A cantilever beam of 4 m span carries three point loads of 1 kN, 2 kN and 2 kN at 2 m, 3 m and 4 m from the fixed end respectively. What is the bending moment at 3 m from fixed support?
- A4 kNm (Anticlockwise)
- B6 kNm (Clockwise)
- C3 kNm (Clockwise)
- D2 kNm (Anticlockwise)Correct
Solution
Concept
BM at a section of a cantilever is the sum of moments of all loads on the free-end side of that section.
Formula
Mx = Σ Wi × (ai − x) for loads beyond the section
Step-by-step solution
1. Section is at 3 m from the fixed end. Take the free (right) side. 2. Loads beyond 3 m: only the 2 kN load at 4 m. The 2 kN load at 3 m lies on the section and gives zero moment. 3. Lever arm = 4 − 3 = 1 m. 4. M = 2 × 1 = 2 kNm (hogging). 5. Only option D has magnitude 2 kNm, so the answer is D.
Explanation
For a cantilever, take moments of the loads on the free side of the section. Only the 2 kN load at the free end acts beyond 3 m, at 1 m distance. So BM = 2 kNm, which is hogging.
Why the other options are wrong
A: 4 kNm is wrong; it wrongly adds loads. B: 6 kNm is wrong; it is the BM at 2 m from the free end region, or wrongly includes extra loads. C: 3 kNm is wrong; it has the wrong magnitude.
Shortcut
BM at a section = load beyond it × its distance. Here 2 × 1 = 2 kNm.
Common mistake
Including the load at 3 m (it has zero arm) or loads on the fixed-end side.
Correct answer: D) 2 kNm (Anticlockwise)