of intersecting cylindrical shells in

Abstract

Equations of elastic intersecting cylindrical shells in displacements are derived for the T-joint of pipes. Three-dimensional mathematical model is constructed within the framework of the membrane theory of shells, and the limitations of this approximation are found. Geometric and force conjugation conditions are set on the pipe intersection line, and boundary conditions are imposed on the end of the pipes. Complete three-dimensional mathematical model is presented in the Cartesian coordinate system, to achieve a unified approach to solving the problem without splitting into subdomains. Reduced statement of the boundary value problem with respect to only two independent variables is found. This result is obtained from the symmetry condition of the mechanical system with respect to the plane of the Tjoint. The conjugation conditions are eliminated from the final formulation of the boundary value problem. Existence of singularity stress field in the vicinity of the junction and permissibility of using bushing connections in the formulation of the problem are illustrated by numerical example.

Keywords

• 1 boundary value problem
• T-shaped shell joint
• numerical methods
• singularity

1. Introduction

2. Problem statement for a membrane cylindrical shell of T-shaped joint in displacements

We consider two pipes with T-shaped joint. The radius of the branch is small compared to the radius of the main pipe, i.e. r R  1 5 . The ratio between the thickness of the pipe and the radius for both pipes does not exceed 1 20 . In Figure 1, we denote: L , l – length of the large (1) and small (2) pipes, respectively, (O; x, y, z ) – Cartesian coordinate system, axis Ox coincides with the axis of large cylinder and the axis Oz coincides with axis of small cylinder. We introduce cylindrical coordinates ( x, ,1 ) and ( z, ,2 ) for both pipes.  r z + r12 v(2) + wr(22) = 1 −Eh 2 p, where  – Poisson ratio, E – modulus of elasticity, H , h – thickness of large and small cylinders, u(1) , v(1) , w(1) and u(2) , v(2) , w(2) – components of the displacement vector, where the index (1) denotes belonging to a large cylinder, and index (2) denotes belonging to a small cylinder, p – uniform internal pressure.

Set the boundary conditions at the cylinders ends. We restrict movements of large cylinder along coordinate x , and we equate shear force to zero: x = − L : u(1) = 0,

2 x =

L : u(1) = 0, 1 u(1) R  1 u(1) + + v(1) x v(1) ( 6 )

Proposition 1. On the intersection line of the two middle surfaces of cylindrical shells, described by equation ( 4 ), the following relations are satisfied: −u(1) sin − v(1) sin cos + w(1) r cos cos = v(2) ,

R u(1) cos − v(1) sin sin + w(1) cos sin = w(2) , v(1) cos + w(1) sin = u(2) , 3. Boundary value problem for two rectangles in Cartesian coordinates

3.1. Converting equations in two-dimensional form

x  EH Then the boundary value problem will take the following form:

To reduce a number of required functions in equations (1), we express

u(1) v(1) 1− 2 u(2) v(2) w(1) = −R − + pR2 , w(2) = −r

− z  + 1− 2

Eh pr2.

(2) ( 3 ) ( 7 ) ( 8 ) r  2 + z The boundary conditions for the large cylinder are the same, while for the small cylinder they take the form: The conjugation conditions are: −u(1) sin − v(1) sin cos − r cos cos

cos cos u(1) x

r −

R v(1) 

+ + r cos cos

pR = v(2) ,

+ 4(r 2 − x2 ) 2xw2z + 2r 2 (1 + ) 2zw2z − 2x wxz = 0,  2 r 2 −r x2 uxx − 2x r 2r− x2 vxy + r wzz = 1 −Eh 2 pr 2.

Proof. To prove this proposition, we use equation of the corresponding cylinder. We consider only the positive part of the cylinder and replace variables using formulas: , , 2 =  y 2 yz , = − 2 = − = r2 − x2  x

, r2 − x2 2 = −  y 2 xy , 2 = − = z 2 z

, R2 − z2  z

, R2 − z2 2 xz

,  x  z 2 = =  x  z , , 2 = x2 x2 xz xz x2 x2 xz xz y22 = R2z−2 z2 z22 − Rz32 z , z22 = z22 . y22 = r 2 x−2 x2 x22 − rx23 x , z22 = z22 . After addition of similar terms, we get equations ( 12 ), as required.

