Gear Tooth Bending Stress Calculator

The Lewis bending estimate with the Barth velocity factor, both published readings of the form factor, the permissible load it implies, and the Hertz contact pressure that usually governs instead — with an explicit list of what Lewis omits and what AGMA 2001 and ISO 6336 add.

Last updated 24 Sep 2026

Gear tooth bending stress

Module, teeth, face and load → Lewis bending stress

Module m (mm)Bending stress goes inversely with module at a fixed tangential load, so a coarser tooth is stronger — at the cost of fewer teeth on the same diameter and a lower contact ratio.

Teeth on the gear being checked zCheck the PINION. It has the smaller form factor, so it is the weaker tooth, and every tooth on it meets the load z₂/z₁ times as often as a wheel tooth does.

Teeth on its mateUsed only for the contact-pressure secondary, which needs both radii of curvature.

Face width b (mm)Stress goes inversely with face width, but only if the load is actually spread across it. A wide face with a flexible shaft carries its load at one end, and this calculation cannot see that. 9 m to 14 m is the usual range.

Tooth form, for the Lewis factorThe three published linear fits, with Y = πy. A 20° tooth is stronger in bending than a 14.5° one at every tooth count, and a stub tooth stronger still.

Manufacturing quality, for the Barth velocity factorFour published forms, here with v in m/s. They are the exact unit conversions of Shigley’s original constants 600 and 1200 ft/min and 50 and 78 √(ft/min).

Power transmitted (kW)

Speed of this gear (rev/min)

Allowable bending stress (MPa)A worked allowable, NOT a material strength. Classical practice takes roughly a third of the ultimate tensile strength for steady loads and less for shock — so 150 MPa is a plausible figure for an ordinary medium-carbon steel gear. A real rating computes this from the material, the life, the reliability and the duty.

Material pair, for the contact pressureUsed only for the Hertz secondary. Contact pressure depends on the combined elastic modulus, so a steel pinion on a bronze or plastic wheel sees a much lower pressure than steel on steel.

Not a circuit: the Lewis form factor Y against tooth count, for whichever of the three published tooth forms you selected, with your own gear marked by the pointer. The shape is the point. The curve is STEEP below about 25 teeth and almost flat above 60, so adding teeth to a small pinion buys real bending strength and adding them to a large wheel buys almost none — which is why the pinion is the part that gets designed and the part that breaks. The bar underneath is the corrected bending stress against your own allowable, with 1.0 marked at two thirds of the scale. Read that bar with the contact pressure in the stack beside it: on ordinary gearing the contact pressure is five or six times the bending stress, and pitting rather than tooth breakage is what usually decides the gear's life.

116.63MPaExample

A 3 mm module 20-tooth 20° pinion, 30 mm face, hobbed, carrying 7.5 kW at 1,000 rev/min against a 40-tooth wheel, with a 150 MPa allowable

A tooth as a cantilever, with a speed correction

σ = F_t / (b·m·Y) · Y = πy · y = A − B/z · σ_corrected = K_v·σ · F_t = 2000·T/d · p_max = √(F’·E*/πR)

F_t: tangential tooth force, from the torque and the pitch radius. The radial force does not enter the bending calculation in this form of it
b: face width. Stress falls inversely with it — IF the load is actually spread across it, which a flexible shaft and a wide face together prevent
m: module. Stress falls inversely with it too, which is the main lever a designer has
Y: Lewis form factor, a function of tooth count and tooth form. It is the ratio that converts the parabola inscribed in the tooth into a stress, and it is dimensionless
K_v: Barth velocity factor, greater than one, MULTIPLYING the load. An empirical allowance for dynamic tooth loading. Note the convention: the reciprocal form C_v divides instead, and mixing them up squares the error
E*, R: for the contact-pressure secondary only: the combined plane-strain modulus, 1/E* = (1−ν₁²)/E₁ + (1−ν₂²)/E₂, and the combined radius, 1/R = 1/ρ₁ + 1/ρ₂ with ρ = (d/2)·sin α

