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 cultivated soil environment changes its structure and is deformed; therefore, the considered model of the processes of interaction of the working body with soil remains understudied. The influence of soil criteria on the working body behavior can be taken into account through its density and tensile strength. To describe the movement of the soil near the leg during finite deformations, a plastic medium model proposed by academician Kh.A. Rakhmatulin and simplified equations obtained on the basis of the hypothesis of flat sections were used. When using the model of linearly elastic and compressible plastic medium, the resistance forces of the soil medium are determined when the legs of the subsoiler, presented in the form of a circular cone, move. It has been established that the magnitude of this force substantially depends on the type of contact conditions between the body and the soil, and its greatest value is achieved in the case of continuous motion. The dependence of the resistance force on time is obtained. According to the results of graph analytical studies, it is obvious that at the initial stage, while the contact area of the circular cone with the soil is variable, the resistance force depending on time changes according to a parabolic law, and then it remains constant. In the case of the movement of the subsoiler leg at a constant speed, it was found that, depending on the coefficient of internal friction and soil traction, a zone of increased soil density can form near the working body of the cultivator, where there is a significant increase in resistance force. With an increase in the angle of internal friction, a slight decrease in the resistance force is observed. The calculations were carried out based on the methods of mechanics of a deformable solid, soil mechanics, and were performed in the Maple-8 programming environment.
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