A cast iron pipe has a $6$ in $.$ outer diameter and a $5$ in $.$ inner diameter. It is subjected to simultaneous bending and twisting. The bending moment is $\backslash (250\backslash$times $10^\{3\}\backslash$mathrm$\{$lb$\}\backslash )-$in$.$ Cast iron is generally treated as a brittle material; its strength, measured in uniaxial tension, is $30$ ksi $.$

$($a$)$ Determine the points on the pipe for which failure under combined bending and twisting would occur.

$($b$)$ Determine the torque that, together with bending, will cause failure.

$($c$)$ Draw Mohr's circle for the location where failure will occur.

$($d$)$ Draw the principal stresses on a square material element at this location, and show the orientation of the failure plane.