Shells as building elements are widely used in various fields of engineering. For example, in industrial and civil construction they are used as roofs and ceilings for structures with large spans (circuses, markets, railway stations, warehouses and hangars), various ramps, awnings and canopies.

Often, thin-walled part of the shell (here and after - the thin envelope) is reinforced by ribs in one or two directions, or has variable holes, bulges, cuts. In such thin envelopes deflections that are commensurate with the thickness of the shell itself, can be formed even under load, which is far from the critical values. The study of these shells containing ribs, linings and openings (here and after - envelopes with step-variable thickness) is of considerable interest in the design of the structures.

In this paper, the nonlinear equations of motion of envelopes with step-variable thickness (in the case of Kirchhoff-Love model) are found. In these equations the geometric nonlinearity, discrete arrangement of ribs and cuts, shear and torsional rigidity of the ribs are taken into account.

The proposed equations allow us to investigate the shell under the action of both dynamic and shock loads. In addition, technique for determining the amplitude-frequency characteristics of nonlinear free vibrations of a shell, considered as a system with n degrees of freedom is developed and realized in the form of computer program.