For straight bevel gears that transmit power between intersecting shafts, this tool computes the three tooth-force components (tangential, axial and radial) in real time. Move the pitch and pressure angles and you can see exactly how much axial load — the load that dictates the choice of thrust bearings — appears on the shaft.
Parameters
Mean radius r_m
mm
Pitch radius at mid-face-width
Pitch angle γ
°
Half-angle of the pitch cone. 45° gives a mitre gear; larger γ raises the axial force
Applied torque T
N·m
Pressure angle α
°
Standard 20°. 14.5° is the older US spec; 25° is used for high-load designs
Bearing span
mm
Distance between the two support bearings, used for a representative reaction estimate
Results
—
Tangential F_t (N)
—
Axial F_a (N)
—
Radial F_r (N)
—
Resultant F (N)
—
F_a / F_t ratio (%)
—
Bearing reaction est. (N)
—
Bevel-gear mesh — three force vectors
Two bevel gears mesh on intersecting shafts. At the mesh point the three force vectors are drawn — tangential (green), axial (blue) and radial (red) — with lengths proportional to the computed magnitudes.
Three tooth-force components of a straight bevel gear. T: torque, r_m: mean pitch radius, α: pressure angle, γ: pitch angle. The axial component requires a thrust-capable bearing (tapered roller, double-row angular contact, etc.) — a plain radial ball bearing cannot support it and will fail rapidly.
$$F=\sqrt{F_t^{2}+F_a^{2}+F_r^{2}}$$
Resultant force F. The total load on the tooth flank used for bending- and contact-stress checks.
What is a bevel-gear tooth-force calculation?
🙋
Bevel gears are the ones inside a car differential or an electric drill, right? What is different about the force calculation compared with an ordinary spur gear?
🎓
Exactly — they are used wherever you have to hand power between two shafts whose axes intersect. The diff in every car, the angle drives of helicopter tail rotors, hand drills, marine propeller-shaft drives, milling-machine heads, even the worm casings of automotive timing mechanisms. The key feature is that the teeth sit on a cone instead of a cylinder. On a spur gear the tooth normal stays in the radial direction; on a bevel gear the cone surface tilts that normal so it now has a component along the shaft. That component is the axial force F_a, and a spur gear simply does not have it.
🙋
OK, so how much does that axial force matter in practice? When I move the pitch-angle slider, F_a really does grow fast.
🎓
Good observation. F_a is proportional to tan α · sin γ. At γ = 20° it is only about 13% of the tangential force F_t, but at γ = 60° it is about 32%, and approaching γ = 80° it asymptotes to roughly tan α ≈ 36%. So as the pitch angle stands up, more and more of the load is pushing the gear off its shaft. That means a bigger reduction ratio (larger γ on the gear) automatically demands more thrust-bearing capacity.
🙋
When you say "thrust bearings", do you mean tapered roller bearings? Why are plain deep-groove ball bearings not enough?
🎓
Tapered roller bearings in a back-to-back arrangement are the classic choice, with double-row angular-contact ball bearings as another common solution. A plain deep-groove ball bearing can carry some axial load, but only about 25% of its dynamic rating. If this tool predicts 500 N of axial force and you mount a bearing with a 50 N axial limit, you get textbook fretting on the inner race after a few hundred hours, then chatter, then seizure. The field rule of thumb is: if F_a / F_t exceeds 10%, move to a thrust-capable bearing.
🙋
The resultant F = √(F_t² + F_a² + F_r²) is also shown. What do I actually use that one for?
🎓
The resultant is what you use to check the tooth itself — bending stress and contact (Hertzian) pressure. It is the total force on one engaged tooth. For bearing selection you split the forces back into shaft coordinates: radial load = vector sum of F_t and F_r, axial load = F_a, and feed those into the bearing manufacturer's life equation. So "tooth strength" and "shaft / bearing strength" use different combinations of the same three forces, and keeping that distinction clear keeps your design tidy.
🙋
What about spiral bevel gears — does the calculation change?
🎓
A spiral bevel has a helix angle β > 0, which adds extra helical terms to F_a and F_r. Worse, depending on the spiral hand and the direction of rotation, the helical axial force can either add to or partly cancel the straight-bevel axial force. So the designer has to track "pinion rotation direction → sign of F_a → bearing mounting direction" as a chain. Straight bevels are simpler — that is why they are still the right starting point for hand calculations and for low-speed industrial drives.
Frequently Asked Questions
The tangential force at the mean radius r_m is F_t = T / r_m, where T is the applied torque. The axial force is F_a = F_t·tan α·sin γ and the radial force is F_r = F_t·tan α·cos γ, with α the pressure angle and γ the pitch angle. The fundamental difference from a spur gear is that F_a is always present, and grows with γ — which is why bevel-gear shafts need thrust-capable bearings. The resultant is F = √(F_t² + F_a² + F_r²).
A bevel-gear shaft always needs bearings that can carry thrust. In practice that means back-to-back tapered roller bearings, double-row angular-contact ball bearings, or a deep-groove plus thrust-ball bearing pair. If the axial force F_a exceeds the allowable axial load of a plain radial ball bearing (typically about 25% of its dynamic rating) the bearing fails quickly. As a field rule of thumb, if this tool's axial/tangential ratio is above 10%, switch to a thrust-capable arrangement.
