Design features of flange connections with hollow elastic metal gaskets used to seal pipelines, hydraulic and pneumatic systems, etc. are considered. The isolation of the working medium in such connection is achieved by the use of sealing elements and further elastic-plastic deformation of the contact surfaces of the connection due to the kinematic convergence of the flanges in the process of tightening the bolts during the assembly of the connected structure. As a result, not only are contact areas formed, but also the intrusion of some surfaces into others, the size and shape of the intrusion being characterised by the degree of contact density. Hence the appears need to determine the required contact pressure. In many cases, the contact pressure is determined by solving the elasto-plastic problem on the example of embedding convex dies in a perfectly rigid-plastic half-space. The contact mechanics of flanged connections with initial contact of elastic metal gaskets with the base surface is considered in accordance with the proposed calculation model of contact interaction of two elastic metal bodies in compression. The full radial displacements are described by integral equations solved by the finite difference method. Assuming that the surfaces of compressible bodies are absolutely smooth and that only normal pressure acts on these contact surfaces, the relationships are obtained which allow the law of distribution of the basic reactions over the contact surface, the size of the contact zone and the contact pressure required to ensure the sealing of the assembly joint to be determined.