Understanding Material Fatigue Analysis: History, Methods, and Real-World Failures

Durable goods and critical infrastructure rely heavily on fatigue analysis. By understanding how and when materials fail, engineers can design safer, longer-lasting products. From early aircraft engines in the 1920s to modern bridges and airplanes, fatigue analysis has shaped how we approach reliability.

A Brief History of Fatigue Analysis

Fatigue analysis began in the 1920s and 1930s with aircraft engine testing. Technicians mounted engines on test towers and ran them until failure.

• Early designs: conservative, producing ~0.5 horsepower per cubic inch.

• Improved designs: reliable output of 1 horsepower per cubic inch and beyond.

Different materials soon revealed different fatigue behaviors:

• Aluminum → no infinite fatigue life. Even light loads eventually cause failure.

• Steel → depending on alloy, can achieve an infinite fatigue limit at ¼–¾ of its yield stress.

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Real-World Fatigue Failures

The de Havilland Comet (1950s)

The first British jet airliner suffered catastrophic fatigue failures. Cracks formed around punched rivet holes in the fuselage. Each pressurization cycle and turbulence event worsened the cracks until several aircraft broke apart mid-flight.

Minneapolis I-35W Bridge Collapse (2007)

Built in the late 1960s, the bridge carried more load than originally designed. Fatigue, combined with corrosion and undersized gusset plates, caused a sudden collapse over the Mississippi River. This tragedy showed that fatigue failures often occur without warning.

Fatigue Analysis Methods

Here are examples of High Cycle Fatigue Analysis methods.

Miner’s Rule

Miner’s rule is a simple cumulative damage model. If there are k different stress levels, and the average cycles to failure at the ith stress, Si, is Ni, then the damage fraction, C, is:

C = Σ(ni/Ni)

where:

• ni is the number of cycles at stress Si.

• C is the fraction of life consumed at different stress levels.

When C reaches 1, failure occurs. This equation determines how much life each stress level uses. It adds the parts together. Often, we can quantify damage as the product of stress and cycles under that stress.

C = Σ(ni/ Wi)

Assuming critical damage is the same across all stress levels:

For instance, if WFailure = 50 for a component, it will fail after 10 cycles at stress level 5 or 25 cycles at stress level 2. Using the critical value of damage, Eqn. (1) becomes:

C shows the cumulative damage proportion to the critical value.

Example:

If a part spends 10% of its life at alternating stress level σ1, 30% at σ2, and 60% at σ3, how many cycles, n, can it undergo before failure? From the S-N diagram, if we know Ni for each stress, we observe that failure occurs when:

C = (n1/N1) + (n2/N2) + (n3/N3) = 1

Note:

1. For “high-low” fatigue tests, where testing occurs sequentially at two stress levels (σ1, σ2), failure happens when C is usually < 1. For “low-high” tests, ***C*** values tend to be > 1.

2. Random loading histories correlate well with the Palmgren-Miner rule.

3. The Palmgren-Miner rule can graphically shift the S-N curve. If N1 cycles are applied at stress level σ1, the S-N curve shifts to a new life value, N’1.

Limitations

The Palmgren-Miner rule has a key limitation: it ignores sequence effects. This means the order of loading doesn’t matter. Many cases observe sequence effects. Another limitation is that it treats damage accumulation as independent of stress level. The modified S-N diagram shows that the entire curve shifts in a uniform manner, regardless of the stress amplitude.

Goodman Relation

The Goodman relation in materials science and fatigue analysis shows how mean and alternating stresses affect a material’s fatigue life.

A Goodman diagram, or Haigh diagram, shows mean stress versus alternating stress. It highlights failure points across cycles. Experimental data on these plots often creates a parabola called the Gerber line. This can be roughly estimated using a straight line known as the Goodman line.

Mathematically, the Goodman relation can be represented as:

σa / σf’ + σm / σut = 1

where:

• σa is the alternating stress,

• σm is the mean stress,

• σf’ is the fatigue limit for completely reversed loading,

• σut is the ultimate tensile stress of the material.

As mean stress goes up, fatigue life goes down for a given applied stress. Plotting this relation shows safe cyclic loading. If the mean and applied stress points are below the curve, the part will last. If they are above, it will fail under those stress conditions.

Low Cycle Fatigue – Coffin-Manson Relation

High stress that leads to plastic deformation makes using stress for loading assessment less effective. The material’s strain offers a simpler, more accurate description. Low-cycle fatigue is often characterized by the Coffin-Manson relation:

Δεp/2 = εf’ (2N)^c

where:

• Δεp/2 is the plastic strain amplitude,

• εf’ is the fatigue ductility coefficient (the failure strain for one reversal),

• 2N is the number of reversals to failure (N cycles),

• c is the fatigue ductility exponent, usually ranging from -0.5 to -0.7 for metals in time-independent fatigue. In cases with creep or environmental interactions, slopes can be steeper.

Why Fatigue Analysis Matters

• Safety → Prevents catastrophic failures in aircraft, bridges, and vehicles.

• Durability → Extends product life and reduces maintenance costs.

• Innovation → Enables lighter, stronger, and more efficient designs.

Fatigue failures may be invisible until the very last moment. That’s why robust analysis — from Miner’s Rule to Coffin-Manson — remains central to engineering practice today.

Norman T.  Neher, P.E.
Analytical Engineering Services, Inc.
Elko New Market, MN
www.aesmn.org