The corner the rolling mill rounds

The slit that costs a factor of six hundred set the open section’s torsion constant beside the closed one’s and found the open one “contains no arrangement at all”: $\Sigma bt^3/3$ adds up thin plates as if they were separate, and it does not matter how they are joined. That is true to the order of accuracy the argument needed, and it is exactly the accuracy at which a rolled section differs from the plates it is drawn as.

A universal beam is not three plates welded together. It is one piece of steel rolled hot between shaped rolls, and where its web meets each flange the rolls leave a fillet, a concave curve of metal a centimetre or so in radius. The fillets are about one per cent of the section’s area and nobody designs with them. This essay asks what they do to the one property of the section that depends on its thickness cubed.

The film over the real outline #

Prandtl’s analogy, which the corner a soap film cannot finish used to follow a keyway, turns the question into a picture. Stretch a film over a hole the shape of the cross-section and blow it up with a small pressure: the film’s slope at any point is the shear stress there, and the volume under it is half the torsion constant. Over a thin strip the film is a long parabolic tent whose height grows as the square of the strip’s thickness, which is where the cube in $bt^3/3$ comes from — a height of $t^2/4$ times a width of $t$.

The film here is relaxed numerically over the actual outline of a 457 × 191 × 67 universal beam — flanges 189.9 by 12.7 mm, web 8.5 mm, root radius 10.2 mm — on a grid a twenty-fourth of the web’s thickness. Because the film over a long strip is the same parabola all the way along, only the junctions and the flange tips need solving: the flanges and web are cut short a few thicknesses from each disturbance, and the lengths cut off are added back at exactly $t^3/3$ a millimetre.

Away from the junction the contours run straight along the flange, the tent of a thin strip. At the junction the film has more room: the circle that fits inside the joint of web, flange and fillet is 19.0 mm across, half as wide again as the flange is thick, and the film domes up into it. With sharp corners the dome is smaller and the contours pinch into the re-entrant corner, where the film’s slope — the stress — has no finite limit.

A tenth of the torsion constant #

Three corrections separate the textbook sum from the section. The first is the tips. A flange’s film cannot be a full parabola right to its edge, because the edge is a boundary too, and the film sags across the last flange thickness or so; each free tip costs about a tenth of $t^4$. The second is the junction, which works the other way: where web meets flange there is more metal than either plate alone has, and the film rises. With sharp corners the two nearly cancel, and the tips win by a little — 34.2 cm⁴ against the sum’s 34.7. The third is the fillet, and it is not small. Fillets that are one per cent of the steel add 8.7 per cent to the torsion constant, 37.2 cm⁴, which is the section tables’ 37.1.

That agreement is the check on the whole calculation, and it is not a coincidence of fitting. The tables’ constant comes from a formula fitted in the 1960s to exactly this problem — two thirds of $b t_f^3$ for the flanges, a third of $(h - 2t_f)t_w^3$ for the web, a junction term $2\alpha D^4$ with $D$ the inscribed circle’s diameter, and a tip correction of $-0.42\,t_f^4$ — and the film, knowing nothing of the formula, lands within a fifth of a per cent of it.

Why one per cent of the area is worth nine of the constant #

The disproportion is the cube at work, and it can be estimated without the film. Over a thin strip the film’s height is set by the local thickness squared, and its contribution to the constant by the thickness cubed per unit length. The junction is a short stretch of section, about one inscribed diameter long, in which the effective thickness is not $t_f = 12.7$ mm but something approaching $D = 19.0$ mm. Over that stretch the constant per unit length is $(19.0/12.7)^3 = 3.3$ times the flange’s. So a patch of metal that adds half as much again to the thickness over a length of about two centimetres adds more than twice the flange’s stiffness over that length — and there are two such patches in an I-section, one at each flange.

The area bookkeeping runs the other way. The four fillets together are $4(1 - \pi/4)r^2 = 89$ mm² of the section’s 8,551. Area counts every square millimetre once; the torsion constant counts each one by the height of the film above it, which grows roughly as the square of its distance from the nearest free surface, and the fillets put metal exactly where the nearest surface is furthest away. The same arithmetic is why the free flange tips cost what they do: the last flange thickness at each tip has metal close to two surfaces at once, and the film there cannot rise to the strip’s full height.

It is also why the fillets matter for torsion and hardly at all for anything else. For bending, the fillets sit near the flanges and add their area times the lever arm squared, about one per cent of the second moment. For shear, they add a little area to the web. Only the torsion constant weights metal by its thickness cubed, and only for torsion is the junction the thickest place in the section.

