A pre-tensioned concrete beam has a tendon of area 100 mm2, initially tensioned to 100 kN. The cross-sectional area of concrete is 50,000 mm2 and the modular ratio is 5. What is the loss of pre-stress (in kN) due to elastic shortening?
- A0.5
- B1Correct
- C1.5
- D2
Solution
Concept
Prestress loss due to concrete elastic shortening
Formula
ΔP = m × fc × As, fc = P / Ac
Step-by-step solution
1. Initial prestressing force P = 100 kN = 100,000 N; concrete area Ac = 50,000 mm2. 2. Compressive stress in concrete at steel level fc = P / Ac = 100,000 / 50,000 = 2 N/mm2. 3. Loss of stress in steel due to elastic shortening = m × fc = 5 × 2 = 10 N/mm2. 4. Tendon cross-sectional area As = 100 mm2. 5. Total loss of prestressing force = Stress loss × As = 10 N/mm2 × 100 mm2 = 1,000 N = 1 kN. Answer: B
Explanation
Elastic shortening loss in pre-tensioned members is calculated by multiplying modular ratio by concrete stress at tendon level. The concrete stress is 100,000 N / 50,000 mm2 = 2 N/mm2, giving a steel stress loss of 10 N/mm2. Multiplying by steel area 100 mm2 gives a force loss of 1 kN.
Why the other options are wrong
A) 0.5 kN underestimates the modular ratio effect. C) 1.5 kN is an incorrect arithmetic scaling. D) 2 kN omits the tendon area proportion.
Shortcut
Loss force = m × (P / Ac) × As = 5 × (100 / 50000) × 100 = 1 kN.
Common mistake
Calculating loss in stress (10 N/mm2) and not converting to total force loss (kN) as requested.
Correct answer: B) 1