第三章平面問題的直角坐標(biāo)解答3—1逆解法與半逆解法3—2矩形梁的純彎曲3—3位移分量的求出3—4簡(jiǎn)支梁受均布荷載3—5楔形體受重力和液體壓力SolutionofPlaneProblemsinRectangularCoordinatesInverseMethodandSemi-InverseMethodPureBendingofaRectangularBeamDeterminationofDisplacmentsBendingofaSimpleBeamUnderUniformLoadsTriangularGravityWall3—2矩形梁的純彎曲xyh/2h/2MMh1例1.圖示矩形截面梁，設(shè)板厚板長(zhǎng)和板寬（平面應(yīng)力情況），若板厚板長(zhǎng)和寬（平面應(yīng)變情況），取厚度為1，在兩端受圖示力偶M作用（單位寬度的力偶），體力不記,求梁內(nèi)各點(diǎn)的應(yīng)力分量,變形及位移分量.PureBendingofaRectangularBeamForconvenience,weconsideronlyaunitwidthofthebeam.LetthebendingmomentontheunitwidthbeM,whichhasadimensionof[force][length]/[length],orsimply[force].1、求應(yīng)力分量a、?。▁，y）=ay3無論a取何值，均能滿足相容方程Thecompatibilityequationisalsosatisfiedforallvaluesoftheconstanta.應(yīng)力分量：Thestresscomponentsareb、利用邊界條件求系數(shù)axyh/2h/2MM上、下面左、右面自然滿足自然滿足M與應(yīng)力形成一致時(shí)取+；反之，取-（1）（2）式（1）自然滿足由式（2）應(yīng)力分量：注意：梁截面慣性矩momentofinertia應(yīng)力分量：上述結(jié)果與材料力學(xué)中的完全相同：梁的各纖維只受按直線分布的彎應(yīng)力Eqscoincidecompletelywiththeelementarysolutiongiveninmechanicsofmaterials.注意：組成力偶的面力按直線規(guī)律分布時(shí)，上述結(jié)果完全精確，如果兩端的面力按其他方式分布，上述結(jié)果有誤差。Thesolutionisexactonlywhenthesurfaceforcesontheendsofthebeamconsistofnormaltractionsproportionaltoy.Ifthecouplesontheendsareappliedinanyothermanner,thissolutionisnolongerexact,becauseitwillnotpreciselysatisfytheboundaryconditionsontheends.按照圣維南原理，只在梁端附近有顯著誤差，在離開梁端較遠(yuǎn)處，誤差可以不計(jì)2、求應(yīng)變分量（平面應(yīng)力問題）straincomponentsplanestressWhenthestresscomponentshavebeenfoundbysolutionintermsofstresses,wecanfindthedisplacementcomponentsbyphysicalandgeometricequations.Thiswillbeillustratedwiththeproblemofpurebendingasanexample.3、求位移分量將u、v代入下式：Substitutingu,vinto：移項(xiàng)得：左邊是y的函數(shù)，右邊是x的函數(shù)，兩邊要相等，只可能等于常數(shù)Itcanberewrittenas：Wenotethattheleftsideisafunctionofyalonewhiletherightsideisafunctionofxalone.

Afunctionofycanbeequaltoafunctionofxonlywhentheyarebothequaltoaconstant,say.上述兩式積分后得：（1）（2）Byintegration,wehave：上述兩式代入u、v的表達(dá)式得：Thus,thedisplacementcomponentscanbeexpressedas：(1)(2)From(1),wecanseethat,nomatterhowthebeamissupported,theangleofrotationofanyverticallineelementinthebeamisOnagivencrosssectionofthebeam,isconstant,sincexisconstant.Thisshowsthatalltheverticallineelementsinthecrosssectionhavethesameangleofrotationduetobendingofthebeam,andconsequentlyprovesthatthecrosssectionremainsplaneafterbending.From(2),weseethatallthelongitudinallines(fibres)ofthebeamwillhavethesamecurvatureafterbending.Thisisthebasicformulaforcomputationofbeamdeflectionsinmechanicsofmaterials.根據(jù)邊界條件求積分常數(shù)將上述純彎曲梁加上約束：xyh/2h/2MM（1）使之成為簡(jiǎn)支梁Ifthebeamissimplysupported：邊界條件：TheconditionsofconstraintareTherewillbeneitherhorizontalnorverticaldisplacementsatthehingedsupport,andtherewillbenoverticaldisplacementeither,atthebarsupport.代入位移表達(dá)式得：最后得簡(jiǎn)支梁的位移分量：FromwhichwehaveInsertionofthesevaluesinto(1)and(2)yields梁軸線的撓曲線方程：xyh/2h/2MMThedeflectionofthebeamaxisis：Whichisthesameastheresultobtainedinmechanicsofmaterials.（2）若梁是懸臂梁Inthecaseofacantileverbeam邊界條件：Theconditionsofconstraint：由此求得：懸臂梁的位移分量：Fromwhichwehave梁軸凈線的優(yōu)撓曲扶線方黃程：上述結(jié)憲論與材頓料力學(xué)歡中的相創(chuàng)同驗(yàn)正靜材料荒力學(xué)訓(xùn)中的綱平面丟假定奪是否排成立采？（1）（2）The停de斥fle曠cti羞on冷of渠the婦be裳a(bǔ)m煤axi民si篩s：Whi熄ch闖is聾als倍ot遼he愉sam喇ea敲st波he圍res捏ult崗ob針tai尊ned常in犁me廁cha歡nic杜so竟fm楊ate紗ria喬ls.由（膠1）由（2宗）無論舍約束川情況右如何范，鉛奸垂直蘭線的懷轉(zhuǎn)角醉都是，恭同一凈截面盛，x為常欺數(shù)，蓮所以潔為筆常數(shù)糟，即馳同一池平面閱轉(zhuǎn)角株相同凈，橫擔(dān)截面鵲保持遠(yuǎn)為平漢面無論約束情況如何，梁的縱向曲率為與材禮料力傲學(xué)求瓣梁的露撓度采時(shí)所屬用的膝基本館公式石一致對(duì)平面應(yīng)變問題，