Torsion in Circular Shafts

Why This Matters

Torsion occurs when a member is subjected to a twisting moment, or torque. This is the primary loading condition for drive shafts in engines, motors, and pumps. Understanding torsion is essential for sizing shafts to safely transmit power without fracturing or twisting excessively.

What It Means

When a torque is applied to a circular shaft, it twists. This twisting action creates shear stresses within the material. The shear stress is zero at the center of the shaft and increases linearly to a maximum at the outer surface. Simultaneously, the shaft experiences a rotational deformation, known as the angle of twist.

Formulas

Torsional Shear Stress

For a solid or hollow circular shaft, the shear stress ($\tau$) at a radial distance $\rho$ from the center is:

$\tau = \frac{T \rho}{J}$

The maximum shear stress occurs at the outer surface (where $\rho = c$, the outer radius): $\tau_{max} = \frac{Tc}{J}$

Where:

• $T$ is the internal twisting torque (N·m)
• $c$ is the outer radius of the shaft (m)
• $J$ is the polar moment of inertia of the cross-section (m⁴)

Note: For a solid circular shaft, $J = \frac{\pi c^4}{2}$.

Angle of Twist

The total angle of twist ($\phi$) for a shaft of constant cross-section and constant material properties is:

$\phi = \frac{TL}{GJ}$

Where:

• $\phi$ is the angle of twist (radians)
• $L$ is the length of the shaft (m)
• $G$ is the shear modulus of the material (Pa)

Engineering Meaning

• Torsional Stiffness: The term $GJ/L$ represents the torsional stiffness of the shaft. To reduce the angle of twist, an engineer can increase the shaft diameter (which drastically increases $J$) or use a stiffer material (higher $G$).
• Material Efficiency: Because shear stress is highest at the outer edge and zero at the center, hollow shafts are much more efficient than solid shafts. They provide nearly the same torsional strength while using significantly less material and weighing less.

Common Mistakes

• Confusing radians and degrees: The formula $\phi = TL/GJ$ strictly outputs the angle in radians. Failing to convert radians to degrees when analyzing practical specifications is a very common error.
• Applying to non-circular cross-sections: The simple formulas $\tau = Tc/J$ and $\phi = TL/GJ$ apply only to solid or hollow circular shafts. Non-circular shafts (like square or I-beams) undergo warping, and require much more complex advanced theories.