$4.$ The cylindrical pressure vessel has an outer radius of $1\mathrm{.}0$ m and a wall thickness of $8$ mm $.$ It is made from steel plates that are welded along the $\backslash (40^\{\backslash$circ$\}\backslash )$ seam $($Fig$.$ P$4).$ The pressure vessel is subjected to an intemal pressure of $5$ MPa $.[30$ pts$]$

$($a$)$ Derive stress components at the surface of the pressure vessel. $[5$ pts$]$

$($b$)$ Construct Mohr's circle based on the stress components determined in $($a$).[10$ pts$]$

$($c$)$ Determine the normal and shear stress on planes parallel and perpendicular to the weld seam. $\backslash ([5\backslash$mathrm$\{$pts$\}]\backslash )$

$($d$)$ Determine the maximum in$-$plane shear stress, the associated orientation of the element and the associated average normal stress. $[5$ pts$]$

$($e$)$ Determine the absolute maximum shear stress and the associated average normal stress. $[5$ pts$]$

Thickness of the pressure vessel made of steel: $\backslash (\backslash$mathrm$\{$t$\}=8\backslash$mathrm$\{$~mm$\}\backslash )$

Figure P$4$