Cantilever under pure end moment: A cantilever of length L is subjected to a constant bending moment M applied at the free end (no shear). What is the vertical deflection at the free end?

Introduction / Context:
Different loadings on cantilevers produce characteristic slope and deflection formulas. A pure moment at the free end creates a uniform bending moment along the length, simplifying integration.

Given Data / Assumptions:

• Cantilever length L; flexural rigidity E I constant.
• Applied end moment M at the free end; no transverse load.
• Small deflection; Euler–Bernoulli beam theory (plane sections remain plane).

Concept / Approach:
The differential equation is E I * y'(x) = M (constant). Integrating twice with fixed-end boundary conditions (zero slope and zero deflection at the built-in end) yields the slope and deflection distributions.

Step-by-Step Solution:
E I y' = M → y' = M / (E I).Integrate: y' → y' dx → y' = (M / (E I)) x + C1.Boundary at fixed end x = 0: slope y'(0) = 0 ⇒ C1 = 0.Integrate: y = (M / (E I)) x^2 / 2 + C2.Boundary at fixed end x = 0: y(0) = 0 ⇒ C2 = 0.Deflection at free end: y(L) = M L^2 / (2 E I).

Verification / Alternative check:
The slope at the free end is θ(L) = M L / (E I); differentiating the deflection confirms consistency.

Why Other Options Are Wrong:
M L / (E I): that is the slope, not the deflection.Other denominators (3 E I) arise for different load cases (e.g., point load).Zero: a pure end moment definitely causes rotation and deflection.

Common Pitfalls:
Applying the point-load formulas instead of the constant-moment case.

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