A cantilever beam of length L carries a couple M at its mid-span. The shear force at quarter span from (i) the free end and (ii) the fixed end is:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 16/11/2022
Detailed Explanation
Shear force comes only from transverse forces. A pure couple contributes a moment, not a force. On a cantilever the fixed end therefore develops only a resisting moment — its vertical reaction is zero.
\( \Sigma V = 0:\; R_A = 0 \) (no transverse load).
Any section: \( V = 0 \). Bending moment: \( M \) (constant) between the fixed end and the couple, zero between the couple and the free end.
Why “(i) Zero, (ii) Zero” Is Correct
With no vertical force anywhere, the shear is zero at both quarter points — option (b).
Why the Other Options Are Wrong
- A. (i) Zero, (ii) M/(2L): There is no vertical reaction, so no shear near the fixed end either.
- C. (i) M/(2L), (ii) M/(2L): M/2L would need a vertical force somewhere — none exists.
- D. (i) M/L, (ii) M/L: M/L arises for a couple on a simply supported beam, where two reactions form the counter-couple.
Key Point
A couple alone on a cantilever: V = 0 everywhere; BM = M between the support and the couple.
Key Concepts for Students
- Only a simply supported (two-support) beam develops reactions ±M/L under a couple.
- An applied couple produces a step in the BMD.
- The fixed-end moment equals the applied couple.
Practice more: this is question 16 in Bending Moment and Shear Force MCQs – Part 2. Try the whole set and check every answer instantly.