# A New Design Method of Shield Tunnel Based on the Concept of Minimum Bending Moment
## 2 Analysis of the Pressure Exerted on the Cross-Section of the Shield Tunnel  
### Hypothesis of the Earth Pressure Acting on the Shield Tunnel  
According to structural mechanics, the rational axis (any section of the axis without bending moment) of the arch is related to the loads that it bears. Since the vertical earth pressure in the stratum is larger than the horizontal earth pressure, when the bending moment of the cross-section of the shield tunnel is zero, the horizontal diameter should not be equal to the vertical diameter. Therefore, when designing the reasonable axis of a shield tunnel section (the bending moment at any section is zero), only the normal condition is considered, which is convenient for analysis and calculation. The hypotheses on the loading of the shield tunnel have been made, as follows:  
(1) There is no horizontal soil resistance.  
There is zero bending moment in the designed shield tunnel cross-section. Hence, it can be approximately considered that no deformation takes place in the shield tunnel and there is no horizontal soil resistance at the two sides of the tunnel.  
Figure 2: The excessive deformation and related water-leakage of longitudinal segment joints: (**a**) The excessive deformation of the segmental joint; (**b**) Longitudinal segment joint water-leakage; (**c**) Segment corner water-leakage.  
(2) Do not consider the weight of the shield tunnel structure.  
As an underground structure bearing the soil pressure, the weight of the shield tunnel is much smaller than the surrounding earth pressure that it bears. According to the design specification of metro [10], the lining thickness should be 0.04-0.06 times larger than the outer contour diameter of the tunnel and the buried depth is much larger than the thickness of the segment. Hence, the bending moment of the tunnel structure that is caused by the weight of tunnel is much smaller than the bending moment of the tunnel structure that is caused by the surrounding earth pressure.  
(3) Calculate the vertical earth pressure based on the earth-pillar theory and do not consider the water pressure.  
The soil is soft in the soft soil area, and it is impossible to make an arch. Also, the permeability coefficient is small.  
Based on the above hypotheses, the earth pressure surrounding the shield tunnel is demonstrated in Figure 3. The pressure can be calculated as Equations (1)-(3).  
\[P_{1}=\gamma H \tag{1}\]  
\[P_{2}=\gamma Hk=P_{1}k \tag{2}\]  
\[P_{3}=\gamma ak \tag{3}\]  
where \(a\) is the vertical diameter of the shield tunnel, \(b\) is the horizontal diameter at the center of the vertical diameter of the shield tunnel, \(P_{1}\) is the vertical earth pressure, \(P_{2}\) is the horizontal earth pressure on top of the tunnel, \(P_{3}\) is the part of earth pressure that occurs where the earth pressure at the bottom of the tunnel exceeds the earth pressure on top of the tunnel, \(\gamma\) is the volume weight of the soil in the stratum. For the layers of soil, take the average volume weight, \(H\) is the buried depth of the tunnel, \(k\) is the lateral earth pressure coefficient.