Problem $4[45$ points$]$

$(1)$ For simply supported concrete beam shown below, show that ${}_{\text{m}\text{a}\text{x}}=5W\frac{L^4}{384}EI$ using double$-$integration method. $(2)$ Calculate the instantaneous deflections and long$-$term deflection after $5$ years, assuming $80\%$ of sustained dead load, $50\%$ of live load are continuously applied for $60$ months. $W_D=1\mathrm{.}3\frac{k}{f}t,W_L=1\mathrm{.}7\frac{k}{f}t,fy=60,000,$ and $f^{\text{'}}c=5000,n=9,L={23}^{\text{'}}.(3)$ Perform the design check for flexural strength as well as maximum permissible deflection per ACI $318.$