For a rectangular cross-section under a transverse shear force F, determine the maximum shear stress at the neutral axis in terms of F and the cross-sectional area A.

Introduction / Context:
The shear stress distribution in a rectangular section is parabolic, with the maximum at the neutral axis. Knowing the factor relating τ_max to the average shear stress F/A is important for beam design under shear.

Given Data / Assumptions:

• Cross-section: rectangle of breadth b and depth d (area A = b*d).
• Shear force: F at the section.
• Small-deflection, linear elasticity.

Concept / Approach:
The general shear formula is τ = V*Q / (I*b), where V = F, Q is the first moment of area about the neutral axis for the portion above (or below) the level considered, I is the second moment of area, and b is the width at that level. At the neutral axis of a rectangle, Q = (b*(d/2)*(d/4)) = b*d^2/8, I = b*d^3/12, and b is the full breadth.

Step-by-Step Solution:
1) For a rectangle: I = b*d^3/12, A = b*d.2) At the neutral axis: Q = b*(d/2)*(d/4) = b*d^2/8.3) τ_max = V*Q / (I*b) = F*(b*d^2/8) / ( (b*d^3/12)*b )? Use the correct formula τ = F*Q/(I*b) with b as breadth at NA = b.4) Simplify: τ_max = F * (b*d^2/8) / ( (b*d^3/12) * 1 ) = F * (12 / 8) * (1 / d*b) = (3/2) * F / (b*d) = 1.5 * (F/A).

Verification / Alternative check:
Average shear = F/A; the parabolic distribution in a rectangle peaks at 1.5*(F/A) at the center and is zero at the top/bottom fibers.

Why Other Options Are Wrong:

• F/A: This is the average, not the maximum.
• 2*(F/A): Overestimates by 33%.
• 0.5*(F/A): Underestimates; corresponds to no standard location.

Common Pitfalls:

• Using τ = F/A as a design check in rectangular beams; must use τ_max = 1.5*(F/A).
• Forgetting that τ varies parabolically, not uniformly.

Final Answer:
1.5 * (F / A).