[J]Estimation of fatigue life of railway bridges u

JournalofSoundandVibration(1980)70(4),527-541ESTIMATIONOFFATIGUELIFEOFRAILWAYBRIDGESUNDERTRAFFICLOADSL.FR+BARailwayResearchInstitute,Prague,Czechoslovakia(Received8October1979,andinrevisedform3January1980)RandommodellingofrailwaybridgeloadingenablesfatiguedamagetobecalculatedonthebasisofthecumulativedamagetheoryofPalmgren-Minerandtheclassificationofthestress-timehistorybymeansofthe“rain-flow”countingmethod.Theresultsofcal-culationsarethemeanvalueofthedamageandthestandarddeviationofthestresses,andthusanestimationofthebridgefatiguelife.Accordinglythebridgelifeisdependentonthenumberofstresscyclesandtheirdistribution,thestandarddeviationofstresses,andontheshapeoftheWijhlercurve.Bridgelifeincreaseswithincreasingspananddecreaseswithanincreasingtrafficload.Resultsarepresentedasobtainedinadetailedstudyoftheeffectsonthebridgelifeofdifferentparameters(vehiclespeed,dampingofbridgevibrations,variabilityinlengthandtimeofthemovingloadanditsmagnitude,numberofstresscyclesandtheirdistribution).Theequivalentdamagemethod(theATmethod)intheintegralformenablesonetocomparetheeffectsofthetrafficloadswiththoseofthestandardloading.1.INTRODUCTIONIthasbeenrecentlyassumedthatthedynamicstressesinrailwaybridgesareofrandomcharacterbecausetheirloadingisinsomecasesastochasticprocess.Thisapproachwasexaminedindetailinreferences[l-3]fortwobasicloadingcases,asshowninFigures1and2.Inthefirstcase(Figure1)theloadingisthatduetothemovementofarandomforcealongabeam;thisisanappropriateidealizationforbridgesofshortspanorforlongitudinalbeams.P(t)PFigure1.MotionofarandomforceP(t)alongthebeam.Thesecondcase(Figure2)isapplicabletothevibrationofbridgesoflargespansbecausetheirloadingcanberepresentedbyaninfinitestripofmovingcontinuousrandomload.Thecaiculatedresultsofreferences[2]and[3],forseveralbasictypesofcovariancesoftheinput,aregivenintermsofthecoefficientofvariationV,forthedeflectionatthebeam5270022--460X/80/120527+1.5$02.00/O@1980AcademicPressInc.(London)Limited528L.FR+BAFigure2.Motionofaninfinitestripofcontinuousrandomloadp(x.t)alongthebeam.spanmid-point:V”=VPY(Z)(1)forthefirstcase,andV”=(Vfyi+VfyP+vfryf+v;y;P)1’2(2)forthesecondcase.Thefollowingnotationisused:VP,VP,V,VP,andV,arethecoefficientsofvariationoftheload;y(z),yP,y,yPrandy,arethecharacteristicparametersoftherandomresponseofthebeam(asdefinedinreferences[2]and[3]);z=et/listhedimensionlessco-ordinate;cisthespeed;tisthetime;1isthespanofthebeam.Theresultofmajorimportance,asgivenbyequations(1)or(2),istheexpressionofthecoefficientofvariationforthedeflectionV,intermsofthecoefficientsofvariationoftheload.ThispermitsthestandarddeviationofthedeflectionD,tobecalculatedfromV,andfromthemeanvalueofthedeflectionu.atbeamspanmid-point:D”=v,v,.(3)Itcanreadilybeprovedthatasimilarrelationship,alsointermsofV,,isapproximatelyvalidforthestress(+inthebridge:D,=V,a,(4)whereD,isthestandarddeviationofthestressandu.isthemeanvalueofthestressatthebeamspanmid-point.Therelation(4)isusedinthefollowingsectionsofthispaper,itbeingassumedthatthecoefficientofvariationV,hasbeencalculatedaccordingtothemethodsforshortorlongspanswhichweredescribedinreferences[2]and[3].Themainaimofthepresentpaperistostudytherelationshipbetweenrandomvibrationofrailwaybridgesduetotrafficloadsandtheirfatigue.Thus,experiencefromaircraftandmechanicalengineering[4-6]canbeappliedincivilengineeringtotheestimationoffatiguelifeofrailwaybridges,asfatiguehasbeenidentifiedasoneofthedecisivecriteriaforthedesignofsteelbridges.2.FATIGUEDAMAGEACCUMULATIONInvestigationofthefatigueofsteelstructuressubjectedtodynamicloadinghaveledtothefollowingresults:1.Thefatiguedamageisessentiallydeterminedbythestressrange[7]s=Au=crmax-urnin,(5)FATIGUELIFEOFBRIDGES529whereamaxandwmi,arethemaximumandminimumstresses,respectively,atagivenpointofthebridgeandatagivenstresscycle.2.TherelationshipbetweenthenumberofloadcyclestofractureNandthestressranges(theWijhlercurveorSNcurve)canbeapproximatedby?Nsk=C,(6)wherel/kistheslopeoftheWiihlercurverepresentedasalineonalog-logscale(seeFigure3)andCisaconstant.Inthispaper,theWiihlercurvesdenotedbySN1andSN3,withtheslopesk=3andk=3.75,respectively,areused(seeTable1).”“,4logNFigure3.WiihlercurveandPalmgren-Minertheoryofcumulativedamage;probabilitydensityf(s)ofstressrangess.TABLE1WiihlercurveconstantsTypeofWGhlercurveSNlSN3Unitsk3C4x1ol22.507x10”1,530x1Ol21.064x10”0,627x10”0.426xlOI23.7511.5x1o14(N/mm2)k5.930x1ol33.440x1o132.288x1013(N/mm2)k1.248x10”0.796xlOI1.5962.5292.6154.3613.7606.55716.38312.0159.40018.8523.TheconstantCinequation(6)isdeterminedbythedetaileddesign(itdependsonstressconcentration,weldjoint,etc.).4.ThefatiguedamageDisgiven,withadequateaccuracyforourcomparativepurposes,bythePalmgren-Minertheoryoflineardamageaccumulation[8],D=Cai/Ni,(7)ItSomeauthors(see,e.g.,reference[12]),haveassumedadoubleslopedW6hlercurve.Nevertheless,thestraightlineWiihlercurveisusedhereforthepurposeofcomparison.530L.FR+BAwhereniisthenumberofstresscyclesofthestressrangeskandNiisthenumberofstresscyclesontheWohlercurve(Figure3).FailureoccursaccordingtothistheorywhenD=l.(8)5.Ther