decision support for conceptual performance

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2006; 35:115–133Published online 17 October 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.536

Decision support for conceptual performance-based design

Helmut Krawinkler1;∗;†, Farzin Zareian1, Ricardo A. Medina2 and Luis F. Ibarra3

1Department of Civil and Environmental Engineering; Stanford University; Palo Alto; CA 94305; U.S.A.2Department of Civil and Environmental Engineering; University of Maryland; College Park;

MD 20742; U.S.A.3CNWRA; Southwest Research Institute; San Antonio; TX 78238; U.S.A.

SUMMARY

Performance assessment implies that the structural, non-structural, and content systems are given and thatdecision variables, DVs, (e.g. expected annual loss, mean annual frequency of collapse) are computedand compared to speci�ed performance targets. Performance-based design (PBD) is di�erent by virtueof the fact that the building and its components and systems �rst have to be created. Good designs arebased on concepts that incorporate performance targets up front in the conceptual design process, so thatsubsequent performance assessment becomes more of a veri�cation process of an e�cient design ratherthan a design improvement process that may require radical changes of the initial design concept. Inshort, the design approach could consist of (a) specifying performance targets (e.g. tolerable probabilityof collapse, acceptable dollar losses) and associated seismic hazards, and (b) deriving engineeringparameters for system selection, or perhaps better, using the relatively simple design decision supporttools discussed in this paper. Copyright ? 2005 John Wiley & Sons, Ltd.

KEY WORDS: performance-based design; conceptual design; performance assessment; collapse; losses;engineering demand parameters

1. INTRODUCTION

Performance assessment, as developed in recent Paci�c Earthquake Engineering Research(PEER) Center studies, implies that for a given system so-called decision variables, DVs,(dollar loss, length of downtime, or number of casualties) are determined whose values shouldful�ll speci�ed performance targets [1–4]. For instance, for life safety=collapse performance,the process of determining DVs is as summarized here: intensity measures, IMs, (e.g. spectralacceleration at the �rst mode period of the structure, Sa(T1)), are determined from seismic

∗Correspondence to: Helmut Krawinkler, Department of Civil and Environmental Engineering, Stanford University,Stanford, CA 94305-4020, U.S.A.

†E-mail: [email protected]

Contract=grant sponsor: Paci�c Earthquake Engineering Research Center

Received 17 February 2005Revised 13 May 2005

Copyright ? 2005 John Wiley & Sons, Ltd. Accepted 13 May 2005

116 H. KRAWINKLER ET AL.

hazard analysis; relevant engineering demand parameters, EDPs, (e.g. storey drifts) are pre-dicted from structural analysis for given values of IMs (and representative ground motions);local collapse fragility curves (e.g. for �oor slabs that may drop because of shear failure atcolumn-to-slab connection) and global collapse fragility curves of the type shown in Figure6 are developed to predict local and global collapse probabilities; and as a last (and notyet resolved) step, predictions are made of the DVs, i.e. the number of lives lost and thenumber of injuries. A similar process is followed for the assessment of direct ($)loss, withcollapse fragility curves replaced by fragility curves for speci�c damage measures (DMs) andassociated loss functions.The mathematical formulation for evaluating decision variables and providing decision sup-

port to the owner=user, considering uncertainties inherent in all parts of the process, is providedby the PEER framework equation expressed as follows:

�(DV)=∫∫∫

G〈DV|DM〉 dG〈DM|EDP〉 dG〈EDP|IM〉 d�(IM) (1)

where �(DV) is a desired realization of the DV, such as its mean annual frequency of ex-ceedance, and the Gs represent complementary cumulative distribution functions.Design is di�erent from performance assessment, by virtue of the fact that the building

and its structural components and system �rst have to be created. One can view design asan iterative assessment process that starts with a judgmental conceptual design for whichperformance assessment is carried out, and the design is improved (tuned) in successiveiterations until the performance targets are met. This design process is an option, but nota very attractive one. A poor initial conceptual design may be tuned to an extent that itful�lls the performance targets, but it likely will never become a good design. Good designsare based on concepts that incorporate performance targets up front in the design decisionprocess, so that subsequent performance assessment becomes more of a veri�cation processof an e�cient design rather than a design improvement process that may require radicalchanges.Conceptual design is greatly facilitated by focusing on discrete performance targets asso-

ciated with discrete hazard levels—similar to the way it is being practiced in most of theperformance-based guidelines presently in use. In the conceptual design phase, engineers areused (and likely will be so for many years to come) to select and rough-proportion structuralsystems for strength, sti�ness (drift limitations), ductility, and perhaps energy dissipation ca-pacity and �oor accelerations. The art of engineering, which should be practiced in this phase,is to use global information on important performance targets in order to come up with a struc-tural system that ful�lls speci�ed performance objectives in the most e�ective manner. Thisimplies exploration of design alternatives, which may be utilizing di�erent structural materialsand systems or advanced technologies such as base isolation or internal energy dissipationdevices.The challenge is to provide the designer with a small set of most relevant criteria for

important EDPs on which good conceptual design can be based. In concept, this meansreversing the information �ow discussed before for performance assessment, and workingtowards quanti�cation of limits for relevant EDPs, given that desired performance can beexpressed in terms of targeted DV values at discrete performance levels. These relevant EDPs,such as storey drifts, �oor accelerations (or velocities), and storey (or component) ductility

Copyright ? 2005 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2006; 35:115–133

