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 i.

the midrange stress must be zero. Fluctuating simple loads It uses a criterion to relate midrange and alternating stresses (modified Goodman, Gerber, ASME-elliptic, or Soderberg). i.

Again, only one type of loading is allowed at a time. 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  

fs torsion

stresses.

 

axial

 

f bending

for the bending stresses, torsional stresses, and for the 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 (k c ? K f ?)

.

Assuming that all stress components are in time phase with each other.

1.

2.

3.

4.

5.

6.

For the strength, use the fully corrected endurance limit for bending, S e .

Apply the appropriate fatigue concentration factors to all stress components.

Multiply any alternating axial stress components by 1/k c,ax Find the principal stresses.

Find the von Miss alternating stress,



’ a and mean stress



’ m .

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

K f

and

K fs .



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: