In current work for the first time, buckling analysis of bidirectional functionally graded (BFG) Euler beam having arbitrary thickness variation rested on Hetenyi elastic foundation is presented. Moreover, a new scheme based on calculus of variations and collocation method for converting the buckling problem to an algebraic system of equations is proposed. The mentioned scheme leads to obtain the buckling characteristic equation of beam and therefore the first buckling loads are obtained. Various conditions including variation of mechanical properties across the thickness and through the axis, arbitrary thickness variation, Hetenyi elastic foundation, special boundary conditions like the shear hinge and classical boundary conditions like the clamped, simply supported, clamped-simply supported and cantilever beams are considered to show the compatibility of proposed scheme with the various circumstances. The fast convergence and compatibility with the various circumstances are the advantages of the proposed technique. Due to lack of similar studies in the literature, the same exercises are conducted by using the Spectral Ritz method for pursuing the validity of the proposed scheme. The same basis is used for Spectral Ritz and proposed methods. There is an excellent agreement between the results of well-known Spectral Ritz method and the results of proposed scheme, which validates the outcomes of proposed technique.