弹性力学课件8

第三章平面问题的直角坐标解答3—1逆解法与半逆解法3—2矩形梁的纯弯曲3—3位移分量的求出3—4简支梁受均布荷载3—5楔形体受重力和液体压力SolutionofPlaneProblemsinRectangularCoordinatesInverseMethodandSemi-InverseMethodPureBendingofaRectangularBeamDeterminationofDisplacmentsBendingofaSimpleBeamUnderUniformLoadsTriangularGravityWall3—2矩形梁的纯弯曲xyh/2h/2MMh1例1.图示矩形截面梁，设板厚板长和板宽（平面应力情况），若板厚板长和宽（平面应变情况），取厚度为1，在两端受图示力偶M作用（单位宽度的力偶），体力不记,求梁内各点的应力分量,变形及位移分量.PureBendingofaRectangularBeamForconvenience,weconsideronlyaunitwidthofthebeam.LetthebendingmomentontheunitwidthbeM,whichhasadimensionof[force][length]/[length],orsimply[force].1、求应力分量a、取（x，y）=ay3无论a取何值，均能满足相容方程Thecompatibilityequationisalsosatisfiedforallvaluesoftheconstanta.应力分量：Thestresscomponentsareb、利用边界条件求系数axyh/2h/2MM上、下面左、右面自然满足自然满足M与应力形成一致时取+；反之，取-（1）（2）式（1）自然满足由式（2）应力分量：注意：梁截面惯性矩momentofinertia应力分量：上述结果与材料力学中的完全相同：梁的各纤维只受按直线分布的弯应力Eqscoincidecompletelywiththeelementarysolutiongiveninmechanicsofmaterials.注意：组成力偶的面力按直线规律分布时，上述结果完全精确，如果两端的面力按其他方式分布，上述结果有误差。Thesolutionisexactonlywhenthesurfaceforcesontheendsofthebeamconsistofnormaltractionsproportionaltoy.Ifthecouplesontheendsareappliedinanyothermanner,thissolutionisnolongerexact,becauseitwillnotpreciselysatisfytheboundaryconditionsontheends.按照圣维南原理，只在梁端附近有显著误差，在离开梁端较远处，误差可以不计2、求应变分量（平面应力问题）straincomponentsplanestressWhenthestresscomponentshavebeenfoundbysolutionintermsofstresses,wecanfindthedisplacementcomponentsbyphysicalandgeometricequations.Thiswillbeillustratedwiththeproblemofpurebendingasanexample.3、求位移分量将u、v代入下式：Substitutingu,vinto：移项得：左边是y的函数，右边是x的函数，两边要相等，只可能等于常数Itcanberewrittenas：Wenotethattheleftsideisafunctionofyalonewhiletherightsideisafunctionofxalone.

Afunctionofycanbeequaltoafunctionofxonlywhentheyarebothequaltoaconstant,say.上述两两式积积分后后得：：（1）（2）Byintegration,wehave：上述两两式代代入u、v的表达达式得得：Thus,thedisplacementcomponentscanbeexpressedas：(1)(2)From(1),wecanseethat,nomatterhowthebeamissupported,theangleofrotationofanyverticallineelementinthebeamisOnagivencrosssectionofthebeam,isconstant,sincexisconstant.Thisshowsthatalltheverticallineelementsinthecrosssectionhavethesameangleofrotationduetobendingofthebeam,andconsequentlyprovesthatthecrosssectionremainsplaneafterbending.From(2),weseethatallthelongitudinallines(fibres)ofthebeamwillhavethesamecurvatureafterbending.Thisisthebasicformulaforcomputationofbeamdeflectionsinmechanicsofmaterials.根据据边边界界条条件件求求积积分分常常数数将上述纯弯曲梁加上约束：xyh/2h/2MM（1）使之成为简支梁Ifthebeamissimplysupported：边界条件：TheconditionsofconstraintareTherewillbeneitherhorizontalnorverticaldisplacementsatthehingedsupport,andtherewillbenoverticaldisplacementeither,atthebarsupport.代入入位位移移表表达达式式得得：：最后后得得简简支支梁梁的的位位移移分分量量：：FromwhichwehaveInsertionofthesevaluesinto(1)and(2)yields梁轴轴线线的的挠挠曲曲线线方方程程：：xyh/2h/2MMThedeflectionofthebeamaxisis：Whichisthesameastheresultobtainedinmechanicsofmaterials.（2）若梁是悬臂梁Inthecaseofacantileverbeam边界条件：Theconditionsofconstraint：由此此求求得得：：悬臂梁的位移分量：Fromwhichwehave梁轴轴线线的的挠挠曲曲线线方方程程：：上述述结结论论与与材材料料力力学学中中的的相相同同验正正材材料料力力学学中中的的平平面面假假定定是是否否成成立立？？（1）（2）Thedeflectionofthebeamaxisis：Whichisalsothesameastheresultobtainedinmechanicsofmaterials.由（（1））由（（2））无论论约约束束情情况况如如何何，，铅铅垂垂直直线线的的转转角角都都是是，，同同一一截截面面，，x为常常数数，，所所以以为为常常数数，，即即同同一一平平面面转转角角相相同同，，横横截截面面保保持持为为平平面面无论约束情况如何，梁的纵向曲率为与材材料料力力学学求求