What is shear stress distribution in a circular shaft?

Short Answer:

The shear stress distribution in a circular shaft refers to how shear stress varies from the center of the shaft to its outer surface when the shaft is subjected to a twisting moment or torque. The shear stress is zero at the center (axis) and increases linearly with the radius, reaching a maximum value at the outer surface of the shaft.

In simple words, when torque acts on a circular shaft, the twisting causes internal shear stresses that vary along the radius. The outermost layer carries the maximum shear stress, while the innermost layer (center) carries none. This variation helps determine the shaft’s strength and efficiency in transmitting torque.

Shear Stress Distribution in a Circular Shaft

Detailed Explanation :

When a circular shaft is subjected to torque, internal shear stresses develop within the material to resist twisting. These stresses are distributed across the cross-section of the shaft, and their variation along the radius is known as the shear stress distribution. This concept is essential in understanding the strength and behavior of shafts under torsional loading.

Definition

The shear stress distribution in a circular shaft can be defined as:

“The variation of shear stress from the center (axis) of the shaft to its outer surface when the shaft is subjected to torque.”

This distribution is not uniform — it changes linearly with the distance from the center. The maximum stress occurs at the surface, and it decreases to zero at the center.

Torsion in a Circular Shaft

Consider a circular shaft of radius  and length  subjected to a torque . When this torque acts, the shaft twists through an angle . The outer surface of the shaft moves through the maximum distance, while points closer to the axis move less.

Due to this twisting, shear strain develops at every radial position , and shear stress () is produced within the shaft material. The relationship between these parameters is obtained from the torsion equation.

Derivation of Shear Stress Distribution

The basic torsion equation for a circular shaft is:

where,

  •  = Applied torque (N·m)
  •  = Polar moment of inertia (m⁴)
  •  = Shear stress at radius  (N/m²)
  •  = Radial distance from the center (m)
  •  = Modulus of rigidity (N/m²)
  •  = Angle of twist (radians)
  •  = Length of shaft (m)

From this relation,

This shows that shear stress  is directly proportional to radius .

At the outer surface, , hence the maximum shear stress is:

Shear Stress Distribution Formula

At any point in the shaft at a distance  from the center:

Thus, shear stress increases linearly from the center (where , ) to the outer surface (where , ).

This linear variation means that every circular layer of the shaft resists torque proportionally to its distance from the center.

Polar Moment of Inertia (J)

The polar moment of inertia  depends on the cross-sectional geometry:

  • For a solid circular shaft:
  • For a hollow circular shaft:

Hence, for both solid and hollow shafts, the stress distribution follows the same principle — zero at the center and maximum at the surface.

Graphical Representation

Although not drawn here, the stress distribution can be described as follows:

  • The shear stress starts from zero at the center of the shaft.
  • It increases linearly as the radius increases.
  • The stress curve forms a straight line from the center to the outer edge.
  • At the outer surface, the stress reaches maximum value (τₘₐₓ).

This linear distribution indicates that outer layers bear the highest share of the applied torque, while inner layers contribute less.

Physical Meaning

  • The outer surface of the shaft resists most of the applied torque.
  • The center region contributes very little to torque resistance because the stress there is almost zero.
  • Hence, in engineering design, material near the center does not contribute effectively to strength — this is why hollow shafts are often preferred over solid shafts, as they save material while maintaining high strength.

Example Calculation

Let a solid circular shaft have:
, , .

Polar moment of inertia,

Maximum shear stress,

Shear stress at :

This calculation clearly shows that stress increases proportionally with radius.

Importance of Shear Stress Distribution

  1. Design of Shafts:
    Helps determine maximum shear stress and choose suitable material and diameter.
  2. Material Optimization:
    Explains why hollow shafts provide nearly equal strength as solid ones with less material.
  3. Failure Prevention:
    Ensures that the maximum shear stress does not exceed the permissible shear strength of the material.
  4. Understanding Torsional Strength:
    Provides insight into how internal layers contribute differently to torque resistance.
  5. Practical Use:
    Used in designing axles, drive shafts, crankshafts, and couplings in machines.
Conclusion

In conclusion, the shear stress distribution in a circular shaft varies linearly from zero at the center to a maximum at the outer surface. The relationship is given by , indicating that outer fibers of the shaft experience the highest stress. Understanding this distribution helps in designing efficient and safe shafts capable of transmitting torque without failure. This concept also justifies the engineering preference for hollow shafts, which provide higher efficiency with reduced weight.