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ReliabilityEngineers,PartO:JournalofRiskandProceedingsoftheInstitutionofMechanicalhttp://pio.
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1177/1748006X13476821publishedonline14February20132013227:109originallyProceedingsoftheInstitutionofMechanicalEngineers,PartO:JournalofRiskandReliabilityDian-QingLi,Shui-HuaJiang,Shuai-BingWu,Chuang-BingZhouandLi-MinZhangcomplexperformancefunctionModelingmultivariatedistributionsusingMonteCarlosimulationforstructuralreliabilityanalysiswithPublishedby:http://www.
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htmlCitations:WhatisThis-Feb14,2013OnlineFirstVersionofRecord-Mar22,2013VersionofRecord>>atNATIONALUNIVSINGAPOREonMarch23,2013pio.
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navDOI:10.
1177/1748006X13476821pio.
sagepub.
comModelingmultivariatedistributionsusingMonteCarlosimulationforstructuralreliabilityanalysiswithcomplexperformancefunctionDian-QingLi1,Shui-HuaJiang1,Shuai-BingWu1,Chuang-BingZhou1andLi-MinZhang2AbstractThesimulationofmultivariatedistributionshasnotbeeninvestigatedextensively.
ThisarticleaimstoproposeMonteCarlosimulation(MCS)-basedproceduresformodelingthejointprobabilitydistributionsandestimatingtheprobabilitiesoffailureofcomplexperformancefunctions.
Twoapproximatemethods,namelymethodPandmethodS,oftenusedtoconstructmultivariatedistributionswithgivenmarginaldistributionsandcovariance,areintroduced.
TheMCS-basedproceduresareproposedtosimulatethetheoreticalmultivariatedistributionsorapproximatemultivariatedistributionsconstructedbymethodsPandS,whicharefurtherusedtocomputetheprobabilitiesoffailureofcomplexperformancefunctions.
Fourillustrativeexampleswithknowntheoreticaljointprobabilitydistributionsareinvestigatedtoexaminetheaccuracyoftheproposedproceduresinmodelingthemultivariatedistributionsandestimatingtheprobabilitiesoffailure.
Theresultsindicatethatthebivariatedistributionscanbeeffectivelysimulatedbytheproposedprocedures,whichcanevaluatethereliabilityofcomplexperformancefunctionsefficiently.
Theseprovideausefultoolforsolvingthereliabilityproblemswithcomplexperformancefunctionsinvolvingcorrelatedrandomvariablesunderincompleteprobabilityinformation.
Theperformanceofthesimulationproceduresassociatedwiththetwoapproximatemethodshighlydependsonthelevelofprobabilityoffailure,theformofperformancefunction,andthedegreeofcorrelationbetweenvariables.
KeywordsJointprobabilitydistributionfunctions,Pearsoncorrelationcoefficient,Spearmancorrelationcoefficient,MonteCarlosimulation,probabilityoffailureDatereceived:24October2012;accepted:4January2013IntroductionItiswellknownthatstructuralreliabilityisoftenexpressedintermsofprobabilityoffailure,whichisgivenbymultipleintegralsoverthefailuredomain.
Theoretically,whenthejointcumulativedistributionfunction(CDF)orjointprobabilitydensityfunction(PDF)ofseveralrandomvariablesinvolvedinthereliabilityanalysisisknown,thedirectintegrationmethodcanbeemployedtocalculatetheprobabilityoffailure.
1,2Inengineeringpractice,however,thejointCDForjointPDFofcorrelatedrandomvariablesisoftenunknownowingtolimiteddata.
Inmostpracticalapplications,onlythemarginalPDFsandthecovar-iancematrixareavailable.
Basedonsuchlimitedinfor-mation,themodelingofjointPDFofcorrelatedvariablesisachallengingproblem.
3,4Ontheotherhand,whenthefailuredomainunderlyingaperfor-mancefunctioniscomplex,thedirectintegrationmethodforcalculatingtheprobabilityoffailuremaynotbeperformedevenifthejointPDFofvariablesis1StateKeyLaboratoryofWaterResourcesandHydropowerEngineeringScience,WuhanUniversity,Wuhan,China2DepartmentofCivilandEnvironmentalEngineering,TheHongKongUniversityofScienceandTechnology,Kowloon,HongKongCorrespondingauthor:Dian-QingLi,StateKeyLaboratoryofWaterResourcesandHydropowerEngineeringScience,KeyLaboratoryofRockMechanicsinHydraulicStructuralEngineering(MinistryofEducation),WuhanUniversity,8DonghuSouthRoad,Wuhan430072,China.
Email:dianqing@whu.
edu.
cnatNATIONALUNIVSINGAPOREonMarch23,2013pio.
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Thisisbecauseevaluatingthemultipleintegralsisverydifficultinmostcases.
