When a rod is healed but prevented from expanding, the stress developed is independent of

${\mathrm{l}}_{\mathrm{t}}={\mathrm{l}}_0(1+\mathrm{\alpha t})$
The extension in length is given by $1={\mathrm{l}}_{\mathrm{t}}-{\mathrm{l}}_0={\mathrm{l}}_0\mathrm{\alpha t}$
If the same rod of length 1o is subjected to stress along its length, then extension in length can be calculated by Hooke's law.
$\mathrm{Y}=\frac{\text{ stress }}{\text{ strain }}=\frac{\text{ stress }}{\left(\mathrm{l}/{\mathrm{l}}_0\right)}=\frac{{\mathrm{l}}_0\text{ stress }}{\mathrm{l}}$
$\therefore \mathrm{l}=\frac{{\mathrm{l}}_0\text{ stress }}{\mathrm{Y}} ....\left(2\right)$
If the rod is prevented by expanding, we have
                    ${\mathrm{l}}_0\mathrm{\alpha t}=\frac{{\mathrm{l}}_0\text{ stress }}{\mathrm{Y}}$
$\therefore$Stress = $\mathrm{Y} \mathrm{\alpha } \mathrm{t}$ (independent of I_o)