英漢雙語彈性力學(xué)92

會(huì)計(jì)學(xué)1英漢雙語彈性力學(xué)922第九章扭轉(zhuǎn)第1頁/共72頁3Chapter9Torsion§9-1TheTorsionofEqualSectionPole§9-2TheTorsionofEllipticSectionPole§9-3Membraneassimilation§9-4TheTorsionofRectangularSectionPole§9-5TheTorsionofRingentThinCliffPoleTorsion第2頁/共72頁4第九章扭轉(zhuǎn)§9-1等截面直桿的扭轉(zhuǎn)§9-2橢圓截面桿的扭轉(zhuǎn)§9-3薄膜比擬§9-4矩形截面桿的扭轉(zhuǎn)§9-5開口薄壁桿件的扭轉(zhuǎn)扭轉(zhuǎn)第3頁/共72頁5

Materialmechanicshassolvedthetorsionproblemsofroundsectionpole,butitcan’tbeusedtoanalyzethetorsionproblemsofnon-roundsectionpole.Forthetorsionofanysectionpole,itisarelativelysimplespatialproblem.Accordingtothecharacteristicoftheproblem,thischapterfirstgivesthedifferentialfunctionsandboundaryconditions,whichthestressfunctionshouldsatisfyofsolvingthetorsionproblems.Then,inordertosolvethetorsionproblemsofrelativelycomplexsectionpole,weintroductionthemethodofmembraneassimilation.Torsion第4頁/共72頁6扭轉(zhuǎn)

材料力學(xué)解決了圓截面直桿的扭轉(zhuǎn)問題，但對(duì)非圓截面桿的扭轉(zhuǎn)問題卻無法分析。對(duì)于任意截面桿的扭轉(zhuǎn)，這本是一個(gè)較簡(jiǎn)單的空間問題，根據(jù)問題的特點(diǎn)，本章首先給出了求解扭轉(zhuǎn)問題的應(yīng)力函數(shù)所應(yīng)滿足的微分方程和邊界條件。其次，為了求解相對(duì)復(fù)雜截面桿的扭轉(zhuǎn)問題，我們介紹了薄膜比擬方法。第5頁/共72頁7§9-1

TheTorsionofEqualSectionPole1.StressFunction

Aequalsectionstraightpole,ignoringthebodyforce,isundertheactionoftorsionMatitstwoendplanes.Takeoneendasthexyplane,asshowninfig.Theotherstresscomponentsarezeroexceptfortheshearstressτzx、τzySubstitutethestresscomponentsandbodyforcesX=Y=Z=0intotheequationsofequilibrium,wegetxMMoyzTorsion第6頁/共72頁8扭轉(zhuǎn)§9-1

等截面直桿的扭轉(zhuǎn)一應(yīng)力函數(shù)

設(shè)有等截面直桿，體力不計(jì)，在兩端平面內(nèi)受扭矩M作用。取桿的一端平面為xy面，圖示。橫截面上除了切應(yīng)力τzx、τzy以外，其余的應(yīng)力分量為零將應(yīng)力分量及體力X=Y=Z=0代入平衡方程，得xMMoyz第7頁/共72頁9

Fromthefirsttwoequations,weknow,τzx、τzyarefunctionsofonlyxandy,theyhavenothingtodowithz.Fromthethirdformula:Annotation:thedifferentialequationsofequilibriumforspatialproblemsare:Accordingtothetheoryofdifferentialequations,theremustexistafunctionx,y,fromitThefunctionx,yiscalledstressfunctionoftorsionproblems.aTorsion第8頁/共72頁10扭轉(zhuǎn)

根據(jù)前兩方程可見，τzx、τzy只是x和y的函數(shù)，與z無關(guān)，由第三式注：空間問題平衡微分方程根據(jù)微分方程理論，一定存在一個(gè)函數(shù)x,y，使得函數(shù)x,y稱為扭轉(zhuǎn)問題的應(yīng)力函數(shù)。a第9頁/共72頁11Thenote:whenthebodyforceiszero,thecompatibilityequationsintermsofstresscomponentsforspatialproblemsare

Substitutestresscomponentsintothecompatibilityequationswhichignoringthebodyforce,wecansee:thefirstthreeformulasandthelastaresatisfied,theothertwoformulasdemandNamelybTorsion第10頁/共72頁12扭轉(zhuǎn)注：體力為零時(shí)，空間問題應(yīng)力分量表示的相容方程

