## 2 Seismic Calculation Method and Important Calculation Parameters of Subway Underground Structures
### Important Calculation Parameters  
(1) Foundation Spring Stiffness  
As an important parameter of the FBM and the DBM, the foundation reaction spring stiffness parameter plays an important role in FBM and the DBM. As the key parameter of the DBM and the DESANM, the horizontal relative displacement parameter of the stratum under seismic action has a significant influence on the seismic calculation results. For the value of foundation reaction spring stiffness parameter, the relevant norms make no clear provisions, and the relevant research has reached no unified conclusion.  
The finite element calculation model has been established, as shown in Figure 4. In Figure 4, the width of the soil model is six to seven times that of the underground structure, and the depth of the soil model extends to the bedrock surface. The influences of soil properties and structural shape characteristics on foundation spring stiffness in loess area were evaluated, as shown in Figure 5. The proposed equation of foundation spring stiffness was fitted using MATLAB, as shown in Equations (2)-(7). The proposed equation can be used for the seismic calculation of underground station structures with various rectangular section sizes in loess sites.  
\[\begin{array}{l}\mathrm{K1}=-0.1041E\Big{(}\frac{1}{h^{2}}-3.6\frac{1}{h}-0.05\Big{)}\big{(}d^{3}-17.682d^{2}+144.992d+1459.612\big{)}\\ \Big{(}\frac{1}{h^{3}}-0.15\frac{1}{h^{2}}-0.003\frac{1}{h}+0.003\Big{)}\frac{ 1}{4.417-v}-9.125\times 10^{4}\end{array} \tag{2}\]  
\[K2=2.696\times 10^{-6}E\bigg{(}\frac{1}{h^{2}}-9.16\frac{1}{h}-0.18\bigg{)} \Big{(}d^{3}-6.912d^{2}-450.422d-9408.926\Big{)}(0.83+v)+1540 \tag{3}\]  
\[K_{3}=-13.92E\frac{1}{H^{1.1}}\bigg{(}\frac{1}{b^{2}}-1.794\frac{1}{b}-0.087 \bigg{)}(\frac{1}{1.128-v})+8.076\times 10^{6} \tag{4}\]  
\[K_{4}=-28.15E\frac{1}{H^{0.3}}\bigg{(}\frac{1}{b^{2}}-0.4\frac{1}{b}+0.009 \bigg{)}(\frac{1}{5.8-v})+1.122\times 10^{7} \tag{5}\]  
\[K_{5}=0.002413E(d^{3}-7.373d^{2}-223.914d+136.272)\]  
\[\begin{array}{l}\left[\left(\frac{1}{b}\right)^{3}-0.329(\frac{1}{b}\right) ^{2}+0.007(\frac{1}{b})-4.9\times 10^{-5}\right]\big{(}\frac{1}{0.7-v}+6.579 \big{)}+805.6\end{array} \tag{6}\]  
\[\begin{array}{l}K_{6}=0.0006815E(d^{3}-18.856d^{2}+149.856d+314.46)\\ \left[\left(\frac{1}{b}\right)^{3}-0.299(\frac{1}{b}\right)^{2}+0.049(\frac{1 }{b})-0.00026\Big{]}(\frac{1}{0.7-v}+17.158)-8769\end{array} \tag{7}\]  
When \(B\leq 30\) m, \(H\leq 30\) m, and \(3\) m \(\leq D\leq 3.5\) m, the following relationship holds:  
\[K_{4}=(0.5\sim 0.6)K_{3} \tag{8}\]  
\[K_{2}=(0.85\sim 0.9)K_{1} \tag{9}\]  
where \(K_{1}\) and \(K_{2}\) represent the normal and tangential stiffness of the side, respectively (Pa); \(K_{3}\) and \(K_{4}\) are the normal and tangential stiffness of the bottom surface, respectively (Pa); \(K_{5}\) and \(K_{6}\) denote the normal and tangential stiffness of the top surface (Pa); \(b\) and \(h\) are the width and height of the structural section, respectively (m); \(d\) refers to the burial depth of the roof (m); \(H\) is the distance between the station floor and bedrock (m); and \(E\) and \(v\) represent the elastic modulus (Pa) and Poisson's ratio, respectively.  
Figure 4: Finite element calculation model of reaction spring parameters of underground structure foundation. (**a**) Meshed model, (**b**) normal spring stiffness calculation model, (**c**) tangential spring stiffness calculation model.  
Figure 5: Cont.  
Figure 5: _Cont_.  
(2) Horizontal Relative Displacement of Strata in a Loess Area  
A numerical model of free field dynamic response considering seismic intensity, ground motion characteristics and loess soil characteristics was established, as shown in Figure 6. Where 1 is an infinite element boundary, 2 denotes a fixed boundary, and 3 is bedrock. The maximum response horizontal displacement of free field was obtained by inputting the Xi'an artificial wave, Taft wave, and Songpan wave in turn (Table 1 and Figure 7).  
Figure 6: A numerical model of free field vibration.