{"id":948,"date":"2026-07-07T10:15:55","date_gmt":"2026-07-07T13:15:55","guid":{"rendered":"https:\/\/qevork.com\/?p=948"},"modified":"2026-07-10T15:20:53","modified_gmt":"2026-07-10T18:20:53","slug":"iso-6336-pitting-resistance-the-complete-sh-calculation-walkthrough","status":"publish","type":"post","link":"https:\/\/qevork.com\/pt\/iso-6336-pitting-resistance-the-complete-sh-calculation-walkthrough\/","title":{"rendered":"Resist\u00eancia ao Pitting conforme a ISO 6336: O Guia Completo do C\u00e1lculo de $S_H$"},"content":{"rendered":"\t\t
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The Calculation Most Engineers Think They Understand<\/h2>

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 \u2014 and four of the most consequential ones are either omitted from secondary sources, explained incorrectly, or applied in ways the standard does not permit.<\/p>

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.<\/p>

Before the first equation: what pitting actually is, and what the $S_H$ calculation does and does not govern.<\/p>

What Pitting Is \u2014 and What ISO 6336 Actually Governs<\/h2>

A gear flank does not fail because a formula returned $S_H < S_{H,\\min}$. It fails because subsurface shear stresses \u2014 generated by repeated Hertzian contact \u2014 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.<\/p>

ISO 6336-2:2019 makes a distinction that almost no secondary source states correctly: not all pitting is failure.<\/strong><\/p>

The standard defines three regimes. Initial pitting that subsequently decelerates (degressive pitting) or ceases entirely (arrested pitting) is considered tolerable \u2014 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.<\/p>

The practical implication: for some slow-speed industrial gears with large teeth \u2014 module 25 or larger \u2014 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.<\/p>

This is not a loophole. It is a documented normative position in Clause 4 that most engineers have never read \u2014 because they locate the formula and never read the scope.<\/p>

The $S_H$ calculation governs destructive pitting only.<\/strong> It does not govern:<\/p>

  • Micropitting<\/strong> (grey staining, surface-initiated sub-asperity fatigue) \u2014 governed by ISO\/TR 15144-1, via the specific film thickness $\\lambda$<\/li>
  • Scuffing<\/strong> (instantaneous adhesive failure) \u2014 governed by ISO\/TR 13989-1\/2, via flash temperature or integral temperature criterion<\/li>
  • Tooth flank fracture<\/strong> (subsurface crack at the case-core boundary) \u2014 governed by ISO\/DTS 19042-1<\/li>
  • Wear<\/strong> (abrasive material removal in mixed\/boundary lubrication) \u2014 no ISO 6336 method<\/li><\/ul>

    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.<\/p>

    Normative context declaration \u2014 required before every calculation report:<\/strong>
    Standard: ISO 6336-2:2019 | Method: B | Lubrication: oil-lubricated | Gear type: external cylindrical | Failure mode: destructive pitting<\/p>

    These are not boilerplate. They define which equations apply.<\/p>

    The Structural Assumption That Silently Invalidates Results<\/h2>

    Every ISO 6336 $S_H$ calculation carries an implicit requirement almost never stated: all factors must belong to the same normative context object.<\/strong> ISO 6336 and AGMA 2001-D04 use symbols that appear interchangeable \u2014 $Z_E$ and $C_p$ are both “elasticity factors”, $S_H$ appears in both standards \u2014 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.<\/p>

    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 \u2014 and will not produce an error message.<\/p>

    The Master Equation \u2014 Complete Factor Inventory<\/h2>

    $$S_H = \\frac{\\sigma_{HP}}{\\sigma_H} \\geq S_{H,\\min}$$<\/p>

    Calculated contact stress at the determinant point:<\/strong><\/p>

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

    Nominal contact stress at the pitch point:<\/strong><\/p>

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

    Permissible contact stress:<\/strong><\/p>

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

    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.<\/p>

    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 $\\sigma_H$ \u2014 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 \u2014 but does not eliminate them.<\/p>

    Part I \u2014 Nominal Contact Stress $\\sigma_{H0}$<\/h2>

    $Z_H$ \u2014 Zone Factor<\/h3>

    Physical meaning:<\/strong> $Z_H$ 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 \u2014 independent of load, material, and lubrication.<\/p>

