A post-tensioned pre-stressed concrete beam (300 mm × 600 mm) is subjected to an effective pre-stressing force of 1,200 kN with an eccentricity of 100 mm below the centroid. Neglect self-weight and external loads. What is the maximum compressive stress at the bottom fibre due to pre-stressing?
- A10.00 MPa
- B11.11 MPa
- C12.22 MPa
- D13.33 MPaCorrect
Solution
Concept
Bottom fibre stress in prestressed beam
Formula
σbottom = P/A + (P·e)/Z
Step-by-step solution
1. Cross-sectional area A = 300 mm × 600 mm = 180,000 mm2. 2. Section modulus for bottom fibre Z = b × D2 / 6 = 300 × (600)2 / 6 = 18 × 106 mm3. 3. Direct compressive stress = P / A = 1,200,000 N / 180,000 mm2 = 6.67 MPa. 4. Bending stress at bottom fibre = P × e / Z = (1,200,000 N × 100 mm) / (18 × 106 mm3) = 6.67 MPa. 5. Total bottom fibre compressive stress = 6.67 + 6.67 = 13.33 MPa. Answer: D
Explanation
The bottom fibre stress in a prestressed concrete beam without external load is the sum of direct compressive stress P/A and flexural compressive stress P·e/Z. Substituting the given beam dimensions and eccentricity yields a total compressive stress of 13.33 MPa.
Why the other options are wrong
A) 10.00 MPa underestimates the flexural component of prestress. B) 11.11 MPa uses an incorrect section modulus. C) 12.22 MPa is an intermediate calculation error.
Shortcut
When eccentricity e = D/6, bottom fiber stress is exactly twice the direct stress: 2 × (P/A) = 13.33 MPa.
Common mistake
Subtracting the eccentricity term instead of adding it at the bottom fiber where the tendon is eccentrically located.
Correct answer: D) 13.33 MPa