# Mathematical modeling of bending stress waves in an aboveground oil pipeline under unsteady seismic action

## Abstract

The problem of numerical modeling of bending waves in an aboveground oil pipeline under nonstationary seismic action is studied. To solve the unsteady dynamic problem of elasticity theory with initial and boundary conditions the finite element method was applied. Using the finite element method in displacements, a linear problem with initial and boundary conditions was led to a linear Cauchy problem. A quasi-regular approach to solving a system of linear ordinary differential equations of the second order in displacements with initial conditions and to approximation of the studied domain is proposed. The technique is based on the schemes: point, line and plane. The area under study is divided by spatial variables into triangular and rectangular finite elements of the first order. According to the time variable, the area under study is divided into linear finite elements with two nodal points. The algorithmic language Fortran-90 was used in the development of the software package. The problem of the effect of a plane longitudinal wave in the form of six triangles on an elastic half-plane to assess physical reliability and mathematical accuracy is considered. A system of equations consisting of 8 016 008 unknowns is solved. The calculation results are obtained at characteristic points. A quantitative comparison with the results of the analytical solution is taken. Furthermore, the problem of the impact of a plane longitudinal seismic wave at an angle of 90° degrees to the horizon on an aboveground oil pipeline is considered. The seismic impact is modeled as a Heaviside function, which is applied at a distance of three average diameters from the edge of the pipe. The calculation results were obtained at the characteristic points of the object under study. A system of equations consisting of 32 032 288 unknowns is solved. Bending waves prevail in the problem under consideration.

## Keywords

### Vyacheslav K. Musayev

Moscow state University of civil engineering; Russian University of Transport; Mingachevir State University

Author for correspondence.
Email: musayev-vk@yandex.ru
ORCID iD: 0000-0003-4336-6785

Doctor of Technical Sciences, Professor of the Department of Integrated Safety in Construction, Moscow State University of Civil Engineering; Professor of Technosphere Safety Department, Russian University of Transport; Professor of the Department of Higher Mathematics, Mingachevir State University

26 Yaroslavskoe Shosse, Moscow, 129337, Russian Federation; 9 Obraztsova St, bldg 9, Moscow, 127994, Russian Federation; 21 Dilyara Alieva St, Mingachevir, AZ4500, Republic of Azerbaijan

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