Both the tangent modulus theory and the double modulus theory are classical theories which can be applied to the elastic-plastic stability analysis of columns. In the traditional tangent modulus theory, numerous iterations are required to calculate the critical buckling stress and this makes the method very time-consuming. In this paper, an explicit formula for establishing a direct correlation between the critical stress and the Euler stress has been proposed to reduce trial calculations. This formula can be applied to spherical shells by simplifying their stiffened plates to the form of beams on elastic foundations. The explicit expressions of both modulus theories can be used to calculate the ultimate strength of a spherical shell under pressure. The results from the proposed expression are compared with experimental results and other numerical results.