Transverse Shear in Circular Sections — Maximum vs Average For a beam with a solid circular cross-section under transverse shear, the maximum shear stress is how many times the average shear stress over the section?

Introduction:Design under transverse shear requires knowledge of how shear stress distributes across various cross-sections. This question targets the ratio between maximum and average shear stress for a solid circular section.

Given Data / Assumptions:

• Solid circular cross-section of diameter D.
• Beam is prismatic and linearly elastic.
• Uniform shear force V across the section.

Concept / Approach:The average shear stress is V / A, where A is the cross-sectional area. For a solid circle, the shear stress distribution is parabolic, and the closed-form solution gives τmax = (4/3) * τavg.

Step-by-Step Solution:Average shear: τavg = V / A = V / (π * D^2 / 4).Using τ = V * Q / (I * b) with circular geometry yields τmax = 4/3 * τavg.Therefore, ratio = 4/3.

Verification / Alternative check:This ratio is a standard result alongside the rectangular section result (τmax = 1.5 * τavg) and thin-walled torsion approximations.

Why Other Options Are Wrong:

• equal to / twice: Do not match the parabolic distribution result.
• 1.5 times / 3/2 times: This belongs to rectangular sections, not solid circles.

Common Pitfalls:Interchanging the circular and rectangular ratios or assuming a uniform shear distribution.

Final Answer:4/3 times