Fatigue Failure Theories
Design of Machine Elements
© Dr Moudar Zgoul, @zgoul_ju, 2011 [extracted from different sources]
Failure Theories
Fatigue Failure Theories
Criteria Equations
Combining Loading Modes
Fatigue problems are classified under three categories:
i.
Completely reversing simple loads
It is handled 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.
i.
Fluctuating simple loads
It uses 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.
i.
Combinations of loading modes
It uses combined bending, torsion, and axial loadings.
Combining Loading Modes
• Completely reversed single stress
which is handled 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.
• Fluctuating loads
It uses 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.
• Combination of different types of loading
such as combined bending, torsion, and axial.
Combining Loading Modes
• Earlier, a load factor was used to obtain the endurance limit, and
hence the result is dependent on whether the loading is axial,
bending, or torsion.
• But, “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
determining the endurance limit depend on the type of loading.
Combining Loading Modes
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 multiple
stresses on a stress element into a single equivalent von Mises stress.
The same approach will be used here.
Combining Loading Modes
1) The first step is to generate two stress elements, one for the
alternating stresses and one for the midrange stresses.
2) Apply the appropriate fatigue stress concentration factors to each of
 
the stresses; apply K f
for the  K fs torsion
stresses.  K f

bending
for the bending stresses,
torsional stresses, and
for the axial
axial
3. Next, calculate an equivalent von Mises stress for each of these two
stress elements,
,
4. Finally, select a fatigue failure criterion (modified Goodman, Gerber,
ASME-elliptic, or Soderberg) to complete the fatigue analysis.
Combining Loading Modes
The equivalent von Mises stress for each of these two stress elements:
Combining Loading Modes
Case of Combined Axial, Bending and Torsion Loading
(kc? Kf?).
Assuming that all stress components are in time phase with
each other.
1. For the strength, use the fully corrected endurance limit for
bending, Se.
2. Apply the appropriate fatigue concentration factors to all
stress components.
3. Multiply any alternating axial stress components by 1/kc,ax
4. Find the principal stresses.
5. Find the von Miss alternating stress, ’a and mean stress ’m.
6. Use any of the theories above to compute the safety factor.
Combining Loading Modes
’a and mean stress ’m are alternating and mean VM stresses.
Both the steady and alternating components are augmented by Kf
and Kfs.

If stress components are not in phase but have same
frequency, the maxima can be found using phase angles and
then summed.

Otherwise assume that the stress components will reach an
in-phase condition so their magnitudes are additive.
Combining Loading Modes
Example: