# A New Design Method of Shield Tunnel Based on the Concept of Minimum Bending Moment
## 1 Introduction  
The cross-sections for most of the existing shield tunnels all over the world are generally circular [1; 2; 3; 4]. Additionally, there are other cross-sections for shield tunnels, such as transverse oval, rectangular, quasi-rectangular (applied in Ningbo Metro Line 3, Zhejiang province, China), semicircle, horseshoe-shaped, double-circle-shaped, and tri-circle-shaped, as shown in Figure 1. The main load for a shield tunnel as an underground structure is the earth pressure [5]. Normally, the tunnel is subjected to larger vertical earth pressure than horizontal earth pressure, and both vertical and horizontal earth pressure increase as the depth of the ground increases. So, the cross-section of a shield tunnel inevitably undertakes bending moment. Especially in the soft soil area, there is little horizontal soil resistance at the side of the tunnel [6; 7; 8]. That means that when even layer transverse oval deformation occurs, the difference between the vertical earth pressure and the horizontal earth pressure is large. According to the design specification of metro [9; 10], the largest oval deformation for the new constructed shield tunnel upon the acceptance check was 5D%, where D represents the outer diameter of the tunnel. However, for some soil conditions, the oval deformation will fail to meet this code, especially in the soft soil area. The tunnel is a slender underground structure, so the shield tunnel can be considered as a curved beam structure from the perspective of plane strain, and the length of the curved beam is far larger than the height of the curved beam. The deformation of the beam structure is mainly caused by bending moment. For the cross-section of the shield tunnel, the rotation of the segment longitudinal joint and the bending moment are the main factors that cause transverse oval deformation [11; 12]. Besides, when the shield tunnel is under the bending moment, the segment longitudinal joints are prone to segment corner damages and will stretch up [13; 14], which will reduce the pressure stress on the waterproof sealing washer or, even worse, the waterproof sealing washer will completely open, causing joint waterproof failure [15; 16; 17; 18]. Figure 2 shows the excessive deformation of the segmental joints and the related water-leakage issues. Besides this, under bending moment, the connecting bolts at the segment longitudinal joints will be stretched. Under the over-large bending moment, plastic deformation will occur for the thread of the connecting bolts, causing damages to the segment longitudinal joints [19; 20; 21; 22]. One study [23] found that the segment joints were the weak parts of the test structure, through a full-scale test analysis of the bearing capacity of the shield tunnel structure assembled by joints, and the plastic development of the segment joints easily caused structural damage. It illustrates that most of the diseases are caused by the bending moment of the cross-section of the shield tunnel [19; 24; 25; 26]. In all of the existing studies, the properties of a determined section of shield tunnel, as well as for other underground structures, such as culvert pipe  
Figure 1: Typical cross-section of shield tunnels:(**a**) Circular; (**b**) Double-circle-shaped; (**c**) Tri-circle-shaped; (**d**) Semicircle; (**e**) Horseshoe-shaped; (**f**) Transverse oval; (**g**) Rectangular; (**h**) Quasi-rectangular. and underground pipe (most of the pipe in the ground is circular), have been analyzed, but whether the cross-section is suitable for resisting transverse deformation has not been taken into consideration. From the above-mentioned diseases caused by bending moment, it is necessary to design a rational cross-section in order to decrease the bending moment as much as possible.