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SAE TECHNICALPAPER SERIES 2004-01-1288

Reliability-Based Fatigue StrengthTesting by the Staircase Method

Pamela M. SnyderRochester Institute Technology

Ming-Wei Lu and Yung-Li LeeDaimlerChrysler Corp.

Reprinted From: Reliability and Robust Design in Automotive Engineering(SP-1844)

2004 SAE World CongressDetroit, MichiganMarch 8-11, 2004

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ABSTRACT

The staircase fatigue testing method is a recognizedmethod for determining the fatigue limit of powertraincomponents. The purpose of this paper is to improve upon existing standards by adding common practices that will ensure a higher degree of statistical accuracy in the data. This includes specifying appropriate sample sizes, stress increments and initial load conditions, as well as making suggestions for appropriate methods of analyzing the data. Two methods (Dixon and Moodmethod and probit analysis method) are selected and compared in terms of relative percent difference on four parameters (mean, standard deviation, B10 fatiguestrength and B50 fatigue strength). The staircase data are obtained by simulations from normal and lognormal fatigue limit distributions.

INTRODUCTION

When designing components the fatigue strength andfatigue limit are the usually factors that need to be taken into account before a part can be approved forproduction. These factors are determined from the S-Ncurve, which is unique for each part or geometry and type or class of material. A sample P-S-N curve is shown in Figure 1.

The P-S-N curve can be created based on several points representing the applied load and number of cycles untilthe part fails. For wrought steels, the curve itself has two distinct regions: the finite life region and the fatigue limit region. The finite life region is the sloped region of the curve. In this region the curve can be used toestimate how many cycles a part will survive at a given loading before failure, or the maximum load that can be applied so the part will survive a given number of cycles, with relative certainty. The methods to statistically

characterize the P-S-N curve can be found elsewhere [ASTM, 1998 and Williams et al, 2003] and are beyond the scope of our discussion in this paper. The fatigue limit region is the flat section of the curve. Here the curve represents the fatigue strength below which an infinite number of cycles can be sustained. Infinite life isconsidered one million cycles for most wrought steels. For materials other than steels, the knee of the S-Ncurve is not so apparent. The fatigue strength at a large number of cycles (107 to 108 cycles) becomes of interest to the design community and is called the endurance limit. The test methods to identify the fatigue limit or the endurance limit include the Staircase Test Method, the Probit Fatigue Testing Method, the Step-Test Method, the Prot Fatigue Testing Method, and AcceleratedTesting Methods [Wirsching, 1983]. Among thesemethods, the Staircase Test method is the preferablestandards [British Standards, 1966, Japanese Standard, 1981, French Standard, 1991] and has been extensively discussed elsewhere [Zhang and Kececioglu, 1998 andLin et al, 2001].

Figure 1 - Sample P-S-N curve, showing the distribution of data points used to generate the curve.

2004-01-1288

Reliability-Based Fatigue Strength Testing by the Staircase Method

Pamela M. Snyder Rochester Institute Technology

Ming-Wei Lu and Yung-Li Lee DaimlerChrysler Corp.

Copyright @ 2004 SAE International

2

Here is a brief description of the Staircase Test method. An initial load is applied to the part and run for a life of interest, say one million cycles. If the part fails prior to the life target, the load is reduced by a stress increment and another part is run for one million cycles. The load continues to be reduced by the stress increment until a part survives the target life. At this point the load isincreased by the stress increment and run for one million cycles. This process continues until the appropriatenumber of parts has been tested, increasing the load if the part survives or reducing it if the part fails. Once theappropriate number of tests have been run the dataneeds to be analyzed to determine the fatigue limit. In general terms, the fatigue limit is roughly the median of the data; roughly half the points should fall above and below the fatigue limit. However, it is hoped that all the points above the fatigue limit are failures and all below are survivals. Since this does not always occur it isnecessary to use more advanced statistical tools. For this paper the Dixon & Mood [Dixon and Mood, 1991] and the probit analysis method [Aldrich and Nelson,1984] were employed for data reduction analyses.

