英汉双语材料力学11.pptx

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1、第1页/共66页CHAPTER 11 ENERGY METHODCHAPTER 11 ENERGY METHOD 111 GENERAL EXPRESSIONS OF THE STRAIN ENERGY112 MOHRS THEOREM(METHOD OF UNIT FORCE)113 CATIGLIANOS THEOREM第2页/共66页第十一章第十一章 能量方法能量方法 111 变形能的普遍表达式112 莫尔定理(单位力法)113 卡氏定理第3页/共66页111 GENERAL EXPRESSIONS OF THE STRAIN ENERGY1、Principle of energy：2、

2、Calculation of the strain energy of rods：1).1).Calculation of the strain energy of rods in tension or compression：Strain energy stored in the elastic body is equal to the work done by external forces，that is:Method to analyze and calculate displacements、deformations and internal forces of deformable

3、 bodies by this kind of relation is called energy method.orDensity of the strain energy:第4页/共66页111 变形能的普遍表达式一、能量原理：二、杆件变形能的计算：1.1.轴向拉压杆的变形能计算：弹性体内部所贮存的变形能，在数值上等于外力所作的功，即 利用这种功能关系分析计算可变形固体的位移、变形和内力的方法称为能量方法。第5页/共66页2.2.Calculation of the strain energy of rods in torsion：3.3.Calculation of strain ene

4、rgy of rods in bending：or Density of the strain energy:orDensity of the strain energy:第6页/共66页2.2.扭转杆的变形能计算：3.3.弯曲杆的变形能计算：第7页/共66页3、General expressions of the strain energy：Strain energy is independent of the order of loading.Deformations due to mutually independent load may be summed up each other.

5、For slender columns，the strain energy due to shearing forces may be neglected.Deflection factor of shear第8页/共66页三、变形能的普遍表达式：变形能与加载次序无关；相互独立的力（矢）引起的变形能可以相互叠加。细长杆，剪力引起的变形能可忽略不计。第9页/共66页Solution：In energy method（work done by external forces is equal to the strain energy）Determine internal forcesDetermi

6、ne internal forcesABending moment:Torque:Example 1 A semicircle rod as shown in the figure is lie in horizontal plane.A vertical force P act at its point A.Determine the displacement of point A in vertical direction.PROQMNMTAAPNBj jTO第10页/共66页MN 例1 1 图示半圆形等截面曲杆位于水平面内，在A点受铅垂力P的作用，求A点的垂直位移。解：用能量法（外力功等

7、于应变能）求内力APROQMTAAPNBj jTO第11页/共66页Work done by external forces is equal to the strain energyWork done by external forces is equal to the strain energyStrain energyStrain energy：Letthen第12页/共66页外力功等于应变能变形能：第13页/共66页Example Example 2 Determine the deflection of point C by the energy method,where the b

8、eam is of equal section and straight.Solution：Work done by external Work done by external forces is equal to the strain energyforces is equal to the strain energyBy using symmetry we get：Thinking：For the distributed load,can we determine the displacement of point C by this method？qCaaAPBfLet第14页/共66

9、页 例2 用能量法求C点的挠度。梁为等截面直梁。解：外力功等于应变能应用对称性，得：思考：分布荷载时，可否用此法求C点位移？qCaaAPBf第15页/共66页112 MOHRS THEOREM(METHOD OF UNIT FORCE)Determine the displacement f A of an arbitrary point A.1、Provement of the theorem：aAFigfAq(x)Figc A0P=1q(x)fAFigb A=1P0第16页/共66页112 莫尔定理(单位力法)求任意点A的位移f A。一、定理的证明：aA图fAq(x)图c A0P=1q(x

10、)fA图b A=1P0第17页/共66页 Mohrs theorem(method of unit force)2、General form of Mohrs theorem第18页/共66页 莫尔定理(单位力法)二、普遍形式的莫尔定理第19页/共66页3、What we must pay attention to as we apply Mohrs theorem：Coordinate of Coordinate of M0(x)must be coincide with that of M(x).For each segment the coordinate may be set up f

11、reely.Mohrs integrationmust be through the whole structure.Mohrs integrationmust be through the whole structure.M0:The internal force of the structure as we act a generalized unit force along the direction,of the generalized displacement that is to be determined,where the applied force is taken out.

