What is left after it ripples

A stocky section reaches yield everywhere and then stops. That is what makes it stocky, and every plastic argument on this site quietly depends on it.

A thin one does something else. Long before the yield stress arrives anywhere, one of its plates goes out of plane — a web ripples, a flange waves — and the stress in the middle of that plate stops rising. It does not stop rising near the edges, where the plate is held by the plate next door, and the load goes on increasing.

So the member has not failed. It has become a different member: the one made of the parts that are still working.

Which free body produced the number #

Take a strip of the buckled plate across its width.

Before buckling the stress is uniform. After it, the middle of the plate has gone out of plane and can carry no more; the strips near the two supported edges are still straight and go on being loaded. The stress distribution across the width becomes a saddle — high at the edges, low in the middle — and the total force is the area under it.

Winter’s device is to stop describing that distribution and describe the force instead. Replace the real plate carrying a non-uniform stress with a narrower plate carrying the edge stress uniformly, of width

$$b_{eff} = \rho\, b, \qquad \rho = \frac{\lambda_p - 0.22}{\lambda_p^2}\;\;\text{for}\;\lambda_p > 0.673$$

where $\lambda_p = (b/t)/(28.4\varepsilon\sqrt{k})$ is the plate’s slenderness written so that $\lambda_p = 1$ means it buckles exactly at yield. Below 0.673 nothing is lost at all.

The two constants are fitted rather than derived, and the shape is not: $\rho$ falls as $1/\lambda_p$ for large slenderness rather than as $1/\lambda_p^2$, which is the statement that a buckled plate keeps a fixed width of working material rather than a fixed fraction of it.

The plates lose their material in different places #

This is where it stops being a strength calculation and starts being a geometry one.

A plate supported on both edges — a web between two flanges — loses its middle. The two effective strips sit at the two ends, and the plate’s own centroid does not move.

A plate supported on one edge — an outstand, a flange with a free edge — loses its free end. The effective strip sits at the supported end, and the plate’s centroid moves toward the support.

So a section made of plates of both kinds does not shrink symmetrically, and its centroid is somewhere else.

The eccentricity is an output #

That last sentence is worth saying slowly, because it inverts the usual order.

Everywhere else in this collection an eccentricity is an input. Somebody puts a load off the centroid; the moment follows. Here the load is applied exactly where it was designed to be applied, along the axis of the member as drawn, and the moment appears because the section moved.

For this channel: the effective area is 506 mm² of a gross 732, so the squash load falls from 256 kN to 177. At that 177 kN, the 8.0 mm shift is worth 1.42 kNm of bending that appears in no load case anywhere. On a 3 m column that is not a rounding error; it is a beam-column problem the member was not designed as.

And the direction is not fixed. Take the lips off the same channel and the flanges become outstands, lose their free edges, and drag the centroid the other way — a shift of −6.8 mm rather than +8.0. The same mechanism, on two sections that differ by 20 mm of lip, moves the load in opposite directions.

An eccentricity that arrives from the section rather than from the load still has to be checked against the region a compression may sit in before the far face is pulled, and it consumes part of a budget the designer thought was untouched. It is also no accident that this happens to a channel: a section with one axis of symmetry and not two already had a centroid that was not in the middle of anything, which is the same asymmetry that makes it move sideways when it is pushed straight down, and there is nothing to stop the effective centroid being somewhere else again.

Thin the same channel and the shift grows, because the plates lose more of themselves.

Twice the metal is more than twice the section #

The capacity of a fully effective section is proportional to its area, which is proportional to its thickness. The capacity of a thin one is not.

A thicker plate is doing two things at once: it is larger, and it is less slender. So $\rho$ rises as well as $A$, and the two multiply. Fitted over a sweep from 0.6 mm to 8 mm on this channel, the squash load goes as

$$N_c \propto t^{1.43}$$

which for the plain channel — whose flanges are outstands and buckle much earlier — is $t^{1.75}$.

The concrete form: going from 1.5 mm to 3.0 mm on the lipped channel takes the capacity from 111 kN to 302, a factor of 2.72 for twice the material. There are very few places in this subject where doubling the metal buys more than double, and they are all places where something was being lost that the extra metal stops losing.

The thickness at which this channel’s plates are all fully effective is 6.15 mm. A 200 mm deep channel would have to be six millimetres thick before its web stopped buckling, which is not a cold-formed section at all. Every cold-formed member in service is on the curved part of that graph, which is why the effective-width calculation is not an occasional refinement in that world but the whole of the section calculation.

The lip is worth twenty millimetres of nothing much and all the difference #

The two channels compared above differ by a 20 mm return on each flange, and the comparison is the clearest demonstration on the page of what an edge support is worth.

