Example 02: Finding the Number of 28-mm Steel Bars of Singly-Reinforced Concrete Cantilever Beam

For Balanced Capacity
$f_c = 0.45f_c' = 0.45(21) = 9.45 ~ \text{MPa}$

$f_s = 165 ~ \text{MPa}$
 

$\dfrac{x_{bal}}{f_c} = \dfrac{d}{f_c + f_s/n}$

$\dfrac{x_{bal}}{9.45} = \dfrac{685}{9.45 + 165/9}$

$x_{bal} = 233 ~ \text{mm}$
 

$C_{bal} = \frac{1}{2}f_c \, b x_{bal} = \frac{1}{2}(9.45)(400)(233)$

$C_{bal} = 440.37 ~ \text{kN}$
 

$M_{bal} = C_{bal}(d - \frac{1}{3}x_{bal}) = 440.37[ \, 0.685 – \frac{1}{3}(0.233) \, ]$

$M_{bal} = 267.45 ~ \text{kN}\cdot\text{m}$
 

M_max < M_bal, thus, the beam is singly-reinforced.
 

For Singly Reinforced Section
$f_s = 165 ~ \text{MPa}$

$f_c = ~ ?$
 

Accurate Solution for finding A_s (Not recommended)

Approximate Solution for finding A_s (Recommended)
$M_{max} = T(d - \frac{1}{3}x_{bal})$

$M_{max} = A_sf_s(d - \frac{1}{3}x_{bal})$

$A_s = \dfrac{M_{max}}{f_s(d - \frac{1}{3}x_{bal})}$

$A_s = \dfrac{232.4(1000^2)}{165[ \, 685 - \frac{1}{3}(233) \, ]}$

$A_s = 2319.13 ~ \text{mm}^2$
 

Number of bars
$N = \dfrac{A_s}{A_b} = \dfrac{2319.13}{\frac{1}{4}\pi (28^2)}$

$N = 3.76$

Use 4 - 28 mm ø bars           answer