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1、,CHAPTER 11 ENERGY METHODS,材料力学,第十一章 能量方法,CHAPTER 11 ENERGY METHOD,111 GENERAL EXPRESSIONS OF THE STRAIN ENERGY112 MOHRS THEOREM(METHOD OF UNIT FORCE)113 CATIGLIANOS THEOREM,第十一章 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,111 GENERAL EXPRESSIONS OF THE STRAIN ENERGY,1、Principle of energy：,2、Calculatio
2、n of the strain energy of rods：,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 bodies
3、by this kind of relation is called energy method.,ENERGY METHOD,or,Density of the strain energy:,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.轴向拉压杆的变形能计算：,能量方法,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。,2. Calculation of the strain energy of rods in torsion：,3. Calculation of strain ene
4、rgy of rods in bending：,ENERGY METHOD,or,Density of the strain energy:,or,Density of the strain energy:,2.扭转杆的变形能计算：,3.弯曲杆的变形能计算：,能量方法,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
5、 other.,For slender columns，the strain energy due to shearing forces may be neglected.,ENERGY METHOD,Deflection factor of shear,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。,能量方法,Solution：In energy method（work done by external forces is equal to the strain energy）,Determine inter
6、nal forces,A,ENERGY METHOD,Bending 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.,MN,例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（外力功等于应变能）,求内力,能量方法,
7、A,P,R,O,Work done by external forces is equal to the strain energy,Strain energy：,ENERGY METHOD,Let,then,外力功等于应变能,变形能：,能量方法,Example 2 Determine the deflection of point C by the energy method,where the beam is of equal section and straight.,Solution： Work done by external forces is equal to the strai
8、n energy,By using symmetry we get：,Thinking：For the distributed load ,can we determine the displacement of point C by this method？,C,a,a,A,P,B,f,ENERGY METHOD,Let,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？,能量方法,C,a,a,A,P,B,f,112 MOHRS THEOREM(METHOD OF UNIT FORCE),Determine the
9、displacement f A of an arbitrary point A.,1、Provement of the theorem：,a,A,Fig,fA,ENERGY METHOD,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：,能量方法,a,A,图,fA,Mohrs theorem(method of unit force),2、General form of Mohrs theorem,ENERGY METHOD,莫尔定理(单位力法),二、普遍形式的莫尔定理,能量方法,3、What we must pay attention to as we apply
10、 Mohrs theorem：, Coordinate of M0(x) must be coincide with that of M(x). For each segment the coordinate may be set up freely.,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 dis
11、placement that is to be determined, where the applied force is taken out., M(x)：The internal force of the structure acted by original loads., The product of the applied generalized unit force and the generalized displacement to be determined determined must be of the dimension of work.,ENERGY METHOD
12、,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 点，沿所求 广义位移 的方向加广义单位力 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。,能量方法,Example 3 Determine the displacement and the angle of rotation of point C by the energy method .,Solution：Plot the diagram of the struc
13、ture acted by the unit load,Determine the internal force,ENERGY METHOD,例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,能量方法,B,q,x,Deformation,x,ENERGY METHOD,变形,能量方法,x,Determine the angle of rotation. Set up the coordinate again (as shown in the figure）,ENERGY METHOD,=0,求转角，重建坐标系（如图）,能量方法,=0,Solution： Plot t
14、he diagram of the structure acted by a unit load,Determine the internal force,5,20,A,300,P=60N,B,x,500,C,x1,ENERGY METHOD,Example 4 A folding rod is 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，Dete
15、rmine the vertical displacement of point B.,例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,能量方法,5,20,A,300,P=60N,B,x,500,C,x1,Determine the deformation,ENERGY METHOD,变形,能量方法,113 CATIGLIANOS THEOREM,Give Pn an increment dPn ，then：,1)First apply forces P1、 P2、 Pn on the b
