The purpose of the present investigation is to solution of compression problem of thin cylindrical polymer shells by two planes by using the finite-element method in the finite element program ANSYS, taking into account the geometrical and physical nonlinearity. Let us consider a thin homogeneous isotropic cylindrical shell of radius R, width B, and length L under the action of a force P that is transmitted through the undeformed upper plane as shown in Figure 1. The lower plane is immobile. For solve of this problem we introduce the dimensionless parameters of displacement α and load β. The finite-element model constructed in the ANSYS program has the form of a quarter of a circular ring owing to symmetry. The boundary conditions correspond to the symmetry conditions. Since we consider thin-walled shells which can be subjected to great displacements and rotations, for different forms of elastic potentials, as a finite element we took a SHELL181 shell element, which corresponds to the requirements adopted. In the contact zone, a CONTA174 contact element is used. The elastic properties of the material were modelled by Hooke’s relations and the Treloar and Mooney–Rivlin potentials. The problem is solved with account of friction in the contact zone. The solution of the nonlinear contact problem is performed by a stepwise procedure with respect to displacements and with an iteration refinement on each step. The buckling of thin cylindrical shells under contact compression was considered additionally. The transition boundary between the first (elliptic form with the central contact zone) and the second (dumbbell form with two peripheral contact zones) deformation forms was examined.