DECISION SUPPORT FOR CONCEPTUAL PERFORMANCE-BASED DESIGN 117

demands relate to structural parameters (strength, sti�ness, ductility, etc.) that drive designdecisions on structural materials and systems.Given EDP limits and associated IM hazards for various performance levels, conceptual

design for multiple performance objectives can be carried out. In general, performance shouldbe concerned with structural and non-structural systems as well as building contents. There isno single design parameter that will control all performance goals at all performance levels.For instance, damage to non-structural components is controlled often by interstorey driftlimitations, which demand large sti�ness. Damage to building contents, on the other hand,is mostly proportional to �oor accelerations, which can be limited by reducing the sti�nessand=or strength of the structure. At the other extreme, life safety and collapse prevention arecontrolled by the inelastic deformation and energy dissipation capacities of ductile elementsand the strength capacity of brittle ones.This discussion indicates that di�erent performance objectives may impose con�icting

demands on strength and sti�ness, and that seismic design is likely to become an itera-tive process in which di�erent performance criteria may lead to trade-o�s between strengthand sti�ness requirements, but in which no compromise should be made on issues of lifesafety and collapse prevention. This iterative process can be accomplished in two phases; aconceptual design phase in which various e�ective structural systems are explored and rough-sized, and a performance assessment phase in which performance of the structural, non-structural, and content systems is evaluated and �nal design decisions and modi�cations aremade—with due consideration given to all important sources of uncertainty.This paper is concerned with the conceptual design phase. Two challenges need to be

addressed in the context of performance-based conceptual design. One is to develop data onEDP limits associated with speci�c performance targets. Once such EDP limits have beenestablished, together with IMs that represent discrete hazard levels for which the EDP limitsapply, the challenge is to create structural systems that e�ciently accommodate these EDPlimits. This paper is concerned with these two challenges.

2. DOMAINS THAT CONTROL PERFORMANCE-BASED DESIGN

From a macro-perspective, PBD comprises three domains; the hazard domain, the structuralsystem domain, and the loss domain, as illustrated in Figure 1. The hazard and loss domainsprovide design constraints, and the structural system domain contains all design alternativesworthy of exploring. In performance assessment, all three domains contain random variablesto be described by a central value and a measure of dispersion to account for epistemic andaleatory uncertainties. In conceptual PBD the objective is to search for e�ective solutions ata time at which the details of the structural system are yet to be determined. Therefore, itappears to be fair to focus on expected (mean) values of all random variables, and delay theuncertainty evaluation and propagation till the performance assessment phase. This facilitatesup-front design decision making by permitting a focus on the most important global behaviouraspects without having to deal with mathematical formulations that are important but obscurebehaviour-based decision making.In the following discussion the content of the three domains, seen from the perspective of

conceptual PBD, is summarized.



( )| &E EDP IM NC

Collapse

|P C IM

( )| &E Loss EDP NC ( )|E Loss C

No Collapse

CollapseNo Collapse

-Direct Loss = Loss$ -Down Time Loss = Loss -Life Loss (Death) = LossD

Mean Hazard Curve(s) for Design

Alternatives

Mean IM-EDP Curves for Design Alternatives

Mean Loss Curves Mean Loss due to Collapse

λ (IM)

Collapse Fragility Curves for Design Alternatives

Hazard DomainHazard Domain

( )| &E EDP IM NC

Collapse

( )P IM

Loss DomainLoss Domain

( )|E Loss EDP NC ( )|E Los

Structural System DomainStructural System Domain

No Collapse

CollapseNo Collapse

-Direct Loss = Loss$-Direct Loss = Loss$ -Down Time Loss = Loss-Down Time Loss = Loss -Life Loss (Death) = LossDD

Mean Hazard Curve(s) for Design

Alternatives

Mean IM-EDP Curves for Design Alternatives

Mean Loss Curves Mean Loss due to Collapse

(IM)

Collapse Fragility Curves for Design Alternatives

Figure 1. Domains controlling performance-based design.

2.1. Hazard domain

The hazard domain contains the return period dependent description of the ground motionintensity plus associated time-history records. The intensity measure could be a scalar or avector quantity [5], with the latter being of particular importance in the case of near-faultground motions. In the numerical example presented later the elastic spectral acceleration atthe �rst mode period of the structural system (Sa(T1)) is used as IM. For a scalar IM thedescription suitable in the context of conceptual PBD is the mean hazard curve �(IM). Itis important to note that the mean hazard curve changes with the �rst mode period of theexplored structural system.The records selected to represent the seismic input for speci�c IM values a�ect the structural

demand curves (mean IM-EDP curves) contained in the structural system domain discussed inSection 2.3. The associated issues of ground motion scaling and near-fault e�ects have beenand still are the subject of research [5–8] and are not discussed further in this paper.

2.2. Loss domain

It is widely accepted that earthquake design decisions should be based—colloquially—onthe three D’s, i.e. dollars, downtime, and deaths. Unfortunately, there are many ambigui-ties in this simple phrase. For one, we have not yet succeeded in quantifying deaths (andmaybe we do not want to quantify it). Thus, in many cases collapse is used as a surro-gate for deaths or casualties (life safety). Furthermore, we still have only vague insight intodowntime and the quanti�cation of associated losses, which may a�ect not only the ownerbut may have more global consequences [9–11]. Therefore, downtime is not discussed ex-plicitly here, even though it can be represented by the same concepts illustrated for direct($)loss.Up front it is necessary to divide losses into two sub-domains, one containing losses if no

collapse occurs (NC domain), and the other containing losses if collapse occurs (C domain).Both sub-domains contribute to the three D’s, as illustrated conceptually in Figure 1.



