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IntroductionThe first investigations focused on the study of the earthquake resistance of underground pipelines were carried out as long ago as [1950][1951][1952][1953][1954][1955][1956][1957][1958][1959][1960]. It was suggested that damage to pipelines is caused chiefly by the action of a seismic wave travelling along the axis of the pipeline [1][2][3].A number of studies describing damage to underground pipelines occurring in the course of earthquakes then appeared. These studies established the degree of damage and specific fault rate of underground pipelines as a function of the occurrence depth, the soil conditions, geometric dimensions, type of joints, and the influence of the pressure of the contents and the quality of the construction. There exist data on the destructions of underground pipelines that pass through active fractures, ruptures in the ground, areas with collapses, cracks, and water-saturated soil [4][5][6][7][8][9].Modeling of interaction in the system "pipe − soil" is one of the major problems of the seismodynamics of underground pipelines. Statement of problemA mathematical model for the case of movement of the points of a rod subjected to the combined action of longitudinal and transverse forces is presented in [10]. A system of equations describing the motion of a linear underground pipeline based on the Hamilton-Ostrogradskii variational principle with arbitrary direction of the seismic wave relative to the axis of the pipeline was obtained in [3,4]. On the basis of certain assumptions [10], the pipeline is modeled in the form of a rod and the displacements are selected as follows:where u 1 and u 2 are the displacements of sections of the pipeline; u and w are longitudinal and transverse displacements; and α 1 is turning angle of a section of pipe.A system of equations describing the oscillations of a rectilinear pipeline that interacts with the surrounding soil in the case of seismic action functioning in an arbitrary direction is presented. A numerical implementation of problems related to the determination of the mode of deformation of a pipeline experiencing seismic movement of the soil is performed.

The effect of real records of several earthquakes on polymer pipeline vibrations are studied and analyzed in the paper. Longitudinal vibrations of straight end section of the underground pipeline are considered using the Kelvin-Voigt model. To solve the equation, the method of finite differences of the second order of accuracy is used. An algorithm has been compiled and on its basis a software product has been developed for the numerical calculation of an underground polymer pipeline under the effect of real seismograms recorded in different cities of the world. The stress-strain state (SSS) of polymer pipelines is determined under various boundary conditions.

A system of equations describing the oscillations of a rectilinear pipeline that interacts with the surrounding soil in the case of seismic action functioning in an arbitrary direction is presented. A numerical implementation of problems related to the determination of the mode of deformation of a pipeline experiencing seismic movement of the soil is performed. The methods of realization of boundary problem of seismodynamics of underground pipelines at arbitrary direction of seismic load are developed, as well as the methods of realization of the problems of seismodynamics of underground pipelines of non-orthogonal configuration, based on Finite deference method.

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