## 2 Theoretical Basis
### Nonlinear Hysteretic Model  
The typical description of nonlinear soil behavior are models that rely on hysteresis stress-strain relations. With NERA [5], the hysteresis curve is modeled using such parameters as maximum shear modulus \(G_{max}\) and so-called shear modulus degradation curves \(G/G_{max}(\gamma)\) (Figure 3). The IM-model [31; 32] implemented in the NERA algorithm assumes the Masing rule [33] and is based on simulating nonlinear hysteretic stress-strain curves using a series of mechanical elements which have different stiffness \(k_{i}\) and sliding resistances \(R_{i}\) (\(R_{1}<R_{2}<\ldots<R_{n}\)). Initially, the residual stresses in all sliders are equal to zero. During a monotonic loading, slider \(i\) yields when the shear stress \(\tau\) reaches \(R_{i}\)[5]. The results from downhole measurements and from empirical ground motion prediction equations [12; 18; 34; 35] lead to the conclusion that the nonlinear behaviors of vertical and horizontal ground motions differ. Therefore, the modulus reduction and damping curves for the horizontal direction should be modified before applying them to the vertical direction [12], i.e., they should be based on constrained moduli for the cases of unsaturated soil and saturated soil. The right side of the governing equations of motion (14)-(16) can be written in terms of the tangential modulus:  
\[\frac{\partial\tau}{\partial z}=\frac{\partial\tau}{\partial\gamma}\ \frac{\partial\gamma}{\partial z}=H_{t}\frac{\partial\gamma}{\partial z} \tag{27}\]  
where the tangential modulus \(H_{t}\) is [5]:  
\[ \begin{gathered} H_{t_{1}}=k_{1},\ if\ 0\leq\ \tau<R_{1}\\ H_{t_{2}}=\left(k_{1}^{-1}+k_{2}^{-1}\right)^{-1},\ if\ R_{1}\leq\ \tau<R_{2}\\ \ldots\\ H_{t_{n-1}}=\left(k_{1}^{-1}+k_{2}^{-1}+\ldots+k_{n-1}^{-1}\right)^{-1 },\ if\ R_{n-2}\leq\ \tau<R_{n-1}\\ H_{t_{\text{r}}}=\left(k_{1}^{-1}+k_{2}^{-1}+\ldots+k_{n-1}^{-1}+k_{n}^ {-1}\right)^{-1},\ if\ R_{n-1}\leq\ \tau<R_{n}\\ 0,\ if\ \tau=R_{n}\end{gathered}  \tag{28}\]  
Figure 3: Hysteresis curve (**a**), schematic representation of stress-strain model introduced by [31; 32] (**b**), degradation curves of shear modulus \(G/G_{max}\) (\(\gamma\)), constrained moduli above water table \(M/M_{max}\) (\(\gamma\)) and under water table \(M^{*}/M^{*}_{max}\) (\(\gamma\)) and corresponding damping ratio curves \(D_{G}(\gamma)\), \(D_{M}(\gamma)\), \(D_{M^{*}}(\gamma)\) (**c**). The tangential modulus \(H_{t_{i}}\) is related to the secant modulus \(G_{i}\) as follows:  
\[H_{t_{i}}=\frac{G_{i+1}\gamma_{i+1}-G_{i}\gamma_{i}}{\gamma_{i+1}-\gamma_{i}}=G_ {max}\frac{G^{\prime}_{i+1}\gamma_{i+1}-G^{\prime}_{i}\gamma_{i}}{\gamma_{i+1}- \gamma_{i}},\;i=2,\;\ldots,\;n-1\;and\;H_{t_{n}}=0. \tag{29}\]  
The sliding resistances \(R_{i}\) can be expressed as follows:  
\[R_{i}=G_{i}\gamma_{i}=G_{max}G^{\prime}_{i}\gamma_{i},\;i=1,\ldots,\;n \tag{30}\]  
\[G^{\prime}_{i}=\frac{G_{i}}{G_{max}}\]  
Taking into consideration the Equations (5) and (6), the constrained moduli degradation curves for unsaturated soil and for saturated soil, respectively, take the following forms:  
\[M^{\prime}_{i}=\frac{M_{i}}{M_{max}}=\frac{2G_{i}\frac{1-\theta}{1-2\theta}}{ 2G_{max}\frac{1-\theta}{1-2\theta}}=G^{\prime}_{i} \tag{31}\]  
\[M^{*\prime}_{i}=\frac{M^{*}_{i}}{M^{*}_{max}}=\frac{M_{i}+\frac{Kf}{\phi}}{M_ {max}+\frac{Kf}{\phi}} \tag{32}\]  
The damping ratio at horizontal strain \(\gamma_{i}\) can be expressed [5]:  
\[D_{G_{i}}=\frac{2}{\pi}\left(\frac{2A_{i}}{G_{i}\gamma_{i}^{2}}-1\right),\;i= 2,\ldots,\;n \tag{33}\]  
\[D_{G_{1}}=0\]  
\[A_{i}=\frac{1}{2}\sum_{j=2}^{i}\bigl{(}G_{j}\gamma_{j}+G_{j-1}\gamma_{j-1} \bigr{)}\bigl{(}\gamma_{j}-\gamma_{j-1}\bigr{)},\;i=2,\ldots,n\]  
\[A_{1}=0\]  
where \(D_{G}\) is the damping ratio related to horizontal movements.