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The staircase method is an efficient method for estimating fatigue thresholds below which no failure will occur. In this article it is explained how such a test is done and how test data are analyzed.

## List of Symbols

Symbol |
Unit |
Description |

ASTM | American Society for Testing and Materials | |

A | Calculation factor | |

B | Calculation factor | |

F | Failure or Number of failures | |

h_{i} |
Number of failures (if F≤P) or passes [1] (if F>P) at step index i | |

i | Step index, i=0 is given to the lowest test value. | |

N | Number of tests | |

P | Pass or Number of passes (or runouts) | |

s | Standard deviation of the sample | |

S_{0} |
Note [2] | Lowest test value (i=0) |

ΔS | Note 2 | Interval between test levels |

x | Mean of the sample. |

Notes:

[1] In case of a fatigue test, no failure prior to the preassigned number of cycles. Special attention should be given to that number of cycles if the staircase test is intended to determine the fatigue limit. The number of cycles depends on the material to be tested.

[2] Depends on the type of test, e.g. MPa for fatigue tests.

## Performing a Staircase Test

The first test is performed at a level which is equal to the estimated mean value of the property to be determined. If failure (In case of a fatigue test, failure prior to the preassigned number of cycles) occurs, the next specimen is tested at a lower level; if the specimen does not fail, the next test is done at a higher level.

The intervals ΔS between the test levels should be approximately equal to the standard deviation, but this is not a strict condition. However, the interval ΔS should not be larger than twice the standard deviation. The interval ΔS should be kept constant throughout the complete test series. Detailed information on ΔS can be found in [1].

## Analysis of Test Data

Dixon & Mood [1] developed both the test method and an analysis method. The disadvantage of the analysis method by Dixon & Mood is that less than half of the test data (only failures or only passes, whichever is least) is used for the statistical analysis. Several alternatives are developed that use all data. The method described by Deubelbeiss [4] is widely accepted as the best alternative.

Both methods give approximately the same mean value, the Deubelbeiss method gives a better (though larger) approximation of the standard deviation.

### Dixon & Mood

The mean μ and the standard deviation σ are estimated by only the failures or only the runouts/passes, whichever has the smaller total. The test values S which are equally spaced with an interval ΔS are given with step indices i, where i=0 is given to the lowest test value S_{0}. At each level i the number n_{i} of failures or runouts/passes are recorded.

In case the failures are counted the estimate of the mean is:

In case the runouts/passes are counted the estimate of the mean is:

The estimate for the standard deviation is:

Parameters A and B are calculated by:

The estimate of the standard deviation is accurate when:

In principle, all data can be used for the analysis, with possible exception of the so-called first slice, see figure below:

Analysis is performed according to following steps:

- Sorting all test values (failures
__and__passes) in descending order. - Assignment of probability of test values for each test result, e.g. using Benard’s median rank:

- Fitting of the cumulative probability plot (P
_{i}test values), i.e. determination of mean m of test values and the standard deviation s, assuming a normal distribution.

Since no distinction is made between failure and pass, it is possible to add a fictitious result to the results. After N tests, the test value (i.e. not the result) for the (not performed) N+1^{th} test is fixed and therefore usable in the analysis.

The first slice is caused by a mismatch between first estimation of the mean and the actual mean (i.e. mismatch between first test level and actual mean) and consists of test levels outside the staircase range and applied in the earliest stage of testing. The Dixon & Mood method already accounts for this mismatch by using only failures or only the runouts/passes, whichever has the smallest total. When using the Deubelbeiss method, the first slice must be filtered by the user.

## Restrictions

The staircase method gives reliable results under the condition that:

- the distribution is normal and,
- the interval between the stress levels is approximately equal to the standard deviation of the distribution.

Note that the standard deviation estimated by doing staircase tests are significantly underestimated. This is caused by the situation that all data points are very close to the mean value.

It is strongly emphasized that the standard deviation (i.e. scatter) only applies to the test series, i.e. the material data in combination with the surface condition applied in the test specimens. Scatter in fatigue life in an actual structure has many more sources of scatter than just material scatter. It is therefore advised not to use scatter determined from tests for further analysis.

## References

[1] W.J. Dixon, A.M. Mood, A method for obtaining and analyzing sensitivity data, J. Amer. Statist. Ass. 43, pp. 109-126, 1948.

[2] W. Weibull, Fatigue testing and analysis of results, Pergamon Press, 1961.

[3] Standard Test Method for Impact Resistance of Flat, Rigid Plastic Specimens by Means of a Falling Dart (Tup or Falling Mass), ASTM D 5628-96 (Reapproved 2001).

[4] E. Deubelbeiss, Dauerfestigkeitsversuche mit einem modifizierten Treppenstufenverfahren. Materialprüfung 16 (1974) 8, pp. 240-244.

[5] W.J. Dixon, F.J. Massey Jr., Introduction to statistical analysis, New York, 1951.

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