TallBuildingsandDamping:AConcept-Based Data-DrivenModel
Abstract: Moderntallbuildingsarecharacterizedbytheirslendernessandsensitivitytoresonantwindeffects.Thisisespeciallytrueconsideringtheacceleration-basedmotionperceptioncriteriaunderwhichtheymustbedesigned.Inlightofthesignificanceoftheresonant response,dampingplaysanimportantroleinthedesignoftallbuildings.Unfortunately,unlikeothermechanicalcharacteristicsofstructures, dampingisfarmoredifficulttoestimate.Thisisduetotheinherentcomplexityandhighnumberofmechanismsresponsiblefordamping.For thisreason,theexperimentaldeterminationofdampinglevelsfortallbuildingsfromfull-scaledatacollectedduringmonitoringprogramshas obtainedatremendousamountofinterestoverthepastyears.Thispaperfirstlyreviewsthepredictivedampingmodelsthatareavailableinthe literaturehighlightingtheirmeritsandshortcomingsinlightoftheextensiveexperimentaldampingdatacollectedoverthepastfewyears. Anovelamplitude-dependentdata-drivenmodelisthenproposedbasedonafullyprobabilisticdescriptionofthemechanismsthatare hypothesizedtogeneratethemajorityofdampingintallbuildings.Finally,theproposedmodeliscalibratedtoanumberofspecificbuildings demonstratingitsrobustness. DOI: 10.1061/(ASCE)ST.1943-541X.0000890. ©2014AmericanSocietyofCivilEngineers.
Authorkeywords: Damping;Amplitudedependency;Data-drivenmodels;Predictivemodels;Tallbuildingdesign;Dynamiceffects.
Introduction
Theresponseoftallbuildingstoenvironmentalloads,suchaswind andseismicevents,dependsontheirdynamicpropertiessuchas mass,stiffness,anddamping.Althoughcharacteristicslikemass andstiffnessarefairlyeasytoestimate,dampingisconsiderably hardertoestimatewithanysortofcertainty.Beingawarethatthis constitutesadifficultyinthedesignofstructures,itisparticularly significantinthecaseoftallbuildingsduetothestringentlimitationsimposedontheirperformancebyacceleration-basedmotion perceptioncriteriatoguaranteeasatisfactoryhabitability.The difficultyinobtainingaccurateestimatesofdampinglevels incivilstructuresmaybetracedbacktothecomplexnatureof themechanismsthatareattherootofthisphenomenon.Indeed, itiswellknownthatastructurewillbedynamicallydampedby mechanismswithdifferentcharacteristics.Examplesofsuchmechanismsarethecomplexmolecularinteraction(materialdamping), Coulombfrictionbetweenmembersandconnectionsor,froma moreglobalperspective,othermechanismsdependingonthetype ofstructuralsystem,foundationtype,contributionsofinteriorpartitions,exteriorcladding,andothernonstructuralinputs.Damping isingeneraldifficulttomodelmathematicallybecauseeachof theaforementionedsourceswillfollowverydifferentlaws(forinstance,materialdampingmaybemodeledusingviscousmodels, whereasaCoulombmodelmaybeadoptedforfriction)causingthe totaldampingtofollowalawwhichmaybeconsideredunique
1ResearchAssistantProfessor,NatHazModelingLaboratory,Dept.of CivilandEnvironmentalEngineeringandEarthSciences,Univ.ofNotre Dame,NotreDame,IN46556(correspondingauthor).E-mail:sspence@ nd.edu
2RobertM.MoranProfessorofEngineering,NatHazModelingLaboratory,Dept.ofCivilandEnvironmentalEngineeringandEarthSciences, Univ.ofNotreDame,NotreDame,IN46556.
Note.ThismanuscriptwassubmittedonJuly3,2012;approvedon June4,2013;publishedonlineonJune6,2013.Discussionperiodopen untilJuly6,2014;separatediscussionsmustbesubmittedforindividualpapers.Thispaperispartofthe JournalofStructuralEngineering,©ASCE,ISSN0733-9445/04014005(15)/$25.00.
foreachstructure.Havingsaidthissomegeneraltrendscanbe identifiedifparameterssuchasbuildingmaterial,structuralsystem, andfoundationtypeareconsidered.
Inlightoftheimportanceanddifficultiesinmodelingdamping, thepossibility,givenbythedevelopmentofdataacquisitiontechnologiesoverthelastthreedecades,toexperimentallydetermine dampingcharacteristicsfromfull-scalemeasurementshasbeen thoroughlyinvestigated.Thesestudieshavebeenmadepossible byanumberofmonitoringprojectsperformedthroughoutthe worldonwindorseismicallyexcitedtallbuildings(Jeary1986; Ohkumaetal.1991; Çelebiand Şafak1992; LittlerandEllis 1992; Çelebi1993; TamuraandSuganuma1996; Lietal.1998; KijewskiandKareem1999; Lietal.2004b; Lietal.2005; Lietal. 2006; Kijewski-Correaetal.2006; Lietal.2011; Guoetal.2012), includingtheongoingprogramsoftheUniversityofNotreDame concerningthemonitoringofthreetallbuildingsinChicago (Kijewski-Correaetal.2006)andanumberoftallbuildingscurrentlybeingmonitoredinChina(Lietal.1998; Lietal.2003a; Lietal.2004a; Lietal.2008; Lietal.2011; Guoetal.2012). Obviously,thecollectionoffull-scaledataisonlyonepartof theequation.Oncethedatahavebeencollected,appropriatedampingestimationtechniquesmustbeapplied.Becauseoftheconsiderablediversitythatexistsintoday’sbuildings,togetherwiththe practicalconstraintsplacedondataacquisitionsystems(e.g.,limitedinstrumentationpoints,noise,sensoraccuracy),thenumberof methodsthathavebeenproposedfordampingestimationisconsiderable(Kijewski-CorreaandCycon2007).
Overthepastthreedecades,oneofthemostinterestingaspects ofdampingintallbuildingstohaveemergedfromtheanalysisof full-scaledataisitsnonlinearnature.Thenonlinearityisseenas adependencyofdampingontheamplitudeofvibration.This dependency,generallyreportedasanincreaseindampingwith amplitude,hasbeenthesubjectofnumerousstudiesandinvestigationsovertheyears(HartandVasudevian1975; Davenportand Hill-Carroll1986; Jeary1986; Jeary1996; Jeary1997; Fangetal. 1999; Lietal.2000b; Lietal.2008; Lietal.2011; Guoetal.2012). Modelshavebeenproposedforpredictingdampinglevelsinfunctionofvibrationamplitude(DavenportandHill-Carroll1986;
©ASCE04014005-1J.Struct.Eng.
Jeary1986; Lagomarsino1993; Jeary1996; Jeary1997; Satake etal.2003).Alltheaforementionedmodelswereproposedthrough theanalysisofdatabasesofdampingvaluescollectedforvarious vibrationamplitudesandbuildings.Indeed,theymaybeconsidered asphenomenologicalmodelsthathavebeengivenacertainphysicalsenseaposteriori.Untilrecentlythisapproachwastheonly realisticpossibility.However,theincreaseofhighqualitydamping estimatesmadeonspecificbuildingsforrelativelylargeamplitude rangeshasmadepossibletheinvestigationofmoreconcept-based dampingmodelsthatwouldhaveabetterchanceofdescribingthe dampingbehavioroftallbuildings,explaining,forinstance,whyin certaincasesdampinghasbeenseentodecreaseafteracertaincriticalamplitude(Tamura2005; SmithandWillford2007; Tamuraand Yoshida2008; AquinoandTamura2012).Anothersystemnonlinearitythathasbeenobservedduringtheextensivemonitoringof tallbuildingsoverthepast12yearsisthetendencyfortheirnatural frequenciestodecreasetoacertainextentwithamplitude(Tamura andSuganuma1996; Lietal.2000a; Kijewski-CorreaandPirnia 2007; Kijewski-Correaetal.2007).Thishasbeenseentogohand inhandwiththegeneralincreaseofdampingwithamplitude.However,althoughthisphenomenonhasbeenobservedandbest-of-fit linesproposedforspecificbuildings,nomodelsortheorieshave beenproposedthatbringthesetwophenomenatogether,asis mostlylikelythecase.
Inlightoftheaforementionedconsiderationsandtheamount ofdatathathasbeencollectedoverthepastfewyearsconcerning tallbuildings,thetimeseemsrighttoinvestigatethepossibilityof definingspecificconcept-baseddata-drivenmodelsthataremore robustandcompletethanthosecurrentlyavailable.Thispaper focusesonthispossibility.
StructuralDamping
Asbrieflymentionedintheintroduction,dampinghasmanysourcesofvaryingcomplexity.Itisthecombinationofvariousphysical phenomenathatcausesdampingtobeparticularlycomplicatedto estimate,comparedtomassorstiffness.Indeed,althoughthestate variablesthatgoverninertialforcesorstiffnessareeasilyidentifiable,inthecaseofdampingthisisnottrue.Traditionallyinthe dynamicresponseanalysisofstructuresdampingismodeledproportionaltovelocity(linearviscousdamping).Inmanycases,however,thischoiceisdictatedmorebyconveniencethanbyanactual physicalmeaningaswiththischoicethesystemwillbegoverned bylinearsecond-orderdifferentialequations.
Broadlyspeaking,dampingaffectingstructuressuchastall buildingsmaybecategorizedintostructuraldampingandnonstructuraldamping.Structuraldampingcomprisesdampingsources suchasintrinsicmaterialdamping,frictionaldamping,andfoundationdampingduetosoil-structureinteraction.Nonstructural dampingreferstoallothersourcessuchasaerodynamicdamping, causedbythevibrationofastructureimmersedinafluid(Kareem andGurley1996),orpossiblenonlinearitiespresentinloadingor structuralsystemthatmaycause,forexample,couplingbetween orthogonalmodeswithequalfrequencies[thisindirectlyaddsto thedampingowingtoanenergytransferbetweenmotiondirections (Kareem1982; KareemandGurley1996)].Althoughforcertain systemsnonstructuraldampingcanbeveryimportant(forexample aerodynamicdampingonflexiblestructures),ingeneral,fortall buildingsitmaybeneglectedforthecurrentgenerationofbuildings(Marukawaetal.1996),butmayhavetobeconsideredfor supertallbuildingsofthefuture.Also,nonlinearitiesduetoother aeroelasticeffectsorparticularmechanicallock-inphenomena, suchasthosecitedbefore,arepresentlyrare,eveniftheymaycome
intoplayinthesupertallbuildingsthatarebeingplanned.This paper,therefore,willfocusonthemodelingandestimationofstructuraldampingthatwillbereferredto,fromthispointon,simplyas damping.
Forlinearviscousdamping,theratio ξ betweenthedamping presentinthesystemandthecriticaldampingmaybeestimated fromtheratiobetweenthedissipatedenergyinonecycleofresonantsteady-stateharmonicoscillationtothemaximumamountof energyaccumulatedinthestructureinthatcycle
Inthecaseofnonresonantharmonicloading,thelinearviscous dampingmodelhasaseriousdrawbackbecausetheratioof Eq.(1)dependsontheexciting/steady-stateresponsefrequency, acharacteristicnotobservedexperimentally.Hence,fornonresonantharmonicloading,otherdampingmodelsareprobablymore appropriate,suchasthehystereticdampingmodel(complexstiffnessdamping)whichisindependentoftheexciting/steady-state responsefrequency(CloughandPenzien2003).Havingsaidthis, Eq.(1)isusefulasageneraldefinitionofthedampingcapacityof astructure.Indeed,independentlyofthedampingmodel,Eq.(1) providesameanstofindwhatiscommonlydefinedastheequivalentviscousdampingratio.
Damping:FromEstimationtoDatabases
Full-ScaleMonitoringPrograms
Thesignificantincreaseinknowledgeconcerningdampingoverthe past30yearsisinparttheconsequenceofthenumerousmonitoring programsthathavebeenperformedonbuildingsallovertheworld. Thisincreasedmonitoringofthebuiltenvironmenthasbeenmade possiblebytherapidadvancementinacquisitiontechnologiesover thepastfewyearsandthegrowthofinterestinstructuralhealth monitoring.Thefirstcasesofsystematicmonitoringofbuildings underenvironmentalloadingoccurredduringthe1970s(Isyumov andBrignall1975; Taokaetal.1975; Isyumovetal.1988; Brown 2003).Thenduringthe1980sprogramssuchastheCalifornia Strong-MotionInstrumentationofStructures(CSMIS)ofthe CaliforniaGeologicalSurveyallowedthecollectionofasignificantquantityofdatarelatedtotheseismicresponseofbuildings (Bongiovannietal.1987; Şafak1989b; Çelebiand Şafak1991; ŞafakandÇelebi1991; Çelebiand Şafak1992; Çelebi1993; Çelebi1996).Thelast20yearshaveseenagrowinginterestinthe monitoringofwind-excitedtallbuildings(Lietal.1998; Xuand Zhan2001; Lietal.2003a; Lietal.2004a; Campbelletal.2005; Lietal.2005; Kijewski-Correaetal.2006; Pirniaetal.2007; Lietal.2007; Kimetal.2008; Lietal.2008; NiandZhou 2010; Guoetal.2012).Thisextensivemonitoringanddataanalysis havebeenespeciallyintenseinthePacificRimwhereanumberof buildingshavebeenmonitored,notonlyinambientwindconditions,butalsoundersevereweatherconditionssuchastyphoons.
TheEstimationofDamping
Thecollectionoffull-scaledataisobviouslyonlythefirststepin theexperimentalestimationofdamping.Oncethedatahasbeen collectedanappropriatedampingestimationtechniquemustbeapplied.Becauseofthecomplexityofdamping,thisisbynomeansa trivialtask.Indeed,intheliteratureavastnumberofmethodscan befound.Oneofthemostimportantfactorsintheselectionofan appropriatemethodiswhethertheinput(excitingfunction)was
measuredduringtheevent.Practically,thisonlyhappensinthe caseofseismicexcitation,whereasfordatacollectedduringwind eventsonlytheresponseofthesystemisknown.Thismakesa significantdifferenceintherobustnessofthetechniquesthat maybeapplied.
EstimationunderSeismicExcitation
Inthecaseofknowninput,anappropriateSystemIdentification (SI)techniquemaybeappliedforestimatingthedamping.Thetype ofproblemwillingeneralbedescribedbyanonlinearoptimization problem,withscopetheminimizationofanobjectivefunctiongivingameasureofthedifferencebetweenthemodelandmeasured responses.Themethodsaredistinguishedbythechoiceoftheobjectiveandtheminimizationschemeadopted(Kijewski-Correaand Cycon2007).Oneofthemorepopularschemesisbasedonthe regressivetime-seriesmodelingoftherecordedinputandoutput accelerationsusingtheleastsquaresminimization.Thedynamic propertiesmaythenbeextractedfromthepolesofthetransfer function(Ljung1987; Çelebi1993).Anotherpopularregression schemeistheAutoRegressivemodelwitheXogenousinput(ARX) whichhasbeenappliedfortheSIofanumberofmediumtotall buildingsunderseismicexcitation(Şafak1989b; Çelebi1993; Çelebi1996; Çelebi2006; RodgersandÇelebi2006).Inalternative,anARMAX(AutoRegressiveMovingAveragemodelwith eXogenousinput)modelmaybeusedasillustratedintheidentificationoftheEmbarcaderobuilding(Şafak1989b).Anotherway toperformSIistoviewthestructureasaDiscrete-TimeFilter (DTF)thatmustbedesignedfromtheknowledgeoftheinput andoutput(Şafak1989a; Şafak1991).Thismethodhasbeensuccessfullyappliedonanumberoftallbuildings(Şafak1989b; Şafak 1991; ŞafakandÇelebi1991; ŞafakandÇelebi1992; Şafak1993).
EstimationunderWindExcitation
Asmentioned,thedifference,fromasystemidentificationviewpoint,betweenwindandseismicallyexcitedstructuresisinthe practicalimpossibilitytomeasuretheinputforcesinthecasewind excitation.Also,owingtotherelativelylowamplitudesofthewind responses,dampingestimationmethodsappliedtowindresponse datamustbecapableofdistinguishingbetweensignalnoiseand response.Generallyspeaking,becausethewindforcemaybeconsideredbroadbandcomparedwiththenarrowbandnatureofthe system,theinputistakenashavingawhitenoisespectrumaround thestructuralfrequencies.Also,itiscommontoassumetheinput asstationaryandergodiccoupledwithalinearsystem,therefore yieldingastationaryandergodicresponse.Thisallowstheapplicationofanumberofoutput-onlydampingestimationschemesthat allowforthedampingestimationusingtemporalaveragesinplace ofensambles(Kijewski-CorreaandCycon2007).
Probablythemostconventionaltechniquethatfallsintothis categoryistheHalfPowerBandWidth(HPBW)method(Bendat andPiersol1987).Thedampingratioissimplyestimatedfrom theautopowerspectraldensityoftheresponseathalftheheight.
Somespecificexamplesofitsapplicationcanbefoundin(Dobryn etal.1987; Brown2003; Lietal.2005; Kijewski-Correaetal. 2006; Kijewski-Correaetal.2007; Pirniaetal.2007).Otherfrequencydomain-basedschemesincludespectralcurvefitting methodsandmethodssuchasmaximumlikelihoodestimators (Breukelmanetal.1993; Montpellier1996; Erwinetal.2007) orsimilar(LagomarsinoandPagnini1995).
Dampingestimationinthefrequencydomainhasbeenreplaced tosomeextentbytimedomainproceduresowingprimarilytoconcernsaboutsignalprocessingwhileapplyingthefastFouriertransformswhichhavebeenassociatedwithinflateddampingestimates. Oneofthefirsttimedomain-baseddampingestimationmethods appliedtofull-scalewindexcitedbuildingsisbasedonthedirect
estimationoftheautocorrelationfunction(Taokaetal.1975; IsyumovandHalvorson1984; Dobrynetal.1987; Masciantonio etal.1987).However,ithasbeenseenthatthismethodcangive quiteunreliabledampingestimatesifthereisalackofdata (DavenportandHill-Carroll1986).Theaforementioneddifficulties arepartlyresponsibleforthepopularityoftheRandomDecrement Technique(RDT)(KareemandGurley1996).Notonlyhasit beenshownthatthismethodisrelativelystableinestimating damping,butalsothatitcontinuestobeusableevenwhenmoderatenonlinearitiesandnonstationaritiesarepresent(Tamuraand Suganuma1996).
Discussion
Themethodologiesbrieflypresentedinthissectionfordamping estimationcansignificantlyaffectthevaluesobtained.Forthisreason,wheninvestigatingpossibletrendsindampingdatacoming fromtheapplicationofdifferentestimationtechniques,thepossibilityofvariationowingsimplytotheestimationmethodshould notbeoverlooked.Indeed,recentstudieshavehighlightedtheadvantagesofworkinginthetime-frequencydomainthroughthe HilbertTransform(HT)ortheWaveletTransform(WT)(Kijewski etal.2003; Xuetal.2003; BashorandKareem2007)becausethey havebeenseentobefarlesssensitivetonoisecontaminationcomparedwithtraditionalapproaches.Theymayalsobeadoptedfor dampingestimationusingshortdurationnonstationarydataas shownin(BashorandKareem2007; Kijewski-CorreaandCycon 2007)wheresingle-valuedecompositionisappliedtogether withWTs.
Databases
Thecontinuedmonitoringanddampingestimationhasledtoa numberofdampingdatabasesbeingestablishedovertheyears. Examplesofsuchdatabasescanbefoundin(Davenportand Hill-Carroll1986; LagomarsinoandPagnini1995; Satakeetal. 2003; YoonandJu2004).
Fromtheanalysisofthesedatabases,someimportantcharacteristicsconcerningthedampingofsteelandreinforced-concrete buildingshavebeenfound.Awidelyreportedcharacteristicof dampingestablishedthroughtheanalysisofdatabasesisthegeneralincreaseofdampingwithfrequencyowingtothegreater participationofnonstructuralcomponentstothedampingofthe system(Jeary1986; Lagomarsino1993; Satakeetal.2003).Over theyearsanumberofresearcheshaveproposedpredictiverelationshipsconcerningthisphenomenon.Inparticular,Jeary(1986)proposedarelationbetweendamping(ξ )andfrequencyinHertz(n) expressedthroughthesimpleformula
ξ ¼ 0 01n ð2Þ
whereasTamuraetal.(2000)proposedthefollowingmodification basedontheresultsderivedfromanalysisoftheJapanesedatabase [ArchitecturalInstituteofJapan(AIJ)2000]
AmorecomplicatedrelationshipwasproposedbyLagomarsino (1993)
where β 1 and β 2 areconstantsthatdependontheprincipalbuildingmaterial(Lagomarsino1993).Byplottingmodalfrequency againstthefirstthreemodaldampingratiosinthelateraland
nomotiontakesplace).Fromtheseearlyworksmoresophisticated modelshavebeendeveloped.Thefollowingsectionswillfocuson arguablythemostimportantoftheseproposals.
PowerLawModel
DavenportandHill-Carroll(1986)suggestedthatthemeanor expecteddampingratiointallbuildingsmaybeestimatedthrough apowerlawas
Fig.1. Modalfrequencyagainstmodaldampingratiosinthefirstthree modes:(a)steelframedbuildings;(b)reinforcedconcretebuildings
where A and α areconstants, x isthestandarddeviationofthedisplacement(inmm)whereas H isthebuildingheight(inm).
Theproposedmechanismbehindthemodelisbasedonthe assumptionthatthebulkofdampinginbuiltupstructuresiscaused bythefrictionbetweenjoints,bearingplates,floorsandbeams, interiorpartitions,exteriorcladdingandstructuralsystemetc. Thefactthatthisassumptionleadstoalawproportionaltoamplitudewithconstantexponent α,whichisincontrasttotheinverse dependencyseenforCoulombdamping(KareemandGurley 1996),isdemonstratedthroughtheadoptionofWyatt’sStiction model(Wyatt1977).Inparticular,itwassuggestedthatthenumber ofstick-slipelementsincreasesintermsofthevibrationamplitude withapowerlawofthetype
fromwhichitcanbedemonstratedthatthedampingratiowillbe givenby
Fig.2. Buildingheightagainstmodaldampingratios;thetrendcurves, fittedthroughlinear(L)andnonlinear(NL)regressionanalysis,showinghowdampingisexpectedtoreducewithbuildingheight:(a)steelframedbuildings;(b)reinforced-concretebuildings
torsionaldirectionsforthebuildingsintheJapanesedatabasethe frequencydependencyisclearlyvisibleforbothsteel-framed buildings,Fig. 1(a),andreinforced-concretebuildings,Fig. 1(b) InFig. 1 thepreviouslyintroduceddamping-frequencyrelationships[Eqs.(2)
(4)]arealsoreported.Anothercharacteristicthat hasemergedisthetrendconcerningthegeneralreductionofdampingintallbuildingswithincreasingheight(Satakeetal.2003; SmithandWillford2007),shownfortheJapanesedatabasein Fig. 2.Thistrendhasledtosomeconsideringthepossibilityof usingheightasadesirableparameterforpredictivedampingmodelsoftallbuildings(SmithandWillford2007),althoughthispossibilityisnotsharedbyall(BentzandKijewski-Correa2008).
PredictiveModelingandAmplitudeDependencyof Damping
Theinvestigationanddefinitionofthemechanismsbehinddampingoftypicalmultistorybuildingsowealottothepioneering worksofHartandVasudevian(1975),whowereamongthefirst topresenttheamplitudedependencyofdampinginreinforcedconcreteandsteelmultistorybuildings,andWyatt(1977)whopresentedamechanismforitsdescriptionthroughtheintroduction ofelementstermed Stiction (StictionstandsforaSTuckfrICTION element,inwhich,intheearlystagesofanincreasingappliedload,
where K isthestiffnessofthesystemwhile D and u areconstants. Theconstants A and α maybeestimatedfromexperimentaldata derivedfromtestscarriedoutonvariousbuildingtypes,e.g.,steel buildingsorconcretebuildings,thereforeidentifyingthemost appropriatevaluestobegiventotheconstants(Davenportand Hill-Carroll1986).
PiecewiseLinearModel
Jeary(1986)proposedwhathasbecome,toacertainextent,the baselinefordescribingthemechanismbehinddampinginstructures.Themodelisbasedonsimilarconsiderationsasthosecited forthepowerlawmodel.Asbefore,themodeltakesinitialinspirationfromtheworkofWyatt(1977)byrecognizingthatthedominantcauseofdampingisfriction-relatedandthatthismaybe modeledthroughtheuseoftheconceptofstick-slipelements. However,themodelissomewhatmorecomplexindescribing theoriginsbehindthesourcesofenergydissipation.
Inparticular,themodelisbasedonthedefinitionofthreedistinctregionsthatdefinetheamplitudedependencyofdamping. Namely,theregionsaretheconstantlowamplitudeplateau,the nonlineartransitionzone,andtheconstanthighamplitudeplateau. Theexistenceofeachzoneisexplainedthroughtheconceptof criticalshearstressassociatedwithwhataretermedasimperfections.Theselastareconsideredatamateriallevelasessentially microcracksand,atastructurallevel,asimperfectionstypified bydimensionsintheorderofmeters(Jeary1986; Jeary1996; Jeary1997).Themacroimperfectionsaretheresultofthepresenceofconstructionjoints,interfaceofstructuralelements,etc.
Theexistenceofalowplateauregionisthenexplainedconsidering theshearenergynecessarytoactivatedissipationmechanismsassociatedwiththemovementofstructuralimperfections.Because thesearecharacterizedbytheirrelativelylargedimensions,they areconsideredtohavealowcriticalshearstressandthereforewill beactivatedalmostimmediately.Beforethematerialimperfections canbeactivated,acertainamplitudemustbereachedowingtothe gapinthedimensionsoftheimperfections.Thiscreatestheconstantlowamplituderegion.Iftheamplitudesarehighenough,the elongationofthematerialimperfectionswillworkasenergysinks causinganincreaseinthedamping.Thehighertheamplitudethe moreoftheseimperfectionswillbeactivatedcausingtheStiction effectofthenonlineardampingregion.Atacertainamplitudeall thematerialimperfectionswillbeactivatedandthedampingwill becomeconstantonceagaindefiningthehighamplitudeplateau. Ingeneraltermsthismodelmaybedescribedbythefollowing piecewiselinearrelationship:
identifyingtheconnectionbetweenamplitude-dependentfrequencyanddamping.
ProposedDampingModel
Theglobalbehavioroftallbuildingscanbemodeledbyanequivalentdynamicsystemconsideringeachfloortohavethreedegrees offreedom(i.e.,twoorthogonaldisplacementsandarotationabout averticalaxis).Underthisassumption,thedynamicresponseofa tallbuildingmaybeestimatedbysolvingthefollowingdynamic equilibriumequation:
where x isthevibrationamplitude(expressedinthesameunits as H ), 0 01n isthelowamplitudefrequency-dependentdamping value(with n expressedinHz), D istheplandimensionofthe building(expressedinm)whereas ξ HA isthehighamplitudedampingvalue.
Modificationsonthepiecewiselinearmodelhavebeenproposedovertheyears.InparticularTamuraetal.(2000)suggested slightmodificationstoEq.(8)totakeintoaccountthemodest differencesseenintheamplitude-dependentdampingcharacteristicsofthebuildingscomprisingtheJapanesedatabase,whereas Lagomarsino(1993)proposedthefollowingmodification:
where M, C and K arethemass,dampingandstiffnessmatrices respectively, z istheLagrangianresponsevectorwhereas fðtÞ isthe vectorofthetimevaryingforcingfunctionsactingatthereference centerofeachfloorandevaluatedasdeemedappropriatebythe analyst(e.g.,ifthewindresponseisofinterest,then fðtÞ could beestimatedbywindtunneltests).Insolvingtheseequations,it iscommontoperformamodalanalysis,truncatedtothefundamentalmodes(firsttwotranslationalmodesandfirsttorsionalmode), andthereforereplacetheprevioussystemwiththefollowingthree uncoupledgeneralizedequationsofmotion:
where λ istheslendernessofthebuildingtakenas H =D and β 3 isa constantwhile
and D areexpressedinconsistentunits.
Discussion
Boththepowerlawandpiecewiselinearmodelswereoriginally calibratedtodatabases.However,ashighqualitydampingdata overrelativelysignificantamplituderangeshasbecomeavailable, attemptshavebeenmadetofitEq.(8)tospecificbuildingresponseswithvaryingdegreesofsuccess(Lietal.2000b; Kijewski etal.2003; Lietal.2003b)whilenospecificattempttofit Eq.(5)hasbeenmade.Atthisjunctureitshouldbenotedthat boththeaforementionedmodelsaretoacertainextentphenomenological.Inparticular,alltheabove-mentionedmodelsconsider dampingasanamplitudeincreasingdynamicpropertyofthe system.However,thereisevidencethat,afterreachingacritical amplitude,insomecasesdampingmayactuallystarttodecrease withamplitude(Tamura2005; SmithandWillford2007; Tamura andYoshida2008; AquinoandTamura2012).Also,togetherwith amplitude-dependentdamping,ithasbeenwidelyreportedthat naturalfrequencywillalsobecomeamplitude-dependentassuming steadilydecreasingvalues(TamuraandSuganuma1996; Lietal. 2000a; Kijewski-CorreaandPirnia2007; Kijewski-Correaetal. 2007).Atpresent,therearenomodelsthatarecapableofdescribinginageneralfashionthesetwophenomena.Theaimofthis paperistheproposalofaprobabilisticconcept-baseddata-driven modelthatiscapablenotonlyofprovidingaphysicalreason whydampingmayattimesdecreasewithamplitude,butalsoof
where ζ j isthegeneralizedviscousdampingratio, qj ðtÞ isthe modaldisplacement, ωj isthe jthcircularfrequencywhereas Qj ðtÞ and mj arethegeneralizedforceandmass,respectively. InderivingEq.(11),classicaldampingisgenerallyassumed. However,ashasbeenextensivelypresented,theassumptionof classicaldampingisnotalwaystruefortheresponseofmostreal buildingswheredampingis,amongotherthings,amplitude-dependent.Also,asalreadymentioned,thenaturalfrequenciesofthe buildingshowatendencytosoftenasthevibrationamplitudeincreases.Thepossibilityofcapturingtheaforementionedresponse characteristicswhilestilldescribingthefundamentaldynamicresponsethroughareducedsystemofequationswouldseemofparticularinterest.Tothisend,thefollowingparagraphswillpresent adampingmodelthatimplicitlyassumesthatthecoupledequations ofmotionofEq.(10)maybereplacedbythreeindependentnonlinearsingledegreeoffreedomdynamicequilibriumequationsof thefollowingform:
where M i istheparticipatingmassofthe ithequation, Ci isthe viscousmaterialdampingcoefficient, xi ðtÞ, xi ðtÞ and xi ðtÞ are the ithacceleration,velocityanddisplacementresponses, K i is thezero-amplitudestiffness, k isthecombinedstiffnesslossresultingfromtheslippingofthesticksurfacesattheenvelopeamplitude of ~ xi , ˆ f i isthetotalfrictionaldampingforceavailabletothe ith equationwhereas pf 0 x0 ð ; Þ isthejointprobabilitydensityfunction betweenthealeatoryfrictionforces f 0 ,generatedbytheslipping surfaces,andtherandomvibrationamplitudes, x0 ,atwhichthese surfacesbecomeactive.ForallintentsandpurposesEq.(12)plays ananalogousroletoEq.(11)indescribingthedynamicresponse ofthebuildingwith,however,someaddedcomplexitythatshould allowthemeaningfulmodelingoftheexperimentallyobserved amplitude-dependentdampingandnaturalfrequency.Froma
