A systematic framework for formulating convex failure envelopes in multiple loading dimensions

The failure envelope approach is widely used to assess the ultimate capacity of shallow foundations for combined loading, and to develop foundation macro-element models. Failure envelopes are typically determined by fitting appropriate functions to a set of discrete failure load data, determined either experimentally or numerically. However, current procedures to formulate failure envelopes tend to be ad hoc, and the resulting failure envelopes may not have the desirable features of being convex and well-behaved for the entire domain of interest. This paper describes a new systematic framework to determine failure envelopes - based on the use of sum of squares convex polynomials - that are guaranteed to be convex and well-behaved. The framework is demonstrated by applying it to three data sets for failure load combinations (vertical load, horizontal load and moment) for shallow foundations on clay. An example foundation macro-element model based on the proposed framework is also described.