The plate the wrong theory gets right

A column in a material with no yield plateau buckles at the stress where Euler’s formula, written with the tangent modulus in place of the elastic one, is satisfied. That is the whole of the column’s inelastic theory, and it works because a column bends in one plane: one fibre stretches, the opposite one shortens, and the only stiffness either sees is the slope of the stress–strain curve at the stress it already carries. A stainless steel column with a rounded curve loses two fifths of its strength at the slenderness where its proof stress and its elastic buckling stress meet, because there the tangent modulus has fallen to a fraction of the elastic one.

Stainless steel and aluminium are mostly used in thin sections — cold-formed channels, extruded tubes, plates welded into boxes — and those fail by their plates buckling locally rather than by the whole member bending. A plate buckles by bending in two directions at once and twisting, and it needs three stiffnesses, not one. The question the column left was whether a plate is lowered by its rounded curve as much as a column is, and whether the theory that gives the lower answer is again the one the tests support.

Three stiffnesses where a column has one #

A long plate, simply supported on its four edges and compressed along its length, buckles into a pattern of bulges each about as long as the plate is wide. For an elastic plate the buckling stress is the familiar $4\pi^2E/12(1-\nu^2)$ times $(t/b)^2$, with four as the buckling coefficient for this support. Written so that the material’s stiffnesses appear separately, it is

$$\sigma = \frac{\pi^2}{6}\left(\frac{t}{b}\right)^2\left[\sqrt{C_{11}C_{22}} + C_{12} + 2C_{66}\right],$$

where $C_{11}$ is the material’s stiffness along the load, $C_{22}$ across it, $C_{12}$ the coupling between them and $C_{66}$ the stiffness in shear, which is what resists the plate’s twisting between bulges. For an isotropic elastic material the bracket comes to $2E/(1-\nu^2)$ and the formula is the elastic one.

Once the material is past the straight part of its curve, the three stiffnesses are no longer one modulus times three constants. They are the material’s instantaneous stiffness in each direction at the stress it carries — and how a material that has softened along one direction responds to being pushed in another is not something a uniaxial stress–strain curve says. A theory of plasticity has to supply it, and there are two classical ones.

Flow theory says that plastic strain develops only in the direction the yield surface points — along the deviator of the stress already there — and only while the stress is pushing outward on the surface. A plate compressed along x has softened along x. Pushed across, or twisted, it responds almost elastically, because those increments do not push outward on the surface. This is the theory a metal’s behaviour actually follows: plastic strain depends on the path taken to reach a stress, and flow theory is built on that.

Deformation theory says that the total strain is a function of the total stress, as for a nonlinear elastic material, so the secant modulus — stress over total strain — softens every direction equally, shear included. It is wrong for any path that turns, and for a metal it is a theory of convenience.

The difference is stark. At the proof stress the stainless steel’s tangent modulus is 9 per cent of its elastic modulus, and that is all a column has to bend against. The plate by flow theory still has 85 per cent of its elastic stiffness. Its stiffness along the load has fallen with the tangent modulus, but its stiffness across the load is nearly elastic, the coupling between them has grown, and its twisting stiffness has not moved at all — the dashed line runs flat at 0.35 of the elastic bracket, because flow theory gives a uniaxially compressed material its full elastic shear modulus. Deformation theory softens all three: 34 per cent of the elastic plate’s stiffness at the proof stress, a third of it twisting.

A stocky plate at twice its proof stress #

Put each theory’s stiffness into the formula and solve for the stress at which it is satisfied, as for the column.

For slender plates all three agree, because the plate buckles while the material is still elastic. As the plate gets stockier they separate, and flow theory leaves the others behind. At λ̄p = 0.6 — a plate about 34 times as wide as it is thick, a common proportion for a stainless flange — flow theory puts the buckling stress at 2.30 times the proof stress; at 0.4, at 5.29. At 2.30 times its proof stress this stainless steel’s curve is at a strain of 30 per cent, far into the range where any real plate has long since folded.

Deformation theory puts the same plate at 0.99 of its proof stress, and the column at 0.76.

