Fatigue reliability assessment of ageing railway truss bridges: Rationality of probabilistic stress-life approach

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Case Studies in Structural Engineering

jo u r n al ho me p ag e :ww w . e l s e v i e r . c o m / l o c a t e / c s s e

Fatigue reliability assessment of ageing railway truss bridges:

Rationality of probabilistic stress-life approach

Nirosha D. Adasooriya

DepartmentofStructuralandMechanicalEngineeringandMaterialsScience,UniversityofStavanger,N-4036,Norway

a rt i c l e i n f o

Articlehistory:

Received20February2016 Receivedinrevisedform2April2016 Accepted30April2016

Availableonline3May2016

Keywords:

Railwaybridge Fatiguelife

Loadingsequenceeffect Reliabilityindex Damagestressmodel

a b s t ra c t

Railauthoritiesallovertheworldarepayingattentiontoextendtheservicelivesofrailway bridges.ThefamousMiner’sruleemployeddeterministicorprobabilisticfatigueassess- mentapproachesaregenerallyusedtopredictremainingfatiguelifeofageingrailway bridges.Undermanyvariableamplitudeloadingconditions,lifepredictionshavebeen found tobe unreliablesince Miner’sruledoesnot properlytakeaccounttheloading sequenceeffect.Therefore,thispaperpresentsacomparisonofanewprobabilisticfatigue assessmentapproachwithdeterministicapproachconsistingofanewdamageindicator, whichcapturestheloadingsequenceeffectofvariableamplitudeloadsmoreprecisely thantheMiner’srule.Thecomparisonisperformedbyapplyingbothfatigueassessment approachestopredicttheremainingfatiguelifeofanageingrailwaybridge.Thiscompar- isonintendstoconcludethepossibilityofcapturinguncertaintybehindloadingsequence effectbyproposedprobabilisticfatigueassessmentapproach.Initiallythepaperpresents thebothapproaches.Thentheproposedapproachesareappliedtopredictthefatiguelives ofanageingrailwaybridge.Finallypredictedfatiguelivesarecomparedandrationality, significanceandvalidityoftheproposedapproachesarediscussed.

©2016TheAuthor(s).PublishedbyElsevierLtd.Thisisanopenaccessarticleunderthe CCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Majorityoftherailwaybridgesintheworldareexceedingtheirdesignlivesandbridgeauthoritiesareworkingonprecise lifeextensionmethods[1–3].Asaresult,asignificantamountofresearchareongoingfordevelopmentofpreciousstructural healthmonitoringandlifeassessmentmethods[2–12].Asrailwaybridgesarevulnerablefortime-dependentfatiguedamage duetocyclicnatureoftrafficloads,theassessmentofremainingfatiguelifeofrailwaybridgesforcontinuingserviceshas becomemoreimportantthanever,especiallywhenmakingdecisionsregardingstructurereplacementandothermajor retrofits.However,thistaskisdifficultduetotheincreaseofaxelloadandcorrosiondeteriorationonbridges.

Thefatigueassessmentofstructuresismainlydonebyeitherdeterministicorprobabilisticapproach.Mostofdetermin- isticfatigueassessmentapproachesofrailwaybridgesaregenerallybasedonthecombinationofmeasuredstresshistories underactualtrafficload[12,13],Miner’srule[14]andrailwaycodeprovidedfatiguecurve(alsoreferredS-NorWöhler curve).Althoughthementioneddeterministicapproachpredictstheremainingfatiguelife,theuncertaintiesinherentin thefatigueevaluationprocessarenotcaptured.Theseuncertaintiesarefoundintheprocessofdeterminationofstresshis- tories(i.e.structuralanalysis,fieldmeasurements,loadtesting,loadingsequenceandrespectivehistories),selectingdetail categories,choosingfatiguedamagetheories[15,16].

