Problem #$1$: $($need to show all calculations in detail$).$

$(50\%)$

For the beam and loading shown:

Draw the shear and bending$-$moment diagrams $($need to show all calculations in detail$).$

Determine the maximum absolute value $($a$)$ of the shear and $($b$)$ of the bending moment.

Knowing that the allowable normal stress for the steel used is ${}_{\text{a}\text{l}\text{l}\text{o}\text{w}\text{a}\text{b}\text{l}\text{e}}=24$ksi, select the most economical S$-$shape beam to support the loading shown.

The ultimate stress of the $S-$shape beam you selected is ${}_{\text{u}\text{l}\text{t}\text{i}\text{m}\text{a}\text{t}\text{e}}=58$ksi, what is the factor of safety of your design? $($Hint: F$.$S$.=\frac{{}_{\text{u}\text{l}\text{t}\text{i}\text{m}\text{a}\text{t}\text{e}}}{{}_{\text{m}\text{a}\text{x}}}).$ Comment on how safe your beam design is$.$