1 Introduction

Currently, due to the depletion of hydrocarbons in their traditional areas of production, the world has begun to work on the development of oil and gas fields in remote areas and offshore. This development involves a set of scientific and technical issues related to the transportation of oil products to areas of their consumption. Because of novelty and complexity, many of these issues remain unsolved, and the traditional approaches presented in the literature on the calculation of pipelines are inefficient for their solutions.

One of the unsolved problems is the problem of rejection of the pipeline from its design position, arising from the motions of soil, vibrations from anthropogenic factors, seismic activity, intrinsic instability, and for other reasons^[1-2]. In this regard, the question arises regarding the study of dynamics of a pipeline in a deformable medium.

A large number of monographs and reviews have been devoted to this subject of dynamics of pipelines. The influence of seismic phenomena on pipelines was studied in detail^[2]. The problems regarding the interaction of engineering designs with the fluid branched in a new direction of computational and applied mechanics are called "fluid-structure interaction problems". The review of fluid-structure interaction problems was found in Ref. [3]. A detailed review and reference material on subject of marine underwater pipelines can be found in Ref. [4].

The traditional approach to research of statics and dynamics of pipelines is based on the theory of rods. A nonlinear model of dynamics of the pipeline on the basis of the rod theory was offered and studied in Ref. [5]. The problem of equilibrium of a pipe with the moving fluid in the field of gravity was solved. The exact solution of the equation for the simplified non-stationary model was constructed.

An analytical method on the basis of the rod theory was introduced and developed in Ref. [6] for research of the upheaval buckling of the pipeline. Effective engineering measures against the buckling of the underwater pipeline were studied. Global lateral buckling was studied in Ref. [7] on the basis of the variational principles. The energy method was introduced to deduce the analytical solution which is suitable for the global modes of a buckling of the idealized underwater pipeline and for the analysis of some characteristics of buckling. The finite element method was used to simulate the lateral global buckling of the pipeline at high temperature and pressure.

In Ref. [8], the finite strains of the pipeline and pipe interaction with the external medium were investigated. The refined analytical methodology of calculation of strain was proposed for studying the failure mechanism more accurately. On the basis of the theory of a beam on elastic foundation, the position of pipe potential destruction and the stress and strain distributions along the pipeline were derived.

New problems of the pipeline theory are linked to the shell theory. The finite element method is applied to the analysis of stress-strain state of a pipe wall, as a rule. For example, in Ref. [9], a new harmonic axisymmetric thick shell element for static and dynamic analysis was suggested. This element based on the modified Hellinger-Reissner variational principle that is the modern research technique of two-dimensional tasks. In Ref. [10], the new axisymmetric shell element was developed for the limit and shakedown analysis. The element was based on variational problem formulation Hellinger-Reissner type with interpolation of velocity and stress fields independently from each other. A new variational approach was applied in Ref. [11] to the analysis of buckling of a cylindrical shell. The influence of geometrical parameters and boundary conditions on the critical loads and buckling modes was discussed in detail, and some ideas in this problem were analyzed.

In Ref. [12], the equations of dynamics of elastic shells subjected to large deformations were formulated in Euler coordinates. The equations of dynamics of nonlinearly elastic shells in Euler coordinates are convenient for formulation of the problems about interaction of pipelines, hoses, and other nonlinearly deformable thin-walled tubular elements of designs with moving fluids and gases.

In a series of papers^[13-17], we investigated the kinematics and dynamics of a bent pipeline. Mathematical models were constructed, algorithms for their numerical analysis were developed, and test problems were solved. In Ref. [13], the geometry and the mathematical model for the internal problem of the quasi-linear hydro-elastic vibrations were studied. In Ref. [16], we verified this mathematical model on the basis of data in the literature and found that the proposed model is more general than the standard models constructed on the basis of the rod theory. In Ref. [15], we studied the nonlinear waves in straight and bent tubes, and in the analysis of these waves, we obtained the Korteweg-de Vries equation and proved that the Klein-Gordon-Fock equation's precision is sufficient to achieve the required accuracy of mathematical modelling.

An external problem regarding slow deformation of the underground pipeline was studied^{[14, 17]}. A paper^[17] was devoted to the analysis of kinematics and validity of the mathematical model of the motion pipeline in the external medium. The algorithm for numerical analysis of the mathematical model was created and tested. The motion of the pipeline as a flexible rod in a viscous medium was studied^[14].

