# Nonlinear Stability Analysis of Shallow-Buried Bias Tunnel Based on Failure Mode Improvement
## 6 Example Analysis
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{6}{c}{**Horizontal Surface (\(\delta\) = 0°)**} & \multicolumn{6}{c}{**Slope Surface Dip Angle (\(\delta\) = 18°)**} \\ \hline \multicolumn{2}{c}{**Reference [50]**} & \multicolumn{2}{c}{**Reference [51]**} & \multicolumn{2}{c}{**Reference [17]**} & \multicolumn{2}{c}{**Limit Analysis**} & \multicolumn{2}{c}{**Limit Analysis Upper Bound Method in This Paper**} \\ \multicolumn{2}{c}{**Limit**} & \multicolumn{2}{c}{**Limit Analysis**} & \multicolumn{2}{c}{**Upper Bound**} & \multicolumn{2}{c}{**Normative**} & \multicolumn{2}{c}{**Recommendation**} & \multicolumn{2}{c}{} \\ \multicolumn{2}{c}{**Equilibrium**} & \multicolumn{2}{c}{**Method**} & \multicolumn{2}{c}{**Method**} & \multicolumn{2}{c}{**Mode A**} & \multicolumn{2}{c}{**Mode B**} & \multicolumn{2}{c}{**Method**} & \multicolumn{2}{c}{**\(m\) = 1.0**} & \multicolumn{2}{c}{**\(m\) = 1.1**} & \multicolumn{2}{c}{\(m\) = 1.2**} & \multicolumn{2}{c}{\(m\) = 1.4**} \\ \hline \(K_{0}\) & \(q_{0}\)/kPa & \(K\) & \(q/\)kPa & \(q/\)kPa & \(q/\)/\(\zeta\)/\(\zeta\) & \(q/\)kPa & \(K\) & \(q/\)kPa & \(q/\)kPa & \(q/\)kPa & \(q/\)kPa \\
1.5 & 264.7 & 1.2 & 148.9 & 147.7 & 158.5 & 30 & 307.1 & 1.2 & 144.3 & 172.7 & 188.9 & 218.1 \\
1.4 & 270.9 & 1.0 & 169.6 & 169.3 & 175.7 & 32 & 308.5 & 1.0 & 166.5 & 192.4 & 225.7 & 257.2 \\
1.3 & 277.3 & 0.8 & 197.8 & 199.1 & 201.1 & 34 & 310.3 & 0.8 & 193.3 & 231.0 & 268.4 & 293.9 \\
1.2 & 283.9 & 0.6 & 238.8 & 243.2 & 243.0 & 36 & 312.5 & 0.6 & 235.2 & 280.3 & 329.1 & 365.7 \\
1.1 & 290.8 & 0.5 & — & — & — & 38 & 315.2 & 0.5 & 272.9 & 315.2 & 361.7 & 414.6 \\
1.0 & 297.9 & 0.4 & — & — & — & 40 & 318.2 & 0.4 & 301.3 & 344.7 & 401.9 & 466.2 \\ \hline \hline \end{tabular} Note: \(K_{0}\) is the lateral pressure coefficient of the tunnel top rock and soil layer. Referring to the relevant research results for the Taisha base theory, the recommended value interval is 1.0–1.5. As shown in Table 1, when the ground surface is horizontal, the surrounding rock pressure calculated using the upper-bound limit analysis method is smaller than that obtained by the Terzaghi ultimate equilibrium method. When the ground surface is inclined, the surrounding rock pressure calculated using the upper-bound limit analysis method is also smaller than that obtained by the code-recommended method. The order of magnitude of the surrounding rock pressure obtained by different calculation methods is as follows: code-recommended method \(>\) Terzaghi ultimate equilibrium method \(>\) upper-bound limit analysis method. In contrast, the standard recommendation method is more conservative, and the upper limit method of limit analysis is more economical. Taking \(m\) = 1.0, that is, the degradation to the linear Mohr-Coulomb damage criterion, the \(q\) value of the surrounding rock pressure calculated by this method agrees well with the results of References [50,51]; thus, the rationality and validity of this method are proved. Considering the property that the value of the surrounding rock pressure \(q\) increases with the value of the nonlinear coefficient \(m\), if the nonlinearity of the strength criterion for geotechnical materials is simplified to linearity in the practice of shallow-tunnel engineering, the support force required to maintain the stability of the surrounding rock in shallow tunnels may often be underestimated, which may lead to potential safety hazards in the project. The results of both the limit analysis upper limit method and the Tyshaki limit equilibrium method show that the value of the surrounding rock pressure decreases with the increase in the value of \(K(K_{0})\); thus, weakening of the side wall support force will cause the increase in the surrounding rock pressure when carrying out the construction of shallow-buried tunnels, which will reduce their overall stability.  
\end{table} 
Table 2: Parameter analysis value range.