Buckling of Columns Explained: Euler, Causes and Limits

Press down on the end of a plastic ruler standing on a desk and, long before the plastic is crushed, the ruler bows sideways and gives way. That sudden sideways failure is buckling, and it is one of the most important ideas in structural engineering. A column can be made of perfectly good material, carry a load well below its crushing strength, and still fail because it is too slender to stay straight.

This article explains buckling from intuition upwards: why it happens, how the Euler formula describes it, what effective length means, a worked example with real numbers, and the mistakes students and homeowners most often make when they think about slender members. The aim is understanding, not a design procedure; real columns are checked by an engineer against the relevant code.

What buckling is, in plain terms

A short, stocky column fails by crushing. The material simply reaches its strength and yields or fractures. A long, thin column behaves differently. Under increasing compression it reaches a point where staying straight is no longer the most stable shape. Any tiny imperfection, a slight bow, an off-centre load, a knot in a timber post, gets amplified, and the column deflects sideways rapidly. That instability is buckling.

The key insight is that buckling is a stiffness problem, not a strength problem. What resists it is the column’s ability to resist bending, which depends on the material’s stiffness and the shape of the cross-section, not on how strong the material is in crushing. That is why a high-strength steel does not necessarily make a slender column much better at resisting buckling: steels of different grades have almost the same stiffness.

Buckling also tends to be sudden. A beam that is overloaded usually sags visibly and gives warning. A slender column near its buckling load can look perfectly straight until it is very close to failure, which is why engineers treat it with particular respect.

The Euler buckling formula

In the mid-eighteenth century the Swiss mathematician Leonhard Euler derived the classical solution for an ideal, perfectly straight, elastic column. His result for the critical load, the load at which the column becomes unstable, is usually written today as:

P_cr = π²EI / (KL)²

P_cr is the critical (Euler) buckling load.
E is Young’s modulus, the stiffness of the material. For structural steel it is roughly 200,000 to 210,000 N/mm²; for normal concrete it is far lower, typically around 30,000 N/mm².
I is the second moment of area of the cross-section about the axis it bends around. See our explanation of moment of inertia for how shape drives this number.
L is the actual length between supports, and K is the effective length factor that depends on how the ends are held.

Two features of the formula carry most of the insight. First, length appears squared in the denominator: double the length and the buckling load falls to a quarter. Second, the material’s strength does not appear at all. Only stiffness (E) and geometry (I and L) matter, which is the mathematical statement of the intuition above.

Effective length and end conditions

The factor K describes how the ends of the column are restrained. The textbook theoretical values are:

Both ends pinned (free to rotate, not to move sideways): K = 1.0.
Both ends fixed against rotation: K = 0.5.
One end fixed, one end pinned: K is approximately 0.7.
One end fixed, the other completely free, like a flagpole: K = 2.0.

Because KL is squared, the difference is large. A flagpole-type column has an effective length twice its real length and buckles at one quarter of the pinned-pinned load. A column fixed at both ends, in theory, carries four times the pinned-pinned load. Real connections are rarely perfectly fixed, so design codes generally recommend more conservative effective lengths than the ideal values.

A worked example

Take a steel column 3 m long, pinned at both ends, with E = 200,000 N/mm² and a second moment of area about its weaker axis of I = 1.0 × 10⁷ mm⁴. Work in newtons and millimetres:

KL = 1.0 × 3,000 = 3,000 mm, so (KL)² = 9.0 × 10⁶ mm².
π²EI = 9.87 × 200,000 × 1.0 × 10⁷ ≈ 1.97 × 10¹³ N·mm².
P_cr = 1.97 × 10¹³ / 9.0 × 10⁶ ≈ 2.19 × 10⁶ N, or about 2,190 kN.

If the same column were 6 m long, the critical load would fall to roughly 550 kN. If it were genuinely fixed at both ends at 3 m, the ideal value would rise to roughly 8,770 kN. These swings show why length and end restraint dominate column behaviour.

