The restraint that works where the twist is

An open-section beam loaded off its shear centre twists, and nothing in the design rules limits how much. A deck is a spring, not a wall found the restraint most beams actually have: a floor or roof deck fastened to the top flange resists its rotation everywhere along the span, as a distributed rotational spring. An 8 m section on forks, carrying 12 kN/m at 75 mm from its shear centre, twists 6.25° with nothing fastened to it; a deck restraining it at 5 kN·m per metre per radian leaves 4.20°. That essay spread the deck’s restraint evenly along the span, and ended on the cases where it is not even: a precast plank bearing on one side, a deck spanning onto the beam from one direction, a secondary beam framing in at mid-span. Each restrains the twist over a short length, or at a point.

The question it asked was whether a restraint concentrated at a point is worth its total stiffness, or less — whether, as with a stiffener whose worth depends on where it is, the deck’s spread matters as much as its amount.

The same total, spread and gathered #

The beam’s twist obeys one equation along its length: its warping stiffness and its Saint-Venant stiffness resist the twisting, the applied torque drives it, and any restraint fastened to it pushes back in proportion to how much it has twisted. A deck enters that equation as a restraint per metre, the same at every station. A connection at one station enters it as a concentrated spring: a rotational stiffness $K$ at a point, which pushes back with a moment $K\theta$ there and nowhere else.

Hold the total fixed. The deck’s 5 kN·m per metre per radian over 8 m is 40 kN·m per radian in all. Put that whole 40 into one spring at mid-span instead.

The beam then twists 3.13° at mid-span, against the deck’s 4.20. The gathered restraint does better than the spread one, by a quarter of the remaining twist. And it does so for the plainest of reasons: a rotational spring resists in proportion to the twist where it is, the twist is largest at mid-span, so a spring at mid-span works against the largest twist there is. The deck’s restraint near the forked ends acts on twists that are small anyway and buys almost nothing.

A point is worth twice its total #

Turn the comparison round and ask how stiff a single mid-span spring has to be to match the deck. The answer is 19.5 kN·m per radian — 49 per cent of the deck’s total. A restraint concentrated at mid-span is worth almost exactly twice the same total spread uniformly along the span.

The factor of two has a tidy reason. The twist of a beam on forks under uniform torque is close to a half-sine along the span, so a uniform restraint does work against an average twist of about $2/\pi$ of the peak, and its effect on the peak is weighted by that shape again: the uniform deck’s effectiveness goes as the mean of $\sin^2$, a half, where the mid-span spring’s goes as $\sin^2$ at its peak, one. Twice as effective per unit of stiffness, in the first mode — which is what the beam mostly is.

So the deck’s question has an answer opposite to the stiffener’s. A stiffener’s worth can collapse when it is in the wrong place; a twisting restraint’s worth is greatest when it is concentrated in the right one. For twist at mid-span, gathering beats spreading.

Spreading over more points #

Between one point and a continuous deck lie the real arrangements: secondary beams at third-points, joists every metre, planks fastened at their bearings. Split the same 40 kN·m per radian among two, three, seven or fifteen evenly spaced points and the largest twist climbs steadily from the single point’s 3.13° toward the deck’s 4.20: 3.64 with two, 3.78 with three, 4.01 with seven, 4.11 with fifteen. Every split does better than the deck, because every split puts some of its restraint nearer mid-span than the deck’s average; none does as well as one point, because each moves some away.

The curve that matters to a designer runs the other way. With one point, that connection carries a moment of 2.18 kN·m; with two, the more heavily loaded carries 1.10; with seven, 0.40; with fifteen, 0.19. The restraint that is most effective per unit of stiffness is the one that asks most of a single connection.

What the point carries #

A rotational restraint holds a beam by taking part of its torque. The uniform deck took 26 per cent of the 7.20 kN·m the eccentric load applies, spread over 8 m as a moment of about 0.23 kN·m per metre on its fasteners. The mid-span spring of the same total takes 30 per cent, 2.18 kN·m, through one connection, and a rigid restraint takes 4.38 kN·m — 61 per cent of the whole torque, which is the most one point can take; the rest reaches the supports along the two halves of the beam regardless.

