The Effect of Geometry on the Stress Behavior of Flexible Tube with the Lateral Deflection

1Jinbong Kim

1Hanseo University, jbkim@hanseo.ac.kr

Abstract As analysis research, the effect of boundary condition combination of the stress of the automobile

flexible tube has been proposed in this paper. FEM solution for a Ω-shaped flexible tube under the action of lateral deflection and angle of rotation are obtained. The research shows the fact that the maximum von-Mises stress decreases to minimum value and the stress exhibit an upward trend after the minimum stress with the increase of the lateral deflection or the angle of rotation. Furthermore, an important suggestion should be made that the maximum von-Mises stress becomes minimum on the condition of lateral deflection (mm) /angle of rotation (degree) =10.

Keywords: Flexible tube, von-Mises stress, Lateral deflection

1. Introduction

Flexible tube is often adopted as an expansion joint in pipeline engineering, pressure measuring

element in precision instrument and automobile exhaust system. Flexible tubes as an element for deformation compensating are subjected to axial displacement, lateral deflection and angle of rotation. It must withstand high temperatures, and should combine high flexibility with high strength and durability. A steel flexible tube type joint is commonly used. It generally consists of a multi-ply bellows to reduce the temperature of the bellows and improve flow conditions.

Experience shows that this joint sometimes causes the complex dynamic behavior of the exhaust system and this has caused car and component manufactures severe problems. A few reports concerning dynamic characteristics have been found in literature [1] [2].

With a sufficient number of bellows, a flexible tube allows practically unlimited elastic displacement and rotations and finally failed as shown in Figure 1. Clark et al.[3] Investigated the linear deformation of flexible tubes. In recent, a few researchers studied on the problem of nonlinear deformation of flexible tube [4-7].

Proper dimensioning requires deep understanding of the characteristics of the bellows and their interaction with the rest of the exhaust system. Off the shelf products seldom fit a specific application, which was experienced when the bellows were introduced into exhaust systems. Failure take place after rather short operation times and substitution of stronger and much more expensive materials do not solve the problem [8].

The most comprehensive and widely accepted text on bellows design is however the Standards of the Expansion Joint Manufacturers Association [9]. A comparison of the ASME code and the EJMA standards is given by Hanna concluding that the two conform quite well in most aspects. Even though, EJMA is beneficial for the design of the bellows, it is difficult to analyze the behavior of bellows in detail because of its complex geometry.

As a deep understanding of the characteristics of the flexible tube with complex geometry and various conditions is required to proper dimensioning, the paper has analyzed the effect of geometry on the stress of flexible tube subjected to lateral deflection.

2. Analysis model

To obtain the flexible tube profile, it was modeled with the finite element code. The flexible tube

was meshed with 8 node shell elements and elastic - plastic non linear analysis was performed. Figure 2 displays the geometry profile for the analysis model. The mesh consists of 112,800 elements. Lateral displacement with 0mm to 21mm angle of rotation with 0 to 0.21 degrees was applied at the end of boundary condition as shown in Figure 2. Material properties and parameters used in analysis are

The Effect of Geometry on the Stress Behavior of Flexible Tube with the Lateral Deflection Jinbong Kim

International Journal of Digital Content Technology and its Applications(JDCTA) Volume 7, Number 11, July 2013 doi : 10.4156/jdcta.vol7.issue11.27

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described in Table 1. ANSYS was used as FE-solver for stress analysis. Analysis model was classified as shown in Table 2. The height of 1st convolution is 8mm and the others are 10mm in model A. In addition, the height of 1st convolution is 6mm, the 2nd convolution is 8mm and the 3rd convolution is 10mm in model B. The pitch of convolution is 6.8mm.

Boundary conditions applied to the analysis model is as shown in Figure 2. Left side is fixed and deflection and rotation is applied to right.

Figure 1. Photograph showing the failed Flexible Tube

Table 1. Material property

Tangent Modulus

(GPa)

Young's Modulus

(GPa)

Inner Diameter of Tube(mm)

Thickness (mm)

Quantities of Flexible tube

Type of Element

1.880 188 64.32 0.315 23 8-node Shell

Table 2. Classification of Analysis Model

Type Height of 1st

Convolution (mm) Height of 2nd

Convolution (mm) Height of 3rd

Convolution (mm) Pitch (mm)

A 8 10 - 6.8

B 6 8 10 6.8

Figure 2. Analysis Model and Boundary Condition

3. Results and discussions

Figure 3(a), (b) represents the von-Mises stress distribution for the case of 1mm and 13mm

deflection without angle of rotation at the end. If the flexible tube is only deflected vertically and there is no angle of rotation, von-Mises stress is distributed symmetrically at four corners. Even though the deflection increases from 1mm to 13mm, this distribution pattern remains continuous and only the

Fixed

↓ Deflection

↓

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stress increases. Figure 4(a), (b) represents the von-Mises stress distribution for the case of 0mm deflection and 0.03º and 0.21º rotation at the end. If the flexible tube is only rotated at the end and there is no boundary condition of lateral deflection, von-Mises stress is distributed asymmetrically as shown in Figure 4(a), (b). Even though the rotation angle increases from 0.03º to 0.21º, this distribution pattern remains continuous and only the stress increases.

