## 2 Methods  
### Effective Stress Method (ESM)  
The effective stress method (ESM) was first proposed by [PERSON] and [PERSON] [25]. Pile skin friction is calculated based on the ESM, the expression formula of maximum skin friction, \(f_{m}\), is shown in Equation (1):  
\[f_{m}=\beta\sigma_{v}^{\prime} \tag{1}\]  
where \(f_{m}\) is the maximum skin friction unit pile depth, \(\beta\) is the empirical factor, which is influenced by the nature of surrounding soil of pile, \(\beta=(1-sin\varphi)tan\delta\) for normally consolidation soil, \(\beta=(1-sin\varphi)OCR^{0.5}tan\delta\) for overconsolidated soil, \(\varphi\)' is the effective friction angle, \(OCR\) is the overconsolidated ratio, and \(\delta\) is the friction angle of the pile-soil interface [14]. They also can be determined based on field test as shown in Table 1. \(\sigma_{v}\)' is the mean vertical effective stress of surrounding soil. The relative settlement, which is the vertical displacement difference between soil and pile, varies linearly with depth, as shown in Figure 1. \(S_{p}\), \(S_{s}\), and \(S_{pl}\) are the settlements of pile, soil, and pile bottom, respectively. \(L\) is the length of pile, while \(h\) is the embed depth. The relationship between the skin friction of pile, \(f\), and the relative settlement of pile soil, \begin{table}
\begin{tabular}{c c} \hline \hline
**Soil** & \(\beta\) \\ \hline Clay & 0.2–0.25 \\ Silty soil & 0.25–0.35 \\ Sand & 0.35–0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The value range of empirical factor \(\beta\). \(u\), is ideally elastoplastic. With the increase of \(u\),\(f\) increases linearly. The elastic modulus, \(k\), is constant. When \(u\) reaches the elastic critical value, \(u_{m}\), \(f\) reaches its ultimate value, \(f_{m}\), \(u\) increases further, skin friction reaches plastic stage, and \(f\) is constant, as shown in Equation (2) and Figure 2.  
\[f=\left\{ \begin{array}{l}ku,u<u_{m}\\ f_{m},u>u_{m}\end{array} \right. \tag{2}\]  
Meanwhile, it assumes that the transition section of skin friction from maximum negative to positive is linear. The distribution of skin friction of a single pile is shown in Figure 3a,b, where \(r\) is the depth of neutral point and \(a\) and \(b\) are the depth range in which the NSF and PSF reach the limit value, respectively. The two cases of pile skin friction are discussed in detail, and the closed-form analytical solutions of axial force and displacement of pile shaft are given. Figure 1: The relative settlement of pile soil. Figure 2: The ideal elastoplastic relationship between skin friction of the pile and the relative settlement of pile soil. Case 1: When the relative settlement of pile soil at pile bottom is small, PSF does not reach the plastic stage and the resistance of pile bottom soil is \(N\). The settlement of the surrounding soil is larger than that of the pile shaft within the depth range of \(r\). The NSF occurs in this section and reaches the ultimate value within the depth range of \(a\), as shown in Figure 3a. In this case, the relative settlement of pile soil in the pile bottom, \(S_{pl}<u_{m}\), is shown in Figure 3a. According to the hypothesis mentioned above, the geometric relationship of pile soil settlement and governing equations can be given as Equation (3):  
\[\left\{ \begin{array}{l}a=\frac{u_{0}-u_{m}}{u_{0}+S_{pl}}L\\ r=\frac{u_{0}}{u_{0}+S_{pl}}L\\ C\beta\gamma^{\prime}L^{2}\Big{(}u_{0}-S_{pl}\Big{)}(u_{0}-u_{m})-2 Nu_{0}\Big{(} u_{0}+S_{pl}\Big{)}=0\end{array} \right. \tag{3}\]  
where \(C\) is the perimeter of pile cross-section and \(u_{0}\) is the relative settlement of pile soil in ground surface. When the settlement and pile bottom reaction force satisfied the governing equations above, the plastic depth of skin friction and the location of neutral point can be determined. According to the previous assumptions and boundary conditions, the axial force of pile shaft can be expressed in segments as Equation (4):  
\[\left\{