That moment has to go somewhere. A secondary beam framing into the beam’s web with a fin-plate connection takes it as a bending moment in the fin plate and as twist in the secondary beam; a deck plank bearing at one point takes it as a bearing couple. Connections designed for shear alone carry a twisting moment of a few kilonewton-metres only by accident. The deck spread that demand over every fastener on the span, and the internal force with no diagram only becomes visible once it is gathered at a point.

A rigid point moves the twist outward #

The rigid mid-span restraint in the first figure removes the twist at mid-span completely — and leaves the beam twisting 0.40° at 1.7 m from each end. Holding mid-span rigidly divides the beam into two beams of 4 m, each forked at one end and held at the other, and each twists under its own share of the torque with its peak nearer the fork. Since twist under uniform torque grows steeply with span, two 4 m beams twist far less than one 8 m beam, which is why 0.40° is so small; but the place to look for it has moved from where the eye goes to a point a fifth of the span in.

Other spans, and ends held against warping #

The factor of two is not a property of this beam. On forks it is the same at every span drawn: a mid-span point matching the deck needs 49 per cent of the deck’s total on a 4 m beam, 49 on an 8 m one, 49 on a 12 m one and 49 on a 16 m one, even though the deck’s own effect changes enormously over that range — removing 6 per cent of the free twist at 4 m and 73 per cent at 16. On forks the twisted shape stays close to a half-sine at every span, and the factor of two belongs to the half-sine.

Hold the ends against warping and the shape changes: the twist rises from zero with zero slope at each end, more like a raised cosine than a sine, and is more sharply concentrated toward mid-span. A mid-span point is then worth more still. It matches the deck with 39 to 41 per cent of the deck’s total, from 4 m to 16 m — about two and a half times its total spread out.

The warping-fixed beam twists far less to begin with — 2.55° free against 6.25 on forks — and the deck, at 2.14°, and the mid-span point of the same total, at 1.71°, keep their order. With the mid-span point rigid the twist peaks at 0.21° at 2.0 m from each end, further in than on forks, because a warping-fixed half-beam peaks nearer its middle.

A real point is nearly rigid #

The figures treat the point’s stiffness as a choice, and the comparisons need that. A real point restraint — a secondary beam framing in at mid-span — has a stiffness set by the secondary beam and its connection, and it is large. An IPE 300 spanning 6 m to a pinned far end resists the primary’s rotation by bending about its major axis, with a stiffness $3EI/L = 3 \times 17{,}600/6 = 8{,}800$ kN·m per radian. Its fin-plate connection is much softer; taking it as 1,500 kN·m per radian, an assumed figure of the right order, the two in series give about 1,300 kN·m per radian — 66 times the 19.5 that matches the whole deck.

At that stiffness the point is, for this beam, rigid. The twist at mid-span vanishes, the beam twists 0.51° at 1.9 m from each end instead, and the connection carries 4.2 kN·m — 59 per cent of the applied torque — through a fin plate designed to carry the secondary beam’s end shear. Even if the connection were ten times softer than assumed, it would still be several times stiffer than the deck it replaces, and would still collect most of that moment. So the practical question about a secondary beam framing in at mid-span is not whether it restrains the primary’s twist; it does, almost completely. It is whether its connection was designed for the moment that restraining takes.

Planks on half the span #

The arrangement that is neither point nor deck is restraint over part of the span — planks bearing on one half, a deck that changes direction half-way. With the deck’s whole total gathered onto the half nearest one end, at twice the density, the peak twist is 4.28°, a shade more than the uniform deck’s 4.20°, and moved a little toward the unrestrained half. With only the deck’s own density on that half — half the total — the peak is 5.05°.

So gathering restraint toward an end, rather than toward mid-span, costs a little rather than gaining; the half-span restraint works on the half of the sine nearer the zero. The rule that comes out of all of it is one rule: a twisting restraint is worth what it is worth at the station it acts on, weighted by how much the beam wants to twist there. Mid-span is the best station; the ends are the worst.

What the deck still does better #

None of this makes a uniform deck the worse restraint in practice; it makes it a different one. The deck asks almost nothing of any single fastener — 0.23 kN·m per metre on average here — so it needs no connection designed for twist, and it keeps working when a fastener or two is missing. It restrains the beam everywhere, so the twist between restraints never has room to grow, and a beam carrying a load that moves along it is held wherever the load happens to be. And it restrains the beam during construction, plank by plank, before any secondary beam has been connected.

