The third root of the cubic

A column that is strong enough and still falls over does so at the smaller of two Euler loads, one for each principal axis. That is the whole of the subject for a hollow section, an I-section and a rectangle, and it is two-thirds of it for everything else.

Three ways to leave the straight configuration #

A prismatic member displaced slightly from straight has three degrees of freedom at each cross-section: it can move in each of two directions and it can rotate about its own axis.

Each of those has a critical load. Two are Euler loads about the principal axes and are familiar. The third is a torsional load, at which the column stops resisting a twist about its shear centre, and it is

$$P_{\theta} = \frac{1}{i_0^2}\left(\frac{\pi^2 E I_w}{L^2} + GJ\right)$$

with $i_0$ the polar radius of gyration about the shear centre. Two features of that expression decide everything on this page.

It contains a term with no length in it. $GJ$ does not grow as the member is shortened, so $P_\theta$ approaches a floor as $L \to 0$ while both Euler loads run away to infinity. A short column is therefore a torsional problem and a long one is not, and the crossover is a length rather than a slenderness.

And it is divided by $i_0^2$, which is measured about the shear centre rather than the centroid. For a section whose two coincide that is the ordinary polar radius; for one where they do not it is larger, and the torsional load is smaller.

Which free body produced the number #

The free body is the whole member, given three small displacement fields at once — $u(z)$, $v(z)$ and $\phi(z)$ — and asked whether it wants to return.

Writing the potential energy of that state produces three equations, and the interesting thing is what couples them. The bending terms are independent: a displacement in one principal direction produces no moment about the other, which is what “principal” means. The twist, however, moves the section’s centroid sideways if the centroid is not on the axis of rotation — and the axial load, acting through that displacement, does work.

So the coupling terms contain $y_0$ and $z_0$, the offsets from the shear centre to the centroid, and they appear nowhere else. The three equations decouple exactly when the shear centre is at the centroid, which is the definition of a doubly symmetric section.

Setting the determinant of the three-by-three system to zero gives a cubic in the load, and the answer is its lowest root. For a section with one axis of symmetry the cubic factorises: one root is a pure flexural load about the axis of symmetry, and the other two come from a quadratic coupling the remaining bending mode with the twist.

The coupling never helps #

An eigenvalue problem’s roots interlace, and that has a consequence worth stating on its own.

Coupling takes from the mode that was lowest and gives to the one that was highest. That is a theorem about eigenvalues rather than a fact about columns: adding coupling to a system spreads its spectrum, so the smallest root falls and the largest rises.

The practical form of it is that there is no section for which the coupled answer is above the lowest uncoupled one, so taking the lowest of the three separate loads is always unconservative and never conservative. How unconservative depends on the offset, and the offset is not a free variable — it is a property of the section, and it cannot be tuned without changing everything else too.

What the twist actually looks like #

It is worth being concrete about the deformation, because “twisting about the shear centre” is easy to say and hard to picture.

In a flexural mode every cross-section stays the shape it was and moves sideways. In a torsional mode every cross-section stays the shape it was and rotates, about a point that need not be inside the material. The member becomes a long shallow helix, its two flanges moving in opposite directions in plan, and its axis staying put.

In the coupled mode both happen at once, in a fixed ratio decided by the eigenvector. The section moves sideways and rotates, and the point on the section that does not move is neither the centroid nor the shear centre but somewhere else entirely — a point whose position is an output of the eigenvalue problem rather than a property of the shape.

That is the observation that makes the mode so hard to recognise in a failed member. A column that has bent is obvious; a column that has twisted has a distortion visible only in plan, and one that has done both looks from most directions like an ordinary bow. Test reports from the 1930s describe failures as flexural that later analysis identified as coupled, and the disagreement is about what a photograph shows rather than about mechanics.

Which sections are at risk #

The size of the effect varies by more than an order of magnitude across ordinary shapes.

Three groups fall out.

Doubly symmetric sections are safe. The shear centre is at the centroid, the modes decouple, and the torsional load has to be compared with the flexural ones but never mixes with them. For a rolled I-section the torsional load is above both anyway, which is why the mode is left out of design so often.

