10TorsionofPrismaticalBars10.1Stressesanddisplacements10.2Membraneanalogy10.3Ellipticbarsintorsion10.4Rectangularbarsintorsion例：圖示為等截面直桿（不計(jì)體力），兩端平面內(nèi)受方向相反的兩個(gè)力偶作用，取如圖所示的坐標(biāo)系，求應(yīng)力分量。xyzMMo如何計(jì)算？Semi-inversemethod10.1StressesanddisplacementsConsiderastraightprismaticalbartwistedbytwooppositecouples,eachwithmomentM,ontheendplanesofthebar.分析如下：由材料力學(xué)可知，桿件橫截面上有剪應(yīng)力zx、zy，其他應(yīng)力分量為零，即x=y=z=xy=0，體力X=Y=Z=0Weshallsolvetheproblemintermsofstressesandapplythesemi-inversemethod.Beinghintedbythestressdistributiongiveninmechanicsofmaterialsforacircularshaft,weassumethatonlythetorsionalstressesaredifferentfromzero,whichotherstresscomponentsvanish.將上述應(yīng)力分量代入平衡微分方程：Substitutingtheseintodifferentialequations（1）（2）（3）由（3）得：由（1）、（2）可知，zx、zy是x、y的函數(shù)ThefirsttwoofEqs.showthatthetorsionalstresszx、zyarefunctionofxandyonly.或?qū)懗桑簩⑸鲜霰磉_(dá)式代入相容方程：Rewrittenas由微分理論可知，一定存在一個(gè)函數(shù)（x，y）使得：Hence,thereexistsacertainfunctionsothat前四式自然滿足，最后兩式為：將代入上式這說明：Hence,mustbeconstant1、桿件側(cè)面：邊界條件xyzMMon=0，X=Y=Z=0代入邊界前兩式恒滿足，第三式為：ConsidertheboundaryconditionsOnthelateralsurfaceThefirsttwoEqs.AreidenticallysatisfiedwhilethethirdequationreducestoSo：所以，以上邊界條件可寫為：Since：ontheboundaryofacross-section,wehaveShowsthatmustbeconstantalongtheboundaryofacross-section常數(shù)如何??？Since所以在應(yīng)力函數(shù)上增加或減少一個(gè)常數(shù)，對(duì)應(yīng)力分量無(wú)影響，可取s=0（在單連域?qū)嵭闹那闆r下）thestresscomponentsarenotaffectedbyadditionofaconstantto.Hence,inthecaseofsimplyconnectedcross-section,forsimplicity,wemaytaketheboundaryvalueofaszero.在多連域情況下，每個(gè)邊界上的為常數(shù)，但每個(gè)常數(shù)并不相同，只能取其中一個(gè)邊界的為零.Inthecaseofmultiplyconnectedcross-section,theconstantvaluesofondifferentboundarycurvesmaydifferfromeachother.Hence,thevalueofcanbetakenaszeroalongonlyoneoftheboundarycurves,whilethevaluesofonallotherboundarycurvesmustbedeterminedbytheconditionofsingle-valueddisplacement.2、桿桿件端端面（（上端端）：：邊界條件xyzMMol=m=0n=-1代入邊邊界條條件得得：且兩端端的面面力必必須合合成為為力偶偶：Oneitherendplane(seetheupperone)SincethesurfaceforcecomponentsmustcomposedacouplewithmomentM,itisrequiredthat驗(yàn)證如下邊界條件：xyMoABA、B為邊界上A、B點(diǎn)的值，取零，所以同理，可證明：WhereA、BarethevaluesofattwopointsAandBontheboundary,whichmustbezero.其中的的一項(xiàng)項(xiàng)用分分部積積分：：xyMoABA、B為邊界上A、B點(diǎn)的值，取零，所以WhereA、BarethevaluesofattwopointsAandBontheboundary,whichmustbezero.同理所以：Thus歸納Tosumup：：

等截面直桿的扭轉(zhuǎn)問題最終歸結(jié)為求應(yīng)力函數(shù)（x，y），使它滿足且s=0，然后由由下列列公式式求應(yīng)應(yīng)力分分量：：Tosumup,inatorsionproblem,ifwecanfindafunctionwhichsatisfies,s=0,,then,thetorsionalstressescanbefound.求位移移分量量：（將應(yīng)應(yīng)力分分量代代入物物理方方程））Nowderivetheformulasfordisplacementcomponents.Substitutionofstresscomponentsintophysicalequations.將幾何方程代入上述表達(dá)式，得：Substitutionofgeometricalequations.u0,v0,x,y,z是積分分常數(shù)數(shù)，代代表剛剛體位位移，，k也是積積分常常數(shù)，，若不不計(jì)積積分常常數(shù)，，則如果用用柱坐坐標(biāo)表表示，，則u0,v0,x,y,zarearbitraryconstantsrepresentingtherigid-bodymovementsofthebar.Assumingthattheupper-endofthebarisconstrainedtopreventtheserigid-bodymovements,wehavethesimpleexpressions.Inpolarcoordinates,thedisplacementcomponentswillbeThusitisseenthateverycross-sectionofthebarrotatesanangle=kzaboutthezaxisandrepresentsthetwistperunitlength,thatis,therelativeangulardisplacementoftwocross-sectionsatunitdistanceapart.由此可可得結(jié)結(jié)論，，每個(gè)個(gè)橫截截面在在xy面上的的投影影不改改變形形狀，，只是是轉(zhuǎn)過過一個(gè)