Beam Cross$-$Section Properties and Direct Stress $(30$ pts$)$

Consider a beam having a thin$-$walled open$-$section $($shown below$).$ It is made from a single material, and the

thickness of the walls, $t,$ is much smaller than the lateral dimension $a.$

a$)$ Determine the $x$ and $y$ coordinate of the centroid and show the location on the figure.

b$)$ Determine the $I_{},I_{yy},$ and $I_{xy}$ for this cross$-$section, in terms of a and $t.$

c$)$ Determine the peak magnitude and location on the cross$-$section of the direct stress, ${}_{zz},$ due to

bending given the following applied moments: $M_x=400N-m,M_y=200N-m.$

d$)$ What are is the tension margin of safety if the beam is made from Aluminum $6061-$T$6?$