## 3 Fragility Surfaces  
This section presents the fragility analysis results of the buried operating steel pipelines subjected to longitudinal PGD, obtained using the methodology described in Section 2.5. The evaluated fragility surfaces for the buried pipeline at a cover depth \(H_{c}\) = 1.5 m are shown in Figures 9a and 9b for the NOL and the PIL performance limit states, respectively. Clearly, the probability of the pipeline exceeding the performance criteria increases for larger soil displacements \(\delta\) and lengths \(L_{b}\), being greater for the NOL, compared to the PIL limit state. The iso-probability lines are defined by two half-lines parallel to the \(\delta\) and \(L_{b}\) axis (Figure 9). This is consistent with the pipeline deformation response observed for the deterministic analysis (Figures 7 and 8), as schematically illustrated in Figure 10. Specifically, the median values of soil displacement (\(\delta\), \(L_{b}\)), corresponding to a 50% probability of reaching the NOL and PIL criteria are (0.13 m, 193.4 m) and (0.48 m, 267.4 m), respectively, as indicated in Table 7. These critical values are less than the ones evaluated deterministically for the unpressurized pipeline (Figure 6), particularly for the NOL performance limit state (Table 7). This may result in under-designed pipeline systems when using the deterministic approach, highlighting the importance of accurate uncertainty analysis for a reliable infrastructure design. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Deterministic Analysis**} & \multicolumn{4}{c}{**Fragility Analysis (Median Value)**} \\ \cline{2-7}  & \(H_{c}\) **(m)** & **1** & **1.5** & **2** & **1** & **1.5** & **2** \\ \hline NOL & \(\delta_{cr 0.7}\) & 0.31 & 0.22 & 0.17 & 0.18 & 0.13 & 0.10 \\  & \(L_{cr1}\) (m) & 356.5 & 254.9 & 198.3 & 270.2 & 193.4 & 151.7 \\ \hline PIL & \(\delta_{cr 0.7}\) & 0.76 & 0.54 & 0.42 & 0.67 & 0.48 & 0.38 \\  & \(L_{cr2}\) (m) & 388.9 & 278.0 & 216.3 & 372.5 & 267.4 & 208.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of pipeline performance based on deterministic and fragility analysis results. Figure 9: Fragility surface of buried pipeline (\(H_{c}\) = 1.5 m) for (**a**) Normal Operability Limit (NOL) and (**b**) Pressure Integrity Limit (PIL). To investigate the effect of the cover depth \(H_{c}\) on pipeline performance, the fragility functions were evaluated considering a minimum cover depth value of \(H_{c}\) = 1 m and a maximum of \(H_{c}\) = 2 m (Figure 11). This results in a proportional variation in the soil friction reaction per unit pipe length \(f_{\text{s}}\)[31] that directly controls the pipeline deformation demand (Table 2). Clearly, the probability of the pipeline exceeding the performance limit states for a given amount of PGD (\(\delta\), \(L_{b}\)) increases with greater burial depths and associated soil friction reaction. Specifically, the median values of ground displacement intensity (\(\delta\), \(L_{b}\)), corresponding to a 50% probability of reaching the NOL criteria are (0.18 m, 270.2 m) and (0.10 m, 151.7 m) for the shallower (\(H_{c}\) = 1 m) and deeper (\(H_{c}\) = 2 m) soil cover depths,  
Figure 11: Fragility surface of buried pipeline for different cover depths and performance limit states: (a) \(H_{c}\) = 1.0 m, NOL; (b) \(H_{c}\) = 1.0 m, PIL and (c) \(H_{c}\) = 2.0 m, NOL; and (d) \(H_{c}\) = 2.0 m, PIL. Figure 10: Schematic representation of the performance assessment of the buried pipeline subjected to the PGD demand (\(\delta\), \(L_{b}\)), using the deterministic and fragility analysis framework. respectively (Table 7). Likewise, a critical PGD demand (\(\delta\), \(L_{b}\)) of (0.67 m, 372.5 m) and (0.38 m, 208.4 m) is needed to achieve a 50% probability of reaching the PIL criteria for the former (\(H_{c}\) = 1 m) and latter (\(H_{c}\) = 2 m) pipe burial condition, respectively.