The slab that catches what the joints let go

The cable that pays the factor back followed a beam over a lost column as it sagged first into a bending mechanism and then into a cable. A beam held at its far ends gains resistance as it sags, so a sudden loss needs less than the factor of two a spring needs — on the bending plateau. Past one depth of sag the cable is a straight line again and the factor climbs back toward two. And the connections at the beam’s far ends are asked to turn through a tenth of a radian on the way, which ordinary connections cannot: good for 0.03 or 0.07 radians, they are exhausted at about 0.94 and 1.19 times the beam’s plastic collapse load, before the cable has formed.

That essay ended with the beam alone. A floor that loses a column has a beam each way crossing over it and a slab spanning between them, and the slab, held at its edges by the beams and the floor that did not lose anything, can hang as a membrane in two directions at once. Whether the slab’s membrane arrives before or after the beams’ connections run out — and so whether a floor’s robustness belongs to its steelwork or to its concrete — is the question.

A floor round a lost column #

The floor is the earlier essay’s beam, doubled: two beams of 8 m spans crossing over the lost interior column, each 600 mm deep with a plastic moment of 800 kN·m, each collapsing at 400 kN at the lost column — 800 kN for the pair. Their resistance as the lost column sags is the earlier essay’s curve: elastic, then the three-hinge mechanism of after the first yield on a plateau that the axial force lifts, then a cable, until their connections at the surrounding columns fracture.

Over the four bays round the lost column is a slab 16 m square, held at its edges by the floor beyond, with a mesh of 252 mm² a metre each way at 500 N/mm² — a standard welded fabric. Its bending capacity is small beside either mechanism here and is left out. What it has is its mesh, which can carry 126 kN per metre of slab in tension each way.

A square membrane hanging from fixed edges under a uniform load has a closed form, Pucher’s: its central sag is $0.0737\,qa^2/T$ under a tension $T$ the same each way. The load it holds is then

$$Q = q\,a^2 = 13.6\,T\,w_0,$$

the tension times the sag, whatever the slab’s size. A slab twice as large holds the same total load at the same sag and tension, because each square metre is held twice as far from its edges and its share of the sag is half as steep. The tension, though, is not free: it comes from the membrane’s stretch. A sag $w_0$ in the cosine shape stretches the slab by an average strain $\pi^2 w_0^2/8a^2$, and the mesh’s tension is that strain times its stiffness until the mesh yields.

The slab hangs only once the beams have sagged further than their joints can follow #

The slab’s curve starts flat and rises as the cube of the sag, because its tension grows as the square of the sag and the load as the tension times the sag. At 0.2 m it holds almost nothing. At 0.56 m, where connections good for 0.07 radians fracture, it holds 579 kN — less than the beams’ own collapse load of 800. Only at 0.72 m does its mesh yield and its curve become a straight line, holding 1,232 kN, and it goes on in a straight line until the mesh fractures at 1.44 m.

So the floor’s resistance is the beams’ rising curve, then a fall where their connections let go, then the slab alone. The membrane arrives after the joints have gone.

The rotation at which the membrane arrives has no span in it #

The mesh yields when the membrane’s average strain reaches $f_y/E$, at a sag of $(a/\pi)\sqrt{8f_y/E}$. The beams crossing over the lost column span half the slab each side, $L = a/2$, so their ends have turned through the sag over $L$:

$$\theta_y = \frac{2}{\pi}\sqrt{\frac{8f_y}{E}},$$

0.090 radians for mesh of 500 N/mm². The span has cancelled, and so has the mesh’s area: a heavier mesh carries more once it has yielded, but it yields at the same strain and so at the same sag. A floor’s slab arrives as a membrane at a rotation set by the steel of its mesh and nothing else — and 0.09 radians is beyond the 0.03 to 0.07 that ordinary beam-to-column connections can turn through before they fracture. Even mesh of 250 N/mm², which yields at 0.064, would need connections at the top of that range.

A bay that is not square changes the arithmetic but not its form. A membrane over a rectangle carries most of its load in the short direction, whose curvature is the steeper, and its mesh yields first along the short span; the beams that matter are then the ones spanning that way, and their ends turn through the same $(2/\pi)\sqrt{8f_y/E}$ when it happens. A longer bay is not a shallower membrane. It is the same membrane with more of its load arriving through the short direction.