Drywall Partitions with Metal Frame Drywall Partitions with Metal Frame

0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.005 0.010 0.015 0.020 0.025 0.030

EDP (IDR)

0.000 0.005 0.010 0.015 0.020 0.025 0.030

EDP (IDR)

Tape, Paste & Paint

Replacement gypsumboards but not frame

Replacement of thewhole partition

P(D

M>

dm |

ED

P)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

E[L

oss

|ED

P]

(a) (b)

Figure 2. Damage and loss in a drywall partition: (a) damage state fragility curves [12]; and(b) expected ($)loss-EDP curve (modi�ed from Figure 5.9 in Reference [13]).

2.2.1. Direct ($)loss due to damage (no collapse). Losses may occur in any of the subsys-tems of a building, and in general need to be aggregated over components and subsystems.A building could be divided into subsystems according to functional use (e.g. structural,non-structural, content) and according to the sensitivity of the subsystem components toengineering demand parameters, EDPs (e.g. interstorey drift, �oor acceleration). The objectiveis to establish, for each subsystem, a relationship between loss (repair or replacement cost)and a ‘most relevant’ engineering demand parameter (EDP), so that the latter can be used bythe engineer to guide design decisions. The EDP has to be well correlated with losses in allcomponents of the subsystem, and it has to be well correlated with global structural responsein order to permit deduction of global design decisions.The process of computing loss-EDP relationships requires the following ingredients and

steps:

1. The availability of a set of fragility curves for each component of the subsystem, whichde�ne, as a function of the EDP, the probability of being in, or exceeding, speci�cdamage states requiring speci�c repair actions. Figure 2(a) shows a typical set of fragilitycurves for drywall partitions with metal frames [12], using interstorey drift ratio (IDR)as an EDP.

2. The availability of cost-of-repair functions for each damage state.3. The ability to integrate fragility and cost-of-repair functions for each component. In thecontext of conceptual design it appears quite acceptable to use expected cost of repairfor each damage state (E[Lj|DM=dmi]) and compute only expected loss as a functionof EDP, i.e.

E[Lj |EDPj= edp]=m∑i=1E[Lj|DM=dmi]P(DM=dmi|EDPj= edp) (2)

The result is an expected (mean) loss-EDP relationship for the speci�c component, asshown in Figure 2(b).

4. The summation of expected losses (as a function of EDP) over all the components of thesubsystem. The result will be a single mean loss-EDP curve for this subsystem (mean



loss curve in Figure 1, i.e. E(Loss|EDP&NC). The same process needs to be carriedout for all other subsystems, which will provide information on the contribution of theindividual subsystems to the total loss, given a value of EDP.

This process is straightforward, in concept, but at this time it is most di�cult or impossibleto implement, simply because of the lack of data. Mean loss-EDP curves for most individualcomponents and for subsystems are not available at this time. But work has started in thePEER Center on the development of mean loss curves, and it is hoped that the developmentof such curves will become a focus of future research. It is di�cult to see how PBD can beimplemented meaningfully without the availability of such curves.

2.2.2. Direct ($)loss due to collapse. Collapse also is a contributor to direct ($)loss. It maybe a major contributor for ‘non-conforming’ structures (e.g. older RC frame structures [14]).For ‘conforming’ structures (i.e. structures designed according to modern standards) most ofthe direct ($)loss comes from damage in relatively moderate but frequent events, rather thanfrom complete or partial collapse. Thus, for the purpose of ($)loss estimation it is assumedthat collapse causes total monetary loss, and its impact on expected loss is estimated as thistotal loss times the probability of collapse.

2.3. Structural system domain

The structural system domain contains information that relates, for selected design alternatives,engineering parameters with the hazard and loss domains.

2.3.1. No collapse sub-domain. As long as the structure does not collapse, the structuralsystem domain contains mean IM-EDP curves, with IM being the intensity measure employedin the hazard domain, and EDP being an engineering demand parameter that correlates wellwith the loss in one of the subsystems de�ned in the loss domain. The EDP could be a storeyparameter (if losses are evaluated on a storey-by-storey basis) or a global parameter suchas the average of the maximum storey drifts (or of maximum �oor accelerations) over theheight of the structure. In the conceptual design phase, mean values of EDPs are appropriatemeasures.Mean IM-EDP curves (E(EDP|IM&NC)) are obtained by subjecting structural systems of

speci�c properties to sets of ground motions representative of speci�c IM values. If it canbe assumed that the frequency content of the ground motions is insensitive to magnitude anddistance within the IM range of primary interest, then incremental dynamic analyses (IDAs)of the type illustrated in Figure 3 can be utilized to obtain mean IM-EDP curves. Such curveshave been developed and stored in a database for many EDPs for a wide range of momentresisting frames [7], and additional curves are under development for reinforced concrete wallstructures. In order to implement the proposed conceptual PBD process, it will be necessaryto have available mean IM-EDP curves for the range of design alternatives to be evaluated.Development of such curves is a relatively straightforward process for regular structures, anda more elaborate but also more critical e�ort for irregular structures.

2.3.2. Collapse sub-domain. This domain contains collapse fragility curves, which portray theprobability of collapse as a function of the intensity measure (P(C|IM)). Such curves can beobtained by subjecting deteriorating structural systems of speci�c properties to sets of ground



Average Interstory Drift Ratio IDA curvesN=9, T1=0.9, γ =0.3, ξ=0.05, Peak-Oriented model, BH, LMSR-N

0

0.5

1

1.5

2

0 0.01 0.02 0.03 0.04 0.05

Average Storey Drift, IDRavg.

Sa(

T1)/

g

Individual responses

Mean

Figure 3. Mean IM-EDP curve for a nine-storey frame with T =0:9 s obtainedfrom incremental dynamic analyses (IDAs).

