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 YFa (which captures the lever arm from the load-application point at the tooth tip to the root section) and the stress-correction factor YSa (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 βb) 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; it is 1.0–1.1 for spur gears and slightly lower for well-made helical gears. For the bending stress path, KFβ and KFα replace KHβ and KHα using slightly different weighting (ISO 6336-3 provides a correction exponent for KFβ, but the conservative simplification KFβ = KHβ is widely used and is what the calculator applies).

Reducing KHβ is the most effective lever for improving the face-load safety factor: increasing the shaft stiffness or shortening the face-width-to-diameter ratio b/d1 both reduce KHβ. Improving the ISO accuracy grade is 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 YFa 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.