Fatigue Failure TheoriesDesign of Machine Elements
© Dr Moudar Zgoul, @zgoul_ju, 2011 [extracted from different sources]
Failure Theories
Fatigue Failure Theories
Criteria Equations
Combining Loading ModesFatigue 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.
ii. 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.
iii. 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 for the bending
stresses, for the torsional stresses, and
for the axial stresses.
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.
f bendingK
fs torsionK
f axialK
Combining Loading Modes
The equivalent von Mises stress for each of these two stress
elements:
Combining Loading ModesCase 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: