Classified by a gradient it does not have

Every coefficient in the essay that derived them belonged to a stress pattern that was known before the plate was looked at. Uniform compression is such a pattern. So is pure shear. A flange outstand in a column has one and a flange outstand in a beam has one, and in every case the gradient across the plate is a consequence of the member’s loading rather than of the plate’s own condition.

A web in bending is the exception, and the way it fails to fit is worth more than the coefficient it produces.

The coefficient is a function of the gradient, and the gradient is steep #

The stress across a web running from compression at one edge to tension at the other is described by one number: $\psi = \sigma_2/\sigma_1$, the stress at the second edge over the stress at the first. Uniform compression is $\psi = 1$. A gradient falling from compression to nothing is $\psi = 0$. Pure bending about the middle of the panel is $\psi = -1$.

The reason for the sixfold range is the one the derivation gives for every coefficient: it measures how short a half-wave the boundary conditions and the stress pattern force the buckle into. Only the compressed part of the web can drive a buckle. In pure bending that part is half the panel, so the buckle is confined to a strip half as wide and is correspondingly stiffer, and the coefficient rises to reflect it.

That is a clean statement while $\psi$ is known. The trouble is that $\psi$ is a property of the section’s elastic neutral axis, and a class 4 section’s neutral axis is not where the drawing puts it.

What the reduction takes, and where it takes it from #

Take an 1800 by 12 mm web between 450 by 30 mm flanges, in S355. Its web is 150 thicknesses deep, which is half again past the limit at which a web stays fully effective, so part of it has to be written off.

The rule is the ordinary one. The slenderness is measured on the whole panel, a reduction factor $\rho$ follows from it, and the effective width $\rho b_c$ is apportioned over the compression zone — 40 per cent of it against the compressed edge and 60 per cent against the line where the stress changes sign. What is left in between is a strip of web that the calculation treats as absent.

The two strips are where they are for a reason that is easy to state and is not the reason usually given. A buckled plate sheds load toward its supported edges, so the strip against the flange is the one still carrying full stress. The strip against the zero-stress line is not there because it is effective — it is there because the stress on it is nearly zero, so removing it would make no difference and keeping it is simpler than arguing about it.

And the hole is not in the middle of the web. It is in the middle of the compression half, entirely above the neutral axis, which is the fact the rest of this essay turns on.

The section is no longer symmetric, so its axis is no longer central #

Removing 278 mm of a 12 mm plate from the compression side takes 3,336 mm² out of the top half of a section that had 13,500 mm² in each flange. The section is now asymmetric, and its elastic neutral axis is wherever the first moment of what is left vanishes.

It drops. The axis falls 37.8 mm toward the tension flange, which deepens the compression zone from 900 mm to 938 and shallows the tension zone by the same amount. The gradient is no longer $-1$ — the two edge stresses are no longer equal and opposite, because they are no longer equidistant from the axis.

So the calculation eats itself. The coefficient was computed at $\psi = -1$; the section that coefficient produces does not have $\psi = -1$; and a coefficient computed at the gradient the section actually has would have removed more, which would have moved the axis further.

A fixed point rather than a formula #

There is no closed form for the answer and there is a very short iteration.

Read $\psi$ from the current axis. Take $k$ from it. Take $\rho$ from $k$. Remove what $\rho$ says to remove. Find the axis of what is left. Repeat.

The contraction is the part worth understanding, because it is what makes the loop a calculation rather than a hazard. What each pass removes is proportional to what the previous pass moved the axis; what the axis moves is proportional to the change in what was removed; and the change is a small fraction of the removal because the coefficient curve, steep as it is, is not that steep. The ratio here is about a tenth, so the series is finished to three significant figures after three terms.

That the loop converges is not a general fact and is worth saying so. A feedback that removed material on the tension side would move the axis the other way, deepen nothing, and settle immediately; one with a gain above unity would run away, and the section would have no effective section at all. The gain is well below one for every web in normal use, and it rises with slenderness — from a fifth at the class boundary toward a ninth for the deepest webs drawn here, which is the wrong direction for anyone hoping it might diverge somewhere interesting.

The class boundary is safe from all of this, and only the class boundary #

There is an obvious objection to everything above, and answering it locates exactly where the trouble starts.

