Angle of Twist Calculator – Shaft Torsion
Calculate the angle of twist for circular shafts under torsional load using θ = TL / (GJ).
Enter the applied torque, shaft length, shear modulus, and either the shaft diameter or polar moment of inertia to instantly compute the angle of twist in radians and degrees.
Angle of Twist Calculator – Shaft Torsion
Calculate the angle of twist for circular shafts under torsional load using θ = TL / (GJ).
If provided, this overrides the diameter-based J calculation.
About the Angle of Twist Calculator
When a shaft is subjected to a torque, it twists along its axis. The angle through which one end rotates relative to the other is called the angle of twist. This quantity is fundamental in mechanical engineering, particularly in the design of drive shafts, axles, torsion bars, and any rotating machinery where structural integrity and precision are critical.
The governing formula is θ = T × L / (G × J), where θ is the angle of twist in radians, T is the applied torque in Newton-metres, L is the length of the shaft in metres, G is the shear modulus of the material in Pascals, and J is the polar moment of inertia of the cross-section in metres to the fourth power. This formula is derived from the theory of elasticity and assumes that the material behaves linearly (Hooke's Law applies), the shaft is straight and uniform, and the cross-section remains plane after twisting.
For a solid circular shaft — the most common geometry in engineering practice — the polar moment of inertia is J = π × d⁴ / 32, where d is the shaft diameter. This calculator automatically computes J from the diameter you provide, so you do not need to calculate it separately. If you already know J (for example, for a hollow shaft, rectangular bar, or other cross-section), you can enter it directly in the optional field, which overrides the diameter-based computation.
The shear modulus G quantifies how stiff a material is against shear deformation. Common values include approximately 80,000 MPa for steel, 26,000 MPa for aluminium alloys, and 37,000 MPa for brass. Selecting the correct G for your material is essential for accurate results. Using an incorrect shear modulus is one of the most frequent sources of error in torsion calculations.
Understanding the angle of twist matters for several engineering applications. In power transmission systems, excessive twist introduces angular error between input and output shafts, which can degrade the performance of precision equipment. In structural applications, torsional stiffness — the inverse of the compliance measured by θ/T — determines how well a structure resists twisting loads. In machine tool spindles, automotive driveshafts, and helicopter rotor masts, engineers must ensure the angle of twist under maximum operating torque remains within allowable limits.
The result is given in both radians and degrees for convenience. Radians are the natural unit for angular quantities in mechanics, but degrees are more intuitive for most practical interpretations. One radian equals approximately 57.3 degrees.
Angle of Twist Examples
Three worked examples demonstrating the angle of twist formula for common engineering materials.
| Input | Result | Notes |
|---|---|---|
| Steel shaft: T = 1500 N·m, L = 1.5 m, G = 80000 MPa, d = 0.03 m | θ ≈ 0.3536 rad ≈ 20.26° | J = π×(0.03)⁴/32 ≈ 7.952×10⁻⁸ m⁴. Standard steel driveshaft under heavy torque. |
| Aluminium shaft: T = 500 N·m, L = 1.0 m, G = 26000 MPa, d = 0.04 m | θ ≈ 0.0766 rad ≈ 4.39° | J = π×(0.04)⁴/32 ≈ 2.513×10⁻⁷ m⁴. Lower G gives larger twist than steel for same dimensions. |
| Brass shaft: T = 800 N·m, L = 2.0 m, G = 37000 MPa, d = 0.025 m | θ ≈ 1.133 rad ≈ 64.9° | J = π×(0.025)⁴/32 ≈ 3.835×10⁻⁸ m⁴. Long slender shaft shows high twist. |
How to use the Angle of Twist Calculator
- Enter the applied torque T in Newton-metres (N·m), shaft length L in metres, and shear modulus G in MPa (e.g. 80000 for steel, 26000 for aluminium).
- Enter the shaft diameter d in metres to have J computed automatically using J = π × d⁴ / 32, or enter the polar moment of inertia J directly in m⁴ to override it.
- Click Calculate to get the angle of twist in radians and degrees.
- Review the result: if θ seems too large, check that G is in MPa and lengths are in metres — unit mismatches are the most common source of error.
- Click Reset to clear all fields and start a new calculation.
Angle of Twist FAQ
What is the angle of twist formula?
The angle of twist is θ = T × L / (G × J), where T is torque (N·m), L is shaft length (m), G is shear modulus (Pa), and J is the polar moment of inertia (m⁴). For a solid circular shaft, J = π × d⁴ / 32.
What is the shear modulus and where do I find values?
The shear modulus G (also called modulus of rigidity) measures a material's resistance to shear deformation. Typical values: steel ≈ 80,000 MPa, aluminium ≈ 26,000 MPa, brass ≈ 37,000 MPa, titanium ≈ 41,000 MPa. Check your material's datasheet for precise values.
What is the polar moment of inertia?
For a solid circular shaft, J = π × d⁴ / 32. For a hollow shaft with outer diameter D and inner diameter d, J = π × (D⁴ − d⁴) / 32. Enter J directly if you have a non-circular or hollow cross-section.
Why is the result very large for thin or long shafts?
The angle of twist is proportional to L and inversely proportional to d⁴. A small reduction in diameter or large increase in length dramatically increases the twist. For example, halving the diameter increases θ by a factor of 16.
Can I use this calculator for hollow shafts?
Yes. Calculate J = π × (D⁴ − d⁴) / 32 for your hollow shaft, then enter that value in the optional Polar Moment of Inertia field. The calculator will use it directly instead of computing J from diameter.
What does a large angle of twist mean in practice?
A large angle of twist indicates that the shaft is torsionally compliant. In power transmission, this introduces angular backlash between input and output. Engineers typically limit the angle of twist per unit length (θ/L) to values like 0.25°/m to 1°/m depending on precision requirements.