## 2 Improved Penzien Model
### Dynamic Response of the Free Field
#### 2.2.2 Dynamic Nonlinearity Model of Soil  
The site soil has strong nonlinear properties, and the dynamic nonlinearity of the site soil was simulated using an equivalent linear model first presented by [PERSON] [61; 62]. The soil in the model is a viscoelastic medium, and the equivalent shear modulus \(G\) and equivalent damping ratio \(D\) are used to express the basic characteristics of the relationship between dynamic stress and dynamic strain. \(G\) and \(D\) are expressed as functions of the dynamic strain. [PERSON] and [PERSON] [63] improved the soil model through experiments and first presented the empirical relationship in the prediction \(G_{d}/G_{max}\)\(\sim\)\(\gamma_{d}\), the Hardin-Drnevich model. The equations are shown in Equations (3)-(5).  
\[\frac{G_{\rm d}}{G_{\rm max}}=1\ -\ {\rm H}(\gamma_{\rm d}) \tag{3}\]  
where  
\[H(\gamma_{\rm d})=\frac{\gamma_{d}/\gamma_{\rm r}}{1+\gamma_{\rm d}/\gamma_{ \rm r}} \tag{4}\]  
\[\gamma_{r}=\tau_{max}/G_{max} \tag{5}\]  
\(G_{\rm d}\), \(G_{\rm max}\), \(\gamma_{\rm d}\), and \(\gamma_{\rm r}\) denote the dynamic shear modulus, maximum dynamic shear modulus, dynamic shear strain, and reference shear strain, respectively. \(G_{\rm max}\) is determined according to Equation (6). When \(\gamma_{\rm d}\) is large enough, the ultimate value of the soil shear strength as the asymptotic line is taken as \(\tau_{\rm max}\). In general, \(\tau_{\rm max}\) is the value when \(\gamma_{\rm d}\) is 0.1%.  
\[G_{\rm max}=\rho v_{s}^{2} \tag{6}\]  
where \(\rho\) and \(v_{s}\) are the density and shear wave velocity of the soil.  
[PERSON] [64] and [PERSON] and [PERSON] [65] improved the Hardin-Drnevich model and proposed a formula to better describe the relationship between the shear stress and the shear strain in soil. The soil model is called the Davidenkov foundation model. The Davidenkov model has been applied to the dynamic nonlinearity simulation of soil in the shaking table test considering the interaction between the soil and structure [59; 60; 66; 67], and the accuracy of the model to simulate the soil nonlinearity was proven. Equation (4) was improved and is shown in Equation (7).  
\[{\rm H}(\gamma_{\rm d})=\left[\frac{\left(\gamma_{\rm d}/\gamma_{1}\right)^{2 B}}{1+\left(\gamma_{\rm d}/\gamma_{1}\right)^{2B}}\right]^{\rm A} \tag{7}\]  
where \(A\), \(B\), and \(\gamma_{1}\) are the fitted parameters according to the soil characteristics. \(\gamma_{1}\) is no longer the reference shear strain in Equation (5) and denotes only a fitted parameter here. When \(A\) and \(B\) are equal to 1.0 and 0.5, respectively, the equations in the Davidenkov foundation model are the same as those in the Hardin-Drnevich model.  
The empirical damping ratio formula proposed by [PERSON] [68] is used to express the relationship between the damping ratio and shear modulus ratio, as shown in Equation (8).  
\[D/D_{\rm max}=\left(1-\frac{G_{d}}{G_{\rm max}}\right)^{\beta} \tag{8}\]where \(D_{\rm max}\) and \(D\) are the maximum damping ratio and damping ratio, respectively, and \(\beta\) denotes a shape factor of the curve \(D{\sim}\gamma_{\rm d}\). Further, Equation (8) is modified as follows.  
\[D=D_{\rm min}+D_{0}\bigg{(}1-\frac{G_{d}}{G_{\rm max}}\bigg{)}^{\beta} \tag{9}\]  
where \(D_{\rm min}\), \(D_{0}\), and \(\beta\) are fitted parameters according to the soil characteristics.