    Formula (ISO 6336-2:2019, Clause 6.2):<\/strong><\/p>

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

    Where:<\/p>

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

      For a standard spur gear ($\\beta = 0\u00b0$, $x_1 + x_2 = 0$, $\\alpha_n = 20\u00b0$):<\/p>

      $$Z_H = \\sqrt{\\frac{2 \\times 1.0 \\times \\cos 20\u00b0}{\\cos^2 20\u00b0 \\times \\sin 20\u00b0}} = \\sqrt{\\frac{2 \\times 0.9397}{0.8830 \\times 0.3420}} \\approx 2.495$$<\/p>

      What engineers miss:<\/strong> $Z_H$ depends on $\\alpha_{wt}$, not $\\alpha_t$. These are equal only when the working centre distance equals the reference centre distance \u2014 i.e., when $\\sum x = x_1 + x_2 = 0$. Any positive profile shift sum increases $a_w$, increases $\\alpha_{wt}$, and decreases $Z_H$.<\/p>
      Design<\/th>$x_1$<\/th>$x_2$<\/th>$\\sum x$<\/th>Effect on $Z_H$<\/th><\/tr><\/thead>
      Symmetric shift<\/td>+0.3<\/td>\u22120.3<\/td>0<\/td>No change \u2014 standard centre distance<\/td><\/tr>
      Positive-sum shift<\/td>+0.3<\/td>+0.2<\/td>+0.5<\/td>$\\alpha_{wt} > \\alpha_t$ \u2192 $Z_H$ decreases<\/td><\/tr><\/tbody><\/table>

      For $m_n = 2.5$ mm, $z_1 = 20$, $z_2 = 40$, $\\alpha_n = 20\u00b0$, $\\sum x = +0.5$: $\\alpha_{wt} \\approx 22.3\u00b0$, giving $Z_H \\approx 2.349$ \u2014 a 5.83% reduction in $Z_H$, which compounds to an 11.3% reduction in $\\sigma_{H0}^2$ and a measurable improvement in $S_H^2$.<\/p>

      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.<\/p>

      $Z_E$ \u2014 Elasticity Factor<\/h3>

      Physical meaning:<\/strong> $Z_E$ 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 \u2014 not a property of either material alone.<\/p>

      Formula (Hertz contact theory, adopted by ISO 6336-2:2019, Clause 7):<\/strong><\/p>

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

      Standard values (ISO 6336-2:2019, Table 1):<\/strong><\/p>
      Material pair<\/th>$Z_E$ $\\left[\\sqrt{\\text{N\/mm}^2}\\right]$<\/th><\/tr><\/thead>
      Steel \/ Steel<\/td>189.8<\/td><\/tr>
      Steel \/ Cast iron GJL<\/td>188.0<\/td><\/tr>
      Steel \/ Nodular iron GJS<\/td>189.8<\/td><\/tr>
      Cast iron GJL \/ Cast iron GJL<\/td>180.0<\/td><\/tr><\/tbody><\/table>

      What engineers miss:<\/strong> $Z_E$ 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 \u2014 it cannot be optimised by material selection within the same class.<\/p>

      $Z_\\varepsilon$ \u2014 Contact Ratio Factor<\/h3>

      Physical meaning:<\/strong> $Z_\\varepsilon$ 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.<\/p>

      Formula (ISO 6336-2:2019, Clause 8, Method B):<\/strong><\/p>

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

      $$Z_\\varepsilon = \\sqrt{\\frac{4 – \\varepsilon_\\alpha}{3}}$$<\/p>

      For helical gears with $\\varepsilon_\\beta \\geq 1$:<\/p>

      $$Z_\\varepsilon = \\sqrt{\\frac{1}{\\varepsilon_\\alpha}}$$<\/p>

      For $0 < \\varepsilon_\\beta < 1$ \u2014 the intermediate case \u2014 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.<\/p>

      Numerical sensitivity \u2014 spur gear:<\/strong><\/p>
      $\\varepsilon_\\alpha$<\/th>$Z_\\varepsilon$<\/th>$\\Delta\\sigma_{H0}$ vs. $\\varepsilon_\\alpha = 1.50$<\/th><\/tr><\/thead>
      1.50<\/td>0.913<\/td>baseline<\/td><\/tr>
      1.60<\/td>0.894<\/td>\u22122.0%<\/td><\/tr>
      1.70<\/td>0.876<\/td>\u22124.1%<\/td><\/tr>
      1.80<\/td>0.856<\/td>\u22126.2%<\/td><\/tr>
      1.90<\/td>0.837<\/td>\u22128.3%<\/td><\/tr><\/tbody><\/table>

