Investigation of the Errors of Fast Trigonometric Interpolation in Solving the Problem of Stresses in a Bar

Using the fast trigonometric interpolation, the problem on stresses in a rectangular bar is solved. The obtained approximate analytical solution is compared with the exact one. During this analysis the relative error of the displacement components, the stress tensor components, the discrepancy between the Lame equilibrium equations, and the discrepancy of the boundary conditions are investigated. It has been established that when using the second-order boundary function in fast expansions and a small number of terms in the Fourier series (from two to six), the maximum relative error dmax of the displacement components and the stress tensor components is less than one percent. With an increase in the order of the boundary function and/or the number of terms N in the Fourier series, d(max) decreases rapidly. Increasing the order of the boundary function is a more effective way to reduce the calculation error of d(max) than increasing the number of terms in the Fourier series. When studying the intensity of stresses s(max) in a bar with different overall dimensions of a rectangular cross-section, but with the same area of all sections, it has turned out that the smallest value of among all s(max) cross-sections is observed for a bar with a square section.