Because the pressure and the shell wall are symmetrical about the axis shown, the tensile stress σa produced in the wall is uniform around the shell's circumference. This normal stress is assumed to be uniformly distributed across the wall thickness (because the wall is thin).
Since the wall is thin, the exposed area of the sphere wall can be approximated by the product of the sphere inner circumference and the wall thickness. Equilibrium in the horizontal direction requires the following:
Therefore, the axial stress created in the wall of the spherical pressure vessel is given by:
The outer surface of the shell is subjected to biaxial stress (i.e., normal stress in two directions). By symmetry, a pressurized sphere is subjected to uniform normal stresses σa in all directions.