Summary: In this paper, we are interested in a shape optimization problem for a fluid-structure interaction system composed by an elastic structure immersed in a viscous incompressible fluid. The cost functional to minimize is an energy functional involving together the fluid and the elastic parts of the structure. The shape optimization problem is introduced in the 2-dimensional case. However the results in this paper are obtained for a simplified free-boundary 1-dimensional problem. We prove that the shape optimization problem is wellposed. We study the shape differentiability of the free-boundary 1-dimensional model. The full characterization of the associated material derivatives is given together with the shape derivative of the energy functional. A special case is explicitly solved, showing the relevancy of this shape optimization approach for a simplified free boundary 1-dimensional problem. The full model in two spatial dimensions is under studies now.