A load space formulation for probabilistic finite element analysis of structural reliability

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ProbabilisticEngineeringMechanics14(1999)73±81Aloadspaceformulationforprobabilistic®niteelementanalysisofstructuralreliabilityX.L.Guan*,R.E.MelchersDepartmentofCivil,SurveyingandEnvironmentalEngineering,TheUniversityofNewcastle,Callaghan,NSW2308,AustraliaAbstractAloadspaceformulationforcalculatingthefailureprobabilityofcomplexstructuresforwhichthelimitstatefunctionsareimplicitisdescribedinthispaper.Thisformulationisusedinconjunctionwithprobabilistic®niteelement(PEE)analysisandemploysadirectionalsimulationtocalculatethestructuralreliability.Apartfromtheadvantagethatalowerorderspaceisused,themainadvantageoftheloadspaceformulationproposedinthispaperisthatthenumberofinversionsofthestructuralstiffnessmatrixand/oritsgradientswithrespecttothematerialpropertyrandomvariablesisreduceddramaticallywhencomparedwiththeusualMonteCarlosimulation(MCS)method.Whenusedina®niteelementreliabilityanalysis,thisprocedurecansavesigni®cantamountsofCPUtime.Numericalexamplesarepresentedtoshowtheef®ciencyandaccuracyoftheproposedapproach.q1998ElsevierScienceLtd.Allrightsreserved.1.Introductioncostly.ToreducethetotalCPUtime,severalalternativemethodshavebeenproposed.Theseincludethe®rst-orderInstructuralreliabilityanalysis,themodellingofthesecond-moment(FORM)method[4,10,22],theresponsestructuralmechanicstoasuf®cientdegreeofre®nementissurface(RS)method[2,5,12,17],thetangent-multi-plane-equallyasimportantasthedeterminationofstructuralfail-surface(TMPS)method[6],andtheinterior-multi-plane-ureprobabilities.Ideally,structuralmechanicsmodellingsurface(IMPS)method[7].However,eachoftheseshouldinvolveastate-of-the-artmethodofstructuralanaly-methodshaslimitations.Inthepresentpaper,acomputa-sis,suchas®niteelementtechniques.When®niteelementtionalapproachbasedontheuseofloadspaceformulationtechniquesareusedinareliabilityanalysis,theyareusuallyisproposed,anditisusedinconjunctionwithPFEanalysisreferredtoasprobabilistic®niteelement(PFE)techniques.toestimatethestructuralfailureprobability.TheloadspaceThisisbecausetheparametersdescribingastructurearemethodhasbeenusedpreviouslyfortimeinvariantrelia-nowmodelledasrandomvariablesand/orrandom®elds.bilityestimationproblemsforstructureswiththeexplicitForexample,thestructuralmaterialpropertieswouldlimitstatefunctions[9,18].Thismethodhasalsobeenusuallybemodelledasrandom®elds.Thisimpliesthatappliedfortimevariantproblemsandforsystemreliabilitytheuncertaintiesinstructuralpropertiesaredistributedinanalysis[15,16].arandommannerspatiallyoverastructure,therebyformingThroughtheuseof®niteelementanalysis,theloadspacearandom®eld.Thesearethenrepresentedbyanumberofapproachhastheadvantagethattheproblemisreducedtoarandomvariablesthroughoneoftherandom®elddiscreti-smallnumberofrandomvariables(orrandomprocesses)inzationtechniques[4,8,10,11,19±21].theloadspace.The(largenumberof)randomvariablesInprinciple,the``exact''valueofthestructuralfailureobtainedfromdiscretizationoftherandom®eldsdescribeprobabilityforotherthansimplestructurescanbeestimatedthestructureandinparticularitsstructuralstrengthinthethroughtheMonteCarlosimulation(MCS)method.Unfor-loadspace