Gear stress analysis: Lewis equation vs AGMA J and I factors

Gear teeth fail in two main ways: they crack at the root from repeated bending, or the tooth faces pit from repeated contact. The Lewis equation estimates the first. The AGMA 2001 method rates both, using two geometry factors, J and I, and fatigue-based allowable stresses.

Two ways a gear tooth fails

• Bending fatigue. Each tooth is a short cantilever loaded once per revolution. The stress peaks in the root fillet, and after enough cycles a crack starts there and the tooth breaks off.
• Pitting (surface fatigue). Where two teeth touch, the contact patch carries very high stress just below the surface. Over many cycles small pieces flake out of the tooth face, the profile degrades, and noise and wear take over.

Which one limits a design depends on material and size. Hardened steel gears are usually limited by pitting; softer or smaller teeth more often by bending.

What is the Lewis bending equation?

In 1892 Wilfred Lewis treated a gear tooth as a cantilever beam with the whole load at its tip. The root bending stress is:

W_t is the tangential load at the pitch circle, P the diametral pitch (or m the module), F the face width, and Y the Lewis form factor, which depends only on the tooth shape. For 20° full-depth teeth (Shigley, Table 14-2):

Lewis is a quick sizing check, but it leaves out three things that matter: the stress concentration in the root fillet, the fact that a second tooth pair shares the load while the load is near the tip, and fatigue. It also says nothing about pitting.

How does AGMA calculate bending stress (the J factor)?

AGMA 2001 keeps Lewis's structure and adds the missing pieces:

• K_o overload: shocks from the driver and the load.
• K_v dynamic: extra load from tooth spacing errors at speed.
• K_s size, K_m load distribution across the face, K_B rim thickness.
• J the bending geometry factor, which replaces Y.

J is computed from the actual tooth (AGMA 908). The load sits at the highest point of single tooth contact, the highest point on the tooth where one pair carries the whole load, rather than at the tip. The critical section is where an inscribed Lewis parabola touches the root fillet. J then divides by a stress concentration factor for the fillet radius (Dolan and Broghamer's photoelastic results). Some AGMA 908 values for 20° full-depth gears cut with a 0.25-module tip radius:

J depends on the mating gear (it sets where single-tooth contact ends), on the cutter's tip radius, and on profile shift. Pinions below about 18 teeth at 20° are missing from the tables because they undercut; see undercut and profile shift.

How does AGMA calculate contact stress (the I factor)?

Contact stress comes from Hertz's theory of two curved surfaces pressed together:

d_P is the pinion's pitch diameter. C_p is the elastic coefficient of the two materials, about 2300 √psi (191 √MPa) for steel on steel. I is the pitting geometry factor, which describes how curved the two profiles are where contact stress peaks. At the pitch point it has a simple form:

Here m_G is the gear ratio (gear teeth over pinion teeth): + for external gears, − for internal. AGMA 908 evaluates the curvatures at the pinion's lowest point of single tooth contact instead, which gives slightly smaller and more accurate values. The minus sign for internal gears matters in planetary sets: a ring's concave teeth wrap around the planet, so I is much larger and contact stress lower than on the sun mesh.

How do you calculate a gear safety factor?

AGMA compares the stresses with fatigue strengths, not yield. For through-hardened steel the allowable stresses rise with Brinell hardness (Grade 1, 10⁷ cycles, 99% reliability):

The safety factors are S_F = S_t / σ for bending and S_H = S_c / σ_c for contact. Because contact stress grows with the square root of load, an S_H of 1.4 is a margin of about 2 on load. Compare S_F with S_H² to see which failure mode governs.

How are planetary gears rated differently?

• Load per planet. The tooth force is the torque divided by the number of planets and the driven radius: Wt = T / (N · r).
• Unequal load sharing. Planets never share load perfectly. A mesh load factor K_γ covers this. Published guide values are 1.10 for 3 planets, 1.25 for 4, 1.35 for 5 and about 1.44 for 6.
• Reversed bending on planets. Each planet tooth is pushed on one flank by the sun and on the other by the ring, so it sees fully reversed bending. AGMA reduces its allowable bending stress to 70%.
• Two meshes per planet. The planet is checked against both. The external sun mesh usually has the higher contact stress.
• The internal ring. AGMA has not established a bending J method for internal gears. A common approximation treats the ring tooth as a rack tooth meshing with the planet; GearStudio does this and marks the result as an estimate.
• Helical sets. AGMA 908 rates a helical tooth as a virtual spur gear in the normal plane, with z / cos³ψ teeth. When the face is longer than one axial pitch (axial contact ratio above 1), several oblique contact lines share the load, which raises both J and I. A narrower face is rated as low axial contact ratio and gains little. GearStudio applies this to helical designs.

Worked example

GearStudio's default set: sun 19, planets 26 (three of them), ring 71, 20 DP, 20°, 0.5 in face width, 50 in·lbf on the sun, K_o 1.25, K_m 1.3, K_v 1.2, K_s 1.0. Sun and planets are 360 HB steel, the ring 150 HB.

1. Tooth force per planet: 50 / (3 × 0.475) = 35.09 lbf, times K_γ 1.10 gives Wt = 38.60 lbf.
2. Sun bending, J = 0.341: σ = 38.60 × 1.95 × 20 / (0.5 × 0.341) = 8.83 ksi. Allowable at 360 HB is 40.6 ksi, so S_F = 4.60.
3. Sun–planet contact, I = 0.0848, d_P = 0.95 in: σc = 2291 × √(38.60 × 1.95 / (0.95 × 0.5 × 0.0848)) = 99.0 ksi. Allowable is 145.0 ksi, so S_H = 1.46, or 2.14 on load.
4. Planet bending, J = 0.368: 8.19 ksi against 70% of 40.6 = 28.4 ksi, so S_F = 3.47.
5. Ring: bending 6.15 ksi (S_F 3.97); contact 53.6 ksi against 77.4 ksi, S_H 1.44 (2.08 on load).

Contact governs: the smallest margin on load is 2.08, at the ring. Lewis alone would have reported a comfortable bending margin and missed that the tooth faces are the limit.

This is a sizing method, not a certified AGMA or ISO 6336 rating. Check critical designs independently before manufacture.

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