Beam design with deflection, 5.5 m span

Given: $L=5.5$ m, $b=230$ mm, $D=550$ mm, superimposed load 50 kN/m (excluding self weight), Fe415, mild exposure, effective cover 50 mm. The grade of concrete is not stated, so M20 (the minimum for mild exposure, IS 456 Table 5) is adopted. $d=550-50=500$ mm.

Step 1: Loads and moment

$$\begin{aligned} \text{Self weight}&=0.23\times0.55\times25=3.16\ \text{kN/m},\quad w=53.16\ \text{kN/m}\\ w_u&=1.5w=79.74\ \text{kN/m}\\ M_u&=\frac{w_uL^2}{8}=\frac{79.74\times5.5^2}{8}=301.53\ \text{kN·m} \end{aligned}$$

Step 2: Limiting moment

$$M_{u,lim}=0.138f_{ck}bd^2=0.138\times20\times230\times500^2=158.66\ \text{kN·m}<M_u$$

The section is doubly reinforced (the size is fixed).

Doubly reinforced design

$$M_{u2}=M_u-M_{u,lim}=301.53-158.66=142.87\ \text{kN·m}$$

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

$$\varepsilon_{sc}=0.0035\left(1-\frac{d'}{x_{u,max}}\right)=0.0035\left(1-\frac{50}{240.0}\right)=0.00277$$

From the design stress-strain curve of Fe415 (IS 456 Fig. 23): $f_{sc}=351.9$ N/mm$^2$. Stress in concrete displaced by the steel: $f_{cc}=0.446f_{ck}=8.92$ N/mm$^2$.

$$A_{sc}=\frac{M_{u2}}{(f_{sc}-f_{cc})(d-d')}=\frac{142.87\times10^6}{(351.9-8.92)(500-50)}=926\ \text{mm}^2$$ $$A_{st1}=\frac{0.36f_{ck}\,b\,x_{u,max}}{0.87f_y}=1101\ \text{mm}^2,\qquad A_{st2}=\frac{A_{sc}(f_{sc}-f_{cc})}{0.87f_y}=879\ \text{mm}^2$$ $$A_{st}=A_{st1}+A_{st2}=1980\ \text{mm}^2$$

Provide: tension 5-25 mm dia ($2454$ mm$^2$), compression 5-16 mm dia ($1005$ mm$^2$), with 8 mm stirrups.

Step 3: Deflection check (IS 456 cl. 23.2)

$$p_t=\frac{100A_{st}}{bd}=2.13\%,\quad f_s=0.58f_y\frac{A_{st,req}}{A_{st,prov}}=194.2\ \text{N/mm}^2,\quad k_t\approx0.95\ \text{(Fig. 4)}$$ $$p_c=0.87\%,\quad k_c=\frac{1.6p_c}{p_c+0.275}=1.22$$ $$\left(\frac{L}{d}\right)_{allow}=20\times k_t\times k_c=20\times0.95\times1.22=23.2,\qquad \left(\frac{L}{d}\right)_{actual}=\frac{5500}{500}=11.00$$

The beam is safe in deflection.

Answer: $M_u=301.53$ kN·m, $A_{st}=1980$ mm$^2$ (5-25 mm), $A_{sc}=926$ mm$^2$ (5-16 mm); safe in deflection.