Spiral bevel gears carry torque between intersecting shafts in aero-engine accessory and main drives, helicopter transmissions, and automotive axles. Their teeth roll and slide simultaneously, so the flank load is concentrated on a small elliptical contact patch and pitting — surface-originated contact fatigue — frequently sets the power limit of the pair. Three rating systems dominate design practice: the Chinese aviation series HB/Z 89, issued by the Ministry of Aviation Industry in 1985 and still applied to military and auxiliary aero drives; ISO 10300:2023; and ANSI/AGMA 2003-D19, published in 2019.
All three descend from the Hertz contact model, yet they choose different reference sections, different coefficient sets, and different reliability conventions. Published comparisons have not converged: some report ISO safety factors above AGMA values, others report the opposite; HB has been found both below and above AGMA depending on the gear set examined. The disagreement is not a matter of computational error but of how each standard books the same physical effects. This article traces those bookkeeping differences through the coefficient structure of the three standards and shows where the resulting stresses and safety margins separate.
Two acceptance criteria and their reliability conventions
Every standard accepts a design on one of two equivalent conditions: the calculated contact stress must not exceed the allowable contact stress, or the contact safety factor must not fall below a specified minimum.
$$\sigma_H \le \sigma_{HP} \qquad (1)$$
$$S_H \ge S_{H\min} \qquad (2)$$
HB and ISO define a minimum safety factor directly. HB assigns 1.12 where high reliability is demanded and 1.00 for low-reliability applications; ISO works in the range 1.0–1.1. AGMA has no minimum-safety-factor concept at all: its reliability factor ZZ enters the allowable stress and is tabulated against failure probability, which makes the reliability requirement far more granular.
| Reliability level | HB SHmin | ISO SHmin | AGMA ZZ |
|---|---|---|---|
| Very high, failure probability < 0.01% | 1.12 | 1.0–1.1 | 1.22 |
| High, < 0.1% | 1.00 | — | 1.12 |
| Moderate, < 1% | — | — | 1.00 |
| Low, < 10% | — | — | 0.92 |
| Very low, < 50% | — | — | 0.84 |
Reference section: gear mid-point versus outer end
The geometric dividing line between the standards is the section at which the Hertz model is applied. HB and ISO convert the bevel pair into an equivalent cylindrical gear at the mid-point of the face width, take the maximum contact stress point as the pitting criterion, and express the load through the tangential force, diameter, and ratio of that virtual gear. AGMA keeps the original large-end parameters, treating the Hertz stress between meshing flanks directly and correcting it for load sharing between teeth and for manufacturing error.
The choice propagates into every subsequent coefficient: an equivalent-gear formulation needs curvature, contact-line, and contact-ratio factors referred to the virtual cylinder, while a large-end formulation folds the same effects into a single geometry factor. Neither route is inherently more accurate, but the mid-point route requires more intermediate quantities and therefore offers more places for the two standards to diverge.
Stress equations and the coefficient families
HB writes the calculated stress as a basic stress multiplied by a load-term correction, and the allowable stress as the fatigue limit reduced by life and temperature factors and divided by the minimum safety factor:
$$\sigma_H=\sigma_{H0}\sqrt{K_A K_V K_{H\beta}},\qquad \sigma_{H0}=Z_H Z_E Z_\varepsilon Z_i Z_\rho Z_C\sqrt{\frac{F_{mt}}{d_{v1}b}\cdot\frac{u_v+1}{u_v}} \qquad (3)$$
$$\sigma_{HP}=Z_N Z_\theta\,\frac{\sigma_{H\lim}}{S_{H\min}} \qquad (4)$$
ISO adds the load-sharing factor between teeth to the load term and replaces the geometric factors by contact-line length, relative curvature, and three zone/load-sharing coefficients:
$$\sigma_H=\sigma_{H0}\sqrt{K_A K_V K_{H\beta}K_{H\alpha}},\qquad \sigma_{H0}=\sqrt{\frac{F_n}{l_{bm}\rho_{rel}}}\,Z_{MB}Z_{LS}Z_E Z_K \qquad (5)$$
$$\sigma_{HP}=\sigma_{H\lim}Z_{NT}Z_X Z_L Z_V Z_R Z_W Z_{Hyp} \qquad (6)$$
AGMA works directly with torque and the large-end pitch diameter, with a single geometry factor inside the radical and the size and crowning factors applied to the stress:
$$\sigma_H=Z_E\sqrt{\frac{2000\,T_1}{b\,d_{e1}^2 Z_I}K_A K_V K_{H\beta}Z_X Z_{XC}} \qquad (7)$$
$$\sigma_{HP}=\frac{\sigma_{H\lim}Z_{NT}Z_W}{S_H K_\theta Z_Z} \qquad (8)$$
Grouping the coefficients by the physical effect they represent exposes how unevenly the three standards distribute their effort.