Seven sections, the same tenth #

The same holds across the tables. For every one of seven sections — four beams and three columns, from 25 to 101 kg/m — the film over the rolled outline gives the published constant to within 0.2 per cent, and the fillets add between 9 and 13 per cent over the sharp outline. The smallest sections gain most: a 152 × 152 × 23 UC has a 7.6 mm root on 6.8 mm flanges, so its fillets are large compared with the plates they join, 1.7 per cent of its area, and they are worth 13 per cent of its torsion constant. The textbook sum is 7 to 12 per cent low for every rolled section in the tables, and always on the same side.

For stiffness, that is an error in the conservative direction and a designer using $\Sigma bt^3/3$ has lost a tenth of the section’s torsional stiffness and nothing else. It matters most where the torsion constant is the whole of a result: a beam that fails sideways has a critical moment that goes as $\sqrt{GJ}$ once its span is long enough for warping to fade, so the fillets’ nine per cent of $J$ is worth about three per cent of the buckling moment of this beam over a 12 m span, and just over half a per cent over 3 m, where warping carries most of the torque.

Where the stress is #

The larger surprise is not in the volume of the film but in its slope.

A design check of an open section in torsion takes the shear stress in each plate as $T t/J$, and the largest in the section as that in its thickest plate: the flange, at $T t_f/J$. Along the flange face, far from the web, the film says the same — its slope is exactly the thin strip’s. It says the same on the web, at two thirds of it, which is the web’s thickness over the flange’s. And on the fillet it says something else: the stress climbs to 1.49 times the flange formula, at the point where the fillet’s curve is most sharply turned towards the web.

The number has a simple reading. The junction is, locally, a strip as thick as the circle that fits inside it — 19.0 mm — and a strip that thick carries $T D/J$ at its surface, 1.50 times the flange’s. The film’s peak on the fillet is within a per cent of that. Across the seven sections the fillet carries 45 to 65 per cent more than the flange formula, always close to $D/t_f$, so the correction a designer needs is one number the section tables already print the ingredients of. The thickest plate in an I-section is the junction, and the design formula does not know it is there.

The check that belongs at the fillet #

A member in torsion is rarely in torsion alone. The torque usually arrives with the bending that the same eccentric load produces, and the design check combines the two: a shear stress from the twist and a normal stress from the bending, compared with the yield stress through von Mises’s criterion. Where that combination is largest is a question about where each stress is largest, and the fillet is badly placed. It sits at the underside of the flange, within a flange thickness of the extreme fibre, so the bending stress there is ninety-odd per cent of the flange’s; and it is where the Saint-Venant shear stress is highest in the whole section. The combined check that governs an I-section in torsion and bending is at the fillet, not at the flange face the formula names and not at the web.

Warping complicates this, and in the other direction. A member restrained against warping carries part of its torque by bending its flanges in opposite directions, and the normal stress from that is largest at the flange tips, not at the junction. Where warping dominates — short members, or members near a warping-restrained support — the flange tips govern and the fillet’s Saint-Venant peak is secondary. Where Saint-Venant torsion dominates — long members twisting freely, which is the case the torsion constant describes — the fillet governs, and the factor of one and a half that the formula leaves out is the whole margin a designer thought was there.

A stress concentration of 1.5 is modest as these things go; a hole in a plate triples the stress beside it. What makes this one worth knowing is that it has no hole to announce it. The section looks smooth, the fillet looks like a strengthening, and the one formula every check uses reads the flange.

A bigger root, a stiffer section, and past a point a worse stress #

Varying the root radius separates the two things a fillet does. It always adds stiffness, and faster as it grows: 1.3 per cent at a quarter of the flange thickness, 7.7 at three quarters, 30 at one and a half. Its effect on the stress has a minimum. A tight root concentrates the stress at a corner — at a quarter of the flange thickness the peak is 1.59, and with no radius at all the re-entrant corner’s stress is unbounded, the same singularity as the keyway’s. A generous root spreads it, but also thickens the junction, and the peak climbs again with $D$: 1.60 at one and a quarter thicknesses, 1.69 at one and a half. The least-stressed junction has a root about half the flange’s thickness, and the rolled sections, at 0.8 to 1.1, sit just past that minimum, where a little more stiffness costs a little more stress.

Rolled sections were not proportioned for torsion; the root radius is set by the rolls, by the flow of metal into the flange during rolling, and by the need to keep a smooth transition for bending and for the weld of any stiffener that has to be cut round it. That the result is within a few per cent of the least-stressed junction is a pleasing accident rather than a design.

