Calculate the angle of twist in a shaft under torsional loading using the standard formula phi = TL / GJ. Enter the applied torque, shaft length, material shear modulus, and cross-section dimensions. Results are shown in both radians and degrees.
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Standard formula
Torsion formula from classical mechanics of materials (Timoshenko, Beer & Johnston).
1. Loading and Shaft
N.m
m
mm
mm
2. Material
GPa
Please check your inputs. Diameter values must be positive, inner diameter must be less than outer diameter, and all values must be greater than zero.
Angle of Twist Results
Angle of Twist (phi)
-
degrees
Angle of Twist
-
radians
Polar Moment J
-
mm⁴
Twist per metre
-
degrees / m
Calculation Breakdown
Torque (T)-
Shaft length (L)-
Shear modulus (G)-
Cross-section-
Polar moment of inertia (J)-
GJ (torsional rigidity)-
Angle of twist (phi)-
Design Reference
phi in radians-
phi in degrees-
Twist per metre-
Typical limit (general shaft)0.25 to 1 deg/m
Typical limit (precision)< 0.1 deg/m
Within limit?-
Worked Example (matching default inputs)
Given: T = 500 N.m, L = 1.0 m, solid shaft d = 50 mm, G = 80 GPa (structural steel).
Step 1 - Polar moment of inertia: J = pi * d⁴ / 32 = pi * (0.05)⁴ / 32 = 6.1359 x 10⁻⁷ m⁴
Step 2 - Angle of twist: phi = TL / GJ = (500 x 1.0) / (80 x 10⁹ x 6.1359 x 10⁻⁷) = 500 / 49,087 = 0.010187 rad
Result: phi = 0.010187 rad = 0.5836 degrees (0.5836 deg/m, within the 1 deg/m general engineering limit).
What Is the Angle of Twist?
When a shaft is subjected to a torque (a twisting force), it deforms by rotating along its length. The angle through which one end of the shaft rotates relative to the other is called the angle of twist, typically denoted by the Greek letter phi. This is a key calculation in shaft design, coupling selection, and drive-train engineering.
The Torsion Formula
The angle of twist for a linearly elastic shaft of constant cross-section is:
phi = TL / (GJ)
Where:
phi = angle of twist (radians)
T = applied torque (N.m)
L = shaft length (m)
G = shear modulus of elasticity of the material (Pa or N/m²)
J = polar moment of inertia of the cross-section (m⁴)
The product GJ is called the torsional rigidity or torsional stiffness of the shaft. A higher GJ means less twist for the same torque.
Polar Moment of Inertia
For a solid circular shaft of diameter d:
J = pi * d⁴ / 32
For a hollow circular shaft with outer diameter dₒ and inner diameter dᵢ:
J = pi * (dₒ⁴ - dᵢ⁴) / 32
Hollow shafts are commonly used to reduce weight while maintaining torsional stiffness, since material near the centre of a solid shaft contributes relatively little to J.
Shear Modulus of Common Materials
Material
Shear Modulus G (GPa)
Structural steel
80
Stainless steel
77
Copper
45
Titanium alloy
44
Cast iron
41
Brass
37
Aluminium alloy
26
Values are typical for common engineering grades. Always confirm with the material data sheet for critical applications.
Design Limits and Practical Guidelines
Most engineering standards do not specify a single universal twist limit; the appropriate limit depends on the application. Common guidelines include:
General engineering shafts: 0.25 to 1 degree per metre of shaft length.
Precision machinery (machine tools, spindles): less than 0.1 degree per metre.
Transmission shafts: typically limited to 1 degree per metre.
Where a shaft exceeds the angular limit before the stress limit, it is said to be stiffness-limited rather than strength-limited. Increasing the shaft diameter, shortening the shaft, or choosing a higher-G material will all reduce the angle of twist.
Sources and method: Timoshenko, S. P. & Goodier, J. N., Theory of Elasticity (3rd ed., 1970); Beer, F. P. & Johnston, E. R., Mechanics of Materials (7th ed., 2015). Formula phi = TL/(GJ) is the standard linear-elastic torsion formula applicable to circular cross-sections within the elastic range.
This calculator applies the classical linear-elastic torsion formula and assumes a uniform circular cross-section, constant torque along the shaft length, and material behaviour within the elastic limit. For non-circular sections, stepped shafts, or plastic deformation, consult a structural or mechanical engineer.
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