How to Size a Shaft for Torque and Bending

A shaft sized to carry its torque snaps after four months. The static calculation was correct and the shaft was never overloaded.

Rotating shafts fail by fatigue, and fatigue is governed by different rules from the static strength calculation most people start with.

Torque from power and speed

The first step, and the only unambiguous one:

T = 9550 × kW ÷ rpm

15 kW at 1,450 rpm gives 98.8 Nm.

Note the inverse relationship with speed. The same power at 145 rpm is 988 Nm — ten times the torque — which is why low-speed high-power shafts are so much larger than high-speed ones of the same rating.

Combined loading

A shaft carrying a pulley or a gear is in torsion from the transmitted power and in bending from the radial load. Both stress the material simultaneously, and the combination is what sizes the shaft.

Using maximum shear stress theory:

d = ∛(16 × √(M² + T²) ÷ (π × τ))

With 98.8 Nm of torque, 85 Nm of bending and 40 MPa allowable shear, that gives about 27 mm.

That figure is the starting point, and three things push it upward.

Fatigue governs, not static strength

A rotating shaft under bending experiences fully reversed stress. Every point on the surface goes from maximum tension to maximum compression once per revolution.

At 1,450 rpm that is 1,450 stress cycles per minute — over two million in a day. Materials fail at stress levels far below their static strength when cycled enough times, and that is what determines a rotating shaft’s life.

Static calculation asks whether the shaft will break now. Fatigue analysis asks whether it will break in a year, and it is the question that matters for anything that rotates continuously.

Fatigue analysis brings in surface finish, size effects, reliability factors and stress concentrations, and it typically requires a larger diameter than the static calculation suggests.

Stress concentrations are where it breaks

Fatigue cracks start at stress concentrations, and a shaft is full of them:

Keyways — a sharp-cornered slot removing material at the point of highest stress. This is the classic shaft failure origin.

Shoulders — a step change in diameter, where a sharp corner concentrates stress severely and a generous fillet radius reduces it substantially.

Grooves for circlips and seals.

Cross-drilled holes.

The practical consequence is that a shaft’s failure location is predictable from its geometry before it is made, and that generous fillet radii at shoulders are among the cheapest reliability improvements available.

What else a real design covers

Beyond fatigue and stress concentrations:

Deflection, which affects bearing life and gear meshing. A shaft strong enough can still deflect too much for the gears it carries.

Critical speed, where the shaft’s natural frequency coincides with its rotational speed and vibration grows dramatically.

Bearing selection and spacing, which determine the bending moments in the first place.

Torsional stiffness, where a shaft transmitting varying torque may need to resist wind-up.

Those interact, which is why shaft design is engineering work rather than a formula.

Deflection frequently governs

A shaft strong enough can still be unusable.

Excessive deflection misaligns whatever the shaft carries. Gears mesh at the wrong angle, which concentrates load on one edge of the teeth. Bearings run misaligned, which shortens their life substantially. Seals wear unevenly and leak.

Deflection depends on the fourth power of diameter, so it is extremely sensitive to size — a small increase in diameter reduces deflection dramatically, which is why shafts governed by deflection are noticeably larger than strength alone requires.

Typical limits are expressed as a maximum deflection per unit of bearing span, and as a maximum slope at the bearings themselves. Both are stricter than the strength requirement in most gear and pulley applications, which is why checking strength alone frequently produces an undersized shaft.

Common mistakes to avoid

• Sizing on static strength for a rotating shaft.
• Cutting a keyway into a shaft sized without allowing for it.
• Sharp corners at shoulders.
• Ignoring deflection, which affects bearings and gears before it affects the shaft.
• Treating a preliminary diameter as a design.

How to do it with Shaft Diameter Calculator

The Shaft Diameter Calculator handles the combined loading.

1. Enter transmitted power and speed to get the torque.
2. Add the bending moment from the shaft’s loading and supports.
3. Read the minimum diameter as a starting point.
4. Add allowances for keyways and stress concentrations, and have a rotating shaft assessed for fatigue.

Other engineering calculators are in the tools directory.

Frequently asked questions

Why does a keyway matter?

Because it removes material and concentrates stress at a sharp corner, which is exactly where fatigue cracks start. A shaft sized without allowing for its keyway is undersized.

Is static calculation enough?

No. A rotating shaft under bending sees fully reversed stress every revolution, so fatigue governs. Materials fail well below their static strength when cycled millions of times.

Can I use this for a real design?

As a preliminary check. Real shaft design covers fatigue, stress concentrations, deflection, critical speed and bearing loads, and needs a qualified engineer.

Final thought

Size it statically, then add for the keyway and check fatigue. The static number is where shaft design starts and it is never where it ends.

Try the free Shaft Diameter Calculator