The paper describes a method to attach tubes to tube plates in heat exchangers. The method is based on the use of annular sealing elements (ring relief). To determine the optimal sizes, shapes and mechanical characteristics of ring relief, the stressed-deformed state of the connection of a tube with a tube plate was numerically investigated. Fatigue testing was made to take into account cyclic loads. It was established that the fatigue strength of tubes fastened to tube plates is determined by the fretting process. For these investigations, 20K steel was used. It is concluded that the use of ring sealing elements, formed from the metal of the tube plate is an effective means to increase air-tightness of connections and provides the stabilization of their qualitative parameters.