Short Answer:

Torsional rigidity is the property of a shaft that represents its resistance to twisting under the action of an applied torque. It is defined as the torque required to produce a unit angle of twist per unit length of the shaft.

In simple words, torsional rigidity tells how stiff a shaft is when twisted. A shaft with higher torsional rigidity will twist less for the same applied torque. It depends on the material’s modulus of rigidity (G) and the shaft’s polar moment of inertia (J). Mathematically, torsional rigidity = GJ.

Torsional Rigidity

Detailed Explanation :

The torsional rigidity of a shaft is an important mechanical property that determines its ability to resist twisting when subjected to torque. It plays a key role in designing shafts, axles, and other machine components that transmit rotational motion or power. The higher the torsional rigidity, the stiffer the shaft, meaning it will deform less under a given torque.

Definition of Torsional Rigidity

The torsional rigidity is defined as:

“The torque required to produce a unit angle of twist per unit length of the shaft.”

It measures the stiffness of a shaft in torsion — how much torque is needed to twist it by one radian over a length of one meter.

Mathematically,

where,

•  = Modulus of rigidity (N/m²)
•  = Polar moment of inertia of the cross-section (m⁴)

Hence, torsional rigidity depends on both the material property (G) and the geometric property (J).

Derivation of Torsional Rigidity

From the torsion equation:

where,

•  = Applied torque (N·m)
•  = Angle of twist in radians
•  = Length of the shaft (m)

Rearranging for torque ,

Now, the angle of twist per unit length is .
Hence, torque per unit twist per unit length is:

Therefore,

This shows that torsional rigidity depends on both the modulus of rigidity of the material and the polar moment of inertia of the shaft cross-section.

Units of Torsional Rigidity

Since  has units of N/m² and  has units of m⁴,

It represents the torque required to twist the shaft through one radian per meter of its length.

Physical Meaning

Torsional rigidity gives an idea of how much a shaft resists twisting.

• If a shaft has high torsional rigidity, it means the shaft is stiffer and twists less for the same applied torque.
• If a shaft has low torsional rigidity, it means the shaft is flexible and twists more easily.

Thus, torsional rigidity defines the torsional stiffness of the shaft — the ability to resist angular deformation due to torque.

Factors Affecting Torsional Rigidity

1. Modulus of Rigidity (G):
   The higher the modulus of rigidity of the material, the higher will be its torsional rigidity. For example, steel has higher torsional rigidity than aluminum due to its higher G value.
2. Cross-sectional Geometry (J):
   The polar moment of inertia  depends on the shape and size of the cross-section.
   • For a solid circular shaft:

1. • For a hollow circular shaft:

1. Larger diameters or hollow sections increase , thereby increasing torsional rigidity.
2. Material Type:
   Different materials have different shear moduli. Steel, with a high G, gives better torsional rigidity compared to lighter materials like aluminum or copper.
3. Length of the Shaft:
   Though length does not directly affect GJ, a longer shaft will show more total angular twist for the same torque because the angle of twist .

Significance of Torsional Rigidity in Design

1. Stiffness of Shafts:
   High torsional rigidity ensures less angular deformation and better alignment in rotating machinery.
2. Power Transmission Efficiency:
   In rotating shafts like crankshafts and drive shafts, high torsional rigidity ensures smooth power transmission with minimal twist.
3. Prevention of Vibration and Failure:
   Low torsional rigidity may lead to torsional vibrations, noise, and fatigue failure. Adequate rigidity prevents such failures.
4. Comparison Between Solid and Hollow Shafts:
   A hollow shaft offers higher torsional rigidity per unit weight compared to a solid shaft of the same material and outer diameter. Hence, hollow shafts are widely used in automobiles and aircrafts to reduce weight without losing stiffness.

Example Calculation

Let’s consider a solid circular steel shaft of 50 mm diameter.

 

Therefore,

This means a torque of 24.56 kN·m is required to twist the shaft by one radian per meter of its length.

Importance in Practical Applications

• Used in design of rotating shafts, propeller shafts, axles, and machine spindles.
• Ensures stability and accuracy in power transmission systems.
• Helps in determining torsional deflection limits to avoid excessive vibration.
• Useful for comparing stiffness between solid and hollow shafts of the same material and weight.
• Critical in the aerospace and automotive industries where lightweight and high-stiffness designs are needed.

Conclusion

In conclusion, torsional rigidity is a measure of a shaft’s resistance to twisting under applied torque. It is given by the product of the material’s modulus of rigidity (G) and the cross-sectional polar moment of inertia (J), expressed as GJ. A high torsional rigidity means the shaft is stiffer and resists twisting effectively. It is a vital parameter in the design of rotating components to ensure safe, efficient, and vibration-free operation in mechanical systems.