On Saint-Venant's principle in the two-dimensional flexural vibrations of elastic beams

Abstract

In this paper we apply the method proposed in our previous paper [Int. J. Solids Struct. 40(13–14) (2003) 3293] to quantitatively estimate the violation of Saint-Venant's principle in the problem of flexural vibration of a two-dimensional strip. A probabilistic approach is used to determine the relative magnitude of the penetrating stress state and the results of computations are presented as a function of frequency. The results are not dependent on material properties except for Poisson's ratio. The numerical results given are appropriate for all isotropic materials with equal Lame' coefficients. Our major conclusion is that over a wide range of frequencies, the maximum propagating stress is always small compared with the maximum applied stress; hence Saint-Venant's principle may be said to apply in this problem. By considering applied loads with increasing spatial frequency content it was concluded that a smooth self-equilibrated load will cause a larger penetrating stress than a more irregular one. Although initially this may seem counter-intuitive, it results from the penetrating branch having the smoothest spatial distribution of all the branches of the solution to the dynamic elasticity equations. An interesting outcome of our study is that the accuracy of engineering theories for flexural vibrations is much higher than for longitudinal vibrations.