Question F15.6: Select the lightest W410·shaped section that can safely supp......
Select the lightest W410-shaped section that can safely support the load. The beam is made of steel having an allowable normal stress of \sigma_{\text {allow }}=150 \mathrm{MPa} and an allowable shear stress of \tau_{\text {allow }}=75 \mathrm{MPa}. Assume the beam is pinned at A and roller supported at B.
Within the overhang,
V_{\max }=150 \mathrm{kN}
At B,
M_{\max }=150 \mathrm{kN} \cdot \mathrm{m}
S_{\text {reqd }}=\frac{M_{\max }}{\sigma_{\text {allow }}}=\frac{150\left(10^{3}\right)}{150\left(10^{6}\right)}=0.001 \mathrm{~m}^{3}=1000\left(10^{3}\right) \mathrm{mm}^{3}
Select W410 x 67\left[S_{x}=1200\left(10^{3}\right) \mathrm{mm}^{3}, d=410 \mathrm{~mm}\right., and \left.t_{w}=8.76 \mathrm{~mm}\right].
\begin{aligned} \tau_{\max } & =\frac{V}{t_{w} d}=\frac{150\left(10^{3}\right)}{0.00876(0.41)} \\ & =41.76 \mathrm{MPa}<\tau_{\text {allow }}=75 \mathrm{MPa}(\mathrm{OK}) \end{aligned}