# Effects of New Tunnelling on a Buried Pipeline with Joints and Its Detachment
## 2 Jointed Pipeline Subjected to Tunneling
### Calculation Method and Procedure  
The Central Difference Method (CDM) is used to obtain the numerical results. The beam and its surroundings can be divided into \(n+1\) real nodes and four fictitious nodes with step \(l=L/n\), as shown in Figure 7. Central differences (CD) at the point \"\(i\)\" can be defined as an approximation of derivatives. \[\frac{d^{2}w_{i}}{dx^{2}}=\frac{w_{i-1}-2w_{i}+w_{i+1}}{l^{2}} \tag{19a}\]\[\frac{d^{4}w_{i}}{dx^{4}} =\frac{w_{i-2}-4w_{i-1}+6w_{i}-4w_{i-1}+w_{i+2}}{l^{4}} \tag{19b}\] \[\frac{d^{2}(w_{i}-S_{i})}{dx^{2}} =\frac{(w_{i-1}-S_{i-1})-2(w_{i}-S_{i})+(w_{i+1}-S_{i+1})}{l^{2}} \tag{19c}\]  
The additional moment at the joint is equivalent to the limit case that three concentrated forces act around the joint while \(h\) approaches0. The three concentrated forces can be applied to the joint node \(x_{j}\) and its adjacent nodes \(x_{j-1}\), \(x_{j+1}\), and \(2h\) equal to \(l\). According to Equation (11) and Equation (15), the rotation angle can be obtained by  
\[M_{e}=\frac{EI\Delta\theta_{j}}{2h}=(1-\beta)EIw_{j}^{\prime\prime} \tag{20}\]  
\[\Delta\theta_{j}=(1-\beta)lw_{j}^{\prime\prime} \tag{21}\]  
As the number of nodes in the difference method increases, Equation (16) can be rewritten as  
\[EI\frac{d^{4}w}{dx^{4}}+\bigg{[}k(w-S)-G\frac{d^{2}(w-S)}{dx^{2}}\bigg{]}H(x)= \gamma g+(1-\beta)IEI\sum\limits_{j=1}^{N}w_{j}^{\prime\prime}\delta_{j}^{\prime\prime} \tag{22}\]  
The three equivalence forces of the additional moment act at the joint node and its adjacent nodes. The finite difference form of Equation (22) is given by  
\[\left\{ \begin{array}{l}EI\frac{d^{4}w_{i}}{dx^{4}}+\Big{[}k(w_{i}-S_{i})-G \frac{d^{2}(w_{i}-S_{i})}{dx^{2}}\Big{]}H_{i}=\gamma g\quad(i\
eq j-1,j,j+1)\\ EI\frac{d^{4}w_{i}}{dx^{4}}+\Big{[}k(w_{i}-S_{i})-G\frac{d^{2}(w_{i}-S_{i})}{ dx^{2}}\Big{]}H_{i}=\gamma g+\frac{(1-\beta)IEI\sum\limits_{i=1}^{N}w_{i}^{ \prime\prime}}\quad(i=j-1)\\ EI\frac{d^{4}w_{i}}{dx^{4}}+\Big{[}k(w_{i}-S_{i})-G\frac{d^{2}(w_{i}-S_{i})}{ dx^{2}}\Big{]}H_{i}=\gamma g-\frac{2(1-\beta)IEI\sum\limits_{i=1}^{N}w_{i}^{ \prime\prime}}\quad(i=j)\\ EI\frac{d^{4}w_{i}}{dx^{4}}+\Big{[}k(w_{i}-S_{i})-G\frac{d^{2}(w_{i}-S_{i})}{ dx^{2}}\Big{]}H_{i}=\gamma g+\frac{(1-\beta)IEI\sum\limits_{i=1}^{N}w_{i}^{ \prime\prime}}\quad(i=j+1)\\ \end{array} \right. \tag{23}\]  
Employing Equations (19a)-(19c), we have  
\[EI\frac{(w_{i-2}-4w_{i-1}+6w_{i}-4w_{i+1}+w_{i+2})}{\mu} \tag{24a}\] \[+\Big{[}k(w_{i}-S_{i})-G\frac{(w_{i-1}-S_{i-1})-2(w_{i}-S_{i})+(w_ {i+1}-S_{i+1})}{l^{2}}\Big{]}H_{i}\] \[=\gamma g\quad(i\
eq j-1,j,j+1)\]  
Figure 7: Divisions of the pipeline. \[ \begin{split}& EI\frac{(\frac{w_{i-2}-4w_{i-1}+6w_{i}-4w_{i+1}+w_{i+2 }}{l})}{l^{2}}\\ +\Big{[}k(w_{i}-S_{i})-G\frac{(w_{i-1}-S_{i-1})-2(w_{i}-S_{i})+(w_{i+1}-S_{i+1 })}{l^{2}}\Big{]}H_{i}\\ &\quad=\gamma g+\frac{(1-\beta)}{l^{2}}IEI\sum\limits_{i=1}^{N} \frac{w_{i-1}-2w_{i}+w_{i+1}}{l^{2}}\quad(i=j-1)\\ &\quad\quad\quad\quad EI\frac{(w_{i-2}-4w_{i-1}+6w_{i-4w_{i+1}+w_{ i+2}})}{l^{2}}\\ +\Big{[}k(w_{i}-S_{i})-G\frac{(w_{i-1}-S_{i-1})-2(w_{i}-S_{i})+(w _{i+1}-S_{i+1})}{l^{2}}\Big{]}H_{i}\\ &\quad=\gamma g-\frac{2(1-\beta)}{l^{2}}IEI\sum\limits_{i=1}^{N} \frac{w_{i-1}-2w_{i}+w_{i+1}}{l^{2}}\quad(i=j)\\ &\quad\quad\quad\quad EI\frac{(w_{i-2}-4w_{i-1}+6w_{i-4w_{i+1}+w_ {i+2}})}{l^{2}}\\ +\Big{[}k(w_{i}-S_{i})-G\frac{(w_{i-1}-S_{i-1})-2(w_{i}-S_{i})+(w _{i+1}-S_{i+1})}{l^{2}}\Big{]}H_{i}\\ &\quad=\gamma g+\frac{(1-\beta)}{l^{2}}IEI\sum\limits_{i=1}^{N} \frac{w_{i-1}-2w_{i}+w_{i+1}}{l^{2}}\quad(i=j+1)\end{split}  \tag{24d}\]  
with boundary conditions  
\[\left\{