ISO 6336 Pitting Resistance: The Complete $S_H​$ Calculation Walkthrough

The Calculation Most Engineers Think They Understand

Open any gear engineering textbook. Find the $S_H$ formula. Count the factors. Most texts list seven or eight. ISO 6336-2:2019 requires eighteen — and four of the most consequential ones are either omitted from secondary sources, explained incorrectly, or applied in ways the standard does not permit.

This article derives every factor from physical first principles, documents the normative conditions under which each applies, and identifies the specific errors that produce calculations which look correct and are not. The standard reference throughout is ISO 6336-2:2019, Method B, oil-lubricated external cylindrical gears.

Before the first equation: what pitting actually is, and what the $S_H$ calculation does and does not govern.

What Pitting Is — and What ISO 6336 Actually Governs

A gear flank does not fail because a formula returned $S_H < S_{H,\min}$. It fails because subsurface shear stresses — generated by repeated Hertzian contact — exceed the material’s fatigue limit at a depth between 0.1 mm and 0.3 mm below the surface. A crack initiates at a stress concentration (inclusion, carbide boundary, or hardness transition). It propagates along shear planes. It eventually intersects the surface. Material detaches. A pit forms.

ISO 6336-2:2019 makes a distinction that almost no secondary source states correctly: not all pitting is failure.

The standard defines three regimes. Initial pitting that subsequently decelerates (degressive pitting) or ceases entirely (arrested pitting) is considered tolerable — the effective bearing area has enlarged and the rate of damage has self-limited. Linear or progressive increase in pit area under unchanged service conditions is not acceptable and constitutes failure.

The practical implication: for some slow-speed industrial gears with large teeth — module 25 or larger — made from low-hardness steel, pitting over 100% of the working flanks can be tolerated. Individual pits up to 20 mm in diameter and 8 mm deep. The tooth flanks become smoothed and work-hardened to the extent of increasing surface Brinell hardness by 50% or more. ISO 6336-2:2019 explicitly permits safety factors below 1.0 in these cases, provided tooth bending safety remains adequate.

This is not a loophole. It is a documented normative position in Clause 4 that most engineers have never read — because they locate the formula and never read the scope.

The $S_H$ calculation governs destructive pitting only. It does not govern:

• Micropitting (grey staining, surface-initiated sub-asperity fatigue) — governed by ISO/TR 15144-1, via the specific film thickness $\lambda$
• Scuffing (instantaneous adhesive failure) — governed by ISO/TR 13989-1/2, via flash temperature or integral temperature criterion
• Tooth flank fracture (subsurface crack at the case-core boundary) — governed by ISO/DTS 19042-1
• Wear (abrasive material removal in mixed/boundary lubrication) — no ISO 6336 method

A gear that passes $S_H \geq S_{H,\min}$ can fail by any of these four modes. A complete gear design analysis requires separate calculation for each applicable failure mode. The $S_H$ calculation is necessary but not sufficient.

Normative context declaration — required before every calculation report:
Standard: ISO 6336-2:2019 | Method: B | Lubrication: oil-lubricated | Gear type: external cylindrical | Failure mode: destructive pitting

These are not boilerplate. They define which equations apply.

The Structural Assumption That Silently Invalidates Results

Every ISO 6336 $S_H$ calculation carries an implicit requirement almost never stated: all factors must belong to the same normative context object. ISO 6336 and AGMA 2001-D04 use symbols that appear interchangeable — $Z_E$ and $C_p$ are both “elasticity factors”, $S_H$ appears in both standards — but their reference conditions, material baselines, and derivation paths are not interchangeable. Substituting one AGMA coefficient into an ISO factor chain produces a result that is neither standard. It will not declare itself invalid. It will produce a safety factor that looks reasonable.

The edition must also be declared and maintained throughout. The 2019 edition changed $Z_W$ material pairing tables, introduced new viscosity grades VG10/VG15/VG22 for $Z_L$, added auxiliary factor $f_{ZCa}$ for helical gear contact factors, and replaced reference tip/root diameters with active tip/root diameters in several clauses. A calculation that uses 2019 $\sigma_{H\lim}$ values from ISO 6336-5 with 2006 edition $Z_W$ factors is internally inconsistent by construction — and will not produce an error message.

The Master Equation — Complete Factor Inventory

$$S_H = \frac{\sigma_{HP}}{\sigma_H} \geq S_{H,\min}$$

Calculated contact stress at the determinant point:

$$\sigma_H = Z_B \cdot \sigma_{H0} \cdot \sqrt{K_A \cdot K_v \cdot K_{H\beta} \cdot K_{H\alpha}}$$

Nominal contact stress at the pitch point:

$$\sigma_{H0} = Z_H \cdot Z_E \cdot Z_\varepsilon \cdot Z_\beta \cdot \sqrt{\frac{F_t}{d_1 \cdot b} \cdot \frac{u+1}{u}}$$

Permissible contact stress:

$$\sigma_{HP} = \sigma_{H\lim} \cdot Z_{NT} \cdot \frac{Z_L \cdot Z_v \cdot Z_R}{S_{H\lim}} \cdot Z_W \cdot Z_X$$

The complete factor inventory: $Z_H$, $Z_E$, $Z_\varepsilon$, $Z_\beta$, $Z_B$, $Z_D$, $K_A$, $K_v$, $K_{H\beta}$, $K_{H\alpha}$, $\sigma_{H\lim}$, $Z_{NT}$, $Z_L$, $Z_v$, $Z_R$, $Z_W$, $Z_X$, $S_{H\lim}$. Eighteen quantities. Every one is derived below.

