Short Answer:

The strain energy due to torsion is the energy stored in a shaft or circular member when it is twisted by an external torque. This energy is a result of the internal shear stresses and strains produced in the material during the twisting action.

When the applied torque is removed, the stored strain energy helps the shaft return to its original shape, provided the material remains within its elastic limit. This concept is very important in mechanical engineering for designing shafts, springs, and rotating components that experience torsional loads.

Detailed Explanation:

Strain Energy due to Torsion

When a circular shaft is subjected to an external torque, it undergoes twisting deformation. Due to this twisting, every cross-section of the shaft rotates relative to the adjacent section. This rotation causes shear stresses and shear strains to develop within the material. The energy stored in the shaft due to these stresses is known as the strain energy due to torsion.

This stored energy represents the potential energy accumulated because of the internal resistance offered by the material against twisting. If the shaft remains within the elastic limit, this energy is completely recoverable when the torque is removed. However, if the material yields, some energy is lost permanently as deformation.

Understanding strain energy due to torsion is essential for the design and analysis of mechanical shafts, coil springs, and rotating components, as it helps predict failure, calculate angular deflection, and ensure safety during operation.

Derivation of Strain Energy due to Torsion

Consider a solid circular shaft of length , diameter , and radius , subjected to a torque .

When the shaft is twisted, a shear stress  is developed on its surface, which varies linearly from zero at the center to maximum at the outer surface.

From the torsion equation:

Where:

•  = Applied torque
•  = Polar moment of inertia ( for solid circular shaft)
•  = Shear stress at radius
•  = Modulus of rigidity
•  = Angle of twist in radians
•  = Length of shaft

The shear strain at radius  is:

The strain energy stored per unit volume is given by:

Substitute the values of  and :

 

To find the total strain energy , integrate over the entire volume of the shaft:

 

But for a solid circular shaft, , hence:

Now, substituting the value of :

Thus, the total strain energy due to torsion in a solid circular shaft is:

Interpretation of the Formula

The formula  shows that the strain energy due to torsion depends on:

1. Torque (T): The energy increases with the square of the applied torque.
2. Length (L): Longer shafts store more strain energy under the same torque.
3. Modulus of Rigidity (G): Stiffer materials (higher G) store less strain energy.
4. Polar Moment of Inertia (J): Larger cross-sectional area resists twisting better and stores less strain energy for the same torque.

This relationship helps engineers determine how much energy is stored and how much twist will occur for given loading conditions.

For Hollow Shafts

For a hollow shaft with outer radius  and inner radius :

The total strain energy due to torsion is:

This same expression applies, but the polar moment of inertia  changes according to the geometry.

Applications of Strain Energy due to Torsion

1. Design of Shafts:
   Helps engineers design shafts that can safely transmit torque without exceeding the elastic limit.
2. Spring Design:
   Torsional strain energy is the principle behind helical springs, which store energy during twisting.
3. Energy Absorption Systems:
   Used in mechanical systems where energy needs to be stored temporarily and released later, such as torsional dampers.
4. Power Transmission:
   Helps determine angular deflection in transmission shafts to ensure proper alignment and minimize vibrations.
5. Material Testing:
   Used in torsion testing of materials to determine modulus of rigidity and failure characteristics.

Example:

For a shaft with , , , and :

 

Thus, the shaft stores 0.00118 joules of strain energy under the given torque.

Conclusion:

The strain energy due to torsion is the elastic energy stored in a shaft when it is twisted by an external torque. It is a key concept in mechanical design, as it determines how much energy a component can store without yielding. The derived formula  is used for both solid and hollow shafts, showing the dependence on torque, shaft geometry, material rigidity, and length. This understanding ensures safe and efficient design of rotating machinery and power transmission systems.