A typical reinforced concrete beam of rectangular cross$-$section is shown in the figure below. The

beam is reinforced with $6-25mm$ dia bars $(4$ bars at bottom and $2$ bars above it$).$ The clear cover

to the centroid of the reinforcement is $70$ mm $.$ The following properties are given: $fc=30$MPa

$($normal$-$density concrete$),fy=400$MPa;$E=200000$MPa. Find the following:

$($a$)$ Cracking moment $($Mer$)$

$($b$)$ Gross moment of inertia $($I$)$ for the uncracked section $($ignore the effect of reinforcement$)$

$($c$)$ Moment of inertia for the uncracked section using the transformed section properties

$($d$)$ Moment of inertia for the cracked section $($Ier$).$ Use both first principles and the equation

below for finding NA$.$

$\frac{?}{\text{b}\text{a}\text{r}}(y)=d(\sqrt[\phantom{2}]{(n{)}^2+2np}-n)$

$($e$)$ What is the stress in the concrete at the top fiber and bottom fiber and stress in the steel

just prior to cracking and just after cracking. Draw the diagrams.

$($f$)$ What are the stresses developed in concrete and steel at $3$ Mcr $.$ If the allowable stress in

concrete is $0\mathrm{.}45$ fc $\text{'}$ and the allowable stress in steel is $0\mathrm{.}6$ fy $.$ Are these values exceeded at

$3$Mcr$.$ What should be the moment at which the stresses will just exceed allowable limits$.$

$($g$)$ What is the nominal moment capacity Mn of the beam and $Mn.$