英汉双语弹性力学9.ppt

1,Chapter9Torsion,2,第九章扭转,3,Chapter9Torsion,9-1TheTorsionofEqualSectionPole,9-2TheTorsionofEllipticSectionPole,9-3Membraneassimilation,9-4TheTorsionofRectangularSectionPole,9-5TheTorsionofRingentThinCliffPole,Torsion,4,第九章扭转,9-1等截面直杆的扭转,9-2椭圆截面杆的扭转,9-3薄膜比拟,9-4矩形截面杆的扭转,9-5开口薄壁杆件的扭转,扭转,5,Materialmechanicshassolvedthetorsionproblemsofroundsectionpole,butitcantbeusedtoanalyzethetorsionproblemsofnon-roundsectionpole.Forthetorsionofanysectionpole,itisarelativelysimplespatialproblem.Accordingtothecharacteristicoftheproblem,thischapterfirstgivesthedifferentialfunctionsandboundaryconditions,whichthestressfunctionshouldsatisfyofsolvingthetorsionproblems.Then,inordertosolvethetorsionproblemsofrelativelycomplexsectionpole,weintroductionthemethodofmembraneassimilation.,Torsion,6,扭转,材料力学解决了圆截面直杆的扭转问题，但对非圆截面杆的扭转问题却无法分析。对于任意截面杆的扭转，这本是一个较简单的空间问题，根据问题的特点，本章首先给出了求解扭转问题的应力函数所应满足的微分方程和边界条件。其次，为了求解相对复杂截面杆的扭转问题，我们介绍了薄膜比拟方法。,7,9-1TheTorsionofEqualSectionPole,1.StressFunction,Aequalsectionstraightpole,ignoringthebodyforce,isundertheactionoftorsionMatitstwoendplanes.Takeoneendasthexyplane,asshowninfig.Theotherstresscomponentsarezeroexceptfortheshearstresszx、zy,SubstitutethestresscomponentsandbodyforcesX=Y=Z=0intotheequationsofequilibrium,weget,Torsion,8,扭转,9-1等截面直杆的扭转,一应力函数,设有等截面直杆，体力不计，在两端平面内受扭矩M作用。取杆的一端平面为xy面，图示。横截面上除了切应力zx、zy以外，其余的应力分量为零,将应力分量及体力X=Y=Z=0代入平衡方程，得,9,Fromthefirsttwoequations,weknow,zx、zyarefunctionsofonlyxandy,theyhavenothingtodowithz.Fromthethirdformula:,Annotation:thedifferentialequationsofequilibriumforspatialproblemsare:,Accordingtothetheoryofdifferentialequations,theremustexistafunctionx,y,fromit,Thefunctionx,yiscalledstressfunctionoftorsionproblems.,a,Torsion,10,扭转,根据前两方程可见，zx、zy只是x和y的函数，与z无关，由第三式,注：空间问题平衡微分方程,根据微分方程理论，一定存在一个函数x,y，使得,函数x,y称为扭转问题的应力函数。,a,11,Thenote:whenthebodyforceiszero,thecompatibilityequationsintermsofstresscomponentsforspatialproblemsare,Substitutestresscomponentsintothecompatibilityequationswhichignoringthebodyforce,wecansee:thefirstthreeformulasandthelastaresatisfied,theothertwoformulasdemand,Torsion,12,扭转,注：体力为零时，空间问题应力分量表示的相容方程,将应力分量代入不计体力的相容方程，可见：前三式及最后一式得到满足，其余二式要求,13,2.Boundaryconditions,Ontheprofilesofthepole,substituten=0andsurfaceforcecomponentswhicharezerointotheboundaryconditions,wegetthatthefirsttwoformulascanalwaysbesatisfied,thethirdformulademands:,Thenote:thestressboundaryconditionsforspatialproblemsare:,Namely,Beingattheboundary:,Torsion,14,扭转,二边界条件,在杆的侧面上，将n=0，及面力分量为零代入边界条件，可见前两式总能满足，而第三式要求,注：空间问题应力边界条件,15,Thenwehave,Thisilluminatesthatattheboundaryofthecross-section,thestressfunctionisaconstant.Becausethestresscomponentsdontchangewhenthestressfunctionsubtractsaconstant,wecansupposewhenitisasinglesuccessionalsection(solidpole):,c,Atthearbitraryendofthepole,theshearstresscomposestorsion,Torsion,16,扭转,于是有,在杆的任一端，剪应力合成为扭矩,17,Integralstepbystep,andnoticethatequalstozeroattheboundary,Atlastweget,d,Torsion,18,扭转,分步积分，并注意在边界上为零,最后得到,d,19,3.DisplacementFormula,Accordingtotherelationsofstresses,strainsanddisplacements,weget,Afterintegral,weget,Torsion,20,扭转,三位移公式,根据应力、应变、位移的关系可以得到,积分后得到,21,Where,Kdenotesthetorsionangleperunitlength.Ignoringthedisplacementoftherigidbody,weget:,Substitutethemintotheabovefirsttwoformulasattheright,Theabovetwoformulascanbeusedtosolvedisplacementcomponentsw。.,e,f,Torsion,22,扭转,其中K表示杆的单位长度内的扭转角.