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physicalstandpoint,Eq.(12)isbasedonthewidelyaccepted assumptionthatthesourceofamplitude-dependentdampingmay betracedbacktothepresenceofanumberoffrictionsurfacesinthe structureexhibitingstick-slipmechanisms.Inparticular,Eq.(12) definesacombineddampingmodelbecauseitalsoincludesthe contributionoftheviscousmaterialdampingthroughthecoefficient Ci .Thecontributionofthistermisthoughttobesmallcomparedtothefrictionaldamping,butneverthelessimportantat loweramplitudes.WhatmakesEq.(12),andthereforetheproposed dampingmodel,differentfromothermodelsthatcanbefoundin theliteratureisinhowthefrictionaldampingforcesaremodeled. Intheproposedmodel,thesearedescribedbythejointprobability densityfunctionbetweentherandomvariables f 0 and x0 .Although theconceptofrandomlydistributedslipforceshasbeenconsidered previously(DavenportandHill-Carroll1986),themodelingof theamplitudesatwhichtheseforceswillbeinitiatedasarandom variablehasbeenuntilnowoverlooked.Traditionally,theamplitudes, x0 ,aredescribedbyadeterministicfunctionsimilartothat showninEq.(6).Theultimateconsequenceoftheaforementioned assumptionisthatthenumberofslipsurfaceswillcontinuetogrow indefinitelywithamplitude,whichgoesagainstphysicallogicthat wouldsuggestthesaturationofavailableslipsurfacesasthevibrationamplitudesincrease(BashorandKareem2008).Hereinitis proposedtoconsidertheamplitudes x0 probabilisticallydistributed withappropriatelawthatwillingeneraldependonthebuilding underconsideration.Thisprobabilisticmodelingof x0 givesa physicalmeaningtotheamplitudesandallowsthemtocompletely saturateasthevibrationamplitudeincreases.Thephysicalreasoningbehindmodeling x0 aswellas f 0 asrandomvariablesisunderstoodbyconsideringthesourcesoffrictionalforcesinstructural systems,i.e.atboltedjoints,atbearingplates,inbuilt-upelements, betweenfloorandbeams,atin-fillpanelsinframes,betweenframe andcladding,inmicroscopicmaterialimperfectionsetc.Allthese sourcesare,toagreaterorlesserextent,inevitableinbuildings. Theirunengineerednatureobviouslyimpliesthattheywillbe probabilisticallydistributed.Inlightofthis, f 0 and x0 canonlybe coherentlymodeledthroughthejointprobabilitydensityfunction pf 0 x0 ð ; Þ illustrativelyrepresentedinFig. 3.Theultimateresultof thisisthatas ~ xi increasesthenumberoffrictionalsourcesactivated
willincreaseuntil,atsaturation,theyareallactive.Thisiseasily understoodbyrecognizingthatthedoubleintegralofthelast termoftheleft-handsideofEq.(12)issimplythecumulativedistributionfunctionof x0 whichwill,bydefinition,tendto1as xi increases.ThegeneralsettingofEq.(12)isshowninFig. 3
Amplitude-DependentDampingRatio
InthissectionanexpressionfordescribingtheamplitudedependentdampingratioofEq.(12)issearched.Forthesake ofclarity,fortheremainderofthispaperthesubscript i willbe dropped.ToevaluatethedampingratioofEq.(12),theexpression reportedinEq.(1)maybeinvoked.Inparticular,forthecombined systemunderinvestigationthetotaldissipatedenergypercycle, ΔEtot ,maybewrittenas
where ΔEvis istheenergydissipatedduetoviscousdamping whereas ΔEfric istheenergydissipatedowingtofriction.Although thecalculationof ΔEvis ,andbyconsequencetheviscouscontributiontothedampingratio ξ vis ,doesnotpresentanyparticulardifficulty, ΔEfric ,andsothefrictionalcontributiontothedamping ratio ξ fric ,issomewhatmoredifficulttoestimate.Thisparagraph willfocusonestimating ξ fric
ConsideringthesingledegreeoffreedomsystemofEq.(12) underasteady-statevibrationofamplitude x,the jthslipping surfaceofthesystemshowninFig. 3 willdissipatepercyclea quantityofenergygivenby
Thetotalenergydissipatedbytheslippingsurfacesatavibrationamplitude x maythenbewrittenas
where N isthetotalnumberofstick-slipelementsinthesystem. Iftherandomvariables f 0 and x0 areconsideredindependent,then theirjointprobabilitydensityfunctionisgivenby pf 0 x0 ð ; Þ¼ pf 0 ð Þpx0 ð Þ where pf 0 ð Þ istheprobabilitydensityfunctionof f 0 whereas px0 ð Þ istheprobabilitydensityfunctionof x0 .By introducingthissimplificationintoEq.(15)andbythentaking advantageoftheintegrationbypartsrule,Eq.(15)mayfirstbe recastas
where f 0 isthemeanvalueof f 0 .ByinvokingEq.(1)thefrictional componentofthedampingratio ξ fric ofthenonlinearsingledegree offreedomoscillatorofEq.(12)maybewrittenas
Ifthestiffnessoftheslipsurfaces kðxÞ isconsideredsmall,asis expected,comparedtothetotalstiffnessofthestructure K andby
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collectingtheindependenttermstogether,thefollowingexpression canbederivedfortheamplitudedependent-dampingratio
Amplitude-DependentFrequency
where A isaconstanttobedetermined.TofixEq.(19)theprobabilitydensityfunction px0 ð·Þ mustbesetasmusttheconstant A andtheviscousdampingratio ξ vis
Hereitisproposedthat x0 bemodeledasalognormalrandom variable.ThischoiceisparticularlymeaningfulinlightofthestructureofEq.(19).Tobetterunderstandthislaststatementitisconvenienttointroducetheconceptofcriticalamplitude, xcr ,defined astheamplitudeatwhichEq.(19)assumesitsmaximumvalue indicatedwith ξ max .Withthisinmind,Fig. 4(a),andmorespecificallyFig. 4(c),illustratehowtheratiobetweenthecriticalamplitude ~ xcr andthemode mx0 of px0 ð·Þ isconstantasthemode mx0 is variedoncethestandarddeviation σ x0 oftheunderlyingnormal distributionisfixed.Thispropertyisparticularlysignificantasit allowsthemodeoftheslipamplitudedensitytobefixedfrom theknowledgeof xcr and σ x0 .Therelationshipbetween σ x0 andthe aforementionedratio, xcr =mx0 ,isshowninFig. 4(d).Fig. 4(b) illustratestheroleof σ x0 asadimensionlessshapeparameterwithin theproposeddampingmodel.
TheaforementionedpropertyallowsthecalibrationofEq.(19) tobeachievedbyfirstestimatingthecriticalamplitude ~ xcr and shapeparameter σ x0 .Fromtheknowledgeof σ x0 and ~ xcr ,theratio ~ xcr =mx0 maybeestimatedfromFig. 4(d) thereforeidentifing mx0 Theaforementionedprocedurefixes px0 ð Þ leavingonly ξ vis and A tobefound.Afterhavingfixedanappropriatevaluefor ξ vis , A maysimplybeidentifiedbysubstitutingintoEq.(19)thecritical amplitude xcr andthecorrespondingdampingratio ξ max andsolvingfor A
Thepreviousparagraphintroducedaconcept-basedprobabilistic modelfordescribingtheamplitude-dependentdampingratioof Eq.(12).Inthissectionarelationshipissearchedfordescribing theamplitude-dependentstiffnessofEq.(12)withtheaimofmodelinghowthevibrationfrequencyislikelytovarywithamplitude forthesystemschematicallyshowninFig. 3.
Astheamplitudeofvibrationincreases,thenumberofslipsurfacesthatareslippingincreases.Thiscorrespondstoadecreasing numberofslipsurfacesbeingabletoprovidestiffness.Foran infinitesimalchangeinamplitudeofvibration dx andfrictionforce df 0 ,theinfinitesimalchangeinstiffness dk maybewrittenas
where N isthetotalnumberofstick-slipsurfaces.Byintegrating overalltheforcelevelsanduptoagivenamplitude,thefollowing expressionisfoundfortheamplitude-dependentstiffness K ofthe systemofEq.(12)
Byfirstseparatingthevariablesinthedoubleintegralofthe left-handsideofEq.(21)andsimplifying,thefollowingexpression fortheamplitude-dependentstiffnessisobtained:
Eq.(22)allowsthefollowingexpressionfortheamplitudedependentfrequency n tobedefinedwhereitisassumedthat thefrequencyattheamplitude x dependsonthestiffnessatthe sameamplitude
(a)(b)
where B1 ¼ K and B2 ¼ N f 0 areconstantstobeestimated.
Asinthecaseoftheexpressionderivedfortheamplitudedependentdampingratio,Eq.(23)dependsontheprobabilitydensityfunction px0 ð Þ oftheslippingamplitudes.Thedependencyof bothEqs.(19)and(23)on px0 ð Þ isparticularlyinterestingasit bringstheamplitude-dependentdampingandfrequencyproperties togetherintoasinglemodel.
ThecalibrationofEq.(23)dependsfirstlyon px0 ð·Þ,whichmay bederivedfromthecalibrationofEq.(19),andonthechoiceofthe constants B1 and B2 .Although B1 may,intheory,beestimated fromaccuratemodelingofthezeroamplitudestiffness, B2 will ingeneralneedestimatingfromexperimentaldata.
Application1
(c)(d)
Fig.4. StructureofEq.(19)for ξ max ¼ 1 5%:(a)behaviorof px0 ð Þ and ξ ðxÞ for mx0 varyingbetween 10 3 mand1m,andwith σ x0 ¼ 4;
(b)behaviorof px0 ð·Þ and ξ ðxÞ for mx0 ¼ 10 3 mand σ x0 varyingbetween1and6;(c)illustrationoftheconstantnatureoftheratio xcr =mx0 forfixedvaluesof σ x0 ;(d)illustrationoftherelationshipbetween σ x0 and xcr =mx0 ,validforanyvalueof mx0
Inthissectiontheproposeddampingmodelwillbecalibratedtoa selectednumberoftallbuildingsforwhichamplitude-dependent dampingratiosandfrequencieshavebeenexperimentallyestimated throughfull-scalemonitoringprograms.Alltheexperimental dampingdataweredeterminedinthetimedomainusingthe RDT(RandomDecrementTechnique).Tocomparebuildingswith arangeofheights,itisconvenienttointroducethenormalized amplitudeindicatedwith x ¼ðx=H Þ where H istheheightof thebuilding.
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2263 ≈60 × 60
3443 68 × 68
Note:B=smallestplandimension;CO=coreandoutrigger;D=largestplandimension;RC=reinforcedconcrete;S=steel;SF=steelframe;SRC= steel/reinforcedconcrete.
CaseStudies
ThreebuildingsareconsideredwhosemaincharacteristicsarereportedinTable 1.Thechoiceofthesethreebuildingswasmade becauseoftheavailabilityofhighqualityandwelldocumented amplitude-dependentanddampingandfrequencydata(KijewskiCorreaetal.2006; Kijewski-CorreaandPirnia2007; Pirniaetal. 2007; AquinoandTamura2012)aswellastheirdramaticallyopposingbehaviorasvibrationamplitudeincreases.Indeed,asisdepictedinFig. 5,buildings2and3showtheclassicalincreaseof thedampingratiowithamplitudewhereasbuilding1,afteraninitial increaseindampingwithamplitude,showsadecreasingdamping ratiowithamplitude.Thispeculiarbehaviorcannotbeexplained throughtheadoptionofclassicaldampingmodelssuchasthose reviewedinthesection “PredictiveModelingandAmplitude DependencyofDamping” andhasbeenthefocusofrecentstudies (AquinoandTamura2012).
CalibrationoftheProposedDampingModeltothe CaseStudies
Fig. 6 showstheresultsofthecalibrationofEq.(19).Ascaneasily beseen,themodelshowsverygoodagreementwiththeexperimentaldata.Also,thecapabilityoftheproposedmodelincapturingthe supposedlycontradictingbehaviorbetweenbuilding1andbuildings2and3iseasilyachieved.Withinthesettingoftheproposed model,thedifferenceseenintheamplitude-dependentcharacteristicsmaysimplybeascribedtothesignificantlydifferentdistributionparametersoftheslipamplitudes.Indeed,asshowninFig. 7, buildings2and3havesimilarslipamplitudedistributions,whereas building1hasasignificantlydifferentdistribution.Table 2 reports themodelparametersobtainedfromthefittingofthethreebuildings.Fromthistableitisinterestingtoobservethesimilaritybetweenthemodelparameters,and,inparticular, mx0 and σ x0 ,of buildings2and3astheseareconsideredtwobuildingsthatexhibit aclassicalamplitude-dependentdampingbehavior.Therefore,in therealmoftheproposedmodelitisexpectedthatmostbuildings willhaveslipamplitudedistributionssimilartothoseofbuildings 2and3.
Itisinterestingtoobservehowtheproposeddampingmodel allowsthepredictionofthe ξ max representingthemaximumexpecteddampingratiowithinthevalidityofEq.(12).FromTable 2 itcanbeenseenthattheproposedmodelissuggestingthat building3willexhibit ξ max aroundtwotimesthatofbuilding2.
modeofthecasestudybuildings
Fig.7. Comparisonbetweenthefittedprobabilitydensityfunctions px0 ðx0 Þ oftheslipamplitudesforthecasestudybuildings
Table2. ModelParametersforFittedExperimentalData
Thiswouldseemincontrasttowhatcouldbeexpectedconsidering thefactthatbuilding2hasprimarilyresidentialuseandreinforced concreteasstructuralmaterialcomparedwithbuilding3,which hasprimarilyofficeuseandsteelasstructuralmaterial(Table 1). Apossibleexplanationforthiscouldbethefactthatbuilding3 wouldseemtohavegreatershearrackinginitslateraldeformation mechanismcomparedwithbuilding2.Indeed,ithasbeensuggestedrecentlythattheamountofshearrackingcomparedwith axialdeformation(greateroverallcantileveraction)inthelateral deformationmechanismcouldplayanimportantroleindeterminingtheoveralldampingcapacityoftallbuildings(Bentzand Kijewski-Correa2008; Bentz2012).
Fig. 8 showsthefittingofEq.(23)totheamplitude-dependent frequencydatacollectedforthecasestudybuildings,whereas Table 2 showsthemodelparametervalues.Onceagainthestrong capabilityoftheproposedmodelinreproducingtheexperimental dataisveryencouraging.Indeed,forallthreebuildingstheamplitude-dependentfrequencywouldseemtobeeasilyreproduced bytheproposedmechanismthatattributesthelossofstructural stiffness,astheamplitudeincreases,tothelossofstuckstick-slip surfaces.Closeinspectionofthefrequency-amplitudebehaviorof building1(Fig. 8)givesparticularlystrongevidenceofthevalidity oftheproposedmodel.Indeed,ifthehypothesisthattheslip-stick surfacesareprobabilisticallydistributedistrue,thenasthedampingbeginstosaturatethefrequencylossshouldbegintostabilizeto aconstantvalueasiseffectivelyseenfromthefrequency-amplitude responseofbuilding1.
Application2
Thissectionfocusesonthecalibrationoftheproposedmodeltothe amplitude-dependentdampingdataof12tallbuildingswhosemain characteristicsareshowninTable 3.Ascanbeenseenthebuildings,allofwhicharelocatedinthePacificRim,apartfromB2, B10,andB11,thatarelocatedintheUnitedStates(B11inChicago andB12inBoston),haveagoodspreadofheightsandof principalconstructionmaterial.Theamplitude-dependentdamping estimatesforthesebuildingswereobtainedusingtheRDT,ensuringacertainconsistencybetweentheestimates.
Fig. 9 showstheproposedmodelfittedthroughanonlinearleast squaresminimizationtothefirsttwoorthogonaltranslational modes(whenavailable)ofthe12buildings.Thefittedmodel parametersforthe12buildingsareshowninTable 3.Ascan beseen,themodelwouldseemeasilycapableofdescribingthe amplitude-dependentdampingcharacteristicsofallthebuildings. Itisinterestingtoobservehowthecriticalamplitude xcr (Table 3)is ingeneralcontainedbetween1and10,indicatingthestabilityof themodeofthedistributionoftheslippingamplitudes mx0 .Inother words,fromthelimitedavailabledata,themodelispredicting thatthemaximumdampingvalueswilloccurinroughlythesame normalizedamplituderange.Thisconditionisobviouslynot satisfiedbybuilding1ofFig. 6.However,itisbelievedthatthe behaviorofbuilding1representsananomalyinthisrespect.As moremonitoringdatacomesavailable,thisaspectwillbefurther investigated.
ThefittedexperimentaldataofbuildingB10isworthcommentingastheextremelydifferentdampinglevelsseenforthesame amplitudesbetweenthetwomodeshasbeenthesourceofanumber ofdiscussions(Kijewski-Correaetal.2007).AsshowninFig. 9, themodelascribesthisdifferencesimplytoadifferenceintheslip amplitudedistributionbetweenthetwoorthogonalswaymodes. Moreover,themodelpredictsthatthemaximumdampinglevelbetweenthetwomodeswillbesimilar.Itisalsointerestingtoobserve howbuildingsB1toB4haveevidentlylower ξ max comparedwith buildingsB5–B12(Fig. 9 andTable 3).Apossibleexplanation forthiscouldbeascribedtothedifferentlateraldeformation
mechanismsbetweenthebuildings.Indeed,itisexpectedthat buildingsB1–B4haveapredominantlycantileveractionduring lateraldeformation,whereasbuildingsB5–B12areexpectedto deformlaterallyfollowingapredominantlyframerackingaction. Asalreadymentioned,recentstudieshavesuggestedthatthis differencecouldpossiblyleadtosignificantlydifferentmaximum dampingvalues(Bentz2012).
