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President of YunTech1Rigid Mechanics and Applications to Nonlinear Structural AnalysisY. B. YangPresident, YunTech中中南大學，長沙南大學，長沙23 May 2010President of YunTech2Rigid body rulenEquilibrium nMagnitudes of acting forcesPresident of YunTech32D beam elementnEquilibrium nMagnitude of acting forces.President of YunTech4Rigid body rule nMinimum criterion for legitimacy of nonlinear elementsnRigid-body qualified geometric stiffness kg can be derivednThe rule for updating initial nodal forces existing on each elementnPatch test is a special case of the rigid body rulePresident of YunTechRigid body rulenThe rigid body rule should be obeyed in all phases of nonlinear analysisnYang & Chiou (1987), “Rigid Body Motion Test for Nonlinear Analysis with Beam Elements,” J. Eng. Mech., ASCE, 113(9), 1404-1419.5President of YunTech6Updated Lagrangian formulationnC0: initialnC1: lastnC2: currentnC1-C0: incremental step2P1P0President of YunTech73D beam element Tzbybxbbbbzayaxaaaawvuwvuu11111111111111 TxayazaxayazaxbybzbxbybzbfFFFMMMFFFMMMPresident of YunTech8Virtual work equation for 3D rigid beamnFunction of member actions 12221221201111011011021111212LyyzzxxxxLzxyxyzLyxzxLxxTIIIIFuvwFwvFdxAAAMwMvMu wMu vdxFwu vFvu wdxMv wMw vdxuff 111101122LzxxzyyxxyzMMMM ) (EnergyStrainPresident of YunTech9Moment propertiesnSemitangentialnQuasitangential Bending momentsTorquesPresident of YunTech10Rigid displacement fieldnRigid rotation (small)nConstraints bauLxuLxu 11xxaxbx Lx LbabawLxwLxwvLxvLxv1,1xbxabauu,zbzaybya,zbzazybyayLxLxLxLx1,1President of YunTech113D rigid beam elementnRigid element: nNonlinear element:nGeometric stiffness matrix 2111egkufkf 2111gkuff 00abTTaaagabTTbbbghghhihkghghhhi 1111110/0/0/ybzbybxbzbxbFLFLgFLFLFLFL 1111000/20/0/2ybxbbzbxbhMLMLMLML 11110000/2/20zbxbbybxbiMMMM President of YunTechComments nGeometric stiffness kg for rigid elementnEasy to derivenNo numerical integrationsnExplicitnNon-symmetry of the matrixnDue to lack of conjugateness between the rotational d.o.f.s and moments (Reissner 1972)12President of YunTech13Inter-element continuitynPrevious derivation is valid for a single elementnEquilibrium of joints at the deformed statenThe anti-symmetric parts of the element matrices cancel out.nThe stiffness matrix is symmetric on the structure levelPresident of YunTech14Key issue innonlinear analysisEquilibrium of joints should be enforced in deformed positionPresident of YunTechRigid displacement vs. natural deformation15President of YunTech,LargeBuckling analysisPresident of YunTechIncremental-iterative analysis17President of YunTech18sPresident of YunTech19Basic idea for incremental nonlinear analysisnDivide and conquer: Make use of the property of each stagenAll we need for structural nonlinear analysis is a linearized theory plus rigid mechanics.President of YunTech20Incremental equationnKe = elastic stiffnessnKg = geometric stiffnessnh.o.t. = higher order termsnU = displacements to be solvedn2P-1P = load increments (given) 2111. . .geKhoKPtUPPresident of YunTech21General procedurenGiven the load increments 2P-1PnPredictor: Solve for the structural displacements Unelement displacements unCorrector: Compute the element forces 2fnCompute unbalanced forces RnGo for next iterationPresident of YunTech22Predictor stagenBy assembly:nIterations are always required nK affects the direction and number of iterationsnUsing an exact K matrix does not help muchnLinearized, appoximatenUse the Kg derived for the rigid element nAll initial actions are duly considered PPUKKge1121President of YunTech23Corrector stagenInitial forces 1f (large in magnitude)nExisting on the elementnRigid body rule simple