The section that cannot reach its own strength

Structural steel sections are sorted into four classes, and the sorting is done by comparing a plate’s width-to-thickness ratio against a number. Nine, ten, fourteen, thirty-eight, forty-two, seventy-two: a table of integers, multiplied by a factor $\varepsilon = \sqrt{235/f_y}$, with no derivation attached anywhere a designer normally looks.

The integers are not arbitrary and they are not empirical fits either. They are the answer to one question — at what width-to-thickness ratio does a plate buckle at exactly the stress at which it would have yielded — and that question has a closed-form answer this site has already computed, for a different purpose.

What a class actually is #

The classes are not about strength. They are about how far a section can bend before its own plates give way, which is a rotation-capacity question wearing a geometry’s clothes.

A class 1 section can reach its plastic moment and go on rotating at it, far enough for a plastic hinge to form and for a collapse mechanism to assemble. A class 2 section can reach the plastic moment but not hold it — the plates buckle shortly afterwards, so the moment is available and the rotation is not. A class 3 section can reach the moment at which its extreme fibre first yields and no further. A class 4 section cannot even do that: parts of it buckle while the whole section is still elastic, and the design has to proceed on an effective width with the middle of the plate deleted.

So the classification is a ladder of increasing local slenderness, and each rung removes one of the things the analysis was assuming it could have. Class 1 permits plastic analysis. Class 2 permits the plastic moment in an elastic analysis. Class 3 permits only the elastic moment. Class 4 permits less than the section appears to contain.

Which raises the question of how one derivation produces four boundaries, since the plate calculation answers only one question and the table has three numbers in it for every plate.

That is the shape of the whole table. One derivation fixes the class 3 boundary, where the section is asked to reach first yield and no further; the class 1 and class 2 boundaries are tighter by amounts that come from rotation demands rather than from plate theory, and the sections below deal only with the boundary the calculation can reach.

Where the numbers come from #

A flat plate in compression buckles at an elastic critical stress

$$\sigma_{cr} = \frac{k\pi^2E}{12(1-\nu^2)}\left(\frac{t}{b}\right)^2$$

with $k$ the buckling coefficient, which depends on how the plate’s edges are held: about 0.43 for an outstand supported along one long edge and free along the other, which is a flange; about 4.0 for an internal plate supported along both, which is a web. That expression is already on this site, where it was used to explain why a wide plate’s buckling stress has nothing to do with the strength of the material.

Set that stress equal to the yield stress and solve for $b/t$:

$$\frac{b}{t} = \pi\sqrt{\frac{kE}{12(1-\nu^2)f_y}}$$

Everything on the right is known. For an outstand at $f_y = 235$, that gives 18.64. For an internal plate in bending it gives 56.84. Those are the ratios at which a perfect plate is exactly on the boundary between yielding and buckling.

The $\varepsilon$ in the code table is now explicable rather than conventional: the expression contains $f_y$ under a square root in the denominator, so the limit scales as $1/\sqrt{f_y}$, and dividing out by $\sqrt{235/f_y}$ leaves a number that does not depend on the grade. That is the entire content of the $\varepsilon$ factor, and it is a consequence of a square root rather than a calibration.

The gap, and why it is a constant #

The derived limits are larger than the quoted ones, which means the code is stricter than the perfect-plate calculation says it needs to be. The ratios are 1.331 for an outstand, 1.353 for a web in bending, and 1.496 for a web in pure compression.

Those are constant to four significant figures across S235, S355 and S460. That constancy is the finding, because it says what kind of a thing the gap is.

A gap that varied with the grade would suggest something that scales with strength — a material effect. A gap that varied with the plate type but not the grade is a fixed knockdown: the quoted limit is the perfect-plate limit multiplied by a factor of about three-quarters, and the factor is a property of the plate’s boundary conditions and imperfections rather than of the steel.

What is in the knockdown is the same pair of things that put the knee in a column curve. A real plate is not flat — rolling and welding leave it with an initial bow, and a plate that is already bowed starts deflecting under the first increment of load rather than waiting for a critical stress. And a real plate carries residual stress, so parts of it are already in compression before the load arrives and reach yield early, losing their stiffness and leaving the rest to buckle sooner.

Free one of those long edges and the critical stress falls by a factor of nine, because $k$ goes from 4.0 to 0.43, so the crossing moves in to a width nearly three times smaller. The whole difference between a flange limit and a web limit is that one ratio of coefficients, under a square root.

