Knockdown factors for buckling of cylindrical and spherical shells subject to reduced biaxial membrane stress

Cylindrical shells under uniaxial compression and spherical shells under equi-biaxial compression display the most extreme buckling sensitivity to imperfections. In engineering practice, the reduction of load carrying capacity due to imperfections is usually addressed by use of a knockdown factor to lower the critical buckling stress estimated or computed without accounting for imperfections. For thin elastic cylindrical shells under uniaxial compression and spherical shells under equi-biaxial compression, the knockdown factor is typically as small as 0.2. This paper explores the alleviation of imperfection-sensitivity for loadings with a reduced circumferential (transverse) membrane stress component. The analysis of Koiter (1963) on the effect of an axisymmetric imperfection on the elastic buckling of a cylindrical shell under uniaxial compression is extended to both cylinders and spheres for loadings that produce general combinations of biaxial membrane stresses. Increases in the knockdown factor due to a reduction of the transverse membrane component are remarkably similar for cylindrical and spherical shells.