Question

A bar of uniform cross-section and having a length $l$ is hanging from the roof. What will be the elongation in bar due to self weight if $\gamma$ is the specific weight of the bar and $Y$ is the young's modulus of elasticity?

Answer 1

Take an element $dx$ at a distance $x$ from lower end.
$\gamma$ = Weight per unit volume of bar


Now, load experience by the element $dx$ due to self weight of the section $X-X$ below it
$P_{x-x}=\text{(Specific weight)}\times \text{(volume)}=\gamma Ax$

$\therefore {\sigma }_{x-x}=\frac{P_{x-x}}{A}=\frac{\gamma Ax}{A}=\gamma x$

Now let the elongation produced in the element $dx$ is $\delta x$

Strain $=\frac{\delta x}{dx}=\frac{{\sigma }_{x.x}}{Y}=\frac{\gamma x}{Y}$

$\Rightarrow \delta x=\frac{\gamma x}{Y}dx$

Integrating both sides,

$\Rightarrow \underset{0}{\overset{{\delta }_{\text{total}}}{\int }}{\delta }_{\text{strip}}=\underset{0}{\overset{l}{\int }}\frac{\gamma x}{Y}dx$

$\Rightarrow {\delta }_{\text{total}}={\left[\frac{\gamma }{Y}\frac{x^2}{2}\right]}_0^l=\frac{\gamma l^2}{2Y}$

Why this question? Tips:- $\delta \propto l^2$ and independent on the cross-section of bar.