A dynamic problem of elastic-viscoplastic deformation of flexible cylindrical closed circular shells is formulated. Traditional criss-cross reinforcement structures along equidistant surfaces and spatial structures are considered. The inelastic deformation of the materials of the composition is described by the constitutive relations of the theory of flow with isotropic hardening, taking into account the dependence of the plastic properties on the rate of deformation. The two-dimensional kinematic and dynamic equations used, as well as the corresponding initial-boundary conditions, make it possible to calculate the mechanical behavior of flexible cylindrical composite shells with an accuracy of different orders. These equations allow simulating the possible weak transverse shear resistance of such structures. In the simplest version, two-dimensional equations, boundary and initial conditions are reduced to equations of Ambartsumyan’s non-classical theory. The numerical solution of the formulated nonlinear initial-boundary value problem is constructed according to an explicit “cross”-scheme. A comparative analysis of elastic-plastic and elastic-viscoplastic deformation of reinforced cylindrical shells dynamically loaded by internal pressure is carried out. The deformation of fiberglass and metal-composite constructions of different relative thicknesses has been investigated. It is shown that if the dependence of the plastic properties of the materials of the composition on the rate of their deformation is not taken into account, then this leads to an inadequate calculation of the dynamic inelastic deformation of such composite shells. It is shown that even in the case of relatively thin reinforced shells, the use of Ambartsumian’s theory can lead to a significant difference from the solution obtained by the refined theory. This can lead to qualitatively incorrect results when solving inverse problems (for example, rational reinforcement) using Ambartsumyan’s theory. Calculations based on the refined theory showed that replacing the traditional criss-cross structure of reinforcement along equidistant surfaces with a spatial structure of reinforcement in the case of long cylindrical shells of different relative thicknesses does not lead to a positive result. The positive effect of such a replacement of reinforcement structures is observed only for relatively thick short fiberglass shells.