$(1)$ The beam section shown below has a simple span of $12m$; determine the prestressing force using Magnel equations to satisfy the allowable stresses in concrete at transfer and service stages. The beam is prestressed by a single tendon having zero eccentricity at ends and maximum eccentricity at mid$-$span. Take $w_{sw}=48k\frac{N}{m},w_{\text{S}\text{I}\text{D}\text{L}}=7k\frac{N}{m},w_Q=15k\frac{N}{m},f_c^{\text{'}}=32$MPa,$f_{cp}=25$MPa, $R=0\mathrm{.}8,$ clear cover $=30mm$ and cable duct diameter $=60mm.$