Transverse shear – comparing cross-sections of equal area The ratio of maximum shear stress in a rectangular beam to that in a circular beam of the same cross-sectional area is:

Introduction / Context:
Transverse shear stress distributions depend on cross-sectional shape. For beams of equal area under the same shear force V, maximum shear differs by shape. Knowing these ratios helps in preliminary sizing and in selecting efficient sections.

Given Data / Assumptions:

• One beam has a rectangular cross-section; the other is circular.
• Both have the same area A and carry the same shear force V.
• Elastic theory of shear in beams applies; small deformations.

Concept / Approach:
Maximum shear stress for a rectangle is τ_max(rect) = 1.5 * V / A. For a circle, τ_max(circ) = 4/3 * V / A. Taking the ratio eliminates V and A, leaving a pure number.

Step-by-Step Calculation:
τ_max(rect) = (3/2) * V / A.τ_max(circ) = (4/3) * V / A.Ratio = τ_max(rect) / τ_max(circ) = (3/2) / (4/3) = (3/2) * (3/4) = 9/8.

Verification / Alternative check:
Numerical example: choose V = 1 and A = 1. Then τ_max(rect) = 1.5 and τ_max(circ) = 1.333..., giving 1.125 = 9/8, confirming the ratio.

Why Other Options Are Wrong:
2/3 and 3/4 are less than 1 and contradict known results; 1 would imply identical maxima; 4/3 in the choices is the circular multiplier itself, not the ratio asked.

Common Pitfalls:
Comparing shapes with unequal areas; confusing average shear V / A with maximum shear.

Final Answer:
9 / 8