Two reserves for one buckled web

The panel that carries more after it fails found the reserve: a slender web that buckles in shear is not a failed web, because the diagonal that went into compression hands its job to the diagonal in tension, and the panel becomes a truss nobody drew. The tension has to pull on something followed the tension to where it ends — on the flanges, on the stiffeners, and finally on the end posts, which anchor the whole girder’s field.

Both essays used one model, a single band of yielding tension whose angle and width were found by maximising what it carries, and both said the same thing about it: that design does not use that model, that it uses one of two others with names attached, and that the named models disagree with each other by around twenty per cent. This essay sets the two named models side by side on the same panel and finds that twenty per cent understates the disagreement badly. They agree about the buckling, because they are given the same panel. They disagree about what the reserve after it is made of, and that disagreement is large in exactly the place a designer has a choice to make: the stiffeners.

One web, two models #

The web is the one the previous essays used: 1,500 mm deep, 8 mm thick, in steel of 355 N/mm², a depth-to-thickness ratio of 188. Its plastic shear — the shear it would carry if every point in it reached the shear yield stress $f_y/\sqrt 3 = 205$ N/mm² — is 2,460 kN. It buckles long before that. A square panel, with stiffeners at the depth, buckles at a shear stress of 50 N/mm², a quarter of the yield stress.

Basler’s model, published in 1961 from tests at Lehigh and the basis of the American specification ever since, splits the web’s resistance in two. Up to buckling the whole panel carries a uniform shear. After buckling a diagonal band of tension forms, running from corner to corner between two stiffeners and yielding, and its vertical component carries the rest. The flanges are taken as too flexible to anchor anything, so the band is anchored only where it meets the stiffeners, and its width is fixed by the panel’s proportions. The result is one line:

$$\frac{V}{V_p} = C_v + (1 - C_v)\,\frac{\sqrt 3/2}{\sqrt{1 + \alpha^2}},$$

with $C_v = \tau_{cr}/\tau_y$ the share the panel carries at buckling and $\alpha = a/d$ the spacing of the stiffeners over the depth. The second term is the band, and it is the whole of the reserve.

The rotated stress field, developed by Torsten Höglund in Sweden in the 1970s and the basis of the European rule, describes the same web differently. After buckling the principal stresses in the web rotate: the compressive one stays near its buckling value and the tensile one keeps growing, over the whole web rather than in a band, and the horizontal pull that results is gathered along the flanges and anchored at the girder’s end post. The reserve is a function of the web’s slenderness alone:

$$\bar\lambda_w = 0.76\sqrt{\frac{f_y}{\tau_{cr}}}, \qquad \chi_w = \frac{1.37}{0.7 + \bar\lambda_w} \text{ for a rigid end post,} \qquad \chi_w = \frac{0.83}{\bar\lambda_w} \text{ for a non-rigid one,}$$

for slender webs, with $V/V_p = \chi_w$. The stiffener spacing appears in it only through $\tau_{cr}$.

Both models are given the same buckling stress here, the classical one for a simply supported panel, $k_\tau = 5.34 + 4/\alpha^2$, which falls to 5.34 for a panel with no intermediate stiffeners at all. Both stop at the plastic shear. The European rule lets the rotated field rise 20 per cent above it for ordinary steels and adds a contribution from the flanges; neither is used, so that what differs between the two curves is only what each model says happens after the buckle.

The figure is the argument. With stiffeners at the depth Basler gives the web 0.71 of its plastic shear and the rotated field 0.50 — Basler 40 per cent higher. With stiffeners three depths apart the two are close, 0.38 and 0.42. With no intermediate stiffeners at all, Basler gives 0.14, which is the buckling shear and nothing more, and the rotated field gives 0.41, nearly three times as much. The two curves cross at a spacing of about 2.6 depths, and on either side of the crossing they disagree by more than a calibration margin. They are not two estimates of one reserve. They are two reserves.

