梁弯曲的内力Internalforceofbeambend

梁弯曲的内力Internalforceofbeambending主讲人：王雅君杆件的基本变形Basicdeformationoftherod剪切与挤压Shearandextrusion扭转reverse弯曲bend拉伸(压缩)Stretch(compression)以弯曲为主要变形的杆件通称为梁。Barswithbendingasthemaindeformationarecommonlyreferredtoasbeams.弯曲变形Bendingdeformation弯曲变形:当杆件受到一组垂直于其轴线的力或位于轴线平面内的外力偶作用时，使杆件的轴线由直线变为曲线。Bendingdeformation:Whenamemberissubjectedtoasetofforcesperpendiculartoitsaxisoranexternalforcecouplelocatedintheplaneoftheaxis,theaxisofthememberchangesfromastraightlinetoacurve.桥式起重机大梁Bridgecranebeam车削工件turningworkpiece12目录CONTENTS基本概念弯曲时的内力BasicconceptsInternalforceduringbending1基本概念BasicconceptsONE一、基本概念Basicconcepts1.对称弯曲:杆件的几何形状、材料性能和外力都对称于杆件的纵向对称面。Symmetricbending:Thegeometricshape,materialproperties,andexternalforcesoftherodaresymmetricalaboutthelongitudinalsymmetryplaneoftherod.对称弯曲必定是平面弯曲。Symmetricalbendingmustbeaplanebending.纵向对称面Longitudinalsymmetryplane一、基本概念Basicconcepts2.梁:以弯曲为主要变形的杆件通称为梁。Beam:Abeamiscommonlyreferredtoasamemberwhosemaindeformationisbending.常见梁的截面:Crosssectionsofcommonbeams:一、基本概念Basicconcepts3.载荷类型:Loadtype:●集中力:作用在梁上一点的载荷Concentratedforce:theloadactingatapointonthebeamP一集中力P-Concentrationforce一、基本概念Basicconcepts3.载荷类型:Loadtype:●集中力偶:作用在梁上某一截面上的力偶Concentratedcouple:acoupleactingonacertainsectionofabeamM一集中力偶M-concentratedcouple一、基本概念Basicconcepts3.载荷类型:Loadtype:●均布载荷:作用在梁上一定长度范围的连续分布的力Uniformload:acontinuouslydistributedforceactingonabeamoveracertainlengthrangeq一均布载荷q-uniformlydistributedload一、基本概念Basicconcepts4.梁的三种基本形式：Threebasicformsofbeams:①简支梁Simplysupportedbeam②悬臂梁cantileverbeamM一集中力偶M-concentratedcoupleq（x）一分布力q(x)-distributedforce一、基本概念Basicconcepts4.梁的三种基本形式：Threebasicformsofbeamsq一均布力q-uniformlydistributedforceP一集中力P-Concentrationforce①外伸梁overhangingbeam梁的一端外伸Oneendofthebeamextendsout梁的两端外伸Bothendsofthebeamextendoutward一、基本概念Basicconcepts5.弯曲的特点:Characteristicsofbending:受力特点:杆件受到垂直于杆轴线的外力(横向力)或外力偶(其矢量垂直于杆轴线)作用。Forcecharacteristics:Therodissubjectedtoanexternalforce(transverseforce)perpendiculartotherodaxisoranexternalforcecouple(withavectorperpendiculartotherodaxis).变形特点:直杆的轴线在变形后变为曲线，任意两横截面绕垂直于杆轴的轴作相对转动。Deformationcharacteristics:Theaxisofastraightrodchangesintoacurveafterdeformation,andanytwocrosssectionsrotaterelativetotheaxisperpendiculartotherodaxis.2弯曲时的内力InternalforceduringbendingTWO梁的横截面上有两个分量一一剪力和弯矩。Therearetwocomponentsonthecross-sectionofthebeam-shearforceandbendingmoment.受力分析ForceAnalysis截面法sectionmethod二、弯曲时的内力Internalforceduringbending1.弯矩(M)Bendingmoment(M)梁的任一截面上的截面的弯矩等于该截面一侧所有外力对该截面形心矩的代数和。Thebendingmomentofasectiononanysectionofabeamisequaltothealgebraicsumofallexternalforcesononesideofthesectionandthecentroidmomentofthesection.方向:使梁变成凹形的为正弯矩;使梁变成凸形的为负弯矩。Direction:Thepositivebendingmomentthatcausesthebeamtobecomeconcave;Thenegativebendingmomentcausesthebeamtobecomeconvex.

——

左顺右逆为正——

Leftclockwiseandrightcounterclockwisearepositive二、弯曲时的内力Internalforceduringbending2.剪力(Q)Shearforce(Q)梁的任一截面上的剪力等于该截面一侧所有竖向外力的代数和。Theshearforceonanysectionofabeamisequaltothealgebraicsumofallverticalexternalforcesononesideofthesection.方向:绕研究对象顺时针转为正剪力;反之为负。Direction:Turningclockwisearoundtheresearchobjecttoapositiveshearforce;Onthecontrary,itisnegative.

——左上右下为正——Upperleftandlowerrightarepositive二、弯曲时的内力Internalforceduringbending3.内力方程Internalforceequation内力与截面位置坐标x间的函数关系式。Thefunctionalrelationshipbetweentheinternalforceandthecoordinatexofthecross-sectionalposition.剪力方程:Q=Q(x)Shearequation:Q=Q(x)弯矩方程:M=M(x)Bendingmomentequation:M=M(x)4.剪力图和弯矩图Shearforcediagramandbendingmomentdiagram剪力图:Q=Q(x)的图线表示Sheardiagram:representedbyalinewithQ=Q(x)弯矩图:M=M(x)的图线表示Bendingmomentdiagram:linerepresentationofM=M(x)二、弯曲时的内力Internalforceduringbending一、材料力学的基本概念Basicconceptsofmaterialmechanics例:图示简支梁C点受集中力F作用。试写出剪力和弯矩方程，并画出剪力图和弯矩图。Example:PointCofthesimplysupportedbeaminthediagramissubjectedtoaconcentratedforceF.Trytowriteequationsforshearforceandbendingmoment,anddrawshearforceandbendingmomentdiagrams.FAyX1X2(1)确定约束力DeterminethebindingforceΣMA=0FByl-Fa=0

FBy=Fa/lΣMB=0FAyl-Fb=0

FAy=Fb/l(2)建立弯矩方程EstablishbendingmomentequationAC段:M（x1）=Fbx1/l(0≤x1<α)CB段:M（x2）=Fa（l-x2）/l(α≤x2≤l）(3)画弯矩图Drawabendingmomentdiagram解:Solution:一、材料力学的基本概念Basicconceptsofmaterialmechanics例:图示简支梁C点受集中力F作用。试写出剪力和弯矩方程，并画出剪力图和弯矩图。Example:PointCofthesimplysupportedbeaminthediagramissubjectedtoaconcentratedforceF.Trytowriteequationsforshearforceandbendingmoment,anddrawshearforceandbendingmomentdiagrams.(4)建立剪力方程Establishshearequation(5)画剪力图Drawasheardiagram解:Solution:AC段:Q(x1)=Fb/l(0≤x1<α)CB段:Q(x2)=-Fa/l(α≤x2<l)1自测题:Selftestquestions:试写出下图所示梁的内力方程，并画出剪力图和弯矩图。Trytowritetheinternalforceequationofthebeamshowninthefigurebelow,anddrawtheshearforcediagramandbendingmomentdi