將應(yīng)力分量代入不計(jì)體力的相容方程，可見：前三式及最后一式得到滿足，其余二式要求即b第11頁/共72頁132.Boundaryconditions

Ontheprofilesofthepole,substituten=0andsurfaceforcecomponentswhicharezerointotheboundaryconditions,wegetthatthefirsttwoformulascanalwaysbesatisfied,thethirdformulademands:Thenote:thestressboundaryconditionsforspatialproblemsare:NamelyBeingattheboundary:Torsion第12頁/共72頁14扭轉(zhuǎn)二邊界條件

在桿的側(cè)面上，將n=0，及面力分量為零代入邊界條件，可見前兩式總能滿足，而第三式要求注：空間問題應(yīng)力邊界條件即由于在邊界上第13頁/共72頁15ThenwehaveThisilluminatesthatattheboundaryofthecross-section,thestressfunctionφisaconstant.Becausethestresscomponentsdon’tchangewhenthestressfunctionsubtractsaconstant,wecansupposewhenitisasinglesuccessionalsection(solidpole):c

Atthearbitraryendofthepole,theshearstresscomposestorsionTorsion第14頁/共72頁16扭轉(zhuǎn)于是有說明在橫截面的邊界上，應(yīng)力函數(shù)φ為常量，由于應(yīng)力函數(shù)減一個(gè)常數(shù)，應(yīng)力分量不受影響，因此在單連通截面（實(shí)心桿）時(shí)可設(shè)c

在桿的任一端，剪應(yīng)力合成為扭矩第15頁/共72頁17Integralstepbystep,andnoticethatφequalstozeroattheboundaryAtlastwegetdTorsion第16頁/共72頁18扭轉(zhuǎn)分步積分，并注意φ在邊界上為零最后得到d第17頁/共72頁193.DisplacementFormula

Accordingtotherelationsofstresses,strainsanddisplacements,wegetAfterintegral,wegetTorsion第18頁/共72頁20扭轉(zhuǎn)三位移公式

根據(jù)應(yīng)力、應(yīng)變、位移的關(guān)系可以得到積分后得到第19頁/共72頁21Where,Kdenotesthetorsionangleperunitlength.Ignoringthedisplacementoftherigidbody,weget:SubstitutethemintotheabovefirsttwoformulasattherightTheabovetwoformulascanbeusedtosolvedisplacementcomponentsw。.efTorsion第20頁/共72頁22扭轉(zhuǎn)其中K表示桿的單位長(zhǎng)度內(nèi)的扭轉(zhuǎn)角.不計(jì)剛體位移代入前面右邊前兩式上兩式可用來求出位移分量w。ef第21頁/共72頁23Differentiatingtheabovetwoformulaswithrespecttoyandx,thensubtractingthesetwo,weget:ObviouslytheaboveformulaWhereC=-2GK.Obviously,inordertoseekthesolutionofthetorsionproblems,weonlyneedtofindthestressfunction.Wemakeitsatisfytheequationsb,cand

d，thenwesolvethestresscomponentsfromformulaaandgivethevalueofthedisplacementcomponentsfromformulaseandf.Torsion第22頁/共72頁24扭轉(zhuǎn)上兩式分別對(duì)y和x求導(dǎo)，再相減，得可見前面公式b中的C=-2GK.

顯然，為了求得扭轉(zhuǎn)問題的解，只須尋出應(yīng)力函數(shù)，使它滿足方程b、c和d，然后由a式求出應(yīng)力分量，由式e和f給出位移分量的值。第23頁/共72頁25§9-2TheTorsionofEllipticSectionPoleThesemi-majoraxesandsemi-minoraxesoftheellipticareaandbrespectively,itsboundaryfunctionis:Thestressfunctionequalstozeroattheboundary,sowefetchSubstituteitinto1xyaboTorsion第24頁/共72頁26扭轉(zhuǎn)§9-2橢圓截面桿的扭轉(zhuǎn)

橢圓的半軸分別為a和b，其邊界方程為應(yīng)力函數(shù)在邊界上應(yīng)等于零，故取代入1xyabo第25頁/共72頁27WegetThenwehaveSubstituteitformula(1),wegetFormTorsion第26頁/共72頁28扭轉(zhuǎn)得求得回代入1式得由第27頁/共72頁29WecangetThenweget