The band model of the earlier essays sits between them. Its panel was 2,000 mm between stiffeners, 1.33 depths, and it carried 1,187 kN; Basler gives the same panel 1,515 kN and the rotated field 1,149. That is the twenty per cent the earlier essays quoted — measured at one spacing, near the middle of the figure, where the curves are closest together. Move the stiffeners and the gap opens.

Where each reserve comes from #

The filled part of every bar is the same length for both models, because both are given the same buckling stress. Everything that differs is in the outlined part: what each model says the web finds after buckling.

Basler’s reserve is made by the stiffeners. His band runs between two stiffeners and is anchored by them, and its width is set by how far apart they are; as they move apart the band becomes a narrower strip across a longer panel, and the factor $1/\sqrt{1 + \alpha^2}$ takes it toward nothing. Remove the intermediate stiffeners and there is nothing to anchor a band to. The web is then a panel that buckles and stops. Two thirds of the stiffened web’s resistance — 65 per cent of it — is the band, and none of the unstiffened web’s is.

The rotated field’s reserve is made by the web. After buckling the web keeps working as a membrane, as a compressed plate keeps carrying load after it ripples: its compressive principal stress cannot rise much beyond the buckling value, but its tensile one can rise toward yield, and as it does the direction of the principal stresses swings round. The web’s in-plane stress field rotates, and the horizontal pull it produces is gathered along the flanges and resisted at the end of the girder by an end post. Nothing about that needs an intermediate stiffener. With none, 65 per cent of the web’s resistance is still reserve; with stiffeners at the depth, 51 per cent is, because the stiffeners have raised the buckling part and the reserve is a smaller share of a larger total.

So the two models differ about what anchors the reserve. Basler anchors it on the stiffeners, panel by panel, with the flanges doing nothing; the rotated field anchors it at the ends of the girder, with the flanges carrying the horizontal pull along it. The earlier essay found that the interior stiffeners pass the horizontal force along rather than anchor it, and that the field is finally anchored at the end posts. That is the rotated field’s picture, arrived at from the band.

What each model asks of the flanges #

The difference is easiest to see in what each model needs from the flanges, because it needs something different and the flanges are very different at the two things.

A tension band at an angle pulls on whatever it ends on, and its pull has two components. Basler’s band ends partly on the stiffeners and partly along the flanges, and where it meets a flange its pull is mostly across the flange — downward on the top flange, upward on the bottom one — which a flange can resist only by bending between the stiffeners. A flange is a plate lying flat, and pushed across its thickness it bends as easily as a ruler laid flat on a desk. Basler assumed it resisted nothing, and drew his band only as wide as the stiffeners alone could anchor. That assumption is what makes the stiffeners indispensable in his model: take them away and nothing is left that the band can pull on across the flange.

The rotated field asks the flanges for something else. A uniform field of rotated stress over a long web has vertical components that balance each other from one strip of web to the next, so the flanges are not asked to bend; what is left over is a horizontal pull along the web’s edges, which accumulates along the girder and changes the flanges’ axial force. A flange is very good at that. It is the stiffest thing in the girder in exactly that direction, and it delivers the accumulated pull to the end of the girder, where an end post has to take it. That is why the rotated field cares so much about the end post — a rigid one is worth nearly a third more resistance in the unstiffened web, 0.41 against 0.31 — and so little about the stiffeners in between.

So the two models do not disagree about whether a buckled web can carry tension. They disagree about which way the tension pulls on the boundary, and therefore about which part of the boundary has to be there. One puts the anchorage where the flange is weakest and needs stiffeners to supply it; the other puts it where the flange is strongest and needs only an end to stop it at.

The thinner the web, the more they disagree #

A stocky web yields before it buckles, and then neither model has a reserve to compute: all four curves start at the plastic shear. The disagreement opens as the web thins. At a depth-to-thickness ratio of 100, the stiffened web reads 0.93 by Basler and 0.77 by the rotated field, and the unstiffened one 0.49 and 0.65 — modest differences, in opposite directions. At 250 the stiffened web reads 0.67 and 0.40, and the unstiffened one 0.08 and 0.32, a factor of four.

