## 2 Methodology
### Validation of the Analytical Model
#### 2.3.2 Comparison Between Numerical and Analytical Solutions  
Figure 5 shows a comparison between the numerical and analytical model results assessing the performance of the pressurized and unpressurized pipelines in terms of the maximum axial strains for tension and compression, as a function of the ground displacement \(\delta\). Overall, the peak pipe strain magnitudes in the tensile and compressive PGD zone increase monotonically with the ground movement, progressively reaching the NOL and PIL performance limit states at critical values of ground displacement \(\delta_{cri}\), based on the operating conditions (Table 4). The results obtained using the conventional model reported in [8; 11; 13] agree well with the numerical analysis only for the case of the unpressurized pipeline under tension, while diverging significantly for the pressurized pipeline, with a percent difference exceeding 5% (Table 4). Conversely, the comparison between the proposed analytical model and the numerical simulation results shows excellent agreement for both cases of pipeline operating conditions. Specifically, the percent difference between the pipeline performance results for these two methods does not exceed 0.9% (Table 4), demonstrating the greater accuracy of this analytical procedure to assess system response, compared to the conventional method. Figure 5: The comparison between the numerical, the conventional [8; 11; 13], and the proposed analytical models, evaluating the pipeline performance under longitudinal PGD (\(L_{b}=300\) m) in terms of maximum tensile and compressive pipe strain as a function of the ground displacement \(\delta\). Finally, the proposed analytical model, evaluating the response of the buried operating pipeline subjected to longitudinal PGD, can be implemented in most programming languages (e.g., Python, Matlab) for further parametric analyses. Critical Soil Displacement (\(\delta_{cr}\)) and Length (\(L_{cr}\)) for the Performance Limit States of the Operating Pipeline  
As the soil displacement \(\delta\) increases (case II), the maximum magnitude of the pipeline axial force and associated strain at both margins of the PGD zone increase monotonically (Equation (1)), until the soil reaction mobilizes fully over the entire soil block length \(L_{b}\) (case I). Herein, the total soil load (\(f_{s}L_{b}\)) over the PGD zone is resisted by the developed pipe axial force that is at a maximum at the head (\(F_{t,max}\)) and toe (\(F_{c,max}\)) of the sliding soil block. Therefore, the critical soil length \(L_{cr}\), associated with full mobilization of the soil reaction (case I) for a given value of soil displacement \(\delta\), is directly proportional to the overall pipe load at the edges of the PGD zone:  
\[L_{cr}=[F_{t,max}(\delta/2)-F_{c,max}(\delta/2)]/f_{s} \tag{5}\]  
Consequently, the pipeline deformation demand will reach a certain limit state only if the soil displacement \(\delta\) and block length \(L_{b}\) are equal or greater than the critical soil displacement \(\delta_{cr,i}\) and length \(L_{cr,i}\) corresponding to that performance criterion, respectively. Figure 6 shows the variation of the critical soil block length \(L_{cr}\) as a function of the ground displacement \(\delta\), indicating the critical values associated with the achievement of the performance limit states in the pressurized (\(P_{i}/P_{max}=0.75\), \(\Delta T=50\)\({}^{\circ}\)C) and unpressurized pipelines (\(P_{i}/P_{max}=0\), \(\Delta T=0\)\({}^{\circ}\)C). Furthermore, the distributions of the axial strain, stress, force, displacement, and soil friction reaction along the pressurized and unpressurized pipelines for increasing values of applied ground movement \(\delta\) are presented in Appendix A, illustrating the different system responses for case I and case II. Clearly, the pipeline response depends on the operating loads, with the performance limit states in compression being reached for lower levels of critical soil displacement and length in the pressurized pipeline, compared to the unpressurized one.