Code-Based Design of Steel Compression Members

Euler's formula explains ideal elastic buckling, but the design resistance of a real steel compression member is determined using code buckling curves. These curves account implicitly for initial crookedness, residual stresses, cross-section geometry, fabrication method and buckling axis.

Eurocode 3 design check

For a uniform member in compression, EN 1993-1-1 expresses the buckling resistance as

\[N_{b,Rd}=\frac{\chi A f_y}{\gamma_{M1}},\]

for Class 1, 2 or 3 cross-sections. For a Class 4 cross-section, the effective area \(A_{eff}\) is used as required by the standard. The design condition is

\[N_{Ed}\le N_{b,Rd}.\]

Elastic critical load and nondimensional slenderness

For each possible buckling plane, determine the elastic critical load

\[N_{cr}=\frac{\pi^2EI}{L_{cr}^2}.\]

The nondimensional slenderness is then

\[\bar\lambda=\sqrt{\frac{A f_y}{N_{cr}}}\]

for Class 1–3 sections. Both principal axes and any relevant torsional or flexural-torsional mode must be considered.

Buckling reduction factor χ

Unlike the Ukrainian DBN notation \(\varphi\), Eurocode 3 uses the symbol \(\chi\). It is calculated from

\[\chi=\frac{1}{\Phi+\sqrt{\Phi^2-\bar\lambda^2}}\le1,\]

\[\Phi=\frac12\left[1+\alpha(\bar\lambda-0.2)+\bar\lambda^2\right],\]

where \(\alpha\) is the imperfection factor associated with the selected buckling curve. The curve is chosen from the code tables according to section type, buckling axis, steel grade and fabrication route.

Eurocode calculation sequence

1. Determine \(N_{Ed}\), material strength and cross-section class.
2. Calculate \(A\), \(I_y\), \(I_z\) and the relevant radii of gyration.
3. Determine the buckling lengths or elastic critical loads for all relevant modes.
4. Calculate \(\bar\lambda\) for each mode.
5. Select the appropriate Eurocode buckling curve and imperfection factor \(\alpha\).
6. Calculate \(\chi\) and \(N_{b,Rd}\).
7. Verify \(N_{Ed}/N_{b,Rd}\le1\) for the governing mode.
8. Check cross-section resistance, local buckling, member slenderness and any interaction with bending.

Relation to the φ method

The DBN coefficient \(\varphi\) and the Eurocode coefficient \(\chi\) serve a similar purpose: both reduce the ideal cross-sectional compression resistance to account for member instability. Their numerical values must not be interchanged because the definitions of slenderness, resistance and safety factors belong to different code systems.

Scope and limitations

This procedure applies to the basic member-buckling check. Beam-columns, built-up members, frames with sway, non-uniform members and torsional or flexural-torsional buckling require the additional clauses of the applicable standard and its National Annex.

European Commission Joint Research Centre — Eurocodes. See EN 1993-1-1, member buckling provisions and the applicable National Annex.