平的签论 下面推导平面应力问题的平衡微分方程,对单元体列平 衡方程: ∑F F.=0 (ax+ax):x1-ax×1+(x+-dhy)dx×1 r…·ax×1+X·a dy 1=0 ∑ F y =0 (a,+d )ax×1-anax×1+(xn+ax)dy×1 y Tr. dyx1+1Yaxd×1=0 16
16 下面推导平面应力问题的平衡微分方程，对单元体列平 衡方程： 1 1 0 ( ) 1 1 ( ) 1 0 : − + = − + + + = dx X dx dy dy dx y dx dy dy x F yx yx x yx x x x 1 1 0 ( ) 1 1 ( ) 1 0 : − + = − + + + = dy Y dx dy dx dy x dy dx dx y F xy xy y xy y y y
Sorting them gets 00s+ +X=0 ax a y++Y=0 These two differential equation include three unknown functions o,o, tyy=Tx. Therefore, deciding the problem of the stress weight is exceedingly and statically determinate; And still must consider the deformation and displacement, then the problem can be solved For the plane strain problem, the faces front and back still haveo But they do not affect completely the establishes of the equation above. So the equation above applies two kinds of plane problem alike
17 Sorting them gets: 0 0 + = + + = + Y y x X x y y xy x yx These two differential equation include three unknown functions .Therefore,deciding the problem of the stress weight is exceedingly and statically determinate;And still must consider the deformation and displacement,then the problem can be solved. For the plane strain problem,the faces front and back still have But they do not affect completely the establishes of the equation above.So the equation above applies two kinds of plane problem alike. z x y xy yx , , =
平的签论 整理得: 0+m+x=0 ax a 0o.0r a,++F=0 这两个微分方程中包含着三个未知函数x可,x=m。因此 决定应力分量的问题是超静定的;还必须考虑形变和位移,才能 解决问题 对于平面应变问题,虽然前后面上还有,但它们完全不影 响上述方程的建立。所以上述方程对于两种平面问题都同样适用
18 整理得: 0 0 + = + + = + Y y x X x y y xy x yx 这两个微分方程中包含着三个未知函数 。因此 决定应力分量的问题是超静定的；还必须考虑形变和位移，才能 解决问题。 对于平面应变问题,虽然前后面上还有 ,但它们完全不影 响上述方程的建立。所以上述方程对于两种平面问题都同样适用。 z x y xy yx , , =
82-3 The stress on the Inclined Plane. Principal stress 1.The stress on the inclined plane Having known the stress weight ox,a,, Txy =t,of any point P inside the elastic body, we try to get the stress which pass the point p on the arbitrarily inclined cross section From neighborhood of point P taking a plane AB, which is in parallel with the inclined plane above, and draws a small set square or three column PAB on two planes which pass point P and have perpendicularity in the shaft of x and y. When the plane aB approaches point P infinitely, the mean stress on the plane ab will become the stress on the inclined plane above C Establish the length of the face AB in the A plane xy is ds, n is the exterior normal direction and its direction cosine is cos(N, x)=l, cos(N, y)=m S N Fig 2-4 19
19 §2-3 The stress on the Inclined Plane.Principal stress 1.The stress on the inclined plane Having known the stress weight of any point P inside the elastic body,we try to get the stress which pass the point P on the arbitrarily inclined cross section.From neighborhood of point P taking a plane AB,which is in parallel with the inclined plane above,and draws a small set square or three column PAB on two planes which pass point P and have perpendicularity in the shaft of x and y.When the plane AB approaches point P infinitely,the mean stress on the plane AB will become the stress on the inclined plane above. x y xy = yx , , Establish the length of the face AB in the plane xy is dS,N is the exterior normal direction,and its direction cosine is: cos(N, x) = l,cos(N, y) = m P A B xy x y N yx N X N YN y S N x Fig.2－4 o
平的签论 §2-3斜面上的应力、主应力 斜面上的应力 已知弹性体内任一点处的应力分量x2Oyy=x,求经 过该点任意斜截面上的应力。为此在P点附近取一个平面AB, 它平行于上述斜面,并与经过P点而垂直于x轴和轴的两个平 面划出一个微小的三角板或三棱柱PAB。当平面A与P点无限 接近时,平面AB上的应力就成为上述斜面上的应力。 设A面在x平面内的长度为dS, A 厚度为一个单位长度,M为该面的外 法线方向,其方向余弦为: cos(N, x)=/, cos(N, y)=m S N 图2-4 20
20 §2-3 斜面上的应力、主应力 一、斜面上的应力 已知弹性体内任一点P处的应力分量 ，求经 过该点任意斜截面上的应力。为此在P点附近取一个平面AB， 它平行于上述斜面，并与经过P点而垂直于x轴和y轴的两个平 面划出一个微小的三角板或三棱柱PAB。当平面AB与P点无限 接近时，平面AB上的应力就成为上述斜面上的应力。 x y xy yx , , = 设AB面在xy平面内的长度为dS， 厚度为一个单位长度，N 为该面的外 法线方向，其方向余弦为： cos(N, x) = l,cos(N, y) = m P A B xy x y N yx N X N YN y S N x 图2－4 o