The load factors appear under a square root because $\sigma_H$ scales with the square root of load in Hertz contact theory. Doubling the combined load factor product does not double $\sigma_H$ — it multiplies it by $\sqrt{2} \approx 1.41$. A combined factor of $K_A K_v K_{H\beta} K_{H\alpha} = 4.0$ increases $\sigma_H$ by a factor of 2.0. The nonlinearity attenuates load factor errors in the stress domain — but does not eliminate them.

Part I — Nominal Contact Stress $\sigma_{H0}$

$Z_H$ — Zone Factor

Physical meaning: $Z_H$ converts the tangential load at the reference cylinder to normal load at the pitch cylinder and accounts for the curvature of the mating involute flanks at the pitch point. It is a purely geometric factor — independent of load, material, and lubrication.

Formula (ISO 6336-2:2019, Clause 6.2):

$$Z_H = \sqrt{\frac{2 \cos\beta_b \cdot \cos\alpha_{wt}}{\cos^2\alpha_t \cdot \sin\alpha_{wt}}}$$

Where:

• $\alpha_t$ = transverse pressure angle at the reference cylinder: $\tan\alpha_t = \tan\alpha_n / \cos\beta$
• $\alpha_{wt}$ = transverse pressure angle at the working pitch cylinder: $\cos\alpha_{wt} = (d_{b1} + d_{b2}) / (2a_w)$
• $\beta_b$ = helix angle at the base cylinder: $\tan\beta_b = \tan\beta \cdot \cos\alpha_t$

For a standard spur gear ($\beta = 0°$, $x_1 + x_2 = 0$, $\alpha_n = 20°$):

$$Z_H = \sqrt{\frac{2 \times 1.0 \times \cos 20°}{\cos^2 20° \times \sin 20°}} = \sqrt{\frac{2 \times 0.9397}{0.8830 \times 0.3420}} \approx 2.495$$

What engineers miss: $Z_H$ depends on $\alpha_{wt}$, not $\alpha_t$. These are equal only when the working centre distance equals the reference centre distance — i.e., when $\sum x = x_1 + x_2 = 0$. Any positive profile shift sum increases $a_w$, increases $\alpha_{wt}$, and decreases $Z_H$.

For $m_n = 2.5$ mm, $z_1 = 20$, $z_2 = 40$, $\alpha_n = 20°$, $\sum x = +0.5$: $\alpha_{wt} \approx 22.3°$, giving $Z_H \approx 2.349$ — a 5.83% reduction in $Z_H$, which compounds to an 11.3% reduction in $\sigma_{H0}^2$ and a measurable improvement in $S_H^2$.

Positive profile shift sum improves $S_H$ through two simultaneous mechanisms: reduced $Z_H$ from the geometry, and improved case utilisation in $\sigma_{H\lim}$ because the hardened case is loaded closer to its optimal depth. Profile shift is a strength parameter, not merely a geometry parameter.

$Z_E$ — Elasticity Factor

Physical meaning: $Z_E$ encodes the combined elastic compliance of both contacting bodies. It translates the Hertzian contact pressure model into the ISO stress formula. It is a material-pair factor — not a property of either material alone.

Formula (Hertz contact theory, adopted by ISO 6336-2:2019, Clause 7):

$$Z_E = \sqrt{\frac{1}{\pi \left(\dfrac{1-\nu_1^2}{E_1} + \dfrac{1-\nu_2^2}{E_2}\right)}}$$

Standard values (ISO 6336-2:2019, Table 1):

What engineers miss: $Z_E$ is invariant to heat treatment, surface hardness, module, and tooth geometry. A quenched-and-tempered pinion meshing with a case-hardened wheel has the same $Z_E = 189.8\ \sqrt{\text{N/mm}^2}$ as two normalised soft-steel gears. Heat treatment alters surface microstructure; it does not change the bulk elastic modulus $E$ or Poisson’s ratio $\nu$. Within the steel gear domain, $Z_E$ is a fixed constant — it cannot be optimised by material selection within the same class.

$Z_\varepsilon$ — Contact Ratio Factor

Physical meaning: $Z_\varepsilon$ accounts for the load-sharing effect of simultaneous tooth pair contact. As $\varepsilon_\alpha$ increases, more tooth pairs share the load at any instant, reducing stress per pair. The overlap ratio $\varepsilon_\beta$ modifies the sharing model for helical gears.

Formula (ISO 6336-2:2019, Clause 8, Method B):

For spur gears ($\varepsilon_\beta = 0$), with $\varepsilon_\alpha \leq 2$:

$$Z_\varepsilon = \sqrt{\frac{4 – \varepsilon_\alpha}{3}}$$

For helical gears with $\varepsilon_\beta \geq 1$:

$$Z_\varepsilon = \sqrt{\frac{1}{\varepsilon_\alpha}}$$

For $0 < \varepsilon_\beta < 1$ — the intermediate case — ISO 6336-2:2019 requires interpolation between the two expressions weighted by $\varepsilon_\beta$. This interpolation is frequently omitted in manual calculations and handled inconsistently in older software implementations.

