GENERALIZED COORDINATE FINITE ELEMENT MODELS
GENERALIZEDCOORDINATE FINITEELEMENT MODELSLECTURE 457 MINUTES4·1
Generalized coordinate finite element modelsLECTURE 4 Classification of problems: truss, plane stress, planestrain, axisymmetric, beam, plate and shell con ditions: corresponding displacement, strain, andstress variablesDerivation of generalized coordinate modelsOne-, two-, three- dimensional elements, plateand shell elementsExample analysis of a cantilever plate, detailedderivation of element matricesLumped and consistent loadingExample resultsSummary of the finite element solution processSolu tion errorsConvergence requirements, physical explana tions, the patch testTEXTBOOK: Sections: 4.2.3, 4.2.4, 4.2.5, 4.2.6Examples: 4.5, 4.6, 4.7, 4.8, 4.11, 4.12, 4.13, 4.14,4.15, 4.16, 4.17, 4.184-2
Generalized coordinate finite eleDlent modelsDERIVATION OF SPECIFICFINITE ELEMENTS Generalized coordinatefinite element models (m) iIn essence, we needB(m)T C(m) B(m) dV (m)aW) JH(m) B (m) C (m)-V(m),-'-H(m)T LB(m) dV (m)V(m)R(m)!!S fSHS(m)T f S(m) dS (m)- Convergence ofanalysis results(m) -etc.AAcross section A-A:TXX is uniform.All other stress componentsare zero.Fig. 4.14. Various stress and strainconditions with illustrative examples.(a) Uniaxial stress condition: frameunder concentrated loads.4·3
Ge.raJized coordiDale finite elementlDOIIeIsHale\I\-61ZII\-\\' Tyy , TXY are uniformacross the thickness.All other stress componentsare zero.TXXFig. 4.14. (b) Plane stress conditions:membrane and beam under in-planeactions.u(x,y), v(x,y)are non-zerow 0 , E zz 0Fig. 4.14. (e) Plane strain condition:long dam subjected to water pressure.4·4
Generalized coordinate finite element modelsStructure and loadingare axisymmetric.j(IIII,I\-IAll other stress componentsare non-zero.Fig. 4.14. (d) Axisymmetric condition:cylinder under internal pressure./(before deformation)(after deformation)SHELLFig. 4.14. (e) Plate and shell structures.4·5
Generalized coordinate finite element modelsDisplacementComponentsProblemuwBarBeamPlane stressPlane strainAxisymmetricThree-dimensionalPlate Bendingu, vu, vu,vu,v, wwTable 4.2 (a) Corresponding Kine matic and Static Variables in VariousProblems.-Strain Vector TProblem(E".,)Bar[IC.,]BeamPlane stress(E"., El'l' )' "7)Plane strain(E., EJ"7 )'.7)Axisymmetric[E., E"77 )'''7 Eu )Three-dimensional [E., E"77 Eu )'''7 )'76Plate Bending(IC., 1(77 1("7).)'.,)auau aua/ )'''7 ayax'awawaoyw , IC., -dx ' IC - OyZ,IC., 2Nolallon:E.au ax' 7 11Z7710xTable 4.2 (b) Corresponding Kine matic and Static Variables in VariousProblems.4·&
Generalized coordinate finite element modelsProblemStress Vector 1:TBarBeamPlane stressPlane strainAxisymmetricThree-dimensionalPlate Bending[T;u,][Mn ][Tn TJIJI T"'JI][Tn TJIJI T"'JI][Tn TJIJI T"'JI Tn][Tn TYJI Tn T"'JI TJI' Tu ][MnMJIJI M"'JI]Table 4.2 (e) Corresponding Kine matic and Static Variables in VariousProblems.ProblemMaterial Matrix. BarBeamPlane StressE1-1':&EEl1 vv 1[o 0 1 .]Table 4.3 Generalized Stress-StrainMatrices for Isotropic Materialsand the Problems in Table 4.2.4·7
Generalized coordinate finite element modelsELEMENT DISPLACEMENT EXPANSIONS:For one-dimensional bar elementsFor two-dimensional elements(4.47)For plate bending elementsw(x,y) Y,2 Y2 x Y3Y Y4xy Y5x .(4.48)For three-dimensional solid elementsu (x,y,z) a, Ozx Y Ci4Z xy .w(x,y,z) Y, y 2x y y y z y xy .345(4.49)4·8
Generalized coordinate finite element modelsHence, in generalu (4.50)ex(4.51/52)(4.53/54)(4.55)ExamplerlpNodal point 69Element005Y.VY.Vla) Cantilever plate@CDV7741X.V8V7X.V(bl Finite element idealizationFig. 4.5. Finite element planestress analysis; i.e. T T TZZZyZX 04·9
