Question F15.1: Determine the minimum dimension a to the nearest mm of the b......
Determine the minimum dimension a to the nearest \mathrm{mm} of the beam’s cross section to safely support the load. The wood has an allowable normal stress of \sigma_{\text {allow }}=10 \mathrm{MPa} and an allowable shear stress of \tau_{\text {allow }}=1 \mathrm{MPa}.
At support,
\begin{aligned} & V_{\max }=12 \mathrm{kN} \quad M_{\max }=18 \mathrm{kN} \cdot \mathrm{m} \\ & I=\frac{1}{12}(a)(2 a)^{3}=\frac{2}{3} a^{4} \\ & \sigma_{\text {allow }}=\frac{M_{\max } c}{I} ; \quad 10\left(10^{6}\right)=\frac{18\left(10^{3}\right)(a)}{\frac{2}{3} a^{4}} \\ & a=0.1392 \mathrm{~m}=139.2 \mathrm{~mm} \end{aligned}
Use a=140 \mathrm{~mm}
\begin{aligned} & I=\frac{2}{3}\left(0.14^{4}\right)=0.2561\left(10^{-3}\right) \mathrm{m}^{4} \\ & Q_{\max }=\frac{0.14}{2}(0.14)(0.14)=1.372\left(10^{-3}\right) \mathrm{m}^{3} \\ & \tau_{\max }=\frac{V_{\max } Q_{\max }}{I t}=\frac{12\left(10^{3}\right)\left[1.372\left(10^{-3}\right)\right]}{\left[0.2561\left(10^{-3}\right)\right](0.14)} \\ & =0.459 \mathrm{MPa}<\tau_{\text {allow }}=1 \mathrm{MPa}(\mathrm{OK}) \end{aligned}