Lewis Bending Stress Formula

The Lewis equation is the fundamental method for estimating the bending stress at the root of a gear tooth. It treats the tooth as a simple cantilever beam loaded at the tip.

The Formula

σ = F_t / (b × m × Y)

Where:

• σ — Bending stress at the tooth root (MPa)
• F_t — Tangential force on the tooth (N) = Power / (pitch line velocity)
• b — Face width of the gear (mm)
• m — Module (mm)
• Y — Lewis form factor (depends on tooth count and pressure angle)

Velocity Factor (Barth Equation)

K_v = 6 / (6 + v) for v in m/s. This dynamic factor accounts for impact loading at higher speeds.

Design Criteria

• The actual bending stress must be less than the allowable stress divided by the safety factor
• Typical safety factors: 1.5–2.0 for general use, 2.5–3.0 for critical applications
• Face width is typically 8–12 times the module for spur gears