The torque that should not be shed

A compatibility torsion goes away if it is allowed to: the spandrel cracks, its torsional stiffness collapses, and the moment it was holding redistributes to the beam that was trying to give it away.

The hero is that redistribution priced. A spandrel of $GJ = 1.35 \times 10^5$ kN·m² takes 51 per cent of the floor beam’s 197 kN·m fixed-end moment when uncracked — 100 kN·m of torque — and 21 per cent at a quarter of that stiffness, which is 41. The flat line beneath is a canopy’s 116 kN·m, which is equilibrium torsion and does not move.

This rung is about the two things the release costs, because neither is in the sentence that licenses it.

What the floor beam picks up #

That identity is the whole accounting. Nothing is lost in the redistribution, and the twenty per cent that arrives at midspan is exactly the sixty per cent that left the support.

Which produces the first threshold, and it is arithmetic rather than judgement. If the floor beam was designed for the shared support moment, it is now under-designed at midspan by twenty per cent. A designer who models the joint as partially fixed, sizes the beam on that model, and then relies on the spandrel shedding its torque has designed the beam for a state it will not be in.

The honest procedure is the one the identity suggests: design the floor beam as simply supported — for the full 197 kN·m free moment, or as near it as the detailing allows — and then treat whatever restraint the spandrel actually provides as a bonus that reduces the crack widths rather than as a capacity.

Which free body produced the sharing #

The distribution factor is a stiffness ratio and it looks like bookkeeping, so it is worth taking the joint apart once.

Cut a small body out around the joint where the floor beam meets the side of the spandrel. Three things enter it: the floor beam’s end moment, and the spandrel’s torque from each side. Moment equilibrium of that body says the beam’s end moment equals the difference between the two torques, which is what makes the spandrel a torsion member at all — a beam framing into the middle of another beam applies a torque to it, and the torque runs both ways to the columns.

Compatibility says the two rotate together. The beam’s end rotation about the spandrel’s axis is the spandrel’s twist at that point, because they are connected. One equation.

Put a stiffness on each. The beam resists rotation at $3EI/L_b$ if its far end is pinned; the spandrel resists twist at $4GJ/L$ for a member fixed against twist at both ends and loaded at midspan. Solve, and the share falls out as the ratio of one stiffness to their sum.

That is a moment distribution with a torsion member as one of its branches, and nothing about it is special. The only unusual feature is that one branch’s stiffness collapses by a factor of four when it cracks and the other’s does not — which is why the joint has two answers rather than one, and why the whole of this essay exists.

The free body also says what a designer can do about it, and it is short: change one of the two stiffnesses, change the beam’s far-end condition, or accept the share. There is no fourth option, because the equation has only those terms in it.

The rotation has to happen #

The second cost is not a force at all.

Getting from 51 per cent to 21 per cent means the spandrel’s torsional stiffness has fallen to a quarter, and it does not fall by decree. It falls because the spandrel cracks in torsion, spirally, along the length between the columns, and the cracks open enough to let the joint rotate.

Three things about that state are worth stating, because the strength calculation contains none of them.

It is a rotation with a size. The joint has to turn by the fixed-end rotation less what the beam’s own stiffness takes back, and on this spandrel that is of the order of a few milliradians over the 8 m — small, and enough to be visible as a step in a soffit line if the spandrel supports anything that has to be level.

It is a crack width. Torsional cracks in a beam without adequate closed links open in an uncontrolled way, because the mechanism is a spiral and there is nothing crossing it. A spandrel on a building’s edge, exposed, is the worst place on the structure to have uncontrolled cracks.

And it is not reversible. The stiffness does not come back when the load goes off, so the redistribution is permanent from the first time the floor is fully loaded — which is usually during construction, before anything about the finished behaviour has been observed.

So the minimum torsion reinforcement is not a strength provision. Codes require closed links and longitudinal steel in a spandrel whose torque has been shed, sized on a fraction of the cracking torque, and the reason is the crack width and the rotation rather than the capacity. It is the same argument the shed moment makes: a redistribution is a permission to compute differently, not a permission to detail differently.

