## 2 Methodology
### Validation of the Analytical Model
#### 2.3.1 Finite Element Analysis of the Soil-Pipeline System  
Within the numerical approach, the pipeline was modeled using the PIPE22H beam element type implemented in Abaqus/Standard [33], which is particularly suitable to model long, slender pipelines with a thin-walled circular cross-section, allowing the possibility of specifying external or internal pressures. Instead, the longitudinal soil-pipeline interaction was modeled with uniaxial spring elements SPRING2 connected at each node of the pipeline on one end, while being assigned the far-field ground motion at the other end through the boundary conditions. The adopted mesh size for the beam pipe elements is 0.10 m, based on the mesh sensitivity study performed herein, assuring efficiency and accuracy of the numerical solution.  
The X42 steel grade pipe material model is defined within the [PERSON] plasticity theory with isotropic hardening, with Young modulus \(E\) = 210 GPa, and yield stress \(\sigma_{y}\) = 290 MPa. The elasto-plastic force displacement relationship of the soil springs is defined by the sliding soil friction force per unit length of the pipeline, \(f_{s}\) = 26.8 kN/m, and the relative soil-pipe displacement at the onset of friction sliding, \(u_{0}\) = 3 mm, calculated according to the ALA guidelines [31], assuming compacted dense sand with friction angle \(\phi\) = 40\({}^{\circ}\) (Table 3).  
The length of the pipeline-soil system is equal to 1000 m, so the system response to the imposed ground displacement is representative of an infinitely long pipeline, unaf  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Units** \\ \hline Pipe diameter, \(D\) & 0.508 & m \\ Pipe wall thickness, \(t\) & 7.1 & mm \\ Pipe elastic modulus, \(E\) & 210 & GPa \\ Poisson’s ratio, \(\
u\) & 0.3 & \\ Pipe yield stress, \(\sigma_{y}\) & 290 & MPa \\ Pipe cover depth, \(H_{c}\) & 1.5 & m \\ Soil density, \(\gamma\) & 18.0 & kN/m\({}^{3}\) \\ Soil friction angle, \(\phi\) & 40 & \({}^{\circ}\) \\ Pipe–soil interface friction angle, \(\delta_{i}\) & 28 & \({}^{\circ}\) \\ Soil pressure at rest, \(K_{0}\) & 1 & \\ Soil friction reaction per unit pipe length, \(f_{s}\) & 26.8 & kN/m \\ Relative soil–pipe displacement at friction sliding, \(u_{0}\) & 3 & mm \\ \hline \hline \end{tabular}
\end{table}
Table 3: Pipe–soil system parameters.  
fected by the far-end boundary conditions. The assumed soil block length is \(L_{b}=300\) m (case II), located at the center of the soil-pipeline system so that its midpoint lies on the pipeline bisector.  
The numerical analysis was conducted in two consecutive steps. First, the internal pressure and temperature variations were applied in the pipeline, while the pipeline ends and the free ends of the soil springs were restrained in the longitudinal direction. Second, the soil block movement \(\delta=4\) m was applied statically with a maximum step increment equal to \(\Delta\delta=1\) mm at the free nodes of the soil springs within the soil block length, while outside of the moving block the soil nodes remain fixed. At each step increment, the nonlinear equilibrium equations were solved iteratively by the Newton-Raphson method, allowing us to assess the system performance at any level of applied ground displacement, until material failure.