a new analytical procedure for the derivation of displacement-based vulnerability curves for...

Engineering Structures 27 (2005) 397–409

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A newanalytical procedure for the derivation of displacement-basevulnerability curves for populations of RC structures

Tiziana Rossettoa,∗, Amr Elnashaib,1

aDepartment of Civil and Environmental Engineering, University College London, Gower Street, London, WC1E 6BT, United KingdombCivil and Environmental Engineering Department, University of Illinois at Urbana-Champaign, IL, USA

Received 12 July 2004; received in revised form 5 November 2004; accepted 5 November 2004Available online 7 January 2005

Abstract

A new procedure is proposed for the derivation of analytical displacement-based vulnerability curves for the seismic assessment opopulations of reinforced concretestructures. The methodology represents an optimum solution compromising between reliability andcomputational efficiency. Adaptive pushover analysis is employed within a capacity spectrum framework of assessment, to deterperformance of a population of building models for increasing ground motion intensity. The building model population is generatedsingle design through consideration of material parameter uncertainty, with design of experiment techniques used to optimise the psize. Uncertainty in ground motion is accounted for through the use of suites of accelerograms with characteristics that are repreof the hazard level associated with the performance level assessed in each vulnerability curve. The new homogeneous reinforced concdamage scale, which is experimentally calibrated to maximum inter-storey drift for different structural systems, is used to determinedamage state of the building at the performance point. The results of the assessments are used to construct response surfaces frodamage statistics forming the basis of the vulnerability curves are generatedthrough re-sampling. The proposed methodology is illustratedfor the case of low-rise, infilled RC frames with inadequate seismic provisions. The derived curves show good correlation with obsepost-earthquake damage statistics.© 2004 Elsevier Ltd. All rights reserved.

Keywords: Vulnerability curves; Capacity spectrum assessment; Adaptive pushover analysis

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1. Introduction

Vulnerability curves relate the probability of exceedenof multiple damage states to a parameter of ground moseverity, and can therefore be regarded as a graphicrepresentation of seismic risk. At any given ground motivalue, the vertical distance between adjacent damage stacurves represents the probability of a building being withthe lower of the two damage states considered. Incase of building populations, use of vulnerability curv

∗ Corresponding author. Lecturer in Civil Engineering. Tel.: +442076794488; fax: +44 2073800986.

E-mail address:[email protected] (T. Rossetto).1 Willett Professor of Engineering, Director of the Mid-America

Earthquake Center.

0141-0296/$ - see front matter © 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2004.11.002

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yields a prediction of the proportion of the exposestock in each damage state after an earthquake. Sevvulnerability relationships for reinforced concrete (RCbuildings have been proposed in the past which are baseanalytically simulated building damage statistics. No uniqmethodology exists for the derivation of these relationshiwith a variety of analysis techniques, structural idealisatioseismic hazard and damage models being used. Thfactors strongly influence the derived vulnerability curveshapes, and different choices have been seen to resin significant discrepancies between the seismic riassessments made by different authorities for the salocation, structure type and seismicity [1]. Regardless ofthese choices, all existing methods for analytical fragilifunction derivation are computationally very intensive, aslargenumber of analyses are required to fully represent t

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398 T. Rossetto, A. Elnashai / Engineering Structures 27 (2005) 397–409

Table 1Threshold ISDmax%values ofthe HRC damage scale for infilled RC frame structures

HRC damage state None Slight damage Light damage Moderate damage Extensive damage Partial collapse Collapse

ISDmax (%) 0.00 0.05 0.08 0.30 1.15 2.80 >4.36

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structural and ground motion uncertainties involved inseismic assessment of building populations. For examSinghal and Kiremidjian [2] carry out non-linear timehistory analyses on finite-element bare reinforced concframe models of varied material properties, usingartificial acceleration records, scaled to 40 seismic intensilevels (4000 analyses in total). The repetition of sucstudy for many different structure classes is impractdue to the time involved and specialised analysis torequired. In order to reduce the analysis time, exisanalytical fragility curve derivation methods commonmake compromises, either as regards the numbestructural variations and earthquake records used, oregards the accuracy of the structural modelling, anaand assessment technique. For example, Mosalam[3] adopt 800 earthquake records covering a wide raof intensities to test 200 material variations of thstructural model, but adopt a single-degree-of-freedsystem idealisation for the analyses. Compromises sucthese may affect the reliability of the final vulnerabilcurves. Therefore, within this paper a new displacembased procedure for the generation of vulnerability curfor RC building populations is proposed. The procedadopts a response surface methodology in the generof the population damage statistics, which allows botreduced number of analyses and a reliable representof the population response uncertainty to be achieThe consequent reduction in analysis time allows accumodels and analysis tools to be used. Furthermorecombined use of an adaptive pushover analysis and capspectrum method of assessment in the proposed cderivation procedure avoids the repetition of analyfor increasing ground motions and further reducescomputational effort.

