Palmgren-Miner equation derivation

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In summary, the equation used in ISO standard ISO 6336-6 helps to consolidate damage caused by multiple torque, cycle bins and calculates a single equivalent torque that produces the total damage for the given number of cycles. Its objective is to derive the equivalent torque formula given in the ISO standard, and this can be found through the use of Palmgren-Miner's rule.
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k.udhay
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TL;DR Summary
This equation is used in ISO standard ISO 6336-6. I want to understand how it is derived.
Summary: This equation is used in ISO standard ISO 6336-6. I want to understand how it is derived.

This equation helps to consolidate damage caused by multiple torque, cycle bins. A bin means a torque applied for a given number of cycle. As an outcome of the equation, a single equivalent torque that produces the total damage caused by multiple bins for the total number of cycles as original bins.

The equation is:
Equivalent torque from ISO standard.PNG


Pl. note that coordinates n1,T1; n2,T2 and n3,T3 do not need to be on the same line.

My objective is to derive the above equation. While I attempted to do that, I could progress to this level:

SN curve fundamental equation.PNG


Can someone help to proceed further to reach until the equivalent torque formula given in the ISO standard? Many thanks in advance!
 
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Hi, I was about to answer this and suggest it would be better in the Engineering topic here (as this is all about engineering approximation rather than scientific derivation), but while searching for refs I realized that you have posted this on another forum where it has been answered: https://www.eng-tips.com/viewthread.cfm?qid=460402
 
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Related to Palmgren-Miner equation derivation

1. What is the Palmgren-Miner equation?

The Palmgren-Miner equation is an empirical formula used to predict the fatigue life of a material subjected to repeated cyclic loading. It takes into account the stress amplitude, mean stress, and the number of cycles to failure.

2. How is the Palmgren-Miner equation derived?

The Palmgren-Miner equation is derived from the Miner's rule, which states that the total damage caused by a combination of stress cycles is equal to the sum of the individual damages caused by each stress cycle. This rule is then modified to account for the effects of mean stress and the material's S-N curve.

3. What are the assumptions made in the derivation of the Palmgren-Miner equation?

The Palmgren-Miner equation assumes that the material's fatigue behavior is linear and that the damage caused by each stress cycle is independent of the previous cycles. It also assumes that the material's S-N curve is valid for all stress amplitudes and mean stresses.

4. How accurate is the Palmgren-Miner equation?

The accuracy of the Palmgren-Miner equation depends on the assumptions made and the quality of the data used to determine the material's S-N curve. It is generally more accurate for materials with a well-defined S-N curve and when used within the range of stress amplitudes and mean stresses for which it was derived.

5. Can the Palmgren-Miner equation be used for all types of materials?

The Palmgren-Miner equation is commonly used for metallic materials, but it may not be applicable to other types of materials such as polymers or composites. It is important to use caution and verify its applicability when using it for non-metallic materials.

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