# Theoretical Analysis on the Effectiveness of Pipe Roofs in Shallow Tunnels
## 3 Evaluation of the Effect of Pre-Support of the Pipe Roof
### Evaluation Index Calculation Method  
When analyzing the interaction between the pipe shed steel tube and surrounding rock, it is assumed that the steel tube and surrounding rock extrude each other. Thus, the convergent deformation of the surrounding rock on the pre-supported boundary can be considered equal to the deflection deformation of the pipe shed steel tube towards the tunnel. The deformation of the surrounding rock on the pre-supported boundary between the pipe shed steel tubes can be obtained via linear interpolation of the deformation of two adjacent steel tubes. The convergence deformation integral for the surrounding rock of any computed section can be used to obtain the formation loss of this section (\(V_{l}\)):  
\[V_{l}=R\theta\sum_{i=1}^{n-1}(\omega_{i}+\omega_{i+1})-\frac{\theta}{4}\sum_{ i=1}^{n-1}(\omega_{i}+\omega_{i+1})^{2}+R\omega_{n}\theta^{\prime}-\frac{\omega_{n} {}^{2}}{4}\theta^{\prime} \tag{15}\]  
where \(R\) is the radius of the pre-support boundary, m; \(\theta\) is the central angle corresponding to the supporting boundary arc between any two steel tubes, rad; \(\theta^{\prime}\) is the central angle corresponding to the arc from the No. \(n\) steel pipe to the arch foot segment, rad; and \(\omega_{i}\), \(\omega_{i+1}\), and \(\omega_{n}\) are, respectively, the bending deformation of steel tubes \(i\), \(i+1\), and No. \(n\) in the calculated section.  
Then, the formation loss rate of the calculated section \(\eta\) is obtained as  
\[\eta=\frac{2V_{l}}{\pi H_{k}{}^{2}} \tag{16}\]where \(V_{l}\) is the formation loss of the calculated section, m\({}^{2}\), and \(H_{k}\) is the excavation height of the palm face, m.  
The surface settlement curve caused by excavation of the underground tunnel is commonly called the \"settlement groove\". The Peck formula [19] is a classical formula widely used globally to estimate settlement curves [20; 21; 22]. [PERSON] analyzed a large quantity of surface settlement data and related engineering data and concluded that the width \(i\) of the settlement trough was as follows:  
\[i=\frac{H+H_{k}}{\sqrt{2\pi}\tan(45^{\circ}-\frac{\varphi}{2})} \tag{17}\]  
where \(H\) is the buried depth of the pre-supported boundary vault, m; \(H_{k}\) is the excavation height of the face, m; and \(\varphi\) is the internal friction angle of surrounding rock, \({}^{\circ}\).  
Then, the settlement value of any point on the surface corresponding to the calculated section can be obtained as  
\[s(y)=s_{\text{max}}e^{-\frac{y^{2}}{2\pi}} \tag{18}\]  
where \(y\) is the distance between the calculation point and the center point of the settlement curve, m; \(i\) is the width of the settling trough, m; and \(s_{\text{max}}\) is the maximum surface settlement value of the calculated section, m. The calculation expression is  
\[s_{\text{max}}=\frac{V_{l}}{\sqrt{2\pi i}} \tag{19}\]  
The control effect of the pipe shed pre-support on stratum deformation can be quantitatively evaluated by calculating the stratum loss rate \(\eta\), the span of the surface settlement trough \(i\), and the maximum surface settlement \(s_{\text{max}}\) by selecting the proper calculation interface along the tunnel's longitudinal direction.