The collector the slab does not need

The floor is a beam lying down: a storey’s wind or earthquake force is carried across the plan by the floor plate, spanning between the walls or braced frames that take it to the ground, and the plate’s shear at each of those lines is the line’s share. Rigid by one code and not by the other found that the share itself is uncertain — whether the plate is treated as rigid or flexible moves load between lines by a great deal, and codes design each line for the larger. It ended on what happens next. A braced bay is a few metres long and a brace line is the whole depth of the floor. However much the line receives, it has to be gathered along the line into the bay.

That is the job of a collector, or drag strut: a member running along the brace line, usually a steel beam already there for gravity, connected to the floor along its length and to the braced bay at its end. It is designed for more than its calculated force — for an overstrength factor, because it must not fail before the bay it feeds — and its calculated force comes from a rule simple enough to do on the back of the drawing.

The rule, and what it assumes #

The rule treats the floor plate as a girder whose chords, along its two long edges, carry all of its bending and whose web, the plate, carries a uniform shear. At a brace line the plate therefore hands over its share $R$ as a uniform shear flow $q = R/D$ along the line’s whole depth $D$. Where the line has no bay, the collector collects it; where the bay starts, the collector delivers what it has gathered. So the collector’s force rises linearly from nothing at the line’s end to a peak where it meets the bay:

$$N_{\max} = R\left(1 - \frac{b}{D}\right)$$

for a bay of length $b$ at one end of the line, and half of that from each side for a bay in the middle.

Take a floor 48 m between two brace lines and 24 m deep, carrying 40 kN per metre of span, so that each line receives 960 kN, with a braced bay 6 m long at one end of each line and a steel collector of 5,000 mm² along it. The rule’s uniform shear is $960/24 = 40$ kN/m, and the collector’s force at the bay is $960 \times (1 - 6/24) = 720$ kN.

The rule is a statement about equilibrium only in its last step. The first step — that the plate delivers its shear uniformly — is a statement about stiffness: about how a plate, a collector and a short rigid bay share the job of moving 960 kN onto 6 m of a 24 m line. That is a question the stiffest path answers, and in a floor the plate and the collector are two paths.

The floor as a membrane #

To answer it the floor has to be solved as what it is: a plate in plane stress, its own shear stiffness and the collector’s axial stiffness both included, with the braced bay as a short length of the line held rigidly and the rest of the line held only by whatever the collector can do. The plate is meshed into small quadrilaterals, the collector and chords are bars along its edges, and the system is solved for the displacement of every point. Supported along its whole lines, the same model reproduces the deflection of a deep beam by Timoshenko’s theory to within 2 per cent, which says the plate is being modelled as the beam the rule thinks it is.

With a steel deck — a profiled sheet whose shear stiffness, around 10,000 kN per metre, comes from the sheet’s own flexibility and its fasteners — the collector carries 689 kN, 96 per cent of the rule. The deck is soft in shear, the collector is stiff by comparison, and the deck hands its shear to the collector almost evenly along the line, as the rule assumes.

With a 150 mm concrete slab the collector carries 171 kN — about a quarter of the rule’s force. The slab is nearly two hundred times stiffer in shear than the deck, so stiff that it does not wait for the collector: it carries its shear across to the braced bay directly, and the collector, an axial member that has to stretch to pick up load, picks up little.

The slab carries the rest beside the bay #

What the collector does not carry, the slab does, and the shear flow beside the line shows where. With the deck it is nearly uniform, a little above the rule’s 40 kN/m everywhere. With the slab it is concentrated beside the bay: 132 kN/m averaged over the bay’s 6 m, 3.3 times the rule’s, falling away to a few kilonewtons per metre at the far end of the line. At the two ends of the bay it peaks sharply, and the peak grows each time the mesh is refined, because a rigid support that stops abruptly part-way along a plate is a stress singularity — in the real slab, the place where it cracks first.

