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Bevel & Worm Gear Calculator — Safety Factors, Forces & Efficiency (ISO 10300 / AGMA 2003)

Governing standard: ISO 10300· ISO 10300-2:2014 (Method B, bevel surface durability) · ISO 6336-3 root stress on virtual cylindrical gear · AGMA 2003-B97 load factors · ISO 14521 / DIN 3996 worm geometry

How ISO 10300 works — the method explained

This tool provides an engineering estimate — it uses an accepted simplified model rather than a single citable governing standard. Use it for preliminary sizing and verify the final design against manufacturer data or a licensed engineer.

The MechanixCalc bevel and worm gear calculator sizes and verifies both gear types in a single tool. For straight bevel gears it applies ISO 10300-2:2014 (Method B) contact stress on the virtual cylindrical gear at the mean section — the Tregold back-cone construction — and ISO 6336-3 root bending stress with AGMA 2003-B97 load factors (overload Ko, dynamic Kv, load-distribution Km). Enter the tooth numbers, mean module, face width, power and speed, and the tool returns contact safety factor SH, root safety factor SF, the three force components, cone distance and efficiency in one pass.

For worm gears the calculator uses ISO 3408 / DIN 3975 geometry (diameter quotient q, lead angle γ) with a simplified-Hertz contact stress estimate keyed to the worm flank radius — the dominant curvature — and the DIN 3996 forward-efficiency formula for self-locking detection. Worm pitting and wear results carry an engineering-estimate badge because the full DIN 3996 mesh-geometry integrals require paywalled chart factors; verify safety-critical worm drives against DIN 3996 or ISO 14521 before sign-off.

What this calculator does

  • ISO 10300-2:2014 (Method B) contact stress on the virtual cylindrical gear (Tregold construction) with bevel factor ZK = 0.85
  • ISO 6336-3 root bending stress via virtual tooth numbers — YFa and YSa fitted to the standard's tabulated curves
  • AGMA 2003-B97 load factors: overload factor Ko, dynamic factor Kv (quality-grade-dependent), mounting factor Km
  • Tangential, radial and axial tooth force components at the mean pitch circle, including spiral-angle correction
  • Worm gear efficiency and self-locking detection (γ < ρ condition per ISO 14521 / DIN 3996)
  • Heat generation from friction losses — critical for worm drives with low efficiency
  • Branded PDF engineering report citing the governing standard and all factors

Method & formulas

Bevel contact stress (ISO 10300-2:2014, Method B)

ISO 10300-2 rates bevel gear surface durability on the virtual cylindrical gear at the mean section — the back-cone equivalent spur gear produced by the Tregold construction. The virtual pinion and gear diameters are d_v = d_m / cos δ, giving a virtual gear ratio u_v that differs from the actual ratio. The nominal contact stress follows the ISO 6336-2 Hertz line-contact form applied to the virtual pair, multiplied by the empirical bevel factor ZK = 0.85 that aligns the virtual-cylindrical model with bevel test data. AGMA 2003-B97 load factors (KA, Kv, Km) are applied to the tangential force before the stress computation.

ISO 10300-2 nominal contact stress
σH = ZH · ZE · ZK · √( Ft · KA · Kv · Km · (u_v + 1) / (b_eff · d_v1 · u_v) )

where ZH = zone factor (2.4946 at αn = 20° straight bevel); ZE = 189.8 √MPa (steel/steel elasticity factor, ISO 6336-2); ZK = 0.85 (ISO 10300-2 bevel factor); Ft = tangential force at mean pitch circle (N); KA, Kv, Km = overload, dynamic, load-distribution factors (AGMA 2003-B97); b_eff = effective face width ≤ R/3 (mm); d_v1 = virtual pinion pitch diameter (mm); u_v = virtual gear ratio (z_v2/z_v1)

Bevel root bending stress (ISO 6336-3 on the virtual gear)

Root bending stress is evaluated at the virtual (back-cone equivalent) pinion tooth number z_v1 = z1 / cos δ1, which is the weaker member. The tooth-form factor YFa and stress-correction factor YSa are read from the ISO 6336-3 (αn = 20°, x = 0 basic rack) tabulated curves. MechanixCalc uses rational/logarithmic fits to those curves to compute YFa and YSa at the exact virtual tooth number, then multiplies by the factored load to obtain σF. The root safety factor SF = σFP / σF, where σFP is the permissible root stress for the chosen material.

