AGMA Gear Pitting Contact Stress (sigma_c) and Elastic C_p Formula

Contrast SQRT(W_t) Hertzian surface pitting against linear Lewis tooth-root bending

Spur and helical gears fail by two distinct mechanisms: root fatigue fracture (governed linearly by tangential load W_t via Lewis bending) and pitch-line surface pitting fatigue (governed by Hertzian cylinder-on-cylinder contact stress sigma_c = C_p * SQRT(W_t * K_all / (b * d_p * I_geom)), where elastic coefficient C_p = SQRT(1 / (PI() * ((1 - nu_1^2)/E_1 + (1 - nu_2^2)/E_2)))). Because load sits under a square root, quadrupling W_t doubles Hertzian contact stress!

Verify steel-on-steel gears (E_1 = E_2 = 207,000 MPa in B2:B3, nu_1 = nu_2 = 0.30 in B4:B5): C_p evaluates to 190.2719 sqrt(MPa). Under tangential load W_t = 3,600 N (B6), combined overload/dynamic/alignment factor K_all = 1.50 (B7), face width b = 25 mm (B8), pinion pitch diameter d_p = 60 mm (B9), and AGMA geometry factor I_geom = 0.10 (B10), the radical equals SQRT(5400 / 150) = SQRT(36) = 6.0000, yielding sigma_c = 1,141.6311 MPa! Pairing the steel pinion with a cast-iron gear (E_2 = 100,000 MPa, nu_2 = 0.21) widens the contact patch and lowers C_p to 151.0282 sqrt(MPa) (sigma_c = 906.1692 MPa).

AGMA Elastic Coefficient C_p (sqrt(MPa)) Formula

Compute C_p directly from pinion and gear Young’s moduli (E_1, E_2 > 0) and Poisson ratios (0 <= nu_1, nu_2 < 0.5) rather than relying on rounded lookup tables.

=LET(e_1, B2, e_2, B3, nu_1, B4, nu_2, B5, IF(OR(e_1 <= 0, e_2 <= 0, nu_1 < 0, nu_1 >= 0.5, nu_2 < 0, nu_2 >= 0.5), "Invalid AGMA Pitting Input", ROUND(SQRT(1 / (PI() * ((1 - nu_1^2) / e_1 + (1 - nu_2^2) / e_2))), 4)))

AGMA Hertzian Pitting Contact Stress sigma_c (MPa) Formula

Multiply exact unrounded C_p by SQRT((W_t * K_all) / (b_face * d_pin * I_geom)) inside LET for full 4-decimal fidelity.

=LET(e_1, B2, e_2, B3, nu_1, B4, nu_2, B5, w_t, B6, k_all, B7, b_fc, B8, d_pn, B9, i_gm, B10, IF(OR(e_1 <= 0, e_2 <= 0, nu_1 < 0, nu_1 >= 0.5, nu_2 < 0, nu_2 >= 0.5, w_t < 0, k_all < 1, b_fc <= 0, d_pn <= 0, i_gm <= 0), "Invalid AGMA Pitting Input", LET(cp_el, SQRT(1 / (PI() * ((1 - nu_1^2) / e_1 + (1 - nu_2^2) / e_2))), ROUND(cp_el * SQRT((w_t * k_all) / (b_fc * d_pn * i_gm)), 4))))

Further reading

• Arte Sheets Lesson — Lewis Gear-Tooth Bending Stress and Barth Velocity Factor
• Arte Sheets Lesson — Dowson-Higginson EHL Line-Contact Minimum Film Thickness