## 1 Introduction  
Buried continuous pipelines are vulnerable to the effects of Permanent Ground Deformation (PGD) resulting from seismic-induced landslides, faulting, and lateral spreading due to liquefaction. The pipeline response depends on its orientation with respect to the direction of the ground movement, which is generally a combination of transverse and longitudinal ground movement, occurring perpendicular and parallel to the pipeline axis, respectively. Transverse PGD induces predominant shear and bending stresses at the margins of the PGD zone, similar to the fault-crossing problem investigated by many researchers during the last 50 years [1; 2; 3; 4; 5]. Conversely, the longitudinal PGD results in axial compression and tension in the pipeline that may ultimately lead to local buckling and tensile rupture, respectively. According to previous investigations [6; 7; 8; 9], buried continuous pipelines are significantly more vulnerable to PGD in the longitudinal direction, requiring a detailed analysis of the system performance under this hazard. The response of buried continuous pipelines subjected to longitudinal PGD depends on the pipe deformation capacity under operating conditions, as well as the soil-structureinteraction, amount of ground movement \(\delta\) and its spatial extent \(L_{b}\), and pattern of ground deformation (Figure 1). The rigid block pattern, defined by a downslope movement \(\delta\) within the soil block length \(L_{b}\), induces localized relative soil-pipeline displacement at the margins, resulting in the largest pipe strains when compared to other distributive geometries considered by [PERSON] and [PERSON] [7]. Consequently, this pattern has been extensively employed within simple analytical or more complex numerical methods to assess pipeline performance under longitudinal PGD [7; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19]. These analytical structural models are more computationally efficient, compared to a complex numerical analysis approach, which requires further expertise of the engineer to analyze the models for use in routine engineering applications [20]. The existing analytical models for assessing the deformation of buried continuous pipelines subjected to longitudinal PGD consider the pipe material either as linear elastic [7; 10] or inelastic, following a Ramberg-Osgood stress-strain relationship that is equal in tension and compression [8; 11; 19]. Herein, two conditions were established for which the pipeline deformation demand was assessed, depending on whether the soil block length \(L_{b}\) is short (case I) or long enough (case II) to fully mobilize the soil reaction along the pipeline under the imposed ground displacement (Figure 2). The pipeline design force was computed as the minimum between the ultimate soil reaction transferable to the pipe over a length of \(L_{b}/2\) (case I) and the force computed assuming that the pipeline is fully compliant with the soil (case II), as recommended by the 2005 ALA guideline [12]. This conventional model predicts that half the total applied soil load is resisted in tension and half in compression, because of the assumed symmetric configuration of the soil-pipeline system, including the material constitutive relationships in tension and compression. Consequently, for a given PGD demand (\(d\), \(L_{b}\)), the estimated peak pipeline strain magnitude in tension is equal to that in compression, as shown in Figure 2. This is not representative of pipelines made of materials with different stress-strain relationships for tension and compression under uniaxial or biaxial stress state conditions, resulting in asymmetric pipeline loading and strain demand in the tensile and compressive ground deformation zone. Therefore, to accurately assess pipeline performance under longitudinal PGD, it is necessary to adopt analytical methods considering the asymmetric nonlinear material behavior of the operating pipeline for tension and compression as well as the elastoplastic axial soil-pipeline interaction [5; 21].