Numerical sensitivity — spur gear:

A 0.3 increase in $\varepsilon_\alpha$ — achievable through addendum height increase or profile modification at no manufacturing cost increase with standard hobbing tooling — reduces $\sigma_{H0}$ by 6.2% without altering mass, material, or module. Contact ratio is the most systematically underutilised strength lever available to a gear designer.

What engineers miss: The model boundary at $\varepsilon_\beta = 1$ is not a physical discontinuity. It is the point at which the helical load-sharing model fully supersedes the spur model. Treating helical gears with $\varepsilon_\beta = 0.7$ as either pure spur or pure helical introduces an error that is largest precisely where the intermediate formula is most needed — near $\varepsilon_\beta = 0.5$.

$Z_\beta$ — Helix Angle Factor

Physical meaning: $Z_\beta$ corrects for the influence of the helix angle on the length and orientation of the instantaneous contact line. In helical gears, the contact line is inclined to the gear axis, distributing load across a longer path than the equivalent spur cross-section.

Formula (ISO 6336-2:2019, Clause 9):

$$Z_\beta = \sqrt{\cos\beta}$$

What engineers miss: $Z_\beta$ captures only part of the helix angle effect. The benefit appears here and in $Z_\varepsilon$ via higher $\varepsilon_\beta$. The cost — increased axial force $F_a = F_t \tan\beta$, which loads thrust bearings — is entirely outside the $S_H$ calculation. A design optimised for minimum $\sigma_{H0}$ by maximising $\beta$ will have axial bearing loads increasing as $\tan\beta$. The system optimisation requires closing this loop. ISO 6336-2 optimises the gear; the engineer optimises the system.

$Z_B$ and $Z_D$ — Single-Pair Tooth Contact Factors

These factors are systematically omitted in textbook examples, set to 1.0 in simplified software, and their physical basis is almost never explained. They require explicit treatment because their omission is not conservative.

Physical meaning: For spur gears with $\varepsilon_\alpha \leq 2$, the determinant contact point — where contact stress is maximum — is not the pitch point. It is the inner point of single-tooth contact (IPSTC): the point where the previous tooth pair disengages and the current pair carries full load alone. The curvature radii at the IPSTC are smaller than at the pitch point, producing higher Hertzian stress.

$Z_B$ (for the pinion) and $Z_D$ (for the wheel) shift the stress calculation from the pitch point to the IPSTC.

For $\varepsilon_\alpha \leq 2$ (ISO 6336-2:2019, Clause 6.3):

$$Z_B = \sqrt{M_1} \quad \text{if } M_1 > 1\text{, otherwise } Z_B = 1$$ $$Z_D = \sqrt{M_2} \quad \text{if } M_2 > 1\text{, otherwise } Z_D = 1$$

Where $M_1$ and $M_2$ are functions of the transverse radii of curvature at the IPSTC, dependent on $\varepsilon_1$, $\varepsilon_2$ (addendum contact ratios of pinion and wheel), and gear ratio $u$.

For $\varepsilon_\alpha > 2$: $Z_B = Z_D = 1.0$ — the contact maximum occurs at the pitch point.

Practical impact: For a spur gear with $\varepsilon_\alpha = 1.6$, $Z_B$ can reach 1.04–1.07 depending on the gear ratio. This is a 4–7% increase in $\sigma_H$, which corresponds to an 8–15% reduction in $S_H^2$. For a design near $S_{H,\min}$, this is the difference between pass and fail. The calculation is performed separately: $Z_B$ applies to pinion ($\sigma_{H1}$), $Z_D$ applies to wheel ($\sigma_{H2}$).

Part II — Load Factors

The four load factors $K_A$, $K_v$, $K_{H\beta}$, $K_{H\alpha}$ are defined in ISO 6336-1:2019, not in ISO 6336-2. Critical distinction: $K_{H\beta} \neq K_{F\beta}$ for the same gear pair. The face load distribution at the flank contact (for $S_H$) differs from the distribution at the tooth root (for $S_F$) because the load path to the root involves tooth bending stiffness. ISO 6336-1:2019 provides separate procedures for both. Applying the same numerical value to both $S_H$ and $S_F$ calculations is an error.

$K_A$ — Application Factor

Physical meaning: $K_A$ absorbs all dynamic load increments originating outside the gearbox — prime mover torque variation, driven machine impact, torsional oscillations. It is the ratio of peak operational load to nominal load, applied to the entire load chain.

Method B guide values (ISO 6336-1:2019, Table 2 — updated in 2019 edition):

What engineers miss: $K_A$ is the factor most frequently inherited from previous projects without verification. Torsionally soft couplings, variable-speed drives with resonance regions, and multi-stage drivetrains with inter-stage dynamics all produce $K_A$ values that require measurement or simulation. $K_A \geq 1.0$ always — there is no drive train configuration that reduces dynamic load below nominal in the normative sense.

$K_v$ — Dynamic Factor

Physical meaning: $K_v$ accounts for internal dynamic load increments generated by gear mesh excitation — pitch errors, profile deviations, and periodic mesh stiffness variation. It is the factor most sensitive to manufacturing quality and pitch line velocity.

Method B requires:

• Pitch line velocity: $v = \pi d_1 n_1 / (60 \times 10^3)$ [m/s]
• ISO 1328-1 accuracy grade or specific deviation $f_{pb}$ [µm]
• Resonance ratio: $N = n_1 / n_{E1}$

Numerical sensitivity:

What engineers miss: $K_v$ is not an application property — it is a manufactured product property. Two geometrically identical gears with different ISO quality grades have different $K_v$ values and therefore different $S_H$ values under identical loads. This is the normative quantification of the return on manufacturing investment: a grade improvement from Q8 to Q6 at $v = 10$ m/s reduces $K_v$ from ~1.22 to ~1.12 — a reduction in the load factor product of approximately 8%.

