## 3 Verification of Proposed Model
### Comparison of Dynamic Responses  
The applicability of the proposed modeling method was verified by comparing various dynamic responses under earthquake loading using two experimental cases. Figure 9 shows time histories of the excess pore water pressure according to the relative density of upper soil layer and depth. In this figure, the excess pore pressure ratio is calculated by dividing the excess pore water pressure by the initial vertical effective stress in soil. The peak value of the excess pore pressure ratio in Case 1 (D\({}_{\text{r}}\) = 55%) reached 1.0 (means liquefaction) at 4.5 m and decreased to 0.5 at 7 m because the soil at such depths had higher confining pressure. However, the peak ratio in Case 2 (D\({}_{\text{r}}\) = 35%) was 1.0 in the entire region of the upper layer. It appears that liquefaction also occurs at greater depth in Case 2, unlike that in Case 1, because the relative density of the upper soil is less than that of Case 1; thus, the confining pressure is relatively lower. The time histories of the ratios computed from the proposed model agreed well with the experimental results in both cases. Thus, it is demonstrated that the proposed model can properly simulate the excess pore pressure response of the liquefiable sand with various relative densities.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **D\({}_{\text{r}}\) = 35\%** & **D\({}_{\text{r}}\) = 55\%** & **D\({}_{\text{r}}\) = 80\%** \\ \hline Porosity & 0.438 & 0.409 & 0.377 \\ Saturated unit weight (kN/m\({}^{3}\)) & 19.38 & 19.87 & 20.41 \\ Friction angle (\({}^{\circ}\)) & 32 & 34 & 36 \\ Poisson’s ratio & 0.32 & 0.31 & 0.30 \\ Volume change constants & \(C_{1}\) = 1.05, \(C_{2}\) = 0.38 & \(C_{1}\) = 0.34, \(C_{2}\) = 1.17 & \(C_{1}\) = 0.13, \(C_{2}\) = 3.07 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Input properties of the model soil used in numerical analysis.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Young’s Modulus (MPa)** & **70** \\ \hline Outside diameter (m) & 0.667 \\ Wall thickness (m) & 0.0724 \\ Moment of inertia (m\({}^{4}\)) & 0.0061 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Input properties of the model pile used in numerical analysis.  
Figure 9: Comparison of excess pore pressure ratio in the free-field zone: (**a**) Case 1 (D\({}_{\text{r}}\) = 55%) and (**b**) Case 2 (D\({}_{\text{r}}\) = 35%).  
Figure 10 shows time histories of the pile bending moment according to the depth and the soil relative density. It shows that the amplitude of the bending moment decreased after 3 s when the excess pore pressure reached the peak values. In addition, the peak values of Case 2 (D\({}_{\rm r}\) = 35%) were smaller than those of Case 1 (D\({}_{\rm r}\) = 55%). The proposed model can accurately predict the peak amplitude and the decreasing trend of the bending moment during liquefaction. Figure 11 shows time history of the horizontal displacement at the pile head. As shown in the figure, the displacement calculated from the proposed model accurately predicted the displacement amplitude and the decrease of the amplitude during liquefaction. Despite the slight discrepancy in phase, the calculated maximum moment and displacement reasonably agreed with the measured values of the experiments with discrepancy less than 10%. Finally, it can be concluded that the proposed model can simulate the dynamic soil-pile-structure interaction in liquefiable sand under various conditions. In the verification results, on the other hand, the maximum bending moment at 1 m depth near the ground surface and the maximum pile displacement occurs at approximately 3.5 s. This almost coincides with the time when the excess pore pressure ratio sharply increases in Figure 9. This appears to be because of the influence of kinematic force caused by softening and deformation of the soil around the pile as well as the inertial force generated by the pile head.  
Figure 10: Comparison of the bending moment in a pile: (**a**) Case 1 (D\({}_{\rm r}\) = 55%) and (**b**) Case 2 (D\({}_{\rm r}\) = 35%).