Shaft Fillet and Stress Concentration Calculator

Kt for the three stress raisers that matter on a shaft — a shoulder fillet, a circumferential groove and a keyway — from published closed-form fits with their sources and their uncertainty named, plus the notch sensitivity that turns Kt into the fatigue-effective Kf. The chart of Kt against fillet radius is the point of the page.

Last updated 24 Sep 2026

Shaft fillet and stress concentration

Feature and radius → Kt, and the Kf a fatigue check uses

FeatureThe three that matter on a machine shaft. Each uses a different published closed-form fit and the page says which.

LoadingTorsion is always the milder case on the same geometry. There is no published axial fit for a groove in this batch’s sources, so that combination is refused rather than guessed.

Larger diameter D (mm) — or the full shaft diameter for a groove

Smaller diameter d (mm) — at the bottom of the step or groove, or the shaft diameter for a keyway

Fillet, groove or keyway corner radius (mm)The one thing you control. Kt falls steeply with it and the series below is the whole page: read off how much a slightly larger radius buys.

Ultimate tensile strength (N/mm²)Sets the notch sensitivity. A soft steel is far less notch sensitive than a hard one, so the same geometry costs it less.

Bending moment (N·m, for the stress figures)

Torque (N·m)

Axial force (N)

Not a circuit: the shoulder in HALF section, with the centreline along the bottom — the material is between the centreline and the profile. Only one thing in the drawing moves, and it is the one the page is about: the fillet radius, drawn at three times scale against the shaft diameter and quantised to two-hundredths of it. Three times scale because a real r/d of 0.05 drawn honestly is a hairline; the step height is schematic for the same reason and does not follow your D and d. Watch the fillet as you change the radius and watch the bars move with it. The bars are drawn on one scale starting at 1, because 1 is what a plain shaft with no stress raiser gets, so the LENGTH of each bar is the stress raising rather than the factor. The upper bar is Kₜ, the geometric factor, which is what a static or brittle check uses; the lower one is K_f, the fatigue-effective factor after notch sensitivity, which is what a fatigue calculation uses. The gap between them is what a ductile material does not actually pay — and on a hard, high-strength steel that gap nearly closes.

1.800Example

A shoulder from 50 mm down to 40 mm with a 3 mm fillet, in bending, in a 700 N/mm² steel carrying 350 N·m

A published fit, and the notch sensitivity that turns it into a fatigue factor

Kt = C₁ + C₂(2h/D) + C₃(2h/D)² + C₄(2h/D)³, C = f(√(h/r)) · keyway: Kt = 1.426 + 0.1643u − 0.0019u², u = 0.1/(r/d) · q = 1/(1 + √a/√r) · Kf = 1 + q(Kt − 1)

Kt: theoretical (geometric) stress concentration factor. Elastic, material-independent, and what a static or brittle check uses
Kf: fatigue-effective factor. What a fatigue calculation uses, and it is materially smaller than Kt for a small notch in a ductile steel
q: notch sensitivity, between 0 and 1. Zero means the material ignores the notch entirely; one means it feels all of it
√a: Neuber’s material constant, fitted to the ultimate tensile strength in kpsi as Shigley prints it. Different fits for bending/axial and for torsion
h, r, D, d: fillet or groove height, radius, and the two diameters. h = (D − d)/2, and h/r decides which branch of the published fit applies