5,6Therefore,themethodstocalculatetheprobabilitiesoffailureforcomplexper-formancefunctionsinvolvingcorrelatedvariablesunderincompleteprobabilityinformationstillneedtobeexplored.
Incurrentpractice,twocommonapproximatemul-tivariateconstructionmethodsareusedformodelingthejointprobabilitydistribution.
7,8OneistomatchthePearsoncorrelationinNelson,9whichisreferredtoasapproximatemethodPhereafter.
TheotheristomatchtheSpearmancorrelationinNelson,9whichisreferredtoasapproximatemethodShereafter.
AspointedoutbyLebrunandDutfoy10andLietal.
,8methodPisessentiallybasedontheNatafdistributionmodel.
11ThejointprobabilitydistributionmodelobtainedfrommethodPisoftenusedforstructuralreliabilityanalysisunderincompleteprobabilityinformation.
12–15TheapproximatemethodScanalsobeusedtomodelthejointprobabilitydistributionofrandomvariables.
3,16IncomparisonwithmethodP,thereislimitedresearchintheliteratureonstructuralreliabilityanalysisassoci-atedwithmethodS.
Forbothapproximatemethods,Lietal.
8studiedthereliabilityofaverysimpleperfor-mancefunctionwherethedirectintegrationmethodisadoptedtocalculatetheprobabilityoffailure.
However,asmentionedpreviously,theprobabilityoffailureforacomplexperformancefunctioninvolvingcorrelatedvariablesunderincompleteprobabilityinfor-mationmaynotbeevaluatedbythedirectintegrationmethod.
Additionally,althoughthesetwoapproximatemethodsarewidelyusedformodelingthemultivariatedistributionsandcalculatingtheprobabilityoffailure,theaccuracyofthesetwomethodsusedtoestimatetheprobabilityoffailurehasnotbeenstudiedextensively.
TheobjectiveofthisstudyistoproposeMonteCarlosimulation(MCS)-basedproceduresforsimulatingthejointprobabilitydistributionsandestimatingtheprob-abilitiesoffailureofcomplexperformancefunctions.
Toachievethisgoal,thisarticleisorganizedasfollows.
In'Twoapproximatemethodsforconstructingmultivariatedistributions',methodPandmethodSareintroduced.
In'MCS-basedproceduresforsimulatingmultivariatedis-tributionsandprobabilitiesoffailure',theMCS-basedproceduresforsimulatingthephysicalsamplesassociatedwiththeexactmethodandtwoapproximatemethodsareproposed.
Theproceduresforsimulatingtheprobabilityoffailurearealsopresentedinthissection.
In'Resultsanddiscussion',fourillustrativeexampleswithknowntheoreticaljointprobabilitydistributionsareinvestigatedtoexaminetheaccuracyoftheproposedproceduresinestimatingtheprobabilityoffailure.
TwoapproximatemethodsforconstructingmultivariatedistributionsInthefollowingdiscussion,fourtypesofdistributions,namely,independentstandarduniformvariables,independentstandardnormalvariables,correlatednor-malvariables,andcorrelatednon-normalvariableswillbereferred.
Toavoidconfusion,symbolsU,Z,X,andYdenoteindependentstandarduniformvariables,independentstandardnormalvariables,correlatednor-malvariables,andcorrelatednon-normalvariables,respectively.
Thesedefinitionsapplythroughouttheentirearticleunlessstatedotherwise.
Therearetwowidelyusedapproximatemultivariatedistributionsconstructionmethods.
8,9Theyareapprox-imatebecausethemultivariatedistributionsarecon-structedbasedonmarginalPDFsandincompletecorrelationinformation.
ThefirstapproximationmethodistomatchthePearsoncorrelation.
Wecallthis''approximatemethodP''forPearson.
ThesecondapproximationmethodistomatchtheSpearmancor-relation.
Wecallthis''approximatemethodS''forSpearman.
Multivariatecorrelatednon-normaldistri-butions,particularlywiththreeormorecomponents,areuncommoncomparedwiththebivariatedistribu-tions.
Forcompleteness,theconstructionofbivariatedistributionsusingthetwoapproximatemethodsispresentedbelow.
ApproximatemethodPThisapproximatemethodisessentiallybasedontheNataftransformation.
11Considerthefollowingiso-probabilistictransformation6F(xi)=FYi(yi)xi=F1FYi(yi)&i=1,21inwhichF()isthestandardnormalCDF;F21()istheinversestandardnormalCDF;FYi(yi)istheCDFofYi.
Thus,correlatednon-normalvariablescanbetransformedintocorrelatedstandardnormalvariablescomponentbycomponent.
