## 3 Methodology of Numerical Analysis  
Pipelines made of API X80-grade line pipe steel were numerically simulated in this study using a tubular FE shell model developed with ABAQUS/Standard, in order to study the stability of steel pipelines when subjected to uniform axial compression, uniform bending, or combined axial compression + bending. The parameters investigated in the FE study were selected according to three factors considered to be most influential to the buckling response and respective limit strains of tubular shell structures, i.e., the dimensional factor (related to the \(D/t\) ratio), the load factor (related to internal or external overpressure), and the strain hardening factor (related to shape of stress-strain curve). Four variations of the \(D/t\) ratio were specified thus: DT1 = 41.152, DT2= 64.078, DT3 = 82.156, and DT4 = 104.622. The \(D/t\) ratio was varied by maintaining the pipe's outer diameter (OD) at a constant nominal pipe size (NPS) of 36 inches (914.4 mm), and respectively changing the wall thickness of the pipe models thus: t\({}_{1}\) = 22.22 mm (for DT1), t\({}_{2}\) = 14.27 mm (for DT2), t\({}_{3}\) = 11.13 mm (for DT3), and t\({}_{4}\) = 8.74 mm (for DT4). In addition to non-pressurized conditions, four levels of internal pressure corresponding to 20%, 40%, 60%, and 80% of the yield pressure in the circumferential direction were applied to the pipe models. To properly account for the influence of the material's strain hardening properties on the CLS, three material-related parameters each were derived for both YPT curves and RHT curves. The two parameters obtained for both RHT and YPT pipes comprised the two model \"shape\" constants (\(K_{nm}\) and \(H_{nm}\)), while the yield plateau length (YPL) was applied as the third parameter for YPT pipes and the ratio of the PLS to the UPS (herein referred to as the \"PLUS\" ratio) was considered as the third parameter for RHT pipes. The pipe geometry was modeled as a 3D deformable shell structure, and for computational efficiency, symmetric boundary conditions were applied in the longitudinal and transverse directions of the pipe, so that only a quarter of the full pipe specimen (half of the specified length and half of the circumference) was modeled. The numerical analysis procedure conducted in this study is a geometric and material nonlinear type; hence, the boundary conditions applied to create the X- and Z-symmetries for the assumed quarter models are able to provide an accurate representation of the actual response of the full pipe models. With the application of the \(Z\)-symmetry boundary condition (symmetry across the \(X\)-\(Y\) plane), the quarter pipe models were therefore extruded to three times the pipe diameter (3.0\({}^{\bullet}\)D). The assigned length of the pipe model was selected according to the recommendations of [PERSON] et al. [58] to ensure that there are no interactions between the end boundary conditions and the strain and stress distribution at the pipe mid-length, where initiation of local buckling deformation is expected to occur. At the loading end of the pipe model (i.e., the end where the rotation or axial displacement was applied), a reference point was created at the center of the pipe cross-section and restrained such that only translation along the \(Z\)-axis was allowed for uniform axial compression, and rotation about the \(X\)-axis plus translation along the \(Z\)-axis was allowed for uniform bending and combined axial compression + bending. A kinematic coupling constraint was then assigned for connecting the nodes at the loading end to the reference point to allow the end of the pressurized pipe to expand in the