## 3 Examples  
### Verification of the Present Formulation  
In order to verify the rationality of the analytical method proposed in this study, the settlement displacement of buried pipelines of two cases calculated using this approach is compared with the results reported in [PERSON] et al. and [PERSON]'s studies based on different methods. Case #1: [PERSON] et al. [29] presented the influences of a buried pipeline caused by a foundation pit excavation, based on the Winkler theory. This case is a 8.8 m deep excavation, which is surrounded by water pipelines, and buried 2 m below the ground surface. The pipeline consists of concrete, and the length from the axis of pipeline to the edge of foundation pit is 2.3 m. The outer diameter of the pipeline is 2000 mm, and the inner diameter is 1680 mm. There are seven settlement observation points on the pipeline, and the interval between two points is 5 m. Based on prospecting data, the value of the subgrade modulus is \(k=10.000\) kN/m\({}^{3}\). According to this present method, the distance from the center of water pipeline to the end of the observation point is 15 m, so the value of pipeline length is \(l\) = 15 m. The bending stiffness of the water pipeline calculated as follows:  
\[EI =3\times 10^{10}\times\frac{\pi(D^{4}-d^{4})}{64}=3\times 10^{10}\times \frac{3.14\times(2^{4}-1.68^{4})}{64}\] \[=11.83\times 10^{9}\mathrm{N}\cdot\mathrm{m}^{2}\]  
Therefore, the initial parameters can be derived as \(w_{0}\) = 33.22 mm, \(M_{0}\) = 5789.13 kN-m. Figure 3 shows that the vertical displacement of the pipeline is induced by foundation pit excavation, which is obtained by different methods. It can be seen that the vertical displacement calculated using the present method is basically close to the observed data. The calculated result proposed by [PERSON] et al. is also shown in Figure 3. From the above figure, it can be seen that the curve shapes of the calculated vertical displacements of this study are, in general, consistent with the result of the method proposed by [PERSON] et al., though there are slight differences between the two results. The maximum settlement displacement is one of the key indexes of pipeline deformation, and it is always concentrated on by the researchers. The observed maximum displacement of the pipeline center is about 29 mm, and it is smaller than the result calculated in this study, and the relative error is about 12.71%. Compared with the result calculated by [PERSON] et al., it can be found that the relative error is about 15.2%. Moreover, the results also show that the accuracy of using the method of this study is better than when using the method proposed by [PERSON] et al. Case #2: [PERSON]. [31] provided another example; the plane figure of foundation pit is rectangular-shaped, and the edge lengths are, respectively, 166 m and 50 m. This case is a 25.3 m deep excavation, which is surrounded by gas pipelines, and is buried 1.3 m below the ground surface. The outer diameter of the gas pipeline is 500 mm, and the wall thickness is 4 mm. The pipeline consists of steel, and the length from the axis of the pipeline to the edge of the foundation pit is 10.5 m. The elastic modulus of the steel pipe is \(E\) = 200 GPa, and the Poisson's ratio is \(\mu\) = 0.3. The foundation pit is excavated in the sand soil, with an elastic modulus \(E\) = 200 MPa and a Poisson's ratio of \(\mu\) = 0.3. The other soil material parameters are as follows: the unit weight of soil is \(\gamma\) = 18 kN/m\({}^{3}\), the cohesion of soil is \(c\) = 2 kPa, and the internal friction angle of soil is \(\varphi\) = 18\({}^{\circ}\). The value of the subgrade modulus is \(k\) = 10,000 kN/m\({}^{3}\). According to this present method, the distance from the center of the water pipeline to the end of the observation point is 90 m, so the value of the pipeline length is \(l\) = 90 m.  
Figure 3: Comparison of calculated result and Case#1.