计算固体力学课件ComputationalSolidMechanics3章节.ppt
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1,CHAPTER3TWODIMENSIONALFEM,2,3-1PLANEELASTICPROBLEM,1.CONSTITUTIVERELATIONS,when,ijiscalledengineeringstrainkisdummyindex,rangeover1,2G,aretheLameconstants,3,2.GEOMETRICALRELATION,4,3EQUATIONSOFEQUILIBRIUM,Dividingthroughbydx1dx2dx3,wehave,5,Cyclicpermulationofsymbols,Iftheredonotexistexternalmomentproportionaltoavolume,thisleadtothestresstensorissymmetric,Inabovederivation,Twoimportantassumptionsarebasedon:
Thereisdefinitioneverywhereinthebody.Thestressiscontinuous.,Forrod,Forbeam,6,4.PLANESTATEOFSTRESS,Assumptions:
(1)Uniformandthinplate,
(2)Twosurfacesarefree,allexternalloadinthexy-planeareindependentofcoordinatez,(3)Elasticpropertiesareindependentofz,7,Question:
doesthedirectstrainzvanish?
No,becausethetwosurfacesarefree!
8,5.PLANESTATEOFSTRAIN,Assumptions:
(1)Uniformandlongcolumn,
(2)allexternalloadparalleltothesectionareindependentofcoordinatez,(3)Elasticpropertiesareindependentofz,9,Question:
doesthedirectstresszvanish?
No,thedeformationalongzdirectionisconstrained.,Attentions:
Theequilibriumandgeometricalequationsaresameforplanestressandplanestrainproblems?
(2)Theuniquedistinctionistheconstitutiverelations,andtransformationisavailable.SameFEModelcanbeused,PLANESTRAIN,10,6.TheMinimumPotentialEnergyPrinciple,Thestrainenergydensityfunctionis,Bu:
fixedboundaryB:
freeboundary,11,Eulerequations,Naturalboundaryconditions,12,7.RayleighQuotient,Kineticcoefficientwithoutconcentratedmass,Maximumpotentialenergywithoutconcentratedsprings,13,8.FiniteElements,
(1).Rectangularelements:
inCartesiancoordinates,suitableforregulardomain.,
(2).Triangularelements:
inareacoordinates,suitableforregularandirregulardomain,ortakingastransitionelementofdifferentsizerectangularelement,(3).Iso-parametricelements:
inCartesiancoordinates,suitableforanydomain.,14,9.RectangularElements,Useonedimensionalfunctions:
Questions(3):
(1)C0orC1?
Completeness?
(2)Thenumberofnodalparametersforeachnode?
(3)Compatibilityatnodeandalongsideline?
15,Bilinearelement:
n=2,similarly,Questions
(2):
(1)why,?
(2)Howtointerprettheshapefunctiongeometrically?
16,Thediscretizationofstrainenergygivestheelementstiffnessmatrix,Attention:
Stiffnessmatrixdependsona/bonly.largeorsmallelementhasthesamestiffness?
(2)Finedmeshisgoodornot?
(3)Thenodaldisplacementareexact?
17,双线性矩形单元,双线性矩形单元的质量矩阵,18,19,Attention:
Massmatrixandloadvectorisproportionaltotheareaa*boftheelement.,
(2)Concentratedloadisasingularloadforplaneproblem,thecorrespondingdisplacementisinfiniteintheory!
Thefinerofthemesh,thelargerofthedisplacement!
20,10.TriangularElements,
(1)Areacoordinate:
(2)RelationofareaandrectangularCartesiancoordinate,Questions
(2):
Twoofthemareindependent,Whytointroduce1,2,3?
(2)1,2,3varieslinearly?
21,(3)Allowabledisplacementfunction,Attentions
(2):
(1)Completepolynomials,impropernodecollocation.,
(2)non-completepolynomialsmaycausecalculationdifficulties!
BecauseitisC0,soLagrangeinterpolationpolynomialscanbeuseddirectly:
22,nistheorderofelement,isthecoordinateofnodei,istotalnumberofnodesoftheelement,when,Attentions:
(1)ConstantStrainTriangularelement,CST,
(2)Lineholdsafterdeformation!