What a class costs, in bending capacity #

The classification decides which moment a section may be designed for, and the steps between the classes are not small.

For the rolled I-section used throughout these essays, the plastic moment is 72.6 kNm and the elastic moment — the moment at which the extreme fibre first reaches yield — is 66.8. The step from class 2 to class 3 therefore costs 8%, which is the shape factor of a compact I-section and is modest, because a section with material in its flanges had little in reserve to lose.

What a class 3 section is stopped at is worth picturing. The stress diagram is a straight line reaching the yield stress at the two surfaces and nowhere else, with the material near the neutral axis working well under capacity; a class 1 or class 2 section is permitted to recruit that material and a class 3 one is not. For an I-section there is very little of it to recruit, which is exactly why the step costs so little.

For a rectangle the step would cost 33%, and for a tee 44%. So the penalty for a poor classification is largest exactly for the sections that gain most from plasticity, which is a compounding rather than a coincidence: a section with a large shape factor has a lot of material near the neutral axis, which is a section with a deep web, which is a section whose web is more likely to be slender.

The step from class 3 to class 4 has no fixed size at all, because it depends on how much of each plate is deleted. That is the one boundary where the loss is a computation rather than a ratio.

Which free body produced the number #

The free body is a rectangular panel of plate, cut out of the flange or web, with the in-plane compressive stress on two opposite edges and the support conditions of its long edges stated rather than assumed. That statement is the whole of $k$: 0.43 says one long edge is held against out-of-plane movement and the other is free; 4.0 says both are held.

The classification of a real section then depends on which of those a given plate is, and that is a judgement about the section rather than a calculation. A flange outstand of an I-section is held at the web and free at the tip, so 0.43. A web is held at both flanges, so 4.0. A box section has no outstands at all, so every plate takes 4.0, which is why box sections classify so much better than open ones at the same material thickness — the same steel, arranged so that no edge is free.

So the decision that separates a section classifying well from a section classifying badly is taken when the shape is chosen, before any thickness is picked and before any grade is specified. It is a decision about which edges are held.

The width $b$ is measured from the point of restraint, not from the centreline, and the difference is the root radius. That is why the code table’s $c/t$ is defined with a diagram beside it and why a section’s tabulated ratio is not what a scale rule on the outline would give.

The requirement depends on the answer #

There is a circularity in how the classes are used that is worth naming, because it is the one place the table stops being a lookup.

A plastic global analysis is permitted only if every section at which a hinge forms is class 1. Sections elsewhere in the same frame need only be class 2, since they have to reach their plastic moment and not hold it, and sections that never yield need only be class 3. So the requirement is not a property of the member; it is a property of the member’s role in a mechanism.

And where the hinges form is an output of the analysis. It is decided by the ratio of the applied moments to the plastic moments along the frame, which depends on the sections chosen, which depends on the classification, which depends on where the hinges are. Nothing in the table breaks the loop.

Two resolutions are used and they are not equivalent. The safe one is to make every member class 1 and stop thinking about it, which is what most design does and which costs a little material in the members that were never going to hinge. The honest one is to run the analysis, read off the hinge locations, and check the classification of those sections against class 1 and the rest against class 2 — and then check that the sections chosen have not moved the hinges somewhere else. That converges quickly in practice and it is a genuine iteration rather than a check.

The trap sits between the two. A frame analysed plastically, with a class 2 member at a location the mechanism needs, has a calculated collapse load that is unavailable: the hinge that was supposed to rotate loses its moment shortly after forming, the mechanism does not assemble, and the load at which the frame actually fails is the one that formed the first hinge. Which is the same failure this site described for a brittle redundant structure, arriving here from a plate thickness rather than from a material.

Where the model stops #

The classification is a proxy for rotation capacity and does not measure it. A class 1 section is presumed able to deliver whatever rotation the plastic analysis demands. What that demand actually is depends on the structure — on how many hinges have to form, in what order, and how far apart the critical sections are — and nothing in the classification asks. A frame requiring an unusual amount of redistribution can exceed a class 1 section’s capacity, and there is no check in ordinary design that would notice.

