## 2 Displacement Field of Soil Caused by the Excavation of Non-Circular Caverns
### Correction of Soil Displacement Owing to Soil Loss  
Based on [PERSON]'s [42] source and sink theory, [PERSON] [21] assumes the soil to be a linear elastic material. The deformation mechanism during tunnel excavation is equivalent to radial shrinkage on the tunnel surface and long-term elliptical deformation under initial in-situ stress. For any Poisson's ratio, the displacement and stress fields in a semi-infinite space result from the superposition of these two effects. The settlement at the existing tunnel due to excavation at a free site can be expressed as follows:  
\[ \begin{split} U_{z}(x)&=\frac{2\varepsilon R^{2}}{ m}\Bigg{(}\frac{(m+1)z_{2}}{r_{2}^{2}}-\frac{mz\big{(}x^{2}-z_{2}^{2}\big{)}}{r_{2}^{4}} \Bigg{)}\\ &\qquad\qquad-2\delta R^{2}h\Bigg{(}\frac{x^{2}-z_{2}^{2}}{r_{2} ^{4}}+\frac{m}{m+1}\frac{2 zz_{2}\big{(}3x^{2}-z_{2}^{2}\big{)}}{r_{2}^{6}} \Bigg{)},\end{split}  \tag{2}\]  
\[r_{1}=\sqrt{x^{2}+(z-h)^{2}};r_{2}=\sqrt{x^{2}+(z+h)^{2}}, \tag{3}\]  
where \(\varepsilon\) is uniform radial contraction, \(\varepsilon=u_{1}/R\); \(u_{1}\) is uniform radial displacement, \(\delta\) is the tunnel elliptical, \(R\) is the tunnel radius, \(h\) is the buried depth of the tunnel, \(x\) is the horizontal coordinate from the tunnel axis, \(\
u\) is the Poisson ratio, and \(z\) is the vertical coordinate, \(z_{2}=z+h\), and \(m=1/(1-2\
u)\). The key parameters for [PERSON]'s [21] formula are the values of the radial shrinkage \(\varepsilon\) and the ovalization parameter \(\delta\). [PERSON] [21] proposes the calculation formula of radial shrinkage \(\varepsilon\) as follows:  
\[\varepsilon=\frac{A}{4(1-\
u)\pi R^{2}}=\frac{\eta}{4(1-\
u)}, \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Shape** & **Oval** & **Arch** & **Rectangle** & **Square** & **Positive Trapezoid** \\ \hline \(k\) & 1.05 & 1.10 & 1.20 & 1.15 & 1.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of correction coefficient \(k\) for different shapes. Figure 1: The circumscribed circle of different tunnel sections. where \(A\) is the area of surface settling trough per unit length, and \(\eta\) represents the rate of formation loss. [PERSON] [42] presents the relationship between the two parameters as shown below:  
\[\delta=4(1-v)(1-2v)\epsilon, \tag{5}\]  
The radial shrinkage \(\epsilon\) and ovalization parameter \(\delta\) proposed by [PERSON] [21] and [PERSON] [42] are often tested for calculating ground settlement. However, these differ from the measured ones after being verified through actual project results. Thus, these two parameters require adjustment. [PERSON] [43] proposed four models for soil movement around tunnels to calculate stratum deformation in shallow buried tunnels, as illustrated in Figure 2. Among these, both B.C-3 and B.C-4 consider tunnel ovalization deformation. In conjunction with measured data and [PERSON]'s tunnel convergence model, it was determined that the tunnel's elliptic convergence aligns with the B.C-4 model. Consequently, the choice of convergence model and the validation of measured data play crucial roles in optimizing elliptic parameters, with their adjustments significantly affecting the accuracy of elliptic parameters in non-circular caverns. By considering the B.C-4 model as a benchmark for the long-term elliptical deformation of the tunnel, the tunnel ellipticity parameter \(\delta\) can be calculated as follows:  
\[\delta=\frac{D_{\text{h}}-D_{\text{v}}}{D}=\frac{1.5u_{0}}{2R}, \tag{6}\]  
where \(D_{\text{h}}\) is the horizontal diameter after deformation, \(D_{\text{v}}\) is the vertical diameter after deformation, and \(D\) is the tunnel design diameter. Regarding the radial contraction \(\epsilon\) suggested by [PERSON] [21], it has been found through verification and calculation that directly applying [PERSON]'s [21] results to non-circular cavities typically results in underestimated soil displacement. Therefore, adjustments are necessary.