Shaft Torsion Calculator (stress & twist)

Understanding the shaft torsion calculator — concept and principle

A shaft that transmits torque is twisted: shear stress builds up in the material and one end rotates relative to the other. This calculator gives the maximum shear stress, the angle of twist and the polar moment of area J for a solid or hollow circular shaft, as used when sizing drive shafts, axles, spindles and torsion bars.

How it is calculated

• Polar moment: J = π(D⁴ − d⁴)/32 in mm⁴, with outer diameter D and inner diameter d (d = 0 for a solid shaft).
• Maximum shear stress: τ = T × r ÷ J, where r = D/2 is the outer radius. Torque in N·m is multiplied by 1,000 to get N·mm, so τ is in MPa.
• Angle of twist: θ = T × L ÷ (G × J) in radians, converted to degrees by × 180/π. G is the shear modulus in GPa (steel ≈ 79), L the length in metres.

The principle

For a circular shaft, plane cross-sections stay plane and radii stay straight as it twists. Shear strain therefore grows linearly from zero at the centre to a maximum at the surface, and with Hooke's law in shear (τ = Gγ) so does stress. Summing the resisting moment of every ring of material gives T = τJ/r, and integrating the twist rate T/(GJ) along the length gives θ. Because material near the centre carries little stress, a hollow shaft can save a large share of the weight while losing only a little strength.

Worked example

1. Inputs: T = 500 N·m = 500,000 N·mm, D = 40 mm, d = 0, L = 1 m, G = 79 GPa.
2. J = π × 40⁴ ÷ 32 = π × 2,560,000 ÷ 32 = 251,327.4 mm⁴.
3. τ = 500,000 × 20 ÷ 251,327.4 = 39.78874 MPa.
4. θ = 500 × 1 ÷ (79 × 10⁹ × 251,327.4 × 10⁻¹²) = 0.025183 rad = 1.442865°.

How to read the result

Compare τ with the allowable shear stress for the material. A common estimate of shear yield strength is about 0.5 to 0.58 times the tensile yield strength (Tresca and von Mises criteria respectively), and design values are lower still after safety factors. Twist matters for precision drives and gear alignment, where a limit per metre of length is often specified.

Common mistakes

• Using the radius in the J formula with the 32 divisor; πr⁴/2 and πD⁴/32 are the same thing, but mixing them gives a factor of 16 error.
• Entering G as Young's modulus E (200 GPa for steel instead of about 79).
• Applying these formulas to square or open sections, which warp and need different equations.

Frequently asked questions

Why is a hollow shaft more efficient?

Stress is highest at the surface; removing low-stressed material from the centre reduces weight much more than it reduces J.

How are G and E related?

For an isotropic material G = E/(2(1 + ν)). For steel, 200/(2 × 1.3) ≈ 77 GPa, close to the usual 79 to 80.

How do I get torque from motor power?

T = P/ω, or T (N·m) ≈ 9,549 × P (kW) ÷ rpm.

Sources: R. G. Budynas and J. K. Nisbett, Shigley's Mechanical Engineering Design, McGraw-Hill · J. M. Gere and B. J. Goodno, Mechanics of Materials, Cengage