Geometrically nonlinear axisymmetric bending analysis of a circular plate on a circular ring support is investigated in this computational study. The thickness of the plate is uniform and the material of the plate is isotropic and homogeneous. The plate is assumed to be under the action of uniform transverse pressure. The plate perimeter is considered to be fully simply supported (S) or fully clamped (C). The intermediate ring support is assumed to prevent deflection. Geometrically nonlinear shallow spherical shell equations are adapted to a circular plate by setting the apex of the shell to zero. The boundary conditions at the edge, and at the ring support, and the conditions at the center of the plate due to the axisymmetric load are satisfied exactly. The system of nonlinear ordinary differential equations is transformed to nonlinear algebraic equations by the finite difference method. The solution is obtained by the Newton-Raphson method. The diagram of the deflection along the diameter of the plate is drawn. The accuracy of the results is verified by checking the deflection with the results of the linear solution obtained by the Ritz method, and good agreement is obtained. The influence of the location of the ring support on the deflection is investigated. Several examples are solved to determine the influence of the thickness on the results.