National University of Science and Technology “MISIS” (Moscow, Russia):

V. N. Shinkin, Dr. Sci. (Phys.-Math.), Prof., e-mail: shinkin-korolev@yandex.ru

The various classical approximations are used to describe the shape of the subwater part of pipelines: for example, the equations of elastic line of slightly-sloping beam, non-stretched ideal flexible thread and slightly-sloping rigid thread. However, all these equations have the significant drawbacks that prevent them from being applied for the accurate calculating of the shape of subwater pipelines at their laying in deep-water seas, which are usually much deeper than lakes and rivers. So, the equations of the elastic line of slightly-sloping beam and slightly-sloping rigid thread are valid only in the case of small inclination angles of the pipeline, and the equation of the non-stretched ideal flexible thread (the equation of the chain line of the cable) does not take into account the resistance of the pipeline to bending and torsion. The subwater main pipelines are laid on the sea-bottom by the special marine ships and can have a laying depth of more than two kilometers. In this case, the longitudinal axis of the pipeline is so much curved during the laying that the excessive bending can lead to the destruction or defects of the pipeline. Therefore, at the laying of pipelines, it is extremely important to know the shape of their free subwater part in water and to correctly apply the pontoons, stingers and pipeline tension with the help of the anchors of the marine ships. The nonlinear equation of fourth order for the elastic flexible rod does not involve restrictions on the inclination angle of the subwater part of the pipeline and takes into account the resistance of the pipeline to bending. Although this nonlinear equation has no the general analytical solution, it can be solved numerically, using the Runge-Kutta method of fourth-order accuracy, or approximately by means of the power series. The approximate solution of the equation of elastic flexible rod by means of a power series is given below. For this solution, the coefficients of the series are found up to the 12^th degree of the series argument. As an example of the use of the found approximate solution, the numerical calculation of the shape and inclination angle of the subwater part of TurkStream at its laying on the sea-bottom are given.

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