## 2 Methods
### Effective Stress Method (ESM)
\begin{array}{l}F(z)=\frac{C\beta\gamma^{\prime}a}{2}z^{2},0<z\leq a \\ F(z)=\frac{C\beta\gamma^{\prime}a}{2(r-a)}\big{(}-z^{2}+2 rz-ar\big{)},a<z\leq r \\ F(z)=\frac{C\beta\gamma^{\prime}}{2}\Big{(}ra-\frac{L}{L-r}(z-r)^{2}\Big{)},r< z\leq L\end{array} \right. \tag{4}\]  
where \(F(z)\) is the axial force of pile in depth of \(z\), \(\gamma^{\prime}\) is the effective weight of surrounding soil, and \(z\) is the depth of soil. From the equation above, it can be seen that when \(z=r\), the axial force reaches the maximum, which is in accord with the distribution characteristics of the axial force of the pile shaft in this condition. With further consolidation of the surrounding soil, NSF and the drag load of the pile increases. With an increase in the pile shaft settlement, the ultimate PSF of the pile bottom occurs in the depth range of \(b\), as shown in Figure 3b. When the relative settlement, \(S_{pl}\)  
Figure 3: Distribution of skin friction of a single pile: (**a**) Case 1; (**b**) Case 2.  
reaches the elastic critical value, \(u_{m}\), the PSF reaches the ultimate value. Similar geometric equations and the governing equation can be expressed as in Equation (5):  
\[\left\{ \begin{array}{l}a=\frac{u_{0}-u_{m}}{u_{0}+S_{pl}}L\\ r=\frac{u_{0}}{u_{0}+S_{pl}}L\\ b=L-2r+a\\ C\beta\gamma^{\prime}(ra-(L-b)(L-r-b)-(2L-b)b)-2N=0\end{array} \right. \tag{5}\]  
Similarly, the governing equation of the axial force can be obtained as Equation (6):  
\[\left\{ \begin{array}{l}F(z)=\frac{C\beta\gamma^{\prime}}{2}z^{2},0<z\leq a \\ F(z)=\frac{C\beta\gamma^{\prime}a}{2(r-a)}\big{(}-z^{2}+2 rz-ar\big{)},a<z\leq r \\ F(z)=\frac{C\beta\gamma^{\prime}}{2}\bigg{(}ra-a\frac{(z-r)^{2}}{r-a}\bigg{)},r<z\leq L-b\\ F(z)=\frac{C\beta\gamma^{\prime}}{2}\bigg{(}ra-a\frac{(L-b-r)^{2}}{r-a}+z(z-L +b)\bigg{)},L-b<z\leq L\end{array} \right. \tag{6}\]  
The calculating parameters involved in the axial force equation of above two cases are easy to obtain at the engineering site. In order to verify the applicability and rationality of the theoretical method, the ESM was used to calculate the skin friction of different single piles in reference [15]. The skin friction of a single pile in situ was calculated and compared to the measured value. There are two kinds of field test piles, one is a conventional single pile and the other is a single pile with an asphalt coated surface. Because asphalt reduces the friction between pile and soil, \(\beta\) = 0.1 for the coated pile, which is lower than the range of normal experience. For a normal uncoated pile, \(\beta\) = 0.25. Meanwhile, a two-dimensional (2D) pile-soil model was established by ABAQUS and the NSF of the single pile was also analyzed. The FEM model is a two-dimensional axisymmetric model with a vertical dimension of 40 m and a horizontal dimension of 10 m. The mesh of the soil near the pile was 0.1 m \(\times\) 1 m, while the mesh of the soil at the border was 1 m \(\times\) 1 m. The mesh size gradually increased outward. The friction coefficient was \(\mu\) = 0.1 for the coated pile, and \(\mu\) = 0.25 for the normal uncoated pile in the FEM model. The parameters involved in the calculation and numerical simulation are shown in Tables 2 and 3. The results are shown in Figure 4. The compressibility of the soil layer in the field test was relatively large, which had an obvious negative friction resistance effect on the pile shaft. Both the test data and calculation results reflect this phenomenon. The compressibility of the soil within the range of 0-10 m should be improved, with methods such as grouting. Furthermore, a general agreement among the results of the ESM results, FEM results and field measurements demonstrates that the calculation method of pile side friction presented in this paper is reliable, so the present method and the 2D FEM model can be used to evaluate and analyze the pile skin friction.