Abstract.The paper suggests an approach of mathematical modeling of composite cylindrical shells under axisymmetric load. The stress-strain state of each layer is characterized by a system of forces, moments, strains, and displacements adopted in the classical theory using Kirchhoff-Love's hypotheses (L. I. Balabuha -I. V. Novozhilov variant). The study of the stress-strain state is accomplished in accordance with the statements and hypotheses of the complex structures deformation theory taking into account shear stresses in joints. There have been built mathematical models for composite structures in the form of a circular cylindrical shell, closed in a transverse direction and bounded by two plane sections perpendicular to the axis of the cylinder. Computation of a two-layered cylindrical hinged shell under axisymmetric load has been performed. 1IntroductionComposite structures formed by layers of materials with different physical and mechanical properties are used as protective constructions in oil and gas industry complexes. The connection of individual layers is made up by continuous or discrete type finite rigidity braces. A shift of layers under loadispossible; therefore, the stress-strain state is studied in terms of multilayered structures with delamination. The above research area is related to the theory of composite structures [1,2,3] for composite rods, plates and shallow shells. The theory of complex structures deformation in terms of shearing stresses in joints is applied to composite cylindrical shells. According to the theory [1] composite shells are structures consisting of several layers with finite rigidity braces pliable in the longitudinal direction and absolutely rigid in the transverse direction. 2Mathematical modelA composite cylindrical structure consisting of separate i+1 layers and joints is under consideration. It is assumed that deflection w is the same for all the package layers, and each individual shell is regarded as i-th layer thicknessh (i) .

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