$(15$ marks$)$ A reinforced concrete beam $($shown in Fig. $4)$ is considered balanced if the maximum

stresses in the steel and concrete are equal, respectively, to the allowable stresses ${}_s$ and ${}_c.$ To achieve

this design, the neutral axis must be located a distance $x$ from the top of the beam, where:

$x=\frac{d}{1+\frac{{}_s{E}_c}{{}_c{E}_s}}$

where $E_c$ and $E_s$ are the Young's moduli of the concrete and steel, respectively, and $x$ and $d$ are shown

in Fig. $4.$

$($a$)$ Provide a proof of Eq$.(1).$

Hint: Remember that the maximum stress in concrete ${}_c$ and in the steel ${}_s$ are:

${}_c=\frac{Mx}{\text{I}},{}_s=\frac{nM(d-x)}{\text{I}}$

where $M$ is the maximum allowable moment, $n=\frac{E_s}{E_c}$ and I is the moment of inertia of the

section.

$($b$)$ Given that the depth of the beam is $d=20$ in and the area of the steel reinforcement is $A_s=$

$3\mathrm{.}1i{n}^2,$ solve for the width of the beam $b$ in order for it to be balanced. The properties of the

concrete and steel including the Young's moduli and allowable stresses, respectively, are given

below:

Concrete: $E_c=3{10}^6,{}_c=1350,$ Steel: $E_s=29{10}^6,{}_s=20$ksi

Hint: Start by looking at how the location of the neutral axis was found for reinforced concrete

beams in Module $1.$