## 2 Theoretical Model
### Soil Arching in the Embankment  
As shown in Figure 3, a local coordinate system is established where the origin is situated at the surface of the embankment and the downwards is the positive direction of the \(z^{*}\). The pile is rigid relative to soft soil. The differential settlement occurs between the inner cylinder and the outer hollow cylinder under the load of fill. Therefore, friction is generated on the interface and acts downwards on the inner cylinder but upwards on the outer hollow cylinder. The friction can be acquired by the following equation:  
\[f=\beta\sigma_{i}(z^{*})k_{ae}\tan\varphi_{f} \tag{2}\]  
where \(f\) is the friction between the inner cylinder and the outer hollow cylinder; \(\beta\) is the mobilization coefficient of friction; \(\sigma_{i}(z^{*})\) denotes the vertical stress in the inner cylinder; \(k_{ae}\) represents active pressure coefficient in embankment; and \(\varphi_{f}\) is the internal friction angle of the fill. Figure 2: The installation pattern of piles: (**a**) a cylindrical unit cell model, (**b**) square pattern, (**c**) rectangular pattern, (**d**) triangular pattern. For the sake of simplicity, \(\beta\) is usually assumed to be 1 [23; 26]. It does not satisfy the stress condition that the friction between the inner cylinder and the outer hollow cylinder ought to zero at the equal settlement plane. In order to satisfy the stress condition and consider the influence of relative displacement, the author assumes that \(\beta\) varies linearly from 0 at the equal settlement plane to 1 at the pile top. Considering the flatness of the road surface, the case of \(h<h_{e}\) is beyond the scope of this paper. \[k=\beta k_{\alpha e}=\left\{ \begin{array}{cc}0&0<z^{*}\leq h-h_{e}\\ \frac{k_{\alpha e}}{h_{e}}(z^{*}+h_{e}-h)&h-h_{e}<z^{*}\leq h\end{array} \right. \tag{3}\]  
where \(h\) is the height of embankment, and \(h_{e}\) is the height of the equal settlement plane. As Figure 3 shows, the inner cylinder is divided into small pieces with a thickness of \(dz^{*}\), and the equilibrium of vertical force can be expressed as  
\[S_{i}\sigma_{i}(z^{*})+\gamma_{f}S_{i}\mathrm{d}z^{*}+\pi Df\mathrm{d}z^{*}=S_ {i}(\sigma_{i}(z^{*})+\mathrm{d}\sigma_{i}(z^{*})) \tag{4}\]  
\[f=\sigma_{i}(z^{*})k\tan\varphi_{f} \tag{5}\]  
where \(S_{i}\) is the cross-sectional area of the inner cylinder, \(\gamma_{f}\) is the unit weight of the fill, and \(D\) is the maximum diameter of the conical head. When \(z^{*}=h-h_{e}\), \(\sigma_{i}(h-h_{e})=\gamma_{f}(h-h_{e})\), Equations (3) and (5) are substituted into Equation (4), then Equation (4) is integrated from \(z^{*}=h-h_{e}\) to \(z^{*}\), the following formula can be obtained:  
\[\sigma_{i}(z^{*})=\gamma_{f}(h-h_{e})H+\frac{H\sqrt{\frac{\pi}{2}}\gamma_{f} Erf[\frac{\sqrt{T}(z^{*}+h_{e}-h)}{\sqrt{T}}]}{\sqrt{T}} \tag{6}\]  
where \(T\) and \(H\) are the intermediate variables: \(T=\frac{4\tan(\varphi_{f})k_{\alpha e}}{Dh_{e}},H=\exp(T\frac{(h_{e}+z^{*}-h)^ {2}}{2})\). At any horizontal sections in the embankment, the equilibrium of vertical force can be denoted as follows:  
\[m\sigma_{i}(z^{*})+(1-m)\sigma_{o}(z^{*})=\gamma_{f}z^{*} \tag{7}\]  
where \(m\) is the area replacement rate, \(m=S_{i}/(S_{i}+S_{o})\), \(\sigma_{o}(z^{*})\) is the vertical stress in the outer hollow cylinder, \(S_{0}\) is the cross section area of the outer hollow cylinder. Figure 3: The cylindrical unit cell model. The vertical stress \(\sigma_{o}(z^{*})\) can be formulated as  
\[\sigma_{o}(z^{*})=\frac{\gamma_{f}z^{*}-m\sigma_{i}(z^{*})}{1-m} \tag{8}\]  
The pile-soil stress ratio (\(n\)) and the load sharing ratio (\(n_{e}\)) are defined as follows, respectively:  
\[n=\frac{\sigma_{i}(h)}{\sigma_{o}(h)} \tag{9}\]  
\[n_{e}=\frac{\sigma_{i}(h)S_{i}}{\gamma_{f}h(S_{i}+S_{o})} \tag{10}\]  
The relative displacement incurs the friction. The outer hollow cylinder is subjected to an upward friction, which causes the outer hollow cylinder rebound. In contrast, a compression deformation is generated on the inner cylinder.