fatigue testing 625 distribution functions for the linear

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FATIGUE TESTING 625

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The database is constructed from test re-sults that contain numbers of cycles until either fracture or crack initiation from me-tallic components. The test results do not include runouts, i. e., tests that were stopped without failure. All test results are taken from previous publications [13, 14, 21-28]. Therefore, only those results that are tabulated are included in the database. Test results that are documented only in charts are excluded.

The database consists of eighty-nine test series with sample sizes of 14 ≤ n ≤ 500, as shown in Figure 2. In total, 6,086 indi-vidual tests cover a range of cycle num-bers ranging from 790 ≤ N ≤ 1.2 × 108, as shown in Figure 3. The database con-tains the results of specimens that are

knowledge, no study has rated the applica-bility of the log-normal and Weibull distri-butions using modern tools for statistical testing, such as the Shapiro-Wilk test [19] or the Anderson-Darling test [20], and is based on a large database.

This study establishes a large database containing the results of fatigue tests from the linear regime of the S-N curve, where the fatigue life is finite (approximately 104 < N < 106), and large sample sizes (n >> 10). Statistical tests based on the methods of Shapiro and Wilk [19] and An-derson and Darling [20] are used to test the applicability of the log-normal distribu-tions to the database. In addition, probabil-ity plots that are easy to interpret are also used to test the distribution function.

Database containing fatigue test results with large sample sizes

Test series with large sample sizes (n >> 10) are needed to obtain reliable re-sults for the statistical tests of the distribu-tion function. Individual tests of one test series must be conducted at the single load level. Because fatigue tests are both time and cost intensive, test results of this qual-ity are scarce.

The database introduced in this study is an extended version of the one used in [21].

Constant amplitude fatigue tests, also re-ferred to as Wöhler tests, are widely used in the field of fatigue analysis. The result of a Wöhler test is the number of cycles that can be endured at a certain load am-plitude until the component fails, e. g., breaks. The results of constant amplitude fatigue tests are subjected to a natural scatter, as shown in Figure 1, and there-fore need to be evaluated using statistical tools. A critical issue during this evalua-tion process is the assumed distribution function. For metallic components, the log-normal distribution is often assumed in the field of fatigue analysis [1-9]. However, the Weibull distribution is also used in many cases [10, 11]. Other distribution functions, such as the arcsine distribution or logit distribution, are of lesser impor-tance in the field of fatigue analysis.

As can be seen in Figure 1, the assumed distribution function can lead to extreme differences if the results are extrapolated to small probabilities of failure, e. g., Pf = 1/1000 = 10-3. The test engineer must typically rely on experience for the predefini-tion of a distribution function. Statistical test-ing of the test series is typically difficult or impossible because of the small sample sizes (n << 15) that are predominant in practice.

Numerous studies have focused on se-lecting distribution functions for metallic components [10, 12-18]. To the authors’

This study establishes a database containing the re-sults of fatigue tests from the linear region of the S-N curve using sources from the literature. Each set of test results originates from testing metallic components on a single load level. Eighty-nine test series with sample sizes of 14 ≤ n ≤ 500 are included in the database, re-sulting in a sum of 6,086 individual test results. The test series are tested in terms of the type of distribution function (log-normal or 2-parameter Weibull) using the Shapiro-Wilk test, the Anderson-Darling test and proba-bility plots. The majority of the tested individual test results follows a log-normal distribution.

Christian Müller, Michael Wächter, Rainer Masendorf and Alfons Esderts, Clausthal-Zellerfeld, Germany

Distribution functions for the linear region of the S-N curve

Article Information

Correspondence AddressDr.-Ing. Christian MüllerInstitute for Plant Engineering and Fatigue AnalysisTU Clausthal, Building C 18 Leibnizstraße 3238678 Clausthal-Zellerfeld; GermanyE-mail: [email protected]

KeywordsDistribution function, S-N curve, statistical testing, Shapiro-Wilk test, Anderson-Darling test, log-normal distribution

Figure 1: Evaluation of Woehler tests using the log-normal distribution and Weibull distribution


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component-like or real components, such as bolts and roller bearings. The majority of the results originate from specimens and components of steel ma-terials, whereas a smaller portion origi-nates from tests on aluminum alloys, as shown in Figure 4. Only one test series is included for cast iron.

