Numerical and analytical calculation of longitudinal bending of prismatic elastic rods under the action of axial compressive load taking into account the authorized weight

Abstract

Abstract The article is devoted to the problem of calculating the stability of compressed rods taking into account their own weight. A resolving differential equation of the second and fourth order with respect to the dimensionless deflection is obtained. A technique for numerically analytical and numerical solution of the obtained equation using the finite difference method is proposed. As a result, the problem was reduced to a generalized eigenvalue problem. Comparison with the solution of other authors is given. The problem of determining the critical values for the rods is posed as the problem of finding the domain of these values depending on the boundary conditions and the coefficients α and .. The objective function is a converging power series with unknown coefficients. The solution of the prismatic rod stability problem is carried out in the MatLab environment. The values of α and ., for which the critical load takes a minimum value, are found. To test the methodology, a number of problems were solved and compared with known solutions. The proposed technique, in contrast to analytical solutions, allows solving the problem with arbitrary fixation of the ends of the rod. It is also suggested to take into account the length-varying stiffness and heterogeneity of the rod. Test problems showed good agreement with literature data. In the future, it is planned to develop a calculation method taking creep into account. The higher quality of the analytical solution is shown in comparison with the existing methods.