The floor that feeds two bays

The collector the slab does not need followed a floor’s storey force into a single short braced bay at one end of each brace line. The rule that sizes the collector along the line assumes the floor hands its shear to the line uniformly. A concrete slab does not: it is so stiff in shear that it sends most of its load straight into the bay, and the collector carries about a quarter of what the rule says. A steel deck, soft in shear, behaves as the rule supposes. That line had one bay, so every kilonewton of the line’s share went into it, and the only question was how it got there.

Many lines have two bays. A long building is braced at both ends of each long line, or has a core wall at one end and a braced frame at the other, or a wall where the stair is and a frame where the plan allows one. Then the floor’s shear has two places to go, and how it divides between them is a question about the floor, not only about the bays.

Two answers, both in the standards #

There are two ways of dividing a line’s force between its bays that every designer knows, and the design standards name them.

A rigid diaphragm moves as a block. The whole brace line moves together, each bay deflects by the same amount, and each takes a share in proportion to its own lateral stiffness. A core wall ten times as stiff as a braced frame on the same line takes ten-elevenths of the line’s force, and the frame one-eleventh: 9%.

A flexible diaphragm has no shear stiffness worth counting. It spans between the lines as a series of simple beams, and along each line it delivers the load to whichever bay is nearer: each bay takes its own tributary length. With a bay at each end of the line, each takes half.

Rigid by one code and not by the other found that the classification itself depends on which code is asked, and that real floors sit between the two. Here the floor is solved as what it is — a plate in plane stress, 48 by 24 m, carrying 40 kN per metre of span, 960 kN to each brace line, with a steel collector along each line and chords along its edges — and the two bays are springs, a braced frame of 200 kN/mm at one end of each line and a bay ten times as stiff at the other.

A stiff floor shares by stiffness #

The uncracked slab is close to the rigid floor. With the other bay ten times as stiff, it sends 16% of the line to the braced frame, against the rigid floor’s 9%. The deck is closer to halves: 37%, against 50%. The cracked slab is between them, at 30%.

So the intuition that a stiff floor sends its shear to the nearer bay is the wrong way round. A floor stiff in shear makes the line move as a block, and a line moving as a block shares by stiffness. The floor that delivers each half of the line to its nearer bay is the floor too soft to move the line as a block — the steel deck, and not quite even that, because the collector along the line is a stiff bar and does part of the moving for it.

The rigid-floor share is the lower bound on the soft bay’s force, and every real floor exceeds it. A design that sizes the braced frame by the rigid-floor share sizes it for 86 kN of a line that will put 155 into it under a perfect slab and 353 under a deck.

What the floor’s shear stiffness decides #

The three floors are three points on one curve.

The share falls steadily as the floor’s shear stiffness rises, over three orders of magnitude, from 37% at the deck’s stiffness to about 12% at three times the slab’s. The steepest part of the curve is between a cracked slab and an uncracked one. That is exactly the range a concrete floor moves through in an earthquake, or under a design wind that cracks it at its re-entrant corners, and the soft bay’s force doubles between the two ends.

The uncracked slab’s shear stiffness is 1,875 MN/m — a 150 mm slab with a shear modulus of 12.5 GPa. A slab cracked across its width keeps its shear stiffness only through aggregate interlock and the reinforcement crossing the cracks, a tenth or less of the uncracked figure. A profiled steel deck gets its in-plane shear stiffness from its fasteners and the warping of its profile, and a deck shear stiffness of the order of 20 MN/m is typical of fastened profiles before any concrete topping is cast — a hundredth of the slab’s.

The collector carries what the floor does not #

The force that does not go straight from the floor into a bay goes along the line, through the collector.

The collector’s force is largest at the inner edge of the stiff bay, where everything that the floor has delivered to the line between the bays, and is now on its way to the stiff bay, enters it. Under the slab that is 78 kN: the slab itself carries the transfer in shear, through its own depth, and the collector is hardly needed. Under the cracked slab it is 265 kN, and under the deck 339. The softer the floor, the more of the transfer the collector carries, the same finding as the single-bay line’s, and here it is the transfer between two bays rather than into one.

