Justification of the Asymptotic Expansion Method for Homogeneous Isotropic Beams by Comparison with De Saint-Venant’s Solutions

The formal asymptotic expansion method is an attractive mean to derive simplified models for problems exhibiting a small parameter, such as the elastic analysis of beam-like structures. Usually this method is rigorously justified using convergence theorems Yu and Hodges, 2004 . In this paper it is illustrated how the Saint-Venant’s solution naturally arises from the lowest order terms of an asymptotic expansion of the elastic state for the case of homogeneous isotropic beams. It is also highlighted that the Saint-Venant solutions corresponding to pure traction, bending and torsion involve the solution of the first-order microscopic problems, while for the simple bending problem, the solution of the second-order microscopic problems is needed. The second-order problems provide therefore a way to characterize the transverse shear behavior and the cross-sectional warping of the beam.