## 3 Probabilistic Analysis
### Reliability Analysis
#### 3.2.2 Sensitivity Analysis  
To better comprehend the impact of each variable on the load-carrying capacities of the structure, a sensitivity analysis was performed using UQlab. Due to the correlation between some variables, the analysis of covariance (ANCOVA[XE \"ANCOVA\")) was used to produce helpful sensitivity indices for both correlated and uncorrelated input variables [51]. As expected, the parameters with the most influence over the load-carrying capacity are the ones used to describe the backfill soil due to its high contribution to the bridge capacity when analyzing the in-plane behavior (and excluding the spandrel walls) [52]. Next in relevance is the scour profile (e.g., shape, depth), which has a considerable impact on the load-carrying. Lastly, the contribution from the remaining parameters can be neglected when compared with other parameters. As shown in Figure 6, the impact from the scour variable is mainly uncorrelated, and the ones related to the backfill are more heavily influenced by the correlation among its input variables (i.e., the magnitude of the correlated indices is closer to the total first-order indices). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**ID** & **Description** & **Element** &
\begin{tabular}{c} **Nominal** \\ **Values** \\ \end{tabular} & **Units** & **CV (\%)** & **Type** & **Reference** \\ \hline \(FA_{S}\) & Friction angle & Soil & 34.61 & \({}^{\circ}\) & 10 & Lognormal & [45] \\ \(SD_{S}\) & Saturated density & Soil & 20.50 & kN/m\({}^{3}\) & 5 & Lognormal & [45] \\ \(DD_{S}\) & Dry density & Soil & 17.50 & kN/m\({}^{3}\) & 5 & Lognormal & [45] \\ \(SD_{B}\) & Saturated density & Backfill & 19.00 & kN/m\({}^{3}\) & 5 & Lognormal & [46] \\ \(C_{B}\) & Cohesion & Backfill & 10.00 & kPa & 10 & Lognormal & [46] \\ \(FA_{B}\) & Friction angle & Backfill & 21.06 & \({}^{\circ}\) & 10 & Lognormal & [24] \\ \(DD_{B}\) & Dry density & Backfill & 17.00 & kN/m\({}^{3}\) & 5 & Lognormal & [24] \\ \(D_{C}\) & Density & Concrete & 25.00 & kN/m\({}^{3}\) & 10 & Gaussian & [47] \\ \(D_{M}\) & Density & Masorny & 16.00 & kN/m\({}^{3}\) & 10 & Gaussian & [48] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Random variables for the reliability analysis. The surrogate model, based on the relevant variables, is used to evaluate an experimental design of 10,000 simulations using the Monte-Carlo (MC[XE \"MC\"]) sampling method, which will capture the capacity curve. Then, a Gumbel minimum distribution was fitted based on the generated histogram of adequacy factors (i.e., number of load increments that the structure can withstand without collapsing based on the acting load), which allowed us to define the resistance curve, \(R\). The loading curve is based on the data collected from measurements performed by the Portuguese statistical institute (INE[XE \"INE\"]) of railway frequency and magnitude. The loading curve, \(S\), was defined using existing data fitted to a Gumbel distribution, which is described by a mean equal to 184.41 kN and a COV of 9.06% [9]. The limit state function (2) was introduced to UQlab to evaluate the reliability of the structure by using the probabilistic distribution of both loading and capacity curves. Additionally, model uncertainties for limit state models (defined by a Gaussian distribution of mean 1 and COV 15%) were also considered based on the recommendations of the Joint Committee on Structural Safety (JCSS[XE \"JCSS\"]) model code for modeling shallow foundation stability with homogeneous soil profiles [45]. To obtain the reliability index of the structure for a given discharge value, traditional methods like MC may require numerous simulations to converge with a satisfactory level of accuracy. Therefore, subset simulation techniques are employed herein to overcome this limitation by solving simpler reliability problems with intermediate thresholds [44]. Once the reliability analysis is completed, the failure probability and reliability index are obtained for each discharge value and flow angle.