# A New Design Method of Shield Tunnel Based on the Concept of Minimum Bending Moment
## 4 Cross-Section Design and Case Analysis for Minimum Bending Moment Shield Tunnel in Soft Soil Area
### Case Analysis for Minimum Bending Moment Shield Tunnel
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Soil Layer** & **Thickness/m** & \begin{tabular}{c} **Unit** \\ **Weight** \\ **/kN/m3** \\ \end{tabular} & \begin{tabular}{c} **Moisture** \\ **Content/\%** \\ \end{tabular} & \begin{tabular}{c} **Void** \\ **Ratio** \\ \end{tabular} & \begin{tabular}{c} **Cohesion/kPa** \\ **Angle/\({}^{\circ}\)** \\ \end{tabular} & \
\begin{tabular}{c} **Internal** \\ **Friction** \\ **1 us/MPa** \\ \end{tabular} &
\begin{tabular}{c} **Compression** \\ **Ratio** \\ \end{tabular} \\ \hline Clay & 4.5 & 20.5 & 23.3 & 0.695 & 44 & 15.5 & 6.58 & 0.30 \\ Mucky silty clay & 6.5 & 19.6 & 39.7 & 1.123 & 9 & 16.5 & 3.36 & 0.30 \\ Mucky clay & 29 & 18.8 & 49.4 & 1.392 & 13 & 10.5 & 2.27 & 0.33 \\ \hline \end{tabular}
\end{table} 
Table 6: Soil mechanical parameters for different stratums. Figure 16: The interaction between the tunnel and the soil of the numerical simulation model. Figure 17: The grid division of the numerical simulation model (unit: m). as the thickness of the overlying soil increased, and the bending moment increased more obviously in the case of the circular cross-section. As for the conditional on the same working condition, compared to the circular cross-section, the vertical elliptical cross-section featured a much smaller bending moment. Also, as the thickness of the overlying soil increased, the decrease in the bending moment of the vertical elliptical cross-section was more obvious. When the thickness of the overlying soil of the tunnel was 25 m, the bending moment diagram of two differently shaped cross-sections is shown in Figure 18. It can be seen from Figure 18 that the maximum values of the two differently shaped cross-sections both appear at the bottom of the tunnel arch, while the minimum bending moment values both appear at the waist of the tunnel. The maximum bending moment value of the circular cross-section tunnel was 995.103 kN\(\cdot\)m and the minimum bending moment value was \(-\)969.411 kN\(\cdot\)m. On the other hand, the maximum bending moment value of the vertical elliptical cross-section tunnel was 422.12 kN\(\cdot\)m and the minimum value was \(-\)350.606 kN\(\cdot\)m. When compared with the circular cross-section, the maximum bending moment of the vertical elliptical cross-section was 57.58% less while the minimum value was 67.83% less. The cross-section shape of the zero bending moment shield tunnel in this article was obtained by methods of rational assumptions. In addition, with the consideration that, normally, metro lines are constructed with only one shield tunnel of one cross-section, the weighted average method was used to obtain the minimum bending moment shield tunnel cross-section shape. As a result, the bending moment of the similar vertical elliptical cross-section shield tunnel obtained in the simulation was not zero. However, it can be learnt from the numerical simulation calculation, by means of the improvement of the cross-section shape of the tunnel, that the bending moment was significantly decreased compared to the circular tunnel, which makes it favorable both in the case of the transverse \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Minimum Bending Moment**} & \multicolumn{2}{c}{**Circular Shield Tunnel**} \\  & & \multicolumn{2}{c}{**Shield Tunnel Cross-Section**} & \multicolumn{2}{c}{**Cross-Section**} \\ \cline{3-6}
**Engineering** & & **H/m** & **Max** & **Min** & **Max** & **Min** \\
**Conditions** & & **Bending** & **Bending** & **Bending** & **Bending** \\  & & **Moment/kN\(\cdot\)m** & **Moment/kN\(\cdot\)m** & **Moment/kN\(\cdot\)m** & **Moment/kN\(\cdot\)m** \\ \hline
1 & 7 & 218.644 & \(-\)164.328 & 362.403 & \(-\)327.64 \\
2 & 13 & 245.091 & \(-\)208.737 & 567.907 & \(-\)538.893 \\
3 & 19 & 331.966 & \(-\)275.233 & 779.613 & \(-\)755.177 \\
4 & 25 & 422.12 & \(-\)350.606 & 995.103 & \(-\)969.411 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Maximum and minimum bending moments of two different bending moment shield tunnel cross-sections for different conditions.