The problem of contact interaction of structures taking into account the deformation and strength properties of the contact seam material is considered. To discretize the contact layer, frame-rod contact finite elements (CFE) are used, by means of which the physical properties of the seam material (initial strength, line and nonlinear deformability) are modelled. By means of CFE are also modelled various contact conditions – separation, clutch, friction-sliding, etc. On the base of the proposed discrete contact model and the method of step-by-step analysis, a numerical algorithm for solving the contact problem has been developed taking into account the deformation and strength properties of the seam material. This approach allows in one step-by-step cycle to perform simultaneous account of the initial strength of the contact seam, as well as the conditions of unilateral deformable constraints and friction-sliding of its surfaces in areas where the strength of the contact seam is broken. The use of frame-rod contact elements allows the physical nonlinearity of the contact layer to reduce to the internal nonlinearity of only the system of contact elements, while the nonlinear properties of the seam are set by means of the nonlinear characteristics of the individual rods of contact elements. Iterative refinement of the nonlinear solution for the current level of loading is performed by the method of compensating loads. With the help of the proposed approach, the numerical solution of the problem of contact of the structure with the base under different conditions of the contact seam has been obtained and analyzed.