Horizontal shear stress distribution in a beam: pick the incorrect proportionality statement for the intensity of horizontal shear at a given layer.

Introduction / Context:
In beam theory, the horizontal (longitudinal) shear stress at a layer is given by the well-known formula τ = VQ / (Ib). Understanding how τ varies with different geometric and loading quantities helps in web/flange design and shear connection detailing.

Given Data / Assumptions:

• Prismatic beam, small deflection, linear elastic behaviour.
• V = transverse shear force at the section.
• Q = first moment of area of the portion about the neutral axis for the layer considered.
• I = second moment of area about the neutral axis for the whole section.
• b = width at the layer where τ is evaluated.

Concept / Approach:
From τ = VQ / (Ib): τ is directly proportional to V and Q, and inversely proportional to I and b. Since Q = Aȳ (area above/below the layer times the distance of its C.G. from the N.A.), τ is directly proportional to both A (of the considered part) and ȳ.

Step-by-Step Solution:
Recognize direct proportionality: τ ∝ V and τ ∝ Q.Note that Q = Aȳ → τ ∝ A and τ ∝ ȳ.Recognize inverse proportionality: τ ∝ 1/I and τ ∝ 1/b.Thus, any statement claiming direct proportionality to I is incorrect.

Verification / Alternative check:
Compute τ for a rectangular section: τmax = 1.5V/(bh); here I is in the denominator, confirming inverse dependence.

Why Other Options Are Wrong:

• Shear force: τ increases with V (directly proportional).
• Area of the section (portion considered via Q): τ increases with A of the part in Q.
• Distance of C.G. from N.A.: τ increases with ȳ via Q.
• Width of beam: τ is inversely proportional to b, not directly; the option did not claim direct proportionality to width.

Common Pitfalls:
Confusing total area with the partial area in Q; ignoring the layer width variation in flanged sections; misinterpreting I's effect.

Final Answer:
moment of the beam section about its neutral axis