Cantilever with end point load – free-end deflection A prismatic cantilever beam of span L with flexural rigidity E I carries a concentrated load P at its free end. Determine the vertical deflection at the free end (use small-deflection Euler–Bernoulli theory).

Introduction:
Standard beam formulas are essential tools for quick checks in structural analysis. The cantilever with a tip load is among the most frequently referenced cases, providing a simple expression that relates load, stiffness, and span to tip deflection.

Given Data / Assumptions:

• Cantilever beam of length L.
• Flexural rigidity constant along the span: E I.
• Concentrated load P acting vertically at the free end.
• Small-deflection Euler–Bernoulli beam theory; shear deformation neglected.

Concept / Approach:

The curvature–moment relationship is d^2y/dx^2 = M(x) / (E I). For a cantilever with a tip load, the moment diagram is linear with maximum at the fixed end. Integrating twice with appropriate boundary conditions (zero deflection and slope at the fixed support) yields closed-form expressions for slope and deflection; evaluating at the free end provides the desired tip deflection.

Step-by-Step Solution:

Verification / Alternative check:

Tabulated beam tables list tip slope θ = P * L^2 / (2 * E * I) and tip deflection δ = P * L^3 / (3 * E * I), confirming the derived result.

Why Other Options Are Wrong:

Options B and C correspond to slope or different load cases; D doubles the correct value; E is the UDL deflection form, not tip load.

Common Pitfalls:

Confusing slope and deflection expressions; applying the simply supported formula; ignoring shear deformation in very deep beams (not significant for slender beams).

Final Answer:

δ = P * L^3 / (3 * E * I)