Variability and Accuracy of Target Displacement From Nonlinear St

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InternationalScholarlyResearchNetworkISRNCivilEngineeringVolume2011,ArticleID582426,16pagesdoi:10.5402/2011/582426

Research ArticleVariability and Accuracy of Target Displacement fromNonlinear Static Procedures

Rakesh K. Goel

Department of Civil and Environmental Engineering, California Polytechnic State University, San Luis Obispo, CA 93407-0353, USA

CorrespondenceshouldbeaddressedtoRakeshK.Goel,[email protected]

Received13December2010;Accepted19January2011

AcademicEditors:R.Jangid,S.Pantazopoulou,andI.Takewaki

Copyright2011RakeshK.Goel.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.

ThispapercomparesthetargetdisplacementestimatefromfourcurrentnonlinearstaticproceduresFEMA-356CM,ASCE41CM,ATC-40CSM,andFEMA-440CSMwiththevaluederivedfromrecordedmotionsoffivestronglyshakenreinforcedconcrete buildings.This comparison provides useful insight into two important questions: (1) how much does the targetdisplacementvaryamongthefournonlinearstaticprocedures?and(2)cantheengineeringprofessionaccuratelypredicttheresponseof a realbuilding during anearthquake event using currentlyavailable modelingtechniques andpushover analysisprocedures?Itisshownthattheseproceduresmayleadtosignificantlydifferentestimatesofthetargetdisplacement,particularlyfor short-period buildings responding in the nonlinear range. Furthermore, various nonlinear static procedures applied tononlinearmodelsdevelopedusinggenerallyacceptedengineeringpracticeprovideeithersignificantoverestimation orunderestimationofthetargetroofdisplacementwhencomparedtothevaluederivedfromrecordedmotions.

1. Introduction

Nonlinear static procedure (NSP) or pushover analysisis widely used for seismic design/evaluation of buildings.The NSP requires nonlinear static pushover analysis ofthe structure subjected to monotonically increasing lateralforceswithspecifiedheight-wisedistributionuntilatargetdisplacementisreached.Thebuildingdesignisdeemedtobeacceptableif seismicdemands(e.g.,plastichingerotations,drifts,etc.)at thetargetdisplacementarewithinacceptable

values.The twowidely usedproceduresto estimatethetarget

displacement in the NSP are: (1) the Coefficient Method(CM) defined in the FEMA-356 document [1], and (2)the Capacity Spectrum Method (CSM) specified in theATC-40 document [2]. The CM utilizes a displacementmodificationprocedureinwhichthetargetdisplacementiscomputed by modifying displacement of a linearly-elastic,single-degree-of-freedom(SDF)systembyseveralempiricalcoefficients. The SDF system has the same period anddampingasthefundamentalmodeoftheoriginalbuilding.The CSM is a form of equivalent linearization in whichthe target displacement is estimated bymultiplying elastic

displacementof anSDFsystembythe fundamentalmodeparticipation factor. The SDF system has larger (effective)periodanddampingthanthe original building. The CSMuses empirical relationships for the effective period anddampingasafunctionofductilitytoestimatedisplacementofanequivalentlinearSDFsystem.

MostpreviousinvestigationsondevelopmentandevaluationoftheCMand/orCSMtocomputethetargetdisplacementusedcomputermodelsofbuildings;anexhaustivelistofreferencesisavailableintheFEMA-440report[ 3].These

investigationsprimarilyfocusedontheaccuracyofvariousnonlinearstaticproceduresinpredictingtargetdisplacementof computer models; the exact value of the target roofdisplacementwastakenasthepeakroofdisplacementcomputedbynonlinearresponsehistoryanalysisofthecomputermodelsubjectedtoselectedearthquakemotionatitsbase.Becausetheexacttargetdisplacementandthevaluefromvariousnonlinearstaticproceduresusedthesamecomputermodel, these investigations eliminated thediscrepancydueto modeling assumptions; errors examined were onlyduetononlinearstaticprocedures.Recentinvestigationsonthistopicfoundsignificantvariabilityinthetargetdisplacementsfromvariousprocedures[4,5].
mailto:[email protected]:[email protected]


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2 ISRNCivilEngineering

Recorded motions of strongly shaken buildings, especially those deformed into the nonlinear range, providea unique opportunity to gain insight into two importantquestions:(1)howmuchdoesthetargetdisplacementvariesamong various nonlinear static procedures? and (2) cantheengineeringprofessionaccuratelypredicttheresponse

of a real building during an earthquake event using currentlyavailablemodelingtechniquesandpushoveranalysisprocedures?While paststudiesthatused either generic orwell-calibrated computermodelshave providedsignificantinsightintothefirstquestion,theadditionaldatageneratedin this investigation using validated computer models ofrealbuildingsisusefulineitherconfirmingorcontradictingthepreviousfindings.Theinsightintothesecondquestioncan only be gained by comparing the roof displacementestimatedfromvariousnonlinearstaticproceduresappliedto a computer model, developed using generally acceptedengineeringpractice,ofthebuildingwiththevalueobservedduring an actual earthquake event. This investigation isspecificallyaimedatfillingthisneed.

Theproceduresconsideredin thisinvestigationare:(1)CMdefinedintheFEMA-356document[1],(2)improvedCMproposedintheFEMA-440report[3] and adopted in theASCE-41standard[6],(3)CSMdefinedintheATC-40document[2],and(4)improvedCSMproposedbyGuyaderandIwan[7]andadoptedintheFEMA-440report[3].

2. Selected Buildings

Recordedmotionsofbuildingsthatwerestronglyshakenandpotentiallydeformedbeyondtheyieldlimitduringtheearthquakearerequiredforthisinvestigation.Forthispurpose,

fiveconcretebuildings,rangingfromlow-risetohigh-rise,havebeenselected(Table1).Thestrong-motiondatausedinthisinvestigationareidentifiedin Table1 foreachbuilding.Thedataselectedinthisinvestigationaretheprocesseddataavailable from the Center for Engineering Strong MotionData (CESMD) (http://www.strongmotioncenter.org). Thedataprocessingprocedureinvolveslow-pass andhigh-passfiltering using Ormsby filters. Further details of the dataprocessingareavailableinapaperbyShakaletal.[8].

