Geometrically nonlinear relations have been developed to relate the deflection of a thin film/substrate system to the intrinsic film stress when these deflections are larger than the thickness of the substrate. Using the Rayleigh-Ritz method, these nonlinear relations were developed by approximating the out-of-plane deflections and midplane normal strains by 2nd order polynomials with unknown coefficients. Solving for the unknown coefficients to minimize the strain energy of the system produces several plate deflection configurations. In an isotropic system, at very low intrinsic film stresses, a single, stable, spherical plate configuration is predicted. However as the intrinsic film stress increases, the theoretical solution bifurcates, predicting one unstable spherical shape and two stable ellipsoidal shapes. In the limit as the intrinsic film stress approaches infinity, the ellipsoidal configurations develop into cylindrical plate curvatures about either one of the two axes. Although similar formulations have been reported before, this new relation is significantly more accurate when compared to a three-dimensional nonlinear finite element solution.