CTR K

Cylindrical Gears Calculator — Tooth Root Bending & Hertz Contact Stress (ISO 6336 / AGMA 2001)

Governing standard: ISO 6336· ISO 6336-1/2/3:2019 (load factors, contact stress Method B, bending stress Method B) · DIN 3990-1 / ISO 6336-1 Method C (KV) · AGMA 2001-D04 / Shigley §14-11 (KHβ / Km) · DIN 3960 / ISO 21771 (profile shift, involute geometry)

How ISO 6336 works — the method explained

The MechanixCalc cylindrical gear calculator sizes and verifies spur and helical gear pairs to ISO 6336 and AGMA 2001 — the two governing international standards for gear-tooth fatigue. Enter the normal module, tooth counts, face width, helix angle, profile shift coefficients, material, speed and power, and the tool returns the tooth-root bending safety factor (SF), the Hertz contact safety factor (SH), the full involute gear geometry, tangential and axial forces, operating centre distance, transverse contact ratio and mesh efficiency in one pass.

It is built for power-transmission and gearbox engineers who need a defensible, standards-cited calculation for a new gear pair or a gear-ratio verification — and who need to hand a reviewer a complete worked calculation rather than a spreadsheet. The animated gear-mesh visualisation, involute tooth-profile diagram, failure-mode analysis cards and tornado sensitivity chart make the results immediately interpretable for design reviews.

What this calculator does

  • ISO 6336 tooth-root bending stress (Method B) with form factor YFa, stress-correction factor YSa, helix factor Yβ and rim factor YB
  • ISO 6336 Hertzian contact stress with zone factor ZH, elasticity factor ZE, contact-ratio factor Zε and helix factor Zβ, including single-pair tooth-contact factors ZB/ZD for spur and partial-overlap helical gears
  • DIN 3990-1 / ISO 6336-1 Method C dynamic load factor KV and AGMA 2001-D04 face-load distribution factor KHβ / Km across ISO accuracy grades Q6–Q12; bending face-load factor KFβ = KHβ^NF per ISO 6336-1
  • Application factor KA as a selectable input (ISO 6336-6 guide table) and finite-life rating via the ISO 6336 life factors Y_NT (bending) and Z_NT (contact) — enter a required number of load cycles, or leave it for the conservative continuous-duty (endurance) design
  • Profile shift analysis to ISO 6336 / DIN 3960: operating centre distance, operating pressure angle and transverse contact ratio εα computed from the involute function
  • Animated gear-mesh visualisation, involute tooth-profile diagram with dimension lines, and DXF export of the full-involute tooth geometry
  • Sensitivity (tornado) analysis — contact safety factor SH vs ±20% variation in module, face width, power and pinion tooth count
  • Failure-mode analysis cards for pitting, root fracture, scuffing, abrasive wear and noise; 20+ built-in materials; branded PDF engineering report

Method & formulas

Tooth-root bending stress (ISO 6336-3 Method B)

ISO 6336-3 evaluates tooth-root bending fatigue at the 30°-tangent section of the fillet. The nominal root stress σF₀ is the specific tangential load Ft/(b·mn) amplified by the form factor YFa (which captures the lever arm from the load application point to the root), the stress-correction factor YSa (which captures the stress concentration at the fillet), the helix angle factor Yβ and the rim factor YB. The combined load factor KF = KA · KV · KFβ · KFα then scales σF₀ to the actual operating stress. The permissible root stress σFG = σFlim · YST · YNT, where σFlim is the material's tooth-root fatigue limit (rated at the reference stress-correction factor YST = 2.0) and YNT is the life factor.

MechanixCalc implements the full ISO 6336-3 §6.4–6.6 geometric chain — including the fixed-point iteration for the root-fillet auxiliaries G, E, H and θ — so YFa and YSa share one consistent geometry. This removes the non-conservative bias that arises when YSa is omitted or when the reference YST is not applied on the permissible side.

Tooth-root bending stress (ISO 6336-3)
σF = (Ft / (b · mn)) · YFa · YSa · Yβ · YB · KF

where Ft = tangential force (N); b = face width (mm); mn = normal module (mm); YFa = tooth form factor; YSa = stress-correction factor; Yβ = helix angle factor; YB = rim factor; KF = KA · KV · KFβ · KFα (combined load factor)

Bending safety factor
SF = (σFlim · YST · YNT) / σF [min SF = 1.4]

where σFlim = tooth-root fatigue limit of the material (MPa); YST = 2.0 (reference stress-correction factor, ISO 6336-3 §6.1); YNT = bending life factor (1.0 for continuous duty / N ≥ 3×10⁶; > 1 for a finite required life N_L); SF_min = 1.4 per ISO 6336

Hertz contact stress (ISO 6336-2 Method B)

The contact (pitting) stress is computed at the pitch point using the Hertz formula for two cylinders in contact, then transformed to the governing single-pair contact point via the single-tooth-contact factors ZB (pinion, point B) and ZD (gear, point D). The zone factor ZH converts the pitch-point geometry into an equivalent-radius ratio, the elasticity factor ZE reflects the modulus of the gear pair (189.8 MPa^0.5 for steel–steel), the contact-ratio factor Zε shares the load across meshing teeth, and the helix factor Zβ accounts for the oblique contact line of a helical gear. For full-overlap helical meshes (εβ ≥ 1), ZB = ZD = 1; for spur and partial-overlap helical gears the single-pair factor can raise the governing flank stress by 3–13%.

