Spherical shells, together with shells of other forms, are widely used in various branches of mechanical engineering and the construction trades; see [35, 39, 73, etc]. The advantages of these structures include the fact that they give maximum useful volume, assigning universality in their use to the construction, they have great strength under various perturbations, and they are simultaneously supporting and enclosing structures.As is known, the stability of thin-walled structures is often the criterion determining their carrying capacity. The first investigations of the stability of smooth spherical shells with static loading by an external pressure appeared around 70 years ago; see [13], for example. Since then, very many works have considered the stability of unreinforced spherical shells. The results of these investigations have been outlined in detail and analyzed in a series of reviews, of which the most complete (at the corresponding time period) are [13-15,
28, 65, 68].The critical external pressure found in most experimental works (see the review [28]) was 2-4 times lower than the theoretical value obtained from the well-known classical formula of upper critical pressure. It also follows from experimental results that the development of stability loss of thin-walled spherical shells compressed by an external pressure is very diverse. Therefore, the construction of a universal mathematical model covering various variants of stability loss is a problem that remains to be solved. The use of traditional methods for investigating combined variants of buckling is also associated with great difficulty. Note that, as a result of experimental investigation of the stability of smooth shells on carefully prepared (by the method of sputtering in vacuum) samples [8,41], the validity of the classical formula has been established [13], and the strong influence of initial imperfections on the critical load has been shown [28]. The trend to increase in level of critical load (which corresponds to decrease in weight of shell structures) stimulated research on ribbed shells. For spherical shells, these investigations began in [63,64], where the stability of hollow spherical shells was considered. Theoretical investigations on the basis of an approximate scheme which reduces the calculation of a ribbed shell to the consideration of a smooth shell with reduced rigidity characteristics (a structurally orthotropic shell) have been complemented by experimental investigations of the stability of metallic reinforced shells.In [63], which began the investigation of spherical reinforced shells, an empirical formula of the following form was proposed for determining the critical external pressure, on the basis of analyzing experimental dataIn Eq. (I), the following notation is used: ~ is the semivertex angle of the shell; h is the thickness of the covering; R is the radius of curvature of the shell; E is the elastic modulus of the material; also ~0 is the semivertex angle of the insert in the strip; d is the diameter of the sup...
