Prestressed concrete beam – stress at the top fibre due to prestress only A rectangular beam 300 mm wide by 900 mm deep is prestressed with an initial force P = 700 kN applied at midspan with an eccentricity e = 350 mm relative to the centroidal axis (eccentricity taken towards the top fibre). Determine the stress at the top of the section due to prestress alone, in N/mm² (state the sign).

Introduction:
Prestressing introduces a uniform direct stress and a bending stress due to eccentric application of the prestress. Superimposing these gives the fibre stresses. Understanding the effect of eccentricity direction is crucial for identifying whether a fibre is in compression or tension under prestress alone.

Given Data / Assumptions:

• Section: b = 300 mm, h = 900 mm.
• Prestress force P = 700 kN applied with eccentricity e = 350 mm toward the top fibre.
• Calculate stress at the top fibre only; ignore other loads.
• Elastic, uncracked section; initial condition (losses neglected).

Concept / Approach:

Total fibre stress = direct stress ± bending stress. Direct stress is uniform: P/A. Bending stress arises from moment M = P * e distributed as ±M/Z at extreme fibres. Since the eccentricity is toward the top, the bending moment compresses the top fibre and tensions the bottom fibre under prestress alone.

Step-by-Step Solution:

Verification / Alternative check:

If the eccentricity had been toward the bottom, the bending component would be tensile at the top and the net top stress would be about 3.46 N/mm² tension, demonstrating the importance of sign convention.

Why Other Options Are Wrong:

2.59 (compression) includes only the direct stress; 46 (tension) is unrealistic for given geometry; zero ignores both components; 5.00 (tension) has wrong sign and magnitude.

Common Pitfalls:

Confusing the direction of eccentricity; using incorrect section modulus; forgetting unit consistency (N/mm²).

Final Answer:

8.64 (compression)