Kc=α cKe

c y r

Residual Strength

Capping (Peak) Point

Ke

Hardening StiffnessPost-Capping Stiffness

Elastic Stiffness

Fy

Fc

F s sKe

Fr =

=α

Fyλ

K

δ δ δ δ

Figure 4. Backbone curve for deteriorating component hysteresis models [16].

motions representative of the range of IMs in which collapse is expected. If all componentdeterioration modes are adequately presented in the analytical model, it should be feasibleto analytically trace structures till collapse by incrementing the IM of the ground motionuntil a minute increment in IM leads to a very large increment in a global EDP, indicatingdynamic instability.Research has been performed recently on the ‘collapse capacity’ of moment resisting frames,

utilizing component hysteresis models that account for strength deterioration in the backbonecurve (see Figure 4) and for cyclic deterioration in strength and sti�ness [15]. The col-lapse capacity is de�ned as that value of the ‘relative intensity’, de�ned here as [Sa(T1)=g]=�(�= base shear strength coe�cient, i.e. the base shear strength of the system normalized by



0

2

4

6

8

10

[Sa(

T1)

/g]/

0

2

4

6

8

10

[Sa(

T1)

/g]/

0

2

4

6

8

10

[Sa(

T1)

/g]/

00000 10 20 30

2

4

6

8

10

[Sa(

T1)

/g]/

γ

αs=0.05, δ δ/ y=4, αcc =-0.10, γs,c,k,a=inf, LMSR-N

Individual responses

MAX. STOREY DUCTILITY vs. NORM.STRENGTH

Maximum Storey Ductility Over the Height, µs,max

N=9, T1=0.9, ξ=0.05, K1, S1, BH, θ=0.015, Peak-Oriented Model,

Figure 5. IDAs till collapse and distribution of collapse capacity [16].

0.2

0.4

0.6

0.8

1

5 10 150

0

[Sa(T1)/g]/γ vs PROBABILITY OF COLLAPSEN=9, T1=0.9, BH, Peak Oriented Model, LMSR-N, ξ=5%,

αs=0.03, δc/δy=Var, αc=Var, γs,c,k,a=Inf, λ=0

Prob

abili

ty o

f C

olla

pse

[Sa(T

1)/g]/γ

δc/δy=6, αc=-0.10

δc/δy=4, αc=-0.10

δc/δy=2, αc=-0.10

δc/δy=4, αc=-0.30

δc/δy=2, αc=-0.30

Figure 6. Collapse fragility curves for nine-storey frame structures with T1 = 0:9 s [16].

its seismic weight, Vy=W ) at which dynamic instability occurs due to deterioration and P−�e�ects. It is noted that [Sa(T1)=g]=� is equivalent to the ductility dependent strength reductionfactor R�. The probability distribution function, PDF (assuming a lognormal distribution) ofthe collapse capacity is obtained as illustrated in Figure 5, and the corresponding cumulativedistribution function, CDF, represents the collapse fragility curve, P(C|IM). Collapse fragilitycurves of the type shown in Figure 6 have been derived for regular frames subjected to a set



EDPλ (IM)E

($L

oss

| ED

P &

NC

)

Structural System DomainHazard Domain

Loss Domain

IMIM

EDP

Mean Hazard Curve(s) for

Design Alternatives

Mean IM-EDP Curvesfor Design Alternatives

Collapse Fragility Curvesfor Design Alternatives

Mean $Loss Curve(s) (No Collapse) Mean $Loss Value (Collapse)

( )E oss C

EDPE

($L

oss

| ED

P &

NC

)


Loss Domain

IMIM

EDP

Mean Hazard Curve(s) for

Design Alternatives



Mean $Loss Curve(s) (No Collapse) Mean $Loss Value (Collapse)

$E L ss |C

P (C | IM)

Figure 7. Conceptual PBD for acceptable ($)loss at discrete hazard levels.

of 40 ground motions [16]. In Reference [16] it has been concluded that the collapse fragilitydepends primarily on the component ductility capacity �c=�y (which is assumed to be the samefor all components in the structure), the post-capping sti�ness ratio �c (see Figure 4), and thecyclic deterioration parameter �s;c;k;a. These parameters, together with the fundamental periodT1 and the base shear strength parameter �=Vy=W , control the design for collapse safety.

3. DECISION SUPPORT FOR CONCEPTUAL DESIGN

In the up-front conceptual design process, decisions have to be made on the type of structuralsystem and its global strength, sti�ness, and ductility properties. The information contained inthe three domains discussed before can be utilized to provide much needed support for thisdesign decision process.

3.1. Design decisions derived from performance targets for direct ($)loss

The conceptual process of making design decisions based on acceptable direct ($)loss isillustrated in Figure 7. One way of expressing desired performance is to specify acceptablelosses at speci�c hazard levels, such as the 50% probability of exceedance in 50 years (50=50)hazard. As discussed previously, losses may occur in several subsystems that are sensitive todi�erent EDPs. Losses in di�erent subsystems could be assessed simultaneously, or the focuscould be placed on the subsystem that contributes most to the value of the building, withother subsystems being evaluated subsequently. The latter approach is illustrated in Figure 7.The lower left portion of the �gure illustrates the mean ($)loss-EDP curve for the domi-

nant subsystem if no collapse occurs, E($Loss|EDP&NC), and the lower right portion shows