Worked example

A 3 mm module 20-tooth 20° pinion, 30 mm face, hobbed, carrying 7.5 kW at 1,000 rev/min against a 40-tooth wheel, with a 150 MPa allowable
The load first. 7.5 kW at 1,000 rev/min is 71.62 N·m, and on a 60 mm pitch diameter that is a tangential force of 2000T/d = 2,387 N. The pitch line velocity is 3.142 m/s
The Lewis form factor for a 20-tooth 20° full-depth tooth: y = 0.154 − 0.912/20 = 0.10840, and the module form of the equation uses Y = πy = 0.34055. Note that the other published reading of this factor — a directly tabulated column rather than this linear fit — gives about 0.322 at 20 teeth, 5.8 per cent lower. The results panel prints both, because neither can be shown to be the right one from anything this page could read
The static Lewis stress: σ = F_t/(b·m·Y) = 2,387/(30·3·0.3405) = 77.89 MPa. What that calculation assumes is worth spelling out: the whole tangential load applied at the TIP of ONE tooth, the tooth treated as a cantilever with a parabola inscribed in it, and no stress concentration at the root fillet at all
Now the speed correction. For a hobbed profile the Barth factor is (3.5637 + √v)/3.5637 = 1.4974, so the corrected bending stress is 116.63 MPa. Against your 150 MPa allowable that is a utilisation of 0.778, or a factor of 1.29 on the stress. Turned round, the tooth could carry 3,070 N of tangential force, which is 92.1 N·m and 9.65 kW at this speed
AND NOW THE NUMBER THAT PUTS THE OTHER ONES IN PERSPECTIVE. The Hertz contact pressure at the pitch point, with the flank radii (d/2)·sin α = 10.26 mm and 20.52 mm and steel on steel, is 668 MPa — 5.7 times the bending stress. That is the quantitative form of the thing every gear textbook says and nobody quite believes: pitting and not tooth breakage is the usual limit. A bending number that looks comfortable tells you very little on its own
So what IS this page's answer good for? Choosing a module and a face width, comparing two candidate designs, and catching a design that is obviously wrong. It is not a rating, and the list of what Lewis omits is long: the stress concentration at the root fillet (1.4 to 1.8 times on ordinary proportions), load sharing between teeth, load distribution across the face, rim thickness, a proper dynamic factor, and surface durability. ANSI/AGMA 2001-D04 and ISO 6336 add all of it, they are out of scope for this site, and a first-pass bending number is not a substitute for either

Lewis form factor Y, from the three published linear fits

These come from the three published linear fits, y = 0.154 − 0.912/z for 20° full depth, 0.124 − 0.684/z for 14.5° and 0.175 − 0.841/z for the 20° stub tooth, with Y = πy. The fits are used in preference to a tabulated Y column for a reason worth stating: every published TABULATION of Y that this page’s research could reach sits inside an image, and the two textual readings obtained disagreed with each other. The fits and the tabulated column agree exactly at 12 teeth (0.2450) and to within 0.3 per cent at the rack limit (0.4838 against 0.485), and diverge by up to about eight per cent in mid-range — so the results panel prints BOTH readings and the difference between them. Eight per cent on a first-pass bending estimate is not the largest uncertainty on this page; the stress concentration Lewis omits altogether is. 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.
Teeth z	20° full depth	14.5° full depth	20° stub	20° stronger than 14.5° by
12	0.2450	0.2105	0.3296	16.4%
14	0.2792	0.2361	0.3611	18.3%
16	0.3047	0.2553	0.3846	19.4%
18	0.3246	0.2702	0.4030	20.2%
20	0.3405	0.2821	0.4177	20.7%
24	0.3644	0.3000	0.4397	21.5%
30	0.3883	0.3179	0.4617	22.1%
40	0.4122	0.3358	0.4837	22.7%
50	0.4265	0.3466	0.4969	23.1%
60	0.4361	0.3537	0.5057	23.3%
100	0.4552	0.3681	0.5234	23.7%
300	0.4743	0.3824	0.5410	24.0%