E-mailaddress:mudiyan.n.adasooriya@uis.no

http://dx.doi.org/10.1016/j.csse.2016.04.002

2214-3998/©2016TheAuthor(s).PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/

licenses/by-nc-nd/4.0/).

Theprobabilisticfatigueassessmentsareoriginatedtocapturetheeffectoftheseuncertaintiesmoreprecisely.This approachisgenerallybasedonprobabilityoffatiguefailureassociatedreliabilityindex.Fatiguereliabilityindexprovidesa toolforpredictingtheremainingfatiguelife[16].Anumberofstudiesonthereliabilityanalysishavebeendoneforfatigue lifepredictionofbridges.Imametal.[17]hasproposedaprobabilisticfatigueassessmentmethodologyforrivetedrail- waybridgesunderhistoricalandpresent-daytrainloading.KwonandFrangopol[18]haveperformedfatiguereliability assessmentofsteelbridgesusingtheprobabilitydensityfunction(PDF)oftheequivalentstressrangesobtainedbyfiled measurementdata.Nietal.[19]hasproposedafatiguereliabilitymodelforfatiguelifeandreliabilityevaluationofsteel bridgeswithlong-termmonitoringdata,whichintegratestheprobabilitydistributionofhotspotstressrangewithacon- tinuousprobabilisticformulationofMiner’sdamagecumulativerule.Recently,Kwonetal.[15]andSolimanetal.[16]have proposedaprobabilisticbilinearstress-lifeapproachforbetterfatigueassessmentofsteelbridges.Miner’srulehasbeen usedasthefatiguedamagetheoryforabovementionedprobabilisticmodels.

TheMiner’sruleisthesimplestandthemostcommonlyusedfatiguelifepredictiontechnique.Oneofitsinteresting featuresisthatlifecalculationissimpleandreliablewhenthedetailedloadinghistoryisunknown.However,undermany variableamplitudeloadingconditions,lifepredictionshavebeenfoundtobeunreliablesinceitdoesnotproperlytake accounttheloadingsequenceeffect[20–22].Therefore,itisuncertaintousetheMiner’sruleforremainingfatiguelife estimationofrailwaybridgesbecausemostoftherailwaybridgesaresubjectedtovariableamplitudeloadings.Noneof researchstudieshaveconfirmedabouttheconsiderationoftheloadingsequenceeffectonprobabilisticfatigueassessment approaches.

Toovercomethisproblemtosomeextent,objectiveofthispaperistocompareprobabilisticfatigueassessmentapproach withdeterministicapproachconsistingofanewdamageindicator(i.e.damagestressmodel)[22],whichcapturestheloading sequenceeffectofvariableamplitudeloadsmorepreciselythantheMiner’srule.Thecomparisonisperformedbyapplying bothfatigueassessmentapproachestopredicttheremainingfatiguelifeofanageingrailwaybridge.Thiscomparison providesanindicationofrationalityofprobabilisticstress-lifefatigueapproachforageingrailwaybridges.

2. Fatiguereliabilityassessmentusingstress-lifeapproach

Thissectionintroducesapreciseprobabilisticfatigueassessmentapproachandarecentlyproposeddeterministicfatigue assessmentapproach.Thefirstapproachgenerallyisbasedonprobabilityoffatiguefailureassociatedreliabilityindex.The secondapproachisbasedonanewdamageindicator,whichcapturestheloadingsequenceeffectmorepreciouslythanthe Miner’srule.

2.1. Fatiguereliabilityindex

ThissectionproposesamethodtodeterminefatiguereliabilityindexofbridgesbasedonprobabilisticbilinearS-N approach.Thefatiguereliabilityofastructuralcomponentoradetailcategoryisrelatedtotheprobabilityofnotviolatinga particularfatiguelimitstate.Basedonthelimitstatefunction(i.e.g(t)=R−S),thefailureprobabilityofastructuralmember oradetailcategoryisdefinedasP_f=P(g(t)<0).

