## 2 Theoretical Derivation
### Solution of Piles under Single Vertical Load
#### 2.2.2 Nonlinear Solution of Single Pile under Vertical Load  
The expansive soil would swell after immersion. The soil around the pile exerts an uplift force above the pile's neutral point (i.e., the vertical friction resistance equals 0), and it exerts a pulldown force below the neutral point. When a vertical load is applied to the pile top, the shear direction remains unchanged above the neutral point. However, the direction of relative movement between the pile and soil is opposite to that during immersion below the neutral point, and reverse shear will occur at the pile-soil interface. Figure 2: Schematic diagram of differential dispersion of single pile after immersion. Reverse shear can be considered as an unloading process. [PERSON] et al. [21] and [PERSON] et al. [22] believed that the unloading stiffness equals the initial tangent stiffness during the unloading process. Therefore, the pile-soil interface model in Figure 1 can be improved. As shown in Figure 3, it can uniformly be expressed the load transfer model of pile-soil under vertical loading after immersion. \(OA\) and \(OB\) are the immersion swelling stages, respectively; \(AC\) is the continuous loading section; \(BD\) is the unloading section; and \(DE\) is the reverse loading section. For the convenience of description, the pile-soil interface load transfer model can be expressed as:  
\[\tau(z)=\left\{ \begin{array}{cc}\frac{G_{s}(z)}{r_{0}\ln(r_{m}/r_{0}-\psi/1- \psi)}&OA,AC\\ -\frac{G_{s}|s(z)|}{r_{0}\ln(r_{m}/r_{0}-\psi/1-\psi)}&OB\\ k_{z}(s(z)-s_{p})&BD\\ \frac{G_{s}(s(z)+|s_{p}(z)|)}{r_{0}\ln(r_{m}/r_{0}-\psi/1-\psi)}&DE\end{array} \right. \tag{17}\]  
where \(w_{load}\) is a new displacement of pile under vertical load after immersion; Here, vertical pile-soil relative displacement \(s(z)=w_{swell}+w_{load}-w_{s}(z)\); and \(s_{p}(z)\) is residual displacement of pile-soil interface. The pile control equation can also be expressed as:  
\[\frac{d^{2}w(z)}{dz^{2}}-\lambda^{\prime 2}s(z)=0 \tag{18}\]  
The boundary conditions of the pile top and pile tip are as follows:  
* For the pile top: \[w_{load}(0)=w_{0}\] (19) where \(w_{0}\) is the known reload displacement. * For the pile tip, the vertical bearing capacity can be expressed as: \[P(l)=\left\{ \begin{array}{cc}\frac{s_{b}(l)}{1/K_{\text{lo}}+s_{b}(l)/q_{ mult}}&(s_{b}(l)\geq 0)\\ 0&(s_{b}(l)<0)\end{array} \right.\] (20)  
Figure 3: Load transfer curve of pile–soil interface under vertical load. where \(s_{b}(l)\) is the vertical pile-soil relative displacement at the pile tip, which is calculated by the following formula:  
\[s_{b}(l)=w_{swell}(l)+w_{load}(l)-w_{s}(l) \tag{21}\]  
[PERSON] et al. [23] and [PERSON] et al. [24] proposed the empirical formula of \(K_{bz}\) (i.e., the pile-tip soil initial stiffness) as follows:  
\[K_{bz}=\frac{4G_{s}r_{0}}{1-v_{s}} \tag{22}\]  
where \(v_{s}\) is the Poisson's ratio of the soil. The tangent stiffness of the pile tip soil is  
\[K_{bz}{}^{\prime}=\left\{ \begin{array}{cc}\frac{1}{1/K_{bz}+s_{b}(l)/q_{alt} }&(s_{b}(l)\geq 0)\\ 0&(s_{b}(l)<0)\end{array} \right. \tag{23}\]  
For saturated viscous soils with poor drainage conditions, the ultimate pile tip resistance \(q_{ult}\) can be expressed as follows [25]:  
\[q_{ult}=5.14c+q \tag{24}\]  
where \(q\) is the average vertical pressure on the side of the pile tip plane, and it can be calculated by the following formula:  
\[q=\frac{(1+2k_{0})\gamma L}{3} \tag{25}\]  
where \(k_{0}\) is the static earth pressure coefficient; and \(\gamma\) is the soil gravity. As shown in Figure 4, the central difference method is used to discretize the pile length into \(n\) equal elements, and a virtual equal division node is added to the pile tip.