# Analysis of the Dynamic Behavior of Multi-Layered Soil Grounds
## 2 Materials and Methods
### Numerical Analysis
#### 2.3.1 DeEPSOIL Program  
DEEPSOIL is a one-dimensional (1D) seismic site response analysis program; it focuses on simulating the effects of seismic events on soil in the vertical direction. This one-dimensional approach allows for a detailed analysis of the interaction of soil layers along the depth of the soil profile [23].  
In this study, four models were established in DEEPSOIL, and their cross-sectional diagrams are shown in Figure 5. The position of the derived results in DEEPSOIL (white circle) was completely consistent with that of the experimental design (Figure 4).  
Figure 4: Layout of soil layers in the 1 g shaking-table test: (**a**) Case 1: silica sand; (**b**) Case 2: weathered soil A; (**c**) Case 3: silica sand and weathered soil A; (**d**) Case 4: silica sand, weathered soil A and B.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Case** & **Soil Type** & **Depth** & **Description** \\ \hline
1 & Silica sand & 0.6 m & Loose \\
2 & Weathered soil A & 0.6 m & Very dense \\  & Silica sand & 0.3 m & Loose \\
3 & Weathered soil A & 0.3 m & Very dense \\  & Silica sand & 0.2 m & Loose \\  & Weathered soil B & 0.2 m & Dense \\  & Weathered soil A & 0.2 m & Very dense \\ \hline \end{tabular}
\end{table}
Table 3: Descriptions of the soil layers for each case.  
The constitutive model employed was the Darendeli mode IIn DEEPSOIL. The Darendeli model characteristically correlates soil properties, such as shear wave velocity, with fundamental parameters like density, effective confining pressure, and plasticity index [24]. It is used in seismic site response analysis to estimate soil stiffness, which is crucial for understanding how structures respond to ground motions during earthquakes. The empirical formula for the Darendeli model is  
Figure 5: Modeling in DEEPSOIL: (**a**) Case 1; (**b**) Case 2; (**c**) Case 3; (**d**) Case 4.  
\[\frac{\mathrm{G}}{\mathrm{G_{max}}}=\frac{1}{1+\left(\frac{\gamma}{\gamma_{\mathrm{r} }}\right)^{\mathrm{a}}}, \tag{1}\]  
where \(\mathrm{G}=\mathrm{shear}\) modulus; \(\mathrm{G_{max}}=\mathrm{small}\)-strain shear modulus; \(\gamma=\mathrm{shear}\) strain; \(\gamma_{\mathrm{r}}=\mathrm{reference}\) strain; \(a=\mathrm{curvature}\) coefficient.  
In the DEEPSOIL modeling, the basic parameters required are thickness, unit weight, shear wave velocity, effective vertical stress, and shear strength. Among these, effective vertical stress is related to the unit weight of the soil and the depth of the soil layer.  
According to the empirical formula proposed by [PERSON] and [PERSON] (2015) and illustrated in the following expression, shear wave velocity is related to soil depth and soil density. \(\mathrm{K_{2}}\) can be considered to be determined mainly by the void ratio or relative density and the strain amplitude of the motion.  
\[\mathrm{V_{s}}=\sqrt{\frac{1000\mathrm{K_{2}}(\sigma_{m}^{\prime})^{0.5}}{ \rho}}, \tag{2}\]  
where \(\mathrm{V_{s}}=\mathrm{shear}\) wave velocity (m/s), \(\sigma_{\mathrm{m}}^{\prime}=\mathrm{effective}\) vertical stress, and \(\gamma=\mathrm{soil}\) density (kg/m\({}^{3}\)).  
Based on the DEEPSOIL User Manual, shear strength is related to the shear wave velocity and soil density. Shear strength can be expressed as  
\[\mathrm{C_{v_{s}}}=\rho\mathrm{V_{s}^{2}}\text{-}0.8\text{-}0.1\%, \tag{3}\]  
where \(\mathrm{V_{s}}=\mathrm{shear}\) wave velocity (m/s), \(\rho=\mathrm{soil}\) density (kg/m\({}^{3}\)).