## 2 Theoretical Research
### The Loads on a Buried Long Pipeline  
#### 2.2.1 The Vertical Compressive Load  
The total vertical compression on the pipeline (\(P\)) consists of the pressure from the rockfall impact force transmitting through the soil above the pipeline (\(q_{\rm v}\)) and the earth pressure above the pipeline (\(P_{\rm v}\)), depicted as follows [32]:  
\[P=fq_{\rm v}+P_{\rm v}, \tag{9}\]  
where \(f\) represents the penetration coefficient which is related to the buried depth from the soil surface to the crown of the pipeline, as shown in Table 1.  
In a collision, the contact area between the rockfall and the soil is smaller than the surface of the soil and, furthermore, the length of the pipeline is relatively infinite; thus, \(F_{\rm max}\) can be regarded as a vertical concentrated load that can be calculated by Equation (7), and its sketch is depicted in Figure 3, in which \(F_{\rm max}\) is the vertical concentrated load which acts on the coordinate origin.  
The vertical pressure (\(q_{\rm v}\)) at a certain position in the soil, i.e., point \(M\) as shown in Figure 3, can be expressed as follows [33]:  
\[q_{\rm v}=\frac{3F_{\rm max}}{2\pi R^{2}}\cos^{3}\beta, \tag{10}\]  
where \(R\) is the distance between point \(M\) and the original coordinate shown as follows:  
\[R=\left(x^{2}+y^{2}+z^{2}\right)^{\frac{1}{2}}=\left(r^{2}+z^{2}\right)^{ \frac{1}{2}}. \tag{11}\]  
In Figure 3, \(\beta\) is the angle between the depth (\(z\)) and the distance (\(R\)), which can be obtained by the following equation:  
\[\cos\beta=\frac{z}{R}. \tag{12}\]  
When the pressure acting on point \(M^{\prime}\) of the buried pipeline reaches up to \(F_{\rm max}\), \(y\) approximates to 0. Therefore, the following equation can be obtained:  
\[R=\left(x^{2}+z^{2}\right)^{\frac{1}{2}}. \tag{13}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline
**The Buried Depth of Pipeline (m)** & **0–0.3** & **0.3–0.6** & **0.6–0.9** & **>0.9** \\ \hline \(f\) & 1.5 & 1.35 & 1.35 & 1.15 \\ \hline \end{tabular}
\end{table}
Table 1: The relationship between the buried depth of the pipeline and \(f\).  
Figure 3: Vertical concentrated load of the soil layer.  
Substituting Equations (12) and (13) into Equation (10), we can obtain the pressure from the rockfall impact force transmitting through the soil above the pipeline as shown below.  
\[q_{\rm v}=\frac{3F_{\rm max}z^{3}}{2\pi(x^{2}+z^{2})^{\frac{5}{2}}}. \tag{14}\]  
The calculation method of the earth pressure of the soil above the pipeline exerted on the pipeline can be divided into two cases based on the position of water line. When the groundwater level is under the pipeline, \(P_{\rm v}\) is a constant load generated by the gravity of the soil, and can be represented as follows [34]:  
\[P_{\rm v}=\gamma\cdot C_{1}, \tag{15}\]  
where \(\gamma\) is the unit weight of the soil, while \(C_{1}\) is the buried depth of pipeline.  
When the groundwater level is over the pipeline, the buoyancy effect of water should be taken into account. The \(P_{\rm v}\) can then be expressed as follows:  
\[P_{\rm v}=\gamma\cdot C_{2}+\gamma_{\rm sat}h_{\rm w}-\gamma_{\rm w}h_{\rm w}, \tag{16}\]  
where \(C_{2}\), \(\gamma_{\rm sat}\), \(\gamma_{\rm w}\), and \(h_{\rm w}\) denote the height between the ground surface and the groundwater level, the saturated unit weight of the soil, the unit weight of water, and the height of the groundwater level over the crown of the pipeline, respectively.