Prestressed concrete—bent tendon for point-load balancing: If a bent (deviated) tendon is required to balance a central concentrated load W on a span L, what is the minimum central dip h in terms of prestressing force P?

Introduction / Context:

In prestressed concrete, profiling a tendon generates equivalent upward forces that can partially or fully balance external loads. For a central point load, a bent/parabolic tendon with a central sag produces an upward component that can be tuned via its dip h and prestressing force P.

Given Data / Assumptions:

• Simply supported span of length L with a concentrated midspan load W.
• Prestressing force magnitude P available.
• Idealized tendon profile creating vertical components at deviators and midspan.

Concept / Approach:

For a parabolic/equivalent profile balancing a uniform load, w_b = 8 * P * h / L^2. For a concentrated midspan load, the balancing condition translates to equating the tendon-induced upward effect to W, leading to the classical result h = (W * L) / (8 * P).

Step-by-Step Solution:

Verification / Alternative check:

Dimensional check: W (force) * L (length) / P (force) gives length, consistent with h. Practical values yield reasonable sags.

Why Other Options Are Wrong:

• Divisors 4 or 6 or 12 lead to excessive or insufficient dip relative to the known standard relation.
• (W * L^2) / (8 * P) has wrong dimensions (length^2).

Common Pitfalls:

• Applying the uniform-load balancing formula directly without adapting to the point-load case.

Final Answer:

h = (W * L) / (8 * P)