Problems associated with the choice of parametersground motion and damage characterisation can be idenin almost all existing vulnerability relationships [4].The parameter chosen to represent ground motiothe construction of vulnerability curves must be brepresentative of the damage potential of earthquand easily quantifiable from knowledge of the earthqucharacteristics. Peak ground values or intensity vaare therefore unsuitable for this purpose. It is widrecognised that a closer relationship exists between obsdamage and structural deformations than applied fodue to the ability of the former to account for nonlinear structure behaviour. This observation is confirmeby Rossetto and Elnashai [4], where spectral displaceme(Sd(T)) is shown togive a better correlation to the dama

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observed in 99 populations of buildings after 19 worldwidearthquakes than spectral acceleration. In view of theseobservations, of the development of displacement-batechniques in earthquake engineering which have rendepossible the use of spectral displacement as a desparameter, and of the recent derivation of reliable attenuatrelationships for Sd(T) by Bommer et al. [5], elasticspectral displacement (at 5% critical damping,Sd5%(T)) ischosen to represent the seismic demand in the propovulnerability curves. The homogenised reinforced concre(HRC) damage scale presented in [4] is used to evaluate thestructural performance from the analyses and to definedamage limit states associated with the developed curThe scale is subdivided into seven damage states ranfrom “No-damage” to “Collapse”, each of which is clearlydefined in terms of the typical structural and non-structudamage expected in the four main types of RC structfound in Europe. The scale is experimentally calibratto the parameter of maximum inter-storey drift respons(ISDmax%) for the different structure types. An example othe ISDmax% valuesobtained for the class of infilled RCframes is presented inTable 1. As the damage state andground motion are assessed using measures of deformaand displacement, respectively, the proposed curvesappropriate for use in a displacement-based assessmframework.

2. Analytical curve generation methodology

The proposed methodology for vulnerability curvederivation prescribes the analysis of a population of Rbuildings subjected to a number of earthquake records wdistinct characteristics. It is thus able to account for the effof variability in seismic input and structural characteristion the damage statistics simulated for the building clas(“system”), and evaluate the associated uncertainty in thevulnerability prediction. The procedure is summarised iFig. 1 and may be regarded as consisting of four masteps. Step 1, “system definition”, consists in the selectand design of a single structure with material, configuratiand seismic resistance characteristics that are representativof the building class being assessed. Deviation in seismresistance of buildings within the class is considered throuthe analysis of a population of building models, generatefrom the general system design by varying its structuproperties. Step 2, “definition of ground motion inputinvolves the selection of suites of earthquake records foranalysis. Within the proposed methodology different suitof accelerograms are adopted in the derivation of each li

T. Rossetto, A. Elnashai / Engineering Structures 27 (2005) 397–409 399

Fig. 1. The flow chart of the proposed analytical vulnerability curve derivation method.


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state curve. These “performance-consistent” record suiteare selected in accordance with spectra that are characteristicof a seismic event with a return period, for which thstructural damage state defining the curve is acceptable or isexpected. Designs of experiment procedures are considein the selection of the population and earthquake recosuite sizes for the analyses. This is done in orderoptimise computational effort and to guarantee convergenof results. Step 3, the “model evaluation”, is carried ousing an innovative adaptive pushover analysis techniqwithin a capacity spectrum framework of assessmeAdaptive pushover analyses can account for the effectground motion characteristics on structural response arequire reduced computational effort compared to timhistory analyses. The maximum inter-storey drift responof each structure within the population is assessed,increasing intensities of ground motion, using a modifiecapacity spectrum method. This assessment method avthe repetition of analyses forincreasing ground motions, andfurther reduces the analysis number from several thousato a few hundred. Step 4, the “statistical processinof analysis results”, involves the definition of responssurfaces, relating the observed inter-storey drift responsethe structural property and ground motion parameter valueResponse surface equations are defined for each hazscenario, through separate consideration of the analystatistics resulting from each suite of performance-consistrecords. A re-sampling technique is adopted to generbuilding damage statistics from the response surfaces,a range of ground motion severities. Hence, vulnerabilicurves are plotted. The response surface equation usedevelop each damage state curve is selected accordinthe hazard level associated with the satisfaction of a desiperformance objective. The generated vulnerability curvmay therefore be defined as being “performance consisteUncertainty in the damage state prediction, and its variatiwith increasing ground motion intensity, is accounted forthrough vulnerability curve confidence bounds. These aredetermined from consideration of the fit of the analyticalobservations of maximum inter-storey drift to the responsurfaces, within the damage histogram generation process.

The choice of building design and material propertieassumed for the model are dependent on the composiof the building stock in the assessed region. The buildingmust be grouped into categories with similar lateral loaresistance and different vulnerability curves must be derivfor each building class. The overall risk to the populationfor an earthquake of given size may then be obtainedcombining the damage state exceedence probabilitieseach structure class according to their relative proportioin the assessed region. It is emphasised that any structtypology may be assessed within the framework of thproposed methodology through an appropriate selectof structural model, structural properties for variatioand their corresponding probability distributions, and theground motion input. However, in order to better illustrat

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the proposed analytical vulnerability curve derivationmethodology, in the following text each step of thprocedure is explained within an example applicationa population of low-rise infilled RC frames of typicaEuropean construction, which are designed to old seiscodes (not including capacity design concepts). Conclusiare then drawn as regards the ability of the resulting curto reproduce observational damage data.

2.1. System definition

A regular, three-storey infilled frame configurationchosen to represent the system (i.e. the low-rise infilRC frame structural class used as an example here).frame is designed according to the prescriptions for loadimaterial, member dimensioning and detailing of the seismand gravity load design codes in place in Italy in 1982. Thcode is chosen as being representative of the seismic deof the existing European building stock. The full designthe infilled frame is presented in [6] and the elevation, beamand column details are illustrated inFig. 2. The structureconsists of four frames with constant inter-storey heights abay widths of 3 m and 4.5 m, respectively. It is symmetricin plan and elevation, with the external and internal framdesigns differing due to changes in the design loads.intermediate seismic zone (Zone 2,S = 9), approximatelycorresponding to apga of 0.07 g (10% exceedence in50 years), is adopted in the design, and results in a debase shear of 8.4% of the structure weight. Concrete wa characteristic compressive strength (fck) of 30 MPa andFeB38 grade ribbed steel bars with characteristic yield str( fyk) of 380 MPa areused for the structure. It is observethat despite the inclusion of seismic actions in the desithe shear reinforcement in all columns andmembers isinsufficient for significant levels of section confinement.

The system is modelled using the Inelastic DynamAnalysisof Structure finite-element package, INDYAS [7].The program allows a 3D finite-element model of thbuilding to be constructed, wherein the distributionreinforcement within sections is modelled bar bybar andthe non-linear behaviour of materials is taken into accountThe seismic behaviour prediction accuracy and stabiof the INDYAS program has been proven by Pinho [8]and Rossetto [6], amongst others. Due to symmetry, onhalf the structure is modelled for the analysis. The sudivision of the frames into finite elements is carrieout considering the spread of plasticity in members. TMander et al. [9] model is used to represent the effecof concrete confinement. However, in the present cwhere members havenegligible confinement, the choicof confinement model may be regarded as an insignificsource of response uncertainty. A bi-linear elasto-plasmodel with kinematic strain hardening is used to represthe reinforcing steel behaviour. The infill panel responis modelled via inclined rectangular struts that actcompression only. The stress–strain curve proposedPanagiotakos and Fardis [10] is modified to account for the


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(b) Internal frame beam and column sections.

(c) External frame beam and column sections.

Fig. 2. Drawings of the low-rise infilled RC frame designed to the 1982Italian seismic code.

effect of openings using the equations of Mosalam [11],and used to represent the non-linear behaviour of the infiValues of 1240, 2515, 0.26 and 2.4 MPa are usedthe infill shear modulus, Young’s modulus, mean diagoncracking strength and mean horizontal compressive strerespectively, according to wallette tests carried out at tUniversity of Pavia, Italy [12].

A population of frames with different dynamic responscharacteristics can be generated through the treatmenselected structural parameters as random variables. Othe variables are selected, they are assigned probabdistribution functions based on experimental observatioValues of the parameters are then extracted from tdistributions using appropriate sampling techniques. Finathe sampled values are combined to define a series of frawith different characteristics, all nominally representingthe same structure. Clearly, consideration of all possiuncertainties in the global and local structural characteristicsyields an extremely large number of permutations fthe analysis. It is recommended that the general planand elevation configuration of the system are treatdeterministically, as changes in these are expected to athe design forces, code prescriptions, member design anddetailing, and hence warrant a separate vulnerability curderivation. If the typical practice of providing minimumsection sizes and reinforcement for the code defined loais considered, therebar configuration, section and membgeometry can also be regarded as constant. Variatin material properties is large within a population, anderives from differences in manufacturing processes alocal construction practices. Therefore, it is proposthat the concreteunconfined compressive strength (fc),the infill compressive strength (fcw) and the beam andcolumn reinforcing bar steel yield strength (fy) are chosento be randomly varied between frames. In the lattcase, a correlation coefficient of 0.7 is assumed to exbetween the sampled parameter values within the elementsof single frames. It is highlighted that the probabilitdistributions and degree of inter-correlation forthe materialparameters should be selected in accordance with the typconstruction practice and reliability of the building materimanufacturing processes used in the assessed area.Table 2summarises the probability distribution functions assignto each material property for the assessment of the examinfilled frame population. These were determined throuconsideration of numerous tests on European construcmaterials dating from the assumed time of constructio(e.g. Pipa and Carvalho [13], Petersons [14], amongstothers).

