Angle of Twist Calculator

Calculate angle of twist from torque, length, polar moment of inertia, and shear modulus, or solve for torque in torsion problems.

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Angle of Twist Formula

The angle of twist for a circular shaft in torsion is calculated from torque, shaft length, polar moment of inertia, and shear modulus.

\theta = \frac{T L}{J G}

Rearranged forms:

T = \frac{\theta J G}{L}

L = \frac{\theta J G}{T}

J = \frac{T L}{\theta G}

G = \frac{T L}{\theta J}

• θ = angle of twist, usually in radians
• T = applied torque
• L = shaft length
• J = polar moment of inertia of the shaft cross-section
• G = shear modulus of the shaft material

The calculator uses the torsion relationship above. If angle of twist is blank, it solves for θ. If torque is blank, it rearranges the same equation and solves for T. The same relationship can also be rearranged to solve for length, polar moment of inertia, or shear modulus when the other values are known.

For consistent engineering units, use torque in N·m, length in m, polar moment in m⁴, and shear modulus in Pa. The resulting angle is in radians, which can be converted to degrees using:

\theta_{deg} = \theta_{rad} \times \frac{180}{\pi}

Common Shear Modulus Values

Use the shear modulus for the shaft material. Actual values vary by alloy, heat treatment, and temperature.

Example Problems

Example 1: Calculate angle of twist

A steel shaft has an applied torque of 500 N·m, a length of 2 m, a polar moment of inertia of 1.2 × 10^-6 m⁴, and a shear modulus of 79 GPa.

\theta = \frac{T L}{J G}

\theta = \frac{500 \times 2}{(1.2 \times 10^{-6})(79 \times 10^9)}

\theta = 0.01055 \text{ rad}

The angle of twist is about 0.01055 rad, or 0.604 degrees.

Example 2: Calculate required torque

A shaft twists 0.02 rad over a length of 1.5 m. Its polar moment of inertia is 2.0 × 10^-6 m⁴, and its shear modulus is 80 GPa.

T = \frac{\theta J G}{L}

T = \frac{0.02(2.0 \times 10^{-6})(80 \times 10^9)}{1.5}

T = 2133.33 \text{ N·m}

The required torque is about 2133 N·m.

FAQ

What does angle of twist mean?

Angle of twist is the amount a shaft rotates because of applied torque. One end of the shaft rotates relative to the other end. It is usually measured in radians, but it is often reported in degrees for easier interpretation.

Why does a longer shaft twist more?

Angle of twist is directly proportional to length. If torque, material, and cross-section stay the same, doubling the shaft length doubles the angle of twist.

What is the difference between polar moment of inertia and shear modulus?

Polar moment of inertia describes the torsional stiffness of the shaft shape and size. A larger shaft diameter usually gives a much larger polar moment of inertia. Shear modulus describes the stiffness of the material itself. Steel has a higher shear modulus than aluminum, so a steel shaft usually twists less than an aluminum shaft with the same dimensions and torque.