Statistical testing

Basics of statistical testing. Using statisti-cal testing tools, a hypothesis can be con-structed, e. g., “The sample originates from a population that follows a log-normal dis-tribution”, and can be tested for validity. The hypothesis can be accepted or rejected, but cannot be proven due to its statistical nature. Statistical testing is always per-formed using the same scheme:

1. Propose a null hypothesis and an alter-native hypothesis, e. g., a) H0: The sample originates from a log-

normal distribution b) H1: The sample does not originate

from a log-normal distribution2. Select a significance level α3. Draw a sample4. Compare the result from the sample to

the rejection-critical region, which is defined through the null hypothesis and significance level α.

The following example of a mean is consid-ered to provide a deeper understanding of statistical testing, as shown in Figure 5. A series of constant amplitude fatigue tests at a single load level shall follow a log-nor-mal distribution (population pop). The test series is repeated theoretically with a pre-defined sample size n for an infinite num-ber of repetitions. Therefore, the mean N50 %, sample of the sample is calculated for each repetition. The means of the sample then follow a separate distribution func-tion; for this example, this is a log-normal distribution with the same mean as the population. In this manner, probabilities of occurrence α can be assigned to the single means, as shown in Figure 5. For example, α = 10 % indicates that 10 % of all expected means of the sample will be smaller than the limit drawn in Figure 5.

For statistical testing, the probability of occurrence α is interpreted to be a signifi-cance level. It describes the value from which the sample result significantly dif-fers from the proposed null hypothesis, and the latter must be rejected. For the example in Figure 5, e. g., if the result of the sample lies on the left side of the drawn limit, the null hypothesis is rejected. The sample would differ significantly from the expecta-tions connected to the null hypothesis.

The significance level α is not a fixed value but must be selected depending on the application. Typical values for the sig-nificance level are α = 1 %, 5 % or 10 %

[29]. The significance level α is also de-scribed as the probability of a type 1 error. In the case of a correct null hypothesis, α describes the probability of rejecting the null hypothesis even though it is correct (type 1 error). Thus, a significance level of α = 10 % is a hard criterion for accepting the null hypothesis. The introduced exam-ple can be transferred from the mean to distribution functions.

Shapiro-Wilk test. The Shapiro-Wilk test [19] is the most powerful test for nor-mal and log-normal distributions [30]. To test a sample, e. g., the number of cycles N, with regard to its type of distribution func-tion, the sample is first logarithmized and sorted in an ascending order, and then, the test statistic W is calculated. The test sta-tistic W (see Equation (1)) is the ratio of dif-ferent estimates for the variance. The term in the numerator calculates the variance using order statistics and weights ai that are distribution specific [31]. The denomi-nator is the well-known estimator for the sample variance.

W =

1n −1

1n −1

⋅

ai ⋅ lg(Ni )i=1

n

∑⎛

⎝⎜

⎞

⎠⎟

2

(lg(Ni )− lg(N50%))2

i=1

n

∑

(1)

Here, lg(N50 %) is the logarithm of the mean of the sample, as shown in Equation (2).

lg(N50%) = 1n

lg(Ni)i=1

n

∑

(2)

In the case that the sample originates from a population that follows a log-nor-mal distribution, the test statistic W as-pires W = 1. Otherwise, W is smaller than 1. If W is small, the null hypothesis tends to be rejected, as shown in Figure 6. For small sample sizes of n ≤ 50, the values for the weights ai and the rejection limit for several significance levels can be de-

Figure 2: Summary of sample sizes in the database

Figure 3: Summary of the available number of cycles, N, in the database

Figure 5: Fundamentals of statistical testing

for the example with mean N50 %

Figure 4: Proportions of different materials and components with respect to the overall number of experiments (89 in total)


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termined as detailed in [19]. An approxi-mation formula is provided in [19] for large sample sizes. The approximation for the weights ai given in [32] is used in this study. Different algorithms are available for calculating the rejection limit for the test statistic W (see, e. g., [33]). The rejec-tion limit can also be determined using Monte-Carlo simulations, which are used for this study (see Figure 6).

Anderson-Darling test. This test, ac-cording to Anderson and Darling [20], is the second most powerful test for a log-normal distribution [30]. It is easier to com-pute than the Shapiro-Wilk test and can also be applied to other distribution func-tions. It is applied only to the log-normal distribution in this study.

