A cantilever wooden beam is 3 m long and carries a UDL of 4 kN/m. The cross-section of the beam is 100 mm width and 200 mm depth. What is the maximum bending stress for this section ?
- A2·7 MPa
- B27 MPaCorrect
- C270 MPa
- D0·27 MPa
Solution
Concept
Flexure formula
Formula
σ = M / Z; M = w L² / 2; Z = b d² / 6
Step-by-step solution
1. M = 4 × 3² / 2 = 18 kN·m = 18 × 10⁶ N·mm. 2. Z = 100 × 200² / 6 = 6.667 × 10⁵ mm³. 3. σ = 18 × 10⁶ / 6.667 × 10⁵ = 27 N/mm² = 27 MPa.
Explanation
The maximum bending moment of a cantilever under a UDL is at the fixed end and equals wL²/2. Dividing by the section modulus gives the extreme fibre stress.
Why the other options are wrong
2.7 MPa and 0.27 MPa are 10 and 100 times too small and 270 MPa is 10 times too large, from slips in unit conversion.
Shortcut
18 kN·m ÷ 0.667×10⁶ mm³ = 27.
Common mistake
Unit slip between kN·m and N·mm.
Correct answer: B) 27 MPa