AccordingtotheNatafdis-tributionmodel,theapproximatejointPDFofcorre-latednon-normalrandomvariablesY1andY2,denotedasfP12y1,y2;rP,Y,canbeexpressedasfP12y1,y2;rP,Y=f1(y1)f2(y2)f12x1,x2;rPP,Xf(x1)f(x2)2wheref1(y1)andf2(y2)arethePDFsofY1andY2,respectively;f(x1)andf(x2)arethestandardnormalPDFsofX1andX2,respectively;f12x1,x2;rPP,XisthejointPDFofbivariatestandardnormaldistributionwiththePearsoncorrelationcoefficientofrPP,XbetweenX1andX2,whichcanbeobtainedfromthefollowingintegralrelationrP,Y=+''+''y1m1s1y2m2s2f1(y1)f2(y2)f12x1,x2;rPP,Xf(x1)f(x2)dy1dy23110ProcIMechEPartO:JRiskandReliability227(2)atNATIONALUNIVSINGAPOREonMarch23,2013pio.
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comDownloadedfromForgivenmarginaldistributionsofY1andY2,andthePearsoncorrelationcoefficientrP,YbetweenY1andY2,therPP,Xinequation(3)canbedirectlysolvedbythetwo-dimensionalGaussian-Hermiteintegralmethod(seeLietal14formoredetailedinformation).
ApproximatemethodSThesecondapproximationideaistomatchtheSpearmancorrelation.
LetrS,XbetheSpearmancorre-lationcoefficientbetweenthetransformedvariablesX1andX2,whichisdefinedasrS,X=rPF(x1),F(x2)4Basedonequations(1)and(4),wehaverS,Y=rS,X5TheapproximatejointPDFofcorrelatednon-normalrandomvariablesY1andY2,denotedasfS12y1,y2;rP,Y,canbeexpressedas7,8fS12y1,y2;rS,Y=f1(y1)f2(y2)f12x1,x2;rSP,Xf(x1)f(x2)6wheref12x1,x2;rSP,XisthejointPDFofbivariatestandardnormaldistributionwiththePearsoncorrela-tioncoefficientofrSP,XbetweenX1andX2,whichcanbeexpressedintermsoftheSpearmancorrelationcoef-ficientrS,Xas9rSP,X=2sinp6rS,X7Itisevidentfromequations(2)and(6)thattheonlydifferenceinjointPDFsbetweenmethodPandmethodSisthePearsoncorrelationcoefficientbetweenthetransformedvariablesX1andX2,whicharedeterminedbyequation(3)andequation(7)formethodsPandS,respectively.
MCS-basedproceduresforsimulatingmultivariatedistributionsandprobabilitiesoffailureTransformationsofindependentstandarduniformvariatestocorrelatednon-normalvariatesOneobjectiveofthisarticleistoprovidesomeclosed-formbenchmarkexamplesofreliabilityanalyseswherethetheoreticaljointprobabilitydistributionsareknown.
Oncethejointprobabilitydistributionsareknowntheoretically,itisrelativelystraightforwardtoconductsimulationsofbivariatedistributionusingtheRosenblatttransformation.
17Wecallthisanalysisbasedoncompletejointprobabilityinformationasthe''exactmethod''inthisstudy.
Fortheexactmethod,thecorrelatednon-normalvariablescanbeobtainedfromtheindependentstandarduniformvariablesasfollows.
SupposeasetofnrandomvariablesY=(Y1,Y2,.
,Yn)withajointCDFofFY(Y).
AsetofstatisticallyindependentstandarduniformvariablesU=(U1,U2,.
,Un)arederivedfromthefollowingequationu1=F1(y1)u2=F2j1(y2y1j).
.
.
.
.
.
un=Fnj1,,n1(yny1jyn1)8>>>>>:8whereu1,u2,,unaresamplesfromtheUspace;Fnj1,,n1(yny1jyn1)istheconditionalCDFofYngivenY1=y1,Y2=y2,.
,Yn-1=yn-1.
Invertingtheaboveequationssuccessively,thedesiredcorrelatednon-normalvariablesYcanbeobtainedasy1=F11(u1)y2=F12j1(u2y1j).
.
.
.
.
.
yn=F1nj1,,n1(uny1jyn1)8>>>>>>>:9inwhichF21()istheinversefunctionofF().
Theoretically,thecorrelatednon-normalvariablesYcanbeobtainedfromtheindependentstandarduni-formvariablesUthroughequations(8)and(9).
However,apracticaldifficultyisthat,unlessthejointCDFofYissimpleinform,theinverserelationsshowninequation(9)mayonlybeobtainednumeri-cally.
Whenthenumbersofsamplesareverylarge,thesolutionofequation(9)couldbetime-consuming.
Inaddition,formultivariatedistributionmodels,itisverydifficulttosolveequation(9)directly.
ThesepotentialshortcomingscanberemovedthroughapproximatemethodsPandSinasubtleway,whicharepresentedasbelow.
