Question $2.$ STATIC FAILURE

A stationary helical gear has a $250-mm$ diameter and is mounted on the overhung end of a shaft, a distance of $50$ mm from the nearest bearing. The shaft of $25-mm$ diameter is made of steel with ${}_{\text{y}\text{i}\text{e}\text{l}\text{d}}=800$MPa. The components of the load on the helical gear are shown in the figure below.

a$)$ Transfer all loads to the critical cross$-$section of the shaft $($Neglect stress concentration$).$

b$)$ On this particular cross$-$section compute stresses at the critical point.

c$)$ Draw the Mohr's circle representing the stress state at this point. Find principal stresses ${}_1$ and ${}_2.$

d$)$ Using maximum distortion energy theory $($von Mises$)$ comment on the static failure of the shaft.

Note that for biaxial stresses: ${}_e=({}_1^2+{}_2^2-{}_1{}_2{)}^{\frac{1}{2}}$