High-speed gearboxes fail not because $K_A$ was wrong, but because $K_v$ was underestimated. At $v > 15$ m/s with quality 9, $K_v$ can exceed 1.5 and dominate the entire load factor product.

$K_{H\beta}$ — Face Load Distribution Factor

Physical meaning: $K_{H\beta}$ accounts for non-uniform load distribution across the facewidth due to shaft bending and torsion, housing distortion, bearing compliance, gear blank distortion, and manufacturing misalignment. It is the most consequential factor for wide-face gears.

Method B (ISO 6336-1:2019, Clause 6.7):

$$K_{H\beta} = 1 + \frac{c_{\gamma}^\prime \cdot F_{m,\beta}}{F_t / b}$$

The equivalent misalignment:

$$F_{m,\beta} = \sqrt{(f_{sh} + f_{ma})^2 + f_{be}^2} – y_\beta + c_\beta$$

Where $f_{sh}$ is shaft deflection (bending + torsion under load), $f_{ma}$ is housing bore alignment error, $f_{be}$ is bearing position error, $y_\beta$ is running-in allowance, and $c_\beta$ is the crowning correction.

What engineers miss — three critical points:

First: $K_{H\beta} = 1.0$ requires calculation justification. For any gear with $b/d_1 > 0.3$, or shaft spans producing meaningful deflection, assuming 1.0 is not conservative — it is incorrect. “Symmetric bearing arrangement” is not a calculation.

Second: Crowning is a design variable, not a manufacturing nicety. A gear designed with $C_\beta = 8$ µm of crowning can achieve $K_{H\beta} \approx 1.2$ for wide-face applications where the uncrowned gear yields $K_{H\beta} = 1.6$ or higher. The normative framework quantifies this directly through $c_\beta$ in the equivalent misalignment formula.

Third: $K_{H\beta} \neq K_{F\beta}$. The face load distribution at the flank and at the root differ for the same gear pair. ISO 6336-1:2019 provides separate computation procedures. Applying the same numerical value to both is an approximation that ISO 6336 does not sanction.

$K_{H\alpha}$ — Transverse Load Distribution Factor

Physical meaning: $K_{H\alpha}$ accounts for uneven load sharing between simultaneously meshing tooth pairs in the transverse plane. Pitch errors and profile deviations cause load concentration on fewer teeth than the contact ratio implies.

Method B (ISO 6336-1:2019):

$$K_{H\alpha} \geq 1.0 \quad \text{always}$$

$$K_{H\alpha} = \frac{\varepsilon_\alpha \left(0.9 + 0.4 \cdot c_{\gamma}^\prime \cdot \frac{f_{pb} – y_{p\alpha}}{F_t / b}\right)}{[\text{bounded expression}]}$$

Where $f_{pb}$ is the base pitch deviation and $y_{p\alpha}$ is the profile running-in allowance.

What engineers miss: $K_{H\alpha}$ and $K_v$ draw from the same physical error inputs — $f_{pb}$ and profile deviations — but model different effects. $K_v$ models dynamic response to mesh excitation. $K_{H\alpha}$ models static load redistribution from the same imperfections. Using measured $f_{pb}$ for $K_v$ while estimating $K_{H\alpha}$ from quality grade tables breaks the self-consistency of the error model. ISO 6336-1 assumes consistent error basis across both factors.

Part III — Permissible Contact Stress $\sigma_{HP}$

$\sigma_{H\lim}$ — Allowable Contact Stress Number

Physical meaning: $\sigma_{H\lim}$ is determined from standardised gear loading tests on reference test gears under defined reference conditions. It is not a classical tensile or yield strength — it is a gear fatigue test result normalised to a specific geometry, from which all real operating conditions are corrected by the factors below.

Reference conditions (ISO 6336-2:2019, Clause 10):

• Module: $m_n = 3$–$5$ mm
• Surface roughness at reference pitch diameter: $R_{z10} = 3$ µm
• Lubricant: mineral oil, $\nu_{50} = 100$ cSt
• Pitch line velocity: $v = 8.3$ m/s
• Material quality grade: MQ per ISO 6336-5

Values from ISO 6336-5:2016 (Method B, MQ quality):

What engineers miss: The range within each class reflects the MQ–ME quality gradient. ME values — the upper end — require documented evidence of material cleanliness, case depth uniformity, and heat treatment process control that exceeds standard production practice. Using ME values without this documentation is non-conservative. MQ is the correct default for standard production gears. ML represents the floor — below which the standard’s material model does not apply.

$Z_{NT}$ — Life Factor

Physical meaning: $Z_{NT}$ scales $\sigma_{H\lim}$ for operation at cycle counts other than the reference endurance $N_{ref} = 5 \times 10^7$ cycles (Method B, MQ). It implements the Wöhler (S-N) curve for contact fatigue.

The most consequential and least-known aspect of $Z_{NT}$:

Case-hardened and nitrided gears do not have a classical fatigue endurance limit. At $N_L > 3 \times 10^8$ cycles (case-hardened), $Z_{NT}$ decreases below 1.0 and continues decreasing. ISO 6336-5 provides the explicit S-N curves.

