Mathematical Models of Pipelines Alternative Stress States

Abstract

A pipeline mathematical model as moment shell with singularities caused by the domain geometry is built. The parameters that differentiate the various stress states of the pipeline are determined. Two cases of pipeline geometry are considered, which differ in the numerical parameter we have selected. The analysis of the limiting states of the pipeline is performed. The existence of singularities in solutions is established. A numerical analysis of the stress state is carried out for a weakly bent pipeline and a pipeline with a singularity.

Keywords

• 1 Pipeline
• stress-strain state
• singularities
• computing experiment

1. Introduction

2. Problem formulation

fluid flow with a velocity  s 0 .

We define the curvature parameter  = R0 max  0 ( s ) , ( 1 ) where  0 (s) is the initial curvature of the line  . Let the basic geometric relation of the theory of ( 2 )

 →  – pipe with singularity. ( 3 )

In the case of ( 2 ), the pipe is considered as a technical Vlasov shell (see [4]). In the case of ( 3 ), the pipe is considered as a moment shell (see [2]).

The pipe geometry at implementation ( 2 ) was examined in [3, 5]. Let's introduce curvilinear coordinates: s – defined above;  and R are the angle and radius of polar coordinates in the section s . We find for the middle surface of the pipe wall following [6]:

 (s, t ) sin k1 = (1 +  (s, t ) R0 sin ), k2 = 1 R0 ; where k1 and k2 are the main curvatures of a median surface,  is the axis curvature.

The geometry of a pipeline with a kink in the profile was studied in [7, p. 7553].

Let's consider case ( 3 ) on the example of two cylindrical pipes connected at right angles. We denote: Ri – inner radius of the pipe; Re – outer radius of the pipe; L1 , L2 – lengths of the first and second pipe sections along the centerline, respectively; R0 = 0.5( Re + Ri ) . The connected pipes are hereinafter referred to as sections ( 1 ) and ( 2 ). We construct cylindrical coordinate systems on sections ( 1 ), ( 2 ), and denote these coordinates by si (axial),  i (angular) and  i (radial), where i is the section number. We fix the numbering of curvilinear coordinates: x1 = s,

x2 =  , x3 =  . 2.1.

Equations of the pipeline statics and dynamics

The dynamics of a pipeline is governed by the equations of an elastic body [6]: tensor components, and  i is the covariant derivative.

In the stationary case, in the absence of external distributed loads, the equilibrium equations are as follows:

i ki =0. ( 7 )

We use the equations ( 6 ) to describe the case ( 2 ), that is, the dynamics of the slow motion of a curved pipeline. We use the equations ( 7 ) to describe a pipeline with singular profile, that is, for the case ( 3 ).

2.1.1. Dynamics of a bent pipeline

In the case ( 2 ), the movement of the inner flow is considered as quasi stationary. The Darcy's law of a friction [8] was chosen as the law of hydraulic resistance. Ibid also given the equations of stationary motion of an incompressible fluid taking into account the Darcy friction force Φ ( s 0 ) . We denoted:  f – fluid density,  f – fluid viscosity, p – fluid pressure.

For a bent pipe, the equations system is obtained [3]: 1 I (0) A s 1 I (0) B  − 1 −  0 + (1 − )  k1k2u − k2 w  = −

B   A s  + 1 −  0 + (1 − )  k1k2v − k1 w  = −

A s  B   1    B       A     .

AB  s  A s  +   B   +

A v R0 

 + v cos  +  1  A

 A  R0

 +  sin  w −  1  w 2  v 2 

 +  s   , 2 A2  s  1 h

 2v Y = − t t 2 −

,

2.1.2. Static of a pipeline with a singular profile

In the case of ( 3 ), we use the equations ( 7 ). The problem statement within the framework of the moment shells theory is investigated in [7]. The mathematical model equations are derived in the coordinates ( 5 ): 2sui2i + 12−R02 2ui2i + 1 +  2vi +  wi − 12hR20 3swi3i + (1 − )h2 si3wii2 + 1 −Eh 2 X i = 0, 2R0 si i R0 si 1 +  2ui + 1  2vi + 1 − 2svi2i + 1 wi − 3 − 2R0 si i R02  i2 2 R02  i 24R03 h2

3wi + 2 12R02  isi2 − 12−R02 si3ui i2  − 1h22 R0  4swi4 i + R202 si24wi i2 + 1  4 w  R04  i4i  + R0 −u2 sin − v2 sin cos + w2 (1 + cos2  ) = = u1 sin − v1 sin cos + w1 (1 + cos 2  ) , (10) Conjugation conditions for force factors: w1 = w2 .  

We denoted: M (i) – bending moments, S (i) – shear forces, Q (i) – cutting efforts, N (i) – normal efforts, (i) stands for section number.

The mathematical model consists of: ( 1 ) system of equations ( 9 ) with fixed constrains as boundary conditions; ( 2 ) geometric conditions on the conjugation line (10); ( 3 ) interface conditions for force factors ( 11 ).

3. Methods of numerical experiments

3.1. Pipe bending problem

Let's introduce dimensionless variables into the equations ( 8 ):  = s , r = R R0 ,  =  ,  =  t . Here ,  are the characteristic length and frequency of processes in the pipeline. Displacements of the pipe middle surface: u = u R0 , v = v R0 , w = w R0 ; fluid pressure: p = p pa .

We obtain the dimensionless form of the equation ( 8 ). Then we represent their solutions in the form: u ( , , ) = u0 ( ) +  u1 ( , ) sin + O ( 2 ); v( , , ) =  v1 ( , ) cos + O ( 2 ); w( , , ) = w0 ( ) +  w1 ( , ) sin + O ( 2 ); p ( , , r ) = p (0) ( , r ) +  p ( 1 ) ( , r ) sin +  p ( 2 ) ( , r ) cos + O ( 2 ).