12、M(x)：The internal force of the structure acted by original loads.The product of the applied generalized unit force and the generalized The product of the applied generalized unit force and the generalized displacement to be determined determined must be of the dimension of displacement to be determi

13、ned determined must be of the dimension of workwork.第20页/共66页三、使用莫尔定理的注意事项：M0(x)与M(x)的坐标系必须一致，每段杆的坐标系可 自由建立。莫尔积分必须遍及整个结构。M0去掉主动力，在所求 广义位移广义位移 点，沿所求 广义位移广义位移 的方向加广义单位力广义单位力 时，结构产生的内力。M(x)：结构在原载荷下的内力。所加广义单位力与所求广义位移之积，必须为功的量纲。第21页/共66页Example 3 3 Determine the displacement and the angle of rotation of

14、point C by the energy method.Solution：Plot the diagram of the structure acted by the unit loadPlot the diagram of the structure acted by the unit load Determine the internal forceBAaaCqBAaaC0P=1x第22页/共66页 例3 3 用能量法求C点的挠度和转角。梁为等截面直梁。解：画单位载荷图求内力BAaaCqBAaaC0P=1x第23页/共66页SymmetrySymmetryDeformationBAaaC

15、0P=1BAaaCqx()第24页/共66页变形BAaaC0P=1BAaaCqx()第25页/共66页Determine the angle of rotation.Set up the coordinate again(as shown in the figureDetermine the angle of rotation.Set up the coordinate again(as shown in the figure）qBAaaCx2x1BAaaCMC0=1 d)()()()()(00)(00+=aBCaABxEIxMxMdxEIxMxM=0第26页/共66页求转角，重建坐标系（如图

16、）qBAaaCx2x1BAaaCMC0=1 d)()()()()(00)(00+=aBCaABxEIxMxMdxEIxMxM=0第27页/共66页Solution：Plot the diagram of Plot the diagram of the structure acted by a unit load the structure acted by a unit load Determine the internal force510 20A300P=60NBx500Cx1510 20A300Bx500C=1P0Example 4 Example 4 A folding rod is

17、shown in the figure.A bearing is at position A and the rod may rotate freely in the bearing but can not move up and down.Knowing：E=210Gpa，G=0.4E，Determine the vertical displacement of point B.第28页/共66页 例4 4 拐杆如图，A处为一轴承，允许杆在轴承内自由转动，但不能上下移动，已知：E=210Gpa，G=0.4E，求B点的垂直位移。解：画单位载荷图求内力510 20A300P=60NBx500Cx

18、1510 20A300Bx500C=1P0第29页/共66页Determine the deformationDetermine the deformation()第30页/共66页变形()第31页/共66页113 CATIGLIANOS THEOREMGive Pn an increment dPn，then：1)First apply forces P1、P2、Pn on the body，then：2).First apply the force dPn on the body，then：1、Provement of the theorem d dn第32页/共66页113 卡氏定理给P

19、n 以增量 dPn，则：1.先给物体加P1、P2、Pn 个力，则：2.先给物体加力 dPn，则：一、定理证明 d dn第33页/共66页Again apply forces P1、P2、Pn，then：d dnn=nPU d dSecond Castiglianos theorem Italian engineer Alberto Castigliano，18471884 第34页/共66页再给物体加P1、P2、Pn 个力，则：d dnn=nPU d d第二卡氏定理 意大利工程师阿尔伯托卡斯提安诺(Alberto Castigliano，18471884)第35页/共66页2、what we

20、must pay attention to as we apply Catiglianos theorem：ULinear elastic strain energy of the whole structure acted by external loads Pn is considered as a variable.The reactions and the strain energy of the structure and so on must be all expressed as the function of Pn.n n is the deformation of the p

21、oint acted by Pn and it isalong the direction of Pn.If there is noIf there is no Pn corresponding to n n we may first act a Pn along n n and determine the partial derivative and then let Pn be zero.d dn第36页/共66页二、使用卡氏定理的注意事项：U整体结构在外载作用下的线 弹性变形能 Pn 视为变量，结构反力和变形能 等都必须表示为 Pn的函数 n n为 Pn 作用点的沿 Pn 方向的变形。当

22、无与 n n对应的 Pn 时，先加一沿 n n 方向的 Pn，求偏导后，再令其为零。d dn第37页/共66页3、Castiglianos theorem for special structures(rods)：第38页/共66页三、特殊结构（杆）的卡氏定理：第39页/共66页Example 5 Example 5 The structure is shown in the figure.Determine the deflection and the angle of rotation of the section A by Catiglianos theorem.Determine th

23、e deformationDetermine the internal forceSolution：Determine the deflection.Set up the coordinateDetermine the partial derivativeDetermine the partial derivative of the internal force with respect to of the internal force with respect to PAALPEIxO()第40页/共66页 例5 5 结构如图，用卡氏定理求A 面的挠度和转角。变形求内力解：求挠度，建坐标系将

