圆柱壳受集中载荷作用的奇异解

Singular solutions for the stresses in cylindrical shells due to concentrated loads

对精确的圆柱壳方程求完备的极坐标解系是薄壳理论的一个难题。该文给出了修正的Morley方程在坐标原点具有奇性的基本解的一种构造方法,以及基于Schwartz广义函数理论的证明。并据此基本解得到了圆柱壳受集中法向载荷、集中温度载荷作用下的奇异解,这些积分形式的解的被积函数具有第二种Hankel函数和指数函数乘积的形式,容易获得数值结果。该文的计算结果与有限元数值结果能很好地符合,利用该文所提出的奇异解作为特解求解圆柱壳开孔接管支管受载荷问题,与采用Timoshenko方程的双三角级数特解相比可以不受小开孔率的限制。

The complete set of solutions for the stresses in cylindrical shells in polar coordinates was obtained by constructing a fundamental solution of the modified Morley equation for a circular cylindrical shell. The theoretical proof is based on the theory of Schwartz generalized functions. The analytical solution for the stresses in a cylindrical shell subjected to a concentrated normal force and thermal loadings can be obtained from this fundamental solution. Numerical results for the singular solutions with integrands expressed in terms of the product of the second Hankel function and the exponential function agree well with FEM results. The present solutions can be used to analyze two intersecting cylindrical shells as particular solutions, since there is no restriction on the size ratio as with the traditional double Fourier series solutions of Timoshenko equation.