Generalized coordinate finite element modelsLJ2. US2--II--.- - - - - - - - -. Element nodal point no. 4nodal pointno. 5 . structureelement Fig. 4.6. Typical two-dimensionalfour-node element defined in localcoordinate system.For element 2 we haveU{X,y)] (2)[ v{x,y)whereuT [U 1-4·10 H(2) u--
Generalized coordinate linite element modelsTo establish H (2) we use:orU(X,y)][ v(x,y) l!.where [ ! }! [1x y xy]andDefiningwe haveQ Aa.HenceH iPA- 14·11
Generalized coordinate finite element modelsHenceH- fll4(1 x ) ( Hy) ::aI Ia:I: (1 x )( 1 y) :andzt':U3 U4 UsU6UJH'ZJ- [00U2I0 : H IJVJU: H ZI(a)U7 Us:HI.H 16HIs: 0H zs : 0Element layout00:Hu :Ha :UIS -assemblage degreeszeroszeros(b)OJ offreedomO2x18Local-global degrees of freedomFig. 4.7. Pressure loading onelement (m)4·12v.U9 U1a0 0: HI.o : H ZJ H 21 : H:: H: 6 : 0 0: H.VI -element degrees of freedomH 17Ull U12 U1 3 U14:H IIu.
Generalized coordinate finite element modelsIn plane-stress conditions theelement strains arewhereE- au . E av.au avxx - ax' yy - ay , Yxy - ay axHencewhereI [ I1 0 y'OI000 1I 001XI10I4·13
Generalized coordinate finite element modelsACTUAL PHYSICAL PROBLEMGEOMETRIC DOMAINMATERIALLOADINGBOUNDARY CONDITIONS1MECHANICAL IDEALIZATIONKINEMATICS, e.g. trussplane stressthree-dimensionalKirchhoff plateetc.MATERIAL, e.g. isotropic linearelasticMooney-Rivlin rubberetc.LOADING, e.g. concentratedcentrifugaletc.BOUNDARY CONDITIONS, e.g. prescribeddisplacementsetc.YIELDS:GOVERNING DIFFERENTIALEQUATIONS OF MOTIONe.g.!!!).!.ax (EA ax - p(x)1FINITE ELEMENT SOLUTIONCHOICE OF ELEMENTS ANDSOLUTION PROCEDURESYIELDS:APPROXIMATE RESPONSESOLUTION OF MECHANICALIDEALIZATIONFig. 4.23. Finite Element SolutionProcess4·14
Generalized coordinate finite element modelsSECTIONdiscussingerrorERRORERROR OCCURRENCE INDISCRETIZATIONuse of finite elementinterpolations4.2.5NUMERICALINTEGRATIONIN SPACEevaluation of finiteelement matrices usingnumerical integration5.8. 16.5.3EVALUATION OFCONSTITUTIVERELATIONSuse of nonlinear materialmodels6.4.2SOLUTION OFDYNAMIC EQUILI-.BRIUM EQUATIONSdirect time integration,mode superposition9.29.4SOLUTION OFFINITE ELEr1ENTEQUATIONS BYITERATIONGauss-Seidel, NewtonRaphson, Quasi-Newtonmethods, eigenso1utions8.48.69.510.4ROUND-OFFsetting-up equations andtheir solution8.5Table 4.4 Finite ElementSolution Errors4·15
Generalized coordinate finite element modelsCONVERGENCEAssume a compatibleelement layout is used,then we have monotonicconvergence to thesolution of the problem governing differentialequation, provided theelements contain:1) all required rigidbody modes2) all required constantstrain states compatibleLWCDincompatiblelayout t: no. of elementsIf an incompatible elementlayout is used, then in additionevery patch of elements mustbe able to represent the constantstrain states. Then we haveconvergence but non-monotonicconvergence.4·16layout
Geuralized coordinate finite e1eJDeDt models,IIIIIiIII7"/(1-- -"--" 'r,;" /(a) Rigid body modes of a planestress element. QIIIIRigid bodytranslationand rotation;element mustbe stress free.(b) Analysis to illustrate the rigidbody mode conditionFig. 4.24. Use of plane stress elementin analysis of cantilever4·17
Generalized coordinate filite elellent .adels1.tI-lIIIII \\\\\\\-- - -I---('\\\Rigid body mode A2 0.--,.".- \\\-------1Rigid body mode Al 0\Poisson'sratio" 0.30II--IIIIII01r------,Young'smodulus 1.010I--\.J. .I'JRigid body mode A3 0IIIf\\,.IIIFlexural mode A4 0.57692Fig. 4.25 (a) Eigenvectors andeigenvalues of four-node planestress element\ -\\.\\'-\\I\. \\.----"'" """- \-,-- Flexural mode As 0.57692,-----,III 0.76923IIIIIIIIIIIIL 0.76923:II.JIUniform extension mode AsFig. 4.25 (b) Eigenvectors andeigenvalues of four-node planestress element4·18\. . \ r-------- 1IStretching mode A7\IIIII\IIIIIIShear mode A." 1.92308