The load has not disappeared; it has moved. In a concrete floor the collector is relieved and the slab around the bay is loaded instead, by the part of the line’s share the rule assigned to the collector. That changes what has to be checked. The connection between the slab and the braced bay’s top beam — studs, dowels, a cast-in angle — must transfer 132 kN/m on average and more at its ends, where the rule would have given it 40 plus the collector’s force at one point.

The stiffness that decides the split #

The two floors are two points on one curve. What decides how much the collector carries is the ratio of its axial stiffness, $E_c A_c$, to the plate’s shear stiffness over the length of the line, $G t D$. The collector here, 5,000 mm² of steel, has $E_c A_c = 1{,}050$ MN; the slab’s $G t D$ is 45,000 MN, the deck’s 240. At a ratio of 0.02 the collector carries a quarter of the rule; at 4.4, nearly all of it. In between the curve is a smooth S, the shape of every stiffness-sharing problem on a logarithmic axis.

It is also, recognisably, shear lag. A flange attached to a web along one edge picks up load from the web over a distance set by the flange’s axial stiffness and the web’s shear stiffness, and a flange that is stiff against a web that is soft — or a web that is stiff against a flange that is soft — does not share evenly. A collector is a flange on the edge of the floor plate, and a concrete slab is a very stiff web.

The rule sits at the far right of the curve, and not quite at its end. It is the limit of a collector infinitely stiff against the plate and chords that carry all the plate’s bending, so that the plate’s shear is uniform across its depth. This floor’s chords are light, the 150 mm slab bends as a deep beam on its own, and a deep beam’s shear is not uniform across its depth but larger in the middle. Make the collector rigid with these chords and it carries 8 per cent more than the rule; give the chords a hundred times the area and the rule is recovered. The rule is a model of a girder with a uniform web, and a floor is that girder only when its chords make it one.

Shorter bays, and bays in the middle #

The rule’s force falls as the bay lengthens, in proportion to the part of the line the bay does not occupy, and halves when the bay moves to the middle of the line, where two collectors share the work. The membrane follows the same trends at its own level: with the slab, 171 kN for a 6 m bay at the end and 85 kN for the same bay in the middle, a quarter of the rule in both cases. The deck tracks the rule closely throughout. Where the bay is placed is worth exactly what the rule says it is worth; what the floor is made of decides how much of that worth the collector actually sees.

The floor before cracking and after #

That seems to say a concrete floor’s collector can be lighter than the rule makes it, and it would — if the slab’s shear stiffness were a property of the floor. It is not. Concrete cracks, under shrinkage before the building is occupied and under the storey’s own lateral load once it is, and a cracked slab is stiffer than its cracked section says but much less stiff than an uncracked one. Every halving of the slab’s shear stiffness moves the floor left-to-right along the curve: at half the uncracked stiffness the collector carries 36 per cent of the rule, at a quarter 50, at a tenth 68, at a twentieth 79. A slab cracked by the first cycles of an earthquake is on its way to being a steel deck.

So a collector in a concrete floor carries a quarter of the rule’s force in the floor as built and most of it in the floor as cracked, and the cracked floor is the one present when the collector matters. The rule is the right design force for the collector, not because it describes the slab but because it bounds every state the slab can be in. And the slab around the bay needs the opposite state: its shear concentration, 132 kN/m averaged over the bay, is largest when the slab is uncracked and stiff, which is when it is first loaded. The collector and the slab-to-bay connection are designed for the larger of two cases, each of them a different floor.

What the overstrength is protecting #

Collectors are designed for their calculated force times an overstrength factor — two to two and a half in the commonest rules — because the braced bay they feed is meant to yield in a large earthquake and deliver its own strength, which is more than its design force, and the collector must still be standing when it does. Read against the membrane, that factor is doing two different jobs in the two floors.

In the steel-deck floor, where the collector carries 96 per cent of the rule, the factor is what it says it is: the margin between the force the analysis predicts and the force the yielding bay can deliver. In the concrete floor as built, the collector carries a quarter of the rule, so a collector sized for two and a half times the rule has a margin of about ten against the uncracked floor — most of it hidden in the slab’s stiffness, and none of it dependable, because the cycles that bring the bay to its strength are the same cycles that crack the slab and hand the load back to the collector. By the time the bay yields, a heavily cracked slab has moved the collector to two thirds of the rule or more, and the margin has shrunk toward the factor the rule intended.

