## 3 Dynamic Analysis of Buried Pressure Pipelines  
### Seismic Response of Buried Pipelines  
Research and seismic damage records show that the axial influence of buried pipelines subjected to seismic loading is much greater than the lateral impact [27; 28; 29]. Therefore, this paper only considers the axial seismic behavior of the pipeline. When the random process is processed as an interval process based on the analysis of the non-random vibration, the response of the structure should have forms of intervals with upper and lower bounds [25]. With limited sample information of excitation, the non-random vibration method can reduce the dependence on the sample and provide feasible guidance for seismic research of pipelines with lack of information. The interaction between soil and pipeline is modeled by using a nonlinear Winkler Foundation model, in which the interaction behavior is represented by nonlinear discrete soil springs and dampers. The basic assumption for the interaction of soil and pipeline can be summarized as follows:  
(1) It is assumed that the site conditions and the soil parameters are unchanged along the pipelines;  
(2) The pipe damping is extremely small compared to the soil damping of the foundation. Therefore, the material damping of the pipeline itself can be ignored;  
(3) The soil around a pipeline is simplified as a continuously uniform distributed damper and spring as shown in Figure 3.  
Figure 3: Simplified model of pipeline and soil. Figure 2: Non-stationary non-random process. Since a pipeline is a long-span structure, it is necessary to consider the temporal and spatial changes of the ground motion. Combined with the previous analysis, the axial dynamic equation for establishing a pipeline suffered from earthquake has the following form:  
\[m\ddot{u}(x,t)+c\dot{u}(x,t)+ku(x,t)-K\frac{\partial u(x,t)}{\partial x^{2}}=ku_ {\rm g}(x,t)+c\dot{u}_{\rm g}(x,t) \tag{12}\]  
where \(m\) is the weight of the pipeline; \(c\) is the damping coefficient of the medium around the pipeline; \(k\) is the stiffness coefficient around the pipeline; \(K\) is the axial stiffness of the pipeline; \(x\) is the axial coordinate of the pipeline; \(u_{g}(x,t)\) is the axial displacement of the site subjected to earthquake; and \(u(x,t)\) is the axial displacement of the pipeline. Research has shown that the axial displacement of a pipeline is the sum of the static displacement caused by the site and the dynamic displacement caused by the inertia and damping of the pipeline [30; 31]. Based on the series solution of buried pipeline with multipoint excitation, dynamic displacement can be represented as:  
\[u_{\rm D}(x,t)=\sum\limits_{i=0}^{\infty}q_{i}(t)\cos\frac{i\pi x}{l},i=0,1,2,3\cdots\infty \tag{13}\]  
where \(u_{\rm D}(x,t)\) is the dynamic displacement changing with time; \(l\) is the length of the pipeline; and \(q_{i}(t)\) is the generalized coordinate, which can be obtained according to the decomposition of vibration mode. The analytical solution of generalized coordinates has the following expression:  
\[\left\{ \begin{array}{l}q_{0}(t)=\frac{1}{2\omega_{0}}\int_{0}^{l}\exp[\xi_{ 0}\omega_{0}(t-\tau)]\tilde{a}_{0}(t)\sin\omega_{0}(t-\tau)d\tau\\ q_{i}(t)=\frac{1}{\omega_{i}}\int_{0}^{l}\exp[\xi_{i}\omega_{i}(t-\tau)]\tilde {a}_{i}(\tau)\sin\omega_{i}(t-\tau)d\tau,i=1,2,3\cdots n\end{array} \right. \tag{14}\]  
where \(\xi_{i}\) is the damping ratio corresponding to the \(i\)-th mode of vibration; and \(\omega_{i}\) is the damped vibration frequency of the pipeline, which can be expressed as \(\sqrt{\left(\frac{\tilde{\alpha}\pi^{2}K}{l_{\rm m}}+\frac{k}{m}\right)\left( 1-\xi_{i}^{2}\right)}\). As mentioned before, in the response of the pipeline, static displacement is caused by site displacement.