Inclosingthissection,itisworthnotingthatthemodelhasbeen fittedtodatacollectedoveralimitedrangeofamplitudeswhich obviouslycreatesacertainamountofvariabilityinthemodelpredictionsforamplitudesoutsidethisrange.Forinstance,thedifferenceseenbetweenthemaximumdampingratiosofthetwolateral translationalmodesofbuildingB5maysimplybeduetoalackof dataathighervibrationamplitudes.Nonetheless,therobustnessof theproposedmodelreportedinthissectionisextremelyencouragingandsuggeststhevalidityofthemodel.
ProposedDampingModelasaPredictorTool
Thissectionfocusesonthepossibilityofusingtheproposeddampingmodelasapredictorandanalysistoolfortheestimationof amplitude-dependentdampingratiosoftallbuildingsofheight above100m.Toinvestigatethispossibility,anappropriatedatabaseofamplitude-dependentdampingvaluesisnecessary.
DatabaseofAmplitude-DependentDampingValues
Thedatabasethatwascompiledforthisstudyconsistsinitially of76buildingsofheightsvaryingbetween100and282mextractedfromtheJapanesedatabase(Satakeetal.2003),therefore ensuringahighqualityofthedampingvaluesandassociated amplitudes,becauseoftheextensivedatapostprocessingthat hasbeencarriedout(AIJ2000; Satakeetal.2003).Togetherwith thesebuildings,thatconsistmainlyofsteel-framedconstruction,an additional19buildings,withheightsrangingfrom100to443m, withgreaterstructuralsystemandconstructionmaterialvariation areincludedinthedatabase.Thedampingdatafortheselasthave beencollectedfromthenumerouspublicationsthatcanbefoundin theliteraturereportingtheresultsoftheextensivemonitoringprogramspresentedinthesection “Full-ScaleMonitoringPrograms” andconcerningsomelandmarkbuildingsintheUnitedStatesas wellasthePacificRimregion.Finally,amplitude-dependentdata ofthreetallbuildingsmonitoredduringtheCSMISprogramofthe CaliforniaGeologicalSurveyarealsoincludedinthedatabase.The datapointsofthesebuildingsconsistofdampingestimatesmadeon notablesteel-framedU.S.WestCoastbuildings,suchastheTransamericaTower(Çelebiand Şafak1991; ŞafakandÇelebi1991), underhighamplitudevibrationduringtheLomaPrietaandSan Fernandoearthquakes(BeckandJennings1980; ŞafakandÇelebi 1991; Çelebi1993).
Eachbuildinginthedatabasehasoneortwoassociateddata pointsindicatingamplitude-dependentdampingestimatesmadefor thefirsttranslationalmodes.Awidevarietyofsystemdamping estimationmethodswereusedunderanumberofnaturallyoccurringandinducedexcitations.Table 4 reportsthebreakdownofexcitationtypesusedforestimatingthedampingratios.Table 4 also reportsthedomaininwhichthedampingwasevaluatedorwhether asystemidentificationmethodwasused.Ascanbeseen,the majorityofthedampingestimatesweremadeinthetimedomain. Inthesecases,theprevalentmethodwastheRDT.Animportant numberofdatapointswereobtainedinsteadthroughfrequency domainestimationtechniques.Inparticular,amixtureofspectral curvefittingmethodsandtheHPBWmethodwereapplied. ConcerningthedatapointsobtainedfromtheapplicationofSI
Note:IV=inducedvibrations;AV=ambientvibrations;EQ=earthquake; FD=frequencydomain;SI=systemidentification;TD=timedomain.
techniques,thevastmajorityofthedatawasobtainedthroughthe implementationofARMAXandARXschemeswithafewhigh amplitudedatapointscomingfromtheapplicationofaDTF scheme.Therelativelybroadspectrumofdampingestimationtechniquesusedforobtainingthepointsofthedatabasewillallowa preliminaryinvestigationintopossibledifferencesseenindamping levelsresultingsimplyfromdifferentestimationschemes.
ModelCalibrationandDiscussion
Fig. 10 showsaLeastAbsoluteSquares(LAR)fitoftheproposed modeltothedatabaseoftheprevioussection,whereasTable 5 reportsthemodelparametersderivedfromtheLARfitofthemodel tothedatabase.Fig. 10 alsocomparestheproposedmodelwiththe piecewiselinearmodel,themodificationproposedbyTamuraetal. (2000),andthepowerlawmodel.Inplottingthepiecewiselinear modelanditsmodification,themeanvaluesofthedatapointsconcerningthenaturalfrequenciesandplandimension D [Eq.(8)]in thedirectionofvibrationareconsideredleadingtosmoothpredictivecurves.Toillustratetherangeofvaluesthatthesemodelsgive consideringthevariationinnaturalfrequencyandplandimension withinthedatabase,themaximumandminimumlimitsofthe aforementionedmodelsarealsoillustratedinFig. 10.Itshouldbe observedalsothatforthemodelproposedbyTamuraetal.(2000) moderatelydifferentcoefficientswereproposedforreinforcedconcretebuildingscomparedwithsteelbuildings.Inthepresent comparison,onlythepredictivemodelconcerningsteel-frame structuresisplottedbecausethelargemajorityofthedatapoints
inthedatabasecorrespondtothisstructuraltype(Table 4).Also, indefiningtheirmodel,Tamuraetal.(2000)limiteditsrangeof applicabilitytoatopdriftratioofnomorethan 2 × 10 3 cm=mand amaximumbuildingheightof200m.Intherepresentationofthe modelpresentedinFig. 10 thelimitondriftratiohasbeenrespected,althoughitshouldbeobservedthatthedatabasedoescontainanumberofbuildingswithheightgreaterthan200m.
Fromthecomparisonoftheaforementionedpredictivedamping models,itisevidenthowtheproposedmodeliswellcapableof describingtheamplitude-dependentcharacteristicsofthedatabase, whereasthepiecewiselinearmodeldoesnotseemcapableinthis respect.Itisalsointerestingtoobservehowfortopdriftratiosof lessthan 2 × 10 3 cm=m,themodelproposedin(Tamuraetal. 2000; Tamura2012)hastheclosestcorrespondencetotheproposed model,whileforhigheramplitudesthepowerlawmodelseemsto describethedatabasebetterthanthepiecewiselinear.Thesimilarity betweentheproposedmodelandthepowerlawmodelin thehighamplituderangecanbeascribedtotheunderlyingsimilaritiesbetweenthephilosophicalstandpointsofthetwomodels. However,afundamentalandimportantdifferencebetweenthe twomodelsliesinthewayinwhichtheproposedmodelbehaves ashigheramplitudesofvibrationareencountered.Indeed,inthe proposedmodelthesaturationofthedampingmechanismscauses thepredictivecurvetostarttoflattenout,whereasthepowerlaw modelpredictseverincreasingdampingvalues.Fig. 10 alsogives somevalidationoftheproposedmodelinthehighamplituderegion.Indeed,inthecalibrationoftheproposedmodelinsection “ProposedDampingModel” therewasverylittledatatowards highamplitudesofvibration.Thedatabaseofthepresentsection, ontheotherhand,doescontainsomehighamplitudedata(Beck andJennings1980; ŞafakandÇelebi1991; Çelebi1993).ThesimilarvaluesofthemodelparametersreportedinTables 2 and 3
(withtheexceptionofbuilding1ofTable 2)comparedwiththose reportedinTable 5 wouldseemencouragingasafirstvalidationof themodelinthehighamplituderegion.
Fig. 11(a) showstheestimatedprobabilitydensityfunctionof theresidualsofthedatapoints.Ascanbeenseen,thesearewell estimatedbyaGaussiandistributionwithzeromean,indicatingthe efficiencyoftheproposedmodelindescribingtheamplitudedependenttrendofthedata.Indeed,thecorrespondinghistogram concerningpowerlawmodel,Fig. 11(b),clearlyshowshowthe residualsofthismodelcannotbefittedbyazeromeanGaussian distribution.
AnalysisoftheResiduals
Theresidualscanbethoughtofaselementsofvariationunexplainedbythefittedmodel,thereforetheiranalysiscanshedlight onfactorsthatmaycauseaparticularbuildingtobeaboveorbelow thedampingvaluepredictedbyamodel.Thissectionwillexplore thispossibilitywiththeaimofexplainingsomeofthespreadseen inthedata.
