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Statistical evaluation of fatigue tests using maximum likelihood

Klaus Störzel and Jörg Baumgartner
From the journal Materials Testing


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


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.





scatter of a S-N curve


cumulative density function


DIN 50100


load level for fatigue tests


load level of the first staircase test


high cycle fatigue regime (N<Nk)


highest load level for fatigue tests


lowest load level for fatigue tests


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


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


long life fatigue regime (N3Nk)


load level method


maximum likelihood


number of cycles


number of test results


number of test results where failure at Ni < Nk occurred


number of test results where failure at Ni ≥ Nk occurred


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


limit number of cycles (end of test)


number of levels for fatigue tests


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


probability density function


probability of occurrence

P s

probability of survival


pearl string method


random number


load or stress


standard deviation


staircase method


standard deviation of the logarithmic load in the S-direction


standard deviation of the logarithmic load in the N-direction


support function


integration variable of the distribution function of the normal distribution





reference point




individual test result, counting parameter


knee point




number of cycles


at the position of the limit number of cycles NG




test result as a runout


load or stress






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

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