The class of a section is not a property of the section. It depends on the stress distribution across the plate, which depends on the loading. A web in pure bending has half of it in tension and classifies well; the same web under axial load is in compression throughout and classifies much worse — 38 rather than 42 in the quoted numbers, and 1.496 rather than 1.353 in the knockdown. So a member’s class changes when the load changes, and a beam-column has to be classified for the combination it carries.

Post-buckling strength is real and the classification is silent about it. A plate that has buckled goes on carrying load — the middle drops out and the edges work harder — and the effective-width treatment of a class 4 section is precisely an accounting for that. The class boundary is not a cliff.

And a ductile material can be assembled into a brittle member. This is the most important consequence and it deserves stating in one sentence: mild steel has 20% elongation and a section made of it can have essentially zero rotation capacity, because what fails is not the material but the arrangement.

The generalisation #

The pattern is that a limit expressed as a number is a calculation somebody has already done, and knowing which calculation is what says when the number stops applying.

A designer who knows only that the outstand limit is 14ε cannot say what happens to it for a plate stiffened along its free edge, for a curved flange, for a plate in shear rather than compression, or for a material with a different modulus. A designer who knows the limit is $\pi\sqrt{kE/12(1-\nu^2)f_y}$ knocked down by a quarter can answer all four, because each of them is a change to one term.

It is the same relationship this site has to the deflection coefficient $5/384$ and to the effective-length factors: the tabulated number is fine until the case is not in the table, and the case is not in the table surprisingly often.

What the picture cannot show #

The figure at the top of this page plots one number per plate per grade, and there are three things it flattens that decide real cases.

The class of a section is the worst of its plates, not an average. A member with a compact flange and a slender web is a class 4 member; the good flange does not compensate. So the classification is a minimum over the parts, and a section can be pushed a whole class by a single plate that is a millimetre too thin.

The buckling coefficient is drawn as a constant and it is not. The 0.43 and 4.0 used here are the values for a long plate under uniform compression with idealised edge restraint. A real flange is elastically restrained by the web rather than simply supported, which raises $k$; a real web under a stress gradient has part of it in tension, which raises $k$ a great deal; and a plate whose length is comparable to its width has a $k$ that depends on the aspect ratio. Each of those moves the derived limit, and the figure has one number where a family belongs.

And the knockdown factor is drawn as if it were measured, which it is not. The 1.331 and 1.353 are the ratio of a calculation to a code number, and the code number is the residue of a test programme with scatter in it. What the constancy across grades establishes is the form of the gap — that it is a fixed multiplier and not a strength effect — and that is a claim the arithmetic supports. The precise value of the multiplier is a claim about somebody else’s tests.

A surprising place this turns up #

The classification is the reason the shape factor argument comes out where it does, and the two arguments push in opposite directions in a way that is worth putting side by side.

The shape-factor argument says: a section with material spread far from the neutral axis is efficient, has a small shape factor, and needs little curvature to reach its plastic moment. The classification says: a section with material spread far from the neutral axis has wide thin plates, classifies badly, and may not be able to reach its plastic moment at all.

Both are true and they meet somewhere. The optimum is not the thinnest plates that can be rolled, and it is not the stockiest arrangement either; it is the section whose plates are just stocky enough to be class 1 while its material sits as far out as that permits. Nearly every standard rolled section in a catalogue is close to that point, which is not a coincidence — the catalogues were designed by people solving exactly this trade, and the reason a universal beam looks the way it does is that the two arguments on this page were balanced against each other.

Where the ladder goes next #

Later rungs on this anchor: the effective-width treatment of class 4 sections in full, and Winter’s expression for it. Rotation capacity measured against demand rather than assumed, and the tests that established the boundaries. Buckling coefficients for plates under stress gradients, in shear, and with stiffeners, which is where $k$ stops being one of two numbers and becomes a function. Interaction between local and global buckling, where a class 4 column loses effective area and thereby shifts its own centroid. Cold-formed sections, which are almost all class 4 by construction and have a design philosophy built around it. And the same argument for concrete, where the equivalent of a class is whether the section is under- or over-reinforced.

Historically the classification arrived with plastic design and for its sake. Once it was accepted in the 1950s that a section could be designed on its plastic moment, it became urgent to say which sections could actually deliver one, and the tests that fixed the boundaries were rotation tests rather than strength tests: beams bent until the moment fell away, with the classification drawn where the available rotation stopped covering the demand. The numbers in the table are the residue of that programme, and they are quoted far more often than the question they answer is asked.