How to Calculate Gear Contact and Tooth-Root Bending Stress (ISO 6336)

The tooth-root bending stress σF is computed at the 30°-tangent section of the root fillet — the point of highest bending stress for a rack-generated tooth. The nominal specific load Ft/(b·mn) is the tangential force per unit face width per unit module; multiplying by the tooth form factor YF (which captures the lever arm from the load-application point to the root section — in ISO 6336-3 Method B the outer point of single-pair tooth contact, where one tooth alone carries the full load) and the stress-correction factor YS (which captures the stress concentration from the fillet curvature) converts this to a nominal root bending stress. The helix angle factor Yβ and rim factor YB then adjust for oblique contact and thin-rim geometries. Finally the combined load factor KF = KA·KV·KFβ·KFα scales the nominal stress to the actual operating stress.

On the permissible side, ISO 6336-3 §6.1 rates the material's tooth-root fatigue limit σFlim at a reference stress-correction factor YST = 2.0 (corresponding to the standard test gear). The allowable root stress is therefore σFG = σFlim·YST·YNT, where YNT is the life factor (1.0 for N > 3×10⁶ cycles). The safety factor SF = σFG/σF must reach at least 1.4. A common non-conservative mistake is to omit YST from the permissible side while also omitting YSa from the stress side — the two errors partly cancel, but they leave the SF calculation inconsistent with the standard.

The Hertz contact (pitting) stress is calculated at the pitch point using the two-cylinder contact model, then transformed to the inner single-pair contact point via the single-tooth-contact factors ZB (pinion, point B) and ZD (gear, point D). For spur gears with εα ≤ 2 — where only one tooth pair carries the full load at the inner contact point — ZB or ZD can raise the governing flank stress by 3–13% compared with the pitch-point value. For helical gears with full axial overlap (εβ ≥ 1), ZB = ZD = 1 because the inclined contact line means no single-pair loading event occurs.

The zone factor ZH converts the pitch-point curvature into the equivalent radius ratio needed by the Hertz formula; its value is determined by the operating (working) transverse pressure angle αtw and the base helix angle βb. For a standard spur gear (αn = 20°, no profile shift, β = 0°), ZH ≈ 2.495. The elasticity factor ZE = 189.8 MPa^0.5 applies to a steel–steel pair (E = 206 GPa, ν = 0.3). The contact-ratio factor Zε shares the load across simultaneously meshing teeth; for spur meshes it equals √((4−εα)/3), which is about 0.89 for a typical contact ratio of 1.64. The helix factor Zβ = 1/√(cos β), on the reference helix angle β accounts for the oblique contact line in helical gears (1.0 for spur).

The load factors are the bridge between the nominal transmitted load and the actual worst-case tooth load that the stress formulas must be evaluated at. KA (application factor) accounts for external shock and inertia from the driven machine; typical values range from 1.0 (uniform electric-motor drive, smooth load) to 1.75 (heavy shock). KV (dynamic factor) accounts for the internal dynamic overload caused by mesh-frequency excitation; it depends on the pitch-line velocity, the pinion tooth count and the ISO accuracy grade — lower quality gears at high speed carry a significantly larger KV. KHβ (face-load distribution factor) accounts for misalignment of the contact line across the face width, caused by shaft deflection, bearing clearance and tooth crowning; it is sensitive to the ratio b/d1 and to whether the pinion is symmetrically mounted. KHα (transverse load-sharing factor) accounts for unequal load sharing between simultaneously meshing tooth pairs; by Method B it grows as the base-pitch deviation rises relative to the specific load, up to the limits set by the total contact ratio (for spur gears up to ≈ 1.27 on contact and ≈ 1.41 on bending at εα 1.64); a finer accuracy grade or a heavier specific load brings it towards 1.0. For the bending stress path, KFβ and KFα replace KHβ and KHα using slightly different weighting: ISO 6336-1 defines KFβ = KHβ^NF with the exponent NF = (b/h)²/(1 + b/h + (b/h)²), which is always < 1, so KFβ ≤ KHβ. The simplification KFβ = KHβ is widely quoted and is conservative (it over-states the root face load), but the MechanixCalc calculator applies the faithful ISO 6336-1 exponent rather than the simplification.

KHβ is usually the factor that decides a lightly loaded pair: with unmodified flanks the helix tolerance puts the load on part of the face. Crowned or end-relieved flanks, a mesh adjusted to a checked contact pattern at assembly and a finer accuracy grade lower it; a stiffer shaft lowers it for an overhung pinion. Narrowing the face raises KHβ's share less than it raises the unit load, so it usually lowers SH. Improving the accuracy grade is also the most effective lever for reducing KV at high speed.

Profile shift (addendum modification) with coefficient x moves the generating rack radially outward (positive x) or inward (negative x) when the gear is cut. A positive shift on the pinion increases the tooth thickness at the root — reducing the form factor YF and improving bending strength — and moves the pitch point away from the undercut zone. The penalty is a change in the operating centre distance (when x1 + x2 ≠ 0) and a slight change in the operating pressure angle αtw, both computed by solving the involute equation inv(αtw) = inv(αt) + 2·tan(αn)·(x1+x2)/(z1+z2).

For the contact stress, a positive total shift (x1 + x2 > 0) increases αtw, which raises ZH slightly — a mild debit on pitting. The dominant benefit of profile shift is therefore on the bending side, particularly for pinions with low tooth counts (z1 < 17) where undercut otherwise weakens the root. The calculator applies profile shift across the full geometry chain — pitch diameters, tip/root radii, contact ratio and all stress factors — so both SF and SH reflect the actual shifted geometry.