# Dynamic Performance of Soil-Tunnel System under Transverse Sinusoidal Excitations
## 4 Response of FF Ground under Sinusoidal Excitation
### Deformation of the Ground  
The deformation of the ground can be evaluated by lateral displacement of the laminar box. As shown in Figure 16, in the cases of Sin-2, Sin-3, and Sin-6, the maximum lateral deformation of the laminar box is relatively large. In the cases of Sin-9, Sin-11, Sin-14, and Sin-17, the maximum lateral displacement is more sensitive to the depth of the model soil. It is concluded that the deformation of ground shows shear-type deformation of first-order vibration modes when the frequency of excitation approaches the fundamental frequency of the model ground. The oscillation of the lateral displacement along depths is becoming active with the increase of input frequency. On the one hand, this is because there are resonance effects when the input frequency approaches the fundamental frequency of the ground. On the other hand, high-order vibration modes may distinctly affect the deformation of the ground. The results mentioned above indicate that frequency characteristics of the input seismic waves affect the ground deformation. In order to analyze the modal contribution to the deformation response of the ground, modal amplitude responses of ground are analyzed by means of mode decomposition method under sinusoidal excitations of different frequency. The sum of vibration modes times each modal amplitude is the result of the displacement of the model ground; that is, the total displacement is decomposed by the modal contribution. Figure 15: Response spectrum of ground surface acceleration under different cases. The modal participating mass ratios [23] for the first four modes of the model ground are listed in Table 7. The modal participating mass ratios can be expressed as follows:  
\[\gamma_{n}=\frac{\left[\int_{0}^{H}m(z)\varphi_{n}(z)dz\right]^{2}}{\left[\int_{0 }^{H}m(z)\varphi_{n}^{2}(z)dz\right]\left[\int_{0}^{H}m(z)dz\right]} \tag{8}\]  
where \(n\) is the mode order, \(\varphi_{n}(z)\) is the lateral deformation of nth order mode of soil at the depth of \(z\), \(m(z)\) is the mass distribution of the soil slice at the depth of \(z\), and \(H\) is the height of soil column. The number of modes considered for the model ground is four since the total modal participating mass ratio is over 90% [24], as tabulated in Table 7.  
According to the classical mode analysis method [23], the contribution of the nth order vibration mode to the node displacement \(u_{n}(z,t)\) is written as follows:  
\[u_{n}(z,t)=\varphi_{n}(z)q_{n}(z,t) \tag{9}\]  
where \(q_{n}(z,t)\) is the modal amplitude of nth order at time of t.  
For a homogeneous soil layer, the vibration mode can be expressed as follows [25]:  
\[\varphi_{n}(z)=\cos\frac{(2n-1)\pi}{2H}z\ \ \ n=1,2,3,\cdots \tag{10}\]  
The total displacement can be obtained by combining the contribution of vibration modes:  
\[u(z,t)=\sum u_{n}(z,t)=\sum\varphi_{n}(z)q_{n}(z,t) \tag{11}\]  
where \(u(z,t)\) is the lateral deformation of soil column at depth z, and time t.  
According to the above formulas, the experimental results of the linear wire displacement gauges deployed along the height of the model box are analyzed, and the amplitude responses of vibration modes can be obtained. The analysis results are shown in Table 8. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Mode Order** & **Modal Participating Mass Ratios** & **Total Ratio** \\ \hline \(n=1\) & 81.1\% & \\ \hline \(n=2\) & 9.0\% & \\ \hline \(n=3\) & 3.2\% & \\ \hline \(n=4\) & 1.7\% & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Modal participating mass ratios. Figure 16: Maximum lateral displacement of the laminar box along depths. As can be seen from Table 8, the first-order modal amplitude is the largest under any excitation frequency, which shows that the first-order vibration mode plays a leading role. The modal amplitude decreases with the increase of mode order, and this is especially obvious under lower frequency excitations. On the other hand, the modal amplitude decreases with the increase of input frequency, and this is especially obvious in the lower mode order.