Gear Tooth Bending Stress (Lewis)
The tooth-strength check the spur-gear-geometry tile leaves out. Lewis beam strength treats the tooth as a cantilever loaded by the tangential load Wt at the pitch line: sigma = Wt / (F pc y), with face width F, circular pitch pc = pi/Pd, and the Lewis form factor y = a - b/T for the tooth system (20 deg full depth a,b = 0.154, 0.912; 14.5 deg full depth 0.124, 0.684; 20 deg stub 0.175, 0.841). The diametral-pitch form Y = pi y gives the same stress as Wt Pd/(F Y). A 500 lb tangential load on an 8-pitch, 1.5 in wide, 20-tooth 20 deg gear runs about 7,800 psi. Static Lewis stress only: the Barth velocity factor and the AGMA geometry (J) and load factors are not modeled, so it runs optimistic at speed. Companion to spur-gear-geometry; AGMA 2001 and the gear maker govern.
Formula and source
sigma = Wt / (F pc y); pc = pi/Pd; y = a - b/T (20 deg full depth a,b = 0.154, 0.912; 14.5 deg full depth 0.124, 0.684; 20 deg stub 0.175, 0.841); Y = pi y gives sigma = Wt Pd/(F Y).
Lewis beam-strength equation (Wilfred Lewis, 1892; public domain) with the standard per-system form-factor closed forms (Shigley / Machinery's Handbook), by name.
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