The membrane also says something the rule cannot: that the factor need not cover the floor’s own load path being wrong in the unsafe direction for the collector. Across every floor stiffness the collector carries no more than the rule, except for a stiff collector in a plate whose own bending makes its shear non-uniform, where it carries up to 8 per cent more — well inside the factor. The rule bounds the collector. It is the slab around the bay, loaded by a concentration the rule does not describe, that has no factor protecting it unless one is applied deliberately.

A composite floor — a concrete topping cast on a steel deck — behaves as the concrete slab does here as long as the topping is continuous past the bay, because its shear stiffness is the concrete’s; the deck underneath matters only where the topping is cut.

The split, by hand #

The collector’s share can be estimated without a mesh, as a shear-lag problem. Treat the collector as a bar picking up shear from the plate over a length $\ell$ near the bay, with the plate’s shear stiffness acting over a width comparable to the line’s own length. The bar picks up load over a decay length

$$\ell \approx \sqrt{\frac{E_c A_c\,D}{G t}},$$

which for the slab is $\sqrt{1.05 \times 10^6 \times 24/1.9 \times 10^6} \approx 3.6$ m and for the deck $\sqrt{1.05 \times 10^6 \times 24/10^4} \approx 50$ m. A decay length short against the line, as the slab’s is, means the collector only gathers load from a few metres beside the bay; a decay length long against it, as the deck’s is, means it gathers from the whole line, as the rule assumes. The ratio of the two, about fourteen, is the gap between a quarter and nearly all.

A membrane, a rigid bay and a floor of one material #

The floor is an elastic membrane. It carries in-plane forces only, as a floor plate does for lateral load; its bending under gravity is a separate problem. Concrete’s nonlinearity enters only through the reduced stiffness above.

The braced bay is rigid. A real braced frame deflects under its share, and a flexible bay attracts less of the slab’s concentration and more of the collector’s force; the bay modelled here bounds the concentration from above.

And the plate is uniform. Openings near a brace line — a stair, a shaft — cut the slab’s direct path to the bay, and a floor whose slab is interrupted beside the bay sends more to the collector than one that is continuous.

Openings, the connection and the cycles #

They cannot show an opening. The commonest reason a slab cannot reach the bay directly is that something is in the way, and a collector beside a stair core is carrying the rule’s force whatever the slab’s stiffness.

They cannot show the connection’s own stiffness. The slab reaches the collector and the bay through studs or dowels with their own slip, which put a third flexibility in series and move the split toward the collector.

And they cannot show cycling. A collector is a capacity-protected member precisely because the part that is meant to be weak — the bay’s brace — must yield first, and the forces those cycles put through the collector are set by the brace’s strength, not by the floor’s stiffness at any one moment.

Where the shear actually goes #

The uniform-shear rule assumes a floor that hands over its shear evenly. For a 6 m bay on a 24 m line it gives the collector 720 kN.

A steel deck nearly does, and its collector carries 96 per cent of the rule. A concrete slab does not: its collector carries about a quarter, 171 kN, and the slab beside the bay carries the rest at 3.3 times the rule’s shear.

The split is set by the collector’s axial stiffness against the plate’s shear stiffness, on one S-shaped curve; the rule is its end only when the chords carry all the bending.

Cracking moves a slab toward the rule — to 68 per cent at a tenth of its stiffness — so the collector needs the rule’s force and the slab around the bay needs the uncracked concentration.

Still open: the floor that reaches two bays at once #

Every line here has one bay. Many have two — a braced bay at each end of a long line, or a core wall and a braced frame on the same line — and then the floor’s shear can reach the line at two places, with the collector between them carrying the difference. Which bay the slab feeds directly and which it reaches through the collector, whether a stiff slab sends its shear to the nearer bay regardless of the bays’ own stiffness, and whether the collector between them is in tension or compression depending on which state of cracking the floor is in, is the question a line with two bays puts to the single bay here.