# Theoretical Analysis on the Effectiveness of Pipe Roofs in Shallow Tunnels
## 2 Establishment of a Pipe Roof Analysis Model
### The Pressure of the Surrounding Rock Acting on the Pipe Roof
The position of the _i_th steel pipe is  
\[\alpha_{i}=(i-1)\theta \tag{1}\]  
where \(i\) is the steel pipe number and \(\theta\) is the center angle corresponding to the arc of the support boundary between any two steel pipes. Then,  
\[\theta{=}\ 2\text{arcsin}\frac{a}{2R} \tag{2}\]  
where \(R\) is the boundary radius of the pre-support protection, m, and \(a\) is the ring spacing of the pipe shed, m.  
When the height \(H\) of the support boundary vault to the ground is known, according to the geometric relationship, the height of the _i_th steel pipe to the ground, \(H_{i}\), can be found as follows:  
\[H_{i}=H+R(1-\cos\alpha_{i}) \tag{3}\]  
Before the tunnel is excavated, the part of the rock mass to be excavated and the surrounding rock form a pair of balanced forces and reaction forces on the support boundary, with sizes equal to the initial ground stress. The initial ground stress of the shallowly buried tunnel is calculated according to the self-gravity stress, and the formation is assumed to  
Figure 2: Schematic diagram of a cross section of the pipe roof pre-support. be isotropic rock and soil. Then, the initial ground stress vertical component \(\sigma_{v}\) and the horizontal component \(\sigma_{h}\) at any point on the pre-support boundary are  
\[\left\{ \begin{aligned} &\sigma_{v}=\gamma H(\alpha)\\ &\sigma_{h}=\lambda\sigma_{v}\end{aligned} \right. \tag{4}\]  
where \(\gamma\) is the weight of the overlying rock of the tunnel, \(\text{kN}/\text{m}^{3}\); \(H(\alpha)\) is the height of any point on the support boundary from the surface, m; \(\lambda\) is the lateral pressure coefficient of the surrounding rock, \(\lambda=\mu/(1-\mu)\); and \(\mu\) is the Poisson's ratio of the surrounding rock. The support boundary between the No. \(i-1\) steel pipe and the No. \(i\) steel pipe is taken as the research object. According to the equilibrium conditions, the moment of point \(i-1\) is taken, and \(\sum M_{i-1}=0\) obtains  
\[\int_{\alpha_{i-1}}^{\alpha_{i}}\sigma_{v}R^{2}(\sin\alpha-\sin\alpha_{i-1})d \alpha+\int_{\alpha_{i-1}}^{\alpha_{i}}\sigma_{h}R^{2}(\cos\alpha_{i-1}-\cos \alpha)d\alpha-q_{i,j}R\sin(\alpha_{i}-\alpha_{i-1})=0 \tag{5}\]  
The moment of point \(i\) is then taken, and \(\sum M_{i}=0\) obtains  
\[\int_{\alpha_{i-1}}^{\alpha_{i}}\sigma_{v}R^{2}(\sin\alpha_{i}-\sin\alpha)d \alpha+\int_{\alpha_{i-1}}^{\alpha_{i}}\sigma_{h}R^{2}(\cos\alpha-\cos\alpha_ {i})d\alpha-q_{i-1,r}R\sin(\alpha_{i}-\alpha_{i-1})=0 \tag{6}\]  
From the control Equations (5) and (6), we obtain  
\[q_{i,l}=\gamma R(H+R)[A_{1}(i)+\lambda A_{3}(i)]-\gamma R^{2}[A_{2}(i)+\lambda A _{4}(i)] \tag{7}\]  
\[q_{i-1,r}=\gamma R(H+R)[A_{5}(i-1)+\lambda A_{7}(i-1)]-\gamma R^{2}[A_{6}(i-1) +\lambda A_{8}(i-1)] \tag{8}\]  
where the \(A_{1}(i)\), \(A_{2}(i)\), \(A_{3}(i)\), \(A_{4}(i)\), \(A_{5}(i-1)\), \(A_{6}(i-1)\), \(A_{7}(i-1)\), and \(A_{8}(i-1)\) positional functions are calculated according to the following equations:  
\[A_{1}(i)=\frac{1}{\sin\theta}\bigg{[}2\sin\frac{\theta}{2}\sin\bigg{(}\alpha_{ i}-\frac{\theta}{2}\bigg{)}-\theta\sin(\alpha_{i}-\theta)\bigg{]}\]  
\[A_{2}(i)=\frac{2}{\sin\theta}\sin^{2}\frac{\theta}{2}\cos^{2}\bigg{(}\alpha_{ i}-\frac{\theta}{2}\bigg{)}\]  
\[A_{3}(i)=\frac{1}{\sin\theta}\bigg{[}\theta\cos(\alpha_{i}-\theta)-2\sin\frac {\theta}{2}\cos\bigg{(}\alpha_{i}-\frac{\theta}{2}\bigg{)}\bigg{]}\]  
\[A_{4}(i)=\frac{1}{\sin\theta}\Bigg{[} \begin{array}{c}\sin\frac{\theta}{2} \sin\left(\alpha_{i}-\frac{\theta}{2}\right)\sin\alpha_{i}\\ +\sin\frac{\theta}{2}\cos(\alpha_{i}-\theta)\cos\left(\alpha_{i}-\frac{\theta} {2}\right)-\frac{\theta}{2}\end{array}