UFDC Home  Search all Groups  UF Institutional Repository  UF Institutional Repository  UF Theses & Dissertations   Help 
Material Information
Thesis/Dissertation Information
Subjects
Notes
Record Information

Full Text 
PAGE 1 ProsandConsofRotatingGroundMotionRecordsto FaultNormal/ParallelDirectionsforResponse HistoryAnalysisofBuildingsErolKalkan,M.ASCE1;andNealS.Kwong2Abstract: AccordingtotheregulatorybuildingcodesintheUnitedStates(e.g.,2010CaliforniaBuildingCode),atleasttwohorizontal groundmotioncomponentsarerequiredforthreedimensional(3D)responsehistoryanalysis(RHA)ofbuildingstructures.Forsiteswithin 5kmofanactivefault,theserecordsshouldberotatedtofaultnormal/faultparallel(FN/FP)directions,andtwoRHAsshouldbeperformed separately(whenFNandthenFParealignedwiththetransversedirectionofthestructuralaxes).Itisassumedthatthisapproachwillleadto twosetsofresponsesthatenvelopetherangeofpossibleresponsesoverallnonredundantrotationangles.Thisassumptionisexaminedhere, forthefirsttime,usinga3Dcomputermodelofasixstoryreinforcedconcreteinstrumentedbuildingsubjectedtoanensembleofbidirectionalnearfaultgroundmotions.Peakvaluesofengineeringdemandparameters(EDPs)werecomputedforrotationanglesrangingfrom 0through180toquantifythedifferencebetweenpeakvaluesofEDPsoverallrotationanglesandthoseduetoFN/FPdirectionrotated motions.ItisdemonstratedthatrotatinggroundmotionstoFN/FPdirections(1)doesnotalwaysleadtothemaximumresponses overallangles,(2)doesnotalwaysenvelopetherangeofpossibleresponses,and(3)doesnotprovidemaximumresponsesforall EDPssimultaneouslyevenifitprovidesamaximumresponseforaspecificEDP. DOI: 10.1061/(ASCE)ST.1943541X.0000845 2013AmericanSocietyofCivilEngineers. Authorkeywords: Nearfaultgroundmotion;Directivity;Responsehistoryanalysis;Seismiceffects.IntroductionIntheUnitedStates,boththeCaliforniaBuildingCode [ InternationalConferenceforBuildingOfficials(ICBO)2010 ] andInternationalBuildingCode( ICBO2009 )refertoChapter16 ofASCE/SEI7( ASCE2010 )whenresponsehistoryanalysis (RHA)isrequiredfordesignverificationofbuildingstructures. Forthreedimensional(3D)analysesofsymmetricplanbuildings, ASCE/SEI7requireseitherspectrallymatchedorintensitybased scaledgroundmotionrecords,whichconsistofpairsofappropriate horizontalgroundaccelerationcomponents.Foreachpairof horizontalcomponents,asquarerootofsumofsquares(SRSS) spectrumshallbeconstructedbytakingtheSRSSofthe5% dampedresponsespectraoftheunscaledcomponents.Eachpair ofmotionsshallthenbescaledwiththesamescalefactorsuchthat themeanoftheSRSSspectradoesnotfallbelowthecorresponding ordinateofthetargetspectrumintheperiodrangefrom 0 2 T1to 1 5 T1(where T1istheelasticfirstmodevibrationperiodofthe structure).Thedesignvalueofanengineeringdemandparameter (EDP) memberforces,memberdeformations,orstorydrifts shallthenbetakenasthemeanvalueoftheEDPoverseven (ormore)groundmotionpairs,oritsmaximumvalueoverall groundmotionpairsifthesystemisanalyzedforfewerthanseven groundmotionpairs.Thisprocedurerequiresaminimumofthree records. AsinputforRHAs,strongmotionnetworksprovideuserswith groundaccelerationsrecordedinthreeorthogonaldirections two horizontalandonevertical.Thesensorsrecordinghorizontalaccelerationsareoften,butnotalways,orientedinthenorthsouth(NS) andeastwest(EW)directions.Theserecordswithstationspecific orientationsarereferredtoasasrecordedgroundmotions.If therecordinginstrumentwasinstalledinadifferentorientation abouttheverticalaxisthantheNSandEWdirections,andthe correspondingpairofgroundmotionswasofinterest,thena twodimensional(2D)rotationtransformationcanbeappliedto theasrecordedmotion.Sincetheinstrumentcouldhavebeeninstalledatanyangle,therotatedversionsarepossiblerealizations. Althoughtheasrecordedpairofgroundmotionmaybeapplied tothestructuralaxescorrespondingtothestructure stransverseand longitudionaldirections,thereisnoreasonwhythepairshouldnot beappliedtoanyotheraxesrotatedaboutthestructuralvertical axis.Equivalently,thereisnoreasonwhyrotatedversionsshould notbeappliedtothestructuralaxes.Whichangle,then,shouldone selectforRHAremainsaquestioninearthquakeengineering practice. Thisnotionofrotatinggroundmotionpairshasbeenstudiedin variouscontexts.AccordingtoPenzienandWatabe( 1975 ),the principalaxisofapairofgroundmotionsistheangleoraxisat whichthetwohorizontalcomponentsareuncorrelated.Usingthis ideaofprincipalaxis,theeffectsofseismicrotationangle,defined astheanglebetweentheprincipalaxesofthegroundmotionpair andthestructuralaxes,onstructuralresponsewasinvestigated ( FranklinandVolker1982 ; FernandezDavilaetal.2000 ; MacRae andMattheis2000 ; TezcanandAlhan2001 ; Khoshnoudianand Poursha2004 ; RigatoandMedina2007 ; Lagaros2010 ; Goda 2012 ).Aformulaforderivingtheanglethatyieldsthepeak elasticresponseoverallpossiblenonredundantangles,called critical(or critical),wasproposedbyWilsonetal.( 1995 ).Other researchershaveimprovedupontheclosedformsolutionof1ResearchStructuralEngineer,USGS,MenloPark,CA94025(correspondingauthor).Email:ekalkan@usgs.gov 2Ph.D.Candidate,Univ.ofCalifornia,Berkeley,CA94709.Email: nealsimonkwong@berkeley.edu Note.ThismanuscriptwassubmittedonFebruary3,2012;approvedon April5,2013;publishedonlineonApril8,2013.Discussionperiod openuntilMarch28,2014;separatediscussionsmustbesubmittedfor individualpapers.Thispaperispartofthe JournalofStructuralEngineering ,ASCE,ISSN07339445/04013062(14)/$25.00.ASCE040130621J.Struct.Eng. PAGE 2 Wilsonetal.( 1995 )byaccountingforthestatisticalcorrelationof horizontalcomponentsofgroundmotioninanexplicitway( Lopez andTorres1997 ; Lopezetal.2000 ).TheWilsonetal.( 1995 )formula is,however,basedonconceptsfromresponsespectrumanalysis an approximateprocedureusedtoestimatestructuralresponsesinthe linearelasticdomain.Focusingonlinearelastic,multidegreeoffreedomsymmetric,andasymmetricstructures,Athanatopoulou ( 2005 )investigatedtheeffectoftherotationangleonstructural responseusingRHAsandprovidedformulasfordeterminingthe maximumresponseoverallrotationangles,giventheresponse historiesfortwoorthogonalorientations.Athanatopoulou( 2005 ) alsoconcludedthatthecriticalanglecorrespondingtopeakresponse overallanglesvariednotonlywiththegroundmotionpairunder considerationbutwiththeresponsequantityofinterestaswell. AccordingtoSection1615A.1.25oftheCaliforniaBuilding Code( ICBO2010 ),atsiteswithin3mi(5km)oftheactivefault thatdominatesahazard,eachpairofgroundmotioncomponents shallberotatedtothefaultnormal(FN)andfaultparallel(FP)directions(alsocalledstrikenormalandstrikeparalleldirections)for 3DRHAs.ItisbelievedthattheanglecorrespondingtotheFN/FP directionswillleadtothemostcriticalstructuralresponse.This assumptionisbasedonthefactthat,intheproximityofanactive faultsystem,groundmotionsaresignificantlyaffectedbythefaultingmechanism,directionofrupturepropagationrelativetothesite, andthepossiblestaticdeformationofthegroundsurfaceassociated withflingstepeffects( BrayandRodriguezMarek2004 ; Kalkan andKunnath2006 ).Thesenearsourceeffectscausemostofthe seismicenergyfromtherupturetoarriveinasingle,coherent, longperiodpulseofmotionintheFN/FPdirections( Mavroeidis andPapageorgiou2003 ; KalkanandKunnath2007 2008 ).Thus, rotatinggroundmotionpairstoFN/FPdirectionsisassumedtobea conservativeapproachappropriatefordesignverificationofnew structuresorperformanceevaluationofexistingstructures. Usinga3Dstructuralmodelofaninstrumentedbuildingand anensembleofnearfaultgroundmotionrecords,thisstudy systematicallyevaluateswhetherFN/FPdirectionsrotatedground motionsleadtoconservative(thetermconservativeisusedhere eitherwithpeakorclosetopeakEDPvalues)estimatesofEDPs fromRHAs.DescriptionofStructuralSystemandComputer ModelThetestbedsystemusedisa3Dcomputermodeloftheformer ImperialCountyServicesbuildinginElCentro,California.This relativelysymmetricalbuildinghadanopenfirststoryandfiveoccupiedstories(Fig. 1 ).Designedin1968,itsverticalloadcarrying systemconsistedof12.7cmreinforcedconcrete(RC)thickslabs supportedbyRCpanjoists,whichinturnweresupportedbyRC framesspanningintheorthogonaldirection.Fig. 2 showsthefoundationandtypicalfloorlayouts.Thelateralresistanceofalllevels inthelongitudinal(EW)directionwasprovidedbytwoexterior momentframesatColumnLines1and4andtwointeriormoment framesonColumnLines2and3.Thelateralresistanceinthetransverse(NS)directionwasnotcontinuous.Atthegroundfloorlevel, itwasprovidedbyfourshortshearwallslocatedalongColumn LinesA,C,D,andEandextendingbetweenColumnLines2 and3only(Fig. 2 ,top).Atthesecondfloorandabove,lateral (NS)resistancewasprovidedbytwoshearwallsattheeast andwestendsofthebuilding.Thiscausedthebuildingtobe topheavywithasoftfirststory,asshowninFig. 1 ( Todorovska andTrifunac2008 ).Thedesignstrengthoftheconcretewas 34.5MPaforcolumns,20.7MPafortheelementsbelowground level,and27.6MPaelsewhere.Allreinforcingsteelwasspecified tobegrade40(Fy 276 MPa).Thefoundationsystemconsisted ofpilesundereachcolumnwithpilecapsconnectedwithRC beams(Fig. 2 top). Thebuildingwasinstrumentedin1976with13sensorsat4 levelsofthebuildingand3sensorsatafreefieldsite.Thesensors inthebuildingmeasurehorizontalaccelerationsatthegroundfloor, secondfloor,fourthfloor,androof;verticalaccelerationwasmeasuredatthegroundfloor;theinstrumentationlayoutofthebuilding isgiveninKalkanandKwong( 2012 ).Therecordedmotionsofthis buildingareavailableonlyfortheMw6.51979ImperialValley earthquake,duringwhichthisbuildingwasdamagedandsubsequentlydemolished.Thepeakrecordedaccelerationsduringthis earthquakewere0.34gatthegroundfloorand0.58gattheroof level.Thisbuildingisararecaseofaninstrumentedbuilding severelydamagedbyanearthquake( GoelandChadwell2007 ). Fig.1. (a)ImperialCountyServicesbuilding(photographbyC.Rojahn,withpermissionfromUSGS);(b)eastendofImperialCountyServices building,showingarowofcolumns(farright)thatfailedduringmainshock;viewisnorth(photographbyC.Rojahn,withpermissionfromUSGS)ASCE040130622J.Struct.Eng. PAGE 3 Fig. 1 (bottom)showstheconcentrationofdamageintheground floorcolumnsasaresultofconcretespallingandbucklingof reinforcingbars.Thedetailsaboutthedesign,recordeddata, andobserveddamagecanbefoundinKojicetal.( 1984 ). The3Dcomputermodelofthisbuildingwascreatedusing OpenSees( 2010 ).Centerlinedimensionswereusedintheelement modeling,thecompositeactionoffloorslabswasnotconsidered, andthecolumnswereassumedtobefixedatthebaselevel.Forthe responsehistoryevaluations,masseswereappliedtoframemodels basedonthefloortributaryareaanddistributedproportionallyto thefloornodes.Thesimulationmodelswerecalibratedtothe responsedatameasuredduringtheImperialValleyearthquake soastovalidateandverifytheanalyticalresultsofthecomparative study. Table 1 liststhelinearelasticperiodsofthefirstseveral modes,alongwiththeirmodalparticipationandcontribution factors( Chopra2007 )fortwoorthogonaldirectionsalongthe structuralaxes.Thefundamentalmodeisprimarilyalongthe momentframe(EWdirection)orXdirectionofthecomputer model.AsshowninTable 1 ,theperiodofthestructurealongthis directionis1.2s,whiletheperiodofthestructureintheYdirection Foundation Plan Floor Plan (a) (b) Fig.2. FoundationandgroundlevelplanandtypicalfloorlayoutofImperialCountyServicesbuilding.Note:Transverse(Yornorthsouth)and longitudinal(Xoreastwest) Table1. LinearElasticDynamicPropertiesofImperialCountyServices Building Modenumber( n )Period( s ) n ; xn ; yMCF,x(%)MCF,y(%) 11.25.30.084.50.0 20.40.04.80.068.4 30.4 1 9 0.010.50.0 40.30.0 0 8 0.01.9 50.2 1 0 0.03.00.0 60.2 0 7 0.01.40.0 Note:Themodalparticipation( )andmodalcontributionfactors(MCFs) areshowntoillustratehowthefirstsixmodescontributetothelinearelastic responsesintwoorthogonaldirections.ASCE040130623J.Struct.Eng. PAGE 4 is0.4s,whichistheperiodofthesecondmode.Theirregularities intheNSstiffnessatthegroundfloorappeartohaveresultedin excessivetorsionalresponseandinsignificantcouplingofthe NSandtorsionalexcitationsandresponses.FortheNS(orY) direction,thestructureisnotfirstmodedominatedbecausethe modalcontributionfactorforthefirstmodeinthisdirectionis only68%.GroundMotionsSelectedForthisinvestigation,20nearfaultstrongmotionrecords,listedin Table 2 ,wereselectedfrom10shallowcrustalearthquakescompatiblewiththefollowingscenario: Momentmagnitude: Mw 6 7 0 2 Closestfaultdistance: 0 1 Rrup 11 km NEHRPsoiltype:CorD ShowninFig. 3 arethe5%dampedresponsespectraforthe XandYcomponentsoftheasrecordedgroundmotions.Also shownisthemedianspectrum,computedasthegeometricmean of20responsespectraineachdirection.Themedianspectrashow significantlylargedemandsatthefirstandsecondmodesofthe buildinginbothdirections.MethodologyforEvaluationofFaultNormal/Parallel DirectionsRestrictingourselvestothelinearelasticversionofthestructural model,weinvoketheprincipleofsuperposition(tobeelaboratedas follows)tocomputestructuralresponsesforalargenumberofseismicrotationangles.Whensubjectedtoasinglehorizontalcomponentofgroundmotion,theequationofmotionfora2Dmultistory buildingis( Chopra2007 ) m ¨ u c u k u Peff m i ¨ ug 1 where m c ,and k N N squarematrices(where N isthenumberofdynamicDOFs)thatdefinethestructuralpropertiesofthe building.Inthiscase,theeffectiveearthquakeforce Peffconsistsof asinglehorizontalcomponentofgroundmotion ¨ ugappliedalong thedirectionspecifiedbytheinfluencevector i .However,when analyzinga3Dmultistorybuilding,subjectedtotwohorizontal componentsofgroundmotion,onemustincludealargernumber ofDOFsinthestructuralmatricesandappropriatelymodifythe effectiveearthquakeforce( GoelandChopra2004 ; Athanatopoulou 2005 ).Ifoneappliesthetwoasrecordedhorizontalcomponentsof groundmotion,denotedas ¨ ugxand ¨ ugy,directlyalongthestructural axes,thentheeffectiveearthquakeforcebecomes Peff M ix¨ ugx iy¨ ugy 2 wheretheinfluencevectors ixand iyrefertothedisplacementofthe lumpedmasseswhenthestructureissubjectedtoaunitground displacementalongthe X and Y structuralaxes,respectively.To obtainthestructuralresponseswhenthebuildingissubjectedto ahorizontallyrotatedversionoftheasrecordedgroundmotion Table2. SelectedNearFaultStrongGroundMotionRecords Pair numberEarthquakenameYearStationname MwRrup(km) VS 30(m/s) Faultnormal component Faultparallel component PGA (g) PGV (cm = s) PGD (cm) PGA (g) PGV (cm = s) PGD (cm) 1Tabas,Iran1978Tabas7.42.17670.8118970.88042 2ImperialValley,CA1979ECMelolandOverpassFF6.50.11860.4115400.32715 3ImperialValley,CA1979ElCentroArray#76.50.62110.5109460.34524 4SuperstitionHills,CA1987ParachuteTestSite6.51.03490.4107510.35022 5LomaPrieta,CA1989Corralitos6.93.94620.545140.5427 6LomaPrieta,CA1989LGPC6.93.94780.997630.57231 7Erzincan,Turkey1992Erzincan6.74.42750.595320.44517 8Northridge,CA1994Newhall WPicoCanyonRd6.75.52860.488550.37522 9Northridge,CA1994RinaldiReceivingSta6.76.52820.9167290.46321 10Northridge,CA1994Sylmar ConverterSta6.75.42510.6130540.89353 11Northridge,CA1994Sylmar ConverterStaEast6.75.23710.8117390.57829 12Northridge,CA1994Sylmar OliveViewMedFF6.75.34410.7123320.65411 13Kobe,Japan1995Takatori6.91.52560.7170450.66323 14Kocaeli,Turkey1999Yarimca7.44.82970.348430.37356 15ChiChi,Taiwan1999TCU0527.60.75790.41692150.4110220 16ChiChi,Taiwan1999TCU0657.60.63060.8128930.68058 17ChiChi,Taiwan1999TCU0687.60.34870.61913710.4238387 18ChiChi,Taiwan1999TCU0847.611.25531.2115320.44421 19ChiChi,Taiwan1999TCU1027.61.57140.3107880.27855 20Duzce,Turkey1999Duzce7.26.62760.462470.58048 Note: Mw= momentmagnitude;PGA = peakgroundacceleration;PGV = peakgroundvelocity;PGD = peakgrounddisplacement; Rrup= closestdistanceto coseismicruptureplane; VS 30= averageshearwavevelocityofupper30mofsite. 0 1 2 3 0 1 2 3 4 Period (s), TnPseudo Acc., Xcomponent (g) 0 1 2 3 0 1 2 3 4 Period (s), TnPseudo Acc., Ycomponent (g) Fig.3. Pseudoaccelerationresponsespectraof20nearfaultstrong groundmotions;dampingratio5%(dashedline=medianspectrum ofallrecords)ASCE040130624J.Struct.Eng. PAGE 5 pair,denotedas ¨ u arb gxand ¨ u arb gy,wemodifytheeffectiveearthquake forceas P arb eff M ix¨ u arb gx iy¨ u arb gy M ix cos ¨ ugx sin ¨ ugy iy sin ¨ ugx cos ¨ ugyy M cos ix¨ ugx iy¨ ugy sin ix ¨ ugy iy¨ ugxg cos M ix¨ ugx iy¨ ugy sin f M ix ¨ ugy iy¨ ugxg cos P 1 eff sin P 2 eff 3 where =anarbitraryseismicrotationangleofinterest.Eq.( 3 ) showsthat P arb effisalinearcombinationoftwoexcitations; P 1 effcorrespondstotheexcitationwheretheasrecordedground motionpairisapplieddirectlytothestructuralaxes,whereas P 2 effcorrespondstotheexcitationwheretheasrecordedground motionpairisfirstrotated90clockwisebeforebeingapplied tothestructuralaxes.Usingtheprincipleofsuperpositionfora fixedresponsequantityofinterest,theresponsehistoryforany arbitraryseismicrotationangle r arb maybecomputedasalinear combinationoftworesponsehistories onecorrespondingto P 1 effandtheothercorrespondingto P 2 eff: r arb cos r 1 sin r 2 4 where r 1 =responsehistoryunderexcitation P 1 eff;and r 2 = responsehistoryunderexcitation P 2 eff. Viewingtheresponseasbothafunctionoftimeandrotation angleenablesustobetterunderstandhowthecriticalangle cr, definedastheanglecorrespondingtothelargestresponseover allnonredundantrotationangles,varieswithbothEDPandground motionpair.Foragivenresponsequantityofinterestandrecord pair,theFN/FPdirectionswillcorrespondtotwovalues(i.e.,FN andFProtatedgroundmotionsareappliedalongthe X and Y axes, thenalongthe Y and X axesofthestructure).Bycomparingthese twovalueswiththestructuralresponsesatallotherpossibleangles, onecanevaluatethelevelofconservatisminsuchdirections.If obvioussystematicbenefitsoftheFN/FPorientationsexisted,they shouldbeobservablebyrepeatingsuchcomparisonsforseveral EDPsandrecordpairs. Evenifnoobvioustrendsareobserved,onecanstillcompare theFN/FPdirectionswiththenorotationcase.RatherthancomparingtheFN/FPdirectionstotheasrecordeddirections,however, theasrecordeddirectionmaybeviewedasanarbitrarilyassigned orientation.Asaresult,onewillbeabletostatethelikelihoodofthe 0 1 3 5 Pair No.1 Drms=0.12 Pair No.2 Drms=0.12 Pair No.3 Drms=0.09 Pair No.4 Drms=0.14 Pair No.5 Drms=0.13 0 1 3 5 Pair No.6 Drms=0.09 Pair No.7 Drms=0.12 Pair No.8 Drms=0.13 Pair No.9 Drms=0.11 Pair No.10 Drms=0.14 0 1 3 5 Pair No.11 Drms=0.11 Pair No.12 Drms=0.11 Pair No.13 Drms=0.11 Pair No.14 Drms=0.15 Pair No.15 Drms=0.07 0 1 2 3 0 1 3 5 Pair No.16 Drms=0.11 0 1 2 3 Pair No.17 Drms=0.12 0 1 2 3 Pair No.18 Drms=0.08 0 1 2 3 Pair No.19 Drms=0.11 0 1 2 3 Pair No.20 Drms=0.11Period (s)Spectral Acceleration (g) Fig.4. Maximumandminimumenvelopesforsquarerootofsumofsquares(SRSS)responsespectrarotatedthroughallanglesfrom0through180 with5interval;rootmeansquare( Drms)metricisshownforeachhorizontalpairofgroundmotiontoindicatedegreeofvariationinrotatedspectra; smallvaluesof DrmsinallpanelsindicatethatvariationofspectralvaluesbyrotatinggroundmotioncomponentsisinsignificantASCE040130625J.Struct.Eng. PAGE 6 FN/FPresponsesbeingconservativeinsteadofsimplystating whetherornotitwasconservative. Iftherotationangle forarecordpairwastheonlysourceof uncertaintyandtheprobabilitydistributionfor wasspecified,then aconditionalprobabilitydensityfunction(PDF)forthestructural responsemaybedefined.Inparticular,if isuniformlydistributed from0through180,thenthePDFfortheEDPmaybeestimated by(1)obtainingarandomsampleof n rotationanglesbasedonthe uniformdistribution,(2)computingtheEDPcorrespondingtoeach ofthe n angles,and(3)formingahistogramwiththecollectionof EDPvalues( Wasserman2004 ).Equippedwithanestimateofthe EDP sprobabilitydistribution,conditionedonagroundmotion pair,onecanapproximatelydeterminetheprobabilityofexceeding theFN/FPresponses.LowprobabilitiesofexceedancewouldsuggestthatthereissomemeritinfocusingourattentionontheFN/FP directions.StructuralResponseVariabilitywithRotationAngleAccordingtotheASCE/SEI7provisionsunderSection16.1.3.2, thehorizontalcomponentsaretobeidenticallyscaledsuchthatthe averageoftheSRSSspectrafromallscaledhorizontalcomponent pairsexceedsthetargetdesignspectrum(definedunderASCE/SEI7,Section11.4.5or11.4.7)overtheperiodrangeof 0 2 T1to 1 5 T1. HowWilltheSRSSSpectrumChangeiftheGround MotionPairisRotated? Byrotatingeachofthe20recordpairsinTable 2 from0to180 witha5intervalintheclockwisedirection,onecancompute37 alternativeSRSSspectra.Fig. 4 showsthemaximumandminimum envelopesboundingsuchrotatedversionsoftheSRSSresponse spectraforeachgroundmotionpairs(noscalingisapplied).Inthis figure, Drmsreferstorootmeansquare,ametricusedtoquantifythe variabilityofspectralaccelerations(Sa)withchangingrotation angle. Drmsiscomputedforeachrotationangleoverallspectral periodsas Drms 1 N XN i 1 ln Samax ; i ln Samin ; i2v u u t 5 where i referstothe i thspectralperiodand N isthetotalnumberof logarithmicallyspacedspectralperiods.Itisvisuallyevidentthat 0 0.2 0.4 0.6 0.8 1 Pair No.1 Pair No.2 Pair No.3 Pair No.4 Pair No.5 0 0.2 0.4 0.6 0.8 1 Pair No.6 Pair No.7 Pair No.8 Pair No.9 Pair No.10 0 0.2 0.4 0.6 0.8 1 Pair No.11 Pair No.12 Pair No.13 Pair No.14 Pair No.15 0 45 90 135 180 0 0.2 0.4 0.6 0.8 1 Pair No.16 0 45 90 135 180 Pair No.17 0 45 90 135 180 Pair No.18 0 45 90 135 180 Pair No.19 0 45 90 135 180 Pair No.20Normalized firststory drifts in Xdirection (EW) Rotation angle, (deg) Fig.5. Normalizedfirststorydriftinlongitudinaldirection(Xoreastwest)asafunctionofclockwiserotationangle for20groundmotionpairs; thenormalizingfactoristhemaximumvalueoverallanglesforthegroundmotionpairbeingconsidered;thisfactordiffersforeachpair;thisfigure showsthatstorydriftcanvarybyafactorof2overthepossibleanglesofinterest.(Note:FNdirectionisnotnecessarilyat0)ASCE040130626J.Struct.Eng. PAGE 7 theSRSSresponsespectrumvarymarginallywithrotationangle. Therelativelysmall DrmsvaluesindicatethatseveralrotatedversionsofthegroundmotionpaircansatisfytheASCEcriteriaand yetprovidestructuralresponsesthataredifferent(asshownlater). Thisfigurealsoimpliesthatrotatinggroundmotionshaveamarginaleffectonthegroundmotionscalingfactorscomputedforeach pairtosatisfytheASCEcriteria. HowMuchVariabilityisthereintheElasticStructural ResponsesastheRotationAngleisVaried? Fig. 5 addressesthisquestionbyshowingthedriftsinthelongitudinal(EWorX)directionforthefirststoryasafunctionof therotationangleforallrecords.Tobetterunderstandtherelative variability,eachsubplotwasnormalizedbythemaximumresponse overallangles.Maximumresponsesforindividualgroundmotion pairswerefoundtooccuratdifferentangles.Withtheexception ofafewpairs,thefirststorydrift(i.e.,interstorydrift)inthe Xdirectioncanvarybyafactorof2overthepossibleanglesof interest.Thisisconsideredtobealargevariation. Althoughthisfigureindicatesthatthefirststorydriftinthe Xdirectiondoesnotvarysignificantlywithrotationanglefor groundmotionPairNo.3,thesamestatementcannotbemade forotherresponsequantities.ConsideringPairNo.3,variousother responsequantitiesareshownasafunctionofrotationanglein Fig. 6 .ItisevidentthatpeakvaluesofotherEDPsoccuratdifferent Fig.6. ForgroundmotionPairNo.3,normalizedengineeringdemandparameters(EDPs)showdifferentdegreeofvariationwithrespecttoclockwiserotationangle .Inthisfigure, Pz, Mx, My,and Tzcorrespondtothefirststorycornercolumn saxialforce,momentsabouttwoorthogonal directions,andtorsion;thenumberfollowingtheXorYdirectionindicatesthefloor,forexample,AccelX6meanssixthflooraccelerationalong Xdirection.(Note:FNdirectionisnotnecessarilyat0)ASCE040130627J.Struct.Eng. PAGE 8 anglesforthesamerecordpair.LargevariationforEDPsotherthan storydriftisalsoobserved.Forexample,thetorsionforanarbitrarilyselectedcolumncanvarybyafactorof2overthepossible angles. Tobetterquantifythisvariationwithrotationangle,thecoefficientofvariation(COV)iscomputedusingEq.( 6 )foreachground motionpairandforeachresponsequantityrelatedtoanarbitrarily selectedcornercolumninthefirststory(thesevaluesareshownin Table 3 ): COV 1 n 1Pn n 1 xi x 2q x 6 TheCOVfor MxislargerforPairNo.1thanforPairNo.2.The reverseistrue,however,whentheresponsequantityofinterestis Myinstead.Here,theCOVislargerforthesecondpairthanforthe firstpair.Theseresultsdemonstratethatonemustconsiderboththe responsequantityofinterestandthegroundmotioncharacteristics whenattemptingtopredictthevariabilitywithrespecttorotation angleinadvance. ThefactthatthevariabilitydependsonboththeresponsequantityandgroundmotionpaircanalsobeobservedinFig. 7 ,where theheightwisedistributionofstorydriftsintheXdirectionover severalanglesisshown.Toillustratethevariabilityintheresponses withineachpair,acommonscalewasnotusedforthedriftaxis. Table3. CoefficientofVariationforForce(P)andMoment(MorT) ParametersalongX,Y,andZDirectionsofaFirstStoryCornerColumn Pair number Coefficientofvariationforarbitraryfirststorycornercolumn Px(kips) Py(kips) Pz(kips) Mx(kipin) My(kipin) Tz(kipin) 10.260.290.280.290.260.23 20.360.150.120.170.350.21 30.080.270.140.280.070.22 40.340.160.170.170.360.16 50.090.340.260.360.050.32 60.170.250.210.270.240.23 70.350.090.120.070.340.14 80.240.120.090.140.220.09 90.290.160.070.180.290.21 100.240.140.090.170.230.18 110.300.240.220.260.310.26 120.220.390.320.390.300.37 130.280.070.200.080.250.10 140.200.170.140.160.200.10 150.410.280.090.260.380.27 160.120.070.150.060.120.06 170.110.260.220.270.120.27 180.400.270.130.270.390.27 190.240.210.120.210.230.19 200.250.280.220.270.250.18 Note:X=longitudinal;Y=transverse;Z=verticaldirectioninplanview. 