Variational approaches to the natural frequencies of a clamped uniform beam-column

Abstract

The transverse flexibility of heavily loaded structural elements may be increased significantly by beam-column effects, particularly when the structures are subjected to large longitudinal accelerations. The determination of the affected vibration and stability characteristics is then of concern to the designers of space-vehicle components and of payload elements, since these may be flexible enough to have critical accelerations in the neighborhood of those attained by modern rockets. For as simple a structural element as the uniform beam subjected to a longitudinal acceleration, the determination of the natural frequencies presents difficulties because the governing differential equation contains variable coefficients. Exact solutions are impossible to obtain. In the present study, bounds for the natural frequencies of a clamped beam carrying a linearly varying compressive load and a constant end load are obtained by three methods based on a variational characterization of the eigenvalues. Upper bounds are calculated using the Rayleigh-Ritz method. Lower bounds are obtained by the method of Kato and by the Bazley-Fox second projection method. The results show that the Rayleigh-Ritz method coupled with the Bazley-Fox second projection method yields excellent bracketing of the eigenvalues, since the ratio of the gap between the bounds over their average is in most instances less than 1%, even for values of the loads close to those creating buckling. The method of Kato, for the trial functions chosen, yields less satisfactory results. The bounds for the natural frequencies are also compared to the approximations obtained by lumping the distributed load at each end. The errors in the determination of frequencies by this lumping process are found to be appreciable in some cases of interest to designers.