What the spandrel is actually reinforced for #

A torque that has been shed still leaves the spandrel with something to carry, and it is worth being precise about what.

The residual torque. 41 kN·m in the hero, at a quarter stiffness — not zero, and it has to be reinforced for.

The minimum. Codes set a floor on torsion reinforcement in a member whose torque was reduced, typically expressed as a fraction of the cracking torque, on the reasoning that a member which has cracked in torsion needs steel crossing the crack whether the calculation asks for it or not.

And the equilibrium part, undiminished. 116 kN·m of canopy in every figure on this page, which no argument reduces.

Those three do not simply add, because they are not simultaneous in the same way — but the reinforcement has to satisfy the largest of the combinations that can occur, and in practice the spandrel ends up with closed links at a spacing set by the crack-control rule rather than by any of the three torques.

Which is the honest description of what shedding buys. It does not remove the links. It removes the quantity of longitudinal steel that an unreduced elastic torque would have demanded, which on a heavily loaded spandrel was the thing that made the section impossible to cast. Detailing is where a redistribution is actually paid for, and the payment here is a closed link cage in a member that would otherwise have had open links.

There is one more provision that belongs here and is easy to lose. The floor beam’s bottom steel has to be carried into the support, because a beam whose support moment has been shed is a beam whose midspan region has grown — and the point of contraflexure has moved towards the support by the same redistribution.

The threshold, and where it sits #

The rung below this one establishes that a compatibility torque can be shed. The question this one is for is when it should not be, and there are three answers, in order of how often they bite.

When the beam that picks it up cannot. The identity above says the midspan moment rises by exactly what the support sheds, so the test is whether the floor beam has that capacity. On a beam sized for its free moment, it does; on a beam sized for a partially fixed model, it may not.

When the rotation is a serviceability problem. A spandrel carrying a façade, a curtain wall bracket, a lift guide, or anything whose alignment matters is a member where a few milliradians is a real defect, and no strength argument reaches it.

And when the torque is large enough that shedding it is a big redistribution rather than a small one. That is the vaguest of the three and it is the one the codes gesture at without a number.

That pair is the clearest statement of the threshold available. The stiffer the torsion member relative to the beam, the more it attracts and the more remains after cracking — so a designer who reaches for a deeper spandrel because it looked overstressed in torsion has made the problem larger in both states.

Which reads as a paradox and is the ordinary behaviour of a shared load path: the member that is stiffened attracts what it was stiffened against. The way out is the opposite move — make the spandrel less stiff in torsion, or accept the crack and detail for it.

There is a fourth threshold that is worth naming because it is the one a code cannot write. Shedding assumes the spandrel is ductile enough to crack and keep going, and a member that cracks in torsion and then fails in torsion has not redistributed anything — it has failed. A spandrel with open links, or with longitudinal bars that are not properly anchored at the corners, does exactly that: the spiral crack forms, the concrete outside it spalls, and there is no cage left to carry the residual torque.

So the permission has a precondition that is a detailing requirement rather than an analysis one. A torque may be shed from a member detailed to survive shedding it, and nothing in the calculation checks that the detailing is there. It is the same structure of argument as a plastic analysis requiring rotation capacity: the method is licensed by a property of the section that the method itself does not compute.

The other stiffness in the ratio #

The distribution factor is $\dfrac{4GJ/L}{4GJ/L + 3EI/L_b}$, so there are two ways to make the spandrel’s share small and only one of them involves the spandrel.

Stiffening the floor beam is the lever nobody reaches for, and it is often available for nothing — a floor beam is usually sized by its own deflection, so it has bending stiffness to spare, and a slightly deeper one takes the torsion problem away without touching the member that has it.

A condition at the other end of the floor beam has halved the torque in the spandrel, and it is a condition eight metres away that nobody checking the spandrel would look at. That is the general hazard of a distribution factor: it is a property of a joint, and a joint’s properties are decided by members that do not meet there.