16、ody ，then：,2). First apply the force dPn on the body，then：,1、Provement of the theorem,ENERGY METHOD,113 卡氏定理,给Pn 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,能量方法,Again apply forces P1、 P2、Pn ，then：,Italian engineer Alberto Castigliano， 18471884,ENERGY METHOD,再给物体加P1、 P2、Pn 个力，则：,能量方
17、法,意大利工程师阿尔伯托卡斯提安诺(Alberto Castigliano， 18471884),2、what we 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 e
18、xpressed as the function of Pn., n is the deformation of the point acted by Pn and it isalong the direction of Pn ., If there is no Pn corresponding to n we may first act a Pn along n and determine the partial derivative and then let Pn be zero.,ENERGY METHOD,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
19、为变量，结构反力和变形能 等都必须表示为 Pn的函数, n为 Pn 作用点的沿 Pn 方向的变形。, 当无与 n对应的 Pn 时，先加一沿 n 方向的 Pn ，求偏导后， 再令其为零。,能量方法,3、Castiglianos theorem for special structures(rods)：,ENERGY METHOD,三、特殊结构（杆）的卡氏定理：,能量方法,Example 5 The structure is shown in the figure. Determine the deflection and the angle of rotation of the section
20、A by Catiglianos theorem.,Determine the deformation,Determine the internal force,Solution：Determine the deflection. Set up the coordinate,Determine the partial derivative of the internal force with respect to PA,A,L,P,EI,ENERGY METHOD,例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,将内力对PA求偏导,能量方法,A,L,P,E
21、I,Determine the angle A of rotation,Determine the internal force,There 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 derivative of the internal force M(x) with re
22、spect to MA and let M A =0.,Determine the deformation（ Note：M A=0）,L,x,O,A,P,M,A,ENERGY METHOD,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）,能量方法,L,x,O,A,P,M,A,Example 6 Determine the deflection curve of the beam shown in the figure by Castiglianos theorem.,So
23、lution：Determine the deflection curvethe deflection of an arbitrary point on the beam f（x）.,Determine the internal forces,Determine the partial derivative of the internal force M(x) with respect to Px and let Px =0.,There is no the generalized force corresponding to f（x）.we may act one.,P,A,L,x,C,EN
24、ERGY METHOD,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。,能量方法,P,A,L,x,C,Determine the deformation（ Note：Px=0）,ENERGY METHOD,变形（ 注意：Px=0）,能量方法,Example 7 A beam with equal section is shown in the figure. Determine the deflection f(x) of point B by Catiglianos theorem.,
25、determine internal forces,Solution：1.Determine redundant reactions according to,Determine the partial derivative of the internal force with respect to RC.,Take a primary beam as shown in the,P,C,A,L,0.5 L,B,ENERGY METHOD,figure.,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,能量方法,P,C,A,L,
26、0.5 L,B,Deformation,ENERGY METHOD,So,变形,能量方法,2.Determine,Determine the partial derivative of the internal force with respect,Determine the internal forces,ENERGY METHOD,to P.,2.求,将内力对P求偏导,求内力,能量方法,Deformation,ENERGY METHOD,变形,能量方法,Determine the deformation,Solution：Plot the diagram of the structure
27、acted by unit load,Determine the internal force,Example 8 A frame is shown in the figure. Determine the distance between section A and section B after the deformation.,P,P,A,B,ENERGY METHOD,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。,P,P,A,B,能量方法,59,Chapter 11 Exercises1. A straight rod with the tension (c
28、ompression) 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 of the beam shown in the figure by Castiglianos theorem.3. As shown in the figure, a rigid frame is subjected to forces. Knowing EI
29、 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).,ENERGY METHOD,60,第十一章 练习题 一、抗拉（压）刚度为EI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EI为常数，试用莫尔定理求A、B两点间的相对位移（忽略CD段的拉伸变形）。,能量方法,61,Solutio
30、n:,ENERGY METHOD,62,解：,能量方法,63,4. 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,ENERGY METHOD,64,四、抗弯刚度为EI的梁如图，B端弹簧刚度为k，试用卡氏定理求力P作用点的挠度。 解： 系统的变形能 C截面的挠度,能量方法,65,END,66,本章结束,
