Correct Answer - Option 1 : (i), (ii) & (iii)
Explanation:
The buckling load of a given material depends on Slenderness ratio, Area of a cross-section, and Modulus of elasticity.
Buckling load:
Analysis of long column is done using Euler’s formula:
Elastic Critical stress (fcr)
\({{\rm{f}}_{{\rm{cr}}}}{\rm{\;}} = {\rm{\;}}\frac{{{{\rm{\pi }}^2}{\rm{E}}}}{{{{\rm{\lambda }}^2}}}\)
where E = Modulus of elasticity of the material, and λ = slenderness Ratio
\({\rm{\lambda \;}} = {\rm{\;}}\frac{{{\rm{l}}{{\rm{e}}_{{\rm{ff}}}}}}{{{{\rm{r}}_{{\rm{min}}}}}}\) & Imin = A × rmin2
where rmin = Minimum radius of gyration of the section, Imin = Minor principle Moment of Inertia, A = Cross-sectional Area, and Leff = It depends on the end conditions of the column section.
Eulers Load is given as
\({\rm{P\;}} = {\rm{\;}}\frac{{{{\rm{\pi }}^2}{\rm{E}}}}{{{\rm{l}}_{{\rm{eff}}}^2}}{\rm{r}}_{{\rm{min}}}^2 \times {\rm{A\;}} = {\rm{\;}}\frac{{{{\rm{\pi }}^2}{\rm{E\;Imin}}}}{{{\rm{l}}_{{\rm{eff}}}^2}}\)
Thus by Euler’s load formula,
\({\rm{Load\;}} \propto {\rm{\;}}{{\rm{I}}_{{\rm{min}}}}{\rm{\;and\;Load}} \propto \frac{1}{{{{\left( {{{\rm{l}}_{{\rm{eff}}}}} \right)}^2}}}\)
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, Imin = Minimum moment of inertia, and Le = Effective length
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End conditions
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Le
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Buckling load |
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Both ends hinged
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Le = L
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\({P_b} = \frac{{{\pi ^2}E{I_{}}}}{{L^2}}\) |
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Both ends fixed
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\({L_e} = \frac{L}{2}\)
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\({P_b} = \frac{{{4\pi ^2}E{I_{}}}}{{L^2}}\) |
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One end fixed and another end is free
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Le = 2L
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\({P_b} = \frac{{{\pi ^2}E{I_{}}}}{{4L^2}}\) |
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One end fixed and another end is hinged
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\({L_e} = \frac{L}{{\sqrt 2 }}\)
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\({P_b} = \frac{{{2\pi ^2}E{I_{}}}}{{L^2}}\) |
We know that MOI for a circular section is \(I = \frac{\pi d^4}{64}\)
