$-22.$ A beam is being designed to support the loads shown in Figure P$7-22.$ The four shapes proposed are $($a$)$ a round bar, $($b$)$ a square bar, $($c$)$ a rectangular bar with the height made four times the thickness, and $($d$)$ the lightest European standard IPE I$-$shape. Determine the required dimensions of each proposed shape to limit the maximum stress due to bending to $80$ MPa $.$ Then, compare the magnitude of the cross$-$sectional areas of the four shapes. Since the weight of the beam is proportional to its area, the one with the smallest area will be the lightest.