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

For static loads, Saint-Venant (1885) proposed the principle that the detailed description of the stress distribution across the end of a beam was unimportant in determining the stress throughout most of the beam. His principle allows most beam analysis to be accomplished using approximate methods rather than requiring the solution of the exact elasticity equations. There is a self-equilibrating stress distribution resulting from the difference between the “engineering” and the exact stress distributions. Toupin (1965) proved that this self-equilibrating stress distribution has an exponential decay over a short distance into the beam. For dynamic loads, Lamb (1916) demonstrated that some fraction of a self-equilibrated load will penetrate into the beam. Because of this, Saint-Venant's principle of exponential decay of stress resulting from a self-equilibrating load is not valid for a dynamic load. This thesis describes the work accomplished in order to propose a quantitative measure for the resultant violation of Saint-Venant's Principle in dynamics. To simplify the technical details we focus on the simplest of dynamic problems; excitation of a semi-infinite elastic isotropic homogeneous strip by a self-equilibrated harmonic force applied at the end. The degree to which Saint-Venant's principle is violated is computed as the ratio of the maximum penetrating stress divided by the maximum stress caused by the end load. As the applied load will usually be of the same order as the self-equilibrated component, this ratio will serve as a measure of the error induced by using Saint-Venant's principle in dynamical problems. The self-equilibrated part of the end load is not usually known in engineering problems, but if nothing is known about the self-equilibrated load, then nothing can be said about the penetrating stress state. To circumvent this limitation we use a probabilistic model. A random selection process defines the self-equilibrated load distribution. The ratio of maximum stresses is then a random variable. Its properties are determined numerically by running a series of analyses, where the non-propagating part of the applied load is constructed using coefficients from a set of Gaussian variables with zero mean and variance of unity. The histogram of the resultant numerical values of the stress ratio is found to be describable by the beta probability distribution. The mean value and range of the ratio at each value of a non-dimensional frequency parameter is obtained. For a given required accuracy the frequency range where the Saint-Venant principle can be used is determined, and thus, where one-dimensional classical beam theory can be applied. We characterize also the increase in this range which is achieved in the refined plate theory proposed by Berdichevsky and Le (1980).