Worked example

A shoulder from 50 mm down to 40 mm with a 3 mm fillet, in bending, in a 700 N/mm² steel carrying 350 N·m
Geometry: h = (50 − 40)/2 = 5 mm, h/r = 5/3 = 1.667, and 2h/D = 10/50 = 0.2. h/r is below 2, so the published fit's FIRST branch applies — and the two branches agree at the boundary to half a per cent, which is how we chose between three published transcriptions of them
Evaluate the polynomial and Kt = 1.800. That is a fit to Peterson's chart, not a measurement, and three published transcriptions of it disagree in the third digit — treat it as good to two figures
Notch sensitivity. At 700 N/mm² — 101.5 kpsi — Shigley's fit gives √a = 0.06100 in^½, and with r = 3 mm = 0.1181 in, q = 1/(1 + √a/√r) = 0.849
So Kf = 1 + 0.849 × (1.800 − 1) = 1.679. Using Kt for the fatigue check instead would over-penalise the shaft by 18% of the stress-raising part of the answer
Nominal bending stress at the 40 mm section: 32M/πd³ = 55.7 N/mm², so the peak elastic stress is Kt × that = 100.3 N/mm² and the figure a fatigue calculation should use is Kf × that = 93.5 N/mm²
Now the point of the page. DOUBLE the fillet radius to 6 mm and Kt falls to 1.550 — a 31% reduction in the stress-raising part, for nothing but a different tool radius. r/d is the variable you control and it is worth more than almost anything else on the drawing. The series above shows the whole curve so you can see where it stops paying
For contrast: the same shaft in TORSION on the same fillet has Kt = 1.453, about 43% less stress raising than bending. And a KEYWAY in the 40 mm section, with a realistic 0.8 mm corner radius, has Kt = 2.200 in bending and 2.617 in torsion — much worse than any fillet here, and not improvable by radius, because a keyway's corner is always a small fraction of the shaft. If the shaft is fatigue-critical, the keyway is the problem and the fillet is not
What to do with Kf is on the fatigue page: multiply the alternating stress by it, then take it through a Goodman or Gerber line at the mean stress. Kf multiplies the ALTERNATING component; the mean component is a separate question and in a ductile material local yielding relieves it

Kt for a 50/40 mm step, or a 40 mm shaft, at five radii

Four things are visible here. Torsion is always milder than bending on the same geometry, by roughly 20 per cent on a shoulder — which is why a shaft in pure torsion tolerates a sharper fillet than one in rotating bending. A groove is always worse than a shoulder of the same depth and radius, because the material on both sides constrains it. The keyway rows barely move with radius compared with the fillet rows, because Peterson’s keyseat formulas are in r/d and a keyway’s corner radius is a small fraction of the shaft in every practical case — a keyway is a bad notch and you cannot fillet your way out of it. And the keyway numbers are around 2.2 in bending and 2.6 in torsion, NOT the “1.6 bending and 1.3 torsion” that circulates: those two are old FATIGUE factors Kf rather than Kt, and RoyMech’s own table is explicitly labelled as carrying Kf and not Kt. The same designation can mean different dimensions in different standards families — ANSI against ISO, inch against metric, one national standard against another. The family used here is named beside every figure; check which one your part was made to.
Feature and loading	r = 0.5 mm	1 mm	2 mm	4 mm	8 mm
Shoulder fillet, bending	3.140	2.498	2.013	1.685	1.471
Shoulder fillet, torsion	2.374	1.907	1.590	1.373	1.222
Shoulder fillet, axial	3.122	2.527	2.082	1.731	1.485
U-groove, bending	4.419	3.296	2.503	2.047	1.688
U-groove, torsion	2.826	2.232	1.823	1.535	1.348
Profiled keyway, bending	2.619	2.053	1.747	1.588	1.508
Profiled keyway, torsion	2.966	2.493	2.231	2.094	2.024

Notch sensitivity, and why designers over-penalise a small fillet in a soft steel

