Q$5\mathrm{.}1\mathrm{.}1$ A cantilever beam $AB$ is subjected to a uniformly distributed load $q$ as shown in Fig. Q$5\mathrm{.}1\mathrm{.}1.$ A cross$-$sectional profile of the beam is rectangular but varies linearly with a constant width $b$ and a varying height $d.$ How should the height $d$ vary as a function of $x$ in order to have a fully stressed beam? $($Assume that an allowable stress is ${}_{\text{a}\text{l}\text{l}\text{o}\text{w}\text{.}})$

Answers: $,d=x\sqrt[\phantom{2}]{\frac{3q}{b{}_{\text{a}\text{l}\text{l}\text{o}\text{w}}}}$

Fig. Q$5\mathrm{.}1\mathrm{.}1$