Gear Contact Stress⚠ unverified
Mechanical / Gears · Compute Hertzian surface contact stress for a gear tooth
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Wt | Wt | N | 1.0 | Transmitted tangential load |
| d | d | — | 1.0 | Pitch diameter of the pinion, in length units |
| F | F | — | 1.0 | Face width of the tooth, in length units |
| I | I | — | 1.0 | Dimensionless geometry factor for surface durability |
| Cp | Cp | — | 1.0 | Elastic coefficient, in units of square root of pressure |
| Kv | Kv | — | 1.0 | Dynamic (velocity) factor. Default is 1.0 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | σc | e.g. Pa | Contact stress, in consistent pressure units (e.g. Pa). Returns 0.0 when ``d * F * I`` is non-positive |
The science & history
Understanding the Parameters
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Tangential load $W_t$ — the transmitted load, but note the square root: contact stress scales as $\sqrt{W_t}$, so it grows slowly with load. Doubling the load raises contact stress only ~41 %, which is characteristic of Hertzian contact.
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Pinion diameter $d$ and face width $F$ — larger pinion and wider face spread the load over a bigger, less-curved contact, lowering stress ($\sigma_c \propto 1/\sqrt{dF}$). Using the pinion diameter is deliberate — the smaller gear has the sharper curvature and higher contact stress, so it governs.
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Geometry factor $I$ — bundles the relative radii of curvature of the meshing teeth, the load sharing, and the gear ratio into one dimensionless number (larger $I$ = gentler contact = lower stress).
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Elastic coefficient $C_p$ — combines both materials' elastic moduli and Poisson's ratios, $C_p = \sqrt{1/\{\pi[(1-\nu_1^2)/E_1 + (1-\nu_2^2)/E_2]\}}$ — the Hertzian material group. Stiffer materials give a smaller contact area and higher stress.
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Velocity factor $K_v$ — the same dynamic-load amplification as in the Lewis equation.
Derivation (Approaching a Proof)
Two meshing involute teeth touch along a line and, in cross-section, behave like two cylinders in contact — the classic Hertzian problem (Hertzian Contact Pressure). Hertz theory gives the maximum contact pressure between two cylinders of effective radius $R$ (from the two curvatures) carrying load per unit length $w = W_t/F$:
$$p_{\max} = \sqrt{\frac{w\, E^*}{\pi R}},$$
where $E^*$ is the combined (reduced) elastic modulus. Casting this in gear terms: the effective radius at the pitch point scales with the pinion diameter $d$ (and the gear ratio, folded into the geometry factor $I$), and grouping the elastic constants into $C_p = \sqrt{E^*/\pi}$ gives
$$\sigma_c = C_p \sqrt{\frac{W_t}{d F I}}.$$
The square-root form is the signature of Hertzian contact (stress $\propto \sqrt{\text{load}}$ because the contact area itself grows with load). Multiplying by the velocity factor $K_v$ for dynamic overload yields the working AGMA-style equation. The stress is compared against the material's surface fatigue strength (allowable contact stress) to predict pitting life.
Dimensional check. With $C_p$ in $\sqrt{\text{Pa}}$ and $W_t/(dFI)$ in $\text{N}/\text{m}^2 = \text{Pa}$, $\sigma_c = \sqrt{\text{Pa}}\cdot\sqrt{\text{Pa}} = \text{Pa}$. ✓
History and Development
Gear surface durability rests on Heinrich Hertz's 1882 contact theory, adapted to gears by Earle Buckingham (1930s–40s) and standardised by AGMA into the pitting-resistance equation with its elastic coefficient $C_p$ and geometry factor $I$. Surface-fatigue (pitting) analysis, together with Lewis/AGMA bending, forms the two-pronged basis of all modern gear rating (AGMA 2001, ISO 6336).
Related Concepts: Gear Bending Stress Lewis, Hertzian Contact Pressure, Contact Stress Herzian, Hertzian Max Pressure, Gear Scoring Index, Fatigue Endurance Limit
Notes: Pitting (surface-fatigue) check; pair with Lewis bending. Uses the pinion diameter (governs). $C_p$ has units $\sqrt{\text{Pa}}$; $d$, $F$ are lengths despite the registry label (see note). Compare against allowable contact stress.