| Influence | HB | ISO | AGMA |
|---|---|---|---|
| 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 |
| Tooth surface condition | — | ZW | ZW |
| Service conditions | ZN, Zθ | ZNT, ZL, ZV, ZR, ZHyp | ZNT, Kθ |
Load coefficients: the dominant source of discrepancy
The application factor KA is defined identically in all three standards — an allowance for externally induced dynamic overload — and all three scale the stress linearly with it. HB offers a two-by-two grid (uniform or lightly vibrating driver against uniform or lightly vibrating driven machine, values 1.00 to 1.35). ISO and AGMA share a four-by-four grid extending to 2.25 and beyond for heavy shock on both sides of the mesh, and AGMA qualifies the upper entries with “or higher”.
| Driven machine | Uniform driver | 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 standards part company. HB and AGMA both express it as a function of pitch-line velocity and an accuracy grade, and their closed forms are structurally similar; AGMA uses
$$K_V=\left(\frac{A}{A+200\,v_{et}}\right)^{-B},\qquad A=50+56(1-B),\qquad B=0.25\,(12-Q_V)^{0.667} \qquad (9)$$
but the grades themselves are inverted: HB treats grade 5 as the finest, AGMA treats grade 11 as the finest. ISO abandons the single-expression approach altogether, computing a dimensionless reference speed N and switching between sub-critical, primary-resonance, intermediate, and super-critical branches, with the mesh stiffness, effective pitch deviation, and equivalent face width entering the expression. The result is a step-like rise of KV with speed rather than a smooth curve, and the local branch can change when the tooth count, module, or helix angle is altered.
The face-load distribution factor KHβ also diverges sharply. HB multiplies a mounting coefficient by a torque-mismatch term:
$$K_{H\beta}=\left(1+\frac{|T_{D1}-T_1|}{T_{D1}}\right)K’_{H\beta} \qquad (10)$$
AGMA adds a face-width term KHβ = Kmb + 5.6 × 10−6 b2, with Kmb = 1.00, 1.10, or 1.25 according to whether both, one, or neither member is straddle-mounted. ISO writes KHβ = 1.5 KHβbe, where the mounting coefficient additionally distinguishes full-load from light-load operation and the theoretical full-load test condition, ranging from 1.00 to 1.50. The leading constant of 1.5 alone lifts the ISO stress by roughly 22% relative to a stress computed with a unit distribution factor, and it is one of the principal reasons ISO stresses sit above the other two.
ISO alone introduces an inter-tooth load-sharing factor, which for total contact ratios up to 2 reads
$$K_{H\alpha}=\frac{\varepsilon_{v\gamma}}{2}\left[0.9+0.4\sqrt{\frac{c_\gamma\,(f_{pt}-y_a)}{K_A K_V K_{H\beta}/b_v}}\right] \qquad (11)$$
with cγ the mean mesh stiffness, fpt the pitch deviation, and ya the running-in allowance. It grows with contact ratio and pitch deviation and falls as the mesh is more heavily loaded.
Geometry coefficients: many small factors versus one large one
HB spreads geometry over five coefficients. The node-region factor ZH accounts for curvature at the pitch point through the equivalent base helix angle and transverse pressure angle; the curvature factor Zρ converts the pitch-point stress to the point of maximum stress as a function of the equivalent tooth ratio and the mesh-line parameter at that point; the contact-ratio factor Zε changes form at a total contact ratio of 2; the inertia factor Zi equals 1 above that threshold and 2/εvγ below it; and the crowning factor ZC is 1.225 for crowned teeth against 1.304 for uncrowned teeth.
ISO replaces these with contact-line length, relative curvature radius, and two zone factors. The effective contact line lbm = lb0(1 − Clbm) shortens the theoretical length to allow for the tip chamfer; the load-sharing factor ZLS weights the contact traces at tip, mid-point, and root of the tooth by a parabolic peak-load distribution whose exponent is 3.0 for uncrowned and 1.5 for crowned flanks; the mid-point zone factor ZMB compares relative curvature at the mean and critical load points and switches its auxiliary coefficients F1, F2 at longitudinal contact ratios of 0 and 1. ISO fixes the size factor ZX at 1 and the bevel factor ZK at 0.85.
AGMA compresses the same physics into a single geometry factor ZI, built from the contact-line length, the load-sharing ratio, the relative curvature radius at the large end, and an inertia term, evaluated throughout with large-end module and pitch diameter. Its size factor is not a constant but rises with face width:
$$Z_X=0.5\;(b<12.7\ \text{mm});\quad Z_X=0.004\,92\,b+0.437\,5\;(12.7\le b\le 114.3\ \text{mm});\quad Z_X=1\;(b>114.3\ \text{mm}) \qquad (12)$$
and its crowning factor is 1.5 for crowned and 2 for uncrowned teeth. Although the coefficient counts differ by a factor of three, all three standards ultimately depend on the same blank parameters: tooth numbers, module, face width, helix angle, pressure angle, and tooth form.
