The corner with one fillet

The corner the rolling mill rounds took the torsion constant of a rolled I-section, taught as the sum of three thin plates, $\sum bt^3/3$, and solved Prandtl’s soap film over the section’s real outline instead. Where the web meets each flange the mill leaves a fillet, a curve of steel about a hundredth of the section’s area, and the film rises into it. Over seven standard sections the film reproduced the published constants to a fifth of a per cent and said the fillets were worth 9 to 13 per cent of them. It also put the section’s largest torsional shear stress on the fillet, at close to $T\cdot D/J$, with $D$ the diameter of the largest circle that fits in the junction. That is half as much again as the flange formula every design check uses.

An I-section’s junction is a T: the web meets the flange in its middle, with a fillet on each side. A channel’s is an L at each flange, and an angle is a single L with one fillet and two leg tips. Angles and channels are the sections most often loaded in torsion without anyone meaning them to be: an angle bracket hung off one leg, a channel purlin whose load passes outside its shear centre, a lintel whose brick sits on one flange. The question is whether what the film found for the T holds for the L.

The film over an angle’s corner #

The film’s volume is half the torsion constant and its steepest slope is the largest shear stress — two volumes, and both of them torques, in the language of the analogy. The angle is an equal angle 100 × 100 × 10 mm with a 12 mm root radius at its inside corner and a 6 mm toe radius on the inside corner of each leg’s tip, roughly the proportions of the European rolled range. The film is relaxed over its real outline exactly as it was over the I-section. The legs are cut a few thicknesses past the fillet with their tips kept, since far from a corner or a tip a thin strip’s film is the one-dimensional parabola and each extra length of leg adds exactly $t^3/3$ a unit length. The lengths cut off are then added back.

Along each leg the contours are the parallel lines of a thin strip, the film rising to a ridge along the leg’s middle. At the corner the two ridges meet and the film swells into the square of metal the two legs share, and further into the fillet. The swelling is on the outside of the L, away from the fillet, because that is where the thickest metal is. The largest circle that fits in the corner, 15.8 mm across against the legs’ 10, sits with its centre on the diagonal and touches both outside faces and the fillet.

The sharp angle’s film does the same with less metal to swell into, and its steepest slope sits in the re-entrant corner. There the true slope grows without limit as the grid is refined, the singularity a soap film meets at a corner it cannot finish.

Six counts of one constant #

An angle has no single $\sum bt^3/3$. A thin-plate sum adds the plates’ lengths, and an L made of two plates of thickness $t$ either counts the corner square twice, once in each leg, or counts each leg to the other’s centreline.

The centreline count, $(2b - t)\,t^3/3$, gives 6.33 cm⁴, and the outside count, $2b\,t^3/3$, gives 6.67. The film over the sharp outline gives 6.18, below both: the angle’s two free tips each lose more than the sharp corner gives back, as the I-section’s flange tips did. With the root fillet the film gives 6.80, above both. The fillet adds 0.66 cm⁴, 10% of the rolled constant, from 31 mm² of steel, 1.6% of the section’s area.

The toe radii work the other way. Each removes a sliver of steel from the inside corner of a leg’s tip, where the film is already falling towards the free end, and together they take 0.7% off. Rolling them in costs the angle a fifteenth of what the root fillet gives.

So for this angle the two sums bracket the sharp outline and the rolled outline sits above both: the centreline sum is 7% low, the outside sum 2% low. Which one a designer was taught decides whether the sum is a fair estimate or a poor one.

Stressed like a strip as thick as its corner #

The I-section’s fillet carried its peak stress at about the stress of a strip as thick as the circle inscribed in its junction. An L’s inscribed circle is smaller than a T’s for the same plates, so the L’s fillet should be stressed less.

The stress along the inside edge is the leg’s own, $T\cdot t/J$, until the edge turns into the fillet. Round the fillet it climbs to 1.57 times that, at the middle of the curve, and comes back down on the other leg. The inscribed circle predicts 1.58. The inscribed-circle rule holds for the L as it did for the T, to within a per cent here and a few per cent across the channels below. The rule is the useful one: a stress at the fillet of $T\cdot D/J$, with $D$ read off a drawing of the corner, needs no film at all.

