Torsional Shear Stress Calculator

What is torsional shear stress?

A moment trying to twist a bar about its axis (a torque) produces shear stress in the section. In a circular section this stress is zero at the centre and grows in proportion to the distance from it: τ = T × ρ / J. The largest value is at the outer surface: τ_max = T × r / J.

J is the polar second moment of area. For a solid circle J = π × d⁴ / 32, for a hollow one J = π × (D⁴ − d_i⁴) / 32. Material near the centre carries little stress, so a tube carries more torque than a solid bar of the same weight.

Angle of twist

The rotation of one end relative to the other is θ = T × L / (G × J). The result is in radians and the tool also shows degrees. G is the shear modulus.

Thin walled closed sections: Bredt's formula

For sections that are not circular but closed and thin walled, such as a box section, the stress is τ = T / (2 × A_m × t). A_m is not the area of the material but the area enclosed by the centre line of the wall. T / (2 × A_m) is the shear flow and it is constant around the perimeter.

Example

A solid shaft of 50 mm diameter carries a torque of 1 kN·m. J = π × 50⁴ / 32 = 613 592 mm⁴, so the stress is 1 000 000 × 25 / 613 592 = 40.7 MPa. If the shaft is 2 m long with G = 80 GPa, the angle of twist is 2.33 degrees.

Limits

The formulas cover circular sections and thin walled closed sections. Open profiles such as I, channel and angle sections are very weak in torsion and are not handled here. A tube slit along its length is also an open section and keeps only a tiny fraction of the stiffness of the closed tube. Stress concentration at keyways and shoulders has to be added separately.