August 2026 13 min read

Gear Tooth Strength Calculator

Free gear tooth strength calculator. Check spur gear tooth bending stress (Lewis) and contact (Hertz) stress, with safety factors and required face width, from module, teeth, power and speed. 20° full-depth.

Gear Tooth Strength Calculator

Gear Tooth Inputs

Unit System

SI — Module (mm) Imperial — DP

Module m (mm)

Tooth size

Teeth z

Face Width b (mm)

Ratio u (z₂/z₁)

Gear/pinion

Speed n (rpm)

Power (kW)

Power Unit

kW hp

Allow σ_Flim (MPa)

Bending allowable

Allow σ_Hlim (MPa)

Contact allowable

Auto-calculate instantly

Bending Safety Factor

4.52

σ_F = 55.3 MPa.

Safe — both safety factors above typical minimums.

Bending σ_F

55.3MPa

Contact σ_H

218.6MPa

Contact SF

3.20

Tangential F_t

1,061N

Pitch Dia d

60mm

Pitch Velocity

4.71m/s

b_req (bending)

4.4mm

b_req (contact)

2.0mm

Bending σ_F

55.3MPa

F_t/(b·m·Y)

Contact σ_H

218.6MPa

Hertz stress

Contact SF

3.20

σ_Hlim/σ_H

Tangential F_t

1,061N

2T/d

Pitch Dia d

60mm

m·z

Pitch Velocity

4.71m/s

π·d·n/60

b_req (bending)

4.4mm

Needed face

b_req (contact)

2.0mm

Needed face

Result Summary

Module	3 mm
Teeth z	20
Face Width b	20 mm
Ratio u	2
Pitch Diameter	60 mm
Tangential Load	1,061 N
Bending σ_F	55.3 MPa
Bending SF	4.52
Contact σ_H	218.6 MPa
Contact SF	3.20
Power	5 kW

Estimated Result — Uses the simplified Lewis bending estimate σ_F = F_t/(b·m·Y) and Hertz contact σ_H = Z_E·√[F_t·(u+1)/(b·d·u)] with Z_E = 189.8 √MPa (steel-steel). The Lewis form factor Y is interpolated from the 20° full-depth table. It omits AGMA load factors (K_a, K_v, K_m, K_Hβ) and assumes low speed and proper lubrication, so treat results as preliminary sizing, not a certified AGMA/ISO 6336 rating.

Interactive Diagram

The tangential load F_t bends the tooth at the root (σ_F) and compresses the flank (σ_H).

Visualizations

Actual vs Allowable Stress (MPa)

55.3MPaσ_F

250MPaσ_Flim

218.6MPaσ_H

700MPaσ_Hlim

Bending Safety Factor

SF300×

Formula

Two failure modes limit a gear tooth: root bending fatigue (σ_F) and flank contact fatigue / pitting (σ_H). The Lewis formula estimates bending stress from the tangential load F_t, face width b, module m and form factor Y. Hertz contact stress uses the elasticity factor Z_E and the mesh ratio u. Comparing each to its allowable gives a safety factor.

σ_F = F_t / (b · m · Y)

Alternative Form

σ_H = Z_E · √[ F_t·(u+1) / (b·d·u) ]

Variables

Symbol	Name	Unit	Description
F_t	Tangential load	N	2T/d = 1000P/v
b	Face width	mm	Tooth length
m	Module	mm	Tooth size
Y	Lewis form factor	—	0.25–0.49 (20°)
u	Ratio	—	z_gear/z_pinion
d	Pitch diameter	mm	m·z
Z_E	Elasticity factor	√MPa	≈189.8 steel-steel

Explanation

Bending stress scales with load and drops with face width, module and tooth shape (Y). Contact stress scales with load and the (u+1)/u ratio but only with √face and √diameter, so widening the face helps both but increasing the module helps bending far more than contact. A safety factor below 1 means failure is likely; aim for SF ≥ 1.25 (bending) and SH ≥ 1.1 (contact) in preliminary design.

