## 3 FEA Model and Validation
### Bending Moment of Lining Segment  
Figure 13 illustrates the changes in the lining bending moment of the segments in different cases. As can be seen in Figure 13, the grouting pressure of the second tunnel advance has significant effect on the bending moment of ER60 in the first tunnel. However, similar to the influence on the first tunnel convergence, the face supporting pressure has little influence on the bending moment of the first tunnel, since the face supporting pressure mainly affects the stress in the axial direction of the tunnel. As the grouting pressure increases, the positive bending moment decreases at the lining sidewall (see Figure 13a). This can be attributed to the unloading effect caused by insufficient grouting pressure in the second tunnel. The maximum positive bending moment in all cases is 113.4 kN\(\cdot\)m in Case 1, with the minimum magnitude of 85.0 kN\(\cdot\)m in Case 4. In all cases, the maximum magnitude of negative bending moment is \(-91.82\) (Case1), and the minimum is \(-63.03\) kN\(\cdot\)m (Case 4). In Figure 14, the relationship between the bending moment \(M\) and the grouting pressure index \(K_{g}\) is fitted with linear functions (see Equation (19)). The similarity \(R^{2}\) in different fitting functions are over 0.99. \[M=a+b\cdot K_{g} \tag{21}\]  
where \(a\) and \(b\) are the linear fitting parameters. Figure 12: Convergence of ER60 in different cases: (**a**) influence of grouting pressure on ER60 convergence; (**b**) influence of face supporting pressure on ER60 convergence. Figure 13: Moment of E60 (C/D = 0.648) under different cases: (**a**) under different grouting pressures; (**b**) under different supporting pressures. In Figure 14a, the max negative bending moment \(M_{n}\) on the top of lining decrease with increasing \(K_{g}\). In addition, with the decrease of C/D, the slop \(b_{j}\) and the magnitude of intercept \(a_{j}\) decrease. Figure 14: Influence of grouting pressure on lining bending moment in different cases: (**a**) the minimum negative bending moment at ER60 top; and (**b**) the maximum positive bending moment at ER60 sidewall. In Figure 14b, the max positive bending moment on the EB sidewall lining decreases with the increase of \(K_{g}\). With the decrease of C/D, the magnitude of slop \(b_{i}\) and the intercept \(a_{i}\) of different fitting lines decrease. Notably, all fitting lines intersect at the point adjacent to \(K_{g}=1\) (i.e., Case 0) in which the grouting pressure during the second tunnel construction is equal to \(G_{0}\), revealing that Case 0 is the most suitable case since the variation of the bending moment in the first tunnel is stable with the TBM advance of the second tunnel. Figure 15 shows that the relationship between face supporting pressure and the bending moment under different tunnel depth can be fitted by Lognormal function:  
\[M=M_{0}+\frac{A}{\sqrt{2\pi}wK_{g}}e^{\frac{-|\ln K_{g}/K_{c}|^{2}}{2w^{2}}} \tag{22}\]  
where \(M_{0},\ A,\ w,\ K_{c}\) are the fitting parameters in the Lognormal function with the details demonstrated in Table 4.  
As shown in Figure 15a, when C/D exceeds 0.56, as the grouting pressure increases, the bending moment on the first tunnel side-wall firstly increases and then continuously decreases with the maximum value at \(K_{s}=0.7\). This can be explained as that the first tunnel bending moment can be affected by the loosened zone of the second tunnel, which formed by the soil arching effect. When the supporting pressure is greatly reduced, it not only affects the soil within the tunnel, but also the soil above the tunnel, resulting in the change of the loosened zone [47]. When C/D is lower than 0.56, the bending moment of the first tunnel continuously decreases as the supporting pressure increases. In Figure 15b, the bending moment can also be fitted by a lognormal curve, as expressed in Equation (20). With the increase of Ks, the bending moment continuously increases. When C/D is greater than 0.58, the bending moment value increases as the C/D decreases.