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.

Question

A reinforced concrete T-beam contains [latex]1.25 imes 10^{-3} m^{2}[/latex] 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.

Accepted Answer

Suppose the neutral axis falls below the underside of the flange. The area of the equivalent concrete beam is

[latex](0.60)n-0.45(n -0.10) + (0.00125)15 = 0.15 n + 0.0638 m^{2}[/latex]

The position of the neutral axis is obtained by taking moments, as follows:

[latex](0.60 n) \left\lgroup\frac{1}{2}n ight group +(0.00125)(15)(0.30) -0.45 (n-0.10) \left\lgroup\frac{1}{2} ight group (n+0.10)[/latex]

[latex]= (0.15 n + 0.0638)n[/latex]

This reduces to

[latex]n^{2} + 0.850n-0.1044 = 0[/latex]

the relevant root of which is [latex]n = 0.109 m[/latex] 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

[latex]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}[/latex]

[latex]= (0.259 + 0.000 + 0.685)10^{-3} m^{4}[/latex]

[latex]= 0.944 imes 10^{-3} m^{4}[/latex]

If the maximum allowable concrete stress is attained, the permissible moment is

[latex]M=\frac{\sigma _{c}I_{c}}{n} =\frac{\left(6.5 imes 10^{6} ight)\left(0.944 imes 10^{-3} ight) }{ 0.109}=56.3 kNm[/latex]

the maximum allowable steel stress is attained, the permissible moment is

[latex]M=\frac{\sigma _{s}I_{c}}{m(0.30-n)}=\frac{\left(115 imes 10^{6} ight)\left(0.944 imes 10^{-3} ight) }{15( 0.191)}=37.9 kNm[/latex]

Thus, steel is therefore the limiting material, and the permissible bending moment is 37.9 kNm.

Question 11.7: A reinforced concrete T-beam contains 1.25 × 10^-3 m² of ste...

Related Answered Questions

A steel I-section, 12.5 cm by 7.5 cm, is encased in a rectangular concrete beam of breadth 20 cm and depth 30 cm to the lower flange of the I-section. Estimate the position of the neutral axis of the composite beam, and find the permissible bending moment if the steel stress is not to exceed 115 ...

Verified Answer:

Verified Answer:

Verified Answer:

Verified Answer:

A timber joist 15 cm by 7.5 cm has reinforcing steel flange plates 1.25 cm thick. The composite beam is 3 m long, simply supported at each end, and carries a uniformly distributed lateral load of 10 kN. Estimate the maximum bending stresses in the steel and timber, and the intensity of shearing ...

Verified Answer:

Verified Answer: