A $16$ foot long beam supports the loading shown; the cross section is shown on the next page.

Points $\backslash (\backslash$mathbf$\{$A$\}\backslash )$ and $\backslash (\backslash$mathbf$\{$C$\}\backslash )($pin joint$),$ and the applied forces and moment are on the centroidal axis of the beam. $\backslash ($ z$=0\backslash )$ at the extreme left end of the beam.

$($a$)$ Determine the centroid location and moment of inertia for the cross section.

$($b$)$ Draw a free$-$body diagram for the beam, and determine the reaction forces.

$($c$)$ Determine the internal forces at the centroid of a section a$-$a located at $\backslash ($ Z$=2\backslash$mathrm$\{$ft$\}\backslash ).$

$($d$)$ Determine the stresses for a point D that is on section a$-$a$.$ located $3$ in DOWN from the top surface. and show the stress state $($in psi or ksi $)$ for the point $\backslash (\backslash$mathbf$\{$D$\}\backslash )$ on a volume element.

$($e$)$ Determine the internal forces at the centroid of a section $\backslash ($ b$-$b $\backslash )$ located at $\backslash ($ Z$=9\backslash$mathrm$\{$ft$\}\backslash ).$

$($f$)$ Determine the stresses for a point E that is on section $\backslash (\backslash$boldsymbol$\{$b$\}-\backslash$boldsymbol$\{$b$\}\backslash ),$ located $3$ in UP from the bottom surface, and show the stress state $($in psi or ksi $)$ for the point $\backslash (\backslash$mathbf$\{$E$\}\backslash )$ on a volume element.

$($g$)$ Determine the internal forces at the centroid of a section c$-$clocated at $\backslash ($ Z$=11\backslash$mathrm$\{$ft$\}\backslash ).$

$($h$)$ Determine the stresses for a point $\backslash (\backslash$mathbf$\{$H$\}\backslash )$ that is on section $\backslash (\backslash$boldsymbol$\{$c$\}\backslash )-$c$.$ located $10$ in UP from the bottom surface. and show the stress state $($in psi or ksi $)$ for the point $\backslash (\backslash$mathbf$\{$H$\}\backslash )$ on a volume element.