This paper addresses the evaluation of thermal and mechanical stresses that occur in assemblies with flat ring flanges, of optional type. The flange rings are fixed by welding to the wall of the cylindrical body of the vessel, with a constant thickness along its length. Regarding of the above, the compatibility of the deformations of the component elements (radial displacements and rotations) is approached. A linear algebraic system is formed in which both external loads (pressure, temperature) and connecting loads (bending unit moments and unit forces) are present. The present analysis discusses the quantitative, cumulative effect of the deformed gasket and the stiffness of the curved/bent screws on the tightness of the system. The methodology is flexible by introducing selection factors so that the mentioned influences can be easily separated.