A prestressed concrete beam of section 120 mm × 300 mm is used over an effective span of 6000 mm to support a uniformly distributed load of 4 kN/m including the self-weight of the beam. The beam is prestressed by a straight cable carrying a force of 180 kN located at an eccentricity of 50 mm. What is the stress due to prestressing force?
- A500 N/mm²
- B50 N/mm²
- C5 N/mm²Correct
- D0.5 N/mm²
Solution
Concept
Direct (axial) stress due to prestress = P/A, uniform over the whole section.
Formula
σ = P/A ; A = b × d
Step-by-step solution
1. Area A = 120 × 300 = 36000 mm² 2. Prestressing force P = 180 kN = 180000 N 3. Direct stress σ = P/A = 180000 / 36000 4. σ = 5 N/mm² (compression)
Explanation
The prestressing force acts on the full concrete area, so the direct stress is P/A = 5 N/mm². The eccentricity of 50 mm also gives bending stress P·e/Z = 9 × 106 / 1.8 × 106 = 5 N/mm². So the direct stress is 5 N/mm².
Why the other options are wrong
A: 500 N/mm² is far too high (unit error, 100 times bigger). B: 50 N/mm² is 10 times bigger (decimal error). D: 0.5 N/mm² is 10 times smaller (decimal error).
Shortcut
180 kN / 36000 mm² = 180000/36000 = 5 N/mm².
Common mistake
Using kN with mm² without converting to N, which gives a wrong power of 10.
Correct answer: C) 5 N/mm²