Barth velocity factors, with v in metres per second

Four published forms, not three: (3.048 + v)/3.048 for a cast profile, (6.096 + v)/6.096 for cut or milled, (3.5637 + √v)/3.5637 for hobbed or shaped and (5.5594 + √v)/5.5594 for shaved or ground. The metric constants here are the EXACT unit conversions of the originals, which are stated in feet per minute: 600 and 1200 ft/min are 3.048 and 6.096 m/s, and 50 and 78 √(ft/min) become 3.5637 and 5.5594. Most sources round them to 3.05, 6.1, 3.56 and 5.56; the difference is under a tenth of a per cent and this page uses the exact values. ONE WARNING ABOUT CONVENTION. K_v here MULTIPLIES the load, so it is greater than one and it makes the stress worse. Several textbooks print a velocity factor C_v = 3/(3+v), 6/(6+v) or 5.6/(5.6+√v) which is LESS than one and multiplies the ALLOWABLE stress instead. The two conventions are reciprocals of each other and mixing them up squares the error — check which one a formula you are copying intends. 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.
v (m/s)	Cast profile	Cut or milled	Hobbed or shaped	Shaved or ground
1.0	1.328	1.164	1.281	1.180
2.5	1.820	1.410	1.444	1.284
5.0	2.640	1.820	1.627	1.402
10.0	4.281	2.640	1.887	1.569
20.0	7.562	4.281	2.255	1.804
30.0	10.843	5.921	2.537	1.985

What Lewis leaves out, and who puts it back

ANSI/AGMA 2001-D04 and ISO 6336 add every one of these and are OUT OF SCOPE for this site — they need a measured accuracy grade, a chosen reliability and life, a load spectrum and a dozen factors that cannot be guessed, and a page that pretended to compute them would be worse than one that does not. What this page gives you is a first-pass SIZING estimate: a number to choose a module and a face width with, and to compare two candidate designs with. **A Lewis bending number is not a rating.** Do not put one on a drawing, do not use it to accept a design whose failure matters, and do not use it at all where the governing limit is surface durability, which it usually is. This page sizes a part; it does not certify one. Where the answer carries a consequence — a load path, a lifting duty, a pressure boundary, a fastener holding something that can fall — confirm it against the design code that governs the application, and against the manufacturer’s own rating, before relying on it.
What is missing	Why it matters	Where the full methods handle it
Stress concentration at the root fillet	Lewis inscribes a parabola in the tooth and treats it as a plain cantilever. The real tooth has a fillet at the root, and the stress there is 1.4 to 1.8 times the nominal bending stress for ordinary proportions	AGMA 2001 and ISO 6336 both carry it, as a stress correction factor computed from the actual fillet geometry
Load sharing between teeth	Lewis puts the whole load at the tip of ONE tooth. In a real mesh with a contact ratio of 1.6 the load is shared for much of the cycle, and the highest-stress condition is the load at the highest point of SINGLE tooth contact, not at the tip	Both methods use the single-tooth-contact point and a load distribution factor. Lewis’s tip assumption is conservative for the load position and unconservative for everything else
Load distribution across the face	The shaft twists and bends, the housing spreads, and the teeth are never quite parallel. A nominally uniform load across a wide face is not uniform	A face load distribution factor, which for a wide face on a flexible shaft can exceed 1.5 on its own. This is why lead crowning exists
Rim and web thickness	A thin-rimmed gear flexes under the tooth load, so the tooth root is not built into a rigid body and the effective root stress rises	A rim thickness factor. Lewis assumes an infinitely stiff body
Dynamic tooth loads properly	The Barth factor on this page is an empirical speed correction from the 1890s. The real dynamic load depends on the mesh stiffness, the transmission error, the accuracy grade and the inertias	A dynamic factor computed from a measured accuracy grade and the operating speed relative to the mesh resonance
SURFACE DURABILITY, which is usually what governs	The contact pressure at the pitch point on the default example here is about six times the bending stress. Pitting, not tooth breakage, is the commonest gear failure	The whole of AGMA 2001’s and ISO 6336’s pitting calculation, with lubricant film, surface finish, work hardening and a separate allowable

