How to use this calculator

Pick solid or hollow, enter the diameters, and give the load either as a torque or as power and rotational speed, which is how motors and gearboxes are rated. Enter the length between the points where torque is applied and the shear modulus of the material.

Formulas

Polar moment of inertiaJ = π (D⁴ − d⁴)32 (d = 0 for a solid shaft)

Shear stress, maximum at the outer surfaceτ = T rJ τ_max = T (D/2)J

Angle of twistφ = T LG J

Torque from power and speedT = Pω ω = 2π n60 (n in rpm)

In US units, T (lbf·in) = 63,025 × P (hp) / n (rpm). The calculator handles the conversion for you.

Worked example

40 mm solid steel shaft, 500 N·m (the default)

J = π × 0.040⁴ / 32 = 2.513 × 10⁻⁷ m⁴ = 251,300 mm⁴.
τ_max = 500 × 0.020 / 2.513 × 10⁻⁷ = 39.8 MPa.
φ = 500 × 1.0 / (79.3 × 10⁹ × 2.513 × 10⁻⁷) = 0.0251 rad = 1.44°.
Switch to power: 15 kW at 1,200 rpm gives ω = 125.7 rad/s and T = 15,000 / 125.7 = 119.4 N·m.

Why hollow shafts are efficient

Shear stress is zero at the center and largest at the surface, so material near the axis carries little load. A tube with an inside diameter of half the outside diameter weighs 25% less than a solid shaft, but its J, and therefore its stiffness and strength in torsion, drops by only about 6%.

Assumptions and limits

Circular cross-section, constant along the length; these formulas do not apply to square or rectangular shafts.
Linear elastic material and small twist angles.
Pure torsion. Real shafts often also carry bending from gears, pulleys and belts; combine the stresses with Mohr’s circle.
Keyways, shoulders and grooves raise local stresses. Apply a stress concentration factor for fatigue design.

Frequently asked questions

What is the difference between G and E?

E (Young’s modulus) governs stretching and bending; G (shear modulus) governs twisting. For steel, E ≈ 200 GPa and G ≈ 79 GPa, related by G = E / 2(1 + ν).

How much twist is acceptable?

It depends on the machine. A common rule for transmission shafting limits twist to about 1° over a length of 20 diameters; precision drives need much less.

How do I size a shaft for a required stress?

For a solid shaft, rearrange to D = (16T / πτ_allow)^1/3, then round up to a standard size and check twist.

For education and preliminary checks only. Have a qualified engineer verify any design before it is built.