Material, surface, and service-condition terms
All three standards use one elastic coefficient, evaluated identically from Young’s moduli and Poisson’s ratios, giving 191.65 √MPa for steel pairs. HB has no hardness-related term at all. ISO and AGMA both use a work-hardening factor of unity for equal hardness and diverge for dissimilar hardness: ISO sets ZW to 1.2 below HB 130 and decreases it linearly to 1.0 at HB 470, whereas AGMA computes
$$Z_W=1+\left(0.008\,98\,\frac{HB_1}{HB_2}-0.008\,29\right)\left(\frac{z_2}{z_1}-1\right) \qquad (13)$$
so that a large tooth-number ratio combined with a harder pinion raises the allowable.
Service-condition coefficients show the widest structural spread. HB carries a life factor ZN = (109/NV)0.056 below 109 cycles and unity above it, plus a temperature factor Zθ = 394/(273 + θM) once the blank temperature exceeds 121 °C. ISO uses a table-based life factor for a wider range of materials, and adds oil-film, velocity, roughness, and hypoid factors, all of which increase with the fatigue limit; for equal helix angles the hypoid factor is 1.045. AGMA uses a life factor ZNT = 3.482 2 NV−0.060 2 above 104 cycles and a temperature factor Kθ that is the reciprocal of HB’s Zθ; because one sits in the numerator of the allowable and the other in the denominator, the two have the same effect. Note that the blank temperature is not the oil inlet temperature: in the worked cases below, a 120 °C oil supply corresponds to roughly 152 °C bulk flank temperature.
Three gear sets, three answers
The coefficients are best compared on a common set of gears. Three representative cases are taken from a series of 24 designs sharing an input power of 5 kW, 8 500 r/min, 109 stress cycles, tooth-flank roughness 4.8 µm, root roughness 20 µm, oil at 120 °C, equal hardness on pinion and wheel, straddle mounting at both members, and a contact fatigue limit of 1 880 MPa.
| Case (z1/z2, mt, b) | Standard | σH (MPa) | σHP (MPa) | SH |
|---|---|---|---|---|
| 5 (36/49, 5.972 mm, 62 mm) | HB | 1 477.2 | 1 556.2 | 1.180 |
| ISO | 2 158.8 | 2 109.8 | 0.977 | |
| AGMA | 1 509.8 | 1 678.8 | 1.112 | |
| 9 (31/42, 6.935 mm, 62 mm) | HB | 1 484.8 | 1 536.7 | 1.159 |
| ISO | 2 151.5 | 2 109.6 | 0.981 | |
| AGMA | 1 564.5 | 1 678.8 | 1.073 | |
| 13 (27/37, 7.960 mm, 62 mm) | HB | 1 518.7 | 1 520.8 | 1.122 |
| ISO | 2 040.2 | 2 110.1 | 1.034 | |
| AGMA | 1 600.7 | 1 678.8 | 1.049 |
ISO stresses exceed the other two by roughly 35–45%, and its allowable stresses exceed them by a similar margin, so the safety factor is not simply scaled. Four mechanisms account for the pattern.
First, the dynamic factor differs by construction: KV is 1.00 in HB, 2.04 (case 5 and 9) to 1.89 (case 13) in ISO, and 1.26 in AGMA. HB and AGMA both tie the factor to a pitch-line velocity and a grade, and their numerical results are close; ISO’s resonance-branch formulation produces a substantially larger value because it treats mesh stiffness and tooth-quality deviations as the excitation source rather than empirical velocity curves.
Second, KHβ is 1.00 in HB, 1.50 in ISO, and 1.01–1.02 in AGMA for these straddle-mounted cases. The AGMA face-width correction is small at 62 mm because its constant term is small; the ISO constant of 1.5 is not.
Third, the allowable stresses are built differently. HB divides by the minimum safety factor 1.12, so the reported SH is the stress ratio multiplied by that value; ISO includes no explicit divisor but carries oil, velocity, roughness, and hypoid factors of 1.02, 1.058, 0.996, and 1.045 that together raise the allowable by about 12%; AGMA’s allowable sits between them.
Fourth, ISO’s extra geometric factors do not cancel: the load-sharing factor ZLS falls from 0.749 to 0.845 as the tooth count decreases, and the bevel factor ZK = 0.85 and mid-point zone factor ZMB ≈ 1.0 shift the result further. Whenever a coefficient that only one standard introduces departs from unity, the numerical agreement between standards degrades.
