The study of material fatigue analysis has become increasingly important as the lifespan of durable goods continues to increase. One of the earliest areas of fatigue analysis took place in the 1920s and 30s in the design and manufacture of aircraft engines. Lacking any other means, engines were mounted on towers, started, and run until something failed. Early engine designs very conservative, that is the amount of horsepower per cubic produced was very low. Early designs were on the order of ½ hp per cubic inch, or less. As experience increased, this changed to the point where well over 1 hp per cubic inch could be extracted reliably.
Specific materials have their own fatigue property peculiarities. For example, aluminum, regardless of how lightly loaded, will eventually result in a fatigue failure. Television documentaries have shown what looked to be complete and undamaged aircraft being flown to scrap yards, where all useful hardware is removed, and the aircraft structure is chopped to pieces. Most of the aircraft structure, being aluminum, is constantly being loaded and unloaded during each take off, flight, and landing cycle. Each airframe can tolerate only so many cycles of this nature.
An early example of catastrophic fatigue failure occurred in the 1950s on an aircraft known as the deHavilland Comet. This was a British jet airliner design with one of the earliest pressurized fuselage designs. Several aircraft broke apart in midflight before the culprit was finally found – cracks emanating from rivet holes in the outside fuselage skin. As it turns out, the rivet holes were punched rather than drilled, which produced fine cracks around each hole. Each pressurization cycle, turbulence load, landing or takeoff put stress on the holes until the hairline cracks expanded to the point that the fuselage burst.
Steel is a bit of a different animal altogether regarding fatigue, depending on the alloy. The rules of thumb regarding fatigue stress would be to take from 1/4 to 3/4 of the yield stress to determine the infinite life fatigue limit.
A recent catastrophic failure occurred in Minneapolis, Minnesota on a bridge on an interstate freeway crossing the Mississippi River. Constructed in the late 1960s, the bridge load was gradually increased by adding lanes. Bridges are typically constructed of fairly ordinary steel alloys, in which you would use value of one fourth of the material yield stress to estimate its fatigue stress, or less. The parts that failed were the gusset plates that connected the large main support beams together. Failure occurred suddenly, and was very deadly. Such is the nature of fatigue failure which usually gives very little – if any – warning. Chemical corrosion on bridges due to salt exposure is another issue entirely. It not only attacks the steel but the concrete as well.
Fatigue Analysis Methods
These are examples of High Cycle Fatigue Analysis methods.
Miner’s Rule
Miner’s rule is probably the simplest cumulative damage model. It states that if there are k different stress levels (with linear damage hypothesis) and the average number of cycles to failure at the ith stress, Si, is Ni, then the damage fraction, C, is:
where:
– ni is the number of cycles accumulated at stress Si.
– C is the fraction of life consumed by exposure to the cycles at the different stress levels.
In general, when the damage fraction reaches 1, failure occurs. The above equation can be thought of as assessing the proportion of life consumed at each stress level and then adding the proportions for all the levels together. Often an index for quantifying the damage is defined as the product of stress and the number of cycles operated under this stress, which is:
Assuming that the critical damage is the same across all the stress levels, then:
For example, let’s say WFailure=50 for a component. So the component will fail after 10 cycles at a stress level of 5, or after 25 cycles to fail at a stress level of 2, and so on. Using Eqn. (2) as the critical value of damage that will result in failure, Eqn. (1) becomes:
C represents the proportion of the cumulative damage to the critical value.
Example:
A part is subjected to a fatigue environment where 10% of its life is spent at an alternating stress level, σ1, 30% is spent at a level σ2, and 60% at a level σ3. How many cycles, n, can the part undergo before failure? If, from the S-N diagram for this material the number of cycles to failure at σi is Ni (i =1,2,3), then from the Palmgren-Miner rule failure occurs when:
Above is shown the solution for n cycles given.
Note:
1. “High-low” fatigue tests where testing occurs sequentially at two stress levels (σ1 ,σ2) where σ1 >σ2 generally shows that failure occurs when,
where C normally is < 1, i.e the Palmgren-Miner rule is non-conservative for these tests. For “low-high” tests, c values are typically > 1.
2. For tests with random loading histories at several stress levels, correlation with the Palmgren-Miner rule is generally very good.
3. The Palmgren-Miner rule can be interpreted graphically as a “shift” of the S-N curve. For example, if N1 cycles are applied at stress level σ1 (where the life is N1 cycles), the S-N curve is shifted so that goes through a new life value, N’1.
Limitations
A major limitation of the Palmgren-Miner rule is that it does not consider sequence effects, i.e. the order of the loading makes no difference in this rule. Sequence effects are definitely observed in many cases. A second limitation is that the Palmgren-Miner rule says that the damage accumulation is independent of stress level. This can be seen from the modified S-N diagram above where the entire curve is shifted the same amount, regardless of stress amplitude.
Goodman Relation
In materials science and fatigue, the Goodman relation is an equation used to quantify the interaction of mean and alternating stresses on the fatigue life of a material.
A Goodman diagram, sometimes called a Haigh diagram[3] or a Haigh-Soderberg diagram,[4] is a graph of (linear) mean stress vs. (linear) alternating stress, showing when the material fails at some given number of cycles.
A scatterplot of experimental data shown on such a plot can often be approximated by a parabola known as the Gerber line, which can in turn be (conservatively) approximated by a straight line called the Goodman line. [4][5]
Mathematical Description
The area below the curve indicates that the material should not fail given the stresses. The area above the curve represents likely failure of the material.
The Goodman relation can be represented mathematically as:
Where 
is the alternating stress, 
is the mean stress, 
is the fatigue limit for completely reversed loading, and 
is the ultimate tensile stress of the material. The general trend given by the Goodman relation is one of decreasing fatigue life with increasing mean stress for a given level of applied stress. The relation can be plotted to determine the safe cyclic loading of a part; if the coordinate given by the mean stress and the applied stress lies under the curve given by the relation, then the part will survive. If the coordinate is above the curve, then the part will fail for the given stress parameters.[6]
Low cycle Fatigue – Coffin-Manson Relation
Where the stress is high enough for plastic deformation to occur, the accounting of the loading in terms of stress is less useful and the strain in the material offers a simpler and more accurate description. Low-cycle fatigue is usually characterised by the Coffin-Manson relation (published independently by L. F. Coffin in 1954 and S. S. Manson 1953):
where,
- Δεp /2 is the plastic strain amplitude;
- εf‘ is an empirical constant known as the fatigue ductility coefficient, the failure strain for a single reversal;
- 2N is the number of reversals to failure (N cycles);
- c is an empirical constant known as the fatigue ductility exponent, commonly ranging from -0.5 to -0.7 for metals in time independent fatigue. Slopes can be considerably steeper in the presence of creep or environmental interactions.
A similar relationship for materials such as Zirconium, is used in the nuclear industry.
Norman T. Neher, P.E.
Analytical Engineering Services, Inc.
Elko New Market, MN
www.aesmn.org