岩土工程外文文献

FORM,SORM,andspatialmodelingingeotechnicalengineeringB.K.Low⇑SchoolofCivilandEnvironmentalEngineering,NanyangTechnologicalUniversity,SingaporearticleinfoArticlehistory:AvailableonlinexxxxKeywords:ReliabilityindexFORMSORMSpatialautocorrelationsGeotechnicalengineeringabstractAnintuitiveellipsoidalperspectiveisdescribedtogetherwiththreespreadsheet-automatedconstrainedoptimizationalFORMproceduresandaSORMapproach.ThethreeFORMproceduresarethencomparedinthecontextofgeotechnicalexamplesofaconfinedsoilelement,arockslope,andanembankmentonsoftgroundwithspatiallyautocorrelatedundrainedshearstrengthinthesoftclayfoundation,theper-formancefunctionofwhichisbasedonareformulatedSpencermethodwithsearchforreliability-basedcriticalnoncircularslipsurface.Twomethodsofmodelingspatialautocorrelationsarepresented,andthemeritsandlimitationsofthethreeconstrainedoptimizationalFORMproceduresarestudied.Thecomple-mentaryrolesandinterconnectionsamongthethreeconstrainedoptimizationalFORMproceduresandSORMapproachareemphasized.ComparisonsarealsomadewithMonteCarlosimulations.2013ElsevierLtd.Allrightsreserved.1.IntroductionTheHasofer-Lind[13]indexforcaseswithcorrelatednormalrandomvariablesandthefirst-orderreliabilitymethod(FORM)forcaseswithcorrelatednonnormalsarewellexplainedinDitlevsen[10],Shinozuka[39],AngandTang[1],Melchers[34],HaldarandMahadevan[12],andBaecherandChristian[2],forexample.ThepotentialinadequaciesoftheFORMinsomecaseshavebeenrecognized,andmorerefinedalternativesproposed,inChenandLind[7],DerKiureghianetal.[9],WuandWirsching[47],andZhaoandOno[49],amongothers.Ontheotherhand,theusefulnessandaccuracyoftheFORMinmostapplicationsarewellrecognized,forinstancebyRackwitz[38].Thefocusofthispaperisonspreadsheet-basedproceduresforFORM(whichextendstheHasofer–Lindindexforcorrelatednor-malsintothenonnormalrealm),SORMonthefoundationofFORMresults,systemFORM,andreliabilityanalysisaccountingforspa-tiallyautocorrelatedsoilproperties.Specifically,asimplegeome-chanicsexampleisfirstexaminedtoillustratespreadsheetbasedSORManalysisonthefoundationofFORMreliabilityindexandFORMdesignpoint.Thisisfollowedbyarockslopewithcorrelatednonnormalrandomvariables,solvedusingtheuspaceapproachforcomparisonwiththeLowandTang[29]nspaceapproach.Finally,spatiallyautocorrelatedshearstrengthismodeledinthereliabilityanalysisofanembankmentonsoftground.Theadvanta-gesandlimitationsofthreeFORMcomputationalapproaches,namelyconstrainedoptimizationwithrespecttotheoriginalrandomvariables,thenormalizedbutunrotatednvector,andthenormalizedandrotateduvector,respectively,areinvestigated.Spatialautocorrelation(alsotermedspatialvariability)arisesingeologicalmaterialbyvirtueofitsformationbynaturalprocessesactingoverunimaginablylongtime(millionsofyears).Thisen-dowsgeomaterialwithsomeuniquestatisticalfeatures(e.g.spa-tialautocorrelation)notcommonlyfoundinstructuralmaterialmanufacturedunderstrictqualitycontrol.Forexample,bythenat-ureoftheslowprecipitation(overmanyseasons)offine-grainedsoilparticlesundergravityinwaterinnearlyhorizontallayers,twopointsincloseverticalproximitytooneanotherarelikelytobemorepositivelycorrelated(likelytohavesimilarundrainedshearstrengthcuvalues,forexample)thantwopointsfurtherapartintheverticaldirection.SystemFORMandSORMareextensionsofFORM.TheclassicalcomputationalapproachofFORMinnormalizedandrotatedu-spaceiselegantbutshroudedinmathematicaldetails.Anintu-itiveperspectiveandtwospreadsheet-automatedFORMcomputa-tionalapproacheswereprovidedinLowandTang[28,29],withtheaimtofacilitateunderstanding.Thetwoapproaches(inthexspaceandnspace,respectively)aresummarizedinthenextsection,togetherwithathirdalternativeofspreadsheetconstrainedopti-mizationintheuspace.Theyaremeanttocomplementtheelegantclassicalu-spaceFORMapproach.TheLowandTang[29]n-spaceapproacheasilyrevertstotheu-space(requiredforSORMcompu-tation)usingasimpleequation.Thespreadsheet-basedreliabilityapproachescanbeappliedtoimplicitfunctionsorstand-alonenumericalpackages,viatheresponsesurfacemethods(RSM),forexample.RSMasabridgebetweenspreadsheet-basedgeotechnicalFORManalysisandstand-alonenumericalpackagesisillustratedbythefollowing:0167-4730/$-seefrontmatter2013ElsevierLtd.Allrightsreserved.⇑Tel.:[phone].E-mailaddresses:[e-mail],[e-mail](2013)xxx–xxxContentslistsavailableatScienceDirectStructuralSafetyjournalhomepage::LowBK.FORM,SORM,andspatialmodelingingeotechnicalengineering.StructSaf(2013),(i)XuandLow[48]conductedFORManalysisonsecond-degreepolynomialresponsesurfacefunctionconstructedfromfiniteelementanalysisofembankmentstability.(ii)ChanandLow[6]constructedsecond-degreepolynomialresponsesurfacebasedonfiniteelementanalysisofalater-allyloadedpile,andconductedFORMandSORManalyses.(iii)LüandLow[32]conductedFORMandSORManalysesonthemovementofahorseshoe-shapedhighwaytunnel,inwhichtheresponsesurfacefunctionwasbasedonnumericalanal-ysesusingtheco