Frequency,Material,andHeightDependency
Asmentionedearlierinthispaper,variationindampinglevelsbetweendifferentbuildingshasoftenbeenlinkedtothedifferencesin naturalfrequencies.ThistendencycanbeseeninFig. 12(a) where thedifferenceinfrequencybetweenthevariousbuildings(minimum0.13Hzandmaximum0.7Hz)wouldseemtobeweakly correlatedwithresidualstakingonvaluesaboveorbelowthetrend curve.Thiswouldindicatethatforthefirsttwotranslationalmodes
oftallbuildingsthedependencyofdampingonfrequencydoesexists,althoughminimally.Obviously,theweakdependencyonfrequencyofthefirsttwotranslationalmodesmaynotholdforhigher modeswheregreaterfrequencydependencymaywellexist.Asimilarresultisseenconcerningheight[Fig. 12(b)]whereasmallbut discerniblerelationshipbetweenheightanddampingwouldseem evident.Theweaknatureofthesetrendscomparedwithwhatis normallyobserved(Tamuraetal.2000; Satakeetal.2003)ismost likelyresultingfromtherestrictionplacedonheight(H > 100 m) forthebuildingsinthepresentdatabase.Indeed,theresultsreportedheredonotconflictwiththosereportedin(Tamuraetal. 2000; Satakeetal.2003)whereforbuildingswithheightabove 100mtherelationshipbetweendampingratioandheightisweaker thanthatseenforbuildingsofheightlessthan100m.Withthisin mind,itwouldseeminterestingtoinvestigateotherparametersthat couldcomplementheight/naturalfrequencyindeterminingappropriatedampingvaluesforbuildingsofheightgreaterthan100m.
Fig. 13 showstheresidualsintermsofthepredominantconstructionmaterial.Ascanbeseen,mostbuildingsinthedatabase areconstructedfromsteel,howevertherewouldseemtobeindicationthatconcretebuildingstendtohavegreaterdampingvalues (seenaspositiveresiduals)ascomparedwiththeirsteelcounterparts.Therewouldnotseemtobeanyparticulardifferencebetween steelandcompositebuildings.
CantileverversusShearBehavior
Recentlyithasbeensuggestedthatinherentdampinglevelsof tallbuildingscanberelatedtotherelativecontributionofshear deformation(frameracking)versusaxialdeformation(cantilever action)(Kijewski-Correaetal.2006; Erwinetal.2007; Bentz andKijewski-Correa2008; Bentz2012).Intermsoftheresiduals, thisconcepttranslatesintobuildingsthathaveapredominantly cantileveractiontakingonnegativevaluesandthereforebeing underthedampingvaluespredictedbytheproposedmodel.This possibilityisinvestigatedinFig. 14 wherebuildingswithdocumented(HalvorsonandIsyumov1986; BentzandKijewski-Correa 2008; Bentz2012)frameorcantileveractionarehighlightedas wellasbuildingsforwhichthemodeshapesareknownallowing theclassificationtobecarriedoutonthebasisoftheprocedures proposedin(BentzandKijewski-Correa2008; Bentz2012).From thisfigureitwouldseemthatbuildingsexhibitingapredominately cantileverlateraldeformationmechanismdoindeedhaveareduced dampingcapabilitycomparedwithbuildingswithastrongershear rackingframeaction.Itisalsointerestingtoobservehowthedividinglinebetweenthetwobehaviorsisrepresentedbymoreofa transitionzoneratherthanadistinctjumpconfirmingwhatwas reportedin(Bentz2012).
EffectsoftheDampingEvaluationTechnique.Ashighlightedinthesection “TheEstimationofDamping,” thereexists
avastnumberofmethodsthathavebeenproposedforestimating dampinginstructures.Thehighestlevelofseparationbetweenthe variousmethodscanbemadeintermsofwhetherdampingisestimatedinthefrequencydomain,timedomainorifinputandoutput areknownallowingasystemidentificationschemetobeused.
Table 6 reportsthemeanvaluesoftheresidualscalculatedon thedampingestimatesmadeundertheaforementionedconditions. Itisinterestingtoobservehowthedampingestimatesmadeinthe frequencydomainwouldseemtobelargerthantheircounterparts madeinthetimedomain.ThisisalsoshowninFig. 15 whereit wouldseemevidentthatthefrequencydomainestimatestend towardsmorepositiveresidualsindicatinghowdampingestimates madeinthisdomainwouldseemtobepronetooverestimate,a concernalsohighlightedinKijewski-CorreaandPirnia(2007).
Table 6 alsoreportsanegativemeanfortheresidualsofdata estimatedusingSImethods.Itishoweverdifficulttodrawanyfirm conclusionsfromthis,astheresidualsareseentotakeonboth positiveandnegativevalues(Fig. 15)indicatingthatthisresult couldsimplybeduetoalackofdatapoints.
AmbientVibration,InducedVibration,andEarthquake Fig. 16(a) showsthedistributionoftheestimateddampingvalues aroundtheproposeddampingmodelintermsoftheexcitationtype, whereasFig. 16(b) showsthedistributionoftheresiduals.Itisinterestingtoobservehowdampingestimatesmadeunderinduced vibrations(IV)andambientvibrations(AV)giveagoodspread ofdatapointsaroundthepredictedvaluesoftheproposedmodel withmeanvaluesoftheresidualsequalto0.11%and 0 01%,respectively.Thisisnotthecasefordampingestimatesmadefrom earthquakerecordswhichhaveameanvalueoftheresidualequalto 0 41%.Thisresultisparticularlyinterestingasitisgenerallyexpectedthatdampingestimatesmadefromseismicrecordswillbe morereliablethanestimatesmadeunderotherexcitationsasSI methodscanbeusedfortheirestimation.Thisresultisinlinewith whatwasreportedintheprevioussection,andcouldbesuggesting anoverestimateofdampingvaluesmadewithoutput-onlyschemes atlowvibrationamplitudes.However,thesmalldimensionofthe datasetmustbekeptinmindwhenmakingsuchanassertion.
Conclusions
Inthispaperanovelconcept-baseddata-drivenprobabilistic dampingmodelwaspresentedfordescribingtheexperimentally observedamplitude-dependentdampingcharacteristicsoftall buildings.Inparticular,themodelmaybeconsideredcombined asittakesintoaccountboththeviscousmaterialdampingaswell asthepredominantfrictionaldampingofbuildings.Themodelwas extendedsoastoalsodescribe,withinasingletheory,thedependencyofnaturalfrequencyonvibrationamplitudethathasbeen widelyreportedintheliteraturetogetherwithamplitude-dependent damping.Expressionswerederivedforestimatingtheamplitudedependentdampingratiosandnaturalfrequenciesoftheproposed model.Initialvalidationofthemodelwascarriedoutonhighfidelityexperimentaldatacollectedonthreetallbuildingswithdramaticallyopposingamplitude-dependentdampingcharacteristicsthat areunexplainablebycurrentmodels.Theproposedmodelwasseen toeasilyfitboththeamplitude-dependentdampingandfrequency data.Thecapabilityofthemodelindescribingthesupposedlycontradictoryexperimentaldatamaybeascribedtoitsconcept-based nature,andinparticulartheideaofeventualsaturationofthedampingsourcesasvibrationamplitudeincreases.Themodelwasfurthervalidatedthroughitscalibrationtotheexperimentaldata derivedon12tallbuildingsfortheirfirsttwoswaymodes.Again themodelwasseentobeexceptionallyrobustandcapableofsheddinglightonsomeseeminglycontradictoryexperimentalresults. Finally,themodelwascalibratedasapredictiveandanalysistool
toaspecificallycompileddatabaseofamplitude-dependentdampingdatafortallbuildingswithheightsvaryingbetween100and 450m.Themodelwasusedasanamplitudecorrectortoolfor thedampingdatathereforeallowingadetailedstudyoftheresultingresiduals.Fromthisinvestigation,itwouldseemthatheightis notagoodpredictiveparameterfortallbuildings.Insteaditwould seemasthoughthepredominantlateraldeformationmechanism, andthereforethestructuralsystem,playsanimportantrolein decidingthedampingcapacityoftallbuildings.Finally,thevariationinexperimentallydetermineddampingvaluesduetodifferent estimationtechniqueswasinvestigated,reaffirmingthetendency forfrequencydomainestimatestobehigherthantimedomain estimates.
Acknowledgments
SupportwasinpartprovidedbytheTallBuildingMonitoring ProgramattheUniversityofNotreDamesupportedbytheNSF GrantNo.CMMI06-01143andtheGlobalCenterofExcellence atTokyoPolytechnicUniversity,fundedbyMEXT.
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