英汉双语弹性力学92

会计学1英汉双语弹性力学922第九章扭转第1页/共72页3Chapter9Torsion§9-1TheTorsionofEqualSectionPole§9-2TheTorsionofEllipticSectionPole§9-3Membraneassimilation§9-4TheTorsionofRectangularSectionPole§9-5TheTorsionofRingentThinCliffPoleTorsion第2页/共72页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第3页/共72页5

Materialmechanicshassolvedthetorsionproblemsofroundsectionpole,butitcan’tbeusedtoanalyzethetorsionproblemsofnon-roundsectionpole.Forthetorsionofanysectionpole,itisarelativelysimplespatialproblem.Accordingtothecharacteristicoftheproblem,thischapterfirstgivesthedifferentialfunctionsandboundaryconditions,whichthestressfunctionshouldsatisfyofsolvingthetorsionproblems.Then,inordertosolvethetorsionproblemsofrelativelycomplexsectionpole,weintroductionthemethodofmembraneassimilation.Torsion第4页/共72页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。第5页/共72页7§9-1

TheTorsionofEqualSectionPole1.StressFunction

Aequalsectionstraightpole,ignoringthebodyforce,isundertheactionoftorsionMatitstwoendplanes.Takeoneendasthexyplane,asshowninfig.Theotherstresscomponentsarezeroexceptfortheshearstressτzx、τzySubstitutethestresscomponentsandbodyforcesX=Y=Z=0intotheequationsofequilibrium,wegetxMMoyzTorsion第6页/共72页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τzx、τzy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xMMoyz第7页/共72页9

Fromthefirsttwoequations,weknow,τzx、τzyarefunctionsofonlyxandy,theyhavenothingtodowithz.Fromthethirdformula:Annotation:thedifferentialequationsofequilibriumforspatialproblemsare:Accordingtothetheoryofdifferentialequations,theremustexistafunctionx,y,fromitThefunctionx,yiscalledstressfunctionoftorsionproblems.aTorsion第8页/共72页10扭转

根据前两方程可见，τzx、τzy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第9页/共72页11Thenote:whenthebodyforceiszero,thecompatibilityequationsintermsofstresscomponentsforspatialproblemsare

Substitutestresscomponentsintothecompatibilityequationswhichignoringthebodyforce,wecansee:thefirstthreeformulasandthelastaresatisfied,theothertwoformulasdemandNamelybTorsion第10页/共72页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，其余二式要求即b第11页/共72页132.Boundaryconditions

Ontheprofilesofthepole,substituten=0andsurfaceforcecomponentswhicharezerointotheboundaryconditions,wegetthatthefirsttwoformulascanalwaysbesatisfied,thethirdformulademands:Thenote:thestressboundaryconditionsforspatialproblemsare:NamelyBeingattheboundary:Torsion第12页/共72页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即由于在边界上第13页/共72页15ThenwehaveThisilluminatesthatattheboundaryofthecross-section,thestressfunctionφisaconstant.Becausethestresscomponentsdon’tchangewhenthestressfunctionsubtractsaconstant,wecansupposewhenitisasinglesuccessionalsection(solidpole):c

Atthearbitraryendofthepole,theshearstresscomposestorsionTorsion第14页/共72页16扭转于是有说明在横截面的边界上，应力函数φ为常量，由于应力函数减一个常数，应力分量不受影响，因此在单连通截面（实心杆）时可设c

在杆的任一端，剪应力合成为扭矩第15页/共72页17Integralstepbystep,andnoticethatφequalstozeroattheboundaryAtlastwegetdTorsion第16页/共72页18扭转分步积分，并注意φ在边界上为零最后得到d第17页/共72页193.DisplacementFormula

Accordingtotherelationsofstresses,strainsanddisplacements,wegetAfterintegral,wegetTorsion第18页/共72页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第19页/共72页21Where,Kdenotesthetorsionangleperunitlength.Ignoringthedisplacementoftherigidbody,weget:SubstitutethemintotheabovefirsttwoformulasattherightTheabovetwoformulascanbeusedtosolvedisplacementcomponentsw。.efTorsion第20页/共72页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。ef第21页/共72页23Differentiatingtheabovetwoformulaswithrespecttoyandx,thensubtractingthesetwo,weget:ObviouslytheaboveformulaWhereC=-2GK.Obviously,inordertoseekthesolutionofthetorsionproblems,weonlyneedtofindthestressfunction.Wemakeitsatisfytheequationsb,cand

