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

$($a$).$ The following equations are known to be valid analytical design tools for thin discs subjected to a

thermal gradient of $T=-kr,$ where ${}_r$ and ${}_H$ represent radial and hoop stress respectively:

${}_r=A-\frac{B}{r^2}+\frac{Ekr}{3},{}_H=A+\frac{B}{r^2}+\frac{2Ekr}{3}$

Develop, showing step$-$by$-$step derivation, specific equations for both ${}_r$ and ${}_H$ for the case of a thin

disc with a central hole of inner radius $R_1$ and outer radius $R_2.$

$(38$ marks$)$