Thedistancefrombuildingtoepicenteroftheearthquake(Table1) indicates that the selected buildings were mostlikelysubjectedto near-faultmotions.Althoughresultsarenot presented here for brevitys sake, response spectra formotions recordedat the base ofseveralof thesebuildingsindicatedcharacteristicscompatiblewiththoseexpectedfornear-fault motions; a more comprehensive discussion onhow toidentify near-fault effects from responsespectra isavailableinChopra[9].

3. Procedures to Compute Target Displacement

3.1. FEMA-356 Coefficient Method. ThetargetdisplacementintheFEMA-356CM[1] is computed from

T2et= C0C1C2C3Sa g, (1) 42

whereSa = responsespectrumaccelerationattheeffectivefundamental vibration period and damping ratio of thebuildingunderconsideration,g=accelerationduetogravity;Te = effective fundamental period of the building in thedirectionunder considerationcomputed bymodifyingthefundamentalvibrationperiodfromelasticdynamicanalysis,

forexample,eigen-valueanalysis,Ti, b y

KiTe= Ti (2)Ke

in which Ki is the elastic stiffness of the building andKe is the effective stiffness of the building obtained byidealizing the pushover curve as a bilinear relationship;C0 = coefficient to relate the elastic response of an SDFsystem to the elastic displacement of the multi-degree-offreedom(MDF) buildingat the control node takenas thefirst mode participation factor or selected from tabulatedvaluesintheFEMA-356documentC1=coefficienttorelatethemaximuminelasticandelasticdisplacementoftheSDF

systemcomputedfrom1.0; Te Ts,C1=

1.0 + (R 1)Ts/TeR ; Te < Ts, (3)1.5; Te


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3ISRNCivilEngineering

Table 1:Fivereinforcedconcretebuildingsselected.

Buildingsname CSMIPStation Stories/basement Earthquake Epic.Dist.(km)

ImperialCountyServicesBuilding 01260 6/0 1979ImperialValley 28.4

ShermanOaksCommercialBldg 24322 13/2 1994Northridge 9

NorthHollywoodHotel 24464 20/1 1994Northridge 19

WatsonvilleCommercialBldg 47459 4/0 1989LomaPrieta 18SantaBarbaraOfficeBldg 25213 3/0 1978SantaBarbara 13

Baseshear

VdVy

0.6VyKe

1Ke P-KeeKe

2Ke

y d Displacement,Figure 1:Idealizedforce-deformationcurveinASCE-41CM.

themotionsrecordedatthebaseofthebuilding.Therefore,theperiodTswasestimatedfromthelinearly-elasticresponsespectrumofthemotionrecordedatthebaseofthebuilding.For this purpose, a smooth spectrum was fitted to the

combined D-V-A response spectrum and the period atthe intersection of the acceleration-sensitive and velocity-sensitive regions wasselectedtobe theperiod Ts. Furtherdetails of this procedure may be found in Chopra [9].

3.2. ASCE-41 Coefficient Method. ThetargetdisplacementintheASCE-41CM[6] is computed from

T2et= C0C1C2Sa g, (6) 42

wherecoefficient C0 relatestheelastic responseof anSDF

systemtotheelasticdisplacementoftheMDFbuildingatthecontrolnodetakenasthefirstmodeparticipationfactor.ThecoefficientC1 isgivenby

C1=

1.0; Te >1.0s, 1.0 +

R 1aT2e ; 0.2 s < Te 1.0s,

1.0 + R 10.04a; Te 0.2 s

(7)

inwhichaisequalto130forsoilsiteclassAandB,90forsoilsiteclassC,and60forsoilsiteclassesD,E,andF(see

ASCE-41fordetailsofvarioussiteclasses),respectively.ThecoefficientC2 isgivenby

1.0; Te >0.7s, C2= 2 (8) 1 R 11 + ; Te 0.7 s.

800 TeFinally, the ASCE-41CM imposes limitation on R to

avoiddynamicinstabilityas

d |e|hR Rmax= + ; h= 1.0 + 0.15ln(Te) (9)y 4

in which d is the deformation corresponding to peakstrength,y istheyielddeformation,and e istheeffectivenegativepostyieldslopegivenby

e= P-+(2 P-), (10)where 2 is the negative postyield slope ratio defined inFigure1,P-isthenegativesloperatiocausedbyP-effects,andisthenear-fieldeffectfactorgivenas0.8forS1 0.6and0.2forS1


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and/ordegradedloops;andTypeBbetweenTypesAandCandprovidesequationsforcomputing foreachofthethreetypesofhystereticbehavior.

Since the equivalent linearization procedure requiresprior knowledge of the displacement ductility ratio (12),ATC-40 document describes three iterative procedures:

Procedures A, B, and C. Procedures A and B are themosttransparentandconvenientforprogramming,whereasProcedureCispurelyagraphicalmethod.DetailsoftheseproceduresareavailableintheATC-40documentandarenotpresentedhereforbrevityssake.

3.4. FEMA-440 Capacity Spectrum Method. ThetargetdisplacementintheFEMA-440CSM[3] is computed from

t= C0Sd(Teff,eff), (13)wherecoefficientC0 isthefundamentalmodeparticipationfactor, and Sd(Teff,eff) is the maximum displacement ofa linearly-elastic SDF system with effective period, Teff,and effective damping ratio, eff. The FEMA-440 CSMincludes improved expressions, compared to the ATC-40CSM,todeterminetheeffectiveperiodandeffectivedampingdeveloped by Guyader and Iwan [7]. Consistent with theoriginal ATC-40 procedure, three iterative procedures forestimatingthetargetdisplacementarealsooutlined.Finally,a limitation on the strength is imposed toavoid dynamicinstability(9).

TheimprovedformulasforeffectiveperiodanddampingratiointheFEMA-440documentare 2 3

0.2 1 0.038 1 +1 To; < 4.0,

0.28+0.13 1 + 1 To; 4.06.5,Teff= 1 0.89 1 +1 To; > 6.5,

1+0.05 2

2 34.9 1 1.1 1 +o; < 4.0, 14.0 + 0.32 1 +o; 4.06.5,eff= 2 0.64 1 1 Teq19 + > 6.5. 2 o;

0.64 1 To(14)

Theseformulasapplyforperiodsintherangeof0.2and2.0s.TheFEMA-440documentalsoprovidesformulaswithconstantsA toL thatarespecifieddependingontheforce-deformationrelationships(bilinear,stiffness-degrading,andstrength-degrading)andthepostyieldstiffnessratio,;theseformulasarenotincludedhereforbrevityssake.