Main purposes of this paper are to (ⅰ) construct a mathematical model of a pipeline as a geometrically nonlinear shell containing flowing fluid and surrounded by an external medium; and (ⅱ) perform computational experiments on the proposed mathematical model. The objectives are to (ⅰ) determine the displacements and strains of the longest pipelines under a constant internal pressure difference in the fluid, (ⅱ) calculate the dynamics of a pipeline with a profile in the form of a catenary containing fast fluid flow, and (ⅲ) determine the displacements of the pipeline for a variable internal pressure difference. The algorithm of a reduction of the equations of the shell theory to one-dimensional statement^[13] developed by us was applied at construction of this mathematical model. In the numerical experiments, we found that the cross sections of a thin-walled curved pipeline experienced warping and strain of the pipe wall that is strongly non-uniform at the vicinities of the points of fixing and in locations of extreme curvature of a profile. The considered mathematical model is demonstrated to describe the known effect of "the transitions of pipes/risers behaviour".

The main background engineering problem of the present paper is the problem of stress and strain distributions in a wall of the bent underground or underwater pipeline. Development of a new research technique of bent pipe dynamics will promote the development of the pipeline monitoring systems.

2 Geometry of the curved pipe

The metal pipeline of length L with a circular cross-section of radius R₀ and a wall of small thickness h is considered. The centre line of the pipe is slightly curved along the flat curve , where s is a natural parameter (arc length). The pipe is immersed in a highly viscous medium and filled with a steady flow of fluid, which is moving with the velocity ϑ_s0 under the influence of a constant pressure drop. The initial radius of curvature of the axial line ρ₀(s) is assumed to be large compared with the radius of the tube.

The systems of coordinates used for the mathematical modelling of the movement of the pipeline were described in Refs. [13] and [17] (see Fig. 1). Curvilinear coordinates {O, s, θ, R} are built on the axis Γ so that s is the length of the arc (OO'), and {O′, θ, R} are the polar coordinates in the cross-section of a pipe.

Curvilinear coordinates are denoted as follows: x₁ =s, x₂ =θ, and x₃ =R. On the basis of the formulae of Ref. [18], the following geometrical parameters are found:

(1)

Following (1), for the middle surface of the pipe wall, the geometric relationships are executed,

(2)

where k₁ and k₂ are the main curvatures of a median surface. Expressions for the Lame's coefficients following from (2) are

(3)

The formula (1) was published only for the toroidal coordinates with the constant curvature κ₀ (see Ref. [19]). (3) was published in Ref. [20].

The geometric properties of the wave operator in the fluid-filled slightly curved tube were investigated^[21] but without formulation of the practical problems of mechanics. The study used a curvilinear coordinate system based on the Frenet relations for a space curve. Analogues of (1) and (2) are obtained.

Unlike the curvilinear coordinates used in Ref. [19], we consider the general geometry with a variable curvature of a profile. In comparison with Ref. [21], here, concrete mechanical tasks are investigated, which is a more difficult problem to solve than the examples that were presented in Ref. [21]. The transition kinematics between the current and initial states of the pipeline was studied in Ref. [17].

3 Formulation of a mechanical task: initial equations for the mathematical model

In this work, the characteristics of movement of the bent pipeline under the influence of an internal fluid stream, the resistance of the medium, and the intrinsic elastic forces of the pipe wall are investigated.

For the study of the large-scale processes, the movement of the inner flow is considered as quasi-stationary. We neglect the oscillatory processes within this article. The Darcy's law of a friction^[22] was chosen as the law of hydraulic resistance. The equations of stationary movement of an incompressible fluid are^[23]

(4)

The resisting force Φ(ϑ_s0), affecting on a fluid stream, is expressed by the ratios^[22-23],

(5)

where

Here, ρ_f is the fluid density inside the pipe, and µ_f is the fluid viscosity. The components of the fluid velocity vector v₀ along the coordinate axes s, θ, and R are denoted as v_s, v_θ, and v_R, respectively. Formulae for the coefficients λ₁ and β were determined by Nikuradse^[22] and were also explored in the work of Loitsyanskii^[23] in which the technique of calculation of the friction of a fluid stream in rough pipes was established. In Ref. [22], a detailed experimental substantiation of (5) was given, and the error of this approximation was found.

The longitudinal surface force affecting from a fluid flow on the internal pipe wall is calculated on the formula,

which is derived from a pipe element equilibrium condition in the lengthwise direction and from (5).

