Finite asymmetric deformation of shallow spherical shells

Abstract

A procedure is developed for the integration of the system of nonlinear partial differential equations governing the asymmetric deformation of shallow spherical shells. A suitable iteration scheme based on a finite-difference approach is shown to yield the asymmetric postbuckling states for the spherical cap under uniform pressure. Thus, the practically important asymmetric lower buckling is systematicall y computed for the first time. Applications to more general asymmetric buckling problems are indicated by the example of the half-loaded spherical cap. VER the past ten years considerable effort has gone into the integration of the equations for the finite deformation of shallow spherical shells. The first phase of the work was concerned with the development of analytic techniques for the accurate mathematical description of the finite axisymmetric deformation of shallow spherical shells.14 The pace of the research was determined in no small degree by the increasing power and availability of high-speed computing devices. The second phase began with the work of Weinitschke5 and Huang6 in which the problem of the clamped spherical cap under uniform pressure was studied to find out if asymmetric buckling of the classical bifurcation type could take place for certain ranges of shell geometries before the symmetric snap-through occurred. Thus the restriction to axisymmetric deformations, which had been imposed in previous work, was relaxed only to the extent that small asymmetric deformations in the neighborhood of the symmetric deformation state were allowed. In both Refs. 5 and 6 it was found that the asymmetric bifurcation buckling loads were less than the symmetric snap-through loads over a wide range of shell geometries, as shown in Fig. 1. However, the results of Refs. 5 and 6 were in serious disagreement as is evident from Fig. 1. In Ref. 7, very close agreement with the loads computed by Huang resulted when the problem was studied within the context of the small asymmetric vibration of the shell in the neighborhood of the symmetric primary state. Also, recent, very carefully conducted experiments by Krenzke and Kiernan8 on the buckling of spherical caps indicate (Fig. 1) a far better correlation between theory and experiment than was previously found, provided the minimum of either the symmetric snap-through load or the asymmetric bifurcation load is chosen for the buckling load. In the light of progress that has been achieved, it would seem that the next phase of work should be concerned with the difficult area of finite asymmetric deformations of shells. This work is necessary both in order to achieve further understanding of the snap-buckling problem by studying the asymmetric postbuckling behavior of a shallow cap as well as to broaden the scope of problems for which stability predictions can be made to include nonsymmetrically loaded and supported shells.