4. Analytical Models

4.1. Modeling Procedure. Needed for estimating the targetdisplacementisthepushovercurveofthebuilding.Forthispurpose,three-dimensionalmodelsoftheselectedbuildingsweredevelopedusingthestructuralanalysissoftwareOpen

SystemforEarthquakesEngineeringSimulation( OpenSees)[10].Twomodelsweredevelopedforeachbuilding:linearly-elastic model for computing the mode shapes and frequencies (or vibration periods), and a nonlinear modelforpushoveranalysis.Gravityloadswereincludedinbothmodelsandwereappliedpriortoeigenanalysistocompute

mode shapes and frequencies or the pushover analysis.Furthermore,P-Deltaeffectswereincludedinbothmodels.The beams, columns, and shearwalls in the linear modelweremodeledusingelasticBeamColumnelementinOpenSeeswith effective (or cracked) section properties as per theFEMA-356 recommendations [1]. The beams, columns,and shear walls in the nonlinear model were modeledwith nonlinearBeamColumn element with fiber section inOpenSees.Contributionstostiffnessduetoflexuralaswellassheareffectswereincludedinbothmodels.

The nonlinear element used fiber sections containingconfinedconcrete,unconfinedconcrete,andsteelreinforcingbars to model the axial-flexural behavior, whereas linear-elastic behavior was assumed for the shear and torsionalbehavior.Thecompressivestress-strainbehaviorofconcrete,bothconfinedandconfined,wasmodeledwithConcrete04materialinOpenSees (Figure2(a))andtensilestrengthwasignored.Furthermore,concretewasassumedtocompletelylose strength immediately after the crushing strain. Thecrushingstrainoftheunconfinedconcretewasselectedtobeequalto0.004andthatforconfinedconcretewasselectedtobe that corresponding tothe rupture ofconfining steelusing thewell-established Mandermodel [11].Thestress-strain behavior of steelwas modeled with ReinforcingSteelmaterial in OpenSees (Figure2(b)). Further details of thematerial models are available in McKenna and Fenves[10]. The nominal strengths of concrete and steel wereselected based on the values specified in the structuraldrawings.

FortwoofthefiveselectedbuildingsWatsonvilleCommercial Building and Santa Barbara Office Buildingthefoundationflexibilitywasexpectedtosignificantlyinfluencetheresponseduringstronggroundshakingbecausebothofthese low-rise buildingscontained longitudinal and transverseshearwalls.Thefoundationflexibilitywasincludedinanalyticalmodelsofthesebuildingsbyattachingsixlinearspringsthreealongthex-,y-,andz-translation,twoaboutthex- and y- rocking,andoneaboutthez-torsionatthebaseaspertheFEMA-356recommendationsforfoundationflexibilitymodeling[1].

4.2. Validation of Analytical Models. Themodelsdevelopedin this investigationare based on generallyaccepted engineeringpractice.Therefore,thedifferencebetweenthetargetdisplacementfromvariousnonlinearstaticproceduresandthevaluederivedfromrecordedmotionsofabuildingwouldbe due to errors due to inaccuracies in modeling as wellasnonlinearstatic procedures.In thisinvestigation, whichutilizes data from actual buildings during an earthquakeevent,itmaynotbecompletelypossibletoisolatetheerrorsfrom thesetwo sources.However, it is important that theanalyticalmodelbeasaccurateaspossibletominimizetheerrorsduetomodeling.Forthispurpose,modelsofselected


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Confined

Unconfined

Stress

Strain Strain

Stress

(a) ConcreteModel (b) SteelModel

Figure 2:Materialmodelsusedfornonlinearanalysis.

buildingsusedin thisinvestigationwerevalidatedbycomparingtheirvibrationpropertiesanddisplacementresponses(due to recorded base motions during actual earthquakeevents)withthoseobservedduringactualearthquakeevents.The following is a description of the procedure used tovalidatethemodeloftheNorthHollywoodHotel.Asimilarprocedurewasusedtovalidatemodelsforotherbuildings.

First,fundamentalvibrationperiodsofthebuildinginthe longitudinal and transverse directions were identifiedfrom recorded motions of the building by using a well-known transfer function approach (Figure3). Next, theseperiodswerecomputedfromeigen-analysisof thelinearly-elastic model (Figure4). Finally, the linearly-elastic modelwasvalidatedbycomparingthevibrationperiodsidentifiedfromrecordedmotionsandcomputedfromeigen-analysis.Although not identical to the periods identified fromrecordedmotions,thevibrationperiodsfromeigen-analysis

are close enough for most practical applications: periodfromeigen-analysisin thelongitudinaldirectionis 2.57seccomparedtotheidentifiedvalueof2.64sec.Itisusefultoemphasizeagainthatthelinear-elasticmodelconsideredinthisinvestigationisbasedongenerallyacceptedengineeringpractice and is not intentionally calibrated to match theperiodsindentifiedfromtherecordedmotions.

Thenonlinearmodelusedinthisinvestigationwasfirstvalidated by comparing the fundamental vibration periodestimatedfromthepushovercurvesagainstthevaluefromeigen-analysisandvalue identifiedfromrecordedmotions.Forthispurpose,thepushovercurvewasconvertedtothecapacity curve of the equivalent inelastic SDF system by

scaling the roof displacement by 1/(1r1) and base shearby1/M where1 isthefirst-modeparticipationfactor, r11is the first-mode component at the roof (or target node),andM1

isthefirst-modeeffectivemass.Thefundamentalvibration period is estimated by recognizing that initialelasticslopeofthecapacitycurveoftheequivalentinelasticSDFsystemisequalto1

2 whichgivesT1= 2/1 [12].The results presented in Figure5 indicate that the

pushover curve also provides estimates of fundamentalvibration periods that are close to the values identifiedfromrecordedmotionsandcomputedfromeigen-analysisof linearly-elastic model. For example, the longitudinalvibration period of 2.78sec (Figure5(a)) from pushover

curve compares quite well withthe value of 2.64 secidentified fromrecordedmotions(Figure3(a))and2.57seccomputedfromeigen-analysisofthelinearelasticmodel(Figure4(a)).

