## 3 Results and Discussion
### Numerical Calculations
#### 3.2.4 The Case of Variable Stress Normal to the Outer Surface of the Pipeline
Therefore, it can be seen that an account for the dynamic component of the stress normal to the outer surface led to a more accurate determination of longitudinal stresses in the underground pipeline. As calculation results show (Figure 13), at \(\sigma_{N}\
eq const\) the dependences \(\tau(u)\) were very complex. Curves 1-3 in Figure 13 refer to pipeline cross-sections \(x\) = 5 m, 10 m, and 15 m. As seen from Figure 13, at \(\sigma_{N}\
eq const\), it is very difficult to determine and evaluate in advance the interaction patterns by trajectories \(\tau(u)\). Moreover, these patterns differ for different points of the contact surface of the pipeline with soil. This difference confirms the correctness of the approach considered in this paper. According to that approach, the interaction patterns were local and determined at each point of the underground structure and soil's outer surface. Further, in the calculations, they appeared according to the initial data of the problem, proceeding from the very nature of the dynamic process of interaction. It should also be noted that the dependences \(\tau(u)\) obtained at \(\sigma_{N}\
eq const\) did not fit into the framework of existing canonical laws of material deformation (Figure 13). Changes in the velocity of pipeline cross-sections at \(x\) = 0 m, 5 m, 10 m, and 15 m are shown in Figure 14 (curves 0-3). The maximum values of the velocity of the pipeline cross-sections decreased with distance. At a distance of 15 m from the initial section of the pipeline, \(v_{cmax}\), they were approximately twice less. With time, the amplitude of the velocity of the pipeline cross-sections increased, and at a certain level, it became a steady-state one (Figure 15). Here curves 4-6 refer to pipeline cross-sections \(x\) = 30 m, 60 m, and 90 m. The difference between the maximum and minimum values of the pipeline's velocity cross-sections decreased with distance from the initial cross-section (Figure 15). Figure 12: Stress change in time in the pipeline cross-sections with accounting of \(\sigma_{N}^{D}\) for \(x\) = 30 m, 60 m, and 90 m (curves 4, 5, and 6, respectively). Figure 14: The change pattern in the pipeline cross-sections velocity in time at \(x\) = 0 m, 5 m, 10 m, and 15 m (curves 0, 1, 2, 3, respectively). Figure 15: The change in the section’s velocity in time at \(x\) = 0 m, 30 m, 60 m, and 90 m.  
section (Figure 12). Tensile stresses did not appear in the pipeline sections at \(x=30\) m, 60 m, and 90 m, and the stress values fluctuated in the area of compressive stresses (Figure 12). A positive stress value corresponded to compressive stresses. Simultaneously, the value of relative displacements in time was increasing all the time (Figure 16). Curves 1-3 refer to pipeline cross-sections \(x=5\) m, 10 m, and 15 m. The growth of relative displacements was explained by the fact that the soil medium under considered loads was deformed more than the pipeline. This led to an increase in the values of relative displacements, and they exceeded the values of \(u_{*}~{}=~{}0.005\) m.  
The results of the above calculations show that an account for the dynamic component of normal stress led to new qualitative and quantitative results. In this paper, we considered the effect of a plane longitudinal seismic wave of a frequency of 50 Hz on an underground pipeline. The calculation results showed that a decrease in the seismic wave frequency led to an even more significant increase in the maximum value of longitudinal stresses in the underground pipeline. This is the subject of further research.