# Study on the Seismic Response of Shield Tunnel Structures with the Preload Loss of Bolts
## 2 Numerical Simulation
### Seismic Wave and Boundary Conditions  
Seismic energy should be dissipated into the far-field foundation under seismic loading. Therefore, it is necessary that outgoing scattering waves do not reflect at the finite domain boundaries of the model. Artificial viscoelastic boundaries can absorb the energy radiating outward from the computation domain. Hence, artificial viscoelastic boundaries not only can effectively simulate the radiation-damping effects of semi-infinite space media,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Preload** & **Preload Loss Rate** & **Temperatures** & **Carrying Capacity** \\
**(kN)** & **(\%)** & **(\({}^{\circ}\)C)** & **Redundancy (kN)** \\ \hline
280 & 0 & \(-\)150.86 & 172.39 \\
240 & 14.29 & \(-\)129.31 & 212.39 \\
200 & 28.57 & \(-\)107.76 & 252.39 \\
160 & 42.86 & \(-\)86.21 & 292.39 \\
120 & 57.14 & \(-\)64.66 & 332.39 \\
0 & 100 & 0 & 372.39 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The preload value corresponds to the temperature applied.  
Figure 2: Load application stage.  
but also provide a good representation of the elastic-restoring behaviour of semi-infinite foundations [31]. Simulating infinite boundaries is achieved by applying three-directional spring dampers to finite element mesh nodes, as illustrated in Figure 3.  
The formulas for calculating the mechanical parameters of springs and dampers are as follows:  
\[\begin{array}{l}K_{BT}=\alpha_{BT}\frac{G}{R},C_{BT}=\rho C_{S}\\ K_{BN}=\alpha_{BN}\frac{G}{R},C_{BN}=\rho C_{P}\end{array} \tag{1}\]  
where \(K_{BT}\), \(K_{BN}\) are the stiffness coefficients of the tangential spring and normal spring, respectively; \(C_{BT}\), \(C_{BN}\) are the damping coefficients of the tangential damper and normal damper, respectively; \(\alpha_{BN}\) is the correction factor for the normal spring stiffness, and its value is 1.33; \(\alpha_{BT}\) is the correction factor for the tangential spring stiffness, and its value is 0.67; \(G\) is the shear modulus of the medium, \(\rho\) is the density of the medium; \(R\) is the distance from the wave source to the artificial boundary node; \(C_{S}\), \(C_{P}\) represent the shear wave velocity and longitudinal wave velocity of the medium, respectively.  
The spring dampers cannot constrain boundary displacements. Therefore, seismic loads need to be applied in a concentrated manner on nodes. The node influence area is determined by applying a unit load to the model, as shown in the following:  
\[F_{B}=\left(K_{B}u_{B}^{ff}+C_{B}\dot{u}_{B}^{ff}+\sigma_{B}^{ff}n\right)A_{B} \tag{2}\]  
where \(K_{B}\) is the spring stiffness; \(C_{B}\) is the damping coefficient; \(\sigma_{B}^{ff}\) is the stress tensor in the free field; \(u_{B}^{ff}=[u\ v\ w]\) is the displacement vector in the free field; \(u_{B}^{ff}=[\dot{u}\ \dot{v}\ \dot{w}]\) is the velocity vector in the free field; \(n\) is the cosine vector in the outward normal direction to the boundary; \(A_{B}\) is the cosine vector in the outward normal direction to the boundary [32; 33; 34; 35].  
Kobe waves with peak ground accelerations (PGA) of 0.1 g, 0.3 g, and 0.5 g were selected for the study based on the current national standard GB 50909-2014 (Seismic Design Code for Urban Rail Transit Structures) [36]. The acceleration time history and Fourier spectrum curve for the 0.1 g peak ground acceleration Kobe wave are shown in Figure 4. The time histories for 0.3 g and 0.5 g accelerations can be obtained via scaling. The seismic waves have a duration of 20 s. The selected Kobe wave has a peak ground acceleration that occurs at 3.46 s into the seismic wave. It should be noted that seismic waves are input at the bottom of the model in the form of shear waves.  
Figure 3: Three-dimensional artificial viscoelastic boundary.