Thenonlinearmodelwasfurthervalidatedbycomparingthe displacement responses of the model subjected tomotions recorded at the base of the building with thedisplacementsderivedfrom recorded motions.TheresultsshowninFigure6fortheNorthHollywoodHotelindicatethatthemodelprovidesdisplacementresponsehistoriesthatmatch quite well with the displacement histories derivedfromrecordedmotions.

Thevibrationperiodsofselectedbuildingsfromthethreesourcessystem identification using recorded motions,eigen-analysis of the linearly-elastic model, and pushoveranalysisofthenonlinearmodelaresummarizedinTable2.Thepresented results indicatethatvibration periods fromsystemidentificationandeigen-analysisofthelinear-elastic

model match quite well for all buildings. The vibrationperiods estimated frompushover analysisof thenonlinearmodel also match quite well with results from the twoother sources for three of the five buildingsImperialCounty,ShermanOaks,andNorthHollywood.However,thepushoveranalysisprovides longerperiodscompared tothetwoothersourcesfortwoshear-wallbuildingsWatsonvilleandSantaBarbara.

Thelongervibrationperiodfromthepushoveranalysisofthetwoshear-wallbuildingsisduetolowerinitialelasticstiffnessofthesystemduringthepushoveranalysiscomparedto that in the model used for eigen-analysis. The lowerstiffnessofthesystemduringpushoveranalysisisapparently

duetolowereffectivemomentofinertiaoftheshearwallscompared to the value of 0.5 times the gross momentof inertia assumed in the model for eigen-analysis of thebuilding.Thisobservationisconsistentwiththatinarecentstudy[13] whichconcluded that the factor toconvertthegrossmomentofinertiatotheeffectivemomentofinertiaissignificantly lowerthanthevalueof 0.5 specifiedin theASCE-41andFEMA-356documents.Thediscrepancycanbeparticularlylargeforlowvaluesofaxialforce;experimentaldatapresentedinElwoodetal.[13]indicatedthatthefactorcan beas low as0.1 for zeroaxial forcelevel. Clearly, thevibrationperiodcomputedbasedontheeffectivemomentof inertia factor of 0.5, the value specified in FEMA-356
http:///reader/full/valueof2.64http:///reader/full/valueof2.64http:///reader/full/valueof2.64


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20 20

15 15

f1= 0.378Hz,T1= 2.64s

NHH-longitudinal

TR

0 1 2 3 4 5

Frequency(Hz)

f1= 0.354Hz,T1= 2.82s

NHH-transverse

0 1 2 3 4 5

Frequency(Hz)

TR 10 10

5 5

0 0

(a) (b)

Figure 3:FundamentalvibrationperiodofNorthHollywoodHotelidentifiedfromrecordedmotions.(a)Longitudinaldirection,and(b)Transversedirection.

ZXY Z XY

(a) 1stLong.Mode:T= 2.572s (b)1stTrans.Mode:T= 2.980sFigure 4:VibrationperiodsandmodeshapesoftheNorthHollywoodhotelcomputedfromeigen-analysisofthelinear-elasticmodel.(a)Longitudinaldirection,and(b)Transversedirection.

200 200

150 150

Ti=2.572sT1=2.782s21 Vb/

M1

(cm

/s2)

0 20 40 60 80

ur/1r1 (cm)

Ti=2.98sT1=2.906s

21

0 20 40 60 80

ur/1r1 (cm)

Vb/M

(cm

/s2)

1100 100

50 50

0 0

(a) (b)

Figure 5:ComputationofthefundamentalvibrationperiodfromcapacitycurveoftheequivalentinelasticSDFsystemsoftheNorthHollywoodHotel.(a)Longitudinaldirectionand(b)Transversedirection.


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Table 2:Comparisonof fundamentalvibration periods(sec) fromsystemidentification,linear-elasticmodel,andpushoveranalysisofnonlinearmodel.

BuildingSystemID

Longitudinaldirection

Linearmodel Nonlinearmodel SystemID

Transversedirection

Linearmodel Nonlinearmodel

ImperialCounty N/A N/A N/A 0.41 0.41 0.48

ShermanOaks 2.93 2.67 2.51 2.93 2.94 2.62

NorthHollywood 2.64 2.57 2.78 2.82 2.98 2.91

Watsonville 0.24 0.27 0.46 0.30 0.31 0.41

SantaBarbara 0.16 0.15 0.23 0.20 0.18 0.28

ATC-40CSM.Furtherdetailsofthisprocedure,denotedasthemodifiedADRSprocedure,areavailableintheFEMA440document[3].

Typically,thelocusofperformancepointsintheATC-40andFEMA-440CSMisplottedonthecapacitycurvefortheequivalentinelasticSDFsystemtoestimatethedisplacementSd. In this investigation, the displacement Sd (or Sa =Sd/(2/ Te)2) is used to compute peak roof displacementusing(1),(6),(11),and(13)forFEMA-356CM,ASCE-41CM,ATC-40CSM,andFEMA-440CSM,respectively,whichisthenplottedonthepushovercurveofthebuilding.SuchaplotpermitsdirectcomparisonoftargetdisplacementfromtheCSMprocedureandtherecordeddisplacement.

The pushover curves needed in implementing theselectedproceduresweredevelopedforfundamental-modeheight-wise distribution of lateral loads defined by s =m1in whichm is the mass matrix, 1 is the vector offundamentalmodeshape,and istheloadmultiplierduringpushover analysis. The fundamental mode distribution isthe only distribution specified in the ATC-40, FEMA

440, and ASCE-41 pushover procedures whereas it is oneof the distributions specified in the FEMA-356 pushoverprocedure.

5.1. Imperial County Services Building. The Imperial County ServicesBuildingisuniqueamongthefiveselectedbuildinginthisinvestigationbecauseitcollapsedduringtheselectedearthquake. The strength and stiffness of this buildingwasprovidedprimarilybymoment-resistingframesinthelongitudinal direction and shear walls in the transversedirection.Thepostearthquakeinvestigation[15]aswellthepushoveranalysis[14]indicatedthatthisbuildingcollapsedin the longitudinal direction due to concrete crushing at

basesofcolumnsinthemoment-resistingframes.Obviously,thefourprocedurescouldnotbeappliedtoestimatepeakdisplacementof thisbuildingin thelongitudinaldirection.Thecollapseof thebuilding in the longitudinal direction,however, did not significantly influence its stiffness andstrength in the transverse direction because shear walls,which provide most of the stiffness and strength in thisdirection, did not exhibit significant damage. Therefore,theseprocedurescouldbeappliedtoestimatethepeakroofdisplacementinthetransversedirection.However,theresultsforthisbuildingshouldbeviewedwithcareastheseresultsincludeerrors associated withthemodelingand analyticalprocedure, mentioned previously, aswellasdue toeffects

offailurein the longitudinaldirection onresponse inthetransversedirection.

