International Conference on Mechanical, Production and Automobile Engineering (ICMPAE'2011) Pattaya Dec. 2011

Stiffness of Pipe Elbow Thiagarajan.N, Dhananjay.R, and Shakti Prasad

The thermal expansion of pipeline is allowed to happen as if there is no constraint to expansion. Every element of the piping frame expands linearly along its axis and as a whole the entire frame work is scaled to the dimensions proportional to the temperature difference. Then the nodes in the pipeline where the supports are present are given a displacement such that they are held by the supports. Thus the thermal problem can be converted to an equivalent displacement problem. Known displacements are given to certain points (where they are supported) and the stress in the system, displaced shape and the support reactions are found out by finite element method. The stiffness of the piping system is contributed mainly by the pipe spool and the pipe bends. Valves can be considered as lumped masses for the analysis when the pipeline stretch is considerable compared to the valve length. While the stiffness matrix for a beam element is widely available in the text book, the stiffness matrix for pipe elbows is not. In this paper the stiffness matrix of the pipe elbow is obtained from the first principles using the castiglianos theorem. First section deals with the finite element formulation and the second section deals with the verification of the results using software.

Abstract—Pipelines expand and contract based on the operating temperature conditions. Unlike structures like railway lines pipeline cannot be cut and expansion joint be installed directly to take care of expansion or contraction. Pipelines carry liquids under pressure and the provision of loose joint to the system removes the ability of the pipeline to hold the liquid under pressure. It tends to open up near the expansion joint. To avoid the pipeline from opening, the pipe has to be held together using end thrust blocks. Thus introduction of expansion joint is associated with the requirement of thrust supports at the pipeline ends. The cost of the end thrust block is dependent on the pressure in the pipeline and also the diameter of pipe. It is always not necessary to use expansion joints to absorb the temperature based expansions. The pipeline itself has inherent flexibility to accommodate thermal expansions and the flexibility of the system is dependent on the piping layout. If the stress created due to the thermal deformation is within the allowable limits the piping system can operate without the use of expansion joint. The advantage of use of the inherent flexibility is that heavy thrust supports are avoided and the integrity of the structure is not lost as there is no loose joint or cut in the piping element. It is the pipe elbow which gives sufficient flexibility to the piping system to expand and contract based on the temperature conditions. Finite Element Method can be used to analyze the deformation, stress in pipe and the support reactions. Stiffness matrix for the entire system has to be computed to analyze the stresses and deformation. This paper primarily deals with the derivation of the stiffness matrix for a pipe elbow from first principles using castigliano’s energy theorems. As a part of verification finite element software is used to model and analyze a curved beam created by a finite number of straight beam elements. The solution of the curved beam element is found to be closely matching with the software result.

II. STRUCTURAL ANALYSIS OF PIPE ELBOW A pipe bend is considered as a beam and the flexural energy is computed in terms of the moment at any arbitrary section. Then applying Castigliano’s theorem the relationship between the applied forces and the nodal displacements are found out. Two cases are developed in this paper to obtain the stiffness matrix of the pipe elbow. In case-1 node 2 is fixed and node 1 is free. Forces are applied on node1 and the displacements of the free end are found as a function of the applied loads. Similar analysis is carried out by fixing node 1 and applying loads at node2. Principle of superposition is used to find the global stiffness matrix.

Keywords—Stress Analysis, FEM, Flexibility of Piping system, Energy Methods

I. INTRODUCTION

P

iping systems transporting fluids experience various temperatures during operation depending on the fluid temperature. The temperature of fluids creates expansion or contraction in the piping frame work and the supports which act as restraints to pipeline movement create proportional stress in the pipeline. The stress in the pipe becomes zero when the restraints are removed. Finite element method can be used for analyzing the stress, deformation and loads on the piping system.

A. Case-1: Node 2 Fixed and Node 1 Free

Thiagarajan.N is with the Engineering Design and Research Center, Larsen and Toubro ECC Division, Chennai-89. (Phone: 7598480775; e-mail: [email protected]). Dhananjay.R is with the Department of Mechanical Engineering, Amrita University, Kollam, India. (Phone: 08129708827 e-mail: dhananjayr@am,amrita.edu). ShaktiPrasadis a PG student with Amrita University, (08907284043 email: [email protected]).

Fig. 1 Node 2 Fixed and Node 1 Free

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International Conference on Mechanical, Production and Automobile Engineering (ICMPAE'2011) Pattaya Dec. 2011

M z (θ ) = F y1 R(1 − cos(90 − θ )) − Fx1 R sin(90 − θ ) − M z1 π /2

∫

U=

0

∂U = ∂Fx1

π /2

∂U = ∂F y1

π /2

u1 = v1 =

M 2 (θ ) Rdθ 2 EI

θ1 =

∫ 0

∂U = ∂M z1

∫ 0

0

B. Case-2: Node 1 Fixed and Node 2 Free

(1 -5)

M (θ )  ∂M   Rdθ  EI  ∂Fx1 

In this analysis the node 1 is fixed and the node 2 is subjected to the forces and the moments as shown.

