For a cantilever beam of length L subjected to a pure couple M applied at the free end, what is the vertical deflection at the free end in terms of E and I?

Introduction / Context:
Beam deflection under various boundary conditions is a staple in structural analysis. A cantilever subjected to a tip moment M experiences constant bending moment along its length, leading to a characteristic deflection and rotation profile.

Given Data / Assumptions:

• Cantilever length L, modulus E, second moment I.
• Pure end couple M at the free tip; no shear or point load.
• Small-deflection, linear elastic theory applies.

Concept / Approach:
Euler–Bernoulli beam theory: d^2y/dx^2 = M(x)/(E I). For a tip couple, M(x) = M is constant. Integrate twice with boundary conditions at the fixed end (deflection and slope zero at x = 0).

Step-by-Step Solution:
1) Curvature: d^2y/dx^2 = M / (E I).2) Integrate once: dy/dx = (M / (E I)) * x + C1.3) Fixed end slope at x = 0 ⇒ C1 = 0.4) Integrate again: y = (M / (E I)) * x^2 / 2 + C2.5) Fixed end deflection at x = 0 ⇒ C2 = 0.6) Free end deflection: y(L) = M * L^2 / (2 * E * I).

Verification / Alternative check:
Dimensional check: M has units of forcelength; multiplying by L^2 and dividing by EI (force/area * length^4) yields length, consistent for deflection.

Why Other Options Are Wrong:
M * L / (E I) is the rotation at the free end, not the deflection.Formulas with 1/3 or L^3 apply to different loading (uniform loads or tip forces), not a pure couple.Zero contradicts bending theory.

Common Pitfalls:
Mixing tip load and tip moment cases; forgetting fixed-end boundary conditions.

Final Answer:
M * L^2 / (2 * E * I)