Prestressed rectangular member with a concentric tendon: If a tendon is prestressed by force P through the centroidal longitudinal axis of a prism with area A, what is the uniform compressive stress induced in the concrete immediately after transfer (neglecting self-weight and losses)?

Introduction / Context:In concentric prestressing, a tendon passes through the centroid. The prestressing force is therefore applied along the centroidal axis, causing a uniform precompression across the section with no bending moment at transfer.

Given Data / Assumptions:

• Rectangular prismatic member with cross-sectional area A.
• Tendon stressed to force P and anchored concentrically.
• Self-weight and time-dependent losses neglected at the instant of transfer.

Concept / Approach:Direct stress = Force / Area. With a concentric force, there is no eccentricity; hence no additional bending stress. The entire section is uniformly compressed by magnitude P/A.

Step-by-Step Solution:Identify area A and prestress force P.Compute uniform compressive stress: sigma_c = P / A.Confirm eccentricity e = 0, so bending stress M/Z = 0.

Verification / Alternative check:Stress distribution is rectangular and uniform; strain compatibility ensures uniform shortening. Any deviation from concentricity would add bending: sigma = P/A ± M/Z; with M = Pe = 0 here.

Why Other Options Are Wrong:

• P/(2A) and 2P/A are incorrect scalings.
• PA and A/P are dimensionally wrong for stress.

Common Pitfalls:Ignoring eccentricity in non-concentric layouts, or forgetting to adjust for immediate losses in real designs; however, the ideal concentric case remains P/A.

Final Answer:P / A.