Non-classical forms of loss stability of cylindrical shells joined by a stiffening ring for certain forms of loading

A structure in the form of two coaxial cylindrical shells with different radii, joined by a stiffening ring either rigidly or by hinges, is considered. Starting out from improved equations of general form constructed earlier, a linearized contact problem is formulated that enables all possible classical and non-classical forms of loss of stability to be investigated in the case of axisymmetric forms of loading of the structure. The initial relations of the problem are transformed to an equivalent system of integro-algebraic equations containing integral Volterra-type operators by integrating along the longitudinal coordinate and representing the two-dimensional and one-dimensional required unknowns introduced into the treatment in the form of the sum of trigonometric functions in the circumferential coordinate that, in changing into a perturbed state, allows the possibility of the shell deforming in antiphase forms. A numerical algorithm for constructing solutions of the resulting equations is proposed, based on the method of finite sums, that enables all the boundary conditions of the problem and the conditions for the joining of the shells with the stiffening ring to be satisfied exactly. Retaining and discarding parametric terms in the relations for the shells, the stability of a structure of the class considered is investigated in the case when an external pressure acts on the stiffening ring and, also, in the case of its axial tension during which the stiffening ring is found to be under wrench deformation conditions and, in a shell of larger diameter, subcritical circumferential compressive stresses are formed.