# Study on the Seismic Response of Shield Tunnel Structures with the Preload Loss of Bolts
## 2 Numerical Simulation
### Method of Applying Preload  
There are several methods to apply preload to the steel bars, including the multi-points constraints method, initial stress method, rebar layer method, assembly load method, and cooling method. The cooling method leverages the thermal expansion and contraction  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multicolumn{2}{c}{**Young’s**} & \multirow{2}{*}{**Poisson’s Ratio**} & \multirow{2}{*}{**Mass Density**} & \multirow{2}{*}{**Damping Ratio**} \\  & **Modulus (MPa)** & & & & \\ \hline Silty clay & 126 & 0.35 & 1.85 & 0.09 \\ Concrete & 34,500 & 0.20 & 2.42 & 0.02 \\ Bolt & 210,000 & 0.3 & 7.85 & 0.05 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material parameters.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{3}{c}{**Uniaxial Compression**} & \multicolumn{3}{c}{**Uniaxial Tension**} \\ \hline \multirow{2}{*}{**Stress (MPa)**} & **Inelastic Strain** & **Damage Factor** & \multirow{2}{*}{**Stress (MPa)**} & **Cracking Strain** & **Damage Factor** \\ \hline
26.892 & 0 & 0 & 3.423939 & 0 & 0 \\
38.455 & 6.6 & 0.208666 & 3.257697 & 0.3 & 0.125466 \\
34.997 & 10.9 & 0.309287 & 2.701560 & 1.1 & 0.349736 \\
28.692 & 16.2 & 0.420016 & 2.169453 & 1.8 & 0.495345 \\
23.006 & 21.2 & 0.513588 & 1.561394 & 3.2 & 0.650423 \\
18.640 & 25.8 & 0.586791 & 1.045034 & 5.8 & 0.778459 \\
15.401 & 30.2 & 0.643377 & 0.675897 & 10.7 & 0.867185 \\
12.985 & 34.3 & 0.687565 & 0.430703 & 20.5 & 0.922839 \\
9.725 & 42.0 & 0.751014 & 0.272294 & 40.0 & 0.955944 \\
6.315 & 56.6 & 0.824267 & 0.171334 & 78.7 & 0.975075 \\
3.624 & 84.5 & 0.890016 & 0.107651 & 156.0 & 0.985949 \\
1.784 & 148.0 & 0.941518 & 0.067876 & 308.0 & 0.992065 \\  & & & 0.032288 & 953.0 & 0.996891 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Uniaxial stress–strain relationship and concrete damage parameters.  
properties of steel. It applies temperature loads to the preload steel bars, causing them to contract. It can provide a certain level of preload to the concrete [29]. The cooling method is used to apply preload to high-strength bolts in this study. The coefficient of thermal expansion for steel is taken as 1.25 \(\times\) 10\({}^{-5}\) m/\({}^{\circ}\)C. A linear relationship is established between the temperature load values applied to the bolts and the preload values. During the elastic yield stage of the preload bolt material, a decrease of 1 \({}^{\circ}\)C corresponds to a preload value of 1855.503 N applied to the high-strength bolts when preload is applied using the cooling method. The preload value specified in the standards for high-strength bolts is 280 kN [30], corresponding to a temperature stress value of 150.902 \({}^{\circ}\)C. The preload values at different temperatures are as shown in Table 3.  
Figure 2 illustrates the progression of applying bolt preload. In the first stage, the high-strength bolt preload is applied under static loading conditions, set at 280 kN (no preload loss has occurred during this stage). The second stage simulates different degrees of preload loss, where the preload values change from initial values to various loss degrees as listed in Table 3. The third stage is the seismic loading stage. The fourth stage is the period after seismic loading, which includes recovery and continued usage.