ISO 6336-3 root bending stress
σF = Ft · KA · Kv · Km · YFa · YSa / (b_eff · m_n)

where YFa = tooth-form factor at virtual tooth number z_v1 (ISO 6336-3 x=0 rack); YSa = stress-correction factor at z_v1 (ISO 6336-3); m_n = normal module (mm); b_eff = effective face width (mm); remaining symbols as above

Worm gear efficiency and forces

Worm drive efficiency is governed by the lead angle γ and the friction angle ρ = atan(μ / cos αn). The forward (worm-drives-wheel) efficiency follows from the helix geometry. When γ < ρ the worm is self-locking — the wheel cannot back-drive the worm — which is useful for load-holding but makes the drive irreversible. The worm tangential force is used to find the separating force Fr and the gear tangential force FtGear, which drives the root and contact stress calculations. Worm σH is a simplified-Hertz estimate flagged with an engineering-estimate badge; full pitting and wear verification should follow DIN 3996 / ISO 14521.

Worm forward efficiency
η = tan γ / tan(γ + ρ)

where γ = lead angle of the worm (degrees); ρ = friction angle = atan(μ / cos αn); μ = sliding friction coefficient (≈ 0.05 for steel worm / bronze wheel); αn = normal pressure angle (degrees). Self-locking condition: γ < ρ.

Worked example

Verify a straight bevel gear pair: z1 = 20, z2 = 40, mean module m = 4 mm, shaft angle Σ = 90°, face width b = 30 mm, power P = 10 kW, input speed n1 = 1500 rpm, pressure angle αn = 20°, material 42CrMo4 (QT), quality grade ISO 7.

Given

  • Pinion teeth z120
  • Gear teeth z240
  • Mean module m4 mm
  • Shaft angle Σ90°
  • Face width b30 mm
  • Power P10 kW
  • Input speed n11500 rpm
  • Pressure angle αn20°
  • Material42CrMo4 (QT)

