Torsion in Circular Shafts

Why it matters

Torsion occurs when a component 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.

Meaning

When a torque is applied to a circular shaft, it twists. This twisting causes internal 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 undergoes an angular 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:

The maximum shear stress occurs at the outer surface (where $\rho = c$, the outer radius):

Angle of Twist

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

Engineering Meaning

• Torsional Stiffness: The term $GJ/L$ represents the torsional stiffness of the shaft. To minimize 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 maximum at the outer edge and zero at the center, hollow shafts are much more efficient than solid shafts. A hollow shaft provides nearly the same torsional strength while using significantly less material and being much lighter.

Common Mistakes

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