A $24-$in$-$long round cantilever beam is loaded repeatedly, but no load reversal occurs. If the bar is made from an AIS $1040$ cold$-$drawn steel to support the

$2000$ Ib ball. Assume a safety fathon polished. Sn$\text{'}$ is unknown. Note that $G_{\text{s}\text{i}\text{e}\text{e}}$ is over $2$ in$.$

$S_{\text{u}\text{i}\text{r}}=24,000$

$M_{\text{m}\text{a}\text{x}}=FL=(2000)(24)=48000$ICb

$M_{\text{m}\text{i}\text{n}}=0$

$S_{\text{m}\text{a}\text{x}}=\frac{M_{\text{m}\text{a}\text{x}}}{Z}=\frac{48000\text{i}\text{n}\text{l}\text{b}}{Z}$

$Z=\frac{D^3}{32}$

$S_{\text{m}\text{e}\text{a}\text{n}}=\frac{s_{\text{m}\text{i}\text{n}}+S_n}{2}=\frac{\frac{48\mathrm{.}000}{2}}{2}=\frac{24000}{2}$

$S_n=(tin)(100)(\mathrm{.}75)=30\mathrm{.}0$ksi

$\frac{1}{2\mathrm{.}5}=\frac{\frac{24000}{2}}{74\mathrm{.}000}+\frac{\frac{24000}{2}}{30,000}$

$2=2\mathrm{.}85m^3$

$D=\sqrt[3]{\frac{32(2\mathrm{.}85)}{}}$

$D=3\mathrm{.}037$