Thereliabilityindexprovidesameasureoffatiguedamageoftheconsidereddetailcategoryofthebridge.Inotherword, reliabilityindexdefinestheprobabilityofviolatingfatiguelimitstate.Thefatiguereliabilityindexisdefinedas,

ˇ=␾⁻¹

1−P_f

(1) where␾⁻¹istheinverseofthestandardnormalcumulativedistributionfunction.Thecorrespondingfatiguelimitstate functioncanbederivedas,

g(t)=−D (2)

whereisMiner’scriticaldamageaccumulationindex,whichisassumedtobelognormaldistributionwithameanvalue of1.0andcoefficientofvariation(COV)of0.3andDistheMiner’sdamageaccumulationindex,whichcanbederivedas,

D={ N(t)

A1

(S^L_re)^m1 forN(t)≤ A1

CAFT^m1 N(t)

(CAFT^m2−m1×A1)(S^B_re)^m2 forN(t)> A1

CAFT^m1

(3)

whereS^L_reandS_re^B areequivalentconstantamplitudestressrangescalculatedusinglinearandbilinearS-Napproach respectivelyasshowninEq.(4).TheCAFTdesignatedasconstantamplitudefatiguethreshold.Them₁andm₂aretheslopes ofstress-lifefatiguecurveaboveandbelowtheCAFT,respectively.TheA₁isthefatiguedetailcoefficientabovetheCAFTof thefatiguecurve.TheA2=A1CAFT^m2−m1,isthefatiguedetailcoefficientbellowtheCAFTofthefatiguecurve.TheN(t)isthe numberofcyclesthatconsidereddetailcategoryhassubjectedatthelifetimeoft.Them1,m2,CAFTandN(t)areconsidered asthedeterministicparameters.ThestressrangeSre,fatiguedetailcoefficientA₁areconsideredasrandomvariables.

Asbridgesaregenerallysubjectedtovariableamplitudestresscycles,theequivalentconstantamplitudestressrangeSre

canbecalculatedforbilinearS-Napproachas[16],

Sre=

⎡

⎣

(n^o_jS_rj^m2) n^o_i

+ n^o_j

⎤

⎦

1/m1

(4)

wheren^o_iisthenumberofcyclesinstressrangebinS_rigreaterthantheCAFTandn^o_jisthenumberofcyclesinstressrangebin S_rjlessthantheCAFT.The n^o_i

+ n^o_j

isthetotalnumberofcycles.Alternatively,theequivalentconstantamplitude stressrangeSrecanbecalculatedusingthePDFofthestressrangesasfollows[16],

Sre=

⎡

⎣

CAFT

0

(CAFT^m1−m2)×S^m2×fs(s)×ds+

S^m1×fs(s)×ds

⎤

⎦

1/m₁

(5)

TheEqs.(2)and(3)canbeusedtocalculatefatiguereliabilityindex(␤)byusingMonteCarlosimulationemployed softwaressuchasR,RELSYS,CALRELoretc.Thefatiguereliabilityindex(␤)versuslifetimeofbridgeshouldbeplottedand comparedwithtargetreliabilityindex(␤target)todeterminethefatiguelifeofeachdetailcategory.

2.2. Damagestressmodel:anewdamageindicator

Thissectionintroducesanewdamagemodelconsistingofanewindicatorwhichpredictsfatiguelifeindeterministicway.

Thedefinitionofthedamageindicator,D_iandthedetaileddescriptionofthedamagestressmodelforvariableamplitude loadingaregiveninthecorrespondingpapers[3,22].Alsotheaccuracyofbothdamagestressmodel(DSM)andprocedure offullyknownS-Ncurvedeterminationinfatiguelifepredictionhavebeenconfirmedbycomparingthetheoreticalfatigue lives(i.e.obtainedbyMiner’sruleandnewDSM)withexperimentallyobservedfatiguelivesforfewmaterialtypes[3,22].

