories) IDR E(LS | IDR) IDR E(LNS | IDR) PFA E(LNS | PFA) IDR E(LS | IDR) IDR E(LNS | IDR) PFA E(LNS | PFA)

0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.240 0.101 0.521 5.05 0.352







0.00

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0.20

0.30

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0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

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0.40

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0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.248 0.101 0.488 5.05 0.368




Floor Type: Typ Floor 0.008 0.012 0.008 0.105 0.40 0.012 0.108 0.256 0.108 0.489 5.40 0.3740.009 0.017 0.009 0.131 0.45 0.015 0.109 0.257 0.109 0.489 5.45 0.3740.010 0.021 0.010 0.157 0.50 0.019 0.110 0.258 0.110 0.490 5.50 0.3750.011 0.025 0.011 0.181 0.55 0.023 0.111 0.259 0.111 0.490 5.55 0.3760.012 0.029 0.012 0.204 0.60 0.028 0.112 0.260 0.112 0.490 5.60 0.3760.013 0.032 0.013 0.226 0.65 0.033 0.113 0.261 0.113 0.490 5.65 0.3770.014 0.034 0.014 0.245 0.70 0.038 0.114 0.262 0.114 0.490 5.70 0.3780.015 0.036 0.015 0.262 0.75 0.044 0.115 0.262 0.115 0.490 5.75 0.3780.016 0.038 0.016 0.278 0.80 0.050 0.116 0.263 0.116 0.490 5.80 0.3790.017 0.039 0.017 0.291 0.85 0.056 0.117 0.264 0.117 0.490 5.85 0.3790.018 0.041 0.018 0.304 0.90 0.063 0.118 0.264 0.118 0.490 5.90 0.3800.019 0.042 0.019 0.314 0.95 0.069 0.119 0.265 0.119 0.490 5.95 0.3800.020 0.042 0.020 0.324 1.00 0.076 0.120 0.266 0.120 0.491 6.00 0.3810.021 0.043 0.021 0.333 1.05 0.083 0.121 0.266 0.121 0.491 6.05 0.3810.022 0.044 0.022 0.341 1.10 0.090 0.122 0.267 0.122 0.491 6.10 0.3820.023 0.045 0.023 0.349 1.15 0.097 0.123 0.267 0.123 0.491 6.15 0.3820.024 0.045 0.024 0.356 1.20 0.104 0.124 0.268 0.124 0.491 6.20 0.3830.025 0.046 0.025 0.362 1.25 0.111 0.125 0.268 0.125 0.491 6.25 0.3830.026 0.046 0.026 0.369 1.30 0.118 0.126 0.268 0.126 0.491 6.30 0.3840.027 0.047 0.027 0.375 1.35 0.125 0.127 0.269 0.127 0.491 6.35 0.3840.028 0.048 0.028 0.381 1.40 0.132 0.128 0.269 0.128 0.491 6.40 0.3850.029 0.048 0.029 0.386 1.45 0.139 0.129 0.270 0.129 0.491 6.45 0.3850.030 0.049 0.030 0.392 1.50 0.146 0.130 0.270 0.130 0.491 6.50 0.3850.031 0.050 0.031 0.397 1.55 0.152 0.131 0.270 0.131 0.491 6.55 0.3860.032 0.051 0.032 0.402 1.60 0.159 0.132 0.271 0.132 0.491 6.60 0.3860.033 0.051 0.033 0.407 1.65 0.165 0.133 0.271 0.133 0.492 6.65 0.3860.034 0.052 0.034 0.412 1.70 0.172 0.134 0.271 0.134 0.492 6.70 0.3870.035 0.053 0.035 0.417 1.75 0.178 0.135 0.271 0.135 0.492 6.75 0.3870.036 0.054 0.036 0.421 1.80 0.184 0.136 0.272 0.136 0.492 6.80 0.3870.037 0.056 0.037 0.425 1.85 0.190 0.137 0.272 0.137 0.492 6.85 0.3880.038 0.057 0.038 0.429 1.90 0.196 0.138 0.272 0.138 0.492 6.90 0.3880.039 0.058 0.039 0.433 1.95 0.202 0.139 0.272 0.139 0.492 6.95 0.3880.040 0.060 0.040 0.437 2.00 0.207 0.140 0.272 0.140 0.492 7.00 0.3890.041 0.062 0.041 0.440 2.05 0.213 0.141 0.273 0.141 0.492 7.05 0.3890.042 0.064 0.042 0.443 2.10 0.218 0.142 0.273 0.142 0.492 7.10 0.3890.043 0.066 0.043 0.446 2.15 0.224 0.143 0.273 0.143 0.492 7.15 0.3890.044 0.068 0.044 0.448 2.20 0.229 0.144 0.273 0.144 0.492 7.20 0.3900.045 0.071 0.045 0.451 2.25 0.234 0.145 0.273 0.145 0.492 7.25 0.3900.046 0.074 0.046 0.453 2.30 0.239 0.146 0.273 0.146 0.492 7.30 0.3900.047 0.077 0.047 0.455 2.35 0.243 0.147 0.273 0.147 0.492 7.35 0.3900.048 0.080 0.048 0.457 2.40 0.248 0.148 0.274 0.148 0.492 7.40 0.3910.049 0.084 0.049 0.459 2.45 0.252 0.149 0.274 0.149 0.492 7.45 0.3910.050 0.087 0.050 0.461 2.50 0.257 0.150 0.274 0.150 0.492 7.50 0.3910.051 0.091 0.051 0.463 2.55 0.261 0.151 0.274 0.151 0.492 7.55 0.3910.052 0.095 0.052 0.464 2.60 0.265 0.152 0.274 0.152 0.492 7.60 0.3910.053 0.099 0.053 0.465 2.65 0.269 0.153 0.274 0.153 0.492 7.65 0.3920.054 0.104 0.054 0.467 2.70 0.273 0.154 0.274 0.154 0.493 7.70 0.3920.055 0.108 0.055 0.468 2.75 0.277 0.155 0.274 0.155 0.493 7.75 0.3920.056 0.113 0.056 0.469 2.80 0.280 0.156 0.274 0.156 0.493 7.80 0.3920.057 0.117 0.057 0.470 2.85 0.284 0.157 0.274 0.157 0.493 7.85 0.3920.058 0.122 0.058 0.471 2.90 0.287 0.158 0.274 0.158 0.493 7.90 0.3930.059 0.126 0.059 0.472 2.95 0.291 0.159 0.274 0.159 0.493 7.95 0.3930.060 0.131 0.060 0.473 3.00 0.294 0.160 0.274 0.160 0.493 8.00 0.3930.061 0.136 0.061 0.474 3.05 0.297 0.161 0.274 0.161 0.493 8.05 0.3930.062 0.140 0.062 0.474 3.10 0.300 0.162 0.274 0.162 0.493 8.10 0.3930.063 0.145 0.063 0.475 3.15 0.303 0.163 0.275 0.163 0.493 8.15 0.3930.064 0.149 0.064 0.476 3.20 0.306 0.164 0.275 0.164 0.493 8.20 0.3930.065 0.153 0.065 0.476 3.25 0.309 0.165 0.275 0.165 0.493 8.25 0.3940.066 0.158 0.066 0.477 3.30 0.311 0.166 0.275 0.166 0.493 8.30 0.3940.067 0.162 0.067 0.478 3.35 0.314 0.167 0.275 0.167 0.493 8.35 0.3940.068 0.166 0.068 0.478 3.40 0.317 0.168 0.275 0.168 0.493 8.40 0.3940.069 0.170 0.069 0.479 3.45 0.319 0.169 0.275 0.169 0.493 8.45 0.3940.070 0.174 0.070 0.479 3.50 0.321 0.170 0.275 0.170 0.493 8.50 0.3940.071 0.177 0.071 0.480 3.55 0.324 0.171 0.275 0.171 0.493 8.55 0.3940.072 0.181 0.072 0.480 3.60 0.326 0.172 0.275 0.172 0.493 8.60 0.3950.073 0.184 0.073 0.481 3.65 0.328 0.173 0.275 0.173 0.493 8.65 0.3950.074 0.188 0.074 0.481 3.70 0.330 0.174 0.275 0.174 0.493 8.70 0.3950.075 0.191 0.075 0.481 3.75 0.332 0.175 0.275 0.175 0.493 8.75 0.3950.076 0.194 0.076 0.482 3.80 0.334 0.176 0.275 0.176 0.493 8.80 0.3950.077 0.198 0.077 0.482 3.85 0.336 0.177 0.275 0.177 0.493 8.85 0.3950.078 0.201 0.078 0.483 3.90 0.338 0.178 0.275 0.178 0.493 8.90 0.3950.079 0.203 0.079 0.483 3.95 0.340 0.179 0.275 0.179 0.493 8.95 0.3950.080 0.206 0.080 0.483 4.00 0.342 0.180 0.275 0.180 0.493 9.00 0.3950.081 0.209 0.081 0.484 4.05 0.343 0.181 0.275 0.181 0.493 9.05 0.3950.082 0.212 0.082 0.484 4.10 0.345 0.182 0.275 0.182 0.493 9.10 0.3960.083 0.214 0.083 0.484 4.15 0.347 0.183 0.275 0.183 0.493 9.15 0.3960.084 0.217 0.084 0.485 4.20 0.348 0.184 0.275 0.184 0.493 9.20 0.3960.085 0.219 0.085 0.485 4.25 0.350 0.185 0.275 0.185 0.493 9.25 0.3960.086 0.221 0.086 0.485 4.30 0.351 0.186 0.275 0.186 0.493 9.30 0.3960.087 0.224 0.087 0.485 4.35 0.352 0.187 0.275 0.187 0.493 9.35 0.3960.088 0.226 0.088 0.486 4.40 0.354 0.188 0.275 0.188 0.493 9.40 0.3960.089 0.228 0.089 0.486 4.45 0.355 0.189 0.275 0.189 0.493 9.45 0.3960.090 0.230 0.090 0.486 4.50 0.356 0.190 0.275 0.190 0.493 9.50 0.3960.091 0.232 0.091 0.486 4.55 0.358 0.191 0.275 0.191 0.493 9.55 0.3960.092 0.234 0.092 0.487 4.60 0.359 0.192 0.275 0.192 0.493 9.60 0.3960.093 0.236 0.093 0.487 4.65 0.360 0.193 0.275 0.193 0.493 9.65 0.3960.094 0.237 0.094 0.487 4.70 0.361 0.194 0.275 0.194 0.493 9.70 0.3970.095 0.239 0.095 0.487 4.75 0.362 0.195 0.275 0.195 0.493 9.75 0.3970.096 0.241 0.096 0.487 4.80 0.363 0.196 0.275 0.196 0.493 9.80 0.3970.097 0.242 0.097 0.488 4.85 0.364 0.197 0.275 0.197 0.493 9.85 0.3970.098 0.244 0.098 0.488 4.90 0.365 0.198 0.275 0.198 0.493 9.90 0.3970.099 0.245 0.099 0.488 4.95 0.366 0.199 0.275 0.199 0.493 9.95 0.3970.100 0.247 0.100 0.488 5.00 0.367 0.200 0.275 0.200 0.493 10.00 0.397



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0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

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0.40

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0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

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0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.001 0.05 0.000 0.101 0.207 0.101 0.453 5.05 0.404







0.00

0.10

0.20

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0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

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0.30

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0.60

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0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

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0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)



