A cantilever beam of length \( L \) is fixed at one end and carries a concentrated downward load \( P \) at its midpoint together with an applied moment \( M = PL/2 \) at the free end, as shown in the figure. Neglecting the self weight of the beam, which one of the following options correctly shows the shear force diagram (SFD) and the bending moment diagram (BMD) for this beam?
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Find the SFD and BMD from the load P alone first, then add the constant moment the end couple M contributes.
A quicker route is to first fix the shear force diagram, since only one shape is physically possible, and then use it to rule out wrong bending moment shapes.
The only transverse force on this beam is $P$ at midspan, since the applied moment $M$ at the tip carries no force. So the shear must equal $P$ from the wall to midspan and drop to zero for the rest of the beam. That fact already rules out any option whose SFD stays at $P$ for the full length.
Option A: SFD matches (drops to zero at midspan), and its BMD rises from 0 at the wall to $PL/2$ at midspan, then holds flat at $PL/2$ to the tip, matching a moment $M=PL/2$ that exactly cancels the load's own fixing moment at the wall.
Option B: SFD also drops correctly, but its BMD starts at $PL$ at the wall, which would mean the applied moment adds to the load's fixing moment instead of cancelling it, contradicting the given moment.
Option C: keeps the shear at $P$ across the whole beam, with no drop at midspan, which cannot be right since there is no force beyond the load point.
Option D: has the same flawed full length shear as option C, so it fails for the same reason regardless of its bending moment shape.
Only option A keeps a physically correct shear diagram and the bending moment shape that matches the given end moment, so it is the answer.