Result

  • Tangential force Ft≈ 1592 N
  • Contact safety factor SH≈ 3.20
  • Root safety factor SF≈ 4.42
  • Output speed750 rpm
  1. Gear ratio u = z2 / z1 = 40 / 20 = 2.0. Output speed = 1500 / 2 = 750 rpm.
  2. Mean pitch diameters: dm1 = 4 × 20 = 80 mm; dm2 = 4 × 40 = 160 mm.
  3. Pitch cone angles (Σ = 90°): δ1 = atan(sin 90° / (2 + cos 90°)) = atan(1/2) = 26.57°; δ2 = 90° − 26.57° = 63.43°.
  4. Cone distance R = dm2 / (2 · sin δ2) = 160 / (2 × 0.8944) = 89.4 mm. Effective face width b_eff = min(30, 89.4/3) = min(30, 29.8) = 29.8 mm.
  5. Tangential force Ft = P × 1000 × 60000 / (π × dm1 × n1) = 10 000 × 60 000 / (π × 80 × 1500) = 600 000 000 / 376 991 ≈ 1592 N.
  6. Load factors (AGMA 2003-B97, ISO grade 7 → AGMA Qv ≈ 10): KA = 1.25; B = 0.25 × (12 − 10)^(2/3) = 0.25 × 1.587 = 0.397; A = 50 + 56 × (1 − 0.397) = 83.8; pitch-line velocity v = π × 80 × 1500 / 60 000 = 6.28 m/s; Kv = ((83.8 + √(200 × 6.28)) / 83.8)^0.397 = ((83.8 + 35.4) / 83.8)^0.397 = 1.4229^0.397 ≈ 1.15; Km = 1.10 + 5.6 × 10⁻⁶ × 30² = 1.105. K = 1.25 × 1.15 × 1.105 ≈ 1.59.
  7. Virtual gear geometry: dv1 = 80 / cos 26.57° = 80 / 0.8944 = 89.4 mm; dv2 = 160 / cos 63.43° = 160 / 0.4472 = 357.8 mm; u_v = 357.8 / 89.4 = 4.0.
  8. Zone factor ZH = √(2 / (cos²20° · tan 20°)) = √(2 / (0.8830 × 0.3640)) = √(2 / 0.3214) = √6.22 = 2.494. Elasticity factor ZE = 189.8 √MPa (steel/steel). Bevel factor ZK = 0.85.
  9. Contact stress: σH = 2.495 × 189.8 × 0.85 × √(1592 × 1.59 × (4.0 + 1) / (29.8 × 89.4 × 4.0)) = 402.4 × √(12 644 / 10 667) = 402.4 × √1.185 = 402.4 × 1.089 ≈ 438 MPa.
  10. SH = σHP / σH = 1400 / 438 ≈ 3.20. Root safety factor: virtual tooth number zv1 = 20 / cos 26.57° = 22.4; using engine rational fits to ISO 6336-3 (x = 0, αn = 20°): YFa ≈ 2.72, YSa ≈ 1.57; σF = 1592 × 1.59 × 2.72 × 1.57 / (29.8 × 4) = 10 787 / 119.2 ≈ 90.5 MPa; SF = 400 / 90.5 ≈ 4.42. Both factors well above minimums (SH ≥ 1.2, SF ≥ 1.5) — design PASSES.

Illustrative example with round numbers — the load factors (KA, Kv, Km) are simplified for clarity. Run the calculator with your actual geometry for the full ISO 10300-2 Method B result.

Frequently asked questions

Which standard does this bevel and worm gear calculator use?

Bevel contact stress follows ISO 10300-2:2014 (Method B) on the virtual cylindrical gear at the mean section, using the Tregold back-cone construction and the AGMA 2003-B97 load factors (Ko, Kv, Km). Root bending stress uses ISO 6336-3 evaluated at the virtual tooth number. For worm gears, geometry is per ISO 3408 / DIN 3975; the forward efficiency and self-locking condition follow ISO 14521 / DIN 3996. Worm contact stress (σH) is a simplified-Hertz engineering estimate and is flagged as such — verify safety-critical worm drives against the full DIN 3996 before sign-off.

What are SH and SF, and what values should I target?

SH is the contact (pitting) safety factor — the ratio of the material's permissible contact stress to the calculated contact stress. SF is the root (bending) safety factor. As a general guide, target SH ≥ 1.2 and SF ≥ 1.5 for general industrial drives. The calculator flags yellow for borderline results and red when either factor falls below these thresholds.

How does the tool compute the three force components for a bevel gear?

The tangential force Ft is computed at the mean pitch circle from the transmitted power and input speed. The radial (separating) force Fr = Ft · tan(αn) · cos(δ1) and axial force Fa = Ft · tan(αn) · sin(δ1), where δ1 is the pinion pitch cone angle. For spiral bevel gears the interactive thrust-force panel accepts a spiral angle ψ and applies the full AGMA spiral bevel force equations.

When is a worm gear self-locking, and is that always desirable?

A worm is self-locking when the lead angle γ is less than the friction angle ρ = atan(μ / cos αn). Self-locking means the load on the wheel cannot back-drive the worm — useful for hoists, jacks and valve actuators that must hold position without a brake. However, self-locking worms also cannot regenerate power during deceleration and tend to have efficiency below 50%, so significant heat is generated — ensure adequate thermal capacity.

Is the bevel and worm gear calculator free?

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

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