Discussion
The effective lengths of the columns with ideal boundary conditions shown in Case-I, Case-II, and Case-III are respectively
Answer: D
The effective lengths are \(L\), \(2L\), \(L/2\).
The effective length is the length of a pin-ended column that buckles at the same load, measured between points of zero moment (inflection points) in the buckled shape:
- Hinged-hinged: the whole column is one half sine wave, so \(L_{e} = L\).
- Fixed-free: the column is half of a pin-ended half wave, so \(L_{e} = 2L\).
- Fixed-fixed: the inflection points lie at \(L/4\) from each end, leaving \(L/2\) between them, so \(L_{e} = L/2\).
The options with \(4L\) or \(L/4\) confuse the effective length with the factor in the buckling load (\(P_{cr}\) changes by 1/4 and by 4 for these cases), and \(L, L, L\) ignores the end conditions.
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