# Dynamic Performance of Soil-Tunnel System under Transverse Sinusoidal Excitations
## 5 Responses of Segmental Tunnel
### Relationship between Sinusoidal Excitations and Maximum Tunnel Responses  
Amplification coefficient is defined as the ratio of maximum acceleration at the top to that of the bottom of the tunnel. As can be seen from Figure 24a, the maximum acceleration at the top of the tunnel first increases and then decreases with the increase of input frequency. In the case of the Sin-3, the acceleration reaches the peak value, 0.21 g. When the frequency of sinusoidal excitation is less than 6 Hz, acceleration at the top of the tunnel is amplified compared with that of the bottom of the tunnel. When the frequency of sinusoidal excitation is more than 6 Hz, acceleration at the top of the tunnel attenuates compared with that of the bottom of the tunnel, which is consistent with the variation of acceleration in model soil at the near depth. It also shows that the response of underground structures is dominated by the seismic response of surrounding soil [27]. As can be seen from Figure 25, the maximum cross-sectional deformation of the tunnel first increases and then decreases with the increase of input frequency. In the case of the Sin-3, the cross-sectional deformation reaches a maximum of 1.54 mm, or the diameter change rate reaches a maximum of 1.7%. As can be seen from Figure 26, the maximum extension/closure of longitudinal joints first increases and then decreases with the increase of input frequency. In the case of the Sin-3, the extension/closure deformation reaches a maximum of 0.09 mm. Maximum dynamic earth pressure is defined as the maximum additional pressure caused by the earthquake. As can be seen from Figure 27, the maximum dynamic normal earth pressure first increases and then decreases with the increase of input frequency. In the case of the Sin-3, the dynamic normal earth pressure reaches a maximum of 4.06 kPa. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Cases**} & \multirow{2}{*}{**Label**} & \multicolumn{2}{c}{**Absolute**} & \multirow{2}{*}{**Cases**} & \multirow{2}{*}{**Label**} & \multicolumn{2}{c}{**Absolute**} \\  & & **Maximum (\(\mu\varepsilon\))** & & & **Maximum (\(\mu\varepsilon\))** \\ \hline \multirow{4}{*}{Sin-2 Hz} & SW1 & 9.70 & \multirow{4}{*}{Sin-3 Hz} & SW1 & 22.63 \\ \cline{2-6}  & SW2 & 12.94 & & SW2 & 25.88 \\ \cline{2-6}  & SW3 & 11.30 & & SW3 & 20.99 \\ \cline{2-6}  & SW4 & 8.91 & & SW4 & 13.76 \\ \hline \multirow{4}{*}{Sin-6 Hz} & SW1 & 12.12 & \multirow{4}{*}{Sin-9 Hz} & SW1 & 7.27 \\ \cline{2-6}  & SW2 & 12.13 & & SW2 & 7.28 \\ \cline{2-6}  & SW3 & 10.49 & & SW3 & 4.85 \\ \cline{2-6}  & SW4 & 8.90 & & SW4 & 8.09 \\ \hline \multirow{4}{*}{Sin-11 Hz} & SW1 & 8.89 & \multirow{4}{*}{Sin-14 Hz} & SW1 & 7.27 \\ \cline{2-6}  & SW2 & 7.28 & & SW2 & 5.66 \\ \cline{2-6}  & SW3 & 5.65 & & SW3 & 6.46 \\ \cline{2-6}  & SW4 & 6.47 & & SW4 & 4.86 \\ \hline \multirow{2}{*}{Sin-17 Hz} & SW1 & 6.46 & \multirow{2}{*}{Sin-17 Hz} & SW3 & 5.65 \\ \cline{2-6}  & SW2 & 4.85 & & SW4 & 8.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Peak dynamic strains of segments. Figure 24: Maximum acceleration response at the top and bottom of the tunnel. (**a**) Maximum. (**b**) Amplification coefficient. Figure 25: Maximum cross-sectional deformations under different cases. As can be seen from Figure 28, the maximum dynamic strain of segments first increases and then decreases with the increase of input frequency. In the case of the Sin-3, the dynamic strain of segments reaches the peak value, 25.88 \(\mu\epsilon\). All the maximum responses of the tunnel aforementioned first increase and then decrease with the increase of input frequency. All the maximum responses of the tunnel reach the peak value when the frequency of excitation approaches but are lower than the fundamental frequency of the model ground. The maximum responses of the tunnel are highly dependent on frequency of input motion.