## 4 Analysis of the Dynamic Response Characteristics of Tunnel Lining
### Analysis of Stress Change of Tunnel Invert with Vibration Load  
Figure 16 shows the stress-time curve of the typical measuring points B3 and B11. It can be seen from the figure that the inner surface of the tunnel invert mainly has compressive stress, while the outer surface has tensile stress. The stress of the measuring point is constantly changing with the excitation load. When the base does not have a void, the amplitude of the compressive stress at point B3 is approximately 3.60 kPa, and that of point B11 is approximately 3.86 kPa. When the base has a void, the change amplitude of the compressive stress of point B3 is approximately 3.80 kPa, and that of point B11 is approximately 4.08 kPa. As is well known, the fatigue life of concrete is determined by the average stress and stress amplitude. The larger the two values become, the smaller the fatigue life of the structure then becomes. Therefore, the tunnel base voids will accelerate the structural fatigue damage.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **No Base Void** & **With Base Void** & **Growth Rate (\%)** \\ \hline A1 & 2.01 & 3.11 & 55.6 \\ A2 & 1.73 & 2.11 & 21.9 \\ A3 & 1.55 & 1.76 & 13.5 \\ A4 & 1.34 & 1.58 & 8.9 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Statistics of peak acceleration of tunnel axial measuring point (m/s\({}^{2}\)).  
Figure 15: Peak value of the acceleration of each measuring point of the tunnel invert (m/s\({}^{2}\)).  
Table 5 lists the statistics of the peak stress at each measuring point of the tunnel invert. When the tunnel base has a void, the stress value of each measuring point of the tunnel invert increases. B2, B3, B4, B10, B11, and B12 are in the base void area, and the peak stress increase in the inverted arch is the most obvious. At this time, the stresses at points B3 and B11 are \(-41.8\) kPa and \(42.6\) kPa, respectively, which are increases of \(21.5\%\) and \(21.4\%\), respectively, from the stresses when the base does not have a void. The increase in the peak stress decreases significantly at the measuring points outside the base void range. The no-base-void measuring points B5 and B12 are the closest sites to the base void. Their peak stresses increase by only \(5.4\%\) and \(5.7\%\). In contrast, for points B8 and B16, furthest from the void area, the peak stress barely increases.  
The tensile strength design value of C30 concrete is \(f_{t}=1.43\) MPa [19], and the \(\sigma_{\max}>1/2f_{t}\) structure suffers fatigue damage with the action of the cyclic load. According to Table 1, the stress similarity ratio is 1:20, and the diagram of the peak stress and allowable stress at the bottom of the tunnel invert is obtained (Figure 17). When the tunnel base does not have a void, the maximum tensile stress values of each measuring point at the bottom of the tunnel invert are all less than \(715\) kPa, and the tunnel structure does not produce fatigue damage with the action of the train load. When the tunnel base has a void, the tensile stress values of measuring points B10, B11, and B12 are greater than the allowable stress value, and the tunnel invert suffers fatigue damage with the long-term action of the train load.  
Figure 16: Stress-time curve of typical tunnel measurement points: (**a**) no base void; (**b**) with base void.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Measuring Point** & **B1** & **B2** & **B3** & **B4** & **B5** & **B6** & **B7** & **B8** \\ \hline No base void & \(-26.5\) & \(-30.8\) & \(-34.4\) & \(-31.1\) & \(-27.6\) & \(-21.9\) & \(-20.1\) & \(-18.1\) \\ With base void & \(-29.0\) & \(-37.4\) & \(-41.8\) & \(-38.2\) & \(-29.1\) & \(-22.2\) & \(-20.6\) & \(-18.3\) \\ \hline
**Measuring Point** & **B1** & **B2** & **B3** & **B4** & **B5** & **B6** & **B7** & **B8** \\ \hline No base void & 25.9 & 31.9 & 35.1 & 31.5 & 28.2 & 22.3 & 20.4 & 18.5 \\ With base void & 29.2 & 38.3 & 42.6 & 38.8 & 29.8 & 22.5 & 20.9 & 18.6 \\ \hline \end{tabular}
\end{table}
Table 5: Peak stress at each measuring point of the tunnel invert (kPa).