Why three rating bases give three answers
Tooth-flank pitting is one of the dominant failure modes of geared turbomachinery drives. Once pit craters link up across the active flank, surface quality degrades, mesh noise and vibration rise, friction losses increase and the service life of the whole transmission shortens. Designers therefore size spiral bevel pairs against a contact-stress criterion, and three documents dominate practice: ISO 10300:2023, which covers the load capacity of bevel gears in separate parts for general influence factors and for surface durability; ANSI/AGMA 2003-D19, issued in 2019 for generated straight, zerol and spiral bevel teeth; and the Chinese aviation standard HB/Z 89.1 through 89.4, issued by the Ministry of Aviation Industry in 1985. The HB series was adapted from ISO with aviation-specific modifications and remains the reference for main and accessory bevel drives in Chinese aero engines.
Feed the same geometry, material and duty cycle into the three procedures and the answers diverge — sometimes by more than 40 % in calculated contact stress. Published cross-checks do not converge either: several studies report ISO contact safety factors below the AGMA value, others report the opposite ranking, and finite-element verification has been used both to defend and to question AGMA conservatism. A whole-standard ranking is thus not a reliable design argument. What is reproducible is the coefficient-level decomposition: which physical effects each standard counts, where they place them in the equation chain, and how much of the final difference each one carries.
Acceptance criteria: stress margin versus safety factor
Two formulations are in use. The stress form compares the calculated contact stress with an allowable value, σH ≤ σHP. The factor form compares a calculated safety factor with a minimum admissible value, SH ≥ SHmin. HB and ISO use both and each defines SHmin in its own way. AGMA 2003 has no minimum-safety-factor concept in the same sense; its reliability factor ZZ performs a comparable role but is keyed directly to an admissible failure probability and sits in the denominator of the allowable stress, so a larger ZZ lowers σHP.
| Failure probability | < 0.01 % | < 0.1 % | < 1 % | < 10 % | < 50 % |
|---|---|---|---|---|---|
| HB, SHmin | 1.12 | 1.00 | — | — | — |
| ISO, SHmin | 1.0–1.1 | — | — | — | |
| AGMA, ZZ | 1.22 | 1.12 | 1.00 | 0.92 | 0.84 |
The AGMA scale is the most explicit: it resolves five failure-probability bands and lets the designer trade reliability against size. HB offers only a two-step choice, and ISO gives a range rather than a table. One practical trap follows from this. In the HB procedure the reported safety factor is referenced to the unfactored fatigue limit, so it equals 1.12 times the ratio σHP/σH; the acceptance threshold is correspondingly SHmin = 1.12, not 1. Comparing an HB safety factor against an ISO or AGMA threshold of unity therefore understates the HB margin by 12 %.
Reference section: blank parameters or mid-face equivalent gear
All three documents rest on the same Hertzian core — the pressure between two curved elastic bodies loaded along a line — but they evaluate it at different sections of the tooth.
AGMA works directly with the outer, or large-end, blank parameters of the real bevel gear. The Hertz contact stress between mating flanks is taken as the strength measure, and the effects of load sharing between teeth and of manufacturing error are gathered into correction factors rather than into a transformed geometry. HB and ISO instead convert the pair into an equivalent cylindrical gear located at the mid-face section, and use the maximum contact-stress point on the flank as the pitting criterion; ISO additionally carries the mid-face contact-line length and the relative curvature radius explicitly into the basic stress term.
The consequence is a systematic offset in what the number means. The AGMA stress is referenced to the section with the largest module and the highest pitch-line velocity, where the absolute Hertzian pressure is highest but the load per unit length is distributed over the longest contact line. The HB and ISO stress is referenced to the mean section, which governs the bulk of the flank. Neither choice is more “correct”; they simply answer slightly different questions, which is why cross-standard comparisons must be made coefficient by coefficient.
