Multiple Scales Solution for a Beam with a Small Bending Stiffness

Abstract

This paper considers the problem of a beam with a small bending stiffness, within the framework of a nonlinear beam model that includes both the classical cable and the linear beam as limiting cases. This problem, treated as a perturbation of the catenary solution, is solved with the multiple scales method. The resulting expressions of the beam deflection and of the internal forces, as well as those obtained with the more commonly applied matched asymptotics method, are compared with numerical results. This comparison indicates that a better accuracy can be achieved with the multiple scales approach, for a similar computational effort. These results also suggest that application of the multiple scales method to the solution of beam problems involving boundary layers extend the range of values of the small parameter, for which accurate analytical solutions can be obtained by a perturbation technique.