Question

The bending moment diagram for a cantilever beam subjected to a couple at the free end would be a

• A. Triangle
• B. Parabola
• C. Cubic parabola
• D. Rectangle

Hint:

For a cantilever beam: \[ \boxed{ \begin{array}{c|c} \text{Loading} & \text{BMD Shape}\\ \hline \text{End Couple} & \text{Rectangle}\\ \text{Point Load} & \text{Triangle}\\ \text{UDL} & \text{Parabola} \end{array} } \]

Answer 1

The Correct Option is D

Step 1: Recall what a couple does to the shear force.
A pure couple applied at the free end of a cantilever does not add any transverse force, so the shear force stays zero all along the beam.
Step 2: Connect shear to the slope of the bending moment diagram.
Since $\dfrac{dM}{dx} = V = 0$ everywhere, the bending moment cannot change from one section to the next.
Step 3: Conclude the shape of the diagram.
A bending moment that stays exactly the same value along the whole length plots as a flat line at constant height, which is a rectangle.
\[ \boxed{\text{Rectangle}} \]