Problem #$1(25$ points$)$

Design for axial tension capacity of single channel section of $10"$ or $12"$ deep $($Lightest possible section$)$ as shown to carry a dead load of $115$ kips and live load of $40$ kips. if you choose a $10"$ deep, gage is $2$ "from the top and bottom and $3$ in between the holes. If you choose a $12"$ the gage is ${2\mathrm{.}5}^{\text{'}\text{'}}$ from the top and bottom and ${3\mathrm{.}5}^{\text{'}\text{'}}$ in between. $($C $12$ gage spacing is shown in the figure$).$ The holes are for $1"$ bolts. Use both the exact and also the tabulated values for the reduction factor for the net area. Fy $=52$Ksi,$Fu=66$Ksi, and ignore the strength of the plate.

a$)$ Using LRFD$,$ determine the total service and ultimate load capacity of the lightest section.