That is why the two mechanisms do not add. The beams’ cable and the slab’s membrane both need large sag, but the beams’ sag is limited by their connections’ rotation and the slab’s resistance by its mesh’s strain, and for any ordinary floor the first limit is reached first.

When the joints let go, the slab catches the floor #

A column lost suddenly lets the floor’s weight fall until the work the floor’s resistance has done equals the work the weight has done, as the earlier essay found for the beam. Here the resistance falls when the connections fracture, but the work the beams did before they fractured stays done: it has been taken out of the falling floor as plastic work in the hinges and stretch in the cable. The slab then has to absorb the rest.

Below the beams’ collapse load the floor stops on the beams, with little help from the slab. Above it, with connections good for 0.07 radians, a load of 1.2 times the beams’ collapse load stops at 0.46 m with the connections intact. From 1.4 times the connections fracture on the way down and the floor falls on to its slab, which catches it 0.91 m down, and the last load the floor holds before the mesh fractures is 1.80 times. With connections good for only 0.03 radians they fracture at every load above the beams’ own capacity, and the slab alone catches 1.2 times the load 1.16 m down; the last load held is 1.50.

A metre of sag is the end of the floor as a floor. It is not the end of the building, which is what robustness is for: the structure that survives losing a member was a structure whose damage stayed local, not one that stayed in service.

What a connection would have to be #

The arithmetic also says what it would take for the two to add. A connection that could turn through 0.09 radians and go on carrying the beams’ cable would keep the beams working until the mesh yields, where the beams alone would be holding 2.25 times their collapse load and the slab another 1.54. Such connections exist — joints with long slotted holes that let the beam’s end slide before it bears, or end plates thin enough to fold — and the connection is not a point found that a joint’s rotation and its strength are two properties a designer can trade, not one. But they are designed for it, and an ordinary fin plate or flush end plate is not: its bolts reach their bearing limit, or its plate tears at its weld, well before.

So a robustness check that counts the slab’s membrane alongside the beams’ catenary is counting two things that cannot happen together in an ordinary frame. It may still be right about the total — the slab does catch the floor — but it is right for the wrong reason, and the deflection at which it is right is a metre rather than the half-metre it implies.

The floor’s own dynamic factor #

Slowly loaded, the floor with 0.07-radian connections holds 2,463 kN, 3.08 times the beams’ collapse load, at the moment its mesh fractures. Suddenly loaded it holds 1.81. The ratio of the two is a dynamic factor of 1.70 — close to the two that the cable that pays the factor back found the beam’s catenary returning to. The reason is the same and sharper: the slab’s resistance is a straight line from the origin once its mesh has yielded, a spring, and a spring loaded suddenly needs twice its static load; and on the way down to that spring the floor has passed through a fall in resistance, where the joints let go, which a static check never meets.

A static assessment of the floor would say it carries three times the beams’ collapse load. The floor falling on to its slab carries less than two. The difference is the energy the falling floor gathered between the joints’ fracture and the slab’s arrival — 0.56 m to 0.72 m in which the floor was carried by a membrane still too slack to stop it.

The slab carries little while the joints hold, and everything after #

The slab’s share is a step, not a ramp. While the joints hold it carries a twelfth of the load at the beams’ collapse load and a fifth at 1.2 times. When the joints fracture its share jumps to all of it. There is no load at which the two share the floor in comparable parts, because the deflections at which each works are a factor of two apart.

The force nobody put in the model found the same split inside a single slab strip, and the edge that cannot hold the arch what happens to it when the strip’s ends give: compressive membrane action at small deflection, while the strip arches between its restrained ends, and tensile catenary action at large deflection, with the two overlapping only in a narrow band. A floor over a lost column has the beams’ cable in the first role and the slab’s membrane in the second, and the band they overlap in is the band of rotations no connection reaches.

A floor’s robustness belongs to its joints until they break, and to its slab after #

The beams alone hold 0.94 times their collapse load with 0.03-radian connections and 1.19 with 0.07-radian ones — the earlier essay’s numbers. The mesh adds to that at every rotation, and it adds most where the connections are weakest: with 252 mm² a metre, 0.03-radian connections hold 1.50 times, a gain of 60 per cent, and 0.07-radian ones 1.81, a gain of 52. With 393 mm² a metre the floor holds more than twice the beams’ collapse load whatever its connections.

