Question 11.P.7: A reinforced concrete T-beam contains 1.25 × 10^-3 m² of ste......
A reinforced concrete T-beam contains 1.25 × 10^{-3} m² of steel reinforcement on the tension side. If the steel stress is limited to 115 MN/m² and the concrete stress to 6.5 MN/m², estimate the permissible bending moment. The modular ratio is 15.
Suppose the neutral axis falls below the underside of the flange. The area of the equivalent concrete beam is
(0.60)n – 0.45(n – 0.10) + (0.00125)15 = 0 15n + 00638 m²
The position of the neutral axis is given by
(0.60n)\left(\frac{1}{2}n\right) + (0.00125)(15)(0.30) – 0.45(n – 0.10)\left(\frac{1}{2}\right)(n + 0.10) = (0.15n + 0.638)n
This reduces to
n² + 0.850n – 01044 = 0
the relevant root of which is n = 0.109 m which agrees with our assumption earlier that the neutral axis lies below the flange.
The second moment of area of the equivalent concrete beam is
I_c = \frac {1}{3} (0.60)(n^3) – \frac{1}{3}(0.45)(n – 0.10)^3 + 0.00125(15)(0.30 – n)^2 \\\\ = (0.259 + 0.000 + 0.685)10^{-3} m^4 \\ = 0.944 \times 10^{-3} m^4If the maximum allowable concrete stress is attained, the permissible moment is
M = \frac{\sigma_cI_c}{n} = \frac{ (6.5 \times 10^6)(0.944 \times 10^{-3}) }{0.109} = 56.3 kNmIf the maximum allowable steel stress is attained, the permissible moment is
M = \frac{\sigma_sI_c}{m(0.30 – n)} = \frac{ (115 \times 10^6)(0.944 \times 10^{-3}) }{15(0.191)} = 37.9 kNmSteel is therefore the limiting material, and the permissible bending moment is 37.9 kNm.