Gear Strength and Design Calculations
Tooth forces, bending stress by the Lewis equation, contact stress, safety factors and sizing a gear pair for a given power.
From power to tooth force
The calculation starts with the torque and the tooth forces. The torque on a gear is T = 9.549 × P ÷ n (N·m, P in watts and n in rpm). The force that pushes the teeth round, the tangential force, is Ft = 2 T ÷ d (with d the pitch diameter), or equivalently Ft = P ÷ V with V the pitch-line speed V = π d n ÷ 60. The radial force Fr = Ft tan φ pushes the gears apart and, together with the thrust of helical teeth, is what the shafts and bearings must carry (see Shafts, Keys and Couplings and Bearings and Bearing Selection).
Bending stress: the Lewis equation
Wilfred Lewis (1892) treated the tooth as a cantilever beam loaded at its tip. In metric form the bending stress at the root is σ = Ft ÷ (b × m × Y), with b the face width, m the module and Y the Lewis form factor that depends on the number of teeth. A tooth with fewer teeth is thinner at the root, so Y is smaller and the stress larger.
| Teeth z | 12 | 15 | 17 | 20 | 24 | 30 | 40 | 60 | 100 | Rack |
|---|---|---|---|---|---|---|---|---|---|---|
| Y | 0.245 | 0.290 | 0.303 | 0.322 | 0.337 | 0.359 | 0.389 | 0.422 | 0.447 | 0.485 |
Dynamic and application factors
The static Lewis stress is raised by two factors. The dynamic factor Kv allows for the impacts of teeth meeting with small errors: for cut or milled teeth Kv = (6.1 + V) ÷ 6.1 (V in m/s), and it rises with speed (a faster gear needs better accuracy). The application (overload) factor Ka allows for the load being greater than the nominal: 1.0 for a uniform load and smooth source, 1.25 for light shock, 1.5 to 1.75 for moderate shock, 2 or more for heavy shock (a crusher, a reciprocating drive). The corrected stress is σ = Kv Ka Ft ÷ (b m Y).
Worked example: checking a 1.5 kW pinion
- Load. A motor of 1.5 kW at 1 500 rpm gives T = 9.549 × 1 500 ÷ 1 500 = 9.55 N·m. The pinion has m = 2 mm and z = 20 (d = 40 mm), driving a 50-tooth gear. Ft = 2 × 9.55 ÷ 0.040 = 477 N.
- Speed and factors. V = π × 0.040 × 1 500 ÷ 60 = 3.14 m/s, so Kv = (6.1 + 3.14) ÷ 6.1 = 1.52. For light shock take Ka = 1.25.
- Bending stress. With a face width b = 8 m = 16 mm and Y = 0.322 (z = 20): σ = 1.52 × 1.25 × 477 ÷ (16 × 2 × 0.322) = 88 MPa for the pinion; the 50-tooth gear (Y = 0.409) sees 69 MPa. The pinion is the weaker element.
- Safety factor. Against an allowable bending stress of 300 MPa (through-hardened alloy steel, see below) the factor is 300 ÷ 88 = 3.4: adequate (the usual minimum is about 1.4 to 2).
Contact stress and surface durability
Bending breaks a tooth; contact (Hertz) stress pits and wears its surface. Two teeth touching are like two cylinders pressed together, and the contact stress is high and acts at the surface. For a pair of steel gears a simplified form is σH = ZE ZH √(Ft (u + 1) ÷ (b d1 u)), with u = z2 ÷ z1, ZE = 189.8 √MPa the elasticity factor of steel on steel, and ZH = √(2 ÷ (sin φ cos φ)) = 2.49 for a 20° pressure angle. The factors Ka and Kv are applied as √(Ka Kv).
Worked example: contact stress of the same pair
- u = 2.5. σH (nominal) = 189.8 × 2.49 × √(477 × 3.5 ÷ (16 × 40 × 2.5)) = 484 MPa.
- With the factors √(1.25 × 1.52) = 1.38: σH = 666 MPa.
- Typical allowable contact stresses (endurance, indicative) are 500 to 700 MPa for through-hardened steel (about 300 HB) and 1 200 to 1 500 MPa for case-hardened and ground teeth. The through-hardened pair is close to its limit (666 vs 650 MPa): contact, not bending, governs. A case-hardened pair has a large reserve (666 vs 1 300 MPa), which is why case hardening allows a much smaller gear.
- Both must be checked. Contact stress rises only with the square root of the load, but the material limits are so close that for hardened gears it is usually the surface that fails first (pitting), while for soft, large-module or plastic gears it is the root.
Allowable stresses and safety factors
| Material | Allowable bending stress (MPa, indicative) | Allowable contact stress (MPa, indicative) |
|---|---|---|
| Case-hardened and ground steel | 350 to 450 | 1 200 to 1 500 |
| Nitrided steel | 300 to 400 | 1 000 to 1 200 |
| Through-hardened steel (about 300 HB) | 200 to 300 | 500 to 700 |
| Normalized or annealed steel | 120 to 180 | 350 to 450 |
| Cast iron (grade 40) | 60 to 90 | 350 to 450 |
| Bronze (worm wheels) | 70 to 100 | Depends on the lubricant and the worm |
| Nylon, acetal (POM) | 20 to 35 (reduce with temperature) | — |
- These are endurance stresses for infinite or very long life, with reliability around 99 percent. They are not the strength at the first load: the full ISO and AGMA methods use the material’s endurance limit and many modifying factors (see Fatigue and Stress Concentration).