The stress and allowable-stress equations
HB separates a nominal stress from a three-factor load chain:
$$\sigma_H=\sigma_{H0}\,K_A K_V K_{H\beta},\qquad \sigma_{H0}=Z_H Z_E Z_\varepsilon Z_i Z_\rho Z_C\sqrt{\frac{F_{mt}}{d_{v1}b}\cdot\frac{u_v+1}{u_v}}$$
$$\sigma_{HP}=Z_N Z_\theta\,\frac{\sigma_{H\lim}}{S_{H\min}}$$
Here Fmt is the tangential force on the equivalent cylindrical gear pitch circle, dv1 its pitch diameter, b the face width and uv the equivalent tooth-number ratio. ISO adds a fourth load factor and shifts the geometry into the square-root term:
$$\sigma_H=\sigma_{H0}\,K_A K_V K_{H\beta}K_{H\alpha},\qquad \sigma_{H0}=\sqrt{\frac{F_n}{l_{bm}\,b\,\rho_{rel}}}\,Z_{MB}Z_{LS}Z_E Z_K$$
$$\sigma_{HP}=\sigma_{H\lim}Z_{NT}Z_X Z_L Z_V Z_R Z_W Z_{Hyp}$$
with Fn the nominal normal force on the equivalent gear, lbm the mid-face contact-line length and ρrel the relative radius of curvature. AGMA writes the same physics through torque and a composite geometry factor:
$$\sigma_H=Z_E\sqrt{\frac{2000\,T_1}{b\,d_{e1}^{2}Z_I}}\,K_A K_V K_{H\beta}Z_X Z_{XC}$$
$$\sigma_{HP}=\frac{\sigma_{H\lim}Z_{NT}Z_W}{S_H K_\theta Z_Z}$$
where T1 is the pinion torque and de1 the large-end pitch diameter. The three expressions share a common structure — a Hertzian nominal term multiplied by a chain of correction factors — and the elastic coefficient is identical in all of them:
$$Z_E=\sqrt{\frac{1}{\pi\left[\dfrac{1-\nu_1^{2}}{E_1}+\dfrac{1-\nu_2^{2}}{E_2}\right]}$$
which gives 191.65 √MPa for a steel-on-steel pair, so material stiffness enters the comparison as a common multiplier and cannot explain any of the divergence.
Correction factors: what each standard counts
Grouping the factors by the physical effect they correct shows where the documents actually differ. The counts are nine in HB, seventeen in ISO (some of them dimensional quantities rather than pure coefficients) and eleven in AGMA — but AGMA’s geometry factor ZI is itself a composite that absorbs several effects which HB and ISO treat separately.
| Category | HB | ISO 10300 | AGMA 2003 |
|---|---|---|---|
| Load | KA, KV, KHβ | KA, KV, KHβ, KHα | KA, KV, KHβ |
| Geometry | ZH, Zρ, Zε, Zi, ZC | ZMB, ρrel, lbm, ZLS, ZX, ZK | ZI, ZX, ZXC |
| Material | ZE | ZE | ZE |
| Flank condition | — | ZW | ZW (hardness ratio) |
| Service condition | ZN, Zθ | ZNT, ZL, ZV, ZR, ZHyp | ZNT, Kθ, ZZ |
Load-side factors
The application factor KA carries the same definition everywhere: external dynamic overload from the prime mover and the driven machine. HB tabulates it only for a two-by-two matrix of uniform and lightly vibrating machines, with values of 1.00, 1.10, 1.25 and 1.35. AGMA and ISO use the same four-by-four grid, extended to moderate and heavy shock, and are numerically identical.
| Driven machine | Uniform | Light shock | Moderate shock | Heavy shock |
|---|---|---|---|---|
| Uniform | 1.00 | 1.10 | 1.25 | 1.50 |
| Light shock | 1.25 | 1.35 | 1.50 | 1.75 |
| Moderate shock | 1.50 | 1.60 | 1.75 | 2.00 |
| Heavy shock | 1.75 or higher | 1.85 or higher | 2.00 or higher | 2.25 or higher |
The dynamic factor KV is where the three documents separate sharply. HB and AGMA both use empirical power laws driven by pitch-line velocity and an accuracy grade, but the grades are numbered in different directions and on different scales — HB treats grade 5 as the finest, AGMA treats grade 11 as the finest — so the same physical gear quality does not map onto the same coefficient. AGMA’s form is explicit:
$$K_V=\left(\frac{A}{A+200\,v_{et}}\right)^{-B},\qquad A=50+56(1-B),\qquad B=0.25\,(12-Q_V)^{0.667}$$
where vet is the large-end pitch-line velocity and QV the transmission accuracy grade. ISO abandons the single power law and introduces a dimensionless reference speed N that separates sub-critical, main-resonance, intermediate and super-critical operation, with a different expression in each band (N ≤ 0.75, 0.75 < N ≤ 1.25, 1.25 < N < 1.5, N ≥ 1.5). The ISO expressions combine the equivalent face width bv, the effective pitch deviation fpeff, the single-tooth mesh stiffness c′, the tangential force and a set of band coefficients cv1…cv7. Because the result is governed by where the operating point falls relative to resonance, the ISO factor rises in a step-like rather than a monotonic manner, and the same change in speed or module can move the design from one band into another and reverse a trend.
The face-load distribution factor KHβ corrects for non-uniform load along the face width. All three standards express it through a mounting coefficient:
$$K_{H\beta}=\left(1+\frac{|T_{D1}-T_1|}{T_{D1}}\right)K’_{H\beta}\ \ \text{(HB)},\qquad K_{H\beta}=K_{mb}+5.6\times10^{-6}b^{2}\ \ \text{(AGMA)},\qquad K_{H\beta}=1.5\,K_{H\beta be}\ \ \text{(ISO)}$$
| Support arrangement | HB, K′Hβ | AGMA, Kmb | ISO, KHβbe |
|---|---|---|---|
| Both gears straddle-mounted | 1.00 | 1.00 | 1.00 |
| One gear straddle-mounted | 1.10 | 1.10 | 1.00–1.32 depending on load state |
| Neither gear straddle-mounted | 1.25 | 1.25 | 1.00–1.50 depending on load state |
Two structural differences matter more than the tabulated values. First, AGMA adds a face-width term, 5.6 × 10−6b2, which for a 60–62 mm face width contributes only about 0.01–0.02, so HB and AGMA stay close. Second, ISO applies a fixed 1.5 multiplier on top of the mounting coefficient. With KHβbe = 1.00 for a fully loaded, straddle-mounted pair, ISO therefore returns KHβ = 1.5 where HB and AGMA return 1.0 — a 50 % step in the stress chain from this factor alone. ISO additionally resolves the mounting coefficient against load state, distinguishing fully loaded, lightly loaded and test-gear conditions. The transverse load-distribution factor KHα, present only in ISO, corrects for uneven sharing between teeth in mesh; it depends on the total contact ratio εvγ, the pitch deviation fpt, the running-in allowance ya and the mean mesh stiffness cγ, and it enters the stress chain as a positive multiplier.
Geometry-side factors
HB builds its geometry chain from five terms. The zone factor ZH converts pitch-point curvature to stress and depends on the equivalent base-helix angle and transverse pressure angle. The curvature factor Zρ shifts the reference from the pitch point to the point of maximum contact stress, so it responds to tooth number, helix angle, pressure angle and face width. The contact-ratio factor Zε has a breakpoint at εvγ = 2, with separate expressions below and above it. The inertia factor Zi is 1 when εvγ > 2 and 2/εvγ otherwise, penalising low-contact-ratio running. The crowning factor ZC takes 1.225 for a crowned flank and 1.304 for an uncrowned one.
ISO replaces these with quantities tied to the actual contact ellipse: the mid-face contact-line length lbm = lb0(1 − Clbm), which accounts for the shortening caused by the tip crown; the relative curvature radius ρrel; the mid-point zone factor ZMB, which captures the curvature difference between the mean load point and the critical point and involves the equivalent addendum and base diameters plus coefficients F1, F2 that change with transverse and overlap ratios; and the load-sharing factor ZLS, which splits load between the tip, mid-point and root regions of the contact trace using a parabolic peak-load description with exponent 3.0 for uncrowned and 1.5 for crowned flanks. ZLS becomes strongly sensitive when the overlap ratio εvβ is small. ISO fixes the size factor ZX at 1 and the bevel factor ZK at 0.85 as reasonable values.
AGMA compresses the same physics into one composite geometry factor ZI, which contains the contact-line length gc, the load-sharing ratio εNI, the relative curvature radius ρy0, the inertia factor and the large-end diameter and module. Its size factor is not a constant but a function of face width, applied to the calculated stress rather than to the allowable:
$$Z_X=\begin{cases}0.5 & b<12.7\ \text{mm}\\ 0.004\,92\,b+0.437\,5 & 12.7\le b\le114.3\ \text{mm}\\ 1 & b>114.3\ \text{mm}\end{cases}$$
and its crowning factor takes 1.5 for crowned and 2.0 for uncrowned flanks — the same qualitative role as HB’s ZC but numerically different. Because a size factor below unity multiplies the stress in the AGMA expression, it partly offsets the larger dynamic factor that AGMA returns, which is why AGMA and HB stresses end up close despite very different intermediate coefficients.
Material, flank and service-condition factors
Material stiffness is handled identically and can be set aside. Flank hardness is not: HB has no such factor, whereas ISO and AGMA both introduce a hardness factor ZW that multiplies the allowable stress. Both take ZW = 1 when the two flanks have equal hardness. For dissimilar hardness ISO uses
$$Z_W=1.2-\frac{HB-130}{1700},\qquad 130\le HB\le470$$
with 1.2 below and above the range, so the ISO factor falls as the flank gets harder. AGMA instead works from the hardness ratio and the tooth-number ratio:
$$Z_W=1+\left(0.008\,98\,\frac{HB_1}{HB_2}-0.008\,29\right)\left(\frac{z_2}{z_1}-1\right)$$
so a larger tooth-number ratio and a harder pinion both raise the factor. The two formulations therefore respond in opposite directions to an increase in pinion hardness, which matters for case-hardened pinion / through-hardened wheel combinations.