Generalized coordiDate finite element lDodels(0 ·11G) 17IT'c.,I -./@ .f:20@ IS@)a) compatible element mesh;2constant stress a 1000 N/cmin each element. YYb)incompatible element mesh;node 17 belongs to element 4,nodes 19 and 20 belong toelement 5, and node 18 belongsto element 6.Fig. 4.30 (a) Effect of displacementincompatibility in stress prediction0yy stress predicted by theincompatible element mesh:PointABCDE2Oyy(N/m )106671635913031303Fig. 4.30 (b) Effect of displacementincompatibility in stress prediction4·19
MIT OpenCourseWarehttp://ocw.mit.eduResource: Finite Element Procedures for Solids and StructuresKlaus-Jürgen BatheThe following may not correspond to a particular course on MIT OpenCourseWare, but has beenprovided by the author as an individual learning resource.For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Generalized coordiDate finite element lDodels ·11 17 'c. IT,I .f: 20 IS a) compatible element mesh; 2 constant stress a 1000 N/cm in each element. YY b) incompatible element mesh; node 17 belongs to element 4, nodes 19 and 20 belong to element 5, and node 18 belongs to element 6. F
Finite element analysis DNV GL AS 1.7 Finite element types All calculation methods described in this class guideline are based on linear finite element analysis of three dimensional structural models. The general types of finite elements to be used in the finite element analysis are given in Table 2. Table 2 Types of finite element Type of .
boundary conditions following the standard finite element procedure. In addition the enrichment functions are easily obtained. 2. GENERALIZED FINITE ELEMENT METHOD The Generalized Finite Element Method (GFEM) is a Galerkin method whose main goal is the construction of a fin
The Generalized Finite Element Method (GFEM) presented in this paper combines and extends the best features of the finite element method with the help of meshless formulations based on the Partition of Unity Method. Although an input finite element mesh is used by the pro- . Probl
Quasi-poisson models Negative-binomial models 5 Excess zeros Zero-inflated models Hurdle models Example 6 Wrapup 2/74 Generalized linear models Generalized linear models We have used generalized linear models (glm()) in two contexts so far: Loglinear models the outcome variable is thevector of frequencies y in a table
Keywords: Global-local, Polynomial enrichment, Stable generalized FEM, Generalized FEM, Nonlinear Analysis 1Introduction The Generalized/eXtended Finite Element Method (GFEM) [1, 2] emerged from the difficulties of the FEM to solve cracking problems due to the need for a high degree of mesh refinem
1 Overview of Finite Element Method 3 1.1 Basic Concept 3 1.2 Historical Background 3 1.3 General Applicability of the Method 7 1.4 Engineering Applications of the Finite Element Method 10 1.5 General Description of the Finite Element Method 10 1.6 Comparison of Finite Element Method with Other Methods of Analysis
Finite Element Method Partial Differential Equations arise in the mathematical modelling of many engineering problems Analytical solution or exact solution is very complicated Alternative: Numerical Solution – Finite element method, finite difference method, finite volume method, boundary element method, discrete element method, etc. 9
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