WhenthejointprobabilitydistributionofrandomvariablesYisconstructedbymethodP(seeequation(2)),equation(9)canbefurthersimplifiedasZ=F1(U)X=LPZyi=F1YiF(xi)8xobtainedfromtheCholeskydecompositionforthePearsoncorrelationmatrixRPP,X=(rPP,X)n3ninwhichrPP,XisthePearsoncorrelationcoefficientintheXspaceobtainedfromequation(3).
Similarly,whenthejointprobabilitydistributionofrandomvariablesYisconstructedbymethodS(seeequation(6)),equation(9)canalsobefurthersimpli-fiedasZ=F1(U)X=LSZyi=F1YiuuijRank(uui)=Rank(xi)8xobtainedfromtheCholeskydecompositionforthePearsoncorrelationLietal.
111atNATIONALUNIVSINGAPOREonMarch23,2013pio.
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comDownloadedfrommatrixRSP,X=(rSP,X)n3ninwhichrSP,XisthePearsoncorrelationcoefficientintheXspaceobtainedfromequation(7);Rank(uui)=Rank(xi)denotesthatthesimulatedsamplesuifromtheUspacearerearrangedasuuiwiththesamerankasxifromtheXspace.
Simulationofcorrelatednon-normalvariatesHavingpresentedthetransformationsoftheindepen-dentstandarduniformvariablestothecorrelatednon-normalvariablesassociatedwiththethreemethods,thenextstepistosimulatecorrelatednon-normalvariatesbythethreemethodsbecauseoneoftheprimaryappli-cationsofjointprobabilitydistributionsissimulation.
TheproceduresforsimulatingthephysicalsamplesfromaspecifiedmultivariatedistributionintheMATLABenvironmentarepresentedbelow.
Simulationofmultivariatedistributionsassociatedwithexactmethod1.
SimulatenindependentstandarduniformrandomvariablesUN3n,inwhichNisthesamplesize.
ThiscanbeobtainedusingU=rand(N,n).
Thefunctionrand('state',0)isusedtofixtheinitialseed.
2.
SubstitutethesimulatedUN3nintoequation(9).
Thenthecorrelatednon-normalsamplesYN3nareobtained.
Inmostcases,thismayonlybeobtainednumericallywherenonlinearequationsolutionsareofteninvolved.
SimulationofmultivariatedistributionsassociatedwithmethodPandmethodS1.
SimulatenindependentstandarduniformrandomvariablesUN3n,obtainsuccessivelyZN3nusingZ=norminv(U).
2.
CalculatethePearsoncorrelationmatrixRPP,X=(rPP,X)n3nfromtheknownPearsoncorrela-tionmatrixRP,Y=(rP,Y)n3nusingequation(3);orRSP,X=(rSP,X)n3nfromtheknownSpearmancor-relationmatrixRS,X=(rS,X)n3nusingequation(7).
3.
PerformtheCholeskydecompositionforRPP,XorRSP,X.
ThelowertriangularmatrixLPorLSareobtainedusingLP=chol(RP)'orLS=chol(RS)'.
4.
SubstitutethesimulatedZN3nandLPintoequa-tion(10)orLSintoequation(11).
Then,thecorre-latednormalsamplesXN3nareobtainedusingX=Z*LP'orX=Z*LS'.
5.
FormethodP,thecorrelatednon-normalsamplesYN3narereadilyobtainedusingY=XtoYConvert(X,mean,std,par1,par2)where''XtoYConvert''isanuser-definedfunction.
FormethodS,therearrangedsamplesUUN3naredeterminedfirstfromUN3nusing[num,index]=sort(X(N,n)),UU(index,n)=sort(U(N,n)).
Then,YN3nareobtainedusingY=XtoYConvert(norminv(UU),mean,std,par1,par2).
SimulationofprobabilitiesoffailureForengineers,theprobabilityoffailuremaybeofthegreatestinterest.
Letg(Y)betheperformancefunction.
SincethephysicalsamplesYN3nfromagivenjointprobabilitydistributionareobtainedusingtheafore-mentionedproceduresasshownin'Simulationofcor-relatednon-normalvariates',theprobabilityoffailureforaspecifiedg(Y)canbereadilyevaluatedbysubsti-tutingthephysicalsamplesintog(Y).
ItiswellknownthattheaccuracyofMCShighlydependsonthenum-berofsamples.
TherequirednumberofsamplesNshouldsatisfythefollowingrelation6N51pfpf3COV2pf12wherepfistheprobabilityoffailureassociatedwithg(Y);COVpfisthecoefficientofvariationofpf.
Inthisstudy,thenumberofsamplesissetas106withacon-siderationofaccuracyandefficiency.
Thus,forcom-monlyusedvaluesofCOVpf=10%,theaccuracyisadequateforpfexceeding10–4.
