3 Application of discontinuity layout optimization

,20130009,published24April20134692013Proc.R.Soc.A SamuelHawksbee,ColinSmithandMatthewGilbert three-dimensionalplasticityproblemsApplicationofdiscontinuitylayoutoptimizationto  References/content/469/2155/20130009.full.html#ref-list-1Thisarticlecites21articles,1ofwhichcanbeaccessedfreeThisarticleisfreetoaccessSubjectcollections(199articles)mechanicalengineering (30articles)civilengineering  ArticlesonsimilartopicscanbefoundinthefollowingcollectionsEmailalertingserviceheretheboxatthetopright-handcornerofthearticleorclickReceivefreeemailalertswhennewarticlescitethisarticle-signupin/subscriptionsgoto:Proc.R.Soc.ATosubscribetoonMay4,2014DownloadedfromonMay4,2014Downloadedfromrspa.royalsocietypublishing.orgResearchCitethisarticle:HawksbeeS,SmithC,GilbertM.2013Applicationofdiscontinuitylayoutoptimizationtothree-dimensionalplasticityproblems.ProcRSocA469:20130009.:4January2013Accepted:28March2013SubjectAreas:civilengineering,mechanicalengineeringKeywords:limitanalysis,plasticity,discontinuitylayoutoptimization,three-dimensionalanalysisAuthorforcorrespondence:MatthewGilberte-mail:[e-mail]uitylayoutoptimizationtothree-dimensionalplasticityproblemsSamuelHawksbee,ColinSmithandMatthewGilbertDepartmentofCivilandStructuralEngineering,UniversityofSheffield,MappinStreet,SheffieldS13JD,UKAnewthree-dimensionallimitanalysisformulationthatusestherecentlydevelopeddiscontinuitylayoutoptimization(DLO)procedureisdescribed.WithDLO,limitanalysisproblemsareformulatedpurelyintermsofdiscontinuities,whichtaketheformofpolygonswhenthree-dimensionalproblemsareinvolved.Efficientsecond-orderconeprogrammingtechniquescanbeusedtoobtainsolutionsforproblemsinvolvingTrescaandMohr–Coulombyieldcriteria.Thisallowstraditional‘upperbound’translationalcollapsemechanismstobeidentifiedautomatically.Anumberofsimplebenchmarkproblemsareconsidered,demonstratingthatgoodresultscanbeobtainedevenwhencoarsenumericaldiscretizationsareemployed.1.IntroductionTheformaltheoremsofplasticlimitanalysisprovidethetheoreticalframeworknecessarytoallowdirectevaluationoftheloadrequiredtocausecollapseofabodyorstructure,withouttheneedforintermediatecalculationsteps.Whileearlyworkinthefieldfocusedonthedevelopmentofhand-typelimitanalysismethods,whicharestillwidelyusedinengineeringpractice,morerecentresearchhasfocusedonthedevelopmentofcomputationalmethods,suchasfinite-elementlimitanalysis[1–3].Computationallimitanalysismethodshavesuccessfullybeenusedtoobtainboundsonthecollapseloadforplanestrainproblems,providingarapidandrobustmeansofevaluatingsafety.However,applicationoflimitanalysistechniquestothree-dimensionalproblemshasbeenmorelimited.2013TheAuthors.PublishedbytheRoyalSocietyunderthetermsoftheCreativeCommonsAttributionLicense:20130009..................................................Thisisunfortunatesinceengineersareincreasinglyseekingtomodeltherealthree-dimensionalgeometryofagivenproblem,ratherthanatwo-dimensionalidealization.Researchonthree-dimensionalproblemshasfocusedprimarilyonexploitingthepotentialoffinite-elementlimitanalysisformulations,makinguseofoptimizationtechniquestoobtainupperorlowerboundsolutions.Theeffectivenessofagivenformulationisnormallycloselylinkedtothatoftheoptimizationtechniqueusedtoobtainsolutions,andtheyieldcriterioniskeyindeterminingwhichoptimizationtechniqueisappropriateforaparticularformulation.Park&Kobayashi[4]wereamongthefirsttodevelopalimitanalysisformulationforthree-dimensionalproblems,referredtoasthe‘rigid-plasticfinite-elementmethod’.Theirapproachinvolvedrelaxingtheflowruletoobtainadirectestimationofthecollapseload.Rigid-plasticfiniteelementshavesuccessfullybeenusedtoinvestigateawiderangeofmetal-formingapplications.However,themethodappearsonlytohavebeenappliedtovonMisesorDrucker–Pragermaterials,andthesolutionsobtaineddonothaveclearupperorlowerboundstatuswithinthecontextofthefundamentaltheoremsofplasticlimitanalysis.Lyamin&Sloan[5,6]proposedathree-dimensionalfinite-elementlimitanalysisformulationthatusednonlinearprogrammingtoobtainsolutions.Theirformulationcouldbeappliedtoproblemsinvolvinganyyieldfunctionthatiseverywheredifferentiable.However,thisprecludesdirectuseoftheMohr–CoulombyieldcriterionowingtothesingularitythatexistsattheapexoftheMohr–Coulombcone.Consequently,Lyamin&Sloan[5]proposedtheuseofahyperbolicsmoothingtechniquetocreateanapproximatedyieldsurface,differentiableeverywhere.MorerecentlyKrabbenhøftetal.[7]andMartin&Makrodimopoulos[8]haveusedsemidefiniteprogramming(SDP)tocreateformulationscapableofhandlingtheMohr–Coulombyieldfunctiondirectly.However,atpresentSDPalgorithmsarerelativelyimmature,limitingthesizeofproblemsthatcanbesolved.Chenetal.[9]usedfullyrigidfiniteelementstoobtainupperboundsolutionsforthree-dimensionalslopestabilityproblems.Adistinguishingfeatureoftheirformulatio