Angle of Twist Calculator

The angular deflection of a uniform shaft under torsion relies on mechanics of materials principles. The core equation is \(\theta = \frac{T \cdot L}{J \cdot G}\), where \(\theta\) is the angle of twist in radians, \(T\) is the applied torque, \(L\) is the length of the shaft, \(J\) is the polar moment of inertia, and \(G\) is the shear modulus of the material.

Imagine a solid aluminum bar that is 3 meters long with a 100 mm diameter, experiencing a torque of 10 kN·m. Given the shear modulus of aluminum is 80 GPa, we first calculate the polar moment of inertia for a solid circle: \(J = \frac{\pi d^2}{32} = \frac{\pi (0.1\text{ m})^4}{32} \approx 9.817 \times 10^{-6}\text{ m}^4\). Substituting these values into our formula: \(\theta = \frac{10000\text{ N}\cdot\text{m} \times 3\text{ m}}{9.817 \times 10^{-6}\text{ m}^4 \times 80 \times 10^9\text{ Pa}}\), yielding an angle of twist of approximately 0.0382 radians, or about 2.19 degrees.

Always ensure your units are consistent before plugging numbers into the formula; mixing millimeters and meters is a common source of calculation errors. Remember that the standard output of the formula is in radians, so multiply by \(180/\pi\) if you need degrees for your final documentation. Additionally, verify that your applied torque does not exceed the material's elastic limit to prevent permanent plastic deformation.