      A 0.3 increase in $\\varepsilon_\\alpha$ \u2014 achievable through addendum height increase or profile modification at no manufacturing cost increase with standard hobbing tooling \u2014 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.<\/p>

      What engineers miss:<\/strong> 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 \u2014 near $\\varepsilon_\\beta = 0.5$.<\/p>

      $Z_\\beta$ \u2014 Helix Angle Factor<\/h3>

      Physical meaning:<\/strong> $Z_\\beta$ 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.<\/p>

      Formula (ISO 6336-2:2019, Clause 9):<\/strong><\/p>

      $$Z_\\beta = \\sqrt{\\cos\\beta}$$<\/p>
      $\\beta$ [\u00b0]<\/th>$Z_\\beta$<\/th>Reduction in $\\sigma_{H0}$ vs. spur<\/th><\/tr><\/thead>
      0<\/td>1.000<\/td>\u2014<\/td><\/tr>
      10<\/td>0.992<\/td>\u22120.8%<\/td><\/tr>
      15<\/td>0.982<\/td>\u22121.8%<\/td><\/tr>
      20<\/td>0.969<\/td>\u22123.1%<\/td><\/tr>
      30<\/td>0.931<\/td>\u22126.9%<\/td><\/tr><\/tbody><\/table>

      What engineers miss:<\/strong> $Z_\\beta$ captures only part of the helix angle effect. The benefit appears here and in $Z_\\varepsilon$ via higher $\\varepsilon_\\beta$. The cost \u2014 increased axial force $F_a = F_t \\tan\\beta$, which loads thrust bearings \u2014 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.<\/p>

      $Z_B$ and $Z_D$ \u2014 Single-Pair Tooth Contact Factors<\/h3>

      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.<\/p>

      Physical meaning:<\/strong> For spur gears with $\\varepsilon_\\alpha \\leq 2$, the determinant contact point \u2014 where contact stress is maximum \u2014 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.<\/p>

      $Z_B$ (for the pinion) and $Z_D$ (for the wheel) shift the stress calculation from the pitch point to the IPSTC.<\/p>

      For $\\varepsilon_\\alpha \\leq 2$ (ISO 6336-2:2019, Clause 6.3):<\/strong><\/p>

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

      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 $u$.<\/p>

      For $\\varepsilon_\\alpha > 2$:<\/strong> $Z_B = Z_D = 1.0$ \u2014 the contact maximum occurs at the pitch point.<\/p>

      Practical impact:<\/strong> For a spur gear with $\\varepsilon_\\alpha = 1.6$, $Z_B$ can reach 1.04\u20131.07 depending on the gear ratio. This is a 4\u20137% increase in $\\sigma_H$, which corresponds to an 8\u201315% 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 ($\\sigma_{H2}$).<\/p>

      Part II \u2014 Load Factors<\/h2>

      The four load factors $K_A$, $K_v$, $K_{H\\beta}$, $K_{H\\alpha}$ are defined in ISO 6336-1:2019<\/strong>, 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.<\/p>

      $K_A$ \u2014 Application Factor<\/h3>

      Physical meaning:<\/strong> $K_A$ absorbs all dynamic load increments originating outside the gearbox \u2014 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.<\/p>

      Method B guide values (ISO 6336-1:2019, Table 2 \u2014 updated in 2019 edition):<\/strong><\/p>
      Prime mover<\/th>Driven machine<\/th>$K_A$<\/th><\/tr><\/thead>
      Uniform (electric motor, turbine)<\/td>Uniform<\/td>1.00<\/td><\/tr>
      Uniform<\/td>Moderate shocks<\/td>1.25<\/td><\/tr>
      Uniform<\/td>Heavy shocks<\/td>1.50<\/td><\/tr>
      Light shocks (multi-cylinder engine)<\/td>Uniform<\/td>1.10<\/td><\/tr>
      Light shocks<\/td>Moderate shocks<\/td>1.35<\/td><\/tr>
      Light shocks<\/td>Heavy shocks<\/td>1.60<\/td><\/tr>
      Heavy shocks (single-cylinder engine)<\/td>Moderate shocks<\/td>1.75<\/td><\/tr>
      Heavy shocks<\/td>Heavy shocks<\/td>$\\geq$ 2.00<\/td><\/tr><\/tbody><\/table>

      What engineers miss:<\/strong> $K_A$ 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 \u2014 there is no drive train configuration that reduces dynamic load below nominal in the normative sense.<\/p>

      $K_v$ \u2014 Dynamic Factor<\/h3>

      Physical meaning:<\/strong> $K_v$ accounts for internal dynamic load increments generated by gear mesh excitation \u2014 pitch errors, profile deviations, and periodic mesh stiffness variation. It is the factor most sensitive to manufacturing quality and pitch line velocity.<\/p>

      Method B requires:<\/strong><\/p>

      • Pitch line velocity: $v = \\pi d_1 n_1 \/ (60 \\times 10^3)$ [m\/s]<\/li>
      • ISO 1328-1 accuracy grade or specific deviation $f_{pb}$ [\u00b5m]<\/li>
      • Resonance ratio: $N = n_1 \/ n_{E1}$<\/li><\/ul>

        Numerical sensitivity:<\/strong><\/p>
        $v$ [m\/s]<\/th>ISO Quality 6<\/th>ISO Quality 8<\/th>ISO Quality 10<\/th><\/tr><\/thead>
        3<\/td>~1.05<\/td>~1.10<\/td>~1.18<\/td><\/tr>
        10<\/td>~1.12<\/td>~1.22<\/td>~1.38<\/td><\/tr>
        20<\/td>~1.20<\/td>~1.38<\/td>~1.65<\/td><\/tr><\/tbody><\/table>

        What engineers miss:<\/strong> $K_v$ is not an application property \u2014 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 \u2014 a reduction in the load factor product of approximately 8%.<\/p>

        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.<\/p>

        $K_{H\\beta}$ \u2014 Face Load Distribution Factor<\/h3>

        Physical meaning:<\/strong> $K_{H\\beta}$ 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.<\/p>

        Method B (ISO 6336-1:2019, Clause 6.7):<\/strong><\/p>

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

        The equivalent misalignment:<\/p>

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

        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.<\/p>

        What engineers miss \u2014 three critical points:<\/strong><\/p>

        First:<\/strong> $K_{H\\beta} = 1.0$ requires calculation justification. For any gear with $b\/d_1 > 0.3$, or shaft spans producing meaningful deflection, assuming 1.0 is not conservative \u2014 it is incorrect. “Symmetric bearing arrangement” is not a calculation.<\/p>

        Second:<\/strong> Crowning is a design variable, not a manufacturing nicety. A gear designed with $C_\\beta = 8$ \u00b5m 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.<\/p>

        Third:<\/strong> $K_{H\\beta} \\neq K_{F\\beta}$. 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.<\/p>

        $K_{H\\alpha}$ \u2014 Transverse Load Distribution Factor<\/h3>

        Physical meaning:<\/strong> $K_{H\\alpha}$ 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.<\/p>

        Method B (ISO 6336-1:2019):<\/strong><\/p>

        $$K_{H\\alpha} \\geq 1.0 \\quad \\text{always}$$<\/p>

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

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

        What engineers miss:<\/strong> $K_{H\\alpha}$ and $K_v$ draw from the same physical error inputs \u2014 $f_{pb}$ and profile deviations \u2014 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.<\/p>

        Part III \u2014 Permissible Contact Stress $\\sigma_{HP}$<\/h2>

        $\\sigma_{H\\lim}$ \u2014 Allowable Contact Stress Number<\/h3>

        Physical meaning:<\/strong> $\\sigma_{H\\lim}$ is determined from standardised gear loading tests on reference test gears under defined reference conditions. It is not a classical tensile or yield strength \u2014 it is a gear fatigue test result normalised to a specific geometry, from which all real operating conditions are corrected by the factors below.<\/p>

        Reference conditions (ISO 6336-2:2019, Clause 10):<\/strong><\/p>

        • Module: $m_n = 3$\u2013$5$ mm<\/li>
        • Surface roughness at reference pitch diameter: $R_{z10} = 3$ \u00b5m<\/li>
        • Lubricant: mineral oil, $\\nu_{50} = 100$ cSt<\/li>
        • Pitch line velocity: $v = 8.3$ m\/s<\/li>
        • Material quality grade: MQ per ISO 6336-5<\/li><\/ul>

