Discussion
Which one of the following vertical columns, of circular cross-section, sustains the highest load without buckling?
Answer: C
The strongest column is the cantilever column with a length L and cross-section diameter 2d.
Euler's buckling load is \(P_{cr} = \pi^{2}EI/L_{e}^{2}\), where \(I = \pi d^{4}/64\) for a circular section and \(L_{e}\) is the effective length: 2L for a cantilever (fixed-free) and L for a column pinned at both ends (hinge and roller).
In units of \(\pi^{2}EI_{d}/L^{2}\), where \(I_{d}\) is the second moment of area for diameter d:
- Cantilever, length L, diameter d: \(1/(2)^{2} = 0.25\).
- Pinned-pinned, length 2L, diameter d: \(1/(2)^{2} = 0.25\).
- Cantilever, length L, diameter 2d: I is 16 times larger, so \(16/4 = 4\).
- Pinned-pinned, length L, diameter d: 1.
Doubling the diameter multiplies I by \(2^{4} = 16\), which outweighs the factor of 4 lost to the cantilever's effective length.
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