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.

破坏包络公式通常用于评估地基在组合载荷下的极限承载力，并用于纳入宏观单元模型。然而，每个载荷分量之间复杂的相互作用，尤其是对于六自由度 (6DoF) 载荷，意味着确定令人满意的公式通常是一个复杂的过程。以前的研究人员已将这一困难确定为在基础工程应用中采用失效包络方法的障碍。为了解决这个问题，本文描述了一个用于推导全局凸失效包络公式的系统程序；该程序应用于圆形表面基础，承载在不排水的粘土上，在 6DoF 负载空间中。公式显示为密切代表从有限元分析确定的各种载荷条件下的失效载荷组合，包括非平面水平力矩载荷。描述了基于所提出的公式的示例宏单元模型；宏观元模型提供了由单独的有限元分析确定的基础行为的近似表示。该论文的一个关键方面是它展示了一个确定性能良好的故障包络公式的自动化过程。与之前用于确定失效包络公式的手动程序相比，该过程的自动化性质具有相当大的优势。描述了基于所提出的公式的示例宏单元模型；宏观元模型提供了由单独的有限元分析确定的基础行为的近似表示。该论文的一个关键方面是它展示了一个确定性能良好的故障包络公式的自动化过程。与之前用于确定失效包络公式的手动程序相比，该过程的自动化性质具有相当大的优势。描述了基于所提出的公式的示例宏单元模型；宏观元模型提供了由单独的有限元分析确定的基础行为的近似表示。该论文的一个关键方面是它展示了一个确定性能良好的故障包络公式的自动化过程。与之前用于确定失效包络公式的手动程序相比，该过程的自动化性质具有相当大的优势。