In this paper, elastic-plastic symmetrical buckling of a thin solid circular plate of variable thickness, under uniform edge pressure, is investigated, based on both Incremental Theory (IT) and Deformation Theory (DT). Two kinds of simply supported and clamped boundary conditions have been considered. A power-law function was assumed for thickness variation. To minimize the integral uniqueness criterion, based on Rayleigh-Ritz method, transversal displacement was approximated by a test function which includes some unknown coefficients and satisfies geometric boundary conditions. Substituting the test function in the stability criterion and minimizing with respect to the unknown coefficients results in a homogeneous algebraic set of equations in terms of unknown coefficients. For non-trivial solution, the determinant of coefficient matrix should be equated to zero. Using this equation, critical buckling load is determined. The results of present study were compared with existing analytical solutions for circular plate of constant thickness and a good agreement was observed. This clearly shows the validity of presented analysis. Then the effect of thickness variation and boundary conditions type on the critical buckling load was investigated, for commercial aluminum and steel 1403 materials. The results show that when the thickness of circular plate center is 10% greater than its edge thickness the buckling load may increase up to 40% comparing with the circular plate for which the center thickness is 10% less than its edge thickness.