Automated procedure to derive convex failure envelope formulations for circular surface foundations under six degrees of freedom loading

Failure envelope formulations are typically employed to assess the ultimate capacity of foundations under combined loading and for incorporation in macro-element models. However, the complex interaction between each load component, especially for six degree of freedom (6DoF) loading, means that determining satisfactory formulations is often a complex process. Previous researchers have identified this difficulty as an obstacle to the adoption of the failure envelope approach in foundation engineering applications. To address this issue, the paper describes a systematic procedure for deriving globally convex failure envelope formulations; the procedure is applied to a circular surface foundation, bearing on undrained clay, in 6DoF load space. The formulations are shown to closely represent the failure load combinations determined from finite element analyses for a variety of loading conditions, including non-planar horizontal-moment loading. An example macro-element model based on the proposed formulation is described; the macro-element model provides a close representation of the foundation behaviour determined from a separate finite element analysis. A key aspect of the paper is that it demonstrates an automated process to determine well-behaved failure envelope formulations. The automated nature of the process has considerable advantages over the manual procedures that have previously been employed to determine failure envelope formulations.