# Internal Force Analysis of Buried-boring Piles in the Yuanzishan Landslide
## 2 Methodology and Validation
### Simplified Calculations of Buried-Boring Piles during a Landslide
#### 2.1.4 Internal Force of the Bored Pile below the Slip Surface  
The lateral load acting on the buried-boring pile below the slip surface was expressed using the following equation [31]:  
\[EI\frac{d^{4}x}{dy^{4}}+xk_{1}B_{P}=0\ \ \ (0<y\leq h_{4}) \tag{14}\]  
where \(k_{1}\) is the modulus of the subgrade reaction of the bedrock to lateral force for the anti-slide pile, which is determined by the compressive strength of the rock [23]; \(E\) is the Young modulus of the pile, and \(I\) is the moment of inertia of the pile, where I=\(\pi^{\rm{o}}\)D\({}^{4}\)/64 and \(D\) is the diameter of the buried-boring pile; \(B_{p}\) is the calculation width of the pile, which was calculated using the following formula [32]:  
\[B_{P}=0.9\times(1.5D+0.5) \tag{15}\]  
The following expressions were obtained by solving the previous equations [33]:  
\[M_{y}=-4x_{0}\beta^{2}Elq_{3}-4\theta_{0}\beta EIq_{4}+M_{0}\varphi_{1}+\frac {Q_{0}}{\beta}\varphi_{2}; \tag{16}\]  
\[Q_{y}=-4x_{0}\beta^{3}EIq_{2}-4\theta_{0}\beta^{2}EIq_{3}-4M_{0}\beta\varphi_ {4}+Q_{0}\varphi_{1}; \tag{17}\]  
\[\theta_{y}=\beta\bigg{(}-4x_{0}\varphi_{4}+\frac{\theta_{0}}{\beta}\varphi_{1 }+\frac{M_{0}}{\beta^{2}EI}\varphi_{2}+\frac{Q_{0}}{\beta^{3}EI}\varphi_{3} \bigg{)}; \tag{18}\]  
\[x_{y}=x_{0}\varphi_{1}+\frac{\theta_{0}}{\beta}\varphi_{2}+\frac{M_{0}}{\beta^ {2}EI}\varphi_{3}+\frac{Q_{0}}{\beta^{3}EI}\varphi_{4}; \tag{19}\]  
where \(Q_{y}\), \(M_{y}\), \(x_{y}\), and \(\theta_{y}\) are the shear force, bending moment, horizontal displacement, and rotation angle for the pile below the slip surface with changing depth, respectively; \(Q_{0}\), \(M_{0}\), \(x_{0}\) and \(\theta_{0}\) are the shear force, bending moment, horizontal displacement, and rotation angle for the pile located at the slip surface, respectively; \(\varphi_{1}\), \(\varphi_{2}\), \(\varphi_{3}\), and \(\varphi_{4}\) are influence moduli that were calculated using the following equations [33; 34]:\[\varphi_{1}=\cos(\beta\cdot y)\cdot\cosh(\beta\cdot y) \tag{20}\]  
\[\varphi_{2}=\frac{1}{2}[\sin(\beta\cdot y)\cdot\cos h(\beta\cdot y)+\cos(\beta \cdot y)\cdot\sin h(\beta\cdot y)] \tag{21}\]  
\[\varphi_{3}=\sin(\beta\cdot y)\cdot\sin h(\beta\cdot y) \tag{22}\]  
\[\varphi_{4}=\frac{1}{4}[\sin(\beta\cdot y)\cdot\cos h(\beta\cdot y)-\cos(\beta \cdot y)\cdot\sin h(\beta\cdot y)] \tag{23}\]  
In Equations (16)\(-\)(19), \(y\) is the calculated depth of the pile below the slip surface and \(\beta\) is the coefficient of deformation, which was calculated by the following formula [33]:  
\[B_{P}=0.9\times(1.5D+0.5) \tag{15}\]0  
Other external forces acting on the pile head were negligible in the calculations [33]. The solution of Equations (16)-(19) was dependent on the values of \(M_{0}\), \(Q_{0}\), \(x_{0}\) and \(\theta_{0}\). Lateral load conditions can be used to determine \(M_{0}\) and \(Q_{0}\), whereas the lower pile boundary condition can be used to determine \(x_{0}\) and \(\theta_{0}\)[29]. By substituting calculated values of \(M_{0}\), \(Q_{0}\), \(x_{0}\) and \(\theta_{0}\) into Equations (16)-(19), \(M_{y}\), \(Q_{y}\), \(x_{y}\), and \(\theta_{y}\) could be obtained at any depth of the buried-boring pile below the slip surface. Validation of the Calculation Method and Determination of the Driving Force Distribution Coefficient  
Lirer [35] conducted a long-term monitoring study on a circular anti-slide pile with a diameter of 0.4 m on the Masseria Marino landslide, Italy, and determined the change of bending moment and shearing force of the pile during the landslide failure. The relevant geotechnical parameters are listed in \[B_{P}=0.9\times(1.5D+0.5) \tag{15}\]1  
The ratio of the driving force acting on the pile head to the driving force acting at the slip surface is defined as the distribution coefficient of the driving force, \(m\), which has an important influence on the internal force of the pile [36].