Design an efficient rectangular concrete beam to support a superimposed service dead load of $80$ psf and a service live load $125$ psf $.$ The structure has beams regularly spaced at ${10}^{\text{'}}-0^{\text{'}\text{'}}$ center to center. The beam spans ${30}^{\text{'}}-0"$ and for simplicity you may analyze the beam as a simply$-$supported member with pin and roller supports at its ends.

Use a beam width of $16$ inches. You need to determine the total height of the beam $($ h $)$ and the depth to the reinforcing $($d$).$

Use a depth to width $(\frac{d}{b})$ ratio of no less than $1\mathrm{.}5,$ no more than $1\mathrm{.}8.$ Assume the beam uses #$4$ stirrups. Use $f^{\text{'}}c=5,000,fy=60,000$ and the unit weight of concrete is $145$ pcf $.$

As part of your design, perform the following steps:

a$.$ Draw the ultimate $($factored$)$ shear and moment diagrams.

b$.$ Determine the beam's dimensions.

c$.$ Size the reinforcing.

d$.$ Prove that the beam lies within the tension$-$controlled range and $=0\mathrm{.}9.$

e$.$ Calculate the final moment capacity $(Mn)$ and compare to demand $().$

f$.$ Calculate the efficiency of the beam you designed. It should fall within this range:

$0\mathrm{.}9\frac{}{Mn}1\mathrm{.}0$

g$.$ Show a sketch of the cross$-$section of the beam, and show that the reinforcing you chose fits within the beam. Assume the largest aggregate is $\frac{3}{4}"$ and the diameter of the concrete vibrating wand is only $1-\frac{1}{4^{\text{'}\text{'}}}$ diameter.

h$.$ Using the #$4$ stirnups, determine the spacing required to resist the maximum shear demand.