The contact pressure figure: what it is and what it is not

This is the one secondary on the page that could be mistaken for something it is not, so it is spelt out. The computation is honest and independent: Hertz line contact with ρ = (d/2)·sin α at each flank, the combined plane-strain modulus from both materials, and the normal tooth force F_t/cos α. What it cannot be is a durability rating, because that needs the geometry and load factors this site does not carry. An AGMA-style surface geometry factor was deliberately NOT used here, for exactly the reason the Lewis table above is used in preference to a tabulated Y: the factor is published inside the standard, the standard is copyrighted, and a half-remembered version of it would be worse than a clean physics calculation with its limits stated. 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.

What the Hertz figure here IS	The maximum contact pressure between two elastic cylinders of the flanks’ local radii, at the PITCH POINT, with the whole tooth load carried by one pair. p_max = √(F’E*/πR), which is Hertz’s own line-contact result — a result in elasticity, not a rating constant. It reproduces Buckingham’s published gear surface-stress formula exactly, because Buckingham’s 0.564 is 1/√π
What it is NOT	It is not an AGMA or ISO pitting rating and must not be compared with a published allowable contact stress. Those numbers carry a geometry factor, a load distribution factor, a dynamic factor, a surface condition factor and a life factor, all absent here
Why it is on the page anyway	Because it is the single most useful piece of context for a bending number. On the default example it is about six times the bending stress, which is the quantitative version of “pitting, not breakage, is what usually limits a gear”. If a design looks comfortable in bending, this is the number that says whether that comfort means anything
How to read it	Comparatively. Doubling the torque raises it by only 41 per cent, because it goes as the square root of load; a bigger MATE barely helps, because the pinion’s own curvature dominates the combined radius; and a softer material pair drops it a great deal, which is why a steel pinion on a bronze or plastic wheel is a durable combination at modest load

A cantilever from 1893, what it omits, and the contact pressure that usually governs

Lewis treats the tooth as a cantilever, and that is both the point and the problem. σ = F_t/(b·m·Y), with Y the form factor: a parabola is inscribed in the tooth, the whole tangential load is applied at its TIP, one tooth pair is assumed to carry everything, and the root fillet is ignored. Wilfred Lewis published it in 1893 and it is still the right first calculation, because it contains exactly the three levers a designer has — module, face width and tooth count — and nothing else. The form factor comes from three published linear fits, y = 0.154 − 0.912/z for 20° full depth, 0.124 − 0.684/z for 14.5° and 0.175 − 0.841/z for the 20° stub tooth, with Y = πy.

The Barth velocity factor, and a convention that squares the error if you get it wrong. Dynamic tooth loads rise with speed, and Lewis has nothing to say about them, so an empirical factor is applied. There are FOUR published forms, not three: (3.048 + v)/3.048 for a cast profile, (6.096 + v)/6.096 for cut or milled, (3.5637 + √v)/3.5637 for hobbed or shaped and (5.5594 + √v)/5.5594 for shaved or ground, with v in metres per second — and those constants are the exact unit conversions of the originals, which were 600 and 1200 ft/min and 50 and 78 √(ft/min). The warning: K_v in this form is greater than one and MULTIPLIES the load. Several textbooks print a velocity factor C_v = 3/(3+v) or 6/(6+v), which is less than one and multiplies the ALLOWABLE stress instead. They are reciprocals; mixing them up is a factor of K_v².

Be explicit about what Lewis leaves out, because the list is long and every item makes the real stress higher. Stress concentration at the root fillet, which is 1.4 to 1.8 times on ordinary proportions and is absent here entirely. Load sharing between teeth, and the fact that the worst case is the load at the highest point of single-tooth contact rather than at the tip. Load distribution across the face, which on a wide face and a flexible shaft is worth 1.5 on its own. Rim and web thickness, since Lewis assumes an infinitely stiff body behind the tooth. A proper dynamic factor from a measured accuracy grade. And surface durability. ANSI/AGMA 2001-D04 and ISO 6336 add all of that and are out of scope for this site — they require a measured accuracy grade, a chosen life and reliability, a load spectrum and a dozen factors that cannot be guessed. A first-pass bending number is not a rating.

