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Fatigue Failure Resulting from Variable Loading 317
6–13 Torsional Fatigue Strength
under Fluctuating Stresses

23
Extensive tests by Smith provide some very interesting results on pulsating torsional
fatigue. Smith’s first result, based on 72 tests, shows that the existence of a torsional
steady-stress component not more than the torsional yield strength has no effect on
the torsional endurance limit, provided the material is ductile, polished, notch-free, and
cylindrical.
Smith’s second result applies to materials with stress concentration, notches, or
surface imperfections. In this case, he finds that the torsional fatigue limit decreases
monotonically with torsional steady stress. Since the great majority of parts will have
surfaces that are less than perfect, this result indicates Gerber, ASME-elliptic, and other
approximations are useful. Joerres of Associated Spring-Barnes Group, confirms
Smith’s results and recommends the use of the modified Goodman relation for pulsat-
ing torsion. In constructing the Goodman diagram, Joerres uses
(6–54)
S su = 0.67S ut
Also, from Chap. 5, S sy = 0.577S yt from distortion-energy theory, and the mean load
factor k c is given by Eq. (6–26), or 0.577. This is discussed further in Chap. 10.

6–14 Combinations of Loading Modes
It may be helpful to think of fatigue problems as being in three categories:

• Completely reversing simple loads
• Fluctuating simple loads
• Combinations of loading modes

The simplest category is that of a completely reversed single stress which is han-
dled with the S-N diagram, relating the alternating stress to a life. Only one type of
loading is allowed here, and the midrange stress must be zero. The next category incor-
porates general fluctuating loads, using a criterion to relate midrange and alternating
stresses (modified Goodman, Gerber, ASME-elliptic, or Soderberg). Again, only one
type of loading is allowed at a time. The third category, which we will develop in this
section, involves cases where there are combinations of different types of loading, such
as combined bending, torsion, and axial.
In Sec. 6–9 we learned that a load factor k c is used to obtain the endurance limit,
and hence the result is dependent on whether the loading is axial, bending, or torsion.
In this section we want to answer the question, “How do we proceed when the loading
is a mixture of, say, axial, bending, and torsional loads?” This type of loading introduces
a few complications in that there may now exist combined normal and shear stresses,
each with alternating and midrange values, and several of the factors used in determin-
ing the endurance limit depend on the type of loading. There may also be multiple
stress-concentration factors, one for each mode of loading. The problem of how to deal
with combined stresses was encountered when developing static failure theories. The
distortion energy failure theory proved to be a satisfactory method of combining the

23 James O. Smith, “The Effect of Range of Stress on the Fatigue Strength of Metals,” Univ. of Ill. Eng. Exp.
Sta. Bull. 334, 1942.