Study of the Convergence of Polynomials and the Finite Element Method Taking Into Account the Plastic Properties of the Material

One of the challenges associated with the development of numerical methods for structural analysis based on the finite element method is justifying the reliability of the obtained results and comparing the effectiveness of new approaches with existing ones. Theoretically, this is often difficult to accomplish, even for specific cases. Therefore, to address these issues, numerical experiment data obtained on benchmark problems are usually employed. This article explores a class of objects representable by a prismatic finite element with a cross-section of arbitrary shape. For this prismatic finite element with variable mechanical and geometric parameters across the section, the solution relationships of the semi-analytical finite element method are presented for calculating nodal reactions and stiffness matrix coefficients, considering the material's plastic properties. The convergence of calculation results based on the semi-analytical finite element method using Fourier series and polynomials is compared with the results obtained using the finite element method (FEM) considering the material's plastic properties. This comparison is conducted on a test problem involving the stress-strain state of an infinitely long strip with a rectangular cross-section subjected to uniformly distributed loading. The FEM solution for this problem is used as a reference result. The calculations are performed with a fixed number of finite elements across the thickness of the crosssection, as the study establishes that further increasing their number has a negligible impact on the calculation results. The obtained calculation results allow concluding about the equal convergence rate of polynomials and Fourier series in the considered problem. At the same time, it is worth noting the significant advantage of the semi-analytical finite element method compared to the conventional finite element method. This is because maintaining only five terms in the series expansion ensures solving the problem with the same accuracy as using a nine-node mesh in the grid domain.