Without it, each flange is an outstand: 63 mm of free plate with $k = 0.43$, giving $\lambda_p = 2.06$ and $\rho = 0.44$. With it, the same 63 mm is a plate held at both ends with $k = 4$, giving $\lambda_p = 0.68$ and $\rho = 0.997$.

The steel is identical apart from the lips. The effective area goes from 44% to 69% of gross, and the squash load from 100 kN to 177 — a 77% increase for 6% more metal, all of it spent making an edge into a support rather than making anything thicker.

That is the same trade the whole subject keeps offering and rarely this starkly: geometry beating material by a factor no amount of material could buy.

It is the local-buckling counterpart of the same steel in a different shape, which is the standing version of the argument for bending. There the material moves; here it barely moves at all, and what has changed is which edges are held.

Classification is the same question, asked coarsely #

The four-class system on section classification sorts sections by what they can reach before a plate buckles: a plastic moment, a yield moment, or something less. Class 4 is the class where something less applies, and this page is what “something less” means.

Read the other way round, the classes are a set of thresholds on $\lambda_p$ and the effective width is what happens past the last of them. The web here at $\lambda_p = 2.11$ is deep into class 4; the lip at 0.61 is fully effective; and the flange at 0.677 is four thousandths of a slenderness unit past the plateau, keeping 99.7% of itself — which is a nice demonstration that the plateau is a real boundary and not a cliff.

Where the effective section is going next #

The area is only half of what a member needs. A column also needs a second moment of area, and the effective section’s is not the drawn section’s either — the web has lost the material nearest its own centroid, which contributes least to $I$ about the strong axis, so $I$ falls much less than $A$ does.

That has an odd consequence worth stating: local buckling makes a thin section relatively more slender in area than in stiffness, so a member’s radius of gyration $\sqrt{I/A}$ actually goes up when its plates buckle. The section is worse in every absolute sense and better in that one ratio, which is why an effective-section column check has to use the effective area with the gross-section slenderness rather than mixing them.

The reason is the one material far from the middle states for every section: $I$ is weighted by the square of the distance from the axis, and what the web lost is exactly the material sitting on the axis. Nearly a third of the area went and a much smaller share of the stiffness went with it, because the third that went was the third contributing least.

A plate is not a column, and that is the whole reason this works #

Everything on this page depends on a plate carrying load after it has buckled, and it is worth being explicit that this is not a property structures generally have.

A perfect column reaching its Euler load is at neutral equilibrium: it can deflect sideways at constant load, and the load does not rise. A real one, with an initial bow, never quite reaches it. A column has no post-buckling reserve at all — the critical load is the end of the story.

A plate has a large one, and the mechanism is geometric. As a plate buckles into half-waves, the strips of material near its supported edges cannot follow the buckle — they are held straight — so the buckled middle has to stretch transversely to accommodate the out-of-plane movement. That stretching develops a membrane tension across the plate, and a membrane tension resists further deflection. The plate stiffens itself as it buckles.

The size of the reserve is substantial and is exactly what the effective-width formula encodes. At a plate slenderness of $\bar\lambda_p = 2$, the critical stress is a quarter of yield while the plate actually carries 0.445 of yield — 1.8 times its own buckling load. At $\bar\lambda_p = 3$ it carries 2.8 times. The more slender the plate, the larger the multiple, because there is more buckled middle to be restrained by the same edges.

Which is why the whole scheme exists for plates and for nothing else in this collection. A slender column is designed to stay below its critical load; a slender plate is designed to go past it and to be counted at what is left, and the difference between the two design philosophies is entirely a transverse strip of material that one of them has and the other does not.

Two effective widths, one phrase #

There is a vocabulary trap here worth naming, because both quantities are called an effective width, both are written $b_{eff}$, and they have nothing to do with each other.

The local-buckling effective width is the subject of this page: a plate that has rippled, with its middle discounted and its edges counted at full stress. It depends on the stress level, on $b/t$, and on the plate’s edge conditions.

The shear-lag effective width is a different phenomenon entirely: a wide flange whose load is introduced along a narrow line — a web, a bolt group, a stiffener — does not reach a uniform stress across its width, because the force takes a finite distance to spread. The flange is fully capable of carrying the stress; it simply has not been given it. That width is a fraction of the span — of the order of $L/8$ each side — and it has no $b/t$ in it at all.

The two can apply to the same flange at once and they multiply: a wide thin flange on a short span may be reduced first for shear lag, and then the reduced width reduced again for local buckling. Getting the order wrong, or applying one and calling it both, is a mistake the shared symbol invites.

