Problem $1(100pts)$ The rectangular beam cross$-$section shown below is used to build a simply$-$

supported beam, spanning $20$ ft $.$ The beam supports a superimposed uniform dead load of $w_D$ of

$2\mathrm{.}0ki\frac{p}{f}t$ and a superimposed uniform live load of $w_L$ of $1\mathrm{.}2ki\frac{p}{f}t.$ Design the beam cross$-$

section, i$.$e$.,(1)$ determine the size of beam $(b,h,$ and d $)$ and $(2)$ determine the number and size

of reinforcement, following the steps shown below.

Use $f_c=5000$ psi and $f_y=60$ksi

$(1)$ Find $M_u$ by assuming the self$-$weight $w_{\text{s}\text{e}\text{l}\text{f}}$ would add $20\%$ more moment.

$(2)$ Compute required $b,d,$ and $h$ in the way we discussed in the class. Provide the answers in

inches. Note that you must assume

$(1)h=d+2\mathrm{.}5$ in$.$

$(2)$ the strain of reinforcement at failure $({}_s)$ is $0\mathrm{.}010$

$(3)d=1\mathrm{.}7b$

$(3)$ Select b$,$ d$,$ and h based on the answers in Step $(2)$

$(4)$ Using the dimensions for the beam cross$-$section selected in Step $(3),$ compute the self$-$weight

of the beam

$(5)$ Add the self$-$weight to the dead load and compute Mu again.

$(6)$ find the amount of tensile reinforcement, $A_s$ required to resist the moment due the factored

loads, i$.$e$.,$ find $A_{s,\text{r}\text{e}\text{q}}.$

$(7)$ find the minimum amount of reinforcement, $A_{s,\text{m}\text{i}\text{n}}.$

$(8)$ select the number and size of the bars based on $A_{s,\text{r}\text{e}\text{q}},$ and $A_{s,\text{m}\text{i}\text{n}},$ i$.$e$.,$ determine $A_{s,}$ pro,

$(9)$ check if the beam cross$-$section with $A_{s,\text{p}\text{r}\text{o}}$ is tension$-$controlled or not

$(10)$ Provide a sketch for the final design of the cross$-$section. $($Use straight edges or a ruler to

draw it$)$

For the calculation of As$,$ use the chart below.

ASTM STANDARD REINFORCING BARS