Cylindrical Gears Calculator — Tooth Root Bending & Hertz Contact Stress (ISO 6336)

Applied in full: the stress-correction factor YSa from the ISO 6336-3 §6.4–§6.6 root-fillet geometry, the helix factors Yβ and Zβ, the contact-ratio factor Zε (including the partial-overlap blend) applied to contact stress, the single-pair tooth-contact factors ZB and ZD, the zone factor ZH, the size factors YX and ZX, the rim factor YB from your rim thickness, the lubricant, velocity and roughness factors ZL·ZV·ZR from your oil viscosity and flank roughness, the application factor KA, the dynamic factor KV by DIN 3990-1 Method C, the face-load factors KHβ and KFβ = KHβ^NF, the life factors Y_NT and Z_NT, and the mean-stress factor Y_M 0.7 when a member is an idler or the drive reverses (selectable). Simplified and stated as such everywhere: the form factor assumes the smallest standard hob fillet (ρfP = 0.20·mn, conservative) and no protuberance or grinding notch; there is no Yε contact-ratio factor in the bending path (ISO 6336-3 does not use one there — only Zε, on the contact side); the transverse load factors KHα and KFα follow ISO 6336-1 Method B with both upper limits (the base-pitch deviation taken as the grade's single-pitch tolerance, the mesh stiffness from the standard's single-stiffness form for solid blanks), or your own Method A value; KHβ follows ISO 6336-1 Method C with no shaft deflection (it rises with a coarser grade and at light unit loads), never below the AGMA enclosed-unit estimate, and you can override it; the work-hardening factor ZW is held at 1.0, which is the conservative direction; the elasticity factor ZE assumes a steel pair, which overstates the contact stress by about 9 percent for the ductile-iron option; scuffing is not rated (no ISO/TS 6336-20/21 temperature check) — only a screening indicator, which caps the verdict at CHECK when HIGH; a pointed tooth tip is refused and a tip thinner than 0.2·mn is flagged and capped at CHECK. Every one of these limits is printed in the PDF report and returned by the API.

Yes. Leave the rim thickness at zero for a solid disc and the rim factor YB stays at 1.0. Enter the rim thickness below the tooth root and the calculator computes the backup ratio mB = sR/hT and applies ISO 6336-3 §7: YB = 1 for mB at or above 1.2, and YB = 1.6·ln(2.242/mB) below it, which raises the root stress and lowers the bending safety factor by the same factor. Below mB 0.5 the standard gives no value, so the calculator refuses to rate the gear rather than extrapolating — that case needs an FEA of the rim and web.

Engine updates have moved the factors in both directions, and each one carries its own engine version, so a saved calculation shows when it was made with an older engine. The 2026-08-05 revision was conservative only: the transverse load factor no longer steps down from 1.10 to 1.05 the moment a helix angle is entered (the step had the wrong sign), the permissible root stress carries the ISO 6336-3 size factor YX (below 1 above a 5 mm module) and the permissible contact stress carries the ISO 6336-2 lubricant, velocity and roughness factors ZL·ZV·ZR. The 2026-10-04 revision went the other way for the tooth root: the form and stress-correction factors moved from the older tip-load approximation to ISO 6336-3 Method B at the outer point of single-pair contact. The tip-load form had read the root safety factor about a third too LOW on the reference mesh, so root safety factors rose; contact was not affected. Later revisions tightened the rating again where it had been too kind, for example on undercut pinions and the single-pair contact factors. To compare an old printout with a new run, re-run it on the current engine and read the PDF's substituted formulas — every factor is printed with its value.

Tooth-root bending stress is evaluated using an ISO 6336-3:2006-based method (Method B form factor YF and stress-correction factor YS from the root-fillet geometry at the outer point of single-pair contact, and helix factor Yβ; no Yε contact-ratio factor, which Method B does not use in the bending path) and the contact (pitting) stress to ISO 6336-2:2006-based (Hertz formula with zone, elasticity, contact-ratio and single-pair-contact factors). The dynamic load factor KV uses the DIN 3990 / ISO 6336 simplified Method C closed-form, and the face-load distribution factor KHβ follows ISO 6336-1 Method C (the grade's helix tolerance, flank modification and assembly adjustment; never below the AGMA 2001-D04 / Shigley Eq. 14-30 enclosed-unit estimate), and you can override KHβ with your own ISO 6336-1 Method C value. The transverse load factors KHα and KFα follow ISO 6336-1 Method B — they grow for a lightly loaded or coarse-quality mesh up to the standard's limits — and a Method A value can be entered instead; the PDF prints every term. Profile shift geometry follows ISO 6336-1 / DIN 3960 / ISO 21771. The full method and every scope limit are shown in the generated PDF engineering report.

Profile shift (addendum modification) moves the tool radially when cutting a gear, increasing the tooth thickness at the root and reducing undercut risk for low tooth-count pinions. The coefficients x1 (pinion) and x2 (gear) shift the pitch point relative to the reference circle, which changes the operating centre distance and operating pressure angle. The calculator solves the involute equation by Newton iteration to find the exact operating pressure angle αtw, derives the operating centre distance a_w and checks each member against the undercut limit x_min — the larger of the textbook (z_min − z)/z_min with z_min = 17·cos³β and the exact rack limit 1 − z·sin²αt/(2·cos β). A member below x_min is flagged undercut: its real involute starts higher, so the contact ratio is taken over the shortened path, its root rating is treated as not valid and the verdict is capped at CHECK.

ISO 6336 does NOT specify minimum safety factors — ISO 6336-1 leaves them to agreement between manufacturer and purchaser. This calculator adopts the DIN 3990-11 industrial-gearbox guide values SF_min = 1.4 for tooth-root bending and SH_min = 1.2 for Hertz contact (pitting), and the verdict is the conjunction of both checks: a design passes only when BOTH factors clear their own minimum, and the governing mode named on screen is the one with the smallest margin relative to its own minimum. For critical or high-cycle applications a licensed engineer should confirm the safety factors are appropriate for the specific duty cycle and consequences of failure.

Yes — the full helical-gear analysis is included: helix angle β up to 45°, transverse module mt = mn/cos β, base helix angle βb, the helical contact-ratio factor Zε (spur, partial-overlap and full-overlap branches per ISO 6336-2), helix factor Zβ, helix angle factor Yβ for bending, and overlap ratio εβ = b·sin β/(π·mn). Profile shift coefficients x1 and x2 are applied across the full involute geometry, load-factor and stress chains.

Yes. The application factor KA is a selectable input following the ISO 6336-1 Annex B / DIN 3990-1 guide table (driving × driven machine shock character; default 1.25 — a value below 1.0 is refused, never silently raised). For finite-life rating, enter the required number of pinion load cycles N_L and the calculator applies the ISO 6336 life factors — Y_NT for tooth-root bending (referenced to 3×10⁶ cycles) and Z_NT for pitting — selected by the material's strength class. The slower wheel is rated for its own N_L/u cycles, and leaving N_L blank keeps the conservative continuous-duty (endurance) design used by default.

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