TheresultspresentedinFigure7fortheImperialCountyServicesBuildingduetothe1979ImperialValleyearthquakeshowthattheFEMA-356CM,ASCE-41CM,ATC-40CSM,and FEMA-440 CSM lead to target roof displacement of6.98cm,7.60cm,5.64cm,and5.46cm,respectively;thepeak

roof displacement derived from recorded motions duringthis earthquake is 5.78cm. The differences between thepeak roofdisplacements fromthe FEMA-356CMandtheASCE-41 CM are clearly due to different values of thecoefficientthatconvertsthepeakdisplacementofalinear-elasticSDFsystemtothatofaninelasticSDFsystembetweenthe two CM procedures. Recall that the factor to convertthe peak displacement of a linear-elastic SDF system tothatofan inelasticSDFsystemis equaltoC1C2C3 fortheFEMA-356 CM (1) and C1C2 for the ASCE-41 CM (6).Furthermore, values of the individual coefficients betweenthetwoproceduresdifferforthesamevalueofRandTe (see(3) and (4)fortheFEMA-356CM,and(7) and (8)forthe

ASCE-41CM).Althoughquitedifferentinimplementationdetails,thetwoCSMproceduresATC-40andFEMA-440ledtoverysimilarestimatesoftargetdisplacementforthisbuilding.

5.2. Sherman Oaks Commercial Building. The presentedresults indicate that the two CM proceduresFEMA-356andASCE-41provideidenticalestimateofthepeakroofdisplacements of the Sherman Oaks building: the roofdisplacementis27.98cm(Figures8(a)and8(b)).SuchisthecasebecausethecoefficientC1 intheFEMA-356CM(3) and C1 andC2 intheASCE-41CM((7) and (8))areallequaltounitybecausefundamentallongitudinalvibrationperiodof

thisbuildingisthelongerthanthethresholdperiodvalue,and C3 intheFEMA-356CM(4)isequaltounityduetopositive postyield stiffness. The ATC-40 CSM and FEMA440CSMprovidepeakroofdisplacementof24.26cmand27.09cm,respectively(Figures 8(c)and8(d)).UnlikethetwoCM procedures, the two CSM procedures lead to slightlydifferentvaluesoftheroofdisplacement.ThisdifferenceisduetodifferentvaluesofeffectiveperiodanddampingratiousedintheseCSMprocedures(see(12) and (14)).Thepeakroof displacement derived from recorded motions of thisbuildingduringtheselectedearthquakeis33.6cm.

Allfourproceduresleadtoidenticalpeakroofdisplacementinthetransversedirection:thepeakroofdisplacement


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ut=5.7

8cm

uc=6.9

85cm

121086420

Roofdisplacement(cm)

0

3000

6000

9000

12000

15000

18000

Bas

eshear

(kN)

ut=5.7

8cm

uc=7.6

01cm

121086420


0

3000

6000

9000

12000

15000

18000

Bas

eshear

(kN)

(a) FEMA-356CM (b) ASCE-41CM

uc=5.6

44cm

ut=5.7

8cm

121086420


0

3000

6000

9000

12000

15000

18000

Baseshe

ar

(kN)

ICSBtransverseu

c=5.4

65cm

ut=5.7

8cm

121086420


0

3000

6000

9000

12000

15000

18000

Baseshe

ar

(kN)

(c) ATC-40CSM (d) FEMA-440CSM

Figure 7:Computationof the roof displacement from theFEMA-356 CM,ASCE-41CM, ATC-40CSM, andFEMA-440CSM in thetransversedirectionoftheImperialCountyServicesBuilding.

is equal to 17.98cm (Figure9). Such is the case becausethe building in the transverse direction remains in thelinearelasticrangeduringthe1994Northridgeearthquake.Recall that the coefficients C1, C2, and C3 in the FEMA356 NSP ((3) and (4)) as well as the coefficients C1 andC2 intheASCE-41NSP((7) and (8))areequaltooneforalinearly-elasticsystem.Furthermore,thevibrationperiodanddampingratiointheATC-40CSMandFEMA-440CSMremainequaltothatofalinear-elasticsystemfor = 1(see(12)and (14)).Thepeakroofdisplacementderivedfromtherecordedmotionsofthisbuildinginthetransversedirectionis22.71cm.

The presented results also indicate that the peak roofdisplacements from the four procedures for the ShermanOaksbuildingare less thanthose fromrecorded motions.Such is the case because these procedures attempt tocapture the response only due to the fundamental mode.Such procedures, obviously, cannot capture the responsedue to higher modes; several higher modes contribute tothe response of the Sherman Oaks Commercial Building[14].

5.3. North Hollywood Hotel. Althoughstronglyshaken,theNorth Hollywood Hotel remainedwithinthe linear-elasticrange in both the longitudinal and transverse directions

duringthe1994Northridgeearthquake.Theresultsforthisbuilding arepresented only for the transverse directionthe direction with the larger roof displacement. As notedpreviouslyfortheShermanOaksbuildingin thetransversedirection, all four procedures provide estimates of peakroof displacement that is identical and equal to 14.33cm(Figure10). The peak roof displacement derived fromrecordedmotionsofthisbuildingis17.46cm.ForreasonssimilartothoseidentifiedpreviouslyfortheShermanOaksCommercial Building, the lower estimate from the fourprocedures is due to the inability of these procedures tocapturehighermodeeffectsthatcontributesignificantlyto

thetransverseresponseofthisbuilding[14].

5.4. Watsonville Commercial Building. TheresultspresentedinFigure11fortheWatsonvilleCommercialBuildingindicate that the estimate of the peak roof displacements inthelongitudinaldirectionfromtheFEMA-356CM,ASCE41 CM, ATC-40 CSM, and FEMA-440 CSM is 3.13cm,3.01cm,3.12cm,and2.98cm, respectively. Thepeakroofdisplacementderivedfromrecordedmotionsofthisbuildingis 3.33cm. As noted previously for the Imperial CountyServices Building, the two CM procedures provide different estimates of peak roof displacement because variouscoefficientsintheseproceduresdifferforshortervibration


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104 104

2 2

1.5 1.5

uc=27.

98cm

ut=33.

6cm

0 10 20 30 40 50 60


uc=27.

98cm

ut=33.

6cm

0 10 20 30 40 50 60


Baseshear

(kN)

Baseshear

(kN)

Baseshear

(kN)

Baseshear

(kN)

1

0.5

0

1

0.5

0


104 104

2 2

1.5 1.5

uc=24.

26cm

ut=33.

6cm

0 10 20 30 40 50 60


SOlongitudinalu

c=27.

09cm

ut=33.

6cm

0 10 20 30 40 50 60


1 1

0.5

0

0.5

0


Figure 8:Computationof the roof displacement from theFEMA-356 CM,ASCE-41CM, ATC-40CSM, andFEMA-440CSM in the

longitudinaldirectionoftheShermanOaksCommercialBuilding.

periods; the fundamental longitudinal vibration period ofthisbuildingis0.27sec(Table2).

The building remained essentially in the linear elasticrangeinthetransversedirection(Figure12).ThetwoCMproceduresprovideestimatesofthepeakroofdisplacementsthatareessentially identical: theFEMA-356CMprovidesavalueof2.75cmandASCE-41CMgivesavalueof2.68cm(Figures 12(a) and 12(b)). The ATC-40 CSM providesan estimate of the peak roof displacement of 2.42cm(Figure12(c))whereasthevaluefromtheFEMA-440CSMis2.76cm(Figure12(d)).Thepeakroofdisplacementofthis

buildingderivedfromitsrecordedmotionis1.93cm.Unlike the peak roof displacements of the Sherman

OaksCommercialBuildingandtheNorthHollywoodHotel,the two CM or the two CSM procedures do not providean identical estimate of the peak roof displacement ofthe Watsonville Commercial Building in the transversedirection even though the building remains within thelinear-elastic range (Figure12). This occurs due to thediscrepancy between the effective fundamental vibrationperiodestimatedfrom(2)andthevibrationperiodestimatedfromthepushovercurve(Table2),andtheactualdampingratioandthedampingratioofthelinear-elasticsystemusedinthisinvestigation.

5.5. Santa Barbara Office Building. Althoughstronglyshakenduringthe1978SantaBarbaraearthquake,theSantaBarbaraOfficeBuildingremainedessentiallywithinthelinear-elasticrangeinbothdirections.Forreasonsofbrevity,thecomputationofthetargetdisplacementfromthefourproceduresispresentedonlyin thelongitudinaldirection.Thepresentedresultsindicatethatestimateoftheroofdisplacementinthelongitudinal direction fromthe FEMA-356 CM, ASCE-41CM,ATC-40CSM,andFEMA-440CSMis0.36cm,0.35cm,0.57cm, and 0.56cm, respectively, (Figure13). The peakvalueoftheroofdisplacementderivedfromrecordedmotion

is 0.68cm. The four procedures do not provide identicalestimates of thepeak roofdisplacements, even though thebuilding remained within linear elastic range because ofreasons noted previously for the Watsonville CommercialBuilding.

6. Variability in Target Displacement

The estimates of the target displacement from the fournonlinearstaticproceduresFEMA-356CM,ASCE-41CM,ATC-40 CSM,andFEMA-440CSMalongwith thepeakroof displacement derived from recorded motions of the


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104 104

2 2

1.5 1.5

uc=17.

98cm

ut=22.

71cm

0 10 20 30 40 50 60


uc=17.

98cm

ut=22.

71cm

0 10 20 30 40 50 60


Baseshear

(kN)

Baseshear

(kN)

Baseshear

(kN)

Baseshear

(kN)

1

0.5

0

1

0.5

0


104 104

2 2

1.5 1.5

uc=17.

98cm

ut=22.

71cm

0 10 20 30 40 50 60


SOtransverseu

c=17.

98cm

ut=22.

71cm

0 10 20 30 40 50 60


1 1

0.5

0

0.5

0


Figure 9:Computationof the roof displacement from theFEMA-356 CM,ASCE-41CM, ATC-40CSM, andFEMA-440CSM in the

transversedirectionoftheShermanOaksCommercialBuilding.

selectedbuildingsaresummarizedinTable3 inthelongitudinaldirectionandTable4inthetransversedirection.Theseresultspermit the following importantobservations aboutvariability of target displacement estimates from variousnonlinearstaticprocedures.

First,variousnonlinearstaticproceduresprovideidenticalestimatesoftargetdisplacementoflong-periodbuildingsthat remain within the linear elastic range (see NorthHollywood Hotel in the longitudinal direction in Table3,andSherman OaksCommercial BuildingandNorth Hol

lywood Hotel in the transverse direction inTable4).Thisis consistent with the expectation that nonlinear staticproceduresshouldprovideidenticalestimatesforbuildingsrespondinginthelinearelasticrange.

Second,variousnonlinearstaticproceduresmayprovidedifferent estimates of target displacement of short-periodshear-wall buildings that remain within the linear elasticrange(seeSantaBarbarabuildinginthelongitudinaldirectioninTable3 andWatsonvilleandSantaBarbarabuildingsinthetransversedirectioninTable4).ThisoccursbecauseofthesensitivityofthepeakdisplacementoftheequivalentSDF system to period and damping in the short-periodrange.Recallthattheremaybeaslightdiscrepancyinthe

effective fundamental vibration period estimated from(2)andthevibrationperiodestimatedfromthepushovercurveforthesebuildings(Table2),andtheactualdampingratioandthedampingratioofthelinear-elasticsystemusedinthisinvestigation.Itisusefultopointoutthatthelargevariabilitynotedhereisduetouseofresponsespectrumforindividualgroundmotion,whichcanbeveryjaggedintheshort-periodrange;thisvariabilitywouldbemuchlessifasmoothdesignspectrumisused.

