T-beam with neutral axis below the slab (flange in compression): which relationship between section variables holds for equilibrium of internal forces?

Introduction / Context:
In T-beams where the neutral axis lies below the flange (slab), the flange is wholly in compression. Force equilibrium between compression in the effective flange thickness and tension in steel leads to a specific relationship linking geometric and material terms.

Given Data / Assumptions:

• B = effective flange width.
• ds = thickness of slab (flange).
• n = depth of neutral axis from the top surface.
• d = effective depth to tension steel centroid.
• At = area of tensile steel; m = modular ratio (Es/Ec).
• Neutral axis lies below the slab, so the compressive block within slab is rectangular.

Concept / Approach:
With NA below the flange, compression is confined to slab thickness ds, giving a compression force equal to stress block intensity multiplied by area. In transformed section analysis (working stress method), compressive force in concrete equals the tension in transformed steel: C = T. The lever arm is not needed for this force equilibrium statement—only force balance is used to derive the presented relation.

Step-by-Step Solution:
Transformed area of steel = m * At at NA depth n.Compression in slab portion = B * ds * (n - ds) in transformed terms consistent with linear strain assumption.Equate compression to transformed steel tension to get B * ds * (n - ds) = m * At * (d - n).

Verification / Alternative check:
Strain diagram confirms linear distribution; centroid of compression block lies at mid-thickness of ds when NA is below slab, conforming to the adopted expression for transformed equilibrium.

Why Other Options Are Wrong:

• Options a, b, c: misuse distances (d + n) or invert terms; dimensional and physical inconsistency.
• Option e: incorrect because a valid relation exists.

Common Pitfalls:
Confusing n with ds location, and mixing ultimate-limit-state rectangular stress blocks with working-stress transformed analysis without consistency.

Final Answer:
B * ds * (n - ds) = m * At * (d - n).