Theconceptisonlysummarizedinthispaperwithanalgorithmforcomprehension.

Forinstance,amemberissubjectedtocertainstressamplitudeorstressrangeof␴iforn_inumberofcyclesatloadlevel iandN_iisthefatiguelife(failurenumberofcycles)correspondingto␴i(Fig.5).Hence,theresiduallifeatloadlevelican beobtainedas(N_i−n_i).Thestress␴(i)eqwhichcorrespondstothefailurelife(N_i−n_i)isnamedasi^thleveldamagestress amplitudeorstressrange(otherwiseitcanbeintroducedasstressamplitudeorstressrangerelevanttotheresiduallife).

Hence,thenewdamageindicator,D_iisstatedas, D_i= _(i)eq−_i

_u−_i (6)

where␴uistheinterceptoftheS-Ncurvewiththeordinateatone-quarteroffirstfatiguecycle.Furthermore,itcanbe statedthat,␴uistheultimatetensilestrengthamplitudeortherangeforrotatingbendingtest-basedS-Ncurvesanditis theultimateshearstrengthamplitudeortherangefortorsionalfatiguetest-basedS-Ncurves.

Samedamageisthentransformedtoloadleveli+1andhencedamageequivalentstressatleveli+1iscalculatedwith therelation,

D_i= _(i)eq−_i

_u−_i = _(i+1)eq−_i+1

_u−_i+1 (7)

FurthersimplificationofEq.(7),

_(i+1)eq=D_i(u−_i+1)+_i+1 (8)

where_(i+1)eqisthedamageequivalentstressamplitudeorthestressrangeattheleveli+1.Thusthecorrespondingequiv- alentnumberofcyclestofailureN_(i+1)RcanbeobtainedfromtheS-NcurveasshowninFig.1.The_i+1istheamplitudeor therangeofappliedstressattheleveli+1andwhenitissubjectedton_(i+1)numberofcycles,thecorrespondingresiduallife attheloadleveli+1,N_(i+1)Riscalculatedas,

N_(i+1)R=N_(i+1)R−n_(i+1) (9)

Hencethedamagestressamplitudeorthestressrange␴(i+1)eq,whichcorrespondstoN_(i+1)Ratloadleveli+1,canbe obtainedfromtheS-NcurveasshowninFig.1.Thenthecumulativedamageatloadleveli+1isdefinedas,

D_(i+1)=_(i+1)eq−_i+1

u−_i+1 (10)

Atthefirstcycle,thedamagestressamplitudeortherange␴(i)eqisequaltoappliedstress␴1andcorrespondingdamage indicatorbecomesD_i=0.Similarlyatthelastcycle,thedamageindicatorbecomesD_i=1when␴(i)eqisequalto␴u.Therefore, thedamageindicatorisnormalizedtoone(D_i=1)atthefatiguefailureofthematerialandthesameprocedureisfollowed

Fig.1.SchematicrepresentationofparametersinfullrangeS-Ncurve.

Fig.2.Generalviewsofconsideredbridge.

(b)

C

BR3 BR3

CG CG CG CG

CG CG

BR3 BR3

ST ST

ST ST ST

ST

ST ST

ST ST ST

ST

ST ST

ST ST ST

ST

ST ST

ST ST ST

ST

EB2 EB1 EB4 EB2 EB1 EB4

C

(a)

MG1

DC2

DC1 DC3 DC4 DC5 DC5

DT2

DT1 DT3 DT4 DT5 DT5

MT1

MT1 MT1 MT2 MT2 MT2 MT3 MT3

MC1

MC1 MC1 MC2 MC2 MC2 MC3 MC3

Fig.3.Membersetcategorization:(a)maintrussgirder,(b)horizontalbridgedeck[3].

untilD_i=1.Here,thedefinedfatiguefailureisthetimetakenfortheoccurrenceofthefirstthrough-thicknesscrackatthe locationofmaximumstressofthestructuralcomponent.