And pitting is usually the limit, not bending. That is why this page also computes the Hertz contact pressure at the pitch point: p_max = √(F’E*/πR), with the flank radii (d/2)·sin α and the combined plane-strain modulus of the two materials. On the default example it comes out at about six times the bending stress. That figure is honest — it is Hertz’s own line-contact result and it reproduces Buckingham’s published gear surface-stress formula exactly, because Buckingham’s 0.564 is 1/√π — and it is deliberately NOT dressed up as a durability rating. An AGMA-style surface geometry factor was not used, for the same reason a tabulated form factor was not: it lives inside a copyrighted standard, and a half-remembered version of it would be worse than a clean physics calculation with its limits stated.

How to use the answer. As a sizing tool. Pick a module and a face width, see where the stress lands, compare two candidate designs, and use the permissible tangential force and torque in the results to see how much headroom a design has. Then look at the contact pressure to find out which failure mode you are actually near, and at the contact ratio to see whether the dynamic behaviour is reasonable. For the tooth geometry itself see the spur gear geometry calculator — particularly the undercut limit, because an undercut tooth has had the bending section cut out of it and no form factor knows that — and for the shaft the gear loads, the shaft deflection and slope calculator and the shaft fillet and stress concentration calculator, which is where the same fillet-concentration argument is made properly for a shaft.

Frequently asked questions

What is the Lewis equation for gear tooth bending stress?

σ = F_t/(b·m·Y) in metric, where F_t is the tangential tooth force, b the face width, m the module and Y the Lewis form factor — a dimensionless function of the tooth count and the tooth form. In inch units the same thing is σ = F_t·P_d/(b·Y). It treats the tooth as a cantilever with a parabola inscribed in it, with the whole load at the tip of one tooth and no stress concentration at the root, which is why it is a sizing estimate rather than a rating.

What is the Lewis form factor for 20 teeth?

It depends which published reading you take, and this page prints both. The linear fit for a 20° full-depth tooth gives y = 0.154 − 0.912/20 = 0.1084, so Y = πy = 0.3405. A directly tabulated Y column gives about 0.322 at the same tooth count — six per cent lower. The two agree exactly at 12 teeth (0.2450) and to within 0.3 per cent at the rack limit, and diverge in mid-range. The fits are used here because they are printed in text and can be checked; every tabulation found sits inside an image.

What is the Barth velocity factor?

An empirical multiplier on the tooth load that allows for dynamic effects at speed. With v in metres per second there are four published forms: (3.048 + v)/3.048 for a cast profile, (6.096 + v)/6.096 for cut or milled, (3.5637 + √v)/3.5637 for hobbed or shaped, and (5.5594 + √v)/5.5594 for shaved or ground. Those constants are the exact conversions of the original 600 and 1200 ft/min and 50 and 78 √(ft/min). Beware the reciprocal convention C_v = 6/(6+v), which is less than one and multiplies the allowable stress instead.

Is a Lewis bending stress a gear rating?

No, and the gap is large. Lewis omits the stress concentration at the root fillet (1.4 to 1.8 times on ordinary proportions), load sharing between teeth, load distribution across the face (worth 1.5 or more on a wide face), rim and web flexibility, a proper dynamic factor from a measured accuracy grade, and surface durability. ANSI/AGMA 2001-D04 and ISO 6336 add all of it and are out of scope for this site. Use a Lewis number to choose a module and a face width and to compare candidates; do not put one on a drawing or use it to accept a design whose failure matters.

Why is the contact pressure so much higher than the bending stress?

Because they are different kinds of quantity. The bending stress is spread over the whole tooth section; the contact pressure acts on a band a fraction of a millimetre wide, so it is far larger numerically and is compared with a much higher allowable. On the default example here the pitch-point contact pressure is about six times the bending stress. The practical point is the ordering: pitting, not tooth breakage, is the commonest gear failure, so a comfortable bending number on its own tells you very little.

