Thin-walled beams with a cross-section of arbitrary geometry: Derivation of linear theories starting from 3D nonlinear elasticity

The subject of this paper is the rigorous derivation of lower dimensional models for a nonlinearly elastic thin-walled beam whose cross-section is given by a thin tubular neighbourhood of a smooth curve. Denoting by h and delta(h), respectively, the diameter and the thickness of the cross-section, we analyse the case where the scaling factor of the elastic energy is of order epsilon(2)(h), with epsilon(h)/delta(2)(h) -> l is an element of [0, + infinity). Different linearized models are deduced according to the relative order of magnitude of delta(h) with respect to h.