## 3 Results and Discussion
### Numerical Calculations
#### 3.2.2 The Case of a Large Critical Relative Displacement  
As the critical relative displacement \(u_{*}\rightarrow\infty\) (\(u_{*}\ =\ 100\ \mathrm{m}\)) and \(\sigma_{N}\ =\ \sigma_{N}^{S}\ =\ const\), at the boundary of the pipeline-soil contact, we obtain a nonlinear interaction model (14). In this case, the direct (translational) and reverse movement of the pipeline relative to soil occurs only according to Equation (14). The dynamic soil stress \(\sigma_{N}^{D}\) normal to the semi-infinite pipeline's outer surface (\(L\ =\ 1000\ \mathrm{m}\)) is not taken into account. However, the relative displacement \(u\) is determined considering the soil motion, that is, \(u\ =\ u_{g}-u_{c}\), where \(u_{c}\) is the absolute displacement of the pipe sections, \(u_{g}\) is the absolute displacement of soil. The value of \(u_{g}\) is determined from the solution of the system of Equation (40). The time change in longitudinal stresses in the pipeline cross-sections \(\sigma_{c}(t)\) and the corresponding dependences of shear stress on relative displacement \(\tau(u)\) are shown in Figures 2 and 3. Curves 0-3 in Figure 2 refer to the pipeline cross-sections x = 0, 5, 10, and 15 m. Here curve 0 shows the change in set load (30) in the initial sections of the pipeline and soil. As seen from Figure 2, a significant increase in the values of the longitudinal stress is observed in the cross-sections of the pipeline with a distance from the initial cross-section. At \(x\) = 5 m (curve 1), the stress amplitude \(\sigma_{cmax}\) exceeds the stress amplitude \(\sigma_{cmax}\) in the initial section by about 12 times, at \(x\) = 10 m by 18 times, and at \(x\) = 15 m by 17 times. Thus, an account for the soil motion around the underground pipeline leads to a multiple increase in the longitudinal stress in the pipeline sections. Such an increase in stress values occurs due to the following circumstances. Under the load acting on the pipeline and soil, changing according to the law (30), a longitudinal wave begins to propagate along the pipe and soil. As the wave velocity propagating in the pipeline is greater than the one in soil, the pipeline sections are the first to start moving. At the initial points of time, the relative velocity value is \(v\ =\ \frac{du}{dt}\ =\ \frac{du_{\kappa}}{dt}\ -\ \frac{du_{\kappa}}{dt}<0\) (Figure 4). After a certain time, the wave in soil reaches these sections of the pipeline. Since the soil rigidity is less than the pipeline stiffness (\(C_{g}<C_{c}\)), the soil was more strained than the pipe, that is, the absolute displacement of soil was greater than the absolute displacement of pipe in the corresponding sections of the pipe and soil. As a result, the sign of the relative velocity, and, hence, of the relative displacements, changes (points A, B, and C in Figures 4 and 5). The sign of shear stress \(\tau\) changed too (Figure 6), that is, the shear stress (the resistance force) turned from a passive force into an active one. The active (driving) force \(\tau\) led to an increase in the stress value in the pipeline, as seen in Figure 2. At the same time, the dependences \(\tau(u)\) remained almost linear (Figure 3). The value of the relative displacement \(u\) did not exceed its permissible value of \(u_{*}\ =\ 0.005\) m, due to the small values of \(\gamma_{g}\ =\ 1.1\). Figure 3: Dependence of shear stress on relative displacement at \(\sigma_{N}\ =\ \sigma_{N}{}^{s}\ =\ const\). The increase in stress in the pipeline cross-sections is a consequence of the interaction force \(\tau\) transform from a passive (resistant) force into an active (driving) one. In the case of  
Figure 4: Changes in relative velocity of pipe sections in time for \(x\) = 0 m, 5 m, 10 m, and 15 m (curves 0, 1, 2, and 3, respectively). Figure 5: Changes in relative displacements in time for \(x\) = 0 m, 5 m, 10 m, and 15 m (curves 0, 1, 2, and 3, respectively). Figure 6: Time dependence of shear stresses for \(x\) = 5 m, 10 m, and 15 m (curves 1, 2, and 3, respectively).