计算固体力学课件ComputationalSolidMechanics3章节优质PPT.ppt

计算固体力学课件ComputationalSolidMechanics3章节优质PPT.ppt

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（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?

（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