Numerical and analytical calculation of the buckling of elastic prismatic rods under the action of axial compressive loading with account for the dead load

Introduction. The article addresses the problem of stability of compressed rods with regard for the dead load. A resolving differential equation of the second and fourth order with respect to dimensionless deflection is obtained. A methodology underlying numerical-analytical and numerical solutions to the obtained equation is proposed. It employs the power series and the finite difference method. As a result, the numerical solution is reduced to the generalized eigenvalue problem. A comparison with the solution developed by other authors is provided. Materials and methods. The problem of determining critical values, applicable to rods, is formulated as the task of identifying the domain of these values depending on the boundary conditions and dimensionless coefficients α and β that correspond to the concentrated force and the distributed load of the dead load. The function, approximating the deflection, represents the power series having indefinite coefficients. The objective function represents converging power series having unknown coefficients. The solution to the stability problem of a prismatic rod is obtained in the MatLab environment. The values of α and β are found; they correspond to the minimal critical load. Results. Several problems were solved and new solutions were compared with the well-known ones in order to test the methodology. Conclusions. Unlike analytical solutions, the proposed methodology allows to solve the problem, if the ends of the rod are fixed arbitrarily. One can also take account of the rod stiffness that varies along the length, as well as the heterogeneity of the rod. Test problems show good convergence with the data provided in the sources. In the future, the methodology will take account of creep. A higher quality analytical solution, surpassing other existing methods, is demonstrated.