Short Answer:

Torsional rigidity is the property of a shaft that indicates its resistance to twisting when a torque is applied. It is defined as the torque required to produce a unit angle of twist in the shaft. In simple terms, it measures how stiff a shaft is under torsion — the higher the torsional rigidity, the smaller the angle of twist for a given torque.

Torsional rigidity depends on the material property (modulus of rigidity), the geometry of the shaft (polar moment of inertia), and its length. It is an important factor in the design of shafts and other rotating mechanical components to ensure strength and durability under working loads.

Detailed Explanation:

Torsional Rigidity

The torsional rigidity of a shaft is a measure of its ability to resist twisting under the action of an external torque. When a shaft is subjected to a twisting moment, it experiences angular deformation along its length. The rigidity of the shaft in torsion expresses how strongly the shaft resists this angular deformation.

Mathematically, torsional rigidity is defined as the product of the modulus of rigidity (G) and the polar moment of inertia (J) of the shaft’s cross-section. It represents the torque required to produce one radian of twist per unit length of the shaft.

This means that a shaft with a larger value of  will twist less for a given torque, making it more rigid and capable of transmitting higher power safely.

Mathematical Relationship

From the torsion equation:

where:

•  = Applied torque
•  = Polar moment of inertia
•  = Modulus of rigidity
•  = Angle of twist (in radians)
•  = Length of the shaft

Rearranging the formula gives:

This shows that the torsional rigidity (GJ) is the measure of torque required to cause a unit angle of twist (θ = 1 radian) in a shaft of unit length.

Hence,

The higher the value of torsional rigidity, the greater the resistance offered by the shaft to twisting under the applied torque.

Factors Affecting Torsional Rigidity

1. Modulus of Rigidity (G):
   This is a material property that defines the material’s resistance to shear deformation. Materials with a higher modulus of rigidity, such as steel, possess higher torsional rigidity compared to softer materials like aluminum.
2. Polar Moment of Inertia (J):
   It depends on the shape and size of the shaft’s cross-section. For a solid circular shaft,

and for a hollow shaft,

where  = diameter of solid shaft,  = outer diameter, and  = inner diameter.
A larger polar moment of inertia increases the shaft’s torsional rigidity.

1. Material and Cross-Section:
   A combination of a material with a higher modulus of rigidity and a larger cross-sectional area increases torsional rigidity.

Significance of Torsional Rigidity

• Strength and Safety: A shaft with high torsional rigidity can transmit larger torques without excessive deformation, ensuring safe and efficient operation of machinery.
• Reduced Vibrations: High torsional rigidity minimizes angular deformation, reducing torsional vibrations in rotating systems like engines and turbines.
• Power Transmission Efficiency: Efficient transfer of torque occurs when the shaft resists twisting, leading to stable power output.
• Design Parameter: It is a crucial factor in designing shafts, couplings, and transmission systems to ensure that they do not fail due to excessive twist.

Practical Example

Consider a steel shaft of 1 m length and 50 mm diameter. The modulus of rigidity for steel is approximately .
The polar moment of inertia for the shaft is:

Hence, torsional rigidity:

This value represents the torque required to produce one radian twist per unit length, showing that the shaft has a high resistance to twisting.

Importance in Design

In engineering design, it is not enough for the shaft to simply transmit the torque without failure; it must also maintain proper alignment and avoid excessive twisting. Shafts with low torsional rigidity may result in misalignment, increased wear, and fatigue in mechanical systems.

Thus, the concept of torsional rigidity ensures that designers choose suitable materials and dimensions to maintain the balance between strength, stiffness, and weight.

Conclusion

Torsional rigidity is an essential mechanical property that measures a shaft’s resistance to twisting under an applied torque. It depends on both material properties (modulus of rigidity) and the geometric properties (polar moment of inertia) of the shaft. A shaft with higher torsional rigidity performs better in torque transmission, experiences less angular deformation, and ensures safe and efficient operation of mechanical systems.