第七章+弯曲变形+.ppt

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1、BilingualEdition,return,exit,Chapter7,Chapter7Deformationinbending,71Basicconceptsandexampleproblemsinengineering弯曲变形的基本概念和工程实例72Approximatedifferentialequationofthedeflectioncurve挠曲线的近似微分方程73Determinedeflectionsandrotationalanglesofthebeambytheintegrationmethod用积分法求弯曲变形,弯曲变形,GO,GO,Chapter7,GO,74Det

2、erminedeflectionsandrotationalanglesofthebeambythesuperpositionmethod用叠加法求弯曲变形75Methodtosolvesimplestaticallyindeterminateproblemsofthebeam简单超静定梁的求解方法76Somemeasurementsimprovingthebendingrigidityofbeam提高弯曲刚度的一些措施,Chapter7,GO,GO,GO,弯曲变形的基本概念和工程实例,71Basicconceptsandexampleproblemsinengineering,1.Pract

3、icalproblemsofdeformationinbending,Chapter7,Becausethedesignofbeamsisfrequentlygovernedbyrigidityratherthanstrength,thecomputationofdeformationisalsoimportantinbeamanalysis.,桥式起重机的横梁变形过大,则会使小车行走困难，出现爬坡现象。,Chapter7,摇臂钻床的摇臂或车床的主轴变形过大，就会影响零件的加工精度，甚至会出现废品。,Chapter7,齿轮传动,轮齿不均匀磨损，噪声增大，产生振动；,加速轴承磨损，降低使用寿命；

4、若变形过大，使传动失效。,Chapter7,但在另外一些情况下，有时却要求构件具有较大的弹性变形，以满足特定的工作需要。,例如，车辆上的叠板弹簧，要求有足够大的变形，以缓解车辆受到的冲击和振动作用。,Chapter7,跳水用的跳板如果太硬，就不能起助跳作用,Chapter7,继电器中的簧片,当变形足够大时，可以有效接通电路；,当变形不够大时，不能有效接通电路；,Inengineering:,Chapter7,2.Basicconcepts,Thedeflectioncurve挠曲线,Thesmoothcurvethattheaxisofthebeamistransformedintoafter

5、deformationiscalledthedeflectioncurve.梁的轴线变形后的曲线。,（连续、光滑、平坦的平面曲线）,Chapter7,Itisdesignatedbyw.Itispositiveifitsdirectionisthesameasy,otherwiseitisnegative.,横截面形心在垂直于轴线方向的位移。,deflection挠度,angleofrotation转角,变形后，横截面相对其原来位置转过的角度。,Itisdesignatedby.Itispositiveiftheangleofrotationrotatesintheanticlockwised

6、irection,otherwiseitisnegative.,Theequationofdeflectioncurve,转角即为挠曲线在该点的切线与x轴的夹角。,Chapter7,Therelationbetweentheangleofrotationandthedeflectioncurve转角与挠曲线的关系,(通常b,Chapter7,wmaxoccursinthesegmentAC.,when,Chapter7,Discussion:,when,From:,Weget,(approximate),(accurate),2.65%,Chapter7,Example3已知梁的EI为常数,今

7、欲使梁的挠曲线在处出现一拐点，则比值为多少？,由梁的挠曲线近似微分方程,在梁挠曲线的拐点处有：,从弯矩图可以看出：,知，,Solution:,Chapter7,determinethereactionforces,Writeoutthedifferentialequationandintegrateit,FromAtoB:,Solution:,Chapter7,Example4Fortheprismaticbeamandloadingshown,determinetheslopeanddeflectionatthefreeendC.,FromBtoC:,Chapter7,determineth

8、econstantsofintegration,Boundaryconditions:,Continuityandsmoothcondition:,ABsegment:,BCsegment:,Chapter7,findCandwC,BCsegment:,Chapter7,Example5Determinetheequationsofthedeflectioncurveandtheangleofrotation,maxandwmax,Becauseof,Solution:,Chapter7,symmetry,weneedtoanalyzeonlyhalfofthebeamACD,Continui

9、tyandsmoothconditions:,Boundaryconditions:,Thesymmetryofbeamimplies：,Chapter7,theequationsofthedeflectioncurveandslope:,Themaximumandw:,Chapter7,积分法求梁的变形关键点：,分段原则：集中力、集中力偶作用点，分布力的起、终点，梁的自然端点为分段点。,边界条件：支承条件、连续条件、光滑条件。有多少积分常数就有且仅有多少个边界条件。,积分法优点：得到挠度方程w(x)和转角方程(x)。因而可求出任意截面的挠度和转角。,积分法缺点：繁、荷载复杂时分段多，因而积分