.However,thereisa(small)pricetopayforthistunately,thecomputationalcostoftheMCSmethodisapproach.Thisisthatthe(probabilistic)relationshipusuallyveryhigh.Thereasonforthisissimplythatabetweentherandom®eldrandomvariablesandtheoverallcomplete®niteelementanalysisisinvolvedforeachrealisa-structuralresponserandomvariablesneedstobeestimated.tionofthevectorofrandomvariables(i.e.foreachsampleThiscanbeachievedthroughtheuseof®niteelementanaly-generated)andbecausea®niteelementanalysisitselfissis.Apartfromthefactthatalowerorderspaceisused,amajoradvantageoftheloadspaceformulationisthatthenumberofthePFEexecutions,whichinvolveanumberof*Correspondingauthor.Tel.:161-49-216-073;Fax:161-49-216-991;e-mail:cerem@cc.newcastle.edu.au.inversionsofthestructuralstiffnessmatrixand/orits0266-8920/99/$-seefrontmatterq1998ElsevierScienceLtd.Allrightsreserved.PII:S0266-8920(98)00017-4 74X.L.Guan,R.E.Melchers/ProbabilisticEngineeringMechanics14(1999)73±81gradientswithrespecttothebasicrandomvariables(i.e.structuralresistancevectorR.TherandomvariablesS,structuralmaterialproperties),canbereduceddramatically.givenbyRS´A1C,isthencompletelyde®nedasaIn®niteelementreliabilityanalysis,thisprocedurecanbefunctionofX:signi®cantinsavingCPUtime.Thiswillbedemonstratedinalatersectionthroughnumericalexamples.SSXuAa(3)BecausethelimitstatefunctionscannotbeexpressedexplicitlyintermsoftherandomvariablesXforcomplex2.Loadspaceformulationinhyperpolarcoordinatesstructures,therelationshipbetweenSandXisnoteasilyde®nedÐamoredetaileddiscussionofthisrelationshipThefailureprobabilityofastructurecanbecalculatedinwillbegiveninthenextsection.Forthepresent,itistheloadspaceinpolarcoordinatesfromthefollowingnotedthatbecauseofthis,theconditionalPDFFS/Ahastoformula[15,16]:bedeterminedapproximatelyasfollows."#fa´Fs=a´fs´a1csm21Forstructureswhicharemodelledbyrandomstrength,AS=AQPfEuES=u(1)dimensions,etc.,thedistributionofSmightbeassumedhuukS=us=uadditiveinnatureandsincethedimensionofXofteniswherefA(a)istheprobabilitydensityfunction(PDF)ofthelarge,thecentrallimittheoremcanbeinvokedtosuggestunitvectorofdirectioncosinesA;Sdenotesthescalarstruc-thatSisapproximatelyGaussiandistributedaccordingtoturalstrengthSs$0;misthenumberofrandomvari-ZS2mSablesforloads;h(u)isanimportancesamplingPDFfrom1sS21v2QFSspe2dv21#s#1(4)whichsamplesuaredrawn.Thenthedirectioncosineais2p21obtainedfromaa(u).AlsokS/u(s/u)isanappropriatelywheremSisthemeanvalueofSandsSdenotesthestandardchosenimportancesamplingPDFintheradialdirectionsdeviationofS,calculatedfrom:andFS/A(s/a)isthecumulativedistributionfunction(CDF)pforS/A.Finally,fQ(s´a1c)istheprobabilitydensityfunc-sSVarS(5)tionfortheloadvectorQ.TheassumptionCmQisoftenmade,inwhichinwhichVar(S)representsthevarianceofS.However,ScontainsXimplicitly,sothatthemeanandQS´A1C(2)varianceofSarenotnecessarilyeasilyobtainedfromthesolutionofthefollowingintegrations:However,whentheunitsofmeasurementforQarenotconsistentforeachqi,alineartransformationisappropriateZ1Z1[14].EvidentlySandAarerelatedtothemdimensionalrealmS¼Sx1;x2;¼;xnfx1;x2;¼;2121mspaceRde®nedintheloadcapacityvectorofthestructureRthroughR2CS´A.Itdescribes,for®xedXx,thexx1;x2;¼;xndxdx¼dx(6)n12ncriticallimitstatefunction(s)forthestructure,suchthatwhentheloadvectorQisgreaterthantheloadcapacityZ1Z12vectorR,thestructurefails(seeFig.1).Hence,thesystemVarS¼Sx1;x2;¼;xn2mSfx1;x2;¼;2121strengthisdescribedbyaprobabilisticallyde®nedsafedomainD.Formoststructuralsystems,Disde®nedthroughxx1;x2;¼;xndxdx¼dx(7)n12noneormorelimitstatefunctionsGi(Q,X)0,(iInsteadoftheintegrations,the®rsttwomomentsofSmight1,2,¼,nl),wherenlisthenumberoflimitstatefunctions,beestimatedapproximatelyasfollows.QandXrepresenttheloadvectorandthematerialpropertyvariables.Inthehyperpolarformulation,itfollowsthatG(Q,X)0impliesthattheloadvectorQequalsthe2.1.Differentiablelimitstatefunction(LoadSpaceFormulation1)Ausefulapproachistoletthe®rsttwomomentsofSbeapproximatedbyexpandingSS(X)inaTaylorseriesaboutthemeansofX[1,13].Bytruncatingtheseriesatlinearterms,the®rsttwomomentsofSareapproximatedas:mS.SmXuxmX;Aa(8)XnrXnrVarScicjCovXi;Xj(9)i1j1Fig.1.Realizationofonerayandonelimitstate(afterMelchers[15]).wherenristhenumberofrandomvariablesformaterial X.L.Guan,R.E.Melchers/ProbabilisticEngineeringMechanics14(1999)73±8175properties,i.e.X,andTheconventional®niteelementmethodforlinearstaticproblemsiswellknown,andneednotberepeatedhere,save2SXci;uxmx;Aa(10)forthereadertorefertospeci®cresultslaterinthepaper.2xiThedisplacementresponseUofalinearstructureunderEvidently,fromEq.(10),thisapproachisonlyfeasibleforstaticloadsisgivenbythefollowinggoverningequationdifferentiablelimitstatefunctions.Inthiscase,afterthe®rst[3]:twomomentsofshavebeenobtainedusingEqs.(8)±(10),K´UF(14)theCDFFs(s)canbecalculatedbyEqs.(4)and(5).whereKistheglobalstiffnessmatrix.Itisafunctionof2.2.Nondifferentiablelimitstatefunction(LoadSpacematerialpropertyrandomvariablesX.AlsoFistheglobalFormulation2)loadvectorwhichisafunctionofloadrandomvariablesQ.TheglobalstiffnessmatrixKisobtainedbyassemblingtheIfthelimitstatefunctionisnotdifferentiable,theaboveelementstiffnessmatricesovertheentirestructuralregionmethodtocalculatethe®rsttwomomentsofSinAaisPteeasKe1k,whereteisthetotalnumberofelementsandnotapplicable.ThemeanandvarianceofScanbeestimatedthesummationimpliestheadditionoftheappropriatebyusingsimulationinwhichthesamplesSÃareobtainedbyelementstiffnessmatricesatappropriatelocationswithingeneratingsamplesXÃfromfx(x)andevaluatingSÃthroughtheglobalstiffnessmatrix.TheglobalnodalforcevectorthestructuralanalysisSS(X).ThemeanandvarianceofSFisobtainedbyassemblingnodalforcevectorsFthenarecalculatedby:Pteeeee1F,andkandFareobtainedby:1XNZmSS^i(11)keBTDeBdv(15)Ni1veZZXNeTTS^2m2FBDe0dv2Bs0dviSvevei1VarS(12)ZZN21TT1NBFdv1NSTds(16)whereNisthesamplesize.Asbefore,theCDFF(S)isveseScalculatedbyusingEqs.(4)and(5).inwhichBisthestrain-displacementmatrix,Deistheelas-Limitedexperiencegainedfromthepresentstudyticitymatrixofelemente,Nistheshapefunction,veisthesuggeststhatatleast800samplesarerequiredtoobtainavolumeofanelementeandseisitssurface,SdenotestheTreasonableestimateforVar(S).Thisexperiencesuggestsexternalsurfacetractions,BFisthebodyforces,ande0andalsothatthisapproachtendstoproduceamoreaccurates0denoteinitialstrainsandinitialstresses.estimateforthe®rsttwomomentsofScomparedwiththeAccordingtotheconventional®niteelementtheory,the®rstapproach.Thisislikelytobeduetothe®rstapproachstrainandstressatanypointwithinanelementcanbeproducingerrorsduetolinearisationatthemeanvalueobtainedfrom:point.eeB´u(17)3.Conventional®niteelementanalysiseesD´B´u(18)eItiswell-knownthatthelimitstatefunctionsG()forwhereuisthenodaldisplacementofelemente.complexstructuresmaynotbeexpressedexplicitlyintermsofthebasicrandomvariables.Further,G()isde®ned4.LoadspaceformulationforcomplexstructuresbyafailuremodewhichisafunctionoftheresponsesRS(suchasmaximumstresss,straine,ordeformationUataAsnotedearlier,inordertoestimateFS(s),therelation-speci®cpoint).SincethemeasurementsandobservationsshipbetweenSandXhastobedetermined®rst.Therela-areusuallymadeonbasicquantitiesratherthanthetionshipbetweenSandXforthreedifferenttypesoflimitresponseswhicharederivedquantities,theresponsesRSstatefunctionswillnowbegiven.mustbetransformedfrombasicvariablesXandQthroughCase1.Considerthedisplacementlimitstatefunction:thestructuralmechanicstransformation:GU02Uip(19)RSRSX;Q(13)whereU0isthethresholddisplacementataspeci®cpoint,Forallbuttrivialstructures,thistransformationisavailableUiprepresentstheipthcomponentofdisplacementvectorUonlyinanalgorithmicsense,e.g.a®niteelementcode.atthesamepoint.FromEq.(14)Therefore,theparametersinloadspaceformulationshavetobederivedwiththeaidofPFEanalysis.21UKF(20) 76X.L.Guan,R.E.Melchers/ProbabilisticEngineeringMechanics14(1999)73±81Xneandsetting21UipKipjFj(21)j1XnbXne21ss1DBmn;ng;kd;lKlj´FAj(28)wherenedof´nn,dofisthedegreeoffreedomofthel1j1structure,andnndenotesthetotalnumberofnodesinthe®niteelementmesh.TheequivalentnodalforcesFcanbeeXnbXneobtainedbeassemblingtheelementnodalforcesFwhichis21ss2DBmn;ng;kd;lKlj´Fmj(29)calculatedusingEq.(16).ThetermsSTandBTarereplacedl1j1byQ.ThenthejthnodalforceFjcanbewrittenas:thelimitstatefunctionofEq.(27)issimpli®edto:FjS´FAj1Fmj(22)Gs02S´ss11ss2(30)whereShasthesamemeaningasbefore,andFmjandFAjarecalculatedastheequivalentnodalforcesbyusingmQAgainsettingG0producesandAastheappliedloads,respectively.SubstitutingEqs.s02ss2(21)and(22)intoEq.(19),gives:S(31)ss1Xne21GU02KipjS´FAj1Fmj(23)Case3.Inthiscase,considertheprincipalstresslimitj1statefunctionandlettingG0,thereisobtained:Gsp2sp(32)0mn;ng;ddXnepU2K21´Fmwheresdenotestheprincipalstressthresholdataspeci®c0ipjj0pj1point,smn;ng;ddistheprincipalstresscomponentattheS(24)Xnesamepoint,mn,nghavethesamemeaningsasthoseusedK21´FipjAjforsmn;ng;dd,dddenotesthecomponentnumberofprinci-j1palstressinwhichdd1anddd2representtheprincipalstressfortensileandcompressioncomponents,respectively.Case2.ConsiderthestresslimitstatefunctiontobeFrombasicmechanics,theprincipalstressescanbedeter-analysedas:minedas:Gs02smn;ng;kd(25)s1spmn;ng;1mn;ng;2smn;ng;dd2wheres0denotesthestressthresholdataspeci®cpoint,mn,ng,kdrepresenttheelementnumber,Gaussianpointssmn;ng;12smn;ng;22number,andthecomponentofthestress(e.g.kd1,2,3^1s2(33)2mn;ng;3representthehorizontal,verticalandshearstresscompo-nent,respectively)atthesamepoint.CombiningEq.(18)andEqs.(21)and(22),leadsto:wheres(mn,ng,kd)(kd1,2,3)canbeexpressedasinEq.(26)(dd21)andtheoperator^canbereplacedby(21).For01XnbXneconvenience,setsS@DBK21´FAmn;ng;kdmn;ng;kd;lljAjl1j1XnbXne21skd;1DBmn;ng;1Klj´FAj(34)XnbXnel1j1211DBmn;ng;kd;lKlj´Fmj(26)l1j1XnbXne21wherenbdof£nd,nddenotesthenumberofnodesperskd;2DBmn;ng;kd;1Klj´Fmj(35)l1j1elementin®niteelementmesh.UponsubstitutingEq.(26)intoEq.(25)Then801