WSM stresses under 50 kNm moment

Given: $b=300$ mm, $d=550$ mm, 4-16 mm dia, $A_{st}=4\times\frac{\pi}{4}(16)^2=804.2$ mm$^2$, $M=50$ kN·m. Permissible: $\sigma_{cbc}=5$, $\sigma_{st}=140$ N/mm$^2$. For M20, modular ratio $m=13.33$ (IS 456, Annex B-1.3).

Step 1: Neutral axis (elastic cracked section)

$$\frac{b x^2}{2}=m A_{st}(d-x)\ \Rightarrow\ \frac{300x^2}{2}=13.33\times 804.2(550-x)$$ $$x=165.72\ \text{mm}$$

Step 2: Lever arm

$$z=d-\frac{x}{3}=550-\frac{165.72}{3}=494.76\ \text{mm}$$

Step 3: Stresses

$$\begin{aligned} \sigma_{c}&=\frac{2M}{b\,x\,z}=\frac{2\times 50\times10^6}{300\times 165.72\times 494.76}=4.07\ \text{N/mm}^2\\ \sigma_{s}&=\frac{M}{A_{st}\,z}=\frac{50\times10^6}{804.2\times 494.76}=125.66\ \text{N/mm}^2 \end{aligned}$$

Both are less than the permissible values (5 and 140 N/mm$^2$), so the section is safe.

Answer: stress in extreme compression concrete $=4.07$ N/mm$^2$; stress in tension steel $=125.66$ N/mm$^2$.