Abstract : In this paper, the authors apply an extension of the direct stiffness method of structural analysis to the linear and nonlinear analysis of shells of revolution under arbitrary static loading with variable thickness properties in the circumferential direction and the meridional direction. The primary difference between this research and previously published research is that all the Fourier harmonics in the circumferential direction are now coupled. The thickness variation in the circumferential direction yields an 8N x 8N element stiffness matrix in which N is the number of harmonics. The resulting computer program is used to conduct a linear, nonlinear, and stability analysis of the Apollo aft heat shield.