0 1 2 3 1 2 3 4 5 6 Pair No.1 0 1 2 Pair No.2 0 0.5 1 1.5 Pair No.3 0 2 4 Pair No.4 0 0.5 1 1.5 Pair No.5 0 1 2 3 1 2 3 4 5 6 Pair No.6 0 1 2 3 Pair No.7 0 1 2 3 Pair No.8 0 2 4 6 Pair No.9 0 2 4 6 Pair No.10 0 1 2 3 1 2 3 4 5 6 Pair No.11 0 1 2 3 Pair No.12 0 5 10 Pair No.13 0 1 2 Pair No.14 0 2 4 Pair No.15 0 1 2 3 1 2 3 4 5 6 Pair No.16 0 1 2 3 Pair No.17 0 2 4 6 Pair No.18 0 1 2 3 Pair No.19 0 1 2 Pair No.20Drifts in Xdirection (EW), (%)Floor number Fig.7. Storydriftprofilesinlongitudinal(Xoreastwest)directionfor20groundmotionpairsrotated0through180clockwisewithanintervalof 10.Toillustratetherelativevariabilitywithrespecttotherotationangle,acommonscalewasnotusedASCE040130628J.Struct.Eng. PAGE 9 0 0.5 1 2 4 6 Pair No.1 0 0.5 1 Pair No.2 0 0.5 1 Pair No.3 0 0.5 1 Pair No.4 0 0.5 1 Pair No.5 0 0.5 1 2 4 6 Pair No.6 0 0.5 1 Pair No.7 0 0.5 1 Pair No.8 0 0.5 1 Pair No.9 0 0.5 1 Pair No.10 0 0.5 1 2 4 6 Floor numberPair No.11 0 0.5 1 Pair No.12 0 0.5 1 Pair No.13 0 0.5 1 Pair No.14 0 0.5 1 Pair No.15 0 0.5 1 2 4 6 Pair No.16 0 0.5 1 Pair No.17 0 0.5 1 Normalized drifts in X/EW direction (%) Pair No.18 0 0.5 1 Pair No.19 0 0.5 1 Pair No.20 Fig.8. Storydriftprofilesinlongitudinal(Xoreastwest)direction;anglescorrespondingtofaultnormalandfaultparalleldirectionsshownby dashedanddasheddottedlines,respectively 0 0.5 1 2 4 6 Pair No.1 0 0.5 1 Pair No.2 0 0.5 1 Pair No.3 0 0.5 1 Pair No.4 0 0.5 1 Pair No.5 0 0.5 1 2 4 6 Pair No.6 0 0.5 1 Pair No.7 0 0.5 1 Pair No.8 0 0.5 1 Pair No.9 0 0.5 1 Pair No.10 0 0.5 1 2 4 6 Floor numberPair No.11 0 0.5 1 Pair No.12 0 0.5 1 Pair No.13 0 0.5 1 Pair No.14 0 0.5 1 Pair No.15 0 0.5 1 2 4 6 Pair No.16 0 0.5 1 Pair No.17 0 0.5 1 Normalized drifts in Y/NS direction (%) Pair No.18 0 0.5 1 Pair No.19 0 0.5 1 Pair No.20 Fig.9. Storydriftprofilesintransverse(Yornorthsouth)direction;anglescorrespondingtofaultnormalandfaultparalleldirectionsshownby dashedanddasheddottedlines,respectivelyASCE040130629J.Struct.Eng. PAGE 10 Thevariabilitywassignificantlylargeforsomegroundmotion pairs(e.g.,PairNos.12,15,and18),ascomparedtosmallervariabilityobservedforPairNos.3,16,and17.Forthefifthpairof groundmotion,thelargerdriftinthefourthstoryindicateshighermodeeffects.Contributionofhighermodeeffectstotheresponse withlargerdemandsatupperstoriesbecomesmorepronouncedat certainanglesonly.SimilarresultsfortheYdirectionareshownin KalkanandKwong( 2012 ).Theseresultsalsoconfirmthefactthat crvarieswithgroundmotionandwithresponsequantityofinterest.Thisisbecause crisaquantitythatishighlydependentonthe completeresponsehistoryoftheEDP.Asaresult,determininga rotationanglethatyieldsaconservativeestimateofstructuralresponsesimultaneouslyforbothalargenumberofresponsequantities(EDPs)andgroundmotionrecordsisdifficult;itiseasy, however,tocompute crforasingleEDPunderasinglepairof accelerograms( Athanatopoulou2005 ).EvaluationofFaultNormal/ParallelDirectionRotated GroundMotionToevaluatetheusefulnessofrotatingarecordpairintheFN/FP directions,apracticecommonlyused,theEDPscorresponding totheFN/FPdirectionsarecomparedagainstthosecorresponding toallotherdirections.Tolimitthecomputationstoareasonable size,eachasrecordedpairisrotatedclockwisebyincrementsof 10insteadof5beforetheEDPsarecalculated.Asaresult, thetwoFN/FPsetsofresponsesarecomparedagainst19othersets. Forexample,the21heightwisedistributionsofstorydriftsin theXdirection,foreachrecordpair,areshowninFig. 8 ;plots showingdriftsintheYdirectionareshowninKalkanandKwong ( 2012 ).ThedistributionofdriftscorrespondingtotheFNdirection ishighlightedinred,whilethatcorrespondingtotheFPdirectionis highlightedingreen.Todisplaythevariabilityinresponseswithin eachpair,thedriftvaluesarenormalizedbythemaximumdrift valueoverall19anglesandovertheentireheight.Forsomepairs (e.g.,PairNos.5,6,and8),themaximumoftheFN/FPdriftsisnot thelargestamongallpossibilities.Visually,themaximumofthe FN/FPdriftsisthelargestamongallpossibilitiesapproximately onlyfor10ofthe20recordpairs.Consequently,theFN/FPdrifts arenotalwaysconservative. WhetherornottheFN/FPdriftsareconservativedepends notonlyonthegroundmotionpairbutalsoontheEDP.Forinstance,althoughtheFNdirectionyieldsthemaximumheightwise distributionofdriftsintheXdirectionforPairNo.18,theFNdirectionyieldstheminimumheightwisedistributionofdriftsinthe Fig.10. Histogramof1,000randomlyobtainedrealizationsoffirststorydriftinXdirection;dashedline=valuecorrespondingtofaultnormal direction;dasheddottedline=valuecorrespondingtofaultparalleldirectionASCE0401306210J.Struct.Eng. PAGE 11 Ydirectionforthesamepair,asdemonstratedinFig. 9 .Asanother example,althoughtheFPdirectionyieldsthelargestroofdriftin theXdirectionforPairNo.5,thesamedirectionforthesamepair doesnotguaranteeaconservativefirststorydriftintheXdirection, asdemonstratedinFig. 8 .Thus,onecannotbecertainthatthe worstcaseresponsesarealwaysobtainedwhenperformingRHAs withgroundmotionsrotatedtotheFN/FPdirections. IftheFN/FPDirectionsDoNotGeneratetheMaximum ResponsesforAllResponseQuantitiesandforAll GroundMotionPairs,isthereStillanAdvantageto RotatinganasrecordedPairPriortoPerforming ResponseHistoryAnalyses? Toaddressthisissue,theFN/FPdirectionrotatedgroundmotions areevaluatedfromastatisticalviewpoint.Supposetheonlysource ofaleatoricuncertaintyinresponsesisduetouncertaintyinthe rotationangleofthegroundmotionpair.Inotherwords,given thestructuralmodelandgroundmotionpair,theEDPwillhave aprobabilitydistributionthatisdirectlyrelatedtotheprobability distributionfortherotationangle.Thisconditionaldistributionfor theEDPcanserveasabenchmarktoevaluatetheusefulnessin rotatingasrecordedgroundmotionstotheFN/FPdirections. BecausethefunctionalrelationshipbetweentheEDPandthe rotationangleisdifferentforeachEDPofinterest,theconditional probabilitydistributionwillbedifferentfordifferentEDPs. Moreover,becausethefunctionalrelationshipisgenerallycomplex (especiallyfornonlinearinelasticsystems),directanalytical determinationoftheprobabilitydistributionisnotfeasible. Consequently,MonteCarlosimulationisusedheretoestimate thesedistributions.Assumingtherotationangleisauniformlydistributedrandomvariable,arandomsampleofanglesisgenerated. Foreachangleintherandomsample,theEDPofinterestisdetermined.Summarizingsuchdataintheformofhistograms,forall recordpairsandforthefirststorydriftintheXdirection,leads totheplotsshowninFig. 10 ;similarplotsfortheYdirection areshowninKalkanandKwong( 2012 ). Thehistogramsinthisfiguremaybeinterpretedasapproximate PDFsforthenormalizedEDPs(normalizedbytheirmaximumvalues).Thenormalizedscalesconfirmthattheresponsevariability dependsonboththerecordpairandtheEDPofinterest.These approximatedensitiesareboundedsincetherangeofpossiblerotationanglesisfinite.AmajorityoftheapproximatePDFssharea commonshape.Specifically,thedistributionsappeartobebimodal, withthemodalvaluesoftenattheextremes,alsovalidforother EDPs.Aroughinterpretationofthisisthatifoneweretodetermine Fig.11. ForagivenpairofgroundmotionandagivenvalueoffirststorydriftsinXdirection,theprobabilityofobservinganengineeringdemand parameter(EDP)valueequaltoorlessthanthegivenEDPvalueisshownbasedon1,000realizations;redline=EDPvaluecorrespondingtofaultnormaldirection;greenline=EDPvaluecorrespondingtofaultparalleldirection;blueline=largerofFN/FPresponsesASCE0401306211J.Struct.Eng. PAGE 12 theEDPcorrespondingtoarandomlychosenangle,theEDP wouldmostlikelybeamaximumoraminimumvalue(ratherthan somewhereinbetween)withrespecttoallpossiblevalues.Ifone weretotaketheEDPasthelargeroftheFN/FPEDPsinstead, Fig. 10 illustratesthatthevaluewouldusuallybelargerthanhalf ofallpossibleresponses. Toquantifythelatterobservations,theconceptofcumulative distributionfunctions[CDFs;theCDFof x ,or F x ,indicates theprobabilityofobservingavalueequaltoorlessthanthevalue of x ]isutilized.ApproximateCDFsforthenormalizedfirststory driftsintheXdirectionareshowninFig. 11 [similarplotsare shownfortheYdirectioninKalkanandKwong( 2012 )].This figureissimplythedatafromFig. 10 replottedinadifferent way.ThesteepslopeneartheendsoftheCDFsisconsistentwith thepreviousobservationthatresponsesneartheextremesofthe possiblerangehavehigherprobabilitiesofoccurringrelativeto othervalues.Tounderstandwhatinformationthelarger(blue) oftheFN(red)andFP(green)responsesprovide,wewillfocus onthefirstsubplotinFig. 11 .Thesubplotindicatesthatthere isapproximatelya65%probabilityofobservingafirststorydrift valuelessthanorequaltotheFPvalueidentifiedinblue(inthis caseitisalsothelargeroftheFN/FPvalues).Equivalently,thereis approximatelya35%probabilitythattheFPvaluewillunderestimatethedriftforpreciselyrecordPairNo.1.Focusingontheblue linesforallrecordpairsnext,oneobservesthattheprobabilityof observingadriftvaluelargerthanthemaximumoftheFN/FPvalue isconsistentlylessthan50%forallrecordpairs.However,this trendisnotperfect,asdemonstratedforPairNos.8and13for theYdirectiondriftsinKalkanandKwong( 2012 ). TheCDFs,andtheassociatedprobabilitystatements,are approximatebecausetheempiricalcumulativedistributionfunctions(ECDFs)wereshowninsteadofthetrueCDFs.Inprobability andstatistics,theECDFisanestimateoftheCDFobtainedusinga randomsamplefromthetrueCDF( Wasserman2004 ).Assigning anequalprobabilitytoeachvalueintherandomsampleofsize n andusingEq.( 7 )leadstoastaircasecurveknownasthe ECDF: Fn t 1 n Xn i 11 f Xi t g 7 where Xi i thvalueintherandomsampleofsize n ;and1=indicatorfunction itis1onlyiftheeventinthebracketsistrueand 0otherwise.Asthesamplesizeincreases,theECDFconverges almostsurelytothetrueCDFbecauseoftheGlivenkoCantelli theorem( Dudley1999 ).ThisisshowninKalkanandKwong ( 2012 )when100,1,000,and5,000differentrandomsamplesof thefirststorydriftintheXdirectionareusedtocomputethe ECDF.Thecurvecorrespondingtotheuseof1,000valuesis virtuallyindistinguishablefromthatassociatedwiththeuseof 5,000values.Asaresult,1,000valueswereusedtoconstruct thehistogramsandECDFsinthepreviousfigure. SincetheconditionalECDFsvarydependingonresponsequantity,thebenchmarkevaluationsoftheFN/FPdirectionsshouldbe performedconsideringseveralresponsequantities.Usingasample sizeof5,000,Table 4 showstheprobabilitiesofexceedingthe largeroftheFN/FPresponsesforstorydriftsinallstoriesand inbothorthogonaldirectionsofthestructure.Theseprobabilities ofexceedancemaybeinterpretedastheamountoferroronemakes indecidingtousethelargeroftheFN/FPresponseastheworstcasevalueamongallpossibilities.Consideringerrorsfromroundoffandtheuseofafiniterandomsample,Table 4 numerically confirmsthatthereisalwayssomeprobabilityofobtainingaresponsevaluelargerthanthatassociatedwiththeFNandFPdirections.Inotherwords,thereisalwayssomeamountoferrormade whendecidingtousetheFN/FPresponseastheworstamongall angles.However,thecellswithprobabilitiessmallerthan15%(indicatedbyboldfont)maybeviewedasinstanceswheretheFN/FP valueisessentiallyconservative(15%isawidelyacceptedthresholdforsafetyforengineeringapplications).ItisnumericallyconfirmedinTable 4 thatsuchconservatismtypicallyvarieswith responsequantitiesandrecordpair. Withsuchnumericalresultsonecanaddresswhetherrotationin theFN/FPdirectionsisworthwhile.Onealternativetodeliberate Table4. ProbabilitiesofExceedingLargerResponseamongFN/FPValuesforSelectedResponseQuantities,Estimatedwith5,000RandomSamples Pairnumber ProbabilityofexceedinglargerresponseamongFN/FPresponses(%) DRx ; 1DRx ; 2DRx ; 3DRx ; 4DRx ; 5DRx ; 6DRy ; 1DRy ; 2DRy ; 3DRy ; 4DRy ; 5DRy ; 6135282619 14 15302524242323 2 0147910 282525252525 34938 15131211 262525252525 4303027252627423838383838 5323842 6558 35444040404040 6464646454443 13109999 7 111214 15 1515 172525262626 8404141403939 57 4645454444 9 03711141412 1616171717 10383841444646454646464646 11403531272028 655555 12252120252829393838383838 13 443334 726160565554 1415 14121214 16303333333333 15474849 50 4949464444444444 162331 1515 3031204545464746 17323437474948262424242424 18 7811121313544444 19 15 1517181919343232323232 20 5404810 242323232323 Note:Storydriftsforbothorthogonaldirectionsofthebuildingareconsidered.Probabilitiessmallerthan15%areindicatedbyboldfont,whereas probabilities largerthan50%areindicatedbyitalicfont(DRx ; n n thstorydriftinXdirection).ASCE0401306212J.Struct.Eng. PAGE 13 rotationistoutilizetheasrecordedorientation,whichcanbe viewedasarandomlyselecteddirection.Theresponsefromsuch anarbitraryorientationmaybelargerorsmallerthantheFN/FP values. HowOftenDoesTheValueofResponseQuantityfrom theArbitraryDirectionExceedthatoftheFN/FP Value? TheprobabilityvaluespresentedinTable 4 providetheanswer.For example,the35%valueforrecordPairNo.1andfirststorydriftin theXdirectionmeansthat,among5,000trials,theresponsecorrespondingtoarandomlychosendirectionexceedstheFN/FP value35%ofthetime.However,thelatterremarkisnotvalid forallrecordpairsandallresponsequantities,asdemonstrated bythecellsindicatedbyitalicfontinTable 4 .Forinstance, the72%valueforrecordPairNo.13andfirststorydriftinthe Ydirectionmeansthattheresponsecorrespondingtoarandomly chosendirectionexceedstheFN/FPvalue72%ofthetime.Thus, theFN/FPdirectionsarelessconservativeinthisparticularcase. Nevertheless,therelativelyfewredcellssuggeststhatusingthe largeroftheFN/FPresponsetypically,butnotalways,leads toavaluelargerthanthatfromarandomlychosen/asrecorded direction.ConclusionsThecurrentstateofpracticeintheUnitedStatesistorotatetheasrecordedpairofgroundmotionstothefaultnormalandfaultparallel(FN/FP)directionsbeforetheyareusedasinputfor3D RHAsofstructureswithin5kmofactivefaults.Itisassumedthat thisapproachwillleadtotwosetsofresponsesthatenvelopethe rangeofpossibleresponsesoverallnonredundantanglesof rotation.Thus,itisconsideredtobeaconservativeapproachappropriatefordesignverificationofnewstructures.Basedona linearelasticcomputermodelofasixstoryinstrumentedstructure, thisstudy,forthefirsttime,evaluatestherelevanceofusingthe FN/FPdirectionsinRHAsanddemonstratesitsprosandcons asfollows: 1.ItwasshownthatrotatedversionsoftheSRSSresponse spectra,followingtheASCE/SEI7provisionsunder Section16.1.3.2,donotvarymuchwithrotationangle themaximumdifferenceobservedislessthan10%.Several rotatedversionsofagroundmotionpaircansatisfytheASCE criteriaandyetprovidestructuralresponsesthatcanvarybya factorof2. 2.Thecriticalangle crcorrespondingtothelargestresponse overallpossibleanglesvarieswiththegroundmotionpair selectedandtheresponsequantityofinterest.Therefore,it isdifficulttodetermineanoptimalbuildingorientation thatmaximizesdemandsforallEDPsbeforeresponsehistory analysesareconducted. 3.TheuseoftheFN/FPdirectionsappliedalongtheprincipal directionsofthebuildingalmostneverguaranteesthatthe maximumresponseoverallpossibleangleswillbeobtained. EventhoughitmayleadtoamaximumforaspecificEDP,it willsimultaneouslybenonconservativeforotherEDPs. Therefore,iftheperformanceassessmentanddesignverificationisconductedagainstworstcasescenarios,thenbidirectionalgroundmotionsshouldbeappliedatvariousangles withrespecttothestructure sprincipaldirectionstocover allpossibleresponses.Althoughthismightnotbeapractical solution,itcouldstillbeworthdoingforcertainprojects. 4.Treatingtheasrecordeddirectionasarandomlychosendirection,itisobservedthatthereismorethana50%probability thatthelargerresponseamongthetwoFNandFPvalueswill exceedtheresponsecorrespondingtoanarbitraryorientation. Thelatterobservationisvalidformost,butnotall,oftherecordpairsandresponsequantitiesconsidered.Therefore,comparedtonorotationatall,useofthelargerresponseamong thetwovaluescorrespondingtotheFNandFPdirectionsis warranted. Althoughtheseobservationsandfindingsareprimarilyapplicabletobuildingsandgroundmotionswithcharacteristicssimilar tothoseutilizedinthisstudy,theyareincloseagreementwiththose reportedinReyesandKalkan( 2013a b ),wheretheinfluenceof rotationangleonseveralEDPswasexaminedinaparametricstudy usingsymmetric(torsionallystiff)andasymmetric(torsionally flexible)moderndesignsinglestoryandmultistorylinearelastic andnonlinearinelasticbuildingssubjectedtoadifferentsetof nearfaultrecords.AcknowledgmentsNealS.KwongwouldliketoacknowledgetheUSGSforproviding himthefinancialsupportforconductingthisresearch.Special thanksareextendedtoRakeshGoelforgenerouslyproviding the OpenSees modeloftheImperialCountyServicesbuilding. RuiChen,AlexTaflanidis,DimitriosVamvatsikos,AysegulAskan, RicardoMedina,andthreeanonymousreviewersreviewedthe materialpresentedhereinandofferedtheirvaluablecomments andsuggestions,whichhelpedimprovethetechnicalqualityand presentationofthispaper.ReferencesASCE.(2010). Minimumdesignloadsforbuildingsandotherstructures ASCE/SEI710 ,Reston,VA. Athanatopoulou,A.M.(2005). Criticalorientationofthreecorrelated seismiccomponents. Eng.Struct. ,27(2),301 312. Bray,J.,andRodriguezMarek,A.(2004). Characterizationofforwarddirectivitygroundmotionsinthenearfaultregion. SoilDyn.EarthquakeEng. ,24(11),815 828. Dudley,R.M.(1999). Uniformcentrallimittheorems ,CambridgeUniversity Press,Cambridge,UK. FernandezDavila,I.,Comeinetti,S.,andCruz,E.F.(2000). Considering thebidirectionaleffectsandtheseismicanglevariationsinbuilding design. Proc.,12thWorldConf.onEarthquakeEngineering Auckland,NewZealand. Franklin,C.Y.,andVolker,J.A.(1982). Effectofvarious3Dseismic inputdirectionsoninelasticbuildingsystemsbasedonINRESB3D82ComputerProgram. Proc.,7thEur.Conf.onEarthquake Engineering Chopra,A.K.(2007). Dynamicsofstructures:Theoryandapplicationsto equationengineering ,2ndEd.,PrenticeHall,EnglewoodCliffs,NJ. Goel,R.K.,andChadwell,C.(2007). Evaluationofcurrentnonlinear staticproceduresforconcretebuildingsusingrecordedstrongmotion data,datainterpretationrep. ,CaliforniaStrongMotionInstrumentationProgram(CSMIP),Dept.ofConservation,Sacramento,CA. Goel,R.K.,andChopra,A.K.(2004). EvaluationofmodalandFEMA pushoveranalysis:SACbuildings. EarthquakeSpectra ,20(1), 225 254. Goda,K.(2012). ComparisonofpeakductilitydemandofinelasticSDOF systemsinmaximumelasticresponseandmajorprincipaldirections. EarthquakeSpectra ,28(1),385 399. InternationalConferenceforBuildingOfficials(ICBO).(2009). Internationalbuildingcode ,Whittier,CA. InternationalConferenceforBuildingOfficials(ICBO).(2010). California buildingcode ,Whittier,CA.ASCE0401306213J.Struct.Eng. PAGE 14 Kalkan,E.,andKunnath,S.K.(2006). Effectsofflingstepandforward directivityontheseismicresponseofbuildings. EarthquakeSpectra 22(2),367 390. Kalkan,E.,andKunnath,S.K.(2007). Effectivecyclicenergyasa measureofseismicdemand. J.EarthquakeEng. ,11(5),725 751. Kalkan,E.,andKunnath,S.K.(2008). Relevanceofabsoluteandrelative energycontentinseismicevaluationofstructures. Adv.Struct.Eng. 11(1),17 34. Kalkan,E.,andKwong,N.S.(2012). Evaluationoffaultnormal/ faultparalleldirectionsrotatedgroundmotionsforresponsehistory analysisofaninstrumentedsixstorybuilding USGSOpenFileReport 2012 1058 ,MenloPark,CA, http://pubs.usgs.gov/of/2012/1058/ (Sep.21,2013). Khoshnoudian,F.,andPoursha,M.(2004). Responsesofthree dimensionalbuildingsunderbidirectionalandunidirectionalseismic excitations. Proc.,13thWorldConf.onEarthquakeEngineering ,Mira DigitalPublishing,Vancouver,Canada. Kojic,S.,Trifunac,M.D.,andAnderson,J.C.(1984). Apostearthquake responseanalysisoftheImperialCountyServicesbuildinginEl Centro Rep.CE8402 ,UniversityofSouthernCalifornia,Dept.of CivilEngineering,LosAngeles. Lagaros,N.D.(2010). Multicomponentincrementaldynamicanalysis consideringvariableincidentangle. Struct.Infrastr.Eng. ,6(1 2), 77 94. Lopez,O.A.,Chopra,A.K.,andHernandez,J.J.(2000). Critical responseofstructurestomulticomponentearthquakeexcitation. EarthquakeEng.Struct.Dyn. ,29(12),1759 1778. Lopez,O.A.,andTorres,R.(1997). Thecriticalangleofseismicincidenceandstructuralresponse. EarthquakeEng.Struct.Dyn. ,26(9), 881 894. MacRae,G.A.,andMattheis,J.(2000). Threedimensionalsteelbuilding responsetonearfaultmotions. J.Struct.Eng. ,10.1061/(ASCE)07339445(2000)126:1(117),117 126 .Mavroeidis,G.P.,andPapageorgiou,A.S.(2003). Amathematical representationofnearfaultgroundmotions. Bull.Seismol.Soc.Am. 93(3),1099 1131. OpenSees [Computersoftware].Berkeley,CA, http://opensees.berkeley .edu (Sep.21,2013). Penzien,J.,andWatabe,M.(1974). Characteristicsof3dimensionalearthquakegroundmotions. EarthquakeEng.Struct.Dyn. ,3(4),365 373. Reyes,J.C.,andKalkan,E.(2013a). Significanceofrotatinggroundmotionsonbehaviorofsymmetricandasymmetricplanstructures:Part1. Singlestorystructures. EarthquakeSpectra ,inpress. Reyes,J.C.,andKalkan,E.(2013b). Significanceofrotatinggroundmotionsonbehaviorofsymmetricandasymmetricplanstructures:Part2. Multistorystructures. EarthquakeSpectra ,inpress. Rigato,A.,andMedina,R.A.(2007). Influenceofangleofincidenceon theseismicdemandsforinelasticsinglestoreystructuressubjectedto bidirectionalgroundmotions. Eng.Struct. ,29(10),2593 2601. Tezcan,S.S.,andAlhan,C.(2001). ParametricanalysisofirregularstructuresunderseismicloadingaccordingtothenewTurkishEarthquake Code. Eng.Struct. ,23(6),600 609. Todorovska,M.I.,andTrifunac,M.D.(2008). EarthquakedamagedetectionintheImperialCountyServicesBuilding.III:Analysisofwave traveltimesviaimpulseresponsefunctions. SoilDyn.Earthquake Eng. ,28(5),387 404. Wasserman,L.(2004). Allofstatistics:Aconcisecourseinstatistical inference ,Springer,NewYork. Wilson,E.L.,Suharwardy,I.,andHabibullah,A.(1995). Aclarificationof theorthogonaleffectsinathreedimensionalseismicanalysis. EarthquakeSpectra ,11(4),659 666.ASCE0401306214J.Struct.Eng. PAGE 1 DETERMINATION OF BRACE FORCES CAUSED BY CONSTRUCTION LOADS AND WIND LOADS DURING BRIDGE CONSTRUCTION By SAMUEL TAYLOR EDWARDS A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT O F THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERING UNIVERSITY OF FLORIDA 2014 PAGE 2 2014 Samuel Taylor Edwards PAGE 3 3 ACKNOWLEDGMENTS This thesis would not be possible without the support of several individuals. I would like to express my sincere gratitude to my advisor, Dr. Gary Cons olazio, for his motivation and guidance throughout my graduate education. He is an excellent role model and has greatly influenced my professional and technical skills as an engineer. Additionally, I would like to thank Dr. H.R. (Trey) Hamilton, Dr. Kurt Gurley, and Dr. Jennifer Rice for serving on my supervisory committee. Finally, I would like to thank my family and friends for providing support throughout this process PAGE 4 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ ............... 3 LIST OF TABLES ................................ ................................ ................................ ........................... 6 LIST OF FIGURES ................................ ................................ ................................ ......................... 7 ABSTRACT ................................ ................................ ................................ ................................ ... 11 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .................. 13 1.1 Overview ................................ ................................ ................................ ........................... 13 1.2 Objectives ................................ ................................ ................................ ......................... 15 1.3 Scope of Work ................................ ................................ ................................ .................. 15 2 STRUCTURAL CONFIGURATION AND LOADING CONDITIONS DURING BRIDGE CONSTRUCTION ................................ ................................ ................................ .. 18 2.1 Overview ................................ ................................ ................................ ........................... 18 2.2 Geometric Parameters ................................ ................................ ................................ ....... 18 2.3 Bearing Pads ................................ ................................ ................................ ..................... 19 2.4 Bracing ................................ ................................ ................................ .............................. 19 2.5 Bridge Construction Loads ................................ ................................ ............................... 20 3 DEVELOPMENT OF STRUCTURAL ANALYSIS MODELS ................................ ........... 27 3.1 Overview ................................ ................................ ................................ ........................... 27 3.2 Modeling of Bridge Girders ................................ ................................ .............................. 27 3.3 Modeling of Braces ................................ ................................ ................................ ........... 29 3.4 Modeling of Overhang Brackets ................................ ................................ ....................... 30 3.5 Application of Construction Loads ................................ ................................ ................... 32 4 DEVELOPMENT OF BRACE FORCE PREDICTIONS ................................ ..................... 43 4.1 Overview ................................ ................................ ................................ ........................... 43 4.2 Limited scope Sensitivity Studies ................................ ................................ .................... 44 4.2.1 Effect of Geometric Imperfections on Brace Forces ................................ .............. 45 4.2.2 Effect of K brace Configuration on Brace Forces ................................ .................. 46 4.2.3 Effect of Partial Ap plication of Construction Loads on Brace Forces ................... 47 ................................ .................... 48 4.4 Calculation of B race Forces by Database Interpolation ................................ ................... 50 4.5 Verification of Database Approach ................................ ................................ .................. 52 PAGE 5 5 5 WIND TUNNEL TESTING ................................ ................................ ................................ ... 62 5.1 Overview ................................ ................................ ................................ ........................... 62 5.2 Background on Drag Coefficients ................................ ................................ .................... 63 5.2.1 Dimensionless Aerodynamic Coe fficients ................................ ............................. 63 5.2.2 Terminology Related to Aerodynamic Coefficients ................................ ............... 66 5.3 Current Wind Design Practice in Florida ................................ ................................ ......... 67 5.4 Testing Configurations ................................ ................................ ................................ ..... 70 5.5 Testing Procedure ................................ ................................ ................................ ............. 72 6 WIND TUNNEL TESTING RES ULTS AND ANALYSIS ................................ .................. 81 6.1 Overview ................................ ................................ ................................ ........................... 81 6.2 Key Findings from the Wind Tunnel Test Program ................................ ......................... 81 6.2.1 Influence of Stay In Place Forms and Overhangs on Drag Coefficients ............... 81 6.2.2 Lift Coefficients for Girders and Overhangs ................................ .......................... 83 6.2.3 Torque Coefficients ................................ ................................ ................................ 85 6.3 Analysis of Wind Tunnel Testing Results ................................ ................................ ........ 86 6.3.1 Calculation of Global Pr essure Coefficient for Systems with I shaped Girders .... 86 6.3.2 Calculation of Global Pressure Coefficient for Systems with Box Girders ........... 92 6.3.3 Recommended Procedure for Calculation of Wind Loads ................................ ..... 95 6.3.4 Alternate Procedure for Calculation of Wind Loads for I shaped Girders ............ 96 6.4 Assessment of Brace Forces due to Wind Loads ................................ .............................. 97 7 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS ................................ ....... 114 7.1 Overview ................................ ................................ ................................ ......................... 114 7.2 Brace Forces due to Construction Loads ................................ ................................ ........ 114 7.3 Wind Pressure Coefficients and Corresponding Lateral Loads ................................ ...... 115 APPENDIX A CROSS SECTIONAL PROPERTIES OF FLORIDA I BEAMS ................................ ........ 117 B DIMENSIONED DRAWINGS OF WIND TUNNEL TEST CONFIGURATIONS .......... 121 C TABULATED RESULTS FROM WIND TUNNEL TESTS ................................ .............. 129 LIST OF REFERENCES ................................ ................................ ................................ ............. 138 BIOGRAPHICAL SK ETCH ................................ ................................ ................................ ....... 140 PAGE 6 6 LIST OF TABLES Table page 3 1 Summary of construction loads applied in parametric studies ................................ ............... 36 4 1 Span length values used in the database production parametric study ................................ .. 54 4 2 Other parameter values used in the database production parametric study ........................... 54 4 3 Range of allowable span lengths for FIBs ................................ ................................ .............. 54 4 4 Brace depths used in construction load parametric study ................................ ....................... 55 5 1 Parameter values used in the database production parametric study ................................ ..... 75 5 2 Pressure coefficients in FDOT Structures Design Guidelines ................................ ............... 75 5 3 Summary of wind tunnel tests ................................ ................................ ................................ 75 5 4 Wind tunnel test scaling ................................ ................................ ................................ ......... 75 6 1 Estimated lift co efficients (C L,OHF ) attributable to overhang formwork ............................... 100 6 2 Pressure coefficient during construction for I shaped girders ................................ .............. 100 6 3 Pressure coefficient during construction for Box girders ................................ ..................... 100 6 4 Recommended pressure coefficient for bridges during construction ................................ ... 100 6 5 Alternate pressure coefficient for bridges during construction ................................ ............ 1 00 A 1 Definitions of cross sectional properties required for use of a warping beam element ...... 119 A 2 Cross sectional properties of Florida I Beams ................................ ................................ .... 119 C 1 Summary of wind tunnel tests ................................ ................................ ............................... 129 PAGE 7 7 LIST OF FIGURES Figure page 1 1 Prestressed concrete girders braced together for st ability ................................ ...................... 17 1 2 Bridge construction loads ................................ ................................ ................................ ....... 17 2 1 Girder system ................................ ................................ ................................ .......................... 22 2 2 Defi nition of grade ................................ ................................ ................................ .................. 22 2 3 Definiti on of cross slop e ................................ ................................ ................................ ........ 22 2 4 D efinition skew angle ................................ ................................ ................................ ............. 23 2 5 Defin ition of camber ................................ ................................ ................................ ............... 23 2 6 Definition of sweep ................................ ................................ ................................ ................ 23 2 7 Girder system with quarter point bracing ................................ ................................ ............... 23 2 8 Perpendicular brace placem ent on skewed bridge ................................ ................................ .. 24 2 9 Common brac e types ................................ ................................ ................................ .............. 24 2 10 Stay in place formwork ................................ ................................ ................................ ........ 25 2 11 Temp orary support brackets used to support deck overhangs during construction .............. 25 2 12 Cantilever overhang supporte d by overhang brackets ................................ ........................... 25 2 13 Details of overhang formwork support brackets and loads ................................ ................... 26 2 14 Typical bridge deck finishing machine in operation ................................ ............................. 26 3 1 Finite element model of a single FIB ................................ ................................ ..................... 36 3 2 Modeling of bearing pad stiffness springs at end of girder ................................ .................... 37 3 3 Modeling of br ace configurations in FIB system models ................................ ....................... 37 3 4 Overhang bracket components and geometry ................................ ................................ ........ 38 3 5 Details of overhang bracket model ................................ ................................ ......................... 38 3 6 Cross sectional view of overall braced girder system model ................................ ................. 39 3 7 Isometric view of overall braced girder system model ................................ ........................... 39 PAGE 8 8 3 8 Summary of construction loads considered ................................ ................................ ............ 40 3 9 Eccentric reaction forces from loads applied to SIP forms, and statically equivalent nodal force and moment applied to top of girder ................................ ............................... 40 3 10 Construction loads converted to equivalent nodal loads ................................ ...................... 41 3 11 Coverage of deck loads as a function of finishing machine location (Bridges with only end span braces; no interior braces) ................................ ................................ .................. 41 3 12 Coverage of deck loads as a function of finishing machine l ocation (Bridges with end span and midspan bracing) ................................ ................................ ................................ 42 3 13 Coverage of deck loads as a function of finishing machine location (Bridges with third point bracing) ................................ ................................ ................................ ............ 42 3 14 Coverage of deck loads as a function of finishing machine location (Bridges with quarter point bracing) ................................ ................................ ................................ ........ 42 4 1 Effect of bridge grade on brace forces ................................ 55 4 2 Effect of cross ................................ ... 56 4 3 Effect of camber ................................ ......... 56 4 4 ................................ ........... 57 4 5 K bra ce configurations analyzed in limited scope sensitivity study ................................ ...... 57 4 6 Conservatism of selected K brace configuration (compared to inverted K brace) ................ 58 4 7 Conservatism of selected K brace configuration (compared to offset ends) .......................... 58 4 8 Definition of brace depth ................................ ................................ ................................ ........ 59 4 9 Accuracy of interpolated database approach for maximum end span K brace force ............. 59 4 10 Accuracy of interpolated database approach for maximum intermediate span K brace force ................................ ................................ ................................ ................................ ... 60 4 11 Accuracy of interpolated database approach for maximum end span X brace force ........... 60 4 12 Accuracy of interpolated database approach for maximum intermediate span X brace force ................................ ................................ ................................ ................................ ... 61 5 1 Two dimensional bridge girder cross section with in plane line loads ................................ .. 76 5 2 Definition of C D C L C SD and C SL ................................ ................................ ......................... 76 PAGE 9 9 5 3 Center of pressure of a bridge girder ................................ ................................ ...................... 76 5 4 Definition of C T and C PT ................................ ................................ ................................ ......... 77 5 5 Velocity pressure exposure coefficient used by FDOT ................................ .......................... 77 5 6 Girder cross sections used in study ................................ ................................ ........................ 77 5 7 Parameters definitions for each testing configu ration ................................ ............................ 78 5 8 Wind angle sign convention ................................ ................................ ................................ ... 78 5 9 Equi valence between wind angle and cross slope for box girders ................................ ......... 79 5 10 Overhang dimensions used in wind tunnel study ................................ ................................ 79 5 11 Formwor k and o verhang attachment methodology ................................ .............................. 80 6 1 Influence on plate girder C D values from addition of SIP forms and overhang s .................. 101 6 2 Influence on FIB78 C D values from addition of SIP forms and overhangs .......................... 101 6 3 Influence on plate girder C D values from addition of SIP forms ................................ .......... 102 6 4 Influence on FIB78 C D values from addition of SIP forms ................................ .................. 102 6 5 Influence on plate girder C D values from addition of overhangs ................................ .......... 103 6 6 Influence on FIB78 C D values from addition of overhangs ................................ .................. 103 6 7 Comparison of WF plate girder torque coefficients ................................ ............................. 104 6 8 Comparison of FIB78 girder torque coefficients ................................ ................................ .. 104 6 9 Projected area method ................................ ................................ ................................ .......... 105 6 10 Modified projected area method ................................ ................................ ......................... 105 6 11 Drag coefficients for wide flange plate girder systems ................................ ...................... 106 6 12 Drag coefficients for FIB78 systems ................................ ................................ .................. 106 6 13 Drag coefficients for FIB45 systems ................................ ................................ .................. 107 6 14 Conservatism of modified projected area calculation procedure for I shaped girde rs ....... 107 6 15 Upper bound formulation of reduction factor ( ) for FIB systems ................................ .... 108 6 16 Conservatism of modified projected area cal culation pr ocedure for I shaped girders ................................ ................................ ... 108 PAGE 10 10 6 17 Conservatism of alternative projected area calculation procedure for I shaped girders .... 109 6 18 Conservatism of alternative projected area calculation procedure for I shaped girders (C P,SIPF = 1.8) ................................ ................................ ................................ ................... 109 6 19 Equivalence of box girder cross slope and wind angle ................................ ...................... 110 6 20 Measured global pressure coefficients for box girders ................................ ....................... 110 6 21 Determination of projected depth fo r box girder bridges. ................................ .................. 111 6 22 Conservatism of projected area calculation procedure for box girders (Using current FDOT C P,SIPF = 1.1 and C P,SIPF+OHF = 1.1) ................................ ................................ ........ 111 6 23 Conservatism of projected area calculation procedure for box girders (Using proposed C P,SIPF = 1.2 and C P,SIPF+OHF = 1.5) ................................ ................................ .... 112 6 24 Upper bound formulation of a reduction factor ( ) for box girders ................................ ... 112 6 25 Conservatism of projected area calculation procedure for box girders (reduction factor ( ) included) ................................ ................................ ................................ ..................... 113 A 1 Coordinate system used in the calculation of cross sectional properties ............................ 120 PAGE 11 11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulf illment of the Requirements for the Degree of Master of Engineering DETERMINATION OF BRACE FORCES CAUSED BY CONSTRUCTION LOADS AND WIND LOADS DURING BRIDGE CONSTRUCTION By Samuel Taylor Edwards May 2014 Chair: Gary Consolazio Major: Civil Engineering T he first objective of this study was to develop procedures for determining bracing forces during bridge construction. Numerical finite element models and analysis techniques were developed for evaluating brace forces induced by construction loads acting on precast concrete girders (Florida I Beams) in systems of multiple girders braced together. A large scale parametric study was performed with both un factored (service) and factored (strength) construction loads (in total, more than 450,000 separate three dimensional (3 D) structural analyses were conducted). The parametric study included consideration of different Florida I Beam cross sections, span lengths, girder spacing s deck overhang widths, skew angles, number of girders, number of braces, and bracin g configurations (K brace and X brace). Additionally, partial coverage of wet (non structural) concrete load and variable placement of deck finishing machine loads were considered. A MathCad calculation program was developed for quantifying brace forces us ing a database approach that employs multiple dimensional linear interpolation. The accuracy of the database program was assessed by using it to predict end span and intermediate span brace forces for parameter selections not directly contained within the database, and then comparing the interpolated predictions to results obtained from finite element PAGE 12 12 analyses of corresponding verification models. In a majority of cases, the database predicted brace forces were found to be less than ten percent (10%) in err or. The second objective of this study was to experimentally determine wind load coefficients (drag, torque, and lift) for common bridge girder shapes with stay in place (SIP) formwork and overhang formwork in place, and then to develop recommended global (system) pressure coefficients (e.g., for strength design of substructures). Wind tunnel tests were performed on reduced scale models of Florida I Beam (FIB), plate girder, and box girder cross sectional shapes to measure aerodynamic forces acting on indiv idual girders in the bridge cross section. Tests were conducted at multiple wind angles, and corresponding tests with and without overhang formwork were conducted. Data from the wind tunnel tests were used to develop conservative procedures for calculating global pressure coefficients suitable for use in bridge design. PAGE 13 13 CHAPTER 1 INTRODUCTION 1 1 Overview During the process of constructing a highway b ridge, both the structure and the applied loading conditions will transition through several distinct stages, each of which warrants consideration from a structural safety perspective. Initially, individual girders are lifted by crane and placed into posit ion atop flexible elastomeric bearing pads located on the bridge supports (e.g., abutments or piers). Placement of all girders into their final positions constitutes one of the earliest distinct structural stages that must be assessed for safety. In this s tructural configuration, it is typical for braces to be installed between the individual girders (Figure 1 1 ). Additionally, one or more girders may also be anchored to the bridge supports. Structurally, the system at this stage consists of individual girders, bearing pads, braces, potentially anchors, and support structures (i.e., substructures). Loading conditions consist primarily of vertical gravity loads and horizontal wind loads (for individual girders as well for the collection of all girders as a whole). Primary structural design and safety concerns at this stage of construction focus on girder stability, adequate strength (and stiffness) of braces, and adequate global capacity of the substructure. Addressing these concerns requires that engineering calculation methods be available for quantifying girder buckling capacity, quantifying wind loads on the individual girders (due to predominantly horizontal wind), quantifying brace forces due to the same wind load s, and quantifying global wind induced lateral loads on the bridge substructure (e.g., for strength design). All of these areas of concern were addressed in a previous study funded by the FDOT (BDK75 977 33, Consolazio et al., 2013) that involved the devel opment of analytical methods for predicting structural forces and capacities, and PAGE 14 14 experimental wind tunnel testing to quantify aerodynamic girder wind load coefficients (drag, torque, and lift), including the effects of aerodynamic shielding. After contin ued construction progress, another key stage will be reached wherein stay in place (SIP) forms have been installed between the girders and overhang formwork (and associated overhang support brackets) have been eccentrically attached to the exterior (fascia ) girders of the bridge. Loading conditions at this stage consist primarily of horizontal wind loads (plastic) concrete deck (Figure 1 2 eccentrically to overhang formwork and to stay in place forms. Consequently, torsional momen ts acting on the girders produce axial forces in the bracing system (diagonal and horizontal elements) that must be considered in the brace design process. Several geometric parameters influence the magnitude of brace forces caused by construction loads. A major component of the research presented in this report was carried out to quantify bracing forces caused by construction loads over a wide range of geometric parameters (i.e., span lengths, girder spacings, deck overhang widths, etc.). Also of concern a t this stage of construction are wind induced lateral loads acting globally on the entire bridge cross section. Such loads generate lateral forces on the bridge substructure that must be considered in the design process. While the issue of global lateral w ind load was addressed in the previous BDK75 977 aerodynamic blockage (between girders) that results from installation of SIP forms and overhangs. Therefore, in the study presented in this report, wind tunnel testing was once again PAGE 15 15 used quantify wind load coefficients (drag, torque, and lift) for multiple girder systems, but this time with SIP forms (and possibly overhangs ) present. The goal of measuring this data was to develop conservative methods for calculating global pressure coefficients and global lateral wind loads for substructure design 1 2 Objecti ves One objective of this research was to use finite element analysis of partially constructed bridge systems consisting of multiple braced prestressed concrete girders (Florida I beams) to develop calculation procedures for quantifying brace forces cause d by eccentric construction loads. A related objective was to experimentally determine wind load coefficients (drag, torque, and lift) for common bridge girder shapes with stay in place (SIP) forms and overhang formwork in place, and then to develop recomm ended global (system) pressure coefficients (e.g., for use in the strength design of substructures). 1 3 Scope of Work Construction loads : Numerical finite element bridge models and analysis techniques were developed for evaluating brace forces induced by construction loads acting on precast concrete girders (Florida I Beams) in systems of multiple girders braced together. The construction loads considered were: wet concrete deck load, stay i n place (SIP) form weight, overhang formwork weight, live load, worker line loads, and concentrated loads representing a deck finishing machine. A large scale parametric study, involving more than 450,000 separate three dimensional structural analyses, was performed to compute maximum brace forces for un factored (service) and factored (strength) construction load conditions. The parametric study included consideration of different Florida I Beam cross sections, span lengths, girder spacings, deck overhang widths, skew angles, number of girders, number of braces, and bracing configurations (K brace and X brace). M aximum end span brace forces and intermediate span brace forces quantified from the brace force prediction was developed and coded into a MathCad program for ease of use. W ind loads : Wind tunnel testing was used to quantify wind load coefficients (drag, torque, and lift) for systems of multiple bridge girders (FIB plate girder, and box) with stay in place (SIP) forms and overhang formwork in place. Tests were conducted at multiple wind angles, and corresponding tests with and without overhang formwork were conducted so that the effects of overhang formwork on drag lift, and torque coefficients could be quantified. Drag coefficients measured at each girder position in bridges with I PAGE 16 16 shaped girders, and in bridges with box girders, were used to develop conservative methods for computing global (system) pressure coef ficients suitable for use in bridge design (particularly, for use in calculating global lateral substructure load due to wind). PAGE 17 17 Figure 1 1 Prestressed concrete girders braced together for stability (photo courtesy of FDOT) Figure 1 2 Bridge construction loads (photo courtesy of Gomaco) PAGE 18 18 CHAPTER 2 STRUCTURAL CONFIGURATION AND LOADING CONDITIONS DURING BRIDGE CONSTRUCTION 2 1 Overview Braces used to stabilize girders during bridge construction must be designed to resist forces that are generated by construction loads and win d loads. The manner in which brace forces are distributed to the elements of the bracing system depends on geometric parameters (span length, girder spacing, deck overhang width, etc.), bracing configurations, and cross sectional properties of the girders. In the present study, the girders under investigation are Florida I Beams (FIBs), a group of standard cross sectional shapes of varying depths that are commonly employed in Florida bridge designs. These beams are typically cast offsite, transported to the construction site by truck, then lifted into position one at a time by crane, where they are placed on elastomeric bearing pads and braced together for stability. Formwork systems are then added to support the wet concrete deck and other construction load s encountered during the deck pouring process. In this chapter, a physical description of the relevant terminology. 2 2 Geometric P arameters The term girder system will be used to refer to a group of two or more FIBs braced together in an evenly spaced row ( Figure 2 1 ) In addition to span length and lateral spacing, there are several geometric parameters that define the shape and placement of the girders within a system: Grade : Longitudinal incline of the girders, typically expressed as a percentage of rise per unit of horizontal length (Figure 2 2 ). Cross slope : The transverse incline (slope) of the deck, expressed as a percentage, which results in girders that are staggered vertically ( Figure 2 3 ). PAGE 19 19 Skew angle : Longitudinal staggering of girders, due to pier caps that are not perpendicular to the girder axes (Figure 2 4 ). Camber : Vertical bowing of the girder (Figure 2 5 ) due to prestressing in the bottom flange ; expressed as the maxi mum vertical deviation from a perfectly straight line connecting one end of the girder to the other. Sweep : Lateral bowing of the girder (Figure 2 6 ) due to manufacturing imperfections, expressed as the maximum horizontal devia tion from a perfectly straight line connecting one end of the girder to the other. 