Accessible Requires Authentication Published by De Gruyter August 18, 2021

Statistical evaluation of fatigue tests using maximum likelihood

Klaus Störzel and Jörg Baumgartner
From the journal Materials Testing

Abstract

The statistical evaluation of fatigue tests can be carried out using the maximum likelihood method. With this method, the influence of run-outs on the S-N curve can be statistically considered. Typically, a bilinear S-N curve (Wöhler curve) in double-logarithmic representation is used. The logarithmic normal distribution is the basis for describing the scatter, which is assumed here to be independent of the number of cycles. For parameter determination via the maximum likelihood method, reliability is examined and compared with the evaluation methods proposed in DIN 50100. While a defined test procedure is required for the application of DIN 50100, any test data can be evaluated according to the maximum likelihood method. In comparison with the methods proposed in DIN 50100, it could be shown through some examples that the maximum likelihood method yields very reliable results for all S-N curve parameters.


Dipl.-Ing. Klaus Störzel Fraunhofer-Institut für Betriebsfestigkeit und Systemzuverlässigkeit LBF Bartningstraße 47 64289 Darmstadt

Acknowledgement

Parts of the results presented here were developed in the IGF project (No. 18198 N) “Component design taking into account stress with variable amplitudes and very high number of load cycles (VHCF-VA II)”. The authors thank the German Federal Ministry for Economic Affairs and Energy (BMWi) and the German Federation of Industrial Research Associations (AiF) for the financial support.

Nomenclature

Δ

increment

1/T

scatter of a S-N curve

cdf

cumulative density function

DIN

DIN 50100

H

load level for fatigue tests

H1

load level of the first staircase test

HCF

high cycle fatigue regime (N<Nk)

Ho

highest load level for fatigue tests

Hu

lowest load level for fatigue tests

k

slope of the S-N curve in the high cycle fatigue regime (N<Nk)

k*

slope of the S-N curve in the long life fatigue regime (N3Nk)

LLF

long life fatigue regime (N3Nk)

LLM

load level method

ML

maximum likelihood

N

number of cycles

n

number of test results

n1

number of test results where failure at Ni < Nk occurred

n2

number of test results where failure at Ni ≥ Nk occurred

nB

number of test results were no failure has occurred (runouts)

NG

limit number of cycles (end of test)

nH

number of levels for fatigue tests

Nk

knee point, i. e. transition point between high cycle and long life fatigue regime

pdf

probability density function

P

probability of occurrence

P s

probability of survival

PSM

pearl string method

RND

random number

S

load or stress

s

standard deviation

SCM

staircase method

sS

standard deviation of the logarithmic load in the S-direction

sN

standard deviation of the logarithmic load in the N-direction

sup

support function

t

integration variable of the distribution function of the normal distribution

Subscripts

a

amplitude

B

reference point

corr

corrected

i

individual test result, counting parameter

k

knee point

max

maximum

N

number of cycles

NG

at the position of the limit number of cycles NG

opt

optimum

runout

test result as a runout

S

load or stress

s

survival

tot

total

References

1 O. H. Basquin: The exponential law of endurance tests, Proc. ASTM 10 (1910), pp. 625-630 Search in Google Scholar

2 C. E. Stromeyer: The Determination of fatigue limits under alternating stress conditions, Proceedings of the Royal Society A 90 (1914), pp. 411-425 DOI:10.1098/rspa.1914.0066 Search in Google Scholar

3 A. Palmgren: Die Lebensdauer von Kugellagern (The fatigue life of bearings), Z. VDI (1924), No. 68, pp. 339-341 Search in Google Scholar

4 E. Castillo, A. Fernandez-Canteli, A. S. Hadi: On fitting a fatigue model to data, International Journal of Fatigue 21 (1999), No. 1, pp. 97-106 DOI:10.1016/S0142-1123(98)00048-6 Search in Google Scholar

5 DIN 50100: Schwingfestigkeitsversuch – Durchführung und Auswertung von zyklischen Versuchen mit konstanter Lastamplitude für metallische Werkstoffproben und Bauteile (Load Controlled Fatigue Testing – Execution and Evaluation of Cyclic Tests at Constant Load Amplitudes on Metallic Specimens and Components), Beuth, Berlin, Germany (2016) Search in Google Scholar

6 H. Mauch, H. Zenner: Lebensdauerstatistik – Leitfaden zur Statistik in der Betriebsfestigkeit (Engl.: Fatigue life – Guideline for Statistics in Fatigue Strength), FVA-Forschungsvorhaben 304, Heft 591 (1999) Search in Google Scholar

7 D. J. Finney: Probit Analysis: A statistical Treatment of Sigmoid Response Curve, Cambridge University Press, Cambridge, UK (1947) Search in Google Scholar

8 W. W. Maenning: Das Abgrenzungsverfahren, eine kostensparende Methode zur Ermittlung von Schwingfestigkeitskennwerten (The delimitation method, a cost-saving method for the determination of fatigue strength parameters), Materials Testing 19 (1977), No. 8, pp. 280-289 DOI:10.3139/120.110390 Search in Google Scholar

