T = torque (N·mm)
Hollow circular shaft torsion
Relates torque to outside/inside diameter for a hollow circular shaft in pure torsion.
FORMULA REFERENCE
T = (πτ/16)((Do⁴ − Di⁴)/Do)
Relates torque to outside/inside diameter for a hollow circular shaft in pure torsion.
Variables and units
Keep each value in the units shown below before substituting it into the displayed relation. Convert first rather than mixing unit systems inside one calculation.
τ = selected allowable shear stress (N/mm²)
Do = outside diameter (mm)
Di = inside diameter (mm)
WORKED EXAMPLE
Calculate the stated condition, then interpret its boundary.
Known values
- T = 98,793 N·mm
- τ = 37.5 N/mm²
- Di = 15 mm
Substitution: Solve 98,793 = (π × 37.5 / 16)((Do⁴ − 15⁴)/Do)
Calculated result: Do ≈ 27.5 mm
Engineering interpretation: The root must be solved numerically; confirm that the proposed outside diameter remains greater than the inside diameter.
Conditions, limitations and review
- Uniform hollow circular section
- Pure torsion
- Outside diameter is solved numerically
Use this formula to understand or screen the stated engineering condition. A final design requires the governing standard, actual loads, material condition, manufacturing features and professional engineering judgement.
Source: J. E. Shigley et al., Mechanical Engineering Design, torsion of circular shafts.