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1、COMPUTERMETHODSINAPPLIEDMECHANICSANDENGINEERING56(1986)61-81NORTH-HOLLANDPROBABILISTICFINITEELEMENTSFORNONLINEARSTRUCTURALDYNAMICS*WingKamLIU,TedBELYTSCHKOandA.MAN1NorthwesternUniversity,DepartmentofMechanicalandNuclearEngineering,Evanston,IL60201,U.S.A.Received14March1985Revisedma
2、nuscriptreceived24July1985Afiniteelementmethodapplicabletotrussstructuresforthedeterminationoftheprobabilisticdistributionofthedynamicresponsehasbeendeveloped.Severalsolutionshavebeenobtainedforthemeanandvarianceofdisplacementsandstressesofatrussstructure;nonlinearitiesduetomateria
3、landgeometricaleffectshavealsobeenincluded.Inaddition,totestthismethod,MonteCarlosimulationshavebeenusedandanewmethodwithimplicitand/orexplicittimeintegrationandHermite-Gaussquadraturehasalsobeendevelopedandused.Allthesemethodologieshavebeenimplementedintoapilotcomputercodewithtwo-
4、dimensionalbarelements.1.IntroductionThetheoryofstochasticprocessesiswellestablishedanditsdevelopmentisnowavailableinstandardtexts.Ingeneral,therandomuncertaintieswhichareincludedinastochasticprocesscanbeclassifiedintothreemajorcategories:(1)physicaluncertainty,(2)statisticaluncert
5、ainty,and(3)uncertaintyinthemodel.Adetaileddiscussionofthesetopicscanbefoundin,forexample,[l-4].Traditionally,uncertaintyanalysisinstructuralmechanicshasconcentratedonproblemsofanalmosttotallystochasticnature.Withinthissetting,evenasingledegreeoffreedomsystemwithnonlinearitiesposes
6、aformidablechallengeandhasnotbeensolvedsatisfactorily.ThemostcommonlyemployedsolutiontechniqueistheMonteCarlosimulation(seee.g.[3]).Ingeneral,thesesimulationproceduresarecomputationallyrepetitiveandthereforeexpensive,eventhoughtheyareeasilyapplicabletobothlinearandnonlinearsystems.
7、Forlinearsystems,nonstatisticalmethodssuchassecondmomentanalysis,areavailable[2].Arelatedsecond-orderperturbationtechniqueappliedtoaspecialclassoflinearstructuralvibrationsisdiscussedin[5].Theemphasisisonthemodaldecouplingoftheequationsofmotionwithuncertaindamping.Thesecondmomentan
8、alysishasalsobeenextendedi
