## 3 Calculation Model  
Because of the high strength of reinforced concrete piles, shield machines used to cut piles are usually equipped with special tools which are designed and modified to remove reinforced concrete efficiently. The height of the special tools is a bit higher than that of the face scrapers in order to protect the face scrapers. The shield machine mainly relies on these special tools to cut reinforced concrete and the face scrapers rarely directly contact the piles. Therefore, the force of the shield machine applied on the piles is mainly the comprehensive result of the penetration force and cutting force of the special tools. A theoretical calculation model is established to study the interaction force between the shield machine cutter head and piles. Because the diameter of piles is much smaller than the cutter head diameter in most circumstances, only a few parts of the tools are in the state of cutting pile at a certain time. Furthermore, tools in different positions have different effects on piles. Therefore, it is necessary to determine before the calculation which tools are in the state of cutting and which tools are not. The Cartesian coordinate system is established as shown in Figure 4. Take the center of the cutter head as the coordinate origin. The right direction is the positive direction of \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Layer** & **Thickness** & **Density** & **Cohesion** & **Friction Angle** \\  & **(m)** & **\(\rho\) (g/cm\({}^{3}\))** & **\(c\) (kPa)** & **\(\varphi\) (\({}^{\circ}\))** \\ \hline \⃝1\({}_{1}\) & 3.0 & - & - & - \\ \⃝2\({}_{3}\) & 5.6 & 1.98 & 0 & 28.5 \\ \⃝2\({}_{2}\) & 3.1 & 2.08 & 0 & 33.8 \\ \⃝2\({}_{4}\) & 4.0 & 1.93 & 28.9 & 20.6 \\ \⃝2\({}_{2}\) & 9.9 & 2.08 & 0 & 33.8 \\ \⃝1\({}_{1}\) & 4.2 & 1.94 & 32.5 & 19.9 \\ \⃝2\({}_{2}\) & 12.3 & 2.08 & 0 & 33.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical and mechanical parameters of each soil layer. Figure 3: Cutter head and layout of tools. the x-axis, and the upward direction is the positive direction of the y-axis. Set the z-axis positive direction as the advancing direction of the shield machine. Suppose that the cutter head rotates clockwise around the center and \(i\) is the tool number, then the tools which are cutting pile at a certain time can be determined by the following method. Define the tool phase angle \(\varphi\) as the angle across which the positive half axis of y axis rotates clockwise to the tool center. The initial phase angle (when \(\theta=0\)) of tool No.\(i\) is \(\varphi_{0}^{i}\) (\(0\leq\varphi_{0}^{i}\leq 2\pi\)). Then, the phase angle of tool No. \(i\) at any time can be written as:  
\[\varphi^{i}(\theta)=\varphi_{0}^{i}+\theta \tag{1}\]  
Then:  
\[\left\{ \begin{array}{l}i\in W_{\theta},\ \ \text{only}\ \ \text{if}\ \ e\ -\ \frac{\beta}{2}\leq r_{i}\cdot\sin\varphi^{i}(\theta)\leq e\ +\ \frac{\beta}{2}\\ i\
otin W_{\theta},\ \ \text{if}\ \ r_{i}\cdot\sin\varphi^{i}(\theta)<e\ -\ \frac{\beta}{2}\ \ \ \text{or}\ \ \ r_{i}\cdot\sin\varphi^{i}(\theta)>e\ +\ \frac{\beta}{2}\end{array} \right. \tag{2}\]  
where \(\theta\) is the rotation angle of the cutter head; \(W_{\theta}\) is the number set of all the tools that are in contact with the pile when the rotation angle is \(\theta\); \(e\) (eccentric distance) is the distance between the pile central axis and the cutter head center; \(B\) (cutting surface with) is the width of contact surface between the cutter head and the pile; \(r_{i}\) is the radius of cutting path of the tool No. \(i\). The force exerted by the cutter head on the pile is the vector sum of the forces exerted by each cutter on the pile. Therefore, the total forces and total moments of the cutter head on pile can be calculated by Equations (3) and (4):  
\[\left\{ \begin{array}{l}F_{x}(\theta)=\sum\limits_{i\in W_{\theta}}F_{i}^{i} \cdot\cos\varphi^{i}(\theta)\\ F_{y}(\theta)=-\sum\limits_{i\in W_{\theta}}F_{i}^{i}\cdot\sin\varphi^{i}( \theta)\\ F_{z}(\theta)=\sum\limits_{i\in W_{\theta}}F_{n}^{i}\end{array} \right.