Derivation

Torque T = 9549·P(kW)/n(rpm) [N·m]; F_t = 2T/(d/1000) [N] with d = m·z [mm]. Lewis: σ_F = F_t/(b·m·Y). Hertz: σ_H = Z_E·√(F_t/d·(u+1)/(b·u)); substituting d = m·z. SF = σ_Flim/σ_F, SH = σ_Hlim/σ_H. Required face: b_req,F = F_t/(m·Y·σ_Flim), b_req,H = [F_t(u+1)/(d·u)]/(σ_Hlim/Z_E)².

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Pitch diameter and torque.

d = m·z , T = 9549·P/n

m:3 mm

z:20

P:5 kW

n:1500 rpm

Intermediate: d = 3×20; T = 9549×5/1500

Result: d = 60 mm, T = 31.8 N·m

Step 2

Tangential load from torque.

F_t = 2T/d

T:31.8 N·m

d:60 mm

Intermediate: F_t = 2×31.83/(60/1000)

Result: 1,061 N

Step 3

Bending (Lewis) stress and safety factor.

σ_F = F_t/(b·m·Y) , SF = σ_Flim/σ_F

b:20 mm

Y:0.320

σ_Flim:250 MPa

Intermediate: σ_F = 1,061/(20×3×0.32)

Result: σ_F = 55.3 MPa, SF = 4.52

Step 4

Contact (Hertz) stress and safety factor.

σ_H = Z_E·√[F_t(u+1)/(b·d·u)] , SH = σ_Hlim/σ_H

u:2

σ_Hlim:700 MPa

Intermediate: σ_H = 189.8·√[1,061×3/(20×60×2)]

Result: σ_H = 218.6 MPa, SH = 3.20

Final Answer

SF = 4.52

Worked Examples

See how the calculator works with different scenarios.

Scenario

m=3, z=20, b=20, u=2, P=5 kW, n=1500, σ_Flim=250, σ_Hlim=700.

Inputs

m:3

z:20

b:20

u:2

P:5 kW

Calculation

d=60mm, T=31.8N·m, F_t=1060N; Y=0.320; σ_F=1060/(20×3×0.320)=55.2MPa, SF=4.53; σ_H=189.8·√(1060×3/(20×60×2))=75.3MPa, SH=9.30

Result

SF 4.5, SH 9.3

Interpretation

Well over-designed for these loads — could downsize.

Understanding Your Result

Your Result: 4.52

Typical range: Typical minimums: SF ≥ 1.25 (bending), SH ≥ 1.1 (contact). Your SF = 4.52, SH = 3.20. Both above 1 — safe with margin recommended.

What This Means

This 20-tooth, 3 mm module pinion under 5 kW at 1500 rpm sees bending σ_F = 55.3 MPa (SF 4.52) and contact σ_H = 218.6 MPa (SH 3.20).

Why It Matters

A gear that is geometrically correct can still break or pit if the teeth are too weak for the load. Checking both bending and contact stress prevents in-service tooth failure, which is catastrophic and hard to repair in the field.

Professional Interpretation

For certification use AGMA 2001/ISO 6336 with load factors. If SF<1, raise module or face (b_req,F = 4.4 mm) or use stronger material (σ_Flim). If SH<1, raise face or diameter (b_req,H = 2.0 mm).

Bending

55.3 MPa, SF 4.52

Contact

218.6 MPa, SH 3.20

F_t

1,061 N

Recommendations

•Bending stress is acceptable.
•Contact stress is acceptable.
•Prefer z ≥ 17 at 20° PA to avoid undercut (low Y).
•Use case-hardened steel for high σ_Flim/σ_Hlim.
•Verify with full AGMA/ISO rating before production.

Educational Content

Introduction

A gear can have the right ratio but still fail if its teeth are too weak. This calculator checks the two classic failure modes — root bending fatigue and flank contact (pitting) fatigue — using the Lewis and Hertz methods, and reports safety factors plus the face width you actually need.

Concept Explanation

Compute torque from power and speed, then tangential load F_t = 2T/d. Bending stress σ_F = F_t/(b·m·Y) uses the Lewis form factor Y (tooth shape). Contact stress σ_H = Z_E·√[F_t(u+1)/(b·d·u)] is the Hertz pressure where flanks touch. Dividing each by its allowable gives a safety factor; below 1 means likely failure.

