AGMA bending stress equation and its factors

The AGMA method calculates the actual bending stress at the tooth root using the Lewis idea, but adds factors for the real operating conditions. In SI units, for spur gears,

$$\sigma = \frac{W_t\, K_o\, K_v\, K_s\, K_m\, K_b}{b\, m\, J}$$

where $W_t$ is the tangential load (N), $b$ the face width (mm), $m$ the module (mm) and $J$ the geometry factor.

Geometry factor $J$

$J$ replaces the Lewis form factor. It accounts for:

• the shape of the tooth (number of teeth, pressure angle, addendum), as the Lewis factor does;
• the stress concentration at the root fillet ($K_f$), because the fillet raises the local stress above the nominal beam stress;
• the load sharing between teeth (for spur gears the critical load acts not at the tip but at the highest point of single-tooth contact, because the load is shared in the double-contact zone).

$J$ is read from charts for the given numbers of teeth on the pinion and gear; it is lower for small pinions. Because the AGMA equation uses the module rather than the circular pitch, $J$ is comparable to $Y = \pi y$ in the Lewis form $W_t = \sigma b m Y$, not to $y$ itself. For a 20-tooth pinion, $J$ is about 0.3 to 0.35, similar to $\pi y \approx 0.34$: the root stress concentration lowers it, while the higher load position (load sharing) raises it.

Other factors

Strength and factor of safety

The allowable bending stress depends on the material, heat treatment and hardness; for example, for Grade 1 through-hardened steel $S_t = 0.533\,\text{HB} + 88.3$ MPa. It is corrected for the required life and the operating conditions by the life factor $Y_N$, temperature factor $K_T$ and reliability factor $K_R$:

$$n = \frac{S_t\, Y_N}{K_T\, K_R\, \sigma}$$

$Y_N$ is about 1 at $10^7$ cycles and falls as the number of cycles increases ($Y_N = 1.3558\,N^{-0.0178}$ for $N > 10^7$). The safety factor $n$ must be above 1; values of 1.5 to 3 are commonly used.