At $N_L = 10^{10}$ cycles for case-hardened steel (Method B): $Z_{NT} \approx 0.85$–$0.90$.

The consequence: a design that appears compliant at $S_H = 1.10$ with $Z_{NT} = 1.0$ is actually at $S_H \approx 0.93$–$0.99$ with the correct value — below or at the minimum. Setting $Z_{NT} = 1.0$ for case-hardened gears at long life is not conservative. It is non-conservative. This error appears in practice wherever “unlimited life” is assumed without reading the S-N curve for the actual material class.

$Z_L$, $Z_v$, $Z_R$ — Lubricant Film Factors

Physical basis: Elastohydrodynamic (EHL) theory predicts a minimum film thickness $h_{min}$ dependent on viscosity, velocity, and elastic properties. The specific film thickness (lambda ratio):

$$\lambda = \frac{h_{min}}{R_{z,eff}}$$

governs the proportion of load carried by asperity contact versus full fluid film. Higher $\lambda$ → less asperity contact → lower surface fatigue damage → higher permissible stress.

$Z_L$, $Z_v$, and $Z_R$ are the normative encoding of $\lambda$ as three separable correction factors:

$Z_L$ — Lubricant factor: Higher kinematic viscosity $\nu_{50}$ at 50°C → thicker EHL film → higher $Z_L$. The 2019 edition extended Method B to cover VG10, VG15, and VG22, absent from 2006.

$Z_v$ — Velocity factor: Higher pitch line velocity $v$ → thicker EHL film → higher $Z_v$. The effect saturates above approximately $v = 10$ m/s — the film is fully established and further velocity provides no additional benefit.

$Z_R$ — Roughness factor: Lower combined surface roughness $R_{z10}$ → higher $\lambda$ → higher $Z_R$. A rough surface ($R_{z10} = 10$ µm) reduces $Z_R$ by approximately 15% relative to a well-finished surface ($R_{z10} = 1$ µm) — a reduction in $\sigma_{HP}$ that no heat treatment can recover, because $\sigma_{H\lim}$ is fixed by material and the film factor is a separate multiplier.

Reference condition: $\nu_{50} = 100$ cSt, $v = 8.3$ m/s, $R_{z10} = 3$ µm → $Z_L = Z_v = Z_R = 1.0$.

Critical normative point (ISO 6336-2:2019, Clause 12.3.3): In the static and upper limited-life range (below the lower knee of the S-N curve), $Z_L = Z_v = Z_R = 1.0$. The EHL film factors do not apply when the governing failure mode is static overload rather than fatigue. Software that applies the full factor set regardless of life regime introduces an error in the static strength calculation.

$Z_W$ — Work Hardening Factor

Physical meaning: $Z_W$ accounts for the increase in surface durability of a through-hardened wheel meshing with a surface-hardened pinion. Repeated Hertzian contact work-hardens the softer wheel surface, increasing its fatigue resistance beyond the initial $\sigma_{H\lim}$.

Conditions for $Z_W > 1.0$ (ISO 6336-2:2019, Clause 13):

• Pinion: surface-hardened (Eh, IF, or NT), surface hardness $\geq 45$ HRC
• Wheel: through-hardened wrought steel, 130–400 HB
• $Z_W$ range: 1.2 (at 130 HB) down to 1.0 (at ≥470 HB) — decreasing with wheel hardness

2019 edition change: A third material pairing case was added for specific through-hardened grade combinations, with $Z_W$ values that differ from the 2006 table in the 300–400 HB range.

The asymmetry: $Z_W > 1.0$ applies to the wheel only. For the pinion: $Z_{W1} = 1.0$ always. The pinion is already surface-hardened; it has received its hardening benefit through heat treatment. It is the softer wheel surface that work-hardens in service. Applying $Z_W > 1.0$ to both pinion and wheel is a non-conservative error in the pinion calculation — and it is the type of error that software can introduce silently if the asymmetry is not enforced.

$Z_X$ — Size Factor

Physical meaning: $Z_X$ accounts for the statistical size effect in contact fatigue — larger material volumes have higher probability of containing critical defects, reducing effective fatigue strength relative to the reference test gear.

Method B values:

For the majority of industrial gears ($m_n \leq 10$ mm), $Z_X$ is between 0.95 and 1.0. For large industrial drives with $m_n > 20$ mm, $Z_X$ reduces $\sigma_{HP}$ by 10–15% — compounding with all other factors and not negligible at that scale.

Part IV — Assembling the Complete Calculation

The following sequence eliminates all forward dependencies.

Step 1 — Basic geometry

$$d_1 = \frac{m_n \cdot z_1}{\cos\beta} \qquad \alpha_t = \arctan!\left(\frac{\tan\alpha_n}{\cos\beta}\right) \qquad d_{b1} = d_1 \cos\alpha_t$$

$$a_w = \frac{m_n}{2\cos\beta}(z_1 + z_2) \cdot \frac{\cos\alpha_t}{\cos\alpha_{wt}} \qquad \alpha_{wt} = \arccos!\left(\frac{d_{b1} + d_{b2}}{2a_w}\right)$$

Note: for standard centre distance ($x_1 + x_2 = 0$): $a_w = m_n(z_1+z_2)/(2\cos\beta)$ and $\alpha_{wt} = \alpha_t$. For non-zero profile shift sum, $\alpha_{wt}$ is solved iteratively from the involute function: $\text{inv},\alpha_{wt} = \text{inv},\alpha_t + 2(x_1+x_2)\tan\alpha_n/(z_1+z_2)$.

