A pressurised thin$-$walled cylindrical tank with flat ends is loaded by torque $\backslash ($ T $\backslash )$ and tensile force $\backslash ($ P $\backslash )($see figure$).$ The tank has radius of $\backslash ($ r$=125\backslash$mathrm$\{$~mm$\}\backslash )$ and wall thickness of $\backslash ($ t$=6\backslash$mathrm$\{$~mm$\}\backslash ).$ The internal pressure $\backslash ($ p$=5\backslash$mathrm$\{$MPa$\}\backslash )$ and the torque $\backslash ($ T$=20\backslash$mathrm$\{$kN$\}\backslash$cdot $\backslash$mathrm$\{$m$\}\backslash ).$

$($a$)$ What is the maximum permissible value of the force

$\backslash ($ P $\backslash )$ if the allowable tensile stress in the wall of the cylinder is $200$ MPa $?$

$($b$)$ If force $\backslash ($ P$=500\backslash$mathrm$\{$kN$\}\backslash ),$ what is the maximum acceptable internal pressure in the tank?

Hint: An explicit analytical solution to this problem might be quite challenging $-$ please use any suitable mumerical procedure $($graphical or the trial$-$and$-$error method$)$ to find the required values for a$)$ and b$).$