For problems $6-9,$ show all calculations and provide final answers to $3$ significant figures and boxed.

$6.(20$ pts$)$ A steel pressure vessel is sealed by a gasketed cap fastened with $8$ preloaded $3/8-16$ X $2"$ UNC Grade $5$ hex head bolts. Regular hexagonal nut height for $\frac{3}{8^{\text{'}\text{'}}},.$ The pressure in the cylinder is $1500$ psi. Two potential designs are being considered. One has a confined gasket and the other has an unconfined gasket. Dimensions: $.$ Assume the half$-$apex angle, $.$ Young's modulus for steel, $.$ Whichever design is used, its gasket configuration will be present on both sides of the centerline. For the confined gasket, the member stiffness can ignore the confined gasket; for the unconfined gasket, the member stiffness must combine the stiffness of the steel pressure vessel with the stiffness of the gasket. The gasket can be assumed to run from the outside diameter of the flange, $,$ to the inside diameter of the pressure vessel, $.$

a$.$ What preload should the fasteners be installed with if it is expected that this pressure vessel will require periodic maintenance?

b$.$ What is the thread length of the bolt?

c$.$ What is the length of the unthreaded portion in the grip?

d$.$ What is the length of the threaded portion in the grip?

e$.$ What is the bolt stiffness?

f$.$ What is the member stiffness when a confined gasket is used?

g$.$ What is the joint stiffness constant, C$,$ for design #$1?$

h$.$ What is the member stiffness when an unconfined gasket is used?

i$.$ What is the joint stiffness constant, C$,$ for design #$2?$

j$.$ Which design would you recommend and why?