## 3 Verifications
### Building Axis Parallel to Tunnel Axis  
As shown in Figure 2, with regard to the building axis parallel to and directly above the tunnel axis (\(\alpha=90^{\circ},s_{2}=0\)), the three-dimensional finite difference analysis using FLAC 3D is conducted to reproduce the building settlements caused by tunneling process. The diameter of the circular tunnel is 6.0 m based on the area equivalent method, and the depth to tunnel axis is 15.0 m (Figure 7). The dimensions of the numerical model are 60.0 m in width, 36.0 m in height, and 80.0 m in length, which are sufficiently large to eliminate the boundary effect. The normal displacements of the four vertical boundaries are fixed while the nodes at the bottom boundary are fixed in both vertical and horizontal directions, and the top boundary of the numerical model is left free. The soil is modeled as a linear elastic-perfectly plastic material with the Mohr-Coulomb failure criterion. The isotropic elastic model is employed for simulations of the 0.3 m thick tunnel lining and the existing building. The existing building is ideally represented by an equivalent beam with length 20.0 m, width 2.0 m, and height 1.0 m. The physical and mechanical parameters of the numerical model are shown in Table 2.  
The numerical simulation consists of the following stages [19]: (1) initialization of the geostatic stress in the ground under greenfield condition; (2) resetting the ground movements to zero, followed by the activation of the equivalent beam; and (3) tunnel excavation and lining installation. The tunneling process is simulated one step advance at a time as a repeated sequence of: (1) removing the unit soil at the excavation step (1.0 m); (2) releasing the soil stress; (3) generating the lining elements at the excavation step.  
Figure 6: Comparison with building settlement from finite element method: (**a**) 4 stories; (**b**) 8 stories [17].  
Figure 8 shows the building settlements above the tunnel axis calculated by FLAC 3D compared with those from the proposed method and Winkler-based method (\(g=30.0\) mm, \(d=0\)). The trends of the building settlement curves estimated by the two analytical methods are similar to the finite difference results in three cases. The maximum and minimum building settlements calculated through these three methods are all located at the left and right ends of the building, respectively. The magnitudes of the building settlements gained from the proposed method agree better with the finite difference results than those from the Winkler-based method, especially in the case of \(s_{1}=-20.0\) m. The results of the Winkler-based method are slightly smaller than those from the proposed method, which implies that the building settlements are not substantially subjected to the shear stiffness of the subgrade. Overall, the proposed method can be adopted to effectively and rapidly predict the building response resulting from shallow tunneling.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & **Unit Weight** & **Elastic Modulus** & **Poisson’s Ratio** & **Cohesion** & **Internal Friction** \\  & \(\gamma\) (kN/m\({}^{3}\)) & \(E\) (MPa) & \(\
u\) & \(c\) (kPa) & **Angle \(\varphi\) (\({}^{\circ}\))** \\ \hline Soil & 20 & 30 & 0.3 & 10 & 30 \\ Lining & 25 & \(3.0\times 10^{4}\) & 0.2 & – & – \\ Building & 25 & \(3.0\times 10^{4}\) & 0.2 & – & – \\ \hline \hline \end{tabular}
\end{table}
Table 2: Physical and mechanical parameters of numerical model.  
Figure 7: Finite difference model.