Tendon profile for uniformly distributed load: For a prestressed rectangular beam carrying a uniformly distributed load over its span, which tendon profile best counterbalances the sagging bending moment diagram for economy and serviceability?

Introduction / Context:
In prestressed concrete, tendon profiles are shaped so the line of action of prestress (pressure line) counteracts the bending moments from external loads. For a uniformly distributed load (UDL), the bending moment diagram is parabolic, so an appropriately parabolic tendon profile is most efficient.

Given Data / Assumptions:

• Rectangular beam section under UDL.
• Objective is to balance moments to minimize tensile stresses and deflection.
• Standard bonded tendon behavior; no special end blocks considered.

Concept / Approach:
The pressure line locus should mirror the external moment curve for best balancing. A UDL causes maximum sagging at midspan and zero at supports. Hence the tendon should have maximum eccentricity at midspan and minimal eccentricity near supports. Geometrically, this is a parabolic profile that is concave up (convexity upward), placing the tendon lowest at midspan to produce a counteracting upward moment.

Step-by-Step Solution:
Identify moment shape from UDL: parabolic sagging, peak at midspan.Select tendon profile that gives maximum downward eccentricity at midspan: parabolic concave up.Confirm anchor elevations: tendon rises toward supports to reduce eccentricity near zero-moment zones.Conclude: a parabolic tendon with convexity upward is appropriate.

Verification / Alternative check:
Balancing load method shows that a parabolic tendon can be proportioned so that prestress induces an equivalent upward uniform load, effectively offsetting the UDL over the span.

Why Other Options Are Wrong:

• Straight profiles cannot match the parabolic moment curve efficiently.
• Parabolic with convexity downward creates the opposite eccentricity trend, increasing tension at midspan.

Common Pitfalls:
Confusing “convex up” with “convex down.” Remember: for UDL, tendon is lowest at midspan and higher at supports.

Final Answer:
Parabolic with convexity upward