Discussion
A pin ended column of length 'L', modulus of elasticity 'E' and second moment of cross sectional area is 'I' loaded concentrically by a compressive load 'P'. The critical buckling load (Pcr) is given by :
Answer: A
The critical load is \(P_{cr} = \frac{\pi^2 EI}{L^2}\).
Euler's formula for buckling is \(P_{cr} = \frac{\pi^2 EI}{L_e^2}\), where \(L_e\) is the effective length. For a column pinned (hinged) at both ends, the deflected shape is a half sine wave over the full length, so \(L_e = L\).
Other end conditions change only \(L_e\): \(2L\) for a cantilever, \(L/\sqrt{2}\) for one end fixed and one hinged, \(L/2\) for both ends fixed. The options with \(\pi^2\) in the denominator, with \(3L^2\), or with \(\pi\) rather than \(\pi^2\) match none of these cases.
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