Numerical solution of problem of longitudinal vibration of underground pipeline by the method of finite differences

. This paper analyzes the dynamic response of a wave propagating to the soil horizontally through a pipe. That is, the state considers a linear viscoelastic model of the interaction of the "pipe-soil" system. This problem is solved using the numerical method of differential equations. The longitudinal wave in the ground is taken as a sinusoid. And also, in the article, we will compare the comparative analysis of the results for some values of the coefficients of elasticity and viscosity of taste, propagation velocity, and pulse duration. In the current flow of the "pipe-soil" system, reflections of waves propagating in an underground pipeline at the boundaries of the pipeline, usually propagating in soils. This leads to a change in the shape of the underground pipeline, the value of which can double. The viscosity coefficient of interaction at the "pipe-soil" system contact leads to attenuation of the wave front in the underground pipeline. For soils with values of viscous interaction coefficient more than 100 kN·s/m 2 , this may lead to complete attenuation of the bursts at the wave front in the pipeline. The ratio of steps along the coordinate and time equal to the propagation of waves in the pipeline, obtaining results that coincide with the exact nature.


Introduction
Underground pipelines are a key component of critical life support systems such as water supply, gas, and liquid fuels, sewerage, electricity, and telecommunications.The interaction with the soil structure caused by seismic waves has an important effect on pipeline behavior, and the integration across the pipeline network affects the entire system's performance [1,2].
In recent decades, much attention has been paid to the impact of wave propagation on segmented underground pipelines.In [3][4][5][6], various models were proposed to analyze the interaction of segmented pipelines during wave propagation.
Damage to underground pipelines during an earthquake could be caused by various hazards: permanent soil deformation (landslides, liquefaction, and seismic settlement) and wave propagation effect.The latter is characterized by transient deformation and ground curvature caused by the traveling wave effect.A simple procedure taking into account one traveling wave with an undamped (traveling) waveform was proposed by T.R. Rashidov and N.M.Newmark [7 -8] to analyze the wave propagation.According to T.R.Rashidov's statement, the static theory was first considered by R.M. Mukurdumov [9] and then given in the monograph by Sh.G.Napetvaridze [10].There, he proposed that during wave propagation along the pipeline, the pipe and the soil move in the same way.N.M.Newmark later proposed a similar assumption that the underground pipeline strictly follows the soil movement, called a static theory.Therefore, the maximum axial deformation of the pipe is the same as the maximum axial deformation of the soil.
However, the above procedures consider infinite pipe lengths and therefore do not consider their effective length and construction work (constraint conditions).In [11], analytical relationships were developed for a pipe of finite length subjected to various combinations of boundary conditions (i.e., the free end, fixed or elastic end) for pipelines of different lengths.In 1962, T.R. Rashidov proposed a differential equation for an underground pipeline, which became the basis of the T.R. Rashidov's dynamic theory [7,11].G. De Martino et al. [12] and V. Corrado et al. [13][14][15] developed models of the pipe-soil interaction, taking into account the finite length of the pipeline.Assuming a linear elastic model of soil motion and ignoring the slip at the pipe-soil contact interface, the model analyzes the dynamic behavior of a finite-length pipeline taking into account the boundary conditions at the ends.It was assumed that the pipeline was continuous; that is, any fluctuations between the characteristics (parameters) of the pipeline and its joints were considered insignificant.A.A. Ilyushin and T.R. Rashidov [2] proposed a viscoelastoplastic model of the underground structure interaction with the soil.
In works [14][15][16], the use of different mechanical, mathematical models was analyzed, and several urgent problems of underground and ground structures were solved.
The effect of the coefficients of elasticity, viscosity, and plasticity of the pipeline interaction with soil on the stress-strain state of an underground pipeline is studied in detail in [17][18][19], the influence of inertial forces on the deformed state of an underground pipeline was studied in detail.

Statement of the problem
Let us consider the problem of longitudinal vibration of a main underground pipeline under linear viscoelastic interaction at the contact with soil, with three types of fastening [16]: where m is the weight per unit length of the pipeline; E is the Young's modulus of the pipe material; F is the cross-sectional area of the underground pipeline; x k is the coefficient of elastic interaction of the "pipe -soil" system [2]; µ is the coefficient of viscous interaction of the "pipe -soil" system, that is, the resistance of the equivalent velocity of interaction of the "pipe -soil" system; H is the laying depth; D is the outer diameter of underground pipeline; , ( ) u x t is the absolute displacement in the section x of the underground pipeline at the point in time t ; , ( ) g u x t is the ground displacement corresponding to the section x of the underground pipeline at the point in time t .
where A is maximum soil displacement; Z is angular velocity of seismic wave vibration determined by the formula 2 /Т Z S ; p C is "apparent velocity" of wave propagation (hereinafter referred to as the wave propagation velocity in soil).The "apparent velocity" of wave propagation in the soil can be greater due to the wave's angle of incidence to the pipeline axis or due to the flexible joints of the pipeline.
Ground motion is written in strains: If an underground pipeline is pliantly fixed at the ends, then the boundary conditions are taken in the following form: (5)

