# Optimization Study on Longitudinal Joints in Quasi-Rectangular Shield Tunnels
## 5 Parametric Study
### Influence of Bolt Properties  
As mentioned before, improving the bolt positions to increase the lever arm for balancing the exerted bending moment has a significant effect on the joint rotation behavior for the second and the third stages. For a fixed distance between the bolts and the core segment edges, the bolt diameter and the number of bolts can be varied to see the influence  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multirow{2}{*}{**Axial Force**} & \multicolumn{2}{c}{**Positive Cases**} & \multicolumn{2}{c}{**Negative Cases**} \\ \cline{3-6}  & & **Type-A** & **Type-BPos** & **Type-A** & **Type-BNeg** \\  & & **(kN\(\cdot\)m/m)** & **(kN\(\cdot\)m/m)** & **(kN\(\cdot\)m/m)** & **(kN\(\cdot\)m/m)** \\ \hline \multirow{3}{*}{End of the first stage} & 333 & 25.4 & 19.9 & 39.4 & 33.9 \\  & 667 & 50.9 & 39.8 & 78.9 & 67.8 \\  & 1000 & 76.3 & 59.7 & 118.3 & 101.7 \\ \hline \multirow{3}{*}{End of the second stage} & 333 & 42.3 & & 55.0 & \\  & 667 & 84.7 & & 110.0 & \\ \cline{1-1}  & 1000 & 127.0 & & 165.0 & \\ \hline \hline \end{tabular}
\end{table}
Table 6: Summary of the bending moments at the ends of different stages.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Rotational Stiffness in Positive Cases**} & \multicolumn{2}{c}{**Rotational Stiffness in Negative Cases**} \\ \cline{2-7}
**Axial Force** & **Type-A** & **Type-BPos** & **Increase Rate** & **Type-A** & **Type-BNeg** & **Increase Rate** \\ \hline
333 & 22,591 & 37,809 & 67\% & 17,030 & 33,490 & 97\% \\
667 & 22,069 & 37,404 & 69\% & 16,076 & 32,598 & 103\% \\
1000 & 21,898 & 37,103 & 69\% & 14,604 & 30,106 & 106\% \\ \hline \hline \end{tabular}
\end{table}
Table 7: Summary of the rotational stiffnesses during the third stage under different axial forces (stiffness unit: kN\(\cdot\)m/m/rad, axial force unit: kN\(/\)m).  
on the joint rotational behavior. The essence of these two methods is to enhance the bolts' deformation resistance to a tensile force. The changes of the bolt diameter or the number of bolts might need shape changes of the DJIPs and redesign of the joint. In order to avoid these changes, the Young's modulus of the bolts in the model is adjusted as the only variable to simulate the effect of the enhancement of the bolts' deformation resistance. The bolt diameter of the studied joint is 33 mm and the Young's modulus is 200 GPa. The [PERSON]'s modulus is reset to 372 and 661 GPa to simulate the elongation of bolts with 45 and 60 mm diameters under the same tensile force conditions. The increase of the Young's modulus could also be regarded as more bolts to resist the same tensile force. It has to be pointed out that this simplified method aims to simulate the effect of the bolt elongation resistance on joint rotations under a normal bending moment range, rather than the effects related to bolt strength or joint bearing capacity. Only the results from positive cases under the constant 667 kN/m axial force are presented in Figure 23 for brevity. The bolts' stress curves for the adjusted moduli are almost identical to that for the original modulus. Due to the modulus increase, identical stresses result in smaller bolt elongations and, consequently, in smaller joint openings. Therefore, the joint stiffness increases, as shown in Figure 23. The average rotational stiffnesses during the third stage with the [PERSON]'s moduli of 372 and 661 GPa are summarized in Table 8. For the positive cases, the average stiffness at the third stage increases by 15% and 23%, respectively. Correspondingly, the stiffness increases for the negative cases are 18% and 30%.