## 2 Materials and Methods
### Interaction Laws
From this, it follows that in the prelimiting section \(0\leq u\leq u_{*}\), the interaction force should depend on the value of \(\sigma_{N}\), and Equation (4) has the form  
\[\tau\ =\ K_{x}(\sigma_{N})u \tag{8}\]  
Ignoring \(\sigma_{N}\) at the prelimit stage of interaction, as is done in most studies, could lead to a completely inaccurate description of the process of interaction of an underground pipeline with soil under seismic effects. Experimental studies have established the dependence of the force of interaction \(\tau\) on the buried pipeline's depth [14]. This pipeline depth \(\sigma_{N}\ =\ \sigma_{N}^{S}\) determines the static stress normal to the outer surface of the pipeline. However, it is known that under seismic wave propagation, even in the cases where the wave front is perpendicular to the pipeline axis, a dynamic normal stress \(\sigma_{N}\ =\ \sigma_{N}^{D}\) occurs, which is comparable to \(\sigma_{N}^{S}\), and sometimes greater than \(\sigma_{N}^{S}\). When an arbitrary seismic wave interacts with an underground pipeline, the following relation takes place  
\[\sigma_{N}\ =\ \sigma_{N}^{S}+\sigma_{N}^{D} \tag{9}\]  
where \(\sigma_{N}^{S}\) is determined by the pipeline depth in soil, \(\sigma_{N}^{D}\) is defined as the stress created by the seismic wave. When a compressional seismic wave propagates parallel to the pipeline axis, the normal stress \(\sigma_{N}^{D}\) is approximately determined by the formula  
\[\sigma_{N}^{D}\ =\ K_{\sigma}\sigma_{g} \tag{10}\]  
where \(K_{\sigma}\) is the coefficient of lateral soil pressure and \(\sigma_{g}\) is the longitudinal seismic stress in soil under longitudinal wave propagation. In the case of strong earthquakes in loess soils, the approximate values of \(\sigma_{p}^{max}\ =\ 0.5\) MPa [35] were observed. Then at \(K_{\sigma}\ =\ 0.3\) we got \(\sigma_{N}^{Dmax}\ =\ 0.15\) MPa, according to (10), which corresponded to the greater depth of the pipeline (\(H\ =\ 7.5\) m). Hence, as can be seen, the value of \(\sigma_{N}^{Dmax}\) might even exceed the value of \(\sigma_{N}^{S}\). This shows that the calculations for seismic resistance of underground pipelines without taking into account \(\sigma_{N}^{D}\) were approximate. According to the dynamic theory, in the law of interaction (4) or similar laws, the value of coefficient \(K_{x}\) is considered constant when solving problems of seismic resistance of underground pipelines. In fact, as shown above, it should be variable in the form (8), taking into account relations (9) and (10). The interaction law in the form (8) is the simplest one. In the case of account for the interaction velocity \(\frac{du}{dt}\), this law becomes more complicated. The laws of interaction with account for \(\frac{du}{dt}\), their parametric analysis, advantages, and disadvantages were considered in [14]. It should be noted that the law of interaction in the form (4) or (8) in all cases of interaction of an underground pipeline with soil does not always holds in reality. When elastic or viscoelastic bonds are not broken between soil particles on the contact layer and the values of relative displacement are less than the value of \(u_{*}\), then the interaction law can indeed be taken in the form (8). It is known that the value of \(u_{*}\), depending on the types of soil, ranges from 0.003 to 0.007 m [14]. The value of the peak relative displacement can be taken approximately equal to \(u_{p}\ =\ 0.5u_{*}\). Then \(u_{p}\ =\ 0.0015\div 0.0035\) m. At the values of relative displacement \(u\leq u_{p}\)the force of interaction under direct and reverse motion of the pipeline relative to soil can be described by Equation (8). As soon as the value of \(u\) exceeds \(u_{p}\), i.e., at \(u>u_{p}\), another law of interaction should be accepted at the reverse motion that takes into account the degree of soil structure damage in the contact layer. After complete destruction of the soil contact layer structure, the force of interaction turns into a force of friction.