The buckling of elastic spherical caps

The John equations are used to model the buckling of a simply-supported elastic spherical cap that is subjected to a constant uniform external load . The Liapunov-Schmidt method is used to solve these equations. We show that solutions possessing circular, pear-shaped, elliptical, triangular, square-shaped, pentagonal and a variety of other symmetries branch from the unbuckled state of the shell. The stability of these solutions is discussed. Some numerics that complement the analytical results are also included.