Problem $1$

The beam section shown below is subjected to positive bending. The beam is constructed with normal weight concrete and Grade $60$ steel reinforcement. The concrete cover is $1\mathrm{.}5$ in thick. No$.4$ bars are used as ties $($transverse reinforcement$).$ Concrete cylinder compressive strength is $f_c^{\text{'}}=5000$ psi. The yielding strength of the reinforcing bars is $f_y=60$ ksi.

$(1)$ Determine the cracking moment $M_{cr}$ and the corresponding curvature ${}_{cr}.$

$(2)$ Determine concrete stress at the top face at $M_{rr}.$

$(3)$ Determine stress of the reinforcing bars at $M_{cr}.$