Between the bays the collector is in compression, pushing load from the soft bay’s end towards the stiff bay’s. Inside the soft bay’s length it is in tension, because the soft bay is shedding load to the stiff one through the line.

Tension or compression #

The sign of the collector’s force between the bays is set by which bay is the stiffer.

The collector pulls load towards whichever bay is stiffer: compression in the direction of the stiff bay, tension away from it. With equal bays it carries nothing at mid-line, because the floor’s halves go to their own bays. The size of the force at a given ratio depends on the floor, by an order of magnitude between the slab and the deck.

What does not happen is a change of sign with cracking alone. For a given pair of bays, cracking the slab moves it from the slab’s curve towards the deck’s, increasing the collector’s force several times over without reversing it. The collector’s force is reversed by a reversal of the storey force, which changes nothing in the bays’ relative stiffness and so reverses everything. A collector must be designed for tension and compression because the storey force reverses, not because the floor cracks.

A longer line, a stiffer collector #

Two more properties of the line move the share, in opposite directions.

The further apart the two bays, the more each one is fed by the floor around it. On a 16 m line both floors are close to sharing by stiffness, because the floor between the bays is a short, deep plate that cannot shear much. On a 56 m line the floor between the bays is a long one, which bends and shears in its own plane, and the deck gives the soft bay 44%, nearly half. A long building braced at its two ends shares its long-line force more nearly by halves than a short one, on the same floor, with the same bays.

A collector stiff enough ties the line into a bar that moves as one, and then the bays share by stiffness whatever the floor between the lines does. A collector of 25,000 mm² of steel brings the deck’s share down from 37% to 23%. For the slab, which already moves the line as a block, the collector changes almost nothing. This is the one property of the line a designer chooses outright, and it changes the answer only for a soft floor, where it decides how much of the floor’s job it takes over.

Why design takes the envelope #

The soft bay’s force is 86 kN by the rigid-floor rule, 480 kN by tributary halves, and 155 to 353 kN by the membrane, depending on the floor’s state. That spread is the reason a floor that is neither rigid nor flexible is often analysed both ways, with each element designed for the larger of its two results: the envelope of the two idealisations.

The membrane shows why both ends of the envelope are needed. The braced frame, the soft element, is governed by the flexible end: its force rises as the floor softens, and its design force is the deck’s or the cracked slab’s, not the uncracked slab’s. The stiff wall is governed by the rigid end: its force is largest when the floor is stiffest. A designer who analyses only one state sizes one of the two bays for the wrong force.

And the floor’s state is not fixed. A concrete floor that is uncracked under the design wind is cracked after a moderate earthquake, and the next storm finds its braced frames carrying twice what they carried in the last. The corner that moves most found the same dependence in plan torsion: what a floor’s stiffness does to the distribution of a lateral load is the whole question, and the floor’s stiffness is a property of its history.

The deck before its concrete #

A composite floor is all three floors drawn, at different times. While the building goes up, each floor is a bare profiled deck fixed to its beams, and the frame is braced against wind and erection loads through that deck. When the concrete is cast and cured the same floor becomes a slab, and a design earthquake or a severe storm later cracks it. A structure that was never complete carries the stresses of every stage it passed through. A floor’s sharing of its line force between two bays has no memory of that kind — each state shares as it is — but each state can govern a different element.

During construction the deck shares the line’s force nearly by halves, and the braced frame at the line’s end, often the first lateral element erected, carries 37% of whatever the wind puts on the line, four times its share in the finished building’s rigid-floor analysis. Erection bracing and temporary connections are sized for that stage, and the frame’s own design for the finished one; a frame whose connections were left to the finished-building analysis has been designed for a quarter of what the construction stage asked of it. Once the slab is cast, the same frame sees 16%. Then the slab cracks, and it sees 30%.