Solution methods
The study in [17] compares the methods of Crank-Nicholson, McCormack, and Courant-Friedrich-Lewy (explicit scheme) and shows the accuracy of the explicit scheme relative to other methods when solving discontinuous problems of vibrations of underground pipelines.In this study, the finite difference method solves the problem in an explicit scheme.Careful numerical calculations are performed to prevent unwanted vibrations near the discontinuity (deformation wave front).The problem is solved using an explicit finite-difference scheme; Courant's conditions must be satisfied to obtain results that coincide with the exact solution.It was shown in [17] that the choice of the step ratio in coordinate and time in the form (6) for the problem (1) allows one to obtain results that coincide with the exact solution: Elastically fixed boundary conditions are: To solve equation ( 8), the method of finite differences of the second order of accuracy is used: 2 Let us substitute the approximations of the differentials of the function in time and coordinate (11) and (12) and the displacement function (13) into the differential equation (8), obtaining: ))/ . Here: We consider both ends of an elastically fixed underground pipeline with boundary conditions (10), a and b.Let us approximate the boundary conditions: ), ). 2 4 Then, for 0 j we have: .
The following system of equations is solved: ]/ , 0 , 2 ( 4 After solving the system of algebraic equations (19), we determine the displacements in dimensionless form.Steel pipe characteristics are modulus of elasticity -

Results and Discussion
The wave propagation in a steel pipeline is 5120 m/s, and in the soil this velocity depends on soil type.Let us assume that a wave in soil moves with an apparent velocity of 2500 m/s. Figure 1a shows the change in the deformation of a main underground pipeline of a length of 1 km along the coordinate, both ends of which are fixed in the ground.From Figure 1, a, it can be seen that at times t = 0.02 s, t = 0.06, t = 0.12 s, and t = 0.18 s, the wave front in the pipeline reaches distances of 103.3, 309.8, 619.6, and 929.3 m, respectively.The maximum deformation at the wave front in the pipeline is approximately two times less than the maximum deformation in soil.Figure 1a and 1b show the change in the deformation of the underground pipeline along the coordinate under elastic interaction k х = 0.5 × 10 7 N/m 3 .It was found that with the wave propagation through an underground pipeline over time, the amplitude of oscillations at the wave front in an underground pipeline slowly decreases.Before the wave front arrival at 929.4 m, the amplitude at the wave front decreases by about 5.7%.This is the phenomenon of the wave front attenuation in a pipe with a zero coefficient of viscous resistance, which appeared due to an error in the computational scheme.Figur 2 shows the change in the absolute (a) and relative (b) displacements of the underground pipeline along the coordinate for the points in time 0.02, 0.06, 0.12, and 0.18 s.

