Exercise $4,$ Shaft design $(20$ points$)$:

A shaft has a vertical downward load being applied to the end gear on the left, see figure $4.$ This load is transferred through the shaft and in turn supplies a torque to a gear at point $C.$ The gear at $C$ generates a force on the bottom gear of $2000$ N in the $y-$direction.

Determine the smallest radius in whole millimetres for the shaft B$-$D$,$ which prevents the stress from exceeding ${}_{\text{a}\text{l}\text{l}\text{o}\text{w}}=80$MPa.

$($First find the reaction forces at A and B $,$ and the internal moment diagrams$)$

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