# Investigation of Earthquake-Induced Pipe Damage in Liquefiable Soils
## 3 PM4 Sand Model and Verification Analyses
### Numerical Model and Analyses  
The numerical analysis of the centrifuge experiment conducted by [PERSON] [22] to investigate the uplift behavior of buried pipes in liquefiable soils was performed using the Plaxis PM45 and material model. In the centrifuge experiment, a pipe with a diameter of 5 m was subjected to an acceleration of 66.7 g, corresponding to a burial depth of 1.1D = 5.5 m. Verification analyses for a single pipe at a depth of 1.1D were conducted considering the boundary conditions and material parameters in the centrifuge experiment. The properties of the pipe used in the model included a specific gravity, Gs = 0.3, and an equivalent wall thickness of 0.35 m.  
The finite element mesh of the numerical model and the earthquake input motion are shown in Figure 4. The finite element mesh consists of 3084 triangular elements with 15 nodes each. In the created model, the average element size is 0.71 m, and the earthquake data were applied to the analysis for 30 s at equal intervals of 0.005 s. Since there was no rock at the base of the experimental model, fixed base conditions (none) were selected, and tied degrees of freedom were used as boundary conditions at the lateral boundaries. During the application of earthquake data, drift correction was selected. For verification, numerical analyses were conducted considering parameters determined from Equations (1) and (2) for 0.9 D\({}_{\rm R}\), 0.8 D\({}_{\rm R}\), and 0.85 D\({}_{\rm R}\) for sand at 45% relative density. The PM45 and constitutive model parameters for the case of 0.9 D\({}_{\rm R}\) and the parameters determined for the pipe are presented in Tables 1 and 2, respectively. Figure 3: Correlations for cyclic resistance ratio (CRR) from SPT data (after [PERSON] and Boulanger [51; 52; 53; 54]). In the modeling process, the initial step was created using the K0 procedure. In the second step, an HSS (Hardening Small Strain) soil model was assigned using parameters appropriate for sandy soil characteristics to create a new step. The main purpose of this was to accurately determine the initial stress state in the created soil profile. While the PM4 Sand constitutive equations can successfully model the behavior of cohesionless soils under dynamic loads, they are inadequate under static conditions [55]. In the third step, dynamic analyses were performed using the PM4 Sand model. At this stage, the drainage type \"Undrained A\" was selected to observe the formation of pore water pressure. \begin{table}
\begin{tabular}{c c c c c c} \hline  & **PM4 Sand** & & & **HSS Model** & \\ \hline D\({}_{\text{R}}\) & \%30 & \%45 & D\({}_{\text{R}}\) & \%30 & 45\% & Conglomerate \\
0.9 D\({}_{\text{R}}\) & 0.27 & 0.405 & \(\gamma_{\text{unsat}}\) (kN/m\({}^{3}\)) & 14.5 & 14.5 & 14.5 \\ G\({}_{0}\) & 391.6 & 529.3 & \(\gamma_{\text{sat}}\) (kN/m\({}^{3}\)) & 18.6 & 18.6 & 18.6 \\ h\({}_{\text{po}}\) & 0.2495 & 0.32 & c (kPa) & 0.1 & 0.1 & 0.1 \\ e\({}_{\text{max}}\) & 1.01 & 1.01 & \(\varphi\) (\({}^{\text{o}}\)) & 30 & 30 & 35 \\ e\({}_{\text{min}}\) & 0.55 & 0.55 & E\({}_{\text{50}}\)\({}^{\text{ref}}\) (kPa) & 3000 & 9000 & 31,000 \\ p\({}_{\text{A}}\) & 101.3 & 101.3 & E\({}_{\text{ood}}\)\({}^{\text{ref}}\) (kPa) & 3000 & 9000 & 31,000 \\ n\({}^{\text{b}}\) & 0.5 & 0.5 & E\({}_{\text{ur}}\)\({}^{\text{ref}}\) (kPa) & 9000 & 27,000 & 93,000 \\ n\({}^{\text{d}}\) & 0.1 & 0.1 & & 0.5 & 0.5 & 0.5 \\ \(\varphi\)\({}_{\text{-cv}}\) & 33 & 33 & P\({}_{\text{ref}}\) (kPa) & 100 & 100 & 100 \\ \(\
u\) & 0.3 & 0.3 & G\({}_{\text{0}}\)\({}^{\text{ref}}\) & 60,000 & 60,000 & 179,000 \\ Q & 10 & 10 & \(\gamma_{0.7}\) & 0.0007 & 0.0007 & 0.0001 \\ R & 1.5 & 1.5 & R\({}_{\text{f}}\) & 0.9 & 0.9 & 0.9 \\ \hline \end{tabular}
\end{table} 
Table 1: _PM4 SAND and HSS Model_ Material parameters used in numerical analysis. Figure 4: (a) Plaxis analysis model; (b) earthquake record used in the analysis. Determining the damping parameters of the soil layer is essential in dynamic analyses conducted with finite elements.