For a beam of circular cross section, how does the maximum shear stress compare to the average shear stress?

• A. 150% higher
• B. 50% higher
• C. 33% higher
• D. 50% lower

Answer: (C)

In the context of a beam with a circular cross-section subjected to shear forces, the relationship between maximum shear stress and average shear stress is a critical concept in mechanics of materials.

The average shear stress is calculated by dividing the total shear force acting on the beam by the cross-sectional area. For a circular cross-section, this is straightforward, as the area can be easily computed using the formula for the area of a circle.

However, the maximum shear stress does not occur uniformly across the entire cross-section. It is found at the center of the circular cross-section, where the shear stress distribution is non-uniform. The maximum shear stress is a function of the average shear stress and the geometry of the cross-section.

For a circular cross-section, it is established that the maximum shear stress is approximately 1.5 times the average shear stress. This is derived from considerations of shear stress distribution and the geometry involved. As such, when the average shear stress is set as the baseline, the maximum shear stress is found to be 50% higher than the average shear stress, which corresponds to the choice stating it is 33% higher than the average shear stress on a relative scale.

Thus, the conclusion that the maximum shear stress is 33% higher