  This paper deals with the introduction of a decomposition of the deformations
of curved thin beams, with section of order $\delta$, which takes into account
the specific geometry of such beams. A deformation $v$ is split into an
elementary deformation and a warping. The elementary deformation is the analog
of a Bernoulli-Navier's displacement for linearized deformations replacing the
infinitesimal rotation by a rotation in SO(3) in each cross section of the rod.
Each part of the decomposition is estimated with respect to the $L^2$ norm of
the distance from gradient $v$ to SO(3). This result relies on revisiting the
rigidity theorem of Friesecke-James-M\"uller in which we estimate the constant
for a bounded open set star-shaped with respect to a ball. Then we use the
decomposition of the deformations to derive a few asymptotic geometrical
behavior: large deformations of extensional type, inextensional deformations
and linearized deformations. To illustrate the use of our decomposition in
nonlinear elasticity, we consider a St Venant-Kirchhoff material and upon
various scaling on the applied forces we obtain the $\Gamma$-limit of the
rescaled elastic energy. We first analyze the case of bending forces of order
$\delta^2$ which leads to a nonlinear inextensional model. Smaller pure bending
forces give the classical linearized model. A coupled extensional-bending model
is obtained for a class of forces of order $\delta^2$ in traction and of order
$\delta^3$ in bending.
