Transverse Vibrations of Underground Pipelines with Different Interaction Laws of Pipe with Surrounding Soils

Studies on the effect of changing the stiffness coefficient along the length of the pipeline on its resonant vibration mode are considered in the paper. A computational model of transverse vibrations of the pipeline located in soil with different properties is created. Theoretical and computational studies to solve the problems of stability of underground pipelines located in the soils with different properties under seismic effects are carried out. It is revealed that the vibratory process of the pipeline can be realized at frequencies close to resonance. The results of the study are presented as curves of distribution of displacements of pipeline sections along the length at dimensionless frequencies. When the pipeline vibrates with a frequency close to the resonant frequency, the displacements of pipeline sections can take very large values. It is shown that at frequencies close to resonance, the values of moments can be large in the pipeline sections, which are the reasons for the loss of pipeline stability.


Introduction
At present time pipelines are used to deliver water, oil, gas and etc. Underground pipelines are buried in the ground, and they stretch over large areas, where they interact with soils of different physical and mechanical properties. It significantly increases their exposure to damage from all sorts of sources, in particular during seismic activity.
Therefore, the study of the stability of underground pipelines during earthquakes is relevant. The solution of the problem of seismodynamics of underground pipelines, taking into account the intersection of soils with different properties along the pipe axis, serves to determine the possible seismic hazard.
To date, in domestic and foreign science, there are theoretical and experimental studies for determining the stress-strain state [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and stability [17][18][19][20][21][22][23][24] of underground pipelines interacting with the surrounding soil under seismic effects. A theory of propagating seismic waves in an underground pipeline and surrounding soil called the wave theory of seismic resistance of underground pipelines is proposed in [12]. According to linear and nonlinear laws, this theory is used to analyze only longitudinal vibrations of a pipeline interacting with homogeneous soil. In all the above investigations, when considering the interaction of the underground pipeline with the surrounding soil, the variability of the interaction parameters along the length of the pipeline is not considered. This is especially important for long pipelines buried in the ground and stretched over large areas, where soil conditions change significantly. In this case, the behavior of the pipeline mainly depends on the choice of the law of its interaction with the surrounding soil, which in the simplest cases is characterized by the interaction coefficients determined from experiments [5,7,11,12].
Transverse vibrations are of interest for long pipelines because they cause a loss of stability and such a phenomenon as buckling of pipelines. A methodology for the study of transverse vibratory processes in underground pipelines is developed in this paper when considering the variability of the interaction parameters along the length of the pipeline to establish the cause of the loss of stability of underground pipelines.

Methods
Let us consider the case when the pipeline contacts the surrounding soil at two interaction areas k 1 (0<x<l 1 ) и k 2 (l 1 <x<l), where l is the total length of the pipeline. Set the origin in the initial section of the pipeline and direct the axis х along the pipeline axis. When the time t 1 =(H-2R)/c (where R is the depth of the pipe; с is the speed of the wave propagation) a wave formed at the depth Z=H falls on the pipeline. The wave front is perpendicular to the pipeline axis. If we do not consider the time of the wave flow around the pipeline and the secondary waves of reflection and diffraction, then at time t≥t 1 the distributed load will act along the length of the pipeline k 1 (w 1 -u 0 ) when 0<x<l 1 and k 2 (w 2 -u 0 ) when l 1 <x<l, where w 1 (x,t), w 2 (x,t) are the transverse displacements of pipeline sections, u=u 0 (t)=u 00 (c t -H+2R) is the displacement of soil particles behind the wave front in the plane Z=H-2R.
The equations of transverse vibrations of the pipeline for each section have the form 0 ) ( Consider the case where R  is the wave length; A is the amplitude of the vibrations. The process is considered to be steady, then the solutions of equations (1) and (2) have the form The functions X 1 (х) and X 2 (х) satisfy the equations , , 2 2 Then the solutions of equations (1) and (2) can be represented in the form: Taking into account the boundary conditions (3), the displacement X 1 (x) has the form We put the function X 2 (x) in the following form: From conditions (5) it follows Substituting the expressions B i into formula (10), we obtain Using these conditions, we will compose an equation to determine the constants А 2 and From this system, we find the constants А 2 and А 4 Substituting the expressions for the constants А 2 and А 4 into formulas (9) and (11), we finally obtain problems solutions where the frequency of vibrations of the external action satisfies the condition ω 2 >k 02 /m.
The solutions of equations (12) we represent in the form  (13) where Z(z) are represented through functions of the Krylov type: . sin cos , sin , sin cos , cos   III  II  I   IV  III  II  I   IV  III  II  I   IV  III  II It is seen that all functions Z i satisfy the equation

From boundary conditions (4), it follows
Substituting expressions В i into formulas (5), we establish the form of the function X 2 (x) Using the boundary conditions (5), we compose equations for determining the constants А 2 and А 4 From this system, we determine А 2 and А 4 .

Results and Discussion
Consider the results of calculations. Figures 1 -3 show the curves of the distribution of deflections (referred to the value A) and moments (referred to the value of EJl 2 /A) along the pipeline axis x/l in the case of ω 2 >k 02 /m depends on various frequency ratios The vibratory process of the pipeline can be realized at frequencies close to the roots of the equation, at which the denominators for the functions X 1 (x,ω) and X 2 (x,ω) equal to zero. In figure 2 shows the curves of the distribution of displacements (referred to the value A) of pipeline sections along the length at dimensionless frequencies close to two resonance frequencies depends on various values of the ratios ω 1 /ω and ω 2 /ω ( In the calculations, it was assumed that l 1 /l=0.5. When the pipeline vibrates with a frequency close to the resonant frequency, the displacements of pipeline sections can take very large values. In the case under consideration, the presence of a section with different stiffness coefficients doesn't significantly affect the resonant vibration mode. Similar curves for moments are shown in figures 1 and 3. It can be seen that at frequencies close to the resonance, the values of moments can be large in the pipeline sections, which are the reasons for the loss of pipeline stability.

Conclusions
The results obtained when studying vibratory processes in underground pipelines using the developed software product allow us to recommend them for specific calculations and design of underground pipelines under seismic conditions. A number of numerical results were obtained depending on the frequencies close to two resonance frequencies. So, we can conclude that at pipeline vibrations close to the resonance, the displacements of pipeline sections can take very large values. The values of moments can be large in the pipeline sections, which are the reasons for the loss of pipeline stability.
The obtained results and conclusions on them are in satisfactory agreement with the results of observations of the behavior of pipelines in real conditions, which have been observed repeatedly, especially recently during strong earthquakes.
The results presented provide the analysis of the behavior of underground pipelines during seismic impacts and allow developing effective methods for studying the effects of an earthquake on the stability of the underground pipeline in seismic areas.