## 2 Research Method
### Cumulative Damage Tests
#### 2.2.1 Test System  
The self-developed test system is composed of a lateral hydraulic loading system, lateral springs, a vertical MST loading system, and vertical springs. The size of the test box is \(400\times 300\times 250\) mm (length \(\times\) width \(\times\) height), as shown in Figure 5.  
The lateral digital jack and springs were used to simulate the actual stress state of the tunnel inverted arch. The springs with different stiffness coefficients were used to simulate the bedrock with different elastic coefficients. In addition, the springs absorb the vibration reflection wave and reduce its reflection at the boundaries. A contact steel plate was used to reduce the local failure possibility of specimens that is caused by stress concentration. A lubricant was applied between the steel plate and the specimen to reduce the size effect in the small-scale test system.  
On the basis of the stiffness similarity principle, the spring can provide equal elastic resistance when the same amount of deformation occurs as the bedrock (\(\delta_{i}=\delta_{0}\)). According to the elastic resistance of the surrounding rock at any point and its radial change, it can be obtained as follows:  
\[k\delta_{i}ab=k_{0}\delta_{0}m \tag{3}\]  
Figure 5: Test system: (**a**) Model diagram of test system: 1, lateral pressure spring; 2, bottom spring plate; 3, hydraulic golden roof; 4, MTS system; 5, contact steel plate; 6, gearbox. (**b**) Test loading diagram.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Reference**} & \multirow{2}{*}{**Train Speed(km/h)**} & \multicolumn{2}{c}{**Displacement (mm)**} & \multicolumn{2}{c}{**Tensile Stress (MPa)**} \\  & & **Numerical** & **Location** & **Numerical** & **Location** \\ \hline Rapid rail transit in & & & & & \\ the UK [31] & 200 & 14 & inverted arch & - & - \\ Zhutting Tunnel in & & & & & \\ China [32] & 120 & 2.3 & Side wall & - & - \\ Centripede Tunnel & & & & & \\ in China [33] & 120 & - & - & 0.309 & inverted arch \\  & 200 & 5.8 & inverted arch & 0.141 & inverted arch \\ Model test [34] & 250 & 7.6 & inverted arch & 0.169 & inverted arch \\  & 300 & 9.2 & inverted arch & 0.170 & inverted arch \\ \hline \hline \end{tabular}
\end{table}
Table 3: Measured data of dynamic response of tunnel support structure under train loads.  
where, \(\delta_{i}\) is the compressive deformation of surrounding rock at a certain action point; \(k\) is the elasticity resistance coefficient; \(a\) is the length at the bottom of the specimen; \(b\) is the width at the bottom of the specimen; \(\delta_{0}\) is the compression deformation of the spring; \(k_{0}\) is the rigidity of the mold spring at the bottom of the specimen; \(m\) is the number of die springs at the bottom of the specimen.  
Three kinds of spring stiffness were used in the experiment. Through test calibration, the spring stiffness is \(2.157\times 10^{6}\), \(0.669\times 10^{6}\), and \(0.361\times 10^{6}\) N/m, respectively. The corresponding elastic resistance coefficients of surrounding rocks were obtained by using Equation (3), which are \(1.941\times 10^{9}\), \(0.602\times 10^{9}\), and \(0.325\times 10^{9}\) N/m\({}^{3}\).  
The data collection system mainly composes of a strain gauge, ceramic piezoelectricity, strain collection instruments, an impedance analyzer, and other relevant software. Figure 6 and Table 4 show the data collection instruments, the layout of monitoring points, and the relevant parameters of the instruments.