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Fatigue life prediction based on the rainflow cycle counting method for the end beam of a freight ca

International Journal of Automotive Technology, Vol. 9, No. 1, pp. 95 101 (2008) DOI 10.1007/s12239 008 0012 y

Copyright ? 2008 KSAE 1229 9138/2008/038 12

S. H. BAEK1), S. S. CHO2) and W. S. JOO3)*
2) 1) School of Mechanical Engineering, Dong-A University, Busan 604-714, Korea Department of Vehicle Engineering, Kangwon National University, Gangwon 245-711, Korea 3) Department of Mechanical Engineering, Dong-A University, Busan 604-714, Korea

(Received 17 May 2007; Revised 15 December 2007)
ABSTRACT This paper presents a system for treating of the actual measured data for load histories. The approach consists of two steps: stress analysis and fatigue damage prediction. Finite element analysis is conducted for the component in question to obtain detailed stress-strain responses. A significant number of failures occurred in a brake end beam which led to economic losses and disruption of service. The cracks appeared to be fatigue cracks caused by the dynamic load produced in the loaded bogie frame. Strain gauge data were analyzed, and fatigue cycles were calculated from this data. Rainflow cycle counting was used to estimate cumulative damage of the end beam under in-service loading conditions. The fatigue life calculated with the rainflow cycle counting method, the P-S-N curve, and the modified Miner’s rule agreed well with actual fatigue life within an error range of 2.7%~31%. KEY WORDS : Fatigue life prediction, Rainflow cycle counting, Cumulative damage, Miner’s rule, P-S-N curve, Censored strain data

In the beginning of 2001, cracks were found in the brake end beam of the bogie frame of freight cars in a particular running section of the South Korean railway. The end beam of a freight car is a structural element that supports the bogie frame and braking system. The location and connection method of the end beam should be considered in view of structural design, because the end beam is built into the lower part of the side frame of the bogie. Usually, important parts such as the bogie frame and car body etc. are designed to last more than 25 years (Goo and Seo, 2003; Baek et al., 2005). In the case of a fractured bogie frame, service life can be affected because the maximum stress is lower than the fatigue limit. However, two-thirds of the total number of end beams failed in service via fatigue cracking in this particular running section. The cracked end beams had either two years (240,000 km) or three years (360,000 km) of service. The cracks appear to be fatigue cracks caused by the dynamic load produced in the loaded bogie frame. In the time-domain analysis of structures subjected to random loading, an appropriate cycle counting technique (Matsuishi and Endo, 1968; Downing and Socie, 1982; Nagpal and Kuo, 1996; Wang et al., 2006; Haq et al., 2007) and a fatigue cumulative damage rule (Fatemi and Yang, *Corresponding author. e-mail:

1998; Barboza et al., 2005; Kang et al., 2007) are used to estimate the fatigue service life. The bogie frame of freight cars has been evaluated by endurance test standards. However, because the South Korean railway has many more curved tracks than railways abroad, there is a high braking load during operation. A design specification that reflects the domestic track in the existing endurance test standard must be developed. In the present paper, the load history was obtained from strain measurements on a bogie frame. A three-dimensional finite element model of a simplified bogie frame was developed for static stress analysis. Miner’s rule was combined with a probabilistic S-N curve (Murty et al., 1995; Zheng and Wei, 2005) and stress results to develop a stressbased fatigue life prediction for the brake end beam of the bogie frame.

A general method for fatigue life estimation of railway vehicles is required, as evidenced by cracking that occurred in the end beams of freight cars. As illustrated in Figure 1, by collecting different load amplitudes using the rainflow cycle counting method, the fatigue damage is linearly accumulated, as is proposed by Miner’s rule.


S. H. BAEK, S. S. CHO and W. S. JOO

ing environment is needed for improving the safety of railway vehicles. The extracted cycle produces stress amplitude and mean stress. Cumulative damage D and number of fractures to cycle N are determined using a histogram of cycle ranges and Miner’s rule. For infinite life design for very high mean stresses, the Buch mean stress correction is selected. Miner’s rule is expressed as follows. Failure is expected to occur if: n 1- n 2- n 3D= ----- + ----- + ----- + N f1 N f2 N f3 =

ni ----N fi



where ni is the number of applied cycles and Nfi is the number of cycles to failure at a specified stress amplitude i, respectively. In this study, the critical cumulative damage value of D is chosen to be 1 in Eq. (1). The fatigue life in the repeated signal is expressed as follows: 1 Life= -------------------n i /N fi


Figure 1. Flow chart for fatigue life prediction. 2.1. Rainflow Cycle Counting Method The end beam of a freight car bogie is subjected to variable amplitude service loading. To predict the fatigue life of the end beam in a freight car bogie, service stress (or strain) history is measured by a uniaxial strain gauge. Signal processing uses a cycle counting algorithm to extract stress-strain hysteresis loops quickly and accurately. In this study, rainflow cycle counting was used as a signal processing method for fatigue analysis. Figure 2 shows the procedure for the cycle counting method as demonstrated by Downing and Socie (1982). (i) Consider the following sequence of peaks/valleys. The notation uses point A as the most recent data point, point B as the previous point, and so on. Range A to B > Range B to C (ii) Because the range from A to B is greater than the range from B to C, a cycle is closed, and is represented by the range from B to C. (iii) Figure 2 (b) shows a new cycle. As before, the range from A to B is greater than that from B to C, so B to C is one cycle. This procedure is repeated until no more cycles are closed by this point. The fatigue cumulative damage rule for the actual runn-

Since, in many cases, the Palmgren-Miner theory (Singh, 2002) leads to non-conservative life predictions, the linear damage rule associated with a critical damage sum D, different from one, has been proposed in many design codes for fatigue damage assessment of structures subjected to variable amplitude loading. 2.2. P-S-N Curve Because of the scatter in fatigue life data at any given stress level, it must be recognized that there is not only one S-N

Figure 3. Photograph of a fractured end beam.

Figure 2. Rainflow cycle counting procedure.

Figure 4. Bogie frame model with coupled effect, load, and boundary conditions.



Figure 6. Distribution of stresses of the end beam with braking load. Figure 5. Distribution of stresses in the bogie frame. curve for a given material, but instead, a family of S-N curves with probability of failure as the variable parameter. These curves are called the P-S-N curves (Zheng and Wei, 2005). A P-S-N curve can be obtained from JSME S002. Fatigue data displayed on a log-log plot of stress versus life for finite life can be expressed as follows for an end beam: logN= ? + ? log S±1.64 ? logN ? logN = 1 -6
8 1/2

(3) Figure 7. Strain gauge layout for the end beam. (4) pivot and end beam, respectively. Figure 5 shows that the von-Mises stress for the bogie frame (243.5 MPa) is located on the center pivot. Figure 6 shows that the maximum von-Mises stress for the end beam (75.4 MPa) is located on the corner of the welded gusset plate. These results are particularly interesting from the viewpoint of the fatigue strength, because tensile stresses alone contribute the most to the fatigue crack initiation and propagation. The location of the stress peak in Figure 6 overlaps the fracture region presented in Figure 3. However, the location of the stress peak in Figure 6 does not overlap with the fracture region. The high level of stress in the end beam area was the main cause of crack initiation. The fatigue load (a combination of the self weight and braking load) caused the successive propagation of the crack to critical size and then resulted in rupture at the welded gusset plate. A considerably lower stress value in the region of the side frame can be observed in Figure 5. The center frame, except for the center pivot, isn’t as highly loaded as the end beam. The maximum von-Mises stress in the region of the center frame is only 243.5 MPa, whereas in the side frame the stresses are around 112.2 MPa (Figure 5). 3.3. Estimation of Load History To determine whether the fatigue life is accurately predicted by the measured stress, it is necessary to compare the fatigue life as calculated by the rainflow cycle counting method with that observed in experimental fatigue data attained under in-service loading. Figure 7 shows attachment locations of six strain gauges

logN i

? + ? logS

where ? logN is the standard deviation of the number of cycle to fracture obtained by the staircase test. S-N curves with failure probability 5% or 95% are determined by translating the S-N curve with failure probability of 50% to the coordinate axis (±1.64 ).

