The axis a column buckles about

A section with no axis of symmetry moves sideways when it is loaded straight down, and the quantity responsible is the product of inertia. That essay is about bending. The same quantity does something sharper to a strut, and the difference is that a bending answer is wrong by a factor a designer can see in the deflection while a buckling answer is wrong in a direction nobody checks.

The second moment is a function of direction #

The property a strut needs is the smallest second moment of area the section has, in any direction at all, and the drawing offers two candidates that are usually neither.

Three numbers on that plot are worth separating.

3.733 × 10⁶ mm⁴ is $I_x$, about a leg. It is what a table gives and what somebody in a hurry uses.

1.523 × 10⁶ mm⁴ is $I_2$, the minimum, about an axis at 45°. It is what the column buckles about.

The ratio between them is 2.45, and a critical load is proportional to I, so a strut designed on the first number is being credited with two and a half times the capacity it has.

That is not a refinement. It is the difference between a member that works and one that does not, and it is available to anybody who takes the section property from the right column of the same table.

There is a fourth number worth adding, and it is the one that makes the trap survive. The two drawn axes of an equal angle give the same second moment — 3.733 × 10⁶ mm⁴ about either leg, by symmetry of the shape about its diagonal. A designer who dutifully computes both and takes the smaller has taken the same number twice, found no disagreement, and concluded that the section has no weak axis worth worrying about. The check that would normally catch the error returns a reassuring answer.

Which free body produced the number #

The free body is a length of the strut, cut at two sections, displaced into a slightly bent shape and asked whether it wants to return.

Its equilibrium in the buckled state is the Euler statement in two directions at once: the internal moment about each principal axis has to balance the axial force times the lateral displacement in that direction. Written about the drawn axes those two equations are coupled by the product of inertia — a curvature about one axis produces a moment about the other — and the coupled pair has no simple solution.

Written about the principal axes the coupling vanishes, by definition: the principal axes are the pair for which $I_{xy}$ is zero. The two equations separate, each is an ordinary Euler problem, and the buckling load is the smaller of the two answers.

So the principal axes are not a convenience but the only frame in which the problem decouples, and the smaller principal second moment is the answer because a structure buckles at the lowest load available to it, not at the one someone computed.

That is also why the neutral-axis tilt in the first figure and the buckling axis in this one are the same phenomenon. Both are the product of inertia refusing to let the drawn axes be independent.

What it costs, on the member people actually use #

It is worth doing the arithmetic on a real strut, because the factor of 2.45 on the second moment does not arrive as a factor of 2.45 on anything a designer sees.

Take the 120 × 120 × 12 angle as a 2.4 m pin-ended strut in grade S275, with an area of 2,760 mm² and a squash load of 759 kN.

Using the drawn axis, $r = \sqrt{3.733\times10^6/2760} = 36.8$ mm, so $\lambda = 2400/36.8 = 65$. That is below the crossover, the member is stocky, and a column curve returns something like 80 per cent of the squash load — about 600 kN.

Using the true minimum, $r = 23.5$ mm and $\lambda = 102$. That is above the crossover, the member is now slender, and the same curve returns about 45 per cent — 340 kN.

The capacity has not fallen by a factor of 2.45; it has fallen by 1.8, and it has crossed from one side of the column curve to the other. The member has changed category — from one whose strength is set by its material to one whose strength is set by its geometry — which changes not only the number but which of its properties is worth improving. Adding steel to a stocky column helps in proportion; adding it to a slender one helps only if it goes where the second moment is smallest, which for an angle means a second angle.

The shape that looks symmetric and is not #

The size of the effect is not proportional to how odd the section looks.

The same section is worth seeing with its axes turned, because the rotation is what the section table has already done and what the drawing has not.

A channel and a zed differ only in which way one flange points, and one of them has a product of inertia of zero while the other has the largest on the page. The visible asymmetry of a section says nothing about the ratio of its principal second moments, which is a genuine trap: an angle looks like a problem and is a mild one, a zed looks like a channel and is severe.

