A rectangular beam has the dimensions shown with $f_c^{\text{'}}=5000$ and $f_y=60$ksi. The beam must resist a factored moment of $M_u=1\mathrm{.}2M_D+1\mathrm{.}6M_L=1500$ kip$-$ft$.$

a$)$ Find the required As using the "quick and dirty" method for computing moment strength $($remember that $M_{n}1\mathrm{.}2M_D+1\mathrm{.}6M_L,$ with $=0\mathrm{.}9).$

b$)$ Find the required $A_s$ using the "exact" method for computing moment strength. Based on this $A_s,$ choose bars for use in the design. Bars should be no larger than No$.11.$

$2$

c$)$ The ACI $318$ Code assumes that when the beam achieves nominal flexural strength, the maximum concrete compressive strain is $-0\mathrm{.}003i\frac{n}{i}n.$ Using this assumption and considering the information provided in class $($week $3),$ determine the depth of the neutral axis and the steel strain when the beam achieves nominal flexural strength, $M_n.$

d$)$ Check that the reinforcement satisfies ACI $318$ Code requirements for minimimum reinforcement $($ACI Code Section $9\mathrm{.}6)$:

$A_{s,\text{m}\text{i}\text{n}}=\frac{3\sqrt[\phantom{2}]{f_c^{\text{'}}}{b}_{w}d}{f_y}\frac{200b_{w}d}{f_y},(variablesin$ units $of$ inches and psi$)$

e$)$ Check that the reinforcement area is not so large that beam response is not tension controlled. This is checked by demonstrating that steel tension strain equals the steel yield strain ${}_{ty}+0\mathrm{.}003i\frac{n}{i}n$ when maximum concrete compression strain $=-0\mathrm{.}003i\frac{n}{i}n,$ per Figure R$21\mathrm{.}2\mathrm{.}2$b $($shown below$)$ from the ACI $318$ Code. Note that in the figure below $$ is not the section curvature, but is instead the strength reduction factor used in design $($e$.$g$.,M_{n}1\mathrm{.}4M_D$ or $1\mathrm{.}2M_D+1\mathrm{.}6M_L)$ and the figure defines a smaller strength reduction factor $($and thus smaller $M_n)$ for compression$-$controlled response, which is less ductile, and a larger strength reduction factor $($and thus larger $M_n)$ for tension$-$controlled response, which is more ductile.

extreme tension reinforcement, ${}_t.$