Shallow Caps under Pressure Distributions* Finite Polar Dimpling of Sub-buckling Axisymmetric

For shallow caps under an axisymmetric external pressure distribution which is inward (with a representative magnitude pO) in a neighborhood of the pole and outward away from the pole, it has been shown previously that a finite axisymmetric dimple state of deformation is possible if p0 is at least of the order of the classical buckling pressure for a complete spherical shell, Pi, and that, asymptotically, the corresponding dimple base is located by the condition of no resultant vertical force over the dimpled region. In the present paper, we treat the more difficult load magnitude range P&Q << 1 by the method of inner-outer (asymptotic) expansions and show that there is a lower bound K~ on p& for the existence of a dimple state. For p&c 2 Kc, we derive a simple condition for the dimple base radius which reduces to the previous result when (p&J’ is 0( 1) at most. While the analysis is carried out for the special case of a quadratically varying pressure distribution, the method applies to more general load distributions with similar qualitative features, including a uniform internal pressure with an inward directed point force or ring load centered at the apex.