The load that moves with the twist

A beam bending in its strong plane, at a moment well below its capacity, can swing sideways and twist. That is lateral-torsional buckling, and the usual account of it involves a length between restraints, a torsional constant and a warping constant.

There is a term in the same expression that the usual account leaves out, and it can be worth a factor of two. It is the height at which the load is attached.

Which free body produced the number #

Take the beam at the instant it begins to buckle. It has rotated by a small angle $\theta$ about its longitudinal axis, and the axis it rotates about is the shear centre — the point that is not in the section — because that is the point about which a transverse force produces no twist.

Now ask where the load is. If it is applied at a height $a$ above the shear centre, then the rotation has carried its point of application sideways by $a\theta$. The load is vertical and it has not changed; but it is now acting at a horizontal offset from the axis of rotation, so it delivers a torque $Wa\theta$ about that axis.

That torque is in the same direction as the rotation that produced it. It is a destabilising moment proportional to the rotation, which is the signature of a stability problem, and it enters the eigenvalue exactly as a negative stiffness.

Hang the same load a distance $a$ below the shear centre and every sign reverses. The load swings back under the axis, the torque opposes the rotation, and the beam is stiffer against twisting than it would be with the load applied at the shear centre.

Two beams, one difference #

The consequence is that “the load on this beam” is not a complete description of the load.

A joist carrying a floor slab bearing on its top flange has its load applied at $+d/2$. The same joist carrying a load hung from its bottom flange — a monorail, a services tray, a ceiling — has it at $-d/2$. Everything else about the two is identical, and the elastic critical moments differ by roughly a factor of two on an ordinary rolled section at an ordinary unrestrained length.

The factor is not a constant, and this is the part that makes the effect awkward. It depends on the ratio of the load height to a length built out of the section’s own torsional and warping stiffnesses, so it is largest on the sections that are already worst — deep, narrow, and torsionally weak.

Where that torsional and warping stiffness comes from is the flanges bending in their own plane when the section is stopped from warping freely, and how much of it there is depends on the length rather than on the material — which is why the factor moves with the span as well as with the section.

The height term itself is easiest to see on a restraint rather than on a load, because a restraint can be moved through the depth while everything else is held. It is the same term with the opposite sign:

The same argument for a restraint #

If the height of a load matters, so does the height of a restraint, and for exactly the same reason: a brace prevents motion at the point it is attached to, and the point it is attached to is moving by the sideways displacement of the shear centre plus $a\theta$.

The three curves in that figure are worth reading one at a time.

On the compression flange the brace works. The critical moment climbs from the unbraced 143 kNm to 447 — the beam has become two shorter beams — and it reaches 99% of that plateau at 366 kN/m of stiffness. Past the plateau it stops helping, because the beam has found a mode with a node where the brace is and the brace no longer moves.

At the shear centre the same plateau is reached, and it costs 2,252 kN/m: 6.2 times the stiffness, for the identical result. The brace is now restraining a point that barely moves in the buckled shape, so it has to be very stiff to do anything at all.

On the tension flange it never arrives. At the stiffness that finished the job on the compression flange it has bought a factor of 1.022 — two per cent — and stiffening it further buys the same nothing, because the buckled shape simply rotates about the braced point and carries on. That is the brace on the wrong flange, and it is the most consequential detailing error available in steelwork: a member exists, it is connected, it is strong, and it does not restrain anything.

The claim that more stiffness does not rescue the wrong flange is worth testing rather than asserting, and the test is to run the same height sweep at the stiffness the shear centre needed — 6.2 times as much:

Why the plateau exists at all #

Both the load-height effect and the brace-height effect are the same term in one energy expression, and the plateau is the evidence for it.

A brace raises the critical load by forcing the buckled shape to have a node where the brace is. Once the beam is buckling in the shape with that node — two half-waves instead of one — additional brace stiffness restrains nothing, because the node does not move. The critical load is then the critical load of the shorter segment and no arrangement of braces can raise it further.

That is the general shape of every restraint problem on this site, and it is why a brace need not be strong. What the brace has to be is stiff enough to reach the knee, and where it has to be is somewhere that moves.

What it costs when the section is torsionally weak #

The height effect scales with how easily the section twists, so it is largest where the trouble already is.

The cleanest way to see it is to hold every section property fixed and change nothing but the depth the brace has to work across, since in this calculation the depth enters as a lever arm and nothing else:

A channel is the extreme case in ordinary construction, because its shear centre is outside the section entirely — here 109.7 mm from the centroid — so every load is applied at an eccentricity from the axis of rotation, whether or not anybody chose to apply it there. The three uncoupled critical loads at 6 m are 4,754 kN, 785 kN and 811 kN; the coupled answer is 754 kN, 3.9% below the lowest of them, and the mode that governs changes with length.

