What is the angle of twist formula?

The angle of twist is given by phi = TL / (GJ), where T is the applied torque (N.m), L is the shaft length (m), G is the shear modulus of the material (Pa), and J is the polar moment of inertia of the cross-section (m⁴). For a solid circular shaft of diameter d, J = pi * d⁴ / 32. For a hollow shaft with outer diameter d_o and inner diameter d_i, J = pi * (d_o⁴ - d_i⁴) / 32. The result phi is in radians; multiply by 180/pi to convert to degrees.

What is the shear modulus for common engineering materials?

The shear modulus (also called modulus of rigidity, G) measures a material's resistance to shear deformation. Typical values: Structural steel: 80 GPa; Stainless steel: 77 GPa; Aluminium alloy: 26 GPa; Copper: 45 GPa; Brass: 37 GPa; Cast iron: 41 GPa; Titanium alloy: 44 GPa. These values vary slightly between alloy grades and should be confirmed with material data sheets for critical engineering applications.

What is the maximum allowable angle of twist for a shaft?

There is no single universal limit; it depends on the application. A common design rule for general engineering shafts is to limit the angle of twist to 0.25 to 1 degree per metre of shaft length. For precision machinery, limits as tight as 0.1 degrees per metre may apply. Transmission shafts are often limited to 1 degree per metre. Where the application is sensitive to angular displacement (such as machine tool spindles or precision drives), the designer should check the maximum permissible twist angle specified for that system.

Angle of Twist Calculator

This calculator works out how much a shaft twists when a torque is applied along its length, using the standard torsion formula phi = TL / GJ. It is a core check in shaft design, coupling selection and drive-train engineering, since excessive twist can throw a system out of alignment or overload couplings even when the material itself is nowhere near its strength limit. You enter the applied torque, the shaft length, and choose whether the cross-section is a solid or hollow circular shaft, then give the outer diameter, plus the inner diameter for a hollow shaft. For the material, pick a preset shear modulus for common metals such as structural steel, stainless steel, aluminium, copper, brass, cast iron or titanium alloy, or enter your own custom value. The calculator returns the angle of twist in both degrees and radians, the polar moment of inertia of the section, the twist per metre of shaft length, and a full breakdown showing the torsional rigidity and how your result compares with typical design limits for general shafts and precision machinery. Use it to check whether a shaft size keeps twist within an acceptable range, or to see how diameter, length or material choice affects stiffness. These figures follow classical linear-elastic torsion theory and are indicative only, so confirm material properties and design limits with a qualified engineer for any critical application.

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Standard formula Torsion formula from classical mechanics of materials (Timoshenko, Beer & Johnston).

1. Loading and Shaft

Applied torque (T) N.m

N.m

Shaft length (L) m

Cross-section type

Outer diameter (d) mm

Inner diameter (d_i) mm

2. Material

Material preset

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)
0.0006°
degrees

Angle of Twist

0.000010

radians

Polar Moment J

613,592.32

mm⁴

Twist per metre

0.0006°

degrees / m

Calculation Breakdown

Torque (T)500 N.m

Shaft length (L)1 m

Shear modulus (G)80000 GPa (80000 x 109 Pa)

Cross-sectionSolid, d = 50 mm

Polar moment of inertia (J)613,592.315 mm4 (6.1359e-7 m4)

GJ (torsional rigidity)4.9087e+7 N.m2

Angle of twist (phi)0.000010 rad = 0.0006°

Design Reference

phi in radians0.000010 rad

phi in degrees0.0006°

Twist per metre0.0006°/m

Typical limit (general shaft)0.25 to 1 deg/m

Typical limit (precision)< 0.1 deg/m

Within limit?Yes, well 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.
Flexible couplings: accommodate larger angular misalignments; consult the coupling manufacturer's data.

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.

Related Calculators

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.