Socompatibilityholds!
23,when,:
elementwithlinearstrain,Usefulincalculationofstructuralmatrices!
24,11.CurvilinearElements,Attentions:
(1)hardtoprovetheconvergenceofcurvilinearelement,
(2)Curvilinearormappingelementfromparentelement,parentelement,curvilinearelement,25,1)GeometricalField,Itisneededtodefinethearbitrarycurvemathematically.,Niistheshapefunctionofparentelement,mappingmisthenumberofthenodesused,Discussions(side14):
(1)Twonodesdefinealine(1,4),
(2)Threenodesdefineaparabola(1,8,4),(3)Alineneedtwonodes,notthree,emphasizingthatitisnotnecessarytouseallnodestodefinethegeometry.,26,2)DisplacementField,Ni:
theshapefunctionofparentelementn,m:
thenumbersofthenodesusedtodefinedisplacementandgeometry.,Whenn=m,theiso-parametricelement等参元,usedwidely,Whennm,thesub-parametricelement亚参元,displacementgradientislarge,Whennm,thesuper-parametricelement超参元displacementgradientissmall,geometryiscomplicate,27,3)ElementStiffnessMatrix,Questions:
(1)Howtocalculate/x,/y?
(2)Howtotransformdxdytodd?
(3)Whatisthedistinctionofcalculationmethodofcurvilinearelementwithothertypeelements?
28,
(1)Howtocalculate/x,/y?
Whynotcalculatedirectly?
(2)Howtotransformdxdytodd?
(3)Thestiffnessmatrix,29,4)TheRelationBetweenTheJacobianMatrixDeterminantAndThePropertyOfCurvilinearElement,
(1)Foursidesarestraightline:
4501800,thendet(J)0if=1800,thendet(J)=0,
(2)Foursidesareparabola,threenodesoneachside:
themiddlenodeshouldbelocatedwithintheinner1/3oftheside.,(3)Theorderofelementislargerthan2:
thepropertyoftheelementdependsonthesignandmagnitudeofJacobianmatrixJ.,30,PracticalApplication(6):
(1)Thestiffnessofbilinearrectangularelementdependsonlyontheratioa/b.,
(2)Theconcentratedloadisasingularload,thecorrespondingdisplacementincreasesasfinenessofthemesh.,(3)TriangularelementwiththreenodesisCST,payattentiontotheaccuracy.,(4)Payattentiontothedifferenceofplacestressandplanestrain.,(5)Shellelementisthecombinationofplaneelementandplateelement.,(6)Theoretically,therearenoexactnodalresultsforplaneproblems,theaccuracyisassuredbythemesh.,31,Homework:
(1)ToanalyzethedisplacementofsideABloadcase1:
concentratedloadactingatpointsA,Bloadcase2:
distributedloadactingonthelineAB,
(2)Tocalculatethefirst10frequencies.,Usingthreemeshes:
(1)1*1
(2)5*5(3)10*10,32,3-2THINPLATEPROBLEM,Assumptions:
(1)Straightnormallineholdsafterdeformation,
(2)Thereisnoextensionorcompressionformiddleplane,(3)Out-of-planestressesaremuchsmallerthein-plane.,and
(2)areKirchhoffhypotheses,33,1Basicformulae,Accordingtoassumption
(1)and
(2):
Accordingtoassumption(3),theconstitutiverelationofplanestressisadopted:
arenecessaryforthebalanceofstresses,34,Relationofgeneralizedstressandgeneralizedstrain,Accordingtothebalanceconditiontheinfinitesimalvolume,wehave:
isN,notN.M,isN/M,notN,Theunitof,Theunitof,35,x,yarecalledasflexurestress,xyastwistingstresstheyareprimary(major)stresses(oddfunctionofz,fromconstitutiverelationanddefinition):
xz,yzastransverseshearingstress,theyaresecondary(minor)stresses(evenfunctionofz,fromequilibriumequation):
zascompressionstress,itissecondary(minor)stresses(fromequilibriumequation):
36,Thedirectionsofmoment,shearforceandtwistingmoment:
37,2Transformationofcoordinates,
(1)Considerationofboundaryconditions,
(2)Derivationofformulae,Tensorofrankone,Tensorofranktwo,38,3TheMinimumTotalPotentialEnergyPrinciple,B:
freeboundaryB:
simply-supportedboundary,w=0Bu:
fixedboundary,w=0,Question:
whynotconsiderboundarytwistingmoment?
39,Followingformulaeareusedinthevariationcalculation:
40,Eulerequilibriumequationbi-harmonicequation,atpointconcentratedloadacting,freeboundaryB,Qn,Mnsarenotindependent,why?
simply-supportedboundaryB,41,4FiniteElement,Attentions:
(1)C1,likebeam;,
(2)completecompatiblerequirements:
(3)partcompatibility:
wiscontinuous,arenotcontinuous,(4)excessivecompatibility:
exceptarecontinuous,arealsocontinuous,42,1)Rectangularelementwith12nodalparameters(Zienkiewicz和Clough),Partcompatibleelement:
(1)wiscontinuousatnodeandside,iscontinuousatnode,butnotatsidetakeside14asanexample,43,2)Rectangularelementwith16nodalparameters双三次板单元(Bogner),excessivecompatibleelement:
(1)wiscontinuousatnodeandside,iscontinuousatnodeandsidetakeside14asanexample,(3)iscontinuousatnode,44,3)Rectangularelementwith24nodalparameters(1971年Popplewell和McDonaldpresentedthenodals,Gopalacharyulu和Watkinpresentedthefunctions),excessivecompatibleelement,45,4)Triangularelementwith6nodalparameters,Partcompatibleelement:
(1)wiscontinuousatnodesonly,iscontinuousatmiddlepointsofeachside,(3)Constantinternalmomentelement,46,5)Triangularelementwith9nodalparameters,Partcompatibleelement:
(1)wiscontinuousatnodesandsides,
(2)iscontinuousatnodesonly,Completetopower3,hasnocontributiontodeflectionandderivativetoallnodes,allzerovales,Bazelysuggests,symmetry,47,6)Triangularelementwith18nodalparameters,excessivecompatibleelement,48,CompleteCompatibleTriangularElementWith6NodesAnd12NodalParameters,
(1)ChoseapointO:
(2)Displacementfunctionsof012,023,031arethecompletepolynomialsw1,w2,w3topower3,whichcanbedeterminedby10nodalparametersincluding,(3)Atsideofeachsmalltriangle,wiscontinuous,thecontinuityofnormalderivativeisguaranteedbyfollowingconditions,thethreetemporarynodalparametersarethusdetermined.,49,3-4MINDLIN(REISSNER)PLATE剪切板,Thinplate,Thickplate,50,FiniteElementOfThickPlate:
(1)C0element;,
(2)Allowabledisplacementfunction,formedasplaneproblem;threeindependentdisplacementparameters,(3)Boundaryeffectsofplate;,3.6二维数值模拟问题讨论,51,在NASTRAN中,二维单元包括膜(membrane平面应力)弯板(bendingpane不考虑剪切变形)剪切板(shearpanel)壳(shell考虑剪切变形)默认单元二维实体(2Dsolid如平面应变,轴对称),52,例1:
四边简支板在均布法向载荷作用下中点和E点的挠度分析,薄板经典理论,长宽分别为40,20,厚度为1,53,薄板单元为非协调单元,剪切板单元是协调单元,54,长宽分别为40,20,厚度为5,对于厚板两种单元结果相差大于20,55,例2:
四边简支板在法向集中载荷作用下中点和E点的挠度分析,薄板单元,剪切板单元,56,薄板,厚板,剪切板单元,剪切板单元,薄板单元,薄板单元,面内剪应力分布(剪切板的边界效应),57,例3:
小变形与大变形问题:
四边简支和四边固支板:
刚度变大悬臂板:
刚度边小,58,例4:
频率与模态(薄板),四边简支薄板,59,四边固支薄板,60,四边简支板各阶模态形状:
11,21,31,21,22,41,重频,61,例5:
平面问题,应力约束和平衡条件为,应力解析解,62,方法1:
位移边界,中线在自由端的挠度,与用Euler梁得到的结果完全相同,63,方法2:
边界条件,中线在自由端的挠度,T