Finally,thevariabilityinthetargetdisplacementoflong-

periodbuildingrespondinginthenonlinearrangeis muchsmaller compared to short-periodbuildings responding inthenonlinearrange.Forexample,thetargetdisplacementofthelong-periodShermanoaksbuildinginthelongitudinaldirection fromthe four nonlinear static procedures variesfrom 27.98cm to24.26cm (Table3),a variation ofabout13%, whereas that for the short-period Imperial CountyServices building in the transverse direction varies from7.60cmto5.46cm(Table4),avariationofabout28%.ThisisthecasebecauseofsensitivityofvariouscoefficientsintheCMprocedureandtheequivalentperiodanddampingintheCSMprocedurestodegreeofnonlinearity,thatis,valueofR.


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12000

uc=14.

33cm

ut=17.

46cm

uc=14.

33cm

ut=17.

46cm

12000

10000 10000

Baseshear

(kN)

Bases

hear

(kN)

8000

6000

4000

2000

Baseshear

(kN)

Bases

hear

(kN)

8000

6000

4000

2000

0 00 20 40 60 80 0 20 40 60 80

Roofdisplacement(cm) Roofdisplacement(cm)


12000 12000

10000 10000

uc=14.

33cm

ut=17.

46cm

0 20 40 60 80


NHtransverseu

c=14.

33cm

ut=17.

46cm

0 20 40 60 80


8000

6000

4000

8000

6000

4000

2000 2000

0 0


Figure 10:ComputationoftheroofdisplacementfromtheFEMA-356CM,ASCE-41CM,ATC-40CSM,andFEMA-440CSMinthetransversedirectionoftheNorthHollywoodHotel.

Table 3:Peak roofdisplacementestimated from four nonlinearstaticprocedures andderivedfrom recorded motions in longitudinaldirection;allvaluesofdisplacementsareincm.

BuildingFEMA-356CM ASCE-41CM

Nonlinearstaticprocedure

ATC-40CSM FEMA-440CSM Recorded

ImperialCounty N/A N/A N/A N/A N/A

ShermanOaks 27.98 27.98 24.26 27.09 33.60

NorthHollywood 10.17 10.17 10.17 10.17 9.75

Watsonville 3.13 3.01 3.12 2.98 3.33

SantaBarbara 0.36 0.35 0.57 0.56 0.68

Table 4:Peakroofdisplacementestimatedfromfournonlinearstaticproceduresandderivedfromrecordedmotionsintransversedirection;allvaluesofdisplacementsareincm.

BuildingFEMA-356CM ASCE-41CM

Nonlinearstaticprocedure

ATC-40CSM FEMA-440CSM Recorded

ImperialCounty 6.98 7.60 5.64 5.46 5.78

ShermanOaks 17.98 17.98 17.98 17.98 22.71

NorthHollywood 14.33 14.33 14.33 14.33 17.46

Watsonville 2.75 2.68 2.41 2.76 1.93

SantaBarbara 1.06 0.92 1.06 1.19 1.28


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ut=1.9

26cm

uc=2.7

49cm

86420


0

5000

10000

15000

Bas

eshear

(kN)

ut=1.9

26cm

uc=2.6

78cm

86420


0

5000

10000

15000

Bas

eshear

(kN)


ut=1.9

26cm

uc=2.4

19cm

86420


0

5000

10000

15000

Baseshe

ar

(kN)

WTtransverseu

t=1.9

26cm

uc=2.7

58cm

86420


0

5000

10000

15000

Baseshe

ar

(kN)


Figure 12:Computationofthe roof displacementfromthe FEMA-356 CM,ASCE-41CM,ATC-40 CSM,andFEMA-440CSM inthe transversedirectionoftheWatsonvilleCommercialBuilding.

prediction for some buildings (see IC-NS and SB-EW inFigure14) but worsefor others (see SO-EW inFigure14)comparedtotheCMprocedure.Forotherbuildings,thetwoproceduresleadtoessentiallysimilarlevelsofaccuracy(seeSO-NS,NH-EW,andNH-NSinFigure14).

8. Conclusions

This investigation compared the target roof displacementcomputed from the four currently used proceduresFEMA-356 CM, ASCE-41 CM, ATC-40 CSM, and FEMA

440 CSMwith the peak roof displacement derived fromrecordedmotionsoffivereinforced-concretebuildingswiththeaimofdevelopinganimprovedunderstandingofthefollowingtwoquestions:(1)howmuchdoesthetargetdisplacement varies among the four nonlinear static procedures?and(2)cantheengineeringprofessionaccuratelypredicttheresponseofarealbuildingduringanearthquakeeventusingcurrentlyavailablemodelingtechniquesandpushoveranalysisprocedures?Themodels of selectedbuildingsutilized in this investigation are developed using generallyacceptedengineeringpractice.Thesemodelswerevalidatedbut not intentionally calibratedagainst the recorded data.Thiscomparisonhasledtothefollowing conclusions.

Thenonlinearstaticproceduresmayleadtosignificantlydifferent estimates of target displacement, particularly forshort-periodbuildings responding in the nonlinearrange;thelargestvariationnotedin thisinvestigationapproached28%fortheImperialCountyServicesbuilding.Thevariationwasmuchsmallerforlong-periodbuildingsrespondinginthenonlinearrange. These observations areunlikely tobeaffectedbytheinaccuraciesassociatedwithmodelingerrorsbecausethesamemodelwasusedduringimplementationoftheseprocedures.

Thecurrentnonlinearstaticprocedures,whenappliedtononlinearmodelofthebuildingdevelopedusinggenerally

acceptedengineeringpracticemayleadtoeithersignificantover-estimation or under-estimation of the targetroof displacementwhencomparedwiththepeakroofdisplacementobserved during a selected earthquake. The error rangedbetween50%underestimationto40%overestimation.

It is useful to note that poor estimates of target displacement for a few of the buildings from the variouspushover analysis procedures may be due to severe near-fault effects as noted previously by Akkar and Metin [5].Additional errors may occur due to loss of accuracy inrecorded roofdisplacementresulting fromdata processingtechniques described previously. Furthermore, nonlinearstaticproceduresaredesignedtoprovideaccurateestimate


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15

15000

ISRNCivilEngineering

15000

10000 10000

uc=0.3

6cm

ut=0.6

81cm

2.50 0.5 1 1.5 2

uc=0.3

52cm

ut=0.6

81cm

2.50 0.5 1 1.5 2

Baseshe

ar

(kN)

Bas

eshear

(kN)

Baseshe

ar

(kN)

Bas

eshear

(kN)

5000 5000

0 0

Roofdisplacement(cm) Roofdisplacement(cm)

(a) FEMA-356CM

2.5

uc=0.5

71cm

ut=0.6

81cm

0 0.5 1 1.5 2


(b) ASCE-41CM

15000 15000

uc=0.5

64cm

SBlongitudinalu

t=0.6

81cm

10000

5000

10000

5000

0 00 0.5 1 1.5 2 2.5



Figure 13:ComputationoftheroofdisplacementfromtheFEMA-356CM,ASCE-41CM,ATC-40CSM,andFEMA-440CSMinthelongitudinaldirectionoftheSantaBarbaraOfficeBuilding.