TheabovenewdamageindicatorwhichconsistedofDSMhasbeenvalidatedagainstfatiguetestingdataoffewmaterials [3,22,23]andithasbeenconfirmedthatDSMprovidescloserpredictiontofatiguetestingdatathanMiner’sprediction.

3. Casestudy-fatiguereliabilityassessmentofanageingrailwaybridge

Thefatiguereliabilityofanageingrailwaybridgeisdiscussedinthissection.Theassessmentwereperformedusing introducednewmodelsinSection2.

3.1. Consideredrailwaybridge

TheselectedbridgeisoneofthelongestrailwaybridgesinSriLankaspanning160m(Fig.2).Itisasixspan-rivetedbridge withdoublelanerailtrackshavingwarrentypesemithroughtrusses,supportedoncylindricalpiers.Thebridgedeckismade ofwroughtironandthepiersaremadeofcastironcasingswithinfilledconcrete.Thebridgewasconstructedin1885andis locatedinmarineenvironment.Thebridgecomponentshavebeencategorizedtoseveralgroupsentitled“memberset”by consideringsimilarcrosssectionalpropertiesasshowninFig.3.Detailsoftrainscarriedbythebridgeandtheirfrequencies illustratethatthebridgeissubjectedtovariableamplitudeloading[24].

Fig.4. TheFEanalysisresultsformovingtrainload:(a)verticaldisplacementwhenthetrainisinthemiddleofthebridge(b)maximumstresstakenover allstresspointsateachcrosssectionswhentrainisinthemiddleofthebridge(c)minimumstresstakenoverallstresspointsateachcrosssectionswhen trainisinthemiddleofthebridge(d)verticaldisplacementwhenthetrainjustbeforeleavethebridge(e)maximumstresstakenoverallstresspointsat eachcrosssectionswhenthetrainjustbeforeleavethebridge(f)minimumstresstakenoverallstresspointsateachcrosssectionswhenthetrainjust beforeleavethebridge.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.) Yellowcolor:tensilestress

Redcolor:compressivestress

Thelaboratorytestingconcludedthatthebridgesuperstructurematerialiswroughtironandtheobtainedvaluesfor elasticmodulus,yieldstrength,ultimatestrengthintension,fatiguestrengthanddensityare195GPa,240MPa,383MPa, 155MPaand7600kg/m³respectively[3].

3.2. Structuralanalysis

Thebridgedeckwasanalysedusingthegeneral-purposesoftwareSAP2000.Athree-dimensional(3D)model(Fig.4)of onecompletemiddlespanofthebridgewasanalysedunderactualloadingtodeterminestressesinmembersanddeflections, aswellasvariationsofstressesundermovingloads.ThematerialpropertiesrecordedinSection3.1andcalculatedsection propertieswereutilizedforthisanalysis.Thebridgedeckwasmodelledwith3Dframeelementsandtherivetedconnections wereassumedtobefully-fixed[5].Dynamicanalysiswasconductedforeachdifferentpastandpresentpassageoftrains specifiedbytheowner.ThevalidationofFEmodelwasdonebycomparingtheresultsoftimehistorydynamicanalysiswith thosefrommeasuredtimehistoriesduringthestructuralappraisalinyear2001[3,24].Thesecomparisonsshowthatthere aregoodagreementamonganalyticalresultsoftheFEmodelandthemeasurementoftheactualbridge.Finally,themodel isusedtoobtainpastandpresentnominalstresshistoriesofeachmembersduetoeachpassageoftrains.