2 3 Bearing P ads FIB bridge girders rest directly on steel reinforced neoprene bearing pads which are the only points of contact between the girder and the substructure. There is generally sufficient friction between the pad and other structural components so that any movement of a girder relative to the substructure (with the exception of vertical uplift) mus t displace the top surface of the pad relative to the bottom surface. As a result, the girder support conditions in all six degrees of freedom (three translations, and three rotations) can be represented as finite stiffnesses that correspond to the equival ent deformation modes of the pad. These deformation modes fall into four categories: shear, compression (axial), rotation (e.g., roll), and torsion. Bearing pad stiffnesses in this study were quantified using calculation procedures developed and experiment ally validated in a previous study (BDK75 977 33, Consolazio et al., 2013) for typical Florida bridge bearing pads. 2 4 Bracing As adjacent girders are erected during the bridge constructio n process girder to girder braces (henceforth referred to simply as braces ) are used to connect the girders together into a single structural unit. At a minimum, braces are installed near the ends of the girders (close to the supporting piers); such brace s are referred to as end span braces In addition, inter mediate span braces spaced at unit fractions (1/2, 1/3, 1/4) of the girder length may also be included For PAGE 20 20 example, quarter point (1/4 span) bracing divides the girder into four equal unbraced length s ( Figure 2 7 ) When skew is present, brace point locations are longitudinally offset ( Figure 2 8 ) between adjacent girders because FDOT Design Standard No. 20005: Prestr essed I Beam Temporary Bracing (FDOT, 2014a) requires that all braces be placed perpendicular to the girders. Braces are typically constructed from timber or steel members, but individual brace designs are left to the discretion of the contractor, so a var iety of different bracing configurations are possible. The most common types of braces used in practice in Florida are X braces ( Figure 2 9 a ) and K braces ( Figure 2 9 b ). Therefore, in the present study, only these brace types are considered. Braces are typically attached to the girders using bolted connections or welded to cast in steel plates. 2 5 Bridge C onstruction L o ads A major objective this study was to determine axial brace forces induced by bridge construction loads. In particular, the bridge deck placement (concrete application and finishing) process was the construction stage considered. Components of the bridg e construction loads considered in this study were as follows: Wind loads : During bridge construction, wind loads are an important design consideration. However, severe wind loads and active construction loads (i.e., deck placement loads) are not likely to be encountered simultaneously. According to the FDOT Structures Design Guidelines (SDG; FDOT, 2013), the basic wind speed for active construction is specified as 20 mph. In this study, wind loads and construction loads were treated as separate design load cases. Wind loads on girders with stay in place forms were quantified experimentally with wind tunnel testing. Concrete deck : Throughout the deck pouring and finishing process, wet (plastic) concrete has negligible stiffness which is beneficial for shapi ng the concrete into a smooth finished surface. Consequently, a non composite girder system must support these construction loads. However, in the final bridge condition, the bridge deck works together with the girders as a composite system to resist and d istribute loads to the supporting girders. Since the wet (nonstructural) concrete load is incrementally applied to bridges in the longitudinal direction, this load is treated as a variable length load in the PAGE 21 21 finite element analyses. Partial application of concrete deck loads to the girder system will be further explained later in this report. Stay in place formwork : Stay in place (SIP) formwork systems support intra girder loads (wet concrete) that span transversely between girder top flanges ( Figure 2 10 ). Stay in place forms consist of corrugated metal panels that are attached to the tips of the top flange of adjacent girders. The connection between the SIP forms and the girder flange is considered to be incapable of transm itting moments, therefore the SIP forms are Overhang formwork : It is typical for the deck of a bridge to extend past the exterior (fascia) girder, thereby producing a cantilevered o verhang ( Figure 2 11 ). During construction, overhang brackets ( Figure 2 12 ) are used to temporarily support the cantilever portion of the wet deck slab that extends beyond exterior girders. These temporary structural bracket systems support the overhang formwork, wet concrete, construction walkway, workers and concrete finishing machine. A survey of representative literature from overhang bracket manufacturers was co nducted to quantify representative cross sectional properties and longitudinal spacing requirements. Most commercially available formwork systems consist of timber joists and sheathing supported on steel bridge overhang brackets ( Figure 2 13 ). It is important to note that all of the gravity loads supported by the overhang brackets are eccentric relative to the exterior girders, and as such apply torque loads to the exterior girders in the overall cross secti onal system. Finishing machine : Bridge deck finishing machines ( Figure 2 14 ) spread, compact, and finish the freshly placed wet concrete deck surface. The finishing machine is an open steel frame that is supported at the extremities of the bridge width on the overhang brackets described above. Drive wheels (commonly referred to as bogies) move the paver longitudinally along the length of the bridge and are eccentrically supported by screed rails (Figure 2 13 ) on each side of the bridge. A suspended paving carriage with augers, drums, and floats finishes the concrete surface as it moves transversely from side to side across the width of the bridge (perpendicular to th e longitudinal movement of the finishing machine along the length of the bridge). Concrete is typically placed just ahead of the travelling finishing machine using separate equipment, such as a pump. The most common finishing machine manufacturers are Tere x Bid Well and Gomaco. Live loads : Live loads that are present during the deck finishing process consist of workers, temporary materials, and supplementary construction equipment. For brace force calculation purposes, these loads are treated as either unif orm pressure loads, or as line loads, as will be discussed in grea ter detail later in this report. PAGE 22 22 Figure 2 1 Girder system Figure 2 2 Definition of grade (elevation view) Figure 2 3 Definition of cross slope (section view) PAGE 23 23 Figure 2 4 Definit ion skew angle (plan view) Figure 2 5 Definition of camber ( elevation view) Figure 2 6 Definition of sweep (plan view) Fi gure 2 7 Girder system with quarter point bracing PAGE 24 24 Figure 2 8 Perpendicular brace placement on skewed bridge (plan view) A B Figure 2 9 Common brace types. A) X brace. B) K brace PAGE 25 25 Figure 2 10 Stay in place formwork (section view) Figure 2 11 Temporary support brackets used to support deck overhangs during construction Figure 2 12 Cantilever overhang supported by overhang b rackets (Photo credit: Clifton and Bayrak (2008)) PAGE 26 26 Figure 2 13 Details of o verhang formwork support brackets and loads Figure 2 14 Typical bridge deck finishing machine in operation (Photo credit: Gomaco) PAGE 27 27 CHAPTER 3 DEVELOPMENT OF STRUCTURAL ANALYSIS MODELS 3 1 Overview To calculate brace forces induced by construction loads, braced systems of FIB girders were modeled and structurally analyzed using the ADINA (2012) finite element analysis code. The models were capable of capturing overall sys tem level behavior of braced FIB systems (including the influence of brace configuration, bearing pad stiffness, etc.), while remaining computationally efficient enough that hundreds of thousands of parametric analyses could be performed for the purpose of quantifying braces forces. In all cases, brace forces were determined using large displacement (geometrically nonlinear) analyses, in which static loads were applied to the models in incremental steps, taking into account the deformed state of the structu re at each step. The parametric construction stage bridge models analyzed in this study were developed in a semi automated fashion by extending a modeling methodology developed in a previous study (BDK75 977 33, Consolazio et al., 2013) to further include the effects of construction loads and overhang brackets. In the following sections, key aspects of the model development process are summarized, and noteworthy modifications to the previously developed modeling methodology are described in detail. 3 2 Modeling of Bridge G irders In the global coordinate system established for each bridge model, the X axis corresponded to the transverse (lateral) direction; the Y axis corresponded to the lon gitudinal (span) direction, and the Z axis corresponded to the vertical direction. Bridge girders were modeled (Figure 3 1 formulation type provide d in ADINA that possess an additional degree of freedom at each end PAGE 28 28 node to represent the torsionally induced out of plane warping of the cross section. Warping beam elements are generally superior to standard Hermitian beam elements in that the bending an d torsional deformation modes of the warping element are fully coupled together at the element formulation level. However, as a consequence, the use of warping beam elements requires the calculation of a comprehensive set of cross sectional properties many more than standard Hermitian beam elements Details relating to the section properties that were cal culated in this study for the catalog of FDOT FIB cross sectional shapes are provided in Appendix A Throughout this study, material properties assumed for the prestressed concrete FIB girders were f c = 8.5 ksi, unit we ight = 150 pcf, and Poisson's ratio = 0.2. Using these values and the PCI Design Handbook (PCI, 2010), the concrete elastic modulus was computed to be E = 5589 ksi S upport conditions at each end of each girder were modeled with six (6) springs to represen t the stiffness es of the bearing pad in each degree of freedom E ach of the six (6) spring s correspond ed to one of four different pad deformation modes: shear, axial, torsion, and roll (Figure 3 2 ). Pad stiffnesses were determined using the calculatio n methods developed and validated in a previous study (BDK75 977 33 Consolazio et al., 2013) The roll stiffness springs (in both the overturning and bending directions) were assigned nonlinear moment rotation curves th at captured the softening effects of partial girder liftoff from the pad. All remaining pad stif fnesses were treated as linear. Seven (7) standard types of elastomeric bearing pad are described in FDOT Design Standard No. 20510 : Composite Elastomeric Beari ng Pads Prestressed Florida I Beams (FDOT 201 4 c ) for use with FIB girders During the design process selection of the type of pad PAGE 29 29 that will be used in a particular bridge project is based on thermal expansion and live load deflection limit states of th e completed bridge, neither of which can be predicted based solely on girder dimensions (cross sectional properties or span length). As such, it is not appropriate to assume that for each unique type of FIB there is a corresponding single type of bearing pad that would always be utilized. In this study, bearing pad selection was instead based on ensuring that conservatively large values of brace force would be obtained for all analyses conducted. As bearing pad rotational stiffness decreases, the portion of the acting eccentric construction loads that is carried by a pad also decreases, thereby moderately increasing the forces that are developed in the braces Consequently, the bearing pad with the minimum practical roll stiffness will produce the most con servative brace forces. Therefore, the FDOT Type J bearing pad was selected for use throughout this study As documented in Consolazio et al. (2013), as the acting axial compressive load on a pad decreases, so does a component of the roll stiffness of the pad. Hence, for each FIB girder type considered in the present study, the minimum practical span length for that girder type was used to compute an axial pad load (equal to half of the total weight of a single girder). For each such axial load, the roll st iffness of the Type J pad was then computed and then subsequently used for all parametric analyses involving that type of FIB girder. Hence a single minimized roll stiffness curve was calculated for each type of FIB, resulting in a total of seven (7) bear ing pad moment rotation curves. (For additional details on the bearing pad stiffness calculation procedures, see Consolazio et al., 2013, Chapter 6). 3 3 Modeling of B races In bridge const ruction, a wide variety of different bracing configurations are used in practice c onsequently it was not possible for every possible configuration of brace to be included in the parametric studies that were conducted in this study After carrying out a su rvey PAGE 30 30 of bracing designs used in the construction of bri dges throughout Florida, two (2 ) representative brace configurations were identified: X brace : Two diagonal bolt typically passes through both members at the crossing point to create a hinge (Figure 3 3 a) K brace : Steel members (typically steel ang shaped frame and welded or bolted to steel plates that are cast into the web s of the concrete girders (Figure 3 3 b). Additionally, only the X brace and K brace configurations are currently recommended in the FDOT Design Standard No. 20005: Prestressed I Beam Temporary Bracing (FDOT, 2014 a ) for end span and intermediate span bracing applications. For structural analysis purposes, all braces were modeled with beam elements, with each brace member represented by a single element. At the girder connection points, rigid links were used to connect th e brace elements to the girder elements (i.e., the warping beam elements positioned at the centroid of the girders ). Pins and hinges were modeled with beam moment end releases and nodal constraints, respectively. Both X brace and K brace members included i n the parametric studies were modeled as 4 in. x 4 in. x in. steel angles, with an e lastic modulus of E = 29000 ksi. 3 4 Modeling of O verhang B rackets Construction loads applied beyond th e lateral extents of an exterior girder are structurally supported during construction by overhang brackets Specifically, the finishing machine, formwork, overhang wet concrete, and construction worker live loads are typical components of the supported ov erhang loads. To define the lateral eccentricity of the overhang construction loads, two offset parameters had to be established. To be consistent with the FDOT Instructions for Design Standard No. 20010: Prestressed Florida I Beams (FDOT, 2014b ), the conc rete finishing machine was offset 2.5 in. from the overhang edge (Figure 3 4 ). In the FDOT Concrete I girder Beam Stability Program in addition to providing calculations for determining bracing PAGE 31 31 adequacy and girder stability, several recommended values for the overhang geometry are specified, including a 2 ft worker platform width. Therefore, for all the parametric studies conducted herein, the worker platform was assumed to extend 2 ft beyond the finishin g machine supports (Figure 3 4 ). In the girder system models, all components of the overhang brackets were modeled with beam elements, with representative cross sectional properties that were obtained fr om a survey of overhang bracket manufacturers. To represent the offset eccentricities between the girder centroid and the bracket connection points, the deformable overhang bracket elements were connected to girder warping beam elements using rigid links ( Figure 3 5 ). In order to model interaction between the overhang bracket and the girder bottom flange, two co located but separate nodes were used: one at the bottom vertex of the metal overhang bracket, and a s econd at the end of the rigid link representing the surface of the girder bottom flange. At this location, the overhang bracket bears against (i.e., is in compressive contact with) the girder bottom flange. To model this behavior structurally, a constraint condition is defined such that the lateral (X direction) translations of the two co located nodes are constrained to match, while permitting independent movements (relative slip) in the vertical direction. Overhang bracket nodes are positioned (Figure 3 5 ) to define: the three corners of the triangular system; and all locations of load discontinuities (i.e., deck overhang edge) and load application points (i.e., finishing machine and worker line load applicatio n points). The worker line load is conservatively applied to the center of the worker platform width. Thus, the load application of the worker line load is laterally offset in the X direction 12 in. from the assumed finishing machine application point and 14.5 in. from the deck overhang edge (Figure 3 5 ). By combining each of the previously mentioned modeling components, an overall illustration of a PAGE 32 32 typical FIB system model is presented in Figure 3 6 Based on a review of literature obtained from typical overhang bracket manufacturers, brackets are commonly spaced between 4 ft and 6 ft on center longitudinally along the span length of a bridge. In the present study, therefore an average longitudinal spacing of 5 ft was used for all brackets (Figure 3 7 ). Also noted in Figure 3 6 and Figure 3 7 are rigid vertical elements (links) extending from girder centroid to girder top surface which are included in the model for application of construction loads on each girder. These rigid elements account for the vertical eccentricity between the girder centroid and the gird er top surface (where loads are applied). It was determined that brace forces induced by construction loads were not sensitive to changes in the longitudinal spacing of the rigid vertical elements; consequently the rigid links were given an arbitrary long itudinal spacing of 1 ft in the span direction. 3 5 Application of Construction L oads In this section, the magnitudes and methods of application for construction loads considered in this st udy are described. In Figure 3 8 a summary of the superimposed construction loads that will be described in more detail below is provided. Additionally, self weight (i.e., gravity loads from girders, braces, and overha ng brackets) were included in the models. An important consideration in this study was the application of the concrete finishing machine loads. Finishing machines are supported near the extremities of the bridge width by several wheels. The common Terex B id Well 4800 machine has a total wheel base of approximately 8 feet in the longitudinal (bridge span) direction. Since this wheel base is small relative to the typical span lengths of prestressed girder bridges, the finishing machine wheel reaction forces were idealized as single concentrated loads (one load on each side of the bridge equal to half the total machine weight). PAGE 33 33 During bridge deck placement, a concentration of live load will be located at the extremities of the bridge deck. To account for this loading, the AASHTO Guide Design Specifications for Bridge Temporary Works (2008) recommends that a worker line load be applied along the outside edge of all deck overhangs. In addition, the line load shall be applied as a moving load but with a fixed leng th of 20 ft so as to not introduce excessive conservatism. In the girder system models analyzed in this study, the worker line load was centered longitudinally at the concrete finishing machine such that the line load extended 10 ft behind and ahead of the finishing machine. Since temporary bracing must be designed for service and strength limit states, two separate parametric studies were conducted: un factored (service) loads and factored (strength) loads. A summary of un factored and factored constructi on loads is provided in Table 3 1 along with applicable references. Construction loads that are applied between adjacent girders (i.e., on the stay in place forms) produce vertical reaction forces that act on the tip s of the girder top flanges. Since all Florida I beams have a top flange width of 48 in., the lateral eccentricity between the girder centroid and the formwork reaction force (Figure 3 9 ) is 24 inches (half of the gir der top flange width). For analysis purposes, each eccentric reaction forces of this type was converted into a statically equivalent combination of force and moment (Figure 3 9 ) which were then applied along the cente rlines of the girders. Consequently all intra girder distributed loads that were applied over the width of the stay in place formwork were converted into equivalent nodal forces and moments. Other types of construction loads, such as the overhang loads ( overhang fo rmwork, worker line load, etc.), PAGE 34 34 were applied directly to nodes in the structural model based on the appropriate tributary areas (Figure 3 10 ). During the process of placing and finishing of a concrete bridge de ck, wet concrete pressure load is applied to the bridge (by way of SIP forms) over incrementally increasing lengths of the structure. Consequently, loading conditions corresponding to the placement of concrete deck loads over partial lengths of the bridge less than the total span length were considered in this study. For all such partial coverage cases, the position of the finishing machine was taken to coincide with the location of the furthest placed concrete. W et concrete is typically placed just ahead o f the moving finishing machine (Figure 2 14 ) using a pump therefore, in vast majority of paving situations, the location of the finishing machine and the end of the concrete coverage will generally coincide. (Alt hough it is feasible for a finishing machine to be moved to a location other than then end of the placed concrete, it was determined that such situations are generally rare and/or non controlling, and thus were not considered in this study). By analyzing s coverage area were moved in small increments along the bridge length, it was determined that maximum end span brace and intermediate span brace forces occurred when the web slab load terminated at one of the bracing points (e.g., end point, 1/2 point, 1/3 point, 1/4 point). However, depending on the bridge configuration parameters (deck overhang width, girder spacing, etc.) the controlling coverage of concrete loads and finishing machine loads that produced the maximum brace forces varied from case to case. For example, full span concrete coverage with the finishing machine at the end span often produced the highest end span brace forces. However, a partially placed d eck terminating at an interior brace point typically produced the largest interior brace forces. To ensure that maximum brace forces were quantified from the parametric studies, PAGE 35 35 multiple load cases were analyzed (Figures 3 11 3 12 3 13 and 3 14 ) for each bridge, depending on the number of interior bracing lines that were present. Note that for each geometric configurat ion (span length, girder spacing, deck overhang width, etc.), maximum end span brace and intermediate span brace forces were quantified for each loading condition. According to AASHTO Guide Design Specifications for Bridge Temporary Works (2008), a worker line load (75 lb/ft [un factored] over a length of 20 ft) should be included as a design load during the deck placement. This line load accounts for additional workers that are standing on the overhang platform during deck placement. For load cases where the finishing machine was at either the start or the end of the bridge, the worker line load was ap plied over the first or last 20 ft of the structure. For all other cases, where the concrete deck terminated at an interior brace point, the 20 ft line load extended longitudinally 10 ft to either side of the brace point (and finishing machine location). PAGE 36 36 Table 3 1 Summary of construction loads applied in parametric studies Construction l oads Load type Un factored loads Load f actor Factored loads Reference Wet concrete deck Permanent 106.25 psf 150 pcf) 1.25 132.8 psf 150 pcf) I Wet concrete build up Permanent 50 lb/ft 1.25 62.5 lb/ft I Stay in place forms Permanent 20 psf 1.25 25 psf I Overhang formwork Temporary 10 psf 1.5 15 psf II Live load Temporary 20 psf 1.5 30 psf III Worker line load Temporary 75 lb/ft for 20 ft 1.5 112.5 lb/ft for 20 ft III Finishing machine Temporary 10 kip total (5 kip ea ch side) 1.5 15 kip total (7.5 kip each side) II I: per FDOT Structures Design Guidelines ( FDOT, 2013) II: per FDOT recommendations III: per AASHTO Guide Design Specifications for Bridge Temporary Works (AASHTO, 2008) Figure 3 1 Finite element model of a single FIB (isometric view) PAGE 37 37 Figure 3 2 Modeling of bearing pad stiffness springs at end of girder A B Fi gure 3 3 Modeling of brace configurations in FIB system models A) X brace. B) K brace. PAGE 38 38 Figure 3 4 Overhang bracket component s and geometry Figure 3 5 Details of overhang bracket model PAGE 39 39 Figure 3 6 Cross sectional view of overall braced girder syste m model Figure 3 7 Isometric view of overall braced girder system model PAGE 40 40 Figure 3 8 Summary of construction loads considere d Figure 3 9 Eccentric reaction forces from loads applied to SIP forms, and statically equivalent nodal force and moment applied to top of girder PAGE 41 41 Figure 3 10 Construction loads converted to equivalent nodal loads Figure 3 11 Coverage of deck loads as a function of finishing machine location (Bridge s with only end span braces; no interior braces) PAGE 42 42 Figure 3 12 Coverage of deck loads as a function of finishing machine location (Bridge s with end span and midspan bracing) Figure 3 13 Coverage of deck loads as a function of finishing machine location (Bridges with third point bracing) Figure 3 14 Coverage of deck loads as a function of finishing machine location ( Bridges with quarter point bracing) PAGE 43 43 CHAPTER 4 DEVELOPMENT OF BRACE FORCE PREDICTIONS 4 1 Overview To investig ate brace forces caused by eccentric construction loads, a large scale parametric study was performed using finite element models of braced systems of Florida I Beams (FIBs). As discussed in the previous chapter, several load cases were considered dependin g on the number of bracing lines present in the bridge. From the parametric study, maximum end span brace forces and intermediate span brace forces (if present) were quantified for both factored and un factored construction loads. In bridge construction, d ifferent types of bracing may be provided at the end span versus at intermediate span (interior) bracing points due to relative differences in brace forces at these locations. However, it is also typical practice to use a consistent type of bracing through out the interior portion of the bridge (i.e., longitudinally away from the ends). Therefore, for each bridge analyzed in the parametric study, maximum brace forces were quantified once for all end span braces, and a second time for all intermediate span (i nterior) braces. For X brace (cross brace) models, the maximum diagonal brace forces were quantified at end span and intermediate span bracing locations. However, K braces typically have smaller diagonal element forces as compared to the top and bottom ho rizontal element forces. Therefore, for K brace models, maximum diagonal forces were quantified separately from the maximum horizontal element forces (and, as noted above, separately for end span and intermediate span bracing locations). Ultimately, the g oal in performing the parametric study was to develop a method by which engineers can rapidly compute brace forces for varying system parameters (girder type, span length, number of bracing locations, etc.). One approach to achieving this goal would be to PAGE 44 44 perform a moderate size parametric study, use the resulting brace force data to attempt to identify key relationships between forces and system parameters, and then form empirical brace force prediction equations. Such an approach must balance several ofte n competing issues: the need for the empirical prediction equations to be mathematically simple in form; conservative in their prediction of brace force; but not overly conservative (since this outcome could lead to uneconomical bracing designs). The alter native approach taken here was to conduct a large scale dimensional (3 D) structural analyses of FIB bridges were conducted); store the summarized results data into a sim ple database; and then subsequently access and interpolate that database whenever brace forces need to be computed (i.e., predicted for design). By including a sufficiently large number of incremental values of each system parameter (e.g., span length, ske w angle, number of braces, etc.), the error introduced in brace force prediction by database interpolation was kept to an acceptably small level. In verification tests demonstrated later in this chapter, it will be shown that the interpolated database appr oach yields results which are, in a majority of cases, less than ten percent (10%) in error. 