On Saint-Venant's principle in the dynamics of elastic beams

In dynamics, Saint-Venant's principle of exponential decay of stress resulting from a self-equilibrating load is not valid. For a beam type structure, a self-equilibrated load may penetrate well inside the beam. Although this effect has been known for a long time, at least since Lamb's paper [Proc. Roy. Soc. Lon. Ser. A 93 (1916) 1141, it was not clear how to characterize it quantitatively. In this paper we propose a "probabilistic approach" to evaluate the magnitude of the penetrating stress state. The key point is that, in engineering problems, the distribution of the self-equilibrated load is usually not known. By assigning to the self-equilibrated load some probabilistic measure one can find probabilistic characteristics of the penetrating stress state. We develop this reasoning for the simplest case: longitudinal vibrations of a two-dimensional semi-infinite, elastic isotropic homogeneous strip, excited by a periodic load at the end. We show the frequency range where Saint-Venant's principle can be used with good accuracy, and thus, one-dimensional classical beam theory still can be applied. We characterize also the increase in this range which is achieved in the refined plate theory proposed by Berdichevsky and Le P. Appl. Math. Mech. (PMM) 42 (1) (1978) 140]. (C) 2003 Elsevier Science Ltd. All rights reserved.