## 3 Linear Analysis  
Linearization of the Foundation (Applicable to Dynamic Storm, Seismic Resistance, and Fatigue Analysis)  
Since the analyses are dynamic in nature and performed using SACS V16 software, it is essential to model the soil-pile interaction in a linear manner, despite its inherently nonlinear behavior. Consequently, the foundation must be linearized. This process involves the creation of fictitious \"super-elements\" at the head of each pile, designed to approximate the equivalent nonlinear behavior of each pile head through a linear stiffness matrix. These matrices are obtained using SACS software. It is important to note that this linearization is exclusively applied for modal extraction analysis (i.e., periods and mass participation factors). However, for final results, the soil-pile interaction is explicitly considered. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**Linear Analysis**} & \multicolumn{3}{c}{**Nonlinear Analysis**} & \multicolumn{1}{c}{**Predictive**} \\ \multirow{2}{*}{**Item**} & \multirow{2}{*}{**Description**} & **5.1** & **5.2** & **5.3** & **5.4** & **5.5** & **5.6** \\  & & **Operational** & **Dynamic** & **Seismic** & **Ultimate** & **Seismic** & **Spectral** \\  & & **Storm** & **Storm** & **Resistance** & **Wave** & **Ductility** & **Fatigue** \\ \hline
3.0 & Analysis static & X & & & & & \\
3.1. & Pile super element & & X & X & & & X \\
3.2. & Obtaining dynamic & & X & X & & & X \\
3.3.1 & Dynamic wave & & X & & & & \\
3.3.2 & Seismic spectrum response & & & X & & & \\
3.3.3 & Transfer function & & & & & & X \\
4.0 & Analysis pushover & & & & & X & X \\
5.0 & Results & X & X & X & X & & X \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Types of analysis. Figure 3: Sequence of execution of analyses in SACS software. Figure 2: Basic analysis to determine structural integrity. Determination of Dynamic Properties (Applicable to Dynamic Storm, Seismic Resistance, and Fatigue Analysis)  
To determine the dynamic properties of the structure, a dynamic analysis was executed within the Dynpac module of SACS software. This module employs a set of master nodes with user-assigned degrees of freedom to extract dynamic characteristics, including Eigenvalues (natural periods) and Eigenvectors (mode shapes), using the standard Householder-Givens extraction technique. The stiffness and mass matrices are reduced to master nodes via standard matrix condensation and Guyan reduction methods under the assumption that stiffness and mass are similarly distributed. Non-inertial degrees of freedom (i.e., those without mass) must be classified as slave nodes. In this study, master nodes were assigned within the structural model at intersection points of the alphabetic and numerical axes at each bracing level. These nodes define the system configuration and mass distribution, ensuring that all stiffness and mass properties associated with slave nodes are included in the Eigen extraction procedure. Several studies have been conducted on the dynamic behavior of platforms, comparing the periods obtained from scaled models to those from computational models [40; 41; 42; 43], as well as instrumentation for damage detection [44] and vibration studies in offshore marine turbines. However, these studies do not address the potential variations in results based on the definition of master nodes. Master nodes play a crucial role in mass concentration within the structural model, affecting the calculation of natural periods, mode shapes, and mass participation factors. This study evaluates three structurally identical models with different master node configurations, highlighting their influence on the results:  
* Model 1: Masses concentrated solely at axis intersections. * Model 2: Masses concentrated at axis intersections and x-brace connections of the jacket. * Model 3: Similar to Model 2, but with additional mass concentrations on diagonal members supporting cantilevers in the topside (see Figure 4).