The secondary beam that twists what it holds

A beam loaded off its shear centre twists, and an open section twists a long way: the 8 m primary beam of these essays, carrying 12 kN per metre at 75 mm from its shear centre on forked ends, turns through 6.25° at mid-span. A deck fastened to its top flange restrains the twist like a distributed spring, and the same restraint gathered at mid-span is worth about twice its total spread along the span, because mid-span is where the twist is. A rotational spring of 19.5 kN·m per radian at mid-span does as much as a whole deck of 40.

The commonest thing found at the middle of a primary beam is a secondary beam framing into it. It is a mid-span rotational restraint of exactly the kind the last essay recommended: when the primary twists, the end of the secondary has to turn with it, and the secondary resists by bending. It is also something else. The secondary carries its own floor and delivers its end reaction to the primary — through a fin plate welded to the primary’s web, or cleats bolted to it — and that reaction does not act through the primary’s shear centre. It acts at the bolt line, several centimetres to one side. A secondary framing in from one side is a concentrated torque at exactly the point where it is a restraint.

A restraint and a torque at one point #

The secondary here is a 356 × 171 × 51 UB spanning 6 m, pinned at its far end, carrying a reaction of 60 kN at the primary. Its bolt line is 60 mm from the primary’s shear centre, so it applies a torque of 3.6 kN·m. On its own, that torque would twist the primary 5.0° at mid-span — almost as much as the whole deck load does.

Against it, the secondary’s restraint. When the primary twists by θ, the secondary’s end rotates by θ about the secondary’s own bending axis, and the secondary resists with a moment equal to its rotational stiffness times θ. Two stiffnesses are in series: the secondary’s bending, 3EI/L for a beam pinned at its far end — 14,453 kN·m per radian, enormous — and its connection to the primary, which is designed to be a pin.

The same secondary does opposite things depending on its connection. Through a connection of 10 kN·m per radian it is a torque with almost no restraint behind it, and the primary twists half as much again as it did alone. Through 1,000 kN·m per radian it is a near-rigid point at mid-span with a torque on it, and the primary barely twists — 0.70° at its worst, which is no longer at mid-span but either side of it, where a point restraint pushes the twist outward.

Where the line falls #

The question the figure raises is where between those two the secondary stops hurting.

The line falls at 33 kN·m per radian, and the number is worth setting against the classification it lives inside. EN 1993-1-8 calls a beam-to-column or beam-to-beam joint nominally pinned when its rotational stiffness is below half the connected beam’s EI/L — 2,409 kN·m per radian for this secondary. The secondary twists the primary only through a connection seventy times softer than the softest joint the classification calls a pin’s upper limit. A connection designed as a pin is still, for the primary’s twist, nearly a fixed point, because the twist asks for a rotation of a degree or two and a fin plate resists that with real moment long before it is anything like the rigid joint it was not designed to be.

The other two curves complete the picture. A secondary on the side away from the deck applies its torque against the deck’s and helps at every stiffness, even zero. A pair framing in at the same station from both sides — the ordinary arrangement under an internal primary — cancels its torques and adds its restraints, and is the best of all at every stiffness: twice the spring and no load.

Why the line is so low #

The threshold is small because the thing it is compared with is small. An open section of this size is very soft in torsion: a torque of 3.6 kN·m at mid-span turns it through 5°, which makes its own torsional stiffness at mid-span about 41 kN·m per radian. That is the whole of what the primary contributes. Anything at mid-span that resists rotation with a few tens of kN·m per radian is already a partner of the same size, and anything with a few hundred dominates the joint completely.

Seen that way, the secondary’s torque does not so much twist the primary as load the secondary. When the connection is stiff, the 3.6 kN·m that arrives at the bolt line is resisted almost entirely by the secondary bending against its own pinned far end, and only the small remainder goes into the primary’s twisting; the primary’s mid-span barely rotates because it is hardly being asked to carry its own torque at all. When the connection is soft, the torque has nowhere to go but the primary. The eccentricity decides how much torque there is; the connection decides who carries it.

A heavier secondary moves the line #

The torque grows with the secondary’s reaction; the restraint does not.