3.1. Visual Examination of the End Beam The fractured end beam was first subjected to visual examination. The failure location of the end beam is presented in Figure 3. As seen from this figure, the end beam first fractured in the welding zone between the C shape beam and the gusset plate. 3.2. Finite Element Analysis A geometric model of the freight car bogie was developed using CATIA and ANSYS. The finite element model of the bogie frame presented in Figure 4 consists of a 10-node tetrahedral element and a 2-node beam element. The coupling element was selected to model the load applied to the bracket hinge of the end beam. Since geometrical shape, load, and boundary conditions are symmetrical, we use the half-model as the effective model. The load condition was determined through JIS E4207 (1984). The end beam and side frame are manufactured out of SS400 and SM490A, respectively. In this analysis, a vertical load of 17,000 kg and a braking load of 2,875 kg were applied to the center


S. H. BAEK, S. S. CHO and W. S. JOO

Figure 8. Blocks of strain-time history.

Figure 10. Apparatus for the Scenk type fatigue test. Figure 9. Comparison of measured stress and FEA result. with direction perpendicular to the fatigue crack. In this study, we assumed that the direction perpendicular to crack propagation is the principal stress direction. A one axis strain gauge (KFG-5-128) was installed on the end beam before loading, and the test track was the Donghae-Jecheon section. The load history in the test track was measured through 60 km/hr over 25 min, from starting to braking. Figure 8 shows the results of the test series with the load history based on six strain gauges. The stresses resulting from the strain measurement are 48.3 MPa and 72 MPa, respectively. The stress ratios at the region are 0.75. Figure 9 shows maximum principal stresses plotted as experimental data and finite element analysis (FEA) results. Compared to the FEA results, the experimental stresses at locations of G2 and G5 are measured within an error range of 12% as compared with analytical stresses. However, experimental stress at location G4 is lower than the analytical stress. The measured stress at this location is very high due to track vibration and braking load. 3.4. Fatigue Life Prediction Knowledge of the material properties at the most critical point of the end beam is needed for correct evaluation of the integrity of the bogie frame. For this reason, 10 mm thick flat specimens were taken from a broken end beam of SS400 steel and investigated under alternating bending stresses (R = 1). Test results for fatigue life given in Baek et al. (2005) were obtained by flat specimens on a Scenk

Figure 11. P-S-N curve for SS400 steel. type twisting and bending fatigue testing machine (Figure 10), and then plotted on the S-N curves with 5%, 50%, and 95% failure probabilities. Figure 11 shows a P-S-N curve for SS400 steel. The expression of the P-S-N curve with 50% failure probability can be given as follows. logN = 6.728 0.0094 S/2 ± 0.405 (5)

The mean of the fatigue limit by the JSME statistical SN testing method is 52.8 MPa. A commercial fatigue analysis program, Fe-safe (2003), is used to calculate fatigue life of the end beam. Miner’s rule was used as the fatigue cumulative damage rule. The first counting data for stress level was determined within



Figure 12. Rainflow cycle counting histogram.

Figure 14. Time-correlated fatigue damage.

Figure 13. Result of the damage histogram. the confidence interval of the P-S-N curve by a correction method for the curve that considers stresses under fatigue limit. Figure 12 shows the distribution of the stress range and mean stress at the location of G4. Figure 13 shows fatigue damage at each stress cycle using Miner’s rule. It can be noted that although the high amplitude stress cycle has low frequency, fatigue damage is relatively large. Figure 14 shows the damage histories over running time. Damage does not occur during running but most damage occurs during braking. Figure 15 shows fatigue life prediction by the S-N curves with a given failure probability. The fatigue life prediction by the S-N curves with 50% failure probability agrees well with actual fatigue life. In contrast, the fatigue life predictions by the S-N curves with 5% or 95% failure probabilities were underestimated or overestimated, respectively. Figure 16 shows fatigue life prediction at failure location G2 using Miner’s and Modified Miner’s rules. Miner’s rule overestimates fatigue life, but the modified Miner’s rule which considers the stress state under the fatigue limit provides an accurate fatigue life prediction within an error range of 2.7%~31%.