- Safety factors: a typical minimum is 1.3 to 1.6 for bending and 1.0 to 1.3 for contact, with more for critical, shock-loaded or uncertain applications. Failure by tooth breakage is more serious than pitting, so the bending factor is larger.
- Plastic gears creep and lose strength with temperature, moisture and time: use the supplier’s curves and a generous safety factor, and design for a lower pitch-line speed (see Gear Types and Selection).
Sizing a gear pair step by step
- 1. Torque and ratio: from the motor and the load, with the service factor Ka (see Sizing a Motion Axis).
- 2. Choose the type and the material (see Gear Types and Selection).
- 3. Guess the module and the tooth counts: a pinion of 17 to 25 teeth, and a first module from a catalog rating or by trying standard modules in the bending check below (start with a face width of 8 m).
- 4. Compute d, V, Ft, Kv.
- 5. Check the bending stress of the pinion and the gear, and the contact stress; compare with the allowable values.
- 6. Adjust: raise the face width (to about 8 to 12 m), the module, the hardness or the accuracy until both safety factors are met with the smallest drive.
- 7. Check the rest: shafts, keys and bearings for the forces (see Shafts, Keys and Couplings and Bearings and Bearing Selection), lubrication and heat (Lubrication, Wear and Gear Failures).
Worked example: the same load in acetal
A plastic gear of the same geometry (acetal, allowable bending 25 MPa) needs the face width b = Kv Ka Ft ÷ (m Y σall) = 1.52 × 1.25 × 477 ÷ (2 × 0.322 × 25) = 56 mm: more than three times the face width of the steel gear (16 mm), and a much larger module in practice. Plastic gears suit light loads; this 1.5 kW drive would need steel.
Standard methods: AGMA and ISO
The Lewis equation is the starting point. Real designs use the full methods of ISO 6336 (calculation of load capacity of spur and helical gears) or AGMA 2001 / 2101, which add many factors: face-load distribution, mounting and shaft deflection, the geometry factor for the real tooth and the load-sharing of the contact ratio, surface finish, lubricant film, size, temperature, reliability and the number of load cycles (the S–N behavior of the material). The results, with the same basic idea, give a rating that is the allowable load for a stated life. Use software (KISSsoft, MITcalc, Romax, Gleason) or a manufacturer’s catalog for a final design; the calculations above are suitable for first sizing, checks and understanding.
Design checks and good practice
- Face width: 8 to 12 times the module is the usual range. A very wide gear loads the ends of the tooth unevenly when the shafts bend (use crowning, or a narrower gear with a larger module).
- Hardness difference: make the pinion harder than the gear (by 30 to 50 HB for through-hardened sets) so that it wears in the gear and balances the life.
- Accuracy and finish: a better accuracy grade lowers Kv and noise; a ground gear can run at much higher speed.
- Shaft and housing stiffness: bending of the shaft misaligns the teeth and concentrates the load.
- Tooth root fillet: a generous root radius reduces the stress concentration (see Fatigue and Stress Concentration).
- Lubrication: the film must be sufficient for the speed and load (see Lubrication, Wear and Gear Failures).
- Test: check the contact pattern with marking compound under load; a good pattern covers 70 to 90 percent of the face width centered on the tooth.
Related
- Gears and Power Transmission: the overview of this guide.
- Gear Basics and Terminology and Gear Types and Selection: geometry and the choice of type.
- Gear Trains, Ratios and Efficiency: multi-stage drives and efficiency.
- Lubrication, Wear and Gear Failures: failure modes: pitting, scuffing and breakage.
- Fatigue and Stress Concentration and the Material Properties page.
- Gear Ratio, Torque and Speed Generator.
FAQ
What is the Lewis form factor?
A number (about 0.25 to 0.48) that depends on the number of teeth and describes the shape of the tooth: the more teeth the thicker the root and the larger the factor. It is used in σ = Ft ÷ (b m Y) to find the bending stress at the root.
Which fails first: bending or contact?
For hardened gears with a good case, usually the surface (pitting). For softer or larger-module gears, plastic gears and shock-loaded drives, the root can fail first. Always check both.
How wide should a gear be?
Typically 8 to 12 times the module. Wider is stronger but needs stiff shafts and accurate mounting, or the load concentrates at one end of the tooth.
Are these formulas good enough to design a gearbox?
They are good for first sizing and for checking a catalog gear. A gearbox that people depend on should be rated with ISO 6336 or AGMA 2001 and tested, with a safety factor appropriate to the risk.
References
- Shigley, J. E. and Mischke, C. R., Mechanical Engineering Design, McGraw-Hill: Lewis equation, dynamic factor, form factor table.
- ISO 6336-1 to -3, Calculation of load capacity of spur and helical gears; AGMA 2001-D04 and 2101-D04.
- Lewis, W., Investigation of the Strength of Gear Teeth, 1892 (the original paper); Dudley, D. W., Handbook of Practical Gear Design.
Last reviewed: October 2026. The calculations are simplified for teaching and first sizing; a final design must follow ISO 6336 or AGMA and be checked by test.