# Seismic Response Analysis of Buried Nuclear Power Plant Structures in Non-Bedrock Sites
## 1 Introduction
### Davidenkov-Chen-Zhao Liquefaction Constitutive Model  
This liquefiable constitutive model adopts the Davidenkov-Chen-Zhao (DCZ) model coupled with a pore pressure model [40; 41]. The DCZ liquefaction model is selected due to its convenient implementation, satisfactory performance, and excellent compatibility with GFE. Therefore, the DCZ liquefaction model is chosen as the soil constitutive model for liquefaction simulation.  
Firstly, an introduction is first provided to the effective stress principle and volume compatibility principle in the pore pressure increment model (pore pressure coupling component).  
Effective Stress Principle: Volumetric compression deformation in saturated soils requires equivalent water volume expulsion from pores. Undrained conditions are assumed under seismic loading due to insufficient drainage time. This principle is mathematically expressed as follows:  
\[\Delta u=-\Delta\sigma^{\prime} \tag{3}\]  
Volume Compatibility Principle: A reduction in net effective normal stress was proposed by [PERSON] et al. to induce soil volumetric rebound (unloading). Permanent volumetric compression caused by cyclic shear loading, pore water drainage, and effective stress reduction was demonstrated to satisfy compatibility conditions (see equation below).  
\[\varepsilon_{v,d}+\varepsilon_{v,r}=\varepsilon_{v,f} \tag{4}\]  
\[E_{r}=\frac{\Delta u}{\Delta\varepsilon_{v,d}} \tag{5}\]  
Consequently, the unloading modulus is derived from the relationship between the pore pressure increment and volumetric strain increment. Under cyclic loading, the shear-volumetric strain increment coupling of liquefiable soils is expressed by the following equation:  
\[\Delta\varepsilon_{vd}=A_{1}(\gamma-A_{2}\varepsilon_{vd})+\frac{A_{3} \varepsilon_{vd}^{2}}{\gamma+A_{4}\varepsilon_{vd}} \tag{6}\]  
Figure 7: Schematic diagram of liquefaction layer.  
[PERSON] simplified the formula proposed by [PERSON], expressed as follows:  
\[\frac{\Delta\varepsilon_{\text{pd}}}{\gamma}=C_{1}\cdot exp\left(-C_{2}\frac{ \varepsilon_{\text{pd}}}{\gamma}\right) \tag{7}\]  
The threshold shear strain is introduced and further modified to the following:  
\[\frac{\Delta\varepsilon_{\text{pd}}}{\gamma^{*}}=C_{1}\cdot exp\left(-C_{2} \frac{\varepsilon_{\text{pd}}}{\gamma^{*}}\right) \tag{8}\]  
Within the formula, \(\gamma^{*}=\left(\gamma-\gamma_{tr}\right)^{C_{3}},C_{1}\cdot C_{2}=0.15\).  
This study adopts [PERSON]'s formula [40; 41] incorporating threshold shear strain to define effective shear strain. The unloading modulus \(E_{r}\) expression was determined through experimental fitting.  
\[E_{r}=100\sigma_{c0}{{}^{\prime}}mn\cdot\exp(-r_{u}/m) \tag{9}\]  
where \(m\) and \(n\) are fitting parameters, \(\sigma_{c0}{{}^{\prime}}\) denotes the initial effective consolidation stress, and \(r_{u}\) represents the pore pressure ratio.  
Finally, with pore pressure adopted as the parameter for soil stiffness degradation, the transient shear modulus parameter considering the stiffness degradation induced by pore pressure accumulation can be expressed by the following:  
\[G_{\text{max}}^{t}=G_{\text{max}}\cdot\left(1-u_{e}/\sigma^{\prime}{{}_{v0}} \right)^{a3} \tag{10}\]  
where \(G_{max}^{t}\) is the tangent shear modulus at strain reversal, \(\sigma^{\prime}_{v0}\) represents the initial effective overburden stress, and \(a_{3}\) denotes a fitting parameter.  
Note:  
\[r_{u}=\frac{u_{e}}{\sigma^{\prime}_{v0}}\]  
\[\sigma^{\prime}_{v0}=100\sigma^{\prime}_{c0}\]