Question

A long structural column (length =L) with both ends higged is acted upon by an axial compressive load P. The differential equation governing the bending of column is given by
$EI\frac{d^{2}y}{d{x}^2}=-Py$
where y is the structural lateral deflection and EI is the flexural rigidity The first critical load on column responsible for its buckling is given by

• A. $\frac{{\pi }^{2}EI}{L^2}$
• B. $\frac{\sqrt{1{\pi }^2}EI}{L^2}$
• C. $\frac{2{\pi }^{2}EI}{L^2}$
• D. $\frac{4{\pi }^{2}EI}{L^2}$

Answer 1

The correct option is A $\frac{{\pi }^{2}EI}{L^2}$

Differential equation governing the bending of column
$EI\frac{d^{2}y}{d{x}^2}=-Py$
$\frac{d^{2}y}{d{x}^2}+\frac{P}{EI}.y=0$
Solution of differential eq.is
$\Rightarrow y=Acos[\times \sqrt{\frac{P}{EI}}]+Bsin[\times \sqrt{\frac{P}{EI}}]$
when x = 0
y = 0
$\Rightarrow$ A = 0

and when
$x^{\prime }=l$
y = 0
$\Rightarrow 0=Bsin\left[l\sqrt{\frac{P}{EI}}\right]$
$l\sqrt{\frac{P}{EI}}=\pi$ (First value)
$\Rightarrow P=\frac{{\pi }^{2}EI}{l^2}$