Q$2$ Lam$$ theory can be extended to include rotation and thermal gradient.

Given, the general equations for rotation with thermal gradient for plane stress conditions:

${}_r=A-\frac{B}{r^2}-\frac{(3+v)}{8}{}^2{r}^2-\frac{E}{r^2}{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}Trdr,{}_H=A+\frac{B}{r^2}-\frac{(1+3v)}{8}{}^2{r}^2-ET+\frac{E}{r^2}{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}Trdr$

Given that the properties of the relevant steel are as follows:

$E=206\mathrm{.}8G\frac{N}{m^2},v=0\mathrm{.}31$ and $=7,468k\frac{g}{m^3},=11\mathrm{.}2\frac{{10}^{-6}}{}C.$

$($a$).$ Assuming that no pressure is applied externally or internally, determine the hoop stresses at the inside

and outside surfaces of a long thick steel cylinder which is rotating at $4,400$ RPM$.$ The cylinder has an

ID of $200mm$ and an OD of $546mm.$

$(40$ marks$)$

$($b$).$ While rotating, the thick cylinder detailed in $($a$)$ above experiences external heating resulting in a

steady temperature of $35C$ at the inside surface. Find the total hoop stress at both the inside and

outside surfaces assuming the thermal gradient is of the form $T=Kr.$

$(60$ marks$)$