In strength of materials, what is the relationship between the maximum shear stress and the average shear stress for a rectangular beam cross-section?

Introduction / Context:In beam theory, shear stress is not uniformly distributed across a cross-section. For a rectangular section (common in mechanical and civil engineering), the maximum shear stress at the neutral axis differs from the cross-section's average shear stress. This question checks your recall of the exact ratio used in design and checks.

Given Data / Assumptions:

• Beam with a rectangular cross-section of breadth b and depth h.
• Transverse shear force V acts on the section.
• Material is linearly elastic and plane sections remain plane.

Concept / Approach:The average shear stress is V / A where A = b * h. The distribution for a rectangle is parabolic, with the maximum at the neutral axis. Using the shear formula, tau = V * Q / (I * b), we compare peak to average to obtain the ratio.

Step-by-Step Solution:

Verification / Alternative check:Standard beam tables list tau_max for a rectangle as 3/2 times the average, confirming the derivation.

Why Other Options Are Wrong:equal to, 4/3 times, twice, 0.866 times: do not match the rectangular section result derived from tau = V * Q / (I * b).

Common Pitfalls:Assuming uniform shear distribution; confusing the 3/2 factor with bending stress relations; applying the rectangular result to non-rectangular shapes.

Final Answer:

1.5 times