A tube-type vessel, such as a heat exchanger, has tubes that extend away from a tubesheet provided with holes which axially align with the tubes and open into the tube interiors. The diameter of each hole in the tubesheet is less than the outside diameter of the tube with which it aligns, but is greater than the inside diameter of the tube, thereby producing an offset between the inside surface of the tube and the surface of the hole. The tubes are attached to the tubesheet by welds, and each weld forms a fillet between exterior surface of its tube and the back face of the tubesheet and further forms a smooth beveled transition between the inside surface of the tube and the surface of the hole. The fillets eliminate the crevices that surround the tubes in conventional structures, and as a consequence no material will build up around the tubes and collapse them. To unite the tubes with the tubesheet, the latter at each hole is provided with a socket into which the end of a tube is fitted. The socket serves to precisely locate the tube with respect to the hole. Then the hole is welded autogenously to the tubesheet from within the hole. The weld completely penetrates the portion of the tube within the socket and further extends into the tubesheet, both radially beyond the circular side wall of the socket and axially beyond the base of the socket. In so doing, the fillet and smooth beveled transition are produced at the exterior and interior surfaces of the tube, respectively.