第三章平面问题的直角坐标解答.ppt

CHAPTER 3 平面问题的直角坐标解答(SOLUTION OF PLANE PROBLEMS IN RECTANGULAR COORDINATES)
 
§ 逆解法与半逆解法多项式解答( Inverse Method and Semi-inverse Method. Solution by Polynomials)
按应力求解体力为常量的平面问题时, 归结为求解一个应力函数Ф, 它必须满足下列条件:
(1) 在区域内的相容方程
()
(2) 在边界上的应力条件:
()
求出应力函数Φ后,可由下式求得应力分量:
()
( the inverse method):先假定各种形式的满足相容方程()的应力函数Φ, 并由() ponents ,然后再根据应力boundary conditions ()和弹性体的形状,看这些应力分量对应于边界上什么样的面力,从而得知所选的 stress function 可以解决的问题.
In the inverse method, some functions satisfying the differential equations are taken and examined to see what boundary conditions these functions will satisfy and thereby to know what problems they can solve. In the case of solution by Airy’s stress function, we select some function Φ satisfying patibility equation (), find the ponents by Eqs.() or Eqs.() and then find the surface force
components by Eqs.(). In this way, we identify the problem which the stress function can be solve.
例1: 假定应力函数Φ为一次式:
可知不论各系数取何值,() ,若设体力为零,可得应力分量:
由(),可知由这个例子可见
(1)线性 stress function 对应于 no body forces, no surface forces and no stress的情况;
(2) 把平面问题的stress function 加上一个 linear function , 并不影响应力.
例2 取应力函数为二次式:
不论各系数取任何值, patibility equation () 总能满足.
在Φ=ax2 的情况, Eqs.() 给出的the ponents 为:
这种应力相当于()所示的应力. 可见Φ=ax2 能解决矩形板在y方向受均布拉力(设a>0)或均布压力(设a<0)的问题.

对应于Φ= bxy的情况, the ponents 为:
这种 stresses与() stress functionΦ= bxy 能解决矩形板受均布剪力的问题.
在stress function Φ= cy2 的情况,它能矩形板在x 方向受均布拉力(设c>0 )或均布压力(设c<0 )的问题. .
取三项式:
不论系数a 取何值,patibility equation () :
, 可见stress function 能解决矩形梁受纯弯曲的问题.

当 stress function 取四次或四次以上多项式时,则其中的系数要满足一定的条件,patibility equation.
半逆解法( the the semi-inverse method):
针对所求
the semi-inverse method, we assume the solution for the stresses or displacements in a given problem, based on the loading condition and boundary conditions of the problem, and then proceed to show tha