(a) 1 (mm) deflection (b) 13 (mm) deflection

Figure 3. von-Mises stress without angle of rotation (Model B)

(a) 0.03 (degree) (b) 0.21 (degree) Figure 4. von-Mises stress without lateral deflection (Model B)

Figure 5 represents the von-Mises stress according to the angle of rotation with 1mm deflection. When the angle of rotation is small, the stress is mainly affected by the deflection and the stress is concentrated at left and right corner as shown in Figure 5 (a). As the angle of rotation increases, the stress developed by rotation and deflection condition is superposed and the stress at the restricted upper and lower area is gradually reduced. The von-Mises stress of all over the length of the upper and lower area is almost uniform at 0.01° rotation as shown in Figure 5 (b). The maximum von-Mises stress gradually reduced from 58Mpa to 10MPa according to the increase of the angle of rotation as shown in Figure 5 (a), (b). In addition, the concentrated stress at local area is reduced and the stress is uniformly distributed at the certain height from the neutral axis. And if the angle of rotation exceeds 0.01°, the von-Mises stress of right corner where the effect of angular rotation on the stress is dominant gradually increases as shown in Figure 5(c), (d).

If the end of the flexible tube doesn’t rotate (+ in Figure 6.), the maximum von-Mises stress increases steadily as the deflection increases. However, the maximum von-Mises stress decreases to the minimum value and exhibits an upward trend after the stress becomes minimum if the angle of rotation is added to the end of a flexible tube as shown in Figure 6.

The maximum von-Mises stress of model B is higher than that of model A at the angle of rotation of less than 0.03 degrees until the maximum von-Mises stress reaches the minimum value. In addition, the maximum von-Mises stress of model A is higher than that of model B at the angle of rotation of more than 0.03 degrees as shown in Figure 6 until the maximum von-Mises stress reaches the minimum

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value. After the minimum value, the maximum von-Mises stress of model B is higher than that of model A at the angle of rotation

(a) 0º rotation (b) 0.01º rotation

(c) 0.1º rotation (d) 0.21º rotation

Figure 5. von-Mises stress at 1mm lateral deflection (Model B)

Figure 6. Deflection versus max. von-Mises stress Figure 7. Deflection versus the minimum value of the maximum von-Mises stress

0 5 10 15 20 25Deflection(mm)

0

100

200

300

Max

. von

-Mis

es S

tres

s(M

Pa)

Model A

Model B

0 4 8 12Deflection(mm)

0

100

200

300

400

Max

. von

-Mis

es S

tres

s(M

Pa)

Angle of Rotation(Degree)

0 (A model)

0.01

0.05

0.07

0.1

0 (B Model)

0.01

0.05

0.07

0.1

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The minimum value of maximum von-Mises stress linearly increases according to the deflection and the angle of rotation. In this case, the stress of model A is higher than the stress of model B as shown in Figure 7. 4. Conclusions

The results for the effects of geometry on the stress behavior of flexible tube with varying

deformation can be summarized as follows (1) The maximum von-Mises decreases to minimum value and the stress exhibit an upward trend after the minimum stress with the increase of the lateral deflection or the angle of rotation. (2) The maximum von-Mises stress becomes minimum on the condition of lateral deflection (mm)/angle of rotation (degree)=10.

Acknowledgments

The author would like to thank for the substantial support from the Hanseo University (Project code: 121GongHang08)

5. References

[1] Verboven Peter, Valgaeren Rudi, Guillaume Patrick, Van Overmeire Marc, “Some Comments on

Model Analysis Applied to an Automotive Exhaust System”, Proceedings of the International Modal Analysis Conference-IMAC, pp. 987-993, 1998.

[2] Belingiardi, G. & Leonti, S., “Model Analysis in the Design of an Automotive Exhaust Pipe”, International Journal of Vehicle Design, vol. 8, pp. 475-484, 1987.

[3] Clark, R. A., “An expansion bellows problem”, Trans ASME, Ser. E, J. Appl. Mech., vol. 37, pp. 61-69, 1970.

[4] Chien Wei-Zang, Wu Ming-De, “The nonlinear characteristics of U-shaped bellows-calculations by the method of perturbation”, Appl. Math. Mech., vol.4, pp.649-665, 1983.

[5] Liu Ren-Huai, Wang Zhi-Wei,”Nonlinear deformation analysis of U-shaped bellows with varying thickness”, archive of applied mechanics,(Springer-Verlag), vol.70, pp.366-376, 2000.

[6] Jaehan Yoo, Joongyoup Lee, Soo Yong Lee, “Structural Evaluations of Bellows for a Gasgenerator Lox shut-off Valve”, Proceedings of 2011 KSPE Spring Conference, pp.279-282, 2011.

[7] Chul Soo Kim, Seoung Ho Ahn, Yong Hwan Kim, Kwang Woo Chung, “Structural Analysis of Gangway Bellows for the High-speed Railway Vehicle”, Proceedings of 2013 Spring Conference of the Korea Society for Railway, pp.1004-1006, 2013.

[8] Johan Wall, “Dynamics Study of an Automobile Exhaust System”, Blekinge Institute of Technology Licentiate Series, pp.9-11, 2003.

[9] EJMA, “Standards of the Expansion Joint Manufacturers Association 9th ed.”, EJMA, New York, USA, 2011.

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