Question

The ratio of Euler’s buckling loads of columns with the same parameters having (i) both ends fixed and (ii) both ends hinged is
1. 2
2. 4
3. 6
4. 8

Answer 1

Correct Answer - Option 2 : 4

Explanation:

The load at which column buckle is termed as buckling load. Buckling load is given by:

\({P_b} = \frac{{{\pi ^2}E{I_{}}}}{{L_e^2}}\)

where E = Young’s Modulus of Elasticity, I = Minimum Moment of Inertia, and Le = Effective length

End conditions	Le	Buckling load
Both ends hinged	Le = L	\({P_b} = \frac{{{\pi ^2}E{I_{}}}}{{L^2}}\)
Both ends fixed	\({L_e} = \frac{L}{2}\)	\({P_b} = \frac{{{4\pi ^2}E{I_{}}}}{{L^2}}\)
One end fixed and another end is free	Le = 2L	\({P_b} = \frac{{{\pi ^2}E{I_{}}}}{{4L^2}}\)
One end fixed and another end is hinged	\({L_e} = \frac{L}{{\sqrt 2 }}\)	\({P_b} = \frac{{{2\pi ^2}E{I_{}}}}{{L^2}}\)

∴ \({{(P_{b})~both~ends~fixed}\over {(P_{b})~ both~ends~hinged}}={{4\pi^2 EI\over {L^2}}\over {\pi^2 EI\over L^2}} =4\)