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?
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Ah = (W * L) / (4 * P)
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Bh = (W * L) / (6 * P)
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Ch = (W * L) / (8 * P)
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Dh = (W * L) / (12 * P)
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Eh = (W * L^2) / (8 * P)
Answer
Correct Answer: h = (W * L) / (8 * P)
Explanation
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:
Set tendon-induced equivalent upward action equal to applied midspan load.Use the standard relation for a point load case: h = (W * L) / (8 * P).Solve symbolically to obtain the required minimum dip.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)