## 2 Methodology
### Analytical Model for Assessment of the Continuous Pipeline Response Under Longitudinal PGD  
This section describes the developed analytical model for assessing the performance of buried continuous pipelines under longitudinal PGD, considering the nonlinear material constitutive relationships, force equilibrium, and displacement compatibility along the pipe-soil system. This algorithm allows implementation of the asymmetric nonlinear stress-strain relationship for the pipe material for tension and compression, as well as the elastoplastic axial soil-pipeline interaction, accurately assessing the response of the system subjected to this geohazard. The ground deformation is idealized as a rigid-block movement, defined by a downslope movement \(\delta\) over a block length \(L_{b}\), resulting in a tension crack of width \(\delta\) at the upslope end and a compression ridge over a distance \(\delta\) on the downslope end (Figure 3a). The soil displacement assumes a constant value \(\delta\) within the PGD zone, while being zero on either side of it. The moving soil block tends to pull the pipe along with it, resulting in localized relative soil-pipeline displacement at the margins of the PGD zone, and associated resistance forces on either side of the sliding block head and toe. This results in a maximum pipeline axial force at the head and maximum axial compression force at the toe, decreasing linearly thereupon due to the sliding soil friction (\(f_{s}\)). Beyond this zone, the relative soil-pipeline displacement is negligible, and the pipeline displacement matches that of the ground (case II). As the soil displacement increases reaching a critical value \(\delta=\delta_{cr}\), the relative soil-pipeline displacement and associated friction reaction are mobilized over the entire soil block length, and the pipeline deformation remains constant thereafter (case I). Evidently, the region of the soil-pipeline system beyond the soil block behaves like a pull-out test under tension (region I) and compression (region IV), with the end displacement (\(\Delta L\)) applied at the pipe points underlying the tension crack and compression bulge, respectively (Figure 3a). The total pipeline elongation at each side of the soil block head i.e., in region I (\(U_{p1}\)) and region II (\(U_{p2}\)), is equal to the magnitude of the overall pipeline contraction at each side of the toe of the sliding block, i.e., in region III (\(U_{p3}\)) and region IV (\(U_{p4}\)). The strain demand was calculated using the analytical model developed by [30], evaluating the structural response of a continuous buried pipeline, subjected to a pullout force on one end (\(x=0\)). The resulting pipeline displacement was obtained by integrating the axial strains associated with the axial force distribution along the pipeline, considering the beam on elastic foundation theory for the static friction length, and the force equilibrium for the frictional sliding length (Figure 3b). This model employs the theory of plasticity for modeling the pipe material, based on the associated flow rule with the von [PERSON] yield criterion and isotropic strain hardening. Compared to the conventional model [8], the proposed analytical solution accounts for the initial axial thermal strains and biaxial stress state in the pipe due to internal pressure, as shown in Figure 4.  
Herein, the pipe material is assumed to have a piecewise stress-strain curve, defined by the engineering strain-stress values (\(\epsilon_{i}\), \(\sigma_{i}\)), either in tension (\(\epsilon>0\)) or in compression (\(\epsilon<0\)), where \(E_{i}=(\sigma_{i}-\sigma_{i-1})/(\epsilon_{i}-\epsilon_{i-1})\) is the slope of the \(i\)-th segment constituting the pipe multi-linear stress-strain relationship (Figure 4).