Verifying the Movable Elastoplastic Boundary Method by Using Galin’s Problem

Galin’s solution for the problem of biaxial tension of a plate with a hole completely covered by the plastic region appears to be a pearl recognized by the world scientific community. This solution serves as a test for all sorts of approximate approaches to solving elastoplastic problems, including the semi-analytical iterative method being developed by the author, focused on solving more complex problems such as the Kirsch problem in the elastoplastic formulation. The proposed iterative approach for a semi-analytical solution involves an explicit analytical expression for stresses in the plastic region and an iterative numerical solution in the elastic region with a refined border. The paper shows the convergence of the results based on the iterative procedure for the elastoplastic region boundary approaching its analytical position, which follows from the analytical solution of Galin’s elastoplastic problem. Consideration has also been given to obtaining results on the determination of the boundary between the elastic and plastic regions using a competing approximate perturbation method. The advantage of the proposed method lays in not limited modifications in parameters due to the requirement for small differences while formulating a problem from the axisymmetric case as seen in the perturbation method.

Buckling Analysis and Stability of Compressed Low-Carbon Steel Rods in the Elastoplastic Region of Materials

This 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. The 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. The problem of stability of the compressed elements in the elastoplastic region was studied according to the stability theory. The 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.