## 3 Results and Discussion
### Numerical Calculations
#### 3.2.4 The Case of Variable Stress Normal to the Outer Surface of the Pipeline  
Now consider the case of variable normal stress in the laws of interaction (26), i.e., \(\sigma_{N}~{}=~{}\sigma_{N}^{S}+\sigma_{N}^{D}\
eq const\). That means that the normal stress to the outer surface of the pipeline is a variable (\(\sigma_{N}^{D}\
eq 0\)). Figure 10 shows changes in stress over time in fixed sections of the pipeline at \(x=0\), \(5\), \(10\), and \(15\) m (curves 0-3), referring to the calculations' results at \(\sigma_{N}\
eq const\). A qualitatively different picture was observed here in comparison with the case at \(\sigma_{N}~{}=~{}const\) (Figure 8). The stress amplitudes in the considered pipeline sections were growing all the time. In terms of quantity, they differed by about two times. An increase in the maximum stress values in the pipeline cross-sections accounted for the dynamic component of normal stress \(\sigma_{N}^{D}~{}=~{}K_{\sigma}\cdot\sigma_{g}\), where the value of the lateral pressure coefficient equaled 0.3. Figure 9: Stress change in time in the sections of soil medium for \(x=0\) m, \(5\) m, \(10\) m, and \(15\) m (curves 0, 1, 2, and 3, respectively). The behavior of increasing the amplitude of stresses in the cross-sections of the pipeline with an increase in the values of normal stress is explained as follows. According to the regularity relationships of interaction (26), an increase in \(\sigma_{N}\) led to an increase in \(\tau\) on the pipeline's contact surface with soil (Figure 11). Curves 1-3 refer to the same pipeline cross-sections \(x\) = 5 m, 10 m, and 15 m. Compared to the case at \(\sigma_{N}~{}=~{}const\) (Figure 6), the values of shear stresses at \(\sigma_{N}~{}\
eq~{}const\) increased by 2-2.5 times (Figure 11). Since the shear stress is an active force in these cases, the stresses in the pipeline sections increased. However, an increase in the values of longitudinal stresses in the pipeline cross-sections did not occur infinitely, as seen from Figure 10.  
Figure 12 shows the changes in stress values in pipeline cross-sections \(x\) = 30 m, 60 m, and 90 m (curves 4, 5, and 6). As seen from Figure 12, the amplitudes of longitudinal stresses in the pipeline sections tended to be the steady-state ones. Stress values \(\sigma_{cmax}\) are: at \(x\) = 30 m, \(\sigma_{cmax}~{}=~{}22.5\) MPa; at \(x\) = 60 m, \(\sigma_{cmax}~{}=~{}26.5\) MPa; and at \(x\) = 90 m, \(\sigma_{cmax}~{}=~{}27\) MPa. Figure 10: The pattern of stress change in time in pipeline cross-sections at \(\sigma_{N}~{}\
eq~{}const\) for \(x\) = 0 m, 5 m, 10 m, and 15 m (curves 0, 1, 2, and 3, respectively). The calculation results show that the stress amplitude did not change and becomes a steady-state one at \(\sigma_{max}~{}=~{}27\) MPa in further sections of the pipeline. This stress value was about 3 times greater than the one at \(\sigma_{N}~{}=~{}const\) (Figure 8). Calculations also show that the values of these stresses in the underground pipeline at \(\sigma_{max}~{}=~{}0.5\) MPa practically did not depend on the load acting on the pipeline end at \(x~{}=~{}0\). When boundary conditions (33) were accepted at \(x~{}=~{}0\), the values of these stresses practically did not change. The values of longitudinal stresses during the Gazli earthquake (Uzbekistan) were estimated in [35]. According to the data given in [35], the maximum values of longitudinal stresses for main underground pipelines were \(\sigma_{cmax}~{}=~{}36.5\div 85\) MPa. Based on this, it could be stated that the calculated values of longitudinal stresses obtained by the authors were 25% less than the stress lower boundary observed in the pipelines during earthquakes. In the calculations, the initial data of the problem coincided with the data of the Gazli earthquake. However, it should be noted that at \(\sigma_{N}~{}=~{}const\), the value of the stress amplitude in the pipeline \(\sigma_{cmax}\) = 9 MPa (Figure 8) was significantly underestimated; it was four times less than the actual value.