In strength of materials and reinforced concrete design: How does the shear stress (shear intensity) distribute across the depth of a rectangular beam section under transverse shear?

Introduction / Context:
Understanding how shear stress varies over a beam's cross-section is fundamental for safe detailing, particularly of shear reinforcement (stirrups) in reinforced concrete and web thickness in steel beams. Rectangular sections have a characteristic distribution that affects where the maximum shear occurs and how average shear compares to peak values.

Given Data / Assumptions:

• Beam with a prismatic rectangular cross-section.
• Subjected to transverse shear due to applied loads.
• Material is linearly elastic within the range of interest.

Concept / Approach:
The general elastic theory of beams yields the shear stress at a distance y from the neutral axis as tau(y) = V * Q(y) / (I * b), where V is the shear force, Q(y) is the first moment of area above (or below) the layer, I is the second moment of area, and b is the breadth at that layer. For a rectangle, this relation simplifies to a parabolic distribution with maximum at the neutral axis and zero at the outer fibers.

Step-by-Step Solution:
Express tau(y) = V * Q / (I * b) for a rectangle.Compute Q as a quadratic function of y, giving tau(y) proportional to (1 − (2y/d)^2).Recognize the resulting curve is parabolic, peaking at the neutral axis and vanishing at the top and bottom surfaces.

Verification / Alternative check:
From the derived relation, tau_max = 1.5 * V / (b * d) at the neutral axis, while the average shear is V / (b * d). This 1.5 factor further confirms the parabolic nature (not linear) of the distribution.

Why Other Options Are Wrong:
Circular/Elliptical: No standard elastic beam theory gives these shapes for rectangles.Straight line: Would imply nonzero shear at the surface or incorrect zero at neutral axis, which is not the case.None of these: Incorrect as the correct curve is known to be parabolic.

Common Pitfalls:

• Assuming linear variation similar to bending normal stress; shear does not follow the same law.
• Using average shear instead of maximum when checking concrete shear capacity.

Final Answer:
a parabolic curve