An I-section is much better behaved, because its shear centre is at the centroid by symmetry — the property a section that will not stay flat trades on, so the load height is measured from the middle of the depth and a load on either flange is the same distance from it in opposite directions. A section that is symmetric about neither axis has no such convenience.

The lifted beam, where the same term has an asymptote #

The clearest available demonstration is not a floor beam. It is a beam hanging from a crane.

Everyone knows the rigid-body rule: hang a body from a point above its centre of gravity and it is a pendulum, stable at any hook height above zero. A beam is not a rigid body. Tilt it, and the component of its own weight acting across its weak axis deflects it sideways by an amount $\bar{z}$, which moves the centre of gravity further out.

The stability then depends on the difference $y_r - \bar{z}$ — a hook height minus a lateral sag — and that difference goes through zero at a finite hook height. For this 30 m beam it happens at $y_r = 0.236$ m, and the answer a rigid-body calculation gives is out by a factor of 1.90.

This is hung from above and still unstable in one line: the attachment height matters because the thing being attached to is flexible, and the flexibility supplies a length that competes with the height.

The other half of a lift is the rigging above the hook. The sling angle decides the force in each leg, and the horizontal components of those forces run through the thing being lifted as a compression along its own length — so the pick points and their height are a stability question as well as a strength one, and a flatter sling angle worsens both at once.

What is really being computed #

Every result above is an eigenvalue, and the load height enters it as a term that is linear in the rotation. That is worth stating plainly because it explains why the effect refuses to be captured by a single factor.

A destabilising term linear in $\theta$ sits alongside a restoring term that is also linear in $\theta$, and their ratio depends on the section, the length, the restraint condition and the shape of the moment diagram. The published expressions handle this with a coefficient — one factor for the moment diagram’s shape, another for the load height — multiplied together as though they were independent. They are not independent, and the product is an approximation to a coupled eigenvalue, accurate to a few per cent over the range it was fitted on and unreliable outside it.

It is worth a reminder of how little of any of this is about materials. The ratio of a beam’s end rotation to its midspan deflection depends only on the shape of the load; the modulus, the second moment and the span all cancel out of it. The same is true of the height term, which is a ratio of two energies and carries no material property at all — which is why a factor of two in critical moment can be bought or lost by a hanger detail and cannot be bought back with a stronger steel.

Where it decides a design #

Four cases, all ordinary, in which the height of an attachment is the design.

A crane runway beam. The wheel load sits on the top flange, at the top of the rail, which is above the top flange again. It is also free to move along the beam, so there is no restraint at the point of application at all, and the beam has to be checked with the load at its worst position and its worst height simultaneously. Runway beams are the reason the load-height term appears in the standards.

They are also usually long, and length changes the arithmetic of the restraint in a direction that is worth knowing:

A portal rafter under wind uplift. Under gravity the purlins restrain the top flange and the top flange is in compression, which is the happy arrangement. Under uplift the moment reverses, the bottom flange goes into compression, and the purlins are restraining the tension flange — the third curve in the figure above, the one that buys two per cent. That is why portal frames carry fly braces from the purlins down to the inside flange, and why a frame checked only for gravity is not checked.

A monorail or a services load hung below. Here the height effect is favourable, and worth about a factor of two on a section that would otherwise be governed by lateral buckling. Almost nobody claims it, because it takes an explicit calculation to claim and the conservative answer is free.

A beam during erection. Before the deck is on, a beam has no restraint anywhere along it and carries its own weight applied at its own centroid. That is the most dangerous day in the life of the member, and the unrestrained length is the whole span rather than the purlin spacing the finished design assumed.

The common feature of all four is that the load height is decided by a detail — a rail, a purlin cleat, a hanger, a lifting lug — drawn by somebody who is not doing the stability check.

The support has a height too, and its sign is the other way #

The whole argument has been about where the load is applied. The reaction has a height as well, and it enters with the opposite geometry and therefore the same effect.

A downward load above the shear centre is destabilising because the load point moves sideways as the section twists and its line of action develops a lever arm that drives the rotation. An upward reaction below the shear centre does exactly the same thing: the bearing point moves sideways with the bottom flange, the reaction stays vertical, and the couple it develops drives the twist in the same direction.

So a beam sitting on bearings at its bottom flange — which is almost every beam — is supported destabilisingly, in the same sense that a top-flange load is applied destabilisingly.