but exactnElastic actions (relatively small)nCan be computed using only kenTotal forces at C2 1212effkuRigidElasticPresident of YunTechPostbuckling response of structures24President of YunTech25Incremental-iterative nonlinear analysisPresident of YunTech26A good path-tracing schemenTo pass the critical pointsnIterations should not be performed at constant loadsnTo reflect the stiffness change using variable load incrementsnReliable indicator is needednTo reverse the loading directionsnA detector is needed to trace the postbuckling responsePresident of YunTech27Generalized displacement control (GDC) method nGeneral stiffness parameternIndicator for stiffness changenLimit pointsnUnloading 1111111iTiTUUUUGSPPresident of YunTech2824-member shallow domenHangai et al. (1971), Jagannathan et al. (1975), Holzer et al. (1980), Papadrakakis (1981)nZero load conditionnVert displ of joint 1 equals 2 cm + vert displ of joint 2President of YunTech29President of YunTech30Two-member trussnPecknold et al. (1985)nAdjacent equilibrium pathsnBench mark for testing path-tracing schemesnPerfect and imperfect loadingsPresident of YunTech31Imperfect case President of YunTech32Horizontal loadPresident of YunTech33Hinged circular arch-120-60060120020406080100120Displ. v (in)Load P (lb)P1C1P2C2P3C1-60-3003060020406080100120Displ. v (in)Load P (lb)P1C1P2C2P3C1(a) Perfect loading (b) Imperfect loading President of YunTech34Angled framePresident of YunTechRelated publicationsnYang & Shieh (1990), “Solution Method for Nonlinear Problems with Multiple Critical Points,” AIAA J., 28(12), 2110-2116. nKuo & Yang (1995), “Tracing Postbuckling Paths of Structures Containing Multi Loops,” Int. J. for Numer. Meth. in Eng., 38(23), 4053-4075.35President of YunTechComments from E.L. Cardoso et al.nCommunications in Numerical Methods in Eng., 23(4), 2006, 263-271: The GDC method ranks among many techniques proposed to solve non-linear equilibrium equations with both limit and snap-back points (Yang & Shieh 1990). is a very reliable path following method, capable of overcoming both limit and snap back points. 36President of YunTechExtension of the rigid element concept to other elementsPresident of YunTech38Triangular Plate Element (TPE)President of YunTech39Shallow domenH = 12.7 mm h = 6.35 mmthickthinPresident of YunTech40Right angled framePresident of YunTech41 President of YunTech42Related publicationsnYang et al. (2007), “Solution Strategy and Rigid Element for Nonlinear Analysis of Elastic Structures Based on Updated Lagrangian Formulation”, Eng. Struct., 29(6), 1189-1200. nYang et al (2007), “Rigid Body Concept for Geometric Nonlinear Analysis of 3D Frames, Plates and Shells Based on the Updated Lagrangian Formulation,” Comp. Meth. in Appl. Mech. & Eng., 196(7), 1178-1192. President of YunTech43Curved beamsPresident of YunTech44Key issueEquilibrium of joints should be enforced in deformed positionPresident of YunTech45Curved beam elementnRigid element: same for straight & curved beamsnTransformation of coordinatesPresident of YunTech46Positive bendingPresident of YunTech47Related publicationnYang, et al. (2007), “Rigid Element Approach for Deriving the Geometric Stiffness of Curved Beams for Use in Buckling Analysis,” J. Struct. Eng., ASCE, 133(12), 1762-1771.President of YunTechConclusionsnRigid body rulenRigid element for 3D beamnIncremental-iterative analysisnGeneral Stiffness Parameter (GSP)nLimit points and postbuckling responsenExtension to other elementsnPlate elementnCurved beam elementnSimplicity & reliability48President of YunTechhttp:/www.ce.ntu.edu.tw/ybyang/ybyangntu.edu.twThanks for your attention!
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