10、常数多。,Chapter7,用叠加法求弯曲变形,74Determinedeflectionsandrotationalanglesofthebeambythesuperpositionmethod,1.Superpositionofloads载荷叠加,Chapter7,Applyingcondition:Relationbetweenthedeformation(includingthedeflectionandangleofrotation)andtheexternalforcesmustbelinear,thatistheysatisfyHookeslaw.,Themethodofsupe

11、rpositionpermitustousetheknowndisplacementsandslopesforsimpleloadstoobtainthedeformationsforthemorecomplicatedloadings.,Tousethemethodefficientlyrequiresremembertable6.1(1,2,4,6,8,10)byheart.(page185),Chapter7,Determinethedeformationsofthebeamundertheactionofsimpleloadsbylookingupthetable.,Example6D

12、eterminethedeflectionofpointC.,Solution:,Chapter7,Example7DeterminethedeflectionofpointA.,Solution:,Chapter7,Example8Trytodetermine,Solution:,Chapter7,Example9已知AC=BC=a，求wB。,BC段仅有刚体位移，保持直线，如图所示,Solution:,Chapter7,Example10DeterminethedeflectionatpointB.,Solution:,Chapter7,rigidizationmethod刚化法,2.Sup

13、erpositionofstructuralforms结构形式叠加,Chapter7,Bythismethod,thebeamwillbedividedintoseveralportions.,Thismethodisveryapplicableforoverhangingbeamsandcantileverbeamswithpiecewiseconstantcrosssections.,Thefollowingsampleproblemswillgivetheexplanationoftherigidizationmethodofsegmentbysegment.逐段刚化法,在内力不变的前提

14、下，将梁分解(或刚化)为几段，求出各段的变形，然后进行叠加。,Example10DeterminethedeflectionatpointCandB.,BC段变形，AC段刚化,AC段变形，BC段刚化,总变形,Solution:,RigidizeAC,RigidizeBC,Determine:,Chapter7,C,rigidizationmethodofsegmentbysegment.逐段刚化法,RigidizethesegmentAC,RigidizethesegmentBC,Determine:,Chapter7,Example11Determine(I2=2I1),Rigidizeth

15、esegmentAC,RigidizethesegmentBC,Solution:,Chapter7,Sumupthetotaldeformation,Chapter7,Example12Determine(I2=2I1),RigidizethesegmentAC,Solution:,RigidizethesegmentBC,Chapter7,Sumupthetotaldeformation,Chapter7,Example13Determine,Solution:,Chapter7,Example14图示梁处为弹性支座，弹簧刚度,求C端挠度wC。,(1)梁不变形，仅弹簧变形引起的C点挠度为:

16、,(2)弹簧不变形，仅梁变形引起的C点挠度为:,(3)C点总挠度为,Solution:,Chapter7,Example15求图示梁C、D两点的挠度wC、wD。,Solution:,Chapter7,Example16求图示梁C点的挠度wC。,Solution:,Chapter7,Example17求图示梁跨中的挠度wC和B点的转角B,弹簧缩短量,Solution:,Chapter7,Solution:,Example18Determine,Chapter7,Example19DeterminewC,(2)RigidizethesegmentCD,(1)Becauseofsymmetry,So

17、lution:,Chapter7,(3)RigidizethesegmentDB,(4)FindwC：,Chapter7,Example20水平面上的直角拐，AB段为圆轴，直径为d，BC段截面为的矩形，在端点C受铅垂力P作用，E，G已知。求C点的铅垂位移。,分析：,AB弯曲+扭转变形,BC弯曲变形,C点的挠度由三部分组成：,AB弯曲引起的B点下沉,AB扭转引起C点位移,Chapter7,采用逐段刚化法,(1)AB刚化,计算BC弯曲变形引起的C点的挠度.,Solution:,(2)将BC刚化,(保留BC对AB的作用力)，计算AB弯曲引起的C点的挠度,Chapter7,(3)将BC刚化计算AB扭转

