Beam supporting RCC slab, DL 10 kN/m, IL 30 kN

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 $L = 6.00$ m.

Loads and design forces

Span $L = 6$ m. The "30 kN" imposed load is read as 30 kN/m (a UDL, like the dead load of 10 kN/m); dead load is taken to exclude the beam's own weight. Factored load $w_u = 1.5(10 + 30) = 60$ kN/m, plus factored self-weight. The beam is "sufficiently stiff against bearing", so the web buckling and bearing checks are not required.

Moment without self-weight $M = 270.0$ kN·m, so $Z_{p,req} \approx M\gamma_{m0}/f_y = 270.0\times10^6\times1.1/250 = 1188$ 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 $w = 60.00 + 1.07 = 61.07$ kN/m.

$$\begin{aligned} M_u &= \frac{wL^2}{8} = \frac{61.07\times6.00^2}{8} = 274.8\ \text{kN·m} \\ V_u &= \frac{wL}{2} = 183.2\ \text{kN} \end{aligned}$$

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)

The compression flange is restrained by the RCC slab along its full length, so lateral-torsional buckling is prevented.

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

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

$$M_d = 348.7\ \text{kN·m} > M_u = 274.8\ \text{kN·m}\quad \text{Safe}$$

Shear capacity (cl. 8.4)

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

$V_u = 183.2$ 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):

$$\delta = \frac{5 w L^4}{384 E I} = \frac{5\times30.00\times(6000)^4}{384\times2\times10^5\times30400\times10^4} = 8.33\ \text{mm}$$

$\delta = 8.33$ mm $< L/300 = 20.00$ mm. OK.

Web bearing/buckling at supports is not checked because the question states the beam is sufficiently stiff against bearing.

Final design

Provide ISMB 450 (72.4 kg/m): $M_d = 348.7$ kN·m $> M_u = 274.8$ kN·m, $V_d = 555$ kN $> V_u = 183.2$ kN, $\delta = 8.3$ mm $< 20.0$ mm.