## 3 Numerical Method and Loading Mode  
### Loading Mode of Ground Surcharge  
During shield tunnel operation, the primary loads include the initial earth pressure from the surrounding ground and an additional load due to ground surcharge. The Boussinesq formula is a mathematical formula applied within the theory of elastic half-space, primarily used for predicting the stress distribution on the ground subjected to point loads. The fundamental assumption of the Boussinesq formula is that the soil is considered an infinite, homogeneous, isotropic elastic half-space. When a point load is applied to the surface of the soil, this formula enables the calculation of the vertical and horizontal stress at any given point. Consequently, this method facilitates the computation of additional loads on tunnels resulting from ground surcharge. As illustrated in Figure 7, the schematic represents a semi-infinite space subjected to a vertical concentrated force (\(F\)). Upon the application of a vertical concentrated force on the surface of a semi-infinite space, an additional stress is generated at point \(M\) within the semi-infinite space. The horizontal additional stress, \(\sigma_{x}\), and the vertical additional stress, \(\sigma_{z}\), can be calculated using Equation (1) and Equation (2), respectively. \[\sigma_{x}=\frac{3F}{2\pi}\left(\frac{x^{2}z}{R^{5}}+\frac{(1-2v)\left[R^{2}- z(R+z)\right]}{3R(R+z)}-\frac{(1-2v)(x-\epsilon)^{2}(2R+z)}{3R^{3}(R+z)^{2}}\right) \tag{1}\]  
\[\sigma_{z}=\frac{3F}{2\pi}\frac{z^{3}}{R^{5}} \tag{2}\]  
\[r=\sqrt{x^{2}+y^{2}+z^{2}} \tag{3}\]  
where \(v\) is the Poisson's ratio of the soil and \(r\) is the distance from the point of concentration to point \(M\). Based on [PERSON]'s formula, a calculation model for the additional loads on tunnels due to surface surcharge can be established. Figure 8 presents a scenario where the surface of the operating tunnel is set up with rectangular ground attachments, whose length and width are \(L\) and \(B\), respectively. The horizontal distance from the center of this surcharge to the tunnel's axis is \(S\), while the magnitude of the ground surcharge is \(P_{0}\), and the tunnel depth is \(H\). Considering the center of the ground surcharge as the origin, the directions parallel and perpendicular to the tunnel are defined as the \(y\)-axis and \(x\)-axis, respectively. We assume that a minute element within the range of surface  
Figure 7: The schematic diagram of the semi-infinite space subjected to a vertical concentrated force. \(P_{0}\)_ded\(\eta\)_, with the center coordinates of the element being (\(\varepsilon\), \(\eta\), \(\zeta\)), and a certain point on the tunnel surface having the coordinates (\(x_{0}\), \(y_{0}\), \(z_{0}\)). According to [PERSON]'s formula, the calculation formula for the additional loads on tunnels due to surface surcharge can be established through integration. The horizontal additional stress \(Q_{x}\) and the vertical additional stress \(Q_{z}\) can be calculated, respectively, according to Formulas (4) and (5). \[Q_{x}=\frac{3P_{0}}{2\pi}\iint\limits_{\Gamma}\left[\frac{\left(x_{0}- \varepsilon\right)^{2}H_{0}}{R^{5}}+\frac{\left(1-2\
u\right)\left[R^{2}-z_{0 }\left(R+z_{0}\right)\right]}{3R^{3}\left(R+z_{0}\right)}-\frac{\left(1-2\
u \right)\left(x_{0}-\varepsilon\right)^{2}\left(2R+z_{0}\right)}{3R^{3}\left(R +z_{0}\right)^{2}}\right]ded\eta \tag{4}\]  
\[Q_{z}=\frac{3P_{0}}{2\pi}\iint\limits_{\Gamma}\frac{z_{0}^{3}}{R^{5}}ded\eta \tag{5}\]  
\[R=\sqrt{\left(x_{0}-\varepsilon\right)^{2}+\left(y_{0}-\eta\right)^{2}+\left( z_{0}-\xi\right)^{2}} \tag{6}\]  
where \(\Gamma\) ranges from \(-B/2\leq\varepsilon\leq B/2\), \(-L/2\)\(\eta\leq L/2\), \(\
u\) is the Poisson's ratio of the soil, and \(R\) is the distance from the minute element to the calculation point. Figure 9 presents the variation curves of surcharge loads for a specific case study. In Figure 9a, the distribution of the vertical surcharge load is shown for a burial depth of 15 m (\(H=15\) m).