Shear stress distribution on the cross-section of the coil wire in a helical compression spring is shown in the figure. This shear stress distribution represents

Correct Answer:
The correct option is: combined direct shear and torsional shear stress along with the effect of stress concentration at inside edge of the coil wire cross-section.

Step-by-Step Explanation:

1. Understanding the Stresses in a Helical Compression Spring:
When a helical compression spring is subjected to an axial compressive load, the wire of the spring experiences three primary types of stresses:

• Torsional Shear Stress (_τt): The axial force produces a twisting moment (torque) on the wire. This torsional stress varies linearly from zero at the center of the wire to a maximum at the outer surface. The stress is positive (upward) on one side of the vertical axis and negative (downward) on the other.
• Direct Shear Stress (_τd): The axial force also causes a direct transverse shear stress across the cross-section of the wire. This stress is uniform in magnitude and direction across the entire cross-section.
• Stress Concentration due to Curvature: Because the wire is wound in a helix, the inner fiber of the coil has a smaller radius of curvature (it is shorter) than the outer fiber. This curvature causes a stress concentration that significantly increases the shear stress at the inner fiber (inside edge) of the coil wire.

2. Analyzing the Stress Distribution Diagram:
Let us examine the provided image of the wire cross-section:
- The circular boundary represents the cross-section of the coil wire.
- The vertical arrows represent the shear stress distribution across the horizontal diameter of the wire.
- On the left side (which represents the inside edge of the coil), the stress arrows are pointing upwards and have the maximum length (highest magnitude).
- On the right side (which represents the outside edge of the coil), the stress arrows are pointing downwards and have a much smaller length.
- The zero-stress transition point (where the stress profile crosses the horizontal axis) is shifted to the right of the central vertical axis.

3. Mathematical Formulation (Wahl's Factor):
To account for both the direct shear stress and the stress concentration caused by the curvature at the inside fiber, Wahl's stress factor ($K$) is used. The maximum shear stress occurring at the inner fiber is given by:

$${\tau }_{\max }=K\left(\frac{8FD}{\pi d^3}\right)$$

Where Wahl's factor ($K$) is defined as:

$$K=\frac{4C-1}{4C-4}+\frac{0.615}{C}$$

Here, $C$ is the spring index ($C=D/d$). The first term of the equation accounts for the curvature effect (stress concentration), and the second term accounts for the direct shear stress.

Conclusion:
The highly asymmetric shear stress distribution shown in the figure, where the stress is maximum at the inside fiber and minimum at the outside fiber, represents the combined direct shear and torsional shear stress along with the effect of stress concentration at the inside edge of the coil wire cross-section.