The shapes say why. Basler’s stiffened curve flattens toward a floor of its own: as the web thins, $C_v$ goes to zero and the resistance tends to the band term alone, $(\sqrt3/2)/\sqrt2 = 0.61$ for a square panel, whatever the thickness. Thinness costs Basler’s stiffened web almost nothing past a point, because the band yields and a yielding band does not care how thin it is. The rotated field keeps falling, as $1/\bar\lambda_w$, because its reserve is a membrane and a thinner membrane carries less. Unstiffened, the order reverses: Basler’s web has only its buckling stress, which falls as the square of the thickness, while the rotated field’s membrane falls only in proportion to it.

What an intermediate stiffener is worth #

A designer does not ask either model for a resistance in the abstract. They ask whether to put stiffeners in, and where, and the models answer differently.

In Basler’s model, stiffeners at the depth multiply this web’s resistance five times. Three depths apart they still multiply it by 2.7. The stiffener is what makes the reserve exist, so it is worth the whole of it.

In the rotated field, stiffeners at the depth multiply the same web’s resistance by 1.24, and three depths apart by 1.03. The stiffener raises the buckling stress — $k_\tau$ from 5.34 to 9.34 for a square panel — and the reserve is computed from the buckling stress, so the stiffener is worth what a higher buckling stress is worth, through a square root. Past two depths it is worth almost nothing. That is the same threshold behaviour a rib on a compressed plate shows: a stiffener acts by changing where the plate is allowed to buckle, and once the panels it makes are long, the buckling coefficient has nearly reached its floor and the stiffener has nothing left to change.

The 5.34 and the 9.34 are worth a moment. They are the same kind of number as the 4 in plate buckling, which was never a fact about plates but the floor of a set of scallops; here the floor is 5.34 and a square panel sits on the scallop above it.

Stiffen the web, or thicken it #

The worth of a stiffener is meaningful only against its cost, and the plainest cost is its steel. A pair of flats on either side of the web, 100 by 10 mm each, is 2,000 mm² of section; with stiffeners a depth apart on a web 1,500 mm deep it weighs the same as 1.3 mm of extra web thickness, and closer stiffeners weigh proportionately more.

For a shear of 1,500 kN, Basler’s model wants stiffeners: at 0.7 depths a 6.3 mm web suffices, and with the stiffeners’ steel counted the girder carries the equivalent of 8.2 mm, against 13.1 mm for an unstiffened web. Stiffening saves 37 per cent of the web’s steel.

The rotated field is nearly indifferent. Its best stiffened arrangement, at half a depth, is 9.95 mm in all; its unstiffened web is 10.05 mm. Stiffening saves one per cent, which is less than the welding of the stiffeners would cost, and less than the fatigue detail at the end of every stiffener weld is worth to the girder’s life. A designer using the European rule leaves the stiffeners out of a girder like this one, and a designer using Basler’s puts them in at close centres — the same web, the same shear, opposite drawings.

Neither is wrong by the standards of its own rule. The difference is a statement about physics, and it is the statement the two models disagree about: whether a slender web without intermediate stiffeners has a post-buckling reserve. Basler’s tests were on stiffened girders, and his model was built to explain them; it says nothing about a web it gives no band to. Höglund’s model was built to cover webs with and without stiffeners, and the European rule applies its curve to both. The American specification has kept Basler’s form and confined it to panels with stiffeners closer than about three depths, and an unstiffened web there carries its buckling shear — so in the region where the two models disagree most, one of them is not being asked.

A stronger steel buys more in one of them #

The models also disagree about a high-strength steel. Basler’s band yields, so its contribution rises in step with the yield stress, and from 235 to 535 N/mm² — 2.28 times — the stiffened web’s resistance rises 2.03 times. The unstiffened web’s does not rise at all, because buckling is elastic and does not know the yield stress.

