Prestressed rectangular beam under bending: A tendon with prestressing force P acts on a rectangular section of area A and section modulus Z. If the beam is subjected to a maximum bending moment M, what is the minimum extreme-fiber stress f (considering the most critical fiber) when the tendon is at or near the centroid?

Introduction / Context:Precompression from prestress counteracts tensile stresses from external bending moments. For a rectangular section with tendon near the centroid, the resultant stress at an extreme fiber equals the algebraic sum of uniform prestress P/A and bending stress M/Z with appropriate sign.

Given Data / Assumptions:

• P = prestressing force in the tendon.
• A = gross cross-sectional area.
• Z = section modulus about the relevant axis.
• M = maximum external bending moment (sagging).
• Tendon at or near centroid so eccentricity is negligible for this expression.

Concept / Approach:Stress at an extreme fiber = direct stress (from P) ± bending stress (from M). For the tension-critical extreme fiber under sagging bending, bending stress is tensile (positive tension). Prestress P/A provides uniform compression, thus algebraic combination is f = P/A − M/Z at the fiber where the bending stress would otherwise be tensile.

Step-by-Step Solution:Compute uniform compressive stress due to prestress: σp = P/A.Compute bending stress at extreme fiber: σb = M/Z (tension under sagging at soffit).Combine algebraically for the tension-critical fiber: f = σp − σb = P/A − M/Z.This gives the minimum (most negative) stress among fibers.

Verification / Alternative check:If M/Z > P/A, net tension occurs; otherwise the fiber remains in compression. This check matches standard prestress serviceability calculations.

Why Other Options Are Wrong:

• P/A + M/Z: would increase compression and does not represent the tension-critical fiber under sagging.
• M/Z − P/A: same magnitude but sign-reversed; not the “minimum” stress at tension-critical side.
• (P/A)(M/Z): incorrect dimensional form.

Common Pitfalls:Forgetting the sign convention; ignoring eccentricity when it is actually significant (then add ± P*e/Z term).

Final Answer:f = P/A - M/Z