= Gear contact-stress load factor
{title2=$\sqrt{K_{H\beta}}$}
The \b[gear contact-stress load factor] $\sqrt{K_{H\beta}}$ is the multiplier associated with the face-load term in a contact-stress calculation where <Hertzian contact> pressure scales with the square root of local line load. It distinguishes a stress ratio from the underlying <gear face load factor>. The historical definition of $K_{H\beta}$ compares maximum line-load intensity with the mean over the evaluation width.<ISO 6336-1:2006>[Section 7.4.2, equation (38), printed p. 41]{ref}
For ideal elastic line contact, hold the effective geometry and material fixed. Write the peak pressure as
$$
p_0=\sqrt{\frac{E^*w}{\pi R^*}},
$$
where $w$ is load per contact length, $E^*$ the effective elastic modulus and $R^*$ the effective radius. If the largest line load is $w_{\max}=K_{H\beta}\overline w$, division by the pressure calculated from the mean line load gives
$$
\frac{p_{0,\max}}{p_{0,\rm mean}}=\sqrt{K_{H\beta}}.
$$
The square root follows from the pressure model; it is not a second independent multiplier on the load.
For an original example, a face-load ratio $K_{H\beta}=2.25$ gives a stress multiplier $1.5$. A mean-line-load reference pressure $600\,\mathrm{MPa}$ would become $900\,\mathrm{MPa}$ under these assumptions. Multiplying pressure directly by $2.25$ would instead give $1350\,\mathrm{MPa}$ and count the load ratio as though it were a stress ratio. Increasing effective line load fourfold doubles this model pressure.
Small relative uncertainties also scale differently:
$$
\frac{d\sqrt K}{\sqrt K}=\frac12\frac{dK}{K}.
$$
Thus a small $8\%$ relative change in $K$ produces about a $4\%$ relative change in its square root to first order. This sensitivity concerns that term alone; uncertainties in material, geometry and the reference load remain.
A local stress scaling does not predict <gear surface pitting> life on its own. Contact position, curvature, load history and material response matter. A primary NASA analysis explicitly combined varying tooth loads and contact geometry with a fatigue-life model.<Predicted Effect of Dynamic Load on Pitting Fatigue Life for Low-Contact-Ratio Spur Gears by David G. Lewicki>[Summary, printed p. 1; Analysis, tooth-contact intervals and varying curvature/load]{ref} A real rating must keep all factors and load conventions consistent with the chosen method.
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