Shear in a T-section beam\nA beam with a T-shaped cross-section is subjected to a vertical shear force F. At which location within the section does the maximum shear stress occur?

Introduction / Context:
Understanding the distribution of shear stress in non-rectangular sections such as T-sections is critical in structural and machine design. Shear flow is highest where the first moment of area about the neutral axis is large and the local thickness is small.

Given Data / Assumptions:

• Prismatic T-section, homogeneous material, linear elastic behavior.
• Vertical shear force F acting on the beam.
• Bernoulli–Euler beam theory (plane sections remain plane).

Concept / Approach:
The shear stress at a level y is τ = V * Q / (I * b), where V is shear force, Q is the first moment of area above (or below) the level, I is the second moment of area about the neutral axis, and b is the local thickness at that level. For most practical T- and I-sections, τ peaks near the neutral axis and within the web where b is small.

Step-by-Step Solution:
Locate the neutral axis (NA) of the T-section by composite area method.Note the web at the NA has minimal thickness b compared to the flange.Compute τ(y) qualitatively using τ ∝ Q / b; Q is significant near the NA, and b is the web thickness, yielding high τ there.Thus the maximum shear stress typically occurs at or very close to the NA within the web.

Verification / Alternative check:
Detailed τ distribution plots for T/I-beams consistently show a maximum at the NA in the web, decreasing toward zero at free surfaces.

Why Other Options Are Wrong:
Top/bottom extremes are free surfaces where τ → 0. The junction of web and flange can have a local change in slope of τ, but not the absolute maximum if NA lies within the web.

Common Pitfalls:
Assuming peak τ at flange–web junction by intuition; forgetting the influence of local thickness b in τ = VQ/(Ib).

Final Answer:
At the neutral axis of the section (within the web)