Contents
The beam
A floor beam, simply supported over L = 6.00 m. The floor holds the compression flange laterally, so lateral-torsional buckling cannot occur. The loads per metre are a permanent load gk = 5.0 kN/m, including self-weight, and a variable load qk = 7.5 kN/m on a floor in category A (residential). The profile is an IPE 300 in S235.
Step 1: the design load
For the ultimate limit state EN 1990 combines the loads with partial factors. With the recommended values of equation (6.10):
The Dutch national annex, consequence class CC2, takes the less favourable of equations 6.10a and 6.10b instead. For a floor in category A, ψ0 = 0.4:
The example continues with 18.0 kN/m; the results with the Dutch annex follow at the end.
Step 2: material and cross-section
S235 with a thickness of at most 40 mm has fy = 235 N/mm² (Table 3.1), so ε = √(235 / fy) = 1.0. The partial factor γM0 = 1.0, both as the recommended value and in the Dutch annex. E = 210 000 N/mm².
| Property | Value |
|---|---|
| Depth h, width b | 300 mm, 150 mm |
| Web tw, flange tf, root radius r | 7.1 mm, 10.7 mm, 15 mm |
| Area A | 53.81 cm2 |
| Second moment of area Iy | 8356 cm4 |
| Plastic section modulus Wpl,y | 628.4 cm3 |
| Shear area Av,z | 25.68 cm2 |
Step 3: the cross-section class
The class decides whether the plastic resistance may be used. Table 5.2 compares the width-to-thickness ratio c/t of each compressed part with limits that depend on ε.
Step 4: bending
Step 5: shear
For a rolled I-section loaded parallel to the web, §6.2.6(3)a gives the shear area.
Two more conditions. The web needs no shear buckling check, because hw / tw = 278.6 / 7.1 = 39.2 ≤ 72ε / η = 60 (§6.2.6(6), with η = 1.2 from EN 1993-1-5; η = 1.0 is allowed as a conservative value). And the shear force does not reduce the bending resistance, because VEd = 54.0 kN ≤ 0.5 Vpl,Rd = 174.2 kN (§6.2.8); at midspan, where the moment is largest, the shear force is zero anyway.
Step 6: deflection
Deflection is a serviceability check, with the characteristic loads and without partial factors.
EN 1990 leaves the deflection limits to the national annex and the client, so L/250 is only an example here.
The result
| Check | Clause | Unity check |
|---|---|---|
| Bending | EN 1993-1-1 §6.2.5 | 0.55 |
| Shear | EN 1993-1-1 §6.2.6 | 0.16 |
| Deflection, L/250 | EN 1990 A1.4 | 0.50 |
Bending governs, with a unity check of 0.55: the IPE 300 is far from fully used. With the Dutch annex the design load is 17.25 kN/m, MEd = 77.6 kNm and the bending check 0.53; shear becomes 0.15.
An IPE 270, also class 1 (Wpl,y = 484.0 cm3, Iy = 5790 cm4), gives a bending check of 81.0 / 113.7 = 0.71 and a deflection of 17.3 mm, 0.72 of L/250.
What this example leaves out
- Lateral-torsional buckling (§6.3.2). Without lateral restraint of the compression flange, the buckling resistance moment Mb,Rd = χLT Wy fy / γM1 (6.55) takes the place of the cross-section resistance.
- Supports and concentrated loads. The web has to carry the reactions without crippling (EN 1993-1-5 §6).
- Vibration of the floor, a separate serviceability criterion.
- Fire resistance (EN 1993-1-2).
The same check in XBeam2D
XBeam2D does this calculation for beams on any number of supports, in steel, concrete and timber, and checks steel members including lateral-torsional buckling. The report shows each check the way this article does: the clause, the formula, the values that went in and the unity check. A reviewer can follow it, and you can recheck any number by hand.
- EN 1990:2002+A1:2005, Basis of structural design: equation (6.10) and Annex A1 (in the Netherlands NEN-EN 1990+A1+A1/C2:2011 with its national annex: 6.10a and 6.10b, consequence classes).
- EN 1993-1-1:2005, Eurocode 3, Part 1-1: Table 3.1, §5.5, Table 5.2, §6.2.5, §6.2.6, §6.2.8, §6.3.2.
- EN 1993-1-5:2006, Plated structural elements: §5 (shear buckling) and §6 (resistance to transverse forces).
- Profile data: hot-rolled IPE sections to EN 10365.