Two walls that agreed to be one

Draw two shear walls, six metres each, with a two-and-a-half metre opening between them for the doors. Ask how they share a wind load and there are two answers, both of them defensible, and they differ by a factor of seven.

The first: they are two cantilevers. Each has a second moment of $tl^3/12$, they deflect together because the floors tie them together horizontally, so the load splits in proportion to their stiffnesses and each carries its share entirely in bending. Total $I = 12.6\ \text{m}^4$, and a 64 m stack under 60 kN a floor deflects 111 mm.

The second: they are one wall, twelve and a half metres wide with a slot in it. Total $I = 86.7\ \text{m}^4$, and the same building deflects 16 mm.

Nothing about the walls differs between those two calculations. What differs is whether the beams over the openings are believed.

Which free body produced the number #

Cut the pair horizontally at the base and take everything above as the free body. The wind above the cut delivers a total overturning moment of 40,300 kNm, and there are exactly two ways for the two wall sections to supply it:

$$M_{\text{total}} = \underbrace{M_1 + M_2}_{\text{bending in the walls}} + \underbrace{N \ell}_{\text{an axial couple}}$$

where $N$ is the axial force in each wall — compression in one, tension in the other — and $\ell$ is the distance between their centroids, 8.4 m here.

The second term is the one worth staring at. A wall’s own bending has a lever arm of at most its own length; the couple between two walls has a lever arm of the distance between them. At six-metre walls two and a half metres apart, that is 8.4 m against something under 6, and the axial route is intrinsically the cheaper one. In this building the couple carries 25,300 kNm of the 40,300 — 63% — and no bending moment diagram drawn on either wall contains any of it.

That fraction has a name, the degree of coupling, and it is the number the whole system is described by:

$$\text{DoC} = \frac{N\ell}{M_{\text{total}}}$$

It runs from zero, where the beams do nothing and the pair is two cantilevers, to something like 0.8, where the beams are as stiff as they can usefully be made. It never reaches one, and the reason is worth having: a coupling beam of finite depth cannot suppress the walls’ curvature entirely, so some bending always stays in the walls. Even with rigid beams this pair only reaches 81%.

Where the axial force comes from, which is the whole mechanism #

There is only one way to put axial force into a wall that nothing is sitting on, and it is through the beams.

Each coupling beam is bent by the difference between what the two walls are doing at that floor. If the walls rotate together, as they must if the beams are stiff, then the wall on the windward side has to rise and the leeward one has to sink — because two members rotating by the same angle about centres 8.4 m apart cannot both stay where they are vertically. The beam resists that relative movement with a shear, and the shear pushes down on one wall and up on the other.

So the axial force in a wall at any level is the running sum of every coupling beam shear above it, and the base axial force is the sum of all of them: 3,007 kN. That is not an approximation. The beams are the only load path that can put axial force into these walls, so the total of their shears is the axial force, exactly. It is the cleanest equilibrium check the system has, and any analysis that fails it has invented a load path.

The consequence nobody expects: the beams are the worst members in the building #

Three thousand kilonewtons of base axial force, delivered by twenty beams, is an average of 150 kN a beam. The worst is 222 kN, in a member 350 mm wide and 600 mm deep spanning 2.4 m clear.

That is a nominal shear stress of about 1.06 MPa, which does not sound alarming until the span-to-depth ratio is noticed: four. A member four times as long as it is deep does not behave as a beam at all. Its shear span is under two depths, there is no section to design in the ordinary sense, and the internal force finds its way across as a diagonal compression field rather than as flexure with stirrups.

This is why every modern coupled wall has diagonally reinforced coupling beams — bundles of bars running corner to corner in cages, rather than longitudinal steel with links. The detail looks like an over-elaboration and is not: it is the reinforcement following the strut-and-tie path the member actually uses, in a member that will be asked to do it repeatedly and in both directions during an earthquake.

And the worst one is not at the bottom #

The base axial force is the largest axial force. It does not follow that the base beam is the largest beam, and it is not: the bottom beam carries 87 kN and the peak is 222, at level six of twenty — 30% of the way up.

The reason is that a beam’s shear depends on the difference between what the two walls are doing, not on the total. Near the base both walls are almost vertical and almost stationary, so there is very little relative movement for a beam to resist. Near the top both walls have rotated a great deal but they have rotated together, so again there is little for a beam to resist. The maximum difference is somewhere in between, and for ordinary proportions it is a third of the way up.

A designer who details the ground-floor beam and repeats it upwards therefore under-reinforces the beams that actually govern, by a factor of two and a half — and does so while over-reinforcing the one at the base. The error is not conservative in either direction, which is the shape several other distributions on this site share.

What the beam’s own stiffness is, which is not what its span suggests #

A coupling beam is stiff for two reasons that are easy to get wrong in opposite directions.

Its effective span is longer than the opening. The beam’s ends are embedded in walls that are rigid over their own half-lengths, so the member that bends is the clear span, but the rotation it has to accommodate is measured over the centre-to-centre distance. Getting this wrong halves the coupling.

Its shear deformation is not negligible. For a beam of span-to-depth four, the familiar $12EI/l^3$ overstates the stiffness substantially: the flexural stiffness here is 164,000 kN/m and the shear stiffness 911,000, so the series combination is 139,000 — 15% below the flexural figure alone. On a shallower beam the correction is smaller; on a deeper one it grows, and past a span-to-depth of about two the shear term dominates and adding depth stops buying stiffness at anything like the rate the cube suggests.

