Abstract The post-buckling response of a simply supported, axially compressed, uniform bar of non-linearly elastic material, is presented. The buckling analysis considers large deflections of the elastica type in which the influence of bar axis shortening is also taken into account. The solution of the strongly non-linear differential equation is obtained analytically by a simple and very efficient approximate technique leading to reliable results. Conditions of stability of the post-buckling path depending on the degree of the material non-linearity are established. Moreover, the individual and coupling effect of the material non-linearity and the bar slenderness ratio on the post-buckling path are thoroughly discussed. Numerical results for both non-linearly and linearly elastic material show the efficiency and reliability of the proposed procedure.