I-section beam for 9 m x 24 m hall slab

Data

• Steel Fe 410 (E250): $f_y = 250$ N/mm², $f_u = 410$ N/mm², $E = 2\times10^5$ N/mm², $\gamma_{m0} = 1.10$ (IS 800:2007, Table 5). Limit state design.
• Effective span $L = 9.00$ m.

Loads and design forces

The hall is 9 m × 24 m with beams at 3 m c/c, so the beams span the 9 m width (effective span $L = 9$ m assumed c/c of supports). RCC unit weight 25 kN/m³ (assumed).

Dead load $= 9.00$ kN/m, live load $= 9.00$ kN/m. Factored $w_u = 1.5(9.00 + 9.00) = 27.00$ kN/m plus self-weight. Lateral support is adequate (given). Support width is not given, so the bearing check is omitted.

Moment without self-weight $M = 273.4$ kN·m, so $Z_{p,req} \approx M\gamma_{m0}/f_y = 273.4\times10^6\times1.1/250 = 1203$ cm³ (self-weight adds a little).

Trial section

Try ISMB 450 (IS 808): $D = 450$ mm, $b_f = 150$ mm, $t_w = 9.4$ mm, $t_f = 17.4$ mm, $R_1 = 15$ mm; $I_{zz} = 30400$ cm⁴; $Z_e = 1350$ cm³; $Z_p \approx B t_f (D - t_f) + t_w (D - 2t_f)^2/4 = 1534$ cm³; mass $72.4$ kg/m.

Self-weight $= 72.4\times9.81/1000 = 0.710$ kN/m, factored $= 1.5\times0.710 = 1.065$ kN/m. Total factored UDL $w = 27.00 + 1.07 = 28.07$ kN/m.

$$\begin{aligned} M_u &= \frac{wL^2}{8} = \frac{28.07\times9.00^2}{8} = 284.2\ \text{kN·m} \\ V_u &= \frac{wL}{2} = 126.3\ \text{kN} \end{aligned}$$

Section classification (Table 2)

$\varepsilon = \sqrt{250/f_y} = 1.00$. Flange: $b/t_f = (150/2)/17.4 = 4.31 < 9.4\varepsilon = 9.4$. Web: $d/t_w = 41.0 < 84\varepsilon = 84$ (with $d = D - 2(t_f + R_1) = 385.2$ mm). The section is plastic, so $\beta_b = 1.0$.

Moment capacity (cl. 8.2.1.2)

Adequate lateral support is given to the compression flange, so lateral-torsional buckling does not govern.

$$\begin{aligned} M_d &= \frac{\beta_b Z_p f_y}{\gamma_{m0}} = \frac{1534\times10^3\times250}{1.1\times10^6} = 348.7\ \text{kN·m} \\ 1.2 Z_e f_y/\gamma_{m0} &= 1.2\times1350\times10^3\times250/(1.1\times10^6) = 368.2\ \text{kN·m} \end{aligned}$$

The smaller value governs: $M_d = 348.7$ kN·m.

$$M_d = 348.7\ \text{kN·m} > M_u = 284.2\ \text{kN·m}\quad \text{Safe}$$

Shear capacity (cl. 8.4)

$$\begin{aligned} A_v &= D t_w = 450\times9.4 = 4230\ \text{mm}^2 \\ V_d &= \frac{A_v f_y}{\sqrt3\,\gamma_{m0}} = \frac{4230\times250}{\sqrt3\times1.1\times10^3} = 555.0\ \text{kN} \end{aligned}$$

$V_u = 126.3$ kN $< V_d$; also $V_u < 0.6V_d = 333.0$ kN, so the shear is low and the moment capacity is not reduced (cl. 9.2.1). Web shear buckling does not govern since $d/t_w = 41.0 < 67\varepsilon = 67$ (cl. 8.4.2.1).

Deflection (Table 6)

Deflection under unfactored service (imposed) loads, limit span/300 (floor and roof members, Table 6):

$$\delta = \frac{5 w L^4}{384 E I} = \frac{5\times9.00\times(9000)^4}{384\times2\times10^5\times30400\times10^4} = 12.65\ \text{mm}$$

$\delta = 12.65$ mm $< L/300 = 30.00$ mm. OK.

The support width is not given, so web bearing is not checked here (provide a bearing length that satisfies $F_w \ge R$, normally 150 to 200 mm).

Final design

Provide ISMB 450 (72.4 kg/m): $M_d = 348.7$ kN·m $> M_u = 284.2$ kN·m, $V_d = 555$ kN $> V_u = 126.3$ kN, $\delta = 12.6$ mm $< 30.0$ mm.