$$\varepsilon_\alpha = \frac{\sqrt{d_{a1}^2 – d_{b1}^2} + \sqrt{d_{a2}^2 – d_{b2}^2} – 2a_w \sin\alpha_{wt}}{2\pi m_n \cos\alpha_t / \cos\beta}$$

$$\varepsilon_\beta = \frac{b \sin\beta}{\pi m_n}$$

Step 2 — Geometry-based stress factors

$$Z_H = \sqrt{\frac{2\cos\beta_b \cos\alpha_{wt}}{\cos^2\alpha_t \sin\alpha_{wt}}} \qquad Z_E = 189.8\ \sqrt{\text{N/mm}^2} \quad\text{(steel/steel)}$$

$$Z_\varepsilon: \text{per Clause 8, with interpolation if } 0 < \varepsilon_\beta < 1$$

$$Z_\beta = \sqrt{\cos\beta} \qquad Z_B, Z_D: \text{from } \varepsilon_1, \varepsilon_2, u \text{ per Clause 6.3}$$

Step 3 — Load

$$F_t = \frac{2T_1}{d_1} \qquad v = \frac{\pi d_1 n_1}{60 \times 10^3}$$

Step 4 — Load factors (ISO 6336-1:2019)

$K_A$: from drive train class. $K_v$: from $v$, $f_{pb}$, resonance ratio $N$. $K_{H\beta}$: from shaft deflection model, $c_\beta$ if crowned. $K_{H\alpha}$: from $f_{pb}$, $y_{p\alpha}$.

Step 5 — Nominal contact stress

$$\sigma_{H0} = Z_H \cdot Z_E \cdot Z_\varepsilon \cdot Z_\beta \cdot \sqrt{\frac{F_t}{d_1 b} \cdot \frac{u+1}{u}}$$

Step 6 — Calculated contact stress at determinant point

$$\sigma_{H1} = Z_B \cdot \sigma_{H0} \cdot \sqrt{K_A K_v K_{H\beta} K_{H\alpha}}$$ $$\sigma_{H2} = Z_D \cdot \sigma_{H0} \cdot \sqrt{K_A K_v K_{H\beta} K_{H\alpha}}$$

Step 7 — Permissible contact stress

Select $\sigma_{H\lim}$ from ISO 6336-5 for declared material and quality grade. Read $Z_{NT}$ from the S-N curve at $N_L$ — not from a single reference point, not assumed to be 1.0 for long life case-hardened gears. Compute $Z_L$, $Z_v$, $Z_R$ at actual $\nu_{50}$, $v$, $R_{z10}$. Apply $Z_W$ to wheel only ($Z_{W1} = 1.0$). Apply $Z_X$ at actual $m_n$.

$$\sigma_{HP} = \sigma_{H\lim} \cdot Z_{NT} \cdot \frac{Z_L Z_v Z_R}{S_{H\lim}} \cdot Z_W \cdot Z_X$$

Step 8 — Safety factors

$$S_{H1} = \frac{\sigma_{HP1}}{\sigma_{H1}} \qquad S_{H2} = \frac{\sigma_{HP2}}{\sigma_{H2}}$$

Both must satisfy $S_{H1}, S_{H2} \geq 1.0$.

Critical note on $S_{H,\min}$ position in the formula: $S_{H,\min}$ appears in the denominator of $\sigma_{HP}$ — the permissible stress is already reduced by the required safety margin before the ratio is computed. The resulting $S_H$ must be $\geq 1.0$, not $\geq S_{H,\min}$ again. Double-applying $S_{H,\min}$ — checking $S_H \geq S_{H,\min}$ where $\sigma_{HP}$ was already divided by $S_{H,\min}$ — produces an over-conservative result with no normative basis. This error is pervasive in hand calculation sheets that mix the ISO formula with AGMA safety factor conventions.

Part V — Worked Example: Spur Gear, Steel/Steel

Given:

• $m_n = 2.5$ mm, $z_1 = 20$, $z_2 = 40$, $\beta = 0°$, $\alpha_n = 20°$
• $x_1 = x_2 = 0$, $b = 25$ mm
• $T_1 = 80$ N·m, $n_1 = 1{,}500$ rpm
• Pinion: case-hardened Eh, MQ quality, $\sigma_{H\lim,1} = 1{,}500$ MPa
• Wheel: through-hardened St, 300 HB, $\sigma_{H\lim,2} \approx 600$ MPa
• ISO quality grade 7, mineral oil VG 150 ($\nu_{40} = 150$ cSt, $\nu_{50} \approx 95$ cSt for VI≈95 mineral oil), $R_{z10} = 3$ µm
• $K_A = 1.25$, $S_{H,\min} = 1.0$, design life $N_L = 10^8$ cycles

Step 1 — Geometry:

$$d_1 = 2.5 \times 20 = 50\ \text{mm} \qquad d_2 = 2.5 \times 40 = 100\ \text{mm}$$

$$a_w = 75\ \text{mm (standard, } x_1 + x_2 = 0\text{)} \qquad \alpha_{wt} = \alpha_t = 20°$$

$$F_t = \frac{2 \times 80{,}000}{50} = 3{,}200\ \text{N} \qquad v = \frac{\pi \times 50 \times 1{,}500}{60 \times 10^3} = 3.93\ \text{m/s}$$

$$\varepsilon_\alpha \approx 1.62 \qquad \varepsilon_\beta = 0 \text{ (spur)}$$

Step 2 — Stress factors:

$$Z_H = 2.495 \qquad Z_E = 189.8\ \sqrt{\text{N/mm}^2}$$

$$Z_\varepsilon = \sqrt{\frac{4-1.62}{3}} = \sqrt{0.793} = 0.890 \qquad Z_\beta = 1.0$$

