Question $3$: $($Optional$,25$ marks$)$ Choose EITHER this question or question $4.$

An inspection cover plate is required for a pipe network system at a chemical processing plant. The anticipated geometry of the joint is shown in Figure $3.$ The cover plate is made from cast iron with a Young's Modulus of $E_1=100$GPa and a thickness of $20$ mm $,$ while the pipe flange, washers and bolts are all made of carbon steel with a Young's Modulus of $E=207$GPa. The pipe flange is $18$ mm thick, and the washers are $2$ mm thick. The internal pressure will vary cyclically from zero to $4\mathrm{.}0$ MPa and will act on an area defined by a gasket of diameter $A=280mm.$ Initially it is estimated that the bolts will be M$1055mm,$ grade $8\mathrm{.}8.$ The joint may be considered to be temporary and washers are used against the nut face only. The mechanical properties of the bolts may be determined from Table $1$ at the end of the question paper. Assume $D=1\mathrm{.}5d$ for member $1$ and for the washer and $=30.$

$($a$)$ Showing all necessary steps, determine the bolt stiffness $k_b.$

$(5$ marks$)$

$($b$)$ Showing all necessary steps, determine the member stiffness $k_m$ and comment on whether the bolt size chosen is suitable.

$(6$ marks$)$

$($c$)$ The geometry of the joint is now modified slightly and an M$8$ grade $8\mathrm{.}8$ bolt is selected, which results in a member stiffness of $k_m=8\mathrm{.}54214{10}^8\frac{N}{m}$ and a bolt stiffness of $k_b=2\mathrm{.}44170{10}^8\frac{N}{m}.$ Showing all necessary steps, and using this information for all subsequent calculations, determine a practical number of bolts that would be required in the joint $($assume a factor of safety $n_f=2).$

$(7$ marks$)$

$($d$)$ Determine the tensile stress in a bolt after installation, and also when the system pressure reaches $4\mathrm{.}0$ MPa. At what system pressure would the joint separate? Show that the pressure will be released from the vessel before the bolts will break.

$(7$ marks$)$