Parabolic tendon in a prestressed beam: If the tendon carries prestressing force P, has maximum dip h at midspan, and the effective span is L, what is the equivalent upward uniform load produced by the tendon curvature?

Introduction / Context:
Curved prestressing tendons exert an equivalent transverse load on the concrete member. For a parabolic profile (common in simply supported members), this load can be represented as a uniform upward load that counterbalances part of the downward external loads, reducing midspan moments and improving service performance.

Given Data / Assumptions:

• Parabolic tendon with maximum eccentricity (dip) h at midspan.
• Prestressing force P (assumed constant along the span, neglecting friction/anchorage for simplicity).
• Effective span L; small deflection theory applies.

Concept / Approach:
The equivalent load per unit length induced by a tendon profile y(x) is w = P * d²y/dx². For a symmetric parabola with zero eccentricity at supports and maximum h at midspan, y(x) = 4h * x * (L - x) / L². The second derivative is constant, giving a uniform equivalent load.

Step-by-Step Solution:
Let y(x) = 4h * x * (L - x) / L².Compute dy/dx and then d²y/dx²; result: d²y/dx² = 8h / L².Equivalent upward load: w = P * d²y/dx² = 8 * P * h / L².

Verification / Alternative check:
Check units: P has units of force, h/L² yields 1/length, so w has force/length as required. Also, the total equivalent upward force w * L equals 8Ph/L, which, combined with the parabolic distribution, reproduces the same internal effect as the curved tendon.

Why Other Options Are Wrong:

• 4Ph/L² and 6Ph/L² underpredict the curvature effect.
• P/(hL) and 2P/L have incorrect functional dependence or units.

Common Pitfalls:
Using first derivative instead of second derivative, applying the formula to non-parabolic profiles, or forgetting that friction and anchorage can change P and thus w along the span.

Final Answer:
w = 8 * P * h / L^2.