Lewis Gear-Tooth Bending Stress and Barth Velocity Factor Formula

Model a loaded spur-gear tooth as a parabolic-inscribed cantilever beam

Wilfred Lewis (1892) revolutionized gear sizing by modeling a spur-gear tooth under tangential pitch-circle force W_t (in Newtons) as a cantilever beam of face width b (mm) and metric module m (mm, where pitch diameter d_p = m * N_teeth). Static root bending stress is sigma_static = W_t / (b * m * Y_Lewis), where dimensionless Lewis form factor Y_Lewis (e.g., 0.320 for a 20-tooth 20-degree full-depth involute pinion) captures tooth root geometry.

Because pitch-line impact loads grow with pitch-circle velocity V_m_s = PI() * (d_p / 1000) * (RPM / 60), Barth’s metric dynamic factor amplifies tooth load via K_v = (6.1 + V_m_s) / 6.1 for cut commercial gears! Verify by hand for module m = 4 mm (B2), N_teeth = 20 (B3, so d_p = 80 mm), face width b = 25 mm (B4), tangential load W_t = 3,200 N (B5), Y_Lewis = 0.320 (B6), and pitch-line velocity V_m_s = 6.10 m/s (B7), where Barth K_v = (6.1 + 6.1) / 6.1 = 2.0000 exact: static stress is 3,200 / (25 * 4 * 0.320) = 3,200 / 32 = 100.00 MPa, and dynamic tooth root stress is 2.0000 * 100.00 = 200.00 MPa!

Static Lewis Cantilever Tooth Root Stress (MPa) Formula

Remember that 1 N / mm^2 = 1 MPa identically, so dividing Newtons W_t by face width b (mm) * module m (mm) * Y_Lewis outputs megapascals directly with zero power-of-10 conversion bugs!

=IF(OR(B2 <= 0, B4 <= 0, B5 < 0, B6 <= 0), "Invalid Gear Geometry", ROUND(B5 / (B4 * B2 * B6), 4))

Barth Velocity-Amplified Dynamic Tooth Bending Stress Formula

Multiply static Lewis stress by Barth’s cut-gear velocity factor (6.1 + V_m_s) / 6.1. Verify quasi-static operation (set pitch-line speed B7 = 0): K_v reduces to 1.0000 and dynamic bending stress equals 100.0000 MPa.

=LET(mod_m, B2, b_fw, B4, wt_n, B5, y_lew, B6, v_ms, B7, IF(OR(mod_m <= 0, b_fw <= 0, wt_n < 0, y_lew <= 0, v_ms < 0), "Invalid Gear Geometry", LET(k_dyn, (6.1 + v_ms) / 6.1, ROUND((k_dyn * wt_n) / (b_fw * mod_m * y_lew), 4))))

Further reading

• Arte Sheets Lesson — Tallian Specific Film Thickness and Stribeck Regime
• Arte Sheets Lesson — Bredt-Batho Closed Thin-Walled Tube Torsion Constant