The existing test standards describe the StaircaseFatigue Testing Method. They, however, lack necessary common practices for determining sample size, initial load conditions, an appropriate stress increment, andappropriate methods for analyzing the data. This test method is considered the standard for testing anddeveloping powertrain components and is used by many companies within the industry currently. However, many others are not compliant with the standard and choose to use other testing methods. It is for these reasons that it has become important to improve upon the existing test standard for the Staircase Fatigue Testing Method by developing common practices and examining variousdata analysis techniques. It is hoped that by developing common practices for the Staircase Fatigue Testingmethod more companies within the industry will bewilling to use the test method and those already using the test method will have more confidence in their data.

The objective of this paper is to evaluate effects ofsample size and stress increment on a given statistical distribution of the fatigue limit by using the Monte Carlo Simulations technique and by the two data reductionmethods.

SIMULATIONS OF THE STARICASE DATA

In developing common practices it was decided thatcomputer simulations would be the fastest and cheapest way to generate large amounts of data and examineseveral different conditions. It was decided to use Monte

Carlo Simulations to generate the data, which can then be used as data for a test method. For this research, the random numbers generated using the Monte Carlotechnique represent the load capacity of the part. This is a reasonable method of determining the load capacity of the parts, since not every part will have the same load capacity due to things like material defects, machining error, surface defects and other issues that can alter the properties and load capacity of the parts. It was also decided to conduct the simulated staircase fatigue test in Excel and use Visual Basic code to optimize the time required to run the simulations.

The procedure for conducting the staircase fatigue test method is slightly different when using Excel with Visual Basic to gather the data rather than physical testing. A sample of the test set up for a normal distribution isshown in Figure 2; it may be helpful to refer to it through the explanation of the test method.

Figure 2 - Sample of the Staircase Fatigue Test as conducted in Excel.

The first step is to specify a mean, standard deviation, stress increment, sample size, and type of distribution.For Figure 2, the mean is 100, the standard deviation is 5, the stress increment is 2% of the mean, the sample size is 10, and the distribution is normal. Then, theappropriate number of random numbers is generated based on the specified sample size. These randomnumbers from uniform distribution (0,1) are generated using the Monte Carlo simulations described above and represent the cumulative distribution function (cdf) value of the load capacity of a given part. Once, the random numbers have been generated the cdf is generatedbased on the mean and standard deviation of thedistribution being examined. The same cdf is used for the load capacity and load applied in order to relate the values. At this point the simulation begins conducting the test and gathering data. The initial load applied is three times the stress increment above the mean, for this phase of the testing. For every load applied there is a corresponding cdf value. The cdf value of the load applied is then compared to the random number

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generated for the same test number, or the cdf of theload capacity. If the cdf value of the load capacity is greater than the cdf value of the load applied, the part will survive two million cycles. If the cdf value of the load capacity is less than the cdf value of the load applied, the part will fail before two million cycles. If the part survives the next load applied is increased by the stress increment, if the part fails the next load applied isreduced by the stress increment. This processcontinues till the appropriate number of tests has been simulated.

For this phase of the paper, two different probabilitydistributions were examined, normal and lognormal.These distributions were chosen after examining thedistributions of physical test data. Two coefficients of variation (COV) were examined, 5% and 50%. The COVis defined as the standard deviation over the mean. The COV of 5% was chosen after examining physical data and observing the most of the data had a COV of 3.5%-6.5%. The COV of 5% was chosen to be a reasonable value and for ease of calculation.

