## 3 Example Analysis and Verification  
In order to verify the rationality of the derived formula, the reference [18] data is cited for verification. For a metro shield tunnel project, stratum information is shown in Table 1, and the geological sections and instrument layout are shown in Figure 4. The horizontal surrounding rock stresses at the crown can be obtained from the earth pressure cells on the left and right sides. It can be seen from Table 2 that the improved Bierbaumer formula can more closely predict surrounding rock stress values obtained from monitoring than other formulas. With an increase in the nonlinear coefficient \(m\), the calculated value is closer to the measured value, which shows the rationality of the improved formula to some extent. Because \(m\) needs to be tested and the nonlinear coefficient \(m\) values of different types of strata should be different, the improved Bierbaumer formula is more adaptable. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Stratum Information** & \
\begin{tabular}{c} **Density** \\ p/g/cm\({}^{3}\) \\ \end{tabular} & \(c\)**/kPa** & \(\varphi^{\text{f}^{\text{c}}}\) &
\begin{tabular}{c} **Lateral Stress** \\ **Coefficient**\(k\) \\ \end{tabular} \\ \hline Mixed backfill soil & 1.90 & 50 & 15 & 0.45 \\ Strong weathered slate & 2.17 & 65 & 27 & 0.30 \\ Moderately weathered slate & 2.33 & 100 & 36 & 0.28 \\ Slightly weathered slate & 2.65 & 122 & 32 & 0.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ground parameters. Figure 4: Geological sections and instrument layout (Unit: m). Comparative Analysis on Statistical Characteristics of Stresses of Loose Class V and VI Surrounding Rocks  
In order to further verify the rationality of the calculation method and considering the discreteness of geotechnical materials and the variations in spatial distributions, the statistical distribution characteristics of the calculated stress results of various rocks surrounding shallow tunnels are explored, which lays the foundation for the reliable design of tunnel lining. In this paper, based on the Monte Carlo method, the statistical laws of stresses of loose surrounding rock obtained by the improved Bierbaumer formula and other shallow tunnel calculation methods are compared. Refer to reference [31] and code TB-10003-2016 for the design of railway tunnels [32]; the physical and mechanical parameters of rock masses corresponding to classes V and VI are shown in Table 3.  
Due to the lack of statistical data of the nonlinear coefficients of class V and VI rocks, the values of the nonlinear coefficient \(m\) is taken as 1.0-1.5 in this paper. When \(m\) is taken as 1.0, 1.2 and 1.5, different \(\tan\varphi\) values can be obtained. When \(m=1.0\), it is the control group, and it is the linear M-C criterion. Assuming the span of the rectangular tunnel is 10 m, the burial depth and span of the rectangular tunnel in different numerical test groups are shown in Table 4.  
According to the statistical distribution shown in Table 3, a certain random number is generated by MATLAB, and the calculation results of different burial depths (5 m, 15 m and 25 m) are obtained. After K-S tests, the calculation results obey the normal distribution, as shown in the Figures 5 and 6, and the sensitivity analysis results are shown in Figure 7.  
Through variance analysis, the mean values and standard deviations of the surrounding rock stresses calculated by individual formulas are significantly different. As shown in Figure 5, in class V surrounding rock, when the burial depth of the tunnel is 5 m, the average values of the surrounding rock stresses calculated by individual formulas follow \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Group** & **Thickness–Span Ratio** & **Depth (m)** & **Span (m)** \\ \hline
1 & 0.5 & 5 & 10 \\
2 & 1.5 & 15 & 10 \\
3 & 2.5 & 25 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Buried depths and spans of tunnels under different coverage ratios.