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

A plate is to be bolted to a support structure using one of two configurations using $3$ bolts, located at A $,$ B and C$,$ or P$,$ Q and R$,$ as shown in Figure $4.$ The plate will experience a constant force F at an angle of $30$ to the horizontal. The bolts to be used are M$16,$ grade $8\mathrm{.}8$ bolts. The plates in the figure are $16$ mm thick and $140$ mm wide, and are made from mild steel with a yield strength $S_y=200$MPa. The bolts and the plate are to have a minimum static factor of safety of $2.$ Assume that the bolts completely fill the holes in the plates.

Dimensions in mm$.$ Not to scale.

Figure $4$: Bolted joint configuration

$($a$)$ Annotate the diagrams in the answer sheets showing: the bolt group centroid; the direction of the vectors for the primary and secondary shear forces; and the resultant shear force acting on each bolt, in each case.

$(5$ marks$)$

$($b$)$ Identify the bolt $(A,B$ or C $)$ with the highest resultant shear force for configuration $1.$ Determine expressions for the primary and secondary shear forces and the resultant shear force, hence determine the maximum force F that can be supported by the bolts in this configuration. $(7$ marks$)$

$($c$)$ Identify two possible bolts $(P,Q$ or R $)$ for the highest resultant shear force for configuration $2$ and expressions for the primary and secondary shear forces and the resultant shear force. Determine the maximum force F that can be supported by the bolts in this configuration for each bolt and hence identify the bolt that experiences the highest resultant shear force.

$(11$ marks$)$

$($d$)$ Which is the better configuration from the maximum applied force point of view?

$(2$ marks$)$