E.φ=ay3,X=0,Y=0 Bah Bah Fig 2 1g statically equivalent systems静力等效a=2M/h3 0x=6ay=12My/h3=My/I h3/12 0v=0 Txy=0 徐汉忠第一版20007 弹性力学第三章 16
徐汉忠第一版2000/7 弹性力学第三章 16 E. =ay3 , X=0, Y=0 • Fig 1 • Fig 2 • statically equivalent systems 静力等效 a=2M/h3 • x=6ay=12My/h3=My/I I=h3 /12 y = 0 xy=0
3.2 Determination of displacements when σ=Myσ、=0τ、=0位移的确定 In the case of plane stress substitution of stresses into the physical equations(2.6. 4) yields 平面应力问题,将应力代入物理方程得 应变 E=ox- uo,/E-My/(ED Eloy- Ho /E=-HMy/(ED G=0 XY 徐汉忠第一版20007 弹性力学第三章
徐汉忠第一版2000/7 弹性力学第三章 17 3.2 Determination of displacements when x=My/I y = 0 xy=0 位移的确定 • In the case of plane stress, substitution of stresses into the physical equations(2.6.4) yields 平面应力问题，将应力代入物理方程得 应变 x=[x - y ]/E =My/(EI) y=[y - x ]/E = - My/(EI) rxy =xy/G =0
Integration of geometrical equations E,=Ou/ox s=ov/oy r=ou/dy+ov/ox 几何方程的积分 substitution of strains into the geometrical equations(2.4.6)yields 将应变代入几何方程 Ou/ax=My/(ED U=MXy/(ED+f(y) av/ay=-HMy/(ED V=-HMy/(2ED+g(x) ou/dy+ov/ox=0 df(y)/dy=dg(x)/dx+MX/ED 徐汉忠第一版20007 弹性力学第三章 18
徐汉忠第一版2000/7 弹性力学第三章 18 Integration of geometrical equations x =u/x y =v/y rxy =u/y+v/x 几何方程的积分 • substitution of strains into the geometrical equations (2.4.6) yields 将应变代入几何方程 u/x =My/(EI) u=Mxy/(EI)+f(y) v/y = -My/(EI) v= -My2 /(2EI)+g(x) u/y+v/x=0 -df(y)/dy=dg(x)/dx+Mx/(EI)
Separation of variables.分离变量 df(y/dy=dg(x/dx+MX/ED=o -df(y/dy=o f(y=-oy+uo °dg(x)/dx+Mx/(ED= g(x)=-MX/(2ED+ox+vo U=MXY/(EI-oy+uo (3.2.5) v=-My2/(2ED)Mx2/(2ED+ox+v0(3.26 B=Ou/dy= MX/(EI-o ----the cross section remains plane after bending横截面弯曲变形后仍 为平面。 1/p=82v/0x2=-M/ED---all the longitudinal lines will have the same curvature曲率相同 徐汉忠第一版20007 弹性力学第三章
徐汉忠第一版2000/7 弹性力学第三章 19 =u/y= Mx/(EI)- ----the cross section remains plane after bending 横截面弯曲变形后仍 为平面。 1/ = 2v/x 2 =- M/(EI) ----all the longitudinal lines will have the same curvature曲率相同 Separation of variables 分离变量 -df(y)/dy=dg(x)/dx+Mx/(EI)= • -df(y)/dy= f(y)=- y+u0 • dg(x)/dx+Mx/(EI)= g(x)= - Mx2 /(2EI)+x+v0 • u=Mxy/(EI)- y+u0 (3.2.5) v= -My2 /(2EI)-Mx2 /(2EI)+x+v0 (3.2.6)
Review: Rigid-body displacements displacements corresponding to zero strains 刚体位移-应变为零时的位移 u=-oy +uo v=ox+V( uo--the rigid-body translation in the x direction x向刚体平动 Vo--the rigid-body translation in the y direction y向刚体平动 0--the rigid-body rotation of the body about z axis绕z轴的刚体转动 徐汉忠第一版20007 弹性力学第三章 20
徐汉忠第一版2000/7 弹性力学第三章 20 Review: Rigid-body displacements-- displacements corresponding to zero strains 刚体位移--应变为零时的位移 • u= - y +u0 v=x+v0 • u0 --the rigid-body translation in the x direction x向刚体平动 • v0 --the rigid-body translation in the y direction y向刚体平动 • --the rigid-body rotation of the body about z axis.绕z 轴的刚体转动