4 2 Limited scope Sensitivity S tudies Fully characterizing a braced multi girder bridge system re quires a large number of geometric parameters Consequently, conducting a parametric stud y in which all possible combinations of these parameters were considered (even if only a few discrete values were selected per parameter) would require millions of ind ividual structural analyses to be performed. To avoid such a situation, se veral limited scope sensitivity studies were performed to help guide the design of an efficient final parametric stud y As a result of these preliminary investi gations several geometric parameters were identified as having negligible influence on bracing forces due to construction loads and were therefore excluded from the PAGE 45 45 database production parametric study Additionally, sensitivity studies were performed to address the influence of K brace configuration type (i.e., inverted K brace, horizontally offset diagonal connections, etc.) on brace forces so that a conservative, representative brace type could be selected for use in the database production parametric s tudy. Also, in order to achieve an efficient number of construction load cases (due to the longitudinal variability of the finishing machine and concrete deck loads), the critical loading conditions to be used in the database production parametric study we re also determined. 4 2 1 Effect of Geometric Imperfections on Brace Forces Geometric deviations (i.e., deviations from perfectly horizontal, straigh t girders) were considered in several limited scope sensitivity studies to quantify the effects of the geometric deviations on maximum brace forces. In total, approximately 500 analyses were conducted for a variety of FIB bridge configurations. Bridge geom etric parameters that did not have a significant effect on brace forces were not included in the database production parametric study. Similar trends of brace force sensitivity for both maximum end span and intermediate span bracing were observed for all t ested cases. Therefore, representative results for an example girder system point K braces, 9 ft girder spacing, 4 ft deck overhang, 0 deg. bridge skew, service construction loads (no load factors)] are provided below fo r each geometric imperfection parameter: Grade : L ongitudinal incline of the girders ( recall Figure 2 2 ), typically expressed as a percentage of rise per unit of horizontal length. A s evident in the example presented in Figure 4 1 the increase of grade by 2% had no effect on brace forces. Consequently, 0% grade was assumed for all cases in the database production parametric study. Cross slope : Transverse incline (slope) of the deck (recall Figure 2 3 ) typically expressed as a percentage, which results in girders that are staggered vertically. A s evident in the example presented in Figure 4 2 an increase of cross slope by 2% ha d a negligible effect on brace forces Other girder systems with higher and lower cross slopes showed similarly negligible influences on brace forces (for both end span and PAGE 46 46 intermediate span brace forces). Therefore, a cross slope of 0% was assumed for all cas es in the database production parametric stud y Camber : Vertical bowing of the girder ( recall Figure 2 5 ) due to prestressing in the bottom flange expressed as the maximum vertical deviation from a perfectly straight line con necting one end of the girder to the other. G irder camber was implemented as a vertical parabolic shape with a maximum vertical deviation at midspan. A s evident in the example presented in Figure 4 3 an increase of camber of 6 inches (at midspan) for all girders in the system ha d no effect on maximum end span brace and interior span brace forces. Note that camber sensitivit ies for other girder systems with higher and lower maximum cambers were similar. Therefore, zero inches o f camber was assumed in the database production parametric study. Sweep : Lateral bowing of the girder ( recall Figure 2 6 ) due to manufacturing imperfections, expressed as the maximum horizontal deviation from a perfectly straig ht line connecting one end of the girder to the other. Construction tolerances for FIBs are specified in the FDOT Standard Specifications for Road and Bridge Construction (FDOT in. for every 10 ft of girder length, bu t not to exceed 1.5 in. Therefore, both maximum end span and intermediate span brace forces were compared between girder systems without sweep and girder systems with the maximum possible sweep of 1.5 inches (applied to all girders). By extending the model ing methodology developed in BDK75 977 33 ( Consolazio et al. 2013), sweep was implemented using a half sin e function, with the maximum allowable sweep at midspan. In the example presented in Figure 4 4 increas ing sweep from z ero to the maximum permitted value increase d the intermediate span brace forces by approximately five percent ( 5% ) In all tested cases, the percent differences of end span and intermediate span brace forces caused by maximum girder sweep were less than 5% D ue to the relatively small influence of sweep on brace forces, girder sweep imperfections were omitted from the database production parametric study (i.e. lateral girder sweep was assumed to be zero at midspan). 4 2 2 Effect of K brace Configuration on Brace F orces Sensitivity studies were performed to assess the influence of K brace configuration (inverted K brace, horizontally offset diagonal connecti ons, etc.) on brace forces so that a conservative, representative brace type could be selected for the database production parametric study. In total, approximately 300 sensitivity analyses were conducted for a variety of different bridge configurations. F rom a survey of typical bracing types used in modern Florida bridges, four typical K brace types were selected (Figure 4 5 ). Each K brace configuration was given a letter designation ( Type A Type B etc .) for comparis on purposes. PAGE 47 47 This limited scope sensitivity study assessed the effects of K bracing configuration on brace forces for a broad range of geometric parameters (e.g., span length, girder spacing, deck overhang width, etc.). In the FDOT Design Standard No. 2000 5: Prestressed I Beam Temporary Bracing (FDOT 2014 a ) the Type A K brace configuration (Figure 4 5 ) is recommended for both end span and intermediate span bracing applications. Therefore, it was used as a reference br ace by which other K brace forces in the sensitivity study could be compared by normalization. As evident in Figure 4 6 the Type A K brace had normalized maximum brace force results that were approxima tely equal to Type C (i.e., normalized values were approximately 1.0 indicating that the brace forces were very similar). Additionally, the Type A K brace had conservatively higher, but not overly conservative, maximum brace forces compared to the offset d iagonal configurations of Type B and Type D ( Figure 4 7 normalized values were slightly greater than 1.0 in most cases). Note that the average conservatism of Type A compared to both Type B and Type D was a pproximately five percent (5%). Therefore, the Type A configuration was selected as the representative K brace configuration to be used for all cases in the database production parametric study. 4 2 3 Effect of Partial Application of C onstruction L oads on Brace Forces As discussed previously, during the process of placing and finishing a concrete bridge deck, wet concrete pressure loads are applied to a bri dge (by way of SIP forms) over incrementally increasing lengths of the structure. Consequently, loading conditions corresponding to the placement of concrete deck loads over partial lengths of the bridge less than the total span length were considered in t he database production parametric study For all such partial deck coverage cases, the position of the finishing machine was taken to coincide with the location of the furthest placed concrete. As discussed in Chapter 3 w et concr ete is PAGE 48 48 typically placed just ahead of the finishing machine ( recall Figure 2 14 ) using a pump, therefore, in the vast majority of paving situations, the location of the finishing machine and the end of the concret e coverage will generally coincide. A limited scope sensitivity study was conducted to determine the critical locations those that cause maximum brace forces of the wet slab and finishing machine. In total, approximately 900 analyses were conducted where the concrete deck coverage area was moved in small increments along the bridge length. For this sensitivity study, the increments corresponded to the bridge 10 th points (i.e., increments of ten percent (10%) of the bridge span length). In all cases, the ma ximum end span brace and intermediate span brace forces occurred when the web slab load terminated close to one of the bracing points (e.g., end point, 1/2 point, 1/3 point, 1/4 point). Conversely, it was determined that partially placed deck loads termina ting between the locations of the braces did not control maximum end span or intermediate span brace forces. Therefore, for purposes of quantifying maximum brace forces in the database production parametric study, it was found to be adequate to consider th e multiple deck load cases illustrated earlier in Figures 3 11 3 12 3 13 and 3 14 for each bridge, with the specifi c number of load cases depending on the number of intermediate span bracing lines present in the bridge. 4 3 Scope tudy To develop a comprehensive d atabase of brace forces for use in the brace force prediction dimensional (3 D) structural analyses were conducted to quantify maximum end span and intermediate span brace forces for combinations of the following parameters: FIB cross section depth (in.) Span length (ft) Girder spacing (ft) Deck overhang width (ft) PAGE 49 49 Skew angle (deg.) Brace type (K brace or X brace) Number of brace points (end span only, 1/2 point, 1/3 point, 1/4 po int ) Number of girders Construction load factors (un factored service loads; factored strength loads) Specific parameter values that were included in the database production parametric study which involved 453,600 separate analyses are listed in Table s 4 1 and 4 2 For the factored construction load analyses, the load factors previously listed in Table 3 1 were used. (Note that seven (7) of the eight (8) standard FIB cross sections were included in the study. The section is so shallow that use of moment resisting X braces and K braces is not likely to be feasible or warranted). Maximum and m inimum span lengths used in the parametric study were based on design aids included in the FDOT Instructions for Design Standard No. 20010: Prestressed Florida I Beams (IDS 20010; FDOT, 2014b), which provide estimated span lengths (Table 4 4 ) for FIBs with different lateral spacings, based on representative bridge design calculations. Maximum lengths were conservatively based on a lateral girder spacing of 6 ft and an environment while minimum lengths assumed a 12 ft spacing and an reasonable beam designs, the basic ranges taken from FDOT IDS 20010 were extended to cover a total span rang e of 50 ft. The parametric study included span lengths chosen at 10 ft intervals over the final ranges. It was also determined from a survey of typical bridges (and FDOT recommendations) that the range of other geometric parameters, such as deck overhang widths, girder spacings, and skew angles, would cover most design scenarios. Additionally, preliminary analyses indicated that as the number of interior brace points increases beyond three (3) interior braces (i.e., 1/4 PAGE 50 50 point bracing), changes in maximum brace forces become small (i.e., the brace forces converged). Similarly, increasing the number of girders in the system beyond nine (9) girders did not significantly change maximum brace forces in the construction load analyses. Therefore, bridges with mor e than nine girders were not expected to have significantly different brace forces induced by construction loads than would a bridge with nine (9) girders. For each Florida I beam in the parametric study, a single bracing depth (Figure 4 8 ) was assigned based on available web height. For FIB sections, as the overall girder depth increases, so does the available web depth (since the top and bottom flange dimensions remain constant). Therefore, deeper girders in the parametric study allowed for larger brace depths. Additionally, as brace depth increases, the bracing system becomes more effective in resisting torsional moments induced by eccentric construction loads. Therefore, the representative brace depths chos en for the parametric study (Table 4 4 ) were selected so as to avoid producing overly conservative brace force data. 4 4 Calculation of Brace F orces by Database I nterpolation As noted earlier, the approach to brace force prediction (calculation) taken in this study was to conduct a large scale parametric study; store the brace force results into a database; and then access and interpolate that database when forces are needed for bracing design. The overall database of information generated by the database production parametric study was stored into two text one for K braced systems, and the other for X braced systems. Within each databa se file, maximum brace forces were stored both for un factored service loads and for factored strength loads. Additionally, for K braced systems, diagonal element forces and horizontal element forces were computed and stored separately. Data contained with in the database files are organized in such a manner that particular cases of interest can be efficiently located. PAGE 51 51 To make the process of accessing and interpolating the database simple and user friendly, a MathCad based program was developed. The program allows the user to specify the following key system parameters: FIB cross section depth (in.) Span length (ft) Girder spacing (ft) Deck overhang width (ft) Skew angle (deg.) Brace type (K brace or X brace) Number of brace points (end span only, 1/2 point, 1/3 point, 1/4 point ) Number of girders and reports back the maximum end span and intermediate span brace forces for un factored ( service ) and for factored (strength) loading conditions. Of the eight (8) parameters listed above, three (3) are never interp olated since intermediate values are not possible FIB depth, number of interior brace points, and brace type (K brace or X brace). For the five (5) remaining parameters (span length, girder spacing, deck overhang width, skew angle, and number of girders), the corresponding data are extracted directly from the database, if an exact match is available. If, however, an exact match is not available, then interpolation (and in some cases, extrapolation) is used to estimate the brace force. To accomplish this ou tcome, the program implements a five dimensional linear interpolation algorithm. When user specified parameter values lie outside the scope of the database production parametric study (Table s 4 1 and 4 2 ), one of three possible actions will o ccur: 1) data extrapolation; 2) data bounding; 3) generation of an error message. For some parameters, limited linear extrapolation outside the scope of the database is appropriate. However, for some parameters, it is more appropriate to report brace forces for the bounding value from Table 4 1 and Table 4 2 rather than to perform extrapolation. For example, when th e number of girders exceeds nine (9), it is more appropriate to estimate the brace forces using data corresponding to a PAGE 52 52 bridge with nine (9) girders than to perform linear extrapolation. Note that whenever data extrapolation or data bounding are necessary to compute brace forces, a message is generated by the program indicating to the user which process was used. Finally, selected parameter values are not permitted to be outside the range indicated in Table s 4 1 and 4 2 Specifically, skew angles larger than 45 degrees and deck overhang widths larger than 6 ft are not permitted and will result in an error message being generated. 4 5 Verification of Database A pproach To evaluate the level of accuracy and conservatism produced by the interpolated database approach (and to verify correct functioning of the associated calculation program that was developed), a limited s cope sensitivity study was conducted. System parameters used in the sensitivity study (e.g., span length, deck overhang width, skew, girder spacing, number of girders) were specifically chosen to fall between the parameter values (Table s 4 1 and 4 2 ) that were used to generate the brace force database. Additionally, for each set of intermediate system parameters that were chosen, an additional and corresponding finite element brid ge model was constructed and analyzed to provide a datum for brace force comparison. Approximately 200 unique geometric configurations of bridges with end span and mid span bracing (involving approximately 600 separate analyses with multiple construction l oad ing conditions) were tested for a broad range of girder types. The level of accuracy and conservatism was quantified for each case by normalizing the database predicted maximum brace forces by the maximum brace forces computed from the additional finite element analyses. Normalized results for K brace cases are illustrated in Figure 4 9 for end span braces and Figure 4 10 for intermediate span braces. Similarly, results for X bra ce cases are illustrated in Figure 4 11 for end span braces and Figure 4 12 for intermediate span braces. For all four sets of data presented, the majority of the database predicte d brace forces were within 10% of the corresponding finite PAGE 53 53 element computed values. Furthermore, in all four sets of data, the distributions of normalized predictions were clearly biased (skewed) toward the side of producing conservative results. Overall, the level and bias of the conservatism produced by the interpolated database approach was considered to be appropriate and reasonable for purposes of designing braces. Note that these analyses included different combinations of intermediate parameters (i .e., parameters not in parametric study scope) from one intermediate parameter to all five parameters (skew, number of girders, span lengths, girder spacings, and deck overhang width). PAGE 54 54 Table 4 1 Span length values used in the database production parametric study Span length, L (ft) 80 100 110 120 130 150 160 90 110 120 130 140 160 170 100 120 130 140 150 170 180 110 130 140 150 160 180 190 120 140 150 160 170 190 200 130 150 160 170 180 200 210 Table 4 2 Other parameter values used in the database production parametric study Deck overhang width, (in.) S kew angle Intermediate span brace points, n i Girder spacing, (ft) Girders, n g 25 0 0 6 3 36 15 1 9 5 48 30 2 12 9 60 45 3 72 Table 4 3 Range of allowable span lengths for FIBs Values from FDOT IDS 20010 Section Min length (ft) Max length (ft) Span length range 126 80 130 113 142 100 150 124 155 110 160 142 173 120 170 151 182 130 180 159 191 150 200 FIB 175 208 160 210 Spacing 12 ft 6 ft Environment Extremely aggressive Moderately Aggressive PAGE 55 55 Table 4 4 Brace depths used in construction load parametric study Girder type Brace de pth, (in.) 18 22 27 36 41 48 60 Figure 4 1 PAGE 56 56 Fig ure 4 2 Effect of cross slope Figure 4 3 Effect of camber on brace forces for a PAGE 57 57 Figure 4 4 Effect of sweep Figure 4 5 K brace config urations analyzed in limited scope sensitivity study PAGE 58 58 Figure 4 6 Conservatism of selected K brace configuration (compared to inverted K brace) Figure 4 7 Conservatism of selected K brace configuration (compared to offset ends) PAGE 59 59 Figure 4 8 Definition of brace depth Figure 4 9 Accuracy of interpolated database approach for maximum end span K brace force PAGE 60 60 Figure 4 10 Accuracy of interpolated database approach for maximum intermediate s pan K brace force Figure 4 11 Accuracy of interpolated database approach for maximum end span X brace force PAGE 61 61 Figure 4 12 A ccuracy of interpolated database approach for maximum intermediate span X brace force PAGE 62 62 CHAPTER 5 WIND TUNNEL TESTING 5 1 Overview In addition to the cons truction loads discussed in previous chapters, wind loads must also be accounted for when quantifying brace forces for bracing system design. In a previously conducted study (BDK75 977 33, Consolazio et al., 2013) wind tunnel tests were conducted to quanti fy girder drag coefficients for partially constructed bridges at the stage where only the girders and bracing are present. When stay in place forms and overhang formwork are subsequently added to the bridge as part of the ongoing construction process they introduce new barriers to wind flow (e.g., between adjacent girders) and therefore have the potential to alter the drag coefficients for the individual girders within the bridge cross section. A goal of the present study was to therefore to quantify drag c oefficients for individual girders in bridges that are at the construction stage where stay in place forms and overhang formwork have been installed. To achieve this goal, wind tunnel tests were conducted on bridges with stay in place forms present (and i n some cases overhangs) to generate data that would complement data previously measured in BDK75 977 33. By comparing data measured in the current study to data from the previous study, the influences of stay in place forms and overhangs on girder drag coe fficients could be quantified and appropriate drag coefficients for wind load brace force design could be determined. Additionally, since practical bridge structures almost always consist of multiple girders positioned side by side, it was also necessary to investigate the effects of shielding (i.e., aerodynamic interference), in which the windward girder acts as a wind break and reduces the total force on subsequent leeward girders. The wind tunnel tests conducted in this study were PAGE 63 63 therefore performed on bridge cross sections with variations in the number of girders, variations in girder spacing, and variations as to the presence of stay in place forms and overhangs. 