The plate that closes the circuit

The point that is not in the section found why a channel twists under a load through its web: the shear flows in its two flanges run in opposite directions and make a couple, and the section is at rest only if the load passes through a point outside it, beside the web, where the couple is balanced. The eccentricity a purlin cannot avoid measured that point on an ordinary purlin and found the torque it causes was nobody’s mistake. Both ended with a remedy in the list of remedies: close the section. A shear flow with a circuit to run round, the expectation goes, has no couple to leave over, and the shear centre comes home.

It does come home, and this essay measures when. The answer is not “when the section is closed”. It is when the closing plate is thick enough to be a real part of the circuit — about a quarter of the channel’s own thickness — and until then almost nothing happens to the shear centre, while something large happens to everything else. A thin closing plate turns a channel into a section that hardly twists, around a shear centre that has hardly moved, and pays for it in the plate.

A channel with a lid #

The section throughout is a channel 300 mm deep with 100 mm flanges, every wall 10 mm thick, closed across the tips of its flanges by a plate whose thickness runs from a thousandth of the channel’s to all of it. The walls are thin enough to treat as lines. A vertical shear passes through the section, and the question is what shear flow carries it, where its resultant acts, and how stiff the section is in torsion.

The open channel’s flow is the familiar one. Down the web it is largest at mid-depth; along each flange it grows linearly from nothing at the toe to its value at the web; and the two flange flows, pointing in opposite directions a depth apart, make the couple that puts the shear centre 33.3 mm outside the web — $3b^2/(6b + h)$, for walls of one thickness.

A closed section cannot be analysed that way alone, because there is no free edge to start the flow from. The standard move is to cut the circuit somewhere — here, at the middle of the closing plate — compute the open section’s flow from the cut, and then add a constant flow circulating round the whole cell, of whatever size makes the cut close up again: the two faces of the cut must not slide past each other, which is a compatibility condition. That constant is

$$q_c = -\frac{\oint q_0\,ds/t}{\oint ds/t},$$

the open flow averaged round the circuit with each piece of wall weighted by its flexibility, $1/t$. The closing plate enters both integrals. And a thin plate is, in the denominator, a very large term.

The flow round a thin lid and a thick one #

With a closing plate a hundredth of the channel’s thickness the flow round the section is, to the eye, the open channel’s. The circulating flow is 1.2 per cent of the largest open flow; the flanges still carry their opposed flows, the web still carries the shear, the closing plate carries almost nothing. The shear centre has moved from 33.3 mm outside the web to 31.2 mm — three per cent of the way to where it will end up.

With a plate as thick as the rest the section is a symmetric box and the flow is symmetric with it. The circulating flow has grown until the closing plate carries as much shear as the web; each flange’s flow now reverses at its middle instead of running the whole way in one direction, the couple has gone, and the shear centre is at the centre of the box.

The shear centre waits #

Between the two extremes the shear centre’s track is the first figure of the argument. On a logarithmic axis of plate thickness it sits almost still for two decades — 0.3 per cent of the way at a thousandth of the channel’s thickness, 3 per cent at a hundredth — and then moves, 23 per cent at a tenth, half way at 0.28, through the web at 0.22 and on to the middle.

The reason is the weighting in the compatibility integral. The circulating flow is the open flow’s average round the circuit, and each wall’s say in that average is its length over its thickness. The channel’s own walls contribute $(2b + h)/t = 50$; a closing plate of thickness $t_c$ contributes $h/t_c$ — 300 for a plate as thick as the others, 3,000 for one a hundredth as thick. A thin plate is most of the circuit’s flexibility, and a circuit that is almost all flexible plate carries almost no circulating flow. The fraction of the way the shear centre moves is, very nearly, the channel’s share of the circuit’s flexibility relative to the share it has in the full box: $50/3{,}050$ against $50/80$, which is 2.6 per cent, at a hundredth of the thickness.

That is the sense in which the shear centre is a property of how the shear divides, not of whether the section is closed. It moves when the closing plate starts to take shear, and a plate takes shear in proportion to its thickness.

