Use and Applications of the Miner’s Rule

In a static load that does not change in time, a structurally safe object can often be easily found in the environment when it receives a dynamic load that fluctuates with time. This phenomenon is more conspicuous when the tensile load and the compressive load alternately act in the dynamic load. The phenomenon that an object subjected to an alternating load is destroyed after a specific load cycle is called fatigue failure.

Figure 1: Time Signal and SN-Curve Correlation [1]

The Miner’s rule is a way to calculate the fatigue damage of a material through a linear damage accumulation model. For example, following fig 1, It shows three different indices, showing time-stress, SN-Curve and damage tally. Two time cycles are applied to the time-stress graph continuously with $n_1$

( $n_1$

= 2) and $n_2$

( $n_2$

= 3). Continuous cycles indicate how many cycles ( $N_1$

= 6, $N_2$

= 4) can take before the material is destroyed in the SN-Curve in the middle. Finally, when calculating Damage tally, it can divide two times signal cycles into 6 cycles to failure in first damage cycles. Therefore, a material of 0.33 can represent the damage (33%) received [1]. At this time, when the total damage which are $D_1$

and $D_2$

, total 1.08, the material would be breaks. Miner’s rule assumes that the stress level of the repeated stresses causing the damage does not change and has the same level. In other words, the damage done by the first stress cycle is equal to the damage done by the last level. It assumes that the fraction of useful fatigue life used by the repeated stress cycles at a given stress is proportional to the total number of cycles of fatigue life. For example, applying 2 stresses at a stress level that can cause 6 cycles of failure in a component consumes 33% of the fatigue life. Repeated stresses at different stress levels consume different parts of a similar fatigue life. Parts can be expected to fail when 100% of the fatigue life is consumed this way. In Miner’s analysis, the order in which these individual stress cycles are applied is considered unimportant.

Equation 1. can be expressed as the above explanation. k is the stress level and i is the cycle of the failure stress. It can assess the stress proportion for cycle to failure and then adding the proportions for all the levels together.

$\sum _{i=1}^k\frac{n_i}{N_i}=C$

           (1)

According to this hypothesis, the rupture occurs when the sum of fractions of damage, defined only by the consumed cycles ( $n_i$

/ $N_1$

), at various load levels, reaches unity. It is synthesized by the simple relationship.

Application of Miner’s Rule

Miner’s rule can analysis gear drives cumulative fatigue damage. It is important factor in operation of equipment with various loading. When various manufacturers’ options are available, especially when the specifications are unknown, it can be difficult to confirm that the device is suitable for the application. However, if the amount and duration of the load is known to the application, the cumulative fatigue analysis, such as the Miner’s rule, can be used to accurately determine the life of the gear drive. For example, wind turbine meets many conditions for measuring cumulative fatigue damage due to unpredictable continuous oscillations [2]. According to Miner’s rule, load and acceleration tests on the shaft of the wind turbine can be used to analyse the fatigue failure phenomenon through the load and acceleration test of the gear of the machine [3]. Other example is railway vehicles. Structural integrity of railway vehicles should last for a long period against various and continuous fatigue loadings, and the body structures of railway vehicle are manufactured by applying multiform welding types for each material. The damage caused by the fatigue load of the railway car body occurs mostly on the basis of the welded part, and various fatigue strength evaluations should be made. Those phenomena are able to analysis cumulative fatigue damage and it is possible to become one of examples [4].

Limitations of Miner’s Rule and alternative method

Miner’s Rule does not take into account the sequencing or order in which in the cyclic loads are applied, however the order greatly affects fatigue life. In some specific situations, the high and low cycles of stress are possible to cause more damage than the damage tally predicted by Miner’s rule [5]. This hypothesis did not take into account the effect of delaying crack growth by inducing compressive residual stress at high stresses. However, the material could have less damage in high stress and low stress, when the residual stress is stayed in the material. Miner’s linear damage rule can be regarded as a relatively good evaluation method when the applied stress is mainly within elastic range of the material and the load range is wide, the stress above the fatigue limit is mostly.

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 This has sometimes been taken as proof that Miner’s rule is unsafe. However, that is not necessarily true, since one could reasonably expect the scatter in variable amplitude test results to mirror that found in constant amplitude results. In other words, if a specimen would have given a low life under constant amplitude loading, it seems reasonable to assume that it would also be expected to give a low life under variable amplitude loading. Furthermore, to a first approximation, it also seems reasonable to assume that the amount of scatter would be the same with the two different types of loading. The most obvious of these is when the loading involves short block lengths – in other words when the peak stress in the spectrum occurs fairly frequently. A typical example might be an overhead travelling crane working on a production process. Similarly, certain types of earth-moving plant, where the machine essentially goes through a continuous series of digging and unloading cycles, might also qualify. There are almost certainly many other examples. The evidence certainly seems to suggest that the ‘area rule1 would be an advance on Miner’s rule for dealing with this particular type of loading. The second major problem area seems to be the tendency for Z— to decrease as the applied stresses decrease. It is a trend which seems to be very evident under tensile loading when the individual stress ranges in the spectrum are applied in random order and when the peak stresses are fairly high. Much work is, however, still needed to confirm the extent to which the trend continues to lower stresses and also whether it also applies at other stress ratios and mean stresses.

1. SIEMENS, Calculating damage with Miner’s Rule, https://community.plm.automation.siemens.com/t5/Testing-Knowledge-Base/Calculating-damage-with-Miner-s-Rule/ta-p/355057 accessed on 17.4.11
2. D. McVittie, R. Errichello. “Application of Miner’s Rule to Industrial Gear Drives”. American Gear Manufacturers Association. Technical paper 88 FTM 9. 1988.
3. G. Shen, D. Xiang, K. Zhu, L. Jiang, Y. Shen, and Y. Li, “Fatigue failure mechanism of planetary gear train for wind turbine gearbox,” Eng. Fail. Anal., vol. 87, no. September 2017, pp. 96–110, 2018.
4. A. Aeran, S. C. Siriwardane, O. Mikkelsen, and I. Langen, “A new nonlinear fatigue damage model based only on S-N curve parameters,” Int. J. Fatigue, vol. 103, pp. 327–341, Oct. 2017.
5. H. Eskandari and H. S. Kim, “A Theory for Mathematical Framework and Fatigue Damage Function for the S-N Plane,” Fatigue Fract. Test Planning, Test Data Acquis. Anal., pp. 299–336, 2017.