Short Answer:

Torsion is the twisting of a structural or mechanical member when it is subjected to a torque or twisting moment about its longitudinal axis. This torque causes shear stresses within the material and results in angular deformation.

In simple words, torsion happens when one end of a shaft or rod is twisted while the other end is fixed. It causes the material to experience internal shear stresses and twist through a certain angle. Torsion is commonly found in shafts, axles, and transmission systems that transmit power through rotation.

Detailed Explanation :

Torsion

In mechanical engineering, torsion is a fundamental concept used to describe the behavior of structural or machine components under twisting loads. When a torque (or moment) is applied to a member, such as a circular shaft, it tends to rotate about its longitudinal axis. The internal resistance developed within the material to oppose this twisting is called the torsional resistance, and the corresponding stress developed is known as shear stress due to torsion.

Torsion is an important factor in the design of machine parts like shafts, axles, couplings, and drive shafts, which transmit power through rotation. If the torsional strength of these components is not properly designed, they may fail due to excessive twisting or shear stress.

Definition of Torsion

Torsion can be defined as:

“The twisting of a structural member due to an applied torque which produces shear stress over the cross-section and angular deformation along its length.”

In simple terms, torsion occurs when a component is subjected to a twisting moment that causes one end to rotate relative to the other. The stress developed is shear stress, and the deformation produced is angular twist.

Example of Torsion

Consider a circular shaft fixed at one end and twisted at the other by applying a torque . The applied torque causes every cross-section of the shaft to rotate relative to the fixed end, producing shear stresses in all circular layers. The outermost layer experiences the maximum shear stress, while the stress reduces gradually toward the center.

Real-life examples include:

• Automobile drive shafts transmitting torque from the engine to the wheels.
• Wrenches and screwdrivers while tightening bolts.
• Helical springs that work under torsional loading.

Torsional Equation

The relationship between the applied torque, shear stress, and angle of twist in a circular shaft is given by the torsion equation:

Where,

•  = Applied torque (N·m)
•  = Polar moment of inertia of the cross-section (m⁴)
•  = Shear stress at a distance  from the center (N/m²)
•  = Radius of the shaft (m)
•  = Modulus of rigidity (N/m²)
•  = Angle of twist in radians
•  = Length of the shaft (m)

This equation is fundamental in analyzing torsional behavior in circular shafts.

Derivation of Torsion Equation

When a circular shaft is subjected to a torque , the shaft twists through an angle .

1. Shear Strain:
   The outermost layer experiences a maximum shear strain , and the strain at any radius  is proportional to :

1. Shear Stress:
   According to Hooke’s law for shear,

1. Torque:
   The torque on the shaft is obtained by integrating the shear stress over the entire cross-section:

 

The term  is known as the polar moment of inertia (J) of the cross-section. Hence,

1. Final Torsion Equation:

Explanation of Terms

• Shear Stress (τ): The stress developed in the shaft material due to twisting.
• Angle of Twist (θ): The total angular rotation between the two ends of the shaft due to applied torque.
• Modulus of Rigidity (G): A material property that relates shear stress to shear strain.
• Polar Moment of Inertia (J): Represents the shaft’s resistance to twisting. For a solid circular shaft:

For a hollow circular shaft:

Shear Stress Distribution in Torsion

The shear stress due to torsion varies linearly from zero at the center (axis of rotation) to a maximum value at the outer surface. The maximum shear stress is given by:

where  is the outer radius of the shaft.

This linear variation means that the outer fibers of the shaft are most stressed and, hence, most likely to fail under excessive torque.

Assumptions in Torsion Theory

1. The material is homogeneous and isotropic.
2. The cross-section remains circular and plane after twisting.
3. The twist along the shaft is uniform.
4. The shear stress varies linearly with the radius.
5. The material obeys Hooke’s law.

These assumptions ensure accurate results within the elastic limit.

Applications of Torsion

1. Transmission Shafts: Transmit power from engines to machinery.
2. Axles: Carry torque and bending moments in vehicles.
3. Springs: Helical springs store and release torsional energy.
4. Couplings: Transfer rotational motion between shafts.
5. Tools: Wrenches and screwdrivers apply torsional loads in use.

Effects of Torsion

When a member is subjected to torsion:

• Shear stresses are developed in the cross-section.
• Angular deformation (twist) occurs along the shaft length.
• If the torque exceeds the elastic limit, plastic deformation or failure may occur.

To prevent failure, engineers design shafts so that the maximum shear stress and twist remain within safe limits.

Design Considerations for Torsion

1. Strength Criterion:
   The shear stress developed must be less than the allowable shear stress of the material.
2. Rigidity Criterion:
   The angle of twist per unit length should not exceed the permissible limit to maintain alignment and efficiency.
3. Material Selection:
   Materials with high modulus of rigidity (like steel) are preferred.
4. Geometry Optimization:
   Hollow shafts are often used to save weight while maintaining strength and stiffness.

Conclusion

Torsion is the twisting action of a shaft or member caused by an applied torque, resulting in shear stress and angular deformation. The relationship between torque, shear stress, and twist is expressed by the torsion equation . Understanding torsion is essential for designing machine components such as shafts, axles, and couplings, ensuring they can safely transmit torque without excessive stress or deformation. Proper design against torsion guarantees mechanical efficiency, strength, and reliability in power transmission systems.