## 4 Application Example
### Assessment of the Vibration Serviceability Limit State of the Cable under Uncertainty Conditions  
As it was expected, according to the numerical natural frequencies obtained (Table 1), this cable was prone to vibrate under wind action due to both the turbulent component of the wind (the first two natural frequencies are lower than 1 Hz [58]) and the rain-wind interaction phenomenon (the six natural frequencies are lower than 3 Hz [6]). For this reason, the assessment of the vibration serviceability limit state of this stay cable was performed herein following the recommendations of the Federal Highway Administration (FHWA) guidelines [6]. On the one hand, in order to avoid the wind-induced vibrations associated with the rain-wind interaction phenomenon, it must be checked that the damping ratio of all the vibration modes, whose natural frequencies are lower than 3 Hz, are greater than a recommended value [6,59]. In order to determine this recommended value, the FHWA guidelines [6] establishes that the rain-wind interaction \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Natural Frequency** & \(f_{1}\) & \(f_{2}\) & \(f_{3}\) & \(f_{4}\) & \(f_{5}\) & \(f_{6}\) \\ \hline Value [Hz] & 0.452 & 0.905 & 1.351 & 1.802 & 2.254 & 2.706 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical natural frequencies of the cable. phenomenon can be neglected if the Scruton number, \(S_{c}\), is greater than 10 for all the considered vibration modes. This condition may be expressed as follows:  
\[S_{c,i}=\frac{m\xi_{i}}{\rho D^{2}}>10 \tag{19}\]  
where \(\xi_{i}\) is the damping ratio of the \(i^{th}\) vibration mode. Thus, this requirement is equivalent to guaranteeing a minimum damping ratio for each considered vibration mode. The minimum required damping ratio may be determined as follows:  
\[\xi_{i}>\frac{10\rho D^{2}}{m} \tag{20}\]  
As expected, due to the results of previous experimental tests, the damping ratio associated with the first six vibration modes of this cable did not meet this condition [56]. Hence, it was necessary to increase the value of these damping ratios. A passive damping device can be designed and installed on the cable for this purpose. On the other hand, in order to analyze the effect of the turbulent component of wind action on the dynamic behavior of the cable, a transient analysis was performed. As a result of this transient analysis, the dynamic response of the cable under wind action can be obtained and the vibration serviceability limit state of the cable can be assessed. According to the FHWA guidelines [6], this limit state is met if the maximum displacement of the cable is lower than an allowable displacement which is defined in terms of the user tolerance. Table 2 shows the allowable displacement of the cable in terms of the design level required [6]. In this study, a recommended design level was established for the vibration serviceability limit state. Additionally, as the dynamic response of the stay cable was sensitive to the variation of its modal properties associated with the change of the operational and environmental conditions during its overall life cycle, a reliability analysis about the compliance of the vibration serviceability limit state was performed. For this purpose, it was assumed that the axial force of the cable is a random variable normally distributed. According to the results provided by [PERSON] et al., it could be also assumed that this random variable has a range of variation of \(\pm\) 10% [60]. A sample of stay cables with different values of the axial force was generated. The vibration serviceability limit state was assessed on this sample. For this purpose, the vibration serviceability limit state must be reformulated in order to take into account the uncertainty conditions. According to this, this limit state is met if a reliability index, \(\beta_{s}(\theta)\), is greater than an allowable reliability index, \(\beta_{t}\).