Theoretical outer diameter
mm
Mathematical minimum before selecting a real size.
Mechanics · Shafts and torsion
Torsion shaft sizing
Determine the theoretical diameter and an upward-rounded recommended diameter, with a check of the resulting shear stress.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
03
Results
Recommended outer diameter
mm
Rounded up to the next millimetre or next 1/16 inch.
Associated inner diameter
mm
Inner diameter obtained from the entered d/D ratio.
Wall thickness
mm
Radial thickness (D−d)/2 for the recommended hollow shaft.
Stress at recommended diameter
MPa
Must remain at or below the entered allowable shear stress.
View calculation detailsFormulas, conversions, numerical substitution and resultsOpen ↓
Calculation note
Full calculation details
Circular prismatic shaft under constant torque. Linear-elastic behaviour with no stress concentration. The d/D ratio is prescribed for a hollow shaft. The proposed rounding is not a standard size series and does not check fatigue, keyways, critical speed or angular stiffness.
01Formulas and symbols
Formulas used
D
D = ∛(16T / πτadm)Minimum solid-shaft diameterD
D = ∛[16T / (πτadm(1−k⁴))]Minimum hollow-shaft outer diameter with k = d/Dd ; e
d = kD ; e = (D−d)/2Inner diameter and wall thicknessτmax
τmax = 16TD / [π(D⁴−d⁴)]Check at the recommended diameter02Assumptions and limits
Scope of validity
- Circular prismatic shaft under constant torque.
- Linear-elastic behaviour with no stress concentration.
- The d/D ratio is prescribed for a hollow shaft.
- The proposed rounding is not a standard size series and does not check fatigue, keyways, critical speed or angular stiffness.
03Validation example
Reference numerical case
- T = 2 kN·m, τallow = 80 MPa, hollow shaft with k = 0.6.
- Dmin = 52.690 mm.
- Recommended diameter = 53 mm; d = 31.8 mm; e = 10.6 mm.
- The recalculated stress at the recommended diameter is approximately 78.61 MPa.
04Frequently asked questions
Questions about the calculation
Why is d/D required for a hollow shaft?
Torque and allowable stress do not uniquely determine both diameters. The ratio k sets the hollow proportion.
Is the recommended diameter a commercial size?
No. It is a simple upward rounding. Select an available size and repeat all checks.
Does this calculation cover fatigue?
No. Real shafts often require fatigue checks, especially with shoulders, grooves, keyways and variable torque.
05References
Technical references
- Budynas & Nisbett, Shigley’s Mechanical Engineering Design, circular-shaft torsion.
- Gere & Goodno, Mechanics of Materials, elastic torsion.
- BIPM, The International System of Units (SI), 9th edition.