$(20$ Points$)$ Two cylindrical bars, of diameters $d1$ and $d2,$ are confined between two large and stiff supports such that two bars go through identical elongation. Use $LI=10cm$ and $L2=30cm$ for lengths of the two bars, and $d_1=1cm$ and $d_2=0\mathrm{.}5cm.$ Both bars are made of IN $718.$ Some relevant properties of IN $718$ can be found in one of the papers uploaded under "Files".

Young's modulus $($ Y $)$ and CTE $($coefficient of thermalexpansion$)$ for IN $718$ can be taken to be $207$ GPa and $14\mathrm{.}4\frac{{10}^{-6}}{}C,$ respectively, with a yield stress of $896$ MPa $($at $0\mathrm{.}2\%$ strain offset$).$ If needed, the stress$-$strain relationship can be approximated as

$lon=\frac{}{Y}+B(||{)}^{31}*,$ where B is $0\mathrm{.}006621$ when $$ is in MPa, in the region of interest. You may like to plot the stress$-$strain relation to get a feel of the equation.

It is observed that under a certain level of loading $($represented by the force F $),$ each bar experiences the same elongation of $0\mathrm{.}63$ mm $.$ CalculatE. Solve parts ii and iii

$($ii$)$ total load $F.$

$($iii$)$ If now the load is released, calculate the residual stresses in each bar.