Abstract: We employ fundamental equations of non-homogeneous elasticity and Fourierintegral transformations to obtain the general solutions of the stress function.On thebasis of these points of view and when the forces on the boundary are arbityary for nonhomogeneous half-plane problems with the Young’s modulus E(x)-E₀θxp[βx].accurate solutions are obtained At last with the degeneracy it is obtained that thefamous Boussnesq solution and this method is successful.

Key words: plastic post-buckling, perturbation, columns of rectangular cross-section, accurate solution, elasticity, non-homogeneous half-plane problem