Shrinkageofthefreerivetsaremostlyrestrictedbytheconnectedplates,whichconsequentlyarecompressedthroughthe thickness.Theresidualtensileforceintherivetandthecompressiveforceintheplatesgetbalancedeachother;i.e.calledas clampingforce.Thereforetheclampingforcefromtherivetgeneratesacomplextri-axialstressstateintheconnectedplatein thevicinityoftherivethole[25].Finally,theclampingforceseemstobeaffectedbythemechanismofdistributionofstresses alongtheconnection.Theexperiencefromthefieldpracticerevealsthatresultingclampingforcecouldvarysubstantially duetonormalconditions.Therefore,itcouldconsequentlynotbegivenareliablevalue.Furthermore,itcanbeassumed acertainrelaxationoftherivetclampingforceduetocreep,frettingoftheinterfacingplatesurfaces,overloading(dueto residualplasticdeformation)andetcwiththetime(whenbridgeisageing).Therefore,rivetsareconservativelyassumed tobehaveasnon-preloadedboltsinordinaryclearanceholes.Hence,netcrosssection,wheretherivetsarelocated,is consideredfordeterminingnominalstresshistoriesofeachmembersduetoeachpassageoftrains[26].

Fig.5. S-NcurvesforWI-rivetdetail:(a)curvegiveninUKrailwayassessmentcode(b)predictedcurveforfullrangeofnumberofcycles.

3.3. Stress-lifefatiguestrengthcurves

Secondarystress(localstressconcentration)effectinrivetedconnectionbetweentheprimarymembersofbridgeswas foundtobeoneofthemainreasonsforfatiguedamageofageingsteelbridges.Furtherithasbeenidentifiedthatthe rotationalfixityofrivetedconnectionandthevariationintheclampingforceofrivets[25]arethemajorcausesleading tofatiguecrackinginrivetedconnection[1].Therefore,theS-Ncurveofdetailcategory(alsoreferredtoasdetailclass) aregenerallyusedwiththenominalstresshistoriestocapturethefatiguedamageduetotheabove-mentionedstress concentrationneartherivetedconnection.ThisdetailcategorybasedS-NcurvesareobtainedbymodifyingmaterialS-N curvebycorrespondingSCForbyresultsofexperimentsonfullscalerivetedmembers.Thedetailcategoryisdetermined byconsideringthequalityoftheworkmanshipandthecurrentconditionoftherivetedconnection.

Fieldinvestigationsrevealedthattherivetedwroughtironconnectionsofthebridgerepresentlappedorsplicedcon- nectionbehaviourwiththenormalclampingforce.Therefore,rivetedconnectionswereclassifiedasWI-rivet(i.e.WI-rivet detailcategoryorclass),whichisproposedbytheUKrailwayassessmentcode[1,27].ThedifferentmeananddesignS-N curvesforWI-rivetdetailclasshavebeenproposedbypreviousresearchersbasedontheresultsofexperimentsonfull scalerivetedmembers[1].TheabovedesignS-NcurveoftheWI-rivetdetail(i.e.meanminustwostandarddeviations, whichhas2.3%probabilityoffailure)wasusedforfatiguereliabilityassessmentofthisbridge.Thecorrespondingslopes ofS-Nlifefatiguecurvem₁,m₂,thefatiguedetailcoefficientsA₁,A₂andconstantamplitudefatiguethresholdCAFTare4,6, 3.117×10¹³,5.489×10¹⁶and42MParespectively.

HencebothS-NcurvesaboveweretransferredtofullrangeS-Ncurvesusingthepreviouslyproposedmethodinliterature [3].TheobtainedfunctionsandthegeometricalshapesofthecurvegiveninUKrailwayassessmentcodeandthepredicted curveforfullrangeofnumberofcyclesareillustratedseparatelyinFig.5.

4. Fatiguereliabilityassessment

Thestressrangesandtheaveragenumberofcyclesperdayateachmemberswerecalculatedforeachperiodusingthe rainflowcountingalgorithm.Themember,whichismostvulnerabletofatiguedamage,isnamedascriticalmemberin consideredmemberset(i.e.showninFig.3).Thestressrangehistogramsforthecriticalmembersanditsprobabilitydensity functionsareplottedasshowninFig.6.TheFig.6illustratesthatthestressrangesofalmostallcriticalmembersfollowthe log-normaldistribution.Henceequivalentconstantamplitudestressranges(S_re)foreachcriticalmembersofeachmember setswerecalculatedbyEq.(5).

