Columns — effective length equivalence: A column of actual length l with both ends fixed is equivalent in buckling strength to a column of what length with one end fixed and the other end free (cantilever)?

Introduction / Context:
Buckling capacity depends on a column’s effective length, which incorporates end restraints. Converting between end conditions using effective length allows direct comparison of Euler loads for different support cases.

Given Data / Assumptions:

• Euler buckling for long, perfectly straight columns.
• Effective length factor K: fixed–fixed K = 0.5; fixed–free (cantilever) K = 2.0.
• Pcr = π^2 * E * I / (K * l)^2.

Concept / Approach:
Set the Euler load for the fixed–fixed column equal to that of a cantilevered column of unknown length Lc. Equality of Pcr implies equality of effective lengths.

Step-by-Step Solution:

Verification / Alternative check:
Equate Euler loads explicitly: π^2 E I / (0.5 l)^2 = π^2 E I / (2 Lc)^2 → Lc = l/4, confirming the result.

Why Other Options Are Wrong:

• l/8, l/2, l, 2l do not satisfy equality of effective lengths between the two cases.

Common Pitfalls:
Mixing K and C notations; remember C = K^2 if P = π^2 E I / (C l^2).

Final Answer:
l/4