The line that does not move #

Every figure on this page has both lines on it, and the whole design decision is which of the two a given torque belongs to.

The test is a question about the structure rather than about the torque: if this member’s torsional stiffness were zero, would the load still be carried? For the floor beam framing in, yes — it becomes simply supported. For the canopy, no — it falls off.

And the two can be present at once, as they are in every figure here. A spandrel carrying both is designed for the equilibrium torque with no reduction whatever, and may shed the compatibility one on top of it — which means the reinforcement it ends up with is decided by the part of its torque that could not have been argued away, and the argument about the rest changes nothing at all.

The same decision in three other places #

A compatibility action is one that exists because two parts of a structure are connected and that vanishes if they are allowed to move relative to each other. The spandrel is the clearest case and it is not the only one.

Restraint to shrinkage or temperature. A slab held between two stiff cores develops an axial force that has nothing to do with any load, and it disappears the moment the slab cracks. An imposed strain is an action that goes away if the structure yields, and the same reasoning that licenses shedding a torque licenses ignoring it — with the same crack as the price.

Differential settlement in a continuous beam. The moments from a support that moved are compatibility moments; a plastic hinge removes them entirely. What they are not is negligible at serviceability, which is where a support that moved is worth checking even when the ultimate analysis says it is not.

And a secondary member’s contribution to a lateral system. A frame that was never intended to resist wind still attracts a share of it, in proportion to its stiffness, and the usual response is to say it will shed. It sheds by drifting, which is the same rotation argument at building scale.

All four have the same shape and the same trap. The action can be argued away at the ultimate limit state and the deformation that argues it away is a serviceability state nobody computed. The discipline the four share is to name the deformation, estimate it, and decide whether it is acceptable — rather than to treat “compatibility” as a word that ends the discussion.

What to carry away #

Shedding moves the load, it does not remove it. Sixty per cent off the spandrel’s torque is twenty per cent onto the floor beam’s midspan, exactly.

The release is bought with a rotation and a crack. The spandrel has to crack in torsion to deliver the stiffness reduction that licensed the redistribution, and that is a serviceability state with no calculation attached.

A stiffer spandrel is a worse problem in both states. It attracts more before cracking and retains more after, so the intuitive fix makes it worse.

And the floor beam’s stiffness is the lever that works. Trebling it takes the spandrel’s share from 51 per cent to 26, for a member that usually has the stiffness to spare.

Where the model stops #

The cracked stiffness is a fraction, not a calculation. A quarter is a convention; the real post-cracking torsional stiffness of a reinforced section depends on the links, the longitudinal steel and the cover, and is closer to a thin-walled tube’s than to the gross section’s.

Warping is not modelled. A spandrel restrained against warping at its columns is much stiffer than $4GJ/L$ says and attracts correspondingly more — an open section’s warping stiffness can dominate its St Venant stiffness over a short span.

The joint is a point. A real floor beam frames into the side of a spandrel over a connection with its own flexibility, and that flexibility is in series with the spandrel’s torsional one.

And nothing here is a fatigue or a repeated-load statement. The redistribution is computed once, on the assumption that cracking happens and stays; a member cycled between states does not have a single stiffness to put in a distribution factor.

The ladder from here #

Later rungs on this anchor: the torsion member as a branch in a moment distribution carried out by hand. Torsion interacting with shear in a cracked section, where the two demands land on the same links and add. Warping in the compatibility case. Spandrels supporting façades, where the rotation rather than the torque is the design criterion. And the case that sits between the two kinds — a torque that is equilibrium torsion for one load case and compatibility torsion for another, on the same member, in the same building.

The permission to shed compatibility torsion is one of the few places where a code explicitly licenses ignoring a computed action, and it took a long time to arrive. Reinforced concrete spandrels were designed for their full elastic torque for decades, at reinforcement quantities that made them impossible to cast, until tests in the 1960s and 70s showed that the members were cracking and redistributing anyway and that the frames were none the worse. What the tests could not show, and what has been left to judgement ever since, is where the load went — because a test rig has no façade hung on it and no floor beam that somebody else designed.