Service-condition factors cover life, temperature and lubrication. HB uses a life factor ZN = (109/NV)0.056 below 109 cycles and 1 above, plus a temperature factor applied once the bulk tooth temperature exceeds 121 °C:
$$Z_\theta=\frac{394}{273+\theta_M}$$
ISO spreads this category across five factors: a life factor ZNT read from tables that cover a wider material set than HB, a lubricant factor ZL for oil type and viscosity, a velocity factor ZV for pitch-line speed, a roughness factor ZR for flank finish, and a hypoid factor ZHyp that takes 1.045 when the two helix angles are equal. ZL, ZV and ZR all increase with the contact fatigue limit. AGMA keeps a life factor, ZNT = 3.4822 NV−0.0602 (and 2 below 104 cycles), and a temperature factor Kθ that is the reciprocal of HB’s Zθ. Because the two sit on opposite sides of the division — numerator in HB, denominator in AGMA — their effect on the allowable stress is the same, but the input temperatures are not: HB’s bulk temperature variable is meant to aggregate oil temperature, mean temperature rise and lubrication method, whereas AGMA does not prescribe how to compute a blank peak temperature.
Case comparison across 24 blank configurations
A systematic comparison over 24 spiral bevel pairs makes the accounting concrete. All cases share a 5 kW input, 8 500 r/min pinion speed, 109 stress cycles, 4.8 µm flank roughness, 20 µm root roughness, 120 °C oil, equal hardness on both members and straddle mounting on both shafts, with the contact fatigue limit fixed at 1 880 MPa so that only the rating basis varies. The blanks span 25–39 pinion teeth, 5.513–8.6 mm mean module, 60–62 mm face width, 30° or 35° mean helix angle and 20° or 22.5° normal pressure angle.
| Case | z1/z2 | mt (mm) | Basis | KV | σH (MPa) | σHP (MPa) | SH |
|---|---|---|---|---|---|---|---|
| 5 | 36/49 | 5.972 | HB | 1.000 | 1 477.2 | 1 556.2 | 1.179 9 |
| ISO | 2.040 9 | 2 158.8 | 2 109.8 | 0.977 2 | |||
| AGMA | 1.257 8 | 1 509.8 | 1 678.8 | 1.111 8 | |||
| 9 | 31/42 | 6.935 | HB | 1.000 | 1 484.8 | 1 536.7 | 1.159 1 |
| ISO | 2.041 8 | 2 151.5 | 2 109.6 | 0.980 5 | |||
| AGMA | 1.257 8 | 1 564.5 | 1 678.8 | 1.073 0 | |||
| 13 | 27/37 | 7.960 | HB | 1.000 | 1 518.7 | 1 520.8 | 1.121 5 |
| ISO | 1.886 7 | 2 040.2 | 2 110.1 | 1.034 2 | |||
| AGMA | 1.257 8 | 1 600.7 | 1 678.8 | 1.048 7 |
The pattern is consistent across the set. ISO returns calculated stresses of roughly 2 040–2 160 MPa and allowable stresses near 2 110 MPa; HB returns 1 477–1 519 MPa calculated and 1 520–1 556 MPa allowable; AGMA returns 1 510–1 601 MPa calculated against a constant 1 678.8 MPa allowable. ISO therefore sits below unity safety for the larger pinions and above it for the smaller ones, whereas HB and AGMA rank the same cases in the opposite direction.
Decomposing the coefficients identifies the sources:
- Dynamic factor. HB returns KV = 1.000 for all three cases, AGMA returns 1.2578, and ISO returns 1.89–2.04. The HB and AGMA formulas are structurally similar but the accuracy-grade scales are not comparable, so the same physical quality does not yield the same number. ISO’s resonance-band construction adds a further, larger step because the operating point falls in a band where the amplifying expression applies.
- Face-load distribution. With KHβbe = 1.00, ISO returns KHβ = 1.50, while HB returns 1.00 and AGMA about 1.01–1.02. This single factor accounts for half the stress gap between ISO and the other two.
- Allowable side. HB’s temperature factor of 0.906–0.927 pulls its allowable down by 7–9 %, while ISO’s hypoid, lubricant and velocity factors (1.045, 1.02, 1.058) push its allowable up by about 13 %. AGMA’s size factor of about 0.68 on the stress side, combined with a crowning factor of 1.5, largely cancels its higher dynamic factor.
