Question

The figure shows a frame $ABCD$ made of rods, initially stress free. The coefficient of linear expansion of rods $AB$ , $CD$ is $7\alpha $ and for rods $BC$ , $AD$ is $2\alpha $ . Coefficient of linear expansion of rods $AO$ , $BO$ , $CO$ and $DO$ is $\gamma $ . The value of $\gamma $ for which there should be no stress in any rod at any temperature will be $n\alpha $ . Find the value of $2n$ ( $\alpha ,\gamma $ are very small).

Answer 1

$(B D)^2=(B C)^2+(D C)^2$
$z^{2} \, = \, y^{2} \, + \, x^{2}$
$2Z \, \frac{d Z}{d T}=2y \, \frac{d Y}{d T}+2x\frac{d X}{d T}$
$2\sqrt{5}l\times \sqrt{5 \, }lr=2l\left(2 l \alpha \right)+2\left(2 l\right)\left(14 l \alpha \right)$
$10r=4\alpha \, +56\alpha \, $
$r=6\alpha \, $