# Calculation Method for Settlement Deformation of Existing Tunnel Induced by Underpass Construction
## 2 Settlement Calculations for Existing Tunnels
### Energy Method  
Because of its simplicity and comparatively high accuracy, the energy technique can rapidly forecast the settlement and deformation of existing tunnels. A sufficient displacement function must be assumed in order to use the energy technique to determine the influence of the excavation. The displacement function employed in this research is based on the Rayleigh-Ritz approach [16] and assumes that the vertical displacement pattern of the existing tunnel follows the normal distribution curve:  
\[w=\sum_{i=0}^{n}a_{i}\cos\frac{i\pi x}{L_{p}}=\{C\}_{1\times(n+1)}\{a\}_{(n+1 )\times 1} \tag{1}\]  
\[\{C\}=\left\{1,\cos\frac{\pi x}{L_{p}},\cos\frac{2\pi x}{L_{p}},\cdots,\cos \frac{n\pi x}{L_{p}}\right\} \tag{2}\]  
\[\{a\}=\{a_{0},a_{1},a_{2},\cdots,a_{n}\}^{T} \tag{3}\]  
where: \(w\) is the existing tunnel's vertical displacement; \(\{a\}\) is the coefficient to be determined; \(n\) is the order of the stiffness matrix in the subsequent calculations; \(L_{p}\) is the half-width of the existing tunnel's settlement slot, which results from tunnel excavation; and \(x\) is the distance along the tunnel's axis. Figure 1: Impact analysis model for new tunnels passing under existing tunnels. Generally speaking, the system's total potential energy may be separated into three components [19]: the work performed by the foundation reaction force, the additional load brought on by the new tunnel's excavation, and the bending strain energy of the existing tunnel. They can be expressed, respectively, as:  
\[E_{1}=\frac{1}{2}\int\limits_{-L_{p}}^{L_{p}}{(EI)_{\text{eq}}(\frac{\partial^{ 2}w}{\partial x^{2}})dx} \tag{4}\]  
\[E_{2}=\frac{D}{2}\int\limits_{-L_{p}}^{L_{p}}{kw^{2}dx}+\frac{D}{2}\int\limits _{-L_{p}}^{L_{p}}{G(\frac{\partial w}{\partial x})}^{2}dx \tag{5}\]  
\[E_{3}=-\int\limits_{-L_{p}}^{L_{p}}{Pwdx} \tag{6}\]  
where: \((EI)_{\text{eq}}\) is the existing tunnel's equivalent bending stiffness; \(D\) is its diameter; k is the foundation's reaction force coefficient; \(G\) is the foundation's shear layer coefficient; \(P\) is the additional load, \(P=D\sigma_{z}\); and \(\sigma_{z}\) is the additional stress. The foundation reaction force coefficient can be obtained through the following formula, which takes into account the impact of the existing tunnel's depth [20]:  
\[k=\frac{3.08E}{\eta(1-\
u^{2})}\sqrt[3]{\frac{ED^{4}}{(EI)_{\text{eq}}}} \tag{7}\]  
\[\eta= \begin{cases}2.18&z/D\leq 0.5\\ 1+\frac{1}{1.7z/D}&z/D>0.5\end{cases}  \tag{8}\]  
where \(E\) represents the soil's modulus of elasticity; \(v\) is Poisson's ratio; and \(z\) is the tunnel's axial depth. The Pasternak foundation model takes into account how the existing tunnel body interacts with the strata. The following formula, when combined with [PERSON]'s study findings [21], may be used to determine the shear layer coefficient \(G\) on the Pasternak foundation:  
\[G=\frac{2.5 DE}{6(1+v)} \tag{9}\]  
The new tunnel is built based on the principle of minimum potential energy. This means that the total energy of the existing tunnel is increased significantly, depending on each unknown factor. This total energy includes the bending strain energy of the existing tunnel, the work performed by the foundation reaction force, and the work done by the additional load resulting from the tunnel excavation. \[\int\limits_{-L_{p}}^{L_{p}}{(EI)_{\text{eq}}\frac{\partial(\partial^{2}w/ \partial x^{2})}{\partial a_{i}}\frac{\partial^{2}\{C\}}{\partial x^{2}}dx}+D \int\limits_{-L_{p}}^{L_{p}}{k\frac{\partial w}{\partial a_{i}}\{C\}\{a\}dx}+D \int\limits_{-L_{p}}^{L_{p}}{G\frac{\partial(\partial w/\partial x)}{\partial a _{i}}\frac{\partial\{C\}}{\partial x}\{a\}dx}=\int\limits_{-L_{p}}^{L_{p}}{\{P \}\{C\}dx} \tag{10}\]  
Associative Equations (1)-(3) and Equation (10) allow the partial differential results to be expressed in matrix form:  
\[\Big{(}[K_{1}]+[K_{2}]\Big{)}\{a\}=\{P\} \tag{11}\]\[[K_{1}]=\frac{\left(EI\right)_{\text{eq}}\pi^{4}}{L_{p}^{3}} \begin{bmatrix}0 \\ &1\\ &&2^{4}\\ &&&\ddots\\ &&&n^{4}\end{bmatrix}  \tag{12}\]  
\[[K_{2}]=kDL_{p}