不计刚体位移,代入前面右边前两式,上两式可用来求出位移分量w。,e,f,23,Differentiatingtheabovetwoformulaswithrespecttoyandx,thensubtractingthesetwo,weget:,Obviouslytheaboveformula,WhereC=-2GK.,Obviously,inordertoseekthesolutionofthetorsionproblems,weonlyneedtofindthestressfunction.Wemakeitsatisfytheequationsb,candd，thenwesolvethestresscomponentsfromformulaaandgivethevalueofthedisplacementcomponentsfromformulaseandf.,Torsion,24,扭转,上两式分别对y和x求导，再相减，得,可见前面公式b中,的C=-2GK.,显然，为了求得扭转问题的解，只须寻出应力函数，使它满足方程b、c和d，然后由a式求出应力分量，由式e和f给出位移分量的值。,25,9-2TheTorsionofEllipticSectionPole,Thesemi-majoraxesandsemi-minoraxesoftheellipticareaandbrespectively,itsboundaryfunctionis:,Thestressfunctionequalstozeroattheboundary,sowefetch,Substituteitinto,1,Torsion,26,扭转,9-2椭圆截面杆的扭转,椭圆的半轴分别为a和b，其边界方程为,应力函数在边界上应等于零，故取,代入,1,27,Weget,Thenwehave,Substituteitformula(1),weget,Form,Torsion,28,扭转,回代入1式得,由,29,Wecanget,Thenweget,Atlastwehave,Torsion,30,扭转,可得,于是得,最后得,31,Wegetthefinalsolutions:,Andfrom,Thetotalshearstressatanypointofthecross-sectionis,Torsion,32,扭转,最后得到解答,于是由,横截面上任意一点的合剪应力是,33,9-3MembraneAssimilation,Fromtheexampleofthelastsection,weknow:forthesimpleequalsectionpoleofelliptic,wejustgivethecalculatingexpressionofshearstressatthecross-section,wehaventpointedoutthepositionanddirectionofthemaximumshearstressatthesection;butforthepoleswithnottoocomplexsectionsuchasrectangularandthincliff,itisconsiderablydifficulttosolveitsprecisesolution,letalonetherelativelycomplexsectionpole.Forthisreason,weintroducethemethodofmembraneassimilation.Thismethodisbuiltatthebasisofsimilativeofthemathematicrelationbetweenthetorsionproblemofpoleandelasticitymembranewhichisundertheactionofequalsidepressureandexaggeratestightaround.,Supposingthereareevenmembrane,spreadingitattheboundarywhichisequaltoorproportionatetothesectionofthetorsionpole.Whenundertheactionofsmallevenpressureontheprofile,theinnerofthemembranewillproduceeventensility,eachpointonmembranewilloccursmalluprightnessanglechangealongzdirectionasshowninfig.,Torsion,34,扭转,9-3薄膜比拟,由上节的例子可以看出，对于椭圆形这种简单等截面直杆，我们给出了横截面上剪应力的计算表达式，但却没有指出截面最大剪应力的位置及其方向；而对于矩形、薄壁杆件这些截面并不复杂的柱体，要求出其精确解都是相当困难的，更不用说较复杂截面的杆件了。为了解决较复杂截面杆件的扭转问题，特提出薄膜比拟法。该方法是建立在柱体扭转问题与受均匀侧压力而四周张紧的弹性薄膜之间数学关系相似的基础上。,设有一块均匀薄膜，张在与扭转杆件截面相同或成比例的边界上。当在侧面上受着微小的均匀压力时，在薄膜内部将产生均匀的张力，薄膜的各点将发生图示z方向微小的垂度。,35,Fetchasmallsegmentabcdofthemembrane,asshowninfig.Itsprojectiononthexyplaneisarectangle,whichsidelengthsaredxanddyrespectively.SupposethepullofthemembraneperunitwidthisT,thenfromtheconditionofequilibriumalongzdirection,weget:,Afterpredigestion,weget,Torsion,36,扭转,取薄膜的一个微小部分abcd图示，它在xy面上的投影是一个矩形，矩形的边长分别是dx和dy。设薄膜单位宽度上的拉力为T，则由z方向的平衡条件得,简化后得,37,Namely,Moreover,obviouslytheuprightnessangleofthemembraneattheboundaryequalstozero,namely,Forq/Tisaconstant,theabovetwoformulascanberewrittenas,a,Andthedifferentialequationandtheboundaryconditionwhichthestressfunctionsatisfiesare:,Torsion,38,扭转,即,此外，薄膜在边界上的垂度显然等于零，即,而应力函数所满足的微分方程和边界条件为,39,Comparingformulabwithformulaa,weseethatandarealldeterminedbythesamedifferentialequationandboundarycondition,sotheyinevitablyhavethesamesolution.Thenwehave:,Torsion,40,扭转,将式b与式a对比，可见与决定于同样的微分方程和边界条件，因而必然具有相同的解答。