The United States Air Force (USAF) and the National Aeronautics and Space Administration (NASA) have coordinated many research programs that are motivated by the desire for fast, safe, reliable and economically viable hypersonic vehicles. Despite the efforts made and the progress achieved since the early 1950s, many technical and scientific challenges still exist. Slender curved structures can often be found as components of complex structures such as hypersonic vehicles. Snap-through buckling is an instability frequently occuring to initially curved structures subjected to transverse loads. The hypersonic vehicles must operate in extreme environments and combined loading conditions where they have to withstand sustained operations. The structural responses in such operating conditions are highly nonlinear and fully coupled. This thesis focuses on understanding the global stability of thermomechanically coupled curved structures and alleviating the computational costs involved in the modeling approaches.

From a modelling perspective, the computational cost is high for complex structures and therefore usage of such structures needs to be fully justified. Quick assessment measuring tools that provide insight into the structural behavior are required, when deciding on an appropriate structural component. The degree of instability influences the dynamic behavior. Knowledge about it allows for a better understanding regarding how the dynamics is organized. In this thesis, the dregree of instability is found through a numerical procedure that combines arclength and branch switching methods. It is also revealed that a connection exists between the Euler buckling loads for initially flat beams and the force-displacement curve under transverse loading for buckled beams. The results are used to develop a very simple metric to estimate the number of unstable static equilibria (the degree of instability) of a buckled structure based only on its geometry with no need for static or dynamic solvers. This simple metric helps the design engineer to quickly decide between structural components, without spending computational resources, thus making the design process more efficient.

Transient simulations are needed to understand the global stability of structures undergoing snap-through instability. Long-time predictions obtained with the finite element method (FEM) are usually computationally expensive. An alternative approach is to employ instead of FEM a reduced order model (ROM). Although ROMs significantly reduce the computational burden, challenges emerge when choosing the appropriate number of modes to closely represent the static and dynamic behavior. This dissertation proposes a ROM that uses the connection between the Euler buckling loads and the force-deflection curve under transeverse loads to predict the total number of modes needed to accurately estimate the static and dynamic behavior.

To safely design slender curved components of hypersonic vehicles, it is of interest for the design engineer to obtain dynamic stability boundaries that separate small amplitude no snap and large amplitude snap responses in the space of forcing parameters. Obtaining stability boundaries with current approaches is computationally very expensive, because it implies running simulations on a full grid that may result in hundreds or thousands of simulations. To ease the cost, an effcient methodology is proposed to quickly determine dynamic stability boundaries. The algorithm choses adaptively the forcing parameters based on simulation results and reduces significantly the total number of simulations required to obtain a dynamic stability boundary, leading to an affordable computational cost even in the case of complex and/or thermomechanically coupled structures.