## 2 Theoretical Derivation
### Solution of Piles under Single Vertical Load
#### 2.2.1 Pile Displacement Caused by Soil Expansion  
The expansive soil foundation will be uplifted under the action of immersion or rainfall. The uplift height of expansive soil is linearly with depth within the range of atmospheric influence [8], and the soil uplift is equal to 0 below the atmospheric influence depth range. Thus, the displacement of the foundation:  
\[w_{s}(z)=\left\{\begin{array}{ll}w_{s0}-\frac{w_{s0}}{h_{0}}z\ \ (0\leq z\leq h_{0})\\ 0\ \ \ \ \ \ \  
According to the equilibrium conditions of pile microelement, we can obtain:  
\[\frac{dP(z)}{dz}=-U_{p}\tau_{v}(z) \tag{5}\]  
where \(U_{p}\) is the perimeter of the pile section; \(P(z)\) and \(\tau_{v}(z)\) are axial force and vertical shaft resistance of the pile at depth \(z\), respectively. The elastic compression of the pile microelement can be written as:  
\[dw_{swell}(z)=-\frac{P(z)dz}{E_{P}A_{p}} \tag{6}\]  
where \(w_{swell}(z)\) is the displacement of the pile top caused by the uplift of expansive soil; \(E_{P}\) and \(A_{p}\) are the elastic modulus and cross-sectional area of the pile, respectively. The pile control equation can be obtained through simultaneous Equations (5) and (6):  
\[\frac{d^{2}w_{swell}(z)}{dz^{2}}-\frac{U_{p}}{E_{p}A_{p}}\tau_{v}(z)=0 \tag{7}\]  
The tangent stiffness of the soil on the pile-side can be expressed as:  
\[k_{v}=\frac{U_{p}\tau_{v}(z)}{s_{v}(z)} \tag{8}\]  
Given \(\lambda=\sqrt{\frac{k_{v}}{E_{p}A_{p}}}\), the pile control equation can be written as:  
\[\frac{d^{2}w_{swell}(z)}{dz^{2}}-\lambda^{2}(w_{swell}(z)-w_{s}(z))=0 \tag{9}\]  
The pile top and tip boundary conditions are as follows:  
* For the pile top, axial force can be written as:  
\[P(0)=0 \tag{10}\]  
For the pile tip, [PERSON] et al. [7] proved that the pile tip and soil are separated after immersion through experiments. Hence, the force at the pile tip is expressed as follows:  
\[P(L)=0 \tag{11}\]  
As shown in Figure 2, 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 top and pile tip. Substituting the boundary conditions of the pile tip and pile top into Equation (9), we can obtain the system of equations:  
\[\left[{{K_{v}}^{\prime}}\right]\left\{{w_{swell}}\right\}=\left\{{{F_{v}}^{ \prime}}\right\} \tag{12}\]  
where \(\left\{{w_{swell}}\right\}\) is the vertical displacement vector of the pile node,  
\[\left\{{w_{swell}}\right\}=\left\{{{w_{swell,0}}{w_{swell,1}}^{\cdots}{w_{swell,i}}^{\cdots}{w_{swell,n-1}}{w_{swell,n}}}\right\}^{T} \tag{13}\]  
\(\left\{{{F_{v}}^{\prime}}\right\}\) is the vertical load vector of the pile node,  
\[\left\{{{F_{v}}^{\prime}}\right\}=\left\{{-{\lambda_{0}}^{2}h^{2}w_{s,0}-{ \lambda_{1}}^{2}h^{2}w_{s,1}\cdots-{\lambda_{1}}^{2}h^{2}w_{s,i}\cdots-{ \lambda_{n-1}}^{2}h^{2}w_{s,n-1}-{\lambda_{n}}^{2}h^{2}w_{s,n}}\right\}^{T} \tag{14}\]\(\left[K_{v}{}^{\prime}\right]\) is the vertical stiffness matrix of the pile and it is shown as follows:  
\[\left[K_{v}{}^{\prime}\right]= \begin{bmatrix}A_{0}&2&&&&\\ 1&A_{1}&1&&\\ &\ddots&\ddots&\ddots&\\ &&1&A_{i}&1&&\\ &&\ddots&\ddots&\ddots&\\ &&&1&A_{n-1}&1\\ &&&&2&A_{n}\end{bmatrix} _{(n+1)\times(n+1)} \tag{15}\]  
where \(A_{i}=-({\lambda_{i}}^{2}h^{2}+2)\). Solving Equation (12), the vertical displacement along the pile can be obtained as  
\[\left\{w_{swelt}\right\}=\left[K_{v}{}^{\prime}\right]^{-1}\left\{F_{v}{}^{ \prime}\right\} \tag{16}\]  
The iterative process for solving Equation (12) using MATLAB is as follows:  
1. Assign the vertical displacement vector of pile node \(\left\{w_{swell}\right\}\) to zero vector. The relative displacement of pile and soil is calculated. Then, the vertical stiffness matrix \(\left[K_{v}{}^{\prime}\right]\) of the pile is obtained. 2. Calculating displacement vector \(\left\{w_{swell}\right\}\) of pile node by Equation (16). 3.