## 2 Numerical Modelling
### Soil Degradation Stiffness Model  
Cyclic loads such as wind and waves bring about variations in the deformation behavior of soil adjacent to the pile. [PERSON] et al. [29] and [PERSON] et al. [30] performed cyclic loading pile tests to investigate cyclic deformation characteristic of soil around the pile and proposed an exponential function to describe how the modulus of soil reaction for the soil surrounding the pile relates to the number of cyclic loading cycles, but the relevant parameters are complicated to obtain. [PERSON] [31] conducted soil cyclic triaxial tests to study variation of soil degradation secant stiffness under long-term cyclic loading. An exponential function of secant degradation modulus was obtained with the number of cyclic, as follows.  
\[\frac{E_{sN}}{E_{s1}}=N^{-a(X)^{b}} \tag{3}\]  
\[X=\frac{\sigma_{1,cyc}}{\sigma_{1,sf}} \tag{4}\]  
Here, E\({}_{sN}\) denotes the soil secant modulus following the Nth cycle (kPa), while E\({}_{s1}\) represents the soil secant modulus after the first cycle, N is the number of cyclic, a and b are the parameters of exponential degradation function (dense sand: a = 0.20 and b = 5.76, medium-dense sand: a = 0.16 and b = 0.38) [32; 33], X represents the ratio of cyclic stresses, \(\sigma_{1,cyc}\) stands for the maximum principal stress under cyclic stress conditions (kPa), and \(\sigma_{1,sf}\) characterizes the maximum principal stress when static failure occurs (kPa).  
In order to apply Equations (3) and (4) to analyze the soil cyclic characteristic, [PERSON] et al [13] proposed a characteristic cyclic stress ratio \(X_{c}\), which is defined as Equation (5).  
\[X_{c}=\frac{X^{(1)}-X^{(0)}}{1-X^{(0)}} \tag{5}\]  
The rate of soil secant modulus is determined by the relative density of soil and the characteristic cyclic stress ratio. [PERSON] et al. [13] found that for sandy soil, an increase in its relative density leads to a decrease in the secant modulus rate, while an increase in the cyclic stress ratio results in an increase in the secant modulus rate. It is noted that the soil secant modulus degrades significantly in the state of high characteristic cyclic stress ratio. Offshore wind turbine structures have an extremely low natural frequency, which is close to the dynamic loading frequency, but the safety redundancy region of its natural frequency is small in the design, so the degradation of the soil secant modulus should be taken into account in the design of the offshore wind turbine. The flow diagram of this process is shown in Figure 7.  
FLAC3D is a general geotechnical analysis software of finite difference principle and is equipped with the friendly pre-processing and post-processing interfaces. There are no special requirements for the performance of the computer. The calculation is convenient and can effectively consider the problem of large deformations. It has been widely applied in the analysis of rock and soil deformation. FLAC3D 3.0 is employed to conduct this study.  
By applying Fish programming functions in FLAC3D 3.0, the numerical model incorporates the calculation method for soil degradation to analyze the effects of long-term cyclic loading.  
The degradation stiffness numerical model's performance was validated by comparing its calculated results with the pile cyclic test data from [PERSON] et al. [34] and [PERSON] et al. [35]. These comparisons are presented in Figure 8. Clearly, the model can accurately capture the pile's accumulated deformation and the soil's cyclic characteristics under cyclic loading.  
Figure 8: Comparison of numerical calculation and results for tests.  
Figure 7: Flow diagram for modulus degradation numerical model.