The angle’s peak, 1.57 times its leg stress, sits inside the range the I-sections’ fillets reached, 1.45 to 1.65 times their flange stress. What it is not is the stress the leg formula gives, and an angle’s torsion check that uses $T\cdot t/J$ is 57% low at the one place a fatigue crack under repeated twisting would start.

The radius that is kindest to the corner #

The root radius is the mill’s choice and a designer’s given, but its effect is worth seeing across the range.

The torsion constant rises steadily with the radius, slowly at first and then quickly — 8% at a radius of one leg thickness, 30% at two — because a larger fillet fills the corner with metal at the place the film is highest. The peak stress does not move with it. With a small radius the stress concentrates at the tight curve, near the re-entrant corner’s singularity. With a large one the curve is gentle but the circle in the corner is large and the stress follows it. Between them, near a radius of 0.6 of the leg thickness, the peak is least, at 1.46 times the leg stress. The I-section’s peak was least at a root radius of half its flange thickness. Rolled angles have root radii of about 0.9 to 1.4 times their leg thickness, past that minimum, where a larger root buys stiffness and costs stress.

An L junction gives back less than a T #

The question was whether the fillet at an L is worth less than a fillet at a T, and a channel and an I-section with the same plates answer it directly.

Both sections have four free flange tips, and each tip costs the same, because a flange’s end does not know what it is attached to at its other end. They differ at their junctions. The channel’s two L junctions with one fillet each add 0.65 cm⁴ a fillet. The I-section’s two T junctions with two fillets each add 1.02 cm⁴ a fillet. The same steel in the same radius is worth half as much again at a T.

The reason is in how the film is held up. At a T the web’s ridge runs into the middle of the flange’s ridge, and the film there is supported by metal on three sides: the flange either way and the web below. At an L the web’s ridge turns the corner into the flange’s, and the film is supported on two. A fillet does not only add its own small volume of film. It moves the boundary back, and the film over the whole junction rises with it, and it rises further where the film is already held up on three sides than on two.

With both junctions counted, the channel’s rolled constant is 0.992 of its sum: the sum is high, by 0.8%, for a section it is usually supposed to understate. The I-section with the same plates is 8.3% above its sum. A designer who has learned from I-sections that $\sum bt^3/3$ is safely low has learned something true of T junctions only.

Which side of the sum #

The angle and the channel above are one size each. Across the rolled ranges the sum falls on different sides of the film for different sections.

For angles, the centreline sum is low for every size, by 11% for the smallest and 3% for the largest. The outside-length sum is low for small angles and high for the largest, crossing near a 150 × 15. Both trends come from the root radius growing more slowly than the leg thickness across the range: a 50 × 5 angle has a root radius of 1.4 thicknesses and a 200 × 20 of 0.9, so the small angle’s fillet adds 13% to its constant and the large angle’s 6%.

For channels the sum is low by 6% for the smallest and within a per cent either way for the rest. Their fillets add 7 to 13% and their flange tips take most of it back. So for an angle the honest statement is that the outside-length sum is within 5%, and for a channel that the usual sum is within 6%, either way. Neither is a safe underestimate across the range, and for a torsional stiffness used to keep a stress or a rotation within a limit, the side matters more than the size.

Where the angle’s constant is used #

An angle’s torsion constant is not only a stiffness for a twisted bracket. It is the number that decides how an angle strut buckles. A single angle in compression has its shear centre at the corner and its centroid off it, so it can fail by twisting about the corner as well as by bending, and for the equal angles in common use the torsional mode governs over a wide range of lengths. A column that twists instead of bending has a torsional buckling load in which $GJ$ appears directly, with nothing to dilute it at short lengths, since an angle has almost no warping stiffness to add. A torsion constant 7% low from the centreline sum is a torsional buckling load 7% low; one 2% high from the outside sum is a buckling load 2% high. For a member whose design is governed by that mode, which sum a program uses is the difference between a safe check and an unsafe one.

Angles are also the members most often loaded through one leg, which costs them area in tension as an angle that uses half of itself found, and costs them a twist in bending: the load passes through a bolt in one leg, and the shear centre is at the corner, so every load on a single angle brings a torque with it.

