## 3 Numerical Simulation
### Result Analysis
#### 3.3.3 Analysis of Pipeline-Soil Interaction  
Excavation of foundation pits can cause the displacement of a pipeline, soil-pipeline detachment, and pipeline suspension. These problems pose considerable risk to pipeline safety. Since the existing pipeline is buried underground, it is difficult to measure the pipeline-soil gap directly. According to the layered excavation model, this paper analyzes whether measures should be taken for the pipeline to predict the pipeline-soil gap by calculating the gap size. Figure 9: Pipeline settlement in stages. Figure 11 shows the state of detachment between the pipeline and soil. According to the numerical analysis results, the deformation of the pipeline itself was small; its displacement was mainly driven by the deformation of the surrounding soil, resulting in gaps between the pipeline and soil. For ease of exposition, this paper defined the vertical upward direction as 0\({}^{\circ}\) direction, and the angle increased clockwise. From the figure, it can be concluded that, at the beginning of the excavation, the pipeline and soil had a large gap at 30\({}^{\circ}\) and 210\({}^{\circ}\), and the pipeline and soil fit closely at 120\({}^{\circ}\) and 300\({}^{\circ}\). As the foundation pit excavation depth increased, the pipeline-soil gap was closed at 30\({}^{\circ}\) and 210\({}^{\circ}\), while a large gap appeared at 120\({}^{\circ}\) and 300\({}^{\circ}\). In order to clearly explain the influence of gap size between the pipeline and soil on the pipeline safety performance, it is necessary to quantitatively analyze the variation pattern of pipeline gap width. However, the numerical model only enabled us to determine the vertical and horizontal displacements and total displacement of each point of the pipeline, and the pipeline-soil gap width could not be obtained directly through the model. To obtain the gap size between the pipeline and soil, the horizontal and vertical displacements were treated as follows:  
\[d=L\cos\left(\alpha+\theta-\frac{\pi}{2}\right), \tag{2}\]  
where \(d\) is gap width (mm), \(L\) is the detachment distance between soil and pipeline (mm), \(L=\sqrt{(y^{\prime}-y)^{2}+(x^{\prime}-x)^{2}}\), and \(\theta\) is the angle (\({}^{\circ}\)) of the pipeline, with the vertical upward direction defined as 0\({}^{\circ}\). The angle increased clockwise. Furthermore, \(\alpha\) is the displacement  
Figure 11: Pipeline contact with soil at each stage. (**a**) Excavation depth of 1.5 m; (**b**) Excavation depth of 4.5 m; (**c**) Excavation depth of 9.2 m; (**d**) Excavation depth of 13.7 m.  
Figure 10: Layered excavation site. angle (\({}^{\circ}\)) of a point, \(\alpha=\arctan\left(\frac{y-y}{x-x}\right)\), in which \(x\) represents horizontal displacement and \(y\) represents vertical displacement (mm). As shown in Figure 12, substituting \(L\) (i.e., detachment distance between soil A\({}^{\prime}\) and pipeline point A) and \(\alpha\) into Equation (2), the gap width between the pipeline and soil could be calculated (see Figure 13). In Figure 13, the value with a dotted line indicates the thickness of the soil squeezed between the pipeline and soil. A smaller dotted line value denotes a greater squeeze. As can be seen from the figure, the peak values of the gaps in the four stages of the excavation process were 1.6 mm, 1.1 mm, 1.2 mm, and 2.3 mm, occurring at 25\({}^{\circ}\), 65\({}^{\circ}\), 108\({}^{\circ}\), and 110\({}^{\circ}\), respectively. It can be found that, as the excavation depth increased, the peak values of the gaps first decreased and then increased, and the angle of the positions with peak values changed from 25\({}^{\circ}\) to 110\({}^{\circ}\). In the whole excavation process, the pipeline and soil exhibited certain relationships. Because of the influence of gravity and soil disturbance, the pipeline-soil gap changed from an open state to a squeeze state at the angles of 25\({}^{\circ}\) and 205\({}^{\circ}\) to the pipeline. When excavation was completed, the soil was squeezed to the maximum.