The mode between the two that get checked

Plot the elastic buckling stress of a thin-walled strut against the half-wavelength it buckles in, and something appears that no single design check contains: a curve with three minima.

The first is local buckling, at a wavelength of the order of a plate width, where the flats ripple between fold lines that stay put. The last is column buckling, where the section keeps its shape entirely and the member goes as a whole. Both have rules, and both are in every textbook.

The one in the middle is the one this essay is about.

Which free body produced the number #

Cut the section at the web-to-flange fold and take the flange with its lip.

In the distortional mode that piece moves as a rigid pair rotating about the junction. The flange swings out of its own plane and the lip swings with it, and the whole of the section’s resistance comes from two places: the flange-lip assembly’s own reluctance to rotate along the member, and the web’s reluctance to let the fold line turn.

The second is a spring. A web bent by a rotation imposed at its edge is a plate strip in single curvature, so its rotational stiffness per unit length of member is

$$k_\phi = \frac{2D}{h_w} = 627\ \mathrm{N\,mm/rad\ per\ mm}$$

for the section here — and a strut restrained continuously by a spring is exactly the object the continuously braced strut turned out to be.

The first is a torsional buckling problem about a fixed pole. The flange-lip pair rotating about the junction has a polar second moment $I_0$ about that line, a St Venant constant $J$, and a sectorial constant $C_w$ — and it is $C_w$ that carries the surprise.

The whole distortional stiffness is the lip #

The sectorial coordinate about a pole is the perpendicular distance from the pole to the tangent of the wall, integrated along it. For the flange that distance is zero, because the flange’s own line passes straight through the junction it is rotating about.

So the flange contributes nothing at all to $C_w$, and the lip contributes all of it:

$$C_w = \frac{t\,b^2 d_l^3}{3}$$

which for the section drawn is $7.13 \times 10^6\,\mathrm{mm^6}$, and which goes as the cube of the lip length.

That is worth sitting with, because it inverts the usual reading of a lipped channel. The lip is normally described as a stiffener for the flange’s local buckling — a way of making the flange a supported plate rather than an outstand — and that is true and is the smaller half of what it does. Its larger job is that it is the only thing preventing the flange from rotating about the web.

Take it away and the distortional stress collapses from 287 N/mm² to 41, which is where local buckling already was.

The closed form is one this collection already has #

Put the two pieces together and the distortional stress at a half-wavelength $L$ is

$$\sigma_d(L) = \frac{\dfrac{\pi^2 E C_w}{L^2} + GJ + \dfrac{k_\phi L^2}{\pi^2}}{I_0}$$

which is a term falling as $1/L^2$, a constant, and a term rising as $L^2$ — and a function of that shape has a minimum:

$$\sigma_{d,\min} = \frac{2\sqrt{E C_w k_\phi} + GJ}{I_0} \qquad\text{at}\qquad L_d = \pi\left(\frac{E C_w}{k_\phi}\right)^{1/4}$$

That is the same closed form as the strut on an elastic foundation — $2\sqrt{kEI}$ at a half-wavelength of $\pi(EI/k)^{1/4}$ — with the sectorial constant where the second moment goes and the web’s rotational spring where the foundation goes. Asking the site’s own strutOnFoundation routine for the continuum answer with those substitutions returns 689 mm, which is the number in the figure.

So the mode nobody checks is not exotic. It is a strut on a foundation, and the foundation is the rest of the section.

Why the lowest elastic stress is not the one that governs #

On this section local buckling is at 41 N/mm² and distortional at 287 — a factor of seven — and it is the distortional mode that decides the member.

The reason is post-buckling reserve, and the two modes have very different amounts of it. When a flat plate buckles locally it sheds its middle and the load migrates to the supported edges, which are still straight and still stiff; the effective-width method is a whole design philosophy built on how much load that migration can carry, and for a slender plate it is several times the elastic buckling load.

When a section buckles distortionally there is nowhere for the load to migrate to. The fold line that would have been the stiff edge is the thing that is moving. So the reserve is small, the curve of strength against slenderness lies below the local one, and a section whose elastic distortional stress is seven times its local one can still be governed by the distortional check.

It is also more sensitive to imperfections, for the same structural reason: the mode has less to fall back on, so the slope of the post-buckling path is flatter and the knockdown larger.

The lip has an optimum, and past it the lip is the problem #

$C_w$ goes as the cube of the lip, so a longer lip raises the distortional stress steeply — from 159 N/mm² at 8 mm to 287 at 15 and 453 at 25.

It cannot go on. The lip is itself a plate, free along one edge, and its own local buckling stress falls as the square of its length: 780 N/mm² at 15 mm, 281 at 25. The two curves cross.

Which is why the guidance in the tables is a ratio and not a minimum. A lip between about 0.2 and 0.3 of the flange width is not a lower bound that could be exceeded for extra safety; it is a maximum of a curve, and both directions from it are downhill.

The same arithmetic explains the return-lip section — a lip with a further short fold at its end — which raises the lip’s own buckling stress without lengthening it, and moves the crossing point to the right.