Kt is geometry and Kf is what the material actually pays. The relation is Kf = 1 + q(Kt − 1), with q between 0 and 1, and q depends on both the notch radius and the material’s strength: a small notch in a soft, ductile steel yields locally, redistributes the stress and does not feel the full concentration. Read the table across a row and q rises with radius; read down a column and it rises with strength. The practical consequence is the one the page title implies: using Kt where Kf belongs over-penalises a small fillet in a soft steel — at 400 N/mm² and a 0.25 mm radius, q is 0.47, so a Kt of 2.0 becomes a Kf of 1.47 and you have thrown away 53% of the penalty you were charging yourself. It also runs the other way, and that is the safety-relevant direction: in a 1,400 N/mm² steel q is above 0.91 at a 1 mm radius, so a hard shaft pays nearly all of the geometric penalty. High-strength steel is only worth having if the geometry is clean. The form is Peterson’s, q = 1/(1 + √a/√r), with the Neuber constant fitted to the ultimate tensile strength as Shigley prints it — a cubic in S_ut in kpsi, which is why the column is in inch^½. This is a first-pass calculation on an idealised part — uniform section, static load, no stress raisers beyond those stated, and room-temperature material properties. Real parts have fillets, keyways, surface finish and duty cycles that a closed-form answer cannot see.
Ultimate tensile strength (N/mm²)	√a, bending (in^½)	q at r = 0.25 mm	q at 1 mm	q at 4 mm	Kf at r = 1 mm for a Kt of 2.0
400	0.11292	0.468	0.637	0.778	1.637
550	0.08284	0.545	0.705	0.827	1.705
700	0.06100	0.619	0.765	0.867	1.765
900	0.04186	0.703	0.826	0.905	1.826
1,100	0.03053	0.765	0.867	0.929	1.867
1,400	0.01965	0.835	0.910	0.953	1.910

What the fits are, where they came from, and how well they are known

Every number on this page comes from a FIT to a chart, and the page says so rather than presenting it as a measurement. Peterson’s and Roark’s charts are copyrighted and were not reproduced; what is used is the published closed-form approximation to them, which is what every calculator quietly uses. The reason to say it out loud is that it changes what the third digit means: three transcriptions of the same fit, fetched in this batch, disagree by up to a per cent, and the underlying data is photoelastic work from the middle of the last century. Treat Kt as good to two significant figures and no better, and do not make a design decision that turns on 2.17 against 2.26. The same designation can mean different dimensions in different standards families — ANSI against ISO, inch against metric, one national standard against another. The family used here is named beside every figure; check which one your part was made to.
Feature and loading	Form of the fit	Source and provenance	How well it is known
Shoulder fillet, bending / axial / torsion	Kt = C₁ + C₂(2h/D) + C₃(2h/D)² + C₄(2h/D)³, each C a function of √(h/r)	Published closed-form fit to Peterson’s charts, attributed to Pilkey (1997). The charts themselves are copyrighted and were not reproduced	Three transcriptions of this fit were fetched in this batch and they disagree in the third digit. The set used is the one whose two h/r ranges AGREE AT THEIR BOUNDARY: at h/r = 2 and D/d = 1.8 its two branches give 1.4578 and 1.4645, a 0.5% step, where another published transcription steps by 4.7% at the same point. Its torsion set also reproduces an independent academic paper’s own worked value of Kts = 1.452 for d = 10, D = 18, r = 1 to four figures, and a third transcription agrees with it to 0.2% in range 1
U-groove, bending and torsion	The same polynomial form with its own coefficients, over 0.25 ≤ h/r ≤ 50	The same published fit	Its two branches agree at h/r = 2 to within about 4%, which is looser than the fillet’s and is stated here rather than hidden
U-groove, axial tension	—	Not obtained from any source this batch could read	REFUSED. The page returns nothing for this combination rather than guessing a coefficient set. A shaft is loaded in bending and torsion; if you genuinely need axial tension on a grooved bar, get it from Peterson directly
Profiled (end-milled) keyway, bending and torsion	Kt = 1.426 + 0.1643u − 0.0019u² in bending and 1.953 + 0.1434u − 0.0021u² in torsion, with u = 0.1/(r/d)	Peterson’s published closed forms, as reproduced in an ASEE paper on keyed joints. The provenance is stated there: “based on photoelasticity measurements for bending and an electroplating method for torsion”, from studies about fifty years old	A finite-element study quoted in the same paper gets 2.26 bending and 2.20 torsion at r/B = 0.0832, against Peterson’s 2.17 and 2.59 at a comparable radius — good agreement in bending and a real disagreement in torsion, which that paper calls “not conservative” of the finite-element result. The bending number is the better known of the two
Sled-runner keyway	—	Not computed here	A finite-element study gives 2.02 bending, 2.22 torsion and 2.16 axial for a sled-runner keyseat at r/B = 0.0832, which is a little milder than a profiled keyway in bending. No closed-form fit for it was sourced, so it is named and not computed. Note that it is MILDER, so using the profiled figures is the conservative choice

