Beam of triangular section (base horizontal) Where does the maximum transverse shear stress occur?

Introduction / Context:
Shear stress in beams is not uniform and depends on the first moment of area Q at the level considered. For non-rectangular sections such as triangles, understanding where the maximum shear occurs is important for checking web stresses and potential shear failures.

Given Data / Assumptions:

• Triangular cross-section with base horizontal; width varies linearly with height.
• Beam subjected to a transverse shear force V.
• Elastic theory applies; small deformations.

Concept / Approach:
Shear stress at a level y from a reference is τ = V Q / (I b), where Q is first moment of the area above (or below) the level about the neutral axis, I is second moment of area, and b is width at that level. For a triangle, b varies linearly with y, and Q is a quadratic function of y, so τ becomes a cubic-like function with a single maximum within the depth.

Step-by-Step Reasoning (qualitative):

Verification / Alternative check:
Carrying out the full τ(y) = V Q / (I b) derivation with b(y) linear in y and integrating to find Q(y) leads to a stationary point at mid-depth, which is the maximum τ.

Why Other Options Are Wrong:
Apex and base are extreme fibers where τ = 0; centroid is not the location of maximum τ for a triangle; one-third from base is not the maximum for this shape.

Common Pitfalls:
Assuming maximum shear always occurs at the neutral axis; for triangles, the neutral axis is at h/3 from the base, but τ_max is at mid-depth, not at the centroidal axis.

Final Answer:
At mid-height of the section