The stability and post-critical behaviour of periodic snapping are investigated experimentally for a buckled elastic sheet with two clamped ends under an external uniform flow. In addition to experimental investigations, low-order numerical simulations are conducted with the elastica model for the deformation of the sheet, which is coupled with the simple quasi-steady fluid force model based on Bollay's lift theory, in order to identify the deformed shape of the sheet in an equilibrium state and the critical velocity where the sheet begins to snap. Continuous exposure to fluid-dynamic loading induces snap-through oscillations from an initial equilibrium state. While the critical flow velocity for bifurcation is inversely related to the ratio of the streamwise distance of the sheet to its length, it is not significantly affected by the mass ratio of the sheet and the surrounding fluid, leading to divergence instability. In the post-equilibrium state, regular oscillations with the same dominant modes persist in the sheet for a broad range of the flow velocity. As the sheet crosses the midline in the snapping process, the bending energy stored in the sheet is released quickly, and the time for energy release is found to be lower than that required for energy storage. Because of the initial buckled shape, the minimum bending energy of the sheet over a cycle remains at least 40% of its maximum magnitude.