$5\mathrm{.}5-25$ A beam with a T$-$section is supported and loaded as shown in the figure. The cross section has width $b=65mm,$ height $h=75mm,$ and thickness $t=13mm.$

$($a$)$ Determine the maximum tensile and compressive stresses in the beam.

$($b$)$ If the allowable stresses in tension and compression are $124$MPa and $82$MPa, respectively, what is the required depth $h$ of the beam? Assume that thickness $t$ remains at $13mm$ and that flange width $b=65mm.$

$($c$)$ Find the new values of loads $P$ and $q$ so that the allowable tension $(124$MPa and compression