TheCOVofSreisconsideredas0.1[15,28].TheparameterA1andarerandomvariablesandcorrespondingCOV’sare 0.45and0.3respectivelyasdiscussedinSection2.1[18,28].Otherparameterssuchasm1,m2CAFTandN(t)areconsidered asthedeterministicparameters.Asalltherandomvariablesfollowthelognormaldistribution,basedonEqs.(2)and(3), fatiguereliabilityindex,␤canbederivedasfollows:

ˇ(t)=

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎩

+_A1−m₁×_SL

re−lnN(t)

²+²_A

1+(m₁×_SL re)² +_A2−m₂×_SB

re−lnN(t)

²+²_A

2+(m₂×_SB re)²

forN(t)≤ A₁ CAFT^m1 forN(t)> A₁

CAFT^m1

(11)

where␭and␨arelognormalparametersofthevariousrandomvariables.

Fig.6. Stressrangehistogramanditsprobabilitydistributionfunction:(a)forcriticalmemberincrossgirdersetCG;(b)forcriticalmemberinstringerset ST;(c)forcriticalmembersinmaingirdersetMT1;(d)forcriticalmembersinmaingirdersetMT2;(e)forcriticalmembersinmaingirdersetMT3;(f) forcriticalmembersintrussdiagonalsetDT1;(g)forcriticalmembersintrussdiagonalsetDT2;(h)forcriticalmembersintrussdiagonalsetDT3;(i)for criticalmembersintrussdiagonalsetDT4.

Fig.7.Fatiguereliabilityindexversuslifeofthebridge:(a)forcriticalmemberincrossgirdersetCG,(b)forcriticalmemberinstringersetST,(c)forcritical membersinmaingirdersetMT1,MT2andMT3,(d)forcriticalmembersintrussdiagonalsetDT1,DT2,DT3andDT4.

ThecumulativenumberofcyclesN(t),lognormalparametersofSre,A1,A2andaresubstitutedtoEq.(11)andhence thefatiguereliabilityprofiles(i.e.variationoffatiguereliabilityindexwiththeageofthebridge)ofthecriticalmembersof eachmembersetofthebridgearegeneratedandplottedinFig.7.Atargetreliabilityindexisdefinedtoevaluateprobability oflimitstatefailure(i.e.Eq.(2))andcorrespondingfatiguelife.

ReferringtoEq.(1),for5%offailureprobabilityoflimitstateg(t)inEq.(2),tableofthestandardnormalcumulative distributionfunction␾(ˇ)givesareliabilityindex␤as1.65.Thatmeansatargetreliabilityindexof1.65isconsidered basedonsurvivalprobabilityof95%forfatiguefailureprobabilityofapproximately5%[18].Insimilarway,for50%of failureprobabilityoflimitstateg(t)inEq.(2),tableofthestandardnormalcumulativedistributionfunction␾(ˇ)givesa reliabilityindex␤as0.Thatmeansatargetreliabilityindexof0isconsideredbasedonsurvivalprobabilityof50%forfatigue failureprobabilityof50%.Thezerovalueoftargetreliabilityindexgivesanindicationofhighestpossibilityoffatiguefailure.

Generally,targetreliabilitylevelshouldbedeterminedaccordingtotheimportancelevelsofrespectivestructuraldetails [16,18].Therefore,inthiscasestudy,twolimitsoftargetreliabilityindexhavebeenconsideredformoregeneralizedfatigue lifeassessment.Thosetargetreliabilityindicesare1.65and0,whichcorrespondingtosurvivalprobabilityof95%and50%

respectivelyasdescribedabove[18].ThecalculatedfatiguelivesareshowninTable1.

