## 3 Linear Analysis
#### 3.3.3 Spectral Fatigue Analysis--Transfer Functions  
A transfer function characterizes the relationship between stress range cycles and wave height as a function of frequency for each wave direction. The stress range, defined as the difference between maximum and minimum stress in each cycle, was determined for 18 wave incidence directions at 22.5\({}^{\circ}\) intervals (see Figure 5).  
Transfer function values were obtained by dividing stress amplitudes by the corresponding wave height, linking sea states (wave and period) to critical stress amplitudes.  
\begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Campeche Bay**} \\ \hline Period (second “s”) & Acceleration “\(\alpha\)” (g’s) \\ \hline
0.010–0.050 & 0.100 \\
0.125–0.504 & 0.250 \\
10 & 0.013 \\ \hline \hline \end{tabular} Numerical data of the acceleration spectrum for a return period of 200 years and a critical damping coefficient of 5%. Values taken from the Specification of Pemex “P.2.0130.01:2015” [36].  
\end{table}
Table 6: Numerical data of the acceleration spectrum.  
Figure 5: Method used to obtain the range of stress.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Angle Deg**} & \multicolumn{3}{c}{**Damping 2\%**} & \multicolumn{3}{c}{**Damping 5\%**} \\ \cline{2-6}  & **Base Shear** & **Overturning** & **Vertical** & **Base Shear** & **Overturning** & **Vertical** \\  & **Tom** & **[PERSON]** & **Tom** & **Tom** & **[PERSON]M** & **Tom** \\ \hline  
maximum & 700.27 * & 28,694.71 ** & 1314.98 & 709.17 * & 31,175.57 ** & 1509.55 \\ \hline \hline \end{tabular}  
* Maximum Base Shear, ** Maximum Overturning Moment.  
\end{table}
Table 7: Shear and moment versus direction: damping 2% vs. damping 5%.  
Wind-induced cyclic loading was also considered, with mechanical transfer functions defined per mode. The transfer function \(H(f)\) quantifies the stress range as a function of wind frequency and natural mode frequency, given by Equation (2).  
\[H(f)=\frac{1}{K_{i}}\left[\left[1-\left(\frac{f}{f_{n}}\right)^{2}\right]^{2}+ \left[2c\left(\frac{f}{f_{n}}\right)^{2}\right]\right]^{\frac{-1}{2}} \tag{2}\]  
where:  
* \(H(f)\) is the frequency-dependent transfer function.
* \(K_{i}\) is the generalized stiffness.
* \(f\) is the excitaion frequency.
* \(f_{n}\) is the natural frequency of the system.
* \(c\) is the damping ratio.  
The transfer functions were computed at increments of 22.5\({}^{\circ}\) and correspond to the total vectors of the oceanographic information provided in Table 10 of the Pemex specification [36], which includes areas such as May, Yum, and Coastal. It is essential to determine the structural response to wave loads for each direction. Consequently, 16 independent transfer functions were required, each corresponding to a specific wave propagation direction. For this purpose, the orientation of the structure relative to both the North direction and the X-Y coordinate system of the model was taken into account.  
A total of 40 wave cases were considered, beginning with six waves characterized by a period of 12.5 s, followed by period decrements of 1.0 s. Subsequently, four additional waves were introduced, starting at 6.5 s and separated by 0.5 s intervals. To capture the dynamic behavior of the structure near its natural frequencies, a refined distribution of wave periods was applied. This includes 4, 7, 7, 6, and 6 waves, with period decrements of 0.2, 0.15, 0.05, 0.1, and 0.2 s, respectively. The corresponding starting periods for these waves are 4.5, 3.7, 2.75, 2.35, and 1.7 s. This approach ensures a comprehensive evaluation of the structural response to wave-induced loading, particularly in the frequency ranges that may induce resonance phenomena.