Table of Contents
Shaft Torsion
When a torque (twisting moment) is applied to a shaft, it deforms by rotating one end relative to the other. The angle of twist measures this rotational deformation. It depends on the applied torque, shaft length, cross-sectional geometry (polar moment of inertia), and the material shear modulus.
Calculating the angle of twist is essential in mechanical engineering for designing drive shafts, gear systems, steering columns, and any rotating machinery where excessive twist could cause misalignment, vibration, or failure. Typical design limits are 0.25 to 1 degree per meter of shaft length.
Angle of Twist Formula
Where phi is the angle of twist in radians, T is torque, L is length, G is shear modulus, J is the polar moment of inertia, tau is shear stress, and r is the shaft radius. For a solid circular shaft, J = pi*d^4/32.
Shear Modulus of Common Materials
| Material | G (GPa) |
|---|---|
| Steel | 79-84 |
| Aluminum | 26-28 |
| Copper | 44-47 |
| Titanium | 41-45 |
| Cast Iron | 32-41 |
| Brass | 35-40 |
FAQ
What is the polar moment of inertia?
The polar moment of inertia J measures a cross-section resistance to torsional deformation. For a solid circular shaft J = pi*d^4/32. For a hollow shaft J = pi*(D^4-d^4)/32. Hollow shafts have a higher J-to-weight ratio, making them more efficient at resisting twist.
How much twist is acceptable?
General engineering practice limits twist to 0.25-1.0 degree per meter of shaft length, depending on the application. Precision machinery may require much tighter limits. Power transmission shafts typically allow up to 1 degree per 20 diameters of length.
What happens if twist is too much?
Excessive twist causes vibration, misalignment of connected components, premature bearing wear, and can lead to fatigue failure. In the worst case, the shear stress exceeds the material yield strength, causing permanent deformation or fracture of the shaft.