The implementation of stone columns is a widely accepted method for improving the stability of liquefiable soil. A comprehensive understanding of the behavior of the composite ground is crucial for accurate design and calculation in practical applications. Several existing mathematical models were established to assess characteristics of the stone column-improved ground by typically ignoring the vertical seepage within liquefiable soil. This negligence will inevitably lead to significant calculation errors, particularly when the vertical permeability of liquefiable sites is high or the installation spacing of stone columns is large. In this context, a new mathematical model which accounts for coupled radial-vertical seepage within liquefiable soil is proposed to determine the reinforcement performance of stone columns. The equal strain assumption and new boundary conditions are incorporated to obtain numerical solutions with the finite difference method. Then the present solution is degenerated to the conventional calculation model to verify the reasonability of the proposed model. Finally, a parametric analysis is conducted to investigate the impacts of crucial parameters on the performance of stone columns for excess pore water pressure variation during soil liquefaction. The results reveal that the peak value of the maximum excess pore water pressure ratio increases with the increment of both the column spacing and cyclic stress ratio. Moreover, the increasing radial and vertical consolidation parameters Tb and Th will accelerate the dissipation rate of the excess pore water pressure of liquefiable sites. Furthermore, the conventional model neglecting the vertical seepage will underestimate the variation rate of the excess pore water pressure, and the calculation error will become larger with the increase of Th.

The implementation of composite piles has emerged as an appealing technology for reinforcing soft ground through fulfilling the requisite bearing capacity and facilitating consolidation. Nonetheless, previous investigations into the consolidation behavior of composite pile-improved ground have neglected the nonlinearity of soil compression and permeability. In this context, the logarithm models of e-lgcr and e-lgk are introduced to establish an analytical model for the nonlinear consolidation of composite ground with composite piles. Based on equal strain assumption and annular equivalent method, detailed solutions under four special loading schemes are then obtained. Additionally, a comprehensive analysis is conducted to assess the influence of various parameters on the nonlinear consolidation behavior of composite ground, and the feasibility of current model is verified by degenerations. The results show that ignoring the nonlinearity will lead to an overestimation of consolidation rate when the soil's compression indices exceed the permeability indices. Moreover, the consolidation rate of composite ground is inversely proportional to cru/cr ' s0 and Cc/Ckh(v), while demonstrating a direct proportionality to Ksp, Ksg, and kg/kv0. However, cru/cr ' s0 mainly influences excess pore water pressure at the upper layer, and the influence of Cc/Ckv on the consolidation rate is limited and can even be ignored in comparison to Cc/Ckh. Finally, the proposed model is successfully applied to an engineering project in situ, where the obtained results exhibit a noteworthy agreement with the measured data.