A simply supported prestressed concrete beam of effective span of 10 m carries a total distributed load of 40 kN/m. The beam is prestressed with a parabolic cable with force 2500 kN. If the beam is designed by load balancing concept, then the central dip of the cable profile will be
- A100 mm
- B200 mmCorrect
- C300 mm
- D400 mm
Solution
Concept
The load balancing concept uses the transverse component of the prestressing tendon force to neutralize an equivalent portion or all of the external transverse loads on the concrete member.
Formula
w = 8Pe / L²
Step-by-step solution
1. Identify the given parameters: Effective span L = 10 m, total distributed load w = 40 kN/m, prestressing force P = 2500 kN. 2. Apply the load balancing formula for a parabolic cable: w = 8Pe / L². 3. Substitute the known values: 40 = (8 × 2500 × e) / (10)². 4. Simplify the equation: 40 = (20000 × e) / 100 => 40 = 200 × e. 5. Solve for e: e = 40 / 200 = 0.20 m = 200 mm.
Explanation
According to the load balancing concept in prestressed concrete, the upward transverse load exerted by a parabolic cable with prestressing force P and central dip e is given by w = 8Pe/L². Equating this upward load to the total downward distributed load (40 kN/m) allows us to solve for the central dip e.
Why the other options are wrong
A: Results from incorrect calculation or power of span. C: Results from an arithmetic error in division. D: Results from missing a factor of 2 or 4 in the denominator.
Shortcut
Remember that for a parabolic cable, the upward equivalent load is w = 8Pe/L²; directly substitute and solve for e.
Common mistake
Forgetting to square the span length L in the denominator or mixing up units between kN/m and meters.
Correct answer: B) 200 mm