Question

A column of height h with a rectangular cross-section of size a x 2a has a buckling load of P. If the cross-section is changed to 0.5a x 3a and its height changed to 1.5h, the buckling load of the redesigned column will be

• A. P/12
• B. P/4
• C. P/2
• D. 3P/4

Answer 1

The correct option is A P/12

Case-1 :
$P_{cr}=\frac{{\pi }^{2}E{I}_{min}}{l_{eff}^2}$
$I_{min}=\frac{2a(a{)}^3}{12}=\frac{a^4}{6}$
$l_{eff}=h^2$
$\therefore P_{cr1}=P=\frac{{\pi }^{2}E\frac{a^4}{6}}{h^2}=\frac{1}{6}\frac{{\pi }^{2}E{a}^4}{h^2}$
$\frac{{\pi }^{2}E{a}^4}{h^2}=6P$

Case-2 :
$P_{cr}=\frac{{\pi }^{2}EI}{L_{eff}^2}$
$I_{min}=\frac{3a(.5a{)}^3}{12}=\frac{a^4}{32}$
$l_{eff}=(1.5h{)}^2=\frac{9}{4}{h}^2$
$\therefore P_{cr2}=\frac{{\pi }^{2}E\frac{1}{32}{a}^4}{\frac{9h^2}{4}}=\frac{1}{72}\frac{{\pi }^{2}E{a}^4}{h^2}$
$P_{cr2}=\frac{1}{72}\times 6P=\frac{P}{12}$
$P_{cr2}=\frac{P}{12}$