Class Exercise $($doubly$-$reinforced beam sections$)$

Given that this beam section is an under$-$reinforced beam section $($as it relates to the tension reinforcement$),$ please answer the following:

$($a$)$ Ignoring the top bars, calculate the nominal flexural section strength, $M_n$

$($b$)$ Including the top bars, calculate the nominal flexural section strength, $M_n.$ Note you have two choices in starting this part of the problem, assume that $f_s^{\text{'}}=f_y$ or $M_n{f}_s^{\text{'}}.In$ either choice you must justify your assumption.

$(c)$ Taking the ratio $of\frac{b}{a}$ above, how much $ofan$ affect did including the compression reinforcement have $on$:

$i.$ The nominal flexural strength, $M_n?$

$ii.$ The ductility $(i.e.,$ inelastic deformation $of$ the beam beyond the reinforcement yield point$)?$

$(d)$ For the beam section $of$ part $(b)$ above, $is$ the section a tension$-$controlled, compressioncontrolled, $or$ transition section?