A rectangular reinforced concrete beam with width $b=300mm,$ and effectiv e depth of $d=530mm,$ and a total depth of $h=600mm$ is reinforced with th ree No$.36(31006m{m}^2)$ bars. The concrete compressive strength, modulus of rupture, and steel yielding strength are $f_c^{\text{'}}=28$MPa,$f_r=3\mathrm{.}6$MPa and $f_y=420$ MPa $.$ The elastic moduli of steel and modulus ratio are $E_s=200,000$MPa and $n=8.$

$(1)$ Find the maximum service load moment that can be resisted without stressing the concrete beyond $0\mathrm{.}4f^{\text{'}},$ or the steel beyond $0\mathrm{.}3f_{r^2}.$

$(2)$ Determine whether the beam will crack before reaching the service load.

$(3)$ Determine a failure mode, tension$-$ or compression$-$controlled.

$(4)$ Determine the nominal strength for the beam. Use $=\frac{f_{av}}{f_c^{\text{'}}}=0\mathrm{.}72,=0\mathrm{.}425$

$(5)$ Compute the design strength.

$*$Given equation : Parameters k and j for elastic, cracked section beam analysis, where $k=\sqrt[\phantom{2}]{2n+(n{)}^2}-n,j=1-\frac{1}{3}k$