The resulting equations depend on two independent variables. In this case, equations (12.1) are applicable in the domain, which is projection of a large cylinder on the symmetry plane, and equations (12.2) are applicable in the domain, which is the projection of a small cylinder. In this regard, the system is divided into two subsystems, which can be considered independently.

(12.2) , ( 13 ) 3.2.

Boundary conditions for equations in Cartesian coordinates

We perform a similar replacement of variables under conditions (2), ( 10 ). We obtain the boundary conditions in sections 1, 4, 7, see Figure 2:

We take into account the load application method relative to the plane xOz and symmetric nature of the displacement distribution vy . Then, at the intersection points of the shells with the plane xOz , we can take vy = 0 . From this condition and (16) we obtain ( 15 ), as required.

Since the system ( 12 ) consists of two independent subsystems, the intersection line AB should be considered as a boundary, and therefore boundary conditions should be set on it. To do this, we transform the conjugation conditions ( 11 ) by expressing the cylindrical coordinates in terms of Cartesian coordinates and replacing the variables. Then, in the second condition, the terms from the 3rd to the 6th coincide with the left part of the first condition, therefore, they can be excluded from the condition. The conjugation conditions are set on the line AB, where z = R . In view of this, the first condition is satisfied identically. The last conjugation condition is the equality of the shear forces S(1) = −S(2) on the intersection line. Two more boundary conditions can be obtained from it using Vekua bushing coupling [14, 15], according to which there are no tangential stresses at the boundary. Then we obtain the boundary conditions on the line AB for the large and small cylinders, respectively: vy = 0, − R ux + x x

To evaluate the possibility of using a bushing coupling at the junction of two pipes, we present results of numerical analysis in the FreeCAD application package. Two models were analyzed: 1) one-piece model of a T-joint of pipes considered as a three-dimensional body; 2) model of a large pipe under internal pressure with a small pipe inserted as a bushing. The following parameters are set:

Figure 3 shows the stress distribution over the grid nodes for a one-piece model. It can be seen that the stress field changes rapidly in the vicinity of the connection line. The maximum value of von Mises’s stresses is 256.29 MPa for the first model. When calculating the second model, the maximum value of von Mises’s stresses is 283.13 MPa. The relative stress error is 9%, which is acceptable. When the thickness of the pipe wall reduces, the stress increases proportionally.

4. Conclusion

The equations of the cylindrical membrane shell theory in displacements are derived for the Tshaped domain. Boundary conditions and conjugation conditions are imposed on the intersection line. At that it is assumed that the radius of the branch pipe is small compared to the radius of the main pipe.

The original problem is represented as a three-dimensional boundary value problem in Cartesian coordinates. The reduced form of the equations is obtained due to the T-shaped region symmetry. The required number of boundary conditions is imposed. Thus, the resulting reduced problem regarding to the displacements ux , vy , wz consists of equations (12.1), (12.2) and boundary conditions ( 14 ), ( 15 ), (17) in rectangular domains.

A numerical analysis is carried out, which showed that the bushing coupling can be used to transform the coupling conditions into boundary conditions.

5. Acknowledgements

The work has been supported by the Russian Science Foundation (grant 21-11-00039). Computational resources were provided by the Shared Services Center "Data Center of FEB RAS".

6. References

[1] W. Pietraszkiewicz, V. Konopińska, Junctions in shell structures: A review, Thin-Walled

Structures 95 (2015) 310–334. doi:10.1016/j.tws.2015.07.010. [2] V. A. Rukavishnikov, S. G. Nikolaev, On the R -generalized solution of the Lamé system with problem with a singularity belongs to the space W2k,++2 /2+k+1(, ) , Differential Equations 45

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