Torsion of Solid and Hollow Circular Shafts

Definition

Torsion refers to the state of stress and deformation induced in a structural member (shaft) when it is subjected to an external twisting moment (torque) applied about its longitudinal axis. This creates shearing stresses across the cross-section of the shaft, causing it to twist.

Main Content

1. The Torsion Equation

• The relationship between the applied torque ($T$), the polar moment of inertia ($J$), the shear stress ($\tau$), the distance from the center ($r$), the shear modulus ($G$), the angle of twist ($\theta$), and the length of the shaft ($L$) is given by: $\frac{T}{J} = \frac{\tau}{r} = \frac{G\theta}{L}$
• This equation assumes the shaft is straight, homogeneous, isotropic, and the torque is applied perpendicular to the axis.

2. Polar Moment of Inertia (J)

• For a solid circular shaft of diameter $D$: $J = \frac{\pi D^4}{32}$
• For a hollow circular shaft with outer diameter $D$ and inner diameter $d$: $J = \frac{\pi (D^4 - d^4)}{32}$
• $J$ represents the geometric resistance of the cross-section to torsion.

3. Shear Stress Distribution

• Shear stress varies linearly from zero at the center of the shaft to a maximum at the outer surface.
• The maximum shear stress ($\tau_{max}$) occurs at the outermost fiber (where $r = R$ or $D/2$).

[Shear Stress Distribution across a shaft cross-section]

      (Zero at center)
          |
    <--   |   -->
  --------+--------  <- Maximum shear stress at surface
    <--   |   -->
          |

Working / Process

1. Determine Input Parameters

• Identify the torque ($T$) in Newton-meters (Nm) applied to the shaft.
• Measure or define the dimensions: Length ($L$), and diameters ($D$ or $d$). Determine the Shear Modulus ($G$) based on the shaft material (e.g., steel).

2. Calculate Geometric Resistance

• Calculate the Polar Moment of Inertia ($J$) using the appropriate formula for solid or hollow sections.
• Ensure units are consistent (e.g., meters) before calculating.

3. Calculate Stress and Deformation

• Use $\tau_{max} = \frac{T \cdot R}{J}$ to find the maximum shear stress to ensure the material stays within its elastic limit.
• Use $\theta = \frac{TL}{GJ}$ (in radians) to determine the angle of twist, which is critical for systems requiring high precision (like robotics or precision spindles).

Advantages / Applications

• Hollow shafts are widely used in automotive drive shafts because they provide a high strength-to-weight ratio, saving material while maintaining torsional stiffness.
• Solid shafts are preferred in heavy-duty machinery where space is limited, but absolute rigidity is required.
• Understanding torsion is essential for designing power transmission systems, such as engine crankshafts, turbine shafts, and drill bits used in construction.

Summary

Torsion is the twisting effect on a shaft caused by external torque, resulting in shear stresses and angular deformation. The behavior of a shaft under load is governed by the torsion equation, which balances external torque against internal geometric resistance and material properties. The key terms to remember are Polar Moment of Inertia ($J$), which defines resistance to twisting; Shear Modulus ($G$), which defines material stiffness; and Torque ($T$), the causative force of the twisting effect.