Singly symmetric sections are at moderate risk. A channel, a tee and a lipped channel all have a shear centre off the centroid along one axis, and their factor at practical lengths is between 1.5 and 2.

Point-symmetric and cruciform sections are the extreme case. A cruciform has its shear centre at its centroid, so it does not couple — and its warping constant is almost zero, because every wall points at the pole. Its torsional load is therefore very low with no coupling at all, which is a different route to the same trouble.

Where the mode changes over #

The crossover from twisting to bending is a length, and it is worth being able to estimate.

Setting $P_\theta = P_z$ and solving for $L$ gives a crossover length that depends on the ratio $I_w/I_z$ and on $GJ$. Roughly, a section with a large warping constant relative to its minor-axis second moment has its crossover at a short length, and one with a small warping constant has it long.

That is why the tee and the cruciform are the worst cases: both have almost no warping constant, because their plates meet at a point and there is no pair of flanges to bend against each other — the same geometric fact that makes warping restraint worth nothing to them.

Reading the crossover as a competition between two terms #

The two terms in $P_\theta$ behave differently enough that it is worth separating them, because between them they explain every curve on this page.

The warping term, $\pi^2 EI_w/L^2 i_0^2$, is an Euler load in disguise: the two flanges bending in opposite directions in their own planes, over the member’s length, exactly as a column bends. It falls as the inverse square of the length, the same way both flexural loads do.

The Saint-Venant term, $GJ/i_0^2$, is not an Euler load at all. It is a shear flow circulating through the thickness of each plate, its resistance per unit length of twist does not care how long the member is, and it contributes the same amount at any length.

So the torsional curve is a hyperbola plus a constant, and both flexural curves are hyperbolas with no constant. Three curves of the same family and one of them has a floor, which is the whole reason the ordering changes with length and why it changes only once.

That also says which sections have the crossover at a long length: those whose $GJ$ is large relative to their $EI_w$. A cruciform has a respectable $J$ and almost no $I_w$, so its torsional load is nearly flat and sits below the flexural ones over a wide range. A deep I-section has a large $I_w$, so its torsional curve is steep and stays above them.

The mode that needs no compression at all #

There is a fourth instability in the same family, and it is worth putting beside the other three because it shows how little the torsional mode has in common with the flexural ones.

A buckling load with no compression in it is the clearest demonstration that instability is about the stiffness of a configuration rather than about a compressive stress. What is remarkable in that expression is the absence of $G$: the torsional stiffness of the member has nothing to do with the torque at which it buckles in torsion.

What a design rule does about it #

Codes handle the mode with an effective slenderness rather than with the cubic.

The device is to compute the flexural-torsional critical load, convert it to an equivalent slenderness $\lambda = \sqrt{Af_y/P_{cr}}$, and put that into the ordinary column curve. The curve itself is not rederived — it borrows the flexural imperfection factors, on the grounds that the imperfection sensitivity of the torsional mode has not been measured separately and is assumed similar.

That borrowing is the weakest link in the treatment. The torsional mode’s imperfection is an initial twist rather than an initial bow, its magnitude is not covered by any straightness tolerance, and there is no reason the same knockdown should apply. It is a defensible expedient and it is not derived from anything.

For single angles the treatment gives up entirely and substitutes an empirical effective slenderness that folds the mode, the connection eccentricity and the end restraint into one fitted expression — which is the same admission the axis a column buckles about ends on.

The check nobody makes, and why #

A designer working from section tables has everything needed to make the check and usually does not. Three reasons, and only one of them is good.

The number is not in the tables. $I_w$ and the shear-centre offset are published for open sections, but the cubic has to be assembled and solved, and no table can print $P_{cr}$ because it depends on the length and the end conditions.

The mode does not exist for the commonest section. Nearly every steel column in nearly every building is an I-section or a hollow section, and both are safe. A designer can work for years without meeting a member where the third root governs, and a check that never governs is a check that stops being made.