18、变形引起的C点的挠度,计算B截面扭转角,所以，C点位移为：,Chapter7,Example21要求滚轮恰恰走一水平路径，试问梁的轴线应预先弯成怎样的曲线？,Solution:,Chapter7,Example22Anequal-sectionbeamwiththelengthLandtheweightPisputonthehorizontalrigidplane.IfoneendofthebeamisliftedbytheforceP/3,buttheotherpartstillstickontheplane.,Trytodeterminethelengthoftheliftedpart.,

19、Solution:,Theradiusof,curvatureatpointAofthebeamis,thatis,Chapter7,Example23悬臂梁下有一刚性曲面，方程为y=ax3+bx2+c,试问梁上作用什么样的载荷，方使梁与曲面恰好叠合？（a0,b0),(1)边界条件：,(2)确定载荷,Solution:,Chapter7,Chapter7,静定梁：,简单超静定梁的求解方法,75Methodtosolvesimplestaticallyindeterminateproblemsofthebeam,Chapter7,1.Thedegreeofstaticindeterminacy,

20、静不定次数,=未知力个数-独立平衡方程数,Combiningthecompatibilityequationofdeformation,physicalequationwithequilibriumequationstodeterminethewholeunknownforces.,2.Treatmentmethod,Thestructureinwhichredundantconstraintsaresubstitutedbyreactionsprimarystructure.,注意：静定基不是唯一的,Chapter7,Geometricequationcompatibilityequatio

21、nofdeformation.,Physicalequationrelationbetweenthedeformationandforces.,Complementaryequation,Chapter7,Example24DeterminethereactionatendBinthestructureasshowninthefigure.,Determinethedegreeofstaticallyindeterminacy,4-3=1,Setuptheprimarybeam,Physicalequationrelationbetweenthedeformationandforces.,Ge

22、ometricequationcompatibilityequationofdeformation.,Chapter7,Solution:,Complementaryequation,Solveotherproblems(reaction,stress,deformationetc.）,Chapter7,(1)若要求MA、YA，即可把YB代入相应方程（静平衡方程）求解即可。,(2)求得所有的支反力后，可列M方程，画M图，进行强度和刚度的计算。,注:,Chapter7,Example25Determinethereactionsofthesupportsinthestructureasshown

23、inthefigure.,Solution:,Thecompatibilityequationofdeformation:,Chapter7,提高弯曲刚度的一些措施,76Somemeasurementsimprovingthebendingrigidityofbeam,1.Rigidityconditionsofthebeam,梁的刚度条件,Ingeneralwecandothreekindsofcalculationsabouttherigiditybythesesconditions.,thepermissibleangleofrotation许可转角,thepermissiblerati

24、oofthedeflectionandthespan.许用挠跨比,Chapter7,Example26Abeamwith-sectionisshowninthefollowingfigure.Knowingl=8m,Iz=2370cm4,Wz=237cm3,w=l500,E=200GPa,=100MPa.Trytocalculatethepermissibleloadaccordingtotherigidityconditionofthebeamandcheckthestrengthofthebeam.,Strengthofthisbeamsatisfiesrequest.,Solution:

25、,Chapter7,2.Somemeasurementsimprovingthebendingrigidityofbeam.提高弯曲刚度的一些措施,影响梁弯曲变形的因素不仅与梁的支承和载荷情况有关，而且还与梁的材料、截面尺寸、形状和梁的跨度有关。所以，要想提高弯曲刚度，就应从上述各种因素入手：,增大梁的抗弯刚度EI,梁的挠曲线微分方程为:,Chapter7,增大梁的抗弯刚度EI,抗弯刚度EI除与截面形状有关外，还与弹性模量有关。,钢材的弹性模量较大，故用钢材制造的构件有较大的抗弯刚度。,大部分钢材的E值是相近的，因此，增大梁的抗弯刚度，主要是增大Iz值选择合理的截面形状。,将截面面积布置在距中性轴较远处，可在面积不变的情况下获得较大的Iz，这样不但能降低应力，还能减小位移。,改变加载方式和支座形式,Chapter7,改变支座形式：,改变加载方式：,Chapter7,减小跨度或增加支承采用超静定结构,Chapter7,Endofthischapter,Chapter7,Goodbye,Thanks!,Page1986.4(d),6.9(a,d),Page2046.20,6.36,Goodbye,教材：刘鸿文编，材料力学（上、下册）高等教育出版社，2010年9月第5版。,第七章教材上的作业,