the expected loss if collapse occurs, E($Loss|C). The upper left portion shows the mean IMhazard curve for the speci�c site (usually the elastic spectral acceleration at the estimated�rst mode period of the structure is used as IM). The upper central portion shows meanIM-EDP relationships for several design alternatives, and the upper right portion shows col-lapse fragility curves for the same design alternatives. The process is to enter the lowergraph with a value of acceptable loss and obtain the associated EDP, and to enter the leftupper portion with the hazard level at which the loss is acceptable and obtain the associ-ated IM. The intersection of the IM value and the EDP value in the graph presenting meanIM-EDP curves for design alternatives can be viewed as a ‘design target’. All design alter-natives that intersect the IM line to the left of the design target are ‘feasible’ solutions, i.e.the associated expected losses are smaller than the target acceptable loss. The ‘best’ solutionwill depend on many considerations, some of them being discussed in the example given inSection 4.1.The feasibility of a design alternative depends also on the potential ($)loss caused by

collapse of the structure. If the IM line associated with the hazard level of interest is extendedto the right, its intersection with the collapse fragility curves will provide insight into thecollapse probability of the design alternatives, P(C|IM). The total expected loss can then beexpressed as

E($Loss|IM)=E($Loss|IM&NC)× P(NC|IM) + E($Loss|C)× P(C|IM) (3)

with P(NC|IM)=1 − P(C|IM) being the probability that collapse will not occur. SinceE($Loss|C) is very large, it is prudent, from an economic perspective, to eliminate designalternatives that have a noticeable probability of collapse at the hazard levels of primaryinterest for ($)loss control.The approach here illustrated can be extended, if so desired, to estimate expected an-

nual loss rather than focus on a speci�c hazard level. For each design alternative the meanIM-EDP curves can be computed for the full range of hazards signi�cantly a�ecting the di-rect ($)loss. Thus, the expected ($)loss can be evaluated for several discrete hazard levels,and the expected annual loss can be computed by numerical integration over the hazardcurve, i.e.

E[$Loss] =∫IME[$Loss|im ]|d� IM(im)| (4)

This is a rather powerful extension as it permits, for selected design alternatives, direct com-parison of expected annual loss for the full range of hazards, rather than committing tospeci�c hazard levels. This approach is more in line with the presently employed PEER per-formance assessment methodology, which considers decision variables (losses, etc.) as contin-uous variables rather than parameters associated with discrete hazard levels. This approach isparticularly attractive for a building system that has ‘competing’ subsystems, i.e. subsystemsof signi�cant value but with di�erent ‘relevant’ EDPs, such as drift and acceleration sensitivesubsystems. In this case an integrated loss measure appears to be much more objective thana measure associated with one speci�c hazard level.There are many questions to be answered, with a few of them addressed in general next,

and a few more addressed speci�cally in the example illustrated later.What is an appropriate EDP? The choice of the EDP is driven by the sensitivity of the

loss to variation in a basic design parameter. For a subsystem that is sensitive to interstorey



drift, and if loss-based design is performed for the full building, the best choice is believed tobe the average of the maximum drifts over the height, which is a measure of the ‘expected’maximum drift. It is larger than the global drift, because maximum storey drifts occur atdi�erent times [7]. If loss-based design is performed on a storey-by-storey basis, then themaximum interstorey drift for the storey under consideration is an appropriate EDP.What are the design alternatives? In concept, design alternatives incorporate all cost-

e�ective structural systems and materials. They also may include systems utilizing innovativetechnologies. The e�ort is in developing mean IM-EDP relationships for all systems. The de-sign variables could be base shear strength Vy, sti�ness or strength variation over the height,or �rst mode period (if T1 is varied, the hazard curve changes accordingly). For regular framestructures many such IM-EDP curves are documented in Reference [7], and more are underdevelopment. Similar curves are being developed for wall structures.Why bother with MDOF IM-EDP curves rather than use approximate SDOF represen-

tations? In the writers’ opinion the usefulness of SDOF representations in seismic design isoverestimated. Their use invites approximations that are justi�ed in some cases but not inothers. Why not avoid their use if appropriate MDOF information is available? This, in fact,is a necessity when the EDP is �oor acceleration, which is an EDP that correlates poorlywith any SDOF parameter.

3.2. Design decisions derived from performance targets for life safety

In most codes and guidelines, it is assumed that adequate collapse safety (and life safety) isprovided by limiting the maximum storey drift at the design earthquake level to a speci�cvalue (e.g. a drift limit of 0.02 at the 10=50 hazard level). The drift at this hazard level isestimated from either an elastic analysis or an inelastic time history analysis. But the latterusually is executed with the use of component hysteresis models that do not account forstrength and sti�ness deterioration. Thus, these EDP predictions provide no insight into theprobability of collapse. With the advent of deterioration models that do account for importantaspects of deterioration it is becoming possible to trace the response of structures to collapse(e.g. [15–18]) and to be speci�c about a collapse performance target. Such a target could beexpressed as a tolerable probability of collapse (say, 10% at the 2=50 hazard level), or moregeneral, as a tolerable mean annual frequency of collapse.Both options can be pursued in the proposed approach. But in both options it is understood

that collapse itself is not the most relevant performance target. Of primary societal interestis to provide adequate life safety, which could be translated into a tolerable loss of life (orcasualties). Figure 8 illustrates the PBD process for life safety, congruent with that for $loss,but with many curves dashed. The dashed curves imply that parts of the process still have tobe developed. There is little doubt that casualties may occur even if the structure does notcollapse (e.g. falling objects hazards), but the associated mean life loss curves and IM-EDPcurves have yet to be developed. Equally, even if collapse occurs, the relationship betweencollapse (in all its various forms) and the expected loss of lives (or casualties) still has to bedetermined. Some information is available on this issue [12], but much more research needs tobe done to establish such relationships. Thus, from the components of the process illustratedin Figure 8, and expressed by the following equation:

