A cantilever of span l has a load P at the free end. The bending moment at the free end is:
Try it first: pick an option, then read the explanation.
Correct answer: A. 0
Asked in SSC JE 2008 / Rajasthan JE 2015
Detailed Explanation
The bending moment at a section is load × lever arm about that section. At the free end, the load acts at the section itself.
\[ M_x = -P\,x \Rightarrow M_{x=0} = 0,\quad M_{x=l} = -Pl \]
Why “0” Is Correct
The lever arm is zero, so the moment at the free end is 0 — option (a). The maximum, −Pl, occurs at the fixed end.
Why the Other Options Are Wrong
- B. Pl: Pl is the magnitude at the fixed end, not the free end.
- C. −Pl: −Pl (hogging) is the fixed-end moment.
- D. Pl/2: Pl/2 is the moment at mid-span.
Key Point
Free end of a cantilever: BM = 0 (unless a couple acts there).
Key Concepts for Students
- Cantilever + end load: triangular BMD, rectangular SFD.
- Maximum BM at the fixed end = Pl.
- Free-end deflection = Pl³/3EI.
Practice more: this is question 52 in Bending Moment and Shear Force MCQs – Part 4. Try the whole set and check every answer instantly.