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BridgeRatingUsingSystemReliabilityAssessment.IIImprovementstoBridgeRatingPracticesNaiyuWang,M.ASCE1BruceR.Ellingwood,Dist.M.ASCE2andAbdulHamidZureick,M.ASCE3AbstractThecurrentbridgeratingprocessdescribedinAASHTOManualforBridgeEvaluation,FirstEditionpermitsratingstobedeterminedbyallowablestress,loadfactor,orloadandresistancefactormethods.Thesethreeratingmethodsmayleadtodifferentratedcapacitiesandpostinglimitsforthesamebridge,asituationthathasseriousimplicationswithregardtopublicsafetyandtheeconomicwellbeingofcommunitiesthatmaybeaffectedbybridgepostingsorclosures.Thispaperisthesecondoftwopapersthatsummarizearesearchprogramtodevelopimprovementstothebridgeratingprocessbyusingstructuralreliabilitymethods.Thefirstpaperprovidedbackgroundontheresearchprogramandsummarizedacoordinatedprogramofloadtestingandanalysistosupportthereliabilityassessmentleadingtotherecommendedimprovements.Thissecondpaperpresentsthereliabilitybasisfortherecommendedloadrating,developsmethodsthatcloselycoupletheratingprocesstotheresultsofinsituinspectionandevaluation,andrecommendsspecificimprovementstocurrentbridgeratingmethodsinaformatthatisconsistentwiththeloadandresistancefactorratingLRFRoptionintheAASHTOManualforBridgeEvaluation.DOI10.1061/ASCEBE.19435592.0000171.©2011AmericanSocietyofCivilEngineers.CEDatabasesubjectheadingsConcretebridgesReinforcedconcretePrestressedconcreteLoadfactorsReliabilitySteelRatings.AuthorkeywordsBridgesratingConcretereinforcedConcreteprestressedConditionassessmentLoadsforcesReliabilitySteelstructuralengineering.IntroductionTheAASHTOManualforBridgeEvaluationMBE,FirstEditionAASHTO2008allowsbridgeratingstobedeterminedthroughthetraditionalallowablestressratingASRorloadfactorratingLFRmethodsorbythemorerecentloadandresistancefactorratingLRFRmethod,whichisconsistentwiththeAASHTOLRFDBridgeDesignSpecifications2007.ThesethreeratingmethodsmayleadtodifferentratedcapacitiesandpostedlimitsforthesamebridgeNCHRP2001Wangetal.2009,asituationthatcannotbejustifiedfromaprofessionalengineeringviewpointandhasimplicationsforthesafetyandeconomicwellbeingofthoseaffectedbybridgepostingsorclosures.Toaddressthisissue,theGeorgiaInstituteofTechnologyhasconductedamultiyearresearchprogramaimedatmakingimprovementstotheprocessbywhichtheconditionofexistingbridgestructuresinGeorgiaareassessed.TheendproductofthisresearchprogramissetofrecommendedguidelinesfortheevaluationofexistingbridgesEllingwoodetal.2009.Theseguidelinesareestablishedbyacoordinatedprogramofloadtestingandadvancedfiniteelementmodeling,whichhavebeenintegratedwithinastructuralreliabilityframeworktodeterminepracticalbridgeratingmethodsthatareconsistentwiththoseusedtodeveloptheAASHTOLRFDBridgeDesignSpecificationsAASHTO2007.Itisbelievedthatbridgeconstructionandratingpracticesaresimilarenoughinothernonseismicareastomaketheinferences,conclusions,andrecommendationsvalidforlargeregionsinthecentralandeasternUnitedStatesCEUS.TherecentimplementationofLRFDanditscompanionratingmethod,LRFR,bothofwhichhavebeensupportedbystructuralreliabilitymethods,enablebridgedesignandconditionassessmenttobeplacedonamorerationalbasis.Notwithstandingtheseadvances,improvedtechniquesforevaluatingthebridgeinitsinsituconditionwouldminimizethelikelihoodofunnecessaryposting.Forexample,materialstrengthsinsitumaybevastlydifferentfromthestandardizedornominalvaluesassumedindesignandcurrentratingpracticesattributabletostrengthgainofconcreteononehandanddeteriorationattributabletoaggressiveattackfromphysicalorchemicalmechanismsontheother.Satisfactoryperformanceofawellmaintainedbridgeoveraperiodofyearsofserviceprovidesadditionalinformationnotavailableatthedesignstagethatmightbetakenintoaccountinmakingdecisionsregardingpostingorupgrading.Investigatingbridgesystemreliabilityratherthansolelyrelyingoncomponentbasedratingmethodsmayalsobeofsignificantbenefit.Properconsiderationofthesefactorsislikelytocontributetoamorerealisticcapacityratingofexistingbridges.ThispaperisthesecondoftwocompanionpapersthatprovidethetechnicalbasesforproposedimprovementstothecurrentLRFRpractice.ThefirstpaperWangetal.2011summarizedthecurrentbridgeratingprocessandpracticesintheUnitedStates,andpresentedtheresultsofacoordinatedbridgetestingandanalysisprogramconductedtosupportrevisionstothecurrentratingprocedures.ThispaperdescribesthereliabilityanalysisframeworkthatprovidesthebasisforrecommendedimprovementstotheMBEandrecommendsspecificimprovementstotheMBEthataddresstheprecedingfactors.1SeniorStructuralEngineer,Simpson,Gumpertz,andHeger,Inc.,41SeyonSt.,Waltham,MA02453formerly,GraduateResearchAssistant,SchoolofCivilandEnvironmentalEngineering,GeorgiaInstituteofTechnology.2Professor,SchoolofCivilandEnvironmentalEngineering,GeorgiaInstituteofTechnology,790AtlanticDr.,Atlanta,GA303320355correspondingauthor.Emailellingwoodgatech.edu3Professor,SchoolofCivilandEnvironmentalEngineering,GeorgiaInstituteofTechnology,790AtlanticDr.,Atlanta,GA303320355.Note.ThismanuscriptwassubmittedonMarch19,2010approvedonAugust2,2010publishedonlineonOctober14,2011.DiscussionperiodopenuntilApril1,2012separatediscussionsmustbesubmittedforindividualpapers.ThispaperispartoftheJournalofBridgeEngineering,Vol.16,No.6,November1,2011.©ASCE,ISSN10840702/2011/6863–871/25.00.JOURNALOFBRIDGEENGINEERING©ASCE/NOVEMBER/DECEMBER2011/863Downloaded21Mar2012to180.95.224.53.RedistributionsubjecttoASCElicenseorcopyright.Visithttp//www.ascelibrary.orgReliabilityBasesforBridgeLoadRatingBridgedesign,ascodifiedintheAASHTOLRFDspecifications2007,isestablishedbymodernprinciplesofstructuralreliabilityanalysis.Theprocessbywhichexistingbridgesareratedmustbeconsistentwiththoseprinciples.Uncertaintiesintheperformanceofanexistingbridgearisefromvariationsinloads,materialstrengthproperties,dimensions,naturalandartificialhazards,insufficientknowledge,andhumanerrorsindesignandconstructionEllingwoodetal.1982Galambosetal.1982Nowak1999.Probabilitybasedlimitstatesdesign/evaluationconceptsprovidearationalandpowerfultheoreticalbasisforhandlingtheseuncertaintiesinbridgeevaluation.ThelimitstatesforbridgedesignandevaluationcanbedefinedinthegeneralformGðXÞ¼0ð1ÞwhereX¼ðX1X2X3XnÞloadandresistancerandomvariables.Onthebasisofbridgeperformanceobjectives,theselimitstatesmayrelatetostrengthforpublicsafetyortoexcessivedeformation,cracking,wearofthetrafficsurface,orothersourcesoffunctionalimpairment.Astateofunsatisfactoryperformanceisdefined,byconvention,whenGðXÞ0.Thus,theprobabilityoffailurecanbeestimatedasPf¼P½GðXÞ0C138¼ZΩfXðxÞdxð2ÞwherefXðxÞjointdensityfunctionofXandΩfailuredomaininwhichGðxÞ0.InmodernfirstorderFOreliabilityanalysisMelchers1999,Eq.2isoftenapproximatedbyPf¼ΦðC0βÞð3ÞwhereΦðÞstandardnormaldistributionfunctionandβreliabilityindex.Forwellbehavedlimitstates,Eq.3usuallyisanexcellentapproximationtoEq.2,andβandPfcanbeusedinterchangeablyasreliabilitymeasuresEllingwood2000.WhenthefailuresurfaceinEq.1iscomplexorwhenthereliabilityofastructuralsystem,inwhichthestructuralbehaviorismodeledthroughfiniteelementanalysis,isofinterest,Eq.2canbeevaluatedefficientlybyMonteCarloMCsimulation.TheAASHTOLRFDBridgeDesignSpecifications2007areestablishedonFOreliabilityanalysis,appliedtoindividualgirdersNowak1999KimandNowak1997TabshandNowak1991.WiththesupportingprobabilisticmodelingofresistanceandloadtermsNowak1993BartlettandMcGregor1996MosesandVerma1987,anexaminationofexistingbridgedesignpracticesledtoatargetreliabilityindex,β,equalto3.5basedona75yearserviceperiodNowak1999,Moses2001.Consistentwithsuchreliabilitybasedperformanceobjective,theAASHTOLRFDspecificationsstipulatethatinthedesignofnewbridges125Dþ15DAþ175ðLþIÞϕRnð4ÞwhereDdeadloadexcludingweightofthewearingsurfaceDAweightofthewearingsurfaceasphaltLþIrepresentsliveloadincludingimpactϕRndesignstrength,inwhichRnnominalresistanceandϕresistancefactorwhichdependsontheparticularlimitstateofinterest.Thisequationisfamiliartomostdesigners.Whenthereliabilityofanexistingbridgeisconsidered,allowanceshouldbemadeforthespecificknowledgeregardingitsstructuraldetailsandpastperformance.Fieldinspectiondata,loadtesting,materialtests,ortrafficsurveys,ifavailable,canbeutilizedtomodifytheprobabilitydistributionsdescribingthestructuralbehaviorandresponseinEq.2.ThemetricforacceptableperformanceisobtainedbymodifyingEq.2toreflecttheadditionalinformationgatheredPf¼P½GðXÞ0jHC138PTð5ÞwhereHrepresentswhatislearnedfromprevioussuccessfulperformance,inserviceinspection,andsupportinginsitutesting,ifany.Thetargetprobability,PT,shoulddependontheeconomicsofrehabilitation/repair,consequencesoffutureoutages,andthebridgeratingsought.IntheAASHTOLRFRmethod2007,thetargetβfordesignlevelcheckingbyusingHL93loadmodelatinventorylevelis3.5,whichiscomparabletothereliabilityfornewbridges,whereasthetargetβforHL93operatinglevelandforlegal,andpermitloadsisreducedto2.5owingtothereducedloadmodelandreducedexposureperiod5yearsMoses2001.ThepresenceofHinEq.5isaconceptualdeparturefromEqs.2and3,whichprovidethebasisforLRFD.Forexample,trafficdemandsonbridgeslocatedindifferentplacesinthehighwaysystemmaybedifferent.Totakethissituationintoaccount,LRFRintroducesasetofliveloadfactorsforthelegalloadrating,whichdependontheinsitutrafficdescribedbytheaveragedailytrucktrafficADTT.Furthermore,thecomponentnominalresistanceinLRFRisfactoredbyasystemfactorφsandamemberconditionfactorφcinadditiontothebasicresistancefactorφforaparticularcomponentlimitstate.Thesystemfactordependsontheperceivedredundancylevelofagivenbridgeinitsrating,whereastheconditionfactoristoaccountforthebridgessitespecificdeteriorationcondition,andpurportstoincludetheadditionaluncertaintybecauseofanydeteriorationthatmaybepresent.ThebasisfortheLRFRtabulatedvaluesforφcwillbefurtherexaminedlaterinthispaper.TheLRFRoptionintheAASHTOMBEextendsthelimitstatedesignphilosophytothebridgeevaluationprocessinanattempttoachieveauniformtargetlevelofsafetyforexistinghighwaybridgesystems.However,theuncertaintymodelsofloadandresistanceembeddedintheLRFRratingformatrepresenttypicalvaluesforalargepopulationofbridgesinvolvingdifferentmaterials,constructionpractices,andsitespecifictrafficconditions.AlthoughtheLRFRliveloadmodelhasbeenmodifiedforsomeofthespecificcasesasdiscussedpreviously,thebridgeresistancemodelshouldalsobecustomizedforanindividualbridgebyincorporatingavailablesitespecificknowledgetoreflectthefactthateachbridgeisuniqueinitsasbuiltcondition.Aratingprocedurethatdoesnotincorporateinsitudataproperlymayresultininaccurateratingsandconsequentunnecessaryrehabilitationorpostingcostsforotherwisewellmaintainedbridges,asindicatedbymanyloadtestsNowakandTharmabala1988BakhtandJaeger1990Mosesetal.1994FuandTang1995Faberetal.2000Barker2001Bhattacharyaetal.2005.Improvementsinpracticalguidancewouldpermitthebridgeengineertoincludemoresitespecificknowledgeinthebridgeratingprocesstoachieverealisticevaluationsofthebridgeperformance.Thisguidancemusthaveastructuralreliabilitybasis.ImprovementsinBridgeRatingbyUsingReliabilityBasedMethodsInthissection,thebridgeratingsinlightofthereliabilitybasedupdatingofinservicestrengthdescribedintheprevioussectionareexamined.Thepossibilitiesofincorporatingavailablesitespecificdataobtainedfrommaterialtests,loadtests,advanced864/JOURNALOFBRIDGEENGINEERING©ASCE/NOVEMBER/DECEMBER2011Downloaded21Mar2012to180.95.224.53.RedistributionsubjecttoASCElicenseorcopyright.Visithttp//www.ascelibrary.orgstructuralanalysis,andsuccessfulserviceperformancetomakefurtherrecommendationsforimprovingratinganalysisareexplored.IncorporationofInSituMaterialTestingThecompanionpapersummarizedtheloadtestofBridgeID1290045,areinforcedconcreteTbeambridgethatwasdesignedaccordingtotheAASHTO1953designspecificationforH15loadingandwasconstructedin1957.Thespecified28daycompressionstrengthoftheconcretewas17.2MPa2,500psi,whereastheyieldstrengthofthereinforcementwas276MPa40ksi.Thescheduleddemolitionofthisbridgeprovidedanopportunitytosecuredrilledcorestodeterminethestatisticalpropertiesoftheinsitustrengthofthe51yearoldconcreteinthebridge.Fourinchdiameterdrilledcoresweretakenfromtheslabofthebridgebeforeitsdemolition.Sevencoresweretakenfromtheslabatsevendifferentlocationsalongboththelengthandwidthofthebridge.Coresalsoweretakenfromthreeofthegirdersthatwereingoodconditionafterdemolitionthesewerecutinto203mm8in.lengthsandthejaggedendsweresmoothedandcapped,resultinginatotalof14girdertestcylinders.Testsofthese102203mm48in.cylindersconformedtoASTMStandardC42ASTM1995andtheresultsarepresentedinTable1.Ananalysisofthesedataindicatednostatisticallysignificantdifferenceintheconcretecompressionstrengthinthegirdersandslab,andthedatawerethereforecombinedforfurtheranalysis.Themeanaveragecompressionstrengthoftheconcreteis33MPa4,820psiandthecoefficientofvariationCOVis12,whichisrepresentativeofgoodqualityconcreteBartlettandMacGregor1996.Themeanstrengthis1.93timesthespecifiedcompressionstrengthoftheconcrete.Thisincreaseincompressionstrengthoveraperiodofmorethan50yearsistypicaloftheincreasesfoundforgoodqualityconcretebyotherinvestigatorsWashaandWendt1975.Iftheseresultsaretypicalofwellmaintainedolderconcretebridges,theinsituconcretestrengthislikelytobesubstantiallygreaterthanthe28daystrengththatiscustomarilyspecifiedforbridgedesignorconditionevaluation.Accordingly,thebridgeengineershouldbeprovidedincentivesintheratingcriteriatorateabridgebyusingthebestpossibleinformationfrominsitumaterialstrengthtestingwheneverfeasibleEllingwoodetal.2009.Itiscustomarytobasethespecifiedcompressionstrengthofconcreteonthe10thpercentileofanormaldistributionofcylinderstrengthsStandard31805ACI2005.Asuitableestimateforthis10thpercentilebasedonasmallsampleofdataisprovidedbyfc¼C22Xð1C0kVÞð6ÞwhereC22XsamplemeanVsamplecoefficientofvariationandk¼plowerconfidenceintervalonthe10thpercentilecompressionstrength.Byusingthe21testsfromBridgeID1290045withp¼75asanexample,k1.520Montgomery1996andfccanbeexpressedasfc¼ð1–1520012Þ4820¼3941psi27.17MPa,avaluethatis58higherthanthe17.2MPa2,500psithatotherwisewouldbeusedintheratingcalculations.IntheFEmodelingofthisbridgethatprecededthesestrengthtests,theconcretecompressionstrengthwassetat17.2MPa2,500psi,whichwastheonlyinformationavailablebeforethematerialtest.Todeterminetheimpactofusingtheactualconcretestrengthinanolderbridgeontheratingprocess,thefiniteelementmodelwasrevisedtoaccountfortheincreasedconcretecompressionstrengthandthecorrespondingincreaseinstiffnessintotheanalysisofthebridge.Onlyamodestenhancementintheestimatedbridgecapacityinflexurewasobtained,buta34increasewasachievedintheshearcapacityratingsforthegirdersbyusingtheresultsofTable1.BridgeSystemReliabilityAssessmentontheBasisofStaticPushDownAnalysisAlthoughcomponentbaseddesignofanewbridgeprovidesadequatesafetyatreasonablecost,componentbasedevaluationofanexistingbridgeforratingpurposesmaybeoverlyconservativeandresultinunnecessaryrepairorpostingcosts.Itispreferabletoperformloadratingregardingbridgepostingorroadclosurethroughasystemlevelanalysis.Aproperlyconductedproofloadtestcanbeaneffectivewaytolearnthebridgesstructuralperformanceasasystemandtoupdatethebridgeloadcapacityassessmentinsituationsinwhichtheanalyticalapproachproduceslowratings,orstructuralanalysisisdifficulttoperformbecauseofdeteriorationorlackofdocumentationSarafandNowak1998.However,aproofloadtestrepresentsasignificantinvestmentincapital,time,andpersonnel,andthetradeoffbetweentheinformationgainandtheriskofdamagingthebridgeduringthetestmustbeconsidered.ProoftestsarerarelyconductedbythestateDOTsWangetal.2009forratingpurposes.OneofthekeyconclusionsfromthecompanionpaperWangetal.2011,inwhichbridgeresponsemeasurementsobtainedfromtheloadtestsofthefourbridgeswerecomparedwiththeresultsoffiniteelementanalysesofthosebridgeswithABAQUS2006,wasthatthefiniteelementmodelingprocedurewassufficientforconductingvirtualloadtestsofsimilarbridges.Thesevirtualloadtestscanprovidethebasisfordevelopingrecommendationsforimprovingguidelinesforbridgeratingsbyusingstructuralreliabilityprinciples.Asnotedintheintroductorysection,suchguidelinesrequirethebridgetobemodeledasastructuralsystemtoproperlyidentifytheperformancelimitstatesonwhichsuchguidelinesaretobebased.Toidentifysuchperformancelimitstatesandtogainarealisticappraisaloftheconservatisminherentincurrentbridgedesignandconditionratingprocedures,aseriesofstaticpushdownanalysesofthefourbridgeswasperformed.Theseanalysesareaimedatdeterminingtheactualstructuralbehavioroftypicalbridgeswhenloadedwellbeyondtheirdesignlimitasasidelight,theyprovideadditionalinformationtosupportrationalevaluationofpermitloadapplicationssection6A.4.5intheManualofBridgeEvaluation.Inapushdownanalysis,tworatingvehiclesareplacedsidebysideonthebridgeinapositionthatmaximizestheresponsequantityofinterestintheevaluatione.g.,maximummoment,shear,anddeflection.Theloadsarethenscaledupwardstaticallyandtheperformanceofthebridgesystemismonitored.Thedeadweightofthebridgestructureisincludedintheanalysis.Theresponseisinitiallyelastic.Asthestaticloadincreases,however,elementsofthebridgestructurebegintoyield,crack,orbuckle,andthegeneralizedloaddeflectionbehaviorbecomesnonlinear.Ifthebridgestructureisredundantandthestructuralelementbehaviorsareductile,substantialloadredistributionmayoccur.Atsomepoint,however,asmallincrementinstaticloadleadstoalargeincrementindisplacement.Atthatpoint,thebridgehasreacheditspracticalloadcarryinglimit,andisatastateofincipientcollapse.Table1.CompressionTestsof48inCoresDrilledfromRCConcreteBridgeID1290045SourceNumberAveragepsiStandarddeviationpsiCoefficientofvariationGirder144,8806030.12Slab74,6985730.12Overall214,8205860.12Note1psi¼69Pa.JOURNALOFBRIDGEENGINEERING©ASCE/NOVEMBER/DECEMBER2011/865Downloaded21Mar2012to180.95.224.53.RedistributionsubjecttoASCElicenseorcopyright.Visithttp//www.ascelibrary.orgThestaticpushdownanalysisisillustratedinFig.1fortheRCTbeambridgeID1290045.TheFEmodelingwasperformedwithABAQUS2006,withrandommaterialpropertiesdeterminedbytheirrespectivemeanvalues.Thepointofinitialyieldingoccursatapproximately4.31timestheHS2044designloadconfiguration,atadeflectionofapproximately36mm1.4in.,whichisequaltoapproximately1345timesthespan.Theultimateliveloadcapacityofthebridgeisapproximately4.8timestheappliedHS2044loads.FromFig.1,this52yearoldbridgeshowsaconsiderabledegreeofductilityinbehavior.ThelevelofloadimposedbythefourfullyloadedtrucksduringtheloadtestdescribedinthecompanionpaperisalsoshowninFig.1thetestloadinmaximumgirdermomentwasapproximately1.3timesthetwosidebysideHS2044loads.Thecapacityofthisbridgesystemissubstantiallyinexcessofwhatagirderbasedcalculationwouldindicate.Similarpushdownanalyseswereperformedontheotherbridgesdescribedinthecompanionpaper,yieldingtheresultssummarizedinTable2.Theelasticrangesofallfourbridgesareinexcessof4timesthedesignloadlevel,indicatingthelevelofconservatismassociatedwithtraditionaldesignandratingprocedures.AspartoftheefforttodeveloptheAASHTOLRFDBridgeDesignSpecifications,extensivedatabasesweredevelopedtodescribethestrengthofindividualbridgegirdersandvehicleliveloadsprobabilisticallyNowak1999Moses2001.TheHL93liveloadmodelisanoutgrowthofthispreviousresearch.Thatresearchfocusedonthecapacityofindividualbridgegirderssystemeffectswereincludedindirectlyandapproximatelythroughnewgirderdistributionfactorsthatweredevelopedinthecourseoftheproject.Thecapacityofabridgestructuralsystemislikelytobedifferentfromthecapacitypredictedfromananalysisofindividualgirders.Todeterminetheadditionallevelofconservatismifanythatarisesfromsystembehavior,afiniteelementbasedsystemreliabilityanalysisofallfourstudybridgeswasconducted.ThissystemreliabilityanalysisprovidesadditionalperspectiveontheunknownlevelofconservatismfurnishedbythecurrentgenerationofreliabilitybasedconditionevaluationandratingproceduresembodiedintheAASHTOManualforBridgeEvaluation,andhasimplicationsfortheuseofsuchmethodsinpermitratingsforextremevehicleloads.ToacceleratetheFEbasedreliabilityanalysis,efficientFEmodelsofthesamplebridgesweredevelopedwiththeopensourceplatform,OpenSeesVersion2.2.2.ThemoredetailedABAQUSmodels,whichhadbeenvalidatedfromtheloadtestresults,wereemployedtoconfirmthebridgestructuralbehaviorpredictedbytheOpenSeesmodelsasthesystemwasloadedbeyonditsdesignlimit.ByusingtheRCTbeambridgeagainasanexample,Fig.1illustratestheconsistencyachievedbetweentheABAQUS2006andtheOpenSeesmodelsthroughacompletepushdownanalysis,inwhichthebridgeisloadedwellintotheinelasticrange.Followingthisvalidation,thesystemperformanceofthesamplebridgeswascharacterizedstatisticallybypropagatingtheuncertaintiesinmaterialstrengths,stiffnesses,andgeometrythroughtheOpenSeesanalysisbyusingaLatinHypercubeSamplingtechniqueImamandConover1980toachieveefficientcoverageofthesamplespacewitharelativelyfewFEanalyses.TherandomvariablesinvolvedintheseFEanalysestocapturebridgestructuralperformancearedescribedwithstatisticsdefinedintheLRFDdatabasesmentionedpreviously.Thelimitstateofperformancewasassumedasthepointatwhichthebridgesystemexitstheelasticrange,asidentifiedfromitsloaddeflectioncurveseeFig.1.Theflexuralcapacitiessodeterminedfromthissystemreliabilityanalysiswererankorderedandplottedonlognormalprobabilitypaper,asillustratedinFig.2forthestraightapproachRCbridgeID1290045.Thelognormaldistributionprovidesagoodfittothesedata.Themeanandcoefficientofvariationinthesystemcapacityofthisbridgeatfirstyieldare4.31timestheappliedFig.1.PushdownanalysisofRCTbeambridgeID12900451in¼254mmTable2.AnalysisofBridgeCapacityDeterminedasthePointofFirstYieldBridgeIDCountyTypeDesignloadLoadfactorondesignloadLoadfactoronHS2012900450GordonRCTstraightnotpostedH157.464.3101501080BartowRCTskewedpostedHS156.004.5022300340PauldingPrestressedstraightnotpostedHS205.945.9408500180DawsonSteelgirderstraightpostedH159.935.37Fig.2.LognormalfitofthebridgesystemresistanceoftheRCBridgeID1290045866/JOURNALOFBRIDGEENGINEERING©ASCE/NOVEMBER/DECEMBER2011Downloaded21Mar2012to180.95.224.53.RedistributionsubjecttoASCElicenseorcopyright.Visithttp//www.ascelibrary.org
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