Find moment of resistance of a reinforced concrete rectangular beam of width $300 \mathrm{~mm}$ and overall depth $600 \mathrm{~mm}$ if reinforcement is $4-16 \mathrm{~mm}$ dia. bars at a distance of $40 \mathrm{~mm}$ from bottom. M20 grade concrete mix and TOR steel (Fe415 grade) have been used.

Solution:

1. Permissible stresses:

For M20 grade concrete $\sigma_{\text {cbc }}=7 \mathrm{~N} / \mathrm{mm}^{2}, \quad \mathrm{~m}=\frac{280}{3 \sigma_{\mathrm{cbc}}}=\frac{280}{3 \times 7}=13.33 \approx 13$

and for Fe415 grade steel, $\sigma_{s t}=230 \mathrm{~N} / \mathrm{mm}^{2}$

2. Neutral axis $A_{s t}=804 \mathrm{~mm}^{2}, \quad b=300 \mathrm{~mm}, \quad d=600-40=560 \mathrm{~mm}$

$\mathrm{p}=\frac{\mathrm{A}_{\mathrm{st}}}{\mathrm{bd}}=\frac{804}{300 \times 560}=0.004786$, and $\mathrm{mp}=13 \times 0.004786=0.06221$

Coefficient of neutral axis (k) shall be found from the relation $k=-m p+\sqrt{m^{2} p^{2}+2 m p}$

or, $k=-0.06221+\sqrt{(0.06221)^{2}+2 \times 0.06221}=0.296$

3. Lever arm coefficient, $j=1-k / 3=1-0.296 / 3=0.901$

4. Moment of Resistance Moment of resistance with respect to compression concrete $=\frac{1}{2} \sigma_{c b c} k j b d^{2}$

$=\frac{1}{2} \times 7 \times 0.901 \times 0.296 \times 300 \t....

Solution:

1. Permissible stresses:

For M20 grade concrete $\sigma_{\text {cbc }}=7 \mathrm{~N} / \mathrm{mm}^{2}, \quad \mathrm{~m}=\frac{280}{3 \sigma_{\mathrm{cbc}}}=\frac{280}{3 \times 7}=13.33 \approx 13$

and for Fe415 grade steel, $\sigma_{s t}=230 \mathrm{~N} / \mathrm{mm}^{2}$

2. Neutral axis $A_{s t}=804 \mathrm{~mm}^{2}, \quad b=300 \mathrm{~mm}, \quad d=600-40=560 \mathrm{~mm}$

$\mathrm{p}=\frac{\mathrm{A}_{\mathrm{st}}}{\mathrm{bd}}=\frac{804}{300 \times 560}=0.004786$, and $\mathrm{mp}=13 \times 0.004786=0.06221$

Coefficient of neutral axis (k) shall be found from the relation $k=-m p+\sqrt{m^{2} p^{2}+2 m p}$

or, $k=-0.06221+\sqrt{(0.06221)^{2}+2 \times 0.06221}=0.296$

3. Lever arm coefficient, $j=1-k / 3=1-0.296 / 3=0.901$

4. Moment of Resistance Moment of resistance with respect to compression concrete $=\frac{1}{2} \sigma_{c b c} k j b d^{2}$

$=\frac{1}{2} \times 7 \times 0.901 \times 0.296 \times 300 \times 560^{2} \quad \mathrm{~N} . \mathrm{mm}$

$=87.82 \mathrm{KN} m$

Moment of resistance of beam section with respect to tensile steel

$=\sigma_{s t} A_{s t} j d$

$=230 \times 804 \times 0.901 \times 560 \quad \mathrm{~N} . \mathrm{mm}$

$=93.30 \mathrm{KN} \cdot \mathrm{m}$

The actual moment of resistance is smaller of the two values. Therefore, moment of resistance of the beam section $=87.82 \mathrm{KNm}$.