ResultsanddiscussionFourexamplesarepresentedbelowtostudythevalidityoftheMCS-basedproceduresforsimulatingthejointprobabilitydistributionsandestimatingtheprobabil-itiesoffailureassociatedwithcomplexperformancefunctionsinvolvingcorrelatedvariables.
BivariateFrankdistributionInthesubsequentthreeexamples,thebivariateFrankdistributionisusedtoexaminetheaccuracyofthepro-posedproceduresinestimatingtheprobabilitiesoffail-ure.
ItstheoreticaljointCDFisgivenby9,18F12(y1,y2)=1uln1+eu1ey11eu1ey21eu1()y1,y25013where–N4u4+Nanduisnotequaltozero.
ThecorrespondingjointPDFisf12(y1,y2)=ueu1ey1y2u2ey1ey2eu1+eu1ey11eu1ey212y1,y25014ThemarginalCDFsofY1andY2arederivedasF1(y1)=1ey1F2(y2)=1ey2&15Basedonequation(15),onecanreadilyseethatthemarginaldistributionsofY1andY2arestandardexpo-nentialdistributions.
Bothmeansandstandarddevia-tionsareequalto1.
0.
HavingobtainedthetheoreticaljointCDFandPDFofY1andY2,theconditionalprobabilitydistributions112ProcIMechEPartO:JRiskandReliability227(2)atNATIONALUNIVSINGAPOREonMarch23,2013pio.
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TheconditionalCDFofY1givenY2=y2isF1j2(y1jy2)=1f2(y2)F12y1,y2y2=eu1ey2eu1ey11eu1+eu1ey11eu1ey2116andtheconditionalCDFofY2givenY1=y1isF2j1(y2jy1)=1f1(y1)F12y1,y2y1=eu1ey1eu1ey21eu1+eu1ey11eu1ey2117ThreetypesofPearsoncorrelationcoefficientsinXspaceThreetypesofPearsoncorrelationcoefficientsinXspaceareinvolvedincomputingtheprobabilityoffail-ure.
Theformulaeforcalculatingthesecorrelationcoef-ficientsarepresentedbelow.
Fortheexactmethod,thePearsoncorrelationcoeffi-cientbetweenthetransformedstandardnormalrandomvariablesX1andX2,rEP,X,isgivenbyrEP,X=+''+''x1x2f12(x1,x2)dx1dx218Accordingtotheisoprobabilistictransformationshowninequation(1),f12(x1,x2)canbederivedasf12(x1,x2)=f12(y1,y2)f(x1)f1(y1)f(x2)f2(y2)19Basedonthefollowingequationdx1=f1(y1)f(x1)dy1dx2=f2(y2)f(x2)dy28>:20Equation(18)canbefurtherexpressedasrEP,X=+'0+'0F1Fy1F1Fy2f12(y1,y2)dy1dy221FormethodP,thePearsoncorrelationcoefficientrPP,XintheXspacecanbecalculatedusingequation(3).
AccordingtothedefinitioninNelson,9thePearsoncor-relationcoefficientbetweenY1andY2inequation(3)canbederivedasrP,Y=Cov(y1,y2)s1s2=+'0+'0(y11)(y21)f12(y1,y2)dy1dy222inwhichf12(y1,y2)isshowninequation(14).
FormethodS,thePearsoncorrelationcoefficientrSP,XintheXspacecanbecalculatedusingequation(7).
FortheconsideredbivariateFrankdistribution,theSpearmancorrelationcoefficientinequation(7)canbederivedasfollowsrS,X=rS,Y=121010C(u1,u2)du1du23=1+12u2u2u0t2et1dt1uu0tet1dt!
23inwhichC(u1,u2)istheFrankcopulafunction.
9,18TheabovethreePearsoncorrelationcoefficientsintheXspaceareshowninFigure1.
ItcanbeobservedthatthePearsoncorrelationcoefficientsbetweenX1andX2usingthethreemethodsaresignificantlydif-ferentfromthePearsoncorrelationcoefficientsbetweenY1andY2,especiallywhenvariablesY1andY2arenegativelycorrelated.
However,theyareoftenassumedtobeequalinstructuralreliabilityanalysiswithoutapropervalidation.
Furthermore,rP,Ycan-notbecloseto–1and1owingtothelimitationthatthePearsoncorrelationcoefficientsintheXspacearerestrictedto[–1,1].
Thisisapotentialpitfallunderly-ingmethodsPandS.
Inaddition,thereexistsasmalldifferencebetweenthePearsoncorrelationcoeffi-cientsintheXspaceusingthetwoapproximatemeth-odsandtheexactsolutions,especiallyfornegativelycorrelatedY1andY2.
Forinstance,themaximumdif-ferenceinPearsoncorrelationcoefficientsbetweentwoapproximatemethodsandtheexactsolutionsisonly0.
07.
Example1.
AsimpleperformancefunctionisadoptedforbenchmarkpurposesbecausethedirectintegrationmethodproposedbyLietal.
