Purlin for roof truss, 10 m span

Approach. Bi-axial bending of a simply supported purlin, IS 800:2007.

Data

• Steel Fe 410 (E250): $f_y = 250$ N/mm², $\gamma_{m0} = 1.10$. Purlin span $L = 4.0$ m (truss spacing); inclination of rafter $\alpha = 26.57^\circ$ ($\cos\alpha = 0.894$, $\sin\alpha = 0.447$).
• Roof span 10 m, pitch 1/4 (rise/span), so $\tan\alpha = 2.5/5 = 0.5$. Truss spacing (purlin span) 4 m. The vertical load of $500$ N/m (including self-weight) and the wind load of $2$ kN/m are taken as loads per metre of purlin. A sag rod at mid-span is assumed.

Loads per metre run of purlin

• Vertical load (including self-weight) $= 0.50$ kN/m; no separate live load is given.
• Wind load $= 2.00$ kN/m acting normal to the roof (taken as pressure for the downward case and as suction for the uplift case).

Trial section and capacities

Try ISMB 100 (IS 808:1989 values): $D = 100$, $b_f = 50$, $t_w = 4.7$, $t_f = 7.0$ mm, $I_{zz} = 183$ cm⁴, $I_{yy} = 12.9$ cm⁴, $Z_{ez} = 36.6$ cm³, mass 8.9 kg/m (self-weight is already included in the given load). Older steel tables (SP 6(1)) list ISMB 100 as a larger section (75 mm flange, 11.5 kg/m, $Z_{ez} = 51.5$ cm³); if those values are used, every check below is met with a bigger margin.

$$\begin{aligned} Z_{pz} &= b_ft_f(D - t_f) + t_w(D - 2t_f)^2/4 = 41.2\ \text{cm}^3,\quad M_{dz} = \frac{Z_{pz}f_y}{\gamma_{m0}} = 9.37\ \text{kN·m} \\ Z_{py} &= 2\cdot\frac{t_fb_f^2}{4} + \frac{(D - 2t_f)t_w^2}{4} = 9.2\ \text{cm}^3,\quad M_{dy} = \min(Z_{py}, 1.2Z_{ey})\frac{f_y}{\gamma_{m0}} = 1.41\ \text{kN·m} \end{aligned}$$

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 is unrestrained between sag rods ($L_{LT} = L/2$), so the lateral-torsional buckling capacity is used for $M_{dz}$. The greatest ratio is 0.98 $\le 1.0$. Safe in bi-axial bending.

Shear: $V_{max} = 7.34$ kN $\ll V_d = D t_wf_y/(\sqrt3\gamma_{m0}) = 61.7$ kN. OK.

Deflection (unfactored wind load, limit span/150 for elastic sheeting, Table 6): $\delta = 5wL^4/(384EI) = 18.21$ mm $< L/150 = 26.7$ mm. OK.

Result

Provide an ISMB 100 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.