And the members where it does govern are usually small. A tee is a bracing member, a strut, a chord of a light truss; a single angle is a lattice member. Those are sized by a rule of thumb or by a standard connection more often than by an analysis, so the calculation that would have found the mode was never done for a different reason.

The last of those is the honest one and it is also the one that causes the failures. A member designed by rule and governed by a mode the rule does not contain is not conservative by any amount that can be estimated — and the factors of 1.7 and 3.9 in the figures above are what that gap looks like.

What to carry away #

A column has three critical loads and the third is a twist. The torsional load contains a term with no length in it, so it governs short members and not long ones.

The shear centre’s offset from the centroid is what couples the modes, and coupling always lowers the smallest root. Taking the lowest of the three uncoupled loads is never conservative.

The sections at risk are the ones with small warping constants — tees, cruciforms, angles, and anything whose plates meet at a point. A rolled I-section is safe and is why the mode is so often left out.

And the crossover is a length rather than a slenderness, so shortening a member can move it into the mode rather than out of trouble.

Where the same cubic turns up #

The three-mode structure is not peculiar to columns, and recognising it saves meeting it twice.

A beam in bending has the same problem. Its lateral displacement and its twist couple through the applied moment rather than through an axial load, giving the lateral-torsional buckling equations — a beam that fails sideways is the two-freedom version of this page’s three.

A beam-column has all four freedoms at once. Axial load couples bending with twist through the shear-centre offset; moment couples them through the Wagner effect; and the two couplings are not the same term. The result is a genuinely four-dimensional eigenvalue problem that no hand method addresses.

And a monosymmetric section adds a term nobody expects. Its stress distribution is not symmetric about the shear centre, so the axial load’s own work through the twist has an extra contribution — the Wagner coefficient β in the figures above, which is 0.60 for both the channel and the tee and would be zero for a doubly symmetric section.

The unifying statement is that instability is a minimum over every displacement field the member can adopt, and adding a freedom can only lower the answer. A calculation that considers two of three freedoms has found an upper bound to something, which is the wrong side to be on.

Where the model stops #

The section is thin-walled and prismatic. The sectorial properties are computed on a midline, and root fillets, tapered flanges and cold-formed corners all move them.

The ends are pinned in all three senses. A real end restrains bending about two axes, twist and warping by four different and unrelated amounts, and there is one effective length factor in the expression for each.

The load is concentric. A member loaded away from its shear centre has a torque applied from the outset, which is a first-order effect the eigenvalue does not contain.

Local buckling is left out. A thin open section can buckle locally or distortionally before any of these three arrive, and when two modes coincide the interaction is dangerous rather than efficient.

The shear-centre offset is taken from the gross section. A member that has buckled locally has lost part of its width, its centroid has moved, and the offset with it — so the coupling in a class 4 section is computed on a shape that no longer exists.

And the whole of it is elastic. Residual stresses lower all three loads by different amounts, because they act on different parts of the section — and the column curve that is supposed to represent that was calibrated on flexural buckling.

The cubic exists because three modes share one member, and two neighbours show what happens when a fourth is added. A section that cannot stay flat is where the warping term in it comes from, and two ways of buckling at once is what a mode interaction does to a root that was already the smallest of three.

The ladder from here #

Later rungs on this anchor: the governing differential equations set out, and where the cubic comes from. Effective lengths for the torsional mode, and the warping-restraint factor that sits alongside them. Monosymmetric sections in general, and the sign convention that decides which flange helps. Fully unsymmetric sections, where all three modes couple and no root is exact. Distortional buckling, and the three-family interaction. Cold-formed lipped sections, where lips exist partly to move the shear centre. Design curves for the flexural-torsional mode, and whether borrowing the flexural imperfection factors is defensible. And built-up and battened members, where the lacing’s shear flexibility enters the same eigenvalue problem.

Wagner published the torsional buckling load in 1929 and the coupled theory belongs to Vlasov and to Timoshenko in the following decade. What is unusual about it is how completely the result has stayed inside the specialist literature: a mode that halves the capacity of a tee is not in most first courses, and a designer meets it as a clause in a code long before meeting it as a piece of mechanics.