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Advanced Materials Development and Performance (AMDP2011)

International Journal of Modern Physics: Conference Series

Vol. 6 (2012) 343-348

World Scientific Publishing Company

DOI: 10.1142/S2010194512003418

343

THE EFFECT OF GEOMETRY ON FATIGUE LIFE FOR BELLOWS

JINBONG KIM

Dept. of Aeronautical & Mech. Eng, Hanseo University,

Seosan, Chungnam 356-706, KOREAjbkim@hanseo.ac.kr

A bellows is a component installed in the automobile exhaust system to reduce or prevent the

impact from engine. Generally, the specifications on the bellows are determined in the system

design process of exhaust system and the component design is carried out to meet the

specifications such as stiffness. Consideration of fatigue is generally an important aspect of design

on metallic bellows expansion joints. These components are subject to displacement loading

which frequently results in cyclic strains. This study has been investigated to analyze the effect of

geometry on fatigue life for automotive bellows. 8 node shell element and non-linear method isemployed for the analysis. The optimized shapes of the bellows are expected to give good

guidelines to the practical designs.

Keywords: Stress Analysis; fatigue analysis; bellows.

1. Introduction

Bellows is adopted as important element to absorb expansion and contraction in order

to reduce stress in automobile exhaust system. Flexible connection between the exhaust

system and the manifold is necessary because of the rolling of engine. Some torsion takesplace because of the curved path of the exhaust system and considerable axial and

bending deflections must be allowed for. Using a rigid joint would give severe vibration

of the exhaust system, with noise and quick failure due to exceeded material strength as

consequences. 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 bellows were

first introduced into exhaust systems.

Unlike most used piping components, the bellows consists of a thin walled shell ofrevolution with a corrugated meridian, in order to provide the flexibility needed to absorb

mechanical movements. Because of geometric complex, it is difficult to analyze the

behavior of bellows. The axi-symmetrical deformation problems of the bellows have

been discussed.1,2 These problems were investigated by the finite difference method.3

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344 J.-B. Kim

Flexible metal bellows have been used for considerable time in other applications.

Numerous papers deal with various aspects of bellows, such as stresses due to internal

pressure and axial deflection, fatigue life estimations,4

column instability and scrim. Agood grasp of bellows research can also be gained from the conference proceedings of the

1989 ASME Pressure Vessels and Piping Conference.5 Andersson6 derived correction

factors relating the behavior of the bellows convolution to that of a simple strip beam.

This approach has subsequently been the basis of standards and other publications

presenting formulae for hand-calculation for bellows design.

Some formulae have been included in national pressure vessel codes, among which

the ASME code is the most well known. The most comprehensive and widely accepted

text on bellows design is however the Standards of the Expansion Joint Manufacturers

Association6 A comparison of the ASME code and the EJMA standards is given by

Hanna,7 concluding that the two conform quite well in most aspects. In addition, the

EJMA standards were compared with finite element and experimental analyses in some

papers.8

Even though, EJMA is benefit for the design of bellows, it is difficult to analyze the

behavior of bellows because of its complex geometry. The aim of this work is to

represent the effect of the geometric parameters on the mechanical behavior of U-shaped

bellows. The loading condition is under deflection at the end of bellows. The results

present optimal dimensions for the model used in the study.

2. Simulation Model

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

was meshed with 8 node shell elements and elastic - plastic non linear analysis was

performed. Figure 1 displays the geometry profile for the analysis model. The mesh

consists of 100,000 nodes and lateral displacement with 6mm was applied at the end for

boundary condition. Material properties used in analysis are described in Table 1 andanalysis parameters are described in Table 2. ANSYS was used as FE-solver for stress

analysis and the stresses and other results were imported from ANSYS into FEMFAT for

fatigue analysis.

Table 1 Material property and Finite Element Model

Thickness(mm)

Quantities ofBellows

Yield Stress(MPa)

Tangential

Modulus

(MPa)

Young's

Modulus

(GPa)

Type of Element

0.25 919 224(500) 2,000 188 8-node Structural Shell

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The Effect of Geometry on Fatigue Life for Bellows 345

Table 2 Analysis Parameters

3. Results and Discussions

Figure 2 represents stress distribution for the analysis result. The maximum stress occurs

at the secondary convolution root from straight tube as shown in Figure 2 and the

number of cycles to failure shown in Figure 3 is decided at this position. The S/N curve

result is as shown in Figure 3. On the basis of stress data from un-notched specimen,

local S/N curves are calculated at FEM nodes, which are influenced by local component

properties and loads. Figure 3 represents the S/N curve result for the model of 9

convolutions with inner radius of 20mm and radius of convolution of 1.7mm. Lower line

of left represents S/N curve for the base material from dumbbell type specimen test and

upper line of left side represents S/N obtained by FEMFAT for the actual bellows model

in the study. S/N curve for base material is obtained using dumbbell-type specimen and

