Buckling

What this calculator checks

The calculator determines the buckling resistance of the compression member, \(N_{b,Rd}\), according to EN 1993-1-1 §6.3.1, expression (6.47):

\(N_{b,Rd} = \frac{\chi \cdot A \cdot f_y}{\gamma_{M1}}\)

and checks that \(N_{Ed} / N_{b,Rd} \le 1.0\) (expr. 6.46).

This is the governing check for a real compression member — the one missing from the cross-section compression check. The difference is the reduction factor \(\chi \le 1\): the more slender the member, the lower the force at which it buckles, below the section capacity.

The three buckling modes

A compression member can fail in three ways, and the calculator checks all of them; the governing one has the smallest \(\chi\):

1. Flexural buckling about the y-y axis — bending about the major axis; slenderness \(\bar\lambda_y = L_{cr,y} / (i_y \, \lambda_1)\)
2. Flexural buckling about the z-z axis — about the minor axis; usually governs for I-profiles, where \(i_z \ll i_y\)
3. Torsional or flexural-torsional buckling — the member twists, possibly combined with bending; relevant for sections with low torsional stiffness (open, cruciform, angle sections)

For each mode, the normalised slenderness \(\bar\lambda\) enters a buckling curve (a₀, a, b, c, d — Table 6.2), selected from the section shape and buckling axis. The curve gives \(\chi\) through expression (6.49):

\(\chi = \frac{1}{\Phi + \sqrt{\Phi^2 - \bar\lambda^2}} \le 1, \qquad \Phi = 0.5\left[1 + \alpha(\bar\lambda - 0.2) + \bar\lambda^2\right]\)

The imperfection factor \(\alpha\) depends on the curve and penalises real deviations from the ideal member: residual stresses from rolling, initial out-of-straightness, load eccentricity.

Buckling length

The key inputs are the buckling lengths \(L_{cr,y}\), \(L_{cr,z}\) and \(L_{ET}\) (for torsion), not the physical length of the member. They depend on the end restraints:

• pinned–pinned: \(L_{cr} = L\)
• fixed–free (cantilever): \(L_{cr} = 2L\)
• fixed–fixed: \(L_{cr} = 0.5L\)

A member may have different lengths about the two axes — for instance a column laterally braced at mid-height about the weak direction has \(L_{cr,z} = L/2\) but \(L_{cr,y} = L\).

Input data

• \(N_{Ed}\) — the design compressive axial force, in kN.
• Profile and section — the area, radii of gyration \(i_y\), \(i_z\) and torsion constants (\(I_T\), \(I_w\)) are taken from the catalogue.
• \(L_{cr,y}\), \(L_{cr,z}\), \(L_{ET}\) — the buckling lengths for each mode, in mm.
• Steel grade — \(f_y\) per EN 1993-1-1 Table 3.1, as a function of thickness.
• \(\gamma_{M1}\) — the partial factor for stability checks; the recommended value is 1.0, but the National Annex takes precedence. Note that stability uses \(\gamma_{M1}\), not \(\gamma_{M0}\) as for the cross-section.

Assumptions and limitations

• Concentric compression. For compression with bending, see bending and compression.
• The cross-section class is assumed known; for Class 4 the effective area is used.
• Buckling lengths are given as input — the calculator does not itself derive the static scheme of the member in a frame.
• Tapered members and members with intermediate elastic restraints are not treated.

Related calculations

• Compression resistance — the section capacity, the upper bound of \(N_{b,Rd}\)
• Bending and compression — the \(N\)–\(M\) interaction for stability
• Cross-section class — decides whether gross or effective area is used
• Lateral-torsional buckling — the equivalent for beams in bending