A mathematical analysis method is employed to solve the bending problem of slip clamped shallow spherical shell on elastic foundation. Using the slip clamped boundary conditions, the differential equations of the problem are simplified to a biharmonic equation. Using the $R$-function, the fundamental solution and the boundary equation of the biharmonic equation, a function is established. This function satisfies the homogeneous boundary condition of the biharmonic equation. The biharmonic equation of the slip clamped shallow spherical shell bending problem on elastic foundation is transformed to Fredholm integral equation of the second kind by using Green’s formula. The vector expression of the integral equation kernel is derived. Choosing a suitable form of the normalized boundary equation, the singularity of the integral equation kernel is overcome. To obtain the numerical results, the discretization of the integral equation of the bending problem is conducted. The treatment of singular term in the discretization equation is to use the integration by parts. Numerical results of rectangular, trapezoidal, pentagonal, L-shaped and concave shape shallow spherical shells show high accuracy of the proposed method. The numerical results show fine agreement with the ANSYS finite element method (FEM) solution, which shows the proposed method is an effective mathematical analysis method.

采用数学分析方法求解弹性地基上滑动夹紧浅球壳的弯曲问题。使用滑动钳位边界条件，将问题的微分方程简化为双调和方程。使用$R$-函数，双调和方程的基本解和边界方程，一个函数成立。该函数满足双调和方程的齐次边界条件。利用格林公式将弹性地基上滑动夹紧的浅球壳弯曲问题的双调和方程转化为第二类Fredholm积分方程。导出了积分方程核的向量表达式。选择合适的归一化边界方程形式，克服了积分方程核的奇异性。为了获得数值结果，对弯曲问题的积分方程进行了离散化。离散化方程中奇异项的处理是采用分部积分。矩形、梯形、五边形的数值结果，L形和凹形浅球壳表明所提出的方法具有很高的准确性。数值结果与ANSYS有限元法（FEM）解法吻合较好，表明该方法是一种有效的数学分析方法。