In this paper we discuss the problem of computing and analyzing the static equilibrium of a non rigid water tank. Specifically, we fix the amount of water contained in the tank, modelled as a membrane. In addition, there are rigid obstacles that constrain the deformation. This amounts to a non convex variational problem. We derive the optimality system and its interpretation in terms of equilibrium of forces. A second order sensitivity analysis, allowing to compute derivatives of solutions and a second order Taylor expansion of the cost function, is performed, in spite of the fact that the cost function is not twice differentiable. We also study the finite elements discretization, introduce a decomposition algorithm for the numerical computation of the solution, ans display numerical results.