Mathematical model of the pipeline as an elastic shell was developed. Two limiting cases of pipeline geometry were considered. An analysis was given of the fundamental diferences between their stress-strain state. A numerical analysis of the deformations of a weakly bent pipeline and a pipeline with a singularity was performed. It was established that the approximate mathematical model of the weakly bent pipeline describes the original problem with high accuracy. The existence of a stress field singularity in the mathematical model of the pipeline with a kink was established.
The pipeline of length with a circular cross-section of radius 0 and a wall of thickness ℎ is considered (see Figure 1 in [4]). The centerline of the pipe is curved along flat curve Γ0 = { 0, 0 ∶ 0 = 0( ), 0 = 0( )}, where is an arc length (natural parameter). The pipe is filled with a steady lfuid flow, which is moving with velocity 0 under the influence of a constant pressure drop.
(
Suppose that the basic geometric relation of the theory of elastic cylindrical shells is satisfied: = 0 max | 0 ( )| ,
This paper assumes ℎ/ 0 ≤ 10; the admissibility of this is indicated in [6].
1. ≪ 1 — weakly bent pipe; 2. → ∞ — kinked pipe.
These two cases must be investigated from two diferent perspectives. Approach for Case 1: the pipe is investigated as a technical Vlasov shell (see [7]). Approach for Case 2: the pipe is investigated as a moment shell (see [3]). 2.1.1. Slightly bent pipe geometry
We introduce the following curvilinear orthogonal coordinates: is the distance along the pipe centerline and and are the angle and radius of polar coordinates in the cross section at point , see Figure 1 in [4]. The Cartesian coordinates of the pipe point are given by = ( ) + ( )
sin , ( ) = ( ) − sin ,
= cos .
Following [9], from these formulas we can determine the components of the metric tensor, the Christoffel symbols, and the Lamé coeficients for the orthogonal coordinate system constructed.
For the middle surface of the pipe’s wall, the following geometric relationships are executed: = 1 + 0 (, ) sin ,
= 0, 1 = (, ) sin /(1 + (, ) 0 sin ), 2 = 1/ 0; 1 = (1 + 1 ) , 2 = (1 + 2 ) , = − 0. where 1 and 2 are the main curvatures of a median surface, is the axis curvature. Lame coeficients are expressed by the formulas: 2.1.2. Kinked pipe geometry We will consider Case 2 by the example of two cylindrical pipes connected at right angles. Both pipes have the same radii and wall thickness. The geometry of this system was studied in [5] (see Figures 1, 2 in [5]).
ℎ — pipe wall thickness; — inner radius of the pipe; — outer radius of the pipe; 1, 2 — lengths of the first and second pipe sections along the centerline, respectively; 0 = 0.5( + ) — radius of the middle surface of the pipe; — line of intersection of the middle surfaces of two pipes.
Connected pipes are referred to below as segments ⃝ 1 and ⃝ 2 .
Let’s introduce the Cartesian coordinates ( ; , , ), the point coincides with the beginning of the first pipe. The ( ) axis coincides with the generatrix of the segment ⃝ 1 . Equation of the segment ⃝ 2 centerline obviously is = 1. Obviously, in Cartesian coordinates, the equation of the plane of intersection of segments is:
+ − 1 = 0.
From (4) and the formulas for the connection of cylindrical and Cartesian coordinates [10], we obtain the following geometric relations.
Segment ⃝ 1 : Segment ⃝ 2 : = 1, = − 1 sin 1, = 1 cos 1;
≤ 1 − , 1 ≤ 1 + 1 sin 1. = 1 + 2 sin 2, = 2, = 2 cos 2; ≥ 1 − , 2 ≥ − 2 sin 2.
Expressions for the radius vectors of the pipe points follow from formulas (5), (6). Let’s fix the numbering of curvilinear coordinates:
1 = , 2 = , 3 = .
Next, we find the unit vectors of the basis of the curvilinear coordinate system using the formulas from the book [9]. The components of the metric tensor and the Lamé coeficients follow from these formulas: 1(1) = 1, 2(2) = 2, 3(3) = 1; 11( ) = 1, 22( ) = 1 , 33( ) = 1;
2
1( ) = 1, 2( ) = , 3( ) = 1; = 1, 2.
(
Formulas for the coeficients of the first quadratic form of the median surfaces curvatures of the median surfaces 1( ), ( ) of cylindrical pipes follow from (8) for = 0: ( ), ( ) and the ( ) = 1, ( ) = 0; 1 ( ) = 0, 2 ( ) = 0; = 1, 2.
