How this instrument works
Stress concentration factor is the ratio of the actual peak stress right at a geometric discontinuity to the nominal stress the same section would carry if that discontinuity were not there: Kt = σmax ⁄ σnom. Both quantities are stresses, so the ratio is a dimensionless number — a Kt of 3 means the material at the root of the hole or notch is working three times harder than a plain, unbroken section under the identical load would. The formula is shaped as a plain division precisely because Kt is meant to be portable: multiply any newly calculated nominal stress by the same Kt and the local peak scales right along with it, as long as the part stays within its linear elastic range.
The number is not a rule of thumb — for the simplest case, a small circular hole in a wide plate under uniaxial tension, it falls out of an exact elasticity solution. Kirsch published the closed-form stress field around such a hole in 1898, and evaluating it at the hole's edge, perpendicular to the applied load, gives exactly Kt = 3 regardless of how large or small the hole is, provided the plate is wide compared to it. Sharper features push the number higher still: a keyway, a fillet with a tight radius, or a transverse hole through a shaft can carry Kt well past 3, which is why design handbooks publish charts of Kt against radius-to-depth and diameter-to-width ratios rather than a single constant.
The theoretical Kt this calculator returns assumes the material stays elastic everywhere, including right at the stress riser, which is optimistic once local yielding sets in — ductile metal at the root of a sharp notch redistributes load to its neighbors instead of climbing to the full theoretical peak, so the true fatigue-relevant factor, Kf, is usually somewhat lower than Kt and needs a separate notch-sensitivity correction. Kt still governs the first step of the calculation: without it, a stress analysis based on average, textbook formulas silently misses the one location on a part that is actually closest to failing.
- Enter the Peak (local) stress field with σmax — the stress measured, simulated, or read from a chart right at the root of the hole, notch, or fillet, in MPa.
- Enter the Nominal (average) stress field with σnom — the plain-section stress an elementary formula like force over area would give, ignoring the discontinuity entirely.
- Read Stress concentration factor, Kt — a dimensionless multiplier such as 3.0, meaning the local peak runs three times the nominal figure.
- Compare that Kt against a published chart for your actual geometry to sanity-check the peak reading before trusting it in a fatigue or yield calculation.
- Multiply any future nominal stress from the same geometry by this Kt to estimate its new peak, without re-running the full analysis.
Worked example — 150 MPa peak against a 50 MPa nominal section
A fatigue engineer is reviewing an FEA result for a steel plate with a small fastener hole, loaded in simple tension. Away from the hole, the plate carries a nominal stress of 50 MPa, the figure an elementary force-over-area calculation gives for the plain cross-section. Right at the edge of the hole, the finite element model reports a peak of 150 MPa. Entering 150 into Peak (local) stress and 50 into Nominal (average) stress, the instrument converts both to pascals — 150,000,000 and 50,000,000 — and returns Kt = 150,000,000 ⁄ 50,000,000 = 3.0.
That result is not a coincidence: 3.0 is the textbook value for a small circular hole in a wide plate under uniaxial tension, matching Kirsch's 1898 elasticity solution almost exactly. It tells the engineer that fatigue cracks on this plate will start at the hole edge, not in the open field where the nominal 50 MPa reading alone would suggest the part is comfortably under-stressed. A sharper feature at the same nominal load, say a notch with a tighter root radius, would report a higher σmax and push Kt past 3 — which is exactly why designers round or eliminate sharp internal corners wherever a load path runs through them.
Questions
What is the practical difference between σmax and σnom?
σnom is what a plain formula like force divided by area predicts for the section, completely ignoring the hole, notch, or fillet. σmax is the real stress right at that feature's root, taken from a strain gauge placed close to it, a photoelastic test, or a fine finite-element mesh. The gap between the two is exactly what Kt is built to capture — σnom alone would tell you the part looks fine while the actual material at the discontinuity is working several times harder.
Why does Kt stay the same no matter how large the applied load gets?
Because Kt is a ratio between two stresses in the same linear elastic part. Double the applied load and both σmax and σnom double together, so their ratio is unchanged — Kt depends only on the geometry of the discontinuity relative to the section, not on how hard the part is currently loaded. That stops being true once local stress near the root exceeds the material's yield point, since plastic flow there no longer scales linearly with load.
Is Kt the same thing as the fatigue notch factor, Kf?
No. Kt is the theoretical, purely elastic ratio this calculator returns. Kf is the factor actually observed in fatigue tests, and it is usually smaller because ductile materials are less sensitive to a sharp notch than perfect elasticity predicts. The two are linked by Kf = 1 + q(Kt − 1), where q is an experimentally measured notch-sensitivity factor between 0 and 1 for the specific material.
Where does the value Kt = 3 for a hole actually come from?
It comes from Kirsch's 1898 closed-form solution to the elasticity equations for a small circular hole in an infinite plate under uniaxial tension. Evaluated at the hole's edge, perpendicular to the load, that solution gives a peak stress exactly three times the nominal stress far from the hole — a result so clean that it became the standard reference case every other Kt chart is compared against.
Does a bigger hole always produce a higher Kt?
Not on its own. For a circular hole in a plate much wider than the hole, Kt sits close to 3 regardless of the hole's absolute size — only the ratio of hole diameter to plate width matters, through a finite-width correction that grows once the hole starts to fill a meaningful fraction of the section. Shape, not size, is what actually drives Kt up: a sharper root radius on a notch or fillet concentrates stress far more than simply scaling a smooth hole larger or smaller.
What do I do with a Kt value once I have calculated it?
Multiply it by any nominal stress computed for that same geometry to estimate the true local peak without rerunning a full simulation, then compare that peak against the material's yield or fatigue limit. If σmax is already known from a test or FEA instead, running it here alongside σnom checks whether the reported concentration matches what a published chart predicts for that geometry.