# Investigation of Earthquake-Induced Pipe Damage in Liquefiable Soils
## 3 PM4 Sand Model and Verification Analyses
### Numerical Model and Analyses
The Plaxis 2D program determines Rayleigh damping parameters using the C damping, M mass, and K stiffness matrices through Equation (3). \[[C]=\alpha[M]+\beta[K] \tag{3}\]  
The Rayleigh damping parameters were determined in Plaxis using two different variables: Target 1 (\(f_{1}\)) and Target 2 (\(f_{2}\)). Here, (\(f_{1}\)) represents the dominant frequency of the entire soil profile and was calculated using Equation (4). In this equation, \(V_{S}\), the average is the average shear wave velocity of the soil layer, and H is the layer thickness. \(f_{2}\) is determined by rounding the value obtained from Equation (5), which depends on the parameters \(f_{eq}\) and \(f_{1}\), to the nearest integer [55]. \[f1=V_{S,\text{ average}}/4H \tag{4}\]  
\[f2=feq/f1 \tag{5}\]  
In the PM4 Sand soil model, the target damping ratio for sand soil was set to 2%, and analyses were conducted using the values \(\alpha_{R}=0.1761\) and \(\beta_{R}=0.001906\) corresponding to Target 1 and Target 2, respectively. Although PM4 Sand was a successful constitutive model for modeling liquefaction, it was inadequate in obtaining initial stress conditions. Therefore, initial stress conditions were determined using the Hardening Small Strain (HSS) constitutive model. In measurements of pore water pressure increase around the pipe, surges originating from structure of the pipe occurred, and it was observed that the liquefaction ratio was calculated more accurately as one moves away from the pipe. The excess pore water pressure ratio (Ru) value reached \(Ru=1\), indicating the occurrence of liquefaction behavior on and around the pipe. It was determined that the calculated pore water pressures at the edge and base of the pipe were consistent with the centrifuge model experiment. Due to the fluctuations around the pipe, the excess pore water pressure ratios calculated in the far field were consistent with those measured in the experiment [22] (Figure 5a-d). Near the ground surface regions, the effect of pore water pressure increments, as shown in Figure 5a, is calculated to be slightly lower (\(Ru=1\)) compared to the centrifuge model. This is because pore water pressures during earthquakes are more easily dissipated in near-surface areas. However, the lower values of pore water pressure increments calculated by the PM4 Sand model are due to the damping effect of the soil. Nevertheless, the actual behavior occurring in the field can be determined by means of on-site measurements during field experiments. In numerical analyses conducted for verification, material parameters were calibrated to model the behavior of liquefiable sand under seismic effects with PM4 Sand, and it was determined that the displacements calculated with 0.9 D\({}_{R}\) were consistent with measurements (Figure 6). The centrifuge test was modeled by [PERSON] [22] using the FLAC 2D finite differences software. In these analyses, the [PERSON] elastic model [56], capable of modeling liquefaction, was used, and a pipe uplift of 25% less than that determined in the test was computed. In Sudevan [23], the centrifuge model was simulated using the FLAC3D Finn-Byrne formulation, and a structure uplift exceeding 1.0 m was calculated. The pipe uplift results calculated in those analyses are collectively presented in Figure 6.  
In this article, it was determined that the uplift value calculated from verification analyses is consistent with [PERSON]'s [22] experimental measurements. Liquefaction behavior could be realistically modeled using the PM4 Sand model in Plaxis 2D. Figure 5: (**a**) Comparison of the Ru value formed at a depth of 1.7 m in the far field with the experimental results; (**b**) comparison of the Ru value formed at a depth of 5.5 m in the far area next to the pipe with the test; (**c**) comparison of Ru value with test results at a depth of 7.5 m in the far field; (**d**) comparison of Ru value with test results at a depth of 16 m in the model base [22].