The zed’s numbers explain something about how purlins are actually used. A zed used as a beam is restrained continuously by the sheeting, which is what makes the 201 per cent sideways movement a non-issue in service. A zed used as a strut — a wind post, a temporary prop, a member in a lattice — has 0.301 × 10⁶ mm⁴ to buckle about, and there is no sheeting.

Radius of gyration, which is the same trap in different units #

Design is usually done with slenderness rather than with second moments, and the substitution does not help.

Since $r = \sqrt{I/A}$ and the area is the same in every direction, the radius of gyration inherits the second moment’s directional dependence exactly. A section has a maximum radius, a minimum radius, and a value for every axis in between, and the slenderness that matters is the largest one, computed from the smallest radius.

For the angle above, $r_{\min} = \sqrt{1.523\times10^6/2760} = 23.5$ mm against 36.8 mm about the drawn axis — so the slenderness a designer would compute is 1.57 times too small, and the capacity read off a column curve is far more than 1.57 times too high, because the curve is steep.

Why the angle is the case everybody meets #

Single angles are the most-used unsymmetric section in structural engineering — lattice towers, bracing, wind girders, light trusses — and they are used almost entirely in compression.

They also come with a second complication that compounds the first. A single angle is usually bolted through one leg, so the load is applied eccentrically to the centroid and the end restraint is different about the two principal axes: the bolted leg is held against rotation in its own plane and the free leg is not.

The result is that a single angle strut’s real capacity is decided by three things — the minimum principal second moment, the eccentricity of the connection, and a torsional mode the two above have not mentioned — and code treatments handle it with an empirical effective slenderness rather than with any of the mechanics on this page. The equations are exact and the design rule is a fit, which is unusual and is a fair admission of how many effects arrive at once.

The mode that is not either of them #

There is a third buckling mode for an unsymmetric section, and it is often the lowest.

An open section whose shear centre does not coincide with its centroid can buckle by twisting as well as by bending, and for a section with one axis of symmetry the twist couples with bending about the perpendicular axis. For a section with no symmetry at all, all three couple, and the critical load is the smallest root of a cubic.

An angle is the extreme case: its shear centre is at the intersection of its two legs, well away from its centroid, and its torsional constant is tiny because it is an open section made of two flat plates. The torsional-flexural load can be below both flexural ones — which means that even a designer who correctly uses $I_2$ may still have found the wrong number.

The mode between the two that get checked is a related trap in a different family of sections, and the shared lesson is the one this page is an instance of: a buckling load is the minimum over all the modes a member has, and the modes a member has are decided by its section rather than by its drawing.

A check that takes ten seconds #

Given how expensive the mistake is and how easy it is to make, it is worth having a test that runs before any calculation.

Does the section have an axis of symmetry? If yes, the drawn axes are principal and nothing on this page applies. Every I-section, channel, tee, hollow section, rectangle and circle passes.

If no, look up $I_2$ or $r_{\min}$ rather than $I_x$ or $I_y$. Every section table for angles and zeds publishes it, usually under the heading “about the v–v axis” or “minor principal”. The number is there; the problem is that it sits in a column the other sections’ tables do not have, so a designer moving between section types reaches for the wrong one out of habit.

And check whether the member is in compression. In bending the effect produces a sideways deflection that a restraint usually removes — sheeting on a purlin, a slab on a beam. In compression there is nothing to restrain, and the effect is the whole answer.

Those three questions separate the cases where the subject matters from the cases where it does not, and the separation is sharp: an unsymmetric section in compression with no continuous restraint is the entire population at risk, and it happens to be one of the most common members in steel construction.

Where the axes come from, briefly #

The construction is worth having because it makes the 45° in the first figure obvious rather than remembered.

For any pair of orthogonal axes, $I_x$, $I_y$ and $I_{xy}$ transform under rotation exactly as a two-dimensional stress state does — which means Mohr’s circle applies unchanged, with $I_{xy}$ on the vertical axis in place of the shear stress. The principal axes are where the circle crosses the horizontal axis, the principal second moments are the two crossings, and the angle is half the angle on the circle.

