## 3 Results and Discussion
### Application of the Proposed Method
This result is understandable because as the PSW increases in width, the number of structural elements in the model also increases, causing the mass and spring constant to increase. Next, we removed the highly correlated variables (i.e., the mass and spring constants) to eliminate multicollinearity issues. Using multilinear regression analysis, we developed a damping coefficient prediction equation as follows:  
\[\xi=0.00025w+0.178 Tn-0.18 \tag{7}\]  
where \(w\) is the PSW's width (m) and \(Tn\) is the PSW's natural period (\(s\)). Equation (7) has a determination coefficient (\(R^{2}\)) equal to 94% and a minimal standard error of 0.002, which indicates that the damping coefficient can be accurately predicted by Equation (7). The probability, or P-value, of the independent variable was less than the standard significance level of 0.05, indicating that the variable was statistically significant. Our study shows that the width of the PSW significantly affects the damping coefficient. Increasing the width of the PSW results in the incorporation of more structural elements (i.e., piles and beams) in the model, which could result in a more significant contribution of material damping to the structural damping, thereby increasing it. To apply the proposed method to FA Scenario 2, we determined the damping coefficient using Equation (7). Then, using Figures 9 and 10, we obtained the SA based on the PSW's natural period from Table 2 and the damping coefficient from Equation (7). The SA obtained was multiplied by the PSW mass from Table 2 to produce the inertial force. The inertial forces in Scenario 2, based on the TE and ME waveforms, are shown in Table 6. The ME waveform generated a greater inertial force than the TE waveform due to its larger SA at the PSW's natural period. Figure 16: Bending moment distribution of the PSW considering the inertial and kinematic forces due to the TE waveform (Scenario 2). Table 4.0 _Cont._  
Figures 16c,d and 17c,d show the differences in the bending moments observed at piles located outside the slope area. This indicates that at a depth of 36-40 m, the JRA method's bending moment becomes excessive because of the triangular distribution of the kinematic force. By contrast, the bending moment calculated using the proposed method produced reasonable results compared to the FEA. For example, as shown in Figure 16c, at 40 m depth, the JRA method produced a bending moment of 1723 kN-m while the proposed method and the FEA produced bending moments of 166 kN-m and 115 kN-m, respectively. It should be noted that the analyses shown in Figures 16 and 17 were calculated considering both inertial and kinematic forces. Thus, although the JRA's kinematic forces (Figure 14) used to produce Figures 16 and 17 were identical, the outcome was not. This is because the inertial force was calculated using different ground surface waveforms (Table 6). The FA's bending moment distributions for Scenario 2 considered using only the inertial force (black line) and the FEA results (red line) due to the TE and ME waveforms, respectively, are compared in Figures 18 and 19. The FA bending moment produced by the inertial force is only about half of the FEA bending moment. This indicates that the incorporation of kinematic force in the analysis is crucial for obtaining an appropriate bending moment in the case of a PSW on a soil slope. Table 4.1 Bending moment distribution of the PSW considering the inertial and kinematic forces due to the ME waveform (Scenario 2). Table 4.2 Bending moment distribution of the PSW considering the inertial forces due to the TE waveform (Scenario 2). The response of the piles subjected only to inertial force was comparable to that produced by either the response of the beam on the Winkler foundation or the response of the cantilever considering the free length of the pile (Figures 18 and 19), where the primary factors are the flexural stiffness of the cantilever and pile's free length (i.e., the sum of the characteristic length and protruding length of the pile).