The same plates, welded #

A welded I-section — a plate girder, or a fabricated beam made from three plates — has no root fillet. Its web is joined to each flange by two fillet welds, triangles a few millimetres in the leg, and where the weld’s face meets the flange the section’s outline turns inward, a re-entrant corner with 225° of metal round it, which concentrates stress more gently than a sharp right-angled corner’s 270° but still without limit. So the plates of the 457 UB, welded rather than rolled, have a torsion constant close to the sharp outline’s 34.2 cm⁴ and a peak stress at the weld toes that the film cannot bound. That is one of the quiet differences between a rolled section and “the same section” fabricated to replace it: nine per cent less torsional stiffness, and its torsional stress peak moved from a smooth curve in the parent metal to the toe of a weld, which is also where a fatigue crack starts.

The tables’ constant, by hand #

The fitted formula can be reproduced on the back of the drawing for the 457 × 191 × 67 UB. The inscribed circle at the junction has a diameter

$$D = \frac{(t_f + r)^2 + t_w(r + t_w/4)}{2r + t_f} = \frac{22.9^2 + 8.5 \times 12.3}{33.1} = 19.0\ \mathrm{mm}$$

and the fitting coefficient $\alpha = -0.042 + 0.2204\,t_w/t_f + 0.1355\,r/t_f - 0.0865\,r t_w/t_f^2 - 0.0725\,t_w^2/t_f^2 = 0.135$. The four terms are then $\tfrac{2}{3} \times 189.9 \times 12.7^3 = 25.9$ cm⁴ for the flanges, $\tfrac{1}{3} \times 428 \times 8.5^3 = 8.8$ for the web, $2 \times 0.135 \times 19.0^4 = 3.5$ for the two junctions and $-0.42 \times 12.7^4 = -1.1$ for the four tips: 37.1 cm⁴. The junctions add 3.5 and the tips take 1.1 back; with sharp corners the junctions would add almost nothing and the tips would still take their 1.1, which is why the sharp outline is below the plain sum.

The fillet stress is quicker still: $D/t_f = 19.0/12.7 = 1.50$, and the peak is that times the flange formula.

What the film assumes #

The torsion is uniform. Saint-Venant’s torsion constant is the stiffness of a member twisting at a constant rate, with every cross-section free to warp. An I-section held against warping at a support carries much of its torque by bending its flanges in opposite directions, which the section that cannot stay flat set out; the fillet’s contribution is to the Saint-Venant part only, and in a short member held at its ends that part may be the smaller one.

The outline is nominal. Rolling tolerances on flange thickness are a few per cent, and $J$ goes as its cube, so a flange 3 per cent thin loses about 9 per cent of the flange’s contribution — the same size as the fillets’ whole gain. The published constant, the film here, and the fitted formula are all for the nominal section.

The material is elastic. At the fillet’s peak the steel yields first; a section twisted towards its plastic torque redistributes stress away from the fillet onto the flanges, and the film becomes a sand heap whose volume the fillets add to only in proportion to their area — one per cent, not nine.

What the pictures cannot show #

That the fillet is where a rolled section is least perfect. It cools last and slowest in the mill, it carries the largest residual stresses of the section, and it is where laminations and inclusions from the web’s rolling tend to lie. A torsional stress half as high again as the flange’s, on the one part of the section with the least-understood material, is a reason to treat the fillet as the critical point in any fatigue or brittle-fracture check of a member in torsion, and the design formula’s $T t_f/J$ will not flag it.

Nor can they show a crane girder’s rail, a purlin’s cleat or a spandrel’s bracket — the details that actually put torsion into I-sections, usually through an eccentricity nobody could avoid — or whether the torque arrives near a stiffener that is welded across the very fillet whose stress is being discussed.

Still open: the channel and the angle #

An I-section’s junction is a T, with the web meeting the flange in the middle and two fillets side by side. A channel’s is an L at each flange, and an angle is a single L with one fillet and a toe radius at each tip. The inscribed-circle rule says an L junction should be stressed like a strip as thick as its own circle, which is smaller than a T’s for the same plates, and the film would say whether the fillet’s share of the constant is correspondingly smaller. Whether the rule holds for an L — where the film’s slope on a single fillet is not shared with a neighbour across the web — and whether the textbook sum’s error for angles and channels, which are used in torsion far more carelessly than I-sections, is on the same side, is the question that follows this one into the lightest sections in the tables.