英汉双语材料力学11PPT推荐.ppt

1、5111 变形能的普遍表达式变形能的普遍表达式一、能量原理：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：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。62.2.Calculation of the strain energy of rods in torsion：3.3.Calculation of strain energy of rods in bending：or Density of the strain energy:orDensity of the st

2、rain energy:72.2.扭转杆的变形能计算：扭转杆的变形能计算：3.3.弯曲杆的变形能计算：弯曲杆的变形能计算：83、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.For slender columns，the strain energy due to shearing forces may be

3、neglected.Deflection factor of shear9三、变形能的普遍表达式：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。10Solution：In energy method（work done by external forces is equal to the strain energy）Determine internal forcesDetermine internal forcesABending moment:Torque:Example 1 A semicircle rod a

4、s 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 jTO11MN 例例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（外力功等于应变能）求内力APROQMTAAPNBj jTO12Work done by external forces is equal to the str

5、ain energyWork done by external forces is equal to the strain energyStrain energyStrain energy：Letthen13外力功等于应变能变形能：14Example 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 Work done by external forces

6、 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？qCaaAPBfLet15 例例2 用能量法求C点的挠度。梁为等截面直梁。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？qCaaAPBf16112 MOHRS THEOREM（METHOD OF UNIT

7、 FORCE）Determine the displacement f A of an arbitrary point A.1、Provement of the theorem：aAFigfAq（x）Figc A0P=1q（x）fAFigb A=1P017112 莫尔定理莫尔定理（单位力法单位力法）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）fA图b A=1P018 Mohrs theorem（method of unit force）2、General form of Mohrs theorem19 莫尔定理莫尔定理（单位力法单位力法）二、普遍形式的莫尔

8、定理二、普遍形式的莫尔定理203、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 freely.Mohrs Mohrs integrationmustintegrationmust be through the whole structure.be through the whole structure.M

9、0: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.M（x）：The internal force of the structure acted by original loads.The product of the applied generalized unit forc

10、e 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 workdisplacement to be determined determined must be of the dimension of work.21三、使用莫尔定理的注意事项：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

11、立。莫尔积分必须遍及整个结构。M0去掉主动力，在所求 广义位移广义位移广义位移广义位移 点，沿所求 广义位移广义位移广义位移广义位移 的方向加广义单位力广义单位力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。22Example 3 3 Determine the displacement and the angle of rotation of point C by the energy method.Solution：Plot the diagram of the structure acted by the un

12、it loadPlot the diagram of the structure acted by the unit load Determine the internal forceBAaaCqBAaaC0P=1x23 例例3 3 用能量法求C点的挠度和转角。画单位载荷图求内力BAaaCqBAaaC0P=1x24SymmetrySymmetryDeformationBAaaC0P=1BAaaCqx（）25变形BAaaC0P=1BAaaCqx（）26Determine the angle of rotation.Set up the coordinate again（as shown in t

13、he figureDetermine the angle of rotation.Set up the coordinate again（as shown in the figure）qBAaaCx2x1BAaaCMC0=1 d）（）（）（）（）（00）（00+=aBCaABxEIxMxMdxEIxMxM=027求转角，重建坐标系（如图）qBAaaCx2x1BAaaCMC0=1 d）（）（）（）（）（00）（00+=aBCaABxEIxMxMdxEIxMxM=028Solution：Plot the diagram of Plot the diagram of the structure ac

14、ted 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 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.29 例例4 4 拐