*Published Paper*

**Inserted:** 3 jan 2014

**Last Updated:** 1 sep 2015

**Journal:** Siam Journal on Mathematical Analysis

**Volume:** 46

**Number:** 5

**Pages:** 3296–3331

**Year:** 2014

**Abstract:**

We consider a beam whose cross-section is a tubular neighborhood, with thickness scaling with a parameter $\delta_\varepsilon$, of a simple curve $\gamma$ whose length scales with $\varepsilon$. To model a thin-walled beam we assume that $\delta_\varepsilon$ goes to zero faster than $\varepsilon$, and we measure the rate of convergence by a slenderness parameter $ \mathfrak{s}$ which is the ratio between $\varepsilon^2$ and $\delta_\varepsilon$. 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 $ \mathfrak{s}\in \{0, 1\}$ we derive two one-dimensional $\Gamma$-limit problems by letting $\varepsilon$ 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.

**Keywords:**
Gamma-convergence, dimension reduction, linear elasticity, Thin-walled beams, curved cross-section, open cross-section

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