Question 12.P.3: A timber beam 150 mm wide by 300 mm deep is reinforced by a ...
A timber beam 150 mm wide by 300 mm deep is reinforced by a steel plate 150 mm wide and 12 mm thick which is securely attached to its lower surface. Determine the percentage increase in the moment of resistance of the beam produced by the steel-reinforcing plate. The allowable stress in the timber is 12 N/mm² and in the steel, 155 N/mm². The modular ratio is 20.
The maximum stress in the timber beam is given by
\sigma_{\mathrm{t}}=\frac{M_{\mathrm{t}} \times 150}{\left(\frac{150 \times 300^{3}}{12}\right)}=12 \mathrm{~N} / \mathrm{mm}^{2}
which gives
M_{t}= 27 kN m
Also, since \sigma_{s} = 20\sigma_{t}, the steel stress is limiting so that \sigma_{t} = 155/20 = 7.75 N/mm².
Referring to Fig. S.12.3
\frac{150(300-n)^{2}}{2}=\frac{150 n^{2}}{2}+20 \times 150 \times 12(n+6)
which gives
n = 80.7 mm
Now taking moments about the resultant of the compressive stress distribution in the timber
M \times 10^{6}=155 \times 150 \times 12(146.2+80.7+6)+\left(\frac{7.75}{2}\right) \times 150 \times 807\left(\frac{2 \times 80.7}{3}+146.2\right)
which gives
M = 74.4 kN m
Therefore, M_{s}/M_{t} = 74.4/27 = 2.76 so that the increase produced by the steel reinforcing plate is 176%.
