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

Copyright © 2008 KSAE1229 9138/2008/038 12

95

FATIGUE LIFE PREDICTION BASED ON THE RAINFLOW CYCLE COUNTING METHOD FOR THE END BEAM OF A FREIGHT CAR BOGIE

S. H. BAEK1), S. S. CHO2) and W. S. JOO3)*

1)School of Mechanical Engineering, Dong-A University, Busan 604-714, Korea2)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 consistsof two steps: stress analysis and fatigue damage prediction. Finite element analysis is conducted for the component in questionto obtain detailed stress-strain responses. A significant number of failures occurred in a brake end beam which led to economiclosses and disruption of service. The cracks appeared to be fatigue cracks caused by the dynamic load produced in the loadedbogie frame. Strain gauge data were analyzed, and fatigue cycles were calculated from this data. Rainflow cycle counting wasused to estimate cumulative damage of the end beam under in-service loading conditions. The fatigue life calculated with therainflow cycle counting method, the P-S-N curve, and the modified Miner’s rule agreed well with actual fatigue life withinan error range of 2.7%~31%.

KEY WORDS : Fatigue life prediction, Rainflow cycle counting, Cumulative damage, Miner’s rule, P-S-N curve, Censoredstrain data

1. INTRODUCTION

In the beginning of 2001, cracks were found in the brakeend beam of the bogie frame of freight cars in a particularrunning section of the South Korean railway. The end beamof a freight car is a structural element that supports thebogie frame and braking system. The location and connec-tion method of the end beam should be considered in viewof structural design, because the end beam is built into thelower part of the side frame of the bogie.

Usually, important parts such as the bogie frame and carbody etc. are designed to last more than 25 years (Goo andSeo, 2003; Baek et al., 2005). In the case of a fracturedbogie frame, service life can be affected because themaximum stress is lower than the fatigue limit. However,two-thirds of the total number of end beams failed inservice 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 appearto be fatigue cracks caused by the dynamic load producedin the loaded bogie frame.

In the time-domain analysis of structures subjected torandom 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,

1998; Barboza et al., 2005; Kang et al., 2007) are used toestimate the fatigue service life. The bogie frame of freightcars has been evaluated by endurance test standards.However, because the South Korean railway has manymore curved tracks than railways abroad, there is a highbraking load during operation. A design specification thatreflects the domestic track in the existing endurance teststandard must be developed.

In the present paper, the load history was obtained fromstrain measurements on a bogie frame. A three-dimen-sional finite element model of a simplified bogie frame wasdeveloped for static stress analysis. Miner’s rule was com-bined with a probabilistic S-N curve (Murty et al., 1995;Zheng and Wei, 2005) and stress results to develop a stress-based fatigue life prediction for the brake end beam of thebogie frame.

2. FATIGUE DAMAGE APPROACH FOR FATIGUE LIFE PREDICTION

A general method for fatigue life estimation of railwayvehicles is required, as evidenced by cracking that occurredin the end beams of freight cars. As illustrated in Figure 1,by collecting different load amplitudes using the rainflowcycle counting method, the fatigue damage is linearlyaccumulated, as is proposed by Miner’s rule.

*Corresponding author. e-mail: wsjoo@dau.ac.kr

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

2.1. Rainflow Cycle Counting MethodThe end beam of a freight car bogie is subjected to variableamplitude service loading. To predict the fatigue life of theend 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 toextract stress-strain hysteresis loops quickly and accurately.In this study, rainflow cycle counting was used as a signalprocessing method for fatigue analysis.

Figure 2 shows the procedure for the cycle countingmethod 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 bythe range from B to C.

(iii) Figure 2 (b) shows a new cycle. As before, the rangefrom A to B is greater than that from B to C, so B to C isone cycle. This procedure is repeated until no more cyclesare closed by this point.

The fatigue cumulative damage rule for the actual runn-

ing environment is needed for improving the safety of rail-way vehicles. The extracted cycle produces stress ampli-tude and mean stress. Cumulative damage D and numberof fractures to cycle N are determined using a histogram ofcycle ranges and Miner’s rule.

For infinite life design for very high mean stresses, theBuch mean stress correction is selected. Miner’s rule isexpressed as follows. Failure is expected to occur if:

(1)

where ni is the number of applied cycles and Nfi is thenumber of cycles to failure at a specified stress amplitude

i, respectively. In this study, the critical cumulative dam-age value of D is chosen to be 1 in Eq. (1). The fatigue lifein the repeated signal is expressed as follows:

(2)

Since, in many cases, the Palmgren-Miner theory (Singh,2002) leads to non-conservative life predictions, the lineardamage rule associated with a critical damage sum D,different from one, has been proposed in many design codesfor fatigue damage assessment of structures subjected tovariable amplitude loading.