The external surface of a pipe is influenced by the distributed resisting force and the pressure from the external medium. Modelling of the external medium surrounding the underground pipeline is executed by means of viscous fluid as it was described^[2]. In Ref. [24], the solution to the problem of the resistance force to the cylinder's motion in a viscous fluid was derived, which was used in our articles^{[14, 17]} for the calculation of the density of the resistance force of the external viscous medium to the cross-sectional movement of the pipeline. For the dynamic pressure of the external medium, the following formula applies^[14]:

where γ =1.781 1 is the number of Euler-Maskeroni^[25], ρ_e is the density of the medium, µ is the viscosity of medium, h₀ is the depth of immersion of the pipeline, and u^* is the velocity of the pipeline's cross-section moving. The numerical value of µ varies over a wide range, from 10² Pa·s to 10⁵ Pa·s^[26], depending on the external conditions for the underground pipeline.

Based on the algorithm of transition from the three-dimensional elastic body to the shell proposed in Refs. [20] and [27], the equations of motion of a pipeline were deduced as equations of technical shells immersed in a viscous medium^[17]. For applicability of these equations, Vlasov's conditions^[20] must be valid,

(6)

The following parameter should be small:

(7)

The value of λ must ensure that the condition (6) is satisfied. Therefore, λ ≤ 1/4. This ratio is a necessary condition for the applicability of the mathematical model. The results of numerical experiments on the oscillations of pipelines for various values of λ were given in Ref. [16].

The equations of a pipe motion have the following form in a shell approximation:

(8)

Here, u, v, and w are the displacements of the median surface of a pipe along the coordinates s, θ, and R, respectively. I⁽⁰⁾ and χ₀ are the first invariant of the strain tensor and the linear torsion of the pipe wall, respectively. X, Y, and Z are the components of the density of forces acting on the shell along the coordinates s, θ, and R, respectively (the forces of inertia are included in these components, according to the principle of d'Alembert). This corresponds to the traditional notation^[20]. (6) is the condition for applicability of (8).

The first three equations of (8) and the formulae for χ₀ and ∇² are the standard relations of the technical shell theory by Novozhilov and Radok^[20]. Formulae for I⁽⁰⁾, X, Y, and Z were proved in Ref. [17]. It was found that the nonlinear terms in I⁽⁰⁾ take into account the axial tension of the tube due to the transverse displacement of its cross sections. In this case, the change in the radius of the pipe in the deformed state should not exceed the thickness of its wall.

The system of (8) is supplemented by the boundary conditions of rigid fixing and by the homogeneous initial conditions,

(9)

Thus, the initial positions and the equations for the creation of the mathematical model are established. The dynamics of the pipeline must be determined after its filling with a fluid stream, which moves with a constant speed through the pipeline.

4 Formulation and simplification of the equations of the mathematical model

For creation of the mathematical model, the transition to the dimensionless variables is performed:ζ=s/ℓ,r=R/R₀, θ =θ, and τ=ωt.

The displacements of the median surface of a pipe wall in the dimensionless form are denoted by the symbols u'=u/R₀, v'=v/R₀, and w'=w/R₀.

The dimensionless components of the velocity of the fluid are v'_s =v_s0 /(ωℓ), v'_θ =v_θ0/(ωℓ), and v'_r =v_R0/(ωℓ). The pressure in the fluid is p′= p/p_a. The length of the axial line of the pipe in a dimensionless form is L'=L/ℓ.

The appropriate dimensionless unknown functions are presented as follows.

The velocity and pressure in the fluid are

(10)

The displacements of the median surface of a pipe wall are

(11)

The possibility of representation (10) and (11) was proved in Ref. [13]. This possibility is caused by two factors: smallness of parameter λ (7) and periodicity of unknown functions on the angular coordinate θ. By means of (10) and (11), from the equations, the independent variable θ is eliminated, and the task is thereby reduced to a one-dimensional problem.

Below is the rationale for this reduction and the assumptions.

The boundary conditions for the equations of fluid flow (4) are

(12)

By direct calculation, it is proven that the solutions (4) and (12) after substitution (10) in the zeroth-order approximation are

Formulae for the functions of the first approximation on the small parameter λ were found in Ref. [13]. (8) contains only a function of the fluid pressure for r=1,

This formula, in accordance with (10), has the accuracy O(λ²)^[13]. Here, is the curvature function.

