For a column of height L that is fixed in position and in direction at both its top and bottom (fixed–fixed end conditions), what is its effective length used for buckling checks?

Introduction / Context:
In column buckling theory for steel and reinforced concrete members, the concept of effective length converts different end restraints into an equivalent pin–pin length for use in slenderness and Euler load calculations. When both ends of a column are fixed in position and restrained against rotation, the restraining stiffness greatly increases the buckling capacity, which is reflected by a shorter effective length than the actual height.

Given Data / Assumptions:

• Actual column height = L.
• Top end: fixed in position (no lateral translation) and direction (no rotation).
• Bottom end: fixed in position and direction as well.
• Member is straight, prismatic, and concentrically loaded until elastic buckling.

Concept / Approach:
Effective length L_eff is the notional length for which a pin–pin column would buckle in the same load as the real column with restrained ends. Classical end-condition factors (K-factors) provide L_eff = K * L. For fixed–fixed, K = 0.5, so L_eff = 0.5 * L. This smaller L_eff raises critical load and reduces slenderness ratio, λ = L_eff / r.

Step-by-Step Solution:
Identify end condition: fixed–fixed.Use K-factor: K = 0.5.Compute effective length: L_eff = K * L = 0.5 * L.Apply in design: slenderness λ = (0.5 * L) / r and P_cr = (pi^2 * E * I) / (L_eff^2).

Verification / Alternative check:
Relative comparison: for pin–pin, K = 1.0; for fixed–free (cantilever), K = 2.0; for fixed–pin, K ≈ 0.7. The fixed–fixed case appropriately gives the smallest K and thus the largest buckling load for a given L and section.

Why Other Options Are Wrong:

• L or “L (same as actual length)”: corresponds to pin–pin, not fixed–fixed.
• 2L: corresponds to a cantilever (fixed–free), which is the least stiff case.

Common Pitfalls:

• Confusing actual length with effective length.
• Mixing up end-condition descriptions (fixed vs pinned vs guided).

Final Answer:
(1/2)*L