Building Height: High-rise (>10 stories) IDR E(LS | IDR) IDR E(LNS | IDR) PFA E(LNS | PFA) IDR E(LS | IDR) IDR E(LNS | IDR) PFA E(LNS | PFA)

0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.309 0.101 0.479 5.05 0.342







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0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.318 0.101 0.448 5.05 0.356







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0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


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0.90

1.00

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E(L | IDR)


0.00

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0.50

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0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.001 0.05 0.000 0.101 0.265 0.101 0.425 5.05 0.398







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0.90

1.00

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E(L | IDR)


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0.90

1.00

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E(L | IDR)


0.00

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0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.171 0.101 0.572 5.05 0.356


Structural System: Space moment-frame 0.004 0.001 0.004 0.027 0.20 0.002 0.104 0.172 0.104 0.572 5.20 0.3580.005 0.002 0.005 0.044 0.25 0.004 0.105 0.172 0.105 0.573 5.25 0.359





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E(L | IDR)


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1.00

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E(L | IDR)


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0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.177 0.101 0.540 5.05 0.374







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1.00

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E(L | IDR)


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1.00

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E(L | IDR)


0.00

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0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.146 0.101 0.478 5.05 0.397







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1.00

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E(L | IDR)


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1.00

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E(L | IDR)


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1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.250 0.101 0.521 5.05 0.352







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1.00

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E(L | IDR)


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1.00

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E(L | IDR)


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1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.258 0.101 0.488 5.05 0.368







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E(L | IDR)


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E(L | IDR)


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1.00

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E(L | PFA)




0.001 0.000 0.001 0.001 0.05 0.000 0.101 0.215 0.101 0.453 5.05 0.404







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E(L | IDR)


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1.00

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E(L | IDR)


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1.00

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E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.321 0.101 0.479 5.05 0.342







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E(L | IDR)


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1.00

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E(L | IDR)


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1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.330 0.101 0.448 5.05 0.356







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E(L | IDR)


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E(L | IDR)


0.00

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1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.001 0.05 0.000 0.101 0.276 0.101 0.425 5.05 0.398







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E(L | IDR)


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E(L | IDR)


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0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.183 0.101 0.572 5.05 0.356

Strucutral Material: Non-ductile reinforced-concrete 0.002 0.000 0.002 0.009 0.10 0.000 0.102 0.183 0.102 0.572 5.10 0.3570.003 0.001 0.003 0.016 0.15 0.000 0.103 0.183 0.103 0.572 5.15 0.358






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E(L | IDR)


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1.00

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E(L | IDR)


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1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.189 0.101 0.540 5.05 0.374







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E(L | IDR)


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0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.155 0.101 0.478 5.05 0.397







0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.267 0.101 0.521 5.05 0.352







0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.275 0.101 0.488 5.05 0.368







0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.001 0.05 0.000 0.101 0.229 0.101 0.453 5.05 0.404







0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.343 0.101 0.479 5.05 0.342







0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.352 0.101 0.448 5.05 0.356







0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.001 0.05 0.000 0.101 0.293 0.101 0.425 5.05 0.398







0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.174 0.101 0.572 5.05 0.356







0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.179 0.101 0.540 5.05 0.374







0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.148 0.101 0.478 5.05 0.397







0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.253 0.101 0.521 5.05 0.352







0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.261 0.101 0.488 5.05 0.368







0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.001 0.05 0.000 0.101 0.218 0.101 0.453 5.05 0.404







0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.326 0.101 0.479 5.05 0.342







0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.002 0.05 0.000 0.101 0.335 0.101 0.448 5.05 0.356







0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)




0.001 0.000 0.001 0.001 0.05 0.000 0.101 0.278 0.101 0.425 5.05 0.398







0.00

0.10

0.20

0.30

0.40

0.50

0.60

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0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20IDR

E(L | IDR)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 2.00 4.00 6.00 8.00 10.00PFA [g]

E(L | PFA)

APPENDIX C C-1 Subcontractor EDP-DV Functions

APPENDIX C

SUBCONTRACTOR EDP-DV FUNCTIONS

This appendix contains the graphs and data points for the subcontractor EDP-DV

functions developed in Chapter 8 to compute the variability of economic loss of the

example 4-story ductile reinforced concrete moment-resisting perimeter frame office

building. Each set of functions are grouped by floor type (1st floor, typical floor and top

floor) and seismic sensitivity (drift-sensitive and acceleration sensitive). Each set of

functions are presented in 2 pages: the first page plots the functions and the second page

reports the data points of each function. The values of economic loss shown here are

normalized by the replacement value of the story. Interstory drift ratio (IDR) is unit-less

and peak floor acceleration is expressed in terms of the acceleration of gravity (g).


Subcontractor EDP-DV Functions


Strucutral Material: Ductile reinforced-concrete

Structural System: Perimeter moment-frame

Occupancy: Office

Floor Type: 1st Floor

Functions for IDR = 0 to 0.05


(Data points on following page)

0.00

0.05

0.10

0.15

0.20

0.25

0.00 0.05 0.10 0.15 0.20

IDR

E( L | IDR )Concrete

Metals

Doors,Windows,Glass

Finishes

Mechanical

Electrical

0.00

0.05

0.10

0.15

0.20

0.25

0.00 0.01 0.02 0.03 0.04 0.05

IDR


Metals

Doors,Windows,Glass

Finishes

Mechanical

Electrical


Concrete Masonry Metals CarpentryMoisture

Protection

Doors, Windows,

GlassFinishes

Mech-anical

Electrical Concrete Masonry Metals CarpentryMoisture

Protection

Doors, Windows,

GlassFinishes

Mech-anical

Electrical

0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.000 0.101 0.167 0.000 0.050 0.000 0.000 0.133 0.242 0.060 0.0860.002 0.000 0.000 0.001 0.000 0.000 0.002 0.006 0.000 0.000 0.102 0.167 0.000 0.050 0.000 0.000 0.133 0.242 0.060 0.086

0.003 0.000 0.000 0.002 0.000 0.000 0.003 0.011 0.001 0.000 0.103 0.168 0.000 0.050 0.000 0.000 0.133 0.242 0.060 0.086

0.004 0.001 0.000 0.003 0.000 0.000 0.004 0.018 0.001 0.000 0.104 0.169 0.000 0.050 0.000 0.000 0.133 0.242 0.060 0.086

0.005 0.002 0.000 0.004 0.000 0.000 0.007 0.029 0.001 0.002 0.105 0.170 0.000 0.050 0.000 0.000 0.133 0.242 0.060 0.086

0.006 0.003 0.000 0.006 0.000 0.000 0.010 0.045 0.002 0.004 0.106 0.171 0.000 0.050 0.000 0.000 0.133 0.242 0.060 0.086

0.007 0.006 0.000 0.008 0.000 0.000 0.014 0.062 0.003 0.008 0.107 0.171 0.000 0.050 0.000 0.000 0.133 0.242 0.060 0.086

0.008 0.008 0.000 0.010 0.000 0.000 0.017 0.080 0.004 0.013 0.108 0.172 0.000 0.050 0.000 0.000 0.133 0.242 0.060 0.086

0.009 0.011 0.000 0.012 0.000 0.000 0.020 0.098 0.005 0.019 0.109 0.173 0.000 0.050 0.000 0.000 0.133 0.242 0.060 0.086

0.010 0.014 0.000 0.014 0.000 0.000 0.023 0.115 0.007 0.026 0.110 0.173 0.000 0.050 0.000 0.000 0.133 0.242 0.060 0.086

0.011 0.017 0.000 0.016 0.000 0.000 0.026 0.130 0.009 0.032 0.111 0.174 0.000 0.050 0.000 0.000 0.133 0.242 0.061 0.0860.012 0.019 0.000 0.017 0.000 0.000 0.029 0.145 0.010 0.039 0.112 0.174 0.000 0.050 0.000 0.000 0.133 0.242 0.061 0.0860.013 0.021 0.000 0.019 0.000 0.000 0.031 0.158 0.012 0.045 0.113 0.175 0.000 0.050 0.000 0.000 0.133 0.242 0.061 0.0860.014 0.023 0.000 0.020 0.000 0.000 0.034 0.169 0.014 0.050 0.114 0.176 0.000 0.050 0.000 0.000 0.133 0.242 0.061 0.086

0.015 0.024 0.000 0.022 0.000 0.000 0.036 0.179 0.016 0.055 0.115 0.176 0.000 0.050 0.000 0.000 0.133 0.242 0.061 0.086

0.016 0.026 0.000 0.023 0.000 0.000 0.038 0.188 0.017 0.060 0.116 0.176 0.000 0.050 0.000 0.000 0.133 0.242 0.061 0.0860.017 0.027 0.000 0.024 0.000 0.000 0.040 0.195 0.019 0.064 0.117 0.177 0.000 0.050 0.000 0.000 0.133 0.242 0.061 0.0860.018 0.027 0.000 0.025 0.000 0.000 0.042 0.202 0.020 0.067 0.118 0.177 0.000 0.050 0.000 0.000 0.133 0.242 0.061 0.0860.019 0.028 0.000 0.026 0.000 0.000 0.044 0.207 0.021 0.070 0.119 0.178 0.000 0.050 0.000 0.000 0.133 0.242 0.061 0.0860.020 0.029 0.000 0.027 0.000 0.000 0.046 0.212 0.023 0.073 0.120 0.178 0.000 0.050 0.000 0.000 0.133 0.242 0.061 0.0860.021 0.029 0.000 0.028 0.000 0.000 0.048 0.216 0.024 0.075 0.121 0.179 0.000 0.050 0.000 0.000 0.133 0.242 0.061 0.0860.022 0.030 0.000 0.028 0.000 0.000 0.050 0.219 0.025 0.077 0.122 0.179 0.000 0.050 0.000 0.000 0.133 0.242 0.061 0.0860.023 0.030 0.000 0.029 0.000 0.000 0.053 0.222 0.026 0.078 0.123 0.179 0.000 0.051 0.000 0.000 0.133 0.242 0.061 0.0860.024 0.031 0.000 0.030 0.000 0.000 0.055 0.225 0.027 0.079 0.124 0.180 0.000 0.051 0.000 0.000 0.133 0.242 0.061 0.0860.025 0.031 0.000 0.031 0.000 0.000 0.058 0.227 0.029 0.080 0.125 0.180 0.000 0.051 0.000 0.000 0.133 0.242 0.061 0.0860.026 0.032 0.000 0.031 0.000 0.000 0.061 0.229 0.030 0.081 0.126 0.180 0.000 0.051 0.000 0.000 0.133 0.242 0.061 0.0860.027 0.032 0.000 0.032 0.000 0.000 0.064 0.230 0.031 0.082 0.127 0.180 0.000 0.051 0.000 0.000 0.133 0.242 0.061 0.0860.028 0.032 0.000 0.033 0.000 0.000 0.068 0.231 0.032 0.083 0.128 0.181 0.000 0.051 0.000 0.000 0.133 0.242 0.061 0.0860.029 0.033 0.000 0.033 0.000 0.000 0.071 0.232 0.033 0.083 0.129 0.181 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.030 0.033 0.000 0.034 0.000 0.000 0.075 0.233 0.033 0.084 0.130 0.181 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.031 0.034 0.000 0.034 0.000 0.000 0.078 0.234 0.034 0.084 0.131 0.181 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.032 0.034 0.000 0.035 0.000 0.000 0.082 0.235 0.035 0.084 0.132 0.182 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.033 0.035 0.000 0.035 0.000 0.000 0.086 0.235 0.036 0.085 0.133 0.182 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.034 0.036 0.000 0.036 0.000 0.000 0.089 0.236 0.037 0.085 0.134 0.182 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.035 0.036 0.000 0.036 0.000 0.000 0.093 0.236 0.038 0.085 0.135 0.182 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.036 0.037 0.000 0.037 0.000 0.000 0.096 0.237 0.039 0.085 0.136 0.182 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.037 0.038 0.000 0.037 0.000 0.000 0.099 0.237 0.039 0.085 0.137 0.182 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.038 0.039 0.000 0.038 0.000 0.000 0.102 0.238 0.040 0.085 0.138 0.182 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.039 0.040 0.000 0.038 0.000 0.000 0.105 0.238 0.041 0.086 0.139 0.183 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.040 0.041 0.000 0.039 0.000 0.000 0.107 0.238 0.041 0.086 0.140 0.183 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.041 0.042 0.000 0.039 0.000 0.000 0.110 0.238 0.042 0.086 0.141 0.183 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.042 0.043 0.000 0.039 0.000 0.000 0.112 0.238 0.043 0.086 0.142 0.183 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.043 0.045 0.000 0.040 0.000 0.000 0.114 0.239 0.043 0.086 0.143 0.183 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.044 0.046 0.000 0.040 0.000 0.000 0.116 0.239 0.044 0.086 0.144 0.183 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.045 0.048 0.000 0.041 0.000 0.000 0.118 0.239 0.045 0.086 0.145 0.183 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.046 0.050 0.000 0.041 0.000 0.000 0.119 0.239 0.045 0.086 0.146 0.183 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.047 0.052 0.000 0.041 0.000 0.000 0.121 0.239 0.046 0.086 0.147 0.183 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.048 0.054 0.000 0.042 0.000 0.000 0.122 0.239 0.046 0.086 0.148 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.049 0.057 0.000 0.042 0.000 0.000 0.123 0.240 0.047 0.086 0.149 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.050 0.059 0.000 0.042 0.000 0.000 0.124 0.240 0.047 0.086 0.150 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.051 0.062 0.000 0.042 0.000 0.000 0.125 0.240 0.048 0.086 0.151 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.052 0.064 0.000 0.043 0.000 0.000 0.126 0.240 0.048 0.086 0.152 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.053 0.067 0.000 0.043 0.000 0.000 0.127 0.240 0.049 0.086 0.153 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.086