S/N curve modified by FEMFAT using S/N curve for base material is obtained for

bellows modeled in the study. Obtained principal stress from FEM is 320MPa and the

number of cycles to failure for the model in Figure 3 is calculated as 4.89 x 105cycles.Experimental result shows that the specimen which is same size used in Figure 3 failed at

4.3 x 105 cycles. The other experimental results also show 10% to 15% difference in the

number of cycles to failure. These differences are resulted from experimental conditions

and materials in fatigue tests. Even though there are some differences between

experimental results and analysis results, the trend is similar in a number of results.

Radius of Convolution (mm) Quantities of Pitch Inner Diameter (mm)

1.42 9 17 4065

Fig. 1. Simulation Model Fig. 2. Stress distribution with deformed shape

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346 J.-B. Kim

Calculated S/N Curve at present Node

Specimen S/N curve

The number cycles to failure decreases linearly from 945,00cycles to 857,000cycles

according to increase of the radius of tube from 20mm to 32.5mm as shown in Figure 4.

It is caused by the increase of bending moment due to the increase of the radius of

bellows tube and the section modulus in the boundary condition of the constant bendingdeflection. As bending moment increases, the principal stress increases as shown in

Figure 5 and the number of cycles to failure decreases.

Fig. 3. S/N for bellows with 9 convolutions( inner radius of tube:20mm, radius of Convolution :1.7mm)

-The Number of failure cycles : 4.89 x 105cycles, Applied stress : 320MPa

Fig. 4. The number of cycles to failure versus

Inner Diameter of Bellows (Numbers o

Convolution: 19ea, Meridional radius of the

convolution crown : 2mm)

Fig. 5. The principal stress versus Inner Diameter

of Bellows (Numbers of Convolution: 19ea,

Meridional radius of the convolution crown :

2mm)

X 105

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The Effect of Geometry on Fatigue Life for Bellows 347

Fig. 8. The number of cycles to failure versus radius of convolution (Radius of Tube:20mm, Numbers of

convolution: 19ea)

The number of cycles to failure increases from 4.5x105cycles to 13x105cycles with the

variation of convolution from 9 ea to 19 ea as shown in Figure 6. As numbers of

convolution increase, the principal stress decreases due to decrease of the bending

moment at the bending condition of same deflection as shown in Figure 7. As the

principal stress decreases, the number cycles to failure decreases. The number of cycles

to failure increases until 1,320,000cycles at 1.7mm of the meridional radius of the

Fig. 6. The number of cycles tofailure versus numbersof convolution(Radius of Tube : 20mm, Meridional

radius of the convolution crown : 1.7mm, pitch :

5.88mm

Fig. 7. The Principal stress versus numbers o

convolution(Radius of Tube : 20mm, Meridional

radius of the convolution crown : 1.7mm, pitch :

5.88mm

X 105

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348 J.-B. Kim

convolution crown and decreases at the radius as shown in Figure 8. As the meridional

radius of the convolution crown increases, the stress concentration effect is decreased and

the number of cycles to failure increases. After the meridional radius of the convolution

crown becomes 1.7mm, the radius of bellows increases and bending moment increases.

As the bending stress increase with increment of bending moment, the number of cycles

to failure decreases.

4. Conclusions

The results on the effect of geometry on fatigue life for automotive bellows can be

summarized as follows;

(1) The number of cycles to failure is the maximum at 1.7mm of the meridional radiusof the convolution crown for the model in the study.

(2) The number of cycles to failure decreases linearly according to the increase of the

bellows radius.

Acknowledgment

The author would like to thank Hanseo University for substantial support (Project code:

111Gong Hang 13).

References

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3. Hamada M, Nakagawa K, Miyata K, et al.,Bulletin of JSME, 14(71), 401-409, (1971).

4. C. Becht IV,International J. of Pressure Vessels and Piping, 77. 843-850, (2000)

5. Becht IV C, Imazu A, Jetter R, Reimus WS, editors. ASME Pressure Vessels and Piping

Conference,(1989).

6.

Anderson WF. Part II mathematical, Atomic International, NAA-SR-4527, United States

Atomic Energy Commission, (1965).

7. Hanna JW. , The 1989 ASME Pressure Vessels and Piping Conference, (1989),p. 7985.

8. Ting-Xin L, Bing-Liang G, Tian-Xiang L, Qing-Chen W. The 1989 ASME Pressure Vessels

and Piping Conference,(1989), p. 139.Int.J.Mod.Phys.

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