That last point is the practical ceiling on the system, and it is the reason the second refutation above is filed as misapplied rather than as false. Deepening the beams from 400 mm to 1,000 mm does exactly what it is supposed to — the coupling goes from 51% to 72% and the drift from 33.5 mm to 18.4 — but the shear in each beam goes from 133 kN to 284. The stiffness is bought at a shear demand that arrives faster than the stiffness does. Past some depth the beam cannot be reinforced to carry what its own stiffness attracts, and the system’s limit is a connection detail rather than a wall.

The reading that made it solvable by hand #

Before there were frame programs, a twenty-storey coupled wall was twenty coupling beams, forty wall elements and a few hundred simultaneous equations. It was solved anyway, and the trick is worth knowing because it explains what the system actually is.

Smear the beams. Replace twenty discrete members at 3.2 m centres by a continuous shear medium of the same stiffness per metre of height, and the problem stops being a frame and becomes a differential equation in one variable — the axial force in the walls. It is second order, its coefficient is a single dimensionless parameter, and it has a closed-form solution.

That parameter, usually written $\alpha H$, is the whole system in one number. Small values are the two-cantilever limit and large values the composite-wall limit, and the interesting range is between about one and eight — which is where nearly every real coupled wall sits. Everything this essay has computed can be read off a pair of curves plotted against it, and for thirty years it was.

The reading also names the mechanism honestly. A coupled wall is not two walls with beams; it is one wall with a slot in it and a shear connection across the slot. The beams are shear connectors, the axial couple is composite action, and the degree of coupling is the efficiency of the connection — the same quantity, with the same meaning, as the interaction ratio in a composite deck.

Where the opening goes, and why a wider one is not simply worse #

Widen the opening from 2.4 m to 4.0 m, keeping the walls the same, and two things happen in opposite directions.

The lever arm $\ell$ grows from 8.4 m to 10.0 m, which makes the axial couple more valuable per unit of axial force. The beam gets longer, which makes it much less stiff — a cube of the span — so there is less axial force to be had.

The second effect wins: the degree of coupling falls from 63% to 53% and the drift rises from 23.2 mm to 30.0. But it wins by less than the cube would suggest, because the growing lever arm is paying part of the bill. A coupled wall degrades gracefully as its openings widen, which is a useful property in a system whose openings are decided by the architecture rather than by the engineer.

The opposite move — narrowing the opening — runs into the shear ceiling described above rather than into any structural limit. There is a range of openings, roughly 1.5 to 4 m for ordinary storey heights, within which the system works well, and it happens to be the range doors and corridors want.

Where this model stops #

Three assumptions, and the first is the one that matters in an earthquake.

The beams are assumed elastic. They are not, in a serious event, and they are not meant to be: the coupling beams are the designated fuse. They yield first, they dissipate energy, and the degree of coupling falls as they do, so the system migrates from a stiff coupled wall towards a pair of ductile cantilevers as the shaking gets worse. That is a deliberate design, and it is why the beams are detailed for rotation capacity rather than only for strength. It also means the elastic degree of coupling is a description of the building at small amplitudes and not of the building at collapse.

The walls are assumed uncracked. The wall on the tension side of the couple is carrying 3,000 kN of tension at its base, and concrete walls crack under that. The cracked wall is softer, which reduces its share, which increases the other wall’s — an interaction the elastic analysis above does not contain.

The floors are assumed rigid in plane. They have to be, for the walls to be forced to move together at all. A flexible diaphragm uncouples them, and a long thin building with a precast floor and no topping can have two shear walls that are structurally unaware of each other.

What the picture cannot show #

The figures here draw one plane. A real core is a tube — lift shafts and stairs, walls in two directions, openings on every face — and its coupling happens about two axes at once with the same beams participating in both. The degree of coupling then stops being one number and becomes a property of a direction, and the worst direction is usually neither of the two the drawing is set out on.

Nor can the figures show the sequence. The axial couple builds up as the wind builds up; the gravity load was there first. The tension wall is being unloaded from a compression it already had, and whether it ever reaches net tension depends on how much building is sitting on it — which is a question about the floor plan and not about the lateral system at all.

There is a third thing no single frame shows, and it is the one that decides how a coupled wall is detailed. The couple reverses when the wind does. The wall that was carrying 3,000 kN of tension carries 3,000 kN of compression an hour later, and every coupling beam has its shear reversed with it. A member designed for a shear in one direction and reinforced accordingly is a member that has been designed for half its load history — which is why the diagonal cages in a coupling beam are always drawn as a cross rather than as a single diagonal, and why the reversal argument matters more here than in almost any other member.

The generalisation #

The habit worth carrying is about what makes two members into one.

A composite beam is the same argument in miniature: two elements bolted together are stiffer than two elements side by side, by a factor that depends entirely on whether the connection between them can carry the shear that composite action demands. Release the studs and the result is two beams; provide them and it is one, with a second moment that includes the $Ad^2$ term and is several times larger.

Coupled walls are that argument at a building’s scale, with the coupling beams as the studs and the axial couple as the $Ad^2$ term. The parallel is exact enough to be useful: the parallel axis theorem is what says the couple is worth 63% here, and the reason the number is so large is the same reason it is large in a composite deck — the distance between the parts is bigger than anything inside either of them.

Which suggests the question to ask of any assembly of vertical elements: not how stiff each one is, but how far apart they are and what is joining them. The first is a property of the members and is usually the thing that was designed. The second is a property of the plan and is usually the thing that was drawn.