The rotated field’s resistance rises 1.68 times stiffened and 1.64 times unstiffened, much less than the yield stress and nearer its square root, because a stronger steel makes the same web more slender: $\bar\lambda_w$ grows as $\sqrt{f_y}$, and the resistance falls with it. So a designer moving a slender girder to a higher grade gets nearly all of the upgrade by one rule and about half of it by the other, and the gap between the two models’ answers grows with the grade.

The numbers for a square panel, by hand #

For the 1,500 × 8 mm web: $\sigma_E = \pi^2 E/12(1-\nu^2)\cdot(t/d)^2 = 189{,}800 \times (8/1500)^2 = 5.40$ N/mm². A square panel has $k_\tau = 9.34$, so $\tau_{cr} = 50.4$ N/mm², and with $\tau_y = 355/\sqrt3 = 205$ the buckling share is $C_v = 0.246$.

Basler: $0.246 + (1 - 0.246) \times 0.866/1.414 = 0.246 + 0.462 = 0.708$, or 1,742 kN.

Rotated field: $\bar\lambda_w = 0.76\sqrt{355/50.4} = 2.02$, past 1.08, so $\chi_w = 1.37/(0.7 + 2.02) = 0.504$, or 1,240 kN; with a non-rigid end post, $0.83/2.02 = 0.411$.

With no intermediate stiffeners: $k_\tau = 5.34$, $\tau_{cr} = 28.8$, $C_v = 0.141$ — Basler’s whole answer, 346 kN — and $\bar\lambda_w = 2.67$, $\chi_w = 1.37/3.37 = 0.407$, or 1,000 kN.

What both models leave out, and what this comparison leaves out of them #

Both models are given the classical buckling stress of a simply supported panel. The flanges restrain the web’s edges a little, which raises the buckling stress and helps most the model that leans on it hardest — Basler’s unstiffened web, which has nothing else.

The rotated field is capped at the plastic shear, where the European rule allows 1.2 times it for steels up to S460, and its flange contribution is left out. A girder with heavy flanges gets an extra share from the flanges’ bending between plastic hinges — the anchorage the earlier essay computed — which the European rule adds as a separate term. Basler’s model assumes the flanges anchor nothing. Both omissions are made so that the comparison is between the two webs’ reserves alone.

Basler’s inelastic correction is his own, $\sqrt{0.8\,\tau_{cr}\tau_y}$ above 0.8 of the yield stress; it matters only for the stocky webs at the left of the slenderness figure.

Shear alone. A real girder has its largest shear where its moment is small and the reverse, and both rules add an interaction between shear and bending that is a subject in its own right — the flanges that hold the ends of a tension field are the same flanges carrying the bending.

A mechanism and a fitted curve #

The rotated field is a different kind of model from Basler’s, and the figures hide it. Basler’s is a mechanism: a band of stated width at a stated angle, yielding, in a state that can be drawn. The rotated field’s design curve is a fit to tests through a theory of the web’s stress state, and the curve is what the rule uses. That is not a defect — every rule for a buckled plate is partly empirical, because the buckled shape depends on imperfections and residual stresses no calculation knows — but it means the European curve cannot be drawn as a band in a panel, and the figures in the earlier essays, which drew the band, were always drawing Basler’s picture. A model’s choice of what to draw is part of what it reports, and two rules that disagree about a reserve this large are, like two codes that classify one diaphragm differently, a sign that the quantity in between is the thing to have measured.

Still open: the end post both models lean on #

The rotated field’s reserve in an unstiffened web depends on one assumption, the rigid end post, and the non-rigid curve shows what it is worth: 0.41 of the plastic shear against 0.31 with no intermediate stiffeners. Basler’s stiffened web leans on its end post too, for its end panel. A rigid end post is two stiffeners close together, welded to the flanges and to each other, sized as a short beam spanning the depth against the horizontal pull of the field. How stiff it has to be for the rotated field to use the rigid curve, whether the horizontal pull it must resist is the same in the two models — Basler’s anchored panel by panel, the rotated field’s accumulated along the flanges — and whether an end post sized for one model is sufficient for the other, is the question that decides how much of either reserve a real girder end can deliver.