Designs of experiment techniques offer an increasethe efficiency of simulation-based assessment of structureliability, and are used in the proposed method to optimise the population size and expedite convergencethe results. The population size is determined via then

factorial composite method [15]. This method prescribes(2n + 2n + 1) parameter combinations for the generation of


Table 2Material parameter probability distribution functions used in the population generation

Material Concrete Steel Masonry infill

Random variable Meanfc COV f yb Mean fcw

Distribution Normal Log-normal Log-normalMean Nominalfc 0.06 Nominal fcwCOVa 0.12 0.25 0.20Design variable Meanfc Mean fy

c Mean fcw

a Coefficient of variation.b Use of the COV offy as the random variable allows a constant characteristic yield strength to be maintained and a realistic variation in mean yield strength

to be obtained.c Calculated forfyk = nominal fy.

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a second-order response surface that fully representsuncertainty associated withn independent random variablesA smaller number of permutations might be used in thepresent case study as correlation exists between two ofparameters. However, a sample size of 25 is seen to ensconvergence of the fragility curves for the four parameteused. Therefore, 25 values are sampled from each ofthree material parameter probability distributions using tLatin hypercube method. During the sampling, a correlaticoefficient of 0.7 is maintained between the column abeam steel yield strength values. The parameter valuesshuffled and combined according to the algorithm adoptby Law and Kelton [16]. Pushover analysis of the thusderived population of 25 infilled frames shows an averacoefficient of variation (COV) of 14%, 37% and 10% fothe initial stiffness, yield displacement and yield period othe system, respectively. The latter results in an averaCOV of 21% for the ground motion response (Sd5%(T)). Acomparable average COV of 22% in the predicted ISDmax%response of the population is also observed to result frommaterial uncertainty.

2.2. Generation of ground motion input

In the generation of vulnerability curves, randomvariability in the ground motion parameter (i.e.Sd5%(T))is not considered, as this should be included in thdetermination of the hazard within the total risk assessmentWithin an analytical curve derivation procedure, the grounmotion parameter isdeterministic, and is evaluated fromthe earthquake records used as input to the analysHowever, the characteristics of records producing ansingle value of the ground motion parameter can vasignificantly and introduce uncertainty in the structurresponse and consequent damage predictions. Suitesaccelerograms with different peak amplitude, frequencontent, cycle number and duration characteristics mtherefore be adopted in the population analysis. Howevno formal guidance exists for the selection of accelerograsuites for use in vulnerability curve generation, whestructural response must be evaluated over a very wrange of ground motion severities and different performanobjectives checked.

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Seismic performance of buildings is assessed conditially to the probability of occurrence of an earthquake even(e.g. “Serviceability” for frequent earthquakes and “Collapse prevention” for very rare events). Hence risk assement tools should assess the attainment of different damlimit states using records that are representative of eventwith return periods that are consistent with the performanobjective. In most existing vulnerability studies (e.g. [2]),sets ofnatural records are selected (or records are artifi-cially generated) to be consistent with a response spectthat represents the seismic hazard in the assessed reThese records are then scaled by various means, to resent the entire range of ground motion severities used forcurve generation. However, record scaling does not accofor the change in the constitutive characteristics of recorderiving from earthquake events of very different magntudes and return periods. It is therefore proposed that th“Target” spectra representative of scenarios correspondinthe FEMA 273 [17] performance states of “Serviceability”“Damage control” and “Collapse prevention” are defined fthe selection of the accelerograms used in the derivatof the “Slight” and “Light”, “Moderate” and “Extensive”,“Partial Collapse” and “Collapse” fragility curves, respectively. Following a review of existing literature (Bentler [18]and FEMA 273 [17] amongst others), the three returperiods of 95, 475 and 2475 years are found to be represtative of the events associated with the three performaobjectives. A back analysis of the Italian hazard mapsAlbarello et al. [19], associated data and attenuation rulresults in the determination of the earthquake charactetics shown inTable 3for the three soil classes of Eurocod8 [20] and three hazard scenarios. These values are evated over the land classified as Zone 2 by the Italian co(moderate–high seismic hazard), in order to be consistwith the seismic loads assumed in the design of the infilbuilding. The procedure followed is explained in greater detail in [6]. The derived meanpgacorresponds well with thevalues of proposed by Campos-Costa and Pinto [21], for thesame hazard scenario in Portugal. The attenuation relation-ships of Ambraseys et al. [22] and Sabetta and Pugliese [23]are used with equal weighting (as per the derivationthe hazard maps), to establish the “target” acceleration


Table 3Characteristics of the earthquake events defining the “Target” spectra

Soil category Rock Firm SoftReturn period (years) 95 475 2475 95 475 2475 95 475 2475

