Gear Bending Stress (Lewis)⚠ unverified

Understanding the Parameters

• Tangential load $W_t$ — the useful tooth force that transmits torque, $W_t = T/(d/2) = 2T/d$. It bends the tooth; stress is directly proportional to it.

• Diametral pitch $P_d$ — teeth per unit pitch diameter; larger $P_d$ means smaller, finer teeth. Since a smaller tooth is a shorter, thinner cantilever with a smaller section, stress rises with $P_d$. (In SI-module form the equivalent is $\sigma = W_t/(F\,m\,Y)$ with module $m = 1/P_d$.)

• Face width $F$ — the tooth width carrying the load; more width spreads the load, so stress falls as $1/F$. Face width is typically limited to 3–5× the circular pitch to avoid uneven load across the tooth.

• Lewis form factor $Y$ — the tooth-shape geometry factor (from Lewis Form Factor); higher $Y$ (more teeth, larger pressure angle) gives a stronger tooth and lower stress.

• Velocity factor $K_v$ — inflates the stress to account for dynamic tooth loads at speed (impact as teeth engage, tooth errors, deflection). $K_v = 1$ at rest and grows with pitch-line velocity; it is the Lewis equation's crude nod to what AGMA treats with a detailed dynamic factor.

Derivation (Approaching a Proof)

Model the tooth as a cantilever beam of uniform strength built into the gear body, loaded by the tangential force $W_t$ at its tip. The bending stress at the critical root section, from the flexure formula (Beam Bending Stress) with critical thickness $t$ and load height $h$, is

$$\sigma = \frac{M}{S} = \frac{W_t h}{F t^2/6} = \frac{6 W_t h}{F t^2}.$$

Lewis inscribed a parabola of constant stress in the tooth to locate the weakest section, then absorbed the geometry into the dimensionless form factor $Y$ ($t^2/6h$ expressed per unit pitch; see Lewis Form Factor). This collapses the stress to

$$\sigma = \frac{W_t}{F\, m\, Y} = \frac{W_t P_d}{F\, Y},$$

using module $m = 1/P_d$. Finally, multiplying by the empirical velocity factor $K_v$ to account for dynamic overload at speed gives the working form

$$\sigma = \frac{W_t P_d K_v}{F\, Y}.$$

Everything of substance is in the cantilever model; $Y$ carries the tooth geometry and $K_v$ the dynamics. Modern AGMA refines this into $\sigma = W_t K_o K_v K_s \dfrac{P_d}{F}\dfrac{K_m K_B}{J}$ with the geometry factor $J$ replacing $Y$ and separate overload/size/load-distribution factors, but the Lewis skeleton is unchanged.

Dimensional check. $\left[\dfrac{W_t P_d}{F Y}\right] = \dfrac{\text{N}\cdot(1/\text{m})}{\text{m}} = \dfrac{\text{N}}{\text{m}^2} = \text{Pa}$ ($K_v$, $Y$ dimensionless). ✓

History and Development

Wilfred Lewis presented this equation to the Engineers' Club of Philadelphia in 1892 — the first rational method for gear-tooth strength, replacing rules of thumb. It remains the pedagogical foundation of gear design and the ancestor of the AGMA/ISO bending-strength standards used today. It is conservative for slow gears and augmented by the velocity/dynamic factor for faster ones.

Related Concepts: Lewis Form Factor, Gear Contact Stress, Beam Bending Stress, Section Modulus, Fatigue Endurance Limit, Thread Bending Stress

Notes: Bending (root) fatigue check — pair with Gear Contact Stress for pitting. Use module form $\sigma = W_t/(F m Y)$ in SI. $Y$ from Lewis Form Factor; $K_v$ grows with pitch-line velocity. Superseded in detail by AGMA (geometry factor $J$).