AtlastwehaveTorsion第28頁/共72頁30扭轉(zhuǎn)可得于是得最后得第29頁/共72頁31Wegetthefinalsolutions:Andfrom

Thetotalshearstressatanypointofthecross-sectionisTorsion第30頁/共72頁32扭轉(zhuǎn)最后得到解答于是由橫截面上任意一點(diǎn)的合剪應(yīng)力是第31頁/共72頁33§9-3MembraneAssimilationFromtheexampleofthelastsection,weknow:forthesimpleequalsectionpoleofelliptic,wejustgivethecalculatingexpressionofshearstressatthecross-section,wehaven’tpointedoutthepositionanddirectionofthemaximumshearstressatthesection;butforthepoleswithnottoocomplexsectionsuchasrectangularandthincliff,itisconsiderablydifficulttosolveitsprecisesolution,letalonetherelativelycomplexsectionpole.Forthisreason,weintroducethemethodofmembraneassimilation.Thismethodisbuiltatthebasisofsimilativeofthemathematicrelationbetweenthetorsionproblemofpoleandelasticitymembranewhichisundertheactionofequalsidepressureandexaggeratestightaround.

Supposingthereareevenmembrane,spreadingitattheboundarywhichisequaltoorproportionatetothesectionofthetorsionpole.Whenundertheactionofsmallevenpressureontheprofile,theinnerofthemembranewillproduceeventensility,eachpointonmembranewilloccursmalluprightnessanglechangealongzdirectionasshowninfig.Torsion第32頁/共72頁34扭轉(zhuǎn)§9-3薄膜比擬

由上節(jié)的例子可以看出，對(duì)于橢圓形這種簡(jiǎn)單等截面直桿，我們給出了橫截面上剪應(yīng)力的計(jì)算表達(dá)式，但卻沒有指出截面最大剪應(yīng)力的位置及其方向；而對(duì)于矩形、薄壁桿件這些截面并不復(fù)雜的柱體，要求出其精確解都是相當(dāng)困難的，更不用說較復(fù)雜截面的桿件了。為了解決較復(fù)雜截面桿件的扭轉(zhuǎn)問題，特提出薄膜比擬法。該方法是建立在柱體扭轉(zhuǎn)問題與受均勻側(cè)壓力而四周張緊的彈性薄膜之間數(shù)學(xué)關(guān)系相似的基礎(chǔ)上。

設(shè)有一塊均勻薄膜，張?jiān)谂c扭轉(zhuǎn)桿件截面相同或成比例的邊界上。當(dāng)在側(cè)面上受著微小的均勻壓力時(shí)，在薄膜內(nèi)部將產(chǎn)生均勻的張力，薄膜的各點(diǎn)將發(fā)生圖示z方向微小的垂度。第33頁/共72頁35Fetchasmallsegmentabcdofthemembrane,asshowninfig.Itsprojectiononthexyplaneisarectangle,whichsidelengthsaredxanddyrespectively.SupposethepullofthemembraneperunitwidthisT,thenfromtheconditionofequilibriumalongzdirection,weget:Afterpredigestion,wegetyTdxTdydxdyabdcxyTTxzqoTorsion第34頁/共72頁36扭轉(zhuǎn)

取薄膜的一個(gè)微小部分abcd圖示，它在xy面上的投影是一個(gè)矩形，矩形的邊長(zhǎng)分別是dx和dy。設(shè)薄膜單位寬度上的拉力為T，則由z方向的平衡條件得簡(jiǎn)化后得TdxTdydxdyabdcxyTTxzqoy第35頁/共72頁37NamelyMoreover,obviouslytheuprightnessangleofthemembraneattheboundaryequalstozero,namelyForq/Tisaconstant,theabovetwoformulascanberewrittenasaAndthedifferentialequationandtheboundaryconditionwhichthestressfunctionsatisfiesare:Torsion第36頁/共72頁38扭轉(zhuǎn)即此外，薄膜在邊界上的垂度顯然等于零，即由于q/T為常量，所以以上兩式可改寫為a而應(yīng)力函數(shù)所滿足的微分方程和邊界條件為第37頁/共72頁39WhereGkisalsoaconstant,sotheycanberewrittenas:bComparingformulabwithformulaa,weseethatandarealldeterminedbythesamedifferentialequationandboundarycondition,sotheyinevitablyhavethesamesolution.Thenwehave:NamelycTorsion第38頁/共72頁40扭轉(zhuǎn)其中Gk也是常量，故也可改寫為b將式b與式a對(duì)比，可見與決定于同樣的微分方程和邊界條件，因而必然具有相同的解答。于是有即c第39頁/共72頁41SupposethevolumebetweenmembraneandtheboundaryplaneisV,andwenoticethatThenwehaveTherebywehavedFromMoreover,wegeteTorsion第40頁/共72頁42扭轉(zhuǎn)

