Plane stress and Plane strain

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1、Chapter5PLANESTRESSANDPLANESTRAINManyproblemsintheoryofelasticityaretwodimensionalinnatureandtheycanbemodeledasplanestressandplanestrain.Ontheotherhandthesolutionofthree-dimensionalelaticityproblemsaregeneralyverydifficult.Ifthestructureenablessomenecess

2、aryconditions,itcanbeanalyzedusingatwo-dimensionalmathematicalmodel.i.PlaneStress:Athinplatesubjectedtoin-planeloadingactinginitsownplane,thestateofstressanddeformationwithintheplateiscalledplanestress,Fig.5.1.Inthiscaseonlytwodimensions(intheplaneofplat

3、e)arerequiredfortheanalysis.ii.PlaneStress:Ifalongbodyissubjectedtotransverseloadinganditscrosssectionandloadingdonotvarysignificantlyinthelongitudinaldirection,asmallthicknessintheloadedareacanbetreatedassubjectedtoplanestrain,Fig.5.2.Now,strain-displac

4、ementandstress-strainrelationshipswillbedeveloped.5.1BASICEQUATIONS5.1.1Strain-DisplacementRelationshipsThedisplacementvectorδhastwocomponentsfortwodimensionalproblems.u{}δ=(5.1)vThestrain-displacementrelationshipsaregivenasfollows.Foratwodimension

5、alproblem,therewillbeonlythreeindependentstraincomponents(εx,εy,γxy)andthestrain-displacementrelations,Eqs.(2.1),reducedto∂0εx∂xεx0∂uεy=0+εy0(5.2)∂yvγxy∂∂γxy0∂y∂x68yyxxzFigure5.1Athinplateunderinplaneloading.yx

6、zyzpFigure5.2Alongcylinderunderinternalpressure.69Herethesecondtermattherighthandsideistheinitialstrainsvector,suchasthermalstrains.Thefirsttwotypes,εxandεy,arenormalstrainsinthexandydirections,andγxyisshearingstrain.Asbefore,theymaybewritteninmatrixform

7、as:{ε}=+[d]{δε}{0}(5.3)where∂0∂x∂[]d=0(5.4)∂y∂∂∂∂yxandinthecaseofthermalstrainsεx01{}ε0=εy0=α∆T1(5.5)γ0xy0forplanestressproblemsandεx01{}εεαν00==y()11+∆T(5.6)γxy00fortheplanestrainproblems.5.1

8、.2Stress-StrainRelationshipsAssuminganisotropicmaterial,weshalldeveloprelationshipsbetweenstressesandstrainsforbothplanestressandplanestrain.(i)PlaneStressσ=τ=τ=0ε=ε=0zzxyzyzzxIfthethermalstrainsaretakenintotheaccount,the3