A channel loaded off its shear centre #

A channel’s shear centre lies outside its web, on the side away from its flanges, and its position is fixed by the web and flanges’ shear flows, not by its torsion constant. A load applied through the web, which is where purlin cleats and lintel bearings put it, is therefore always eccentric to the shear centre, and the channel twists. Where the twist is resisted by what the channel is connected to — a roof sheet, a floor, a wall — the torque is a compatibility torque, the kind that goes away if the structure lets it, and the channel’s own torsional stiffness decides how much of it the channel takes. Where nothing else resists, the torque is in equilibrium and the channel’s fillets carry its peak stress at $T\cdot D/J$, by the same inscribed-circle rule as the angle’s.

Neither case is unusual. A channel is twisted more often than it is not, and both the stiffness it offers and the stress it carries depend on the two junctions that the sum treats as two plates meeting at a line.

The angle, by hand #

The centreline sum for the 100 × 100 × 10 angle is $(2 \times 100 - 10) \times 10^3/3 = 63\,300$ mm⁴, and the outside sum $2 \times 100 \times 10^3/3 = 66\,700$. A rectangle’s tip loses about $0.105\,t^4$ to its free end, 1,050 mm⁴ for each 10 mm leg, so two tips cost 2,100 mm⁴ off the outside sum: 64,600, against the film’s sharp-outline value with its toe radii of 61,400 and without them of 61,800. The corner of a sharp L counted once in the outside sum is counted roughly right, which is why the sharp outline and the outside sum less its tips agree to within a few per cent.

The root fillet’s area is $(1 - \pi/4) \times 12^2 = 31$ mm². It adds 6,600 mm⁴. The same 31 mm² added to a leg as 3.1 mm of extra length would add $3.1 \times 10^3/3 = 1\,030$ mm⁴, so the fillet is worth six times its own area of plain leg. It is worth that because it does not only carry film of its own: it moves the corner’s boundary back and lets the film over the whole corner rise.

The inscribed circle has its centre on the corner’s diagonal, $c$ from each outside face, touching the fillet: $\sqrt 2\,(t + r - c) - r = c$, so $c = (\sqrt 2\,(t + r) - r)/(1 + \sqrt 2) = 7.92$ mm and $D = 15.8$ mm. The peak shear stress under a torque $T$ is about $T \times 15.8/68\,000$ — 1.58 times the leg’s $T \times 10/68\,000$.

Open sections, elastic and uniform #

Saint-Venant torsion only. The film is the uniform twisting of a long member whose sections are free to warp. An angle’s warping constant is nearly nothing and the film is the whole of its torsional stiffness; a channel’s is not, and a channel restrained at its ends resists twist by warping as well, which the slit that costs a factor of six hundred measured as the gap between open and closed sections.

Nominal dimensions. Rolled sections vary in thickness and radius within tolerances of several per cent, and $J$ goes as the cube of thickness. The comparisons here are between the film and the sum for the same nominal outline, and those comparisons are tighter than any table could be.

Channels with parallel flanges. Older channels and many angles abroad have tapered flanges, whose film is not a parabola of constant thickness along the flange and whose sum is ambiguous in a third way: which thickness to count.

The toe radii are the mill’s. Angles in some ranges have sharp toes; the toe radius takes less than a per cent away either way.

No stress concentration at the toe. The toe’s rounded corner is a convex corner and lowers the film’s slope there, unlike the root.

Two plates meeting, three plates meeting #

One rule explains every number here: metal added at a junction raises the film over the whole junction, and raises it most where the film is held up on the most sides. The most metal surrounds the junction where the most plates meet. A T junction has three plates around it and an L two; a cruciform, four. The torsion constant of a rolled section is therefore not the sum of its plates corrected by a fillet allowance. It is the sum of its plates corrected by an allowance for each kind of junction, and a section of L junctions and free tips — an angle, a channel, a zed — can sit on either side of its sum where an I-section always sits above.

Still open: the angle that is loaded through one leg #

Every number here is the twisting of the whole section about its own shear centre. An angle is usually twisted because it is loaded through one leg — a bracket bolted at one leg’s toe, a lintel carrying brick on its horizontal leg — so that the twist arrives at a point well away from the corner, through a leg that bends across its thickness as it carries the load to the corner. The leg then carries a plate’s bending stress at its root as well as the torsional stress the film gives, both largest at the fillet. Whether the two add at the root fillet — a bending peak on one face and a torsional peak on the fillet — or peak in different places, and whether the fillet’s radius should then be chosen for bending rather than for torsion, is the question the angle’s own use puts to its corner.