$5\mathrm{.}7-4$ The spokes in a large flywheel are modeled as

beams fixed at one end and loaded by a force $P$ and a

couple $M_0$ at the other $($see figure$).$ The cross sections

of the spokes are elliptical with major and minor axes

$($height and width, respectively$)$ having the lengths

shown in the figure part a$.$ The cross$-$sectional

dimensions vary linearly from end $A$ to end $B.$

Considering only the effects of bending due to the

loads $P$ and $M_0,$ determine the following quantities.

$($a$)$ The largest bending stress ${}_A$ at end $A.$

$($b$)$ The largest bending stress ${}_B$ at end $B.$

$($c$)$ The distance $x$ to the cross section of maximum

bending stress.

$($d$)$ The magnitude ${}_{\text{m}\text{a}\text{x}}$ of the maximum bending

stress.

$($e$)$ Repeat part d if a uniform load $q(x)=10\frac{P}{3}L$

is added to loadings $P$ and $M_0,$ as shown in the

figure part $b.$

$b_A=60mm$

$b_B=80mm$

$($a$)$

$($b$)$