## 2 Methodology
### Validation of the Analytical Model
#### 2.3.2 Comparison Between Numerical and Analytical Solutions
Interestingly, the pressurized and unpressurized pipelines satisfy all the performance criteria for soil block lengths \(L_{b}\) that are shorter than 145 m and 254 m, respectively, for any value of the applied ground displacement, \(\delta\). Conversely, the pipeline will fail to satisfy any limit state for values of the soil block length \(L_{b}\) and ground displacement \(\delta\) exceeding 280 m and 0.79 m, respectively (Figure 6). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multirow{2}{*}{\begin{tabular}{c} **Analysis** \\ **Method** \\ \end{tabular} } & \multicolumn{2}{c}{**Normal Operability Limit**} & \multicolumn{2}{c}{**Pressure Integrity Limit**} \\ \cline{3-6}  & & **Tension** & **Compression** & **Tension** & **Compression** \\  & & \(\delta_{t1}\)**(m)** & \(\delta_{c1}\)**(m)** & \(\delta_{t2}\)**(m)** & \(\delta_{c2}\)**(m)** \\ \hline \multirow{4}{*}{\
\begin{tabular}{c} Pressurized \\ pipeline \\ \end{tabular} } & numerical & 0.5706 & 0.0996 & 0.7835 & 0.4192 \\  & analytical & 0.5710 & 0.1001 & 0.7854 & 0.4226 \\  & \% difference & 0.06\% & 0.51\% & 0.24\% & 0.81\% \\  & conventional & 0.3440 & 0.2327 & 0.5179 & 0.3988 \\  & \% difference & 65.90\% & \(-\)57.18\% & 51.30\% & 5.09\% \\ \hline \multirow{4}{*}{
\begin{tabular}{c} Unpressurized \\ pipeline \\ \end{tabular} } & numerical & 0.3656 & 0.2200 & 0.5420 & 0.5738 \\  & analytical & 0.3655 & 0.2199 & 0.5415 & 0.5773 \\  & \% difference & \(-\)0.02\% & \(-\)0.06\% & \(-\)0.08\% & 0.61\% \\  & conventional & 0.3511 & 0.2069 & 0.5249 & 0.3917 \\  & \% difference & 4.12\% & 6.35\% & 3.24\% & 46.48\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The critical ground displacement \(\delta_{cr,i}\) corresponding to the achievement of the normal operability (NOL) and pressure integrity (PIL) limit states, for tension and compression, according to the numerical and analytical methods. Figures 7 and 8 show the maximum pipe strain magnitude as a function of the ground deformation demand (\(\delta\), \(L_{b}\)), for the pressurized and unpressurized pipelines, respectively, highlighting the critical PGD values (\(\delta_{\sigma,i}\), \(L_{\sigma,i}\)) corresponding to the achievement of the performance limit states. Figure 6: The variation of the critical soil block length, \(L_{\sigma}=(F_{t,max}-F_{c,max})/f_{s}\), as a function of the ground displacement \(\delta\), with an indication of the critical values (\(\delta_{\sigma,i}\), \(L_{\sigma,i}\)) associated with the achievement of the pipeline performance limit states. Figure 7: The peak axial strain magnitude in the pressurized pipeline (\(P_{i}/P_{max}=0.75\), \(\Delta T=50\)\({}^{\circ}\)C) as a function of the PGD length \(L_{b}\) and displacement \(\delta\) for (**a**) tension and (**b**) compression. The dashed horizontal curves represent the strain isolines corresponding to the NOL and PIL performance limit states. Clearly, the curve of the critical soil block length \(L_{cr}\) as a function of the ground displacement \(\delta\) (Figure 6) separates the PGD demand characterizing case I of short soil block (\(\delta>\delta_{cr}\lor L_{b}<L_{cr}\)) from that of case II of long soil block (\(\delta<\delta_{cr}\lor L_{b}>L_{cr}\)). Hence, the strain isolines are defined by two half-lines running parallel to the \(\delta\) and \(L_{b}\) axis, intersecting on the critical curve, as shown in Figures 7 and 8.