## 2 Theoretical Model  
In this paper, the research focuses on the recycling process of single-plate a helical pile in dense sand. It is necessary to establish a systematic analysis about the force conditions of single-plate helical pile during the recycling process for formulating a theoretical model to predict the recycling torque of single-plate helical pile in dense sand. Inspired by the assumption proposed by [PERSON] [7] for predicting the installation torque of a single-plate helical pile in sand, the helical pile is assumed fixed at a given depth and prevented upwards movement from the sand. While the application of the recycling torque is continued, the helix should bear the weight of the sand within the cylinder area which is on top of the helix, as shown in Figure 2. The torque applied from outside is resisted by a system of forces acting on the upper surface of the helix. As shown in Figure 3, the vertical force (\(V\)) that caused by the recycling process of helical pile is equal to the sum of the weight of the sand in the cylinder area above the helix (\(W\)), the frictional resistance on the central shaft (\(F_{\mathrm{S}}\)), and the force acting on the central shaft from outside (\(F_{\mathrm{push}}\)):  
\[V=F_{\mathrm{push}}+W+F_{\mathrm{S}} \tag{1}\]Figure 3 shows a developed helix and the forces affecting the applied torque. The forces that develop due to anchor's rotation without upwards movement are, vertical force (\(V\)), frictional force (\(\mu\)\(N\)), a force (\(R\)) parallel to the base of the triangle to overcome the frictional force (\(\mu\)\(N\)), and a force (\(N\)) normal to the inclined plane of the helix to satisfy the condition of the static equilibrium of the system. It should be mentioned that the forces mentioned above are distributed pressures on the helix; however, to simplify the theoretical analysis, these forces are considered concentrated forces acting on a point. According to the force conditions shown in Figure 3, the corresponding equations could be shown as:  
\[R-N\cdot\sin\theta-\mu\cdot N\cdot\cos\theta=0 \tag{2}\]  
\[V+\mu\cdot N\cdot\sin\theta-N\cdot\cos\theta=0 \tag{3}\]  
where  
\(\theta\) = the helix angle;  
Figure 3: The force analysis and physical parameters of helix. Figure 2: The vertical force acting on helix. \(\mu\) = the frictional coefficient between helical pile and sand;  
\(R\) = a force parallel to overcome the frictional force. Based on Equations (2) and (3), \(R\) could be calculated as below:  
\[R=V\cdot\left(\frac{\tan\theta+\mu}{1-\mu\cdot\tan\theta}\right) \tag{4}\]  
Considering the influence of the shape of helix, the weight of sand in the cylinder area above the helix could be expressed as:  
\[W=\frac{1}{2}\cdot\gamma^{\prime}\cdot(2z-2l-p)\cdot\left(\frac{D_{\mathrm{H }}-D_{\mathrm{S}}}{2}\right)\cdot\sqrt{\left(\pi\cdot d_{\mathrm{c}}\right)^{2 }+p^{2}} \tag{5}\]  
where  
\(D_{\mathrm{H}}\) = outer diameter of helix. \(D_{\mathrm{S}}\) = outer diameter of the central shaft. \(\gamma^{\prime}\) = effective unit weight of the sand. \(p\) = pitch of helix. \(z\) = embedment depth. \(l\) = the distance from the bottom of helix to the bottom of shaft. \(d_{\mathrm{c}}\) = the represented diameter where concentrated forces acting on. The torque acting on the helix (\(T_{\mathrm{H}}\)) can be calculated as follows:  
\[T_{\mathrm{H}}=R\cdot\frac{d_{\mathrm{c}}}{2} \tag{6}\]  
\(d_{\mathrm{c}}\) could be expressed as:  
\[d_{\mathrm{c}}=\frac{2}{3}\cdot\left(\frac{D_{\mathrm{H}}^{3}-D_{\mathrm{S}}^ {3}}{D_{\mathrm{H}}^{2}-D_{\mathrm{S}}^{2}}\right) \tag{7}\]  
Based on the equations mentioned above, the torque acting on the helix (\(T_{\mathrm{H}}\)) can be expressed below:  
\[V=F_{\mathrm{push}}+W+F_{\mathrm{S}} \tag{1}\]0  
The torque acting on the central shaft is expressed below:  
\[V=F_{\mathrm{push}}+W+F_{\mathrm{S}} \tag{1}\]1  
where, \(k\) is the coefficient of earth pressure, which should be confirmed according to the stress state of soil.