The torsion constant does not wait #

The torsion constant has the same denominator and a very different numerator. A closed cell resists a torque by a shear flow circulating round it, and its constant is Bredt’s, $4A^2/\oint ds/t$, where $A$ is the area the circuit encloses. The open channel resists by shear stresses reversing across each wall’s thickness, with a constant of $\sum b\,t^3/3$ — the cube of a thin thickness. Those are the two numbers being compared, and the second is tiny: 167,000 mm⁴ for this channel. The first, even with a closing plate a hundredth as thick, is $4 \times 30{,}000^2/3{,}050 = 1.18 \times 10^6$ mm⁴.

The circuit wins as soon as it exists. A plate a thousandth of the channel’s thickness — a hundredth of a millimetre — makes the section 1.7 times as stiff in torsion. A hundredth makes it 8.1 times, a tenth 63 times, and the full box 272. On a scale where the shear centre has barely begun to move, the torsion constant has already gone up by an order of magnitude, because it is measured against a quantity that goes as a thickness cubed and the shear centre is measured against one that goes as a thickness.

That contrast is the finding. It also separates two things the torque that has nowhere to go set side by side as the closed and the slit box — a factor of 1,449 in stiffness between a closed section and the same section cut — into what a partial closure does to each: the stiffness behaves like the closed section almost at once, the shear centre like the open one for a long time.

The twist falls long before the eccentricity does #

What a designer cares about under an eccentric load is the twist, which is the torque about the shear centre divided by the torsional stiffness. The torque’s lever depends on the shear centre; the stiffness on the torsion constant. They have just been found to move on different schedules.

For a load through the web, the lever is 33.3 mm with no plate and still 31.1 mm with a plate a hundredth as thick — 93 per cent of it — but the twist has fallen to 0.12 of the open channel’s, because the stiffness has gone up eightfold. With a plate a tenth as thick the lever is still 14.3 mm, almost half of the original, and the twist is 0.0068 of the open channel’s. At 0.22 of the thickness the shear centre reaches the web and a load there does not twist the section at all; beyond it the lever changes sign and grows again, slowly, against an enormous stiffness.

So the remedy on the list works, but not for the reason the list gives. A boxed channel does not stop twisting because its shear centre has come to the load; it stops twisting because it has become stiff enough that the torque it still carries hardly turns it. The torque is still there — 93 per cent of it for the hundredth-thickness plate — and it has to be carried somewhere.

The plate pays for the circuit #

It is carried round the circuit, as a circulating shear flow of $T/2A$, and a circulating flow passes through every wall of the circuit, including the thinnest.

The stress in each wall is the flow over its thickness. In the channel’s own 10 mm walls, the flow from bending and the circulating torque flow together make a peak of about 15 N/mm² under a 50 kN load through the web, and that hardly changes with the plate. In the plate itself the circulating flow is almost all there is, and its stress is the torque over $2A t_c$: 32 N/mm² at a tenth of the thickness, 282 at a hundredth — past the shear yield of structural steel, in a plate that has moved the shear centre three per cent.

That is the price of the order-of-magnitude stiffness, and it is exactly proportional to how cheaply the stiffness was bought. The thinner the closing plate, the larger the share of the torque it carries per unit of its own thickness, so the plate that makes a boxed section stiff in torsion is also the plate most likely to yield or buckle in shear, and its connection to the channel carries the same flow along its whole length. A thin cover plate on a channel, fixed by screws or intermittent welds at a spacing $s$, has to transfer $q\,s$ at every fixing; a connection that slips lets the plate carry less, closes the circuit less effectively, and gives up the stiffness in proportion. The equivalent thickness of a real closing element — a plate that buckles, a sheet on fasteners, a lattice of battens — is its shear stiffness, not its steel.

The bridge girder that is closed by a lattice #

The largest structures that depend on this arithmetic are steel tub girders: an open U of two inclined webs and a bottom flange, whose top is closed eventually by a concrete deck. Until the deck is cast, the U is an open section with a shear centre well below its bottom flange and almost no torsional stiffness, and on a curved alignment — where bending arrives as twist — it would rotate under its own weight and the wet concrete. So the top is closed temporarily with a lattice of lateral bracing between the two top flanges.

A lattice is a closing plate of very small equivalent thickness. Its diagonals resist the shear across the open top by their axial stiffness, and converted to a plate of the same shear stiffness they are typically a millimetre or two against webs and flanges of tens. On the figures above that is a closing plate of a few hundredths to a tenth of the section’s thickness: enough to multiply the torsional stiffness by tens and turn a section that would twist visibly into one that hardly twists, while leaving its shear centre most of the way to where the open U had it.

