## 1 Introduction
In the case of strain and destruction of the contact soil layer under shear stresses, its limiting value is determined based on the Coulomb law. * The processes of relative displacements \(u\) formation on the contact layer, under its elastic, elastic-plastic strain (\(u<u_{*}\)) and destruction (\(u\geq u_{*}\)), are the two stages of one process--the process of deformation of the contact soil layer before and after destruction. The existing theories of seismic resistance of underground pipelines [1; 6; 7; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32] partially take into account these factors, identified by numerous researchers while analyzing the consequences of severe earthquakes and conducting experimental studies of the underground pipeline interaction with soil under static and dynamic loads. The problem of determining or assessing the seismic resistance of underground pipelines is reduced, in one way or another, to the stress determination in the pipe body by various methods [1; 6; 7; 20; 27; 28; 29; 30; 31; 32]. The stress state of an underground pipe under seismic forces is rather complicated [16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31]. For simplicity reasons, stresses are divided into longitudinal (acting along the pipeline axis), transverse (perpendicular to the pipeline axis), bending, radial, annular ones, etc. [16; 17; 18; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32]. Many researchers noted that the most dangerous stress is the longitudinal one [6; 7; 16; 17; 18; 19; 20; 21; 22; 23; 24]. It is known that the stress state of an underground pipeline under seismic effects arises as a result of its interaction with the surrounding soil environment [1; 6; 7; 27]. Therefore, the interaction of the underground pipeline with soil is also divided into longitudinal and transverse ones. In each of the interaction types, the above factors can be detected under strong ground motions. The longitudinal interaction of an underground pipeline with soil is simpler from a mechanical point of view and was studied in more detail. The transverse interaction of an underground pipeline with soil is more complex and has not yet been sufficiently studied. Under the transverse pipeline-soil interaction, the pipelines of a circular cross-section interact with soil ambiguously. When a circular section moves in soil, greater pressure is generated on the section's front side than on its back side. The side surfaces of this section interact with soil in a more complex way. It has not been studied yet, how, and according to what laws, each point of the circle's outer surface interacts with soil. In seismic resistance calculations of underground pipelines, the most simplified schemes are usually accepted, which could be far from the real situation. In [10], the algorithm of the 3D analysis developed to solve an external boundary problem by applying the combined method based on incorporating the FEM and Somigliana's integral formula is considered. However, such combination methods have not been applied for solving the pipeline subjected to seismic impact. When determining longitudinal stresses in buried pipelines, the following technical theories are used. 1. A simplified theory based on the hypothesis of the equality of longitudinal strains in soil and pipeline under longitudinal seismic loads [6; 7]. This theory determines the soil strain, considered an elastic half-space, under plane wave propagation; the front of the wave is perpendicular to the pipeline axis. The soil strain is determined without considering the pipeline, i.e., it is believed that the buried pipeline does not influence the soil strain formation. In this case, the strain is  
\[\varepsilon_{g}\ =\ \frac{v_{g}}{C_{g}} \tag{1}\]  
where \(\varepsilon_{g}\) is the longitudinal soil strain along the pipeline axis, \(v_{g}\) is the velocity of soil particles in the longitudinal direction, and \(C_{g}\) is the velocity of longitudinal wave propagation in soil.