# Fatigue Life Under Alternating Stress

A part under alternating stress can result in failure from fatigue at stress levels below the yield strength of a material.  We can use the stress life approach to fatigue analysis to develop designs that guard against fatigue failure.

Machine elements that are subjected to cyclic loads that change over time with the same amplitude are under alternating stress.  Consider the stress vs. time plot shown below.

### Alternating Stress

The stress is cyclic (it repeats) and the maximum and minimum stress is the same for each cycle.  This is in contrast with spectral loading where maximum and minimum stress is changing with time and is possibly random.  Spectral loading requires special cycle counting techniques to account for changing stress amplitude and mean stress.  Refer to the Miner's rule article for spectral loading.

The stress amplitude is defined as:

amplitude = (max stress - min stress)/2

The mean stress is defined as:

mean stress = (max stress + min stress)/2

For the example above:

amplitude = (22 - 2)/2 = 10

mean stress = (22 +2)/2 =12

### S-N Diagram

With the stress life approach to fatigue analysis, we need an S-N diagram that describes a material's failure from fatigue for a given number of cycles.  Stress amplitude is plotted on the y-axis and cycles to failure is plotted on the x-axis.  An example S-N diagram is shown below.

An S-N diagram is created by testing many specimens of a material at various stress amplitudes and recording the cycles to failure.  It should be noted that when several specimens are tested at the same amplitude, they will all not fail at the exact same cycle.  Therefore, the line on an S-N diagram is an approximation.

### Fatigue Limit

Fatigue limit is the maximum stress level that the material can be subjected to for a very large number of cycles without failure.  The term endurance limit is often used for the stress that a material can withstand for an unlimited number of cycles.  Some materials, aluminum for example, do not have an endurance limit, and a fatigue limit is assumed for a large number of cycles, such as 107 or 108 cycles.  For this article, fatigue limit means both endurance limit and an artificial limit.

### Correcting for Non-Zero Mean Stress

Generally, an S-N diagram is generated with zero mean stress.  In this case, stress ratio, (min stress/max stress) = -1.  If our situation does not have fully reversed loading where mean stress is not zero, we can correct for mean stress to use the S-N diagram.

#### Algebraic Correction

One way to do this is to calculate an effective stress amplitude that is based on the combination of actual amplitude and mean stress.  There are several theories for mean stress correction.

Once effective stress is determined, if it is less than the fatigue limit, then it is OK.  If effective stress is greater than the fatigue limit,  we go into the S-N diagram and find the number of cycles to failure at the effective stress level.

#### Graphic Method

Another way to check the safety of fatigue resistance is to draw a chart using the theories described earlier.  Consider the Goodman diagram shown below.  The yield line is also drawn because yielding may be the mode of failure instead of fatigue.

For our example, the material properties are:

Ultimate tensile strength = 58

Yield strength = 36

Fatigue limit = 25

The area below the Goodman line is safe from fatigue.  The area below both the Goodman and yield lines are safe from fatigue and yielding.  The point is from the example where amplitude is 10 and mean stress is 12.  It lies in the OK zone.

MEboost can quickly create these charts to visually check if a combination of amplitude and mean stress is in the safe zone.  A diagram with all three lines plus the yield line was created with MEboost.

Each failure criterion is drawn according to:

Soderberg  A line from fatigue limit on the y-axis to the yield strength on the x-axis.  Any point to the left of the line is the safe region.  This is the most conservative model.

Goodman  A line from fatigue limit on the y-axis to the ultimate strength on the x-axis.  Any point to the left of both the Goodman line and the yield line is the safe region.

Gerber A parabolic line from fatigue limit on the y-axis to ultimate strength on the x-axis.  Any point to the left of both the Gerber line and the yield line is the safe region.  This is the least conservative model.

Yield  A line from yield strength on the y-axis to yield strength on the x-axis.  This is used in conjunction with the Goodman and Gerber lines when yielding is a criterion of failure.  For example, any point to the left of both the Goodman and yield lines could be considered safe, while a point between the Goodman and yield line is not safe because of yielding.

### Some Notes of Caution

While we have described the basic method of fatigue life under alternating stress, there are additional considerations.

• As mentioned earlier, the S-N curve is not exact.  It is a fit of scattered data and can lead to inexact results.
• We haven't mentioned modifying factors that affect fatigue limit.  Surface finish, stress concentatrations, etc. will lower a part's fatigue limit.  These factors must be taken into account to adjust the fatigue limit.