Slender curved structures may experience a loss of stability called snap-through, causing the curvature on part or all of the structure to invert inducing fatigue damage. This paper presents a framework for analyzing the transient responses of slender curved structures. A numerical study of snap-through in a shallow arch-like model under periodic excitations is performed on a simplified model and on a detailed finite element model. The boundaries that separate the snap-through and no snap-through regions in the forcing parameters space are identified. Various post-snap responses are analyzed. The effects of initial conditions on the snap-through boundaries and post-snap responses are examined. Forcing parameters that lead to chaotic response are identified.