8canbeemployedtocal-culatetheprobabilityoffailureinthiscase.
Itsperfor-mancefunctionisgivenbyFigure1.
ComparisonamongPearsoncorrelationcoefficientsbetweencorrelatednormalvariablesX1andX2obtainedfromthreedifferentmethods.
Lietal.
113atNATIONALUNIVSINGAPOREonMarch23,2013pio.
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comDownloadedfromg(Y)=y1+y20:724Applyingtheaforementionedproceduresassociatedwiththethreemethods,theprobabilitiesoffailurecanbeobtainedbysimulation.
Fortheexactmethod,substitutingequations(15)and(17)intoequation(8)leadstou1=1ey1u2=eu1ey1eu1ey21eu1+eu1ey11eu1ey218xplicitrelationsbasedonequation(25)y1=ln1u1y2=ln1+1uln1+u2eu1euu1u2euu11!
&'8xample2.
UnlikeExample1,acomplexperformancefunctionisemployedhereintoexaminethevalidityoftheproposedprocedures.
Itsperformancefunctionisgivenbyg(Y)=C1jy1j1=2y1=32y2=31jy2j3=4+exp(y1y2)27whereC1isaconstant.
Thereliabilitylevelcanbevar-iedwhenC1takesdifferentvalues.
Itisevidentthattheprobabilityoffailureassociatedwithequation(27)can-notbecalculatedbythedirectintegrationmethod.
Inthissituation,theproposedprocedurescanbeusedtocalculatetheprobabilityoffailure.
Figure2comparestheprobabilitiesoffailureonlogscaleassociatedwiththethreemethodsforvariousval-uesofC1intheperformancefunction(equation(27)).
Thedifferenceintheprobabilitiesoffailurebetweenthetwoapproximatemethodsandtheexactmethodincreasesastheprobabilityoffailuredecreases.
Whentheyarenegativelycorrelated,bothapproximatemeth-odsunderestimatetheprobabilityoffailure,whichisunconservativeforstructuralsafetyassessment.
WhenY1andY2arepositivelycorrelated,bothmethodsPandSoverestimatetheprobabilityoffailure,whichmeanstheyareconservativeforstructuralsafetyassessment.
Figure3showstheprobabilitiesoffailureonlogscaleforvariousPearsoncorrelationcoefficientsbetweenY1andY2.
TheconstantC1isassumedtobe5.
0.
Notethatthedifferenceintheprobabilitiesoffail-urefromthetwoapproximatemethodsandtheexactmethodissignificantforastrongnegativecorrelationbetweenY1andY2.
Inaddition,thereisasmalldiffer-enceintheprobabilitiesoffailurebetweenmethodPandmethodS.
ForthecaseofrP,Y=–0.
60,theprob-abilitiesoffailureassociatedwiththeexactmethod,methodP,andmethodSare2.
11E-03,2.
54E-04,and2.
52E-04,respectively.
Table1.
ComparisonbetweenprobabilitiesoffailureusingMCS-basedproceduresanddirectintegrationmethod.
rP,YMCS-basedproceduresDirectintegrationmethodRelativeerror(%)EMPMSMEMPMSMEMPMSM–0.
61.
01E-31.
45E-35.
20E-49.
74E-41.
41E-34.
97E-43.
403.
244.
53–0.
50.
0220.
0300.
0240.
0220.
0300.
0240.
370.
681.
22–0.
40.
0520.
0630.
0560.
0520.
0630.
0550.
260.
610.
97–0.
30.
0810.
0910.
0850.
0800.
0910.
0850.
370.
630.
56–0.
20.
1080.
1160.
1110.
1070.
1150.
1100.
560.
520.
59–0.
10.
1330.
1370.
1350.
1320.
1370.
1340.
500.
500.
5100.
1560.
1560.
1550.
1560.
1560.
1540.
440.
440.
470.
10.
1790.
1740.
1770.
1780.
1730.
1760.
380.
400.
330.
20.
2000.
1900.
1960.
1990.
1900.
1950.
330.
360.
330.
30.
2200.
2060.
2140.
2190.
2050.
2130.
360.
320.
310.
40.
2390.
2200.
2310.
2380.
2200.
2300.
250.
310.
250.
50.
2560.
2340.
2470.
2550.
2330.
2460.
350.
300.
270.
60.
2720.
2480.
2620.
2710.
2470.
2610.
320.
340.
280.
70.
2840.
2600.
2750.
2830.
2590.
2740.
330.
330.
260.
80.
2920.
2730.
2860.
2910.
2720.
2850.
280.
340.
22Note:EM,PM,andSMrepresenttheexactmethod,methodP,andmethodS,respectively.
114ProcIMechEPartO:JRiskandReliability227(2)atNATIONALUNIVSINGAPOREonMarch23,2013pio.
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Tofurthervalidatetheproposedproce-dures,thefollowingperformancefunctiondiscussedinGavinandYau19isusedg(Y)=C2+0:16y113y20:04cosy1y228whereC2isaconstant.
ThereliabilitylevelcanalsobevariedwhenC2takesdifferentvalues.
Again,theprob-abilityoffailureassociatedwiththecomplexperfor-mancefunction(equation(28))cannotbecalculatedbythedirectintegrationmethod,whichcanbeevaluatedbytheproposedprocedures.
Figure4comparestheprobabilitiesoffailureonlogscaleassociatedwiththethreemethodsforvariousval-uesofC2intheperformancefunction(equation(28)).
SimilartoExample2,whenY1andY2arepositivelycorrelated,thedifferenceintheprobabilitiesoffailurebetweenthetwoapproximatemethodsandtheexactmethodincreasesastheprobabilityoffailuredecreases.
UnlikeExample2,theprobabilitiesoffailurefromthethreemethodsarealmostthesamefornegativelycorre-latedY1andY2.
Forpositivelycorrelatedvariables,thetwoapproximatemethodsunderestimatetheprobabil-ityoffailure,whichisunconservativeforstructuralsafetyassessment.
Figure2.
ComparisonamongprobabilitiesoffailureobtainedfromthreemethodsforvariousPearsoncorrelationcoefficientsbetweencorrelatednon-normalvariablesY1andY2:(a)rP,Y=–0.
6;(b)rP,Y=–0.
2;(c)rP,Y=0.
2;(d)rP,Y=0.
8(Example2).
Figure3.
EffectofPearsoncorrelationcoefficientsonprobabilitiesoffailureobtainedfromthreemethods(Example2).
Lietal.
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comDownloadedfromSimilarly,theprobabilitiesoffailureonlogscaleforvariousPearsoncorrelationcoefficientsbetweenY1andY2areplottedinFigure5.
TheconstantC2isassumedtobe4.
0.
ComparedwithExample2,thedifferenceintheprobabilitiesoffailurebetweenthetwoapproxi-matemethodsandtheexactmethodincreasesasthepositivecorrelationbetweenY1andY2becomesstron-ger.
Furthermore,theprobabilitiesoffailureassociatedwithmethodParesignificantlydifferentfromthoseassociatedwithmethodSforastrongpositivecorrela-tionbetweenY1andY2.
ForrP,Y=0.
80,theprobabil-itiesoffailureassociatedwiththeexactmethod,methodP,andmethodSare7.
33E-03,2.
88E-03,and3.
29E-04,respectively.
Example4.
Trussstructure:unlikethepreviousthreeexamplesonlyinvolvingtworandomvariables,thereliabilityofatrussstructurewithtennon-normalran-domvariablesstudiedinKimandNa20andBlatmanandSudret21isinvestigatedtoexaminetheperfor-manceoftheproposedprocedures.
Thetrussstructurewith23membersshowninFigure6isusedherein.
ThestatisticsofthebasicrandomvariablesaresummarizedinTable2.
BasedonstudiesinBucherandBourgund22Figure4.
ComparisonamongprobabilitiesoffailureobtainedfromthreemethodsforvariousPearsoncorrelationcoefficientsbetweencorrelatednon-normalvariablesY1andY2:(a)rP,Y=–0.
6;(b)rP,Y=–0.
2;(c)rP,Y=0.
2;(d)rP,Y=0.
8(Example3).
Figure5.
EffectofPearsoncorrelationcoefficientsonprobabilitiesoffailureobtainedfromthreemethods(Example3).
116ProcIMechEPartO:JRiskandReliability227(2)atNATIONALUNIVSINGAPOREonMarch23,2013pio.
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comDownloadedfromandGuanandMelchers,23theloadingsF1andF2,F3andF4,andF5andF6,areassumedtobepositivelycorrelatedwitheachother.
Forillustrativepurposes,itisfurtherassumedthatthecorrelationcoefficientsbetweentheloadings,rF1,F2,rF3,F4,andrF5,F6,arethesame.
TheunsatisfactoryperformanceunderlyingthisproblemisdefinedastheverticaldisplacementatthemidpointMofthetrussexceedingthemaximumallow-abledisplacementvmax,whichistakenasvmax=0.
11m.
20,21Thustheperformancefunctionofthecon-sideredtrussstructureisexpressedasG(X,vmax)=vmaxD(X)29whereXisarandomvectorrepresentingtheinputran-domvariables;D(X)istheverticaldisplacementofthemidpointM,whichiscalculatedbyfiniteelementanal-ysis.
Itisevidentthattheperformancefunctionshowninequation(29)cannotexplicitlybeexpressedasafunctionofinputrandomvariables,whichcannotbeevaluatedbythedirectintegrationmethodasmen-tionedpreviously.
However,theprobabilityoffailureassociatedwiththisimplicitandhighdimensionalrea-listicproblemstillcanbeevaluatedbytheproposedprocedures.
Toproducesufficientlyaccuratereliabilityresults,asamplesizeof53105isadoptedinthiscase.
Inthemeantime,theGumbeldistribution8,9,18isusedtoexaminetheperformanceoftheproposedprocedures.
Figure7showstheprobabilitiesoffailureforvari-ousPearsoncorrelationcoefficientsbetweentheload-ings.
ItcanbeobservedthattheprobabilityoffailureassociatedwithmethodPisslightlydifferentfromthatassociatedwithmethodS,buttheformerisclosertotheexactmethodthanthelatter.
BothmethodsPandSunderestimatetheprobabilityoffailure.
Whentheloadingsarecompletelyindependentorfullycorrelated,theprobabilitiesoffailureassociatedwiththreemultivariateconstructionmethodsappeartobethesame.
Furthermore,thedifferenceintheprobabilitiesoffailurebetweentwoapproximatemethodsandtheexactmethodwillapproachitsmaximumvalueforanintermediatePearsoncorrelationcoefficient.
Theseresultsindicatethattheproposedprocedurescanalsoevaluatethereliabilityofrealisticproblemsinvolvingarelativelylargenumberofrandomvariables.
ConclusionsTwomultivariateconstructionmethods,namely,theapproximatemethodPandtheapproximatemethodS,areintroduced.
TheMCS-basedproceduresarepro-posedforsimulatingthejointprobabilitydistributionsandestimatingtheprobabilitiesoffailureofcomplexperformancefunctions.
FourexampleswiththeoreticalFigure6.
Thecalculationdiagramoftrussstructure.
Figure7.
EffectofPearsoncorrelationcoefficientsonprobabilitiesoffailureassociatedwiththreemethods(Example4).
Table2.
Statisticalparametersofbasicrandomvariables.
RandomvariablesDistributiontypeMeanStandarddeviationCoefficientofvariationE1ofhorizontalmember(kN/m2)Lognormal2.
131082.
131070.
10A1ofhorizontalmember(m2)Lognormal2.
0310-32.
0310-40.
10E2ofdiagonalmember(kN/m2)Lognormal2.
131082.
131070.
10A2ofdiagonalmember(m2)Lognormal1.
0310-31.
0310-40.
10F1~F6(kN)GumbelMax507.
50.
15Lietal.
117atNATIONALUNIVSINGAPOREonMarch23,2013pio.
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comDownloadedfromjointprobabilitydistributionsarepresentedtoexaminetheaccuracyoftheproposedproceduresincomputingtheprobabilitiesoffailureforcomplexperformancefunctions.
Severalconclusionscanbedrawnfromthisstudy.
1.
ThephysicalsamplesunderlyingajointprobabilitydistributioncaneffectivelybesimulatedbytheMCS-basedprocedures,whichcanfurtherproducereliabilityresultsefficientlyandaccurately.
Theyprovideausefultoolforsolvingthereliabilityproblemswithcomplexperformancefunctionsinvolvingcorrelatedvariables.
2.
Thesimulationproceduresassociatedwiththetwoapproximatemethodsaresignificantlysimplerthantheexactmethod,especiallyforcomplexjointprobabilitydistributionfunctionsofrandomvari-ablesormultivariatedistributions.
Itisrecom-mendedthatwhenthetheoreticaljointprobabilitydistributionofrandomvariablesisunknown,thesimulationproceduresassociatedwiththetwoapproximatemethodsbeusedforreliabilityanaly-sisinvolvingcorrelatedvariables.
3.
ThePearsoncorrelationcoefficientsinthecorre-latedstandardnormalXspaceusingthethreemethodscouldbesignificantlydifferentfromthoseinthephysicalYspacefortheconsideredbivariateFrankdistribution.
Suchdifferenceshouldbekeptinmindinstructuralreliabilityanalysisinvolvingcorrelatednon-normalvariablesunderincompleteprobabilityinformation.
4.
Thedifferenceintheprobabilitiesoffailureamongthethreemethodshighlydependsonthelevelofprobabilityoffailure,theformofperformancefunction,andthedegreeofcorrelationbetweenvariables.
Generally,thisdifferenceincreasesastheprobabilityoffailuredecreases.
Itshouldbenotedthatwhentheprobabilityoffailureisverylow,theerrorsassociatedwiththetwoapproximatemeth-odsmaybemorethanoneorderofmagnitude.
FundingThisworkwassupportedbytheNationalScienceFundforDistinguishedYoungScholars[ProjectNo.
51225903],theNationalNaturalScienceFoundationofChina[ProjectNo.
51079112]andtheFoundationfortheAuthorofNationalExcellentDoctoralDissertationofPRChina[ProjectNo.
2007B50].
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