r/d is what you control, and Kt is a fit to a seventy-year-old chart

r/d is the variable you control, and it is worth more than almost anything else on the drawing. A shoulder from 50 down to 40 mm with a 2 mm fillet has a bending Kt of about 1.65; open the fillet to 4 mm and it falls to about 1.44, which removes a third of the stress-raising part of the number for the price of a different tool radius. Nothing else on a shaft is that cheap. The chart on this page plots Kt and Kf against radius across the whole range so you can see where the curve stops paying: it is steep at small radii and flat at large ones, so the first fraction of a millimetre buys most of what is available. Where a big radius will not fit against a bearing face, the answers are an undercut fillet, two smaller steps, or a shoulder ring.

These are fits to charts, and the page says so. Peterson’s and Roark’s stress-concentration charts are copyrighted and are not reproduced here. What is used is the published closed-form approximation to them — a cubic in 2h/D whose four coefficients are functions of √(h/r) — which is what every stress-concentration calculator quietly uses. Saying it out loud changes what the digits mean. Three transcriptions of this fit were fetched during this batch and they disagree in the third digit; the set used here is the one whose two h/r branches agree at their boundary, to 0.5 per cent where another steps by 4.7 per cent, and whose torsion coefficients reproduce an independent academic paper’s own worked value to four figures. Underneath all of it is photoelastic work from the middle of the last century. Treat Kt as good to two significant figures.

Notch sensitivity is where designers lose capacity, and it goes both ways. Kt is geometry; Kf = 1 + q(Kt − 1) is what the material actually pays, and q depends on both the notch radius and the material’s strength. A small notch in a soft, ductile steel yields locally and redistributes, so it does not feel the full concentration: at 400 N/mm² and a quarter-millimetre radius q is about 0.5, so a Kt of 2.0 is really a Kf of 1.5 and half the penalty you were charging yourself was imaginary. The same arithmetic runs the other way and that direction matters more: in a 1,400 N/mm² steel q is above 0.9 at a 1 mm radius, so a hard shaft pays essentially all of the geometric penalty. High-strength steel is only worth buying if the geometry is clean — and a case-hardened surface is notch sensitive even when the core is not.

The keyway is usually the answer to “where will this shaft break”. Peterson’s published closed forms for a profiled keyseat give a bending Kt of about 2.2 and a torsional Kt of about 2.6 at a realistic corner radius — worse than any shoulder fillet on this page, and barely improvable, because a keyway’s corner radius is a small fraction of the shaft diameter in every practical case. Two notes on the figures people quote. The pair “1.6 bending, 1.3 torsion” that circulates for keyways is not Kt: those are old FATIGUE factors, and the pair is normally printed the other way round, with 1.6 for a profiled keyway and 1.3 for a sled-runner. And a sled-runner keyseat is MILDER than a profiled one in bending — about 2.0 against 2.26 in one finite-element study — so using the profiled figures for a sled-runner is the conservative choice, which is what this page does because no closed-form fit for the sled-runner was sourced. Key dimensions themselves are on the parallel key and keyway dimensions calculator.