The mathematical model (8) is transformed to a form including three differential equations with respect to the dimensionless functions u′, v′, w′, and all the necessary additional relations for asymptotic analysis are

(13)

(14)

Here, α=R₀ /ℓ is a small parameter, and E^*=E/ (1-ν²) is a compound coefficient. (13) and (14) are obtained for small strains and finite displacements. The movement of the centerline of the pipeline must be in the plane of the curve Γ₀. Formulae for x, y, and z are derived from the definition of the pipe centerline as a line connecting the geometric places of the centers of gravity of the cross sections.

Next, a solution to (13) and (14) is represented in the form (11). In the equations obtained, the terms with the same powers of λ in the left and right parts are equated with each other. Then, the factors are singled out for the same trigonometric functions.

The final formulation of the equations of the mathematical model^[17] is

(15)

(16)

(17)

(18)

(19)

The system of (15)-(19) is supplemented by the boundary and initial conditions which are received from (9) to solve the problem of dynamics of the pipeline,

(20)

Current coordinates of the centerline x, y, its curvature κ, and the velocity of the transverse pipe motion u₁^* are expressed through the unknown functions,

(21)

(15) and (16) are designated as the equations of the zeroth-order approximation, and (17)-(19) are designated as the equations of the first-order approximation. In the zeroth-order approximation, the pipeline behaves as a straight and cylindrical object, and the solutions to (15), (16), and (20) can be found precisely in Ref. [17]. Using direct substitution into (15) and (16), the solution at a large (compared with R₀) distance from the pipe ends can be verified to be approximated by the following functions:

The initial boundary value problem of the first-order approximation is solved by an explicit three-level difference scheme with the recalculation of values, and the method of its construction and theory were stated in Ref. [28]. In the offered algorithm, v₁ and w₁ are calculated by a strictly explicit scheme, whereas u₁ is calculated by the implicit scheme. If we are restricted by the explicit difference scheme, it is impossible to achieve stability for any arbitrarily small time step. This effect is known in the theory of difference schemes of gas dynamics^[28]. Elimination of this numerical instability is the main advantage of the chosen difference scheme.

The calculation formulae of the difference scheme have the forms of

(22)

After these calculations on the current time step, u_i^j+1 is calculated by the implicit scheme,

(23)

Here, h_τ is the time step, h_ζ is the step to the coordinate, , and . The boundary conditions are taken into account in the standard way^[28]. At each time step, the curvature is recalculated by (21) to take into account nonlinearity of the problem. The initial time step is chosen from the stability condition^[28]. Then, the step on time decreases before achievement of the given accuracy, namely, until when the solution does not change at the decrease of a step twice.

From (11), it follows that the physical sense of λu₁R₀ is a displacement of points from the cross-sectional plane perpendicular to the axial line of the pipe, i.e., warping^[1]. This fact is verified by direct calculation with u₁ =const. Sectional warping of the cross-section of a cylindrical pipe was observed in the experiments of Vlasov^[29].

5 Numerical experiment for the model problem

This section presents results of a numerical experiment on the problem (17)-(19). A comparison of the numerical solution found by the difference scheme (22) and (23) with the exact solution to the model problem is performed.

5.1 Statement of the model problem

The problem of finding the stationary position of the pipe, which is described by the system of (13) and (14), is considered. As a solution to this problem, the following components of the displacement vector w=(u', v', w') are chosen:

(24)

In this case, the boundary conditions for (13) and (14) take the forms of

(25)

The initial position of the center line of the tube y₀ (x) and its vertical displacement after deformation y-y₀ under the condition of selected solution (24) are described by functions^[30-31],

It follows from (24) that the solution to the corresponding one-dimensional problem (15)-(19) has the form of

(26)

It follows from (24) that the boundary conditions (20) are

(27)

Substitution of the solution (26) into (17)-(19) yields the right-hand sides,

To perform a numerical experiment, the following parameters are selected: b₁ =3.0 m, h=0.01 m, L=100 m, R₀ =0.15 m, E=2.07× 10¹¹ Pa, ν =0.24, ϑ_s0 =1 m/ s, ρ_f =998 kg/m³, ρ_t =7 200 kg/m³, ρ_e =1 000 kg/m³, and µ =1 000 Pa· s.

Iterations over time are performed in steps of h_t = 0.000 005 s. The step along the spatial variable is h_s = 0.05 m. It is established that the numerical solution to the nonstationary problem tends to the same solution of the stationary problem in the entire range of values of the parameters of the external medium 10³ Pa·s ≤ µ ≤ 10⁴ Pa·s and 10³ kg/m³ ≤ ρ_e ≤ 2 × 10³ kg/m³.

