An equilibrium phase diagram for the shape of compressively strained free-hanging films is developed by total strain energy minimization. For small strain gradients $\Delta \epsilon$, the film wrinkles, while for sufficiently large $\Delta \epsilon$, a phase transition from wrinkling to bending occurs. We consider competing relaxation mechanisms for free-hanging films, which have rolled up into tube structures, and we provide an upper limit for the maximum achievable number of tube rotations.