Question

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?

• A.

  

• B.

  

• C.

  

• D.

  

Hint:

Find the SFD and BMD from the load P alone first, then add the constant moment the end couple M contributes.

Answer 1

The Correct Option is A

Step 1: Set up the free body using the load P alone.
Take the fixed wall as \( x=0 \) and the free end as \( x=L \), with the downward load \( P \) at \( x=L/2 \).
For a cantilever with a point load at midspan, the shear stays at \( P \) from the wall up to the load, then drops to zero beyond it, since nothing pushes the beam past that point except the end moment.

Step 2: Get the bending moment due to P alone.
Cutting at any \( x \) between \( 0 \) and \( L/2 \) and looking at the segment beyond the cut, the moment needed to balance \( P \) grows linearly from \( 0 \) at \( x=L/2 \) to \( PL/2 \) at the wall \( (x=0) \).
Beyond the load point, from \( x=L/2 \) to \( x=L \), there is no more transverse force, so this part of the moment stays at \( 0 \).

Step 3: Add the effect of the applied end moment \( M=PL/2 \).
A pure moment applied at the tip adds no shear anywhere, since it has no force component, so the SFD stays exactly as in Step 1: \( P \) from \( 0 \) to \( L/2 \), then \( 0 \) to \( L \).
A pure end moment does add a constant bending moment across the whole span, and here it is sized and directed so it exactly cancels the \( PL/2 \) moment that the load alone creates at the wall.

Step 4: Add the two bending moment contributions.
At the wall, \( PL/2 \) (from \( P \)) minus \( PL/2 \) (from \( M \)) gives \( 0 \).
At the load point and beyond, the \( P \)-only moment is already \( 0 \), so only the constant \( PL/2 \) from \( M \) remains, all the way to the tip.

Final Answer:
The bending moment rises from \( 0 \) at the wall to \( PL/2 \) at midspan, then stays flat at \( PL/2 \) to the free end, while the shear stays \( P \) up to midspan and \( 0 \) after, matching option A. \[ \boxed{\text{Option A}} \]