According to Ref. [30], the coefficient a₁ is calculated by

where J is the moment of inertia of the pipe section relative to Oz, and q_n is the transverse load created by the internal fluid flow. This load was established in Ref. [14]. As a result, we obtain

with the chosen numerical parameter a₁ =0.145 908 85 m.

In the numerical experiment, the error in approximating the exact solution (26) by the numerical solution of the boundary value problem (17)-(19) and (27), found by the difference scheme (22) and (23) under the boundary condition (27) is investigated.

5.2 Results of the numerical experiment

For the approximate solution (u_1h, v_1h, w_1h) found with the help of (22) and (23), the relative errors (ε_u, ε_v, ε_w) = (|u₁ − u_1h|/ max |u₁|, |v₁ − v_1h|/ max|v₁|, |w₁ − w_1h|/ max|w₁|) are calculated. Errors are also calculated for the dimensional quantities (λR₀u₁, λR₀v₁, λR₀w₁) which are numerically equal to the displacements of the middle surface of the tube u for sin θ =1, v for cos θ =1, w at sin θ =1, respectively. These errors are denoted by (e_u, e_v, e_w) = with a dimension of length (meters).

In Tables 1 and 2, for specified limit errors, the numbers of nodes in which the relative and absolute errors belong to the specified range and the percentage of this number to the total number of nodes are indicated.

The distribution of relative errors (ε_u, ε_v, ε_w) along the ζ-coordinate is shown in Fig. 2.

A numerical experiment has shown that the difference scheme (22) and (23) allows us to find a numerical solution to the problem (17)-(19) with high accuracy. In Fig. 2, we can see that the values of the relative error modulus for all unknowns do not exceed 5 × 10⁻⁴.

Obviously, the solution (24) to the model shell calculation problem (13)-(14) for fixed values of the angular coordinate differs in grid nodes from the numerical solution by not more than 7 × 10⁻⁵ m at h_s =0.05 m.

6 Numerical experiments for mechanical problems

The solution of modelling tasks establishes compliance of the constructed mathematical model to known results of mechanics. The existence of warping of the cross-section of a pipe is proven under conditions of its sufficient length, rigidity of seal, or existence of sites of non-uniform curvature.

6.1 Physical and geometrical parameters of the pipeline

We consider three various profiles of the pipeline.

Task 1   A cubic parabola y=10^-8 x(x-6 000)(x-12 000) when 0≤ x≤ 12 000. Values of the main geometrical parameters are L=12 500 m and min |ρ₀ |≈ 3 951.5 m. A uniform current velocity ϑ_s0 =1 m/ s is calculated over the interval of time T_end =864 000 s or 10 days.

Task 2  A fractional-rational function y=40(1-0.001x)/(1+10^-6 x²) when -6 000≤ x≤ 6 000. Values of the main geometrical parameters are L=12 000 m and min |ρ₀|≈ 1 194.5 m. A uniform current velocity ϑ_s0 =1 m/s is calculated over the interval of time T_end=691 200 s or 8 days.

Task 3  A catenary when 0≤ x≤ 3 000. Values of the main geometrical parameters are L=3 009 m and min |ρ₀|≈ 17 092.4 m. A uniform current velocity ϑ_s0 =10 m/ s is calculated over the interval of time T_end =86 400 s or 1 day. Additional parameters of this task are h = 0.023 m, R₀ = 0.23 m, µ_f = 0.1 N·s/m², and ρ_f = 998 kg/m³.

Other physical and geometrical parameters in all tasks coincide and are equal. ρ_e = 1 700 kg/m³, µ = 10 000 N · s/m², ρ_t = 7 200 kg/m³, h = 0.005 m, E = 2.07 × 10¹¹ N/m², ν = 0.24, R₀ = 0.3 m, µ_f = 0.667 N · s/m², and ρ_f = 850 kg/m³. These parameters approximately correspond to a stream of light oil in a steel pipe. The differences in Task 3 are caused by the specifics of an example and are explained below.

The mathematical model for all tasks is the system of (15)-(19) with initial and boundary conditions (20) and the additional relationship (21).

As a result of the numerical experiments, the following functions are found: the displacements of the axial line w_n(t, s), the longitudinal and angular strains of a wall ε₁ (s, θ) and ω (s, θ) at the final moment of time t=T_end, and the coordinates of the axial line x(t, s) and y(t, s). For Tasks 1 and 2, the functions u₁(t, s) are calculated, reflecting the magnitude of the cross sections of warping. In all calculations, the steady state is reached.

6.2 Strains and displacements of long pipelines

In Tasks 1 and 2, rather extended profiles are modelled. Figure 3 shows the displacements of the axial line, the coordinates of the pipeline profile, and its longitudinal strain and warping in Task 1.

Displacements of the axial line shown in Figs. 3(a) and 3(b) illustrate the coherence of the numerical calculations for the offered mathematical model with the fundamental laws of mechanics. The concluding displacement of a profile is directed towards the equally distributed loading from a fluid stream on a pipe wall. Figure 3(c) shows that the longitudinal strain of the wall varies greatly in the general vicinity of a point of fixing, which indicates a possible warping of the cross-sections of the pipe. This assumption is confirmed by the graph of the longitudinal wall displacements in Fig. 3(d), the geometric meaning of which is the warping of the cross-section. Thus, the numerical experiment indicates the existence of the cross-sectional warping of the long thin-walled pipeline of the order 0.02R₀.

In Fig. 4, the coordinates of a profile of the pipeline, its angular strain, and warping of cross-sections in Task 2 are represented. Similar to Task 1, the change in the coordinates of the profile at the start (dotted line) and at the end of the calculation in Fig. 4(b) indicates the coherence of the numerical experiment with the laws of mechanics. The angular strains shown in Fig. 4(c) illustrate the distortion of the cross-sections in the general vicinity and at the fixing points of the profile. Cross-sectional warping also occurs in this case, with a warping on the order of 0.003R₀.

6.3 Strain and displacement of the pipeline with a profile-like catenary

An inextensible thread takes the form of a catenary sagging freely under gravity. We can approximately assume that this is a form of the centre line of a marine riser^[4]. The research on pipelines with a profile in the form of a catenary is of great practical importance because it is connected with the rapid development of marine technology.

The aim of solving Task 3 is to analyze the interaction of the extended tube with a fast inner flow of fluid and to compare the solution with the results of Ref. [32], which found the conditions of the existence of the transitions of pipes/risers behaviour. The solution is meant to be a profile movement, in a direction opposite to the direction of its bending.

The results of the numerical solution of Task 3 are represented in Fig. 5. The centerline's coordinates at 10 min, 1 h, and 24 h are represented in Fig. 5(b). A graph of the normal displacement of the axial line is shown in Fig. 5(a).

In Ref. [32], the main characteristic of the tensile properties of a pipe was described by Irvin's parameter in cable mechanics,

where F is an external tensile force that is distributed along a pipe, and N_a is the internal force of a tension. In this task, the only tensile force is the force of friction of an internal fluid stream on a pipe's wall,

S_tb is the area of the internal surface of a pipe. Direct calculation gives E_irv≈ 37.6. This value is far enough from the value E_irv =286 from Ref. [32] at which the transitions of pipes/risers behaviour phenomenon is not valid.

In Fig. 5, the pipe is observed to be displaced in a direction opposite to an initial deflection, which illustrates the phenomenon of the transitions of pipes/risers behaviour. Thus, the results of the calculation for the studied model agree with the known results.

7 Conclusions

In this paper, a new mathematical model of dynamics of the curvilinear pipeline on the basis of the shell theory is proposed. The problem reduction algorithm to one-dimensional formulation is offered. Numerical experiments according to the difference scheme (22) and (23) for the proof of adequacy of mathematical model are performed.

The task about reaching the equilibrium position of a pipe for various types of the pipeline profile is solved numerically. The task about pipe dynamics at a variable internal pressure difference for some of these pipeline profiles is solved numerically. This mathematical model is established to be coordinated with the known results of mechanics of pipelines^{[2, 29, 32]}.

Novelty of the present paper is that for the bent pipeline, we propose a new mathematical model on the basis of the nonlinear shell theory. The form of the solution of the mathematical model at which the problem is reduced to one-dimensional formulation is proposed. The solution to the modelling tasks proves the existence of warping of the cross-section of a pipe under conditions (6) and (7), in analogy to the experimental data^[29]. Warping of the cross-sections of a pipe is calculated in model tasks. Angular and longitudinal strains of the pipeline's wall are calculated when the pipe reaches a state of equilibrium. It is found that the irregular strains occur in the vicinity of the pinning points or extreme camber.

The performed computational experiments demonstrate that the proposed mathematical model describes a wide class of problems of pipeline mechanics. Geometrical parameters of these problems can change in the wide range of values, and the mathematical model at this change keeps its applicability.