III. $[$total: $15\backslash \%]$ A thin$-$walled steel cylindrical vessel with two semi$-$spherical shells at its end has an inner diameter $\backslash ($ D$=800\backslash$mathrm$\{$~mm$\}\backslash )$ and a wall thickness $\backslash ($ t$=5\backslash$mathrm$\{$~mm$\}\backslash ).$ High$-$pressure gas was inlet inside. The vessel is subjected to a pair of concentrate forces $\backslash ($ F $\backslash ),$ and an interior pressure $\backslash ($ p $\backslash ),$ as shown in Figure $2.$ The ends of the tube are closed with a semi$-$spherical shell. Assume the material of the pressure vessel is in a plane$-$stress state $($i$.$e$.,$ the normal stress along with radius direction is $0).$ The yield stress of the steel alloy in the simple tensile test is $\backslash (\backslash$sigma$\_\{\backslash$mathrm$\{$Y$\}\}=220\backslash$mathrm$\{$MPa$\}\backslash ).$

$($a$)$ If $\backslash ($ F$=0\backslash )$ and inner pressure exist, find the principal stresses of elements at point $\backslash ($ A $\backslash )($in the cylindrical section$)$ and point $\backslash ($ B $\backslash )($in the semi$-$spherical section$).$ Assume the elements are in a plane$-$stress state.

Hint: You can check Reference $3.[4\backslash \%]$

$($b$)$ If $\backslash ($ F$=0\backslash ),$ find the pressure $\backslash ($ p$\_\{$y$\}\backslash )$ under which the vessel begins to yield according to Tresca criterion.

$($c$)$ If $\backslash ($ F$=0\backslash ),$ find the pressure $\backslash ($ p$\_\{$y$\}\backslash )$ under which the vessel begins to yield by von Mises criterion. $[3\backslash \%]$

$($d$)$ If $\backslash ($ F$=100\backslash$mathrm$\{$kN$\}\backslash ),$ find the pressure $\backslash ($ p$\_\{$y$\}\backslash )$ under which the vessel begins to yield according to von Mises criterion. $[5\backslash \%]$