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

We consider a beam whose cross section is a tubular neighborhood of a simple closed curve γ. We assume that the wall thickness, i.e., the size of the neighborhood, scales with a parameter δε while the length of γ scales with ε. We characterize a thin-walled beam by assuming that δε goes to zero faster than ε. Starting from the three-dimensional linear theory of elasticity, by letting ε go to zero, we derive a one-dimensional Γ-limit problem for the case in which the ratio between ε2 and δε is bounded. The limit model is obtained for a fully anisotropic and inhomogeneous material, thus making the theory applicable for composite thin-walled beams. Our approach recovers in a systematic way, and gives account of, many features of the beam models in the theory of Vlasov.