From addendum contact ratios for standard gear ($\varepsilon_1 \approx 0.77$, $\varepsilon_2 \approx 0.85$, $u = 2$):

$M_1$ and $M_2$ derive from transverse radii of curvature at the IPSTC per Clause 6.3. The full curvature-radius expressions are not reproduced here — the values below are illustrative for this geometry, not independently re-derived from the clause text, and must be verified against the standard before use in a production calculation:

$$M_1 \approx 1.035 \rightarrow Z_B = \sqrt{1.035} = 1.017$$

$$M_2 \approx 0.98 < 1 \rightarrow Z_D = 1.00$$

In production calculations, $M_1$ and $M_2$ must be computed from the full curvature radius expressions in Clause 6.3, not estimated.

Step 3 — Nominal contact stress:

$$\frac{F_t}{d_1 b} = \frac{3{,}200}{50 \times 25} = 2.560\ \text{N/mm}^2 \qquad \frac{u+1}{u} = \frac{3}{2} = 1.500$$

$$\sigma_{H0} = 2.495 \times 189.8 \times 0.890 \times 1.0 \times \sqrt{2.560 \times 1.500}$$ $$= 421.4 \times \sqrt{3.840} = 421.4 \times 1.960 = 826.0\ \text{MPa}$$

Step 4 — Load factors (approximate for illustration):

$$K_v \approx 1.12 \quad (Q7,\ v = 3.93\ \text{m/s}) \qquad K_{H\beta} \approx 1.15 \quad (b/d_1 = 0.5,\text{ stiff shaft})$$ $$K_{H\alpha} \approx 1.04$$

$$K_A K_v K_{H\beta} K_{H\alpha} = 1.25 \times 1.12 \times 1.15 \times 1.04 = 1.677$$

Step 5 — Calculated contact stress:

$$\sigma_{H1} = 1.017 \times 826.0 \times \sqrt{1.677} = 840.0 \times 1.295 = 1{,}088\ \text{MPa}$$ $$\sigma_{H2} = 1.000 \times 826.0 \times 1.295 = 1{,}070\ \text{MPa}$$

Step 6 — Permissible contact stress:

Pinion (case-hardened, $N_L = 10^8 > N_{ref}$): from ISO 6336-5 S-N curve for Eh, $Z_{NT,1} \approx 0.955$

Film factors at operating conditions: $Z_L \approx 1.02$ (VG 150, $\nu_{50} \approx 95$ cSt — marginally below the reference 100 cSt, small positive correction), $Z_v \approx 1.00$ ($v = 3.93$ m/s, below EHL saturation), $Z_R = 1.00$ ($R_{z10} = 3$ µm, at reference condition).

$$\sigma_{HP1} = 1{,}500 \times 0.955 \times \frac{1.02 \times 1.00 \times 1.00}{1.0} \times 1.00 \times 1.00 = 1{,}462\ \text{MPa}$$

Wheel (through-hardened 300 HB, $Z_{NT,2} \approx 1.0$ — St has endurance limit): $Z_W = 1.10$ [ISO 6336-2:2006 Clause 13.2.1.2 formula, $Z_W = 1.2-(HB-130)/1700$ at reference roughness; 2019-edition constants for the 300–400 HB band not independently verified]

$$\sigma_{HP2} = 600 \times 1.0 \times \frac{1.02 \times 1.00 \times 1.00}{1.0} \times 1.10 \times 1.00 = 673\ \text{MPa}$$

Step 7 — Safety factors:

$$S_{H1} = \frac{1{,}462}{1{,}088} = 1.344 \quad \checkmark$$

$$S_{H2} = \frac{673}{1{,}070} = 0.629 \quad \times\text{ FAIL}$$

The wheel fails. $S_{H2} = 0.63 < 1.0$. The pinion is comfortable at 1.34, but the pairing is inadequate. This is the correct and expected result for a case-hardened pinion against a through-hardened wheel at this torque — the wheel surface durability is the limiting constraint.

The corrective interventions, in order of engineering effectiveness:

1. Harden the wheel surface (Eh or IF) — largest $\sigma_{H\lim}$ gain
2. Increase $d_1$ — reduces $F_t/(d_1 b)$ directly
3. Increase $b$ — reduces $F_t/(d_1 b)$ and potentially improves $Z_\varepsilon$ if $\varepsilon_\beta$ increases
4. Reduce $T_1$ — reduces $F_t$ but is outside the gear designer’s control

Each intervention is quantifiable from the equations above before any hardware is changed.

Part VI — The Five Calculation Errors That Appear in Practice

These are not theoretical concerns. They appear in engineering reports produced by experienced practitioners.

Error 1: Setting $Z_{NT} = 1.0$ for case-hardened gears at long life. Through-hardened steel has a true endurance limit; $Z_{NT} = 1.0$ for $N_L > N_{ref}$ is correct. Case-hardened steel does not. At $N_L = 10^{10}$ cycles, the correct $Z_{NT}$ for Eh (Method B) is approximately 0.85–0.90. A design at $S_H = 1.10$ with assumed $Z_{NT} = 1.0$ is actually at $S_H \approx 0.93$–$0.99$. It does not pass.