0

0.02

0.04

0.06

0.08

60 80 100 120 140

Random variable, X

PD

F

Figure 3(a) Normal distribution with mean = 100 andCOV = 5%

0

0.02

0.04

0.06

0.08

60 80 100 120 140

Random variable, X

PD

F

Figure 3(b) Lognormal distribution with mean = 100 and COV = 5%

Figure 3 (a) Shows the normal probability distribution function when COV is 5%. Figure 3 (b) shows thelognormal probability distribution function when COV is 5%

As shown in Figure 3(a) and 3(b), the normal andlognormal probability functions are identical when theCOV is 5%. Also a larger COV of 50% was chosen for this work.

0

0.002

0.004

0.006

0.008

-100 -50 0 50 100 150 200 250 300

Random variable, XP

DF

Figure 4 (a) Normal distribution with mean = 100 andCOV = 50%.

0

0.002

0.004

0.006

0.008

0.01

0.012

0 50 100 150 200 250 300

Random variable, X

PD

F

Figure 4 (b) Lognormal distribution with mean = 100 and COV = 50%.

Figure 4 (a) Shows the normal probability distribution function when COV is 50%. Figure 4 (b) shows thelognormal probability distribution function when COV is 50%

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To gain the maximum understanding both distributions were tested when the COV was 5% and 50%. Themean was held constant at 100 for all tests to date. The standard deviations used were 5 and 50, to generate the COV of 5% and 50% respectively. Five different samplesizes were tested; 10, 15, 20, 25, and 30. Additionally, three different stress increments were tested; 2%, 5%, and 8% of the mean. For each combination of sample size, stress increment, probability distribution, and COV, 30 tests like the one in Figure 3 were conducted. The findings contained in this paper are based on more than 1600 tests and approximately 50 combinations.

DATA REDUCTION METHODS

Once the test data was gathered it was groupedaccording to the load applied and whether the part failed or survived. This data was then analyzed using twodifferent statistical tools, the Dixon & Mood Method and the probit analysis method. The methods werecompared based on each’s ability to predict the mean, standard deviation, B50 Fatigue Strength, and B10fatigue strength. The Dixon & Mood method is anempirical approximation of mean and standard deviation.When using the Dixon & Mood method there are three required assumptions. First, the fatigue limit data needs to be normally distributed. Second, the stress increment needs to be held constant. Finally, the stress increment needs to be between half and twice the standarddeviation. Since, the second condition is the only one that can truly be controlled when doing physical testing; it was also the only assumption that was alwaysmaintained when conducting this research. The probit analysis method is used to analyze response data,particularly success/failure and response/frequency data at various stress levels. This method can be used to determine mean and standard deviation of a fatigue limit distribution.

The two methods (the Dixon and Mood method and the probit analysis method) are briefly discussed below:

Dixon and Mood method

The Dixon and Mood method provides approximateformulas to calculate the mean, µS, and the standarddeviation, σS, of a fatigue limit, eS . The method

assumes that the fatigue limit follows a normaldistribution. It requires that the two statistical properties be determined by using the data of the less frequentevent (i.e., either only the failures or only the survivals).The stress levels, Si, spaced equally with a chosen

increment, d, are numbered i where i = 0 for the lowest stress level, So. It should be noted that the stressincrement should be in the range of half to twice the standard deviation of the fatigue limit.

Denoting by nDM,i the number of the less frequent event at the numbered stress level i, two quantities ADM and BDM can be calculated:

∑ ×= iDMDM niA , (1)

iDMDM niB ,2 ×= ∑ (2)

The estimate of the mean is then

±×+=

∑ 2

1

, iDM

DMoS n

AdSµ (3)

( )

+

−××=

∑∑

029.062.1 2

,

2

,

iDM

DMiDMBM

S

n

AnBdσ (4)

( ) 3.02

,

2

, ≥−

∑∑

iDM

DMiDMDM

n

AnBif

ordS ×= 53.0σ (5)

if ( ) 3.02

,

2

, <−

∑∑

iDM

DMiDMDM

n

AnB

With the determined mean, Sµ , and standard deviation,

Sσ , for a normal distribution, the lower bound valueassociated with a reliability and confidence levels,

CReS ,, , can be determined by the one-side lower-bound

tolerance limit K factor.

