Simplified calculation method of bearing capacity for super-long piles under multiple combined loads based on envelope line/surface in different loading spaces

Super-long piles in ocean engineering are commonly subjected to multiple combined loads, including complex environmental and operational loads. The properties of these loads are diverse and intricate, and the continuous increase in pile length and free segments enhances coupling between load effects of multiple combined loads. However, no available method currently exists to accurately evaluate the bearing behavior of super-long piles under such multiple combined loading conditions. To address this, a three-dimensional finite element model is developed and validated. The swipe and probe methods are employed to determine failure envelope lines and surfaces under different loading conditions, enabling characterization of bearing behavior. In addition, elliptic theory is applied to fit and normalize the bearing capacity envelope, and corresponding theoretical formulas are derived under multiple combined loading. A systematic sensitivity analysis of key influencing parameters is also conducted. The results indicate that the failure envelope lines resemble classical or inclined ellipses in the V-H (vertical-horizontal), V-M (vertical-moment), and H-M (horizontal-moment) loading spaces. In the V-H-M loading space, the failure envelope forms a closed surface extending outward. Furthermore, the coefficients in the proposed formulation decrease with increasing vertical load.