0.054 0.070 0.000 0.043 0.000 0.000 0.128 0.240 0.049 0.086 0.154 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.086

0.055 0.073 0.000 0.044 0.000 0.000 0.128 0.240 0.050 0.086 0.155 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.056 0.076 0.000 0.044 0.000 0.000 0.129 0.240 0.050 0.086 0.156 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.057 0.079 0.000 0.044 0.000 0.000 0.129 0.240 0.050 0.086 0.157 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.058 0.082 0.000 0.044 0.000 0.000 0.130 0.240 0.051 0.086 0.158 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.059 0.085 0.000 0.044 0.000 0.000 0.130 0.240 0.051 0.086 0.159 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.060 0.088 0.000 0.045 0.000 0.000 0.130 0.240 0.052 0.086 0.160 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.061 0.091 0.000 0.045 0.000 0.000 0.131 0.240 0.052 0.086 0.161 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.062 0.094 0.000 0.045 0.000 0.000 0.131 0.241 0.052 0.086 0.162 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.063 0.097 0.000 0.045 0.000 0.000 0.131 0.241 0.053 0.086 0.163 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.064 0.100 0.000 0.046 0.000 0.000 0.131 0.241 0.053 0.086 0.164 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.065 0.103 0.000 0.046 0.000 0.000 0.131 0.241 0.053 0.086 0.165 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.066 0.106 0.000 0.046 0.000 0.000 0.132 0.241 0.054 0.086 0.166 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.067 0.109 0.000 0.046 0.000 0.000 0.132 0.241 0.054 0.086 0.167 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.062 0.0860.068 0.112 0.000 0.046 0.000 0.000 0.132 0.241 0.054 0.086 0.168 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.069 0.114 0.000 0.046 0.000 0.000 0.132 0.241 0.054 0.086 0.169 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.070 0.117 0.000 0.047 0.000 0.000 0.132 0.241 0.055 0.086 0.170 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.071 0.119 0.000 0.047 0.000 0.000 0.132 0.241 0.055 0.086 0.171 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.072 0.122 0.000 0.047 0.000 0.000 0.132 0.241 0.055 0.086 0.172 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.073 0.124 0.000 0.047 0.000 0.000 0.132 0.241 0.055 0.086 0.173 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.074 0.126 0.000 0.047 0.000 0.000 0.132 0.241 0.056 0.086 0.174 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.075 0.129 0.000 0.047 0.000 0.000 0.132 0.241 0.056 0.086 0.175 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.076 0.131 0.000 0.047 0.000 0.000 0.133 0.241 0.056 0.086 0.176 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.077 0.133 0.000 0.048 0.000 0.000 0.133 0.241 0.056 0.086 0.177 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.078 0.135 0.000 0.048 0.000 0.000 0.133 0.241 0.056 0.086 0.178 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.079 0.137 0.000 0.048 0.000 0.000 0.133 0.241 0.057 0.086 0.179 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.080 0.139 0.000 0.048 0.000 0.000 0.133 0.241 0.057 0.086 0.180 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.081 0.140 0.000 0.048 0.000 0.000 0.133 0.241 0.057 0.086 0.181 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.082 0.142 0.000 0.048 0.000 0.000 0.133 0.241 0.057 0.086 0.182 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.083 0.144 0.000 0.048 0.000 0.000 0.133 0.241 0.057 0.086 0.183 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.084 0.146 0.000 0.048 0.000 0.000 0.133 0.241 0.058 0.086 0.184 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.085 0.147 0.000 0.048 0.000 0.000 0.133 0.241 0.058 0.086 0.185 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.086 0.149 0.000 0.048 0.000 0.000 0.133 0.241 0.058 0.086 0.186 0.184 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.087 0.150 0.000 0.049 0.000 0.000 0.133 0.241 0.058 0.086 0.187 0.185 0.000 0.051 0.000 0.000 0.133 0.242 0.063 0.0860.088 0.152 0.000 0.049 0.000 0.000 0.133 0.241 0.058 0.086 0.188 0.185 0.000 0.051 0.000 0.000 0.134 0.242 0.063 0.0860.089 0.153 0.000 0.049 0.000 0.000 0.133 0.241 0.058 0.086 0.189 0.185 0.000 0.051 0.000 0.000 0.134 0.242 0.063 0.0860.090 0.154 0.000 0.049 0.000 0.000 0.133 0.242 0.058 0.086 0.190 0.185 0.000 0.051 0.000 0.000 0.134 0.242 0.063 0.0860.091 0.156 0.000 0.049 0.000 0.000 0.133 0.242 0.059 0.086 0.191 0.185 0.000 0.051 0.000 0.000 0.134 0.242 0.063 0.0860.092 0.157 0.000 0.049 0.000 0.000 0.133 0.242 0.059 0.086 0.192 0.185 0.000 0.051 0.000 0.000 0.134 0.242 0.063 0.0860.093 0.158 0.000 0.049 0.000 0.000 0.133 0.242 0.059 0.086 0.193 0.185 0.000 0.051 0.000 0.000 0.134 0.242 0.063 0.0860.094 0.159 0.000 0.049 0.000 0.000 0.133 0.242 0.059 0.086 0.194 0.185 0.000 0.051 0.000 0.000 0.134 0.242 0.063 0.0860.095 0.160 0.000 0.049 0.000 0.000 0.133 0.242 0.059 0.086 0.195 0.185 0.000 0.051 0.000 0.000 0.134 0.242 0.063 0.0860.096 0.162 0.000 0.049 0.000 0.000 0.133 0.242 0.059 0.086 0.196 0.185 0.000 0.051 0.000 0.000 0.134 0.242 0.063 0.0860.097 0.163 0.000 0.049 0.000 0.000 0.133 0.242 0.059 0.086 0.197 0.185 0.000 0.051 0.000 0.000 0.134 0.242 0.063 0.0860.098 0.164 0.000 0.049 0.000 0.000 0.133 0.242 0.059 0.086 0.198 0.185 0.000 0.051 0.000 0.000 0.134 0.242 0.063 0.0860.099 0.165 0.000 0.049 0.000 0.000 0.133 0.242 0.060 0.086 0.199 0.185 0.000 0.051 0.000 0.000 0.134 0.242 0.063 0.0860.100 0.166 0.000 0.050 0.000 0.000 0.133 0.242 0.060 0.086 0.200 0.185 0.000 0.051 0.000 0.000 0.134 0.242 0.063 0.086

EDP-DV Function

IDR IDR

EDP-DV Function (cont'd)






Occupancy: Office

Floor Type: Typical Floor




0.00

0.05

0.10

0.15

0.20

0.25

0.00 0.05 0.10 0.15 0.20

IDR


Metals

Doors,Windows,Glass

Finishes

Mechanical

Electrical

0.00

0.05

0.10

0.15

0.20

0.25

0.00 0.01 0.02 0.03 0.04 0.05

IDR


Metals

Doors,Windows,Glass

Finishes

Mechanical

Electrical



Protection

Doors, Windows,

GlassFinishes

Mech-anical


Protection

Doors, Windows,

GlassFinishes

Mech-anical

Electrical

0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.000 0.101 0.172 0.000 0.052 0.000 0.000 0.110 0.226 0.062 0.0890.002 0.000 0.000 0.001 0.000 0.000 0.001 0.006 0.000 0.000 0.102 0.173 0.000 0.052 0.000 0.000 0.110 0.226 0.062 0.089

0.003 0.000 0.000 0.002 0.000 0.000 0.002 0.011 0.001 0.000 0.103 0.174 0.000 0.052 0.000 0.000 0.110 0.226 0.062 0.089

0.004 0.001 0.000 0.003 0.000 0.000 0.003 0.017 0.001 0.000 0.104 0.174 0.000 0.052 0.000 0.000 0.110 0.226 0.062 0.089

0.005 0.002 0.000 0.005 0.000 0.000 0.005 0.028 0.001 0.002 0.105 0.175 0.000 0.052 0.000 0.000 0.110 0.226 0.062 0.089

0.006 0.003 0.000 0.008 0.000 0.000 0.008 0.042 0.002 0.004 0.106 0.176 0.000 0.052 0.000 0.000 0.110 0.226 0.062 0.089

0.007 0.006 0.000 0.010 0.000 0.000 0.010 0.059 0.003 0.008 0.107 0.177 0.000 0.053 0.000 0.000 0.110 0.226 0.062 0.089

0.008 0.009 0.000 0.013 0.000 0.000 0.013 0.076 0.004 0.014 0.108 0.177 0.000 0.053 0.000 0.000 0.110 0.226 0.062 0.089

0.009 0.012 0.000 0.015 0.000 0.000 0.015 0.092 0.005 0.020 0.109 0.178 0.000 0.053 0.000 0.000 0.110 0.226 0.062 0.089

0.010 0.015 0.000 0.018 0.000 0.000 0.018 0.108 0.007 0.026 0.110 0.179 0.000 0.053 0.000 0.000 0.110 0.226 0.062 0.089

0.011 0.017 0.000 0.020 0.000 0.000 0.020 0.123 0.009 0.033 0.111 0.179 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.0890.012 0.020 0.000 0.022 0.000 0.000 0.022 0.136 0.011 0.040 0.112 0.180 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.0890.013 0.022 0.000 0.024 0.000 0.000 0.024 0.148 0.013 0.046 0.113 0.180 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.0890.014 0.024 0.000 0.025 0.000 0.000 0.026 0.159 0.014 0.052 0.114 0.181 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.089