Section classification decides whether a web needs an effective section at all, and it is made on the gross section — at $\psi = -1$ for a symmetric girder, with $k = 23.92$, against a limit of 124ε. If the gradient a class 4 web really has is not $-1$, then the classification that sent it to a class 4 calculation was made at a gradient it does not have either, and the boundary itself is in the wrong place.

It is not, and the reason is that the feedback has no gain at the boundary. A web exactly at the limit has $\rho = 1$: nothing is written off, so the section stays symmetric, so the axis does not move, so $\psi$ really is $-1$ and the check was made at the right gradient. The loop’s gain starts at zero and grows with slenderness, continuously, from the boundary outward.

The two limits agree to within two per cent from opposite directions, which is worth noting because it is the only independent check available on any of this. Reading the boundary out of the reduction — the slenderness at which $\rho$ first falls below one — gives a width-to-thickness ratio of 121.4ε. The tabulated class 3 limit for a web in bending is 124ε. One is an eigenvalue with an empirical curve on top of it and the other is a number in a table, and they land 2.1 per cent apart.

So the classification is sound and the calculation on the far side of it is the thing that drifts. At 100 thicknesses the axis moves 0.8 mm; at 150 it moves 42.5; at 250 it moves 179. The girder that needs the loop least is the one nearest the boundary, and the one that needs it most is the deep welded web that was never going to be checked by hand.

Two errors that differ by a factor of twenty #

The one-pass answer is what almost every calculation reports, and how wrong it is depends entirely on which quantity is asked about.

The width written off: 278 mm against 323, an understatement of 16 per cent. The moment capacity: 10,079 kNm against 9,999, an overstatement of 0.8 per cent. Same calculation, same girder, same single omission, and two errors a factor of twenty apart.

The reason is where the extra 45 mm sits. It is a strip of web straddling the middle of the compression zone, 533 mm from the converged neutral axis on a section 1,860 mm deep. The compression flange is 958 mm from the same axis, and a contribution to the second moment of area goes as the square of the distance — so that strip is worth 31 per cent of what the same area of flange would be worth, and the 537 mm² of it that the first pass missed alters the second moment by 0.6 per cent. The material far from the middle does nearly all the work, and the converse is what is operating here: material near the middle can be lost almost for free.

That asymmetry decides what the correction is for. It is not a strength correction. It is a correction to the description of the section, and it matters wherever the description rather than the modulus is what is being used — a shear check on the web area that is left, a fatigue check at the edge of the write-off, a stiffener design that assumes it is restraining a panel of a stated width.

The effective width barely knows how deep the girder is #

The most useful result in this calculation is one the iteration exposes by accident, and it is visible in the numbers above: the write-off grows and the effective width does not. Over the whole loop, $b_c$ runs from 900 mm to 943 while $\rho b_c$ runs from 622 to 620.

That is not a coincidence, and it is short to show. With the slenderness measured on the whole panel, $\bar\lambda \propto 1/\sqrt{k}$, and in the deep-gradient range $k = 5.98(1-\psi)^2$, so $\bar\lambda \propto 1/(1-\psi)$. Meanwhile $b_c = b/(1-\psi)$. The leading term of $\rho$ is $1/\bar\lambda$, so the leading term of $\rho b_c$ is

$$b_{\text{eff}} \approx 2.445 \times 28.4\,\varepsilon\,t = 69.5\,\varepsilon\,t$$

in which neither the depth of the web nor the gradient appears at all. For a 12 mm web in S355 that is 678 mm, before a second-order correction of about fifty.

The measured values follow it. Holding the thickness at 12 mm and the flanges fixed, a web 1,200 mm deep has an effective compression width of 593 mm, one of 1,800 mm has 620, one of 2,400 mm has 633 and one of 3,600 mm has 646. Tripling the depth of the web buys 53 mm of working compression material, nine per cent, while the compression zone it has to be found in grows from 601 mm to 2,067.

The deep web travels further along the curve for the same reason it writes off more: the two are the same movement. Its converged gradient is −0.742 rather than −0.910, which is a quarter of the way from pure bending to a web with no tension in it at all, on a girder that is loaded in pure bending and drawn symmetric. Nothing about its loading is asymmetric; the asymmetry is entirely manufactured by the calculation.

The design consequence is blunt. Past the class boundary, making a plate girder deeper adds web that carries bending only in its tension half. That is the reason a deep web gets a longitudinal stiffener rather than more thickness: a rib stiff enough to be a node halves the panel and so multiplies the term above by two, which is the only move available that changes the leading term rather than the correction.

