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: