  We consider a family of linearly elastic shells with thickness $2\varepsilon$
(where $\varepsilon$ is a small parameter). The shells are clamped along a
portion of their lateral face, all having the same middle surface $S$, and may
enter in contact with a rigid foundation along the bottom face. We are
interested in studying the limit behavior of both the three-dimensional
problems, given in curvilinear coordinates, and their solutions (displacements
of covariant components $u_i^\varepsilon$) when $\varepsilon$ tends to zero. To
do that, we use asymptotic analysis methods. On one hand, we find that if the
applied body force density is $O(1)$ with respect to $\varepsilon$ and surface
tractions density is $O(\varepsilon)$, a suitable approximation of the
variational formulation of the contact problem is a two-dimensional variational
inequality which can be identified as the variational formulation of the
obstacle problem for an elastic membrane. On the other hand, if the applied
body force density is $O(\varepsilon^2)$ and surface tractions density is
$O(\varepsilon^3)$, the corresponding approximation is a different
two-dimensional inequality which can be identified as the variational
formulation of the obstacle problem for an elastic flexural shell. We finally
discuss the existence and uniqueness of solution for the limit two-dimensional
variational problems found.
