The incorporation of truncated fourier series into finite difference approximations of structural stability equations

Abstract

A new trigonometric approach to the finite difference calculus was applied to the problem of beam buckling as represented by virtual work and equilibrium equations. The trigonometric functions were varied by adjusting a wavelength parameter in the approximating Fourier series. Values of the critical force obtained from the modified approach for beams with a variety of boundary conditions were compared to results using the conventional finite difference method. The trigonometric approach produced significantly more accurate approximations for the critical force than the conventional approach for a relatively wide range in values of the wavelength parameter; and the optimizing value of the wavelength parameter corresponded to the half wavelength of the buckled mode shape. Thus, selection of the wavelength parameter is a simple process if the half wavelength is known. Methods for selecting this parameter in the more general case are also presented. It was found from a modal analysts that the most accurate solutions are obtained when the approximating function closely represents the actual displacement function and matches the actual boundary conditions. It is more difficult to select a satisfactory value of the wavelength parameter for the equilibrium equation which makes the virtual work equation more attractive for practical applications. A comparison of the virtual work and the Galerkin approaches identified marked similarities between the two methods.