The shear stress at the center of a circular shaft under torsion is zero.

Explanation:
When a circular shaft is subjected to torsion, it experiences twisting deformation along its length. Torsion is a type of shear stress, where a force is applied tangentially to the cross-section of the shaft, causing it to twist.

To understand why the shear stress at the center of the shaft is zero, let's consider the distribution of shear stress across the cross-section of the shaft.

1. Shear Stress Distribution:
When a shaft is subjected to torsion, the shear stress distribution follows a linear pattern. It is maximum at the outer surface of the shaft and decreases linearly towards the center. In other words, the shear stress is highest at the outermost fibers and decreases as we move towards the center.

2. Circular Cross-section:
In a circular cross-section, the center of the shaft is the point farthest from the outer surface. As per the shear stress distribution, the shear stress decreases as we move away from the outer surface towards the center. Therefore, at the center of the shaft, the shear stress becomes zero.

3. Symmetry:
Another reason why the shear stress is zero at the center is the symmetry of the circular shaft. In torsion, the twisting deformation is uniform around the centerline of the shaft. Since the shear stress distribution is linear and symmetric about the centerline, the shear stress at the center must be zero.

4. Mathematical Analysis:
The mathematical analysis of torsion in a circular shaft also confirms that the shear stress at the center is zero. By applying the principles of mechanics and using the torsion formula, it can be derived that the shear stress at the center is zero.

In conclusion, the shear stress at the center of a circular shaft under torsion is zero due to the linear distribution of shear stress and the symmetry of the shaft.