5 2 Background on Drag Coefficients In order to calculate wind load on a bridge girder, it is necessary to know the drag coefficient for the girder cross sectional shape. The drag coefficient is a type of aerodynamic coefficient : a dimensionless factor that relates the magnitu de of the fluid force on a particular geometric shape to the approaching wind speed. Drag coefficients are typically a function of the relative orientation of the object relative to the direction of the impinging wind. 5 2 1 Dimensionless A erodynami c C oefficients Fluid forces arise when a solid body is submerged in a moving fluid. As the fluid flow is diverted around the body, a combination of inertial and frictional effects generates a net force on the body. It is observed that this force called aerodynamic force ( F ) when the fluid under consideration is air is directly proportional the dynamic pressure ( q ) of the fluid: ( 5 1 ) where is the mass density of the fluid and V is the flow velocity (engel and Cimbala, 2006). Dynamic pressure can be considered as the kinetic energy density of the fluid. This of fers an intuitive explanation for its proportional relationship to aerodynamic force, which is, at the most fundamental level, the cumulative effect of innumerable microscopic collisions with individual fluid particles. Similarly, if the dimensions of the body are scaled up, it is observed that the aerodynamic force increases quadratically, reflecting the fact that the increased surface area results in a greater total number of collisions. These proportional relationships can be combined and expressed as: PAGE 64 64 ( 5 2 ) w here L 0 and L 1 are arbitrary reference lengths and C F is a combined proportionality factor, called a force coefficient The selection of L 0 and L 1 does no t affect the validity of Equation 5 2 as long as they both scale with the structure. However, it is important to be consistent; force coefficients that use different reference lengths are not directly comparable, and a coefficient for which the reference lengths are not explicitly known is useless for predicting aerodynamic forces. In structural applications, it is common for the product L 0 L 1 to be expressed in the form of a reference area, A which is typically taken a s the projected area of the structure in the direction of wind. By an analogous process, it is possible to derive a moment coefficient ( C M ), which normalizes aerodynamic moment load in the same way that the force coefficient normalizes aerodynamic force. T he only difference is that aerodynamic moment grows cubically with body size rather than quadratically (because the moment arms of the individual collisions grow along with the surface area). Therefore, the moment proportionality expression is: ( 5 3 ) As with the force coefficient, the reference lengths must be known in order to properly interpret the value of C M However, with moment coefficients, it is equally imp ortant to know the center of rotation about which the normalized moment acts. Together, C F and C M are called aerodynamic coefficients and they can be used to fully describe the three dimensional state of aerodynamic load on a structure (for a particular w ind direction). When working with bridge girders, or other straight, slender members, it is often convenient to assume that the length of the girder is effectively infinite. This simplifies engineering calculations by reducing the girder to a two dimension al cross section subjected to PAGE 65 65 in plane aerodynamic line loads (Figure 5 1 ). Depending on the direction of wind, out of plane forces and moments may exist, but they generally do not contribute to the load cases th at control design and can therefore be considered negligible. In two dimensions, the proportionality expressions for the aerodynamic coefficients become: ( 5 4 ) ( 5 5 ) w here is a distributed force (force per unit length) and is a distributed torque (moment per unit length). Note that two dimensional aerodynamic coeffi cients can be used interchangeably in the three dimensional formulation if the reference length L 0 is taken to be the out of plane length of the girder. All further discussions of aerodynamic coefficients in this report will use the two dimensional formula tion unless stated otherwise. The remaining reference lengths ( L 1 and L 2 ) will always be taken as the girder depth, D so that the force and moment coefficients are defined as: ( 5 6 ) ( 5 7 ) Aerodynamic coefficients are sometimes called shape factors because they represent the contribution of the geometry of an object (i.e., the way airflow is diverted around it), independent of the scale of the object or the intensity of the flow. Because of the complexity of the differential equations governing fluid flow, the aerodynamic coefficients of a structure are not calculated from first principles but can, instead, be measured directly in a wind tunnel using reduced scale models. PAGE 66 66 5 2 2 Terminology Related to Aerodynamic C oefficients Aerodynamic force on a body is typically resolved into two orthogonal components, drag and lift. These components have corresponding force coefficients: the drag coefficient ( C D ) and lift coefficient ( C L ). In this report, drag is defined as the lateral co mponent of force and lift is defined as the vertical component of force, regardless of the angle of the applied wind. In several subfields of fluid dynamics, it is more conventional to define drag as the component of force along the direction of the wind stream and lift as the component perpendicular to the wind stream. However, this is inconvenient when evaluating wind loads on stationary structures (e.g. bridge girders) because the angle of the wind stream can change over time. Where necessary in this r eport, the names stream drag ( C SD ) and stream lift ( C SL ) (Figure 5 2 ) will be used to refer to the force components that are aligned with, and perpendicular to, the wind stream. Finally, the term pressure coeffi cient ( C P ), is an alternative name for C D and is often used in design codes to indicate that it is to be used to calculate a wind pressure load ( P ) rather than a total force, as in: ( 5 8 ) This is advantageous because it obviates the need to explicitly specify the characteristic dimensions that were used to normalize the coefficient. Instead, denormalization occurs implicitly when the pressure load is applied ove r the projected surface area of the structure. Unfortunately, this approach breaks down when working with drag and lift coefficients together. If drag and lift are both represented as pressure loads, then the areas used to normalize the coefficients will d iffer (unless by chance the depth and width of the structure are equal). As a result, the magnitudes of the coefficients are not directly comparable that is, equal coefficients PAGE 67 67 will not produce loads of equal magnitude and they cannot be treated mathematic ally as components of a single force vector, which complicates coordinate transformations and other operations. For this reason, the term pressure coefficient is not used in this report, except when in reference to design codes that use the term. In this report, the term torque coefficient ( C T ) refers to the in plane moment that acts about the centroid of the cross section. This is a convenient choice of axis because it coincides with the axes of beam elements in most structural analysis software. Loads ca lculated from C D C L and C T can be applied directly to beam nodes (located at the centroid of the cross section) to correctly model the two dimensional state of aerodynamic load. However, most design codes represent wind load as a uniform pressure load th at produces a resultant force acting at a location called the center of pressure (Figure 5 3 ), which is typically assumed to correspond to the mid height of the cross section. For such circumstances, the term press ure torque coefficient ( C PT ) acting about the center of pressure will be used to differentiate it from the C T which always acts about the centroid (Figure 5 4 ). A summary of the different types of aerodynamic coefficient used in this report is presented in Table 5 1 5 3 Current Wind Design P ractice in Florida Bridge structures in Florida are desig ned in accordance with the provisions of the FDOT Structures Design Guidelines (SDG ; FDOT, 2013 ). As with most modern design codes, the wind load provisions in the SDG are based on Equation 5 8 with additional sca le factors included to adjust the intensity of the wind load according to the individual circumstances of the bridge. Specifically, Section 2.4 of the SDG gives the equation: ( 5 9 ) PAGE 68 68 where P Z is the design wind pressure (ksf kip per square foot ), K Z is the velocity pressure exposure coefficient, V is the basic wind speed (mph), and G is the gust effect factor. The constant term, 2.56 10 represents the quantity 1 2 from equation 5 8 expressed in derived units of (ksf)/(mph) 2 Each county in Florida is assigned a basic wind speed, V adapted from wind maps published by the American Society of Civil Engineers (ASCE 2006), which are based on statistical analyses of historical wind speed records compiled by the National Weather Service. Statistically, V represents the peak 3 second gust wind speed for a 50 year recurrence interval. In other words, if the average wind speeds during every 3 second time interval were recorded over a period of 50 years, V is the expected value of the maximum speed that would be recorded. It is important to note that this does not mean that Florida bridges a re only designed to resist 50 year wind loads. Different load combinations use load factors for wind that effectively adjust the recurrence interval up or down. For example, the Strength III limit state, as stipulated by the SDG, includes a wind load facto r of 1.4, which increases the recurrence interval to approximately 850 years (FDOT 2009). Basic wind speeds published by ASCE are based on measurements taken at an elevation of 33 ft and are not directly applicable to structures at other elevations. Wind that is closer to ground level is slowed by the effect of surface friction, resulting in a vertical wind gradient called the atmospheric boundary layer (Holmes, 2007). The purpose of the velocity pressure exposure coefficient, K Z is to modify the wind pr essure load to account for differences in elevation. Because surface roughness of the terrain is known to reduce the steepness of the gradient, ASCE divides terrains into three exposure categories, B, C, and D, and provides PAGE 69 69 equations for each category. How ever, for simplicity, the SDG conservatively assumes that all Florida structures are in the Exposure C category. As a result, the equation for K Z in Florida is: ( 5 10 ) where z is the elevation above ground (ft). Note that K Z is equal to unity at an elevation of 33 ft (corresponding to the wind speed measurements) and that wind speed is assumed to be constant for elevations of 15 ft or less ( Figure 5 5 ). Wind is characteristically gusty and turbulent, producing dynamic structural loads that can fluctuate significantly over short periods of time. However, it is simpler and more efficient to design structures to resist static loads. Furt hermore, wind tunnel measurements of static force coefficients are typically performed in steady flow (with a major exception being site specific wind tunnel testing, which models a proposed structure along with its surrounding terrain for the express purp ose of capturing turbulent loads). The gust effect factor, G modifies the static design wind pressure so as to envelope the effects of wind gustiness and dynamic structural response on peak structural demand. For aerodynamically rigid bridge structures, d efined as those with spans less than 250 ft and elevations less than 75 ft, the SDG prescribes a gust effect factor of 0.85. By this definition, the vast majority of precast prestressed concrete girder bridges in Florida are considered aerodynamically rigi d. It is noted that G actually reduces the design wind pressure on rigid bridges, reflecting the fact that peak gust pressures are unlikely to occur over the entire surface area of such structures simultaneously (Solari and Kareem, 1998). The SDG further p rovides specific guidance on the calculation of wind loads during the bridge construction stage (as opposed to the calculation of wind loads on the completed bridge structure). If the exposure period of the construction stage is less than one year, a reduc tion factor of 0.6 on the basic wind speed is allowed by the SDG. PAGE 70 70 Calculation of wind pressure using Equation 5 9 requires that an appropriate pressure coefficient ( C P ) be determined for the structure under consideration Pressure coefficients are provided by the SDG for several broad categories of bridge component as indicated in Table 5 2 In the present study the pressure coefficients of interest are those for girders with stay in place form s in place. As Table 5 2 indicates, the SDG provides a single value of C P regardless of girder shape, when deck forms are in place : C P = 1.1 The wind tunnel testing conducted in this study was performed to help determine whethe r C P = 1.1 is an appropriate value, and whether it should depend on the girder type. 5 4 Testing C onfigurations To maximize the potential for comparing results from wind tunnel tests with S IP forms to those with out forms, the test configurations used in a previous study (BDK75 977 33 Consolazio et al ., 2013, no formwork) were used as a guide in determining the wind tunnel test program scope in the present study (with formwork). Consequently testing configurations with the sole difference being the presences of SIP forms were included in the present study. Four different girder cross sectional shapes (Figure 5 6 ) were selected as being representative of a wide range of modern Florida bridges: 78 inch deep Florida I beam ( ) : One of the most common FIB shapes used in bridge design. All FIB shapes have identical flanges, with the differences in girder depth arisin considered in this study. 45 inch deep Florida I beam ( ) : quantify the effect of changing the FIB depth, and to ensure that the resulting design loads would be applicable to a range of FIB shapes. Wide flange plate girder ( WF plate ) : Drag coefficients of I shaped girders have been studied, in the literature, for width to depth ratios ranging from 1:1 to 2:1. However, built up s teel plate girders commonly used to support bridge decks tend to be much deeper than they are wide. The WF Plate girder considered in this study has an 8 ft deep to depth ratio of 3:1, representing the appro ximate lower bound for bridge girders. PAGE 71 71 Box girder ( Box ) : A survey of existing box girder bridges was used to develop a representative 6 ft deep cross section. All of these girder sections were tested in multiple girder configurations and with addition stay in place formwork present (Figure 5 7 a). Additionally, to quantify the influence of overhangs, all bridge cross sections were tested both with and without maximum feasible overhangs (Figure 5 7 b) Fully dimensioned drawings of the girder cross sections and schematics of each test configuration conducted in this study are included in Appendix B W ind tunnel test configurations were defined by type of girder employed ( Figure 5 6 ), and by the following parameters (Figure 5 7 ): Number of girders : Wind tunnel tests wer e performed on 2 girder, 5 girder, and 10 girder configurations. Spacing : Spacing refers to the horizontal center to center distance between girders. Results from previously conducted wind tunnel testing (Consolazio et al., 2013) indicated that girder con figurations with larger spacing produced less shielding (i.e., less aerodynamic interference, and thus larger forces) on leeward girders. Therefore, to yield conservative wind tunnel results in the present study, a characteristic maximum spacing was determ ined for each type of girder based on a survey of existing bridge designs and consultations with the FDOT. Each testing configuration for a given type of girder then used only the maximum spacing. Cross slope : Most bridge decks are designed with a cross sl ope that is 2% or greater in magnitude, and the girders are usually aligned vertically along that slope so that they can evenly support the deck. Therefore, the FIBs and plate girders were tested with a cross slope that was 2% in magnitude, but negative in sign (Figure 5 7 ): i.e., a cross slope of 2 %. A negative cross slope was used because when SIP forms are attached to the top flanges of the girders, the exposed bottom flanges of the girders produce a worst case (maximum) condition in terms of drag forces generated on the shielded leeward girders. Generally, s teel girder bridges can have a greater amount of horizontal curvature than FIB bridges so higher cross slopes are often included to improve vehic le handling. To account for the larger magnitude of cross slope the WF Plate girders were tested in configurations with 8% cross slope. In contrast to I shaped FIB and plate girders, box girders are not generally aligned vertically when supporting a cross sloped deck. Instead, the girders are typically inclined to follow the bridge cross slope. As a result, in this study, box girders were only tested in a 0% (un sloped) configuration however the range of tested wind angles was increased (relative t o the I shaped girder wind angles) as described below. PAGE 72 72 Wind angle : In practical bridge construction situations, the direction of wind flow will not, in general, always be perfectly horizontal. To account for the natural variation in wind angle (and at the recommendation of a commercial wind tunnel test facility), each FIB and plate girder bridge configuration was tested at five (5) different wind angles 5 8 ). In the case of the box girder, a change in wind angle is geometrically equivalent to rotating the girders to match the deck cross slope (Figure 5 9 ). As a result, the box girder configurations were tested at combined effects of 5 wind angle and 5 (8.7%) of cross slope. Overhangs : In most economical bridge designs, the bridge deck extends transversely beyond the extents of the exterior girders, t hus creating an overhang of each side. Since it was desirable to quantify the effects of such overhangs on wind coefficients, overhang formwork was included in many of the configurations tested in the wind tunnel. It should be noted that the overhang formw ork support brackets described earlier ( see Section 3 4 ) were excluded from the wind tunnel tests because they were not expected to influence the wind coefficients. However, the top surface of the overhang formwor k, which was W OHT ) of 5 ft (Figure 5 10 ), as measured from the centerline of the top girder flange to edge of the over hang formwork ( not the edge of the concrete deck), was used for all girder types. However, due to differences in girder top flange widths, the formwork ( W OHF ), which was the extension of formwork beyond the edge of the girder top flange (F igure 5 10 ), varied for the girder types tested. A summary of the scope of the wind tunnel test program is provided in Table 5 3 Note that it was not feasible to ins trument (measure wind forces) at every girder position in every configuration tested. Instead, the girder positions (G1, G2 ... G10.) that were instrumented were strategically chosen to maximize the usefulness of the measured data. 5 5 Testing P rocedure The Boundary Layer Wind Tunnel Laboratory at the University of Western Ontario (UWO) was contracted to fabricate the test specimens and to perform all wind tunnel measurements. Based on the s ize of the UWO wind tunnel, the girder models were constructed at reduced scale ( Table 5 4 ) with air flow properties similarly adjusted so that the resulting forces would be applicable at full scale. All testing was perform ed in smooth flow, with turbulence intensities less than 0.5%. Because the tested cross sections were sharp edged, it was expected that the measured wind forces would not be sensitive to Reynolds number and the force PAGE 73 73 coefficients are applicable over a broa d range of wind speeds In a previous study (Consolazio et al ., 2013), the assertion that wind forces would not be sensitive to Reynolds number was verified by UWO by performing selected tests at multiple Reynolds numbers. Results from those tests did not reveal any obvious Reynolds number sensitivities. The scaled girder models were all 7 ft long (equivalent to 175 ft and 196 ft girders at full scale) and were constructed to be fully rigid, without exhibiting any aeroelastic effects. An adjustable frame w as used to keep the girders properly oriented relative to each other in each test configuration. To measure wind induced girder forces at varying wind angles of attack, the entire bridge cross sectional assembly was rotated in place relative to the wind st ream To maximize the utility of the data collected during the wind tunnel testing, it was desirable to individually quantify the aerodynamic forces (drag, lift, torque) that acted on each girder within the bridge cross section. In order to accomplish thi s goal, each girder in the bridge had to be structurally independent from the rest of the girders i.e., transmission of lateral load from one girder to the next had to be prevented. Simultaneously, however, air flow between adjacent girders also had to be prevented in order to model the blockage effects associated with the presence of SIP forms. The approach used to satisfy both of these requirements was to attach bent plates ( Figure 5 11 ) to the top flang es of the girders in the bridge cross section. The plates represented both the SIP forms and, where present, the overhang formwork. Structural independence was achieved by only extending the SIP form plates to the midpoint between adjacent girder s ( Figure 5 11 ) and leaving a small gap so that force transmission to the adjacent plate was not possible As such, each SIP form plate cantilevered from a girder top flange to the midpoint of the girder spacing on e ach side (for an interior girder). To model the air flow blockage effects of PAGE 74 74 the SIP forms, but without transmitting force across the gaps, a flexible adhesive tape (i.e., an adhesive membrane) was used to seal and span across the gaps. In tests where over hangs were present, the top plates were further extended ( cantilevered ) out to the extents of the overhang formwork Wind forces on the girders in each test configuration were measured individually with a high precision load balance that recorded the tim e averaged horizontal load (drag), vertical load (lift), and torque (overturning moment). These loads were then normalized to produce the aerodynamic coefficients for drag ( C D ), lift ( C L ), and torque ( C T ). Finally, the torque coefficient was adjusted so th at it represented the torque about the centroid of the section, rather than the torque about the point of measurement (which was at mid height for the I shaped girders and at an arbitrary point for the box girders). PAGE 75 75 Table 5 1 Parameter values used in the database production parametric study Symbol Coefficient name Description C D Drag Component of force in horizontal (lateral) direction C L Lift Component of force in vertical directio n C SD Stream Drag Component of force parallel to wind stream C SL Stream Lift Component of force perpendicular to wind stream C P Pressure Alternative name for C D C T Torque Torque measured about centroid C PT Pressure Torque Torque measured about center of pressure Table 5 2 Pressure coefficients in FDOT Structures Design Guide lines ( FDOT, 2013 ) Bridge c omponent C P Substructure 1.6 Girders with deck forms 1.1 Completed s uperstructure 1.1 I shaped b ridge g irders 2.2 Box and U shaped g irders 1.5 Table 5 3 Summary of wind tunnel tests Section Overhangs included Cross slope Spacing (ft) Number of girders Instrumented girder position Test Angles 2% 13 10 1, 2, 3 0, 2.5, 5 Yes 2% 13 10 1, 2, 3, 5, 10 0, 2.5, 5 Box 0% 22 2 1, 2 0, 5, 10 Box Yes 0% 22 2 1, 2 0, 5, 10 WF Plate 8% 14 5 1, 2, 3 0, 2.5, 5 WF Plate Yes 8% 14 5 1, 2, 3, 4, 5 0, 2.5, 5 2% 13 5 1, 2 0, 2.5, 5 Yes 2% 13 5 1, 2, 3 0, 2.5, 5 Table 5 4 Wind tunnel test scaling Section Model scale Reynolds number WF P late 1:25 77000 1:28 56000 1:28 33000 Box 1:25 58000 PAGE 76 76 Figure 5 1 Two dimensional bridge girder cross section with in plane line loads Figure 5 2 Definition of C D C L C SD and C SL (shown in positive direction except when noted) Figure 5 3 Center of pressure of a bridge girder PAGE 77 77 Figure 5 4 Definition of C T and C PT (shown in positive direction) Figure 5 5 Velocity pressure exposure coefficient used by FDOT Figure 5 6 Girder cross sections used in study PAGE 78 78 A B Figure 5 7 Parameters definitions for each testing configuration A) with SIP formw ork. B ) with SIP formwork and overhangs Figure 5 8 Wind angle sign convention PAGE 79 79 Figure 5 9 Equivalence between wind angle an d cross slope for box girders Figure 5 10 Overhang dimensions used in wind tunnel study PAGE 80 80 A B Figure 5 11 Formwork and ove rhang attachment methodology. A) Typical construction schematic B ) Wind tunnel testing setup PAGE 81 81 CHAPTER 6 WIND TUNNEL TESTING RESULTS AND ANALYSIS 6 1 Ov erview Wind tunnel tests were performed on the bridge girder test configurations described in Chapter 5 Several groups of laterally spaced girders were tested to quantify shielding effects, identify trends, and evaluate the ae rodynamic influence of stay in place formwork and overhangs. The complete set of wind tunnel test data, reported using terminology defined in Chapter 5 is available in Appendix C From analysis of the results, simplified calculation procedures were developed for determining global drag coefficients for I shaped girder bridges ( i.e., FIBs and plate girders) and bridges constructed using box girders. 6 2 Key F indings from the Wind T unnel Test P rogram After processing the wind tunnel data into a form consistent with the terminology defined in Chapter 5 which is also consistent with terminology used in a pr evious wind tunnel study (BDK75 977 33, Consolazio et al. 2013) the following key findings and data trends were identified. 6 2 1 Influence o f Stay In Place Forms and Overhangs on Drag C oefficients Representative example comparisons of drag coefficients ( C D ) for systems consisting only of bare girders and drag coefficients for systems consisting of girders with SIP forms and with overhang formw ork are presented in Figure 6 1 for wide flange (WF) plate girders, and in Figure 6 2 for FIB78 girders. Data shown in these figures which represent variations in wind angle, girder type, and bridge width (number of girders) serve to illustrate the influence that the addition of both SIP forms and overhang formwork ha d on drag coefficients. (Note: data for FIB45 and box girder sections exhibit similar trends to tho se illustrated for the WF plate girder and FIB78 sections, but are omitted here for brevity.) PAGE 82 82 In Figure 6 1 all data presented correspond to a magnitude of cross slope equal to 8%. As noted in Chapter 5 in the present study only negative cross slopes were investigated since these conditions produce the most conservative drag coefficients when SIP forms are prese nt. In contrast, in study BDK75 977 33, the cross slope was a positi ve value +8%. However, since these earlier tests were conducted without SIP forms or overhang formwork (bare girders only), and because the WF plate girders have doubly symmetric cross sectional shapes the results obtained in BDK75 977 33 for +8% also cor respond to a cross slope of 8%. Hence, the C D data presented in Figure 6 1 from the previous study (at +8%) and the present study (at 8%) are, in fact, comparable. In general the data presented in Figure 6 1 and Figure 6 2 indicate that the introdu ction of SIP forms and overhang formwork does not alter the fundamental C D trend tha t was first identified in BDK75 977 33, that is: a large positive C D value at windward girder position G1; one or more leeward (shielded) girder positions (G2, G3, ...) with negative C D values; and then subsequent increases of C D values typically producing + C D values for girders farther dow nstream. With regard to the windward girder at position G1, introducing SIP forms and overhang formwork always produced an increase in the C D value when corresponding cases identical in every way except for the pre sence of SIP forms and overhang formwork w ere compared. For leeward (shielded) girder positions, introducing SIP forms and overhang formwork generally produced a slight decrease in the C D values. In Figure 6 3 and Figure 6 4 the effects on C D values produced only by adding SIP forms, but not overhang formwork are illustrated. For a majority of the data shown, adding SIP forms has the effect of reducing the C D values for the windward girder (position G1) and for the first shielded leeward girder (position G2); both of these trends will tend to reduce the total PAGE 83 83 (global) drag force on the bridge cross section. Results for shielded girders farther downwind, however, are mixed. In Figure 6 5 and Figure 6 6 the effects on C D values produced by adding overhang formwork to systems that already have SIP forms are illustrated. That is, Figure 6 5 and Figure 6 6 isolate solely the effects of adding overhang formwork. (This is in contrast to Figure 6 1 and Figure 6 2 which illustrated the combined effects of adding both SIP forms and overhang formwork ). With regard to the windward girder at po sition G1, introducing overhang formwork always produced an increase in the C D value when matched cases identical in every way exce pt for the presence of overhang formwork were compared. In con trast, however, adding overhang formwork had only minor effects on the C D values for shielded downwind girders. 