## 1 Introduction  
Underground pipelines that transport oil, gas, oil products, water, and other liquid and gas products are very important engineering communications worldwide [1; 2; 3]. Some of these underground communications are built and operated in earthquake-prone areas [2; 3; 4; 5]. This circumstance requires the reliable provision of seismic safety of underground pipelines [1; 2; 3]. Seismic safety or seismic stability (the strength under seismic influence) of underground pipelines began to be intensively studied since the 50 s of the last century [6; 7]. It is necessary to distinguish the destruction of pipelines caused by seismic effects from the effects of soil inhomogeneities. In this publication, we will assume soil uniformity. Non-uniform soil support conditions resulting from the erosion of the surrounding soil by the water escaping from leaks may consider in [8]. The article also does not consider the impact of seismic-induced soil liquefaction or landslides on pipelines. The failure of pipelines due to seismic seismic-induced soil liquefaction is considered, for example, in [9; 10; 11], and seismic-induced landslides are considered in [12]. Undercrossing tunneling on adjacent pipelines, which could cause pipelines to collapse [13], is also outside this article's scope. The theories of seismic resistance of underground pipelines are based on the data collected when examining the aftermath of destructive earthquakes [1; 2; 3; 4; 5], laboratory and field experimental studies, and the resulting facts and hypotheses [6; 7; 14; 15]. The point of these hypotheses is as follows:  
* Under strong ground motion, the underground pipeline system is destroyed; these destructions are different: the destruction in the pipeline itself, in butt joints, the pipeline buckling from the ground; soil separation from the pipeline on the surfaces of their contact, etc. ;
* Seismic forces are transmitted to underground pipelines through soil. Therefore, these forces' values directly depend on the physical-mechanical and strain properties of the soil environment surrounding the underground pipeline. * The assumption of an infinitely long straight horizontal pipeline in homogeneous soil simplifies the solution of the problem of seismic impact on the pipeline. This assumption is used in most theoretical studies. An exception is in [11]. The unsteady wave problem of a longitudinal monochromatic wave propagation and reflection from a rigid stationary barrier solved in a one-dimensional statement. * During earthquakes, seismic forces are transmitted to underground pipelines through soil strain, i.e., the soil deforms under seismic wave effect and forces underground pipelines to deform. The pipelines inflict the most severe damage when the route is codirectional to the seismic strain vector [15]. * In the case of strong ground motions, the amplitudes of absolute displacements of soil particles reach 0.1-0.4 m at an oscillation period T = 0.05-20 s; under soil mass vibrations together with the underground pipeline, a difference between the vibrational motion of the pipeline and soil is formed (for various reasons) in vibration amplitude and phase; a displacement of the pipeline relative to the soil is observed, i.e., a relative displacement. * There is a critical value of the relative displacement \(u_{*}\) of the pipeline cross-section, below which there is an elastic bond between the soil particles and the outer surface of the pipeline in contact with the soil. At values of the relative displacement exceeding the critical value, the elastic bond between the pipeline and the ground is ruptured [14]. * After pipeline construction, due to the cohesion phenomena, a relative shear displacement occurs in the soil layer \(\delta_{g}\) thick, called the contact soil layer, between the underground pipeline and soil on their contact surfaces. * At an increase in relative displacement value greater than \(u_{*}\), the contact soil layer undergoes significant shear strain and may collapse; the soil outside the contact layer may remain intact.