The movement with no limit against it

Open any code of practice at the serviceability chapter and there is a table of deflection limits: span over 250 for a beam under total load, span over 360 for the imposed part, ten millimetres for a mullion under wind, and a page of notes about what the limit is protecting.

Look for a twist limit and there is nothing. Not a small number, not a note, not a reference — the quantity does not appear. And a beam loaded anywhere other than through its shear centre twists.

The arithmetic, which is not difficult #

A distributed load $w$ at an eccentricity $e$ from the shear centre applies a distributed torque $m = we$. If the section resisted twist only by shear circulating round its thin walls — Saint-Venant torsion, with the flanges’ resistance to being bent apart left out — the twist at mid-span of a uniformly loaded member would be

$$\theta = \frac{m L^2}{8 G J}.$$

Every term in that is available in a section table. What decides the answer is $J$, and $J$ is the property that varies over three orders of magnitude between sections that look similar.

For the member drawn, $m = 900$ Nmm/mm, $L = 8{,}000$ mm, $G = 81{,}000$ N/mm², $J = 5.6\times10^5$ mm⁴:

$$\theta_{SV} = \frac{900 \times 64\times10^6}{8 \times 81{,}000 \times 5.6\times10^5} = 0.159\ \text{rad} = 9.09°.$$

Nine degrees. That is not a serviceability number; it is a member lying on its side. And it is wrong, by a factor of three and a half, because the flanges resist being bent apart and the ends are not free to warp.

Warping restraint, which is not a refinement #

An I-section resists torque two ways. It shears round its own thin walls — Saint-Venant torsion, the $GJ$ term — and its two flanges bend in opposite directions, which is warping torsion and depends on $EI_w$. Which of the two carries the torque is decided by one dimensionless group:

$$kL = L\sqrt{\frac{GJ}{EI_w}} = 4.50$$

for this member. Small $kL$ means warping does everything; large means Saint-Venant does. At 4.5 the two share, and the restrained answer is

$$\theta = 2.55°,$$

a factor of 3.57 below the hand calculation. That is not a correction. Ignoring warping restraint here makes a poor member look unusable, which is a different kind of wrong from the usual conservative approximation and leads to a different bad decision — specifying a hollow section that was not needed.

That figure is one end restrained and a torque applied at the other, which is the cleanest way to see the split; it stiffens the member by 1.290. The 3.57 above is the same effect on the member this essay is about — both ends held, the torque spread along the span — and the two numbers differ because the restraint is at both ends and its benefit accumulates over the whole length rather than decaying out of one.

The number that matters is not the angle #

Two and a half degrees is hard to have an opinion about. What it does is not.

The flange tip sits half a depth from the axis of rotation, so it moves sideways by $\theta \times h/2 = 8.9$ mm. That is a movement somebody can see, photograph and measure with a tape, and it is the quantity everything downstream of the beam actually experiences: the cladding rail bolted to the flange, the ceiling grid hung from it, the partition running under it, the roller shutter guide fixed to its web.

Put beside the checks that do exist:

The twist produces 62 per cent as much movement as the bending does, and it is checked by nobody.

The movement that does get a limit is the smaller of the two here, and the comparison is the only handle available: the vertical deflection is in a direction the structure was designed for and has a table entry, and the horizontal one is larger, in the direction nothing was designed for, and has none.

Why nobody computes it #

Three reasons, and the third is the interesting one.

The first is that the eccentricity is usually assumed away. A beam is drawn with its load through its centreline, and the centreline of an I-section is its shear centre, so the drawn structure has no torque in it. The eccentricity arrives on site — a purlin cleat on one flange, a slab bearing on one side, a services trapeze hung from a flange tip.

The second is that the calculation is awkward. Warping torsion needs $I_w$, a hyperbolic function and a boundary condition about whether the end can warp — which depends on the connection detail, not on the member. Nothing about it fits on a line, and a quantity that needs half a page is a quantity that gets left until somebody asks for it.

It is worth measuring how awkward. The full expression for the mid-span twist of a member with warping-fixed ends is

$$\theta = \frac{m}{GJ}\left[\frac{L^2}{8} - \frac{L}{2k}\tanh\frac{kL}{4}\right]$$

which at $kL = 4.50$ evaluates to 28 per cent of the Saint-Venant answer. With forks at the ends instead — twist held, warping free — the bracket becomes $L^2/8 - (1 - \operatorname{sech}(kL/2))/k^2$ and the answer 69 per cent of it. There is nothing difficult in it — one hyperbolic cosine and two section properties — and it is nonetheless the reason it does not get done, because it does not look like the other four lines on the calculation sheet.

