## 3 Results and Discussion
### The Horizontal Cyclic Stiffness of the Pile Group  
In this section, the pile group deformation in the perspective of foundation stiffness will be evaluated. Referring to the article of [PERSON] et al. [10], the ratio of the loading amplitude (\(H_{\text{C}}\)) to the difference between the maximum displacement (\(\gamma_{max}\)) and the minimum displacement (\(\gamma_{min}\)) in each cycle is taken as the horizontal cyclic stiffness (\(K_{\text{C}}\)), which can be expressed in Equation (2):  
\[\text{K}_{\text{C}}=\frac{H_{\text{C}}}{\gamma_{max}-\gamma_{min}} \tag{2}\]  
The cyclic load-displacement curves for CLR = 0.25, CLR = 0.5, and CLR = 0.75, with the pile spacing set at 4 times the pile diameter and the loading cycle set at 10 times, are depicted in Figure 15a (where CLR represents the cyclic load ratios). It was observed that the horizontal cyclic stiffness of the pile group decreased as the loading amplitude increased, and the pile group stiffness was reduced from 56.67 \(\text{KN}/\text{m}\) to 21.25 \(\text{KN}/\text{m}\) when CLR increased from 0.25 to 0.75, and its reduction could be up to 62.5%. The incorporation of the horizontal cyclic stiffness of the pile group not only reflects the nonlinear behavior of the soil, but also supports the trend that the accumulative horizontal deformation of the pile group increased with the increase of the load amplitude.  
In addition, when the pile spacing was set at 4 times and CLR = 0.5, the load-displacement curves of the pile group foundation under different cycles are illustrated in Figure 15b. The number of cycles increased from 10 to 5000 times, and the pile group stiffness increased from 28.33 \(\text{KN}/\text{m}\) to 42.7 \(\text{KN}/\text{m}\). As depicted in Figure 15b, it is evident that the horizontal cyclic stiffness of the pile group escalated with the progression of cycles. Notably, the disparity between the maximum and minimum displacements within a single cycle diminished as the cycles increased, which led to a gradual stabilization of the horizontal cumulative deformation of the pile group.  
The findings of this investigation hold some possibilities for augmenting the stability and safety of diverse structures, including bridges, wind turbines, and offshore platforms that are subjected to cyclic loads. Furthermore, these findings could shed light on the optimization of the pile group design under seismic loads. It is worth noting that there exists a substantial correlation between the cumulative deformation of piles and the pore water pressure around the structure. Therefore, in similar research and engineering endeavors, meticulous attention should be paid to the development of pore water pressure around  
Figure 15: The variation in cyclic load with displacement. (**a**) The loading cycle is set at 10 times. (**b**) The pile spacing is set at 4 times and CLR = 0.5 (where CLR is the cyclic load ratio and N is the number of cycles).  
the piles, particularly during the initial cycles, in order to accurately evaluate the dynamic response and deformation of the soil. Regular monitoring after project completion is highly recommended.  
In addition, in the design and monitoring of pile groups, it is imperative to consider the loading conditions of different rows of piles, and the burial depth and arrangement of piles should be reasonably chosen. Moreover, it should be noted that the stiffness of the group pile foundation is not a static parameter throughout the cyclic process, despite conventionally being treated as such in the design phase. Hence, it is prudent to recommend that in future projects of a similar nature, the stiffness of materials should be reasonably selected, to ensure the secure and stable performance of the structure throughout its entire lifespan.