Euler/column basics: The equivalent (effective) length of a column of actual length L with both ends hinged is equal to:

Introduction / Context:
Effective length is the notional length of a column used in buckling calculations to account for end restraints. Recognizing effective length factors is essential for critical load predictions using Euler or design codes.

Given Data / Assumptions:

• Prismatic column of length L.
• Both ends hinged (pinned), allowing rotation but no translation.
• Elastic buckling considered (no inelastic effects).

Concept / Approach:
For a pin–pin column, the effective length factor K = 1.0. Thus the effective length Le = K * L = L. This is the reference case in most code tables, with other end conditions expressed relative to it (e.g., fixed–fixed K ≈ 0.5, fixed–free K ≈ 2.0, fixed–pinned K ≈ 0.7).

Step-by-Step Solution:
Select end condition: both ends hinged.Use K = 1.0 for pin–pin supports.Compute effective length: Le = K * L = L.

Verification / Alternative check:
Euler buckling mode shape for pin–pin is a half-sine wave over the full column length, confirming that the effective buckling half-wave equals the actual length.

Why Other Options Are Wrong:

• 2L and L/2 correspond to fixed–free and fixed–fixed analogs, not pin–pin.
• L/√2 is not a standard effective length for column end conditions.

Common Pitfalls:

• Mixing up end-condition factors, especially when lateral bracing exists at intermediate points.

Final Answer:
L.