# Prediction of Ground Surface Settlements Induced by EPB Shield Tunneling in Water-Rich Soft Strata
## 4 Prediction of Short-Term Ground Deformation
### Deformation Induced by Shield Working Loads  
The well-known [PERSON]'s solution [22], i.e., the solution for the concentrated load-induced displacement in a semi-infinite solid, is used for predicting the deformation caused by shield working loads. An integral method based on [PERSON]'s solutions was used to calculate the deformation response of the EPB shield working loads, including the support pressure, shield shell friction, and cutter head rotation.  
In general, the support pressure applied by the EPB shield should be as close to the in situ earth and water pressure to reduce deformations. However, it is worth noting that the applied support pressure is not equal to the sum of earth and water pressure in most circumstances. This pressure gap between them can be viewed as an evenly distributed load on the tunnel face. By double integrating along the Green function generated from [PERSON]'s solution for the support face under the polar coordinate system, the ground vertical deformation induced by the pressure gap \(\Delta p\) can be written as:  
\[u_{z}^{p}=\int_{0}^{R}\!\!\int_{0}^{2\pi}\!\!\Delta p\xi(x-\rho\cos\theta,y,z- \rho\sin\theta)d\theta d\rho \tag{8}\]  
where \(\Delta p\) denotes the pressure gap between the support pressure and in-situ earth and water pressure, \(u_{z}^{p}\) denotes the ground vertical deformation induced by \(\Delta p\), \(\zeta\) denotes the Green function generated from [PERSON]'s solution for the horizontal force, \(\theta\) denotes the angular coordinate, and \(\rho\) denotes the radial coordinate.  
When calculating the ground deformation induced by shield shell friction, the friction is assumed to be an evenly distributed load along the shield shell. Coulomb law of friction, which is by far the most popular friction model, is suggested for calculating the shield shell friction. Similar to the deformation induced by \(\Delta p\), the shell friction-induced vertical deformation can be obtained by integrating along the Green function over the shield shell area:  
\[u_{z}^{f}=\int_{0}^{S}\!\!\int_{0}^{2\pi}\!f\zeta(x-R\cos\theta,y-\zeta,z-R \sin\theta)d\theta d\zeta \tag{9}\]  
where \(f\) denotes the shield shell friction and can be expressed as \(f=\mu F_{n}\), in which \(\mu\) is the friction coefficient, and \(F_{n}\) denotes the normal stress between shield shell and soil. A summary of the friction coefficient \(\mu\) can be obtained from [23] for both clayey and sandy soils.  
According to the cutter head torque collected from the data acquisition system, the load stress acting on the tunnel face can be derived as:  
\[q=\frac{3T}{2\pi R^{3}} \tag{10}\]  
where \(q\) denotes the load stress acting on the tunnel face, and \(T\) denotes the cutter head torque. Then the displacement induced by cutter head rotation can be written as:  
\[\begin{array}{ll}u_{z}^{q}=&\int_{0}^{R}\!\!\int_{0}^{2\pi}\!q\rho\sin\theta \zeta(x-\rho\cos\theta,y,z-\rho\sin\theta)\\ &+4\rho\cos\theta\xi(x-\rho\cos\theta,y,z-\rho\sin\theta)d\theta d\rho\end{array} \tag{11}\]  
where \(\xi\) denotes the Green function generated from [PERSON]'s solution of the vertical force.  
Finally, by superimposing all of the preceding components, the short-term ground deformation during shield driving can be obtained:  
\[U_{z}=u_{z}^{V}+u_{z}^{p}+u_{z}^{f}+u_{z}^{q} \tag{12}\]