E(DDDDD Loss|IM) = E(DDDDD Loss|IM&NC)× P(NC|IM) + E(DDDDD Loss|C)× P(C|IM) (5)



λ (IM) EDP


Loss Domain

IM IM

EDP

E(

Los

s| E

DP

& N

C)

D

|E ss CD


DMean Loss Curve(s) (No Collapse) D Loss (Collapse)Mean


Mean HazardCurve(s) for

Design Alternatives

EDP


Loss Domain

IM IM

EDP

E(

Los

s| E

DP

& N

C)

DD

|E ss CD |oL ss CD


DMean Loss Curve(s) (No Collapse)DMean DDMean Loss Curve(s) (No Collapse) D Loss (Collapse)Mean D Loss (Collapse)DD Loss (Collapse)Mean


Mean HazardCurve(s) for

Design Alternatives

P (C | IM)

( )

Figure 8. Conceptual PBD for life safety at discrete hazard levels.

only the probability of collapse given IM, P(C|IM), can be evaluated with somecon�dence [16]. Even in that context, much more research needs to be performed to assessvarious local collapse modes, partial collapses, and progressive collapse.Thus, the desirable process is as shown in Figure 8, but at this time it only appears to

be feasible to express a life safety performance target in terms of a tolerable probability ofcollapse, or more general, as a tolerable mean annual frequency of collapse. For the former,the centre portion of Figure 8 is to be skipped, and the intersection of the line denotingthe IM value at the speci�ed hazard level with the line denoting the tolerable probability ofcollapse divides the design alternatives into a feasible and an unfeasible solution space. Asan additional feature, the mean annual frequency of collapse can be computed by integratingthe collapse fragility curve with the hazard curve, i.e.

� c =∫IMP(C|im)|d� IM(im)| (6)

Both approaches are illustrated in the following PBD examples.

4. EXAMPLE IMPLEMENTATIONS OF DECISION SUPPORT FOR PBD

The following examples attempt to illustrate the implementation of the concepts discussedso far. It soon will become evident that parts of the implementation are hypothetical, simplybecause some or much of the data needed for full implementation is missing. We hope thatthis does not discourage the potential utilization of the process, but to the contrary, encouragesengineers and researchers to give the process some thought and a chance for implementation,which will lead to the creation of the needed information through research and development.



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( )( )

Figure 9. Example of conceptual PBD for acceptable ($)loss at discrete hazard levels.

4.1. Example of design for acceptable direct ($)loss

An example of design decisions based on acceptable direct ($)loss is illustrated in Figure 9.The example addresses a nine-storey o�ce building, located in Southern California at a sitefor which the spectral acceleration hazard curves for the periods of 0.9 and 1.8 s are asshown in the upper left portion of the �gure. It was decided to use reinforced concrete as



the primary structural material. Design alternatives comprise moment frame, wall, and dualwall-frame systems. Only moment frame alternatives are evaluated here because of spaceconstraints in the �gure (and because mean IM-EDP curves for wall and wall-frame systemshave not yet fully matured). Again because of space constraints, only moment frames withT1 = 0:9 and 1.8 s and with a base shear coe�cient �=Vy=W =0:1, 0.2, and 0.3 are consideredas alternatives.It is assumed that for ($)loss estimation purposes the building can be divided into three

subsystems, a structural subsystem (SS), a non-structural drift sensitive subsystem (NSDSS),and a non-structural and content subsystem that is sensitive to �oor acceleration (NSASS).(The subsystem classi�cation may vary with the use of the structure, e.g. there also may be a�oor velocity sensitive subsystem.) It is assumed that the NSDSS and NSASS subsystems areknown and can be quanti�ed before structural design decisions have to be made, i.e. their valueis essentially independent of the structural system (this is not a necessary assumption). Thecost of the SS subsystem is design dependent, but it usually is a relatively small contributor tototal investment (in a loss estimation study on a Californian hotel building the SS contributedless than 20% to the total investment [12, 14]).The mean loss-EDP curves for the three subsystems are as shown in the lower portion.

The EDP for the NSDSS subsystem is the average of the maximum interstorey drifts overthe height (IDRavg), the EDP for the NSASS subsystem is the average of maximum �ooraccelerations over the height (FAavg), and the EDP for the SS subsystem is the average ofthe maximum storey ductility ratios over the height (�avg). (The implication is that the lossevaluation is done for the full building and not on a storey-by-storey basis.) At this time thereis little hard data behind these loss-EDP curves; they are based on judgement. Noteworthyis the jump in the mean loss-�avg curve for the SS subsystem from a relatively small valueto the value of total loss (usually the replacement cost in present dollars). This jump occurswhen the owner decides to demolish even though the structure has not collapsed. Jumps alsomay be present in other loss-EDP curves if large losses are associated with the attainment ofspeci�c EDP values.Because di�erent EDPs control the loss in the three subsystems, Figure 9 shows three

di�erent sets of mean IM-EDP curves for the same design alternatives. These are shown inthe centre portion of the upper half of the �gure. Based on relative monetary value of thesubsystems, in this example ‘designing for acceptable ($)loss’ needs to focus on the NSDSSsubsystem. An owner could target an acceptable loss in the NSDSS subsystem of about 0.5million dollars at the 50=50 hazard level. Thus, design targets are created by entering the �gurewith the EDP associated with the target acceptable loss and with the 50=50 mean Sa hazard (inthis example Sa(T1) is used as IM) for appropriate periods. For illustration, periods of 0.9 and1:8 s are selected. The upper central portion of the �gure contains mean Sa(T1)-EDP curvesfor various design alternatives (obtained from statistical studies reported in Reference [7]).From the left of the three IM-EDP graphs it is evident that any solution with T1 = 1:8 s wouldcause losses in the NSDSS subsystem that by far exceed the targeted acceptable losses at the50=50 hazard level. In fact, the three T1 = 0:9 s design solutions barely meet the loss target,i.e. solutions with T1¿0:9 s are discarded. In the three presented T1 = 0:9 solutions the baseshear strength coe�cient �=Vy=W is varied from 0.3 to 0.2 and 0.1. (The fact that the threesolutions overlap for a sizeable range is a consequence of adherence to the equal displacementrule.) Only the �=0:3 solution is attractive because for solutions with smaller � values theaverage storey ductility �avg (third IM-EDP graph) becomes clearly larger than 1.0, indicating



