Euler buckling comparison: What is the ratio of the Euler crippling load of a column with both ends fixed to that of an identical column with both ends hinged?

Introduction / Context:
Euler’s critical load P_cr for elastic buckling depends on the effective length L_e determined by end restraint. Increased fixity shortens L_e and raises P_cr. This question asks for the ratio of capacities for the two classic end conditions.

Given Data / Assumptions:

• Same column (E, I, length L) in both cases.
• Euler formula applies: slender, straight, elastic, ideal end conditions.
• End conditions: fixed–fixed versus hinged–hinged.

Concept / Approach:

P_cr = π^2 E I / L_e^2. For hinged–hinged, L_e = L. For fixed–fixed, L_e = L/2. Ratio is set by the square of L_e.

Step-by-Step Solution:

Verification / Alternative check:

Effective length factors: K = 1.0 (hinged–hinged), K = 0.5 (fixed–fixed). Since P_cr ∝ 1/K^2, ratio = (1/0.5^2) = 4.

Why Other Options Are Wrong:

Other ratios do not match the effective length relationship.

Common Pitfalls:

Confusing fixed–free (cantilever, weakest) with fixed–fixed (strongest).

Final Answer:

4