# Effects of New Tunnelling on a Buried Pipeline with Joints and Its Detachment
## 2 Jointed Pipeline Subjected to Tunneling
### Mathematical Model of joints and Detachment  
The jointed pipeline can be considered as a continuous pipeline with additional local rotations at the joints, which can be achieved by applying additional moments at the joints as shown in Figure 2. The additional moment can be written as distributed loads with the pulse function employed  
\[q_{e}(x)=A\delta^{\prime\prime}(x-x_{j}) \tag{8}\]  
where \(\delta(x)\) is the pulse function, as shown in Figure 6a, \(\delta^{\prime\prime}(x-x_{j})\) denotes second-order derivatives with respect to \(x\), and \(x_{j}\) is the abscissa of the joint. \(\delta^{\prime\prime}(x-x_{j})\) will be simplified to \(\delta^{\prime\prime}\) in the following paper. The additional moment is equivalent to the limit case that three concentrated forces act around \(x_{j}\) while h approaches 0, as shown in Figure 6b, where  
\[P_{0}=(\frac{A}{8h^{3}})\cdot 2h=\frac{A}{4h^{2}} \tag{9}\]  
Figure 6: Additional local rotation can be realized by introducing an additional moment. (**a**) Pulse function; (**b**) Equivalent concentrated forces; (**c**) Additional moment at joints; (**d**) Related rotation at joints.  
As also shown in Figure 6c, \(2P_{0}\) is decomposed into two \(P_{0}\) and combined into couples with the other \(P_{0}\) on the left and right sides, where  
\[M_{0}=P_{0}\cdot 2h=\frac{A}{2h} \tag{10}\]  
The distance between the two couples is \(2h\).  
With additional moments applied, additional local rotations are generated while the equilibrium of the pipeline maintains unchanged. The angle of rotation is shown in Figure 6d.  
\[\Delta\theta=\frac{M_{e}}{EI}\cdot 2h=\frac{A}{2h}\cdot\frac{2h}{EI}=\frac{A}{ EI} \tag{11}\]  
where \(M_{\text{e}}\) is the additional moment, hence  
\[A=EI\Delta\theta \tag{12}\]  
and  
\[q_{e}=EI\Delta\theta\delta_{j}^{\prime\prime} \tag{13}\]  
while the actual rotation of the pipeline determines the additional local rotation, it can be estimated by adding a stiffness reduction factor to the continuous pipeline at each joint, and the actual moment for the jointed pipeline is  
\[M(x)=EIw^{\prime\prime}(x)-M_{e}(x)=\left\{\begin{array}{ll}\beta EIw^{ \prime\prime}(x)&x=x_{j}\\ EIw^{\prime\prime}(x)&x\
eq x_{j}\end{array}\right. \tag{14}\]  
where \(\beta\) is the stiffness reduction factor. In that case,  
\[M_{e}(x)=\left\{\begin{array}{ll}(1-\beta)EIw^{\prime\prime}(x)&x=x_{j}\\ 0&x\
eq x_{j}\end{array}\right. \tag{15}\]  
Taking only the self-weight of pipelines and covered soil into account, the external forces can be considered as line loads. Meanwhile, detachment, as well as additional moments, are under consideration. Equation (6) becomes  
\[EI\frac{d^{4}w}{dx^{4}}+\left[k(w-S)-G\frac{d^{2}(w-S)}{dx^{2}}\right]H(x)= \gamma g+\sum_{j=1}^{N}EI\Delta\theta\delta_{j}^{\prime\prime} \tag{16}\]  
with the boundary conditions  
\[\theta(0)=\theta(L)=0,Q(0)=Q(L)=0,S^{\prime}(0)=S^{\prime}(L)=0 \tag{17}\]  
where \(\gamma\) is the linear density, \(g\) is the gravity acceleration, and \(H(x)\) is the step function defined as  
\[H(x)=\left\{\begin{array}{ll}1&w\geq S\\ 0&w<S\end{array}\right. \tag{18}\]