Contents:Topic 20Beam, Plate, andShellElements-Part II•Formulationofisoparametric(degenerate)beamelementsforlargedisplacementsandrotations• Arectangularcross-sectionbeamelementofvariablethickness;coordinateanddisplacementinterpolations•Useofthenodaldirectorvectors•Thestress-strainlaw•Introductionofwarpingdisplacements•Exampleanalysis:180degrees,largedisplacementtwistingofaring•Exampleanalysis:Torsionofanelastic-plasticcross-section•Recommendationsfortheuseofisoparametricbeamandshellelements•Thephenomenaofshearandmembranelockingasobservedforcertainelements•Studyofsolutionsofstraightandcurvedcantileversmodeledusingvariouselements•Aneffective4-nodeshellelement(theMITC4element)foranalysisofgeneralshells•Thepatchtest,theoreticalandpracticalconsiderations•Exampleanalysis:Solutionofathree-dimensionalsphericalshell•Exampleanalysis:Solutionofanopenbox•Exampleanalysis:Solutionofasquareplate,includinguseofdistortedelements•Exampleanalysis:Solutionofa30-degreeskewplate•Exampleanalysis:Largedisplacementsolutionofacantilever20-2Beam,PlateandShellElements-PartIIContents:(continued)Textbook:Example:References:•Exampleanalysis:CollapseanalysisofanI-beamintorsion•Exampleanalysis:CollapseanalysisofacylindricalshellSections 6.3.4, 6.3.56.18The displacement functionstoaccount for warpingintherectangularcross-section beamareintroduced inBathe,K.J.,andA.Chaudhary, "OntheDisplacement FormulationofTorsionofShaftswithRectangular Cross-Sections," InternationalJournal for Numerical MethodsinEngineering, 18, 1565-1568, 1982.The 4-nodeand8-node shell elements based on mixed interpolation(i.e.,theMITC4andMITC8 elements)aredevelopedanddiscussed inDvorkin, E.,andK.J.Bathe,"AContinuum Mechanics Based Four-Node Shell Element for General Nonlinear Analysis,"EngineeringComputations,1,77-88,1984.Bathe,K.J.,andE.Dvorkin,"AFour-NodePlateBending ElementBased on Mindlin/Reissner Plate Theoryanda Mixed Interpolation,"InternationalJournalforNumerical MethodsinEngineering, 21,367-383,1985.Bathe, K.J.,andE.Dvorkin,"AFormulationofGeneral Shell Ele-ments-TheUseofMixed InterpolationofTensorial Components,"InternationalJournalforNumerical MethodsinEngineering,inpress.The I-beam analysis isreportedinBathe,K.J.,andP.M.Wiener, "On Elastic-Plastic AnalysisofI-BeamsinBendingandTorsion," Computers & Structures, 17,711-718,1983.The beam formulation is extendedtoa pipe element, including ovali-zation effects, inBathe,K.J.,C.A.Almeida,andL.W.Ho,"ASimpleandEffective PipeElbowElement-SomeNonlinear Capabilities," Computers & Struc-tures,17,659-667,1983.FORMULATION OFISOPARAMETRIC(DEGENERATE) BEAMELEMENTS• The usual Hermitian beam elements(cubic transverse displacements,linear longitudinal displacements) areusually most effectiveinthe linearanalysis of beam structures.• Wheninthe following discussionwerefer to a "beam element"wealwaysmean the "isoparametric beamelement."• The isoparametric formulation canbeeffective for the analysis of- Curved beams- Geometrically nonlinear problems- Stiffened shell structures(isoparametric beam and shellelements are coupled compatibly)• The formulation is analogous to theformulation of the isoparametric(degenerate) shell element.Topic 'l\venty20-3Transparency20-1Transparency20-220-4Beam,PlateandShellElements-PartIITransparency20-3Transparency20-4Consider a beam element with arectangular cross-section:ak= thicknessatnode kint-directionbk= thickness atnode kins-directionConsider a beam element with arectangular cross-section:ak= thickness atnode kint-directionbk= thicknessatnode kins-directionTopic Twenty20-5Consider a beam element with arectangular cross-section:Transparency20-5ak= thicknessatnode kint-directionbk= thicknessatnode kins-directionX2t~= director vectorins-directiont~= director vectorint-directionak= thickness atnode kint-directionbk= thickness atnode kins-directionConsider a beam element with arectangular cross-section:X2t~~= director vectorins-directiont'i~= director vectorint-directionTransparency20-620-6Beam,PlateandShell Elements -PartIIwheretv~= direction cosines of the directorvectorinthe t-direction, of nodek at time ttV~i= direction cosines of the directorvectorinthe s-direction, of nodek at time tThe coordinates of the materialparticles of the beam are interpolated asN Nt~htkt~htVkXi=~kXi+ 2~akktik=1k=1N+~k~1bkhktV~iTransparency20-7Transparency20-8N t NtUi=}:hktur + 2}:akhkCV~-°V~)k=1 k=1N+ 2sLbkhkCV~i-°V~i)k=1The vectors°V~and°V~can becalculated automatically from the initialgeometry of the beam element if theelementisassumed to lie initiallyinaplane.AlsoU. -t+dtX· -tx·1-I IN t N S N=~hku~+-2~akhkV~+-2~bkhkV~ik=1 k=1 k=1wherev~andV~iare incrementsinthedirection cosines of the vectorstvrandtv~.These increments are givenTnterms of the incremental rotationsftk,about the Cartesian axes,asVr=ftkxtvr;V~=ftkXtv~• Using the above displacement and geometryinterpolations, we can develop the strain-displacement matrices for the Cartesian straincomponents. A standard transformation yieldsthe strain-displacement relations correspondingto the beam coordinatesTI,E,,.Topic1\venty20-7Transparency20-9Transparency20-102o-BBeam,PlateandShell Elements -PartIITransparency20-11• The stress-strain relationship used forlinear elastic material conditions isTJ~TJ~- componentso0]Gk 0o GkTransparency20-12k = shear correction factorsince only the one normal and twotransverse shear stresses are assumedtoexist.• The material stress-strain matrix foranalysis of elasto-plasticity or creepwouldbeobtained using also thecondition that only the stresscomponents(T1TJ),(TJ~)and(TJ~)arenon-zero.Topic'I\venty20-9Transparency20-13exactwarpingdisplacementsforsquaresectionexactwarpingdisplacementsfor infinitelynarrow section•NotethatthekinematicassumptionsInthebeamelementdonotallow-sofar-forcross-sectionalout-of-planedisplacements(warping).Intorsionalloading,allowingforwarpingisimportant.•Wethereforeamendthedisplacementassumptionsbythefollowingdisplacements:t.u,Torsion constant k in formula,T=kGOa3bTransparency20-14kaAnalytical value(Timoshenko)ADINA1"02"04"010·0100"00'1410"2290"2810"3120'3330"1410"2300"2890'3230"33320-10Beam,PlateandShellElements-PartIITransparency20-15Example:TwistingofaringAll dimensions in inchesthickness= 0.2E=3x 105psiv=0.30.2Transparency20-16Finite element mesh: Twelve 4-nodeiso-beam elements4-nodeelementexprescribedTopic1\venty20-11DemonstrationPhotograph20-1Close-upof;'ringdeformationsUse the T.L. formulation to rotate the ring180