As γ approaches 90° the bevel gear becomes almost a flat ring, sin γ ≈ 1, and F_a grows to nearly F_t·tan α. At the same time cos γ ≈ 0, so F_r shrinks. Most of the load flows along the shaft, so thrust-bearing capacity has to be maximized. Conversely a shallow γ ≈ 20° gives a small F_a that some radial ball bearings can handle. The usual flow is: choose the ratio and pitch angle from the available space, then size the bearings around the resulting axial load.
This tool covers the straight bevel case (helix angle β = 0). A spiral bevel (β > 0) adds a helical term to F_a and F_r. Depending on the combination of spiral hand and direction of rotation, the spiral component can either reinforce or partly cancel the straight-bevel axial load. Spiral bevels are quieter and handle higher loads; straight bevels are cheaper and used at lower speeds. Car differentials almost always use spiral bevel or hypoid gears for noise and load reasons.
Real-World Applications
Automotive differentials and angle drives: The differential of a rear- or four-wheel-drive car turns the engine's longitudinal torque through 90° to the wheels, which is the natural home of the bevel gear — most often a hypoid variant. The axial load on a passenger-car ring gear is several kN, always reacted by a back-to-back pair of tapered roller bearings. Set γ = 45° (a mitre gear) and a high torque in this tool and F_a climbs to about 36% of F_t, exactly the range that drives those bearing choices.
Helicopter tail-rotor drives: Power transmission from the main rotor gearbox to the tail rotor goes through several bevel gearboxes mounted along the bottom and rear of the fuselage. Weight is critical, the bearings are state of the art, and the sign and direction of F_a have direct safety consequences. Designs go through hand calculation, FEM and rig test in three deliberate steps before flight clearance.
Machine tools and power tools: Milling-machine and drill-press angle heads, the right-angle gearbox in cordless drills and impact drivers, agricultural PTO drives — anywhere you have to turn the power flow through a corner in a small package. The right-angle head of a cordless drill uses a small straight bevel pair and small radial ball bearings, and setting γ = 45° in this tool and watching F_a / F_t makes it obvious why those bearings develop play so quickly: the allowable axial load is tiny.
Marine and industrial power-split applications: Ship propeller shafts, gearbox power splits, paper-machine roll drives — anywhere a reduction stage also has to rotate the shaft by 90°. The standard solution is a spiral bevel with tapered roller bearings, and the straight-bevel calculation in this tool is the right "first pass": if the design does not work for a straight bevel with reasonable safety margins, it usually does not work for a spiral bevel either.
Common Misconceptions and Pitfalls
The biggest pitfall is "selecting bearings for a bevel-gear shaft as if it were a spur-gear shaft". A spur gear only produces tangential and radial loads, so a pair of plain radial ball bearings at each end is fine. A bevel gear always adds an axial force F_a, and using the same plain ball bearings means quickly exceeding their allowable axial load and developing fretting, then seizure. On a new bevel-gear design always estimate F_a with this tool first and check that it stays below about 25% of the dynamic radial rating of any plain ball bearing you were planning to use — otherwise switch to a thrust-capable arrangement.
Second, "confusing the pitch radius with the mean radius". The teeth of a bevel gear sit on a cone, so the pitch radius is different at the inner end and the outer end of the face width. The torque calculation uses the mean radius r_m at mid-face, not the outer-end pitch radius R_e. Using R_e under-estimates F_t and therefore under-sizes the bearings. Drawings often specify an "outer-end module", so a documented routine for converting from outer-end module to r_m using face width and the face-width coefficient avoids repeated mistakes.
Finally, "using the resultant F as the bearing load". The resultant F that this tool reports is the total force on one tooth flank — appropriate for bending-stress and contact-stress checks on the tooth, but not for bearing selection. For bearings you have to decompose the load back into shaft coordinates (radial = combination of F_t and F_r, axial = F_a) and split it between the two support bearings according to the mounting arrangement (back-to-back or face-to-face). Using the resultant directly as the bearing equivalent load over-estimates the load and leads to over-specified, expensive bearings. The bearing-reaction estimate here is only a representative order-of-magnitude figure; the final bearing selection always belongs in the manufacturer's life-calculation software.
How to Use
Enter the mean cone radius (rmNum) in millimeters—typical range 20–150 mm for industrial gearboxes.
Set the bevel angle (gammaNum) between 5° and 85°; 45° is common for equal-speed intersecting shafts.
Input the driving torque (torqueNum) in N·m; for a 5 kW motor at 1500 rpm, expect approximately 32 N·m.
Specify the pressure angle (alphaNum), typically 20° for involute profiles.
Click Calculate to resolve tangential, axial, and radial components acting on the tooth flank.
Worked Example
Steel bevel gear pair with mean radius rm=75 mm, bevel angle γ=45°, applied torque=50 N·m, and pressure angle α=20°. The simulator yields: tangential force Ft≈667 N (primary load), axial force Fa≈280 N (thrust on bearing), radial force Fr≈249 N (separating load), resultant force≈756 N. The Fa/Ft ratio of 42% indicates significant axial loading requiring preloaded thrust bearings. Estimated bearing reaction approaches 850 N under these conditions.
Pressure angle α=25° (instead of 20°) increases all force components by 10–15%; validate tooth-root bending stress separately using ISO 10300 formulas.
For high-speed gearboxes exceeding 5000 rpm, dynamic magnification factors of 1.3–1.8 should be applied to static tangential forces.
Straight-cut bevel gears generate no self-aligning torque; bearing misalignment over 0.5° introduces edge-loading and fatigue risk.