And the diagonals then carry the circulating flow. The torque that the eccentric load still applies goes round the quasi-closed cell, and through the lattice it is a set of axial forces in the diagonals, largest where the torque is largest — at the supports of a curved span, and during the stage that is never complete, when the deck is only partly cast. The bracing members are small because the equivalent plate is thin, and they are loaded in exactly the proportion the thinness implies. When the deck hardens it becomes the real closing plate, the section changes while it is loaded, and the lattice’s job passes to a slab which, counted as steel of the same shear stiffness, is tens of times thicker than the lattice it replaces.

Channels of other proportions #

The proportions change the schedule only a little. With flanges from 60 to 300 mm on a 300 mm web, the shear centre is half way home at between a fifth and a third of the channel’s thickness. A wider channel’s plate does its work slightly sooner, because the channel’s own walls are a longer and so more flexible part of the circuit, which leaves the plate a smaller share to dominate. But in every case the plate has to be a substantial fraction of the channel’s thickness before the shear centre moves much, and in every case the torsion constant has multiplied long before.

The hundredth-thickness plate, by hand #

For the channel closed by a plate 0.1 mm thick: the channel’s walls give $\oint ds/t = (100 + 300 + 100)/10 = 50$ and the plate $300/0.1 = 3{,}000$, so the circuit’s total is 3,050. The enclosed area is $100 \times 300 = 30{,}000$ mm², so Bredt’s constant is $4 \times 30{,}000^2/3{,}050 = 1.18 \times 10^6$ mm⁴. The open channel’s constant is $(100 + 300 + 100) \times 10^3/3 = 1.67 \times 10^5$ mm⁴, and the closed section’s is the sum, $1.35 \times 10^6$: 8.1 times the open.

The shear centre’s progress is roughly the channel’s share of the circuit’s flexibility, $50/3{,}050 = 1.6$ per cent, against its share in the full box, $50/80 = 62.5$ per cent: 2.6 per cent of the way, against 3 per cent from the full calculation, which also counts the plate’s small addition to the second moment. So the lever of a load through the web falls from 33.3 mm to about 31 mm.

A 50 kN load through the web then applies a torque of $50 \times 31.1 = 1{,}555$ kN·mm about the shear centre. The circulating flow is $T/2A = 1{,}555{,}000/60{,}000 = 25.9$ N/mm, and in a plate 0.1 mm thick that is 259 N/mm² — with the plate’s small share of the bending flow on top, the 282 of the figure.

Thin walls, St Venant torsion and a rigid cross-section #

The walls are thin and the section keeps its shape. Shear flows are uniform through each wall’s thickness and the cross-section does not distort; a box whose closing plate is very thin and not stiffened can distort instead of twisting, a mode that a box with too few diaphragms has in abundance.

The torsion is St Venant’s. The open channel also resists twist by warping — its flanges bending in their own planes — and over short lengths or near restrained ends that resistance is much larger than its St Venant constant suggests, which the section that cannot stay flat set out. The comparison of twist here is between two St Venant stiffnesses, which is the right comparison far from any restraint and flatters the closing plate near one; a closed section’s warping is small, so the closed side of the comparison is sound either way.

The closing plate is continuous and fully connected. A plate on fasteners, a perforated plate or a set of battens closes the circuit with an effective shear thickness smaller than its steel, and its effect is read off these figures at that effective thickness.

What the pictures cannot show #

That a section is often closed by something that is not a plate at all. Cladding, a floor deck, a precast slab bearing on both flanges, even a ceiling fixed to the toes can close a channel’s circuit partially, with an effective thickness that nobody computed. The figures say such a closure will make the member stiffer in torsion than its section tables say — possibly by an order of magnitude — while leaving the load’s eccentricity nearly where the tables put it, and loading the closing element in shear. Whether that element is strong enough to be relied on, or merely stiff enough to attract the torque until it fails, is a question about something the section’s designer did not draw.

Still open: the cell with two weak walls #

Every figure here closes one side of a channel. A multi-cell section — two cells sharing a web — has several walls whose thicknesses enter several compatibility conditions, and a thin wall in one cell changes the circulating flows in both. Whether a thin interior web behaves like a thin closing plate — irrelevant to the shear centre and decisive for the torsion — or whether its position between two cells lets it matter to both, is the question the same weighting asks of a box girder with a light internal diaphragm or a slender middle web.