Asymptotic closed-form solutions including boundary-layers for orthotropic beams

Abstract

In this paper, asymptotic closed-form solutions with boundary-layers are explicitly obtained. The virtual work principle is applied to the problem by employing the method of multiple-scales, which allows one to systematically separate two-dimensional elasticity problems into the interior and boundary-layer problems. From the interior problem, we calculate the warping displacements first and then construct boundary-layer displacements by employing them as a basis. Saint-Venant's principle is applied to the boundary-layers so that they satisfy the surface boundary conditions in a weak sense. For stress prescribed boundaries, the minimization process is enforced to satisfy the conditions, which allow one to predict the free-edge boundary layers. Cantilevers with three types of materials are taken as numerical examples. Asymptotic closed-form decay rates, displacements, and stresses are derived and compared to a two-dimensional finite element analysis. The results obtained herein clearly show that the present approach yields a reliable prediction of boundary-layers for orthotropic beams.