I-section purlin for industrial building

Approach. The purlin is a simply supported beam bent about both axes, because the loads are inclined to its principal axes. Design to IS 800:2007 (cl. 8.2 and 9.3), with loads from IS 875.

Data

• Steel Fe 410 (E250): $f_y = 250$ N/mm², $\gamma_{m0} = 1.10$. Purlin span $L = 5.0$ m (truss spacing); inclination of rafter $\alpha = 30.00^\circ$ ($\cos\alpha = 0.866$, $\sin\alpha = 0.500$).
• Sag rods are provided at mid-span to restrain the purlin in the plane of the roof. The 1 kN/m² wind load is taken as the net design pressure normal to the roof (acting either towards or away from the roof). Spacing of purlins along the slope = 1.5 m; trusses at 5 m.

Loads per metre run of purlin

• Sheeting: $0.130\times1.5 = 0.195$ kN/m (dead, vertical), plus the self-weight of the purlin (trial section below).
• Live load (IS 875 Part 2, roof slope above 10°): $0.75 - 0.02(\alpha - 10) = 0.35$ kN/m², but not less than 0.40 kN/m² on plan, so use $0.40$ kN/m². Per metre of purlin: $0.40\times1.5\cos\alpha = 0.520$ kN/m (vertical).
• Wind: given wind pressure $1.0$ kN/m² $\times 1.5$ m $= 1.500$ kN/m, acting normal to the roof.

Trial section and capacities

Try ISMB 125: $D = 125$, $b_f = 70$, $t_w = 5.0$, $t_f = 8.0$ mm, $I_{zz} = 445$ cm⁴, $I_{yy} = 38.5$ cm⁴, $Z_{ez} = 71.2$ cm³, mass 13.3 kg/m (self-weight $0.130$ kN/m).

$$\begin{aligned} Z_{pz} &= b_ft_f(D - t_f) + t_w(D - 2t_f)^2/4 = 80.4\ \text{cm}^3,\quad M_{dz} = \frac{Z_{pz}f_y}{\gamma_{m0}} = 18.27\ \text{kN·m} \\ Z_{py} &= 2\cdot\frac{t_fb_f^2}{4} + \frac{(D - 2t_f)t_w^2}{4} = 20.3\ \text{cm}^3,\quad M_{dy} = \min(Z_{py}, 1.2Z_{ey})\frac{f_y}{\gamma_{m0}} = 3.00\ \text{kN·m} \end{aligned}$$

Section class: $b/t_f$ and $d/t_w$ are within the plastic limits ($9.4$ and $84$).

Load combinations and moments (IS 800 Table 4)

Normal component $w_n$ and tangential component $w_t$ (kN/m), with $M_z = w_nL^2/8$ and $M_y = w_t(L')^2/8$, where $L' = L/2$ (one sag rod at mid-span):

For the uplift case the bottom flange is in compression and unrestrained between sag rods ($L_{LT} = L/2$), so the lateral-torsional buckling capacity is used for $M_{dz}$. The greatest ratio is 0.59 $\le 1.0$. Safe in biaxial bending.

Shear: $V_{max} = 6.70$ kN $\ll V_d = D t_wf_y/(\sqrt3\gamma_{m0}) = 82.0$ kN. OK.

Deflection (unfactored wind load of 1.5 kN/m normal to the roof, limit span/150 for elastic sheeting, Table 6): $\delta = 5wL^4/(384EI) = 13.72$ mm $< L/150 = 33.3$ mm. OK.

Result

Provide an ISMB 125 purlin, simply supported on the rafters, with a sag rod (12 mm) at mid-span and cleat connections with two M12 bolts at each end.