Question F15.4: Determine the minimum dimension h to the nearest 1/8 in. of ......
Determine the minimum dimension h to the nearest \frac{1}{8} in. of the beam’s cross section to safely support the load. The wood has an allowable normal stress of \sigma_{\text {allow }}=2 \mathrm{ksi} and an allowable shear stress of \tau_{\text {allow }}=200 \mathrm{psi}.
At the supports,
V_{\max }=4.5 \mathrm{kip}At the center,
M_{\max }=6.75 \mathrm{kip} \cdot \mathrm{ft}
I=\frac{1}{12}(4)\left(h^{3}\right)=\frac{h^{3}}{3}
\sigma_{\text {allow }}=\frac{M_{\max } c}{I} ; \quad 2=\frac{6.75(12)\left(\frac{h}{2}\right)}{\frac{h^{3}}{3}}
Top half of rectangle,
\begin{aligned} & Q_{\max }=y^{\prime} A^{\prime}=\frac{h}{4}\left(\frac{h}{2}\right)(4)=\frac{h^{2}}{2} \\ & \tau_{\max }=\frac{V_{\max } Q_{\max }}{I t} ; 0.2=\frac{4.5\left(\frac{h^{2}}{2}\right)}{\frac{h^{3}}{3}(4)} \\ & \qquad h=8.4375 \mathrm{in} \text {. (controls) } \\ & \text { Use } h=8 \frac{1}{2} \text { in. } \end{aligned}