Gear Strength Calculator — Lewis Bending Stress, AGMA & Surface Durability
Calculate gear tooth bending stress using the Lewis equation and AGMA method. Check tooth strength against material allowable and predict surface pitting life.
Quick Answer
For a 20-tooth spur gear, module 4, 20° pressure angle, transmitting 10 kW at 1450 RPM: Lewis Bending Stress ≈ 98 MPa, AGMA Bending Stress ≈ 112 MPa. For case-hardened 8620 steel (σ_FP = 450 MPa), safety factor ≈ 4.0. Tooth bending passes; check surface pitting separately.
How Gear Strength Calculations Work
Gear teeth fail two ways: bending fatigue (tooth breaks off) and surface pitting (material flakes from contact stress). Both must be checked.
1. Lewis Bending Equation (1893, Still Used)
σ_b = F_t / (b × m × Y), where F_t = tangential force, b = face width, m = module, Y = Lewis form factor (depends on tooth count). The Lewis equation treats the tooth as a cantilever beam — it’s simple but neglects stress concentration at the root fillet and dynamic effects.
2. AGMA Bending Stress
σ = F_t/(b×m) × K_v × K_o × K_s × K_m / (J), where K_v=dynamic factor, K_o=overload, K_s=size, K_m=load distribution, J=geometry factor (enhanced Lewis). AGMA is the industry standard — it accounts for manufacturing, dynamics, and misalignment.
3. Surface Pitting (Contact Stress)
The Hertzian contact stress at the tooth flank determines pitting life. For through-hardened gears, this often governs over bending strength. For case-hardened gears, bending fatigue usually governs because the case resists pitting.
Common Mistakes
- Using Lewis without dynamic factor — Lewis is static. Real gears have dynamic loads from tooth errors and impact. AGMA adds K_v (0.5-1.5× depending on quality and pitch line velocity). Skip K_v and you overestimate strength.
- Not checking both bending and pitting — They are independent failure modes. A gear that passes bending may fail surface pitting at 10⁷ cycles. For through-hardened gears <350 HB, pitting almost always governs. Check both.
- Assuming face width carries load evenly — Misalignment concentrates load on one end of the tooth. AGMA’s K_m factor (1.0-2.0+) accounts for this. For wide face widths (b/m > 12), K_m > 1.5 — crown the teeth (lead modification) to compensate.
- Using wrong material allowable stress — Allowable bending stress depends on hardness, not just alloy. 4140 at 300 HB has different σ_FP than at 400 HB — the hardenability matters. Use AGMA material tables, not generic alloy specs.
- Ignoring number of load cycles — Gear strength is fatigue-based. A gear designed for 10⁷ cycles may have half the capacity of one designed for 10⁴ cycles. Be explicit about required life; don’t default to infinite.
Frequently Asked Questions
What is the difference between the Lewis and AGMA methods?
Lewis (1893): static bending only, single factor Y, simple — good for preliminary sizing. AGMA (current): dynamic + overload + size + load distribution + stress concentration (J factor instead of Y), plus surface durability. AGMA is for detailed design; Lewis for “rough size.”
How do I estimate the dynamic factor K_v?
K_v = (A + √v) / A, where A = 50 + 56(1 − B), B = (12 − Q_v)^(2/3)/4, Q_v = AGMA quality number (3-12). For Q_v=8 (typical industrial): v=5 m/s → K_v≈1.12, v=20 m/s → K_v≈1.35. Higher quality gears have lower K_v — investment in quality pays back in higher capacity.
What face width should I use?
Rule of thumb: 8m ≤ b ≤ 16m for spur gears, 20m-30m for helical. Narrower: wasted capacity. Wider: load distribution problems, edge loading, manufacturing difficulty. For very high power density, go wide but add lead crowning.
How does pressure angle affect gear strength?
20° pressure angle is standard — good balance of strength and contact ratio. 25°: thicker tooth base, ~15% higher bending strength, but higher bearing loads. 14.5°: obsolete, thinner teeth, used only for replacement gears. Increasing pressure angle strengthens teeth but increases radial bearing load.
What gear material and heat treatment should I choose?
Through-hardened 4140/4340 (300-350 HB): general industrial, good machinability. Case-hardened 8620/9310 (58-62 HRC case, 30-42 HRC core): high strength + tough core, for high power density. Nitrided 4140: minimal distortion, lower case depth, for precision gears. Induction hardened 1045: cheap, for low-cost gears. The choice depends on required σ_FP (bending allowable) and σ_HP (contact allowable).
How do I estimate gear life in revolutions?
Bending: N = (σ_FP / σ)^m × N₀, where m=8.7 for through-hardened, m=15.7 for case-hardened (steeper S-N curve). Contact: N = (σ_HP / σ_c)^n × N₀, where n≈6.6. At 1/2 allowable stress, life extends by factor 2^8.7 = 400× for bending. That’s the power of overdesign.
Should I use spur or helical gears?
Helical: smoother (gradual engagement), quieter, ~20% stronger for same size (longer contact line), but generates thrust load (needs thrust bearing). Spur: simpler, no thrust, cheaper, but noisier and lower capacity. For >5 m/s pitch line velocity, helical almost always wins. For low-speed positioning, spur is fine.
How do I use this with our Gear Ratio Calculator?
Gear ratio gives you the tooth counts and speeds. Gear strength takes those inputs and checks whether the teeth can handle the torque. Use ratio first to set tooth counts and module, then strength to verify they survive. The calculators complement each other — ratio for kinematics, strength for durability.