## 1 Introduction
Further, the pipeline strain \(\varepsilon_{c}\) is taken to be equal to the soil strain  
\[\varepsilon_{c}\ =\ \varepsilon_{g} \tag{2}\]With a known value of pipeline strain \(\varepsilon_{c}\), the longitudinal stress of the pipeline is determined from [PERSON]'s law, considering the pipeline as an elastic body  
\[\sigma_{c}\ =\ \varepsilon_{c}E_{c} \tag{3}\]  
where \(E_{c}\) is the modulus of elasticity of the pipeline material. The seismic strength of the pipeline is estimated from the value of longitudinal stress in the pipeline material. It should be noted here that the simplified theory formed the basis of the normative methods for calculating seismic resistance of main underground pipelines and other types of pipelines. Many correction factors were introduced into Equation (3), and several parameters were replaced for convenience when used by designers-engineers. However, hypothesis (2), underlying the simplified theory, is implausible. On the other hand, in this case, the maximum possible longitudinal stress of the pipeline is obtained by calculation. Hypothesis (2) provides a greater margin of safety for the buried pipeline. The equality of pipeline strain and soil strain underlies the first, initial stage in developing the theory of seismic resistance of underground pipelines. 2. The second stage in the theory development led to the creation of a dynamic theory of seismic resistance of underground pipelines (in the middle of the twentieth century). The basis of this theory is the forces of pipeline interaction with soil, which arise due to the difference in strains in the pipeline and soil [1; 5; 6; 7; 16; 17; 18; 19; 20; 21; 22; 23; 24]. In the dynamic theory, hypothesis (2) is considered false. Here, the seismic load on an underground pipeline acts through soil and is determined under longitudinal interaction of the pipeline with soil by the following relation  
\[\tau\ =\ K_{x}u \tag{4}\]  
where \(\tau\) is the shear stress on the contact surface of the pipeline with soil (MPa); \(K_{x}\) is the coefficient of longitudinal interaction or the stiffness coefficient of soil particles connection with the outer surface of the pipeline (MPa/m); \(u\ =\ u_{g}-u_{c}\) is the relative displacement (m), \(u_{g}\) is the absolute longitudinal displacement of soil in the direction of the pipeline axis (m), and \(u_{c}\) is the absolute longitudinal displacement of the pipeline. Further, taking into account (4) or its more complicated versions [14], the differential equations of the pipeline's longitudinal motion are compiled in one-dimensional, two-dimensional, and three-dimensional statements. The obtained equations of motion are solved with the corresponding closing equations of deformation of the pipeline material and the equations of continuum mechanics with certain boundary and initial conditions. In the dynamic theory, in most cases, seismic resistance problems are reduced to stationary problems of the theory of vibrations. The use of the stationary vibration theory is due to the desire to obtain simple engineering solutions for seismic resistance of underground pipelines. The problems of transverse, bending, and torsional vibrations of underground pipelines under seismic loads are considered similarly. The main difference between the problems of the dynamic theory of seismic resistance of underground pipelines and the conventional theory of vibrations is the presence of an interaction force of the type (4) in the equation of motion. Therefore, the reliability of the results obtained according to the dynamic theory is completely determined by the interaction law's reliability (4) or by similar, more complex laws. On the whole, dynamic theory was and still is a significant achievement in developing the theory of seismic resistance of underground pipelines. Based on the foregoing, this research aims to assess the advantages and disadvantages of the dynamic theory and the reliability of its solutions, and to develop an alternative theory of seismic resistance of underground pipelines.