Problem $2$

The simply supported pre$-$stressed concrete beam in the figure is subjected to an evenly distributed load $q$ between supports A and C$.$ The beam span is $L=12,0m.$

The cross section for the beam is rectangular, with the width $b=300mm$ and the height $h=550mm.$

Optimize the alignment of the pre$-$stress cables for this distributed load, given that the cables should be anchored at mid$-$height at the ends of the beam and that the centre of the cables should be $45mm$ above the lower surface of the beam at location B$.$

The cable alignment should be described in the coordinate system shown in the figure, with $y=0$ located at mid$-$ height of the beam.

A complete solution should include a diagram showing the alignment of the cable, and the analytical expression for the alignment given in the coordinate system shown in the figure.