## 2 Calculation Method for Surface Settlement
### Surface Settlement from Tunnel Excavation (\(u_{1}\))  
When an extra-large cross-section tunnel passes beneath airport runways, railway embankments, and  
The following assumptions are made: the overburden load on the tunnel is considered to be the self-weight of the soil and is evenly distributed; the lateral pressure is assumed to be uniformly distributed; and the surface settlement is equal to the settlement at the tunnel vault. 1. Calculation of transverse settlement in tunnel  
The tunnel primary support with a unit width is assumed to be a frame structure. The bottom is modeled as an elastic foundation beam, while the top and sides are subjected to earth pressure. The calculation model is depicted in Figure 1.  
The bending differential equation is established as follows:  
\[\frac{d^{2}y(x)}{dx^{2}}=\frac{q}{2E_{1}I_{1}}(ax-x^{2})-\frac{\lambda qb^{2}} {2} \tag{2}\]  
where q is the overburden load on the tunnel; E\({}_{1}\)I\({}_{1}\) is the primary support stiffness; a is the tunnel width; b is the tunnel height; \(\lambda\) is the lateral pressure coefficient. Integrating Equation (2) and applying the boundary conditions yields:  
\[y(x)=\frac{q}{24E_{1}I_{1}}(a^{3}x-2 ax^{3}+x^{4})+\frac{\lambda qb^{2}}{4E_{1} I_{1}}(x^{2}-ax) \tag{3}\]  
Substituting x = a/2 into Equation (3) to obtain the maximum deflection:  
\[u_{1-Cross-section}=\frac{q}{E_{1}I_{1}}\bigg{(}\frac{5a^{4}}{384}-\frac{\lambda a ^{2}b^{2}}{16}\bigg{)} \tag{4}\]  
The load from the unclosed primary support section is shared equally by the soil in front of the tunnel face and the closed primary support structure. The calculation model is illustrated in Figure 2.  
Figure 1: Mechanical calculation model of tunnel cross-section. Figure 2: Load calculation model of pipe-roof. The calculation and analysis demonstrate that the influence range of the load transferred from the pipe roof to the primary support and the surrounding rock in front of the tunnel face is approximately 5 m. The additional load applied to the primary support with a unit width is 0.1 yhL. Therefore, there is q = \(\gamma\)h + 0.1\(\gamma\)hL. \[u_{1-Cross-section}=\frac{\gamma h}{E_{1}I_{1}}\left[\left(1+\frac{L}{10}\right) \times\left(\frac{5a^{4}}{384}-\frac{\lambda a^{2}b^{2}}{16}\right)\right] \tag{5}\]  
where L is the distance from the tunnel face to the closed primary support structure. If temporary supports are installed in the tunnel, they can be considered as an enhancement to the stiffness of the primary support. The calculation formula for the equivalent stiffness is as follows:  
\[EI=\left(1+\frac{3 nE_{1}I_{1}}{a^{3}k_{1}}\right)\times E_{1}I_{1} \tag{6}\]  
where EI is the equivalent stiffness of temporary and primary supports; \(n\) is the number of vertical temporary supports; k\({}_{1}\) is the stiffness of temporary support, which is calculated using the column stability method with the formula k\({}_{1}\) = (3E\({}_{3}\)I\({}_{3}\))/b\({}^{3}\), where E\({}_{3}\)I\({}_{3}\) is the stiffness of vertical temporary supports. Calculation of longitudinal settlement in tunnel  
Figure 2 illustrates the mechanical calculation model of the pipe roof along the tunnel's longitudinal direction, where the additional load applied to the soil per unit width is expressed as 0.1\(\gamma\)hL. Based on the Winkler hypothesis, the soil settlement at the tunnel face can be calculated using Equation (7):  
\[u_{1-ahead}=\frac{\gamma h}{10k} \tag{7}\]  
where \(\gamma\) is the unit weight of the soil; h is the tunnel depth; k is the elastic resistance coefficient of the soil. The pipe roof in the unclosed primary support section is assumed to be a beam with both ends fixed. Based on this assumption, the maximum deflection can be calculated as follows:  
\[\frac{d^{2}y(x)}{dx^{2}}=\frac{q}{2E_{1}I_{1}}(ax-x^{2})-\frac{\lambda qb^{2}} {2} \tag{2}\]0  
\[\frac{d^{2}y(x)}{dx^{2}}=\frac{q}{2E_{1}I_{1}}(ax-x^{2})-\frac{\lambda qb^{2}} {2} \tag{2}\]1  
where E\({}_{2}\)I\({}_{2}\) denotes the stiffness of the pipe roof.