$3\mathrm{.}5$ For a rectangular section, determine the required

effective size $($width $b$ rounded to whole inches

and effective depth $d)$ and theoretical steel area

A$,$ to carry uniformly distributed dead and live

loads on a simply supported span as given. Do

the case assigned by the instructor. Do not make

the practical decisions of choosing actual over$-$

all beam dimensions and selecting bars. Assume

the dead load includes the beam weight. Use

basic principles $($i$.$c$.,$ no formulas$)$ to obtain

minimum $($ACl$-9\mathrm{.}6\mathrm{.}1\mathrm{.}2)$ and maximum $($ACI$-$

$9\mathrm{.}3\mathrm{.}3\mathrm{.}1)$ reinforcement ratio limits$,$ as well as the

maximum reinforcement ratio corresponding to

a tension$-$controlled section $($verify the latter by

comparing with Table $3\mathrm{.}6\mathrm{.}1).$ Derive the formula

for $R_q$ in terms of $p$ and then use it to get the

limiting values.

$($a$)$ Obtain the largest effective size permitted

by ACI$-9\mathrm{.}6\mathrm{.}1\mathrm{.}2.$

$($b$)$ Obtain the smallest effective size permitted

by ACI$-9\mathrm{.}3\mathrm{.}3\mathrm{.}1.$