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 THE EFFECT OF GEOMETRY ON FATIGUE LIFE FOR BELLOWS JINBONG KIM Dept. of Aeronautical & Mech. Eng, Hanseo University, Seosan, Chungnam 356-706, KOREA jbkim@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 is employed 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 takes place 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 of revolution 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. 3 343
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. A good grasp of bellows research can also be gained from the conference proceedings of the 1989 ASME Pressure Vessels and Piping Conference. 5 Andersson 6 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 Association 6 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 and analysis 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 of Bellows Yield Stress (MPa) Tangential Modulus (MPa) Young's Modulus (GPa) Type of Element 0.25 9 19 224(500 ) 2,000 188 8-node Structural Shell
The Effect of Geometry on Fatigue Life for Bellows 345 Fig. 1. Simulation Model 3. Results and Discussions Table 2 Analysis Parameters Fig. 2. Stress distribution with deformed shape Radius of Convolution (mm) Quantities of Pitch Inner Diameter (mm) 1.4 2 9 17 40 65 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 10 5 cycles. Experimental result shows that the specimen which is same size used in Figure 3 failed at 4.3 x 10 5 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.
346 J.-B. Kim Calculated S/N Curve at present Node Specimen S/N curve 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 10 5 cycles, Applied stress : 320MPa 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 bending deflection. As bending moment increases, the principal stress increases as shown in Figure 5 and the number of cycles to failure decreases. X 10 5 Fig. 4. The number of cycles to failure versus Inner Diameter of Bellows (Numbers of 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)
The Effect of Geometry on Fatigue Life for Bellows 347 X 10 5 Fig. 6. The number of cycles to failure versus numbers of convolution(radius of Tube : 20mm, Meridional radius of the convolution crown : 1.7mm, pitch : 5.88mm Fig. 7. The Principal stress versus numbers of convolution(radius of Tube : 20mm, Meridional radius of the convolution crown : 1.7mm, pitch : 5.88mm 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 4.5x10 5 cycles to 13x10 5 cycles 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
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 radius of 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 1. CHIEN Wei-Zang, WU Ming-de, Applied Mathematics and Mechanics, 4(5), 649-655, (1983). 2. HUANG Qian, Applied Mathematics and Mechanics 7(6), 573-585, (1986). 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. 79 85. 8. Ting-Xin L, Bing-Liang G, Tian-Xiang L, Qing-Chen W. The 1989 ASME Pressure Vessels and Piping Conference,(1989), p. 13 9.