## 3 Offshore Platform Overview and Finite Element Model
### Soil-Structure Interaction  
Existing simulation techniques for pile-soil interaction forces primarily include direct consolidation modeling, equivalent fixed-base modeling, m-method simulation, and P-y curve analysis. Among these, the direct consolidation approach overlooks pile-soil interaction, while the equivalent fixed-base method ignores soil effects [27]. Considering the soil properties in the East China Sea, this study employs P-y, T-z, and Q-z curve methodologies for simulation. The P-y curve method is rooted in elastic subgrade reaction theory: as depicted in Figure 5, it substitutes soil with nonlinear soil springs, overcoming the limitations of earlier linear elastic spring models. This approach accurately captures the nonlinear response of soil surrounding piles under loading conditions. The PySimple1 material model in OpenSees captures nonlinear pile-soil interactions, theoretically based on the three-component series framework proposed by [PERSON] et al. [27]. This model breaks down soil resistance into three serial components--elastic (p-y\({}^{\text{e}}\)), plastic (p-y\({}^{\text{P}}\)), and gap (p-y\({}^{\text{g}}\))--governed by displacement superposition and force equilibrium principles, as detailed in Equation (11). \[ \begin{cases}y=y^{\text{e}}+y^{\text{P}}+y^{\text{g}}\\ \quad p=p^{\text{d}}+p^{\text{c}}\end{cases}  \tag{11}\]  
where y\({}^{\text{e}}\) denotes the elastic displacement component, modeled via linear springs to capture the far--field soil's elastic response; y\({}^{\text{P}}\) represents the plastic displacement component, capturing post--yield soil stiffness decay through the kinematic hardening criterion; and y\({}^{\text{g}}\) signifies the gap displacement component, accounting for pile-soil separation and re-contact via a nonlinear closure spring coupled with a trailing spring. This decomposition overcomes the limitations of traditional P-y curves in modeling stiffness degradation and gap effects under cyclic loading. The model's core constitutive equations are detailed in Equations (12)-(14). Spanning from \(-\)C\({}_{\text{r}}\)P\({}_{\text{pull}}\) to C\({}_{\text{r}}\)P\({}_{\text{pull}}\), the initial stiffness range demarcates the elastic response regime. Under kinematic hardening, the plastic component equations adhere to a  
Figure 5: Schematic diagram of P-y curves for soft clay. modified hardening criterion, where C\({}_{r}\) (the initial yield ratio) assumes 0.2 for sandy soils and 0.35 for clayey soils. \[\mathrm{p}=\mathrm{p}_{\mathrm{p}utt}-\left(\mathrm{p}_{\mathrm{p}utt}-\mathrm{p }_{0}\right)\left[\frac{\mathrm{c}_{\mathrm{y}50}}{\mathrm{c}_{\mathrm{y}50}+ \left|\mathrm{z}_{\mathrm{p}}-\mathrm{z}_{0}^{\mathrm{p}}\right|}\right]^{n} \tag{12}\]  
Within this formulation, p stands for the total soil resistance, P\({}_{\mathrm{p}utt}\) signifies the ultimate bearing capacity, c and n shape the curve profile, z\({}_{\mathrm{p}}\) tracks the current plastic displacement, and z\({}_{0}^{\mathrm{p}}\) marks the initial displacement of the active plastic loading cycle. For API sand, optimal fitting to the specification-prescribed sand backbone curve is achieved with default parameters c = 0.5 and n = 2. The gap closure spring's contact nonlinearity is characterized by a hyperbolic function, where p\({}^{\mathrm{c}}\) denotes the closure force governing the nonlinear compressive response during pile-soil recontact. \[\mathrm{p}^{\mathrm{c}}=1.8\mathrm{p}_{\mathrm{p}utt}\left[\frac{\mathrm{y}_{5 0}}{\mathrm{y}_{50}+50\left(\mathrm{y}_{\mathrm{o}}^{+}-\mathrm{y}^{6}\right)} -\frac{\mathrm{y}_{50}}{\mathrm{y}_{50}+50\left(\mathrm{y}_{\mathrm{o}}^{-}- \mathrm{y}^{6}\right)}\right]^{n} \tag{13}\]  
Memory variables \(\mathrm{y}_{\mathrm{o}}^{+}\) (capturing the peak positive displacement during pile-soil separation as the maximum historical gap opening in the positive direction) and \(\mathrm{y}_{\mathrm{o}}^{-}\) (tracking the peak negative displacement in the negative direction) ensure stiffness continuity during load reversal.