The analytical solutions for the stress distributions within elastic hollow spheres under the diametrical point loads

This paper presents an exact analytical solution for the stress distributions within an elastic hollow sphere subjected to diametrical point loads. The solution is suitable for both thin and thick hollow spheres. New variables are introduced in order to uncouple the system of governing equations so that explicit differential equations are obtained for displacement components and stress components. Moreover, Fourier-Legendre expansion technique is employed in order to determine the unknown coefficients in the analytical solutions for hollow spheres. The present solution can be considered as an extension of the classical solution by Hiramatsu and Oka (Int J Rock Mech Min Sci 3:89-99, 1966) for solid spheres under the point loads, which provided the theoretical basis for the point load strength test. Unlike in solid spheres, the stress concentrations within the hollow spheres under the point loads are usually developed at the joint point of the inner surface and the loading axis, and the thinner the hollow sphere, the larger the tensile stress concentrations developed at the inner surface. This numerical result indicates that the failure of the hollow spheres usually starts at the inner surface, and the normalized tensile stress at the inner surface increases with the increase in Poisson's ratio and internal pressure, but decreases with the increase in the size of the loading area. Moreover, significant shear stress zone is usually developed in the areas immediately inside the outer surface, and the maximum shear stress is often developed at the point immediately inside the outer surface jointing the edge of the loading area and the center of the hollow sphere. The present solution can be used to analyze the failure mechanism of bulk foams made up of hollow spheres in engineering.