Angle of Twist Calculator — Shaft Torsion (φ = TL/GJ)
Find the angle of twist (torsional rotation) of a circular shaft under an applied torque — for both solid and hollow cross-sections — along with the polar moment of inertia and the maximum shear stress at the surface.
N·m
m
mm
GPa
Cross-section
Total rotation of the free end relative to the fixed end
- 1
Polar moment J (solid circle)
π × (50 × 10⁻³)⁴ ÷ 32 = 613,592.3 mm⁴J is the shaft's resistance to torsional deformation. - 2
Angle of twist (radians) — φ = T·L ÷ G·J
500 × 1 ÷ (80 × 613,592.3 × 10⁻³) = 0.010186 - 3
Angle of twist (degrees)
0.010186 × (180 ÷ π) = 0.5836
How does this calculator work?
Angle of twist φ = TL/(GJ), where J = πd⁴/32 for a solid shaft and J = π(d⁴−dᵢ⁴)/32 for hollow. φ in radians; multiply by 180/π for degrees. Max surface shear stress τ = Tr/J. Example: 50 mm solid steel shaft, 1 m long, T = 500 N·m, G = 80 GPa → J ≈ 614,000 mm⁴, φ ≈ 0.58°, τ ≈ 20 MPa.
Formula
How this is calculated
When a torque T is applied to a circular shaft of length L, every cross-section rotates slightly relative to its neighbours. The total rotation between the two ends — the angle of twist φ — follows from the fundamental torsion formula φ = TL/(GJ), where G is the material's shear modulus (resistance to shear deformation in Pa or GPa) and J is the polar second moment of area of the cross-section.
For a solid circular shaft of outer diameter d, J = πd⁴/32. For a hollow shaft (tube) with outer diameter d and inner diameter dᵢ, J = π(d⁴ − dᵢ⁴)/32. Removing the material near the centre (which carries almost no shear stress in torsion) saves significant weight while barely reducing torsional stiffness — the reason drive shafts, structural tubes, and aerospace components use hollow sections.
The maximum shear stress occurs at the outer surface: τ_max = T·r/J (r = d/2). The model assumes a circular cross-section, uniform isotropic material, small angles of twist, and loading within the elastic limit. Non-circular sections, stress concentrations (keyways, steps), high-cycle fatigue, and plastic deformation are not covered. Always apply appropriate safety factors in structural design.
Frequently asked questions
Common values: structural steel ≈ 79–80 GPa, stainless steel ≈ 77–80 GPa, aluminium alloys ≈ 25–26 GPa, titanium ≈ 41 GPa, copper ≈ 44–48 GPa, brass ≈ 37–39 GPa. Always verify with your material's data sheet for precision engineering; values vary by alloy and heat treatment.
Shear stress in torsion varies linearly from zero at the centre to maximum at the outer surface. The central material contributes very little to torsional stiffness (J) but still adds weight. Removing it (making a tube) cuts mass significantly while J — and hence stiffness — decreases only by the small amount contributed by the removed inner section.
The formula φ = TL/GJ assumes purely elastic behaviour. It ceases to apply when the shear stress τ = Tr/J exceeds the material's shear yield strength — at that point the shaft starts to yield plastically and twist more than the formula predicts. Also, for non-circular sections (square, rectangular, I-beams) a different analysis (Saint-Venant torsion theory) is required.
Also known as
TG we-Calculate Editorial Team. (2026). Angle of Twist Calculator — Shaft Torsion (φ = TL/GJ) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/angle-of-twist-calculator
TG we-Calculate Editorial Team. "Angle of Twist Calculator — Shaft Torsion (φ = TL/GJ)." TG we-Calculate. 2026. https://we-calculate.com/calculator/angle-of-twist-calculator.
TG we-Calculate Editorial Team, "Angle of Twist Calculator — Shaft Torsion (φ = TL/GJ)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/angle-of-twist-calculator
@misc{wecalculate_angle_of_twist_calculator, title = {Angle of Twist Calculator — Shaft Torsion (φ = TL/GJ)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/angle-of-twist-calculator}}, year = {2026}, note = {TG we-Calculate} }
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