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 \u3b3. We assume that the wall thickness, i.e., the size of the neighborhood, scales with a parameter \u3b4\u3b5 while the length of \u3b3 scales with \u3b5. We characterize a thin-walled beam by assuming that \u3b4\u3b5 goes to zero faster than \u3b5. Starting from the three-dimensional linear theory of elasticity, by letting \u3b5 go to zero, we derive a one-dimensional \u393-limit problem for the case in which the ratio between \u3b52 and \u3b4\u3b5 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