If one end of a hinged column is made fixed and other end free, how much is the critical load compared to the original value?

Question

Question

This question was previously asked in

BPSC Lecturer ME Held on July 2016 (Advt. 35/2014)

Answer 1

Explanation:

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

\({P_b} = \frac{{{\pi ^2}EI}}{{L_e^2}}\)
where E = Young's modulus of elasticity, Imin = Minimum moment of inertia, and Le = Effective length

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

\({P_{b1}} = \frac{{{\pi ^2}EI}}{{L_e^2}}\)------------(1)

\({P_{b2}} = \frac{{{\pi ^2}EI}}{{4L_e^2}}\)------------(2)

Dividing equation (2) by (1), we get

\(\frac{{{P_{b2}}}}{{{P_{b1}}}} = \frac{1}{4}\)

Hence the critical load will be One-fourth of the original value.