Spiral bevel gears carry high torque with smooth, low-vibration motion in aerospace, vehicle, mining and industrial drivetrains. Their most common bending failure is a fatigue crack at the tooth root, where the fillet concentrates tensile stress under the loaded flank. ISO 10300-3:2023 offers two calculation routes for root bending strength, designated B1 and B2. Both use load factors and a virtual cylindrical gear, but they choose different reference sections and different geometry factors. That choice changes the predicted root stress by several percent and can change the ranking of a design against its required safety factor.
Two virtual-gear models, two reference sections
B1 replaces the bevel tooth with a virtual cylindrical gear at the mid-face section. It takes the root critical section at the point where a 30° tangent touches the fillet, then applies separate factors for tooth form, root stress correction, contact ratio, spiral angle and load sharing. B2 also uses a virtual gear and a 30° tangent root section, but places the reference section at the outer end. Its stress derivation follows the Lewis cantilever model and collects most of the tooth geometry into a single geometry factor.
For B1, the root stress is written as
$$
\sigma_{F-B1}=\frac{F_{vmt}}{b_v m_{mn}}Y_{F\alpha}Y_{S\alpha}Y_{\varepsilon}Y_{BS}Y_{LS}K_AK_VK_{F\beta}K_{F\alpha}
$$
$$
F_{vmt}=\frac{F_{mt}}{\cos\beta_v\cos\beta_{m1}},
\qquad
F_{mt}=\frac{2000T}{d_m}
$$
For B2, the corresponding expression is
$$
\sigma_{F-B2}=\frac{F_{mt}}{b\,m_{mn}}\frac{Y_A}{Y_J}\frac{m_{mt}m_{mn}}{m_{et}^2}K_AK_VK_{F\beta}K_{F\alpha}
$$
Here \(T\) is the nominal torque, \(d_m\) the mid-face pitch diameter, \(b\) the face width, \(b_v\) the virtual-gear face width, \(m_{mn}\) the mid-face normal module, \(m_{mt}\) the mid-face transverse module, \(m_{et}\) the outer transverse module, \(\beta_v\) the virtual spiral angle and \(\beta_{m1}\) the pinion mid-face spiral angle. With \(T\) in N·m and \(d_m\) in mm, the factor 2000 gives a force in N.
The allowable stresses are calculated from the same base material strength but with method-specific surface and notch factors:
$$
\sigma_{FP-B1}=\sigma_{F,\lim}Y_{ST}Y_{NT}Y_{\delta,relt-B1}Y_{R,relt-B1}Y_X
$$
$$
\sigma_{FP-B2}=\sigma_{F,\lim}Y_{ST}Y_{NT}Y_{\delta,relt-B2}Y_{R,relt-B2}Y_X
$$
Both routes rate the gear by comparing the calculated safety factor with a minimum value:
$$
S_F=\frac{\sigma_{FP}}{\sigma_F}\ge S_{F,\min}
$$
ISO 10300 sets \(S_{F,\min}\ge1.3\) for spiral bevel and hypoid gears, and \(S_{F,\min}\ge1.5\) for straight bevel gears or spiral bevel gears with a mid-face spiral angle \(\beta_m\le5^\circ\).
Factor architecture and allowable stress
The two methods share the external and internal load factors \(K_A\), \(K_V\), \(K_{F\beta}\) and \(K_{F\alpha}\). These account for duty-cycle overloads, dynamic increments, face load distribution and transverse load distribution. In the worked case below, \(K_A=1.0\), \(K_V=1.0975\), \(K_{F\beta}=1.50\) and \(K_{F\alpha}=1.10\). The difference lies mainly in how tooth geometry, load sharing and material condition enter the rating.
