The Usefulness of Elementary Theory for the Linear Vibrations of Layered, Orthotropic Elastic Beams and Corrections Due to Two-Dimensional End Effects

With the aid of formal asymptotic expansions, we conclude not only that elementary (Euler-Bernoulli) beam theory can be applied successfully to layered, orthotropic beams, possibly weak in shear, but also that, in computing the lower natural frequencies of a cantilevered beam, the most important correction to the elementary theory—of the relative order of magnitude of the ratio of depth to length—comes from effects in a neighborhood of the built-in end. We compute this correction using the fundamental work on semi-infinite elastic strips of Gregory and Gladwell (1982) and Gregory and Wan (1984). We also show that, except in unusual cases (e.g., a zero Poisson’s ratio in a homogeneous, elastically isotropic beam), Timoshenko beam theory produces an erroneous correction to the frequencies of elementary theory of the relative order of magnitude of the square of the ratio of depth to length.