Fatigue strength is the stress level a material can endure for a specified number of cycles before it breaks. To calculate it with an S-N curve, you plot stress amplitude on the vertical axis against the number of cycles to failure on the horizontal axis, then read the stress value at your target cycle count. The curve turns that target into a number you can design against.
This method comes from materials engineering, not medicine. It matters to anyone following health news about “fatigue” in the medical sense only as a contrast. In medicine, fatigue is a symptom. In engineering, fatigue is a failure mode. The S-N curve belongs to the second meaning, and the math is well established.
What Is an S-N Curve and What Does It Show?
An S-N curve is a graph built from laboratory tests. Each test takes one specimen, applies a repeated stress cycle of known amplitude, and counts how many cycles it survives. The result is a single data point: stress S, cycles N.
Run enough of those tests across different stress levels and a pattern appears. High stress breaks the specimen in few cycles. Low stress lets it run for millions. Plot the points and you get a descending curve that flattens as it moves right.
The S stands for stress amplitude. The N stands for number of cycles to failure. The curve is also called a Wöhler curve, named after August Wöhler, the German engineer who ran systematic fatigue tests on railway axles in the 1850s and 1860s.
What the axes actually measure
- Vertical axis: stress amplitude, usually in megapascals (MPa) or kilopounds per square inch (ksi).
- Horizontal axis: cycles to failure, plotted on a logarithmic scale because the range spans from thousands to millions.
- Each point: one test specimen, one stress level, one failure count.
The logarithmic scale on the horizontal axis is not cosmetic. Cycle counts span several orders of magnitude, so a linear axis would compress most of the useful data into a sliver at the left edge.
How To Calculate Fatigue Strength With S N Curves Step by Step
The calculation itself is straightforward once you have a curve. The work is in getting a curve that applies to your material and your loading conditions.
Step one: define your target cycle count. This is the number of load repetitions the part must survive. A rotating shaft in a motor might see millions of cycles. A bracket that flexes once per day might see a few thousand over its service life.
Step two: locate that cycle count on the horizontal axis. If your target is 100,000 cycles, find 10⁵ on the log scale.
Step three: move vertically from that point until you hit the curve.
Step four: read horizontally to the vertical axis. That stress value is the fatigue strength at your target life.
The number you read is a stress amplitude, not a stress range. Confusing the two is a common error. Stress range is the difference between maximum and minimum stress. Stress amplitude is half that difference. If a part cycles between 20 MPa and 80 MPa, the range is 60 MPa and the amplitude is 30 MPa. S-N curves are typically plotted against amplitude, though some sources use range. Always check which one the curve uses.
That distinction matters more than most people realize. A curve plotted against range will give you a number twice as large as the same curve plotted against amplitude. Reading the wrong axis produces a design that is off by a factor of two.
What Is the Endurance Limit and Why Does It Matter?
For some materials, the S-N curve does not keep descending. It flattens into a horizontal line at a specific stress level. Below that line, the material appears to survive an unlimited number of cycles. That threshold is the endurance limit, also called the fatigue limit.
Steels commonly show this behavior. Many aluminum alloys do not. Their curves keep sloping downward, so there is no stress below which infinite life is guaranteed. For those materials, engineers pick a design life instead — often 10⁷ or 10⁸ cycles — and define fatigue strength at that point.
This is the single most important practical difference between material families in fatigue design. A steel part can be designed for infinite life against a fixed stress. An aluminum part cannot, at least not in the same way.
Some sources also note that the endurance limit is not perfectly flat for steel under all conditions. Very high cycle fatigue testing, beyond 10⁸ cycles, has shown failures below the traditional limit in some cases. The effect is real but the conditions are specific, and standard design practice still relies on the flat-line model for steels in most applications.
Where Do S-N Curves Come From?
No one calculates an S-N curve from first principles. They are measured. That is the honest answer, and it is the reason fatigue design depends so heavily on testing standards.
Two testing approaches dominate. In load-controlled testing, the machine applies a fixed stress amplitude and the specimen fails at some cycle count. In strain-controlled testing, the machine applies a fixed strain amplitude, which is more common when studying low-cycle fatigue where stresses are high enough to cause plastic deformation.