Probit analysis method

The regression model places no restrictions on thevalues that the independent variable (X) take on. X can be continuous, can be only positive, or can be discrete (integers). However, the dependent variable (Y) isassumed to be continuous. But, if Y can take on only two values (say 0 = success and 1 = failure), theviolation of this “continuous” assumption is obvious.Here, Y is known as Bernoulli random variable and has associated with parameter, say P, that represents the probability of Y = 1. That is P = Prob {Y = 1).

The probit model [Aldrich and Nelson, 1984] is simply a nonlinear probability model, which is defined by

P = F(a + b X) (6)

where P is the probability of a failure at the X stress level

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F(y) is the distribution function, a is the constant term, b is unknown coefficient associated with the stressvariable, and X is the stress variable.

The two distribution functions that will be used in this simulation study are given below:

Distribution Distribution function F(y)Normal F(y) = Φ(y)Lognormal F(y) = Φ(ln y)

where Φ is the distribution function of a standard normal random variable.

The two probit parameters a and b are typicallyestimated by a method called Maximum LikelihoodEstimation (MLE) in contrast to ordinary regressionmodels which are estimated by the method of LeastSquares Estimation (LSE).

The distribution function you choose should depend on your data. You want to choose a distribution function that results in a good fit to your data. Goodness-of-fitstatistics can be used to compare fits using different distributions. Certain distributions may be used forhistorical reasons or because they have a specialmeaning in a discipline.

The probit analysis method will estimate the fatiguestrength distribution (from equation 1) at a specific life by using the test data from the staircase method. Thestatistical software MINITAB [MINITAB, 1998] will beused for the estimation of the two probit parameters a and b.

RESULTS

Once all the test data was gathered and analyzed using both the Dixon & Mood method and the probit analysis method, the relative percent difference for each test was calculated based on mean, standard deviation, B10fatigue strength and B50 fatigue strength. The errormeasurement is defined as

valueActual valueActual- valueEstimated

. The average

relative percent difference was found over the thirty tests based on these four factors. The results are based on the trends observed from the average percent relative difference for each combination.

When conducting the test, there were problems with the normal probability distribution when the COV was 50%.The test simulation was creating applied loads with

negative values, which would not be possible in real life.As a result, this testing was stopped and the results are only based on the normal and lognormal probabilitydistributions at COV of 5% and lognormal probabilitydistribution at COV of 50%.

Based on the results of the average percent relative difference in mean and B50 fatigue strength at the COVof 5%, the Dixon & Mood method is sufficient, less than 5% error for all combinations. However, the probitanalysis method was slightly better for all combinations, usually less than 2% improvement. For mean and B50 fatigue strength all the sample sizes had less than 2.5% error for all combinations, so a sample size of 10 issufficient. Also based on mean and B50 fatiguestrength, a stress increment of 2 - 5% of the meanprovides the most accurate results when using the Dixon & Mood method. When using the probit analysis methoda stress increment of 5-8% provides the most accurate results.

Based on the B10 fatigue strength at the COV of 5%, the sample size needs to be a minimum of 15 to obtain less than 5% error for all stress increments. Also the most accurate results were obtained using the Dixon & Mood method and when the stress increment was between 5% and 8% of the mean. Also, when the COV was 5% neither method was accurate for either probabilitydistribution to determine the standard deviation within5%. However, since the standard deviation was 5, if amethod returned 4.5 is was off by 10%. So the actualvalues were reasonable, but the relative difference is still very large and outside the expected range established.

When examining the COV of 50% all the results are based on the lognormal distribution. In determining the mean, Dixon & Mood method provided slightly better results than the probit analysis method. The stressincrement of 2% of the mean was the only one thatprovided less than 5% for any sample size. When using the stress increment of 2% a sample size of 10 issufficient for mean; for the other stress increments the sample size would need to be larger than 30 to have less than 5% error in mean. As far as standarddeviation, B10 fatigue strength and B50 fatigue strength neither method is accurate enough to provide less than 5% error under the conditions that have been tested.