Parameter sensitivity: where the trends invert
Increasing the pinion tooth count, the module, the design speed, or the face width lowers the calculated contact stress in all three standards, since each of these changes raises load capacity per unit of Hertz pressure. The helix angle and the pressure angle are another matter.
For HB, a larger helix angle raises the equivalent pitch diameter and lowers the contact-ratio factor, reducing the calculated stress; it also raises blank temperature, which lowers the temperature factor and therefore the allowable stress. HB is the only standard in which the allowable falls with helix angle. Increasing the pressure angle reduces total contact ratio, which raises the contact-ratio factor and lowers the calculated stress, while the blank temperature falls and the allowable rises.
For ISO, the calculated stress passes through a minimum near 30° helix angle in the cases examined, reflecting the sensitivity of the mid-point contact trace and hence of the load-sharing factor to helix angle. Because the dimensionless reference speed also shifts when tooth count or module changes, the shape of the ISO curve is not transferable between gear sets — a change of blank parameters can move the operating point across a resonance branch and reverse a local trend. The ISO allowable rises with helix angle and falls with face width, the latter because a wider face reduces the relative curvature radius, so the roughness factor drops.
For AGMA, the calculated stress reaches a maximum near 35° helix angle, since the contact-line length inside the contact ellipse and the load-sharing ratio in ZI move in opposite directions on either side of that value. The pressure-angle trend is nearly flat, rising only slightly. Face width reduces the calculated stress by shortening the contact lines within the ellipse. When pinion and wheel hardness are equal, the AGMA allowable is essentially independent of all blank parameters except tooth number, because the hardness-ratio factor collapses to unity.
Safety factors over a seven-case series
Seven geometrically similar pairs, all with 35° mean helix angle, 20° normal pressure angle, 8 500 r/min, and the same operating conditions, show how the ordering of the standards depends on the design point.
| Group | z1 | z2 | mt (mm) | b (mm) | SH HB | SH ISO | SH AGMA |
|---|---|---|---|---|---|---|---|
| 1 | 39 | 53 | 5.513 | 60 | 1.173 | 0.966 | 1.133 |
| 2 | 36 | 49 | 5.972 | 60 | 1.172 | 0.969 | 1.101 |
| 3 | 31 | 42 | 6.935 | 62 | 1.145 | 0.979 | 1.074 |
| 4 | 27 | 37 | 7.960 | 62 | 1.122 | 1.034 | 1.049 |
| 5 | 26 | 35 | 8.270 | 62 | 1.099 | 1.085 | 1.028 |
| 6 | 25 | 34 | 8.600 | 62 | 1.084 | 1.136 | 1.016 |
| 7 | 21 | 29 | 10.000 | 62 | 0.977 | 1.200 | 0.988 |
HB and AGMA move together, declining monotonically as tooth counts fall and module rises, with HB consistently about 0.04–0.06 higher. ISO runs in the opposite direction, crossing unity between groups 3 and 4, and it is markedly more sensitive to tooth number and module — its range of 0.966 to 1.200 is roughly three times that of the other two. No standard is uniformly conservative: HB and AGMA both fall below unity in group 7 where ISO is highest, and ISO falls below unity in groups 1–3 where both others exceed 1.07.
Choosing a framework
The practical consequence is that the three standards are not interchangeable normalisations of one another; they encode different assumptions about how load, geometry, and surface condition should be weighted, and a gear designed to pass one may fail another. Three observations follow from the coefficient structure.
AGMA is the most economical to apply. Working from large-end original parameters removes the equivalent-gear transformation, and the single geometry factor absorbs contact-line length, load-sharing ratio, relative curvature, and inertia in one tabulated value. HB requires the mid-point conversion but retains a compact coefficient set with tabulated values for most terms. ISO demands the most input data — mesh stiffness, pitch deviation, running-in allowance, oil grade and viscosity, surface roughness, and blank temperature — and returns a stress that is highly sensitive to the resonance branch it happens to occupy.
Where the operating conditions are severe and the power density is high, the ISO coefficient set provides more levers for refining a design, and its explicit treatment of oil film, velocity, and roughness makes it the natural choice for aero-engine main drives. For general commercial gearing, where the duty is moderate and the geometry conventional, HB and AGMA give closely comparable results at a fraction of the computational effort, with AGMA offering better resolution of reliability targets through its tabulated reliability factor.
What none of the three standards provides is a design-independent ranking. Since the safety factors of the three frameworks cross over within a single family of geometrically similar gears, a design review that reports only “safety factor” without naming the standard and the coefficient values behind it carries limited information. Reporting the dominant coefficients — dynamic factor, face-load distribution factor, and for ISO the load-sharing factor — alongside the safety factor makes the comparison between candidate designs, and between standards, traceable.