A rectangular cross section of a pre$-$tensioned concrete beam is $300$ mm wide and

$900$ mm deep. It is loaded so that it bends about its major axis. The beam is

made of concrete with characteristic cylinder strength in compression of $40$ MPa.

Assume that the minimum and maximum unfactored bending moments that the

beam may need to carry are $500$ kNm in sagging and $100$ kNm in hogging. The

beam must satisfy no$-$tension condition and the maximum compressive stress in

the cross section must not exceed $45\%$ of the characteristic cylinder strength in

compression.

i$.$ Calculate the maximum prestressing force for this beam.

ii$.$ Plot the Magnel diagram for this cross section and estimate graphically from the

diagram the eccentricities of the maximum prestressing force. Check the value

of the calculated maximum prestressing force calculated under Q$2($b$)$i$.$