The third is that in the finished structure it usually is not there. A beam with a slab cast on it, or a metal deck screwed to it, has continuous torsional restraint along its length and cannot twist at all; the twist calculation is about a condition that exists only before the deck is fixed.

Continuous restraint is worth a great deal and it arrives late. A deck screwed to the top flange makes the twist calculation on this page irrelevant — which is why the case that bites is the construction stage, with the steel bare, the decking laid loose and a concrete pump on it.

That third reason is why the quantity is genuinely awkward rather than merely neglected. The calculation is right for a stage that lasts a fortnight, and wrong for the fifty years afterwards. A code cannot easily write a limit for a condition that is temporary, and a designer cannot easily write off a condition in which people are standing on the member.

The eccentricity sweep, which is the design chart nobody draws #

The twist is linear in the eccentricity, so the whole family is one straight line and it can be read off in the direction anybody would want it.

For the member drawn, the flange tip’s movement reaches the span/250 vertical limit — 32 mm, the only number available to compare it with — at an eccentricity of 270 mm. That is about a flange width and a half: a services trapeze hung 270 mm off the centreline, or a precast unit bearing that far to one side, and the sideways movement of the beam exceeds the limit that would have been applied to its sag.

Two hundred and seventy millimetres is not an unusual eccentricity. It is roughly what a 400 mm wide bearing on one side of a 178 mm flange produces, and it is what a cantilevered edge angle carrying a facade produces, and neither of those is drawn as a torque anywhere.

The eccentricity that reaches the limit is not a property of the detail either. It falls with the span, and quickly, because the twist accumulates as $L^2$ while the limit it is being compared against grows as $L$.

That second reading is the one worth carrying, because it inverts the usual expectation. Warping restraint is a short-member effect, and a long member is the one that needs it — so the two curves diverge in the direction that makes a long open beam worse than a hand calculation using the restrained factor would suggest.

Whether the torque is equilibrium torsion or compatibility torsion is therefore the first question, and it is answered by asking what happens if the beam rotates: if the load stays where it is, the torque is real; if the load redistributes, most of it goes away.

Which free body produced the number #

Cut the beam at mid-span and take one half as the free body. On the cut face are a bending moment, a shear, and a torque — the accumulated $we$ over half the span, $we L/2 = 3.6$ kNm. That torque is the internal force with no diagram in any set of calculations, because the analysis that produced the shear and the moment was two-dimensional and had nowhere to put it.

The eccentricity is what puts it there, and it is worth noticing that the eccentricity is not a load. It is a detail: where the cleat is, which side the slab bears on, how far out the trapeze hangs. The torque is generated by a decision made on a drawing that has no forces on it at all.

And the point the eccentricity is measured from is not the centroid. For a channel it is outside the section entirely, so a channel loaded through its web is loaded eccentrically by construction and twists under a load that looks perfectly central — an eccentricity of tens of millimetres that no drawing shows and no schedule records.

What a limit would have to protect #

A deflection limit is not an arbitrary number. Span over 360 for the imposed part exists because plaster cracks at about that curvature; ten millimetres on a mullion exists because sealed glazing units fail past it; span over 250 exists because floors that sag more than that look wrong and doors stop closing. Every limit in the table names a thing being protected.

A twist limit would have to name its own, and the candidates are different in kind:

• A cladding rail bolted to the flange experiences a differential sideways movement between its ends, which opens the joints in the sheeting.
• A ceiling grid or a raised floor cares about the rotation itself, because a grid that is not level shows.
• A lift guide or a shutter track cares about the plan position of the fixing, to a millimetre or two.
• A crane rail cares a great deal, and is the one case where a limit does exist — because a rail that has rotated derails the crane, and the crane’s manufacturer wrote the number.

That last one is the exception that shows the shape of the rule. Where a limit exists it was written by whoever owned the thing that failed, not by the structural code, and it is expressed as a tolerance on a position rather than as an angle.

Where the model stops #

The ends are torsionally pinned. They have to be held against rotation somehow, and a beam sitting on a bearing with two bolts through its bottom flange is not obviously held. If the ends can rotate, the member does not twist — it rolls off its supports, which is a stability problem rather than a serviceability one and has a much lower failure load.

The torque is uniformly distributed. A single eccentric point load is a concentrated torque and gives a different, larger twist for the same total; a torque applied near a support is nearly harmless and one at mid-span is worst, exactly as a bending moment would be.

The warping restraint is total or absent. Real end connections are neither, and $I_w$’s contribution depends on a detail — an end plate welded across both flanges restrains warping well, a pair of web cleats hardly at all. With forks at both ends — twist held, warping free — this member twists 6.25°; with both ends held against warping, 2.55°. The truth is somewhere between, and the connection detail decides where.

