Linear Models for Composite Thin-Walled Beams by $Gamma$-Convergence. Part I: Open Cross Sections

We consider a beam whose cross section is a tubular neighborhood, with thickness scaling with a parameter δε, of a simple curve γ whose length scales with ε. To model a thin-walled beam we assume that δε goes to zero faster than ε, and we measure the rate of convergence by a slenderness parameter s which is the ratio between ε2 and δε. In this Part I of the work we focus on the case where the curve is open. Under the assumption that the beam has a linearly elastic behavior, for s ∈ {0, 1} we derive two one-dimensional Γ-limit problems by letting ε go to zero. The limit models are obtained for a fully anisotropic and inhomogeneous material, thus making the theory applicable for composite thin-walled beams. The approach recovers in a systematic way, and gives account of, many features of the beam models in the theory of Vlasov.