## 4 Discussion
### Cyclic Stiffness Degradation  
Saturated clay soils with installed pile foundations are susceptible to softening under cyclic loading, which can result in plastic deformation of the soil and the pile-soil system [40; 41]. This softening and plastic deformation can significantly affect the performance of the foundation, potentially leading to structural damage or failure. To gain a deeper insight into the pile-soil system's response to cyclic loading, it is crucial to analyze its cumulative cyclic properties alongside its load-deflection behavior. The cyclic stiffness of the pile-soil system pertains to its capacity to resist deformation and sustain its load-bearing capacity under cyclic loading conditions. This property can be measured by conducting cyclic loading tests on the pile-soil system and analyzing the resulting load-deflection curves. The normalized load-deflection curves are presented in Figure 12. To enhance clarity and facilitate visualization, the rapidly developing stage, slowly developing stage, and steadily developing stage are separated into three sets of load-displacement curves. These three sets correspond to the 1 st to 10 th cycles, the 1490 th to 1500 th cycles, and the 4490 th to 4500 th cycles, respectively. As depicted in Figure 12, the hysteresis curves of the pile-soil  
Figure 11: Comparison diagram of pile bending moment for different skirt diameters in different cycles: (**a**) N = 10; (**b**) N = 100; (**c**) N = 1500; (**d**) N = 4500.  
system under cyclic loading showcase both translational and rotational movements. These phenomena signify the occurrence of plastic deformation and stiffness degradation in the system as the number of cyclic loads increases. To provide a more comprehensive analysis of the cumulative deformation patterns of skirt-piles under horizontal cyclic loading, this section evaluates the deformation by examining foundation stiffness. The cyclic stiffness is calculated by dividing the load amplitude by the displacement amplitude of the pile foundation in each cycle [42]. The calculation formula for cyclic stiffness is Equation (4):  
\[K_{c}=\frac{H_{c}}{(y_{cmax}-y_{cmin})} \tag{4}\]  
where \(K_{c}\) is the cyclic stiffness, \(H_{c}\) is the horizontal cyclic load amplitude, and \(y_{cmax}\) and \(y_{cmin}\) are the maximum and minimum values of horizon cyclic displacement, respectively. This study investigates the variation in cyclic stiffness to gain insights into the evolution of horizontal cyclic stiffness in pile foundations in relation to the number of loading cycles. The results illustrated in Figure 13 reveal a consistent decline in the horizontal cyclic stiffness of the monopile as the number of loading cycles increases. Moreover, the discrepancy between the maximum and minimum load values in the Nth loading cycle consistently decreases with the growing number of cycles. This trend corresponds to an increasing strain level in the soil and a continuous reduction in plastic deformation. These observations suggest that the accumulated deformation tends to stabilize with the increase in the number of cycles. The skirt-pile seat undergoes rapid degradation during the initial loading process but tends to stabilize with increasing loading cycles. In terms of overall stiffness, it was found that III \(>\) II \(>\) I \(>\) monopile. This implies that as the skirt size increases, its cyclic  
Figure 12: Variations in horizontal loads over horizontal displacements for different piles: (**a**) monopile; (**b**) skirt-pile I; (**c**) skirt-pile II; (**d**) skirt-pile III. stiffness becomes greater and its resistance to deformation becomes stronger, which is the same as the conclusion drawn from the cumulative turning angle. At N = 3000, the cutline stiffness variation tends to stabilize, indicating a balance between the softening effect of remodeling and the stiffness effect of reconsolidation.