## 2 Simplified Mechanical Model
### Analytical Model  
Based on the analysis of the deformation phenomenon of buried pipelines, the deformation of pipelines is suggested to be divided into two regions: subsidence area and non-subsidence area. A simplified mechanical model of pipelines is developed, and is shown in Figure 2. This simplified model involves an assumption that the deformation of pipelines occurs in the subsidence area and no deformation of pipelines occurs in the non-subsidence area. Figure 1: Buried pipeline failure case [25]. For the length range of the subsidence area, some researchers provide the reference basis, which is generally taken as twice the longitudinal length of the foundation pit [29]. [PERSON] et al. [30] proposed that the relationship between the length of the outmost boundary of the subsidence area and the height of the retaining wall, mathematically, is  
\[z_{0}=H_{\text{g}}\tan\left(45^{\circ}-\frac{\varphi}{2}\right) \tag{1}\]  
where, \(z_{0}\) is the length of outmost boundary of the subsidence area, m; \(H_{\text{g}}\) is the height of retaining wall, m; \(\varphi\) is the average value of the internal friction angle of soil, \({}^{\circ}\). The Gaussian normal distribution function can be described as the ground settlement curve by the underground excavation in earlier studies, but the process of calculation is very complicated and slow, which is not suitable for practical application. Under ideal conditions, the settlement of pipelines is distributed symmetrically from the center to both ends, with the characteristics of a mathematical even function. It is widely known that one of the advantages of trigonometric functions is that they have simple controlled parameters. Thus, to simplify the complex calculation above for portable engineering application, a proximate calculation method is proposed, which innovatively uses the trigonometric function to instead express the Gaussian normal distribution function. The soil displacement curve \(y(x)\) can be expressed as  
\[y(x)=A\cos(\omega x+\alpha_{0}) \tag{2}\]  
where, \(A\) is the amplitude of the settlement displacement curve, m; \(\omega\) and \(\alpha_{0}\) are undetermined coefficients. Figure 2: Mechanic model of buried pipeline. In this study, assume that the ground settlement corresponding to the center of the pipeline is \(\delta\), and the boundary between the subsidence area and the non-subsidence area is defined as full fixed. The boundary conditions can be expressed as  
\[\left\{ \begin{array}{ll}x=0,&y=\delta\\ x=l,&y=0\\ x=-l,&y=0\end{array} \right.\]  
Thus, the solution of three undetermined coefficients in Equation (2) can be, respectively, written as follows:  
\[\left\{ \begin{array}{ll}A=\delta\\ \alpha_{0}=0\\ \omega=\pi/2l\end{array} \right.\]  
Then, Equation (2) is given by  
\[y(x)=\delta\cos\Bigl{(}\frac{\pi}{2l}x\Bigr{)} \tag{3}\]  
During the excavation, the key reason for the vertical displacement of the buried pipelines is the ground soil disturbance that causes the pipelines' settlement. The Winkler model assumes that the pressure intensity at any point on the surface of the foundation is directly proportional to the settlement, and there is no doubt that the analytical method based on this model is a suitable choice [30]. Thus, the deformation of buried pipelines nearby the foundation pit is calculated using the Winkler model; the differential equation for the deflection of buried pipelines can be expressed as  
\[EI\frac{\mathrm{d}^{4}w(x)}{\mathrm{d}x^{4}}=-kDw(x)+kD\Bigl{[}\delta\cos \Bigl{(}\frac{\pi}{2l}x\Bigr{)}\Bigr{]} \tag{4}\]  
where, \(w(x)\) is vertical displacement equation of pipelines caused by adjacent excavation, m; \(k\) is subgrade modulus, kN/m\({}^{3}\); \(D\) is the outer diameter of pipelines, m; \(EI\) is the bending stiffness of pipelines, kN\(\cdot\)m\({}^{2}\). Equation (4) can be solved by using the initial parameters method. Then, the solution of Equation (4) is divided into two parts: one part is the general solution of vertical displacement; the other is the correction term of vertical displacement.