Short Answer:

Torsional stiffness is the property of a shaft or circular member that defines its ability to resist twisting when subjected to torque. It represents how much torque is required to produce a certain angle of twist in the shaft. In simple words, it shows the rigidity of a shaft against torsional deformation.

It is denoted by the symbol K and is mathematically expressed as K = T / θ, where T is the applied torque and θ is the angle of twist in radians. A higher torsional stiffness means the shaft is more rigid and resists twisting more effectively.

Detailed Explanation :

Torsional Stiffness

Torsional stiffness is an important mechanical property that indicates how resistant a shaft or any circular component is to twisting under the action of torque. When a torque or twisting moment is applied to a shaft, it causes angular deformation known as the angle of twist. The torsional stiffness determines the relationship between the applied torque and this angle of twist.

In engineering applications such as power transmission systems, crankshafts, turbines, and drive shafts, torsional stiffness is one of the most critical factors for performance and safety. If the shaft is not sufficiently stiff, it may twist excessively, leading to vibration, misalignment, or even failure due to fatigue.

Mathematically, the torsional stiffness (K) is defined as:

Where:

• K = torsional stiffness (N·m/rad)
• T = applied torque (N·m)
• θ = angle of twist (radians)

This formula shows that torsional stiffness is the ratio of applied torque to the resulting angular deformation. Hence, it expresses how much torque is needed to cause one radian of twist in the shaft.

1. Derivation of Torsional Stiffness Formula

Let us consider a solid circular shaft of length L, radius R, subjected to a torque T.

From the theory of torsion, the relation between torque and angle of twist is given by:

Where,

• G = modulus of rigidity of the material (N/m²)
• J = polar moment of inertia of the cross-section (m⁴)
• L = length of the shaft (m)
• θ = angle of twist (radians)

Rearranging the equation gives:

Thus,

This is the standard equation of torsional stiffness. It shows that torsional stiffness depends on three main factors:

1. Modulus of rigidity (G): The material’s resistance to shear deformation.
2. Polar moment of inertia (J): The geometric property that represents the distribution of area around the shaft’s axis.
3. Length of the shaft (L): The distance over which the torque acts.

2. Factors Affecting Torsional Stiffness

1. Material Property (Modulus of Rigidity, G):
   Materials with a higher modulus of rigidity (such as steel) provide greater torsional stiffness because they resist shear deformation more effectively.
2. Cross-sectional Shape and Size (Polar Moment of Inertia, J):
   • A larger diameter or a hollow section with more area distributed away from the center increases J, thereby increasing torsional stiffness.
   • For a solid circular shaft,
   • For a hollow circular shaft,
3. Length of the Shaft (L):
   The torsional stiffness is inversely proportional to the shaft length. A shorter shaft will be stiffer and twist less, while a longer shaft will twist more under the same torque.
4. Type of Shaft (Solid or Hollow):
   Hollow shafts generally provide higher stiffness-to-weight ratio because most of the material is concentrated away from the center, where it contributes more to J.

3. Physical Significance

Torsional stiffness plays an essential role in ensuring that a rotating shaft performs safely and efficiently.

• A high torsional stiffness means the shaft will twist very little under applied torque, which ensures accurate power transmission and alignment between connected components.
• A low torsional stiffness causes more twisting, which can lead to mechanical vibration, noise, and fatigue failure.

For example, in a crankshaft of an internal combustion engine, variations in torque during each power stroke create torsional oscillations. If the crankshaft is not stiff enough, these oscillations can grow and cause damage or resonance. Therefore, crankshafts are designed with high torsional stiffness to handle such dynamic loads.

4. Importance in Design Applications

1. Power Transmission:
   In automotive drive shafts and turbines, the torque must be transmitted efficiently without causing excessive twisting. High torsional stiffness ensures smooth torque delivery.
2. Machine Tools:
   Shafts used in lathes and milling machines require high stiffness to maintain precision during operation.
3. Vibration Control:
   Torsional stiffness helps determine the natural frequency of rotating systems. A properly designed system avoids resonance conditions by maintaining suitable stiffness values.
4. Structural Safety:
   In structures such as transmission shafts and couplings, torsional stiffness ensures strength and minimizes deformation under load.

5. Units of Torsional Stiffness

The unit of torsional stiffness depends on the torque and angle of twist used in the equation:

If torque (T) is measured in N·m and angle (θ) in radians, then the unit of torsional stiffness is N·m/rad.

It can also be expressed as N·mm/rad in smaller mechanical systems.

Conclusion:

Torsional stiffness is a fundamental property that defines the rigidity of a shaft or component against twisting under torque. It is mathematically expressed as K = GJ/L, showing its dependence on material rigidity, cross-sectional geometry, and shaft length. High torsional stiffness ensures stability, reduces vibration, and prevents failure in mechanical systems such as engines, turbines, and power transmission shafts. Therefore, understanding and designing for appropriate torsional stiffness is crucial in mechanical engineering.