What decides which mode a section is in #

The three minima move independently, and the design of a cold-formed section is largely a matter of arranging them.

Thickness helps local buckling as $t^2$ and distortional as roughly $t^{1.5}$ — $C_w$ and $k_\phi$ both contain $t$, and $I_0$ contains it too, so the net dependence is weaker than the plate’s. Taking the section from 1.5 mm to 2.5 mm takes local buckling from 41 N/mm² to 115 and distortional from 287 to 512, which closes the gap between them.

Web depth softens $k_\phi$, because a deeper web is a longer plate strip and a longer strip is a softer spring. A deep section is therefore more distortionally slender than a shallow one of the same flange, which is the opposite of the direction depth usually helps.

And member length decides only the global curve. That is the practical signature of the middle mode: brace a member harder and the third minimum does not move at all, because $L_d$ is a property of the cross-section. A member braced at 500 mm centres still has a distortional half-wavelength of 689, and whether the bracing interferes with the mode depends on whether it restrains the flange rather than the section as a whole.

The mode is also why a purlin is checked twice #

Cold-formed sections are not usually struts. They are purlins and rails, in bending, restrained along one flange by the sheeting they carry — and the distortional question arrives there in a form that looks like a different question entirely.

Under gravity load the sheeting is fixed to the compression flange, which restrains it; under wind uplift the compression flange is the free one, and it has nothing but the web’s rotational stiffness holding it. The two load cases therefore have completely different distortional capacities, and the uplift one is worse by a wide margin.

That is why a purlin’s capacity tables have two columns, and why the smaller number is usually the wind case rather than the snow case even though the loads are similar in size. The sheeting is not doing a small amount of good in one direction and a small amount in the other; it is removing a buckling mode in one direction and leaving it in the other.

It also explains a detail that looks like belt and braces. Anti-sag ties, cleats and the sheeting’s own fixings all restrain the purlin at intervals, and the interval that matters is not the one that controls the global mode — it is whether anything is holding the free flange over a length comparable with a distortional half-wavelength.

What a finite-strip analysis is really doing #

The curve at the top of this page is what a finite-strip analysis produces, and it is worth saying what that computation is, because the picture is so unlike an ordinary buckling result.

The section is divided into strips along its length, and each strip’s displacement is written as a half sine wave along the member times an unknown shape across it. That reduces a plate-buckling problem in two dimensions to an eigenvalue problem in one — the cross-section — for each assumed half-wavelength separately. Sweep the wavelength and each solve returns the lowest stress at which the section can buckle with that wavelength imposed.

So the curve is not a load path and not a sequence: it is a family of independent answers, and the minima are the wavelengths the section is willing to buckle at cheaply. Reading it as a curve a member travels along is the commonest mistake made with it, and the reason the three minima are usually described as three “modes” rather than as three local minima of one continuous function.

Where the model stops #

The flange and lip are treated as rigid in their own planes. They are, very nearly, in the distortional mode — the mode is a rotation — but a real distortional shape has some plate bending in the flange too, which softens it and lowers the stress by a few per cent.

The web’s spring is taken as $2D/h_w$, which assumes the two flanges rotate symmetrically outward and the web bends in single curvature. An antisymmetric mode, where one flange goes in and the other out, sees a different stiffness; and a web carrying its own compression is softer still, because its axial stress reduces its bending stiffness — an effect that matters most for exactly the deep webs where distortional buckling is worst.

The local curve is computed element by element with simply supported junctions. That underestimates the local minimum, because the flange restrains the web and vice versa, and the underestimate is why the gap between the first two minima here is wider than a finite-strip analysis would give.

And the whole essay is about uniform compression. A section in bending has a different stress distribution over its plates, a different local coefficient, and a distortional mode that involves only the compression flange — which is a different eigenvalue problem with the same mechanism.

What the pictures cannot show #

The signature curve is a plot of eigenvalues against an assumed half-wavelength, and every point on it is a different buckling problem. It is not the response of any one member: a real strut of a given length does not travel along it.

What a real member does is buckle in whichever mode its length permits, at whichever stress that mode reaches — and a member long enough for the distortional wavelength to fit will find it. The curve is a map of what is available, and the member’s length decides what it may reach.

Nor can the deformed shapes show what a distortional failure looks like on a test rig, which is a section whose flanges have opened outward like a book over a length of half a metre, with the lips folded and the web still nearly flat. It is not a subtle failure. It is only subtle in the calculation.

The assumption the figure rests on #

Every number here assumes the fold line between web and flange is a hinge with a known rotational stiffness and no other freedom.

That is an idealisation of a corner with a radius of several thicknesses, formed by bending, work-hardened, and carrying residual stresses from the forming operation. The radius softens the spring; the work hardening raises the local yield stress at the one point of the section that carries the most curvature; and the residual stresses are self-equilibrating through the thickness and mean the section starts life bent.

The consequence is that distortional buckling is the mode where a section’s manufacturing history matters most, because it is the mode whose stiffness lives entirely in the corner and in the lip — the two parts of the profile that the roll-forming line, rather than the designer, decides.