# Analysis of the Impact of the New Two-Lane Shield Tunnel Underpass on the Existing Tunnels
## 2 Theoretical Analysis of Vertical Settlement in Underpass Tunnels
### Modification of [PERSON]'s Formula  
In predicting surface settlement caused by subway tunnel construction, the classical [PERSON]'s formula assumes that the surface settlement resulting from underground excavation follows a normal distribution curve. However, due to the idealized assumptions underlying the formula (e.g., homogeneous linear elastic soil), its accuracy may be affected by the complexity of geological conditions in practical applications. To enhance the applicability of [PERSON]'s formula, many researchers have proposed revisions and improvements. [PERSON] et al. supplemented [PERSON]'s formula according to some measured settlement data; they superimposed the ground settlement caused by the shield construction of the left and right tunnels on each other to ascertain the settlement curves as shown in Figure 2, and obtained the formula for the ground settlement induced by the construction of the two tunnels at the same depth of burial [18]:  
\[S(\mathrm{x})=S_{\mathrm{max}}\exp\left[-\frac{(x-L/2)^{2}}{2i^{2}}\right]+S_{ \mathrm{max}}\exp\left[-\frac{(x+L/2)^{2}}{2i^{2}}\right] \tag{1}\]  
where \(S(x)\) is the settlement value at the location where the horizontal distance between the center of symmetry of the axis of the new twin-lane tunnel and the soil cross-section of the strata is, mm; \(S_{\mathrm{max}}\) is the maximum value of surface settlement caused by the construction of the shield in the single-lane tunnel, which is calculated as, \(S_{\mathrm{max}}=\frac{\pi D^{2}V_{1}}{4\sqrt{2\pi i}}\), mm; \(D\) is the diameter of the excavated tunnel, m; \(V_{1}\) is the rate of loss of the strata and \(i\) is the width of the sinkhole, m; and \(L\) is the spacing of the center of the new twin-lane tunnel, m.  
In terms of predicting subsurface settlement, [PERSON] et al. proposed that the width of a sinkhole \(i\) at any depth \(z\) can be expressed as [19]:  
\[i=K(z_{0}-z) \tag{2}\]  
Figure 1: Schematic diagram of the calculation model. Figure 2: Schematic diagram of ground settlement underneath two-lane shield tunnel. where \(K\) is the width parameter of the sedimentation tank, a dimensionless parameter mainly related to the soil property, and \(z_{0}\) is the depth of tunnel, m.  
It can be seen from formulas (1) and (2) that under different geological conditions, the width parameter \(K\) of the sedimentation tank and the formation loss rate \(V_{1}\) are important parameters affecting the settlement value, and the values can be adjusted to meet the engineering needs of specific areas. For the width parameter of the sedimentation tank, [PERSON] et al. obtained an empirical formula in the form of a hyperbola by fitting all the test and measured data of clay and sandy soil strata [20]:  
\[K(z)=\frac{K_{s}-c\cdot z/z_{0}}{z/z_{0}} \tag{3}\]  
where \(K_{s}\) is a parameter for the width of the surface settlement trough, \(c=\partial i/\partial x\) is a parameter related to the ground conditions, and the fitting of the empirical parameters derived in clay soil resulted in \(c=0.323\) and in sandy soil \(c=0.197\). A tunnel construction ground loss coefficient is introduced, which is related to the directional angle between the new tunnel and the existing tunnel, i.e.,:  
\[V_{1}^{\alpha}=\lambda^{\alpha}\lambda^{\delta}V_{1} \tag{4}\]  
where \(V_{1}\) represents the formation loss coefficient of the settlement curve of the existing line and \(\lambda^{\alpha}\) is the geometric correction factor for the formation loss rate of natural subsidence by considering the incline angle of the strike. The geometric correction coefficient of formation loss rate considering the influence of the angle between the tunnel and the existing tunnel can be defined and calculated by the following formula: \(\lambda^{\alpha}=\frac{1}{\cos\alpha};\lambda^{\delta}\) refers to the reduction coefficient of formation loss rate caused by external subsidence reduction measures such as grouting and lifting of existing tunnels.