## 3 Theoretical Calculation of the Additional Stress Caused by Shield Construction
### Theoretical Calculation Method
#### 3.2.3 Additional Stress Caused by Shield Ground Loss  
[PERSON] [33] proposed a \"source-sink method\" to calculate the variety of stress fields caused by ground loss in linear elastic semi-infinite bodies. The displacement component generated by an area with a radius a at the point \((x_{0},\,y_{0},\,z_{0})\) is Equation (5) at the point \((x,\,y,\,z)\), and the displacement component generated by mirror image is defined in Equation (6) as follows:  
\[\left\{\begin{array}{l}S_{x1}=-\frac{a^{3}(x-x_{0})}{3r_{1}{}^{3}}\\ S_{y1}=-\frac{a^{3}(y-y_{0})}{3r_{1}{}^{3}}\end{array},\right. \tag{5}\]  
\[\left\{\begin{array}{l}S_{x1}=-\frac{a^{3}(x-x_{0})}{3r_{2}{}^{3}}\\ S_{y1}=-\frac{a^{3}(y-y_{0})}{3r_{2}{}^{3}}\end{array},\right. \tag{6}\]  
where \(r_{1}=\left[\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0} \right)^{2}\right]^{1/2}\)\(r_{2}=\left[\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z+z_{0} \right)^{2}\right]^{1/2}\), \(a\) is radius of void at the point \((x_{0},\,y_{0},\,z_{0})\), and \(S_{x1}\), \(S_{x1}\), \(S_{x3}\), and \(S_{x4}\) are components of displacement.  
Combined with the basic equation for elastic mechanics, the stress generated by these two steps can be calculated by Equation (7) as follows:  
\[\left\{\begin{array}{l}\sigma_{x}=\frac{E_{\mu}}{(1+\mu)[1-2\mu]}\left( \varepsilon_{x}+\varepsilon_{y}+\varepsilon_{z}\right)+2G\varepsilon_{x}\\ \sigma_{y}=\frac{E_{\mu}}{(1+\mu)(1-2\mu)}\left(\varepsilon_{x}+\varepsilon_{y}+ \varepsilon_{z}\right)+2G\varepsilon_{y}\end{array},\right. \tag{7}\]  
where \(E\) is the elastic modulus, \(\mu\) is the Poisson ratio, and \(G\) is the shear modulus of the soil.  
compressive effect of cutterhead on soil is underestimated [27; 28]. [PERSON] and [PERSON] [34; 35] pointed out that the extrusion effect is divided into two parts--one part is produced by the roll of cutterhead, and the other is formed by extrusion force produced by cutting into the soil. The calculated formula is as follows:  
\[P_{1}=\frac{10.31(1-\mu)E_{u}\pi\
u(1-\xi)^{2}}{(1+\mu)(3-4\mu)Dk\omega}+\Delta p ^{\prime}, \tag{16}\]  
where \(P_{1}\) is the frontal additional thrust, \(\mu\) is the Poisson ratio, \(E_{u}\) is the undrained elastic modulus, \(\
u\) is the aperture opening ratio of cutterhead, \(D\) is the diameter of cutterhead, \(k\) is the amplitude number of closed part of cutterhead, \(\omega\) is the rotational speed of cutterhead of the shield machine, and \(\Delta p^{\prime}\) is the extrusion force produced by cutting into the soil.  
2) The friction between the shield shell and the soil (P\({}_{2}\))  
For the current study results, the friction between the shield shell and the soil is assumed to be uniform distribution along the radial direction of the shield, and the vertical pressure of the center of the tunnel is defined as the mean pressure. There is a larger difference with the actual condition. Based on the research results of the friction of the pile-soil interface and considering the friction softening between the shield shell and the soil, the expression is as follows [27; 28]:  
\[P_{2}=\eta\sigma_{\theta}\tan\theta, \tag{17}\]  
where \(\eta\) is the softening coefficient, \(\sigma_{\theta}\) is normal stress at the \(\theta\) angle of the shield shell, and \(\phi\) is the internal friction angle.