Shear in a transversely loaded rectangular beam When a rectangular beam is subjected to transverse loading (for example, vertical loads), on which planes does shear stress act within the beam depth?

Introduction / Context:
In strength of materials, transverse loading on beams induces both bending and shear. For rectangular sections, the distribution of shear stress through the depth is a classic result that every engineer should know when checking web shear and designing reinforcements or web thicknesses.

Given Data / Assumptions:

• Beam has a rectangular cross-section.
• Loading is transverse (producing shear force V and bending moment M).
• Material is linear-elastic; small deformation theory applies.

Concept / Approach:
The shear stress distribution in a rectangular section under transverse shear is parabolic:
tau(y) = (3/2) * (V / A) * (1 - (2y/h)^2)where y is measured from the neutral axis and h is the depth. The stress is zero at the extreme fibres (top and bottom) and maximum at the neutral axis. Therefore, shear acts on every horizontal plane, not just at a single fibre.

Step-by-Step Solution:

Verification / Alternative check:
Average shear V/A equals two-thirds of the maximum shear for a rectangle, consistent with the parabolic law (tau_max = 1.5 * V/A).

Why Other Options Are Wrong:

• Top or bottom fibre only: shear is zero there.
• Only the middle fibre: although maximum at the neutral axis, it is nonzero at adjacent planes too.
• Only on vertical planes: vertical shear stress accompanies complementary horizontal shear; the statement is incomplete.

Common Pitfalls:
Assuming uniform shear stress; forgetting that design for web shear must consider maximum at the neutral axis.

Final Answer:
Every horizontal plane across the depth (maximum at neutral axis)