Standards bodies publish the procedures. ASTM International maintains standards for fatigue testing, and the International Organization for Standardization publishes its own. These documents specify specimen geometry, surface finish, loading frequency, and how to define failure — because a specimen that has cracked but not separated is a different failure point than one that has broken in two.
Published curves exist for common materials in handbooks and standards. Using a published curve is normal engineering practice. Generating a new one is expensive and slow, sometimes requiring weeks of machine time for a single material condition.
Why Real Parts Fail Sooner Than the Curve Predicts
A laboratory S-N curve describes a polished, carefully prepared specimen tested under controlled conditions. Real parts are not that. Several factors shift the curve, usually downward, meaning the part fails sooner than the baseline predicts.
- Surface finish. Rough surfaces create stress concentrations where cracks start. A machined surface performs differently from a forged or cast one.
- Stress concentration. Holes, notches, fillets, and sharp corners raise local stress above the nominal value.
- Size effect. Larger parts tend to have lower fatigue strength, partly because a larger volume of material increases the chance of a critical defect.
- Mean stress. A tensile mean stress reduces fatigue life. A compressive mean stress tends to extend it.
- Environment. Corrosion and elevated temperature both degrade fatigue performance.
These adjustments are applied through correction factors. The modified fatigue strength is the baseline curve value multiplied by those factors. The factors themselves come from established engineering references, not from guesswork, but they carry uncertainty. Design codes build in margins partly for that reason.
How S-N Curves Compare With Other Fatigue Methods
The S-N approach is one of several ways to assess fatigue. It is the oldest and still the most widely used for high-cycle applications where stresses stay in the elastic range.
| Method | Best suited for | Key limitation |
|---|---|---|
| Stress-life (S-N) | High-cycle fatigue, elastic stresses | Poor accuracy when plastic strain is significant |
| Strain-life (ε-N) | Low-cycle fatigue, plastic strains | Requires more complex testing |
| Fracture mechanics | Crack growth prediction from a known flaw | Needs an initial crack size assumption |
The dividing line between high-cycle and low-cycle fatigue is conventionally placed at roughly 10⁴ to 10⁵ cycles. Below that, plastic deformation matters and strain-life methods are more appropriate. Above it, stresses are mostly elastic and the S-N approach works well.
Fracture mechanics answers a different question. Instead of asking when a crack will start, it asks how fast an existing crack will grow. That is useful for inspection intervals and damage tolerance, not for initial design against crack initiation.
What Are the Limits of This Method?
S-N curves are statistical. The same specimen tested at the same stress will not always fail at the same cycle count. Scatter of a factor of two or more between identical tests is common and expected.
That scatter is why design codes do not use the mean curve. They use curves shifted to lower stress, often based on a statistical lower bound. The goal is a probability of failure low enough to be acceptable for the application, not a guarantee of survival.
The method also assumes the loading is constant amplitude. Real service loads vary. Rainflow counting and Miner’s rule are used to convert a variable load history into an equivalent constant-amplitude damage estimate, but that conversion introduces its own approximations. Miner’s rule in particular is known to be approximate and can be non-conservative in some load sequences.
None of this makes the S-N method unreliable. It makes it a model with known boundaries. Used within those boundaries, with appropriate correction factors and safety margins, it has supported decades of safe machine and structure design.
Frequently Asked Questions
What does the S stand for in an S-N curve?
S stands for stress amplitude, plotted on the vertical axis. N stands for the number of cycles to failure, plotted on the horizontal axis.
How do you find fatigue strength from an S-N curve?
Find your target cycle count on the horizontal axis, move up to the curve, then read across to the vertical axis. The stress value you land on is the fatigue strength at that life.
Do all materials have an endurance limit?
No. Many steels show a flat endurance limit below which cycles appear unlimited, but many aluminum alloys do not. For those materials, fatigue strength is defined at a chosen design life instead.
Why do real parts fail before the S-N curve predicts?
Laboratory curves come from polished specimens tested under ideal conditions. Surface finish, notches, part size, mean stress, and environment all shift real performance, usually downward, so correction factors must be applied.