于是有,41,SupposethevolumebetweenmembraneandtheboundaryplaneisV,andwenoticethat,Thenwehave,Therebywehave,d,From,Moreover,weget,e,Torsion,42,扭转,43,Adjustthepressureqofwhichthemembraneisunder,andmaketherightsofformulasc,d,eequaltoone,thenwecangainsomeconclusionsasfollows:,Thusitcanbeseen,themaximumshearstressatcross-sectionoftheellipticsectionwringedpoleexistsattwoendpointsofthesemi-minoraxes,itsdirectionisparalleltothesemi-majoraxes.,Torsion,44,扭转,调整薄膜所受的压力q，使得c、d、e三式等号的右边为1，则可得出如下结论：,1扭杆的应力函数等于薄膜的垂度z。,2扭杆所受的扭矩M等于该薄膜及其边界平面之间的体积的两倍。,3扭杆横截面上某一点处的、沿任意方向的剪应力，就等于该薄膜在对应点处的、沿垂直方向的斜率。,由此可见，椭圆截面扭杆横截面上的最大剪应力发生在短轴的两端点处，方向平行于长轴。,45,9-4TheTorsionofRectangularSectionPole,一TheTorsionofNarrowandLongRectangularSectionPole,Supposethesidelengthsoftherectangularsectionareaandb.Ifaislargethanb(asshowninfig),wecallitnarrowandlongrectangle.Fromthemembraneassimilation,wededucethatthestressfunctionalmostdoesntchangealongwithxatmostcross-section,thenwehave,Torsion,46,扭转,9-4矩形截面杆的扭转,一狭长矩形截面杆的扭转,设矩形截面的边长为a和b(图示)。若ab，则称为狭长矩形。由薄膜比拟可以推断，应力函数在绝大部分横截面上几乎不随x变化，于是有,则,成为,47,Thestresscomponentsare:,Fromthemembraneassimilation,weknow,themaximumshearstressexistsatthelongsideoftherectangularsection.Itsdirectionisparalleltoxaxis,anditsvalueis,Torsion,48,扭转,应力分量为,由薄膜比拟可知，最大剪应力发生在矩形截面的长边上，方向平行于x轴，其大小为,49,2.PolewithRectangularSection,Atthebasisofthestressfunctionfornarrowandlongrectangularsectionpole,wechoosethestressfunctionforanyrectangularsectionpoleasfollow,Substituteintothedifferentialfunction:,Andmakesatisfytheboundaryconditions:,Torsion,50,扭转,代入微分方程,并使满足边界条件,51,Weget,Spreadtherightoftheaboveformulaintotheprogressionofattherangeofy-b/2,b/2,thencomparethecoefficientofbothsides,weget:,SubstituteAminto,wegetthecertainstressfunction:,Torsion,52,扭转,得到,将上式右边在y-b/2,b/2区间展为的级数，然后比较两边的系数，得,将Am代入，得确定的应力函数,53,Fromthemembraneassimilation,weknow,themaximumshearstressexistsatmidpointofthelongsideoftherectangularsectionifab,Wherethewringanglekisobtainedfrom,Torsion,54,扭转,由薄膜比拟可以推断，最大剪应力发生在矩形截面长边的中点若ab,其中扭角k由,55,Torsion,56,扭转,求得,57,9-5TheTorsionofRingentThinCliffPole,Actuallywealwaysfaceringentthincliffpolesfromengineerproblems,suchasangleiron,trough,Ishapedironandsoon.Thecross-sectionsofthesethincliffpolesarealwayscomposedofnarrowrectanglewhichhastheequalwidth.Whateverstraightorbent,frommembraneassimilation,weknow,ifonlythenarrowrectanglehasthesamelengthandwidth,thenthetorsionandtheshearstressatthecross-sectionoftwowringedpolearealmostthesamevalues.,Torsion,58,扭转,9-5开口薄壁杆件的扭转,实际工程上经常遇到开口薄壁杆件，例如角钢、槽钢、工字钢等，这些薄壁件其横截面大都是由等宽的狭矩形组成。无论是直的还是曲的，根据薄膜比拟，只要狭矩形具有相同的长度和宽度，则两个扭杆的扭矩及其横截面剪应力没有多大差别。,59,Supposingaiandbidenotethelengthandthewidthoftheinarrowrectangleofthecross-sectionforthewringedpole,Midenotesthetorsionwhichtherectangularsectionisundergone,Mdenotesthetorsionofallthecross-section,Idenotestheshearstressnearthemidpointofthelongsideoftherectangle,kdenotesthewrestangleofthewringedpole.Fromtheresultofthenarrowrectangle,weget:,Fromthelaterformula,weget,Torsion,60,扭转,设ai及bi分别表示扭杆横截面的第i个狭矩形的长度和宽度，Mi表示该矩形截面上承受的扭矩，M表示整个横截面上的扭矩，i代表该矩形长边中点附近的剪应力，k代表扭杆的扭角。则由狭矩形的结果，得,由后一式得,61,Also,Sowehave:,Consequentlywehave:,Itisnoticeablethat:theshearstressofthemidpointofthelongsideofthenarrowrectangleisconsiderablyprecise.However,becauseoftheexistenceofstressconcentration,thelocalshearstressmaybei