Applications

▸Spur gear design sizing

▸Failure analysis

▸Material selection

▸Preliminary AGMA checks

▸Teaching gear strength

▸Retrofit / uprate studies

Benefits

•Both bending and contact
•Safety factors included
•Required face width output
•Lewis Y table (20°)
•SI/imperial consistent

Limitations

•Simplified (no K factors)
•20° full-depth only
•Low-speed assumption
•Steel-steel Z_E fixed
•Preliminary, not certified

Best Practices

•Aim SF ≥ 1.25, SH ≥ 1.1
•Avoid undercut (z ≥ 17)
•Use proper lubrication
•Validate with AGMA/ISO
•Widen face before module

Common Misconceptions

Myth: Right ratio = safe gear.

Truth: Teeth must also survive the load.

Myth: Module only sets size.

Truth: It strongly sets bending strength.

Myth: Contact stress is minor.

Truth: It causes pitting, a common failure.

Myth: This is full AGMA.

Truth: It is a simplified Lewis/Hertz estimate.

Practical Tips

•SF 4.52, SH 3.20.
•Increase module to boost bending strength fastest.
•Widen face to help both stresses.
•Use z ≥ 17 to avoid undercut (low Y).
•Validate with AGMA/ISO for production.

Common Mistakes

SF < 1

Raise module, face or material allowable.

Ignoring contact

Check SH too — pitting limits life.

Undercut pinion

Use z ≥ 17 or profile shift.

Wrong units

Keep b, m in mm; F_t in N; stress in MPa.

Treating as certified

Apply AGMA/ISO load factors for final design.

Assumptions

This calculator makes the following assumptions to provide estimates:

20° full-depth: Lewis Y from 20° table.

Low speed: K_v ≈ 1 (no dynamic factor).

Steel-steel: Z_E = 189.8 √MPa.

Proper lube: EHL film assumed.

Preliminary: No AGMA load factors.

Limitations

Be aware of these limitations when using this calculator:

No K factors: Application/load factors omitted.

One material pair: Z_E fixed for steel.

Spur only: No helix benefit modeled.

No misalignment: Ideal load share assumed.

Frequently Asked Questions

Answers to common questions about this calculator.

What is gear tooth strength?

Tooth strength is the load a gear tooth can carry before bending fatigue (root) or surface fatigue (pitting) occurs. It is checked by bending stress and contact stress.

What is the Lewis formula?

The classic bending estimate: σ = F_t / (b·m·Y), where F_t is the tangential load, b the face width, m the module and Y the Lewis form factor of the tooth.

What is the Lewis form factor Y?

A dimensionless number (≈0.25–0.49) describing tooth shape; larger teeth (higher z) have higher Y. This calculator interpolates the standard 20° full-depth table.

What is bending stress?

The tensile stress at the tooth root from the transmitted load: σ_F = F_t / (b·m·Y), in MPa when F_t is in N and b, m in mm.

What is contact (Hertz) stress?

The compressive stress where two tooth flanks touch: σ_H = Z_E·√[F_t·(u+1)/(b·d·u)], with u the ratio and Z_E a material factor (~190 for steel-steel).

What is a safety factor?

SF = allowable stress / actual stress. SF > 1 means the tooth is safe; 1.0–1.5 is typical for bending, 1.0–1.3 for contact in preliminary design.

What is the tangential load F_t?

The force at the pitch circle: F_t = 2T/d = 1000·P/v, where T is torque, d pitch diameter and v pitch-line speed.

How is torque from power and speed?

T = 9549·P/n (N·m) with P in kW and n in rpm. Then F_t = 2T/(d/1000).

What is the pitch-line velocity?

v = π·d·n/60 (m/s) with d in m; it sets the dynamic (velocity) factor in full AGMA rating.

What does module m mean?

The tooth size: pitch diameter = m·z. Larger module → bigger, stronger teeth but a larger gear.

What is face width b?

The tooth length along the axis. Wider faces carry more load but can misalign; b is usually 8–12·m.