The plate tests reported since the 1940s — on aluminium alloy plates for aircraft first, stainless steel later — sit close to deformation theory and nowhere near flow theory. That is the plastic buckling paradox: the theory whose stress–strain law is right predicts the buckling load wrongly, and the theory whose law is wrong predicts it well. The column’s two theories had a disagreement of their own — the tangent and reduced moduli — and there the lower one was the one tests supported, and Shanley explained why: the column starts to bend at the tangent-modulus load. The plate’s lower answer is again the one the tests support, but the reason is different, and it can be located.

Most of the disagreement is the twist #

Take flow theory and change one thing: give it deformation theory’s shear modulus, and leave its stiffnesses along and across the load as they were.

Two thirds of the gap closes. The plate’s twisting stiffness — the shear modulus in its plane — accounts for about two thirds of the difference between the two theories at every slenderness where they differ, and the stiffness across the load accounts for most of the rest.

That locates the paradox. Flow theory’s yield surface is smooth, so a material compressed in one direction and then sheared responds elastically to the shear: the shear increment is tangent to the surface and produces no plastic strain. A real metal’s yield surface is not smooth at the current stress. It forms a corner there, or behaves as if it did, because the slip on many crystal planes that makes up plastic flow is triggered by shear in many directions at once, and a small shear added to a large compression does produce plastic strain immediately. Deformation theory’s reduced shear modulus is close to what such a corner gives. So deformation theory is right about the one stiffness that matters most for a plate, for a reason unconnected with why it was adopted.

The other half of the explanation, which the literature since Onat and Drucker gives, is imperfection. A plate that is not perfectly flat begins to twist from the first load, and under flow theory a small twist taken in the plastic range is resisted by much less than the elastic shear modulus, because the path has turned. A flow-theory plate with an imperfection of a fraction of its thickness falls most of the way to deformation theory’s load — the plastic counterpart of a structure that reaches a third of what its perfect theory promised, where the perfect structure’s answer is the one no real structure reaches. Neither of these arguments is computed here; what the figure shows is that the shear modulus is where either argument must act.

Two lower answers for two different reasons #

It is worth setting the plate’s paradox beside the column’s, because they look alike and are not.

The column’s two theories — tangent modulus and reduced modulus — disagree about what happens at the instant of buckling. Both use the same material law. The reduced-modulus theory assumes the load stays constant while the column bends, so that the fibres on the convex side unload elastically and stiffen the section; the tangent-modulus theory assumes the load keeps rising, so that no fibre unloads. Shanley’s observation was that a real column starts to bend at the tangent-modulus load and has to keep gaining load to keep bending, so the lower answer is the load at which bending begins and the higher one is a ceiling it approaches. The disagreement was about the loading path, and the tests settled it.

The plate’s two theories disagree about the material. Both assume no fibre unloads; what differs is how a material already compressed responds to a new increment in a different direction. That is a question a uniaxial test cannot answer at all — a stress–strain curve measured on a coupon is a single path, and both theories fit it exactly by construction. The column needed only the slope of that curve; the plate needs the material’s response off the curve, and the two theories extrapolate it differently. Here the lower answer is right because the material’s real response to a turning path is softer than flow theory’s smooth surface allows, not because of anything about the loading.

So the same rule — use the lower theory — holds for both, and for reasons that have nothing in common. That matters when the rule is extended. A column’s tangent-modulus answer is a lower bound on its bifurcation load in every material, because it rests on a statement about loading. A plate’s deformation-theory answer is not a bound on anything; it is an approximation that happens to match the corner of a real yield surface, and a material whose surface behaves differently — some heavily textured alloys, composites with plastic matrices — need not follow it.

The plate’s support conditions do not change any of this. A coefficient that is not four, for a flange with one free edge or a web in bending, changes the weight given to each of the three stiffnesses: an outstand flange, held on one edge only, buckles mainly by twisting about its supported edge, so its inelastic buckling depends even more on the shear modulus and its paradox is wider.

The rounded curve costs a plate less #

The question the column left can now be answered with the theory the tests support.

Both curves dip where the sharp envelope has its corner — where the elastic buckling stress equals the proof stress, and a material with a plateau would be at its best. The column falls to 0.59 of the envelope there, a loss of 41 per cent. The plate falls to 0.72, a loss of 28. A rounded curve costs a plate about two thirds of what it costs a column at the same slenderness.

The reason is in the stiffness figure. A column has one stiffness and it is the tangent modulus, the steepest-falling measure of a rounded curve. A plate has three, and under deformation theory only the part along the load is the tangent modulus; the stiffness across the load and the twisting stiffness soften with the secant modulus, which falls much more slowly. At the proof stress this stainless steel’s tangent modulus is 9 per cent of E and its secant modulus is 37 per cent. The plate’s buckling stress is a mixture of the two, and the secant pulls it up.

The same mixture explains the stocky end. Below a slenderness of about 0.6 the plate by deformation theory buckles above its proof stress, which a column of the same material does only below about 0.27. A plate in a rounded material keeps gaining from the rising curve long after the column has stopped, because its secant moduli are still large when its tangent modulus has collapsed — which is why a stocky stainless flange can carry more than its proof stress before it buckles, and why the newer design methods for stainless sections, the continuous strength method among them, let the plates of stocky cross-sections reach stresses on the strain-hardened part of the curve.

Stainless steel and aluminium #

The two materials differ in the shape of their knee, which is what the Ramberg–Osgood exponent measures: about 6 for an austenitic stainless steel, 25 or so for a heat-treated aluminium alloy.

The heat-treated aluminium alloy has a sharp knee, and both its column and its plate lose little to it: 16 per cent and 9. The stainless steel’s knee is the rounded one, and its losses are 41 and 28. In both materials the plate loses a little over half to two thirds of what the column loses. The ratio is not a property of either material; it is a property of having three stiffnesses rather than one, two of them governed by the secant.

The flow-theory curve for the aluminium shows the paradox shrinking with the knee.

With a sharp knee the material is nearly elastic until close to its proof stress and nearly flat beyond it, so a plate either buckles elastically, where the theories agree, or buckles on the plateau, where both give little more than the proof stress — except flow theory, which on the stockiest plates still finds an elastic shear modulus to stand on and runs away. The paradox lives where a material spends a long stretch of its curve partly yielded, which is exactly where stainless steel spends it.

What a designer takes from it #

Design rules for stainless steel plates are written as an effective width: the plate is assumed fully effective up to a plate slenderness a little above 0.6, and beyond that it carries load on two strips at its edges whose width falls with slenderness. Those rules are calibrated to tests, not to either theory of plasticity, and they come out close to what deformation theory gives for the buckling stress, plus the post-buckling reserve a plate has and a column does not.

Two things follow from the comparison with the column. The first is that importing a column’s inelastic reduction into a plate check is conservative by a wide margin — about a third of the reduction is unnecessary. The second is that using the theory of plasticity a finite-element package defaults to, which is flow theory, for a perfect stocky plate gives an answer that can be more than twice the true one. A plate analysed that way must be given an imperfection, and a realistic one, before its buckling load means anything. That is the general rule the column that had yielded before it was loaded taught about columns, and for plates in rounded materials it is not a refinement but the difference between a right answer and one twice too high.

What the plate leaves out #

A long plate with simple supports. Real plates in sections are restrained by their neighbours at the edges, their buckling coefficient is not four, and the class a section is put in depends on how the stress across the plate varies, which here is uniform. The theories’ disagreement, and the twist’s share of it, carry over to other supports; the numbers do not.

A single exponent. The Ramberg–Osgood curve with one exponent fits a stainless steel well up to the proof stress and poorly beyond it, where the real curve rises more steeply. Stocky plates that buckle past the proof stress are sensitive to that, and to the two-stage curves used for stainless steel in design.

Isotropy. Cold-rolled stainless sheet is stronger along the rolling direction than across it, and cold-formed corners are much stronger than flats. A plate’s stiffness across the load is then not the same function as along it, which this model assumes.

No imperfection and no post-buckling. Every number is the bifurcation stress of a perfect plate, which is not the plate’s strength. A plate carries load after it buckles, by stretching across its middle, and that reserve is what an effective width measures.

Still open: whether the cold-formed corner is the section’s column #

A cold-formed stainless section has flat plates joined by corners bent to a radius a few times the thickness, and the bending work-hardens the corners to well above the flats’ proof stress, with a sharper knee. The corners do not buckle locally; they are short, thick, curved and strong, and they carry their share of the load like small columns along the section’s edges. Whether a stocky cold-formed section’s local buckling is then governed by its flats, which have the rounded curve and buckle by deformation theory, or held back by its corners, which carry stress the flats have shed — and whether that makes the section stronger than the sum of its plates — is the question that the corner’s different curve puts to the plates beside it.