# Bending stresses in Gears

Gear Stresses

Gears experience two principal types of stresses; bending stress at the root of the teeth due to the transmitted load and contact stresses on the flank of the teeth due to repeated impact, or sustained contact, of one tooth surface against another. A simple method of calculating bending stresses is presented below.

Bending Stresses

The calculation of bending stress in gear teeth can be based on the Lewis formula

$\sigma&space;=\frac{W_{t}}{FmY}$

where Wt = transmitted load (N), F = face width (m or mm), m = module (m or mm), and Y = the Lewis form factor and can be found from the below table.

When teeth mesh, the load is delivered to the teeth with some degree of impact. The velocity factor is used to account for this and is given, in the case of cut or milled profile gears, by the Barth equation.

$K_{v}=\frac{6.1}{6.1+V}$

where V is the pitch line velocity which is given by

$V=\frac{d}{2}\times&space;10^{-3}n\frac{2\pi&space;}{60}$

where d is in mm and n is in rpm.

Introducing the velocity factor into the Lewis equation gives

$\sigma&space;=\frac{W_{t}}{K_{v}FmY}$

This equation forms the basis of a simple approach to the calculation of bending stresses in gears.

 N, Number of Teeth Y Φ = 20° a = 0.8 m b = m Y Φ = 20° a = m b = 1.25 m 12 0.33512 0.22960 13 0.34827 0.24317 14 0.35985 0.25530 15 0.37013 0.26622 16 0.37931 0.27610 17 0.38757 0.28508 18 0.39502 0.29327 19 0.40179 0.30078 20 0.40797 0.30769 21 0.41363 0.31406 22 0.41883 0.31997 24 0.42806 0.33056 26 0.43601 0.33979 28 0.44294 0.34790 30 0.44902 0.35510 34 0.45920 0.36731 38 0.46740 0.37727 45 0.47846 0.39093 50 0.48458 0.39860 60 0.49391 0.41047 75 0.50345 0.42283 100 0.51321 0.43574 150 0.52321 0.44930 300 0.53348 0.46364 Rack 0.54406 Note: a = addendum; b = dedendum; Φ = pressure angle, and m = module

References:

CHILDS, PETER R. N. MECHANICAL DESIGN ENGINEERING HANDBOOK. BUTTERWORTH-HEINEMANN LTD, 2018.