Question $1(40$ pts$)$:

The C$-$shaped beam shown above is supporting two concentrated loads with a magnitude of $P.$ The beam has an allowable bending stress of ${}_{\text{a}\text{l}\text{l}\text{o}\text{w}}=250$MPa and an allowable shear stress of ${}_{\text{a}\text{l}\text{l}\text{o}\text{w}}=100$MPa.

a$)$ Determine the distance to the neutral axis $(\frac{?}{\text{b}\text{a}\text{r}}(y)),$ second moment of area $(I),$ and the section modulus $)$ of the cross$-$section.

b$)$ Draw the shear force diagram $($SFD$)$ and bending moment diagram $($BMD$)$ of the beam. On your diagrams, express the values of shear and moment in terms of the applied load $P.$

c$)$ Determine the maximum value of $P$ such that bending failure will not occur.

d$)$ Determine the maximum value of $P$ such that shear failure will not occur.

e$)$ Based on your answers to $($c$)$ and $($d$),$ what is the maximum load $P$ that can be applied to the beam? Is this beam bending or shear governed?