M (θ )  ∂M  Rdθ EI  ∂F y1 

π /2

∫

Another case will be developed where the node 1 is fixed and forces are applied on node2.

M (θ )  ∂M    Rdθ EI  ∂M z1 

If the relation between the vectors is written in the matrix form, we have

  0 . 7854 0 . 5 1 − F   u1     x 1   R3    0.3562 − 0.5708 Fy1   v1  =  − 0.5   EI θ R   1 − 0.5708 1.5708   M z1   1   R 

Fig. 2 Node 1 Fixed and Node 2 Free

M z (θ ) = Fx 2 ( R − R cos θ ) − Fy 2 R sin θ − M z 2 π /2

The inverse of the coefficient matrix gives the partial stiffness matrix relating the forces and the displacements at node-1.

U=

0

   F   42.8750 39.3711 − 12.9884  u1  x1  EI      Fy1  = 3  39.3711 42.8750 − 9.4845   v1   M z1  R − 12.9884 − 9.4845 5.4588  θ1 R      R 

u2 = v2 =

θ2 =

From the matrix we are able to relate the nodal displacements of node 1 with the forces and moments at that node. The support reactions can be related to the applied forces as Fx2 = - Fx1 Fy2 = -Fy1 Mz2 = - Fx1R + Fy1R - Mz1

∂U = ∂Fx 2

π /2

∂U = ∂Fy 2

π /2

∫ 0

∂U = ∂M z 2

∫ 0

M (θ )  ∂M   Rdθ  EI  ∂Fx 2  M (θ )  ∂M  Rdθ EI  ∂Fy 2 

π /2

∫ 0

M (θ )  ∂M    Rdθ EI  ∂M z 2 

Applying the limits and converting to matrix form,

   u2   0.35619 − 0.5 − 0.5708  Fx 2  3   R   F  0.7854 1 − 0.5  v2  =  y2   θ R  EI − 0.5708 1 1.5708   M z 2   2    R 

EI {−42.8750u1 − 39.3711v1 + 12.9884θ1 R} R3 EI Fy 2 = 3 {−39.3711u1 − 42.8750v1 + 9.4845θ1 R} R M z2 M = − Fx1 + Fy1 − z1 R R M z 2 EI = 3 {9.4845u1 + 12.9884v1 − 1.9549θ1 R} R R Fx 2 =

  F  x2  Fy 2  M z2   R

∫

M 2 (θ ) Rdθ 2 EI

The inverse of the coefficient matrix gives the partial stiffness matrix relating the forces and the displacements at node-1.

  F  39.3711 − 9.4845   u 2   42.8750 x2  EI   42.8750 − 12.98839  v 2   Fy 2  = 3  39.3711 R  M z2  − 9.4845 − 12.98839 5.4588  θ 2 R     R 

  − 42.8750 − 39.3711 12.9884   u1   EI     = 3  − 39.3711 − 42.8750 9.4845   v1   R  9.4845 12.9884 − 1.9549 θ1 R    

The support reactions can be related to the applied forces as Fx1 = - Fx2 Fy1 = -Fy2 Mz1 = Fx2R - Fy2R - Mz2

Thus the forces in node1 and node 2 are functions of the nodal displacements u1, v1 and Θ1. As node2 is fixed all the forces are dependent on the displacements of node-1.

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International Conference on Mechanical, Production and Automobile Engineering (ICMPAE'2011) Pattaya Dec. 2011

Fx1 =

EI {−42.8771u2 − 39.3718v2 + 9.4840θ 2 R} R3 EI Fy1 = 3 {−39.3718u2 − 42.8742v2 + 12.9875θ 2 R} R M z1 = Fx 2 R − Fy 2 R − M z 2

and analyzed with the following boundary conditions.Support Boundary Condition: Bottom fixed Apex Node: a) Rotations are arrested b) Horizontal Force = Fy = 10000 N c) Vertical Force = Fz = -10000 N

M z1 M = Fx 2 − Fy 2 − z 2 R R M z1 EI = 3 {12.9893u2 + 9.485v2 − 1.9549θ 2 R} R R

Number of node points = 135 Number of elements = 136 Radius of the Bend = 6.0 m Coordinate System: y – z Coordinate axes used. (The equation is developed for ‘x-y’ system of axis.)

