What is strain energy due to torsion?

Short Answer:

Strain energy due to torsion is the energy stored in a circular shaft or any solid body when it is twisted by an applied torque within the elastic limit. When a shaft is subjected to torque, it experiences an angular deformation, and the work done by the torque is stored as elastic strain energy in the material.

In simple terms, when a shaft transmits power, it twists slightly, and this twist produces shear stress in the material. The energy stored in the shaft because of this twisting is known as strain energy due to torsion. It helps engineers design safe and efficient shafts, springs, and couplings.

Detailed Explanation :

Strain Energy Due to Torsion

When a shaft is subjected to torque, it undergoes angular deformation, which causes the material fibers to experience shear stress and shear strain. The work done by the applied torque during this deformation is not lost; instead, it is stored within the shaft in the form of strain energy due to torsion.

This energy remains stored as long as the torque applied is within the elastic limit of the material. If the torque is removed, the shaft returns to its original shape, and the stored energy is released. This property is very important in power transmission systems, torsional springs, and mechanical couplings, where shafts must twist slightly without permanent deformation.

Definition

The strain energy due to torsion can be defined as:

“The energy stored per unit volume or in the entire shaft when it is subjected to a twisting moment (torque) within the elastic limit is known as strain energy due to torsion.”

It represents the elastic potential energy accumulated because of the shear stress induced by twisting.

Derivation of Expression

Let a circular shaft of

  • Length =
  • Radius =
  • Polar moment of inertia =
    be subjected to a torque .

The shaft twists through an angle , and the torque produces a shear stress  at a radial distance  from the axis.

From torsion equation, we have:

where,

  •  = Modulus of rigidity,
  •  = Shear stress at radius .

Now, consider a small cylindrical element at a distance  from the center with thickness .
The shear strain at this radius is given by:

The shear stress is related to the strain by:

Work Done by Torque

When torque  is applied gradually, the work done by it equals the average torque multiplied by the angular displacement.
Thus,

This work done is stored as strain energy (U) within the shaft.

Now, from the torsion relation,

Substitute this into the expression for :

 

Hence, this is the total strain energy stored in the shaft due to torsion.

Strain Energy per Unit Volume

If we divide the total strain energy by the total volume of the shaft , we get the strain energy per unit volume:

But since , we can replace  and  to express it in terms of shear stress:

This represents the strain energy due to torsion per unit volume of the shaft.

Explanation of Terms

  •  = Total strain energy due to torsion (J)
  •  = Torque applied on the shaft (N·m)
  •  = Length of the shaft (m)
  •  = Polar moment of inertia  for a solid shaft
  •  = Modulus of rigidity (N/m²)
  •  = Shear stress (N/m²)

Physical Significance

  1. The strain energy due to torsion represents how much energy a shaft can store elastically when twisted.
  2. This energy is used in the design of torsional springs, which store and release energy efficiently.
  3. The formula shows that strain energy is directly proportional to the square of the torque  and inversely proportional to the rigidity .
  4. Stiffer materials (higher ) store less strain energy for the same torque, while more flexible materials can store more energy.
  5. In practical design, the energy absorption capacity determines whether the shaft can withstand shocks and dynamic loads without failure.

Units

  • In SI system: Joules (J)
  • In CGS system: Ergs
    Since strain energy is a form of work or energy, its units are the same as mechanical work (N·m or J).

Applications

  1. Power Transmission Shafts:
    Used in engines and turbines to ensure shafts can twist without permanent deformation.
  2. Torsional Springs:
    Used in watches, measuring instruments, and automotive suspensions for energy storage and release.
  3. Couplings and Rotors:
    To resist torsional stresses and absorb energy safely during power transfer.
  4. Flywheels and Rotating Machinery:
    Helps in analyzing dynamic stresses and preventing torsional vibration.
  5. Failure Analysis:
    Strain energy helps in determining safe torque limits and avoiding fatigue or yielding.

Example

A solid circular steel shaft of diameter 40 mm and length 1 m is subjected to a torque of 200 N·m.
Given:

Moment of inertia:

Strain energy stored:

Hence, the total strain energy due to torsion in the shaft is 0.00124 Joules.

Importance of Strain Energy Due to Torsion

  • Ensures safe design of mechanical components under twisting loads.
  • Helps predict torsional stiffness and angle of twist.
  • Determines energy absorption capacity under dynamic or impact torques.
  • Used in vibration analysis of rotating systems.
  • Plays a vital role in energy-based methods like Castigliano’s theorem for deflection analysis.
Conclusion

The strain energy due to torsion is the elastic energy stored in a shaft or component when twisted by a torque within its elastic limit. It is given by the formula

and per unit volume,

This concept is fundamental in designing shafts, springs, and couplings that transmit torque efficiently without failure. Understanding strain energy due to torsion ensures that materials and components perform safely under twisting loads, maintaining both strength and elasticity.