What to do with Kf. Multiply the ALTERNATING stress component by it and take the result through a Goodman or Gerber line at the mean stress; the mean component is a separate question, and in a ductile material local yielding relieves much of it. The bolt fatigue under alternating load calculator on this site does that construction, and although its subject is a bolt the method is the same one a shaft needs. For where the load that causes the bending comes from, the shaft deflection and slope calculator takes the same transverse load and tells you what it does at the bearing.

Frequently asked questions

What is the stress concentration factor of a shaft shoulder?

It depends almost entirely on the fillet radius relative to the smaller diameter. For a 50 to 40 mm step, a 2 mm fillet gives about 1.65 in bending and 1.34 in torsion; a 4 mm fillet gives about 1.44 and 1.24. The calculator evaluates a published closed-form fit to Peterson’s charts and reports which branch of the fit applies. Treat the answer as good to two significant figures: three published transcriptions of the same fit disagree in the third digit.

What is the difference between Kt and Kf?

Kt is geometric — the elastic stress concentration a notch produces, independent of material. Kf is what the material actually pays in fatigue, and it is smaller: Kf = 1 + q(Kt − 1), where q is the notch sensitivity, between 0 and 1. Use Kt for a static or brittle check and Kf for a fatigue check. Using Kt where Kf belongs is conservative but can waste a lot of capacity — for a small fillet in a soft steel it can double the penalty.

How much does a bigger fillet radius actually buy?

More than most people expect, and the curve is steep exactly where you are. Doubling a 2 mm fillet to 4 mm on a 50/40 step removes about a third of the stress-raising part of Kt. Doubling again from 4 to 8 removes much less. The chart on this page plots the whole curve, which is the most useful single thing here: it shows where a larger radius stops paying and lets you spend the space where it counts.

What is the stress concentration factor of a keyway?

About 2.2 in bending and 2.6 in torsion for a profiled (end-milled) keyseat at a realistic corner radius, from Peterson’s published closed forms. It is worse than any shoulder fillet and it barely improves with radius, because a keyway corner is always a small fraction of the shaft. Beware the figures “1.6 bending and 1.3 torsion” that circulate — those are old FATIGUE factors rather than Kt, and the pair is usually printed the other way round, with 1.6 for a profiled keyway and 1.3 for a sled-runner.

Is torsion or bending worse for a stress raiser?

Bending, on the same geometry, by roughly 20 per cent of the stress-raising part for a shoulder fillet. That is why a shaft in pure torsion tolerates a sharper fillet than one in rotating bending — and why most shaft fatigue failures are at shoulders on rotating-bending shafts. Note the exception: for a keyway, torsion is WORSE than bending, about 2.6 against 2.2, because the keyway’s flanks are aligned badly for a torsional shear field.

Is a groove worse than a shoulder?

Yes, always, at the same depth and radius, because the material on both sides of a groove constrains the notch root where a shoulder has material on one side only. That is why a retaining ring groove is one of the worst things you can put on a fatigue-loaded shaft, and why the groove should be as shallow and as generously radiused as the ring allows, and placed where the bending moment is low.

Does notch sensitivity apply to a case-hardened shaft?

Not to the case. The notch sensitivity fit here is for the bulk material and it says that a soft, ductile steel does not feel the full geometric concentration because it yields locally. A case-hardened or nitrided surface cannot do that: it is hard, it is brittle relative to the core, and it is notch sensitive even when the core is not. Treat q as close to 1 for a hardened surface layer, which means Kf is close to Kt. The compensation is that case hardening also puts a compressive residual stress into the surface, which helps fatigue in a way none of these factors sees.

