Angle of Twist Calculator
Torsion of circular shafts — compute twist angle, or solve for any missing variable.
φ = T × L / (G × J)T = torque · L = shaft length · G = shear modulus · J = polar moment of inertia. Valid for linear-elastic torsion of uniform circular shafts.
Angle of twist — φ = TL/(GJ)
All inputs are converted to SI internally; results shown in both your chosen unit and SI. The formula assumes small, linear-elastic deformation of a uniform circular shaft under pure torsion.
Polar moment of inertia J
Shear modulus G — typical values (room temperature)
| Material | G (GPa) |
|---|---|
| Steel | 79.3 |
| Stainless steel | 75 |
| Carbon steel | 79.5 |
| Cast iron | 41 |
| Aluminium alloy | 26 |
| Copper | 44.7 |
| Brass | 40 |
| Bronze | 44 |
| Titanium | 41.4 |
Approximate values; actual G varies with alloy, temper, and temperature. Sources: Wikipedia "Shear modulus" (steel 79.3, copper 44.7), matmake.com, engineersedge.com.
FAQ — angle of twist
What is the formula for the angle of twist?
The angle of twist of a circular shaft under torque is φ = T × L / (G × J), where T is the applied torque, L the shaft length, G the shear modulus of the material, and J the polar moment of inertia of the cross-section. φ comes out in radians; multiply by 180/π for degrees. Example: a solid steel shaft (G = 79.3 GPa) with diameter 60 mm and length 1 m under T = 100 N·m has J = 1.2723×10^-6 m⁴, giving φ = 9.9111×10^-4 rad = 0.056786°.
How do I calculate torque from a known twist angle?
Rearrange the torsion formula: T = φ × G × J / L. Pick "Solve for: Torque T" in the calculator and enter φ, L, G and J. Example: an aluminium shaft (G = 26 GPa), solid diameter 40 mm, length 0.5 m, twisted by φ = 0.05 rad needs T = 653.45 N·m.
What is the polar moment of inertia J for solid and hollow shafts?
For a solid circular shaft J = π·d⁴/32, and for a hollow shaft J = π·(D⁴ − d⁴)/32, with D the outer and d the inner diameter. Example: a hollow shaft with D = 80 mm and d = 60 mm has J = 2.7489×10^-6 m⁴. Use the "Polar moment J" tab to compute either case.
What is shear modulus and what values should I use?
Shear modulus G (modulus of rigidity) measures a material's resistance to shear deformation. Typical room-temperature values: steel 79.3 GPa, stainless steel 75 GPa, aluminium alloy 26 GPa, copper 44.7 GPa, brass 40 GPa, titanium 41.4 GPa. Pick a preset in the calculator or type a custom value; G must be entered in the same consistent unit system as T, L and J (the tool converts everything to SI internally).
When is the angle-of-twist formula valid?
The formula φ = TL/(GJ) assumes linear-elastic behaviour of a straight shaft with a uniform circular cross-section under pure torsion, with small twist angles. It does not apply past yielding, to non-circular sections (which need different torsion constants), or to large plastic deformations. The calculator warns when φ exceeds 10° as a sanity check on the small-angle assumption.
How do I find the maximum shear stress too?
Torsion shear stress is τ = T × r / J, largest at the outer fibre r. The calculator reports it automatically whenever the outer radius is known. Example: T = 300 N·m on the hollow shaft above (outer radius 40 mm) gives τ = 4.365 MPa at the surface.
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