Targetpga(g cm/s2) 6.70 12.20 21.70 6.10 11.90 20.70 6.10 11.80 22.30Lower boundpga(g cm/s2) 4.10 7.60 13.80 3.50 7.30 12.80 3.50 7.20 14.40Upper boundpga(g cm/s2) 9.30 18.50 34.90 8.70 18.20 33.90 8.70 18.10 35.50Surface magnitude (Ms) 5.75 6.05 6.25 5.55 5.85 6.15 4.95 5.35 5.45Fault distance (r , km) 22 14 8 30 18 12 16 10 4

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spectra from the mean values ofpga, surface magnitude(Ms) and fault distance (r ) for firm soil conditions presentedin Table 3. Equivalent displacement “target” spectra are alderived using the attenuation relationship of Bommer et[5] and are shown inFig. 3. It can be inferred from the workof Wen and Wu [24] that ten accelerograms are sufficiefor the representation of theground motion variability asso-ciated with a given spectral shape. Three suites of ten nural records corresponding to the three damage state stra are, therefore, selected from the European Strong-MotDatabase [25]. The records are chosen for different site soconditions, considering earthquakes with values ofMs andr within ±0.5Ms and±10 km of those defining the targecurves, such that the average spectral shape of the selesuite of records closely approximates the target spectruRanges ofpgaare also used in the record selection; thesedefined by considering the deviation inpgavaluescharacter-ising the hazard in the Italian seismic Zone 2 (Table 3). Therecords thus selected are presented in full in [6]. Notwith-standing the close fit of the mean spectrum of the recorwith the “target” shape (Fig. 3), a large variation in the timehistories and spectra used to represent each of the earthqreturn period scenarios is seen. Uncertainty in ground mtion introduces an average coefficient of variation of 27.0in the seismic spectral displacement demand imposed on thinfilled structure population, for all scenarios. This leadsan average variation in the ISDmax%predicted for each struc-ture, at a given level ofpga, of up to37.2% (COV). The lat-ter value is greater than that resulting from material propevariation within the structural population, and emphasisthe importance of including ground motion uncertaintyanalytical vulnerability evaluations.

Although the adopted record selection techniquetime-consuming, the resulting record suites may be usto carry out vulnerability analyses in all areas witsimilar seismic activity and fault mechanisms to thoconsidered (i.e. medium European seismicity, shallcrustal faulting). For areas with very different seismicior characterised by subduction earthquakes, ranges of pground accelerations, earthquake surface magnitudes andfault distances can be found from historical earthquacatalogues or from local hazard maps using a method simto that shown above. Target spectra can then be develousing appropriate attenuation relationships and suites

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Fig. 3. The “Target” and ensemblerecord suite mean acceleration anddisplacement spectra for the 95, 475 and 2475 year return period earthquakescenarios and firm soil conditions.

records for analysis chosen from catalogues of earthquawith similar fault mechanisms.

2.3. Model evaluation

In order to generate vulnerability curves, the buildingpopulation determined above must be assessed for damfor increasing levels of seismic load. Despite the useof experiment design techniques to limit the number oaccelerograms and building models required for the furepresentation of structural and ground motion uncertaintia large number of analyses are still required to assesspopulation over a wide range of ground motions. Withithe vulnerability curve derivation methodology proposed


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here, the number of analyses is reduced through the usean adaptive pushover analysis procedure within a capacityspectrum framework of assessment.

Pushover analyses are associated with analysis timthat are a fraction of those required for full non-lineadynamic time-history analyses and are thus amenableuse in cases where numerous analyses are required.conventional pushover procedures, a constant distributionforces is applied to a structural model incorporating inelastmaterial properties, and is incremented to structural failurThe analysis is unable to account for the characteristiof earthquake records and the variation in applied seismdemand with increasing structural degradation. Furthermoa poor representation of the deformed shape of structuris often seen if these do not respond predominantly in thfirst mode. Conventional pushover analyses can therefonot be used in the present assessment, where groumotion uncertainty is considered, buildings with inadequatseismic design are addressed and ISDmax% is used as thedamage measure. An adaptive pushover (APO) analymethod is proposed in [26], which follows the sameprocedure as conventional pushover analysis, but at eaload increment updates the lateral applied load distributioto take into account the instantaneous structural stiffnesmodal properties and consequent ground motion demand.The validity of the adaptive pushover technique is verifieby Rossetto [6] with respect to the dynamic time-historyanalysis of seismically designed eight-storey structureand an irregular, non-seismically designed RC frame.reasonable correlation between the structural top-drift, base-shear and inter-storey drift responses predicted by the twmethods is observed and the sequence of formation of locand global collapse mechanisms is satisfactorily predicteThe APO method of Antoniou [26] implementedwithinthe framework of the INDYAS finite-element package [7]is adopted in the analysis of the structure population herThe result of the APO analysis is a set of 750 base-sheversus top-displacement curves, which describe the capacof each of the 25 buildings for the 30 seismic eventconsidered. A single adaptive pushover curve is sufficiefor the evaluation of the structure vulnerability, over allground motion severities, for any earthquake of a givespectral shape. The repetition of structural analyses feach ground motion scale factorincrement is eliminatedand the number of analyses required for the determinatiof the vulnerability curves is considerably reduced (75analyses, compared to 11 250 for time-history analyseThe consequent reduction in analysis time and cost rendthe adaptive pushover technique a highly desirable tool foapplication in vulnerability studies.

