For the statically indeterminate beam below:

$($a$)$ Sketch the deflected shape and moment diagrams. Draw the moment diagrams on the compression side of the beam. Clearly indicate approximate locations of inflection points, points of zero moment and label the compression side of the deflected shape.

$($b$)$ Analyze the beam using the stiffness method. Determine the displacement under the point load. For this part assume the point load is actually $75$ kips from dead load and $25$ kips from live load, the beam has the cross section shown and the length, $L,$ is $40ft.$ Assume the live load is not reducible.

$($c$)$ Determine if the beam's maximum moment demand, from factored loads, has exceeded its factored plastic moment capacity. Use $=0\mathrm{.}9$ and assume the yield strength of the steel is $50$ksi.

$($d$)$ Determine if the beam's moment demand has exceeding its factored yield moment capacity. Use $=0\mathrm{.}9.$ Use $=0\mathrm{.}9$ and assume the yield strength of the steel is $50$ ksi.

$($e$)$ Draw the stress profile for the beam due to the maximum moment from factored loads $($i$.$e$.,$ how the bending stress varies along the cross section$).$ Indicate the values of the stresses at the top and bottom of the cross section.