Question 12.P.1: A timber beam 200 mm wide by 300 mm deep is reinforced on it...
A timber beam 200 mm wide by 300 mm deep is reinforced on its top and bottom surfaces by steel plates each 12 mm thick by 200 mm wide. If the allowable stress in the timber is 8 N/mm² and that in the steel is 110 N/mm², find the allowable bending moment. The ratio of the modulus of elasticity of steel to that of timber is 20.
The second moments of area of the timber and steel are, respectively
I_{\mathrm{t}}=\frac{200 \times 300^{3}}{12}=450 \times 10^{6} \mathrm{~mm}^{4} \\ I_{\mathrm{s}}=2 \times 12 \times 200 \times 156^{2}=117 \times 10^{6} \mathrm{~mm}^{4} \text { (treating the plates as thin) }
Then, from Eq. (12.7) \sigma_{\mathrm{t}}=-\frac{M y}{I_{\mathrm{t}}+\frac{E_{\mathrm{s}}}{E_{\mathrm{t}}} I_{\mathrm{s}}},
8=\frac{150 M \times 10^{6}}{(450+20 \times 117) \times 10^{6}}
which gives
M = 148.8 kN m
From Eq. (12.8) \sigma_{\mathrm{s}}=-\frac{M y}{I_{\mathrm{s}}+\frac{E_{\mathrm{t}}}{E_{\mathrm{s}}} I_{\mathrm{t}}},
110=\frac{162 M \times 10^{6}}{\left(117+\frac{450}{20}\right) \times 10^{6}}
from which
M = 94.7 kN m
Therefore the allowable bending moment is 94.7 kN m.