Problem $5\mathrm{.}6-6.$ A pontoon bridge $($see figure$)$ is constructed of two longitudinal wood beams, known as balks, that span between adjacent pontoons and support the transverse floor beams, which are called chesses. For purposes of design, assume that a uniform floor load of $7\mathrm{.}5$kPa acts over the chesses. $($This load includes an allowance for the weights of the chesses and balks.$)$ Also, assume that the chesses are $2\mathrm{.}5m$ long and that the balks are simply supported with a span of $3\mathrm{.}0m.$ The allowable bending stress in the wood is $15$MPa.

$($a$)$ If the balks have a square cross section, what is their minimum required width $b_{\text{m}\text{i}\text{n}}?$

$($b$)$ Repeat part $($a$)$ if the balk width is $1\mathrm{.}5b$ and the balk depth is b; compare the cross$-$sectional areas of the two designs.

Note The UDL on each balk is given by the product of $7\mathrm{.}5$kPa and half the length of the chesses