d，thenwesolvethestresscomponentsfromformulaaandgivethevalueofthedisplacementcomponentsfromformulaseandf.Torsion第22页/共72页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。第23页/共72页25§9-2TheTorsionofEllipticSectionPoleThesemi-majoraxesandsemi-minoraxesoftheellipticareaandbrespectively,itsboundaryfunctionis:Thestressfunctionequalstozeroattheboundary,sowefetchSubstituteitinto1xyaboTorsion第24页/共72页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1xyabo第25页/共72页27WegetThenwehaveSubstituteitformula(1),wegetFormTorsion第26页/共72页28扭转得求得回代入1式得由第27页/共72页29WecangetThenweget

AtlastwehaveTorsion第28页/共72页30扭转可得于是得最后得第29页/共72页31Wegetthefinalsolutions:Andfrom

Thetotalshearstressatanypointofthecross-sectionisTorsion第30页/共72页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第31页/共72页33§9-3MembraneAssimilationFromtheexampleofthelastsection,weknow:forthesimpleequalsectionpoleofelliptic,wejustgivethecalculatingexpressionofshearstressatthecross-section,wehaven’tpointedoutthepositionanddirectionofthemaximumshearstressatthesection;butforthepoleswithnottoocomplexsectionsuchasrectangularandthincliff,itisconsiderablydifficulttosolveitsprecisesolution,letalonetherelativelycomplexsectionpole.Forthisreason,weintroducethemethodofmembraneassimilation.Thismethodisbuiltatthebasisofsimilativeofthemathematicrelationbetweenthetorsionproblemofpoleandelasticitymembranewhichisundertheactionofequalsidepressureandexaggeratestightaround.

Supposingthereareevenmembrane,spreadingitattheboundarywhichisequaltoorproportionatetothesectionofthetorsionpole.Whenundertheactionofsmallevenpressureontheprofile,theinnerofthemembranewillproduceeventensility,eachpointonmembranewilloccursmalluprightnessanglechangealongzdirectionasshowninfig.Torsion第32页/共72页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。第33页/共72页35Fetchasmallsegmentabcdofthemembrane,asshowninfig.Itsprojectiononthexyplaneisarectangle,whichsidelengthsaredxanddyrespectively.SupposethepullofthemembraneperunitwidthisT,thenfromtheconditionofequilibriumalongzdirection,weget:Afterpredigestion,wegetyTdxTdydxdyabdcxyTTxzqoTorsion第34页/共72页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TdxTdydxdyabdcxyTTxzqoy第35页/共72页37NamelyMoreover,obviouslytheuprightnessangleofthemembraneattheboundaryequalstozero,namelyForq/Tisaconstant,theabovetwoformulascanberewrittenasaAndthedifferentialequationandtheboundaryconditionwhichthestressfunctionsatisfiesare:Torsion第36页/共72页38扭转即此外，薄膜在边界上的垂度显然等于零，即由于q/T为常量，所以以上两式可改写为a而应力函数所满足的微分方程和边界条件为第37页/共72页39WhereGkisalsoaconstant,sotheycanberewrittenas:bComparingformulabwithformulaa,weseethatandarealldeterminedbythesamedifferentialequationandboundarycondition,sotheyinevitablyhavethesamesolution.Thenwehave:NamelycTorsion第38页/共72页40扭转其中Gk也是常量，故也可改写为b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。于是有即c第39页/共72页41SupposethevolumebetweenmembraneandtheboundaryplaneisV,andwenoticethatThenwehaveTherebywehavedFromMoreover,wegeteTorsion第40页/共72页42扭转

设薄膜及其边界平面之间的体积为V，并注意到则有从而有d由又可得e第41页/共72页43Adjustthepressureqofwhichthemembraneisunder,andmaketherightsofformulasc,d,eequaltoone,thenwecangainsomeconclusionsasfollows:

1Thestressfunctionofwringedpoleequalstotheuprightnessangleofthemembrane

2ThetorsionMwhichwringedpolereceivedequalstotwotimesofthevolumebetweenthemembraneandtheboundaryplane.

