弹性力学课件31

2.7BOUNDARYCONDITIONInanalysingplaneproblem,thefollowingeightequationsmaybeconsideredasbasicequation:

twodifferentialequationsofequilibrium;threegeometricalequations;

threephysicalequations.Thebasicequationsinvolveeightunknownfunctionsofxandy:

threestresscomponents;

threestraincomponents;twodisplacementcomponents.Underproperboundaryconditions,theeightbasicequationscanbesolvedfortheeightunknownfunctions.Accordingtoboundarycondition,elasticityproblemsareclassifiedasdisplacementboundaryproblems,stressboundaryproblemsandmixedboundaryproblems.Foraplaneproblem,wehavexybaxabyus=uvs=va.Inadisplacementboundaryproblem,thesurfacedisplacementsofthebodyarespecified.b.Inastressboundaryproblem,thesurfaceforcesactingonthebodyareprescribedandsuchaconditioncanbetransformedintoaconditionaboutthestresscomponentsattheboundary.xyqAXN=lx+mxyYN=my+lxyBPyyxxyxsYNXNThesurfaceforcesareXN=XYN=Ylx+mxy=Xmy+lxy=YThesetwoequations,expressingtherelationsbetweentheboundarystresscomponentsandthesurfaceforcecomponents,arethestressboundaryconditionsofaplaneproblem.Whentheboundaryisnormaltoacoordinateaxis,thestressboundaryconditionsaresimplified2xy3141and3：l=0m=+1y=Y;xy=X++2and4：m=0l=+1x=X;xy=Y++Usepositiveornegativesignaccordingastheoutwardnormalisalongthepositiveornegativedirectionofthecoordinateaxis.c.Inamixedboundaryproblem,someportionoftheboundaryisspecifiedwithknowndisplacementswhiletheotherportionissubjectedtoknownsurfaceforces.Forinstance：xyABCAB：v=0AC：x=0；xy=0Mixedboundaryproblem解：AB边：y=-q;yx=0BC边：x=0;xy=0CD边：y=0;yx=0AD边：u=0；v=0例1、写出图示悬臂梁的边界条件，板厚为1ABCDyxq解：BC边：x=0;xy=0CD边：y=0;yx=0AD边：u=0；v=0AB边：y=yx=0例2、写出图示悬臂梁的边界条件，板厚为1xABCDyq0l解：BC边：y=0;yx=0DE边:AB边：x=-gyxy=0例3、写出图示结构AB、BC、DE的应力边界条件水的重度为ExyABCDNcos(N,x)=cos=lcos(N,y)=cos(90+)=-sin=mlx+mxy=Xmy+lxy=Ycosx-sinxy=0-siny+cosxy=02.8SAINT-VENANT’’SPRINCIPLEInsolvinganelasticityproblem,itisrathereasytoobtainthestresses,strainanddisplacementswhichsatisfyallthebasicequations.However,weoftenencounterdifficultiesinhavingalltheboundaryconditionscompletelysatisfied.Moreover,ithappensfrequentlyinthestresscalculationforastructuralormachineelementthatweknowonlytheresultantofsurfaceforcesonasmallportionoftheelement,butnotthedistributionoftheforces.Undersuchcircumstances,Saint-Venant’sprinciplemaybeofmuchhelptous.Theessenceoftheprinciplecanbestatedasfollows:Ifasystemofforcesactingonasmallportionofthesurfaceofanelasticbodyisreplacedbyanotherstaticallyequivalentsystemofforcesactingonthesameportionofthesurface,theredistributionofloadingproducessubstantialchangesinthestressesonlyintheimmediateneighborhoodoftheloading,andthestressesareessentiallythesameinthepartsofthebodywhichareatlargedistancesincomparisonwiththelineardimensionofthesurfaceonwhichtheforcesarechanged.By““staticallyequivalentsystems””wemeanthatthetwosystemshavethesameresultantforceandthesameresultantmoment.Forinstance：PPPP/2P/2P/2P/AP/APPPPP/2P/2P/2P/AP/APThesolutionforstressesinthecasedisrathersample,asthestressboundaryconditionsareverysimple.AccordingtoSaint-Venant’sprinciple,itssolutionforthestressescanbeappliedtothecasea,bandc.Forcasee,wehavedisplacementb