Note that the weak axis was used. A steel I-section has a much larger I about one axis than the other, and it will buckle about whichever axis gives the lowest load, unless bracing prevents it. Checking only the strong axis is a classic error.

Why real columns fail below the Euler load

Euler’s column is ideal: perfectly straight, perfectly centred load, perfectly elastic material. Real columns have initial bow, residual stresses from rolling or welding, eccentric connections and material that yields. All of these reduce the load a column can actually carry, often substantially, for columns of intermediate slenderness.

For that reason, modern codes do not design columns with the raw Euler formula. Steel design to the Eurocode, which Singapore adopted as its structural design basis, uses buckling curves that blend crushing and buckling behaviour and include imperfection allowances. Concrete columns are checked for additional moments caused by slenderness. The Euler load remains the reference point that these methods are built around, and the concept of slenderness ratio, effective length divided by radius of gyration, is how engineers judge which regime a column sits in.

Short, intermediate and slender columns

Short columns fail essentially by crushing or yielding. Buckling is not the governing concern.
Intermediate columns fail by a combination of yielding and instability. Most practical building columns sit here, and imperfections matter most.
Slender columns fail close to the elastic Euler prediction, well before the material reaches its strength.

Buckling beyond columns

Columns are the textbook case, but buckling appears wherever thin elements are compressed. The compression flange of a long steel beam can twist and buckle sideways, known as lateral torsional buckling, unless it is restrained by a slab or bracing. Thin webs of plate girders can buckle under shear. Thin steel plates, cold-formed sections and the struts of a truss can all buckle locally. Even reinforcement bars in a concrete column can buckle outwards if the links holding them are spaced too far apart, which is one reason link spacing is controlled in RC column design.

For homeowners, the practical relevance usually arises during renovation. Removing a floor, a wall or a bracing member can increase the unbraced length of a column or post, and because length is squared, a modest change can have a large effect on capacity.

Common mistakes

Assuming stronger material solves buckling. For slender members, stiffness and geometry matter far more than strength.
Checking the wrong axis. Members buckle about their weakest unrestrained axis.
Assuming connections are fully fixed. Using K = 0.5 for ordinary connections is unconservative.
Ignoring sway. If the whole frame can move sideways, effective lengths increase, sometimes beyond the real length.
Removing bracing during works. Temporary removal of a slab, wall or brace changes the restraint the original design relied on.
Applying Euler directly to design. It overestimates the capacity of real, imperfect columns of ordinary slenderness.

Frequently asked questions

What causes buckling in a column?

Buckling occurs when a compressed member becomes unstable and deflects sideways because its bending stiffness is not enough to keep it straight. Slenderness, small imperfections, eccentric loads and poor end restraint all make it more likely, and it can happen at stresses well below the material’s crushing strength.

What is the Euler buckling formula?

The Euler formula gives the critical load for an ideal elastic column as P_cr = π²EI / (KL)², where E is Young’s modulus, I is the second moment of area, L is the length and K is the effective length factor for the end conditions. It is a theoretical upper reference that real design methods reduce to account for imperfections and yielding.

Does a fixed column really carry four times the load of a pinned one?

Only in theory. The ideal fixed-fixed effective length factor of 0.5 assumes perfectly rigid end restraint, which real connections rarely achieve. Design codes recommend more conservative effective lengths, so the practical gain from end fixity is real but smaller than the textbook factor of four.

Can a concrete column buckle?

Yes, although typical building columns in reinforced concrete are relatively stocky and are more often governed by crushing combined with bending. Tall or thin concrete columns are checked for slenderness effects, and individual reinforcing bars can buckle locally if the links restraining them are too widely spaced.

Is buckling something a homeowner should worry about?

Not in an existing, properly designed house left as it is. It becomes relevant when works remove floors, walls or bracing, lengthen a post or add load. In those situations an engineer should check the affected members before work begins.

Planning works that change how a column or post is braced or loaded? Talk to our consultancy or get in touch for an accountable structural check.