0.015 0.025 0.000 0.027 0.000 0.000 0.027 0.168 0.016 0.057 0.115 0.182 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.089

0.016 0.026 0.000 0.028 0.000 0.000 0.029 0.176 0.018 0.062 0.116 0.182 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.0890.017 0.027 0.000 0.029 0.000 0.000 0.031 0.183 0.019 0.066 0.117 0.183 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.0890.018 0.028 0.000 0.030 0.000 0.000 0.032 0.189 0.021 0.069 0.118 0.183 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.0890.019 0.029 0.000 0.031 0.000 0.000 0.034 0.194 0.022 0.072 0.119 0.183 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.0890.020 0.030 0.000 0.032 0.000 0.000 0.036 0.199 0.023 0.075 0.120 0.184 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.0890.021 0.030 0.000 0.033 0.000 0.000 0.037 0.202 0.025 0.077 0.121 0.184 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.0890.022 0.031 0.000 0.034 0.000 0.000 0.039 0.205 0.026 0.079 0.122 0.185 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.0890.023 0.031 0.000 0.035 0.000 0.000 0.041 0.208 0.027 0.081 0.123 0.185 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.0890.024 0.032 0.000 0.035 0.000 0.000 0.043 0.210 0.028 0.082 0.124 0.185 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.0890.025 0.032 0.000 0.036 0.000 0.000 0.046 0.212 0.029 0.083 0.125 0.185 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.0890.026 0.032 0.000 0.037 0.000 0.000 0.048 0.214 0.031 0.084 0.126 0.186 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.0890.027 0.033 0.000 0.037 0.000 0.000 0.051 0.215 0.032 0.085 0.127 0.186 0.000 0.053 0.000 0.000 0.110 0.226 0.063 0.0890.028 0.033 0.000 0.038 0.000 0.000 0.054 0.217 0.033 0.085 0.128 0.186 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.029 0.034 0.000 0.038 0.000 0.000 0.057 0.218 0.034 0.086 0.129 0.187 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.030 0.034 0.000 0.039 0.000 0.000 0.060 0.218 0.035 0.086 0.130 0.187 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.031 0.035 0.000 0.039 0.000 0.000 0.063 0.219 0.036 0.087 0.131 0.187 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.032 0.035 0.000 0.040 0.000 0.000 0.066 0.220 0.036 0.087 0.132 0.187 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.033 0.036 0.000 0.040 0.000 0.000 0.069 0.220 0.037 0.087 0.133 0.187 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.034 0.037 0.000 0.041 0.000 0.000 0.072 0.221 0.038 0.088 0.134 0.188 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.035 0.037 0.000 0.041 0.000 0.000 0.075 0.221 0.039 0.088 0.135 0.188 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.036 0.038 0.000 0.041 0.000 0.000 0.078 0.222 0.040 0.088 0.136 0.188 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.037 0.039 0.000 0.042 0.000 0.000 0.081 0.222 0.041 0.088 0.137 0.188 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.038 0.040 0.000 0.042 0.000 0.000 0.083 0.222 0.041 0.088 0.138 0.188 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.039 0.041 0.000 0.043 0.000 0.000 0.086 0.222 0.042 0.088 0.139 0.188 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.040 0.042 0.000 0.043 0.000 0.000 0.088 0.223 0.043 0.088 0.140 0.189 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.041 0.043 0.000 0.043 0.000 0.000 0.090 0.223 0.043 0.089 0.141 0.189 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.042 0.045 0.000 0.044 0.000 0.000 0.092 0.223 0.044 0.089 0.142 0.189 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.043 0.046 0.000 0.044 0.000 0.000 0.094 0.223 0.045 0.089 0.143 0.189 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.044 0.048 0.000 0.044 0.000 0.000 0.095 0.223 0.045 0.089 0.144 0.189 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.045 0.050 0.000 0.045 0.000 0.000 0.097 0.223 0.046 0.089 0.145 0.189 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.046 0.052 0.000 0.045 0.000 0.000 0.098 0.224 0.047 0.089 0.146 0.189 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.047 0.054 0.000 0.045 0.000 0.000 0.099 0.224 0.047 0.089 0.147 0.189 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.048 0.056 0.000 0.045 0.000 0.000 0.101 0.224 0.048 0.089 0.148 0.189 0.000 0.053 0.000 0.000 0.110 0.226 0.064 0.0890.049 0.058 0.000 0.046 0.000 0.000 0.102 0.224 0.048 0.089 0.149 0.189 0.000 0.054 0.000 0.000 0.110 0.226 0.064 0.0890.050 0.061 0.000 0.046 0.000 0.000 0.102 0.224 0.049 0.089 0.150 0.189 0.000 0.054 0.000 0.000 0.110 0.226 0.064 0.0890.051 0.064 0.000 0.046 0.000 0.000 0.103 0.224 0.049 0.089 0.151 0.189 0.000 0.054 0.000 0.000 0.110 0.226 0.064 0.0890.052 0.066 0.000 0.046 0.000 0.000 0.104 0.224 0.050 0.089 0.152 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.064 0.0890.053 0.069 0.000 0.047 0.000 0.000 0.105 0.224 0.050 0.089 0.153 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.064 0.089

0.054 0.072 0.000 0.047 0.000 0.000 0.105 0.224 0.051 0.089 0.154 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.064 0.089

0.055 0.075 0.000 0.047 0.000 0.000 0.106 0.224 0.051 0.089 0.155 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.064 0.0890.056 0.078 0.000 0.047 0.000 0.000 0.106 0.224 0.052 0.089 0.156 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.064 0.0890.057 0.082 0.000 0.048 0.000 0.000 0.107 0.224 0.052 0.089 0.157 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.064 0.0890.058 0.085 0.000 0.048 0.000 0.000 0.107 0.224 0.052 0.089 0.158 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.064 0.0890.059 0.088 0.000 0.048 0.000 0.000 0.107 0.224 0.053 0.089 0.159 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.064 0.0890.060 0.091 0.000 0.048 0.000 0.000 0.107 0.225 0.053 0.089 0.160 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.064 0.0890.061 0.094 0.000 0.048 0.000 0.000 0.108 0.225 0.054 0.089 0.161 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.064 0.0890.062 0.097 0.000 0.048 0.000 0.000 0.108 0.225 0.054 0.089 0.162 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.064 0.0890.063 0.100 0.000 0.049 0.000 0.000 0.108 0.225 0.054 0.089 0.163 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.064 0.103 0.000 0.049 0.000 0.000 0.108 0.225 0.055 0.089 0.164 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.065 0.106 0.000 0.049 0.000 0.000 0.108 0.225 0.055 0.089 0.165 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.066 0.109 0.000 0.049 0.000 0.000 0.109 0.225 0.055 0.089 0.166 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.067 0.112 0.000 0.049 0.000 0.000 0.109 0.225 0.056 0.089 0.167 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.068 0.115 0.000 0.049 0.000 0.000 0.109 0.225 0.056 0.089 0.168 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.069 0.118 0.000 0.050 0.000 0.000 0.109 0.225 0.056 0.089 0.169 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.070 0.120 0.000 0.050 0.000 0.000 0.109 0.225 0.056 0.089 0.170 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.071 0.123 0.000 0.050 0.000 0.000 0.109 0.225 0.057 0.089 0.171 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.072 0.126 0.000 0.050 0.000 0.000 0.109 0.225 0.057 0.089 0.172 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.073 0.128 0.000 0.050 0.000 0.000 0.109 0.225 0.057 0.089 0.173 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.074 0.130 0.000 0.050 0.000 0.000 0.109 0.225 0.057 0.089 0.174 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.075 0.133 0.000 0.050 0.000 0.000 0.109 0.225 0.058 0.089 0.175 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.076 0.135 0.000 0.050 0.000 0.000 0.109 0.225 0.058 0.089 0.176 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.077 0.137 0.000 0.051 0.000 0.000 0.109 0.225 0.058 0.089 0.177 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.078 0.139 0.000 0.051 0.000 0.000 0.109 0.225 0.058 0.089 0.178 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.079 0.141 0.000 0.051 0.000 0.000 0.109 0.225 0.059 0.089 0.179 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.080 0.143 0.000 0.051 0.000 0.000 0.109 0.225 0.059 0.089 0.180 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.081 0.145 0.000 0.051 0.000 0.000 0.109 0.225 0.059 0.089 0.181 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.082 0.147 0.000 0.051 0.000 0.000 0.109 0.225 0.059 0.089 0.182 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.083 0.148 0.000 0.051 0.000 0.000 0.110 0.225 0.059 0.089 0.183 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.084 0.150 0.000 0.051 0.000 0.000 0.110 0.225 0.059 0.089 0.184 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.085 0.152 0.000 0.051 0.000 0.000 0.110 0.225 0.060 0.089 0.185 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.086 0.153 0.000 0.051 0.000 0.000 0.110 0.225 0.060 0.089 0.186 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.087 0.155 0.000 0.051 0.000 0.000 0.110 0.225 0.060 0.089 0.187 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.088 0.156 0.000 0.051 0.000 0.000 0.110 0.225 0.060 0.089 0.188 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.089 0.158 0.000 0.052 0.000 0.000 0.110 0.225 0.060 0.089 0.189 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.090 0.159 0.000 0.052 0.000 0.000 0.110 0.225 0.060 0.089 0.190 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.091 0.161 0.000 0.052 0.000 0.000 0.110 0.225 0.061 0.089 0.191 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.092 0.162 0.000 0.052 0.000 0.000 0.110 0.225 0.061 0.089 0.192 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.093 0.163 0.000 0.052 0.000 0.000 0.110 0.225 0.061 0.089 0.193 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.094 0.164 0.000 0.052 0.000 0.000 0.110 0.225 0.061 0.089 0.194 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.095 0.166 0.000 0.052 0.000 0.000 0.110 0.225 0.061 0.089 0.195 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.096 0.167 0.000 0.052 0.000 0.000 0.110 0.226 0.061 0.089 0.196 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.097 0.168 0.000 0.052 0.000 0.000 0.110 0.226 0.061 0.089 0.197 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.098 0.169 0.000 0.052 0.000 0.000 0.110 0.226 0.061 0.089 0.198 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.099 0.170 0.000 0.052 0.000 0.000 0.110 0.226 0.061 0.089 0.199 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.0890.100 0.171 0.000 0.052 0.000 0.000 0.110 0.226 0.062 0.089 0.200 0.190 0.000 0.054 0.000 0.000 0.110 0.226 0.065 0.089

EDP-DV Function

IDR IDR







Occupancy: Office

Floor Type: Top Floor




0.00

0.05

0.10

0.15

0.20

0.25

0.00 0.05 0.10 0.15 0.20

IDR


Metals

Doors,Windows,Glass

Finishes

Mechanical

Electrical

0.00

0.05

0.10

0.15

0.20

0.25

0.00 0.01 0.02 0.03 0.04 0.05

IDR


Metals

Doors,Windows,Glass

Finishes

Mechanical

Electrical



Protection

Doors, Windows,

GlassFinishes

Mech-anical


Protection

Doors, Windows,

GlassFinishes

Mech-anical

Electrical

0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.000 0.101 0.141 0.000 0.026 0.000 0.000 0.094 0.174 0.079 0.0910.002 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.000 0.000 0.102 0.142 0.000 0.026 0.000 0.000 0.094 0.174 0.079 0.091