The lightest mesh drawn, 142 mm² a metre, adds nothing at 0.03 radians: its membrane, alone at a metre of sag, cannot catch what the beams have let fall before its mesh fractures. There is a threshold, and below it the slab is not a second line of defence at all.

So the answer to the earlier essay’s question is both, in sequence. A floor’s robustness belongs to its steelwork while its joints hold and to its concrete’s mesh after they have broken, and because the slab arrives late it is the mesh, not the joints, that decides how far beyond the beams’ capacity the floor can be loaded. A connection that turns further buys robustness gradually; a heavier mesh buys it in a step, by giving the floor something to fall on to.

That has a design consequence the robustness rules for steel frames do not usually state. They ask for tying forces in the beams and their connections, which is the first line of defence. The slab’s mesh is usually sized for shrinkage and fire, and its continuity across beams and round columns — laps, anchorage into the edge beams, mesh carried over the supports — is what decides whether the second line exists at all. A mesh that is lapped short over a beam is a membrane with a seam in it.

The membrane by hand #

The mesh carries $252 \times 500 = 126$ kN per metre each way. It yields at a strain of $500/200{,}000 = 0.0025$, which the cosine membrane’s stretch $\pi^2 w_0^2/8a^2$ reaches at

$$w_0 = \frac{a}{\pi}\sqrt{8 \times 0.0025} = \frac{16}{\pi} \times 0.141 = 0.72\ \text{m},$$

an end rotation of $0.72/8 = 0.090$ radians for the beams below it. The slab then holds $13.6 \times 126 \times 0.72 = 1{,}232$ kN, 1.54 times the two beams’ collapse load. At the 0.56 m where 0.07-radian connections fracture, the stretch is $0.56^2/0.72^2 = 0.61$ of the yield strain, the tension 76 kN/m, and the slab holds $13.6 \times 76 \times 0.56 = 579$ kN.

What the count leaves out #

The slab’s edges hold. The membrane hangs from the floor beyond the four bays, which must take its pull — a ring of compression in the slab round the damaged area, and tension in the edge beams’ connections at the surrounding columns. A floor at the edge of a building has no slab beyond on one side, and its membrane has nothing to pull against there.

The mesh’s average strain is its fracture. A slab hanging as a membrane cracks, and the mesh’s strain concentrates at the cracks, so a mesh whose bars could stretch 5 per cent may fracture at a much smaller average strain across the bay. The 1 per cent taken here is a choice; halving it halves the sag at which the slab can catch the floor, and the threshold mesh rises.

The beams and the slab are not connected. A composite floor’s studs tie the slab to the beams’ top flanges, so the slab’s membrane and the beams’ cable share their stretch along the beam lines, and the beams’ fracture is not quite the clean step drawn here.

The column vanishes in an instant. A column struck or buckled loses its load over some tens of milliseconds, which is not short against the floor’s own period once it has sagged into a cable and a membrane. A slower loss is partly a static one, and the floor’s dynamic factor falls from 1.70 towards one as the release slows; the sudden case is the bound, not the expectation.

The mesh’s two directions share nothing. A welded fabric’s bars cross at welds, and a membrane stretching both ways at once loads each bar along its length and each weld across it. The count treats the two directions as independent layers, which is right while the welds hold and optimistic where a bar’s crack runs across a weld.

The load is the floor’s own. A sudden loss is a release of the load the column was carrying, and that load is what the floor carries afterwards; a floor above that collapses on to it is a different event, an impact, which the factor of two belongs to one mode found depends on which mode is struck.

Still open: the floor at the edge of the building #

The slab here hangs from the floor all round it. Lose a column at a building’s edge and the slab over the two bays beside it has floor on three sides and the façade on the fourth, where the only thing to pull against is the edge beam — itself spanning over the lost column, and itself a cable. A membrane with one soft edge sags into a trough rather than a dish, and its tension along the soft edge is what the edge beam’s connections have to carry on top of their own. Whether an edge bay’s slab can catch anything once its edge beam’s joints have gone, or whether the edge column is the one whose loss the floor cannot survive, is what a determinate truss has no robustness at all would recognise as the question of the member with no alternative.