Abstract: An improved scheme is proposed by using the singularity theory to study the effect of compressibility of an elastic rod on its critical loads of buckling modes. It is indicated that the critical load of the first buckling mode calculated by the improved scheme is increased compared with that calculated by the classical Euler scheme. For the first buckling mode of an elastic rod, Euler scheme predicts that it's always the supercritical fork bifurcation, but the improved scheme shows the existence of different bifurcations (supercritical fork bifurcation, subcritical fork bifurcation and no buckling mode) in accordance with different value ranges of the rod dimensionless characteristic length. The conformity of the improved scheme to test results is evaluated qualitatively. Besides, it is explained by the analysis of an example based on the improved scheme that why rods with shorter lengths made from traditional materials always occur yield failure rather than buckling of subcritical fork bifurcation.