Should I check the pinion or the wheel?

The pinion, almost always. Its form factor is smaller, because it has fewer teeth and the factor climbs steeply below about 25 teeth, so its tooth is the weaker of the two at the same module and face width. And every tooth on it meets the load z₂/z₁ times as often as a wheel tooth does, so it accumulates cycles that much faster. If the two gears are different materials the comparison needs both, but with a matched pair the pinion governs.

How do I reduce the bending stress on a gear tooth?

A coarser module and a wider face both reduce it inversely, and they are the two big levers. More teeth raises the form factor, steeply below about 25 teeth and hardly at all above 60 — but note that more teeth at the same diameter means a finer module, which works against you, so the useful move is more teeth at a larger diameter. A stub tooth is about 20 per cent stronger at the cost of contact ratio. A better manufacturing grade reduces the velocity factor directly. And check the contact pressure too, because it responds differently: it goes as the square root of load and barely notices a bigger mate.

Related calculators

References

Wilfred Lewis, Investigation of the Strength of Gear Teeth, Proceedings of the Engineers’ Club of Philadelphia, 1893. The original of the formula on this page. Its assumption — the whole tangential load applied at the tip of one tooth, the tooth treated as a cantilever with a parabola inscribed in it, and no stress concentration at the fillet — is the reason a Lewis number is a sizing estimate and not a rating.
R. S. Khurmi and J. K. Gupta, A Textbook of Machine Design. The source for the three linear Lewis form factors used here: y = 0.124 − 0.684/z for the 14.5° full-depth form, y = 0.154 − 0.912/z for 20° full depth and y = 0.175 − 0.841/z for the 20° stub tooth, with Y = πy. Used in preference to a tabulated Y column because every published tabulation this batch could find sits inside an image, and the two textual readings obtained disagreed with each other by six per cent in mid-range.
Richard Budynas and Keith Nisbett, Shigley’s Mechanical Engineering Design, chapters 14 and 15. Cited for the four Barth velocity factors and for the bevel face-width rule b ≤ min(0.3 A_o, 10/P_d). The velocity factors were verified rather than transcribed: an independent implementation of them returns K_v = 2.311 for a cast profile at 4 m/s, which is (3.05 + 4)/3.05 to four figures, and the metric constants 3.05, 6.1, 3.56 and 5.56 are the exact unit conversions of the original 600 and 1200 ft/min and 50 and 78 √(ft/min).
ANSI/AGMA 2001-D04, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth. Cited and explicitly out of scope, for the same reasons as ISO 6336 and with the same consequence: a Lewis number is a sizing estimate and not a rating.
ISO 6336 (all parts), Calculation of load capacity of spur and helical gears. Cited and explicitly OUT OF SCOPE for this site. It is named so that the reader knows what a bending number from this page is not: ISO 6336 adds the tooth form factor and stress correction factor for the real root fillet, load distribution across the face, the dynamic factor from a measured accuracy grade, rim thickness, and — the part that usually governs — surface durability.
Heinrich Hertz, Über die Berührung fester elastischer Körper (1882), in the standard line-contact form p_max = √(F′E*/πR). Used here for the surface pressure at the pitch point, in preference to any gear geometry factor: it is a result in elasticity rather than a rating constant, and it reproduces Buckingham’s gear surface-stress formula exactly — Buckingham’s 0.564 is 1/√π, which is the check this page ran.
Stock Drive Products / Sterling Instrument, Elements of Metric Gear Technology (the technical section of catalogue D805). The source for two things taken as printed here: the minimum tooth count free of undercut — “for 14.5° the value of z_c is 32, and for 20° it is 18” — and the contact ratio guidance, “it is good practice to maintain a contact ratio of 1.2 or greater. Under no circumstances should the ratio drop below 1.1, calculated for all tolerances at their worst case values.” Also the source for the self-locking condition in the form (cos 20° sin γ − μ cos γ) ≤ 0, which is what puts the normal pressure angle into the threshold.