CivilReinforced ConcretehardCourseworkPE / Advanced

Required Tension Steel in a Singly Reinforced Beam

A singly reinforced rectangular concrete beam must resist a factored moment $M_u = 200\ \text{kN·m}$ .

Width $b = 300\ \text{mm}$ , effective depth $d = 500\ \text{mm}$
$f'_c = 25\ \text{MPa}$ , $f_y = 415\ \text{MPa}$
Strength reduction factor $\phi = 0.90$ (tension-controlled)

Find the required area of tension steel $A_s$ .

Given

Mu	200 kN*m	Factored moment
b	300 mm	Beam width
d	500 mm	Effective depth
f'c	25 MPa	Concrete strength
fy	415 MPa	Steel yield strength

Hint 1

Two unknowns, A_s and the stress-block depth a, are coupled through two equations.

Hint 2

Start by assuming a lever arm of about 0.9d, find A_s, then compute a and refine.

Hint 3

a = A_s·f_y/(0.85·f'_c·b). Two iterations converge to within 1%.

Worked solution — try the problem first

Equilibrium of the rectangular stress block gives two coupled equations:

$M_u = \phi A_s f_y\left(d - \frac{a}{2}\right), \qquad a = \frac{A_s f_y}{0.85 f'_c b}$

Since $a$ depends on $A_s$ and vice versa, iterate.

Iteration 1. Guess $a \approx 0.1d = 50$ mm:

$A_s = \frac{200\times10^{6}}{0.90(415)(500 - 25)} = \frac{200\times10^{6}}{177{,}413} = 1127\ \text{mm}^2$

$a = \frac{1127(415)}{0.85(25)(300)} = \frac{467{,}705}{6375} = 73.4\ \text{mm}$

Iteration 2. With $a = 73.4$ mm, so $d - a/2 = 463.3$ mm:

$A_s = \frac{200\times10^{6}}{0.90(415)(463.3)} = \frac{200\times10^{6}}{173{,}022} = 1156\ \text{mm}^2$

$a = \frac{1156(415)}{6375} = 75.2\ \text{mm}$

Iteration 3. $d - a/2 = 462.4$ mm:

$A_s = \frac{200\times10^{6}}{0.90(415)(462.4)} = 1158\ \text{mm}^2$

Converged.

$\boxed{A_s \approx 1158\ \text{mm}^2}$

In practice: 4 No. 20 bars ( $4 \times 314 = 1256\ \text{mm}^2$ ) would be selected — always rounding up.

Check the minimum. $A_{s,min} = \max(1.4/f_y, \sqrt{f'_c}/(4f_y)) \cdot b d = (1.4/415)(300)(500) = 506\ \text{mm}^2$ . Our answer comfortably exceeds it.

Concepts:Reinforced concrete designWhitney stress blockFlexural strengthIterative design

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