可靠性

ReliabilityBasedDesignOptimizationOutlineRBDOproblemdefinitionReliabilityCalculationTransformationfromX-spacetou-spaceRBDOFormulationsMethodsforsolvinginnerloop(RIA,PMA)MethodsofMPPestimationTerminologiesX:vectorofuncertainvariablesη:vectorofcertainvariablesΘ:vectorofdistributionparametersofuncertainvariableX(means,s.d.)d:consistsofθandηwhosevaluescanbechangedp:consistsofθandηwhosevaluescannotbechangedTerminologies(contd..)Softconstraint:dependsuponηonly.Hardconstraint:dependsuponbothX(θ)andη[θ,η]=[d,p]Reliability=1–probabilityoffailureRBDOproblemOptimizationproblemminF(X,η)objectivefi(η)0gj(X,η)0RBDOformulationminF(d,p)objectivefi(d,p)0softconstraintsP(gj(d,p)0)PthardconstraintsComparisonb/wRBDOandDeterministicOptimizationDeterministicOptimumReliabilityBasedOptimumFeasibleRegionBasicreliabilityproblemfR(xR)fS(xS)=const.xSXSXSxRfR(xR)fS(xS)failuredomain:xRxSfSsafedomain:xRxSlimitstatesurfaceg=xRxS=0ProbabilityoffailureReliabiltyCalculationReliabilityindexReliabilityIndexFormulationofstructuralreliabilityproblemx1x20sfg(x)=0fX(x)=const.Vectorofbasicrandomvariablesrepresentsbasicuncertainquantitiesthatdefinethestateofthestructure,e.g.,loads,materialpropertyconstants,membersizes.LimitstatefunctionSafedomainFailuredomainLimitstatesurfaceGeometricalinterpretationuRfailuredomainDfDSsafedomainuS0limitstatesurfaceTransformationtothestandardnormalspaceDistancefromtheorigin[uR,uS]tothelinearlimitstatesurfaceCornellreliabilityindexHasofer-Lindreliabilityindex•Lackofinvariance,characteristicfortheCornellreliabilityindex,canberesolvedbyexpandingtheTaylorseriesaroundapointonthelimitstatesurface.Sincealternativeformulationofthelimitstatefunctioncorrespondtothesamesurface,thelinearizationremainsinvariantoftheformulation.•Thepointchosenforthelinearizationisonewhichhastheminimumdistancefromtheorigininthespaceoftransformedstandardrandomvariables.Thepointisknownasthedesignpointormostprobablepointsinceithasthehighestlikelihoodamongallpointsinthefailuredomain.Forthelinearlimitstatefunction,theabsolutevalueofthereliabilityindex,definedas,isequaltothedistancefromtheoriginofthespace(standardnormalspace)tothelimitstatesurface.Geometricalinterpretationu*G1(u)=0G2(u)=0G3(u)=0DfDSfailuredomainsafedomain0u1u2Hasofer-LindreliabilityindexRBDOformulationsRBDOMethodsDoubleLoopDecoupledSingleLoopDoubleloopMethodObjectivefunctionReliabilityEvaluationFor1stconstraintReliabilityEvaluationFormthconstraintDecoupledmethod(SORA)DeterministicoptimizationloopObjectivefunction:minF(d,µx)Subjectto:f(d,µx)0g(d,p,µx-si,)0InversereliabilityanalysisforEachlimitstatedk,µxkk=k+1si=µxk–xkmppxkmpp,pmppSingleLoopMethodLowerlevelloopdoesnotexist.min{F(µx)}fi(µx)≤0deterministicconstraintsgi(x)≥0wherex-µx=-βt*α*σα=grad(gu(d,x))/||grad(gu(d,x))||µxl≤µx≤µxuInnerLevelOptimization(CheckingReliabilityConstraints)ReliabilityIndexApproach(RIA)min||u||subjecttogi(u,µx)=0ifmin||u||βt(feasible)PerformanceMeasureApproach(PMA)mingi(u,µx)subjectto||u||=βtIfg(u*,µx)0(feasible)MostProbablePoint(MPP)Theprobabilityoffailureismaximumcorrespondingtothempp.ForthePMAapproach,-grad(g)atmppisparalleltothevectorfromtheorigintothatpoint.MPPliesontheβ-circleforPMAapproachandonthecurveboundaryinRIAapproach.ExactMPPcalculationisanoptimizationproblem.MPPesimationmethodshavebeendeveloped.MPPestimationinactiveconstraintactiveconstraintRIAMPPPMAMPPPMAMPPRIAMPPUSpaceMethodsforreliabilitycomputationFirstOrderReliabilityMethod(FORM)SecondOrderReliabilityMethod(SORM)Simulationmethods:MonteCarlo,ImportanceSamplingNumericalcomputationoftheintegralindefinitionforlargenumberofrandomvariables(n5)isextremelydifficultorevenimpossible.Inpractice,fortheprobabilityoffailureassessmentthefollowingmethodsareemployed:u2G(u)=0DsDfl(u)=0*u*0u1regionofmostcontributiontoprobabilityintegraln(u,0,I)=constFORM–FirstOrderReliabilityMethodu2Gv(v)=fv(v)–vn=0DsDf0u1vnvnv~~vn=sv(v)v*~SORM–SecondOrderReliabilityMethodGradientBasedMethodforfindingMPPfindα=-grad(uk)/||grad(uk)||uk+1=βt*αIf|uk+1-uk|ε,stopuk+1isthempppointelsegotostartIfg(uk+1)g(uk),thenperformanarcsearchwhichisauni-directionaloptimizationAbdo-Rackwitz-FiessleralgorithmRackwitz-FiessleriterationformulafindsubjecttoGradientvectorinthestandardspace:whereisaconstant1,istheotherindicationofthepointintheRFformula.foreveryandConvergencecriterionVeryoftentoimprovetheeffectivenessoftheRFalgorithmthelinesearchprocedureisemployedMeritfunctionproposedbyAbdoAbdo-Rackwitz-FiessleralgorithmAlternateProblemModelsolutionto:min‘f’s.tatleast1ofthereliabilityconstraintisexactlytangenttothebetacircleandallothersaresatisfied.Assumptions:minimumoffoccursattheaforesaidpointAlternateProblemModelReliabilitybasedoptimumβ1β2x1x2ScopeforFutureResearchDevelopingcomputationallyinexpensivemodelstosolveRBDOproblemThemethodsdevelopedthusfararenotsufficientlyaccurateIncludingrobustnessalongwithreliabilityDevelopingexactmethodstocalculateprobabilityoffailure