For the Anderson-Darling test applied to a log-normal distribution, the sample, e. g., the number of cycles Ni, is logarithmized and then sorted in ascending order. From the sorted sample, the empirical distribu-tion function is computed and transformed into a uniform distribution. In the same manner, the predefined distribution func-tion (null hypothesis) is transformed into a uniform distribution. The Anderson-Dar-ling test determines the test statistic A2 (see Equation (3)), which is the squared distance between the two uniformly dis-tributed dimensions, whereas the outer ranges are more strongly weighted.

A2 =

(2i −1) ⋅ {ln(zi )+ ln(1− zn+1−i )}i=1

n

∑⎡

⎣⎢⎢

⎤

⎦⎥⎥

n− n

(3)

with zi: quantile of the standard normal dis-tribution for the random variable.

In this case, the number of cycles N was transformed to a standard normal distribu-tion, as shown in Equation (4).

zi =lg(Ni )−

1n

lg(Ni)i=1

n

∑1

n −1(lg(Ni)

i=1

n

∑ − lg(N50%))2

(4)

For the Anderson-Darling test, a feasible approximation formula exists for the rejec-tion limit for the test statistic A2 [34] that is applied in this study, as shown in Figure 7. The null hypothesis tends to be rejected during the Anderson-Darling test for large values of A2, as shown in Figure 7.

Probability plots. Probability plots are not part of classic statistical tests because no level of significance can be chosen. In probability plots, the probability of failure

is drawn over the quantile of a sample that was sorted according to its size. Both axes show characteristic scaling. If the sample originates from a population that follows the assumed distribution function, the sample forms a straight line in the proba-bility plot. A quality criterion for the ap-proximation of the sample by a straight line within the probability plot can be found using the coefficient of determina-tion r2 to form the linear regression [35]. The coefficient of determination exhibits values between 0 ≤ r2 ≤ 1. Therefore, e. g., r2 = 0.8 means that 80 % of the random re-sults can be explained by the assumed prob-ability function and that 20 % cannot [29, 35].

There are several estimation formulas available to estimate the probability of fail-ure Pf for the sample values [36-42]. Equa-tion (5) is used for the log-normal distribu-tion, originally proposed by Rossow [41] and the estimation formula based on Weibull [42] in Equation (6) is used for the 2-parameter Weibull distribution.

PA,log−NV = 3 ⋅ i −13 ⋅n +1

(5)

PA,WBL =i

n +1 (6)

Statistical testing of the database

The Anderson-Darling and Shapiro-Wilk tests are applied to the developed database. Significance levels of α = 5 % und α = 10 % (medium and hard criteria, respectively) are chosen.

Furthermore, the probability plots for the log-normal and 2-parameter Weibull distri-butions and the corresponding coefficients of determination, r²log-ND and r²WBL, are in-vestigated. The chosen quality criterion is r² > 0.95. This means that at least 95 % of the observed random events must be ex-plainable by the assumed distribution func-tion for the null hypothesis to be accepted.

The log-normal and 2-parameter Weibull distributions are both defined by a location and a scatter parameter. Furthermore, both are continuously defined between zero and infinity for random variables. Therefore, they can be compared directly and in a fair manner.

Figure 8 illustrates that the hypothesis “The sample originates from a population that follows a log-normal distribution” is accepted for more than 84 % of all samples (α = 5 %) by both the Shapiro-Wilk test and

Figure 6: Rejection limits for test statistic W of the Shapiro-Wilk test for different significance levels of α [21]

Figure 7: Rejection limits for test statistic A² of the Anderson-Darling test for different signifi-cance levels of α

Figure 8: Results of the Shapiro-Wilk test and a) the Anderson-Darling test, b) testing for the log-normal distribution with α = 5 %


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the Anderson-Darling test. Even for a high significance level of α = 10 %, the hypothe-sis is accepted for more than 76 % of all samples, as shown in Figure 9.

A similar result was obtained for the evaluation of the coefficient of determina-tion of the probability plots, as shown in Figure 10. The log-normal distribution ful-fills the criterion of r² > 0.95 for approxi-mately 79 % of all cases. The 2-parameter Weibull distribution fulfills the criterion of r² > 0.95 for approximately 36 % of all cases, as shown in Figure 10a. Figure 10b illustrates that the log-normal distribution approximates the test results better com-pared to the 2-parameter Weibull distribu-tion in approximately 88 % of all cases.