Which supplies the reasoning behind a requirement that is usually presented as a boundary condition rather than as a mechanism. Every code requires torsional restraint at the supports of a beam, and the requirement looks like a formality about what “simply supported” means for lateral-torsional buckling. It is not: without it, the support itself is a destabilising load-height case, at the one point along the member where the reaction is largest.

A cleat, a stiffened end plate, a pair of web stiffeners, a beam bolted through both flanges — each of those removes the freedom the reaction would otherwise exploit, and each is worth far more than its size suggests for exactly the reason a brace is.

And the height is measured from the shear centre #

One more precision, because it moves the answer on the sections where it matters most.

The lever arm in every expression above is the distance from the load to the shear centre, not to the centroid. For a doubly symmetric I-section the two coincide and nobody has to think about it. For a monosymmetric one they do not: the shear centre sits nearer the flange with the greater minor-axis stiffness — the larger flange.

So a beam with a large compression flange and a small tension flange, which is the sensible arrangement for a member where the compression side is the one at risk, has its shear centre pulled up toward the top. A load applied on that top flange therefore has a shorter lever arm than the same load on a symmetric beam of the same depth, and the destabilising term is smaller.

Two useful things follow. The section chosen for its bending efficiency happens to be the one least penalised by top-flange loading, which is a rare case of two arguments agreeing. And a designer measuring the load height off a drawing has measured it from the wrong datum unless the section is symmetric — by an amount that, on a heavily monosymmetric beam, is a substantial fraction of the depth.

Where the model stops #

Everything above is elastic and perfect. A real beam is bowed, twisted and has residual stresses, so it never reaches an elastic critical moment; it follows a curve that approaches it asymptotically, exactly as a bowed column does. Load height changes the asymptote and therefore changes the whole curve, but the design capacity is somewhere below it and the ratio is not preserved.

An imperfect member deflects from the first increment of load and approaches its critical value without arriving at it, so the elastic critical moment is an asymptote rather than an event — and the design capacity sits somewhere below it on a curve whose shape was fitted rather than derived.

The load may not stay where it was put. A load applied by a slab bearing on the top flange stays on the top flange while the beam twists a little and does not once it twists a lot, because the slab is stiff and the bearing rotates. Everything here assumes the load’s point of application is carried around with the section, which is right for a hanger and questionable for a bearing.

Restraint is rarely a spring at a point. Decking, a slab, or a purlin line restrains a beam continuously and partially, and at a height that varies with the detail. The single-brace model is a stand-in for that, and the useful part of it is the ordering rather than the numbers.

None of this can be settled by a load factor, either, and the reason is the same one that runs through the whole of stability: a limit that goes as the cube of a geometric quantity moves further under a small change of shape than it does under any change of material. The height of an attachment is such a quantity, and it is set on a detail drawing.

What the picture cannot show #

None of the figures shows the beam twisting, because a critical moment is a statement about the instant before anything happens. The buckled shape drawn on an eigenvalue plot has an arbitrary amplitude — it is a mode vector — and it says which way the beam goes and nothing about how far.

Nor do they show the thing that decides most real cases: whether the restraint that is drawn is actually connected to anything that can take a force away. A brace on the compression flange is worth a factor of three, and only if the other end of it reaches a stiff point. A line of braces all connected to each other and to nothing else restrains the beams to one another and lets the whole set buckle together, which is a mode no single-beam calculation contains.

Nor do they show what residual stress has already taken out of the flange before any load arrives. About the weak axis a flange’s stiffness follows the cube of the width of its elastic core, so a residual pattern that costs a few per cent in strong-axis bending costs a great deal more in the mode this page is about — and it is the mode this page is about that the residual pattern was never measured against.

The generalisation #

The habit worth carrying is a question to ask of any restraint or any load: what point of the section is it attached to, and does that point move in the mode being prevented?

It is a question about the mode rather than about the member, and it has a general answer. A restraint is worth something in proportion to how far its attachment point moves in the shape it is trying to prevent; a load is destabilising in proportion to how far its attachment point moves in the same shape. They are the same quantity with opposite signs, which is why one figure in this essay answers both.

The rule extends past beams. A tuned mass damper works because it is attached where the mode has its largest amplitude and would do nothing at a node. A guy is worth more on a mast at the height where the mast wants to move. And an outrigger on a tall building makes the columns work precisely because it connects a core to columns at a level where the core’s rotation is large — the same argument, three orders of magnitude up.

The failure mode is always the same too. Something has been provided, it is adequate in every check made of it, and it is attached to a point that does not move.