## 3 Results
### Study on Seismic Failure Modes of Cable-Stayed Pipe Bridge Oil and Gas Pipeline Structure
#### 3.2.2 Influence of Support on the Interaction between Pipelines  
Based on the investigation of the additional mass of the pipe bridge system, the relationship between the pipe and the support was transformed into a weak constraint. The friction coefficient was adjusted to 0.2, while considering the peak acceleration of the EI-Centro wave as an intensity index. The seismic wave was inputted longitudinally along the bridge, with gradually increasing peak accelerations ranging from 0.2 g to 1.0 g, in increments of 0.2 g. Nonlinear time-history analysis was conducted on the pipe bridge structure under various peak seismic accelerations to investigate damage outcomes resulting from the interaction between the support and the pipeline. Displacement response of the support  
Under seismic activity, excessive longitudinal displacement between the support and the pipeline can result in localized damage, and ultimately, lead to pipeline failure. Therefore, we numerically analyzed the relative displacements of supports under different peak accelerations. Figure 5 illustrates the distribution of supports with the highest equivalent deformation. It is evident from the figure that due to the structural system's complexity, the position of maximum equivalent deformation varied; however, it predominantly occurred near both ends of the import and export sections at approximately one-fourth of the span length from either side. Therefore, the bearing experiencing the most significant equivalent deformation during the seismic process within the initial 30 s was selected. Subsequently, an analysis of the longitudinal displacement time-history curve for this particular bearing was conducted, as depicted in Figure 6. The maximum relative longitudinal displacement value is presented in Table 5.  
The longitudinal displacement of the support gradually increased with the increase in peak acceleration, as evident from Figure 6. At a peak acceleration of 0.2 g, the longitudinal displacement of the support was measured at 0.134 m, while it reached 0.325 m at a peak acceleration of 1.0 g. Consequently, an increasing tilt angle was observed in the support  
Figure 5: Distribution of supports with the largest equivalent deformation. Maximum deformation of the pipeline structure under different working conditions: (a) 0.2 g maximum equivalent deformation support, (b) 0.4 g maximum equivalent deformation support, (c) 0.6 g maximum equivalent deformation support, (d) 0.8 g maximum equivalent deformation support, and (e) 1.0 g maximum equivalent deformation support. until separation occurred between the pipeline and the support, resulting in a noticeable gap forming between them. This indicates that under longitudinal seismic waves, slippage occurs longitudinally between the pipe and its supporting structure, which intensifies with higher peak seismic accelerations. It should be noted that collision between the support and the pipeline can cause localized damage to the latter, hence highlighting the significance of considering their interaction. \(\bullet\) Pipeline stress distribution  
It is evident from the aforementioned analysis that the interaction between the support and the pipeline induced localized failure in the pipeline. We further elucidated the occurrence of local failure in the pipeline due to stress concentration by examining its maximum longitudinal position. Figure 7 depicts a stress cloud map illustrating equivalent stresses experienced by the pipeline at this critical longitudinal displacement. The increase in the peak seismic acceleration resulted in a gradual increase of the equivalent stress at the local location of the pipeline, leading to stress concentration phenomena, as depicted in Figure 7. When the peak seismic acceleration ranged from 0.2 to 0.4 g, the equivalent stress experienced by the pipeline remained below its yield strength of 425 MPa.