## Q. 2.15

A prismatic column of square cross section (dimensions b × b) is subjected to a compressive load P at the top. The material of the column has specific weight w and the allowable compressive stress $\sigma_c$. Obtain a formula for the maximum permissible height h of the column, considering both the load P and the column height.

## Verified Solution

The column is sketched in Figure 2.26. The weight of the column and the applied load P are axial forces. First we write down an expression for axial stress and later we obtain permissible height of the column.

Weight of the column = Volume × Unit Weight

= b²h . w

Total load on the column = (P + b²h . w)

Stress in the column = $\sigma_c$

$\sigma_{ c }=\frac{\left(P+b^2 h w\right)}{b^2}=\frac{P}{b^2}+h w$

∴        $h=\frac{1}{w}\left\lgroup\sigma_{ c }-\frac{P}{b^2} \right\rgroup=\frac{b^2 \sigma_{ c }-P}{w b^2}$