Linear buckling

Critical load factors and buckling shapes — what α_cr means, why it is an upper bound, and how to use it for second-order effects.

Analysis types 3 min read

Linear buckling analysis (also called linear eigenvalue or Euler buckling analysis) solves

(K + α K_{g}) φ = 0

where $K_{g}$ is the geometric stiffness matrix formed from the axial forces under a reference load state. The eigenvalues $α$ are critical load factors: multiply the reference loading by $α_{cr}$ and the structure becomes unstable in the corresponding buckling shape $φ$ .

$α_{cr}$ belongs to one load state and nothing else. A combination that resolves to several concrete states (an envelope, or an EN 1990 rule with each variable action leading in turn) contributes one run per permutation. The envelope's answer is the governing permutation's, and the only way to know which that is, is to solve them.

What it tells you

$α_{cr} = 8$ means the applied loading would have to increase eightfold before the structure buckles elastically in that mode. It measures margin against instability and is a direct input to the code's rules on second-order effects.

For an isolated column the first factor reproduces the Euler load, with an effective length set by the end conditions:

P_{cr} = \frac{π ^{2} E I}{L _{cr}^{2}}

First buckling mode and effective length for the four classical end conditions. The fixed–fixed column's top is held against rotation but free to move down with the load.

What it does not tell you

This is why codes do not let you design to $α_{cr}$ . They use it as a screening quantity; member resistance comes from buckling curves that already account for imperfection and yielding — see Steel members.

The standard use: does second-order behaviour matter?

EN 1993-1-1 classifies a frame by its critical load factor:

$α_{cr}$	Classification	Consequence
≥ 10	Non-sway	Second-order effects may be neglected
3 to 10	Moderately sway-sensitive	Amplify sway effects, or analyse second-order
< 3	Very sway-sensitive	A proper second-order analysis is required

The threshold depends on the analysis:

α_{cr} \geq 10 (elastic analysis), α_{cr} \geq 15 (plastic analysis)

Below the threshold the code permits two options: the amplification rules applied by hand, with the assumption stated in the report, or a second-order analysis. That is in this release, and it can take the buckling shape you just computed as its §5.3.2(11) imperfection.

Running it

Solve a linear static analysis first, and be satisfied with it. The geometric stiffness is built from those axial forces, so a wrong static result gives a meaningless buckling result.
Choose the reference load case or combination. This matters more than anything else: $α_{cr}$ is relative to that loading, and the same structure gives different factors under different combinations. Use the ULS combination that maximises compression.
Choose the number of modes. The first three to five are usually enough to see the pattern.
Leave the completeness check on. It is a Sturm-sequence proof that the modes returned really are the lowest ones. A skipped first mode is an $α_{cr}$ reported too high, which is unsafe rather than merely inaccurate.
Solve and look at the shapes.

Reading the modes

Shape	Interpretation
Whole frame sways sideways	Global sway instability — this is the one for the $α_{cr}$ classification
One column bows between floors	Member buckling; compare with the member's own check
A single slender brace deflects	Often a member that should be tension-only
Local rippling of a plate	Plate buckling; note that member checks may not cover it

A very low first factor attached to one insignificant member usually means a modelling issue rather than a real stability problem: a member with almost no restraint, or a tension-only member, such as a bracing rod, modelled as a beam that can take compression.

Negative and very large factors

Negative $α_{cr}$ means the structure buckles when the reference loading is reversed. Usually harmless and ignorable, unless load reversal is real.
Very large $α_{cr}$ (hundreds) means stability is simply not a concern for that loading — a stiff, braced structure.

Reporting

Modal analysis solves a similar eigenvalue problem with the mass matrix in place of the geometric stiffness matrix; Second-order analysis is what you run when $α_{cr}$ says the screening number is not enough; Linear static explains what the underlying analysis assumes.

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