Deflection of a Cantilever with an End Moment For a cantilever beam of length l, flexural rigidity EI, subjected to a pure couple M applied at the free end, what is the vertical deflection at the free end?

Introduction / Context:
Deflection formulas for basic load cases are central to structural analysis and design checks. A useful case is a cantilever subjected to an end moment (no transverse load), which produces constant bending moment along the span and a characteristic rotation and deflection profile.

Given Data / Assumptions:

• Cantilever length l with fixed end and free end.
• Flexural rigidity EI is constant.
• Applied load: pure moment M at the free end; no shear load.

Concept / Approach:

Use the Euler–Bernoulli beam equation EI * y' = M(x). For a free-end applied moment, the internal bending moment is constant along the beam: M(x) = M (sign per convention). Integrating twice and applying fixed-end boundary conditions yields the slope and deflection functions.

Step-by-Step Solution:

Verification / Alternative check:

Standard beam tables list for this case: slope at free end θ = M l / (EI) and deflection at free end δ = M l^2 / (2 EI), matching the derivation.

Why Other Options Are Wrong:

• Other denominators (1, 3, 4) do not satisfy integration with the correct boundary conditions.

Common Pitfalls:

Assuming zero deflection for a pure moment; mixing up with the tip load case where δ = W l^3 / (3 EI).

Final Answer:

M l^2 / (2 EI)