The dynamics of a pipeline is governed by the equations of an elastic body [9]: 1 (0) 1 (0) − + −( 1 + 2) ⋅ (0) + 2 1 2 + ( 2 ) + ( 1 ) −
∇2∇2 − 1 − 0 + (1 − ) ( 1 2 −
1 − 0 [ 2 ℎ 12 1 + (1 − ) ( 1 2 − ∇2 [( 1 + 22) ] = −
2
[ ( ⋅ ⋅ ) + tensor components, and ▽ is the covariant derivative. where is the density of the pipe material, are the acceleration components, are the stress
In the stationary case, provided there are no external distributed loads, the equilibrium equations are: 2.2.1. Mathematical model of the dynamics of a weakly bent pipeline We use equations (10) to describe the limiting Case 1, that is, the dynamics of the slow motion of a weakly bent pipeline. Equations (11) are used to describe a pipeline with a kink under the action of internal pressure, that is, the limiting Case 2.
For the study of the large-scale processes in Case 1, the movement of the inner flow is considered as quasi stationary. The Darcy’s law of a friction [11] was chosen as the law of hydraulic resistance.
( 0, ∇) 0 = −∇ − ( 0) , (∇, 0) = 0, = const.
The resisting force
( 0), afecting on a fluid stream was described in [11].
We denoted: - fluid density inside the pipe,
- fluid viscosity, - fluid pressure. The components of the fluid velocity vector 0 along the coordinate axes , , are denoted as , , , respectively.
= ▽ , ▽
= 0. )] . 2 ∗ cos ℎ 0 (0.5 − ln ||| 4 ∗
| 0||) 2
1 2 + ℎ ( − ).
, 2 2 [( ) +
( ) ] , external pressure. Functions ,
are defined by the formulas (
Here is denoted: , ,
– displacements of the median pipe surface along the coordinates , , ; (0), 0 – the first invariant of the strain tensor and the linear torsion of the pipe wall; , , components of the forces density acting on the shell along the coordinates , , , respectively; – – the
The system of equations (14) is supplemented by the boundary conditions of rigid fixing and by the homogeneous initial conditions: 2 2 + 1 − 2 2 + 2 02 1 + 2 2 0 + − 0 12ℎ2 0 3 3 + (1 − )ℎ2 3 24 03 1 + 2 2 0 + 1 2 02 2.2.2. Mathematical model of a pipeline with a singular profile
In this case, we restrict ourselves to studying the stationary equations (11). The action of the internal medium on the pipe is reduced to the inclusion of known pressure in the equilibrium equations of the shell. a solid body (11):
The statement of the problem within the framework of the moment shells theory for case (16) was studied in detail in [5]. Based on the introduced coordinate systems (5)–(7) and parameters (8), (9), two systems of equations of a two-dimensional mathematical model were derived from equations for → ∞.
(15) (16) (17) −
Here, , , are the components of the surface external force acting on the shell; , , are the components of the displacement vector of the middle surface of the -th segment.
The boundary conditions number at each end of the bent shell is four, as shown in [12]. This condition can be rigidly fixed as in [4]: 1 = 1 = 1 = 0; 1 = 0, for 1 = 0;
1 (18) 2 = 2 = 2 = 0; 2 = 0, for 2 = 2.
2
Next, we impose the conjugation conditions on the shell connection line to close the boundary value problem (17), (18). The formulas for the conjugation conditions were obtained in [5].
The geometric conditions at the contact are as follows: 2 − 2 cos = 1 + 1 cos , 2 + 2 cos + 2 sin = 1 − 1 cos − 1 sin , − 2 sin − 2 sin cos + 2 (1 + cos2 )= = 1 sin − 1 sin cos + 1 (1 + cos2 ), (19) 1 = 2 .
Equalities (19) contain a complete set of geometric conjugation conditions on the connection line of the median surfaces of cylindrical pipes.
(
(
Here denoted: ( ) are bending moments, ( ) are shear forces, ( ) are cutting eforts, ( ) are normal eforts, ( ) stands for segment number. Expressions of force factors (20) through the wall displacements is given in [5].
The mathematical model of connected pipes as the moment shell consists of the components: (20)
This section discusses a method for solving the problem of Case 1. In this case the condition (12) is satisfied. Let’s introduce dimensionless variables in equations (14): the pipeline. = / , = / 0, = , = . Here , are the characteristic length and frequency of processes in The displacements of the pipe median surface in the dimensionless form are denoted by symbols: ′ = / 0, ′ = / 0, ′ = / 0.
/
in the fluid:
′ =
. The length of the pipe centerline in a dimensionless form: ′ =
The dimensionless components of the fluid velocity: ′ = 0/
, ′ = 0/
,
′ = 0/ ; pressure
Let us bring equations (14) to dimensionless form. Then we represent their solutions in the form:
/
.
