Purlins, girts and cold-formed sections

Thin cold-formed sections buckle in ways hot-rolled ones do not. This lesson works by hand to EN 1993-1-3: effective widths of the flat parts, distortional buckling of the lipped flange, the effective section and its resistance, and the rules for purlins restrained by sheeting. The STATIX cold-formed calculators work to SANS 10162-2 only.

A lesson by hand. The STATIX cold-formed purlin and cold-formed member calculators work to SANS 10162-2 (AS/NZS 4600, the Direct Strength Method) and have no EN 1993-1-3 route. This lesson therefore has no calculator step on its Eurocode route; its SANS twins, Steel Design 2 lesson 2 and lesson 3, use the pages.

1. Learning outcomes

After this lesson you can:

name the local, distortional and global buckling modes of a thin section;
work out the effective width of a flange and a lip (5.5.2, EN 1993-1-5 4.4);
find the spring stiffness, σ_cr,s and χ_d for an edge stiffener (5.5.3);
build the effective section and its compression resistance;
say when sheeting restrains a purlin and what EN 1993-1-3 10.1 checks.

2. Before you start

Lesson 1 and Steel Design 1 (Eurocode) lesson 2 (buckling curves).
Your own copy of EN 1993-1-3 and EN 1993-1-5; this lesson gives clause numbers only.
A pocket calculator (or a spreadsheet) and about 60 minutes.
The same ground on the SA code: Steel Design 2, lessons 2 and 3 (SANS 10162-2).

3. Cold-formed sections

Cold-formed sections. Purlins, girts and light members are rolled cold from thin galvanised strip, typically 1.2 to 3 mm. The lips stiffen the free edges of the flanges; zeds of different sizes nest at laps to make continuous purlins.

Three ways a thin section buckles. Local buckling: the flat plates ripple in short waves. Distortional buckling: the compressed flange and its lip rotate about the web-flange corner, in waves a few times the section depth. Global buckling: the whole member bends or twists, as in Steel Design 1 (Eurocode). EN 1993-1-3 5.5 covers the first two.

4. Geometry

Working on the midline. The calculation uses the dimensions to the middle of the thickness; the notional flat width b_p of each plate runs between the midpoints of the corners. If the inside bend radius is small (5.1(3)), the corners may be taken sharp, as in this lesson.

Geometry limits (5.2, Table 5.1). The design rules apply within these proportions; outside them the resistance must come from tests. A lip that is too short does not stiffen the flange (c/b ≥ 0.2).

5. Local buckling: effective widths

Effective width (5.5.2 with EN 1993-1-5 4.4). A slender plate in compression sheds load from its middle once it buckles; the effective width method replaces it by two strips next to its supported edges, each working at the full stress. For a plate supported on both edges under uniform compression, k_σ = 4.

The lip (5.5.3.2(5)). The lip is a plate with one free edge, so its buckling factor is much lower; for a short lip (b_p,c/b_p ≤ 0.35) k_σ = 0.5. Most lips of purlins are short and stocky enough to stay fully effective.

ε = √(235 / f_yb); λ̅_p = (b_p / t) / (28.4 ε √k_σ) internal plate (both edges supported), uniform compression: k_σ = 4; ρ = (λ̅_p − 0.22)/λ̅_p² ≤ 1 when λ̅_p > 0.673 outstand (lip), short: k_σ = 0.5; ρ = (λ̅_p − 0.188)/λ̅_p² ≤ 1 when λ̅_p > 0.748

6. Distortional buckling

Distortional buckling (5.5.3.1, 5.5.3.2). The compressed flange with its lip behaves like a small strut, held sideways by the web bending as a spring. Its elastic critical stress σ_cr,s comes from the spring stiffness K and the stiffener’s own second moment of area I_s.

K₁ = E t³ / [4(1 − ν²)] × 1 / (b₁² h_w + b₁³ + 0.5 b₁ b₂ h_w k_f) (5.10b) k_f = 0 when the other flange is in tension (bending); 1 for a symmetric section in compression σ_cr,s = 2 √(K E I_s) / A_s (5.15); λ̅_d = √(f_yb / σ_cr,s) χ_d = 1.0 (λ̅_d ≤ 0.65); 1.47 − 0.723 λ̅_d (0.65 < λ̅_d < 1.38); 0.66/λ̅_d (λ̅_d ≥ 1.38) (5.12)

The distortional reduction factor χ_d (5.12). The stiffener’s area is reduced by χ_d, expressed as a reduced thickness t_red = χ_d t of the flange part and the lip (5.5.3.2(12)). The calculation may be iterated with the reduced stress; one pass is on the safe side.

7. The effective section and its resistance

The effective cross-section. Its area A_eff (in compression) or its section modulus W_eff (in bending) times the basic yield strength f_yb gives the cross-section resistance. If the effective centroid moves, a compression member also carries the moment N e_N (6.1.3(3)). Global buckling then uses A_eff in λ̅ (6.2, as EN 1993-1-1 6.3).

8. Purlins and girts restrained by sheeting

Purlins restrained by sheeting (EN 1993-1-3 10.1). Trapezoidal sheeting fixed through its troughs restrains the connected flange laterally when its shear stiffness meets (10.1a), and partly restrains twist. Under gravity the restrained flange is in compression and the cross-section resistance governs; under uplift the free flange is in compression and bends sideways, which 10.1.3 and 10.1.4 check with a lateral spring. The rules apply within h/t, c/t and d/t limits (10.1.1(1)).

Side rails (girts) follow the same rules, with wind pressure acting like gravity on a purlin and wind suction like uplift (10.1.1(4)). The free flange check under uplift (10.1.3, 10.1.4) needs the lateral spring of the sheeting and the purlin’s distortion; that, and the simplified method of Annex E for Z and C purlins, are beyond this lesson’s hand examples.

Values used in this lesson: S350GD (f_yb 350 MPa), E 210 000 MPa, ν 0.3, and the recommended γ_M0 = 1.0 of EN 1993-1-3 2(3). National annexes may differ.

9. The order of work

The order of work, by hand. The STATIX cold-formed purlin and cold-formed member calculators work to SANS 10162-2 (the Direct Strength Method) only; they have no EN 1993-1-3 route, so this lesson works by hand. Steel Design 2 lessons 2 and 3 use those pages on the SA code.