Intermediate beam design for BM and deflection

Assumptions: M20 and Fe415; beam simply supported with effective span $L=5$ m; slab panels 4 m x 4 m, so an intermediate beam carries slab load from 2 m on each side (width 4 m); beam web $b_w=300$ mm, overall depth $D=500$ mm, cover 50 mm, so $d=450$ mm. The slab and beam act together as a T-beam.

Step 1: Loads per metre run of beam

$$\begin{aligned} \text{Slab dead load}&=0.125\times25=3.125\ \text{kN/m}^2\\ \text{Floor finish}&=0.7\ \text{kN/m}^2,\quad\text{Live load}=4\ \text{kN/m}^2\\ \text{Total on slab}&=7.825\ \text{kN/m}^2\ \Rightarrow\ 7.825\times4=31.30\ \text{kN/m}\\ \text{Rib self weight}&=0.30\times(0.50-0.125)\times25=2.812\ \text{kN/m}\\ w&=34.11\ \text{kN/m},\qquad w_u=1.5w=51.17\ \text{kN/m} \end{aligned}$$

Step 2: Bending moment

$$M_u=\frac{w_uL^2}{8}=\frac{51.17\times5^2}{8}=159.90\ \text{kN·m}$$

Step 3: Effective flange width (IS 456 cl. 23.1.2, T-beam)

$$b_f=\frac{l_0}{6}+b_w+6D_f=\frac{5000}{6}+300+6\times125=1883\ \text{mm}\ (\le4000\ \text{mm spacing})$$

Step 4: Position of neutral axis. Moment capacity when $x_u=D_f$:

$$M_{u,f}=0.36f_{ck}b_fD_f(d-0.42D_f)=673.76\ \text{kN·m}>M_u$$

So $x_u<D_f$; design as a rectangular beam of width $b_f$.

Step 5: Tension steel

$$M_u=0.87f_yA_{st}d\left(1-\frac{A_{st}f_y}{b_fdf_{ck}}\right)\ \Rightarrow\ A_{st}=1009\ \text{mm}^2$$

Minimum steel $0.85b_wd/f_y=277$ mm$^2$ is satisfied. Provide 4-20 mm dia ($1257$ mm$^2$).

Step 6: Deflection check (IS 456 cl. 23.2). $b_w/b_f=0.16<0.3$, so the flange factor (Fig. 6) is 0.8. For flanged beams, the steel percentage for Fig. 4 is based on $b_fd$:

$$p_t=\frac{100A_{st}}{b_fd}=\frac{100\times1257}{1883\times450}=0.15\%,\quad f_s=0.58f_y\frac{A_{st,req}}{A_{st,prov}}=193.2\ \text{N/mm}^2,\quad k_t=2.0\ \text{(maximum)}$$ $$\left(\frac{L}{d}\right)_{allow}=20\times2.0\times0.8=32.0,\qquad \left(\frac{L}{d}\right)_{actual}=\frac{5000}{450}=11.11$$

The beam is safe in deflection. (Shear: $V_u=w_uL/2=127.9$ kN; stirrups are designed separately.)

Answer: T-beam, $b_f=1883$ mm, $b_w=300$ mm, $D=500$ mm; $A_{st}=1009$ mm$^2$ (4-20 mm bars); safe in deflection.