Fig. 2. Change in the absolute (a) and relative (b) displacements of the underground pipeline along the coordinate
As seen from Figure 3, a, the wave front in soil reaches the 450th meter point at t = 0.18 s from the left end, with an increase in the coefficient of elasticity of interaction k х from 0.5 × 10 4 to 4.5 × 10 4 kN / m 3 , the deformation of the underground pipeline approaches to the value of soil deformation.Figure 3, b shows the change in deformation behind the wave front in the pipeline at a value of the elastic interaction coefficient of 0.5 × 10 7 and 4 × 10 7 N / m 3 .It can be seen here that the coefficient of elastic interaction affects not only the frequency of oscillations behind the wave front of the pipeline but also the amplitude of deformation at the front of the pipeline.
With an increase in the coefficient of elastic interaction, the value of the pipeline deformation at the wave front decreases, and the oscillation frequency behind the pipeline wave front increases.Immediately behind the wave front in the pipeline, the soil resists so that the deformation, at a certain distance behind the front, changes sign, and gradually, with distance from the wave front, the deformation oscillations damp out.At the same time, the soil wave propagates through the pipeline's lateral surface and excites a wave in the pipeline.
Here, with an increase in the value of the elastic resistance of soil, the frequency of oscillations behind the front increases.Fig. 3. Change in the deformation of the underground main pipeline along the coordinate with elastic properties of the "pipe-soil" system interaction at t = 0.18 s: 1 -k x = 0.5 • 10 7 N / m 3 ; 2k x = 4 10 7 N / m 3 ; 3 -wave in soil Figure 4a shows the change in relative displacements along the length of the underground main pipeline; the interaction between the pipeline and the ground is considered elastic.It can be seen here that the relative displacement reaches its maximum value at the wave front in the soil.At t = 0.18 s, the wave reaches the 450th meter of 1000 m long underground pipeline.With an increase in the coefficient of elastic resistance, the maximum value of the relative displacement greatly decreases.Behind the wave front of the pipeline, high-frequency oscillations appear, the amplitude of which increases with distance.Figure 4b shows the influence of the value of elastic resistance on the amplitude of oscillation of the relative displacement behind the wave front in the pipeline.Figure 6 shows the case in which the reflected wave front at the rigidly fixed end of the pipeline coincides with the wave front in the pipeline.When these waves coincide, the maximum deformation value in the underground pipeline increases up to one and a half times.In Figure 6, the maximum deformation value in the underground pipeline is one and a half times less than the maximum deformation in soil.This is due to the presence of the viscosity of the interaction and the error of the numerical scheme.To determine the time of the maximum value of the pipeline deformation, we use the following formula t=(2L+T•C p )/(C p +a) and obtain t = 0.32625 s.Hence, it can be seen that with an increase in the coefficient of elastic interaction, the maximum deformation value in the underground pipeline tends to be the maximum deformation value in soil.
Figure 7b shows the maximum values of the relative displacement for different values of the coefficient of elastic interaction.As established, with an increase in the coefficient of elastic interaction from 0.5 × 10 4 to 4 × 10 4 kN / m 3 , the maximum relative displacement decreases threefold.Consider an underground main pipeline, when both ends are elastically anchored in the ground.The elastic fixing coefficients are denoted by kn 1 and kn 2 .Let us assume that the coefficients of elastic interaction kn 1 and kn 2 are equal to each other and equal to kn. Figure 8 shows the influence of the compliance coefficient of fastenings on the deformation values at the wave front in an underground main pipeline.With an increase in the fixing compliance coefficient, the deformation values at the wave front increase linearly.The compliance coefficient was found to affect the deformation values at the compliantly attached boundaries (Figure 9).

Conclusion
When the ground moves in the form of a traveling sine wave, an underground pipeline undergoes deformations close to the deformation in soil.An increase in the elastic E3S Web of Conferences 365, 01020 (2023) https://doi.org/10.1051/e3sconf/202336501020CONMECHYDRO -2022 coefficient of interaction leads to an increase in deformation in the buried pipeline.The reflection of the wave front at the fixed end increases the deformation of the underground pipeline by approximately one and a half times.The viscosity coefficient of the interaction contributes to the attenuation of the wave front, depending on the value of the coefficient. sin

k
are the coefficients of compliance of the fastening at the left and right ends of the underground pipeline.Initial conditions are zero:

Fig. 1 .
Fig. 1.Change in the underground pipeline deformation along the coordinate (a) and wave front deformation of an underground pipeline (b) at times t = 0.02, 0.06, 0.12, and 0.18 s

Figure 5 ,
Figure5, a, shows the change in the underground main pipeline with visco-elastic interaction properties.As seen, with the distance, the amplitude at the wave front in the pipeline strongly decreases due to the viscosity of the interaction.Figure5, b shows a graph of the deformation change at the pipeline wave front according to the coefficient of viscosity of the interaction for four points in time.At viscous resistance values greater than 100 kN•s/m 2 , the maximum value at the wave front greatly decreases and is approximately zero (see Figure5, b).

Fig. 7 .
Fig. 7. Change in the maximum value of deformation (a) and relative displacement (b) according to the coefficient of elastic interaction at the contactFigure7,a shows the change in the maximum deformation in terms of the coefficient of elastic interaction for the points in time 0.12 and 0.18 s.Hence, it can be seen that with an increase in the coefficient of elastic interaction, the maximum deformation value in the underground pipeline tends to be the maximum deformation value in soil.Figure7bshows the maximum values of the relative displacement for different values of the coefficient of elastic interaction.As established, with an increase in the coefficient of elastic interaction from 0.5 × 10 4 to 4 × 10 4 kN / m 3 , the maximum relative displacement decreases threefold.

Fig. 8 .Fig. 9 .
Fig. 8. Change in deformation at the wave front in the pipeline according to the coefficient of elastic fastening of the ends of the underground main pipeline