設(shè)薄膜及其邊界平面之間的體積為V，并注意到則有從而有d由又可得e第41頁/共72頁43Adjustthepressureqofwhichthemembraneisunder,andmaketherightsofformulasc,d,eequaltoone,thenwecangainsomeconclusionsasfollows:

1Thestressfunctionofwringedpoleequalstotheuprightnessangleofthemembrane

2ThetorsionMwhichwringedpolereceivedequalstotwotimesofthevolumebetweenthemembraneandtheboundaryplane.

3Theshearstressatsomepointandalongarbitrarydirectionofthewringedpolejustequalstotheslopeatthecounterpointandalongperpendiculardirectionofthemembrane.Thusitcanbeseen,themaximumshearstressatcross-sectionoftheellipticsectionwringedpoleexistsattwoendpointsofthesemi-minoraxes,itsdirectionisparalleltothesemi-majoraxes.xyaboTorsion第42頁/共72頁44扭轉(zhuǎn)

調(diào)整薄膜所受的壓力q，使得c、d、e三式等號(hào)的右邊為1，則可得出如下結(jié)論：1扭桿的應(yīng)力函數(shù)等于薄膜的垂度z。2扭桿所受的扭矩M等于該薄膜及其邊界平面之間的體積的兩倍。3扭桿橫截面上某一點(diǎn)處的、沿任意方向的剪應(yīng)力，就等于該薄膜在對(duì)應(yīng)點(diǎn)處的、沿垂直方向的斜率。

由此可見，橢圓截面扭桿橫截面上的最大剪應(yīng)力發(fā)生在短軸的兩端點(diǎn)處，方向平行于長(zhǎng)軸。xyabo第43頁/共72頁45

§9-4TheTorsionofRectangularSectionPole一TheTorsionofNarrowandLongRectangularSectionPoleSupposethesidelengthsoftherectangularsectionareaandb.Ifaislargethanb(asshowninfig),wecallitnarrowandlongrectangle.Fromthemembraneassimilation,wededucethatthestressfunctionalmostdoesn’tchangealongwithxatmostcross-section,thenwehave,ThencanbewrittenasyaxboTorsion第44頁/共72頁46扭轉(zhuǎn)

§9-4

矩形截面桿的扭轉(zhuǎn)一狹長(zhǎng)矩形截面桿的扭轉(zhuǎn)

設(shè)矩形截面的邊長(zhǎng)為a和b(圖示)

。若a?b，則稱為狹長(zhǎng)矩形。由薄膜比擬可以推斷，應(yīng)力函數(shù)在絕大部分橫截面上幾乎不隨x變化，于是有則成為yaxbo第45頁/共72頁47Thestresscomponentsare:Fromthemembraneassimilation,weknow,themaximumshearstressexistsatthelongsideoftherectangularsection.Itsdirectionisparalleltoxaxis,anditsvalueisAfterintegral,andnoticethatontheboundaryWegetSubstituteintoAfterintegral,wegetSoTorsion第46頁/共72頁48扭轉(zhuǎn)積分，并注意在邊界上即得將代入積分后得故應(yīng)力分量為

由薄膜比擬可知，最大剪應(yīng)力發(fā)生在矩形截面的長(zhǎng)邊上，方向平行于x軸，其大小為第47頁/共72頁492.PolewithRectangularSectionAtthebasisofthestressfunctionfornarrowandlongrectangularsectionpole,wechoosethestressfunctionforanyrectangularsectionpoleasfollowSubstituteintothedifferentialfunction:Andmakesatisfytheboundaryconditions:Torsion第48頁/共72頁50扭轉(zhuǎn)二矩形截面桿