          Values from ISO 6336-5:2016 (Method B, MQ quality):<\/strong><\/p>
          Material class<\/th>Heat treatment<\/th>$\\sigma_{H\\lim}$ [MPa]<\/th><\/tr><\/thead>
          St \u2014 wrought steel<\/td>Through-hardened, 180 HB<\/td>380\u2013450<\/td><\/tr>
          St \u2014 wrought steel<\/td>Through-hardened, 350 HB<\/td>550\u2013620<\/td><\/tr>
          GJL \u2014 grey cast iron<\/td>\u2014<\/td>180\u2013280<\/td><\/tr>
          GJS \u2014 nodular iron<\/td>\u2014<\/td>340\u2013460<\/td><\/tr>
          Eh \u2014 case-hardened<\/td>56\u201362 HRC surface<\/td>1,500\u20131,650<\/td><\/tr>
          IF \u2014 flame\/induction hardened<\/td>52\u201358 HRC surface<\/td>1,100\u20131,300<\/td><\/tr>
          NT \u2014 nitrided steel<\/td>60\u201365 HV case<\/td>1,050\u20131,250<\/td><\/tr><\/tbody><\/table>

          What engineers miss:<\/strong> The range within each class reflects the MQ\u2013ME quality gradient. ME values \u2014 the upper end \u2014 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 \u2014 below which the standard’s material model does not apply.<\/p>

          $Z_{NT}$ \u2014 Life Factor<\/h3>

          Physical meaning:<\/strong> $Z_{NT}$ 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 W\u00f6hler (S-N) curve for contact fatigue.<\/p>
          Life regime<\/th>$N_L$ range<\/th>$Z_{NT}$<\/th>Physical meaning<\/th><\/tr><\/thead>
          Static \/ low cycle<\/td>$< 10^4$<\/td>$> 1.3$<\/td>Short life tolerates higher stress<\/td><\/tr>
          Limited life<\/td>$10^4$\u2013$N_{ref}$<\/td>$> 1.0$<\/td>Decreasing benefit as $N_L$ rises<\/td><\/tr>
          Reference endurance<\/td>$5 \\times 10^7$<\/td>$1.0$<\/td>The definition point of $\\sigma_{H\\lim}$<\/td><\/tr>
          Long life \u2014 through-hardened St<\/td>$> N_{ref}$<\/td>$1.0$<\/td>True endurance limit assumed<\/td><\/tr>
          Long life \u2014 case-hardened Eh<\/td>$> N_{ref}$<\/td>$< 1.0$<\/td>No true endurance limit; damage accumulates<\/td><\/tr>
          Long life \u2014 nitrided NT<\/td>$> N_{ref}$<\/td>$< 1.0$<\/td>Same as case-hardened<\/td><\/tr><\/tbody><\/table>

          The most consequential and least-known aspect of $Z_{NT}$:<\/strong><\/p>

          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.<\/p>

          At $N_L = 10^{10}$ cycles for case-hardened steel (Method B): $Z_{NT} \\approx 0.85$\u2013$0.90$.<\/p>

          The consequence: a design that appears compliant at $S_H = 1.10$ with $Z_{NT} = 1.0$ is actually at $S_H \\approx 0.93$\u2013$0.99$ with the correct value \u2014 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.<\/p>

          $Z_L$, $Z_v$, $Z_R$ \u2014 Lubricant Film Factors<\/h3>

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

          $$\\lambda = \\frac{h_{min}}{R_{z,eff}}$$<\/p>

          governs the proportion of load carried by asperity contact versus full fluid film. Higher $\\lambda$ \u2192 less asperity contact \u2192 lower surface fatigue damage \u2192 higher permissible stress.<\/p>

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

          $Z_L$ \u2014 Lubricant factor:<\/strong> Higher kinematic viscosity $\\nu_{50}$ at 50\u00b0C \u2192 thicker EHL film \u2192 higher $Z_L$. The 2019 edition extended Method B to cover VG10, VG15, and VG22, absent from 2006.<\/p>

          $Z_v$ \u2014 Velocity factor:<\/strong> Higher pitch line velocity $v$ \u2192 thicker EHL film \u2192 higher $Z_v$. The effect saturates above approximately $v = 10$ m\/s \u2014 the film is fully established and further velocity provides no additional benefit.<\/p>

          $Z_R$ \u2014 Roughness factor:<\/strong> Lower combined surface roughness $R_{z10}$ \u2192 higher $\\lambda$ \u2192 higher $Z_R$. A rough surface ($R_{z10} = 10$ \u00b5m) reduces $Z_R$ by approximately 15% relative to a well-finished surface ($R_{z10} = 1$ \u00b5m) \u2014 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.<\/p>