What is the gear ratio u?

u = z_gear / z_pinion (the mesh ratio). Contact stress uses (u+1)/u, so a higher ratio (big gear) lowers pinion contact stress.

What is pitting?

Surface fatigue where tiny pits form on the flanks from repeated Hertz contact; it limits contact stress σ_H.

What is tooth bending fatigue?

Root cracks from repeated bending that can break a tooth; limited by bending stress σ_F.

What material allowables to use?

Through-hardened steel ≈ σ_Flim 200–350 MPa, σ_Hlim 550–800 MPa; case-carburized ≈ σ_Flim 300–500, σ_Hlim 1000–1500. Check the material data sheet.

Is this AGMA rating?

It is a simplified Lewis + Hertz estimate (no K_a, K_v, K_Hβ factors). Use AGMA 2001 / ISO 6336 for a certified design.

Why does my required face width matter?

b_req = F_t/(m·Y·σ_allow) for bending; if your b is below b_req the tooth is under-sized. Widen the face or increase module.

How do I increase strength?

Increase module (strongest lever), widen face, raise tooth count slightly (higher Y), or use a stronger/heat-treated material.

What is the difference from gear ratio calculator?

That gives speed/torque; this checks whether the teeth survive the load.

What about 25° pressure angle?

A 25° PA gives a slightly stronger tooth (higher Y and root) than 20°; this tool uses 20° full-depth values.

What is Z_E?

The elasticity factor in Hertz contact: √(1/π·((1−ν₁²)/E₁ + (1−ν₂²)/E₂)). About 190 √MPa for steel-steel.

Can I use this for helical gears?

Not directly — helical teeth have a modified (higher) capacity from the helix overlap; use the helical calculator or AGMA for those.

What is a good safety factor?

1.25–1.5 for bending and 1.1–1.3 for contact in normal industrial service; higher for shock or uncertain loads.

What causes tooth breakage?

Overload, excessive bending stress, a fault (inclusion), or a broken tooth from a jammed drive. Keep SF > 1 with margin.

What is undercut?

A weak root notch on small pinions (z < 17 at 20°) that reduces strength; avoid by more teeth or profile shift.

How does ratio affect contact stress?

σ_H ∝ √((u+1)/u); as u grows the term → 1, so very high ratios barely change pinion contact stress.

What lubrication matters?

An EHL film prevents scoring; the contact stress limit assumes proper lubrication and surface finish.

What is the dynamic factor?

At high pitch-line speed, tooth impact raises the effective load; full AGMA adds K_v. This tool assumes low-speed (K_v≈1).

What standards cover gear strength?

AGMA 2001 / 2101 (US) and ISO 6336 (international) for bending and contact rating.

How accurate is this estimate?

Within ~10–20% of a full AGMA calculation for low speed; it is for preliminary sizing, not final certification.

What if SF < 1?

The tooth will likely fail — increase module, face width or material allowable until SF > 1 (ideally with margin).

Does the calculator include both failure modes?

Yes — bending (root) and contact (pitting) with separate safety factors and required face widths.

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Related Guides

Deepen your understanding with these educational resources.

AGMA — Gear rating

Bending and contact rating.

ISO 6336 — Tooth strength

International gear rating.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Module (m): Tooth size; d = m·z.
Lewis Y: Tooth shape factor (0.25–0.49).
F_t: Tangential load at pitch circle.
σ_F: Bending (root) stress, MPa.
σ_H: Contact (Hertz) stress, MPa.
Z_E: Elasticity factor, ~190 √MPa (steel).
Safety Factor: Allowable / actual stress.

Authoritative References

Trusted sources and standards referenced by this calculator.

AGMA — Gear rating (2001/2101) ISO 6336 — Gear tooth strength MIT OpenCourseWare — Machine design Bühl / Niemann — Machine design Khurmi — Theory of Machines

Applicable Standards

AGMA 2001-D04AGMA 2101-D04ISO 6336DIN 3990

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Last Updated:

August 2026

Version:

2.1.0

References:

License:

Free to use for educational and professional purposes.