In this talk we are concerned with a two-penalty boundary obstacle problem for the bi-Laplace operator in the upper unit ball. This problem arises in connection with unilateral phenomena for flat elastic plates. It can also be seen as an obstacle-type problem for the fractional Laplacian $(−\Delta)^{3/2}$. Our goals are to establish the well-posedness and the optimal regularity of the solution, and to study the structure of the free boundary. The proofs are based on monotonicity formulas of Almgren- and Monneau-type. This is joint work with Alaa Haj Ali.