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# Calculating damage with Miner’s Rule

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According to Miner's Rule, when damage is equal to “1”, failure occurs. The definition of failure for a physical part varies. It could mean that a crack has initiated on the surface of the part. It could also mean that a crack has gone completely thru the part, separating it.  In this article, a conservative approach to failure will be used: a crack starts to appear on the surface of the part.

Applying a constant amplitude, cyclical stress to a metal coupon causes it to fail (a crack appears) after a specific number of cycles. Stress is defined as Force (F) divided by Area (A) or F/A.

By repeating this test at different stress levels, one could develop a material SN-curve.  For example the SN-curve* may look like this:

SN-Curve: Cyclic Stress Level versus Number of Cycles to Failure

SN-Curves are developed with testing machines that apply constant amplitude loads.  They can be axial loads, torsional loads, bending loads, etc.  Different stress levels are tested and the number of cycles to failure are recorded.

Axial fatigue test machine for material coupons

Damage Accumulation

When a physical part undergoes stress cycles**, Miner’s Rule works like this:

On the left graph, there is a loading time history.  The SN-Curve is the middle graph, and a damage tally is kept on the right side.

In this case, two cycles at a specific amplitude are applied to the part. At this amplitude the part could take 6 cycles before it would fail. Dividing two cycles by six cycles, the accumulated damage is 0.33.  A third of the life of the part has been used.

Two more cycles of a higher amplitude are applied.  At this higher amplitude, four cycles would be required for failure to occur.  Dividing two cycles by four cycles, an additional 0.5 of damage has occured.  The total accumulated damage is now 0.83.  According to Miner’s Rule, no failure has occurred.

One more cycle of the higher amplitude is now applied.  The accumulated damage is now 1.08.  Failure has occurred!

History

In 1945, M. A. Miner popularized a rule that had first been proposed by A. Palmgren in 1924. The rule is variously called Miner's rule or the Palmgren-Miner linear damage hypothesis.

Limitations of Miner's Rule

Miner's Rule does not take into account the sequencing or order in which in the cyclic loads are applied.  For example, if loads are applied in the plastic region, the endurance limit is no longer in effect.

Other Notes

*Note: Real SN curves for metals are log-log curves and easily range into millions of cycles.

**Note: The loads applied to product in the real world are usually not constant amplitude cycles.  To break down a real world load history into cycles, a cycle counting method called 'Rainflow Analysis' is used.

Questions? Email john.hiatt@siemens.com

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Phenom

Has anyone ever compared the Steinberg’s vs. the Narrow Band (NB) method to calculate fatifue damage?

By plotting the Ratio ‘Damage NB’/’Damage ST’ for a range of inverse slope if seems that the ST method is pessimistic (higher damage) up to b ~7.55 . Does this make sense? I was lead to believe that Steinberg method is always pessimistic.

Thanks

Regards

Siemens Genius

Dear

eventhough the latter comment is a little misplaced here in this section:

Steinberg assumes that every zero crossing impies a cycle and distiguishes only 3 types of cycles. It is therefore even more conservative than a narrow band solution, which also typically leads to an overestimate of large cycles. Fatigue methods than "punish" this behavior with factos in the damage.

There are many other methods to estimate the amplitude cycle distribution in a more realistic way (let me here only name Dirlik).

So yes your observation is due to the basic methodology

BR

Michael

Phenom

Thanks for the reply Michael.

Consider the basic 2 equations for each method giving total damage and plot the ratio for a range of b-values and you'll see that the Steinberg's method is only more conservative up to b ~ 7.55. This is what I am trying to get to the bottom of. So a blank statement does not hold! I must have something wrong somewhere!

Thanks

Regards

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