Question 26.15: If the thin-walled box beam of Fig. P.26.13 carries a bendin......
If the thin-walled box beam of Fig. P.26.13 carries a bending moment of 1 kNm applied in a vertical plane, determine the maximum direct stress in the cross-section of the beam. Answer: 85.8 N/mm²
Since {I^{\prime}}_{X Y}=0\,\mathrm{and}\,M_{Y}=0, Eq. (26.68) reduces to
\sigma_{Z}=E_{Z,i}\Biggl[\left({\frac{M_{Y}I_{X X}^{\prime}-M_{X}I_{X Y}^{\prime}}{I_{X X}^{\prime}I_{Y Y}^{\prime}-I_{X Y}^{2}}}\right)X+\left({\frac{M_{X}I_{Y Y}^{\prime}-M_{Y}I_{X Y}^{\prime}}{I_{X X}^{\prime}I_{Y Y}^{\prime}-I_{~X Y}^{2}}}\right)Y\Biggr] (26.68)
\sigma_{Z}=E_{Z, i}{\frac{M_{X}}{I_{X X}^{\prime}}}Y
where
I_{XX}^{\prime}=2\times60000\times\frac{2.0 \times 100^{3}}{12}+2\times20000\times1.0\times150\times50^{2}i.e.,
I_{X X}^{\prime}=3.5\times10^{10}\mathrm{Nmm}^{2}Then
\sigma_{Z}=E_{Z,i}\times{\frac{1 \times 10^{6}}{3.5 \times 10^{10}}}Y=2.86\times10^{-5}E_{Z,i}Y (i)
The direct stress will be a maximum when Y is a maximum, i.e., at the top and bottom of the webs and in the covers. But E_{\mathrm{Z},i} for the webs is greater than that for the covers, therefore
\sigma_{Z}(\mathrm{max})=\pm2.86\times10^{-5}\times60000\times50i.e.,
\sigma_{z}({\mathrm{max}})=\pm85.8\,{\mathrm{N/mm}}^{2} (at the top and bottom of the webs)