## 1 Introduction
With the increase in seismic acceleration under high seismic intensity, the anti-liquefaction performance of the composite foundation stabilized with stone columns was obvious in shaking table model tests ([PERSON] et al. [17]). [PERSON] [18] found that gravel piles reduced the amplitude of the peak pressure, caused by piling, and quickly dissipated the excess pore water pressure in the liquefiable layer. Based on the results of 1 G laboratory-scale shaking table tests, [PERSON] et al. [19] showed that the compaction and drainage effects of gravel piles were the main factors to reduce the liquefaction of the foundation. The compaction of weak foundation by gravel piles not only improved the deformation modulus and bearing capacity but also promoted the rapid dissipation of dynamic pore pressure because of the drainage effect in permeable gravel piles. In this way, the likelihood of liquefaction was greatly reduced. According to theoretical and numerical evaluations, the bearing capacity of soft clay soil, which has been improved through stone columns, with both static and earthquake load effect reinforced with the increase in the diameter and depth of the stone column ([PERSON] et al. [20]). Moreover, laboratory tests and numerical simulations found that the bearing capacity ratio increased, and the settlement reduction factor decreased upon increasing the number and length of stone columns ([PERSON] et al. [21]). [PERSON] et al. [22] pointed out that stone columns, in saturated sandy soils, could significantly improve the soil's resistance to liquefaction through the dynamic response. Meanwhile, [PERSON] et al. [23] performed a numerical analysis on shallow foundations underlain by a stone column, and found that the granular column could effectively mitigate liquefaction. [PERSON] et al. [24] provided a mathematical model for determining the drainage capacity of stone columns, considering their stiffness and limited permeability, during liquefaction. Their modeling results indicated that the soil's susceptibility to liquefaction would increase with the reduction in the stiffness and permeability of the stone column. A three-dimensional finite element method was employed to simulate a centrifuge experiment; in this simulation, if the permeability of stone columns exceeded a threshold, the liquefaction hazard dramatically decreased in a silty sand foundation ([PERSON] et al. [25]). Furthermore, [PERSON] et al. [26] showed that using stone columns was an effective way to mitigate lateral deformation in uniform and stratified liquefiable soil strata. The seismic response of the coupled soil-structure system has been further examined in terms of peak bending moments along the pile foundation, emphasizing the possibility of a kinematic interaction on piles induced by the seismic action ([PERSON] et al. [27]). [PERSON] et al. [28] deduced the analytical solution of consolidation between gravel piles and soil suffered the variable load, based on the premise of considering the influence of replacement ratio and modular ratio between gravel pile and soil. In view of the lateral expansive deformation of gravel piles in composite foundation, [PERSON] et al. [29] proposed an analytical solution of settlement in composite foundation, reinforced with gravel piles. Based on the dynamic constitutive model of sand, a dynamic numerical analysis was performed for gravel pile reinforcement in the South-to-North Water Diversion Project; the excess pore pressure of saturated sand foundation at the bottom of the main canal were obviously reduced due to the drainage effect of gravel piles ([PERSON] et al. [30]). Adopting gravel piles, [PERSON] et al. [31] carried out a loading plate test and a super-heavy dynamic penetration in situ test and evaluated the bearing capacity and liquefaction resistance in treatment subgrade.