# Prediction of Surface Deformation Induced by Ultra-Shallow-Buried Pilot Tunnel Construction
## 1 Introduction
### Calculation of Surface Deformation Considering Non-Uniform Convergence
Equation (10) can be rewritten as  
\[W_{1}(x)\ =\ \int\limits_{a}^{b}\int\limits_{c}^{d}w(x,\xi,\eta)\mathrm{d}\xi \mathrm{d}\eta-\int\limits_{c}^{f}\int\limits_{g}^{h}w(x,\xi,\eta)\mathrm{d}\xi \mathrm{d}\eta \tag{12}\]  
The derived dual integration bounds \(a\)-\(h\) are listed in Table 1, where \(H\) is the burial depth of the pilot tunnel, the tunnel center is calculated by \((B+C)/2\), \(B\) is the height of the rectangular portion, \(A\) is the width of the rectangular portion and \(C\) is the height of the arch portion (see Figure 2). \(\Delta R\) denotes the uniform shrinkage displacement of the section (see Figure 4). The surface displacement caused by the second deformation of the cross-section is calculated by Equation (13):  
\[W_{2}(x)\ =\ \int\limits_{c}^{f}\int\limits_{g}^{h}w(x,\xi,\eta)\mathrm{d}\xi \mathrm{d}\eta-\int\limits_{c^{\prime}}^{fr}\int\limits_{g^{\prime}}^{h}w(x, \xi,\eta)\mathrm{d}\xi\mathrm{d}\eta \tag{13}\]  
The derived upper and lower bounds of the double integration \(e\)-\(h^{\prime}\) are listed in Table 2, where \(B^{\prime}\) is the height of the rectangular portion after uniform shrinkage, \(A^{\prime}\) is the width of the rectangular portion after uniform shrinkage and \(C^{\prime}\) is the height of the arch portion after uniform shrinkage. The height \(C^{\prime}\) of the arnded portion after uniform shrinkage is calculated by Equation (14):  
\[C\ =\ C-\Delta R-\delta \tag{14}\] \begin{table}
\begin{tabular}{c c c} \hline
**Integration** & **Arch Part** & **Rectangular Part** \\
**Bound** & **Arch Part** & **Rectangular Part** \\ \hline \(a\) & \(H-(B+C)/2\) & \(H-(B-C)/2\) \\ \(b\) & \(H-(B-C)/2\) & \(H+(B+C)/2\) \\ \(c\) & \(-\sqrt{R^{2}-\left(H-\eta-\left(\frac{B+C}{2}-R\right)\right)^{2}}\) & \(-\frac{A}{2}\) \\ \(d\) & \(-c\) & \(-c\) \\ \(e\) & \(H-(B+C)/2+\Delta R\) & \(H-(B-C)/2\) \\ \(f\) & \(H-(B-C)/2\) & \(H+(B+C)/2-\Delta R\) \\ \(g\) & \(-\sqrt{(R-\Delta R)^{2}-\left(H-\eta-\left(\frac{B+C}{2}-R\right)\right)^{2}}\) & \(-\left(\frac{A}{2}-\Delta R\right)\) \\ \(h\) & \(-g\) & \(-g\) \\ \(R\) & \(\frac{A^{2}}{8C}+\frac{C}{2}\) & \(-g\) \\ \hline \end{tabular}
\end{table} 
Table 1: Table of uniform shrinkage integral limits for horseshoe-shaped sections. \begin{table}
\begin{tabular}{c c c} \hline
**Integration Bound** & **Arch Part** & **Rectangular Part** \\ \hline \(e^{\prime}\) & \(H-(B+C)/2\) & \(H-(Bt-C)/2\) \\ \(f^{\prime}\) & \(H-(Bt-C)/2\) & \(H+(Bt+C)/2\) \\ \(g^{\prime}\) & \(-A_{\mathrm{oval}}\sqrt{1-\left(\frac{H-\left(\frac{Bt+C}{2}-C\right)-\eta}{B _{\mathrm{oval}}}\right)^{2}}\) & \(-\frac{At}{2}\) \\ \(h^{\prime}\)where \(C\) is the height of the initial arched portion, \(\Delta R\) is the uniform shrinkage displacement of the section and \(\delta\) is the second shrinkage displacement of the section. \(A^{\prime}\) and \(B^{\prime}\) are determined by ensuring the cross-sectional area remains equal before and after the second deformation. The final surface settlement induced by the excavation of the pilot tunnel is given by the formula  
\[W(x)\ =\ W_{1}(x)+W_{2}(x) \tag{15}\]