′( , , ) = 0( ) + 1( , ) sin + ( 2);
′( , , ) = 1( , ) cos + ( 2);
′( , , ) = 0( ) + 1( , ) sin + ( 2);
′ ( , , ) = (0) ( , ) + (
Within the framework of this work, the dynamics of the fluid is considered known; its study is available in [8, 4]. The zero-order solutions (21) are also assumed to be known.
The equations of the first approximation have the form (see [4]): + [2 0 + (1 − )
1 ⎢ = ∗ℎ∗ ⎢⎢ 0 −
2 ⎡ ⎣
0 1 2 0 ] − 2 2 1∗
0 (0.5 − ln | 1∗ 0|)⎥ 4
The main geometric characteristic of the solution is the function: 0 = − + + ⎤ ⎦ 1 2 02 2 2 1 1 +
2 ∗ − 1 +
0 ( ∗ 1
+ 2 ; ) 2
= ⎥⎥ − 02 2 2 1
2 . ∗ (21) (22) (23) (24) 2 2 1 2 −
1 + 0 1
+ ] − − 3 1 2 0 ( 2 + 0 2 1 2 ) 0 2 20 = 02 2 2 1
2 ; + 3 3 ⋅
2 1∗ 0 ) − 2
0 1 = The physical meaning of this function is the pipe centerline displacement in its plane. Knowing the displacements of the pipe walls, it is easy to calculate the longitudinal and angular strains of its wall.
In the kinked pipe problem, we have a connection line between pipes. Singularity points of the stress ifeld appear in shells with such lines, as noted in [3]. The problem of calculating such a stress field is close in a mathematical sense to the problem of calculating stresses in an L -shaped domain, see [13]. Therefore, to solve the problem under condition (16), it is necessary to develop a new computational algorithm. We plan to create this algorithm based on the approach developed in [13, 14, 15, 16, 17, 18, 19].
In this work, the aim was set to illustrate the existence of a stress field singularity and to estimate the limiting stress values. This objective can be achieved by using existing software applications. Therefore, numerical experiments on calculating the stress-strain state of a pipeline with a break in the profile were performed in the FreeCAD software package. An overview of the software package is given in [20].
We have created a pipeline modeling algorithm in the FreeCAD software package. It provides for the creation of a solid 3D model, building a mesh of the finite element method, entering data, setting up the solver, setting up computational experiments, and visualizing the calculation results. The algorithm uses the CalculiX finite element method package and the NetGen meshing package.
Several physical and geometrical parameters in all problems coincide and are equal: ℎ = 0.005 m, = 1700 kg/m3, = 10000 N ⋅ s/m2, = 7200 kg/m3, = 2.07 ⋅ 1011 N/m2, = 0.24, 0 = 0.3 m, = 0.667 N ⋅ s/m2, and = 850 kg/m3. These parameters approximately correspond to a light oil stream in a steel pipe.
= 40(1 − 0.001 ) / (1 + 10−6 2), − 6000 ≤ ≤ 6000.
The pipeline length: = 12000 m. Uniform current velocity 0 = 1 m/s; calculated interval of time = 691200 s.
= 10−8 ( − 6000) ( − 12000) , 0 ≤ ≤ 12000.
The pipeline length: = 12500 m. Uniform current velocity 0 = 1 m/s; calculated over the interval of time = 864000 s.
The following functions were found, as the numerical experiments results: displacements of the centerline (, ), angular wall strains (, ) at the time final moment = , and the coordinates of the centerline (, ), (, ).
From expressions (21) follows that the physical sense of 1 0 is a displacement of points from the cross-sectional plane perpendicular to the pipe centerline, i.e. warping, see [6]. This fact is verified by direct calculation with 1 = c . Sectional warping of the cross-section of a cylindrical pipe was observed in the experiments of V.S. Vlasov [7]. Functions 1(, ) were calculated, reflecting the magnitude of the cross sections warping.
Problem 1 solution.
For problem 1 coordinates and displacements of the pipe centerline shown in Figure 1 (a), (b) respectively. This illustrate the coherence of the numerical calculations for the ofered mathematical model with the mechanical fundamental laws: the displacement of a profile is directed towards the distributed loading from a fluid flow.
In Figure 1 (c), (d) angular strain and the cross-sections warping in problem 1 are represented. The angular strains shown in Figure 1 (c) illustrate the cross-sections distortion in the wide vicinity and at the fixing points of the profile. The cross-sectional warping occurred in this problem, with a warping on the order of 0.003 0, see Figure 1 (d).
Problem 2 solution.
The coordinates of a profile of the pipeline and its warping of cross-sections in problem 2 represented in Figure 2 (a), (b), respectively. Similar to problem 1, the change in the coordinates of the profile at the start (dotted line) and at the end of the calculation in Figure 2 (a) indicates the numerical experiment coherence with the mechanics laws. This numerical experiment indicated the existence of the cross-sectional warping of the long thin-walled pipeline of the order 0.02 0 as shown in Figure 2 (b).