0.003 0.000 0.000 0.001 0.000 0.000 0.001 0.009 0.001 0.000 0.103 0.142 0.000 0.026 0.000 0.000 0.094 0.174 0.079 0.091

0.004 0.001 0.000 0.001 0.000 0.000 0.002 0.015 0.001 0.000 0.104 0.143 0.000 0.026 0.000 0.000 0.094 0.174 0.079 0.091

0.005 0.001 0.000 0.002 0.000 0.000 0.002 0.024 0.002 0.002 0.105 0.144 0.000 0.026 0.000 0.000 0.095 0.174 0.079 0.091

0.006 0.003 0.000 0.003 0.000 0.000 0.004 0.035 0.002 0.004 0.106 0.144 0.000 0.026 0.000 0.000 0.095 0.174 0.079 0.091

0.007 0.005 0.000 0.004 0.000 0.000 0.005 0.049 0.003 0.008 0.107 0.145 0.000 0.027 0.000 0.000 0.095 0.174 0.079 0.091

0.008 0.007 0.000 0.005 0.000 0.000 0.006 0.062 0.005 0.014 0.108 0.146 0.000 0.027 0.000 0.000 0.095 0.174 0.080 0.091

0.009 0.009 0.000 0.006 0.000 0.000 0.007 0.075 0.006 0.020 0.109 0.146 0.000 0.027 0.000 0.000 0.095 0.174 0.080 0.091

0.010 0.012 0.000 0.007 0.000 0.000 0.008 0.087 0.009 0.027 0.110 0.147 0.000 0.027 0.000 0.000 0.095 0.174 0.080 0.091

0.011 0.014 0.000 0.008 0.000 0.000 0.010 0.098 0.011 0.034 0.111 0.147 0.000 0.027 0.000 0.000 0.095 0.174 0.080 0.0910.012 0.016 0.000 0.009 0.000 0.000 0.011 0.109 0.013 0.041 0.112 0.148 0.000 0.027 0.000 0.000 0.095 0.174 0.080 0.0910.013 0.018 0.000 0.010 0.000 0.000 0.012 0.117 0.015 0.048 0.113 0.148 0.000 0.027 0.000 0.000 0.095 0.174 0.080 0.0910.014 0.020 0.000 0.011 0.000 0.000 0.013 0.125 0.017 0.054 0.114 0.149 0.000 0.027 0.000 0.000 0.095 0.174 0.080 0.091

0.015 0.021 0.000 0.011 0.000 0.000 0.014 0.132 0.019 0.059 0.115 0.149 0.000 0.027 0.000 0.000 0.095 0.174 0.080 0.091

0.016 0.022 0.000 0.012 0.000 0.000 0.016 0.138 0.021 0.064 0.116 0.149 0.000 0.027 0.000 0.000 0.095 0.174 0.080 0.0910.017 0.022 0.000 0.013 0.000 0.000 0.017 0.143 0.023 0.068 0.117 0.150 0.000 0.027 0.000 0.000 0.095 0.174 0.080 0.0910.018 0.023 0.000 0.013 0.000 0.000 0.018 0.148 0.025 0.071 0.118 0.150 0.000 0.027 0.000 0.000 0.095 0.174 0.080 0.0910.019 0.024 0.000 0.014 0.000 0.000 0.019 0.152 0.027 0.074 0.119 0.151 0.000 0.027 0.000 0.000 0.095 0.174 0.081 0.0910.020 0.024 0.000 0.014 0.000 0.000 0.021 0.155 0.028 0.077 0.120 0.151 0.000 0.027 0.000 0.000 0.095 0.174 0.081 0.0910.021 0.025 0.000 0.015 0.000 0.000 0.022 0.157 0.030 0.079 0.121 0.151 0.000 0.027 0.000 0.000 0.095 0.174 0.081 0.0910.022 0.025 0.000 0.015 0.000 0.000 0.024 0.160 0.032 0.081 0.122 0.151 0.000 0.027 0.000 0.000 0.095 0.174 0.081 0.0910.023 0.025 0.000 0.016 0.000 0.000 0.026 0.162 0.033 0.083 0.123 0.152 0.000 0.027 0.000 0.000 0.095 0.174 0.081 0.0910.024 0.026 0.000 0.016 0.000 0.000 0.028 0.163 0.035 0.084 0.124 0.152 0.000 0.027 0.000 0.000 0.095 0.174 0.081 0.0910.025 0.026 0.000 0.016 0.000 0.000 0.030 0.165 0.036 0.085 0.125 0.152 0.000 0.027 0.000 0.000 0.095 0.174 0.081 0.0910.026 0.027 0.000 0.017 0.000 0.000 0.033 0.166 0.037 0.086 0.126 0.153 0.000 0.027 0.000 0.000 0.095 0.174 0.081 0.0910.027 0.027 0.000 0.017 0.000 0.000 0.035 0.167 0.039 0.087 0.127 0.153 0.000 0.027 0.000 0.000 0.095 0.174 0.081 0.0910.028 0.027 0.000 0.017 0.000 0.000 0.038 0.168 0.040 0.088 0.128 0.153 0.000 0.027 0.000 0.000 0.095 0.174 0.081 0.0910.029 0.028 0.000 0.018 0.000 0.000 0.041 0.169 0.041 0.088 0.129 0.153 0.000 0.027 0.000 0.000 0.095 0.174 0.081 0.0910.030 0.028 0.000 0.018 0.000 0.000 0.044 0.169 0.043 0.089 0.130 0.153 0.000 0.027 0.000 0.000 0.095 0.174 0.081 0.0910.031 0.029 0.000 0.018 0.000 0.000 0.047 0.170 0.044 0.089 0.131 0.154 0.000 0.027 0.000 0.000 0.095 0.174 0.081 0.0910.032 0.029 0.000 0.019 0.000 0.000 0.050 0.170 0.045 0.090 0.132 0.154 0.000 0.027 0.000 0.000 0.095 0.174 0.081 0.0910.033 0.029 0.000 0.019 0.000 0.000 0.054 0.171 0.046 0.090 0.133 0.154 0.000 0.027 0.000 0.000 0.095 0.174 0.081 0.0910.034 0.030 0.000 0.019 0.000 0.000 0.057 0.171 0.047 0.090 0.134 0.154 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.035 0.031 0.000 0.019 0.000 0.000 0.060 0.171 0.049 0.090 0.135 0.154 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.036 0.031 0.000 0.020 0.000 0.000 0.062 0.171 0.050 0.091 0.136 0.154 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.037 0.032 0.000 0.020 0.000 0.000 0.065 0.172 0.051 0.091 0.137 0.154 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.038 0.033 0.000 0.020 0.000 0.000 0.068 0.172 0.052 0.091 0.138 0.155 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.039 0.033 0.000 0.020 0.000 0.000 0.070 0.172 0.053 0.091 0.139 0.155 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.040 0.034 0.000 0.021 0.000 0.000 0.073 0.172 0.054 0.091 0.140 0.155 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.041 0.035 0.000 0.021 0.000 0.000 0.075 0.172 0.055 0.091 0.141 0.155 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.042 0.036 0.000 0.021 0.000 0.000 0.077 0.172 0.055 0.091 0.142 0.155 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.043 0.038 0.000 0.021 0.000 0.000 0.078 0.173 0.056 0.091 0.143 0.155 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.044 0.039 0.000 0.021 0.000 0.000 0.080 0.173 0.057 0.091 0.144 0.155 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.045 0.041 0.000 0.022 0.000 0.000 0.082 0.173 0.058 0.091 0.145 0.155 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.046 0.042 0.000 0.022 0.000 0.000 0.083 0.173 0.059 0.091 0.146 0.155 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.047 0.044 0.000 0.022 0.000 0.000 0.084 0.173 0.060 0.091 0.147 0.155 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.048 0.046 0.000 0.022 0.000 0.000 0.085 0.173 0.060 0.091 0.148 0.155 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.049 0.048 0.000 0.022 0.000 0.000 0.086 0.173 0.061 0.091 0.149 0.155 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.050 0.050 0.000 0.022 0.000 0.000 0.087 0.173 0.062 0.091 0.150 0.155 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.051 0.052 0.000 0.023 0.000 0.000 0.088 0.173 0.062 0.091 0.151 0.156 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.052 0.054 0.000 0.023 0.000 0.000 0.089 0.173 0.063 0.091 0.152 0.156 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.053 0.057 0.000 0.023 0.000 0.000 0.089 0.173 0.064 0.091 0.153 0.156 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.091

0.054 0.059 0.000 0.023 0.000 0.000 0.090 0.173 0.064 0.091 0.154 0.156 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.091

0.055 0.062 0.000 0.023 0.000 0.000 0.091 0.173 0.065 0.091 0.155 0.156 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.056 0.064 0.000 0.023 0.000 0.000 0.091 0.173 0.065 0.091 0.156 0.156 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.057 0.067 0.000 0.023 0.000 0.000 0.091 0.173 0.066 0.091 0.157 0.156 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.058 0.069 0.000 0.024 0.000 0.000 0.092 0.174 0.066 0.091 0.158 0.156 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.059 0.072 0.000 0.024 0.000 0.000 0.092 0.174 0.067 0.091 0.159 0.156 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.060 0.075 0.000 0.024 0.000 0.000 0.092 0.174 0.068 0.091 0.160 0.156 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.061 0.077 0.000 0.024 0.000 0.000 0.093 0.174 0.068 0.091 0.161 0.156 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.062 0.080 0.000 0.024 0.000 0.000 0.093 0.174 0.068 0.091 0.162 0.156 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.063 0.082 0.000 0.024 0.000 0.000 0.093 0.174 0.069 0.091 0.163 0.156 0.000 0.027 0.000 0.000 0.095 0.174 0.082 0.0910.064 0.085 0.000 0.024 0.000 0.000 0.093 0.174 0.069 0.091 0.164 0.156 0.000 0.027 0.000 0.000 0.095 0.174 0.083 0.0910.065 0.087 0.000 0.024 0.000 0.000 0.093 0.174 0.070 0.091 0.165 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.066 0.090 0.000 0.024 0.000 0.000 0.093 0.174 0.070 0.091 0.166 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.067 0.092 0.000 0.024 0.000 0.000 0.094 0.174 0.071 0.091 0.167 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.068 0.094 0.000 0.025 0.000 0.000 0.094 0.174 0.071 0.091 0.168 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.069 0.097 0.000 0.025 0.000 0.000 0.094 0.174 0.071 0.091 0.169 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.070 0.099 0.000 0.025 0.000 0.000 0.094 0.174 0.072 0.091 0.170 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.071 0.101 0.000 0.025 0.000 0.000 0.094 0.174 0.072 0.091 0.171 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.072 0.103 0.000 0.025 0.000 0.000 0.094 0.174 0.072 0.091 0.172 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.073 0.105 0.000 0.025 0.000 0.000 0.094 0.174 0.073 0.091 0.173 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.074 0.107 0.000 0.025 0.000 0.000 0.094 0.174 0.073 0.091 0.174 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.075 0.109 0.000 0.025 0.000 0.000 0.094 0.174 0.073 0.091 0.175 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.076 0.111 0.000 0.025 0.000 0.000 0.094 0.174 0.074 0.091 0.176 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.077 0.112 0.000 0.025 0.000 0.000 0.094 0.174 0.074 0.091 0.177 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.078 0.114 0.000 0.025 0.000 0.000 0.094 0.174 0.074 0.091 0.178 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.079 0.116 0.000 0.025 0.000 0.000 0.094 0.174 0.075 0.091 0.179 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.080 0.117 0.000 0.025 0.000 0.000 0.094 0.174 0.075 0.091 0.180 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.081 0.119 0.000 0.025 0.000 0.000 0.094 0.174 0.075 0.091 0.181 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.082 0.120 0.000 0.026 0.000 0.000 0.094 0.174 0.075 0.091 0.182 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.083 0.122 0.000 0.026 0.000 0.000 0.094 0.174 0.076 0.091 0.183 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.084 0.123 0.000 0.026 0.000 0.000 0.094 0.174 0.076 0.091 0.184 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.085 0.125 0.000 0.026 0.000 0.000 0.094 0.174 0.076 0.091 0.185 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.086 0.126 0.000 0.026 0.000 0.000 0.094 0.174 0.076 0.091 0.186 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.087 0.127 0.000 0.026 0.000 0.000 0.094 0.174 0.076 0.091 0.187 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.088 0.128 0.000 0.026 0.000 0.000 0.094 0.174 0.077 0.091 0.188 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.089 0.130 0.000 0.026 0.000 0.000 0.094 0.174 0.077 0.091 0.189 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.090 0.131 0.000 0.026 0.000 0.000 0.094 0.174 0.077 0.091 0.190 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.091 0.132 0.000 0.026 0.000 0.000 0.094 0.174 0.077 0.091 0.191 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.092 0.133 0.000 0.026 0.000 0.000 0.094 0.174 0.077 0.091 0.192 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.093 0.134 0.000 0.026 0.000 0.000 0.094 0.174 0.078 0.091 0.193 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.094 0.135 0.000 0.026 0.000 0.000 0.094 0.174 0.078 0.091 0.194 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.095 0.136 0.000 0.026 0.000 0.000 0.094 0.174 0.078 0.091 0.195 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.096 0.137 0.000 0.026 0.000 0.000 0.094 0.174 0.078 0.091 0.196 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.097 0.138 0.000 0.026 0.000 0.000 0.094 0.174 0.078 0.091 0.197 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.098 0.139 0.000 0.026 0.000 0.000 0.094 0.174 0.078 0.091 0.198 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.099 0.139 0.000 0.026 0.000 0.000 0.094 0.174 0.078 0.091 0.199 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.0910.100 0.140 0.000 0.026 0.000 0.000 0.094 0.174 0.079 0.091 0.200 0.156 0.000 0.027 0.000 0.000 0.095 0.175 0.083 0.091

