Exploring the design space of nonlinear shallow arches with generalised path-following

The classic snap-through problem of shallow arches is revisited using the so-called generalised path-following technique. Classical buckling theory is a popular tool for designing structures prone to instabilities, albeit with limited applicability as it assumes a linear pre-buckling state. While incremental-iterative nonlinear finite element methods are more accurate, they are considered to be complex and costly for parametric studies. In this regard, a powerful approach for exploring the entire design space of nonlinear structures is the generalised path-following technique. Within this framework, a nonlinear finite element model is coupled with a numerical continuation solver to provide an accurate and robust way of evaluating multi-parametric structural problems. The capabilities of this technique are exemplified here by studying the effects of four different parameters on the structural behaviour of shallow arches, namely, mid span transverse loading, arch rise height, distribution of cross-sectional area along the span, and total volume of the arch. In particular, the distribution of area has a pronounced effect on the nonlinear load-displacement response and can therefore be used effectively for elastic tailoring. Most importantly, we illustrate the risks entailed in optimising the shape of arches using linear assumptions, which arise because the design drivers influencing linear and nonlinear designs are in fact topologically opposed.