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330 Mechanical Engineering Design
In this way everyone who is party to the communication knows what a design factor
(or factor of safety) of 2 means and adjusts, if necessary, the judgmental perspective.

6–17 Stochastic Analysis 28
As already demonstrated in this chapter, there are a great many factors to consider in
a fatigue analysis, much more so than in a static analysis. So far, each factor has been
treated in a deterministic manner, and if not obvious, these factors are subject to vari-
ability and control the overall reliability of the results. When reliability is important,
then fatigue testing must certainly be undertaken. There is no other way. Consequently,
the methods of stochastic analysis presented here and in other sections of this book
constitute guidelines that enable the designer to obtain a good understanding of the
various issues involved and help in the development of a safe and reliable design.
In this section, key stochastic modifications to the deterministic features and equa-
tions described in earlier sections are provided in the same order of presentation.
Endurance Limit
To begin, a method for estimating endurance limits, the tensile strength correlation
¯
method, is presented. The ratio = S /S ut is called the fatigue ratio. 29 For ferrous
e
metals, most of which exhibit an endurance limit, the endurance limit is used as a
numerator. For materials that do not show an endurance limit, an endurance strength at
a specified number of cycles to failure is used and noted. Gough 30 reported the sto-
chastic nature of the fatigue ratio for several classes of metals, and this is shown in
Fig. 6–36. The first item to note is that the coefficient of variation is of the order 0.10
to 0.15, and the distribution varies for classes of metals. The second item to note is that
Gough’s data include materials of no interest to engineers. In the absence of testing,
engineers use the correlation that represents to estimate the endurance limit S from

e
¯
the mean ultimate strength S ut .
Gough’s data are for ensembles of metals, some chosen for metallurgical interest,
and include materials that are not commonly selected for machine parts. Mischke 31
analyzed data for 133 common steels and treatments in varying diameters in rotating
32
bending, and the result was
= 0.445d −0.107 LN(1, 0.138)
where d is the specimen diameter in inches and LN(1, 0.138) is a unit lognormal vari-
ate with a mean of 1 and a standard deviation (and coefficient of variation) of 0.138. For
the standard R. R. Moore specimen,

0.30 = 0.445(0.30) −0.107 LN(1, 0.138) = 0.506LN(1, 0.138)

28 Review Chap. 20 before reading this section.
29 From this point, since we will be dealing with statistical distributions in terms of means, standard
deviations, etc. A key quantity, the ultimate strength, will here be presented by its mean value, ¯ S ut . This
means that certain terms that were defined earlier in terms of the minimum value of S ut will change slightly.
30 In J. A. Pope, Metal Fatigue, Chapman and Hall, London, 1959.
31 Charles R. Mischke, “Prediction of Stochastic Endurance Strength,” Trans. ASME, Journal of Vibration,
Acoustics, Stress, and Reliability in Design, vol. 109, no. 1, January 1987, pp. 113–122.
32 Data from H. J. Grover, S. A. Gordon, and L. R. Jackson, Fatigue of Metals and Structures, Bureau of
Naval Weapons, Document NAVWEPS 00-2500435, 1960.