In this paper, the elastoplastic buckling of rectangular plates over the Pasternak foundation has been analyzed with the fixed and simply supported boundary conditions. Associated with the uniform loading conditions on the plate by the in- plane compression and tension, the influence of the elastic foundation is investigated in terms of two stiffness parameters; including the Winkler spring and the Pasternak shear coefficients. In order to extract governing equations, two theories are used from the plasticity: deformation theory (DT) with the Hencky constitutive relations and the incremental theory (IT) based on the Prandtl-Reuss constitutive relations. By implementing the generalized differential quadrature method to discrete the differential equations, influences of loading ratio, length to width ratio, plate thickness, and the elastic foundation characters are studied. By comparing the obtained results with the data reported in references, the accuracy of the model is verified. Consideration of results shows that applying the elastic foundation causes to increase critical buckling load. In addition, enhancing the elastic foundation parameters leads to amplifying the difference between buckling loads obtained from two theories, especially in the larger thicknesses. Moreover, according to increasing the plate thickness in the tensile state of the loading, application of the elastic foundation causes to reach plate stress to a value more than the ultimate stress of the specimen.