Discussion
For a long slender column of uniform cross section, the ratio of critical buckling load for the case with both ends clamped to the case with both ends hinged is
Answer: C
The ratio is 4.
Euler's buckling load is \(P_{cr} = \pi^{2}EI/L_{e}^{2}\), where Le is the effective length. A column hinged at both ends has \(L_{e}=L\); one clamped at both ends has \(L_{e}=L/2\). So:
\[ \frac{P_{clamped}}{P_{hinged}} = \frac{L^{2}}{(L/2)^{2}} = 4 \]
A ratio of 2 compares the effective lengths rather than their squares, and 8 has no basis in the Euler formula. 1 would mean end fixity does not matter, but clamping the ends shortens the buckled half-wave and so raises the load.
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