- Factor count. Whenever an ISO-only or HB-only factor departs from unity, the stress results diverge by a corresponding amount; when they happen to equal unity, the two methods converge.
Sensitivity to blank parameters
Because the three documents attach different physical meanings to the same blank parameter, the direction in which the contact stress moves can differ. Sweeping face width, helix angle, pressure angle, module and speed one at a time around the 24-case baseline gives the trends below.
| Parameter increased | HB σH | ISO σH | AGMA σH | HB σHP | ISO σHP | AGMA σHP |
|---|---|---|---|---|---|---|
| Face width b | falls | falls | falls | rises | falls | unchanged |
| Helix angle βm | falls | minimum near 30° | maximum near 35° | falls | rises | unchanged |
| Pressure angle αn | falls | rises | nearly flat, slight rise | rises | rises | unchanged |
| Module mt | falls | falls | falls | rises | — | unchanged |
| Pinion speed n | falls | falls | falls | rises | unchanged | unchanged |
The common ground is that increasing module, tooth number, speed or face width raises load capacity and therefore lowers the calculated stress in all three procedures — the 1/√b dependence of the nominal term guarantees the face-width behaviour. The divergence concentrates on the helix angle, because it is the parameter that most strongly changes the overlap ratio εvβ and hence the contact-line geometry, and on the allowable side, where AGMA’s allowable is essentially independent of the blank once the two hardnesses are equal.
The practical consequence is a ranking flip. Running the seven-gear series from 39/53 teeth and 5.513 mm module down to 21/29 teeth and 10 mm module, all at 35° helix, 20° pressure angle, 60–62 mm face width and 8 500 r/min, gives the safety factors below.
| Group | z1/z2 | mt (mm) | HB, SH | ISO, SH | AGMA, SH |
|---|---|---|---|---|---|
| 1 | 39/53 | 5.513 | 1.172 8 | 0.966 4 | 1.132 6 |
| 2 | 36/49 | 5.972 | 1.171 8 | 0.968 9 | 1.101 0 |
| 3 | 31/42 | 6.935 | 1.144 5 | 0.979 2 | 1.073 7 |
| 4 | 27/37 | 7.960 | 1.121 6 | 1.034 2 | 1.048 8 |
| 5 | 26/35 | 8.270 | 1.098 8 | 1.084 9 | 1.028 2 |
| 6 | 25/34 | 8.600 | 1.083 9 | 1.136 3 | 1.016 3 |
| 7 | 21/29 | 10.000 | 0.976 8 | 1.200 3 | 0.987 8 |
HB and AGMA decline monotonically together as the pinion shrinks and the module grows; ISO rises monotonically over the same range and crosses the other two between groups 3 and 4. Judged against its own admission threshold of 1.12, HB passes groups 1–4 and fails from group 5 onward. ISO, with a threshold near 1.0, fails groups 1–3 and passes from group 4. AGMA at a 1 % failure probability (ZZ = 1.00) fails only group 7. The binding standard therefore depends on where the design sits in the tooth-number and module space, and no single document is uniformly conservative.
Choosing a basis in practice
AGMA is the most economical to apply: it stays in the large-end blank parameters, so no equivalent-gear transformation is needed, and the composite geometry factor collapses several effects into one chart-readable value. HB is comparable in effort and returns results within a few percent of AGMA across the cases examined, which is why the two are widely used for general commercial gearing. ISO demands the most input data — mesh stiffness, effective pitch deviation, running-in allowance, oil viscosity grade, flank roughness, bulk temperature — and its resonance-band dynamic factor makes the result sensitive to a parameter that the other two treat as a smooth function of pitch-line velocity.
That sensitivity cuts both ways. It makes ISO the most complete description of the physics and the appropriate basis for severe duty, high specific load and long-life aerospace applications, but it also means that ISO results cannot be reproduced without a careful determination of the input quantities, and that a favourable ISO number obtained with optimistic roughness or lubricant factors should not be read as evidence that the design is conservative. Conversely, the fact that AGMA returns a safety factor above unity where ISO does not should not be read as a licence to downsize, since the two numbers are referenced to different sections and different failure probabilities.
For design work the defensible strategy is to run the primary basis required by the certification or customer specification, then repeat the calculation with a second basis on the same blank and duty cycle, and reconcile the difference at the coefficient level — dynamic factor first, face-load distribution second, allowable-stress chain third. Those three items account for nearly all of the divergence observed here. Where the two bases disagree on the direction of a trend, as they do for helix angle, the disagreement itself is diagnostic: it marks a design region where the overlap ratio and the resonance position are both changing, and where the blank should be chosen on grounds other than the stress number alone.