3Theshearstressatsomepointandalongarbitrarydirectionofthewringedpolejustequalstotheslopeatthecounterpointandalongperpendiculardirectionofthemembrane.Thusitcanbeseen,themaximumshearstressatcross-sectionoftheellipticsectionwringedpoleexistsattwoendpointsofthesemi-minoraxes,itsdirectionisparalleltothesemi-majoraxes.xyaboTorsion第42页/共72页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。xyabo第43页/共72页45

§9-4TheTorsionofRectangularSectionPole一TheTorsionofNarrowandLongRectangularSectionPoleSupposethesidelengthsoftherectangularsectionareaandb.Ifaislargethanb(asshowninfig),wecallitnarrowandlongrectangle.Fromthemembraneassimilation,wededucethatthestressfunctionalmostdoesn’tchangealongwithxatmostcross-section,thenwehave,ThencanbewrittenasyaxboTorsion第44页/共72页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a»b，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yaxbo第45页/共72页47Thestresscomponentsare:Fromthemembraneassimilation,weknow,themaximumshearstressexistsatthelongsideoftherectangularsection.Itsdirectionisparalleltoxaxis,anditsvalueisAfterintegral,andnoticethatontheboundaryWegetSubstituteintoAfterintegral,wegetSoTorsion第46页/共72页48扭转积分，并注意在边界上即得将代入积分后得故应力分量为

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，其大小为第47页/共72页492.PolewithRectangularSectionAtthebasisofthestressfunctionfornarrowandlongrectangularsectionpole,wechoosethestressfunctionforanyrectangularsectionpoleasfollowSubstituteintothedifferentialfunction:Andmakesatisfytheboundaryconditions:Torsion第48页/共72页50扭转二矩形截面杆

在狭矩形截面扭杆应力函数的基础上，取任意矩形截面杆应力函数为代入微分方程并使满足边界条件第49页/共72页51WegetSpreadtherightoftheaboveformulaintotheprogressionofattherangeofy∈-b/2,b/2,thencomparethecoefficientofbothsides,weget:SubstituteAminto,wegetthecertainstressfunction:Torsion第50页/共72页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，得确定的应力函数第51页/共72页53Fromthemembraneassimilation,weknow,themaximumshearstressexistsatmidpointofthelongsideoftherectangularsectionifa≥bWherethewringanglekisobtainedfromTorsion第52页/共72页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a≥b其中扭角k

由第53页/共72页55Torsion第54页/共72页56扭转求得第55页/共72页57

§9-5TheTorsionofRingentThinCliffPoleActuallywealwaysfaceringentthincliffpolesfromengineerproblems,suchasangleiron,trough,I–shapedironandsoon.Thecross-sectionsofthesethincliffpolesarealwayscomposedofnarrowrectanglewhichhastheequalwidth.Whateverstraightorbent,frommembraneassimilation,weknow,ifonlythenarrowrectanglehasthesamelengthandwidth,thenthetorsionandtheshearstressatthecross-sectionoftwowringedpolearealmostthesamevalues.a1b1a2a1a1b2a3a2a1a3b2Torsion第56页/共72页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。a1b1a2a1a1b2a3a2a1a3b2第57页/共72页59Supposingai

andbidenotethelengthandthewidthoftheinarrowrectangleofthecross-sectionforthewringedpole,Midenotesthetorsionwhichtherectangularsectionisundergone,Mdenotesthetorsionofallthecross-section,I

denotestheshearstressnearthemidpointofthelongsideoftherectangle,kdenotesthewrestangleofthewringedpole.Fromtheresultofthenarrowrectangle,weget:Fromthelaterformula,wegetTorsion第58页/共72页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，得由后一式得第59页/共72页61AlsoSowehave:Consequentlywehave:Itisnoticeablethat:theshearstressofthemidpointofthelongsideofthenarrowrectangleisconsiderablyprecise.However,becauseoftheexistenceofstressconcentration,thelocalshearstressmaybeismorelargerthanthementionedatthejointoftwonarrowrectangle.Torsion第60页/共72页62扭转而故有从而有

值得注意的是：由上述公式给出的狭矩形长边中点的剪应力已相当精确，然而，由于应力集中的存在，两个狭矩形的连接处，可能存在远大于此的局部剪应力。第61页/共72页63Exercise9.1

Onewringedpolewiththecross-sectionofequilateraltrianglehasitshighofa,thecoordinateisshownasfig.ThethreesidesAB,OA,OBofthetrianglesatisfyequations:Pleaseprovethestressfunctionsatisfiesanycondition,andsolvethemaximumshearstressandtwistyangleSolution:substituteintotheequationsofcompatibility

WegetNamelyoxyBAaTorsion第62页/共72页64扭转习题9.1