The results of a stress analysis of the vertical buttresses of a prestressed concrete nuclear containment vessel are presented. The computations were basically performed by a plane strain finite element computer program which employs the Gauss-Seidel iterative procedure to solve the 900 by 900 direct stiffness simultaneous equations. A subroutine for finding an optimum over-relaxation factor for accelerating the convergence rate has been developed which has reduced considerably the trial and error computing time needed for guessing such a factor. The loading conditions include prestress forces of hoop tendons and thermal forces of liner and concrete due to temperature gradients, which were the steady state and transient state solutions of heat distributions in the vessel wall and buttresses under winter normal operating and accident temperature conditions. For the winter accident temperature condition, cracks due to high tensile stresses were investigated.