Asymmetric behavior of a sloping spherical shell under finite deflections

Abstract

The current status of the theory and applications of calculating thin-wall constructions makes it possible to determine the stress‐strain states of elastic shells of various shapes that are in equilibrium under external loads with an accuracy sufficient for engineering practice. This statement is true both for small deflections of shells and for finite and large deflections when the behavior of shells is described in various nonlinear mathematical models. The status of calculations of the stability of elastic shells is distinctly different. The same mathematical models considerably overestimate measured critical loads, and advanced skills are required to determine the critical loads with sufficient accuracy by semiempirical methods. This situation is most clear for well-studied elastic spherical shells. The instability of a spherical shell was likely first observed by Bach [8] in 1902. In 1915, Zolly devoted his dissertation to the first calculation of the stability of thin elastic spherical shells in the linear approximation. Much later, in 1939, Boley and Seckler (see [11]) experimentally found that the critical pressure of a copper hemisphere was one-quarter the theoretical value calculated by Zolly’s formula. Since that time, experimental information on the critical loads of spherical shells has increased continuously, and numerous calculations are carried out to determine the causes of discrepancy between measured and calculated data and to remove this discrepancy [7]. The authors of these calculations took into account the moment stress‐strain state of a shell before the loss of stability, used geometrically nonlinear equations of the theory of shells, and considered shells with initial imperfections, including the imperfect shape of the shell, method of its fixing to the contour, possible elastoplastic deformation of the material of the shell, etc. We consider a thin elastic sloping spherical shell that is rigidly fixed to the contour, has the modulus of elasticity E and Poisson’s ratio ν , and is loaded by transverse pressure (see Fig. 1). The critical loads that are measured and calculated for this shell are shown in Fig. 2, where