A C$8$x$13\mathrm{.}75$ channel section $($Grade A$36)$ is used as a brace member. The end connection to a $\frac{3}{8}"$ thick A$36$ gusset plate is shown in the figure below. Six $\frac{7}{8}"$ diameter A$325$ N bolts are used to make the connection and are located as shown. Compute the design strength $(P_n)$ for each of the following failure mechanisms: $($Section property and bolt shear tables are provided for your use.$)$

$15$

a$)$ Channel member capacity based on yielding of the gross area.

b$)$ Channel member capacity based on rupture on the effective net area.

c$)$ Bolt shcar capacity.

d$)$ Bolt bcaring capacity on channel web.

e$)$ Block shear of the channel member.

f$)$ What is the controlling design strength for this connection?