ThesequentiallawassociatedproposedmethodshowninSection2.2,obtainednominalstressrangesinSection3.2 andfullrangeS-NcurvesshowninFig.5wereusedtogethertoobtainremainingfatiguelivesofcriticalmembersofeach membersetsofthebridge.Theobtainedfatiguelivesoffatiguecriticalmembersofeachmembersets(i.e.whicharepossible tofatiguedamage)areshowninTable1.Itisassumedthatfuturesequenceofpassageissimilartothatofthepresentday.

5. Discussionandconclusions

Aprobabilisticfatigueassessmentapproachandadeterministicapproachconsistingofanewdamageindicator,which capturestheloadingsequenceeffectofvariableamplitudeloadsmorepreciselythanMiner’srule,wereintroducedtoassess thefatiguelifeofanageingrailwaybridge.Obtainedfatigueliveswerecomparedforcriticalmembersofeachmember setsasshowninTable1.TheTableshowsthatboththedeterministicandprobabilisticapproachesprovidealmostcloser fatiguelivesforbridgedeckmembers(i.e.crossgirdersCGandstringersST).However,itisoppositeforthemaingirder trussmembers(i.e.maingirderchordsandtrussdiagonals).

Table1

Summaryoffatiguelivesforcriticalmembersofeachmembersets.

Bridgecomponent Memberset Fatiguelife(years)

DeterministicApproach ProbabilisticApproach

Damagestressmodel ˇtarget=1.65 ˇtarget=0

Crossgirders CG 133 119 170

Stringers ST 134 135 191

Maingirderbottomchord MT1 444 128 183

Maingirderbottomchord MT2 286 102 150

Maingirderbottomchord MT3 290 106 154

Trussdiagonal(tensionmember) DT1 312 165 235

Trussdiagonal(tensionmember) DT2 292 118 170

Trussdiagonal(tensionmember) DT3 259 108 157

Trussdiagonal(tensionmember) DT4 283 111 161

TableshowsthathighlystressedmemberofmaingirderbottomchordMT2isthemostvulnerabletofatiguefailureand thevulnerablemembersarelocatedinthemaingirderconsistingoftrussmembers.Further,itseemsthattherearenomore remaininglivesformajorityofvulnerablemembersofmaingirdertruss(i.e.MT2,MT3,MT1,DT2,DT3&DT4)underthe 95%ofsurvivalprobability.Underthe50%survivalprobability,consideredbridgehasabout20moreyearsofremaining fatiguelife.However,bridgeisstillinservicewithoutanyrecordeddamageorfracture.Thedeterministicapproachpredicts themostvulnerablememberforthefatiguefailureasthecriticalmemberincrossgirdermembersetCG.Accordingtothe deterministicapproach,theremainingfatiguelifeoftheconsideredbridgeisthreemoreyears.

Thedeviationsoffatiguelivesofbothapproachesillustratethatintroducedprobabilisticfatigueassessmentapproach maynotpreciselycapturetheloadingsequenceeffect.However,itcanbeconcludedthatapplicationofintroducedproba- bilisticmodelprovidesaconservativefatigueassessmentforrailwaybridges.Therefore,itisdoubtfultoconcludethatthis introducedprobabilisticmodelandcorrespondingmodalparametersprovideapreciseremaininglifeforageingrailway bridges.Authorsarecurrentlypayingtheirattentiononexperimentalvalidationoftheprobabilisticapproachanddamage stressmodelpredictedfatiguelives.

Acknowledgements

TheauthorswishtoexpresstheirsinceregratitudetoEmeritusProfessorM.P.Ranaweeraandtheteamofexpertswho workedintheRailwayBridgeproject,fortheirgreatadvice,whichlaidthefoundationforthisresearch.Thekindsupport givenbytheRailwaydepartmentisalsoappreciated.

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