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The widespread use of ribbed shells as constructional elements subjected to compressive loads has prompted the intensive development of investigations focusing on their stability and there have appeared large numbers of works which discuss various aspects of the problems. Since a detailed review of works on the stability of ribbed shells published up to 1968 was given in [46] (some questions relating to this theme were also outlined in [5, 7,30,106,139]), attention here is directed mainly to works appearing since 1967, and especially theoretical and experimental investigations in which the influence of discrete spacing of the ribs is discussed.* This constraint is adopted in the present review so as to emphasize the importance of taking this factor into account; it is also the case that, in contrast to pre-1967 works, relatively few of which addressed this question, the investigations appearing in the last 15-20 years have, as a rule, taken account of the discrete spacing of the ribs, in some formulation, in analyzing and determining both the critical loads and the optimal reinforcement parameters.In the present review of recent work, consideration is given to the initial assumption of the theoretical investigations, the method of calculation, the results of analysis of the theoretical data, questions of optimizing ribbed-shell construction, methods and results of experimental investigation, and the results of comparing experimental and theoretical data.In conclusion, problems that are urgent, in our view, and should be investigated in the next few years are formulated.Theoretical investigations of ribbed-shell stability as a rule employ the theory of elastic thin shells based on the Kirchhoff--Love hypotheses to describe the stress-strain state of the casing and Kirchhoff--Clebsch theory of thin rods to describe the stress--strain state of the ribs. It is assumed in almost all the works that the ribs are added to the casing along lines of principal curvature and transmit reactions distributed along these lines to the casing. In works that are known to the present authors, a static stability criterion isemployed, and the problem reduces to the solution of systems of differential or integral equations of neutral equilibrium (or equivalent equations based on the direct use of extremal properties of the total potential energy), where the components of the displacement of a "perturbed" state close to the initial "unperturbed" state are taken as the unknown (so far, only works in which the subcritical state of the shell is assumed to be momentless have been considered).With the given assumptions, the most general formulation of ~he problem involves taking account of all the rigidity parameters of the ribs. In this case, the differential equations of neutral equilibrium may be written in the form (see [5]

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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...