Capacity Spectrum Methods (CSM) are proposed ithe literature for the seismic assessment of buildinperformance using the results of static pushover analys(Fajfar [27] amongst others). They are incorporated incodes of practice such as FEMA 273 [18], and are seenby many to represent the future of seismic assessment

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buildings, as they have simple visual interpretations aninvolve elements of both acceleration and displacemeresponses. A direct comparison is made between earthquspectra representing the seismic demand and a transformforce–displacement curve representing the seismic capacityof the structure. Both demand and capacity curves are plottein spectral displacement–spectral acceleration (SASspace, with the “performance point” (PP), defining thmaximum response of the structure for the given earthquabeing determined from the location of their intersection.The implicit assumption of a fundamental mode of responfor the building is the main impediment to the adopting othe existing capacity spectrum method in the assessmof structural damage using the results of APO analysis.the case of APO analysis a single transformation cannotapplied to the pushover curve as the relative contributiof each mode changes with each applied load incremeHence, an approximate method for the transformation iproposed here, where the instantaneous displaced shape anstorey forces at each increment step of the APO methodused to transform the force–displacement curves into SAspace. The same expressions as for the single-degreefreedom transformation are adopted:

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However, the current displaced shape of the structure nmalised to the top displacement(Φn) replaces the funda-mental mode shape. Consequently,φi is the component ofΦn corresponding to thei th storey,mi is the lumped massat the i th floor anduN and Vb are the top displacemenand base shear at the current load increment, respectively.The reasoning behind this transformation method is thatforce distribution and resulting displacement distribution implicitly incorporate the modal combinations. This assumtion may not be theoretically justified, but is observedyield reasonable assessment results [6]. Another drawbackof existing capacity spectrum procedures isthat they are it-erative, graphical methods of assessment. Due to the lanumber of assessments necessary for the simulation of damage statistics in vulnerability curve generation, the adoptingof such a procedure is impractical. A modified approachthe visualisation of CSM using inelastic earthquake spetra is therefore proposed, which both aids the automatiand increases the efficiency of the assessment procedFollowing transformation of theAPO base-shear–top-driftresponse curves to SASD space, according to the produre outlined above, the transformed pushover curve is idalised as a multi-linear curve. This curve is then discretisinto a series of points, denominated by the capacity–demanchecking points (CDCP). The idealised curve shape up


Fig. 4. Illustration of the proposed method for performance point solution.

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each CDCP location defines the elastic period, ductility anon-linear response curve characteristics of a correspoing single-degree-of-freedomsystem (SDOF). The resultingseries of SDOFs are analysed dynamically for the appliscaled earthquake records, to obtain a set of inelastic specacceleration and displacement values. These demand valuemay be visualised as corresponding to the structure capaalong the radii of the SASD plot that intersect the CDCPUsing these results, a single inelastic demand curve may bedrawn and the performance point defined directly from tcrossing of this curve with the capacity curve (Fig. 4). Themaximum inter-storey drift ratio(ISDmax%) response is de-termined at the PP from the results of the adaptive pushoveranalysis and used to determinethe damage state of the building. The same capacity curve is used to assess a strucover the full range of ground motions required for the vunerability curve generation, through record scaling.

A capacity assessment program (CAsP) is created forimplementation of the proposed procedure in the assessmof the infilled frame population. CAsP is coded in VisuaBasic 5.0 [28] and ispresented fully in [ 6]. In the evaluationof populations of building models, a large number ocapacity curves of differing shape must be dealt with (7in the case of the infilled frame population assessedthis study). CAsP is designed to automatically detect thlocation of yield and the ultimate point locations that bedescribe the curve shape, according to the behavioumodel and yield criteria specified by the user. Three curidealisation models can be used to model the pushoverresponse of the structures: the elastic–perfectly plastic (Emodel, the linear strain hardening (LSH) model and trlinear model (TLM). Several options exist for definition

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of the global yield point and strain hardening slopeeach case. An adaptive strainhardening option is alsoincluded, where the strain hardening varies according tothe point on the curve being assessed. The choicecurve idealisation model strongly affects the performanpoint predictions for infilled frames, where plateaus in thpushover response are seen to occur due to the sudfailure of infill panels. Neither the LSH model nor theEPP model can model this behaviour, the latter modresulting in errors of up to 60% in the performancpoint spectral displacement location. All current capacityspectrum assessment procedures adopt EPP- or LSH-tcurves in the determination of the structural capacityand seismic demand. The above observations therefraise concern over the adequacy of these methodsthe assessment of infilled frames, and promote the uof programs such as CAsP that allow more complicatcapacity curve modelling. Furthermore, in the proposassessment method, consistent curve shapes are usedetermine both the demand and capacity, unlike in previostudies (e.g. [29] and [27]). It is concluded that the TLMwith variable strain hardening gives the best representatof the infilled structure non-linear pushover response andis therefore adopted in the example population assessmhere. A simpler bi-linear curve idealisation may be adequatefor bare frame response idealisation.

