# Analysis of Multilayered Ground Amplification Characteristics by Scaled-Down Model Tests
## 3 Multilayered Flat Model Ground Test Results and Analysis
### One-Dimensional Ground Response Numerical Simulation Model  
In order to calibrate and validate the overall reliability of the shaking table test results utilized in this study, a one-dimensional ground response analysis was performed for the flat ground model created in the laminar shear box. The ground response analysis was performed using the open-source software DEEPSOL, version 7 of the University of Illinois at Urbana-Champaign (UIUC) and analyzed in a nonlinear way [16]. The compositional models were the general quadratic/hyperbolic (GQ/H) ground model and the non-masing hysteresis model.  
Figure 11: Spectral response acceleration for measured ground motion at different depths: (**a**) SA by sine 10 Hz; (**b**) AF of dense ground by sine 10 Hz; (**c**) AF of loose ground by sine 10 Hz; (**d**) AF of boundary ground by sine 10 Hz; (**e**) SA by artificial wave; (**f**) AF of dense ground by artificial wave; (**g**) AF of loose ground by artificial wave; (**h**) AF of boundary ground by artificial wave; (**i**) SA by sine sweep; (**j**) AF of dense ground by sine sweep; (**k**) AF of loose ground by sine sweep; (**l**) AF of boundary ground by sine sweep.  
The parameters contained in the model by DEEPSOIL software are shown in Table 4 (because there are twelve layers in total, the first layer is the main display data and the input data of each layer are also displayed below). The Darendeli constitutive model is used for modeling and nonlinear analysis, and the acceleration-time history and 5% damped spectral acceleration according to ASCE standard were obtained [17], which can be used to explore the dynamic characteristics of soil.  
The model was characterized by a total cross-sectional depth of 60 cm, according to the actual size of the experimental model, and 12 layers with a thickness of every 5 cm from the surface to the bottom were defined. Six layers from the surface to a depth of 30 cm were modeled with loose ground with a unit weight of 14.715 \(\mathrm{kN/m^{3}}\), while six layers from 30 cm to 60 cm were modeled with dense ground with a unit weight of 19.62 \(\mathrm{kN/m^{3}}\).  
The shear strength was calculated by depth according to the Mohr-Coulomb failure line [18] according to Equation (3).  
\[\tau_{f}=c+\sigma\tan\varnothing \tag{3}\]  
where, \(\tau_{f}\) is the shear strength, \(c\) is the cohesion, \(\sigma\) is the normal stress, and \(\varnothing\) is the internal friction angle.  
The specific modeling interface and parameters of each layer are shown in Figure 12.  
In 1972, [PERSON] and [PERSON] [19] proposed a hyperbolic shear modulus reduction function and an approximate shape for the material damping curve for soil. Based on Hardin-Drnevich model, [PERSON] [20] proposed a modified hyperbolic model as Equation (4). The curvature coefficient a is integrated into the normalized modulus reduction curve.  
\[\frac{G}{G_{max}}=\frac{1}{\left(1+\frac{\lambda_{T}}{\tau_{f}}\right)^{a}}, \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline
**Thickness (m)** & 0.05 & **Unit weight (\(\mathrm{kN/m^{3}}\))** & 14.715 (loose soil) \\
**Shear wave velocity (m/s)** & 46.01 & **Effective vertical stress (\(\mathrm{kPa}\))** & 0.367875 \\
**Shear strength (\(\mathrm{kPa}\))** & 0.212393 & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameter of the DEEPSOIL used in this study.  
Figure 12: DEEPSOIL software: (**a**) Modeling in DEEPSOIL; (**b**) parameters of each layer.  
where, \(G_{max}\) is the smallstrain shear modulus, \(\gamma_{r}\) is the reference strain, \(\gamma\) is the shearing strain and \(\alpha\) is the regression parameter.  
Figure 13 shows normalized shear modulus reduction and material damping ratio curves of loose soil and dense soil by Darendeli constitutive model in DEEPSOIL.