The students in the Aerospace Engineering concentration are concerned that the Boeing $737$ was not designed properly. When pressurized at cruising altitude the internal pressure in the fuselage is $0\mathrm{.}5$ bar relative to the outside pressure, resulting a hoop $($circumferential$)$ stress. The fuselage has a diameter of $4$ m and the skin has a thickness of $8$ mm$.$ The dimensions of the window and the corner curvature are shown in the Figure. For this shape, the critical geometry, c$,$ is half of the window width, The engineers determine that the aluminum used is $6061-6$ and looked used EduPack to find that the yield strength is $300$ MPa and the fracture toughness is $1\mathrm{.}3$ MPa.m$^(1)/(2).$ Per FAA guidelines, the maximum allowable stress is $100$ MPa. The critical dimension, c$,$ is $(1)/(2)$ of the window width.

a$.$ Calculate the nominal hoop stress. Is $5$ mm a proper radius for the corners of the window $($meet the FAA guidelines$)?$

b$.$ To be safer, the students recommend that the radius of the corner be changed to $20$ mm$.$ How much below the allowable stress is the proposed design?

c$.$ The students don't want passengers to fall to their death, so they assume that a crack can form at the corner of the window that has an effective length of $0\mathrm{.}08$ m$.$ What stress can it withstand without fast fracture of the crack? Is it safe enough for passengers?