chap 2 introduction to the stiffness：2章介绍了刚度

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1、9CE631Chapter6PlaneFrameandGridEquationspage6.4GridEquations1.Grid:isastructureonwhichloadsareappliedperpendiculartotheplaneofthestructure,asopposedtoaplaneframewhereloadsareappliedintheplaneofthestructure.·AddTorsionalEffectsinabasicbeamelement:99CE631

2、Chapter6PlaneFrameandGridEquationspage2.UsingDirectStiffnessMethodtodeveloptorsionalstiffnesscoefficients:·Basicunknowndisplacementsfortorsion–Angle-of-twistfExercise5:Writebasicformulasforacircularbarintorsion:·RecallMechanicsofMaterials:foracircularcr

3、osssection(Seenextpage)Shearstrainatanyradialr:g=(rdf)/dx=r(df/dx)df/dx:therateofchangeoftheangleoftwistfwithrespecttoxShearstresst:(forlinearmaterial)t=GgG:shearmodulusTorqueinacrosssectionmx:mx=tr=RJ/Rtr=R=mxR/JJ:thepolarmomentofinertiaR:radiusoftheci

4、rcularbarmember·99CE631Chapter6PlaneFrameandGridEquationspageTorsionaldeformationofabarsegment99CE631Chapter6PlaneFrameandGridEquationspageBasicunknownsforacircularbarintorsion:Angle-of-twistfatnodes·Elementstype:Choosetwonodesforanelement:Node1,node2,Þ

5、2DOF:Angle-of-twistf1xandf2x·Shapefunctions:2DOFÞLinearinterpolationfunctions:ÞN1(x)forf1xatnode1:N1=1-(x/L)N2(x)forf2xatnode2:N2=x/L·FieldAngle-of-twistf(x)isafunctionofnodalAngle-of-twistf1andf2:f(x)=N1(x)f1x+N2(x)f2x=[N1(x)N2(x)]{f1xf2x}T·Exercise6:D

6、evelopaformulaofanodaltorqueasafunctionofnodalangle-of-twistf1xandf2xandwriteitinmatrixform:UsingtheformulasfromMechanics:f1x,f2xÞf(x)Þg=r(df/dx)Þt=GgÞmx=tr=RJ/RÝShapefunctionsN1(x),N2(x)99CE631Chapter6PlaneFrameandGridEquationspageÞg,t,mxareallfunction

7、soff1x,f2x:m1x=(f1x-f2x)GJ/Lm2x=(f2x-f1x)GJ/L·Inmatrixform:·Stiffnessmatrixforthetorsionbar:·99CE631Chapter6PlaneFrameandGridEquationspageLocalstiffnessmatrixofagridelementincludingtorsionaleffects:·Globalstiffnessmatrixofagridelement:coordinatetransfor

8、mation:kG=TTGkGTGExercise7:WhichtermsinkGreflecteffectsfromtorsion?·(Example6.5onpage224)99CE631Chapter6PlaneFrameandGridEquationspage6.5BeamElementArbitrarilyOrientedinSpace:-3DBeamElement99CE631Chapter6PlaneFrameandGridEquation