Problem I: A solid bronze shaft $(2)$ connected to a hollow steel shaft $(1)$ at flange B$.$ The outside diameter of $(1)$ is $80$ mm $,$ and the inside is $65$ mm $.$ The outside diameter of $(2)$ is $80$ mm $.$ The allowable shear stress of the steel and the bronze, respectively, $100$ MPa and $60$ MPa $.$ Determine:

$1.$ The maximum torque can be applied to flange $\backslash ($ B $\backslash )$

$2.$ The stresses $\backslash ($ r$\_\{1\}\backslash )$ and $\backslash ($ r$\_\{2\}\backslash )$ developed in the steel and bronze shafts

$3.$ The angle of rotation at flange $\backslash ($ B $\backslash )$

Given: $\backslash ($ G $\backslash )$ of steel $75$ GPa $,$ and $\backslash ($ G $\backslash )$ of bronze $38$ GPa