signi�cant structural damage. (Moreover, for �¡0:3 the right-most collapse fragility curvesindicate measurable collapse probabilities at relatively small Sa(T1)=g values.)The merits of the T1 = 0:9 and �=0:3 solution can be assessed further by inspecting the

expected NSDSS, NSASS, and SS losses at other hazard levels, such as the 10=50 and 2=50levels, as illustrated in Figure 9. For the case illustrated, these losses appear to be acceptable.Moreover, the upper right-most graph shows that for the selected design alternative the prob-ability of collapse is negligible unless the IM becomes very large (very long return periodhazard). But the picture could change radically if the building were a museum (or a hospi-tal), in which case the NSASS mean ($)loss-EDP curve likely will be much steeper and maybe the dominant contributor to total loss. In this case, longer period structures may becomemore attractive (note that the hazard curve changes as the �rst mode period of the structurebecomes longer).The versatility of the proposed approach, which is best implemented in graphical form of

the kind illustrated in Figure 9, lies in the ability to evaluate various design alternatives, con-sidering simultaneously all important subsystems and various hazard levels of interest. Thevisualization aspect permits a behaviour assessment that should be of much value to engineerspracticing this approach. The approach would clearly point out bene�cial trade-o�s in strengthand sti�ness, based on the shape of loss curves for individual subsystems. It also providespreliminary insight into the collapse issue, although that one is usually controlled by col-lapse (life) safety issues discussed in the next section. The data provided in Figure 9 permitalso the evaluation of the expected annual loss according to Equation (4). For the T1 = 0:9 sand �=0:3 and 0.2 design alternatives, the expected annual ($)loss amounts to $44 000, and$51 000, respectively.As an additional advantage, the proposed approach provides a tool for assessing objectively

the costs and bene�ts of various innovative technologies such as base isolation or internalenergy dissipation devices. The costs of implementing the technology would show up in theSS costs, and the bene�ts would show up in the reduction of losses.

4.2. Example of design for a tolerable probability of collapse

Providing collapse safety implies adherence to capacity design concepts, and it implies de-sign for ductility. The latter is implicitly considered in present design approaches with thejudgmental response modi�cation (R) factor or behaviour (q) factor. These factors are tiedto component detailing (ductility) requirements, and in the design process they are used toreduce the strength design level to a fraction of the elastic demand associated with the spectralacceleration at the �rst mode period. To this date it is not known whether or not this R (orq) based design process provides a quanti�able, or for that matter even remotely consistent,factor of safety against collapse. Provided one can develop con�dence in the collapse fragilitycurves of the type illustrated in Figure 6, the process illustrated conceptually in Section 3.2can be utilized to perform designs that target a speci�c tolerable probability of collapse. Thisis illustrated next on the same example for which design for ($)loss has been illustrated.

4.2.1. Design for tolerable probability of collapse at a speci�c hazard level. Desired per-formance at the collapse prevention level could be expressed in terms of a tolerable prob-ability of collapse at a speci�ed hazard level, as for instance, a tolerable probabilityof collapse of 0.1 at the 2=50 hazard level. For the T1 = 0:9 s and �=0:3 structure, the



5

Figure 10. Collapse spectra for frames with beam hinges: (a) for 10% probability of collapse; and(b) for 50% probability of collapse [16].

collapse fragility curve in the right upper portion of Figure 9 indicates that the actual probabil-ity of collapse is greater than 0.1, rendering this solution undesirable for collapse safety. Onecould plot collapse fragility curves for stronger or more ductile structures, or, perhaps betterfrom the perspective of behaviour, take advantage of collapse spectra of the type shown inFigure 10 [16].These spectra show the relative intensity [Sa(T1)=g]=� associated with a certain proba-

bility of collapse (10% in Figure 10(a), and 50% in Figure 10(b)) for frame structureswith T1 = 0:1N (N =number of stories) and several combinations of system parameters (seeFigure 4). The spectra illustrate the e�ect of component ductility capacity (�c=�y) on thisintensity, assuming �c= − 0:1 and no cyclic deterioration (�s;c;k;a= in�nite). For a tolerableprobability of collapse of 10% in a 2=50 event, data of the type shown in Figure 10(a)provides the necessary design decision support (similar spectra are available for other com-binations of system parameters). For instance, if T1 is selected as 0.9 s and the componentductility capacity is 4.0, the [Sa(T1)=g]=� value for a 10% probability of collapse is 4.6, whichfor the 2=50 hazard of the example problem (Sa(0:9)=1:7g) results in a required base shearstrength coe�cient of �=1:7=4:6=0:37. Thus, collapse prevention would control the requiredstrength, unless a larger ductility capacity (better detailing) is utilized or a more �exible struc-ture is used. For instance, for T1 = 0:9 s and �c=�y=6, the [Sa(T1)=g]=� value is 5.4, whichwould result in �=1:7=5:4=0:31. Alternatively, a more �exible structure could be selected(albeit this would be a poor solution based on direct ($)loss, see Figure 9). For T1 = 1:8 s and�c=�y=4, the [Sa(T1)=g]=� value is 3.5, which for the 2=50 Sa value of 0:86g at 1.8 s resultsin a required base shear strength coe�cient of �=0:86=3:5=0:25.These are the kind of trade-o�s that can be evaluated through the use of collapse probability

spectra of the type shown in Figure 10, presuming that a tolerable probability of collapse isspeci�ed at a speci�c hazard level.

