## 5 Conclusions  
This study develops a new analytical model that accurately and efficiently evaluates the performance of buried operating pipelines under longitudinal PGD, considering the asymmetric pipeline response- to tension and compression under varying operational loads.  
A further comparison of the proposed analytical model to the detailed finite element analysis results showed excellent agreement, demonstrating its capacity to accurately assess the pipeline response as a function of the system parameters, including the operational pressure and temperature variations. The analytical model was efficiently implemented within a robust fragility function calculation framework based on MCS, allowing us to assess the probability of exceedance of the pipeline performance limit states conditioned to the PGD demand (\(\delta\), \(L_{b}\)), considering the system uncertainties.  
The evaluated fragility surfaces showed that the probability of the pipeline reaching the performance criteria increases for larger soil displacement \(\delta\) and lengths \(L_{b}\), as well as cover depths \(H_{c}\), because of the greater mobilized soil reaction counteracting pipeline deformation. This requires implementation of proper engineering design solutions that minimize the risk of pipeline damage, for example, by adopting shallow soil cover depths, light weight backfill, and low-friction pipe coating.  
The performed GSA allowed us to quantify the uncertainty of the pipeline performance assessment attributed to each random system parameter, considering the sensitivity over the entire input space, including the nonlinear interaction effects between variables. The PGD length (\(L_{b}\)) was the most influential parameter with respect to the exceedance of the NOL and PIL performance limit states, followed by the soil friction per unit pipe length (\(f_{s}\)) and ground displacement (\(\delta\)), based on the first-order and total-order Sobol indices. This is consistent with the expected system performance, since the intensity of the PGD and soil-pipeline interaction directly control the pipeline deformation demand.  
The significant total sensitivity indices of the temperature variation (\(\Delta T\)) and internal pressure (\(P_{i}\)) for the NOL limit state demonstrated the importance of the effects of their interaction with the other input variables. Both the deterministic and uncertainty analysis results highlighted the impact of operational loads on pipeline performance, which need to be accurately considered in pipeline system modeling and design. The comparison between the deterministic and fragility analysis results showed that neglecting the variability of the system parameters, including operational loads, may result in under-designed pipelines, highlighting the importance of uncertainty analysis for a reliable infrastructure design.  
Overall, the proposed analytical fragility function calculation framework provides a useful methodology for effectively assessing the performance of operating pipelines under longitudinal PGD, quantifying the effects of the critical parameters impacting system response.  
This research received no external funding.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
This publication is made possible in part by support from the Berkeley Research Impact Initiative (BRII) sponsored by the UC Berkeley Library, that is greatly acknowledged. The author would like to thank [PERSON], [PERSON]. [PERSON], [PERSON] and [PERSON] at the Department of Civil and Environmental Engineering of UC Berkeley for valuable discussions.  
The authors declare no conflict of interest.