Thus, in this section the problem about reaching of equilibrium position of a pipe for two types of the pipeline profile was solved numerically. The coherence of the numerical calculations for the ofered mathematical model with the mechanical fundamental laws were illustrated. The angular strains were calculated; this parameter illustrate the cross-sections distortion in the wide vicinity and at the fixing points of the profile. The existence of the pipe cross-sections warping was proved. Angular 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.
Here we investigate several problems under condition (16). Problems were solved for two pipes of the same material and with similar geometry, but these pipes were subjected to diferent loads.
To numerically illustrate the presence of a singularity in the solution, the free software package FreeCAD (see [20]) with the built-in program for calculating the finite element method CalculiX was used. These software packages allow us to see the growth of stresses in the vicinity of the joint line.
FreeCAD was first tested using the Lamé problem for a thick-walled pipe under internal pressure. The exact solution to this problem is given in [9]. The test calculation showed agreement between the approximate solution obtained by FreeCAD and the exact solution.
Problem 1. Geometric parameters: ℎ = 20 mm, 0 = 90 mm, 1 = 500 mm, 2 = 400 mm. Material: S335JO steel, material parameters: = 7800 kg/m3, = 210 GPa, = 0.3, = 343 MPa, [ ] = 490 MPa.
Rigid constraints were specified on the outer ends of the pipes; this corresponds to boundary conditions (18). A uniformly distributed pressure was imposed on the entire inner surface of the pipes with a value of = 100 MPa.
Problem 2. Geometric parameters: ℎ = 5 mm, 0 = 47.5 mm, 1 = 250 mm, 2 = 250 mm. Material:
Rigid constraints were specified on the outer ends of the pipes. A uniformly distributed pressure was imposed on the entire inner surface of the pipes with a value of = 10 MPa.
Problem 1 solution.
The mesh generated by the NetGen mesh generator built into FreeCAD and the von Mises stress limits are shown in Figure 3. In Figure 3 (a) you can see a slight thickening of the mesh in the vicinity of the pipe joint line. This is caused by setting the precision as average when generating the mesh.
In Figure 3 (b) you can see a significant variation in von Mises stresses between maximum and minimum stresses. This illustrates that stresses have domains of rapid growth, in other words, the stress field has singularities. The maximum stress reaches 2098 MPa, which significantly exceeds the ultimate strength of S335JO steel. This value is achieved on the inner side of the pipe wall in the reentrant corner of the domain at = 3 /2, at the joint line, only at a few nodes of the FEM grid. However, the stress drops to 25 MPa at the nodes of the FEM grid located far from the pipe connection line.
Problem 2 solution.
The mathematical model of the finite element method for Problem 2 was built in the FreeCAD software package. FEM mesh parameters: number of nodes equal 31514, number of element surfaces equal 10526, number of volume elements equal 15651. Mesh parameters have been set to high precision. As a result of calculations by the finite element method, the stress distribution was found, see Figure 4.
In Figure 4 (a) shows the stress gradient from maximum red to minimum green. In Figure 4 (b) shows a histogram of stress distribution by the number of grid nodes. Limiting stress values in Problem 2: the maximum stress is 395.5 MPa, the average stress is 89.5 MPa, the minimum stress is 2.9 MPa.
So, let’s summarize the results of the paragraph. An algorithm for mathematical modeling of the pipeline in the FreeCAD software package has been created. The stress fields were found by the ifnite element method, and the corresponding results were presented and visualized. Diferences in the solutions of problems 1, 2 are caused by the diference in the loading data and the diference in geometric parameters. But these diferences are not fundamental.
The main thing in the results obtained is the presence of a singularity in the stress field. When comparing the numerical solutions of problems 1, 2 with the numerical solution of a mathematically close problem of the elasticity theory with an reentrant angle, considered in [21], we can conclude that these solutions are close in a qualitative sense.
A numerical criterion is determined that allows one to diferentiate between two limiting states of the pipeline geometry. Two new mathematical models of the pipeline for each of these limits are presented: model (13), (14) for calculating the dynamics of a smoothly bent pipe; model (17)–(20) for calculating the stress-strain state of a pipeline with a singularity. Methods for the approximate solution of boundary value problems for the constructed mathematical models are proposed. For model (13), (14), a new algorithm for the reduction of a three-dimensional problem to its one-dimensional analog is proposed.
Numerical experiments have been performed. It was established that the proposed reduction algorithm for Case 1 yields results consistent with the data of other researchers. It was established that in Case 1, the cross section warping occur in the pipe.
In Case 2, the presence of a stress singularity on the pipe junction line was established. The calculation was performed by the finite element method in the FreeCAD design environment.
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