For a column of length L with flexural rigidity EI and the loaded end free (cantilever), the Euler critical buckling load is derived using the effective length concept.

Introduction / Context:
Euler buckling theory provides the theoretical elastic buckling load for slender columns. The critical load depends strongly on end conditions through the effective length factor, which modifies the column length in the formula.

Given Data / Assumptions:

• Column length = L.
• Flexural rigidity = EI.
• One end fixed, other end free (cantilever).
• Column is straight, prismatic, centrally loaded, and remains elastic up to buckling.

Concept / Approach:
Euler's formula for critical load is P_cr = (pi^2 * EI) / (L_eff^2). The effective length L_eff is obtained from boundary conditions. For a cantilever (fixed–free), L_eff = 2L.

Step-by-Step Solution:
Identify end condition: fixed–free (cantilever)Use effective length: L_eff = 2LApply Euler formula: P_cr = (pi^2 * EI) / (L_eff^2)Compute: P_cr = (pi^2 * EI) / (2L)^2 = (pi^2 * EI) / (4 * L^2)

Verification / Alternative check:
Relative to a pinned–pinned column (L_eff = L), a cantilever has one-fourth the buckling load for the same L and EI, aligning with the factor 1/4 in the denominator.

Why Other Options Are Wrong:

• (pi^2 * EI) / (L^2): correct for pinned–pinned, not fixed–free.
• (pi^2 * EI) / (2 * L^2): would imply L_eff = sqrt(2) * L, which is not a valid end condition.
• (pi^2 * EI) / (16 * L^2): corresponds to L_eff = 4L, not applicable here.

Common Pitfalls:

• Using actual length instead of effective length.
• Confusing fixed–free with fixed–pinned or fixed–fixed cases.

Final Answer:
(pi^2 * EI) / (4 * L^2)