Second-order effects are ignored. A member that has twisted 2.6° has moved its compression flange sideways, which is the initial condition for lateral-torsional buckling; the two interact, and past a few degrees the twist grows faster than this linear calculation says.

The section is drawn at its nominal dimensions. $J$ for a rolled I-section is dominated by the flange thickness cubed, so a rolling tolerance of half a millimetre on a 12.8 mm flange is 12 per cent on the torsion constant and therefore on the whole answer. Of all the section properties in a table, this is the one with the widest real scatter and the one quoted to three figures.

And the drawing shows one section at one station. The twist varies along the span, the flange tip’s movement varies with it, and the thing attached to the flange is attached along its whole length — so what a cladding rail experiences is a differential movement between its ends, which is a fourth quantity nobody limits either.

The torque is an integral, so the eccentricities can cancel #

The distributed torque $m = we$ was written as though the eccentricity were a single number for the whole member. It is not: it is whatever each piece of load happens to be doing, and the torque at any section is the integral of it along the span.

That makes the sign of each eccentricity as important as its size, and it is a detailing variable rather than a load one.

Cleats alternating sides. Purlins framing into a rafter, one on each side in turn, apply torques of opposite sign at alternate stations. Their sum over the span is nearly zero and the net twist is small — much smaller than a designer adding up magnitudes would compute.

Cleats all on one side. The same purlins all fixed to the same face accumulate, and the member gets the full $weL/2$ at its supports.

The two arrangements are indistinguishable on a framing plan, cost the same, and differ by an order of magnitude in the movement they produce. Which is a rare thing in this collection: a design lever that is free, invisible in every calculation, and available to whoever draws the connections.

There is a trap in the cancellation and it is worth stating, because the obvious reading of it is wrong. Zero net torque does not mean zero twist. A member with the eccentricity one way over its first half and the other way over its second has no torque at mid-span at all, and a torque diagram that peaks at the quarter points with opposite signs — so it twists one way over one half and the other way over the other, with the flange tips moving in opposite directions and the largest movement nowhere near the middle. A cladding rail fixed along such a beam is being racked from both ends toward the centre.

The strength check and the movement check are at opposite ends #

The bending analogy runs one step further than the essay has taken it, and it puts the two questions in different places.

A uniformly distributed torque produces a torque diagram exactly like a shear diagram: largest at the supports, at $weL/2$, and zero at mid-span. The twist, being the accumulation of the rotation the torque causes, is largest at mid-span and zero at the ends.

So the two checks a designer might make on this member are at opposite stations. The stress check — shear stress from the torsion, the warping normal stress in the flanges, and their combination with the bending stress — is at the supports, where the torque peaks and where the bending moment is nothing. The movement check is at mid-span, where the torque is zero and the twist is everything.

That is worth carrying because it defeats the ordinary habit of checking a member at its worst section. There is no worst section: there is a worst section for one quantity and a different one for the other, and a check made at mid-span because that is where the bending moment is will find no torque there at all and conclude, correctly and uselessly, that torsion is not a problem at the section examined.

The section choice, restated as a movement #

The 60-to-1 ratio between the open and the closed section is worth taking out of the arithmetic and putting into the terms a decision is made in.

A 400 mm I-section and a 400 × 200 hollow section of similar weight have very nearly the same bending stiffness — the material is in the same places for the purpose of resisting a moment. They differ by two orders of magnitude in one property only, and that property does not appear in any check unless somebody asks about torsion.

So the two members are indistinguishable on every calculation that gets done and differ by a factor of sixty on one that does not. Which is exactly the condition under which a design decision gets made for the wrong reason: the open section is cheaper, easier to connect to and easier to bolt through, so it is chosen, and the property on which the two differ never enters the comparison.

The cases where this has been paid for are recognisable afterwards and not before: an edge beam carrying a facade on brackets one side, a beam supporting a cantilevered balcony, a spandrel under a curtain wall, a runway beam for a monorail hoist. In each of them the load is permanently off the shear centre, and in each of them the symptom is not a failure but a complaint — the cladding joints are open at one end of the bay, the balcony slopes, the hoist runs to one side.

The ladder from here #

Later rungs on this anchor: the concentrated torque and the position along the span that is worst, which is not mid-span for the twist at mid-span. Torsional restraint from a deck, quantified as a distributed rotational spring, and the stiffness at which the twist stops mattering — a threshold of exactly the kind a stiffener’s rigidity is. The construction-stage case in full, with the decking laid and unfixed and the pump on it, which is where the calculation earns its keep. The interaction with lateral-torsional buckling, where the twist is no longer a serviceability quantity at all. And the question this essay cannot answer: what a twist limit ought to be — because unlike a deflection limit, which is calibrated against cracking finishes and human perception, nobody has ever asked what angle a floor may rotate through before anyone minds.