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The second-order analysis method is a simulation-based design method but less utilized for pile-supported structures due to the current inaccurate and unstable analysis methods for piles, especially for those under complex loading conditions in pile-supported structures. A refined Three-dimensional Pile Element (3DPE) formulation is proposed to promote the application of the second-order analysis method in pile-supported structures. The proposed refined 3DPE formulation can efficiently capture the nonlinear Soil-Structure Interaction (SSI) that is over-simplified in the conventional analysis method. Compared with the existing Pile Element formulation, the proposed refined 3DPE formulation can more accurately capture the structural behaviors of piles under complex loading conditions, including the shear effect and axial-torsional coupled response. Furthermore, three soil stiffness matrices are firstly proposed to …