9 D. J. Dixon, A. M. Mood: A method of obtaining and analyzing sensitivity data, Journal of the American Statistical Association 43 (1948), No. 241, pp. 108-126 DOI:10.1080/01621459.1948.10483254 Search in Google Scholar

10 E. Deubelbeiss: Dauerfestigkeitsversuche mit einem modifizierten Treppenstufenverfahren (Fatigue tests with a modified staircase method), Materials Testing 16 (1974), No. 8, pp. 240-244 Search in Google Scholar

11 M. Hück: Ein verbessertes Verfahren für die Auswertung von Treppenstufenversuchen (An improved method for the evaluation of staircase tests), Materialwissenschaft und Werkstofftechnik 14 (1983), No. 12, pp. 406-417 DOI:10.1002/mawe.19830141207 Search in Google Scholar

12 K. Brownlee, J. Hodges, M. Rosenblatt: The up-and-down method with small samples, Journal of the American Statistical Association 48 (1953), pp. 262-277 Search in Google Scholar

13 T. Svensson, B. Wadman, J. de Maré, S. Lorén: Statistical models of the fatigue limit, Swedish National Testing and Research Institute: Online Project Paper, 2000 Search in Google Scholar

14 R. Little: Estimating the median fatigue limit for very small up-and-down quantal response tests and for S-N data with runouts, In: Heller RA, editor. Probabilistic aspects of fatigue. Philadelphia, PA: American Society for Testing and Materials, 1972 Search in Google Scholar

15 R. Rennert, E. Kullig, M. Vormwald, A. Esderts, D. Siegele: FKM Guideline – Analytical Strength Assessment of Components, Frankfurt a. M., Germany 2013 Search in Google Scholar

16 A. F. Hobbacher: Recommendations for Fatigue Design of Welded Joints and Components, IIW, Springer, Heidelberg, Germany (2016) Search in Google Scholar

17 DIN EN 1993-1-9: Design of Steel Structures – Part 1 – 9: General Rules and Rules for Buildings, Beuth, Berlin, Germany (2010) Search in Google Scholar

18 C. M. Sonsino: Course of SN-curves especially in the high-cycle fatigue regime with regard to component design and safety, International Journal of Fatigue 29 (2007), No. 12, pp. 2246-2258 DOI:10.1016/j.ijfatigue.2006.11.015 Search in Google Scholar

19 C. Müller, M. Wächter, R. Masendorf, A. Esderts: Distribution functions for the linear region of the S-N curve, Materials Testing 59 (2017), No. 7-8, pp. 625-629 DOI:/10.3139/120.111053 Search in Google Scholar

20 C. Müller: Zur statistischen Auswertung experimenteller Wöhlerlinien (Engl.: On the statistical evaluation of experimental S-N curves), PhD-Thesis, TU Clausthal, Clausthal-Zellerfeld, Germany (2015) Search in Google Scholar

21 C. Müller, M. Wächter, R. Masendorf, A. Esderts: Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques, International Journal of Fatigue 100 (2017), No. 1, pp. 296-307 DOI:10.1016/j.ijfatigue.2017.03.030 Search in Google Scholar

22 E. Spindel, E. Haibach: The Method of Maximum Likelihood Applied to the Statistical Analysis of Fatigue Data Including Run-Outs, International Journal of Fatigue 1 (1979), No. 2, pp. 81-88 DOI:10.1016/0142-1123(79)90012-4 Search in Google Scholar

23 W. Nelson: Applied Life Data Analysis, John Wiley, New York, USA (1982) Search in Google Scholar

24 W. Nelson: Fitting of fatigue curves with nonconstant standard deviation to data with runouts, Journal of Testing and Evaluation 12 (1984), No. 1, pp. 69-77 DOI:10.1520/JTE10700J Search in Google Scholar

25 F. Pascual, W. Meeker: Analysis of fatigue data with runouts based on a model with noncostant standard deviation and fatigue limit parameter, Journal of Testing and Evaluation 25 (1997), No. 3, pp. 292-301 DOI:10.1520/JTE11341J Search in Google Scholar

26 R. Pollak: Analysis of Methods for Determining High Cycle Fatigue Strength of a Material with Investigation of Ti-6Al-4 V Gigacycle Fatigue Behavior, PhD Thesis, Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, USA (2005) Search in Google Scholar

27 M. Matsumoto, T. Nishimura: Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator, ACM Transactions on Modeling and Computer Simulation 8 (1998), No. 1, pp. 3-30 DOI:10.1145/272991.272995 Search in Google Scholar

28 A. Martin, K. Hinkelmann, A. Esderts: Zur Auswertung von Schwingfestigkeitsversuchen im Zeitfestigkeitsbereich (For the evaluation of fatigue tests in high cycle fatigue regime), Materials Testing 53 (2011), No. 9, pp. 502-512 DOI:10.3139/120.110255 Search in Google Scholar

Published Online: 2021-08-18
Published in Print: 2021-08-31

© 2021 Walter de Gruyter GmbH, Berlin/Boston, Germany