## 2 Restraint Mechanism of Isolation Pile
### Analytical Solution  
In this study, the Loganathan formula [19] is adopted to calculate the surface settlement trough without considering the influence of isolation piles. The vertical displacement of soil can be obtained by:  
\[\omega_{L}=R^{2}\left\{-\frac{y-h}{x^{2}+\left(y-h\right)^{2}}+\frac{(3-4\mu) (y+h)}{x^{2}+\left(y+h\right)^{2}}-\frac{2y\left[x^{2}-\left(y+h\right)^{2} \right]}{\left[x^{2}+\left(y+h\right)^{2}\right]^{2}}\right\}\frac{4 gR+g^{2}}{4 R^{2}}\cdot\exp\left[-\frac{1.38x^{2}}{\left(h+R\right)^{2}}-\frac{0.69y^{2}}{h^{2}}\right] \tag{1}\]  
where \(\omega_{L}\) is the vertical displacement of soil induced by shield tunnelling without considering the effect of isolation pile (m), \(h\) is the depth of the tunnel axis from the ground surface  
Figure 1: Pile-soil-tunnel interaction mechanism.  
(m), \(R\) is the radius of the tunnel (m), \(g\) is the gap parameter, \(x\) is the lateral distance from the tunnel axis (m), and \(y\) is the depth measured from the ground surface (m).  
The vertical displacement of soil for a vertical line loading upward beneath the surface of a semi-infinite plane can be calculated by:  
\[\omega_{M}=\frac{P}{8\pi G(1-\mu)}\Bigg{[}(3-4\mu)\ln\frac{r_{1}}{r_{2}}+\frac{ x^{2}}{r_{1}^{2}}-\frac{x^{2}}{r_{2}^{2}}\Bigg{]}-\frac{P(1-\mu)}{\pi G}\ln r_{1}+ \frac{P(y-a)^{2}}{2\pi Gr_{1}^{2}}+\frac{Pya\Big{[}x^{2}-(y-a)^{2}\Big{]}}{4\pi G (1-\mu)r_{1}^{4}}+u_{0} \tag{2}\]  
where \(\omega_{M}\) is the vertical displacement of soil caused by the vertical line load (m), \(P\) is the vertical line load per unit length (N/m), \(G\) is the shear modulus (N/m\({}^{2}\)), \(a\) is the distance from load application point to ground surface (m), \(u_{0}\) is the constant related to rigid body displacement (m), \(\mu\) is the Poisson's ratio, \(r_{1}\) is the distance from the observation point to load application point (m), and \(r_{2}\) is the distance from the observation point to load application image point (m).  
\(r_{1}\) and \(r_{2}\) can be expressed as:  
\[r_{1}=\Big{[}x^{2}+(y-a)^{2}\Big{]}^{\frac{1}{2}} \tag{3}\]  
\[r_{2}=\Big{[}x^{2}+(y+a)^{2}\Big{]}^{\frac{1}{2}} \tag{4}\]  
Equation (2) can be expressed as Equation (5) after coordinate transformation.  
\[\begin{array}{l}{\omega_{M}}^{\prime}=\frac{P}{8\pi G(1-\mu)}\bigg{[}(3-4\mu )\ln\frac{R_{1}}{R_{2}}+\frac{x_{1}^{2}}{R_{1}^{2}}-\frac{x_{1}^{2}}{R_{2}^{2 }}\bigg{]}-\frac{P(1-\mu)}{\pi G}\ln R_{1}\\ +\frac{P(y-a)^{2}}{2\pi Gr_{1}^{2}}+\frac{Pya\Big{[}x_{1}^{2}-(y-a)^{2}\Big{]} }{4\pi G(1-\mu)R_{1}^{4}}+u_{0}\end{array} \tag{5}\]  
where \(\omega_{M}^{{}^{\prime}}\) is the vertical displacement of soil caused by restraint effect of isolation pile (m), \(x_{1}\) is the lateral distance from the tunnel axis after coordinate transformation (m), \(x_{1}=x-l\), \(l\) is the distance from isolation pile to tunnel axis (m), \(R_{1}\) is the distance from the observation point to load application point after coordinate transformation (m), and \(R_{2}\) is the distance from the observation point to load application image point after coordinate transformation (m).  
Figure 2: Calculation diagram of Melan solution: (**a**) Melan solution and (**b**) this study.  
\(R_{1}\) and \(R_{2}\) can be expressed as:  
\[R_{1}=\left[\left(x_{1}-l\right)^{2}+\left(y-a\right)^{2}\right]^{\frac{1}{2}} \tag{6}\]  
\[R_{2}=\left[\left(x_{1}-l\right)^{2}+\left(y+a\right)^{2}\right]^{\frac{1}{2}} \tag{7}\]  
The vertical displacement of soil induced by shield tunnelling considering the restraint effect of adjacent isolation pile can be expressed as Equation (8). When \(y\to 0\), the results degenerate into surface settlement trough.  
\[\omega=\omega_{L}+\omega_{M}{}^{\prime} \tag{8}\]