Cantilever with a pure end couple: For a cantilever beam whose free end is subjected to a constant bending moment (pure couple) and no transverse load, the bending-moment diagram along the span will be a:

Introduction / Context:
Knowing the shapes of shear-force and bending-moment (BM) diagrams for standard load cases is essential for quick checks and sizing. A pure couple applied at the free end of a cantilever creates a distinctive internal action distribution.

Given Data / Assumptions:

• Cantilever, fixed at one end, free at the other.
• Applied at the free end: a constant bending moment M only; no point load or UDL.
• Linear elastic behavior, prismatic member.

Concept / Approach:
With no transverse load, the internal shear force is zero everywhere. The internal bending moment equals the applied couple everywhere (equilibrium of each section), hence it is constant along the span.

Step-by-Step Solution:
Transverse load w(x) = 0 ⇒ shear V(x) = 0 (constant zero).dM/dx = V(x) ⇒ dM/dx = 0 ⇒ M(x) is constant.Therefore, the BM diagram is a rectangle of height M extending over the full length.

Verification / Alternative check:
Section any location: the resisting end moment at the cut must balance the applied couple, giving the same magnitude everywhere.

Why Other Options Are Wrong:
Triangle/Parabola/Cubic parabola: these arise with point loads or distributed loads, not with a pure moment.Semicircle: not a standard BM shape from statics.

Common Pitfalls:
Assuming a linearly varying moment because that holds for a point load; here, with zero shear, the moment cannot vary.

Final Answer:
Rectangle.