EDP-DV Function

IDR IDR







Occupancy: Office

Floor Type: 1st Floor

Functions for PFA = 0 to 5g



0.00

0.05

0.10

0.15

0.20

0.25

0.0 2.0 4.0 6.0 8.0 10.0

PFA [g]

E( L | PFA )

Finishes

Mechanical

Electrical

0.00

0.05

0.10

0.15

0.20

0.25

0.0 1.0 2.0 3.0 4.0 5.0

PFA [g]

E( L | PFA )

Finishes

Mechanical

Electrical



Protection

Doors, Windows,

GlassFinishes

Mech-anical


Protection

Doors, Windows,

GlassFinishes

Mech-anical

Electrical

0.05 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 5.05 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.195 0.0910.10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 5.10 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.196 0.092

0.15 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 5.15 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.196 0.092

0.20 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.000 5.20 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.197 0.092

0.25 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.001 0.000 5.25 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.197 0.092

0.30 0.000 0.000 0.000 0.000 0.000 0.000 0.005 0.001 0.001 5.30 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.198 0.093

0.35 0.000 0.000 0.000 0.000 0.000 0.000 0.006 0.002 0.001 5.35 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.198 0.093

0.40 0.000 0.000 0.000 0.000 0.000 0.000 0.008 0.003 0.001 5.40 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.199 0.093

0.45 0.000 0.000 0.000 0.000 0.000 0.000 0.011 0.004 0.002 5.45 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.199 0.093

0.50 0.000 0.000 0.000 0.000 0.000 0.000 0.013 0.005 0.002 5.50 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.200 0.094

0.55 0.000 0.000 0.000 0.000 0.000 0.000 0.015 0.007 0.003 5.55 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.200 0.0940.60 0.000 0.000 0.000 0.000 0.000 0.000 0.017 0.008 0.004 5.60 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.200 0.0940.65 0.000 0.000 0.000 0.000 0.000 0.000 0.019 0.010 0.005 5.65 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.201 0.0940.70 0.000 0.000 0.000 0.000 0.000 0.000 0.021 0.012 0.006 5.70 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.201 0.094

0.75 0.000 0.000 0.000 0.000 0.000 0.000 0.023 0.015 0.007 5.75 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.201 0.094

0.80 0.000 0.000 0.000 0.000 0.000 0.000 0.026 0.017 0.008 5.80 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.202 0.0950.85 0.000 0.000 0.000 0.000 0.000 0.000 0.028 0.020 0.009 5.85 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.202 0.0950.90 0.000 0.000 0.000 0.000 0.000 0.000 0.030 0.023 0.011 5.90 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.203 0.0950.95 0.000 0.000 0.000 0.000 0.000 0.000 0.032 0.026 0.012 5.95 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.203 0.0951.00 0.000 0.000 0.000 0.000 0.000 0.000 0.034 0.029 0.014 6.00 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.203 0.0951.05 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.033 0.015 6.05 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.204 0.0951.10 0.000 0.000 0.000 0.000 0.000 0.000 0.038 0.036 0.017 6.10 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.204 0.0961.15 0.000 0.000 0.000 0.000 0.000 0.000 0.040 0.039 0.018 6.15 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.204 0.0961.20 0.000 0.000 0.000 0.000 0.000 0.000 0.041 0.043 0.020 6.20 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.204 0.0961.25 0.000 0.000 0.000 0.000 0.000 0.000 0.043 0.046 0.022 6.25 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.205 0.0961.30 0.000 0.000 0.000 0.000 0.000 0.000 0.045 0.050 0.023 6.30 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.205 0.0961.35 0.000 0.000 0.000 0.000 0.000 0.000 0.046 0.054 0.025 6.35 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.205 0.0961.40 0.000 0.000 0.000 0.000 0.000 0.000 0.048 0.057 0.027 6.40 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.205 0.0961.45 0.000 0.000 0.000 0.000 0.000 0.000 0.049 0.061 0.029 6.45 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.206 0.0961.50 0.000 0.000 0.000 0.000 0.000 0.000 0.050 0.064 0.030 6.50 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.206 0.0971.55 0.000 0.000 0.000 0.000 0.000 0.000 0.051 0.068 0.032 6.55 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.206 0.0971.60 0.000 0.000 0.000 0.000 0.000 0.000 0.052 0.072 0.034 6.60 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.206 0.0971.65 0.000 0.000 0.000 0.000 0.000 0.000 0.054 0.075 0.035 6.65 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.207 0.0971.70 0.000 0.000 0.000 0.000 0.000 0.000 0.055 0.079 0.037 6.70 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.207 0.0971.75 0.000 0.000 0.000 0.000 0.000 0.000 0.055 0.082 0.038 6.75 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.207 0.0971.80 0.000 0.000 0.000 0.000 0.000 0.000 0.056 0.085 0.040 6.80 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.207 0.0971.85 0.000 0.000 0.000 0.000 0.000 0.000 0.057 0.089 0.042 6.85 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.207 0.0971.90 0.000 0.000 0.000 0.000 0.000 0.000 0.058 0.092 0.043 6.90 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.208 0.0971.95 0.000 0.000 0.000 0.000 0.000 0.000 0.059 0.095 0.045 6.95 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.208 0.0972.00 0.000 0.000 0.000 0.000 0.000 0.000 0.059 0.099 0.046 7.00 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.208 0.0982.05 0.000 0.000 0.000 0.000 0.000 0.000 0.060 0.102 0.048 7.05 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.208 0.0982.10 0.000 0.000 0.000 0.000 0.000 0.000 0.061 0.105 0.049 7.10 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.208 0.0982.15 0.000 0.000 0.000 0.000 0.000 0.000 0.061 0.108 0.051 7.15 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.209 0.0982.20 0.000 0.000 0.000 0.000 0.000 0.000 0.062 0.111 0.052 7.20 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.209 0.0982.25 0.000 0.000 0.000 0.000 0.000 0.000 0.062 0.114 0.053 7.25 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.209 0.0982.30 0.000 0.000 0.000 0.000 0.000 0.000 0.063 0.117 0.055 7.30 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.209 0.0982.35 0.000 0.000 0.000 0.000 0.000 0.000 0.063 0.119 0.056 7.35 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.209 0.0982.40 0.000 0.000 0.000 0.000 0.000 0.000 0.063 0.122 0.057 7.40 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.209 0.0982.45 0.000 0.000 0.000 0.000 0.000 0.000 0.064 0.125 0.058 7.45 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.209 0.0982.50 0.000 0.000 0.000 0.000 0.000 0.000 0.064 0.127 0.060 7.50 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.210 0.0982.55 0.000 0.000 0.000 0.000 0.000 0.000 0.064 0.130 0.061 7.55 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.210 0.0982.60 0.000 0.000 0.000 0.000 0.000 0.000 0.065 0.132 0.062 7.60 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.210 0.0982.65 0.000 0.000 0.000 0.000 0.000 0.000 0.065 0.135 0.063 7.65 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.210 0.098

2.70 0.000 0.000 0.000 0.000 0.000 0.000 0.065 0.137 0.064 7.70 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.210 0.099