Through use of the proposed assessment approach,a single inelastic dynamic analysis of the equivalent SDOsystem is required at each equivalent period considerecompared to the series of analyses for different ductilivalues implied by a graphical solution. Consequently, asignificantly lower computation time is achieved. Within


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CAsP, a full Newton–Raphson iterative scheme is useto solve the dynamic non-linear equilibrium equatiofor the response evaluation of each SDOF. In the castudy presented, each of the 25 APO curves definingpopulation is assessed approximately twenty times forincreasingly scaled accelerogramused in their analysis, untilan inter-storey drift value is obtained which exceeds thatthe most severe damage state of interest. The program isrun for a new earthquake and a corresponding set of Aanalysis results.

2.4. Statistical processing

The capacity spectrum assessment for each framedefined properties( fc, fcw, fy, fyb) yields values of themaximum inter-storey drift response(ISDmax) for increasingvalues of ground motion intensity. The results of thpopulation assessment are used to construct second-oresponse surfaces of the form of the following equation:

ISDmax

Sd5%(T)=a1 f 2

c + a2 f 2cw + a3 f 2

yc + a4 f 2yb + a5 fc

+ a6 fcw + a7 fyc + a8 fyb + a9 fc fcw+ a10 fc fyc + a11 fc fyb + a12 fcw fyc

+ a13 fcw fyb + a14 fyc fyb + C. (2)

A different response surface is generated for each damstate scenario, using the results of the population assessmfor the corresponding “performance-compatible” recosuite. These are each regressed from an average3750 points using the non-linear regression moduleSTATISTICA [30]. An example of the derived scenarioresponse curves, and their fit to the regression data, is shin Fig. 5. The coefficients of correlation(R2) of the curveswith the data range between 0.59 and 0.71. This scatter in thedata is later accounted for in the confidence bound derivatof the vulnerability curves. The quantity of regressiodata exceeds the minimum prescribed by the 2n factorialcomposite method [15] for the derivation of a second-orderesponse surface from the five random variables involv(i.e. the four material parameters and ISDmax). This quantityof data is deemed sufficient for the detection of sevediscontinuities or singularities. Values of the elastic spectdisplacement (for 5% critical damping) corresponding tothe mean of the yield periods for the population (defineon the basis of the point of first deviation from elastbehaviour of each model and evaluated asT = 0.125 s)are used to characterise the seismic demand in the respsurfaces and resulting vulnerability curves. The effectsmaterial variability and structure inelastic behaviour on bothe building capacity and seismic demand are includedthe proposed method for structure performance assessmHence, the influence of these factors on the populativulnerability will implicitly be included in the y-axis of thefragility curves developed. Use of an effective period fothe calculation of the demand parameter characterisingx-axis of the vulnerability curves is therefore superfluous.

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Fig. 5. Illustrations of the 2475YRP response surface (top) and its fithe regression data (bottom; the dashed lines indicate the 5th andpercentile observed/predicted ISDmax%bounds).

Damage statistics are generated for a system throughmultiple selection of material parameter value combinatiofor input into the response surfaces. Values of ISDmaxresponse are evaluated at a series of spectral displacemfor each material parameter combination. Frequency plotsare obtained for each spectral displacement value andbe used to define the shape parameters of probabidistribution functions for ISDmax. Damage state exceedencprobabilities are calculated from the latter functionconsidering the threshold ISDmax values for the HRCdamage states presented inTable 1. In thecase of the infilledframe population, 650 materialparameter combinationsare used to generate the ISDmax histograms from eachscenario response surface, at 100 spectral displacemvalues. The proportion of buildings exceeding each damastate is calculated at each ground motion level and plotagainst the corresponding spectral displacement value forthe vulnerability curve shape regression. The non-linearegression tool of STATISTICA [30] is used to regressforthe parameters of the log-normal cumulative probabilfunctions characterising the vulnerability curve shape


Table 4Summary of the infilled frame population vulnerability curve equation parameters

HRC damage state Meanc 95% upper boundd 5% lower bounde

µa σb µ σ µ σ

Slight damage −7.80 0.60 −8.25 0.60 −6.30 0.42Light damage −7.15 0.40 −7.82 0.60 −5.76 0.34Moderate damage −5.78 0.21 −6.32 0.20 −4.51 0.22Extensive damage −4.44 0.21 −4.92 0.21 −3.12 0.22Partial collapse −3.49 0.22 −3.98 0.22 −2.12 0.27Collapse −2.99 0.22 −3.50 0.22 −1.67 0.24

a Mean values defining the cumulative log-normal vulnerability relationships.b Standard deviation values defining the cumulative log-normal vulnerability relationships.c Mean confidence bound curves.d Upper 95th percentile confidence bound curves.e Lower 5th percentile confidence bound curves.

