Large amplitude free vibration behavior of thin, isotropic square plate configurations resting on Winkler type of elastic foundation are expressed in the form of simple closed-form solutions by using the Rayleigh-Ritz (R-R) method based on coupled displacement fields (CDF). Geometric non-linearity of von-Kármán type is taken into consideration. The in-plane displacement field variations used in the formulation of the R-R (CDF) method are derived by using the governing in-plane static differential equations of the plate which in turn simplifies the difficulty of assuming an in-plane displacement field variations of the plate. Accuracy and robustness of proposed closed-form solutions is compared to the available finite element formulation results. Proposed closed-form solutions are corrected for the simple harmonic motion (SHM) assumption using the well established harmonic balance method (HBM) which is applicable for the homogeneous form of cubic non-linear Duffing equation.