《i弯曲变形》PPT课件

4.梁的变形,DeformationsofBeams,4.1弯曲变形的概念,挠度以y轴正向为正,Conceptofdeformationsforabeam,(1)挠度(deflection),梁有两种变形形式,挠度函数或挠曲线方程,Acoordingtotheassumpationofsmalldeformationswecanget:,转角以x轴正向逆时针旋转为正,(2)转角(slope),Relationbetweenand:,转角函数或转角方程,注意:材料力学所说的梁的变形实质上指的是梁的位移!是梁变形的累加效应（包括梁可能存在的刚体位移）。,注意:材料力学所说的梁的变形实质上指的是梁的位移!是梁变形的累加效应（包括梁可能存在的刚体位移）。,（1）梁存在变形，一般也存在位移（既挠度和转角）。（2）梁存在位移（既挠度和转角），有可能不存在变形。,4.2挠曲线的近似微分方程,Approximatelydifferentialequationofthedeflectioncurve,中性层曲率与弯矩的关系:,曲率计算公式:,Becausedeformationsofthebeamisverysmall,wecanget:,So:,Thepositiveandnegativesymbolofcurvatureisjustcoincidentwiththatofbendingmoment,sowehavetheresultshownfollows:,Itiscalledtheapproximatelydifferentialequationofthedeflectioncurve.,挠曲线近似微分方程,曲率的正负号与弯矩的正负号一致,4.3积分法求梁的变形,Usingintegrationmethodtodeterminethedeformationsofabeam,Basicequationis:,(1)M(x)/EI在梁中为单一函数(asinglefunctioninwholebeam),积分常数C和D由约束条件确定：,Fixedpin,Movablepin,Fixedend,Spring,Anotherbar,Anotherbeam,Therearetwoboundaryconditionsforabeam.,(2)M(x)/EI在梁中是分段函数(isnotsamefunctioninonepartofthebeamfromanother),b.梁的连续条件(Continuousconditionsofthebeam),Atthedividedpointsofthebeamthedeflectionandslopemustbecontinuous,sowecanget:,Thereare2n-2equations.,Continuousconditions:2n-2,Boundaryconditions:2,(i=1,2,n),在梁的分界点处,梁的挠度和转角都应连续,Example:如图所示简支梁受均布载荷作用，右端由拉杆支承，左端由固定铰支承。求梁左端截面的转角以及梁中点的挠度。,Solve:,(1)Determinethereactionatfixedpinandtheaxialforceofthebar,(2)Determinethefunctionofbengingmomentinthebeam,(3)Determinetheequationofdeflectioncurve,(4)DeterminetheunknownconstantsCandD,w(0)=0,boundaryconditions:,D=0,Sothefunctionofslopeanddeflectionwillbe:,(5)DeterminetheslopeatendAanddeflectionatmiddlepointC,（1）梁的内力和变形与载荷及抗弯刚度之间的关系:,转角,集中力F,均布载荷q,挠度,弯矩,剪力,L为梁的特征长度.,4.4梁变形的一些重要特点,3分钟小练习,填空:,(1)当梁的直径减小一倍而其它条件不变时:最大弯曲正应力是原来的倍,最大挠度是原来的倍.,(2)当梁的长度增大一倍而其它条件不变时:最大弯曲正应力是原来的倍,最大挠度是原来的倍.,8,16,2,8,（2）刚性地基或平台上的梁,刚性地基或平台上的梁在接触点可简化为：（1）弯矩为零的固定端。（2）转角为零的简支端。,（3）对称梁和反对称梁,Symmetricalandanti-symmetricalbeams,1)Forabeam,ifloadingsandsupportsaresymmetrical,itiscalledsymmetricalbeam.,对称梁,2)Forabeam,ifloadingsandsupportsareanti-symmetrical,itiscalledanti-symmetricalbeam.,反对称梁,Forthesymmetricalbeam,wecangettheresultsatthemiddlepointofthebeamasfollows.,Fortheanti-symmetricalbeam,wecangettheresultsatthemiddlepointofthebeamasfollows.,Example:DeterminethedeflectionatpointDofthebeamshownfollows.,Solve:,Becausethethebeamisanti-symmetrical,wecanget:,SothepartACofthebeamjustlikeasimplesupportedbeam.,4.5叠加法计算梁的变形,Principleofsuperpositiontodeterminetheslopeanddeflectionofbeams,问题:,如何计算图示各梁在一些特殊点处的挠度和转角?,求A点挠度,积分法太麻烦!,求中点的挠度和B点转角,求A点挠度和转角,求梁中点A点挠度,宜采用叠加法求解,叠加原理,Principleofsuperposition,在小变形线弹性条件下,任何因素引起材料力学问题的内力应力变形都是可以叠加的,这称为叠加原理.,叠加法计算梁的变形,在线弹性小变形条件下,任何因素引起的梁的变形都是可以叠加的.,Theprincipleofsuperpositionisoftenusedtodeterminethedeflectionsorslopesatsomespecialpointsofthebeam.,Someresultsofsimplebeamswillbeneededtodeterminethedeflectionsorslopesofthebeamwhenusingtheprincipleofsuperposition.,注意:叠加法主要用于求梁在某些特殊点处的挠度和转角,应用叠加法时,一些简单梁的变形必须要已知.,叠加法(Principleofsuperposition),（1）载荷因素：多个载荷引起的梁的变形都是可以叠加的.,Example:DeterminethedeflectionatthemiddlepointAofthebeamshownfollows.,Solve:,+,+,a,a/4,EI,A,a/4,=,=,Solve:,例如图外伸梁C点的竖向位移为零时,求载荷P和q间的关系。,Example:DeterminethedeflectionatthemiddlepointCofthebeamshownfollows.,Solve:,Theoriginalbeamcanbesuperpositionwithtwobeamsshownfollows.,+,B,Then:,Example:,+,Solve:,Example:,+,Solve:,Example:,Solve:,+,例求图示自由端的挠度。,例,求图示自由端的挠度。,载荷的分解和重组,例,求图示自由端的挠度。,微分载荷积分法,（向下）,例,求图示自由端的挠度。,逐段刚化求解,例,求图示自由端的挠度。,逐段刚化求解,Example:DeterminetheslopeofsectionB.,Solve:,例题:,例如图，无限长的梁放置在刚性平台上，梁单位长度重量为q，C处有作用力P=qa，求C处的挠度。,梁的变形如图。设平台上梁抬起的长度为L，,2分钟小练习:梁抬起的长度L?,例如图，无限长的梁放置在刚性平台上，梁单位长度重量为q，C处有作用力P=qa，求C处的挠度。,梁的力学模型可简化为左方固定端铰处弯矩为零的悬臂梁。,梁的变形如图。设平台上梁抬起的长度为L，,例求C点竖向位移v。,5.梁的刚度,Stiffnessofbeams,5.1梁的刚度条件,-Allowedmaximumslope,Stiffnesscriteriaofabeam,-Allowedmaximumdeflection,Inengineeringtheallowedmaximumdeflectionwasofengivedbytheformasfollows:,容许转角,容许挠度,5.2刚度条件的应用(Applicationsofstiffnesscriteria),(1)校核刚度(Toexaminethestiffnessofabeam),(2)计算许可载荷(Todeterminethemaximumexternalforces),(3)计算许可截面尺寸(Todeterminethesize