Figure 15. Fatigue life distribution for confidence region.

Figure 16. Comparison of experimental fatigue life by Miner’s rule and predicted fatigue life. Table 1 shows the fatigue life and damage at all strain gauge locations using modified Miner’s rules. The shortest fatigue life and damage are expected to occur at 1,410 cycles and 7.14×10-5 at the location of G4. Considering that one cycle of the load history is 25 min, the fatigue life of end beam is predicted to be 5,837.5 hrs. If


S. H. BAEK, S. S. CHO and W. S. JOO

Table 1. Fatigue life and damage prediction by Miner’s and modified Miner’s rule. Location number Mean stress correction G1 G2 G3 G4 G5 G6 G1 G2 G3 G4 G5 G6 Life No damage 2,533,000 1,015,000 545,800 17,190,000 1,100×106 120,000,000 81,640 80,900 27,040 113,800 52,130,000 Miner’s rule Damage 0 3.95 10 7 9.85 10 7 1.83 10 6 5.82 10 8 9.09 10 10 8.33 10 9 1.22 10 5 1.24 10 5 3.7 10 5 8.79 10 6 1.92 10 8 Year Unlimited 120.48 48.28 25.96 817.637 52,321 25,000 17 16.85 5.63 23.71 10,860 Life No damage 983,700 360,400 209,800 5,146,000 170,000,000 23,200,000 35,350 36,550 14,010 50,780 11,510,000 Modified Miner’s rule Damage 0 1.02 10 6 2.77 10 6 4.77 10 6 1.94 10 7 5.88 10 9 4.31 2.83 2.74 7.14 1.97 8.69 10 8 10 5 10 5 10 5 10 5 10 8 Year Unlimited 46.79 17.14 7.98 244.78 8,086 4,833 7.36 7.61 2.92 10.58 2,397



a freight car speed is 60 km/hr, its life expectancy is 350,250 km. This corresponds to 2.92 years, assuming the endurance life of a freight car as 25 years (3×106 km). But, because the location of G4 is fixed at the center beam, and stress intensity is concentrated at the welded gusset plate, discussion in regard to twisting shear stress is needed. For more accurate fatigue life prediction, further research is required for stress concentration at the welded gusset of the end beam.

The present work proposed a fatigue life estimation method for freight cars based on the rainflow cycle counting method, P-S-N curve, and modified Miner’s rule. Further improvements may be made to the procedure by incorporating a more representative hazard function with cumulative failure probability rather than the cumulative damage rule used in this paper. (1) The measured stress at the end beam agrees well with the FEA result, within a 12% error range. (2) Fatigue data displayed on a log-log plot of stress versus life for finite life can be expressed as follows: logN=6.728 0.094 S/2±0.405 (3) The fatigue damage and life calculated with the stress spectrum during 25 min are 7.14×10 5, and 2.92 years, on the basis of rainflow cycle counting method, P-S-N curve, and modified Miner’s rule.
ACKNOWLEDGEMENT This paper was supported by DongA university research fund in 2006.

Barboza, W., Raminelli, L. F. and Antonelli, J. (2005). Cumulative fatigue damage in the diesel engines appli-

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reliability evaluation. Int. J. Fatigue 17, 2, 85 89. Nagpal, R. and Kuo, E. Y. (1996). A time-domain fatigue life prediction method for vehicle body structures. SAE Paper No. 960567. Singh, A. (2002). The nature of initiation and propagation S-N curves at and below the fatigue limit. Fatigue Fract. Eng. Mater. Struct., 25, 79 89. Wang, H., Kim, N. H. and Kim, Y. J. (2006). Safety

envelope for load tolerance and its application to fatigue reliability design. ASME J. Mech. Des. 128, 4, 919 927. Zheng, X. and Wei, J. (2005). On the prediction of P-S-N curves of 45 steel notched elements and probability distribution of fatigue life under variable amplitude loading from tensile properties. Int. J. Fatigue, 27, 601 609.


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