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) 114], 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 [J. Appl. Math. Mech. (PMM) 42 (1) (1978) 140].