在狹矩形截面扭桿應(yīng)力函數(shù)的基礎(chǔ)上，取任意矩形截面桿應(yīng)力函數(shù)為代入微分方程并使?jié)M足邊界條件第49頁/共72頁51WegetSpreadtherightoftheaboveformulaintotheprogressionofattherangeofy∈-b/2,b/2,thencomparethecoefficientofbothsides,weget:SubstituteAminto,wegetthecertainstressfunction:Torsion第50頁/共72頁52扭轉(zhuǎn)得到

將上式右邊在y∈-b/2,b/2區(qū)間展為的級(jí)數(shù)，然后比較兩邊的系數(shù)，得將Am代入，得確定的應(yīng)力函數(shù)第51頁/共72頁53Fromthemembraneassimilation,weknow,themaximumshearstressexistsatmidpointofthelongsideoftherectangularsectionifa≥bWherethewringanglekisobtainedfromTorsion第52頁/共72頁54扭轉(zhuǎn)

由薄膜比擬可以推斷，最大剪應(yīng)力發(fā)生在矩形截面長(zhǎng)邊的中點(diǎn)若a≥b其中扭角k

由第53頁/共72頁55Torsion第54頁/共72頁56扭轉(zhuǎn)求得第55頁/共72頁57

§9-5TheTorsionofRingentThinCliffPoleActuallywealwaysfaceringentthincliffpolesfromengineerproblems,suchasangleiron,trough,I–shapedironandsoon.Thecross-sectionsofthesethincliffpolesarealwayscomposedofnarrowrectanglewhichhastheequalwidth.Whateverstraightorbent,frommembraneassimilation,weknow,ifonlythenarrowrectanglehasthesamelengthandwidth,thenthetorsionandtheshearstressatthecross-sectionoftwowringedpolearealmostthesamevalues.a1b1a2a1a1b2a3a2a1a3b2Torsion第56頁/共72頁58扭轉(zhuǎn)

§9-5開口薄壁桿件的扭轉(zhuǎn)

實(shí)際工程上經(jīng)常遇到開口薄壁桿件，例如角鋼、槽鋼、工字鋼等，這些薄壁件其橫截面大都是由等寬的狹矩形組成。無論是直的還是曲的，根據(jù)薄膜比擬，只要狹矩形具有相同的長(zhǎng)度和寬度，則兩個(gè)扭桿的扭矩及其橫截面剪應(yīng)力沒有多大差別。a1b1a2a1a1b2a3a2a1a3b2第57頁/共72頁59Supposingai

andbidenotethelengthandthewidthoftheinarrowrectangleofthecross-sectionforthewringedpole,Midenotesthetorsionwhichtherectangularsectionisundergone,Mdenotesthetorsionofallthecross-section,I

denotestheshearstressnearthemidpointofthelongsideoftherectangle,kdenotesthewrestangleofthewringedpole.Fromtheresultofthenarrowrectangle,weget:Fromthelaterformula,wegetTorsion第58頁/共72頁60扭轉(zhuǎn)

設(shè)

ai

及bi分別表示扭桿橫截面的第i

個(gè)狹矩形的長(zhǎng)度和寬度，Mi表示該矩形截面上承受的扭矩，M表示整個(gè)橫截面上的扭矩，i代表該矩形長(zhǎng)邊中點(diǎn)附近的剪應(yīng)力，k代表扭桿的扭角。則由狹矩形的結(jié)果，得由后一式得第59頁/共72頁61AlsoSowehave:Consequentlywehave:Itisnoticeablethat:theshearstressofthemidpointofthelongsideofthenarrowrectangleisconsiderablyprecise.However,becauseoftheexistenceofstressconcentration,thelocalshearstressmaybeismorelargerthanthementionedatthejointoftwonarrowrectangle.Torsion第60頁/共72頁62扭轉(zhuǎn)而故有從而有

值得注意的是：由上述公式給出的狹矩形長(zhǎng)邊中點(diǎn)的剪應(yīng)力已相當(dāng)精確，然而，由于應(yīng)力集中的存在，兩個(gè)狹矩形的連接處，可能存在遠(yuǎn)大于此的局部剪應(yīng)力。第61頁/共72頁63Exercise9.1

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

WegetNamelyoxyBAaTorsion第62頁/共72頁64扭轉(zhuǎn)習(xí)題9.1