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3-DCurvedBeamProblemSpecificationTheproblemconsideredhereisthecurvedbeamofuniformtrapezoidalcross-sectioninexample6.15ofCooketal.ThebeamisbentinitsownplanebymomentsM.Theproblemisnotaxisymmetricbecausedisplacementshavecircumferentialaswellasradialandaxialcomponents.Soweuse3Dsolidelementsratherthanaxisymmetricelements.Thegeometrycanneverthelessbedescribedincylindricalcoordinates.Wewouldliketoobtainthestressesforthetrapezoidalcross-sectionAAshownabove.Stressesinthecurvedbeamdonotvarywithθ,sowecanreducethemodelandanalyzeonlyatypicalslicebetweentwocloselyspacedradialplanesasshownbelow.TheanglebetweenABandCDistakentobe5deg.assuggestedbyCookelal.ThebendingmomentMmustbeappliedindirectlyinthereducedmodelsincewedon'tknowapriorithecircumferentialstressdistributionitproducesonthecross-section.Instead,we'llprescribedisplacementssuchthatradialplanesectionsremainplaneandapuremomentloadactsonthemodeli.e.nonetforceactsonit.ThemomentMcanbecomputedfromthestressdistributiononthecross-sectionobtainedfromFEA.Stressesscalelinearlywiththeappliedmoment.SothestressesassociatedwithaprescribedmomentMpcanbeobtainedbymultiplyingthecomputedstressesbytheratioMp/M.Thez-constantplanecontainingA,B,CandDisasymmetryplane.Soonlyhalfthecross-sectionneedstobemodeled.BoundaryConditionsThenodald.o.f.intheradial(u),circumferential(v),andaxial(w)directionsareconstrainedasfollows:Face1Face2u=0atnodeAv=0atallnodesv=0.0001(rc-r)atallnodesw=0alongABw=0alongCDAllremainingd.o.f.areunrestrained.Settingu=0atApreventsrigidbodymotioninther-direction.Settingv=0onface1nodespreventscircumferentialmotionofface1.Settingw=0onABCDimposessymmetryaboutthemiddler-θplane.TheaboveBConface2nodescausesface2toremainplaneasitrotatesaboutaz-parallelaxisatr=rc.Thefactor0.0001isarbitrarilychosen.Attheoutset,theappropriatevalueofrcisnotknown.TherightvalueofrcwillgiveapurebendingloadsothattheradialreactionRAatnodeAiszero.TwopreliminaryFEanalysiswithguessvaluesofrc=60mmandrc=70mmweredone.TherespectiveRAvaluesturnouttobe2001Na
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