## 3 Ground Motion Propagation of Nonlinear Soil
### Fitting Parameters and Implementation of Davidenkov Model in ABAQUS
#### 3.1.2 Justification of the Davidenkov Model in Ground Motion Propagation  
Before starting the ground motion propagation analysis, a comparison analysis was conducted between the Davidenkov model and the commonly used equivalent linear method, highlighting the necessity of using a nonlinear dynamic constitutive model. Taking bedrock ground motion and a representative soil site as an example [45], the analyzed results of the two methods were obtained, of which the acceleration time history and the Fourier spectrum of input ground motion are shown in Figure 6. Figure 7 illustrates a comparison between acceleration and relative displacement time history responses at the ground surface. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & \(\mathbf{A}\) & \(\mathbf{B}\) & \(\mathbf{\gamma_{0}}\) \\ \hline Clay & 0.91 & 0.52 & 0.00087 \\ Muddy silty clay & 0.78 & 0.58 & 0.00082 \\ Muddy clay & 0.71 & 0.61 & 0.00096 \\ Sand & 1.08 & 0.47 & 0.00081 \\ Silty clay & 1.07 & 0.46 & 0.00071 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fitting parameters for different types of soil. Figure 6: Input bedrock ground motion: (**a**) acceleration time history; (**b**) Fourier spectrum. Figure 7: Comparison of (**a**) ground surface acceleration time history responses and (**b**) ground surface relative displacement responses. In addition, the differences of peak responses and relative differences for five typical ground motions selected from different earthquakes are further calculated and listed in Table 2. As shown in Table 2, the peak ground acceleration responses calculated by the Davidenkov model are less than those by the equivalent linear method, and the mean relative difference of the five records is 17.58%. The peak ground relative displacement responses calculated using the Davidenkov model are also less than those by the equivalent linear method, and the mean relative difference of the five records is 16.17%. The calculation results of the nonlinear dynamic constitutive model are generally smaller than those of the equivalent linear method, which is consistent with previous research [35; 50; 51] and is necessary to represent the actual behavior of the soil. Given the need for the Davidenkov model in this study, its accuracy was further justified via a comparison, in which a real bedrock sample (\"in-borehole,\" 100 m depth below ground surface) and its corresponding ground surface acceleration time history (east-west component of TKCH 080) recorded during the Tokatchi-oki earthquake (2003, Japan) [52] were selected for analysis, as shown in Figure 8. The bedrock acceleration time history was used as input to predict the ground surface acceleration time history, and the computed history is given in Figure 9. It can be seen that the computed time history can predict the recorded time history well. The wave shape of the computed time history is basically the same as that of the recorded time history. Specifically, the peak ground acceleration values of the computed and recorded time history are 0.46 \(g\) and 0.49 \(g\), respectively, and the relative difference is 6.12%. In addition, the normalized acceleration response spectrum (characterized by acceleration amplification factor \(\beta\)) of the computed time history was compared with the recorded data, shown in Figure 10. The results show satisfactory agreement with the data, especially the predominant period of the ground surface acceleration time history, as well as the corresponding resonant peaks, where both are well predicted. These results are consistent with previous research [52] and further justify that the use of the Davidenkov model to represent soil behavior is reasonable and accurate.