# 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
### Analysis for Cross-Section Key Parameters of Zero Bending Moment Shield Tunnel  
For the properties of soft soil, the vertical earth pressure on the crown of the tunnel is considered as the earth-pillar theory earth pressure, calculated by Equation (1). Assume that the thickness of soil above the tunnel crown is 12 m, the vertical diameter of the cross-section axis of the tunnel is 6 m (the central axis position of the outer and inner diameter of the tunnel cross-section is taken as the design diameter). The average unit weight of soil is 19.5 kN/m\({}^{3}\). According to the proposed design method for the cross-section of the zero bending moment shield tunnel, cross-section key parameters of the zero bending moment are obtained based on different lateral earth pressure coefficients, as shown in Table 2. When the lateral earth pressure coefficient \(k\) increases, the eccentric distance \(\Delta\) of the zero bending moment shield tunnel remains the same, and the center horizontal diameter \(b\) and the maximum horizontal diameter \(c\) both increase. Assume that the lateral earth pressure coefficient is 0.46, cross-section key parameters of the zero bending moment shield tunnel a different thickness of soil above the tunnel crown is shown in Table 3. With the increase in the thickness of the soil above the tunnel crowns, the eccentric distance \(\Delta\), the center horizontal diameter \(b\), and the maximum horizontal diameter \(c\) all decrease. The data in Tables 1 and 2 indicate that both the center horizontal diameter \(b\) and the maximum horizontal diameter \(c\) are smaller than the vertical diameter \(a\). The calculation results indicate that there is zero shear force for any section of the cross-section of the zero bending moment, which means that there is no shear deformation at the segment longitudinal joints. The minimum axial force \(N_{1}\) is apparent at the crown of the tunnel, and the maximum axial force \(N_{3}\) is apparent at the level of maximum horizontal diameter. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Boil Layer** & **Unit** & **Moisture** & **Void** & **Liquid** & **Plastic** & **Cohesion/kPa** & **Internal** & **Compression** & **Poisson** & **Ratio** \\  & **Weight/kN/m\({}^{3}\)** & **Content/\%** & **Ratio** & **Limit/\%** & **Limit/\%** & **Cohesion/kPa** & **Angle\({}^{c}\)** & **Us/MPa** & **Poisson** & **Ratio** \\ \hline \hline
3 &
\begin{tabular}{c} Mucky \\ silly clay \\ \end{tabular} & 19.6 & 39.7 & 1.123 & 35.4 & 20.5 & 9 & 16.5 & 3.36 & 0.3 \\ \hline
4 &
\begin{tabular}{c} Mucky \\ clay \\ \end{tabular} & 18.8 & 49.4 & 1.392 & 43.2 & 23 & 13 & 10.5 & 2.27 & 0.33 \\ \hline
5 &
\begin{tabular}{c} Silty clay \\ \end{tabular} & 20.1 & 34.8 & 0.996 & 36.9 & 20.7 & 17 & 14 & 4.27 & 0.31 \\ \hline
6 &
\begin{tabular}{c} Clay \\ \end{tabular} & 20.5 & 23.3 & 0.695 & 34 & 18.5 & 44 & 15.5 & 6.58 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Shanghai geological soil parameters. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**H/m** & \(k\) & \(P_{1}\)/kPa & \(P_{2}\)/kPa & \(P_{3}\)/kPa & \(a\)/m & \(b\)/m & \(c\)/m & \(N_{1}\)/kN & \(N_{2}\)/kN & \(N_{3}\)/kN & \(\Delta\)/m \\ \hline
12 & 0.42 & 222.00 & 93.24 & 46.62 & 6.00 & 4.347 & 4.350 & 326.34 & 372.96 & 482.83 & 0.0997 \\
12 & 0.44 & 222.00 & 97.68 & 48.84 & 6.00 & 4.450 & 4.452 & 341.88 & 390.72 & 494.19 & 0.0997 \\
12 & 0.46 & 222.00 & 102.12 & 51.06 & 6.00 & 4.550 & 4.552 & 357.42 & 408.48 & 505.30 & 0.0997 \\
12 & 0.48 & 222.00 & 106.56 & 53.28 & 6.00 & 4.648 & 4.650 & 372.96 & 426.24 & 516.17 & 0.0997 \\
12 & 0.50 & 222.00 & 111.00 & 55.50 & 6.00 & 4.743 & 4.746 & 388.50 & 444.00 & 526.81 & 0.0997 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Key parameters for no bending moment shield tunnels with different coefficients of lateral earth pressure. The corresponding cross-sections for Tables 1 and 2 are in Figure 12, which are similar to a vertical ellipse.