6 2 2 Lift Coefficients for G irders and O verhangs When girder lift coefficients ( C L ) from the current study, which included SIP forms in all cases, were compared to corresponding lif t coefficients from s tudy BDK75 977 33, which included only bare girders, it was found that the addition of SIP forms increased the measured lift coefficients at every girder position measured, and for every condition tested. From this observation, it is clear that the additio n of SIP forms altered the flow of wind around the bridge cross section. Further, when lift coefficients ( C L ) for systems with SIP forms and overhangs were compared to lift coefficients for systems having only SIP forms (but without overhangs), it was evid ent that the addition of overhangs further increased the lift coefficients, especially for the windward girder (position G1). In addition to quantifying lift coefficients for the girders, it was also of interest to quantify lift coefficients for the overh ang formwork i.e., the width of formwork ( ) extending beyond the girder flange tip; recall Figure 5 10 Unfortunately, including direct PAGE 84 84 measurements of uplift forces on the overhang form work was not feasible within the scope of the wind tunnel testing program. However, it was possible to estimate the overhang formwork lift coefficients ( ) from the wind tunnel data that were measured. For each condition tested (gi rder type and wind angle), the lift coefficient attributable to the presence of the overhang formwork ( ) was computed as: ( 6 1 ) whe re was the lift coefficient measured at the windward girder (position G1) when SIP forms and overhangs were included, and was the lift coefficient measured at the windward girder when SIP forms were inc luded, but overhangs were omitted. Overhang lift coefficients estimated in this manner are summarized in Table 6 1 Since the overhang formwork lift coefficients ( ) reported in Table 6 1 have been estimated by taking differences of girder lift coefficients ( C L ), and since the girder lift coefficients reported throughout this study are normalized relative to the girder depth ( D ), by definition the o verhang formwork lift coefficients are then also normalized by the girder depth ( D ). Therefore, to compute overhang formwork lift forces from the coefficients reported in Table 6 1 the va lues must first be de normalized by the girder depth ( D ) as: ( 6 2 ) or alternately, and more conveniently, expressed as: ( 6 3 ) PAGE 85 85 where is the overhang formwork lift force per ft of girder span length, 2.56 10 represents the quantity in Eqn. ( 6 2 ) expressed in units of (ksf)/(mph) 2 V is the basic wind speed (mph), K Z is the velocity pressure exposure coefficient, G is the gust effect factor, is taken from Table 6 1 and D is the girder depth in ft. It is important to note that since only a single overhang formwork width ( ) was tested for each girder type (Table 6 1 ), the lift forces computed using the coefficients provided in Table 6 1 are specific to widths tested. Additionally, the type of data measured in the wind tunnel test program do es not provide i nsight regarding the form of the lift pressure distributions (e.g., uniform, triangular, nonlinear, etc.) that acted on the overhang formwork during testing. 6 2 3 Torque C oefficients Representative example comparisons of torque coefficients ( C T ) for systems consisting only of bare girders and torque coefficients for systems consisting of girders with SIP forms and with overhang formwork are presented in F igure 6 7 for wide flange (WF) plate girders, and in Figure 6 8 for FIB78 girders. (S ign convention : a positive torque induces a clockwise girder rotation for wind moving from left to right. ) T he most significant trend exhibited by the data was that the addition of overhang formwork significantly increased the torque on the windward girder (position G1) For the leeward (shielded) girder positions (G2, G3, ...) torque coefficients w ere considerably smaller than for the windward girder. When moving from bare girders to girders with SIP forms, moderate increases in torque coefficients were produced, but they were not nearly as pronounced as when overhang formwork was added. PAGE 86 86 6 3 Analysis of W ind T unnel T esting R esults To quantify the full (global) wind force acting on a bridge system, the total of all girder drag coefficients must be considered. A calculation procedure wa s therefore developed for determining a global pressure (drag) coefficient defined as the sum mation of the drag coefficients of all girder s in the bridge cross section Both I shaped girder systems (FIBs and plate girders ) and b ox girder systems were consi dered in the development process 6 3 1 C alculation of Global Pressure Coefficient for Systems with I shaped Girders Current standard practice specif ied in the FDOT Structures Design Guidelines (SDG; FDOT, 2013) involves determining a global pressure coefficient (for a system of multiple girders with SIP forms and possibly overhang formwork), computing an applied pressure using Eqn. ( 5 9 ), and then applying that pressure to the projected area of the bridge. This method, As such, the horizontal wind pressure is applied to the ver tical projected depth ( Figure 6 9 ) of the bridge The global pressure coefficient used in this process, for bridge girders with SIP formwork in place (referred to as in this section), is specified in the FDOT SDG as 1.1 (Table 6 2 ). Magnitudes of the wind loads on shielded girders are highly dependent on the interaction between the system cross slope angle ( cross sl ope ) and the wind angle ( wind ). As the absolute difference between those angles increases, a greater portion of the shielded girders are exposed to direct wind flow, resulting in a roughly proportional increase in girder drag force C onsequently, a strong predictor of total (global) wind load on a girder system is the projected area of the system (i.e., the total unshielded area). To appropriately capture this trend, global drag coefficients must be a function of the projected depth ( ). PAGE 87 87 Because the projected depth is a function of wind which fluctuates randomly over time, engineering judgment must be used in selecting a design value of wind such that it represents the maximum expected angle during the exposure period. For cons ervatism, the sign of wind must be chosen to be in opposition to that of cross slope so that the maximum angle difference ( max ) is computed as: ( 6 4 ) max ca n then be used to calculate the projected depth D proj of the girder system, as: ( 6 5 ) where D is the girder depth, n is the number of girders in the system, and S is the girder spacing (Figure 6 10 ). In this formulation, wind streamlines are assumed to be straight and the shielding effects of girder flanges are ignored as they are not expected to signifi cantly shiel d leeward girders. In contrast to the study BDK75 977 33 ( Consolazio et al. 2013), the presence of SIP forms in the girder systems influences the controlling wind direction. For bridges with a negative cross slope configuration, positive wind angles produ ce larger system level drag coefficients than do negative wind angles. ( This is due to the SIP forms shielding downstream girders when the wind angles of attack are more negative than the cross slope ) Consequently, only wind angles that were in opposition to the cross slope were considered in the cases shown below. For example, in the 8% cross slope ( 4.57 degrees) WF plate girder systems, only 2.5, 0, +2.5, and +5 degree wind angles were included. In the 2% cross slope ( 1.15 degrees) FIB systems, 0, + 2.5, and +5 degree wind angles were included. PAGE 88 88 Furthermore, in addition to SIP forms, the presence of overhang formwork was found to marginally increase the total (global) system level drag forces for I shaped girders. To account for this increase the vert ical project ed depth of the windward overhang formwork was included in the projected depth formulation, as: ( 6 6 ) where is the horizont al width of the overhang formwork (recall Figure 5 10 ) In the remainder of this report using Eqn. ( 6 6 ) to compute the projected depth (instead of the illustrated in Figure 6 9 ), will be referred to as the modified projected area method Within the scope of the wind tunnel tests, it was not feasible to instrument every girder positions for drag coefficient measurement. For reference, the following data were measured directly during the wind tunnel tests: WF plate girder (5 girders) : Fully instrumented for the cases with overhang formwork Positions G1 G3 were instrumented for cases without overhang formwork. FIB78 (10 girders) : Positions G1 G3 were instrumented in both the non overhang and overhang formwork setups. Positions G5 and G10 were instrumented in the systems with overhang formwork FIB45 (5 girders) : Positions G1 G2 were instrumented in both the non overhang and overhang formwork setups. Position G3 was instrumented in the systems with overhang formwork The following process was used to estimate drag coefficients at non measured positions: WF plate girder : Girder positions that were measured in over hang formwork cases were used as an estimate for non instrumented positions in systems without overhang formwork. An example case, at zero degree wind angle, is provided in Figure 6 11 to illustrate this estimation process. FIB78 : Positions G5 and G10 were experimentally measured in the systems with overhang formwork and used as estimates in the non measured positions in systems without overhang formwork. Other intermediate drag coefficients were linearly interpolat ed between the range of G3 G5 and G5 G10. An example case, at zero degree wind angle, is provided in Figure 6 12 to illustrate the linear interpolation process. PAGE 89 89 FIB45 : Position G3 was measured in the systems with ove rhang formwork and was used to estimate the corresponding position for the systems without overhang formwork. The last (most leeward) position in 5 girder FIB45 systems was estimated using the proportionality between the first to the last position (i.e., C D,Last /C D,First ) experimentally measured in the FIB78 systems. When transitioning from a 5 degree wind angle to a +5 degree wind angle, the ratio of first girder to last girder drag coefficients decreases; that proportionality was reflected in the FIB45 e stimations. The following calculation was performed: ( 6 7 ) where the last position (G5) in a FIB45 system was estimated using the proportionality between the firs t and last measured drag coefficient in the FIB78 systems. Similar to the FIB78 estimation process, Position G4 was linearly interpolated between G3 G5 (Figure 6 13 ). To assess the level of conservatism produced by use of the modified projected area calculation method (using Eqn. ( 6 6 ) to compute ) for I shaped girders divided (normalized) by the summation of girder drag forces measured during the wind tunnel study (or estimated, for non instrumented girders positions, as described above). Defined in this manner, the conservatism ratio was whe n the modified projected area method was conservative, and when the method was unconservative. However for convenience, it was desired to compute the conservatism ratio in terms of pressure (drag) coefficients rather than the tota l wind forces corresponding to those coefficients. To do so, it was recognized that in the modified projected area method, is used to compute a pressure that is then applied to the projected depth ( ) of the structure. In contrast, the girder drag coefficients reported from wind tunnel testing ( ) are normalized (referenced) to the girder depth ( D ), not the projected depth ( ). Therefore, to compute a PAGE 90 90 prope r conservatism ratio based on the ratio of pressure (drag) coefficients (rather than forces), it was necessary to define: ( 6 8 ) Using this definition, the conserv atism ratio could be properly defined in terms of pressure (drag) coefficients as Ratios computed in this manner for all I shaped girders are presented in Figure 6 14 It is ev ident in the figure that using a pressure coefficient of = 1.1 (from Table 6 2 ) produced unconservative results in many cases Note that in some cases, the systems were tested in ho rizontal wind (zero degree wind angle), meaning that the modified projected area approach used the same assumption as currently recommended by the SDG (Figure 6 9 ). To ensure that conservative force predictions were ob tained (i.e., normalized values greater than 1.0), it was determined via calibration that the pressure coefficient for girders with formwork in place ( ) need ed to be revised to 1.4 ( see again, Figure 6 14 ). The goal of this method was to provide conservative predictions of global pressure coefficients. However, w ith a design pressure coefficient of 1.4, FIB girder global pressure coefficients are overly conservative in comparison to WF plate girder systems (Figure 6 14 ). To produce a more refined prediction of FIB girder global pressure coefficients, a reduction factor was developed for use in the modified projected area approac h. An ideal global reduction factor ( ) could be calculated as: ( 6 9 ) where the sum of experimentally determin ed girder drag coefficients in a girder system ( C D,exp ) is normalized by the predicted global pressure coefficient. In Figure 6 15 ideal reduction factor s PAGE 91 91 are plotted against maximum angles (absolute differe nce s between the wind angle and cross slope). T o envelope the data a n upper bound linear curve fit was applied: ( 6 10 ) where is the reduction factor to be appl ied in the calculation of FIB girder global pressure coefficients and has units of degrees Additionally, since is a reduction factor computed value must be less than 1.0 thus creating a bi line ar curve. Application of the reduction factor ( ) to the prediction of FIB global pressure coefficients produces conservatism levels that are appropriate for design purposes (Figure 6 16 ) with an average conserva tism ratio (across all three girder types : WF plate girder FIB45 and FIB78) of 1.16. Note that the revised design pressure coefficient = 1.4 was used in these calculations and wind angles ( wind ) were included when computing the maximum angle difference ( max ). An alternative calculation procedure for determining global drag coefficients was also developed by implicitly including wind angle in the projected depth calculation. Similar to the currently prescribed method in the FDOT SDG for global drag coeff icients, wind load was taken as the pressure of the wind acting horizontally on a vertical projection over the exposed area of the structure. In other words, it was assumed that the maximum difference angle ( max ) is equivalent to the cross slope angle ( c ross slope ) and that the wind angle ( wind ) was taken as zero degrees. Then : ( 6 11 ) By normalizing predicted global drag coefficients (determined using Eqn. 6 11 to define max ) by measured global drag coefficients at variable wind angles, the level of conservatism was PAGE 92 92 evaluated. As evident in Figure 6 17 the level of conserv atism produced by this approach was not desirable when a design pressure coefficient = 1.1 was used Therefore, t o ensure that conservative force predictions were achieved (i.e., normalized values greater than 1.0 Figure 6 18 ), it was determined through calibration that the pressure coefficient needed to be = 1.8 (rather than the values of 1.1 or 1.4 previously noted) if wind angles are not explicit ly included in the determination of maximum difference angle max 6 3 2 C alculation of Global Pressure C oefficient for Systems with Box G irders Currently the FDOT Structures Design Guidelines (SDG, FDOT, 2013) specif y that a = 1.1 be applied to both I shaped girder and b ox girder superstructures when deck forms are in place (Table 6 3 ). Similar to the I shaped girder global pressure coefficient calculation procedure described in the previous section the accuracy of the projected area method was compared to the experimentally determined global drag coefficients for box girders. Recall fr om Chapter 5 that box girders were tested in the wind tunnel at 0, 5 and 10 wind angles with the girders aligned with the cross slope ( Figure 6 19 ) Given the matching al ignment of the girder s and the cross slopes, these test configurations were geometrically equivalent to zero degree (horizontal) wind angles with 0% (0 degree ), 8.7% (5 degree ), and 17.6% (10 degree ) cross slopes. Fully measured global drag coefficie nts for box girders with SIP forms are plotted as a function of wind angle in Figure 6 20 For I shaped girders, positive wind angles (that opposed the negative cross slope) always produced higher global drag force s. In contrast, clearly defined trends were not evident for the box girder data in terms of wind angle (i.e. positive wind angles did not necessarily produce global drag forces that exceed those produced at negative wind PAGE 93 93 angles). Consequently, all box gir der cases (positive and negative angles) with SIP forms were included in the development of a global pressure coefficient prediction method To develop such a prediction method, p rojected depths were calculated for each case by considering the entire supe rstructure. In the I shaped girder global pressure prediction method described earlier flange shielding was considered negligible and girders were assumed to behave as flat, vertical plates. In contrast, each box section tested was seven (7) ft across and therefore could not be idealized as a vertical plate Consequently, the projected depth could not be defined simply in terms of girder spacing (S) and angle. Instead, the projected depth of box g irder systems was defined by projecting the entire geometry of the boxes (including bottom flange width) onto a vertical plane (Figure 6 21 a ). Additionally, when overhang formwork was present in the cross section ( Figu re 6 21 b), the vertical projection of the windward overhang was included in the projected depth [analogously to the term previously noted in Eqn. ( 6 6 ) ]. The leeward overhang was omitted Using this projection method, and = 1.1 per Table 6 3 both cases (without and w ith formwork overhangs) at zero degree maximum difference angles ( max ) were under predicted (Figure 6 22 ) i.e., unconservative relative to measured wind tunnel data Additionally, the predicted global drag force was significantly more unconservative (greater error) for the box girder bridge with overhang formwork present. This is important since at zero degree wind and 0% cross slope, the box girder depth ( D ) and the projected depth ( D proj ) and equal S ince the global drag force (and pressure coefficient) was larger when overhang formwork was present despite the fact that the projected depth ( D proj ) was no different than the girder depth ( D ) this PAGE 94 94 indicated that use of projected depth alone was not adequa te to predict the global pressure coefficient (as was the case earlier for I shaped girders). Instead, the global pressure coefficient for box girder bridges was formulated to account for the presence, and width, of the overhang formwork. I t was found tha t the minimum global pressure coefficients needed to produce conservative results at zero wind angle were = 1.19 (SIP forms only) and =1.48 (SIP forms and overhang formwork). For convenience in design, th ese values were rounded to = 1.2 and = 1.5, respectively. In the wind tunnel test program, it was only feasible to conduct tests at a single overhang formwork width: =4.33 ft ( recall Figure 5 10 and Table 6 1 ) Therefore, to account for intermediate overhang widths that are likely to be encountered in practice the pressure coefficient for box gi rder bridges with SIP forms and overhang formwork was defined using linear interpolat ion as: ( 6 12 ) where is defined in units of ft Us ing the box girder together with projected depth ( ), to account for wind angles other than zero degrees, the normalized predicted global pressure coefficients were computed for all conditions tested in t he wind tunnel ( Figure 6 23 ) and were found to be conservative (greater than 1.0). However, for large wind angles, the degree of conservatism was greater than desirable, therefore, a reduction factor ( ) was developed in a manner similar to that previously developed for FIB systems As before, an ideal reduction factor ( ) for each test case was determined using Eqn. ( 6 9 ) where was computed from Eqn. ( 6 12 ) and as defined in Figure 6 21 In Figure 6 24 the ideal reduction factor s are plotted as a function of maximum PAGE 95 95 difference angle ( i.e., absolute value of the difference between the wind angle and cross slope). T o envelope the data a n upper bound linear curve fit was applied: ( 6 13 ) where is defined in Eqn. ( 6 4 ) and has units of degrees, and is the reduction factor for calculation of box girder global pressure coefficients. Use of the reduction factor ( ) produce d conservatism levels that are deemed appropriate for design purposes (Figure 6 25 ). 6 3 3 Recommended P rocedure for C alculation of Wind L oads Using the pressure coefficients for I shaped girders and box girders that were developed above, an overall procedure for computing lateral wind loads was devel oped : 1. Establish wind angle Establish the angle of wind ( ) that will be considered. If wind will be assumed to be horizontal, then set = of differ slope: ( 6 14 ) 2. Determine pressure coefficient : For a partially constructed bridge cons isting of multiple girders with SIP forms ( and possibly overhang formwork) determine the pressure coefficient from Table 6 4 For box girder bridges, the calculation of involves the use of the overhang formwork width (in units of ft.) which is defined in Figure 5 10 3. Determine the pressure coefficient : Compute the pressure coefficient as: ( 6 15 ) where is a reduction factor that takes into account the effects of wind angle : ( 6 16 ) where has units of degrees. PAGE 96 96 4. Compute the design wind pressure Per current FDOT practice compu te the design wind pressure using Eqn. ( 5 9 ) repeated here for convenience: ( 6 17 ) where P Z is the design wind pressure (ksf), K Z is the velocity pressure exposure coefficient, V is the basic wind speed (mph), G is the gust effect factor and t he constant term 2.56 10 6 is in units of (ksf)/(mph) 2 [For additional details, see the FDOT Structures Desig n Guidelines ( FDOT, 201 3)] 5. Apply the design wind pressure over the projected area of the structure. To compute the projected area for box girder bridges, the projected depth ( ) should be determined as indicated in Figure 6 21 To compute the projected area for I shaped girders, the following definition of projected depth ( ) should be used: ( 6 18 ) where D is the girder depth, n is the number of girders, S is the girder spacing, and is the overhang formwork width (in units of ft.) as defined in Figure 5 10 6 3 4 Alternate P rocedure for Calculation of Wind L oads for I shaped G irders For bridges constructed from I shaped girders, th e following alternate procedure for computing lateral wind loads implicitly accounts for the effects of variable wind angles (but without the need for explicitly quantifying ). 