# Example 01: Required Steel Area of Reinforced Concrete Beam

**Problem**

A rectangular concrete beam is reinforced in tension only. The width is 300 mm and the effective depth is 600 mm. The beam carries a moment of 80 kN·m which causes a stress of 5 MPa in the extreme compression fiber of concrete. Use *n* = 9.

- What is the distance of the neutral axis from the top of the beam?
- Calculate the required area for steel reinforcement.
- Find the stress developed in the steel.

**Solution**

Distance of the neutral axis from the top of the beam
$M = Cy$
$Q_{\text{above NA}} = Q_{\text{below NA}}$
$\dfrac{f_s/n}{d - x} = \dfrac{f_c}{x}$

$M = \frac{1}{2}f_c bx(d - \frac{1}{3}x)$

$80(1000^2) = \frac{1}{2}(5)(300x)(600 - \frac{1}{3}x)$

$250x^2 - 450\,000x + 80\,000\,000$

$x = 1600 \, \text{ and } \, 200$

Use $x = 200 ~ \text{mm}$ *answer*

Required steel area

$300x(\frac{1}{2}x) = nA_s(d - x)$

$150x^2 = nA_s(d - x)$

$150(200^2) = 9A_s(600 - 200)$

$A_s = 1666.67 ~ \text{mm}^2$ *answer*

Stress developed in the steel

$\dfrac{f_s/9}{600 - 200} = \dfrac{5}{200}$

$f_s = 90 ~ \text{MPa}$ *answer*

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