Within the scope of this paper, the dynamics of the fluid is considered known and its study is available in [5, 3]. The zero-order solutions ( 12 ) are also supposed to be known.

The equations of the first approximation (see [3]):  2 2u21 − 1 − u1 − 1 +  v1 + w1 +

2 2   1 − + f   2

2 u0 − 2 2 u20 +  (1 − ) w0  − − 3  w1 2w20 + w0 2w21  + 3 3 f  w0 2w20 =  t RE02* 2 2u21 ; 1 −  2  2v1 − v1 − 2  2

1 E *h*

2u1* R0  0.5 − ln |  e u1* R0 | 

  4 + 1 + 2  u1 +

  + w1 + f  w0 −  3 − 2

 u0  −  2 w0 w1 =  t RE02* 2 2v21 ; w1 + h1*22  4 4w41 −  2 2w21  + u1 − v1 + + f 2 w0 + (1 − ) u0  −  2 w0 w1 +  2 f  w0 2    2

   =  1  = E *h*  f s20 f −  

R0  0.5 − ln |  e u1* R0 |    4 2u1*    R 2 2  2 w  − t 0

E *  2 1 .

( 11 ) ( 12 ) ( 13 ) (14) (15) Thus, the three-dimensional problem ( 8 ) is reduced to a one-dimensional formulation. A difference scheme for numerical solution of equations ( 13 )–(15) was constructed in [3]. 3.2.

Numerical method for a pipeline with a singular profile

In the problem with a kinked pipe, there are points of singularity of the stress field, as indicated in [2]. The problem of calculating such a stress field in a mathematical sense is close to the problem of calculating stresses in the L-shaped domain, see [9]. Therefore, to solve the problem under the condition ( 3 ), it is necessary to develop a new computational algorithm. We plan to create this algorithm based on the approach developed in [9, 10, 11, 12].

Numerical experiments on calculating the stress-strain state of a pipeline with a break in the profile were performed in the FreeCAD software package (see [13]) to illustrate the existence of a singularity and estimate the limiting stress values.

We created a pipeline modeling algorithm in the FreeCAD software package. It provides for the creation of a solid 3D model, mesh generation by the finite element method, data entry, solver setup, and visualization of calculation results. The CalculiX finite element method package and the NetGen meshing package are used.

4. Numerical results 4.1.

The pipeline bending problem

 = 10000 N  s m 2 ,

Physical and geometric parameters of the test problem: h = 0.005 m ,  e = 1700 kg m 3 ,  t = 7200 kg m3 , E = 2.07 1011 N m 2 ,  = 0.24 , R0 = 0.3 m ,  f = 0.667 N  s m 2 ,  f = 850 kg m3 , L = 12000 m ,  s0 = 1 m s . Calculation time Tend = 691200 s . The centerline of the pipeline is described by a fractional rational function:

y = 40 (1 − 0.001x ) (1 + 10−6 x 2 ) , − 6000  x  6000.

In numerical experiments, the following are found: displacements of the centerline, angular deformations of the walls, coordinates of the centerline x(t, s) , y (t , s) , longitudinal displacements of the first approximation u1 .

From ( 12 ) it follows that the physical meaning of  u1R0 is warping of pipe cross sections. Warping of the cross-sections of a cylindrical tube were observed in the experiments of V.S. Vlasov [4].

The coordinates of the centerline at the beginning and end of the calculation are shown in Figure 1 (a). This illustrates the consistency of the numerical solution with the mechanics laws: the profile displacement is directed towards distributed load from the fluid flow.

Figure 1 (b) shows the graph of  u1 , repeating the cross-sections warping. Warping has a maximum of about 0.003R0 . In another numerical experiment for a cubic parabola profile, the warping reached 0.02 R0 . 4.2.

The pipeline with singularity

Physical and geometric parameters of the test problem: h = 5 mm, R0 = 47.5 mm, L1 = 250 mm, L2 = 250 mm. Material: S335JO steel,  t = 7800 kg m3 , E = 210 GPa,  = 0.3 ,  t = 343 MPa. Rigid fixing conditions are imposed on the outer ends. Pressure p = 10 MPa is applied to the inner pipes surface.

Mesh parameters are set to high precision. As a result, the stress distribution was found, see Figure 2. Figure 2 (a) shows the von Mises stresses. Figure 2 (b) shows a histogram of stress distribution by the number of mesh nodes. Limiting stress values: maximum stress 395.5 MPa (reached on the inner side in the reentrant corner of the domain), average stress 89.5 MPa, minimum stress 2.9 MPa.

5. Conclusion

A mathematical model of the pipeline as a moment shell of irregular geometry, with singularities caused by the domain geometry, has been built. The parameters that differentiate the various stress states of the pipeline are determined.

A constructive algorithm for the asymptotic solution of the original three-dimensional boundary value problem for a pipeline is created, and on the basis of the obtained asymptotics, reduced mathematical models are constructed. The parameters of the problem are found for which various models are applicable. A numerical criterion has been determined that makes it possible to distinguish between two limiting states of the pipeline geometry.

Numerical experiments were performed. In the case ( 2 ), the results consistency of numerical experiments on the proposed mathematical model with the mechanics laws is shown. The existence of the pipe cross-sections warping is proved.

In the case ( 3 ), the pipeline modeling algorithm in the FreeCAD package is created. The stress fields are found by the finite element method. The presence of a singularity in the stress field is proved.

6. Acknowledgements

The work has been supported by the Russian Science Foundation grant № 21-11-00039, https://rscf.ru/en/project/21-11-00039/. Computational resources were provided by the Shared Services Center "Data Center of FEB RAS".

7. References

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