CONCLUSIONS

The Dixon & Mood method and the probit analysismethod are equally suited to determine the mean for all COVs. At the smaller COV both methods returnexpected values for determining B10 and B50 fatigue

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strength. Neither method is accurate enough to return the standard deviation within 5% of the actual value at any COV. Neither method is accurate enough to return the B10 and B50 fatigue strength at the higher COV for the lognormal probability distribution. Since the twomethods have basically the same accuracy, it isrecommended that the Dixon & Mood method beemployed because it is easier to use and not software dependent.

When examining data that follows a normal probability distribution larger COVs are very unsuccessful, due to the nature of the distribution.

FUTURE WORK NEEDED

At this point there are still several things that will need to be done before any final common practices can bewritten. First, tests need to be run in the ranges already suggested in the standard and checked for statisticalaccuracy. The standard says that only 6 samples need to run, however, there is no statistical data to back this up, and this research did not investigate sample sizes of less than 10. So it would be suggested in the future that samples size of 5-10 also be investigated. Also when investigating the COV of 50% or a large COV, it would be helpful to try stress increments around 50% of the standard deviation, say 20%, 25% and 30% of the mean, better result might be found by trying to hold to some of the Dixon & Mood assumptions. It might be helpful to also investigate a mid-range COV of 20-40%, this would provide a better understanding of how the lognormal and normal distributions compare when the lognormaldistribution is not fully developed but also no longer resemble the normal distribution. And finallyinvestigation into various initial load conditions will beneeded once common practices can be established for stress increment and sample size.

REFERENCES

1. Aldrich, J.H. and Nelson, F.D., Linear Probability,Logit, and Probit Models, Quantitative Applications in the Social Science, SAGE Publications, Inc.,Nerbury Park, CA, 1984.

2. ASTM, “A Guide for Fatigue Testing and theStatistical Analysis of Fatigue Data,” ASTM Special Technical Publication No. 9I-A (2nd Edition), 1916.

3. ASTM, “Standard Practice for Statistical Analysis of Linear or Linearized Stress-Life (S-N) and Strain-Life(e-N) fatigue data.” ASTM E739-91, 1998, pp. 631-637.

4. British Standard Institution, British Standard 3518: Methods of Fatigue Testing: Part 5: Guide to theApplication of Statistics, Linford Wood, MiltonKeynes, British Standard Institution, 1966.

5. Dixon, W.J. and Mood, A.M., A Method for Obtaining and Analyzing Sensitivity Data, J. Amer. Stat. Assoc, 43, 1948, pp. 109-126.

6. Japan Society of Mechanical Engineers, Standard Method of Statistical Fatigue Testing, JSME S 002, 1981.

7. Lassociation Francaise de Normalisation,Normalisation Francaise A 03-405, Essais deFatigue, Tour Europe, Paris La Defense:Lassociation Francaise de Normalisation, 1991.

8. Lin, S.K, Lee, Y. L., and Lu, M.W., „Evaluation of the Staircase and the Accelerated test methods forfatigue limit distribution,“ International Journal ofFatigue, 23, 2001, pp. 75-83.

9. MINITAB statistical software (Release 12), MINITAB Inc., State College, PA, 1998.

10. Williams, C. R., Lee, Y. L., and Rilly, J. T., “APractical Method for Statistical Analysis of Strain-LifeFatigue Data,” International Journal of Fatigue, 25, 2003, pp. 427-436.

11. Wirsching, P.H., Statistical Summaries of fatigueData for Design Purposes, NASA Contractor Report 3697, N83-29731, 1983.

12. Zhang, J. and Kececioglu, D.B., “New Approachesto Determine the Endurance Strength distribution,” 4th ISSAT International Conference on Reliability and Quality in Design, Seattle, Washington, 12-14August, 1998, pp. 297-301.