Short Answer:

Torsional shear stress distribution refers to how shear stress is spread across the cross-section of a shaft when it is subjected to torque or twisting. The shear stress is zero at the center (axis) of the shaft and increases linearly towards the outer surface, where it reaches the maximum value.
This means the outer surface of the shaft experiences the highest stress, while the core remains almost free from torsional stress. Understanding this stress distribution helps in designing shafts that can safely transmit torque without failure.

Detailed Explanation :

Torsional Shear Stress Distribution

When a shaft is used to transmit torque or twisting moment, every layer of the shaft experiences shear stress due to the twisting action. This stress varies from the center of the shaft to its outer surface. The way this stress changes along the radius is known as torsional shear stress distribution. It is one of the most important topics in mechanical engineering because it helps in understanding how a shaft carries the load and where it is most likely to fail.

1. Concept of Torsion in Shafts
   Torsion means twisting of a shaft about its longitudinal axis due to an applied torque. When torque is applied, one end of the shaft rotates relative to the other. This rotation causes the material fibers of the shaft to form a helical shape, creating shear stress within the material.
   Each layer of the shaft (from the center to the surface) experiences a different amount of shear stress depending on its distance from the axis. The farther a layer is from the center, the greater the stress it experiences. This variation gives rise to the torsional shear stress distribution.
2. Shear Stress in a Circular Shaft
   Consider a circular shaft of radius subjected to torque . The shear stress at any point on the shaft, at a distance from the center, is given by the torsion equation:

where,
= Applied torque,
= Polar moment of inertia of the shaft cross-section,
= Shear stress at distance ,
= Modulus of rigidity of the material,
= Angle of twist,
= Length of the shaft.

From this equation, the shear stress can be expressed as:

This shows that the shear stress () is directly proportional to the distance  from the center. Hence, the stress is zero at the center (r = 0) and maximum at the outer surface (r = R).

3. Stress Distribution Pattern
   The distribution of shear stress across the shaft’s cross-section is linear. This means that as we move from the center toward the outer edge, the stress increases in a straight-line manner.

• At the center of the shaft:
• At the outer surface:

If we draw a diagram of this distribution, it would show a straight line starting from zero at the center and rising linearly to the maximum value at the circumference.

4. For Solid and Hollow Shafts
   For a solid circular shaft, the stress increases linearly from zero at the center to maximum at the outer radius.
   For a hollow shaft, the stress at the inner surface is not zero because there is no material at the center. In this case, shear stress starts from a finite value at the inner surface and increases linearly to a maximum at the outer surface.
   This makes hollow shafts more efficient, as the highly stressed outer region is retained while the less stressed inner portion is removed, reducing weight without reducing strength significantly.
5. Importance in Design
   Understanding torsional shear stress distribution helps engineers in designing safe and efficient shafts.

• It allows determining the maximum stress the shaft can safely handle.
• It helps in deciding the material and dimensions of the shaft based on required torque transmission.
• It aids in avoiding mechanical failures such as twisting, cracking, or fatigue.

By analyzing the stress distribution, shafts can be optimized for strength, durability, and cost.

6. Polar Moment of Inertia and its Role
   The polar moment of inertia represents the shaft’s resistance to twisting. It depends on the geometry of the cross-section:

• For a solid shaft,
• For a hollow shaft,

A higher value of  means the shaft can resist higher torque with less angular twist. Thus, by changing the cross-section or making the shaft hollow, we can control the stress distribution and improve performance.

7. Graphical Representation (Conceptual Description)
   If we plot shear stress () versus the radius ():

• The curve is a straight line starting at zero (for solid shaft) and rising linearly to the maximum value at the outer surface.
• The slope of the line represents how fast the stress increases with radius.
  This linear distribution helps engineers easily calculate stress at any radius and ensures proper design.

Conclusion:

Torsional shear stress distribution describes how shear stress varies across a shaft when it is twisted by torque. It increases linearly from the center (zero) to the outer surface (maximum). In hollow shafts, the stress starts from the inner surface and reaches maximum at the outer. Understanding this distribution is essential for designing strong and reliable shafts that can efficiently transmit power without failure. It ensures proper material utilization, safety, and high performance in mechanical systems.