1. Establish Set the maximum slope: 2. Determine pressure coefficient For a partially constructed bridge consisting of multiple I shaped girders with SIP forms ( and possibly overhang formwor k) determine the pressure coefficient from Table 6 5 3. Determine the pressure coefficient Set the pressure coefficient: ( 6 19 ) 4. Compute the design wind pressure Per current FDOT practice compute the design wind pressure using Eqn. ( 5 9 ) repeated here for convenience: PAGE 97 97 ( 6 20 ) where P Z is the design wind pressure (ksf), K Z is the velocity pressure exposure coefficient, V is the basic w ind speed (mph), G is the gust effect factor and t he constant term 2.56 10 6 is in units of (ksf)/(mph) 2 [For additional details, see the FDOT Structures Design Guidelines ( FDOT, 201 3)] 5. Apply the design wind pressure over the projected area of the structure. To compute the projected area use the following definition of proje cted depth ( ): ( 6 21 ) where D is the girder depth, n is the number of girders, S is the girder spacing, and is the overhang formwork width (in units of ft.) as defined in Figure 5 10 6 4 Assessment of Brace F or ces due to Wind L oads In the previous section, a methodology was developed for computing the global pressure (drag) coefficient and associated global wind load on a multiple girder bridge cross section with SIP forms (and possibly overhang formwork) in place. Such wind lo ads will typically be used in a global strength limit state evaluation for the determination of wind load reactions on the substructure. However, applied wind loads will also induce forces in the individual brace components (diagonals and horizontal elemen ts) and, as such, have the potential to affect the brace design process. It was therefore important to determine how the magnitudes of brace forces caused by wind loads compared to those caused by construction gravity loads (e.g., eccentric construction lo ads, etc.). If wind load induced brace forces were consistently smaller in magnitude than construction load induced brace forces, then there would be no need to formulate wind pressure coefficients specifically for use in designing the bracing elements whe n SIP forms are present. To carry out this assessment, structural analysis models of bracing systems were created for purposes of analyzing brace forces due to wind loads. Two dimensional (2 D) bracing PAGE 98 98 models were analyzed by adapting the modeling methodol ogy previously developed in BDK75 977 33 (see Section 5.6 of Consolazio et al. 2013). In the most windward bracing panel (i.e., between girders G1 and G2), the largest differences between the G1 and G2 drag coefficients ( ) were observed. Since such conditions produce the most severe wind induced brace forces, only braces forces in this panel were compared to forces induced by construction loads. As previously noted (e.g., in Section 6 2 3 ), wind tunnel test ing revealed that the addition of overhang formwork significantly increase d the wind induced magnitude of torque on the windward girder (G1) However, for every wind angle tested, the direction of wind induced torque ( related to ) on the overhang was found to be in opposition to the torque (i.e., moment) produced on the overhang by the downward acting gravity (self weight) of the overhang formwork (denoted ). Hence, for all feasible sc enarios, the effect of including wind induced overhang torque would be to reduce (i.e., offset) the effect that moment would have on the development of brace forces. Consequently, for simplicity and conservatism, the worst case l oading condition for brace force assessment was taken as the moment acting without any reduction attributable to wind induced torque. That is, wind induced torque was conservatively omitted. [Note that because the bottom of each overhang bracket bears against (makes contact with) the bottom flange of the exterior girder ( recall Section 3 4 ), but is not struc turally connected to the girder, it is not possible for wind induced torque to exc eed and cause a net increase in torque (and associated brace force). Instead, the maximum effect that wind induced torque can achieve is to fully cancel out .] Consequently, in the simplified 2 D model s used to assess brace forces, the maximum differences in wind tunnel measured drag coefficients ( ) for G1 and G2 were converted into an equivalent maximum PAGE 99 99 horizontal wind force which was simultaneously applied to th e model in conjunction with the maximum gravity induced overhang moment To be particularly conservative in including the effects of on brace forces the overhang formwork dead load (causing ) w as assumed to be 20 psf (rather than the 10 psf value noted earlier in Table 3 1 ), and only worst case geometric configurations were considered. Maximum feasible braced lengths were chosen for B wind loads ( drag forces) for application to the structural analysis models by multiplying them by a tributary length equal to one half the maximum feasible girder span length. Additionally, a SDG (FDOT 2013) allows a reduction factor (R e ) of 0.6 to be applied for structures with an exposure speed in Florida (150 mph), and applying the reduction factor, a construction design wind speed of 90 mph was obtained and used in all analyses. Maximum brace forces pro duced by the combined application of wind induced drag force and maximum feasible overhang moment ( ) were found to be smaller than the maximum brace forces caused by the application of the full set of construction loads listed in Table 3 1 That is, construction loads not wind loads w ere found to produce brace forces that would control the brac e design process. Therefore, pressure coefficients for use specifically in determining wind induced b race forces were not developed in this study. Instead, it is recommended that construction loads be considered the controlling load case for the design of construction stage girder bracing when SIP formwork is in place. PAGE 100 100 Table 6 1 Estimated lift coefficients ( C L,OHF ) attributable to overhang formwork Girder Min. across all wind angles Max. across all wind angles at zero degree wind angle Width of overhang formwork ( ) (see Figure 5 10 ) WF plate 1.32 1.51 1.51 3.66 ft 0.48 1.43 1.38 3.00 ft 0.12 1.80 1.68 3.00 ft Box 0.15 1.55 1.55 4.33 ft Table 6 2 Pressure coefficient during construction for I shaped girders (FDOT SDG) Construction Con dition Pressure Coefficient Deck forms not in place = 2.2 Stay in place (SIP) deck forms in place = 1.1 Table 6 3 Pressure coefficient during construction for Box girders (FDOT SDG) Construction Condition Pressure Coefficient Deck forms not in place = 1.5 Stay in place (SIP) deck forms in place = 1.1 Table 6 4 Recommended p ressure coefficient for bridges during construction Component type Pressure Coefficient I shaped girders with SIP formwork = 1.4 Box girders with SIP formwork Table 6 5 Alternate p ressure coefficient for bridges during construction Component type Pressure Coefficient I shaped girders with SIP f ormwork = 1.8 PAGE 101 101 Figure 6 1 Influence on plate girder C D values from addition of SIP forms and overhangs (All tested wind angles shown) Figure 6 2 Influence on FIB78 C D values from addition of SIP forms and overhangs (All tested wind angles shown) PAGE 102 102 Figure 6 3 Influence on plate girder C D values from addition of SIP f orms (All tested wind angles shown) Figure 6 4 Influence on FIB78 C D values from addition of SIP forms (All tested wind angles shown) PAGE 103 103 Figure 6 5 Influence on plate girder C D values from addition of overhangs (All tested wind angles shown) Figure 6 6 Influence on FIB78 C D values from addition of overhangs (All tested wind angles shown) PAGE 104 104 Figure 6 7 Comparison of WF plate girder torque coefficients (All data are for zero degree wind angle) Figure 6 8 Comparison of FIB78 girder torque coefficients (All data are for zero degree wind angle) PAGE 105 105 Figure 6 9 Projected area method Figure 6 10 Modified projected area method PAGE 106 106 Figure 6 11 Drag coefficients for wide flange plate girder systems (zero degree wind angle) Figure 6 12 Drag coefficients for FIB78 systems (zero degree wind angle) PAGE 107 107 Figure 6 13 Drag coefficients for FIB45 systems (zero degree wind angle) F igure 6 14 Conservatism of modified projected area calculation procedure for I shaped girders PAGE 108 108 Figure 6 15 Upper bound formula tion of reduction factor ( ) for FIB systems Figure 6 16 Conservatism of modified projected area calculation procedure for I shaped girders PAGE 109 109 Figure 6 17 Conservatism of alternative projected area c alculation procedure for I shaped girders Figure 6 18 Conservatism of alternative projected area calculation procedure for I shaped girders ( C P,SIPF = 1.8 and wind angle not included in cal culation of maximum difference angle; i.e., max = ( cross slope )) PAGE 110 110 Figure 6 19 Equivalence of box girder cross slope and wind angle Figure 6 20 Measured global pressur e coefficients for box girders PAGE 111 111 A B Figure 6 21 Determination of projected depth for box girder bridges. A) without overhang formwork. B ) with overhang formwork. Figure 6 22 Conservatism of projected area calculation procedure for box girders (Using current FDOT C P,SIPF = 1.1 and C P,SIPF+OHF = 1.1) PAGE 112 112 Figure 6 23 Conservatism of projected area calculation procedure for box girders ( Using proposed C P,SIPF = 1. 2 and C P,SIPF+OHF = 1.5 ) Figure 6 24 Upper bound formulation of a reduction factor ( ) for box girders PAGE 113 113 Figure 6 25 Conservatism of projected area calculation procedure for box girders (reduction factor ( ) included) PAGE 114 114 CHAPTER 7 SU MMARY, CONCLUSIONS AND RECOMMENDATIONS 7 1 Overview In this study, issues relating to the application of construction gravity loads and lateral wind loads, to bridges under construction, were investigat ed. For construction gravity loads, a brace force prediction methodology was developed. For lateral wind loads, drag coefficients were measured using wind tunnel testing and a methodology for computing global pressure coefficients and applying the associat ed lateral wind pressures to bridges under construction was developed. 7 2 Brace Forces due to Construction L oads Numerical finite element bridge models and analysis techniques were develo ped for evaluating brace forces induced by construction loads acting on precast concrete girders (Florida I Beams) in systems of multiple girders braced together. Construction loads considered in this study included: wet concrete deck load, stay in place ( SIP) form weight, overhang formwork weight, live load, worker line loads, and concentrated loads representing a deck finishing machine. Preliminary limited scope sensitivity studies indicated that brace forces were not particularly sensitive to bridge gra de, bridge cross slope, girder camber, or girder sweep; therefore, variations of these parameters were not included in subsequent parametric analyses. Additional sensitivity studies indicated that the typical configuration of K brace recommended for use th e FDOT generally produced marginally larger brace forces than did three alternative K brace configurations. Consequently, for brace force determination purposes, only the FDOT recommended K brace configuration was used in the remainder of the study. PAGE 115 115 A larg e scale parametric study, involving more than 450,000 separate three dimensional structural analyses, was performed to compute maximum brace forces for un factored (service) and factored (strength) construction load conditions. Maximum end span brace force s and intermediate span brace forces quantified from the parametric study were stored into a database. The parametric study included consideration of different Florida I Beam cross sections, span lengths, girder spacings, deck overhang widths, skew angles, number of girders, number of braces, and bracing configurations (K brace and X brace). Additionally, partial coverage of wet (non structural) concrete load and variable placement of deck finishing machine loads were considered. To make the process of acce ssing and interpolating the brace force database simple and user friendly, a MathCad based program was developed that employed automated data retrieval (from the database) and multiple dimensional linear interpolation. The accuracy of the database interpol ation approach to brace force prediction was found to be suitable for use in design (less than ten percent (10%) error was present in a majority of verification cases assessed). It is therefore recommended that the brace force database and database interpo lation program developed in this study be deployed as a methodology for computing brace forces for bracing design in bridges under construction. 7 .3 Wind Pressure Coefficients and Corresponding Lateral L oads Wind tunnel te sting was used to quantify wind load coefficients (drag, torque, and lift) for systems of multiple bridge girders (FIB, plate girder, and box) with stay in place (SIP) forms and overhang formwork in place. Tests were conducted at multiple wind angles, and corresponding tests with and without overhang formwork were conducted so that the effects of overhang formwork on drag, lift, and torque coefficients could be quantified. PAGE 116 116 Wind tunnel tests indicated that adding SIP forms to systems of bare girders [as were investigated in a previous study (BDK75 977 33, Consolazio et al., 2013)] had only an incremental influence on individual girder drag coefficients, rather than fundamentally changing the distribution of drag coefficients across the bridge. However, it was found that adding overhang formwork significantly increased the wind induced torque on the windward girder. Additionally, by making use of lift force data measured for the windward girder, estimates of uplift forces acting on overhang formwork were produc ed. Drag coefficients measured at each girder position in bridges with I shaped girders, and in bridges with box girders, were used to develop conservative methods for computing global (system) pressure coefficients suitable for use in bridge design (parti cularly, for use in calculating global lateral substructure load due to wind). The developed methodology involves computing global pressure coefficients (using newly proposed values and expressions), computing design wind pressures (using established FDOT methods), and then applying the computed wind pressure to the projected area of the bridge using a newly proposed definition of projected bridge depth. Finally, by comparing brace forces (note: not global substructure forces) caused by construction loads t o brace forces caused primarily by wind load, it was found that the construction loads produced significantly larger brace forces and would therefore be very likely to control the design of bracing systems and bracing elements. PAGE 117 117 APPENDIX A CROSS SECTIONAL PROPERTIES OF FLORIDA I BEAMS In this study, finite element models Florida I Beams (FIBs) were analyzed to evaluate temporary brac ing forces caused by construction loads In each model, the FIBs were modeled using warping beam elements a specialized beam element available in the ADINA finite element code, which require the calculation of a comprehensive set of cross sectional properties. This appendix provides mathematical definitions of all such properties and corresponding numeric values that were calcu lated for each FIB cross sectional shape. Definitions of the cross sectional properties that are required to use the warping beam element in ADINA are listed in Table A 1 Each property requires the evaluati on of an integral over the area of the cross section, in which the integrands are written in terms of coordinates x and y, referenced to the geometric centroid of the section (Figure A 1 ). Some properties also require knowledge of the warping function ( x,y ), which represents the torsionally induced out of plane warping displacements per rate of twist at every point on the cross section. (The units of are therefore in/(rad/in) or in 2 .) For general cross sectional shapes (e.g., an FIB), analytical (clo sed form) solutions for ( x,y ) do not exist; instead the warping field ( x,y ) must be solved numerically. In this study, the calculation of ( x,y ) for each FIB shape was accomplished by discretizing the cross sectional shape into a high resolution mesh of thousands of two dimensional triangular elements, and then employing a finite element approach to solve the governing differential equation. In general, solutions for ( x,y ) change depending on the assumed location of the center of twist. In the literature in Table A 1 ) corresponding to a state of pure torsion i.e., torsion about the shear center. As a result, prior knowledge of the lo cation of the shear center is required to compute several of the PAGE 118 118 warping beam properties. However, it is possible to calculate the coordinates of the shear center, x s and y s (Table A 1 ), using an alternative solution to the warping function ( c ) where the center of twist is assumed to be located at the centroid of the section. Therefore, two different warping functions were computed for each FIB section: first the section centroid was used to compute c and then the location of the shear cent er, obtained from c was used to compute as well as the remaining cross sectional properties. Because all FIB cross sections are symmetric about the y axis, I xy x s I xr and I have a value of zero (0) by definition. The remaining cross section al prop erties calculated for each FIB shape are summarized in Table A 2 PAGE 119 119 Table A 1 Definitions of cross sectional properties required for use of a warping beam element Property Integral form Units Description A in 2 Cross sectional area I yy in 4 Strong axis moment of inertia I xx in 4 Weak axis moment of inertia I xy in 4 Pr oduct of inertia x s in X coordinate of shear center y s in Y coordinate of shear center J in 4 St. Venant torsional constant C in 6 Warping cons tant I xr in 5 Twist/strong axis bending coupling term I yr in 5 Twist/weak axis bending coupling term I in 6 Twist/warping coupling term I rr in 6 W agner constant Table A 2 Cross sectional properties of Florida I Beams Section A (in 2 ) I yy (in 4 ) I xx (in 4 ) y s (in) J (in 4 ) C (in 6 ) I yr (in 5 ) I rr (in 6 ) 81 283 3.00 30,864 81 540 3.46 31,885 81 798 3.81 32,939 82 055 4.07 33,973 1059 82 314 4.27 35,041 1101 82 484 4.38 35,693 1,314,600,000 1143 1,087,800 82 657 4.46 36,421 104,350,000 10,504,000 1,781,400,000 FIB 1227 1,516,200 83,002 4.56 37,859 142,280,000 15,336,000 3,107,900,000 PAGE 120 120 Figure A 1 Coordinate system used in the calculation of cross sectional properties PAGE 121 121 APPENDIX B DIMENSIONED D RAWINGS OF WIND TUNNEL TEST CONFIGURATIONS This appendix includes dimensioned drawings of every girder configuration that was subjected to wind tunnel testing. PAGE 122 122 PAGE 123 123 PAGE 124 124 PAGE 125 125 PAGE 126 126 PAGE 127 127 PAGE 128 128 PAGE 129 129 APPENDIX C TABULATED RESULTS FROM WIND TUNNEL TESTS This appendix contains results fro m all of the wind tunnel tests that were performed, including drag, lift, and torque coefficients. The wind tunnel testing scope is given in Table C 1 Note that the wind coefficients in this appendix are converted to measur ements at the girder centroid. Results for each test configuration are given an ID code consisting of a letter and a number. The letter describes the cross section of the girders, and the number indicates if overhangs were included A second number followi ng a dash is the girder being measured. For example, the designation refers to the fifth (5 ) in a configuration with overhangs (indicated by 2) in a group of ten (10) Table C 1 Summary of wind tunnel test s Configuration Name Section Overhangs included Cross slope Spacing (ft) Number of girders Instrumented girder position A1 2% 13 10 1,2,3 A2 Yes 2% 13 10 1,2,3,5,10 B1 Box 0% 22 2 1,2 B2 Box Yes 0% 22 2 1,2 C1 WF Plate 8% 14 5 1,2,3 C2 WF Plate Yes 8% 14 5 1,2,3,4,5 D1 2% 13 5 1,2 D2 Yes 2% 13 5 1,2,3 PAGE 130 130 Testing Configuration A1 Cross section: Spacing: 13 ft Cross slope: 2% Overhangs : No Number of girders : 10 Instrumented gi rders: 1, 2, 3 Drag coefficient ( C D ) A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 Wind Angle 1.86 0.02 0.13 1.66 0.10 0.32 +0. 0 1.47 0.42 0.17 + 2.5 1. 14 0.52 0.08 + 5.0 0.90 0.31 0.09 Lift coefficient ( C L ) A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 Wind Angle 0.34 1.96 1.91 0.23 0.28 0.68 +0. 0 0.69 1.07 1.53 + 2.5 1.20 2.12 2.50 + 5.0 1.44 2.69 2.70 Tor que coefficient ( C T ) A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 Wind Angle 0.41 0.10 0.05 0.11 0.14 0.07 +0. 0 0.10 0.18 0.01 + 2.5 0.36 0.23 0.01 + 5.0 0.48 0.20 0.04  PAGE 131 131 Testing Configuration A2 Cross section: Spacing: 13 ft Cross slope: 2% Overhangs : Yes Number of girders : 10 Instrumented girders: 1, 2, 3, 5, 10 Drag coefficient ( C D ) A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 Wind Angle 1.90 0.01 0.10 0.18 0.16 1.76 0.05 0.20 0.10 0.21 +0. 0 1.61 0.11 0.42 0.06 0.27 + 2.5 1.43 0.46 0.12 0.03 0.34 + 5.0 1.25 0.53 0.06 0.01 0.46 Lift coefficient ( C L ) A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 Wind Angle 0.14 2.16 1.95 1.18 0.06 1.47 0.90 1.10 0.05 0.04 +0. 0 2.07 0.56 1.03 0.62 0.04 + 2.5 2.64 1.59 2.48 1.50 0.02 + 5.0 2.85 2.05 2.75 2.41 0.24 Torque coefficient ( C T ) A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 Wind Angle 1.47 0.01 0.03 0.04 0.11 1.11 0.18 0.05 0.08 0.14 +0 0 0.98 0.01 0.11 0.09 0.15 + 2.5 0.82 0.26 0.07 0.05 0.13 + 5.0 0.67 0.28 0.03 0.04 0.17 PAGE 132 132 Testing Configuration B1 Cross section: Box Spacing: 22 ft Cross slope: 0% Overhangs : No Number of girders : 2 Instrumented girders: 1, 2 Drag coefficient ( C D ) B1 B1 2 Wind Angle 1.68 0.05 1.74 0.20 1.71 0.52 1.67 0.28 10.0 1.55 0.22 Lift coefficient ( C L ) B1 B1 2 Wind Angle 0.98 1.34 0.01 1.54 1.04 0.97 1.19 0.88 10.0 1.16 0.97 Torque coefficient ( C T ) B1 B1 2 Wind Angle 1.82 0.71 1.56 0.97 0.64 0.60 0.53 0.29 10.0 0.40 0.36 PAGE 133 133 Testing Configuration B2 Cross section: Box Spacing: 22 ft Cross slope: 0% Overhangs : Yes Number of girders : 2 Instrumented girders: 1, 2 Drag coefficient ( C D ) B B Wind Angle 1.73 0.01 1.80 0.08 1.80 0.32 1.92 0.55 10.0 1.88 0.06 Lift coefficient ( C L ) B B Wind Angle 0.83 1.98 0.71 1.77 2.59 1.34 2.08 0.39 10.0 1.89 0.75 Torque coefficient ( C T ) B B Wind Angle 2.76 0.35 3.54 0.41 2.70 0.42 2.75 0.08 10.0 2.58 0.07 PAGE 134 134 Testing Configuration C1 Cr oss section: WF Plate Spacing: 1 4 ft Cross slope: 8 % Overhangs: No Num. of girders 5 Instrumented girders: 1, 2, 3 Drag coefficient ( C D ) C1 C1 C1 C1 C1 Wind Angle 1.97 0.12 0.40 1.88 0.34 0.25 + 0. 0 1.78 0.47 0.12 + 2.5 1.68 0.47 0.25 + 5.0 1.55 0.42 0.39 Lift coefficient ( C L ) C1 C1 C1 C1 C1 Wind Angle 0.08 0.06 0.39 0.30 0.57 1.08 +0. 0 0.38 0.91 1.49 + 2.5 0.39 1.01 1.53 + 5.0 0.45 1.31 1.50 Torque coefficient ( C T ) C1 C1 C1 C1 C1 Wind Angle 0.00 0.00 0.03 0.08 0.09 0.01 +0. 0 0.12 0.13 0. 08 + 2.5 0.12 0.13 0.09 + 5.0 0.15 0.11 0.13  PAGE 135 135 Testing Configuration C2 Cross section: WF Plate Spacing: 1 4 ft Cross slope: 8 % Overhangs: Yes Num. of girders 5 Instrumented girde rs: 1, 2, 3, 4, 5 Drag coefficient ( C D ) C2 C2 C2 C2 C2 Wind Angle 2.20 0.07 0.30 0.34 0.02 2.12 0.09 0.40 0.13 0.25 +0. 0 2.06 0.28 0.30 0.28 0.40 + 2.5 2.02 0.41 0.01 0.41 0.50 + 5.0 1.95 0.46 0.17 0.4 6 0.59 Lift coefficient ( C L ) C2 C2 C2 C2 C2 Wind Angle 1.44 0.69 1.30 0.85 0.11 1.76 0.33 0.13 0.12 0.06 +0. 0 1.88 0.67 1.27 1.02 0.27 + 2.5 1.79 0.81 1.52 1.24 0.37 + 5.0 1.76 0.93 1.55 1.24 0.41 Torque coefficient ( C T ) C2 C2 C2 C2 C2 Wind Angle 0.83 0.10 0.09 0.10 0.05 0.79 0.03 0.04 0.03 0.05 +0. 0 0.75 0.09 0.02 0.16 0.12 + 2.5 0.72 0.13 0.03 0.16 0.14 + 5 .0 0.69 0.14 0.06 0.14 0.15 PAGE 136 136 Testing Configuration D1 Cross section: Spacing: 13 ft Cross slope: 2% Overhangs: No Num. of girders 5 Instrumented girders: 1, 2 Drag coefficient ( C D ) D1 D1 D1 D1 D1 Wind Angle 1.65 0.11 1.51 0.23 +0. 0 1.33 0.38 + 2.5 1.05 0.22 + 5.0 0.89 0.01 Lift coefficient ( C L ) D1 D1 D1 D1 D1 Wind Angle 1.09 2.78 0.23 1.60 +0. 0 0.95 1.02 + 2.5 1.68 2.86 + 5.0 1.93 3.21 Torque coefficient ( C T ) D1 D1 D1 D1 D1 Wind Angle 1 .44 0.07 0.25 0.07 +0. 0 0.36 0.06 + 2.5 0.89 0.32 + 5.0 1.14 0.17  PAGE 137 137 Testing Configuration D2 Cross section: Spacing: 13 ft Cross slope: 2% Overhangs: Yes Num. of girders 5 Instrumented girders: 1, 2, 3 Drag coefficient ( C D ) D2 D2 D2 D2 D2 Wind Angle 1.69 0.13 0.18 1.66 0.18 0.20 +0. 0 1.59 0.28 0.16 + 2.5 1.49 0.32 0 .15 + 5.0 1.41 0.38 0.36 Lift coefficient ( C L ) D2 D2 D2 D2 D2 Wind Angle 0.97 2.96 2.13 0.90 2.34 1.20 +0. 0 2.63 0.58 0.16 + 2.5 3.48 1.69 1.91 + 5.0 3.64 2.30 2.86 Torque coefficient ( C T ) D2 D2 D2 D2 D2 Wind Angle 3.03 0.01 0.05 3.48 0.18 0.09 +0. 0 2.59 0.19 0.16 + 2.5 2.27 0.31 0.39 + 5 .0 2.06 0.55 0.47  PAGE 138 138 LIST OF REFERENCES AASHTO ( American Association of State Highway and Transportation Officials) (20 08 ). Guide Design Specification for Bridge Temporary Works AASHTO, Washington, D.C AASHTO ( Ameri can Association of State Highway and Transportation Officials) (2010 ). LRFD Bridge Design Specifications: 5 th Edition AASHTO, Washington, D.C ADINA (2012 ). Theory and Modeling Guide, Volume 1: ADINA Solids & Structures ADINA R&D, Inc Watertown, MA ASC E ( American Society of Civil Engineers) (2006 ). ASCE 7 05: Minimum Design Loads for Buildings and Other Structures ASCE, New York NY engel, Y. and Cimbala, J. (2006 ). Fluid Mechanics: Fundamentals and Applications McGraw Hill Higher Education, Boston MA Clifton, S. and Bayrak O. (2008) Bridge Deck Overhang Construction Technical Report IAC: 88 5DD1A003 2, University of Texas, Austin, TX. Consolazio, G., Gurley K ., and Harper Z (201 3 ). Bridge Girder Drag Coefficients and Wind Related Bracing Rec ommendations Structures Research Report No. 2013/87322 University of Florida, Gainesville, F L FDOT (Florida Department of Transportation) (2009 ). FDOT, Tallahassee, FL FDOT (Florida Department of Transportation) (2010 ) Standard Specifications for Road and Bridge Construction FDOT, Tallahassee, FL. FDOT (Florida Dep artment of Transportation) ( 2013 ). Structures Manual Volume I: Structures Design Guidelines FDOT, Tallahassee, FL. FDOT (Florida Department of Transportati on) ( 2014 a ). Design Standard No. 20005: Prestressed I Beam Temporary Bracing FDOT, Tallahassee, FL. FDOT (Florida Department of Transportation) ( 2014b ). Instructions for Design Standard No. 20010: Prestressed Florida I Beams FDOT, Tallahassee, FL. FDOT ( Florida Department of Transporta tion) ( 2014c ). Design Standard No. 20510: Composite Elastomeric Bearing Pads Prestressed Florida I Beams FDOT, Tallahassee, FL. Holmes, J. (2007 ). Wind Loading of Structures: 2 nd Edition Taylor & Francis, New York NY P CI (2010 ). PCI Design Handbook: 7 th Edition Precast/Prestressed Concrete Institute, Chicago, IL. Solari G. and Kareem A. (1998 ). Journal of Wind Engineering and Industrial Aerodynamics Vol. 77, pp. 673 684. PAGE 139 139 Young, W. C. and Budynas, R. G. (2002 ). : 7 th Edition McGraw Hill, New York NY PAGE 140 140 BIOGRAPHICAL SKETCH Sam Edwards was born in Amelia Island, Florida, in 1989. In August 2007, he started his career at the Uni versity of Florida, where he received the de gree of Bachelor of Science in civil e ngineering i n May 2012. He then enrolled in graduate school at the University of Florida where he received a Master of Engineering in c ivil e ngineering in May 2014, with an e mphasis in civil structures. 