2.75 0.000 0.000 0.000 0.000 0.000 0.000 0.066 0.139 0.065 7.75 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.210 0.0992.80 0.000 0.000 0.000 0.000 0.000 0.000 0.066 0.141 0.066 7.80 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.210 0.0992.85 0.000 0.000 0.000 0.000 0.000 0.000 0.066 0.143 0.067 7.85 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.210 0.0992.90 0.000 0.000 0.000 0.000 0.000 0.000 0.066 0.145 0.068 7.90 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.211 0.0992.95 0.000 0.000 0.000 0.000 0.000 0.000 0.066 0.148 0.069 7.95 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.211 0.0993.00 0.000 0.000 0.000 0.000 0.000 0.000 0.067 0.149 0.070 8.00 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.211 0.0993.05 0.000 0.000 0.000 0.000 0.000 0.000 0.067 0.151 0.071 8.05 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.211 0.0993.10 0.000 0.000 0.000 0.000 0.000 0.000 0.067 0.153 0.072 8.10 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.211 0.0993.15 0.000 0.000 0.000 0.000 0.000 0.000 0.067 0.155 0.073 8.15 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.211 0.0993.20 0.000 0.000 0.000 0.000 0.000 0.000 0.067 0.157 0.074 8.20 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.211 0.0993.25 0.000 0.000 0.000 0.000 0.000 0.000 0.067 0.158 0.074 8.25 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.211 0.0993.30 0.000 0.000 0.000 0.000 0.000 0.000 0.068 0.160 0.075 8.30 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.211 0.0993.35 0.000 0.000 0.000 0.000 0.000 0.000 0.068 0.162 0.076 8.35 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.211 0.0993.40 0.000 0.000 0.000 0.000 0.000 0.000 0.068 0.163 0.077 8.40 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.211 0.0993.45 0.000 0.000 0.000 0.000 0.000 0.000 0.068 0.165 0.077 8.45 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.212 0.0993.50 0.000 0.000 0.000 0.000 0.000 0.000 0.068 0.166 0.078 8.50 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.212 0.0993.55 0.000 0.000 0.000 0.000 0.000 0.000 0.068 0.168 0.079 8.55 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.212 0.0993.60 0.000 0.000 0.000 0.000 0.000 0.000 0.068 0.169 0.079 8.60 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.212 0.0993.65 0.000 0.000 0.000 0.000 0.000 0.000 0.068 0.170 0.080 8.65 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.212 0.0993.70 0.000 0.000 0.000 0.000 0.000 0.000 0.068 0.172 0.080 8.70 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.212 0.0993.75 0.000 0.000 0.000 0.000 0.000 0.000 0.068 0.173 0.081 8.75 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.212 0.0993.80 0.000 0.000 0.000 0.000 0.000 0.000 0.068 0.174 0.082 8.80 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.212 0.0993.85 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.175 0.082 8.85 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.212 0.0993.90 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.176 0.083 8.90 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.212 0.1003.95 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.178 0.083 8.95 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.212 0.1004.00 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.179 0.084 9.00 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.212 0.1004.05 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.180 0.084 9.05 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.212 0.1004.10 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.181 0.085 9.10 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.212 0.1004.15 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.182 0.085 9.15 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.20 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.183 0.086 9.20 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.25 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.184 0.086 9.25 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.30 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.184 0.087 9.30 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.35 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.185 0.087 9.35 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.40 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.186 0.087 9.40 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.45 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.187 0.088 9.45 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.50 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.188 0.088 9.50 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.55 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.189 0.088 9.55 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.60 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.189 0.089 9.60 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.65 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.190 0.089 9.65 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.70 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.191 0.089 9.70 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.75 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.191 0.090 9.75 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.80 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.192 0.090 9.80 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.85 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.193 0.090 9.85 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.90 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.193 0.091 9.90 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1004.95 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.194 0.091 9.95 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.1005.00 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.195 0.091 10.00 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.213 0.100

EDP-DV Function

PFA PFA







Occupancy: Office

Floor Type: Typical Floor




0.00

0.05

0.10

0.15

0.20

0.25

0.0 2.0 4.0 6.0 8.0 10.0

PFA [g]

E( L | PFA )

Finishes

Mechanical

Electrical

0.00

0.05

0.10

0.15

0.20

0.25

0.0 1.0 2.0 3.0 4.0 5.0

PFA [g]

E( L | PFA )

Finishes

Mechanical

Electrical



Protection

Doors, Windows,

GlassFinishes

Mech-anical


Protection

Doors, Windows,

GlassFinishes

Mech-anical

Electrical

0.05 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 5.05 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.202 0.0950.10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 5.10 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.202 0.095

0.15 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 5.15 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.203 0.095

0.20 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.000 5.20 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.203 0.095

0.25 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.001 0.000 5.25 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.204 0.096

0.30 0.000 0.000 0.000 0.000 0.000 0.000 0.005 0.001 0.001 5.30 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.204 0.096

0.35 0.000 0.000 0.000 0.000 0.000 0.000 0.007 0.002 0.001 5.35 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.205 0.096

0.40 0.000 0.000 0.000 0.000 0.000 0.000 0.010 0.003 0.001 5.40 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.205 0.096

0.45 0.000 0.000 0.000 0.000 0.000 0.000 0.012 0.004 0.002 5.45 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.206 0.096

0.50 0.000 0.000 0.000 0.000 0.000 0.000 0.014 0.005 0.003 5.50 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.206 0.097

0.55 0.000 0.000 0.000 0.000 0.000 0.000 0.017 0.007 0.003 5.55 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.207 0.0970.60 0.000 0.000 0.000 0.000 0.000 0.000 0.019 0.009 0.004 5.60 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.207 0.0970.65 0.000 0.000 0.000 0.000 0.000 0.000 0.021 0.011 0.005 5.65 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.207 0.0970.70 0.000 0.000 0.000 0.000 0.000 0.000 0.024 0.013 0.006 5.70 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.208 0.097

0.75 0.000 0.000 0.000 0.000 0.000 0.000 0.026 0.015 0.007 5.75 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.208 0.098

0.80 0.000 0.000 0.000 0.000 0.000 0.000 0.029 0.018 0.008 5.80 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.209 0.0980.85 0.000 0.000 0.000 0.000 0.000 0.000 0.031 0.021 0.010 5.85 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.209 0.0980.90 0.000 0.000 0.000 0.000 0.000 0.000 0.034 0.024 0.011 5.90 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.209 0.0980.95 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.027 0.013 5.95 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.210 0.0981.00 0.000 0.000 0.000 0.000 0.000 0.000 0.038 0.030 0.014 6.00 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.210 0.0981.05 0.000 0.000 0.000 0.000 0.000 0.000 0.041 0.034 0.016 6.05 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.210 0.0991.10 0.000 0.000 0.000 0.000 0.000 0.000 0.043 0.037 0.017 6.10 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.211 0.0991.15 0.000 0.000 0.000 0.000 0.000 0.000 0.045 0.041 0.019 6.15 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.211 0.0991.20 0.000 0.000 0.000 0.000 0.000 0.000 0.047 0.044 0.021 6.20 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.211 0.0991.25 0.000 0.000 0.000 0.000 0.000 0.000 0.049 0.048 0.022 6.25 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.211 0.0991.30 0.000 0.000 0.000 0.000 0.000 0.000 0.050 0.052 0.024 6.30 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.212 0.0991.35 0.000 0.000 0.000 0.000 0.000 0.000 0.052 0.055 0.026 6.35 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.212 0.0991.40 0.000 0.000 0.000 0.000 0.000 0.000 0.054 0.059 0.028 6.40 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.212 0.1001.45 0.000 0.000 0.000 0.000 0.000 0.000 0.055 0.063 0.029 6.45 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.213 0.1001.50 0.000 0.000 0.000 0.000 0.000 0.000 0.057 0.067 0.031 6.50 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.213 0.1001.55 0.000 0.000 0.000 0.000 0.000 0.000 0.058 0.070 0.033 6.55 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.213 0.1001.60 0.000 0.000 0.000 0.000 0.000 0.000 0.059 0.074 0.035 6.60 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.213 0.1001.65 0.000 0.000 0.000 0.000 0.000 0.000 0.060 0.078 0.036 6.65 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.213 0.1001.70 0.000 0.000 0.000 0.000 0.000 0.000 0.061 0.081 0.038 6.70 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.214 0.1001.75 0.000 0.000 0.000 0.000 0.000 0.000 0.062 0.085 0.040 6.75 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.214 0.1001.80 0.000 0.000 0.000 0.000 0.000 0.000 0.063 0.088 0.041 6.80 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.214 0.1001.85 0.000 0.000 0.000 0.000 0.000 0.000 0.064 0.092 0.043 6.85 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.214 0.1011.90 0.000 0.000 0.000 0.000 0.000 0.000 0.065 0.095 0.045 6.90 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.215 0.1011.95 0.000 0.000 0.000 0.000 0.000 0.000 0.066 0.099 0.046 6.95 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.215 0.1012.00 0.000 0.000 0.000 0.000 0.000 0.000 0.067 0.102 0.048 7.00 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.215 0.1012.05 0.000 0.000 0.000 0.000 0.000 0.000 0.068 0.105 0.049 7.05 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.215 0.1012.10 0.000 0.000 0.000 0.000 0.000 0.000 0.068 0.108 0.051 7.10 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.215 0.1012.15 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.111 0.052 7.15 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.215 0.1012.20 0.000 0.000 0.000 0.000 0.000 0.000 0.069 0.115 0.054 7.20 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.216 0.1012.25 0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.117 0.055 7.25 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.216 0.1012.30 0.000 0.000 0.000 0.000 0.000 0.000 0.071 0.120 0.056 7.30 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.216 0.1012.35 0.000 0.000 0.000 0.000 0.000 0.000 0.071 0.123 0.058 7.35 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.216 0.1012.40 0.000 0.000 0.000 0.000 0.000 0.000 0.071 0.126 0.059 7.40 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.216 0.1012.45 0.000 0.000 0.000 0.000 0.000 0.000 0.072 0.129 0.060 7.45 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.216 0.1012.50 0.000 0.000 0.000 0.000 0.000 0.000 0.072 0.131 0.062 7.50 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.217 0.1022.55 0.000 0.000 0.000 0.000 0.000 0.000 0.073 0.134 0.063 7.55 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.217 0.1022.60 0.000 0.000 0.000 0.000 0.000 0.000 0.073 0.137 0.064 7.60 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.217 0.1022.65 0.000 0.000 0.000 0.000 0.000 0.000 0.073 0.139 0.065 7.65 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.217 0.102

2.70 0.000 0.000 0.000 0.000 0.000 0.000 0.074 0.141 0.066 7.70 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.217 0.102

2.75 0.000 0.000 0.000 0.000 0.000 0.000 0.074 0.144 0.067 7.75 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.217 0.1022.80 0.000 0.000 0.000 0.000 0.000 0.000 0.074 0.146 0.068 7.80 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.217 0.1022.85 0.000 0.000 0.000 0.000 0.000 0.000 0.074 0.148 0.069 7.85 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.217 0.1022.90 0.000 0.000 0.000 0.000 0.000 0.000 0.075 0.150 0.070 7.90 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.218 0.1022.95 0.000 0.000 0.000 0.000 0.000 0.000 0.075 0.152 0.071 7.95 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.218 0.1023.00 0.000 0.000 0.000 0.000 0.000 0.000 0.075 0.154 0.072 8.00 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.218 0.1023.05 0.000 0.000 0.000 0.000 0.000 0.000 0.075 0.156 0.073 8.05 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.218 0.1023.10 0.000 0.000 0.000 0.000 0.000 0.000 0.076 0.158 0.074 8.10 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.218 0.1023.15 0.000 0.000 0.000 0.000 0.000 0.000 0.076 0.160 0.075 8.15 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.218 0.1023.20 0.000 0.000 0.000 0.000 0.000 0.000 0.076 0.162 0.076 8.20 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.218 0.1023.25 0.000 0.000 0.000 0.000 0.000 0.000 0.076 0.164 0.077 8.25 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.218 0.1023.30 0.000 0.000 0.000 0.000 0.000 0.000 0.076 0.165 0.078 8.30 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.218 0.1023.35 0.000 0.000 0.000 0.000 0.000 0.000 0.076 0.167 0.078 8.35 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.218 0.1023.40 0.000 0.000 0.000 0.000 0.000 0.000 0.076 0.169 0.079 8.40 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.219 0.1023.45 0.000 0.000 0.000 0.000 0.000 0.000 0.077 0.170 0.080 8.45 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.219 0.1033.50 0.000 0.000 0.000 0.000 0.000 0.000 0.077 0.172 0.081 8.50 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.219 0.1033.55 0.000 0.000 0.000 0.000 0.000 0.000 0.077 0.173 0.081 8.55 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.219 0.1033.60 0.000 0.000 0.000 0.000 0.000 0.000 0.077 0.175 0.082 8.60 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.219 0.1033.65 0.000 0.000 0.000 0.000 0.000 0.000 0.077 0.176 0.083 8.65 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.219 0.1033.70 0.000 0.000 0.000 0.000 0.000 0.000 0.077 0.177 0.083 8.70 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.219 0.1033.75 0.000 0.000 0.000 0.000 0.000 0.000 0.077 0.179 0.084 8.75 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.219 0.1033.80 0.000 0.000 0.000 0.000 0.000 0.000 0.077 0.180 0.084 8.80 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.219 0.1033.85 0.000 0.000 0.000 0.000 0.000 0.000 0.077 0.181 0.085 8.85 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.219 0.1033.90 0.000 0.000 0.000 0.000 0.000 0.000 0.077 0.182 0.086 8.90 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.219 0.1033.95 0.000 0.000 0.000 0.000 0.000 0.000 0.077 0.183 0.086 8.95 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.219 0.1034.00 0.000 0.000 0.000 0.000 0.000 0.000 0.077 0.185 0.087 9.00 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.219 0.1034.05 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.186 0.087 9.05 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.219 0.1034.10 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.187 0.088 9.10 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.15 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.188 0.088 9.15 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.20 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.189 0.089 9.20 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.25 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.190 0.089 9.25 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.30 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.191 0.089 9.30 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.35 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.192 0.090 9.35 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.40 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.192 0.090 9.40 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.45 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.193 0.091 9.45 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.50 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.194 0.091 9.50 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.55 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.195 0.091 9.55 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.60 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.196 0.092 9.60 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.65 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.196 0.092 9.65 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.70 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.197 0.092 9.70 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.75 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.198 0.093 9.75 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.80 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.198 0.093 9.80 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.85 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.199 0.093 9.85 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.90 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.200 0.094 9.90 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1034.95 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.200 0.094 9.95 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.1035.00 0.000 0.000 0.000 0.000 0.000 0.000 0.078 0.201 0.094 10.00 0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.220 0.103

