A rectangular reinforced concrete beam has dimensions $b=10in.,d=21in.,$ and $h=24$ in$.,$ and is reinforced with $3$#$10$ bars, Fig. $2.$ Materials strengths are $f_y=60,000$ and $f_c^{\text{'}}=4000.$

$($a$)$ Find the moment that will produce the first cracking at the bottom surface of the beam, basing your calculation on $I_g,$ the moment of inertia of the gross concrete section.

$($b$)$ Repeat the calculation using $I_{ut},$ the moment of inertia of the uncracked transformed section. Comment on the differences between your answers to $($a$)$ and $($b$).$

$($c$)$ Find the stresses in concrete and steel in $($a$)$ and $($b$).$

Figure $2$: Concrete beam section in Problem $6.$