## 2 Theoretical Derivation
### Displacement Solution of Single Piles under \(V\)\(\rightarrow\)\(T\) Loading Path
#### 2.3.2 Displacement Solution of Single Pile under \(V\)\(\rightarrow\)\(T\) Loading Path  
The [PERSON] et al. [17] gives the calculation method of a single pile under torsion loading. The load transfer function of the pile-soil interface still adopts the method proposed by [PERSON] et al. [20]. \[\tau_{l}(z)=\frac{G_{\text{s}}s_{l}(z)}{r_{0}ln(r_{m}/r_{0}-\psi/1-\psi)} \tag{33}\]  
where \(\tau_{l}(z)\) is the circumferential frictional resistance of the pile side; \(s_{l}(z)\) is the relative torsional displacement of the pile-soil at depth \(z\), \(s_{l}(z)=r_{0}\theta(z)\), Note that in the equation, \(\psi=\tau_{l}(z)R_{f}/\tau_{tf}(z)\). The displacement control equation of a single pile under torque loading is [17]:  
\[\frac{d\theta^{2}(z)}{dz^{2}}=\alpha^{2}\theta(z) \tag{34}\]  
where \(\alpha=\sqrt{\frac{k_{B}\pi D^{3}}{4G_{Pl}}}\); and \(k_{B}(z)=\frac{\tau_{l}(z)}{s_{l}(z)}\). Using the finite difference method, the Equation (34) can be rewritten as:  
\[\left[K_{l}{}^{\prime}\right]\{\theta\}=\left\{T_{l}{}^{\prime}\right\} \tag{35}\]  
Figure 5: (**a**) Discrete model of pile surface (**b**) Shear element model. where \(\{\theta\}\) is pile node torsion angle vector, \(\{\theta\}=\{\theta_{0}\theta_{1}\ldots\theta_{i}\ldots\theta_{n-1}\theta_{n}\}^{T}\); and \(\{T_{t}\prime\}\) is the torque load vector, \(\left\{T_{t}{}^{\prime}\right\}=\left\{-2 Th/\left(G_{p}J_{p}\right)0\ldots 0 \ldots 0-3T_{b}h/\left(8G_{L}r_{0}^{3}\right)\right\}^{T}\); \(\left[K_{t}{}^{\prime}\right]\) is torque stiffness matrix,  
\[\left[K_{t}{}^{\prime}\right]= \begin{bmatrix}B_{0}&2&&&&\\ 1&B_{1}&1&&&\\ &\ddots&\ddots&\ddots&&\\ &&1&B_{i}&1&&\\ &&&\ddots&\ddots&\ddots&\\ &&&&1&B_{n-1}&1\\ &&&&&&&B_{n}\end{bmatrix}  \tag{36}\]  
where \(T\) is the torque at the pile top; \(G_{p}\) is the shear modulus of the pile; \(J_{p}\) is the polar moment of inertia of pile section; \(G_{L}\) is the shear modulus of the soil at the pile tip; and \(T_{b}\) is the pile tip torque. However, the torque load vector of the pile node cannot be expressed by Equation (35) in expansive soil. The pile tip is still separated from the soil when the vertical load is small, and torque is equal to zero. The pile tip exerts pressure on the soil when the vertical load is large, and the pile tip torque is not equal to zero. Therefore, the torsional load vector can be written as:  
\[\left\{T_{t}{}^{\prime\prime}\right\}=\left\{ \begin{array}{ll}\left\{-2 Th/ \left(G_{p}J_{p}\right)0\ \ldots\ 0\ \ldots\ 0\ 0\right\}^{T}&s_{b}(l)<0\\ \left\{-2 Th/\left(G_{p}J_{p}\right)0\ldots 0\ldots 0-3T_{b}h/\left(8G_{L}r_{0}^{3 }\right)\right\}^{T}s_{b}(l)\geq 0\end{array} \right. \tag{37}\]  
The Equation (35) is rewritten as:  
\[\left[K_{t}{}^{\prime}\right]\{\theta\}=\left\{T_{t}{}^{\prime\prime}\right\} \tag{38}\]  
Solving Equation (38) can obtain the pile node torsion angle vector:  
\[\left\{\theta\right\}=\left[K_{t}{}^{\prime}\right]^{-1}\{T_{t}{}^{\prime \prime}\} \tag{39}\]  
The torsion angle of a single pile under torque loading can be obtained by Equation (39), and the limit torque can be determined by plotting the torque-torsion angle curve at the pile top. The process of solving Equation (39) using MATLAB is as follows:  
1. Calculating the vertical displacement of the pile element according to Section 2.2, then the relative displacement \(s(z)\) of the pile and soil can be obtained. Substituting it into the Equation (17), the vertical friction resistance \(\tau_{sv}\) can be obtained. 2. The circumferential ultimate friction of the pile \(\tau_{tf}\) is achieved by solving Equation (32). 3. Assign the pile node torsion angle vector \(\{\theta\}\) to zero vector, the relative torsional displacement of the pile-soil at depth z is calculated. Then, the torque stiffness matrix \(\left[K_{t}{}^{\prime}\right]\) is obtained. 4. Calculating displacement vector \(\{\theta\}\) of pile node by Equation (39). 5.