Nonlinear thin-walled beams with a rectangular cross-section - Part I

Our aim is to rigorously derive a hierarchy of one-dimensional models for thin-walled beams with rectangular cross-section, starting from three-dimensional nonlinear elasticity. The different limit models are distinguished by the different scaling of the elastic energy and of the ratio between the sides of the cross-section. In this paper we report the first part of our results. More precisely, denoting by h and δ h the length of the sides of the cross-section, with δ h ≪ h, and by ε 2h the scaling factor of the bulk elastic energy, we analyze the cases in which δ h/ε h → 0 (subcritical) and δ h/ε h → 1 (critical).