Using a system of differential equations for whole the structure in which the weak zones are taking into account by employing the term containing the delta-functions, we may perform an analysis of layered systems with weak zones. We study stability of layered orthotropic cylindrical shells discretely supported by circular ribs having weak zones and connected to a hollow elastic cylinder. The structure are under action of an external pressure and an axial compressing load. We present displacement functions for the shell, cylinder, and ribs in terms of trigonometric series. As a result, the problem under consideration may be reduced to the problem for a system of algebraic equations with respect to discontinuities in the first derivative of the displacement function for the ribs at the points corresponding to weak zones. For the case of uniformly located similar ribs and hinges, we obtain the solution in explicit form.