Gear Tooth Bending Stress (Lewis)

ScopeLewis 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, so it runs optimistic at speed; the Barth velocity factor has its own solver, and the AGMA geometry (J) and load factors are not modeled. Companion to spur gear geometry; AGMA 2001 and the gear maker govern.

Formula or tablesigma = 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).

Source and editionLewis beam-strength equation (Wilfred Lewis, 1892; public domain) with the standard per-system form-factor closed forms (Shigley / Machinery's Handbook), by name.

Free accessThe Lewis equation and the form-factor closed forms are standard published gear-design theory (public domain / Machinery's Handbook).

What governsEstimate. AHJ and licensed professional govern.

API field namestransmitted_load_lb, diametral_pitch_1_in, face_width_in, number_of_teeth, tooth_system, bending_stress_psi, lewis_form_factor_y, circular_pitch_in, lewis_Y_diametral

• Form factor y = a - b/T closed form (circular-pitch convention); Y = pi y is the diametral-pitch formLewis / Shigley
• Static stress only no Barth velocity factor (gear-dynamic-tooth-stress applies it), no AGMA J or load-distribution factorsscope of this tile
• Tooth system 20 deg full depth, 14.5 deg full depth, or 20 deg stubstandard involute proportions