A point restraint inverts each of those. It is the most effective use of stiffness and the least tolerant of its own connection: it attracts the torque the way a stiff path attracts load, and if its connection is soft or slips, the beam is back to twisting as if it were free between its ends. The choice between them is the familiar one between a few strong restraints and many weak ones, and the figures make its terms exact: a point is worth twice its stiffness, and charges for it in one connection.

The restraint’s place in the beam’s own torsion is worth one more sentence. On forks the beam resists twist mostly by Saint-Venant shear at long spans and mostly by warping of its flanges at short ones, and a mid-span restraint changes which: holding the middle turns one long beam into two short ones, in which warping does more of the work. That is why the rigid point’s residual twist is so small — 0.40° against 6.25 — out of all proportion to halving the span.

The factor of two, by hand #

The equivalence can be estimated in two lines. Take the beam’s twist as a half-sine of peak $\theta_0$, and its own torsional stiffness in that shape as $k_1 = GJ(\pi/L)^2 + EI_w(\pi/L)^4$ per unit length. A uniform restraint $k$ adds to it directly: the twist becomes $\theta_{\text{free}} \cdot k_1/(k_1 + k)$. A point spring $K$ at mid-span adds, by the same energy argument, $K\sin^2(\pi/2)$ against $k_1 L/2$ — the half from $\int_0^L \sin^2 = L/2$ — so it is equivalent to a uniform $k = 2K/L$. A point of total $K = kL/2$ therefore matches a deck of $k$ per metre: half the deck’s total, which is the 49 per cent found.

For this beam $k_1$ is about 10.2 kN·m per metre per radian on forks, which is the stiffness of the deck that halves the twist; the deck of 5 is half of that, and a single point of 20 kN·m per radian — the same as a uniform 5 — does the same work.

Elastic, one torque, and connections that are springs #

Everything is elastic. The beam’s torsion, the deck’s restraint and the point’s spring are all linear; a connection that slips or yields under 2 kN·m of twisting moment restrains less than its elastic stiffness says, and less the more it is asked.

The torque is uniform. A load concentrated near mid-span puts more of its torque where a mid-span restraint is, and the point’s advantage grows; a load near the ends, less.

And a connection is a spring of known stiffness. A secondary beam’s resistance to the primary beam’s rotation depends on its own torsion and bending, its far-end support and its connection’s flexibility, and is rarely calculated. The figures say what it would be worth if it were.

Bracing, buckling and the secondary beam’s own twist #

They cannot show lateral–torsional buckling. The same restraints that limit twist under load also brace a beam against buckling, and there a point restraint at mid-span changes the buckled shape rather than reducing an existing twist — a brace need not be strong to do it, but it must be on the right flange.

They cannot show the secondary beam’s response. A secondary beam that restrains the primary’s twist is itself twisted, or bent about its weak axis, by the moment it takes; its own serviceability is the cost of the primary’s.

And they cannot show a purlin’s case, where the restraint and the load come through the same sheeting and the eccentricity a purlin cannot avoid is resisted by the same fasteners that apply it.

Gather it where the twist is #

With the same total, a mid-span restraint beats a uniform deck. On an 8 m beam twisting 6.25°, the deck’s 40 kN·m per radian leaves 4.20° spread and 3.13° gathered.

A point is worth twice its total spread out. 19.5 kN·m per radian at mid-span matches the whole deck, because a spring works against the twist where it is and mid-span is where the twist is.

The point pays in one connection. It carries 2.18 kN·m, 30 per cent of the torque; a rigid one 61 per cent, and moves the worst twist out to 1.7 m from each end.

Splitting the total among more points approaches the deck from below, and gathering it toward one end does slightly worse than spreading it — a twisting restraint is worth the twist at its own station.

Still open: the secondary beams that twist the primary #

Every restraint here resists the twist. A secondary beam framing in at mid-span also loads the primary: its end reaction arrives at the primary’s web, off the shear centre, and adds a concentrated torque at the very point where it also restrains the twist. Whether a secondary beam, loaded and framing in on one side only, twists the primary more than it holds it — and whether a pair of secondary beams framing in from both sides, whose torques cancel and whose restraints add, is the arrangement that makes the mid-span point as good as these figures say — is the question a floor’s framing plan puts to the restraint drawn here as a spring.