Angle of Twist Calculator
Calculate shaft angle of twist from torque, length, polar moment of inertia, and shear modulus for torsion analysis.
What Is the Angle of Twist?
When a shaft carries torque, it twists along its length. The angle of twist \(\phi\) measures how far one end rotates relative to the other because of torsion. Engineers use it to check that drive shafts, axles, and structural members stay within allowable deformation limits.
Torsion Formula
$$ \phi = \frac{T L}{J G} $$\(T\) is applied torque in newton-meters, \(L\) is shaft length in meters, \(J\) is the polar moment of inertia in m\(^4\), and \(G\) is the shear modulus in pascals. The result \(\phi\) is in radians.
For a solid circular shaft of diameter \(d\):
$$ J = \frac{\pi d^4}{32} $$Steel often has \(G \approx 80\) GPa. A longer shaft or higher torque increases twist. A larger diameter or stiffer material reduces it.
How to Use This Calculator
Enter torque and shaft length, then choose whether to supply diameter (for a solid circular section) or the polar moment directly. Set the shear modulus in GPa or Pa. The tool reports twist in radians and degrees.
Related tools: Torque Calculator and Bending Stress Calculator.
Frequently Asked Questions
What units should I use for shear modulus?
Use pascals in SI form. Steel is about 80 GPa, which equals 80 billion pascals. The calculator accepts GPa or Pa directly.
When should I enter J instead of diameter?
Use direct J for hollow shafts, non-circular sections, or when you already know the polar moment from a table or CAD model.
Is the angle of twist the same as shear strain?
They are related but not identical. Twist is the total rotation over length L. Shear strain at the outer fiber also depends on radius and geometry.
Does this formula apply to plastic deformation?
No. The formula assumes linear elastic torsion. Beyond the proportional limit, permanent deformation and nonlinear behavior require more advanced analysis.
How do I convert radians to degrees?
Multiply by 180 divided by pi. One radian equals about 57.3 degrees. This calculator shows both units in the result.
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