Question 12.P.5: A reinforced concrete beam is of rectangular section 300 mm ...
A reinforced concrete beam is of rectangular section 300 mm wide by 775 mm deep. It has five 25 mm diameter bars as tensile reinforcement in one layer with 25 mm cover and three 25 mm diameter bars as compression reinforcement, also in one layer with 25 mm cover. Find the moment of resistance of the section using elastic theory if the allowable stresses are 7.5 N/ mm² and 125 N/mm² in the concrete and steel, respectively. The modular ratio is 16.
The area of the compression steel is 3 × π × 25²/4 = 1472.6 mm² and of the tensile steel is 5 × π × 25²/4 = 2454.4 mm² . Then, from Eq. (12.17) \frac{b n^{2}}{2}+(m-1) A_{\mathrm{sc}}\left(n-d_{2}\right)=m A_{\mathrm{st}}\left(d_{1}-n\right),
\frac{300 n^{2}}{2}+(16-1) \times 1472.6(n-25)=16 \times 2454.4(750-n)
from which
n = 242.8 mm
Using the compatibility of strain condition, i.e. Eq. (12.13) \sigma_{\mathrm{s}}=-\sigma_{\mathrm{c}} \frac{E_{\mathrm{s}}}{E_{\mathrm{c}}}\left(\frac{d_{1}-n}{n}\right)=-\sigma_{\mathrm{c}} m\left(\frac{d_{1}-n}{n}\right),
\begin{aligned}&\sigma_{\mathrm{s}}=\frac{16 \sigma_{\mathrm{c}}(750-242.8)}{242.8} \\&\sigma_{\mathrm{s}}=33.4 \sigma_{\mathrm{c}}\end{aligned}
Therefore the limiting stress is the steel stress and \sigma_{c} = 125/33.4 = 3.7 N/mm². Then, taking moments about the compressive steel
M \times 10^{6}=125 \times 2454.4(750-25)-\left(\frac{3.7}{2}\right) \times 300 \times 242.8\left(\frac{242.8}{3}-25\right)
i.e.
M = 214.5 kN m