The twist rises linearly with the reaction for every connection, from a starting point set by the restraint alone. So the 33 kN·m per radian of the previous figure is not a property of the connection: it belongs to a 60 kN reaction. Double the secondary’s load and its torque doubles while its restraint stays the same, and the line moves to roughly twice the stiffness. A connection that makes a light secondary helpful can make a heavy one a net twister — the curve for 30 kN·m per radian crosses the no-secondary line at 56 kN. For every connection there is a secondary heavy enough to make it twist what it holds, which is the reason the comparison belongs in the calculation rather than in a rule of thumb.

At the stiffnesses a real bolted fin plate has, of the order of hundreds to a couple of thousand kN·m per radian, the line is a long way off: at 1,000 kN·m per radian the 150 kN secondary still leaves the primary at 0.79°.

The free play in the bolts #

That conclusion assumed the connection’s bolts bear from the start, and in a bolted fin plate they do not.

Bolts sit in holes 2 mm larger than themselves, and a connection made with clearance holes can rotate through a small angle before any bolt touches the side of its hole. For a bolt group whose outer bolts are 100 mm from its centre, 2 mm of clearance is 0.02 radians, 1.15°, of rotation for nothing.

The torque does not wait for the play. The reaction is there from the moment the secondary is loaded, and it twists the primary until the play is taken up; only then does the restraint begin. So the play arrives at mid-span nearly whole: 0.02 radians of it takes the twist from 0.47° to 1.57°, which is 0.47° plus almost all of the 1.15°. The connection’s stiffness, which decided everything in the earlier figures, is barely relevant here — it acts on whatever twist is left after the play.

This is the same arithmetic as a slip that has to be taken up before a joint carries anything, turned to rotation, and the same remedies apply: fitted bolts, or preloaded bolts whose friction takes the moment before the clearance can be used. A twist limit checked with the secondary as a rigid point and its bolts in clearance holes is checked on a structure that does not exist.

The moment a pin is asked to carry #

A connection that restrains twist carries moment, and this one was designed not to.

The stiffer the connection, the more of the primary’s twist it holds, and the more moment it carries: 7.6 kN·m at 1,000 kN·m per radian on the deck’s side, rising towards 7.9 as the connection approaches rigid. That is twice the secondary’s own torque, because the connection is holding the deck’s twist too — the secondary has become the primary’s torsional support. A fin plate designed for 60 kN of shear on a 60 mm lever already carries 3.6 kN·m of its own eccentricity moment, which the shear design allows for; the extra 4 kN·m is a moment nothing in its design asked about.

In service that is small beside the plate’s capacity. It is not small beside the bolts’ slip resistance, if the joint was meant to be slip-resistant, and it reverses whenever the deck load on the primary is removed while the secondary’s floor is still loaded. The pair is again the better arrangement: each connection of a pair carries 2.1 kN·m, a quarter of the single connection’s, because the torques have cancelled and each is holding only its half of the deck.

Where to frame in #

The secondary’s position along the primary is usually set by the floor plan, but it changes the answer as much as the connection does.

A stiff secondary is most useful at mid-span, where the twist it is holding is largest, and its value falls off steeply towards the supports: at a quarter-span it leaves 2.35°, three times the mid-span figure, because the twist simply moves to the half of the span it cannot reach. A flexible one is most harmful at mid-span, for the same reason the other way round — its torque is applied where the beam is least able to resist it. Two secondaries at the third points, which is a common grid, do better than one at mid-span with the same connections: each a third of the way along, each with its own 3.6 kN·m, and together they leave 0.44° against the single secondary’s 0.70°, because they hold the two places to which a single mid-span restraint pushes the twist. The comparison reverses with flexible connections. Two secondaries bring two torques, 7.2 kN·m between them, and at 10 kN·m per radian they leave the primary at 10.85°, worse than one secondary’s 9.12°; at 30 they are still at 7.10°, above the beam with none. A grid of secondaries on soft connections multiplies the harm, and the line between helping and hurting moves with the count of secondaries as well as with their load.

The edge beam, where both torques point the same way #

The arrangement in which the secondary is most likely to hurt is also the commonest place for a primary to twist. An edge beam carries the slab on one side only, so the slab’s load is off its shear centre towards the inside of the building; and every secondary framing into it comes from the inside too, delivering its reaction on the inside face of the web. The deck’s torque and the secondaries’ torques are in the same sense. That is the “deck’s side” curve in every figure above, and it is the curve with a threshold.