| Role | Method B1 | Method B2 |
|---|---|---|
| Reference section | Mid-face virtual cylindrical gear | Outer-end virtual cylindrical gear |
| Common load factors | \(K_A, K_V, K_{F\beta}, K_{F\alpha}\) | \(K_A, K_V, K_{F\beta}, K_{F\alpha}\) |
| Geometry | \(Y_{F\alpha},Y_{S\alpha},Y_{\varepsilon},Y_{BS},Y_{LS}\) as multipliers | \(Y_J\) as a divisor and \(Y_A\) as a multiplier |
| Life and size | \(Y_{ST},Y_{NT},Y_X\) | \(Y_{ST},Y_{NT},Y_X\) |
| Surface condition | \(Y_{R,relt-B1}\): material- and roughness-dependent | \(Y_{R,relt-B2}=1.025\) for \(R_z\le16\,\mu m\) |
| Notch sensitivity | \(Y_{\delta,relt-B1}\): function of slip-layer thickness and size factor | \(Y_{\delta,relt-B2}=1.0\) if \(q_s\ge1.5\); \(1.5\) if \(q_s<1.5\) |
B1 resolves the tooth shape into five separate factors. \(Y_{F\alpha}\) describes the tooth form, \(Y_{S\alpha}\) corrects the root stress, \(Y_{\varepsilon}\) accounts for contact ratio, \(Y_{BS}\) for spiral angle and \(Y_{LS}\) for load sharing among teeth. B2 instead uses \(Y_J\) to combine tooth form, the position of the worst loaded point, the effective face width after lengthwise modification and load sharing between adjacent teeth; \(Y_A\) then adjusts the root stress. Because \(Y_J\) appears in the denominator, a larger \(Y_J\) lowers the calculated B2 stress.
The allowable-stress treatment also differs. For B1, \(Y_{R,relt}\) depends on material and root roughness: for carburized or quenched-and-tempered steel at \(R_z<1\,\mu m\), it is 1.12; for structural steel, 1.07; for gray cast iron, nitrided steel or carbonitrided steel, 1.025. At higher roughness, material-specific decay functions are used. B2 assigns 1.025 for \(R_z\le16\,\mu m\). B1’s \(Y_{\delta,relt}\) uses slip-layer thickness and size factor, while B2 uses a step change at a root fillet coefficient \(q_s\) of 1.5. B1 therefore discriminates more finely between material conditions; B2 is more global.
Representative 16/27-tooth case
A Gleason circular-arc spiral bevel pair with 16 pinion teeth and 27 wheel teeth provides a concrete comparison. The design data are listed in Table 2. Three torque levels, 200, 300 and 400 N·m, were rated with both methods.
| Parameter | Pinion | Wheel |
|---|---|---|
| Number of teeth | 16 | 27 |
| Outer transverse module \(m_{et}\) | 4.25 mm | |
| Mean spiral angle \(\beta_m\) | 35° | |
| Normal pressure angle \(\alpha_n\) | 20° | |
| Tangential modification coefficient | 0.005 | −0.005 |
| Radial modification coefficient | 0.252 | −0.252 |
| Face width \(b\) | 17 mm | |
| Material | 20CrNiMo | |
| Accuracy grade | 7 | |
| Tooth system | Gleason circular-arc tapered tooth | |
| Load (N·m) | Method B1 (MPa) | Method B2 (MPa) | ||
|---|---|---|---|---|
| Wheel | Pinion | Wheel | Pinion | |
| 200 | 280.46 | 289.13 | 292.61 | 302.61 |
| 300 | 420.69 | 433.70 | 438.92 | 453.92 |
| 400 | 560.92 | 578.26 | 585.22 | 605.23 |
| Load (N·m) | Method B1 | Method B2 | ||
|---|---|---|---|---|
| Wheel | Pinion | Wheel | Pinion | |
| 200 | 4.47 | 4.34 | 4.01 | 3.88 |
| 300 | 2.98 | 2.89 | 2.67 | 2.58 |
| 400 | 2.24 | 2.17 | 2.00 | 1.93 |
B1 returns 95.5–95.9% of the B2 root stress across the table, a mean of about 95.6%. The difference is not merely a uniform 4.4% stress offset: the resulting B1 safety factors are about 12% higher than the B2 values. In this case, B2 is the more conservative rating route because it predicts higher root stress and a lower safety margin. B1’s more detailed factor set does not automatically make it conservative; the numerical outcome depends on the material and geometry factors selected.
The pinion is the critical member in both methods. Its calculated root stress is about 3.1–3.4% above the wheel, and the finite element results below show a similar 2–4% difference. Two effects contribute: the pinion loaded flank is concave, so its root fillet is tighter than the wheel’s convex flank, and the smaller tooth number reduces the root section thickness. A strength-balanced design should therefore not apply the same root-stress target to both members without checking this asymmetry.
The stress values also scale almost exactly with torque. Doubling the load from 200 to 400 N·m doubles the calculated stress and halves the safety factor. That linearity follows directly from the ISO equations and is a key limitation when the method is applied outside the rated full-load condition.
Finite element model and peak extraction
A static contact analysis was built from tooth-surface data generated by the cutting parameters. A simulated roll test checked the contact pattern before loading. The assembly used 20CrNiMo properties, frictional contact with a coefficient of 0.03, pinion tooth surfaces as the contact side and wheel surfaces as the target side, and rotational joints at both gears. The pinion was rotated through 180°, and torques of 200, 300 and 400 N·m were applied to the wheel. The mesh was predominantly hexahedral with a global size of 1 mm and local refinement to 0.1 mm at the roots. The solution used large deflection and weak springs, with 1000 initial substeps, a maximum of 2000 substeps and a 60 s analysis time.
Equivalent stress is unsigned, so each tooth trace shows two positive peaks. For the wheel, the first peak corresponds to tensile root stress as the tooth enters mesh; the second arises from compressive loading as the following tooth comes into contact. For the pinion, the small peak comes first and the main peak later, so the second peak is the relevant tensile value. Extracting the wrong peak can shift the comparison by several percent. Table 5 gives the peak root stresses for three sampled teeth.
| Load (N·m) | Wheel range (MPa) | Pinion range (MPa) |
|---|---|---|
| 200 | 293.6–298.4 | 296.5–310.1 |
| 300 | 392.6–396.8 | 408.6–413.9 |
| 400 | 495.3–499.4 | 505.1–512.2 |
Using the mean of the three sampled teeth as the finite element reference, the deviations of the ISO estimates are summarized in Table 6. Positive values mean that the ISO method is above the finite element mean; negative values mean it is below.
| Load (N·m) | Method B1 | Method B2 | ||
|---|---|---|---|---|
| Wheel | Pinion | Wheel | Pinion | |
| 200 | −5.3% | −4.5% | −1.2% | 0.0% |
| 300 | +6.4% | +5.7% | +11.0% | +10.6% |
| 400 | +12.9% | +13.6% | +17.8% | +18.9% |
The comparison is load-dependent. At 200 N·m, B2 is closest to the finite element result, with deviations of 0–1.2%. At 300 and 400 N·m, B1 is consistently closer, but its deviation grows from about 6% to about 13–14%. A single figure such as “within 6%” therefore describes the mid-load range, not the whole table. B2 remains farther from the finite element values at the higher torques and is the more conservative of the two ISO routes.
Why the methods part company
Three mechanisms explain the differences. First, the reference section changes the module, diameter and face width used in the stress denominator. B1 evaluates the mid-face virtual gear; B2 evaluates the outer-end virtual gear and then converts the result through the module ratio. Second, B1 distributes geometry and material effects across several transparent factors, while B2 compresses them into \(Y_J\) and \(Y_A\). A change in root fillet, face-width modification or adjacent-tooth sharing therefore enters the two ratings with different sensitivity. Third, the two methods treat the root surface and notch condition differently: B1 responds to steel type and roughness, whereas B2 uses fixed values or a step function.
The finite element model adds a further effect that the standard does not resolve directly. ISO 10300 includes load sharing among teeth through \(Y_{LS}\) or \(Y_J\), but it applies that sharing as a fixed modifier at the rated condition. The contact analysis resolves the actual contact pattern, tooth deflection and adjacent-tooth sharing at each torque. As torque rises, more of the load is carried by neighboring teeth, so the finite element stress grows more slowly than the linear ISO estimate. At low torque, the real contact can be more concentrated, so the ISO value may fall below the finite element peak. This produces the sign change seen in Table 6 and means that linear extrapolation from a rated point is unreliable.
Practical interpretation
B1 and B2 are not interchangeable numerical labels for the same rating. For the 16/27-tooth case, B1 gives root stresses about 4.4% lower than B2 and safety factors about 12% higher. B2 is therefore the more conservative screen. B1 is the more detailed route: it separates tooth form, stress correction, contact ratio, spiral angle, load sharing, surface condition and notch sensitivity, and it is closer to the finite element peak at 300 and 400 N·m. B2 is simpler and, at 200 N·m, happened to be closer to the contact solution.
The appropriate choice depends on the design stage and the available data. B1 is preferable when material type, heat treatment and root roughness are known and a differentiated root-strength assessment is required. B2 is useful for preliminary sizing or for comparative rating when a single geometry factor is sufficient. Neither method should be extrapolated linearly to part-load, overload or impact conditions. For those cases, a contact finite element analysis or a dedicated fatigue test is needed, with care taken to extract the tensile root peak on the loaded flank rather than the unsigned equivalent-stress maximum. Because the pinion root is consistently more severely loaded, tooth-number selection and fillet geometry should be balanced against the wheel rather than judged by a single safety-factor target.