4.2.2. Design for tolerable mean annual frequency of collapse. A di�erent way to expressdesired collapse performance is to target a tolerable mean annual frequency (MAF) of collapse,�C. This performance target is more general (it permits the estimation of the probability of



collapse over an expected life time), but it is more di�cult to implement because the compu-tation of a MAF requires integration over the Sa hazard curve. An approximate implementationis possible by means of the simpli�ed closed form expression proposed in Reference [19],which estimates the MAF of collapse as follows:

�C =∫SaP(C|Sa)|d�Sa(Sa)|= �Sa(�C) exp(12 k2�2RC) (7)

The simpli�ed expression on the right-hand side contains the MAF of the spectral accelerationassociated with the 50% probability of collapse, �Sa(�C), and a term that accounts, in an ap-proximate manner, for the uncertainties inherent in the computation of the collapse ‘capacity’(the [Sa(T1)=g]=� value causing collapse). This term contains the slope of the hazard curveat the referenced spectral acceleration value, k, and the dispersion(s) in the collapse fragilitycurve, � (the standard deviation of the natural logarithm of the data if a log-normal dis-tribution is assumed for the probability of collapse given the spectral acceleration). In theexample illustrated here only record-to-record (RTR) variability is considered, which is ex-plicitly contained in the fragility curves shown in Figure 6 (these fragility curves are obtainedby using a deterministic structural model subjected to 40 ground motions). Thus, the term�RC in Equation (7) expresses the e�ect of RTR variability only (‘Randomness in collapseCapacity’), and is found to be on the order of 0.4 to 0.5 (except for long period structuresfor which it is smaller because of the dominance of P −� e�ects).As an example, let us target a tolerable mean annual frequency of collapse of 0.0002 (i.e.

a tolerable probability of collapse of approximately 0:0002×50=0:01 in a 50 year life span).This criterion could be used, together with [Sa(T1)=g]=� spectra for a 50% probability ofcollapse (see Figure 10(b)), to arrive at e�ective design solutions. Again using the exampleof the nine-storey frame structure, the following design alternatives could be explored. If aperiod of 0.9 s and a component ductility capacity of �c=�y=4 are targeted, then the median[Sa(T1)=g]=� value from Figure 10(b) is 7.7. For the site speci�c hazard curve the slopeof the Sa hazard curve in the neighbourhood of a MAF of 0.0001 to 0.0004 is about 2.2,and the �RC value is about 0.4 (from collapse fragility analyses [16]). Thus, from Equation(7), the MAF of the Sa associated with the 50% probability of collapse, �Sa(�C), is equal to(0:0002)= exp(0:5×2:22×0:42)=0:000136. From the Sa hazard curve for the site of the exampleproblem, the corresponding Sa is 2:8g, and the corresponding � value is 2:8=7:7=0:36. Again,this is a larger value than that obtained from ($)loss-based performance targets. Alternativesare to increase the component ductility capacity (if it is increased from 4 to 6, [Sa(T1)=g]=� is8.9, and for the same �Sa(�C) of 0.000136 the � value becomes 2:8=8:9=0:31), or to increasethe structure period. If, for instance, T1 is 1.8 s, the [Sa(T1)=g]=� value from Figure 10(b) is5.4 (for �c=�y=4), and, using the site speci�c k value of 2.4 for the T =1:8 s hazard curve,�Sa(�C) becomes 0.000126, the Sa value for this MAF is 1:4g, and the base shear strengthparameter � becomes 1:4=5:4=0:26.

5. CONCLUDING REMARKS

Conceptual PBD implies a decision process that leads to the selection of one or severale�ective design alternatives based on performance targets for acceptable losses and a tolerableprobability of collapse. In this context, three challenges have to be addressed. One is to develop



data on EDP limits associated with selected performance targets. For this purpose the needexists to develop mean ($)loss-EDP curves for building subsystems for various occupancy anduse categories, either for individual stories or for the building as a whole. Such curves wouldtake the sting out of elaborate probabilistic formulations that appear to make performance-based design so complex. The second challenge is to develop collapse fragility curves thatpermit relatively simple design for collapse safety. Such curves are already available forregular and soft storey moment resisting frame structures. The third challenge is to developmean IM-EDP relationships that permit rapid evaluation of various structural systems. Suchrelationships are available for moment resisting frame structures and are under developmentfor reinforced concrete shear wall structures.If these ingredients are available, e�ective conceptual PBD can be performed with the

semi-graphical process discussed in this paper. The proposed process permits design decisionmaking based on constraints imposed by multiple performance objectives, considering trade-o�s between di�erent structural system choices and between strength, sti�ness, and ductilitycharacteristics. The proposed process also permits a consistent evaluation of the costs andbene�ts derived from the use of innovative technologies, such as base isolation and internalenergy dissipation devices.

ACKNOWLEDGEMENTS

This research was carried out as part of a comprehensive e�ort at Stanford’s John A. Blume EarthquakeEngineering Center to develop basic concepts for PBEE and supporting data on seismic demands andcapacities. This e�ort is supported by the NSF sponsored Paci�c Earthquake Engineering Research(PEER) Center.

REFERENCES

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