Error 2: Assuming $K_{H\beta} = 1.0$ without calculation for wide-face gears. Valid only for narrow, stiff gears with well-aligned shafts and symmetric bearing spans. For $b/d_1 > 0.3$ or shaft spans producing meaningful deflection, $K_{H\beta}$ requires explicit derivation from shaft geometry. Assuming 1.0 is not conservative — it is incorrect by omission.

Error 3: Omitting $Z_B$ and $Z_D$ for spur gears. For $\varepsilon_\alpha \leq 2$, $Z_B > 1.0$ for the pinion in most practical configurations. The omission produces a 4–15% underestimate of $\sigma_{H1}$, which is a corresponding overestimate of $S_{H1}$. A design that appears at $S_H = 1.10$ without $Z_B$ may be at $S_H = 1.02$ with it.

Error 4: Mixing 2006 and 2019 edition factors. Specific items that changed: $Z_W$ tables for 300–400 HB range, $Z_L$ for VG10/VG15/VG22, auxiliary factor $f_{ZCa}$ for helical contact factors, active vs. reference tip/root diameters. A calculation that uses 2019 $\sigma_{H\lim}$ from ISO 6336-5:2016 with 2006 $Z_W$ values is internally inconsistent and will not self-report the inconsistency.

Error 5: Double-applying $S_{H,\min}$. ISO 6336 places $S_{H,\min}$ in the denominator of $\sigma_{HP}$. The resulting ratio $S_H = \sigma_{HP}/\sigma_H$ must be $\geq 1.0$. Checking instead that $S_H \geq S_{H,\min}$ where $\sigma_{HP}$ already contains $S_{H,\min}$ in the denominator applies the safety margin twice. This error appears in hand calculation sheets that follow AGMA conventions where $S_H$ is compared directly to $S_{H,\min}$ without it being embedded in the allowable.

Part VII — What $S_H$ Does Not Cover

A calculation that produces $S_H \geq 1.0$ has established resistance to destructive pitting under the assumed conditions. Four other gear flank failure modes require separate normative analysis:

Micropitting (ISO/TR 15144-1): Sub-surface fatigue initiating at asperity scale, governed by the specific film thickness $\lambda = h_{min}/R_{z,eff}$. Most frequent in slow, high-torque applications with inadequate EHL film. A gear can pass $S_H$ and fail by micropitting — the two calculations share $Z_R$ (roughness factor) but micropitting is governed by the full film thickness model, not the Hertz contact stress.

Scuffing (ISO/TR 13989-1/2): Thermally-driven adhesive failure. Governed by flash temperature (Blok’s criterion) or integral temperature criterion. Not a fatigue phenomenon — it can occur at first engagement. Not related to $S_H$: a gear designed for high contact fatigue safety can still scuff under inadequate lubrication at high PV (pressure-velocity product).

Tooth flank fracture (ISO/DTS 19042-1): Subsurface crack initiating at the case-core boundary in surface-hardened gears, propagating to a flank fracture that detaches a large fragment. The critical subtlety: a gear designed for high $S_H$ by using a thick hardened case and high $\sigma_{H\lim}$ is more susceptible to flank fracture, not less — the case-core boundary is loaded more severely when the case is deep. Pitting resistance and flank fracture resistance are partially opposing design objectives for case-hardened gears. This is known in research but rarely acknowledged in standard gear design practice.

Wear (no ISO 6336 method): Abrasive and adhesive material removal in the mixed or boundary lubrication regime. The dominant failure mode in slow-speed, contaminated-lubricant industrial gearboxes. $S_H$ appears high in these applications because $K_v$ is low and the Hertz stress is moderate — yet the gear fails by wear. The absence of a wear calculation method in ISO 6336 is a documented limitation of the standard.

The $S_H$ calculation is necessary. For most industrial applications, it is not sufficient alone.

Conclusion

The contact safety factor $S_H$ is not a measure of how far a gear is from failure. It is the ratio between the material’s resistance at actual operating conditions and the applied stress at the determinant contact point — where every factor in the chain is a correction from the reference test condition to the real condition.

Every factor that is estimated rather than calculated is an unknown in that ratio. In practice, the weakest estimated input is almost always $K_{H\beta}$ — the factor that requires shaft deflection analysis but is most frequently assumed. The second most frequently estimated input is $Z_{NT}$ for long-life case-hardened gears, where the assumption of $Z_{NT} = 1.0$ is non-conservative by definition.

The minimum required safety factor $S_{H,\min}$ is not specified by ISO 6336. The standard specifies the calculation method and the factor chain. $S_{H,\min}$ is specified by the application standard, the customer contract, or the engineer’s professional judgement — and it must account for every uncertainty in every factor that was not directly measured.

A calculation is only as reliable as its most uncertain input. Knowing which input that is, and why, is what separates a calculation from a number.

Qevork automates the complete ISO 6336-2:2019 $S_H$ and $S_F$ factor chain — including $Z_B$/$Z_D$ for the correct determinant contact point, $Z_W$ per the updated 2019 material pairing table, $Z_{NT}$ from the full S-N curve for each material class, and $K_{H\beta}$ from shaft geometry input rather than assumption. Join the early access list to be notified at launch.