The United States Air Force and the National Aeronautics and Space Administration have made great efforts and spent untold resources to develop reusable hypersonic vehicles since the early 1950s. In spite of great progress, many scientific and technical challenges still exist. This thesis focuses on developing a robust and efficient computational framework for analyzing snap-through, which is a particular concern for the commonly used slender curved structural components of reusable hypersonic vehicles since it can significantly exacerbate fatigue failure. Snap-through is a type of instability where a curved structure suddenly jumps to a remote configuration. This behavior is highly nonlinear involving sudden and large deformations. Snap-through is a dynamic instability triggered by the loss of stability of an equilibrium state. Examining equilibria and their stability is useful and necessary before costlier transient simulations of snap-through. Curved structures undergoing snap-through can have equilibrium states that cannot be captured by path following algorithms. Two types of ``hidden" equilibria are identified: secondary equilibrium branches bifurcated from the primary path and coexisting equilibria unconnected with the primary path. A numerical procedure that combines branch-switching and arclength methods is proposed to retrieve bifurcated secondary branches, and an analytical approach is introduced to obtain unconnected equilibria. With knowledge of the entire equilibrium manifold, transient simulations of snap-through are then investigated. Time integration of snap-through is very challenging because it is a highly nonlinear behavior involving sudden jumps. Even state-of-the-art schemes fail to provide accurate and efficient long-time predictions. This dissertation extends the preliminary work on an efficient composite scheme with significantly enhanced numerical accuracy and computational efficiency in simulating snap-through. In the design of slender curved components of reusable hypersonic vehicles, it is beneficial to efficiently identify the stability boundaries that separate non-snap from post-snap responses for different designs and loading conditions. Obtaining stability boundaries directly from parametric studies is computationally costly even with the most efficient algorithms. To alleviate the cost, an alternative approach to quickly approximate dynamic stability boundaries is proposed. This approach significantly decreases the number of transient simulations needed and therefore greatly accelerates the exploration of dynamic stability boundaries.