An internal primary is the opposite case. Its slab bears on both sides and its secondaries frame in from both, so the slab’s torque is small and the secondaries’ torques cancel in pairs; the only torque left is from unequal loading of the two sides, which is what a pattern load produces. Such a primary twists little, and its secondaries hold what twist there is with no load of their own to undo.

So the twist that matters is concentrated at the building’s perimeter, on beams that carry the façade as well, where a twist of a degree is a cladding bracket out of line by several millimetres. For those beams the comparison in the second figure is the design check, and the useful number from it is not the twist with the secondaries rigid but the stiffness — and the free play — at which they stop helping. Two things move that number against the designer: secondaries that carry more than the 60 kN assumed here, as edge secondaries often do when they also carry the façade’s own weight, and connections detailed with long slotted holes for erection tolerance — holes made bigger so the steel would fit — which turn 0.02 radians of free play into several times that.

The connection, by hand #

The threshold can be found without the beam solver, from a single number the earlier essay derived. A mid-span rotational spring of 19.5 kN·m per radian does as much for this primary as the whole deck, and a spring of $K$ at mid-span reduces the twist roughly in proportion to $1/(1 + K/K_b)$, where $K_b$ is the beam’s own torsional stiffness at mid-span. The torque $Re$ adds a twist of $Re/(K_b + K)$.

The beam’s own stiffness follows from the unrestrained case: 3.6 kN·m twists it 5.0°, 0.087 radians, so $K_b \approx 41$ kN·m/rad for a mid-span torque. The deck’s distributed torque gives 6.25°, 0.109 radians. With a spring $K$ and the torque together, the mid-span twist is about $(0.109 \times 41 + 3.6)/(41 + K)$ radians, and it equals 0.109 when $K = 3.6/0.109 = 33$ kN·m/rad. The secondary breaks even when its spring times the twist it is fighting equals its own torque — which is the whole of the argument in one line, and the beam solver’s 33 to the nearest unit.

A pin that is stiff, bolts that are tight and a primary that is open #

The calculation rests on four choices.

The connection’s stiffness is a number assumed, not measured. A fin plate’s rotational stiffness depends on the plate’s thickness and length, the bolts and their holes, and the web it is welded to; tests put it anywhere from tens to thousands of kN·m per radian. The figures sweep it rather than claim it, and the finding — that the line sits far below the pin classification — holds across that range for this secondary’s load.

The secondary is pinned at its far end. If its far end frames into another primary that twists too, the secondary’s restraint is shared between two twisting beams, and each sees less of it.

The primary’s section is open. A closed or concrete-encased primary is so stiff in torsion that neither the secondary’s torque nor its restraint matters; the argument is for the open sections whose twist is worth limiting.

The deck does nothing. Here the deck is the load and not a restraint, so that the secondary’s effect can be read alone. A deck fastened to the primary restrains it as well, and the secondary’s torque is then shared between two restraints, which makes a flexible secondary less harmful and a stiff one less necessary.

The secondary’s own twist, and the day it was bolted #

The figures cannot show the secondary’s own twist. A secondary loaded off its own shear centre — a floor bearing on one flange, an edge beam — twists as well, and at its end that twist rotates the connection about a different axis from the one the primary’s twist turns it about. Two twisting beams meeting at a connection that is a pin for one rotation and a spring for the other is a joint whose behaviour no single beam’s calculation describes.

They also cannot show construction. During erection the secondaries go in before the deck, often bolted finger-tight, with their full free play and their floor loads still to come; the primary’s twist at that stage has nothing to do with the finished floor’s. The essay on the restraint concentrated where the twist is found the construction stage to be where a twist calculation earns its keep, and the secondary’s free play is a large part of why.

Still open: the primary that is braced by its secondaries against buckling #

Everything here is serviceability: a twist of a degree or so, under service loads, against a limit nobody writes down. The same secondaries are what a designer counts on to brace the primary’s compression flange against lateral–torsional buckling, and a buckling brace has to be stiff and strong in exactly the rotation this essay has been measuring — with the difference that a buckling mode grows from any twist, including the play in the bolts. Whether a secondary on a fin plate with clearance holes is an effective torsional brace for a primary near its buckling load, or whether the free play that adds a degree to the service twist also removes the brace until the beam has already begun to go, is the question the service twist leaves for the ultimate one.