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Fig. 6. Comparison of the infilled frame population fragility curves witheight observed post-earthquake damagedistributions for like populations ofstructures.

A very close fit of the regressed curves to the analyticadatais achieved in all cases(R2 > 0.98). Thevulnerabilityrelationships for the “Slight” and “Light”, “Moderate”and “Extensive”, and “Partial Collapse” and “Collapsedamage states are selected from the curve sets derifrom the 95YRP, 475YRP and 2475YRP response surfacrespectively. These relationships are combined to form“performance-consistent” set of vulnerability curves. Thcurves are illustrated inFig. 6 and their equations aresummarised inTable 4.

The fit of the analysis data to the response surfaces idirect indication of the uncertainty in response predictiodue to input parameter variability. The 5th and 95tpercentile values of the ratio of analytically observed tpredicted ISDmax response are therefore applied to thmean ISDmax predicted by the response surface for eacsystem variation. New exceedence probability statistics agenerated using the extreme ISDmax estimates. These areplotted against the corresponding ground motion values aused in the regression of 90% confidence bounds for tvulnerability curves (Table 4). The confidence bounds arewide and increase in width for higher damage states. Itobserved that similar sized confidence bounds are generawhere the response surfaces are derived from data relato a single scenario and when they arederived from the

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combined data from all scenarios. This indicates that thevariation in material and ground motion is fully representein the analyses for each earthquake scenario.

3. Comparison with observational damage

In Fig. 6, the analytical performance-consistent vulnerability curves for the three-storey infilled RC frame designeto the Italian standards of 1982 are compared to damagstatistics deriving from eight post-earthquake surveys caried out on populations of infilled RC structures (1154 buildings in total). These data consist of a subset of the observa-tional damage distribution database adopted in [4] for thegeneration of empirical vulnerability curves. Although thquantity of data for comparison is limited, the analyticacurves are seen to give a reasonable, slightly conservativeto the empirical damage data, with an average correlationefficient(R2) of 0.62. The analytical curves show improveprediction of observed damage compared to other existingRC structure curves [4] and compared to the infilled framefragility curves of Mosalam et al. [3] (with R2 = 0.39).Comparison of the empirical vulnerability curves derived bRossetto and Elnashai [4] for low-rise, old seismic code, in-filled RC frames with those derived analytically shows thlatter to give a better correlation to the observed damadata (R2 = 0.62 compared to 0.42). This is caused by to thscarcity and highly scattered nature of the post-earthquadamage surveys available forinfilled RC frames, which re-sult in unreliable curve shapes being derived for the latempirical relationships. The width of the confidence bounassociated with the analytical relationships, in terms of maimum exceedence probability interval (at constant spectdisplacement), is observed to be comparable to that obserfor the empirical curves. The width in terms of maximumspectral displacement interval(for constant damage state exceedence probability) is instead a fraction of that observfor the empirical curves. The difference is mainly due to thuncertainty introduced in the derivation of the latter curvedue to scarcity of observational data for high levels of groumotion. These observations give rise to substantial doas regards the reliability of observation-based vulnerabil


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functions and confirm the importance of analytical methodfor thegeneration of fragility curves.

4. Conclusions

The main accomplishment of the study described abovis a clear methodology for the derivation of vulnerabilitycurves using analytical damage statistics. The proceduaddresses problems of uncertainty in the input grounmotion and damage state identification, and yields curvethat are appropriate for use within a displacement-based assessment framework. The proposed methodolorepresents the best possible inelastic static analysis solutiongiven current knowledge and capabilities and has the addebenefit that, through appropriate consideration in modellingit can be applied to any structural type and seismotectonic environment. Comparison of the curves derived fora population of low-rise infilled RC frames of inadequateseismic design with observational data and empirical curvesfor these structures shows that the proposed methodolois capable of producing analytical vulnerability curves thatgive reasonable predictions of observed post-earthquabuilding damage. Further verification of the vulnerabilitycurve derivation procedure for different structural systemand greater quantities of observational data are needed.

Acknowledgements

The work in this paper was funded by the European Community as part of the ‘Safety Assessment For EarthquakRisk Reduction’ (SAFERR) Research Training Network(Contract: HPRN-CT-1999-00035). The contribution of thesecond author, and the tenure of the first author in November 2002 at the University of Illinois, were funded by theMid-America Earthquake Center, a UA National ScienceFoundation Engineering Research Center (grant referencEEC-9701785). Dr R. Pinho of the University of Pavia(Italy) and Dr S. Antoniou are thanked for their technicahelp.

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