## 2 Methods
### Simplified Static-Equivalent Model  
The simplified model is based on a discretization of the system's geometry into beam elements, a concentrated mass representing the rotor-nacelle assembly, and the assumption of static-equivalent loads for the environmental excitation arising from wind and wave action. For this particular study, in which wind and wave loads are assumed to be aligned, a two-dimensional approach to simulate the lateral behavior of the structure was considered. A sketch of the simplified model is depicted in Figure 1. The code of the simplified model is included in the Supplementary Materials.  
The wind turbine's hollow tower and monopile were divided into two-node Euler-Bernoulli beam elements. Based on a convergence study, a sufficiently large number of elements was used to accurately reproduce the conical shape of the tower. Each node has two degrees of freedom: a lateral displacement (\(u_{n}\)) and a rotation (\(\theta_{n}\)), as shown in Figure 1.  
The interaction between the structure and the foundation was modeled using the foundation impedance matrix (\(K_{SSI}\)), which represents the stiffness of the soil-monopile subsystem. As a static-equivalent analysis was carried out, the static component was considered. As shown in Equation (1), this matrix consists of horizontal (\(K_{H}\)), rocking (\(K_{R}\)), and cross-coupled (\(K_{RH}\) and \(K_{HR}\)) terms:  
\[K_{SSI}=\begin{pmatrix}K_{H}&K_{HR}\\ K_{RH}&K_{R}\end{pmatrix} \tag{1}\]  
To compute the foundation impedance matrix, a numerical model [18] previously developed for the dynamic analysis of pile foundations was used. This model utilizes the integral expression of the Reciprocity Theorem along with specific Green's functions for a layered half-space to represent soil behavior. In this formulation, piles are modeled as load lines in terms of soil-pile interaction forces. Additionally, the additional stiffness introduced by the piles is represented using a beam finite element model. Soil and pile variables are coupled by enforcing compatibility and equilibrium conditions at the soil-pile interface in terms of displacements and interaction forces. The main advantage of this model is its ability to accurately reproduce the linear response of the soil-foundation system without discretizing soil variables, resulting in a compact and efficient formulation. The effectiveness of this model in reproducing foundation behavior for OWT monopile and bucket geometries was validated in [19].  
Figure 1: Sketch of the real problem (**left**) and simplified static-equivalent model (**right**). Adapted from [17].  
In assembling the stiffness matrices of all beam elements, including the additional stiffness introduced by the foundation, the following system was defined in order to obtain the structural static response:  
\[\begin{pmatrix}K_{ss}&K_{sb}\\ K_{bs}&K_{bb}+K_{SSI}\end{pmatrix}\begin{pmatrix}\vec{u}_{s}\\ \vec{u}_{b}\end{pmatrix}=F_{ext}^{*} \tag{2}\]  
where the vector \(\vec{u}_{s}\) refers to the displacements and rotations of the active degrees of freedom of the superstructure, while the vector \(\vec{u}_{b}\) represents the displacement and rotation at its base, i.e., the pile head at the mudline level. Finally, \(\vec{F}_{ext}\) denotes the static-equivalent external loads applied to the structure. In the proposed model, only wind and wave loads are considered. Their treatment is outlined in Sections 2.1.2 and 2.1.3, respectively. It is important to notice that, due to the isostatic nature of the model, the internal forces at a specific point of the structure can be obtained directly as the sum of the corresponding forces/moments acting above the studied point.