          Reference condition:<\/strong> $\\nu_{50} = 100$ cSt, $v = 8.3$ m\/s, $R_{z10} = 3$ \u00b5m \u2192 $Z_L = Z_v = Z_R = 1.0$.<\/p>

          Critical normative point (ISO 6336-2:2019, Clause 12.3.3):<\/strong> 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.<\/p>

          $Z_W$ \u2014 Work Hardening Factor<\/h3>

          Physical meaning:<\/strong> $Z_W$ 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 $\\sigma_{H\\lim}$.<\/p>

          Conditions for $Z_W > 1.0$ (ISO 6336-2:2019, Clause 13):<\/strong><\/p>

          • Pinion: surface-hardened (Eh, IF, or NT), surface hardness $\\geq 45$ HRC<\/li>
          • Wheel: through-hardened wrought steel, 130\u2013400 HB<\/li>
          • $Z_W$ range: 1.2 (at 130 HB) down to 1.0 (at \u2265470 HB) \u2014 decreasing with wheel hardness<\/li><\/ul>

            2019 edition change:<\/strong> 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\u2013400 HB range.<\/p>

            The asymmetry:<\/strong> $Z_W > 1.0$ 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 \u2014 and it is the type of error that software can introduce silently if the asymmetry is not enforced.<\/p>

            $Z_X$ \u2014 Size Factor<\/h3>

            Physical meaning:<\/strong> $Z_X$ accounts for the statistical size effect in contact fatigue \u2014 larger material volumes have higher probability of containing critical defects, reducing effective fatigue strength relative to the reference test gear.<\/p>

            Method B values:<\/strong><\/p>
            Module $m_n$ [mm]<\/th>Case-hardened Eh<\/th>Through-hardened St<\/th><\/tr><\/thead>
            $\\leq 5$<\/td>1.00<\/td>1.00<\/td><\/tr>
            10<\/td>~0.95<\/td>~0.97<\/td><\/tr>
            25<\/td>~0.85<\/td>~0.90<\/td><\/tr><\/tbody><\/table>

            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\u201315% \u2014 compounding with all other factors and not negligible at that scale.<\/p>

            Part IV \u2014 Assembling the Complete Calculation<\/h2>

            The following sequence eliminates all forward dependencies.<\/p>

            Step 1 \u2014 Basic geometry<\/strong><\/p>

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

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

            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: $\\text{inv},\\alpha_{wt} = \\text{inv},\\alpha_t + 2(x_1+x_2)\\tan\\alpha_n\/(z_1+z_2)$.<\/p>

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

            $$\\varepsilon_\\beta = \\frac{b \\sin\\beta}{\\pi m_n}$$<\/p>

            Step 2 \u2014 Geometry-based stress factors<\/strong><\/p>

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

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

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

            Step 3 \u2014 Load<\/strong><\/p>

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

            Step 4 \u2014 Load factors<\/strong> (ISO 6336-1:2019)<\/p>

            $K_A$: 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 $f_{pb}$, $y_{p\\alpha}$.<\/p>

            Step 5 \u2014 Nominal contact stress<\/strong><\/p>

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

            Step 6 \u2014 Calculated contact stress at determinant point<\/strong><\/p>

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

            Step 7 \u2014 Permissible contact stress<\/strong><\/p>

            Select $\\sigma_{H\\lim}$ from ISO 6336-5 for declared material and quality grade. Read $Z_{NT}$ from the S-N curve at $N_L$ \u2014 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 $m_n$.<\/p>

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

            Step 8 \u2014 Safety factors<\/strong><\/p>

            $$S_{H1} = \\frac{\\sigma_{HP1}}{\\sigma_{H1}} \\qquad S_{H2} = \\frac{\\sigma_{HP2}}{\\sigma_{H2}}$$<\/p>

            Both must satisfy $S_{H1}, S_{H2} \\geq 1.0$.<\/p>

            Critical note on $S_{H,\\min}$ position in the formula:<\/strong> $S_{H,\\min}$ appears in the denominator of $\\sigma_{HP}$ \u2014 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 $S_{H,\\min}$ \u2014 checking $S_H \\geq S_{H,\\min}$ where $\\sigma_{HP}$ was already divided by $S_{H,\\min}$ \u2014 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.<\/p>

            Part V \u2014 Worked Example: Spur Gear, Steel\/Steel<\/h2>

            Given:<\/strong><\/p>