80

60

40

20

0

IC-NS SO-EW SO-NS NH-EW NH-NS WT-EW WT-NS SB-EW SB-NS

Building

Error

(%)

20

40

60

80

FEMA-356CM ATC-40CSM

ASCE-41CM FEMA-440CSM

Figure 14:PercenterrorinpeakroofdisplacementsfromtheFEMA-356CM,ASCE-41CM,ATC-40CSM,andFEMA-440CSM.

of the median response. Therefore, it is not surprising staticproceduresandinaccuraciesassociatedwithnonlinearthat large errors are noted when these procedures are modeling.appliedto predict target displacement of buildings during Thedatapresentedinthisinvestigationalsoprovidesaindividualgroundmotions.Thelargeerrorsnotedhereare comparativepredictioncapabilityofvariousnonlinearstaticalso because of a combination of errors due to nonlinear procedures. Although limited in size, this data indicates


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that (1) the ASCE-41 CM, which is based on recentimprovementsto theFEMA-356CM suggested in FEMA440document,doesnotnecessarilyprovidebetterpredictionofroofdisplacement,(2)theimprovedFEMA-440CSMalsomaynotprovidebetterpredictionofpeakroofdisplacementscomparedtotheATC-40CSM,and(3)thereisnoconclusive

evidencethattheCMprocedures(FEMA-356orASCE-41)provide better predictions of the peak roof displacementcomparedtotheCSMprocedure(ATC-40orFEMA-440)orvice-versa.

Acknowldegments

This investigation is supported by the California DepartmentofConservation,CaliforniaGeologicalSurvey,StrongMotion Instrumentation Program (SMIP), Contract no.1005-832. This support is gratefully acknowledged. Theauthors aregrateful toDrs.AnthonyShakal, Moh Huang,andErolKalkanofSMIPforprovidingtherecordedmotionsandstructuralplansoftheselectedbuildings.TheauthorsalsoacknowledgethecontributionstothisresearchinvestigationbyMatthewHazenandJoeyGivens,undergraduatestudentsatCalPoly,SanLuisObispoandbyDr.Dae-HanJun,VisitingProfessorfromDongseoUniversity,Korea.

References

[1] FEMA 356, Prestandard and Commentary for the SeismicRehabilitation of Buildings, FEMAPublicationno. 356, TheAmericanSocietyofCivilEngineersfortheFederalEmergencyManagementAgency,Washington,DC,USA,2000.

[2] ATC-40, Seismic evaluation andretrofitof concretebuildings,volumes1and2,Tech.Rep.ATC-40,AppliedTechnologyCouncil,RedwoodCity,Calif,USA,1996.

[3] FEMA-440,Improvement of Nonlinear Static Seismic AnalysisProcedures, AppliedTechnology Council for DepartmentofHomelandSecurity,FederalEmergencyManagementAgency,Washington,DC,USA,2005.

[4] E. Miranda and J. Ruiz-Garca, Evaluation of approximatemethodsto estimatemaximuminelasticdisplacementdemands, Earthquake Engineering and Structural Dynamics,vol.31,no.3,pp.539560,2002.

[5] S. Akkar andA. Metin, Assessment of improved nonlinearstatic procedures in FEMA-440,Journal of Structural Engineering,vol.133,no.9,pp.12371246,2007.

[6] ASCE/SEI-41, Seismic rehabilitation of existing building,

ASCE Standard no. ASCE/SEI 41-06, American Society ofCivilEngineers,Reston,Va,USA,2007.

[7] A.C.GuyaderandW.D.Iwan,Determiningequivalentlinearparametersforuseinacapacityspectrummethodofanalysis,

Journal of Structural Engineering,vol.132,no.1,pp.5967,2006.

[8] A. F. Shakal, M. J. Huang, and V. Graizer, Strong-motiondata processing, in International Handbook of EarthquakeEngineering Seismology, Part B, W. H. K. Lee, H. Kanamori, P.C.Jennings,andC.Kisslinger,Eds.,pp.967981,AcademicPress,Amsterdam,TheNetherlands,2003.

[9] A.K.Chopra,Dynamics of Structures: Theory and Applicationsto Earthquake Engineering,Prentice-Hall,UpperSaddleRiver,NJ,USA,3rdedition,2007.

[10] F. McKenna and G. Fenves, The Opensees Command Language Manual: 1.2, Pacific Earthquake Engineering Center, University of California, Berkeley, Calif, USA, 2001,http://opensees.berkeley.edu/.

[11] J. B. Mander, M. J. N. Priestley, and R. Park, Theoretical stress-strainmodel forconfinedconcrete,Journal of Structural Engineering,vol.114,no.8,pp.18041826,1988.

[12] R.K.GoelandA.K.Chopra,EvaluationofmodalandFEMApushoveranalyses:SACbuildings,EarthquakeSpectra,vol.20,no.1,pp.225254,2004.

[13] K.J.Elwood,A.B.Matamoros,J.W.Wallaceetal.,UpdatetoASCE/SEI41concreteprovisions,Earthquake Spectra,vol.23,no.3,pp.493523,2007.

[14] R.K.GoelandC.Chadwell,Evaluationofcurrentnonlinearstaticproceduresforconcretebuildingsusingrecordedstrong-motion data, Data Utilization Report, California StrongMotionInstrumentationProgram,CDMG,Sacramento,Calif,USA,2007,http://digitalcommons.calpoly.edu/cenv fac/172/.

[15] ATC-9,Anevaluationof theimperialcountyservices building:earthquakeresponseandassociateddamage,Tech.Rep.ATC-9, AppliedTechnology Council,PaloAlto, Calif, USA,

1984.
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Variability and Accuracy of Target Displacement From Nonlinear St