Summary

A database with fatigue tests from the lin-ear region of the S-N curve is established based on the literature. Therefore, the fo-cus lies on sample sizes n >> 10 for one load level. Eighty-nine test series with sample sizes 14 ≤ n ≤ 500 from different metallic materials are added to the data-base. A total of 6,086 individual test re-sults are included.

The test series of the database are inves-tigated in terms of the log-normal distribu-tion using both the Shapiro-Wilk test and the Anderson-Darling test. Furthermore, probability plots are established for the log-normal and 2-parameter Weibull distri-butions. A noticeable majority of the test results belongs to populations that follow a log-normal distribution. The 2-parameter Weibull distribution leads to better approx-imations in only a few cases.

Individual test results in the linear re-gime of the S-N curve of metallic compo-nents follow a log-normal distribution in the majority of cases. In the case when the user does not have more reliable informa-tion about the distribution function, the log-normal distribution should be assumed. However, if a positive experience was made using another distribution function for certain components, this experience should be trusted in.

Acknowledgement

The authors wish to express their gratitude to Professor Dr.-Ing. Harald Zenner and Dipl.-Ing. Manfred Hück for their support and for supplying literature as database.

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Figure 10: Coefficients of determination r² in the probability plots for the log-normal distribution and 2-parameter Weibull distribution; a) absolute coefficients, b) relative coefficients


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DOI 10.3139/120.111053Materials Testing59 (2017) 7-8, pages 625-629© Carl Hanser Verlag GmbH & Co. KGISSN 0025-5300

The authors of this contribution

Dr.-Ing. Christian Müller, born in 1984, studied Mechanical Engineering at Clausthal University of Technology (TUC), Germany and was a scien-tific employee at the Institute for Plant Engi-neering and Fatigue Analysis (IMAB) of TUC be-tween 2010 and 2015. He completed his PhD thesis on the statistical evaluation of S-N curves in 2015. Since 2015, he is responsible for the structural durability of high voltage battery packs of electric battery vehicles at Audi AG in Ingolstadt. Dr.-Ing. Michael Wächter, born in 1986, stud-ied Mechanical Engineering at Clausthal Univer-sity of Technology (TUC), Germany and has been a scientific employee at the Institute for Plant Engineering and Fatigue Analysis (IMAB) of TUC since 2011. He completed his PhD thesis on the determination of cyclic material properties and S-N curves for damage parameters in 2016. Dr.-Ing. Rainer Masendorf, born in 1964, stud-ied Mechanical Engineering at Clausthal Univer-sity of Technology (TUC), Germany and has been a scientific employee at the Institute for Plant Engineering and Fatigue Analysis (IMAB) of TUC since 1994. His PhD thesis (2000) considered the influence of prestraining cyclic material proper-ties of thin sheets. He has been a leading engi-neer at IMAB since 2000. The focus of his work is fatigue testing of materials and components for determining fatigue properties. Prof. Dr.-Ing. Alfons Esderts, born in 1963, studied Mechanical Engineering at Clausthal Uni-versity of Technology (TUC), Germany and com-pleted his PhD thesis in 1995. Between 1995 and 2003, he was Head of the “Fatigue Analysis” Department at Deutsche Bahn AG in Minden, Ger-many. Since 2003, he has been Professor at TUC and Head of the Institute for Plant Engineering and Fatigue Analysis (IMAB). In addition, he has been Vice President for research and technology transfer at TUC since 2015.

Abstract

Verteilungsfunktionen des linearen Bereichs der Wöhlerkurve. Mit Hilfe von Literaturwerten wird eine Datenbasis mit Wöhlerversuchen im Zeitfestigkeitsgebiet aufgebaut. Die Datenbasis enthält 89 Versuchsreihen mit Stichprobenumfängen zwischen 14 ≤ n ≤ 500 je Lasthorizont. In Summe werden 6086 Einzelversuchsergebnisse in der Datenbasis aufge-nommen. Jede Versuchsreihe wird mit dem Shapiro-Wilk-Test, dem Ander-son-Darling-Test sowie Wahrscheinlichkeitspapieren ausgewertet. Der Shapiro-Wilk-Test und der Anderson-Darling-Test werden für den Test auf logarithmische Normalverteilung angewendet. Mit den Wahrscheinlich-keitspapieren wird die Datenbasis auf logarithmische Normalverteilung und zweiparametrische Weibullverteilung untersucht. Die große Mehrheit der Versuchsergebnisse folgt einer logarithmischen Normalverteilung.

fatigue testing 625 distribution functions for the linear

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