EDP-DV Function

PFA PFA







Occupancy: Office

Floor Type: Top Floor




0.00

0.05

0.10

0.15

0.20

0.25

0.0 2.0 4.0 6.0 8.0 10.0

PFA [g]

E( L | PFA )

Finishes

Mechanical

Electrical

0.00

0.05

0.10

0.15

0.20

0.25

0.0 1.0 2.0 3.0 4.0 5.0

PFA [g]

E( L | PFA )

Finishes

Mechanical

Electrical



Protection

Doors, Windows,

GlassFinishes

Mech-anical


Protection

Doors, Windows,

GlassFinishes

Mech-anical

Electrical

0.05 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 5.05 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.263 0.0970.10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 5.10 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.264 0.097

0.15 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 5.15 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.264 0.098

0.20 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.000 5.20 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.265 0.098

0.25 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.000 5.25 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.266 0.098

0.30 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.002 0.001 5.30 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.266 0.098

0.35 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.003 0.001 5.35 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.267 0.099

0.40 0.000 0.000 0.000 0.000 0.000 0.000 0.004 0.004 0.001 5.40 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.268 0.099

0.45 0.000 0.000 0.000 0.000 0.000 0.000 0.006 0.005 0.002 5.45 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.268 0.099

0.50 0.000 0.000 0.000 0.000 0.000 0.000 0.007 0.007 0.003 5.50 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.269 0.099

0.55 0.000 0.000 0.000 0.000 0.000 0.000 0.008 0.009 0.003 5.55 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.269 0.1000.60 0.000 0.000 0.000 0.000 0.000 0.000 0.009 0.011 0.004 5.60 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.270 0.1000.65 0.000 0.000 0.000 0.000 0.000 0.000 0.010 0.014 0.005 5.65 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.271 0.1000.70 0.000 0.000 0.000 0.000 0.000 0.000 0.011 0.017 0.006 5.70 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.271 0.100

0.75 0.000 0.000 0.000 0.000 0.000 0.000 0.012 0.020 0.007 5.75 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.272 0.100

0.80 0.000 0.000 0.000 0.000 0.000 0.000 0.014 0.023 0.009 5.80 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.272 0.1010.85 0.000 0.000 0.000 0.000 0.000 0.000 0.015 0.027 0.010 5.85 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.273 0.1010.90 0.000 0.000 0.000 0.000 0.000 0.000 0.016 0.031 0.011 5.90 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.273 0.1010.95 0.000 0.000 0.000 0.000 0.000 0.000 0.017 0.035 0.013 5.95 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.273 0.1011.00 0.000 0.000 0.000 0.000 0.000 0.000 0.018 0.039 0.015 6.00 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.274 0.1011.05 0.000 0.000 0.000 0.000 0.000 0.000 0.019 0.044 0.016 6.05 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.274 0.1011.10 0.000 0.000 0.000 0.000 0.000 0.000 0.020 0.048 0.018 6.10 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.275 0.1021.15 0.000 0.000 0.000 0.000 0.000 0.000 0.021 0.053 0.020 6.15 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.275 0.1021.20 0.000 0.000 0.000 0.000 0.000 0.000 0.022 0.058 0.021 6.20 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.276 0.1021.25 0.000 0.000 0.000 0.000 0.000 0.000 0.023 0.063 0.023 6.25 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.276 0.1021.30 0.000 0.000 0.000 0.000 0.000 0.000 0.024 0.067 0.025 6.30 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.276 0.1021.35 0.000 0.000 0.000 0.000 0.000 0.000 0.025 0.072 0.027 6.35 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.277 0.1021.40 0.000 0.000 0.000 0.000 0.000 0.000 0.025 0.077 0.028 6.40 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.277 0.1021.45 0.000 0.000 0.000 0.000 0.000 0.000 0.026 0.082 0.030 6.45 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.277 0.1021.50 0.000 0.000 0.000 0.000 0.000 0.000 0.027 0.087 0.032 6.50 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.278 0.1031.55 0.000 0.000 0.000 0.000 0.000 0.000 0.027 0.092 0.034 6.55 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.278 0.1031.60 0.000 0.000 0.000 0.000 0.000 0.000 0.028 0.096 0.036 6.60 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.278 0.1031.65 0.000 0.000 0.000 0.000 0.000 0.000 0.028 0.101 0.037 6.65 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.279 0.1031.70 0.000 0.000 0.000 0.000 0.000 0.000 0.029 0.106 0.039 6.70 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.279 0.1031.75 0.000 0.000 0.000 0.000 0.000 0.000 0.029 0.111 0.041 6.75 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.279 0.1031.80 0.000 0.000 0.000 0.000 0.000 0.000 0.030 0.115 0.043 6.80 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.279 0.1031.85 0.000 0.000 0.000 0.000 0.000 0.000 0.030 0.120 0.044 6.85 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.280 0.1031.90 0.000 0.000 0.000 0.000 0.000 0.000 0.031 0.124 0.046 6.90 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.280 0.1031.95 0.000 0.000 0.000 0.000 0.000 0.000 0.031 0.129 0.048 6.95 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.280 0.1042.00 0.000 0.000 0.000 0.000 0.000 0.000 0.032 0.133 0.049 7.00 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.280 0.1042.05 0.000 0.000 0.000 0.000 0.000 0.000 0.032 0.137 0.051 7.05 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.281 0.1042.10 0.000 0.000 0.000 0.000 0.000 0.000 0.032 0.141 0.052 7.10 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.281 0.1042.15 0.000 0.000 0.000 0.000 0.000 0.000 0.032 0.145 0.054 7.15 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.281 0.1042.20 0.000 0.000 0.000 0.000 0.000 0.000 0.033 0.149 0.055 7.20 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.281 0.1042.25 0.000 0.000 0.000 0.000 0.000 0.000 0.033 0.153 0.057 7.25 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.281 0.1042.30 0.000 0.000 0.000 0.000 0.000 0.000 0.033 0.157 0.058 7.30 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.282 0.1042.35 0.000 0.000 0.000 0.000 0.000 0.000 0.033 0.161 0.059 7.35 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.282 0.1042.40 0.000 0.000 0.000 0.000 0.000 0.000 0.034 0.164 0.061 7.40 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.282 0.1042.45 0.000 0.000 0.000 0.000 0.000 0.000 0.034 0.168 0.062 7.45 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.282 0.1042.50 0.000 0.000 0.000 0.000 0.000 0.000 0.034 0.171 0.063 7.50 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.282 0.1042.55 0.000 0.000 0.000 0.000 0.000 0.000 0.034 0.175 0.065 7.55 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.283 0.1042.60 0.000 0.000 0.000 0.000 0.000 0.000 0.034 0.178 0.066 7.60 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.283 0.1052.65 0.000 0.000 0.000 0.000 0.000 0.000 0.035 0.181 0.067 7.65 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.283 0.105

2.70 0.000 0.000 0.000 0.000 0.000 0.000 0.035 0.184 0.068 7.70 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.283 0.105

2.75 0.000 0.000 0.000 0.000 0.000 0.000 0.035 0.187 0.069 7.75 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.283 0.1052.80 0.000 0.000 0.000 0.000 0.000 0.000 0.035 0.190 0.070 7.80 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.283 0.1052.85 0.000 0.000 0.000 0.000 0.000 0.000 0.035 0.193 0.071 7.85 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.284 0.1052.90 0.000 0.000 0.000 0.000 0.000 0.000 0.035 0.196 0.072 7.90 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.284 0.1052.95 0.000 0.000 0.000 0.000 0.000 0.000 0.035 0.199 0.073 7.95 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.284 0.1053.00 0.000 0.000 0.000 0.000 0.000 0.000 0.035 0.201 0.074 8.00 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.284 0.1053.05 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.204 0.075 8.05 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.284 0.1053.10 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.207 0.076 8.10 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.284 0.1053.15 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.209 0.077 8.15 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.284 0.1053.20 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.211 0.078 8.20 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.285 0.1053.25 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.214 0.079 8.25 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.285 0.1053.30 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.216 0.080 8.30 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.285 0.1053.35 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.218 0.081 8.35 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.285 0.1053.40 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.220 0.081 8.40 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.285 0.1053.45 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.222 0.082 8.45 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.285 0.1053.50 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.224 0.083 8.50 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.285 0.1053.55 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.226 0.084 8.55 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.285 0.1053.60 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.228 0.084 8.60 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.285 0.1053.65 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.230 0.085 8.65 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.286 0.1063.70 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.231 0.086 8.70 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.286 0.1063.75 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.233 0.086 8.75 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.286 0.1063.80 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.235 0.087 8.80 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.286 0.1063.85 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.236 0.087 8.85 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.286 0.1063.90 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.238 0.088 8.90 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.286 0.1063.95 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.239 0.088 8.95 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.286 0.1064.00 0.000 0.000 0.000 0.000 0.000 0.000 0.036 0.241 0.089 9.00 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.286 0.1064.05 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.242 0.090 9.05 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.286 0.1064.10 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.244 0.090 9.10 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.286 0.1064.15 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.245 0.091 9.15 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.286 0.1064.20 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.246 0.091 9.20 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1064.25 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.247 0.091 9.25 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1064.30 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.249 0.092 9.30 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1064.35 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.250 0.092 9.35 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1064.40 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.251 0.093 9.40 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1064.45 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.252 0.093 9.45 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1064.50 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.253 0.094 9.50 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1064.55 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.254 0.094 9.55 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1064.60 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.255 0.094 9.60 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1064.65 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.256 0.095 9.65 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1064.70 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.257 0.095 9.70 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1064.75 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.258 0.095 9.75 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1064.80 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.259 0.096 9.80 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1064.85 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.260 0.096 9.85 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1064.90 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.261 0.096 9.90 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1064.95 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.261 0.097 9.95 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.1065.00 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.262 0.097 10.00 0.000 0.000 0.000 0.000 0.000 0.000 0.037 0.287 0.106

EDP-DV Function

PFA PFA


building-specific loss estimation methods & tools for simplified ...

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