$8\mathrm{.}25$ A beam with a rectangular cross section, as in Fig. $8\mathrm{.}17($c$),$ is made of $2219-$T$851$ aluminum and must withstand a bending moment of $M=160N*m.$ The thickness is $b=10mm,$ and a quarter$-$circular corner crack as large as $a=2mm$ may be present.

$($a$)$ What beam depth $t$ is required for a safety factor $3\mathrm{.}0$ against brittle fracture?

$($b$)$ For the beam depth $t$ as calculated in $($a$),$ is the design adequate with respect to possible fully plastic failure? $($Suggestion: Make a conservative estimate of $M_o$ by assuming that the crack extends across the full thickness $b.)$