Beam steel at midspan, limit state

Given: $b=300$ mm, $d=500$ mm, span $L=7$ m, dead load 45 kN/m (excluding self weight), M20, Fe415, effective cover to compression steel $d'=40$ mm. Limit state of collapse in flexure.

Assumption: overall depth $D=550$ mm ($d$ plus 50 mm cover); the effective span is taken as 7 m.

Step 1: Loads

$$\begin{aligned} \text{Self weight}&=0.30\times0.55\times25=4.12\ \text{kN/m}\\ w&=45+4.12=49.12\ \text{kN/m}\\ w_u&=1.5\times49.12=73.69\ \text{kN/m} \end{aligned}$$

Step 2: Design moment at mid-span

$$M_u=\frac{w_uL^2}{8}=\frac{73.69\times7^2}{8}=451.34\ \text{kN·m}$$

Step 3: Limiting moment of resistance (Fe415, $x_{u,max}=0.48d$)

$$M_{u,lim}=0.138f_{ck}bd^2=0.138\times20\times300\times500^2=206.95\ \text{kN·m}$$

Since $M_u>M_{u,lim}$ and the size is fixed, the section is doubly reinforced.

Doubly reinforced design

$$M_{u2}=M_u-M_{u,lim}=451.34-206.95=244.39\ \text{kN·m}$$

Limiting depth $x_{u,max}=0.48d=240.0$ mm. Strain in compression steel (similar triangles):

$$\varepsilon_{sc}=0.0035\left(1-\frac{d'}{x_{u,max}}\right)=0.0035\left(1-\frac{40}{240.0}\right)=0.00292$$

From the design stress-strain curve of Fe415 (IS 456 Fig. 23): $f_{sc}=353.2$ N/mm$^2$. Stress in concrete displaced by the steel: $f_{cc}=0.446f_{ck}=8.92$ N/mm$^2$.

$$A_{sc}=\frac{M_{u2}}{(f_{sc}-f_{cc})(d-d')}=\frac{244.39\times10^6}{(353.2-8.92)(500-40)}=1543\ \text{mm}^2$$ $$A_{st1}=\frac{0.36f_{ck}\,b\,x_{u,max}}{0.87f_y}=1436\ \text{mm}^2,\qquad A_{st2}=\frac{A_{sc}(f_{sc}-f_{cc})}{0.87f_y}=1471\ \text{mm}^2$$ $$A_{st}=A_{st1}+A_{st2}=2907\ \text{mm}^2$$

Provide:

• Tension steel: 6-25 mm dia ($2945$ mm$^2$), arranged in two layers.
• Compression steel: 5-20 mm dia ($1571$ mm$^2$) at $d'=40$ mm.
• Stirrups (8 mm, 2-legged) at spacing not more than $16\phi_c$, 300 mm and $b$ to hold the compression bars (shear design separately).

Checks: minimum steel $0.85bd/f_y=307$ mm$^2$ OK; $p_t=1.96\%<4\%$ and $p_c=1.05\%$ OK.

Answer: Mid-span $A_{st}=2907$ mm$^2$ (6-25 mm) and $A_{sc}=1543$ mm$^2$ (5-20 mm).