  In this paper, we consider rods whose thickness vary linearly between $\eps$
and $\eps^2$. Our aim is to study the asymptotic behavior of these rods in the
framework of the linear elasticity. We use a decomposition method of the
displacement fields of the form $u=U\_e + \bar{u}$, where $U\_e$ stands for the
translation-rotations of the cross-sections and $\bar{u}$ is related to their
deformations. We establish a priori estimates. Passing to the limit in a fixed
domain gives the problems satisfied by the bending, the stretching and the
torsion limit fields which are ordinary differential equations depending on
weights.