24、内力对PA求偏导ALPEIxO()第41页/共66页Determine the angle A of rotationDetermine the internal forceDetermine the internal forceThere is no the generalized force corresponding to A.we may act one.“Negative sign”expresses that A is contrary to the direction of the acted generalized force MA()Determine the partial

25、 derivative of the Determine the partial derivative of the internalinternal force force MM(x x)with respect to)with respect to MA and let M A=0.Determine the deformation（Note：M A=0）LxO APMA第42页/共66页求转角 A求内力没有与A向相对应的力（广义力），加之。“负号”说明 A与所加广义力MA反向。()将内力对MA求偏导后，令M A=0求变形（注意：M A=0）LxO APMA第43页/共66页Example

26、 6 Determine the deflection curve of the beam shown in the figure by Castiglianos theorem.Solution：Determine the deflection curvethe deflection of an arbitrary point on the beam f（x）.Determine the internal forcesDetermine the partial derivative of the internalDetermine the partial derivative of the

27、internal force force MM(x x)with respect to)with respect to Px and let Px=0.There is no the generalized force corresponding to f（x）.we may act one.PALxBPx CfxOx1第44页/共66页 例6 结构如图，用卡氏定理求梁的挠曲线。解：求挠曲线任意点的挠度 f（x）求内力将内力对Px 求偏导后，令Px=0没 有 与f（x）相 对 应 的 力，加 之。PALxBPx CfxOx1第45页/共66页Determine the deformationD

28、etermine the deformation（Note：Px=0）第46页/共66页变形（注意：Px=0）第47页/共66页Example 7 A beam with equal section is shown in the figure.Determine the deflection f(x)of point B by Catiglianos theorem.determine internal forcesSolution：1.Determine redundant reactions according toDetermine the partial derivative of

29、Determine the partial derivative of the internal force with respect to the internal force with respect to RC.Take a primary beam as shown in theTake a primary beam as shown in thePCAL0.5 LBfxOPCAL0.5 LBRCfigure.figure.第48页/共66页 例7 等截面梁如图，用卡氏定理求B 点的挠度。求内力解：1.依 求多余反力，将内力对RC求偏导取静定基如图PCAL0.5 LBfxOPCAL0.

30、5 LBRC第49页/共66页DeformationSo第50页/共66页变形第51页/共66页2.DetermineDetermine the partial derivative of the internal force with respectDetermine the partial derivative of the internal force with respectDetermine the internal forcesDetermine the internal forcesto to P.第52页/共66页2.求将内力对P求偏导求内力第53页/共66页Deformati

31、on()第54页/共66页变形()第55页/共66页Determine the deformationDetermine the deformationSolution：Plot the diagram of the structure acted by unit loadDetermine the internal forceDetermine the internal forceExample 8 A frame is shown in the figure.Determine the distance between section A and section B after the d

32、eformation.PPAB11第56页/共66页变形解：画单位载荷图求内力 例8 结构如图，求A、B两面的拉开距离。PPAB11第57页/共66页58 Chapter 11 Exercises1.A straight rod with the tension(compression)rigidity EI is subjected forces shown in the figure.May the strain energy be expressed as 2.Try to explain how to determine the deflection of the free end o

33、f the beam shown in the figure by Castiglianos theorem.3.As shown in the figure,a rigid frame is subjected to forces.Knowing EI is a constant.Try to determine the relative displacement between point A and point B by Mohrs theorem(neglecting the tensile deformation of Section CD).第58页/共66页59 第十一章 练习题

34、 一、抗拉（压）刚度为EIEI的等直杆，受力如图，其变形能是否为：二、试述如何用卡氏定理求图示梁自由端的挠度。三、刚架受力如图，已知EIEI为常数，试用莫尔定理求A A、B B两点间的相对位移（忽略CDCD段的拉伸变形）。第59页/共66页60 Solution:第60页/共66页61解：第61页/共66页62 4.A beam with the bending rigidity EI is shown in the figure.The rigidity of the spring at the end B is k.Try to determine the deflection of the point where the force P is applied by Castiglianos theorem.Solution:The strain energy of the system is The deflection of Section C is 第62页/共66页63 四、抗弯刚度为EIEI的梁如图，B B端弹簧刚度为k k，试用卡氏定理求力P P作用点的挠度。解：系统的变形能 C C截面的挠度第63页/共66页64第64页/共66页65第65页/共66页66感谢您的观看！第66页/共66页