2.2. P-S-N CurveBecause of the scatter in fatigue life data at any given stresslevel, it must be recognized that there is not only one S-N

D= n1

Nf1

------+ n2

Nf2

------+ n3

Nf3

------+ =i

ni

Nfi

------ 1

Life= 1

ini/Nfi

---------------------

Figure 1. Flow chart for fatigue life prediction.

Figure 2. Rainflow cycle counting procedure.

Figure 3. Photograph of a fractured end beam.

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

FATIGUE LIFE PREDICTION BASED ON THE RAINFLOW CYCLE COUNTING METHOD 97

curve for a given material, but instead, a family of S-Ncurves 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 lifefor finite life can be expressed as follows for an end beam:

(3)

(4)

where is the standard deviation of the number ofcycle to fracture obtained by the staircase test. S-N curveswith failure probability 5% or 95% are determined bytranslating the S-N curve with failure probability of 50% tothe coordinate axis (±1.64 ).

3. FATIGUE LIFE PREDICTION FOR A BRAKE END BEAM OF A FREIGHT CAR BOGIE

3.1. Visual Examination of the End BeamThe fractured end beam was first subjected to visual exami-nation. The failure location of the end beam is presented inFigure 3. As seen from this figure, the end beam firstfractured in the welding zone between the C shape beamand the gusset plate.

3.2. Finite Element AnalysisA geometric model of the freight car bogie was developedusing CATIA and ANSYS. The finite element model of thebogie frame presented in Figure 4 consists of a 10-nodetetrahedral element and a 2-node beam element. The coupl-ing element was selected to model the load applied to thebracket hinge of the end beam. Since geometrical shape,load, and boundary conditions are symmetrical, we use thehalf-model as the effective model. The load condition wasdetermined through JIS E4207 (1984). The end beam andside frame are manufactured out of SS400 and SM490A,respectively. In this analysis, a vertical load of 17,000 kgand a braking load of 2,875 kg were applied to the center

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 6shows that the maximum von-Mises stress for the endbeam (75.4 MPa) is located on the corner of the weldedgusset plate. These results are particularly interesting fromthe viewpoint of the fatigue strength, because tensile stressesalone contribute the most to the fatigue crack initiation andpropagation. The location of the stress peak in Figure 6overlaps the fracture region presented in Figure 3. How-ever, the location of the stress peak in Figure 6 does notoverlap with the fracture region.

The high level of stress in the end beam area was themain cause of crack initiation. The fatigue load (a combi-nation of the self weight and braking load) caused thesuccessive propagation of the crack to critical size and thenresulted in rupture at the welded gusset plate. A consider-ably lower stress value in the region of the side frame canbe observed in Figure 5. The center frame, except for thecenter pivot, isn’t as highly loaded as the end beam. Themaximum von-Mises stress in the region of the centerframe is only 243.5 MPa, whereas in the side frame thestresses are around 112.2 MPa (Figure 5).

3.3. Estimation of Load HistoryTo determine whether the fatigue life is accurately predict-ed by the measured stress, it is necessary to compare thefatigue life as calculated by the rainflow cycle countingmethod with that observed in experimental fatigue dataattained under in-service loading.

Figure 7 shows attachment locations of six strain gauges

logN= ˆ + ˆ log S±1.64 ˆ logN

ˆ logN = 16---

1

8

logNi ˆ + ˆ logS1/2

ˆ logN

Figure 5. Distribution of stresses in the bogie frame.

Figure 6. Distribution of stresses of the end beam withbraking load.

Figure 7. Strain gauge layout for the end beam.

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

with direction perpendicular to the fatigue crack. In thisstudy, we assumed that the direction perpendicular to crackpropagation is the principal stress direction. A one axisstrain gauge (KFG-5-128) was installed on the end beambefore loading, and the test track was the Donghae-Jecheonsection. The load history in the test track was measuredthrough 60 km/hr over 25 min, from starting to braking.

Figure 8 shows the results of the test series with the loadhistory based on six strain gauges. The stresses resultingfrom 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 ex-perimental data and finite element analysis (FEA) results.Compared to the FEA results, the experimental stresses atlocations of G2 and G5 are measured within an error rangeof 12% as compared with analytical stresses. However,experimental stress at location G4 is lower than the analy-tical stress. The measured stress at this location is very highdue to track vibration and braking load.

3.4. Fatigue Life PredictionKnowledge of the material properties at the most criticalpoint of the end beam is needed for correct evaluation ofthe integrity of the bogie frame. For this reason, 10 mmthick flat specimens were taken from a broken end beam ofSS400 steel and investigated under alternating bendingstresses (R = 1). Test results for fatigue life given in Baeket al. (2005) were obtained by flat specimens on a Scenk

type twisting and bending fatigue testing machine (Figure10), and then plotted on the S-N curves with 5%, 50%, and95% failure probabilities.