That gives three facts for free. The principal axes are always orthogonal. The sum $I_x + I_y$ is invariant, so the two principal values always add to the same total whatever axes were used. And any axis of symmetry is a principal axis, because reflecting the section leaves $I_{xy}$ unchanged and negates it at once, so it must be zero.

The last of those is the practical test. A section with any axis of symmetry has principal axes on the drawing; a section with none does not, and there are only a handful of such sections in common use — the angle, the zed, and any built-up shape somebody has assembled asymmetrically.

The same quantity, three jobs #

The product of inertia is doing three separate things in this collection and it is the same number each time, which is worth saying because it is usually met three times without being recognised.

In bending it tilts the neutral axis, so a load applied straight down produces a deflection that is not. That is the previous rung on this anchor and it is a serviceability effect with a strength consequence at the corners.

In buckling it rotates the axes about which the two Euler problems separate, so the minimum second moment is smaller than either drawn value. That is this page.

And in torsion it does not appear at all — the torsion constant of a thin open section is a sum over its plates and knows nothing about their arrangement, which is why an angle, a zed and a channel of the same plate lengths and thicknesses have very nearly the same J and wildly different I₂.

The third of those is the useful contrast. Bending and buckling are directional and torsion is not, so the section properties that are worst affected by asymmetry are exactly the ones the drawing invites a designer to read off its own axes — and the one that is unaffected is the one nobody reads off anything.

Where the model stops #

The strut is pin-ended and prismatic. Real end conditions differ about the two principal axes, which is exactly the case a single effective length cannot describe.

Torsion is left out until the last section. For a thin open unsymmetric section it should not be, and the full treatment is a three-by-three eigenvalue problem rather than the smaller of two Euler loads.

The load is assumed concentric. A bolted angle’s load acts on the bolt line, and the resulting moment is a first-order effect that the buckling calculation does not contain.

The section is assumed compact. A thin angle’s legs are outstands and may buckle locally before anything else happens, which is a fourth mode and interacts with the third.

And the geometry is the nominal one. A cold-formed zed has rounded corners, a rolled angle has a root fillet, and both change $I_2$ by a few per cent — small against a factor of 2.45, and not against a design that is marginal.

What to carry away #

A strut buckles about the smallest second moment the section has in any direction. For a symmetric section that is one of the drawn axes; for an unsymmetric one it is neither, and the drawn value can be more than twice too large.

The ratio of the principal second moments is the measure, and it is not visible. A zed at 18.5 looks tamer than an angle at 3.90. The only reliable test is whether the section has an axis of symmetry, and the only reliable number is the one the table prints for the minor principal axis.

And the error changes which regime the member is in. The material far from the middle does the work about one axis and is close to the centroid about the other, so an unsymmetric section is stocky in one direction and slender in the perpendicular one — and a design done about the wrong axis has answered a question about a different member.

The section’s own geometry keeps producing this shape of surprise. Loaded straight down, an asymmetric section moves sideways; the point a shear has to pass through is outside the section; and a column can twist instead of bending. All three are consequences of the same fact — that a section’s principal directions are properties of its shape, not of the drawing’s axes — and none of them is visible in the second moment about a horizontal line.

The ladder from here #

Later rungs on this anchor: the torsional-flexural mode set out properly, with the cubic and the three roots. Unsymmetric bending under a moment that is not about a principal axis, where the neutral axis obeys neither the load nor the moment. The single angle’s design rules, and what the empirical effective slenderness is standing in for. Built-up sections assembled from angles, where the pair has principal axes of its own that neither member has. The shear centre of an unsymmetric section, and why a load through the centroid twists it. Sections with one axis of symmetry, where two of the three modes couple and the third does not. And the general theory of thin-walled beams, in which the centroid, the shear centre and the principal axes are three aspects of one formulation.

The transformation of second moments under rotation was known to Euler and the buckling problem is his too, which makes the gap between them slightly comic: the two halves of this page were available to one person in the eighteenth century and were put together only when iron made slender unsymmetric struts common enough to fail.