Contents:Textbook:References:Topic 19Beam, Plate, andShellElements-Part I•Briefreviewofmajorformulationapproaches•Thedegenerationofathree-dimensionalcontinuumtobeamandshellbehavior•Basickinematicandstaticassumptionsused•Formulationofisoparametric(degenerate)generalshellelementsofvariablethicknessforlargedisplacementsandrotations•Geometryanddisplacementinterpolations•Thenodaldirectorvectors•Useoffiveorsixnodalpointdegreesoffreedom,theoreticalconsiderationsandpracticaluse•Thestress-strainlawinshellanalysis,transformationsusedatshellelementintegrationpoints•Shelltransitionelements,modelingoftransitionzonesbetweensolidsandshells,shellintersectionsSections 6.3.4, 6.3.5The (degenerate)isoparametricshellandbeam elements, includingthetransitionelements,arepresentedandevaluatedinBathe,K.J.,andS.Bolourchi,"AGeometricandMaterial NonlinearPlateandShell Element," Computers & Structures, 11,23-48,1980.Bathe,K.J.,andL.W.Ho,"Some Results intheAnalysisofThin ShellStructures,"inNonlinearFiniteElementAnalysisinStructuralMechanics, (Wunderlich, W.,etal., eds.), Springer-Verlag, 1981.Bathe,K. J.,E.Dvorkin,andL.W.Ho,"OurDiscrete KirchhoffandIso-parametricShell Elements for NonlinearAnalysis-AnAssessment,"Computers & Structures, 16,89-98,1983.•19-2Beam,PlateandShellElements-PartIReferences:(continued)Thetriangularflatplate/shellelement ispresentedandalsostudiedinBathe,K.J.,andL.W.Ho,"ASimpleandEffective ElementforAnal-ysisofGeneral ShellStructures,"Computers & Structures, 13,673-681, 1981.STRUCTURAL ELEMENTS• Beams• Plates• ShellsWenote thatingeometrically nonlinearanalysis, a plate (initially "flat shell")develops shell action,andisanalyzedasa shell.Various solution approaches havebeenproposed:• Use of generalbeamandshelltheories that include the desirednonlinearities.- With the governing differentialequations known, variationalformulationscanbederivedanddiscretized using finite elementprocedures.- Elegant approach, but difficultiesariseinfinite element formulations:• Lack of generality• Large numberofnodal degreesof freedomTopicNineteen19-3Transparency19-1Transparency19-219-4 Beam,PlateandShellElements-PartITransparency19-3•Useof simple elements, but a largenumber of elementscanmodelcomplex beamandshell structures.-Anexampleisthe use of 3-nodetriangular flat plate/membraneelementstomodel complex shells.- Coupling between membraneandbending actionisonly introducedatthe element nodes.- Membrane actionisnot very wellmodeled.bendingf membraneartificial.ISsstiffnessI\~3/ degree of freedom with\/_~artificial stiffness~'/..zL5}~--xlX1Stiffness matrixinlocal coordinatesystem(Xi).Example:Transparency19-4• Isoparametric (degenerate) beamandshell elements.- These are derived from the 3-Dcontinuum mechanics equationsthatwediscussed earlier, but thebasic assumptionsofbeamandshell behavior are imposed.- The resulting elementscanbeused to model quite generalbeamandshell structures.Wewill discuss this approachinsomedetail.Basic approach:•Usethe totalandupdated Lagrangianformulations developed earlier.TopicNineteen19-5Transparency19-5Transparency19-619-6Beam,PlateandShellElements-PartITransparency19-7Transparency19-8Werecall, for theT.L.formulation,fHAtS..~HAtE..0dV_HAt(jJ}Jov0II'U0II'-;:ILLinearization~£'vOCiirsoers80ei~°dV +f,vJSi~8o"ly.°dV=HAtm- f,vJSi~8oey.°dVAlso, for theU.L.formulation,JVHA~Sij.8HA~Eij.tdV=HAtmLinearization~JvtCifsters8tei}tdV+Jv~i~8t'T\ij.tdV= HAt9R-f",t'Tij.8tei}tdV• Imposeonthese equations the basicassumptions ofbeamandshell--action:1)Material particles originallyonastraight line normaltothe mid-surface of thebeam(or shell)remainonthat straight linethroughout the response history.For beams, "plane sections initiallynormaltothe mid-surface remainplane sections during the responsehistory".The effect of transverse sheardeformationsisincluded,andhence the lines initially normaltothe mid-surfacedonot remainnormaltothe mid-surface duringthe deformations.TopicNineteen19-7Transparency19-9Transparency19-1019-8Beam,PlateandShellElements-PartITransparency19-11 time 0not90°ingeneraltime tTransparency19-122)The stressinthe direction "normal"tothe beam (or shell) mid-surfaceiszero throughout the response history.Note that here the stress along thematerial fiber thatisinitially normaltothe mid-surfaceisconsidered;because of shear deformations, thismaterial fiber does not remainexactly normaltothe mid-surface.3)The thickness of the beam (or shell)remains constant(weassume smallstrain conditions but allow for largedisplacements and rotations).TopicNineteen19-9FORMULATION OFISOPARAMETRIC(DEGENERATE) SHELLELEMENTS•Toincorporate the geometricassumptions of "straight lines normal tothe mid-surface remain straight",andof"the shell thickness remains constant"weuse the appropriate geometricanddisplacement interpolations.•Toincorporate the condition of "zerostress normal to the mid-surface"weuse the appropriate stress-strainlaw.Transparency19-13Transparency19-14rX2tv~= director vector at node kak= shell thickness at node k(measured into direction oftv~)Shell element geometryExam~:9-nodeelement19-10Beam,PlateandShellElements-PartITransparency19-15Elementgeometrydefinition:•Inputmid-surfacenodalpointcoordinates.•InputallnodaldirectorvectorsattimeO.•Inputthicknessesatnodes.Transparency19-16r---X2- material particle(OXi)• Isoparametric coordinate system(r,s, t):- The coordinatesrands aremeasuredinthe mid-surfacedefinedbythe nodal pointcoordinates(asfor a curvedmembrane element).- The coordinate tismeasuredinthe direction of the director vectorat every pointinthe shell.sTopicNineteen19-11Interpolation of geometry at time0:Transparency19-17N Nk~'hk°Xr,+~k~'a~hk°V~imid-surface effect of shellonly thicknessmaterialparticlewith isoparametriccoordinates(r,s,t)hk= 2-D interpolation functions (asfor 2-D plane stress, planestrain and axisymmetric elements)°X~= nodal point coordinates°V~i= components of°V~Similarly, at timet,0t-coordinatet~h t k<D~htvkXi=L.JkXi+ 2L.Jakk ni~k=1vvvk=1\N\IV~I ITransparency19-18The nodal point coordinates and directorvectors have changed.X30vk_n}-----+-------/-'---X219-12Beam,PlateandShellElements-PartITransparency19-19Toobtain the displacements of anymaterial particle,t t 0Ui=Xi-XiHenceN NtUi=k~1hktu~+~k~1akhkCV~i-°V~i)wheretu~-tx~-°x~I - I I(disp.ofnodalpointk)Transparency19-20tV~i-°V~i= changeindirection cosinesof director vector at node kThe incremental displacements fromtime t to timet+atare, similarly, forany material