THE INFLUENCE OF STATIC FORCES AND INITIAL CONDITIONS ON GAS PIPELINE DEFLECTIONS DURING THE PASSAGE OF A CLEANING PISTON

Abstract

When cleaning pistons pass through gas pipeline beam crossings, forced oscillations of the pipe axis occur. It is important to take into account all the force factors associated with these vibrations, starting with static forces. In this paper, only static forces are taken into account, namely the weight of the pipes at the beam transition and the weight of the cleaning piston. The aim of the study was to determine the deflections of the pipeline axis and the bending moments of their pipes at the beam transition during the passage of the cleaning piston through it, with and without the initial condition. The initial condition is caused by the weight of the pipes at the transition and causes the deflection of the pipeline axis and bending moments of its pipes before the cleaning piston enters the beam transition. This condition was obtained as a result of solving a fourth-order differential equation with consideration of the boundary conditions corresponding to the pinching of the ends of the beam transition pipes. The problem was solved by the Fourier method. The eigenfunctions of the problem of free oscillations of the beam were used. The right-hand side of the inhomogeneous differential equation was represented as an infinite series of the product of the eigenfunction of the free oscillations of the beam and an unknown time function. After finding this unknown time function, the Fourier method was used to obtain the solution of the problem in the form of deflections of the axis of the beam transition, taking into account the initial condition. In the obtained solution, the deflections of the axis of the pipes of the beam transition caused by the initial condition are represented by a polynomial, and the deflections caused by the passage of the cleaning piston through the transition are obtained as an infinite sum. By performing a double differentiation of the deflection function of the axis of the beam transition, the function of bending moments of pipes at the gas pipeline transition is found, taking into account the initial condition. Using the analytical formulas for the deflections of the pipe axis at the beam transition and the bending moments of their pipes, the deflections of the gas pipeline axis at the beam transition and the bending moments were calculated both with and without taking into account the initial condition. Analysing the results, it should be noted that the deflections caused by one initial condition are three times higher than the deflections caused by the passage of the cleaning piston through the transition. The maximum deflections correspond to the cross-sectional area of the crossing x = 15 m. The maximum deflection caused by one initial condition is 6.2 mm, and the total maximum deflection is 8.2 mm. The maximum deflection caused by the cleaning piston itself is 2 mm. The curves in the bending moment plot for different time points are close together. This indicates that the passage of the cleaning piston through the transition has little effect on the value of the bending moments of the pipes at the transition. The maximum modulus values of the bending moments of the pipes at the transition, with and without taking into account the initial condition, are -183.2 kN∙m and -50.02 kN∙m, respectively.

Keywords: beam transition; differential equation; bending moments of pipes at the transition.