Two bays as a lateral system #

The bays here are springs, and in a real building they are not alike in kind. A core wall bends as a cantilever up the building’s height and a braced frame shears storey by storey, and how a tall building stands still depends on that difference: tied together at every floor, the wall holds the frame back near the top and the frame holds the wall back near the base, and the storey shear passes between them through the floors at every level. The floor’s sharing of its load between the bays is therefore not only a matter of this one storey’s force. Each floor is also the link that transfers the interaction force between a wall and a frame on the same line, and a soft floor transfers it less completely, so the wall and the frame drift more nearly as they would alone.

That makes the floor’s state matter twice. It sets how the storey force divides between the bays at each floor, and it sets how much the bays help one another up the height. A cracked slab moves both towards the flexible end.

The line, by hand #

Each brace line carries $40 \times 48/2 = 960$ kN. A rigid floor moves the line as one, so the two bays deflect together and share by stiffness: the braced frame takes $200/(200 + 2\,000) = 9.1\%$, 87 kN, and the wall 873. Each bay’s tributary half of a 24 m line is the bay’s own 6 m and half the 12 m between them, 12 m: 480 kN each.

The membrane’s 155 kN under the slab is 16%. The difference from the rigid floor, 68 kN, is the load the slab delivers to the frame’s end of the line that the slab’s own shear stiffness cannot carry 18 m along the line to the wall. Under the deck, 353 kN, the floor carries less of that transfer in shear and the collector more, and the frame keeps more of its tributary half. The collector’s largest force under the deck, 339 kN, can be checked the same way. If each bay were fed only by its own half of the line, the collector at the wall’s inner edge would carry the 6 m of line between it and mid-line, $960 \times 6/24 = 240$ kN. The frame takes 127 kN less than its half, and that passes along the collector to the wall: $240 + 127 = 367$ kN, against the membrane’s 339, because a deck still carries a little of the transfer in its own shear.

A membrane and two springs #

The bays are springs. A braced frame and a wall have a lateral stiffness at the floor that depends on the floors above and below and on the storey force’s distribution up the height; a single number per bay stands for all of that.

The floor is one material. A slab partly cracked, with cracked bands at its re-entrant corners and over its supports and uncracked elsewhere, is not one of the three floors drawn. Its share lies between them, and where the cracking is decides which way.

The load is uniform across the span. A storey force concentrated near one brace line — a heavy plant room, a mezzanine — loads that line’s floor unevenly, and the share at the near end of the line rises.

Both lines are identical. Each line has its soft bay at the same end, so the building has no plan torsion. A plan whose stiff bays are at opposite ends of the two lines twists, and the share then depends on the floor’s torsional stiffness as well, which is the plan’s own question more than the line’s.

Two bays on a line #

The single-bay line asked how the floor delivers its shear to the line. The two-bay line asks where along the line it goes, and the answer turns the usual intuition round. Stiffness attracts load, as the stiffest path takes the load in a structure with redundant paths, and a floor stiff in shear is what lets stiffness attract it: it makes the line one path. A floor soft in shear breaks the line into two paths, each fed by its own half of the floor, and stiffness then attracts load only as far as the collector can carry it.

The floor is a beam lying down, and a beam lying down across two supports of different stiffness shares its reaction by their stiffness only if the beam is stiff enough to hold its own shape over the distance between them. A deep slab is; a deck, over a long line, is not.

Still open: the floor whose two lines disagree #

Here the two brace lines are alike, each with its soft bay at the same end. A building whose cores sit off centre has them otherwise: a wall at the north end of the west line and at the south end of the east one. The floor then has to twist in its own plane to deliver its load, and how much it twists depends on its shear stiffness exactly as the share along each line did. A rigid floor resists the twist and puts it into the bays as plan torsion. A flexible floor delivers each line’s load locally and the building does not twist at all. Whether a semi-rigid floor makes such a building twist more or less than either idealisation, and whether the envelope of the two still covers the bays at the corners, is the question that a pair of disagreeing lines puts to the floor between them.