The distinction is easy to keep if the question is what the material is failing to do. In shear lag the plate is not being asked; in local buckling it is being asked and has stopped answering.

Where the model stops #

The plate is assumed to redistribute toward its supported edges. It does when the edges are genuinely held. A flange whose lip is too small to act as a support does not — the lip buckles with it, the whole assembly goes as one, and the calculation has to be done for the flange-and-lip together as a distortional mode rather than plate by plate. That mode is not on this page and it governs a great many real cold-formed sections.

Each plate is treated alone. They are not alone: a buckle in the web and a buckle in the flange have to be compatible at the corner, and the true local mode is a coupled one whose half-wavelength is common to both. Treating them separately with fixed $k$ values is a decoupling that is convenient rather than true, and the error runs both ways.

And the material is elastic–perfectly plastic at the edges. The effective strip is assumed to reach $f_y$ and stop, which for a cold-formed section is doubly awkward: the corners are cold-worked and stronger than the flats, and the flats retain residual stresses from the forming.

The scale at which all of this arrives #

There is a reason this subject belongs to cold-formed steel and not to rolled steel, and it is a scaling argument rather than a manufacturing one.

Plate slenderness is $b/t$, and buckling stress goes as $(t/b)^2$. Rolling produces sections whose thickness is set by what a mill can handle — 8 mm and up for anything structural — so a 200 mm rolled web has $b/t$ around 25 and is nowhere near the plateau. Cold forming starts from sheet, which arrives at 1 to 3 mm, so the same 200 mm web has $b/t$ around 100 and is four times as slender, which is sixteen times less buckling stress.

The consequence is that the two families of steel section fail in entirely different ways for entirely geometric reasons. Hot-rolled members yield, buckle overall, or fail at a connection. Cold-formed members lose part of their own section first, always, and every other check is made on what is left.

What slenderness costs at the member scale is the same graph one level up. A plate is a column of its own and $b/t$ is its slenderness, so the reason a thin plate buckles early is precisely the reason a long column does — and a member made of thin plates has both slendernesses at once, which is the situation the last figure on this page is about.

What the pictures cannot show #

The effective section is a bookkeeping device and not a photograph. There is no part of the real plate that is doing nothing and no line across it where the material stops working. What is drawn as two strips of full-strength steel is really one plate carrying a saddle-shaped stress distribution whose integral happens to equal theirs.

Nor can the drawing show the buckle. The plate is out of plane by something of the order of its own thickness, in half-waves whose length is about the plate width, and every section in this page’s figures is drawn flat.

The order the checks have to be made in #

One consequence of the section being an output rather than an input is that the usual sequence of a member check no longer works, and it is worth stating because it catches people.

Ordinarily: compute the section properties, compute the actions, compare. Here the section properties depend on the stress the section is carrying, the stress depends on the actions, and one of the actions — the moment from the centroid shift — depends on the section properties. The calculation is circular, and it is resolved by iterating: assume the section is fully effective, find the stress, reduce the plates, find the new centroid, recompute the stress at the new extreme fibre, reduce again.

For a member in pure compression the loop converges in one pass, because the stress is uniform and the reduction does not change it. For one in bending it does not: reducing the compression zone moves the neutral axis, which changes the stress the reduction was computed at, which changes the reduction. Two or three passes is usual and the answer moves by a few per cent between them.

A section property that depends on the load is not a section property, and everything a designer knows about the order of a calculation has to be set aside for this one family.

The assumption the figure rests on #

$k$ — the buckling coefficient — is a number chosen for each plate from the support conditions assumed at its edges: 4 for a plate simply supported on both, 0.43 for an outstand simply supported at one. Both assume the neighbouring plate offers simple support, which is a lower bound: a thick flange partly clamps the web, and a real web’s $k$ is somewhere between 4 and 7. That single conservative choice runs through $\lambda_p$, $\rho$, the areas, the shift and every number on this page.

The ladder from here #

Later rungs on this anchor: the distortional mode, where a flange and its lip buckle as a unit and no plate-by-plate calculation applies. The iteration for a section in bending, where reducing the compression zone moves the neutral axis and changes the stress the reduction was computed at. Effective sections in the interaction with overall buckling, where the shift computed here becomes the eccentricity in a beam-column check. Cold-work of forming and the corner strength enhancement, which gives back some of what the flats lose. The direct strength method, which drops the plate-by-plate decomposition entirely and works from the whole section’s elastic buckling stresses. And the historical question of where the 0.22 came from, which is a curve fitted to tests on stiffened plates in the nineteen-forties and has been in every code since.