# Optimization Study on Longitudinal Joints in Quasi-Rectangular Shield Tunnels
## 5 Parametric Study
### Internal Forces in the Joint's Vicinity  
The axial force in the considered type of quasi-rectangular tunnel ranges from 500 to 1200 kN for a segment width under the targeted overburden from 7 m to 17 m [30; 31; 32]. Therefore, the axial force is kept constant at 400, 800, and 1200 kN in the model calculations, namely 333, 667, and 1000 kN/m after normalization. Figure 20 presents the joint rotation curves of both original and improved bolt position joints. Figure 21 presents the bolt stress curves for the improved joints. The bolt stress curve of the original joint under 667 kN/m axial force is added to compare the effect of the improvement. Figure 22, taking the example of a positive case, shows the stress distributions at the joint section. At the first stage, when the bending moment is small, the rotation curves are similar. This means that joint rotations are not influenced by the axial forces at this stage. It is worth noting that, at this stage, the rotational stiffnesses are quite large and the bolts do not take any force. During this period, the eccentricity \(h_{e}\) resulting from the combined effect of the exerted moment and axial force, as shown in Figure 22, is small, and the application point of the resultant force is close to the central axis of the joint section. The whole joint core section is under compression, but its outer side is more stressed than the inner side (stress distribution-1 in Figure 22). As the bending moment keeps increasing, the eccentricity \(h_{e}\) increases, and the whole core section is not compressed anymore, leading to the joint opening (stress distribution-2 in Figure 22). The variation of the joint opening is mainly caused by the compressive deformations. This means that the joint behavior at this stage is determined by the height of the joint core section. At the second stage, when the neutral axis moves above the position of the bolt axis, the bolts start being tensioned and are involved in the joint rotation (stress distribution-3 in Figure 22). From Figures 20 and 21, it can be seen that the bolt stress increases quickly while the rotational stiffness decreases gradually. At the third stage, the eccentricity \(h_{e}\) increases further and exceeds the distance between the edge of the outer section and the joint central axis, namely \(h_{1}-h_{c}\) (stress distribution-4 in Figure 22). However, the stress distribution in the compression zone at the joint section cannot exceed this distance. From the rotation and bolt stress curves, at this stage, the rotation curve appears to be almost linear, while the bolt stress also follows a linear trend. For the same joint type under different axial forces, the stiffnesses at the this stage appear to be the same. Figure 20: Evolution of the joint rotation with increasing bending moment under constant axial force: (**a**) positive cases; (**b**) negative cases. From Figure 22, the bending moment values at the end of the first stage and the second stage can be evaluated. As the stress level at the first stage is not large, based on the assumption of a plane strain distribution, the stress distribution is assumed to have a triangular shape, and it balances the exerted axial force \(N\) and bending moment \(M\). When the neutral axis is at the bolt axis, the first stage ends. The corresponding moment value can be calculated with Formula (2). For the end of the second stage, the eccentricity \(h_{e}\) equals \(h_{1}-h_{c}\), and the corresponding moment value can be calculated from Formula (3). The bending moments at the ends of these two stages are summarized in Table 6 for all calculated cases. Only the results for the 667 kN/m axial force cases are added to Figure 21 as examples to show the trend of the stress curves at different stages. Figure 21: Evolution of the bolt stresses with increasing bending moment under constant axial force: (**a**) positive cases; (**b**) negative cases. Figure 22: Stress distributions at the joint core section in a positive case.