Slender curved structures can often be found as components of complex structures in civil, mechanical, and aerospace systems. Under extreme loadings, the structure might undergo snap-through buckling, i.e., the structure is forced to its inverted configuration, inducing fatigue. The focus of this research is the development of a reliable and accurate model for simulating the nonlinear response of shallow arches under transient loading and characterizing these responses to assess the structure's ability to survive if the structure undergoes instabilities. Since no analytical solutions for general systems with snap-through exist, numerical models are needed in order to predict the response of the structure. The finite element method provides the most generality and can be applied to systems with arbitrarily complex geometries.

Unfortunately there are barriers to the numerical prediction. First, the structures exhibit a very complex dynamic response. Coexisting responses are identified under different initial conditions. Chaotic responses are also observed. A framework for analyzing the dynamic responses of slender curved structures is proposed by identifying the relevant features useful in characterizing the transient behavior of shallow arches.

State of the art time integrators are often unable to retrieve long time records of the response after a physical instability event. The performance of several time-stepping schemes is analyzed by identifying the important features that affect the numerical accuracy and robustness. We also identify the region where the schemes are stable for such simulations. The interactions between the time-stepping schemes and the spatial discretizations are examined. This investigation results in recommendations for finite elements and time integrators that give the best performance. A new time integrator that is robust and accurate for long-term simulations is proposed.

The established numerical framework is validated against experimental data. Fabrication imperfections in the experimental arch and prestressing due to the applied boundary conditions are accounted for. A methodology to determine the boundaries of the stability regions in the parameter space under consideration is proposed.

Finally, the effect of initial temperature variation on the transient behavior of shallow arches is also studied. The changes in the snap-through boundaries as the temperature increases are examined.