Related calculators

References

Walter D. Pilkey and Deborah F. Pilkey, Peterson’s Stress Concentration Factors. The charts are copyrighted and were not reproduced. What this page uses is the published closed-form FIT to them, Kt = C₁ + C₂(2h/D) + C₃(2h/D)² + C₄(2h/D)³ with each coefficient a function of √(h/r), for a shoulder fillet in bending, tension and torsion and for a circumferential U-groove in bending and torsion. It is a fit and the page says so.
Amesweb. Stress Concentration Calculator — Shoulder Fillet in a Shaft. The transcription of the Pilkey coefficients used here. It was chosen over two others because its two h/r ranges AGREE AT THEIR BOUNDARY: at h/r = 2 the range-1 and range-2 polynomials give 1.4578 and 1.4645 for a D/d of 1.8, a 0.5% step, where a second published transcription steps by 4.7% at the same point. Its torsion set is identical to the one an independent academic paper attributes to Pilkey (1997), and that set reproduces that paper’s own worked value of Kts = 1.452 for d = 10, D = 18, r = 1 to four figures.
Niels L. Pedersen. Aspects of stress in optimal shaft shoulder fillet, Technical University of Denmark. The independent verification of the torsion coefficient set: it prints the same C₁ to C₄, attributes them to Pilkey (1997), and works a case — d = 10 mm, D = 18 mm, r = 1 mm — whose answer of 1.452 this batch reproduces exactly from the coefficients. That is what settled which transcription to trust.
Calculation of Stress Concentration Factors for Round Shafts with Shoulder Fillets, ASEE conference paper. A third transcription of the same fit, attributed to Roark. Its bending range-1 coefficients differ from Amesweb’s in the third digit (0.927 against 0.947 on C₁’s constant term) and the two give 1.4544 and 1.4578 for the same fillet — a 0.2% difference, quoted on this page as the honest measure of how well a curve fit to a fifty-year-old photoelastic chart is known.
Louis J. Everett / Notes on Design of Keyed Joints, ASEE paper 39427. The source for Peterson’s published closed-form keyseat factors, Kt = 1.426 + 0.1643u − 0.0019u² in bending and Kts = 1.953 + 0.1434u − 0.0021u² in torsion with u = 0.1/(r/d), and for their provenance: “based on photoelasticity measurements for bending and an electroplating method for torsion”, from studies about fifty years old. It also records that Le and Le’s finite-element study shows “good agreement with Peterson for bending but lower values for torsion”, which it calls “not conservative”, and that Pedersen’s work on DIN keyseats “corresponds well to those given by Peterson”.
Stress Concentration Factors Due to Typical Geometric Discontinuities for Shaft Design by Numerical Simulation, ASEE paper 6755. A finite-element study quoting Kt = 2.26 bending, 2.20 torsion and 2.81 axial for a profile keyseat at r/B = 0.0832, and 2.02 / 2.22 / 2.16 for a sled-runner keyseat. Cited for those four values as corroboration of magnitude. Its printed quadratic fits are NOT used: evaluating the bending one at its own r/B = 0.0832 gives 6.33 rather than the 2.26 it states beside it, so something in the extraction or the normalisation is wrong and nothing was taken from them.
Richard G. Budynas and J. Keith Nisbett, Shigley’s Mechanical Engineering Design, notch sensitivity. The source for q = 1/(1 + √a/√r) and for the Neuber constant fitted to the ultimate tensile strength in kpsi: √a = 0.246 − 3.08×10⁻³S_ut + 1.51×10⁻⁵S_ut² − 2.67×10⁻⁸S_ut³ for bending and axial, and 0.190 − 2.51×10⁻³S_ut + 1.35×10⁻⁵S_ut² − 2.67×10⁻⁸S_ut³ for torsion, both in inch^½. Kf = 1 + q(Kt − 1) follows.
RoyMech. Stress Concentration Factors — Kt, Kf Tables & Engineering Guide. Read for the Kf = 1 + q(Kt − 1) relation with its bound 1 ≤ Kf ≤ Kt, and for the note that where q is unknown “using Kt in place of Kf provides the conservative approach”. Its keyway section is explicitly labelled as carrying Kf values and not Kt, which is exactly the distinction the figures “1.6 bending and 1.3 torsion” lose when they are quoted as though they were Kt.