In matrix form,

   F  − 42.8750 − 39.3711 9.4845   u 2  x1  EI      Fy1  = 3  − 39.3711 − 42.8750 12.9884   v 2   M z1  R  12.9884 9.4845 − 1.9549 θ 2 R      R  III. PRINCIPLE OF SUPERPOSITION: By the principle of superposition, the forces applied at node 1 and node 2 in both the cases are superimposed to find a general relationship between the forces and the nodal displacements. Full Matrix is given at the end of the paper. Thus from the energy principles force – displacement relationship is obtained for the pipe elbow. With this stiffness matrix the global stiffness matrix of the entire piping system can be obtained to find out the stress, displacements and the support reactions of the piping system subjected to thermal load. The force displacement model obtained above is verified by solving a finite element model in algor software using finite number of straight beam elements.

Using Castigliano’s energy method stiffness matrix for a 90° pipe elbow is obtained. This matrix is essentially required for analysis of piping systems where pipe elbow forms a major role in providing flexibility to piping system.The stiffness matrix is verified using FEA software where 136 beam elements are used to model a pipe elbow. The deflection at the point of application of load was found to be 0.176761 and 0.176761 meters along y and z direction using software for the applied loads. The loads corresponding to these deflections were found manually to be 10020 N and -10020 N using the force displacement relationship matrix derived in the paper. The values are close to the loads applied in the FEM model. The stiffness matrix of the elbow can be used along with the other normal piping elements to determine the behavior of the overall piping system under static loads.

A finite element model of a curved beam with radius 6.0 m is created using finite number of straight beam elements. Linear isotropic static analysis module of ALGOR student version software is used. TABLE I MATERIAL: ISOTROPIC MATERIAL WITH YOUNG’S MODULUS 200GPA, 0.2191

m

Thickness of Pipe

0.0045

m

Internal Diameter

0.2101

m

Metal Area

0.003033836

Sq.m

Moment of Inertia

1.74724E-05

m^4

Polar M.I (J)

3.49448E-05

m^4

B. Manual Verification: The bottom node is fixed and all the displacements associated with the node are zero. Hence u1, v1, θ1 are zero. The apex node which forms node 2 is restrained to rotate and hence θ2 is zero. The horizontal displacement and the vertical results from the software are found to be 0.176761 and 0.176761 meters. By using the force displacement relation matrix the applied loads are found to be manually 10020 N and -10020 N along horizontal and vertical direction which closely agrees with the applied load in ALGOR software. The small percentage of variation is due to the truncation of values in the stiffness matrix. V. CONCLUSION

IV. FINITE ELEMENT MODEL OF CURVED BEAM:

Outside Diameter

A. Analysis Results: The deformed shape of the curved beam is shown in Appendix-‘B’ and Appendix-‘C’. The apex node is found to deflect 0.176761m downward along negative z direction and 0.176761m along positive x direction for the applied loads and boundary conditions.

REFERENCES [1] [2] [3]

A curved beam made of finite number of straight beams is modeled in Algor Finite Element software (student version)

232

Richard G.Budynas,“Advanced Strength and Applied Stress Analysis”, McGraw Hill International Edition, second edition Daniel L.Schoder, “Structures”, Pearson Education J.N.Reddy, “ An Introduction to the FINITE ELEMENT METHOD”, Tata McGraw Hill, second edition.

International Conference on Mechanical, Production and Automobile Engineering (ICMPAE'2011) Pattaya Dec. 2011

APPENDIX – ‘A’ Force Displacement Relationship for a pipe elbow

 FX 1  39.3711 − 12.9884 − 42.8750 − 39.3711 9.4845   u1   42.8750  F   39.3711  Y1  42.8750 − 9.4845 − 39.3711 − 42.8750 12.9884   v1    M Z1  5.4588 12.9884 9.4845 − 1.9549  θ1 R   R  EI  − 12.9884 − 9.4845 =     FX 2  R 3 − 42.8750 − 39.3711 12.9884 42.8750 39.3711 − 9.4845   u 2      − 39.3711 − 42.8750 9.4845 39.3711 42.8750 − 12.9884  v 2   FY 2     M Z2  9.4845 12.9884 1.9549 − 9.4845 − 12.9884 5.4588  θ 2 R  −    R 

APPENDIX – ‘B’ The following figure shows displacement of the apex due to the application of the nodal forces. The color code shows the ‘z’ or the downward displacement of the beam due to the applied load. Support conditions for the model is discussed in section IV. Z-DISPLACEMENT

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International Conference on Mechanical, Production and Automobile Engineering (ICMPAE'2011) Pattaya Dec. 2011

APPENDIX – ‘C’ The following figure shows displacement of the pipe elbow modeled as a curved beam due to the application of the nodal forces. The color code shows the ‘y’ or the horizontal displacement of the beam due to the applied load. Support conditions for the model are discussed in section IV.

Y-DISPLACEMENT:

234