Figure 11 shows a P-S-N curve for SS400 steel. Theexpression of the P-S-N curve with 50% failure probabilitycan be given as follows.

logN = 6.728 0.0094 S/2 ± 0.405 (5)

The mean of the fatigue limit by the JSME statistical S-N 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’srule was used as the fatigue cumulative damage rule. Thefirst counting data for stress level was determined within

Figure 8. Blocks of strain-time history.

Figure 9. Comparison of measured stress and FEA result.

Figure 10. Apparatus for the Scenk type fatigue test.

Figure 11. P-S-N curve for SS400 steel.

FATIGUE LIFE PREDICTION BASED ON THE RAINFLOW CYCLE COUNTING METHOD 99

the confidence interval of the P-S-N curve by a correctionmethod for the curve that considers stresses under fatiguelimit.

Figure 12 shows the distribution of the stress range andmean stress at the location of G4. Figure 13 shows fatiguedamage at each stress cycle using Miner’s rule. It can benoted that although the high amplitude stress cycle has lowfrequency, fatigue damage is relatively large. Figure 14 showsthe damage histories over running time. Damage does notoccur during running but most damage occurs during brak-ing.

Figure 15 shows fatigue life prediction by the S-N curveswith a given failure probability. The fatigue life predictionby the S-N curves with 50% failure probability agrees wellwith actual fatigue life. In contrast, the fatigue life predic-tions by the S-N curves with 5% or 95% failure probabi-lities were underestimated or overestimated, respectively.

Figure 16 shows fatigue life prediction at failure locationG2 using Miner’s and Modified Miner’s rules. Miner’s ruleoverestimates fatigue life, but the modified Miner’s rulewhich considers the stress state under the fatigue limitprovides an accurate fatigue life prediction within an errorrange of 2.7%~31%.

Table 1 shows the fatigue life and damage at all straingauge locations using modified Miner’s rules. The shortestfatigue life and damage are expected to occur at 1,410cycles 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

Figure 12. Rainflow cycle counting histogram.

Figure 13. Result of the damage histogram.

Figure 14. Time-correlated fatigue damage.

Figure 15. Fatigue life distribution for confidence region.

Figure 16. Comparison of experimental fatigue life byMiner’s rule and predicted fatigue life.

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

a freight car speed is 60 km/hr, its life expectancy is350,250 km. This corresponds to 2.92 years, assuming theendurance life of a freight car as 25 years (3×106 km). But,because the location of G4 is fixed at the center beam, andstress intensity is concentrated at the welded gusset plate,discussion in regard to twisting shear stress is needed. Formore accurate fatigue life prediction, further research isrequired for stress concentration at the welded gusset of theend beam.

5. CONCLUSIONS

The present work proposed a fatigue life estimation methodfor freight cars based on the rainflow cycle counting method,P-S-N curve, and modified Miner’s rule. Further improve-ments may be made to the procedure by incorporating amore representative hazard function with cumulative failureprobability rather than the cumulative damage rule used inthis 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 stressspectrum during 25 min are 7.14×10 5, and 2.92 years,on the basis of rainflow cycle counting method, P-S-Ncurve, and modified Miner’s rule.

ACKNOWLEDGEMENT This paper was supported by Dong-A university research fund in 2006.

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Table 1. Fatigue life and damage prediction by Miner’s and modified Miner’s rule.

Location number Miner’s rule Modified Miner’s ruleMean stress correction Life Damage Year Life Damage Year

G1

None

No damage 0 Unlimited No damage 0 UnlimitedG2 2,533,000 3.95 10 7 120.48 983,700 1.02 10 6 46.79G3 1,015,000 9.85 10 7 48.28 360,400 2.77 10 6 17.14G4 545,800 1.83 10 6 25.96 209,800 4.77 10 6 7.98G5 17,190,000 5.82 10 8 817.637 5,146,000 1.94 10 7 244.78G6 1,100×106 9.09 10 10 52,321 170,000,000 5.88 10 9 8,086G1

Goodman

120,000,000 8.33 10 9 25,000 23,200,000 4.31 10 8 4,833G2 81,640 1.22 10 5 17 35,350 2.83 10 5 7.36G3 80,900 1.24 10 5 16.85 36,550 2.74 10 5 7.61G4 27,040 3.7 10 5 5.63 14,010 7.14 10 5 2.92G5 113,800 8.79 10 6 23.71 50,780 1.97 10 5 10.58G6 52,130,000 1.92 10 8 10,860 11,510,000 8.69 10 8 2,397

FATIGUE LIFE PREDICTION BASED ON THE RAINFLOW CYCLE COUNTING METHOD 101

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