is paper presents new approaches for solving a problem of the stability of compressed rods in the elastoplastic working region of materials. It is known that the columns of buildings, supports of engineering devices, drill rods of oil, and gas extraction industry may be subjected to significant risk of stability loss. Nowadays, there are design methods based on test results defining the relations (e.g., critical stresses-slenderness) to avoid this risk due to stability loss, but the precision and limits of definition are not always known. e main objectives of the study were to develop new approaches that would allow specifying the values of critical stresses of compressed elements beyond the proportional limit. e problem of stability of the compressed elements in the elastoplastic region was studied according to the stability theory. e authors suggested an original approach to the issue; in particular, the determination of values of the critical stresses and the finding of the points of the bifurcation were carried out by the tangent established by experimental results and by the approximation of the so-called double modulus. Comparative analysis showed the advantage of the proposed approach, particularly that the new critical curves were located below the curves of Engesser-Karman and Shanley and above the critical curves established by building codes. A new approach for the determination of critical stresses in the elastoplastic region was developed through which the structural reliability and economic efficiency was increased by almost 12% compared to the existing approaches.