GATE 2025 CE (CE2) – Question 40
Consider a reinforced concrete beam section of 350 mm width and 600 mm depth. The beam is reinforced with the tension steel of 800 mm$^2$ area at an effective cover of 40 mm. Consider M20 concrete and Fe415 steel. Let the stress block considered for concrete in IS 456:2000 be replaced by an equivalent rectangular stress block, with no change in (a) the area of the stress block, (b) the design strength of concrete (at the strain of 0.0035), and (c) the location of neutral axis at flexural collapse. The ultimate moment of resistance of the beam (in kN.m) is __________________ (round off to the nearest integer).
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Correct answer: (B) 148
Explanation
The effective depth is $d = 600 - 40 = 560$ mm. The neutral axis depth is $x_u = \frac{0.87f_yA_{st}}{0.36f_{ck}b} = \frac{0.87 \times 415 \times 800}{0.36 \times 20 \times 350} = 114.6$ mm, which is less than $0.48d = 268.8$ mm, so the section is under-reinforced and steel yields. The IS 456 block has the area $0.36f_{ck}x_u$ and the design strength at the maximum strain is $0.446f_{ck}$, so the equivalent rectangle has the depth $a = \frac{0.36}{0.446}x_u = 0.807x_u = 92.5$ mm. The lever arm is $d - \frac{a}{2} = 560 - 46.2 = 513.8$ mm and $M_u = 0.87 \times 415 \times 800 \times 513.8 = 148.4 \times 10^6$ N mm $\approx 148$ kN m.