Linear damage accumulation

The linear damage accumulation is used to assess the influence of a load spectrum on the service life of a component and goes back to the engineers Arvid Palmgren (1924), B. F. Langer (1937) and Milton Miner (1945).

description

Normally, a component is not only subject to a vibration load with constant amplitudes, i. That is, a rectangular load spectrum , as used in the Wöhler experiment, for example , but the amount of the load is variable. To calculate the service life, the collective amplitude is subdivided (stepped) into individual rectangular collectives with constant amplitude and a number of partial cycles . According to the method of linear damage accumulation, a partial damage is now calculated for each subgroup by dividing the number of partial cycles by the maximum tolerable number of cycles for a Wöhler curve . The partial damages of all partial collectives are added up and result in the total damage D of the component. ${\ displaystyle S_ {a}}$ ${\ displaystyle n_ {i}}$ ${\ displaystyle N_ {i}}$ ${\ displaystyle S_ {a}}$

${\ displaystyle D = \ sum {\ frac {n_ {i}} {N_ {i}}}}$

If the damage exceeds the value 1, a break or crack in the component under the load collective considered is to be expected.

In clear terms, after the linear damage accumulation, it does not matter what load level a certain fraction of the maximum tolerable number of cycles was spent on. Damage to a part of the collective can be converted into a different part of the collective by ${\ displaystyle i}$ ${\ displaystyle j}$

{\ displaystyle {\ frac {n_ {i}} {N_ {i}}} = {\ frac {n_ {j}} {N_ {j}}}}

.

If one imagines a two-stage exposure, it does not matter in which order the loads come after the linear damage accumulation. Sequence effects cannot therefore be explained.

Modifications to the miner's rule

There are numerous modifications of the Miner's Rule that assess damage from vibrations below the so-called fatigue strength . The curve of the Wöhler curve to which the sub-collectives are compared is always modified.

The original miner rule is called the original miner and does not take into account any sub-collectives whose load amplitudes are below the fatigue limit. Designing components with this rule can lead to underdimensioning, since oscillation cycles below the so-called fatigue strength can also cause damage.

{\ displaystyle S_ {a}> S_ {aD}: N = N_ {D} \ cdot \ left ({\ frac {S_ {a}} {S_ {aD}}} \ right) ^ {- k}}

As a conservative variant, the elementary Miner rule according to Palmgren applies . A kink in the Wöhler line is completely neglected, so that all sub-collectives have a damaging effect.

{\ displaystyle S_ {a} \ leq \ geq S_ {aD}: N = N_ {D} \ cdot \ left ({\ frac {S_ {a}} {S_ {aD}}} \ right) ^ {- k }}

Another important modification is the Miner's rule, modified according to Haibach . A decrease in fatigue strength due to a changed incline is taken into account: ${\ displaystyle k * = 2k-1}$

{\ displaystyle S_ {a} \ leq S_ {aD}: N = N_ {D} \ cdot \ left ({\ frac {S_ {a}} {S_ {aD}}} \ right) ^ {- (2k- 1)}}

,

{\ displaystyle S_ {a}> S_ {aD}: N = N_ {D} \ cdot \ left ({\ frac {S_ {a}} {S_ {aD}}} \ right) ^ {- k}}

J. Liu and H. Zenner ( Miner's rule modified from Liu-Zenner ) turned the Wöhler line at the level of the collective maximum value and then continued with the gradient:

{\ displaystyle k ^ {*} = {\ frac {k + m} {2}}}

suggested. The inclination of the crack propagation Wöhler line "m" is added as an additional influencing factor. Furthermore, the beginning of the fatigue limit range is characterized by:

{\ displaystyle S_ {aD} ^ {*} = {\ frac {S_ {aD}} {2}}}

Individual evidence

↑ A. Palmgren: The life of ball bearings . In: Journal of the Association of German Engineers . tape 68 , no. 14 , 1924, pp. 339-341 .
^ BF Langer: Fatigue failure from stress cycles of varying amplitude . In: Journal of Applied Mechanics . tape 59 , 1937, pp. A160-A162 .
^ MA Miner: Cumulative damage in fatigue . In: Journal of applied mechanics . tape 12 , no. 3 , 1945, p. 159-164 .
↑ Ngoc Linh Tran: Calculation model for the simplified estimation of the fatigue behavior of spring plates in prefabricated girder bridges. Dissertation at the Technical University of Darmstadt, Darmstadt 2011, pp. 35–37. Online (accessed September 21, 2018).

Web links

Qualitative and quantitative analyzes of the linear and non-linear damage accumulation hypotheses including the statistical test planning (accessed on September 21, 2018)
Investigations into the validity of Miner's hypothesis (accessed September 21, 2018)
Lifetime prediction of multi-axis loaded elastomer components with special consideration of rotating directions of loading (accessed on September 21, 2018)
Test limits for water pipes made from PE pipes (accessed on September 21, 2018)
Damage accumulation according to Miner (accessed September 21, 2018)

[Palmgren-1] A. Palmgren: The life of ball bearings . In: Journal of the Association of German Engineers . tape 68 , no. 14 , 1924, pp. 339-341 .

[Langer-2] BF Langer: Fatigue failure from stress cycles of varying amplitude . In: Journal of Applied Mechanics . tape 59 , 1937, pp. A160-A162 .

[Miner-3] MA Miner: Cumulative damage in fatigue . In: Journal of applied mechanics . tape 12 , no. 3 , 1945, p. 159-164 .

[4] Ngoc Linh Tran: Calculation model for the simplified estimation of the fatigue behavior of spring plates in prefabricated girder bridges. Dissertation at the Technical University of Darmstadt, Darmstadt 2011, pp. 35–37. Online (accessed September 21, 2018).