Effect of a “Slight” Curvature of the Axis of a Thin-Walled Nonprismatic Beam on its Free Vibration

Abstract

This paper presents an analysis of the effect of a “slight” curvature of the axis of a thin-walled beam on its eigenfrequencies and forms. By “slight” we understand such a beam curvature at which no theory of curved beams (arches, horizontally curved girders), but classical beam theory is applied. The subject of the analysis is the open nonprismatic thin-walled beams with any geometrical parameters, and prismatic beams. The thin-walled beam model used in the analysis was derived on the basis of the momentless theory of plates, using the Vlasov assumptions. The free vibrations of the beams were analysed. Calculations were performed for four groups of beams. Each of the groups comprised two beams with the same configuration of flanges and identical geometrical cross section characteristics, but differing in web geometry. One of the beams had a symmetrical web (a beam with a rectilinear axis in the web plane), whereas the web of the other beam was asymmetrical (a beam with its axis curved in the web plane). The algorithm based on Chebyshev polynomials was used to solve the variable coefficient equations describing the model’s vibration. According to this algorithm, solutions are sought in the form of Chebyshev series and the closed analytical recurrence formulas generated by the algorithm are used to calculate the coefficients of the equations. It has been shown that even a “slight” curvature of the beam’s axis has a significant effect on its eigenfrequencies and forms, as confirmed by a comparison of the results with the ones obtained using Finite Element Method (FEM).