Compression members and columns

From pulling to pushing. A column can fail long before its steel yields, by buckling. This lesson shows why, how the buckling length, the slenderness and the buckling curves of EN 1993-1-1 set the resistance, and how bracing one axis helps, by hand and on the Eurocode 3 route of the STATIX steel member calculator.

1. Learning outcomes

After this lesson you can:

explain why real columns buckle below the squash load and the Euler load;
choose buckling lengths for each axis from the end restraints and the bracing;
work out the non-dimensional slenderness λ̅, pick the buckling curve and find χ;
find the buckling resistance N_b,Rd about both axes and say which governs;
check a column in the STATIX steel member calculator, and say which national choices it makes.

2. Before you start

Lesson 1 (tension members) and the second moment of area.
The free STATIX steel member calculator, set to Eurocode 3 (EN 1993-1-1).
A pocket calculator and about 50 minutes.
The same lesson on the SA code: Steel Design 1, lesson 2 (SANS 10162-1).

3. What a strut does

What a strut does. A short, stocky member is squashed until the whole section yields. A long, slender one bows sideways and buckles at a much lower load, while most of the steel is still elastic. Most columns are somewhere in between.

The elastic critical force N_cr = π²EI/L_cr² (the Euler load) is an upper limit that a real column never reaches. Real columns are not perfectly straight, their loads are not perfectly central, and rolling or welding leaves residual stresses in them; so they fail below both N_cr and the squash load. The buckling curves of EN 1993-1-1 build these effects in.

4. Buckling length

The buckling length L_cr depends on the end restraints. These are the theoretical values; real bases and joints are never fully fixed, so designers use larger values for “fixed” ends. EN 1993-1-1 has no table of buckling lengths: they come from the engineer’s judgement or from a frame analysis. The calculator takes L_cr per axis, as a factor K × L or typed.

5. Slenderness

Slenderness. Each axis has its own radius of gyration i. The non-dimensional slenderness λ̅ compares the squash load with the elastic critical force; it is the ratio L_cr/i divided by λ₁, which depends only on the steel grade (6.3.1.3).

N_cr = π² E I / L_cr² E = 210 000 MPa (3.2.6) λ̅ = √(A f_y / N_cr) = (L_cr / i) / λ₁ λ₁ = π√(E/f_y) = 93.9 ε (6.3.1.3)

6. The buckling curves

The buckling curves. The reduction factor χ falls as the slenderness rises. Curve a0 (not drawn) and curve a are for sections with low residual stresses and imperfections, d for the worst; the imperfection factor α sets which curve. Up to λ̅ = 0.2 there is no reduction.

Why the curves differ. A column is never perfectly straight, and the flange tips of a rolled section cool first and keep compressive residual stresses. Both make it yield and buckle earlier. Heavy, thick or welded sections have more of both and get a lower curve.

The buckling resistance in four steps (6.3.1.1 and 6.3.1.2). Work out λ̅ for each axis, then Φ with the imperfection factor α of the curve, then χ, and the buckling resistance N_b,Rd = χ A f_y/γ_M1 (class 1, 2 and 3 sections). The calculator uses the recommended γ_M1 = 1.0.

Φ = 0.5 [1 + α(λ̅ − 0.2) + λ̅²] χ = 1 / [Φ + √(Φ² − λ̅²)] ≤ 1.0 (6.49) N_b,Rd = χ A f_y / γ_M1 (class 1, 2, 3) (6.47) α = 0.13, 0.21, 0.34, 0.49, 0.76 for curves a0, a, b, c, d (Table 6.1)

Choosing the curve (Table 6.2). The curve depends on the type of section, its proportions, the axis and, for S460, the grade. The calculator’s “auto” reads Table 6.2: rolled I and H sections by h/b and the flange thickness; hollow sections by how they are made (its input “Hollow section made”: hot-finished curve a, cold-formed curve c); other shapes curve c. You can also choose a curve per axis.

7. Two axes and bracing

Two axes. An H column is much stiffer about its strong axis y than about its weak axis z, so it usually buckles about z. A brace at mid-height that stops movement about the weak axis halves L_cr,z; the strong axis is unchanged. Check both axes: the smaller N_b,Rd governs.

8. Local buckling and the class

Local buckling and the section class (5.5, Table 5.2). Thin flanges or webs can buckle locally before the column buckles as a whole. The limits on c/t scale with ε. Most rolled H columns are class 1 to 3 in compression and use their whole area; a class 4 section uses an effective area from EN 1993-1-5. For class 4 I, H and hollow sections the calculator works out A_eff (EN 1993-1-5, 4.4) and uses it in N_Rk and in λ̅ (6.50).

What the steel member calculator’s Eurocode 3 route uses for columns: γ_M1 = 1.0 (recommended; some national annexes use 1.1), E = 210 000 MPa, f_y by the thickest element (EN 10025-2: 355 MPa up to 16 mm for S355), flexural buckling about each axis on its own curve (6.3.1.2), and Table 5.2 for the class. It also checks torsional buckling of doubly symmetric I and H sections (6.3.1.4), which rarely governs for rolled columns. EN 1993-1-1 sets no slenderness limit for columns, and the Eurocode route shows none.

9. In STATIX

The compression route in the STATIX steel member calculator: set the code to Eurocode 3, pick the grade and section, enter N as a positive number, set the buckling lengths (one length, K factors, or a typed length per axis) and the curves (auto or chosen), and read N_b,Rd for both axes.