Pitch-point Hertz contact stress (ISO 6336-2)
σH = ZH · ZE · Zε · Zβ · √( Ft · K / (b · d1) · (u + 1) / u )

where ZH = zone factor; ZE = elasticity factor (MPa^0.5); Zε = contact-ratio factor; Zβ = helix factor; Ft = tangential force (N); K = KA · KV · KHβ · KHα (combined load factor); b = face width (mm); d1 = pinion pitch diameter (mm); u = gear ratio z2/z1

Contact safety factor
SH = (σHlim · ZNT) / σH [min SH = 1.2]

where σHlim = allowable contact stress (MPa); ZNT = pitting life factor (1.0 for continuous duty; > 1 for a finite required life N_L, ISO 6336-2); SH_min = 1.2 per ISO 6336

Involute geometry and profile shift (ISO 6336 / DIN 3960)

Profile shift (addendum modification) with coefficients x1 and x2 changes the operating centre distance and operating pressure angle from their standard values. MechanixCalc solves the involute function inv(αtw) = inv(αt) + 2·tan(αn)·(x1+x2)/(z1+z2) by Newton iteration to find the exact operating transverse pressure angle αtw, then derives the operating centre distance a_w and the transverse contact ratio εα from the line-of-action geometry. Undercut risk is checked against the minimum tooth count z_min = 17/cos³β for each member.

Transverse contact ratio (ISO 6336-1)
εα = ( √(ra1² − rb1²) + √(ra2² − rb2²) − a_w · sin(αtw) ) / p_bt

where ra1, ra2 = tip radii of pinion and gear (mm); rb1, rb2 = base-circle radii (mm); a_w = operating centre distance (mm); αtw = operating transverse pressure angle; p_bt = transverse base pitch = π · mt · cos(αt) (mm)

Worked example

Find the pitch diameters, centre distance, gear ratio and tangential force for a spur gear pair: normal module mn = 5 mm, pinion teeth z1 = 20, gear teeth z2 = 40, no profile shift (x1 = x2 = 0), power P = 5 kW, pinion speed n1 = 960 rpm.

Given

  • Normal module mn5 mm
  • Pinion teeth z120
  • Gear teeth z240
  • Helix angle β0° (spur)
  • Profile shift x1, x20, 0
  • Power P5 kW
  • Pinion speed n1960 rpm

Result

  • Pinion pitch diameter d1100 mm
  • Gear pitch diameter d2200 mm
  • Gear ratio i2.0
  • Centre distance a150 mm
  • Tangential force Ft994 N
  • Pitch-line velocity v≈ 5.03 m/s
  1. For a spur gear (β = 0°) with no profile shift, the transverse module equals the normal module: mt = mn = 5 mm.
  2. Pinion pitch diameter: d1 = mt · z1 = 5 × 20 = 100 mm. Gear pitch diameter: d2 = mt · z2 = 5 × 40 = 200 mm.
  3. Gear ratio: i = z2 / z1 = 40 / 20 = 2.0 (the gear turns at half the pinion speed: n2 = 960 / 2 = 480 rpm).
  4. Standard centre distance (x = 0): a = (d1 + d2) / 2 = (100 + 200) / 2 = 150 mm.
  5. Pinion torque: T1 = P × 9550 / n1 = 5 × 9550 / 960 = 47750 / 960 ≈ 49.7 N·m.
  6. Tangential force at the pitch circle: Ft = 2000 · T1 / d1 = 2000 × 49.7 / 100 = 994 N.
  7. Pitch-line velocity: v = π · d1 · n1 / (60 × 1000) = π × 100 × 960 / 60000 ≈ 5.03 m/s. The calculator then applies load factors KA, KV and KHβ before evaluating σF and σH against the material limits.

Illustrative geometry example only — the calculator continues to compute load factors, root stress σF, contact stress σH and safety factors SF / SH from your actual inputs and material. Verify all outputs against your own design requirements.

Frequently asked questions

Which standard does this cylindrical gear calculator use?

Tooth-root bending stress is evaluated to ISO 6336-3:2019 Method B (form factor YFa, stress-correction factor YSa, helix factor Yβ) and the contact (pitting) stress to ISO 6336-2:2019 Method B (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 AGMA 2001-D04 (Shigley Eq. 14-30). Profile shift geometry follows ISO 6336-1 / DIN 3960 / ISO 21771. The full method is shown in the generated PDF engineering report.

What is profile shift and how does this tool handle it?

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 the undercut limit x_min = (z_min − z)/z_min for each member.

What is the minimum safety factor for gear tooth-root bending and contact?

ISO 6336 specifies minimum safety factors of SF_min = 1.4 for tooth-root bending and SH_min = 1.2 for Hertz contact (pitting). The calculator flags designs below these limits in red and issues warnings when the margins are tight. 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.

Can it analyse helical gears with profile shift?

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.

Can I set the application factor KA and rate a finite (limited) life?

Yes. The application factor KA is a selectable input following the ISO 6336-6 / DIN 3990-1 guide table (driving × driven machine shock character, floored at 1.0; default 1.25). 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.

Is the cylindrical gear calculator free?

You can use it during a free 30-minute preview with no sign-up, and a free 14-day account trial unlocks every calculator with no credit card required. The branded PDF engineering report with the full calculation trail and saved calculations are part of a paid plan.

Related calculators

Run the Cylindrical Gears on your own numbers

Free 30-minute preview — no sign-up. A free 14-day account trial unlocks every tool and the branded PDF report, no credit card required.

Start free