Free vibrations of a thin-walled curvilinear section of an oil-trunk pipeline during ground laying

In this article authors obtained solutions to determine the ground curved sections free vibrations of thin-walled large-diameter pipelines with a liquid flow, based on a geometrically nonlinear version of the semi-momentless toroidal shells medium bending theory by V.Z. Vlasov and V.V. Novozhilov. The pipeline is a toroidal shell, the design scheme is presented in the form of a half torus. Angle β = 1800. The shell is laid on an elastic foundation and makes contact with the ground along a narrow strip. The problem of the soil pressure influence on the shell along a narrow strip is solved as a contact problem, using Fourier's series and an impulse function. The shell is exposed to the cooperative effect of the internal operating pressure, the liquid flowing pressure, the elastic soil foundation, and changes in the geometric characteristics. Motion equations of the middle shell surface are obtained taking into account the geometric and mechanical characteristics, and all the components of the shell material inertial forces. The hinged fastening of the shell ends is taken as the limiting condition. Using the semi-momentless shell theory assumptions, displacements in the longitudinal and circumferential directions are obtained. The solution to the problem of determining the free vibrations comes down to solving the problem of determining AB matrix values. The solutions obtained make it possible to determine the free vibrations frequencies at various wavenumbers values in the longitudinal and circumferential directions, and also make it possible to determine the internal operating pressure contribution, the soil bed coefficient, and geometric characteristics to the pipeline free vibrations frequencies.