## 4 Application Example
### Assessment of the Vibration Serviceability Limit State of the Cable under Uncertainty Conditions
In order to compute the reliability index, \(\beta_{s}(\theta)\), the maximum displacement of the stay cable (obtained from the different transient analyses performed on the sample of stay cables), which constitutes the demand of the wind action, \(D_{a}(\theta)\), and the allowable displacement of the stay cable (established by the FHWA guidelines [6]), which constitutes the capacity of the structure, \(C_{s}\), were determined. Additionally, as the wind action is defined according to a return period of 50 years, the corresponding value of the allowable reliability index is \(\beta_{t}=1.35\), according to the European guidelines [34]. As a numerical method in order to both determine the sample and compute the reliability index, \(\beta_{s}(\theta)\), the Monte Carlo method was considered herein. A convergence analysis was performed to \begin{table}
\begin{tabular}{c c} \hline \hline
**Design Level** & **Allowable Displacement [m] 1** \\ \hline preferred & 0.5\(D\) \\ recommended & 1.0\(D\) \\ not to exceed & 2.0\(D\) \\ \hline \hline \end{tabular} \({}^{1}\)\(D\) is the outer diameter of the cable.  
\end{table} 
Table 2: User tolerance limits for the different design levels [6]. determine the size of the sample [61]. As a result of this convergence analysis, the size of the sample was established at 100. Finally, in order to evaluate the demand of the wind action, \(D_{a}(\theta)\), the wind forces must be determined. For this purpose, simulations of the wind velocities were generated. The simulation of these wind velocities was addressed employing the wave superposition spectral-based method [8]. Both the von Karma spectra and a coherence function, as they are defined by the European guidelines [54], were employed herein. The following design parameters were considered for the wind simulation [54]: (i) basic wind velocity, \(v_{b,0}=26\) m/s; (ii) a directional factor, \(c_{dir}=1\); (iii) a season factor, \(c_{sen}=1\); (iv) a orography factor, \(c_{oro}=1\); (v) an terrain type III category (which involves a terrain factor, \(k_{r}=0.216\); a roughness length, \(z_{o}=0.3\) m; and a minimum height, \(z_{min}=5\) m); (vi) a duration of each simulation of 300 s; and (vii) a time step of \(5\times 10^{-3}\) s [62]. In this study, the wind velocities were generated at ten different heights of the cable (resulting from dividing the cable into ten equal-length segments), as Figure 5 depicts. This mesh density was considered for all the simulations conducted in the paper, in order to ensure that all the obtained results were consistent. Although preliminary analyses performed by the authors concluded that the meshing in Figure 5 was adequate for our aims (illustrating the performance of the proposed motion-based approach), the reader should be aware of the fact that the numerical simulation of the structural response under wind excitation depends on such mesh density, so that further analyses are recommended. A graphical user interface [63] was developed in the commercial software Matlab [64] to evaluate the wind action following the above guidelines. The application of Equations (11) and (12) allows the wind-induced forces in terms of the wind velocities to be computed. For this purpose, the following values for the characteristic parameters were adopted: (i) a density of the air, \(\rho=1.23\) kg/m\({}^{3}\); (ii) a drag coefficient, \(C_{D}=1.2\)[2]; and (iii) a lift coefficient, \(C_{L}=0.3\)[6]. Finally, a transient analysis (time history simulation) was performed for each element of the sample. The nonlinear geometrical behavior of the stay cables was considered for this analysis. A Newmark-beta method (an unconditionally stable method with parameters \(\beta_{m}=1/4\) and \(\gamma_{m}=1/2\)) was considered to solve the transient analysis. Hence, the reliability index, \(\beta_{s}(\theta)\), was computed from the results of this set of transient analysis. Subsequently, the vibration serviceability limit state of the stay cable under uncertainty conditions was assessed.