Strength of Materials – Cantilever with End Load A prismatic cantilever beam of length L, flexural rigidity EI, carries a concentrated load W at the free end. What is the maximum vertical deflection at the free end, expressed in terms of W, L, E, and I?

Introduction / Context:
In strength of materials, classical Euler–Bernoulli beam theory provides closed-form formulas for deflections under standard load cases. A cantilever with a point load W at its free end is a fundamental case used to validate understanding of curvature–slope–deflection relations and boundary conditions.

Given Data / Assumptions:

• Beam type: cantilever, fixed at one end and free at the other.
• Length = L, flexural rigidity = E I, end load = W.
• Small deflection theory, linear elastic material, prismatic section, plane sections remain plane, constant E and I.

Concept / Approach:

The governing relation is E I * d^2y/dx^2 = M(x). Integrate twice to obtain slope and deflection, applying cantilever boundary conditions at the fixed end: y = 0 and dy/dx = 0 at x = 0. Maximum deflection occurs at the free end x = L.

Step-by-Step Solution:

Verification / Alternative check:

Matches standard tables for end-loaded cantilevers and energy methods using Castigliano theorem.

Why Other Options Are Wrong:

W L^3 / (8 E I) and W L^3 / (48 E I) correspond to other loading and support cases. W L^2 / (2 E I) has wrong dimension. 2 W L^3 / (3 E I) overestimates by a factor of 2.

Common Pitfalls:

Mixing simply supported and cantilever formulas, dropping boundary conditions, or misplacing origin for M(x).

Final Answer:

W L^3 / (3 E I)