1、第三章 平面问题的直角坐标解答,31 逆解法与半逆解法,32 矩形梁的纯弯曲,33 位移分量的求出,34 简支梁受均布荷载,35 楔形体受重力和液体压力,Solution of Plane Problems in Rectangular Coordinates,Inverse Method and Semi-Inverse Method,Pure Bending of a Rectangular Beam,Determination of Displacments,Bending of a Simple Beam Under Uniform Loads,Triangular Gravity W
2、all,3.5 TRIANGULAR GRAVITY WALL,Exp. Consider a dam or a retaining wall with triangular section subjected to the action of gravity and the pressure of impounded liquid. Let the density of the wall material be and that of the liquid be .,35 楔形体受重力和液体压力,图示楔形体，左面铅直，右面与铅直成角，下端无限长，承受重力和液体压力，楔形体密度为 ，液体密度为
3、，计算应力分量,Triangular gravity wall,例.,取图示坐标，应力由两部分引起：,（1）重力：与g成正比,（2）液压：与g成正比,即，应力可能是gx、gy、gx、gy 的组合项,At any point in the wall,each of the stress components musy consist of two parts：,produced by gravity：is proportional to g,produced by the pressure：is proportional to g,Hence, if the stress components
4、can be expressions in the form of polynomials, they must be combinations of the expressions in the forms of Agx, Bgy, Cgx, Dgy.,可假设应力函数是x、y的纯三次式，即,We may assume the stress function is a polynomial of third degree,These expressions have already satisfied the differential equations of equilibrium and
5、the compatibility equation.,It remains to inspect whether the boundary conditions can also be satisfied by proper values of the arbitrary constants a, b, c, d.,The distribution of stress components along a horizontal section of the wall is shown in following Fig.：,应力分量x沿水平方向无变化这个结果在材料力学中得不出来,应力分量y沿水
6、平按直线规律变化,在左面：,在斜面：,与材料力学中偏心受压公式一样,应力分量xy沿水平按直线规律变化,在左面：,在斜面：,上述解答就是三角形重力坝中应力的基本解答,本章小结,（1）按应力求解平面问题得方法有逆解法 半逆解法,（2）几种简单的平面问题的多项式解答： a、一次式 =ax+by+c 对应无体力、无面力、无应力的情况,b、二次式 =ax2 解决矩形板在y方向受均布拉或均布压的问题,c、二次式 =bxy 解决矩形板受均布剪力的问题,c、三次式 =ay3解决图示矩形板纯弯曲问题,当取为坐标x、y的三次或三次以上的多项式时，应力分量将不是常量，而是坐标的的函数，此时，对于同一弹性体，在不同的坐标下，对应的应力分量不同，所解决的问题也不同,（3）求解弹性力学的一般思路或步骤a、确定应力函数（满足双调和方程）b、利用应力函数与应力分量之间的关系求应力分量（满足平衡微分方程、相容方程和边界条件）c、应用物理方程求应变分量d、应用几何方程求位移分量,