One side of the section got stronger #

A detail in the converged numbers is worth keeping, because it is the opposite of what a reduction is expected to do.

The gross section has a modulus of 30.58 × 10⁶ mm³ at both faces. The effective section has 28.17 at the compression face and 30.86 at the tension face — higher than it was before any material was removed. Taking area out of the compression half moved the centroid toward the tension flange, so the tension flange is now nearer the axis and its lever arm is shorter, which raises the modulus on that side even as the second moment of area falls.

It changes nothing about the answer, because the smaller modulus governs and that is the compression one. It matters for a section that is not symmetric to begin with, where the two moduli were already different and the reduction can push the governing face from one to the other — so a girder with a large tension flange may find that its effective section is governed at a face that its gross section was comfortable at.

The same loop, in a material that has no buckling in it at all #

The shape of this calculation — material stops working, the axis moves, which changes what stops working — is not peculiar to steel plates, and the other instance is far more familiar.

The structure of the two problems is identical. Something is written off; what is written off depends on the position of the neutral axis; the position of the neutral axis depends on what was written off. The whole of the cracked-section calculation is that loop, and every textbook solves it in closed form because the loop happens to reduce to a quadratic: the depth of concrete in compression appears squared in the first-moment equation and the equation can be written down.

The web problem has no such luck. Its loop passes through $k(\psi)$, which is a piecewise fit rather than an algebraic expression, and through $\rho(\bar\lambda, \psi)$, which is Winter’s empirical curve. Neither is invertible, so the fixed point has to be walked to rather than solved. The difference between the two calculations is not the physics — it is whether the feedback happens to be algebraic, and that is a property of the design rules rather than of the structure.

Which is also why the concrete version is taught as one equation and the steel version is taught as one pass. Both are the first move in the same argument; one of them finishes.

Where the loop is asked to do more than it can #

The coefficient curve has a step in it. The two expressions meeting at $\psi = -1$ give 23.88 and 23.92 — a discontinuity of 0.17 per cent in a quantity that is being iterated toward. It is far smaller than the convergence tolerance and it is a reminder that $k(\psi)$ is a fitted table rather than a function: the whole loop is walking over a curve nobody derived.

The stress distribution is assumed linear throughout. It is linear on a gross section and it is linear on an effective one, and it is not linear on the real plate, which has shed load toward its edges in exactly the pattern the effective width is an accounting device for. Plane sections staying plane is doing work here that it was never asked to do in a section with a hole in it.

Nothing here is a failure calculation. The effective section gives a moment at which the extreme fibre reaches yield on a fictitious section. What the girder actually does at that moment involves the flange’s own stability, the web’s post-buckling reserve and whatever the shear is doing at the same station, and none of the three is in this arithmetic.

Shear is absent, and a real web has some. A panel carrying shear as well as bending buckles in a combined mode whose coefficient is neither of the two, and the tension field that follows the buckle pulls on the flanges in a way this calculation has no representation of.

The write-off is treated as a hole, and it is not one. The material is there, it carries stress, and it carries membrane stress transverse to the span that nothing above accounts for. Treating it as absent is a device that reproduces the measured stiffness and strength; it is not a description of the plate.

And the loop assumes the section stays elastic while it iterates. Each pass computes an elastic neutral axis. A girder close to its capacity has yielded somewhere, at which point the axis moves again for an entirely different reason — and the axis a yielded section settles on is a different construction altogether.

Still open: the curve the loop walks over, and where its exponent came from #

Everything above takes Winter’s reduction as given, and it is the least derived object in the chain. The expression $\rho = (\bar\lambda - 0.055(3+\psi))/\bar\lambda^2$ is a curve fitted through tests on cold-formed sections in the 1940s, and the inverse square in it is not an accident — it comes from assuming the effective width carries the edge stress and that the plate’s post-buckling stiffness is a fixed fraction of its initial one. Where that fraction comes from, and why the fitted constant moves with the stress gradient at all, is a question about the post-buckling surface rather than about the eigenvalue these essays have been climbing toward.

After that, the same loop with a longitudinal stiffener in it, where the stiffener is inside the zone being written off and its own effectiveness is part of the fixed point; and the shear panel, where the buckle is diagonal, the aspect ratio stops dropping out, and the coefficient depends on a dimension that every plate essay so far has been able to ignore.