Loaded twice over before it is a month old

The strength it had on the day argued that the programme of a concrete frame is itself a load case: formwork is struck at three days, the next storey is cast at seven, and the strength that matters is the strength the concrete had when each of those loads arrived, not the 28-day number it was specified by. It named one case and settled it in a paragraph: “a slab struck early is propped from the floors below, and how much load it carries depends on the relative stiffnesses of a series of slabs of different ages” — and because young concrete is nearly as stiff as old, “the load shares out nearly equally between the propped floors”.

That is true of the share and says nothing about the total. This essay follows the total, floor by floor, through a building going up a storey a cycle, and finds that the heaviest load most floors ever carry arrives before they are a month old.

Two assumptions, and a slab’s weight as the unit #

The arithmetic is Grundy and Kabaila’s, from 1963, and it rests on two assumptions. The props are rigid, so slabs connected by props deflect together and share any load that arrives after the connection is made in proportion to their stiffness. And the ground is rigid, so props standing on it pass their load straight through. Loads are measured in units of one slab’s own weight, $w$; a 250 mm slab weighs 6.25 kPa and every number below scales with it.

The sequence is the one on any concrete frame. Floor $k$ is cast on formwork held up by shores standing on floor $k-1$. After the cycle — seven days here — the lowest level of shores is taken out and moved to the top to hold up the formwork for floor $k+1$. With one level of shores, the shores under the wet slab stand on the slab below, which carries them alone. With several levels, the shores stand on shores, and the whole stack down to the lowest level shares the weight.

The two stacks are built from the same number of props and carry the same wet slab, and their loads are different. On the left, the slab at the base of the two-level stack carries 2.02 times its own weight at fourteen days old: its own weight, plus the wet slab above almost entirely, handed down two storeys of shores. On the right, the same two storeys share the wet slab evenly and each carries one and a half times its weight. The whole of this essay is the difference between those two pictures.

One level of shores: twice its weight at a week old #

With a single level of shores, the arithmetic needs no stiffness at all. The slab below the wet slab is the only thing the shores stand on, so it carries the wet slab’s weight on top of its own: twice its own weight, at the age of one cycle. At seven days, at 20 °C, a normal concrete has 78 per cent of its 28-day strength.

That load is large by any measure that matters to the slab’s design. A 250 mm office floor with 1.5 kPa of finishes and 2.5 kPa of imposed load is designed for 1.64 times its own weight in service and 2.27 times factored. The construction load on a one-week-old slab exceeds the whole service load it will ever carry in use, and comes within an eighth of its factored design load — unfactored, on concrete with three quarters of its strength.

Followed through its life, the sixth floor’s load is a short, sharp episode. It is cast and carries nothing, its weight standing in the shores below it. When the next floor is cast on top of it, it takes that floor’s weight, the peak; when its own shores are removed, the load it was holding up through them lands on it, and when the shores above it are removed it is unloaded back to its own weight, which it will carry, with finishes and occupants added, for fifty years. The peak of that history is in its first month, and it is the peak of its whole life for every arrangement except the lightest.

More props, heavier slabs #

The obvious remedy is more props: two or three levels of shores, so that the wet slab’s weight is spread over more floors. It does spread the weight. It does not lower the peak.

With the extra levels as shores the worst load rises: 2.00 on one level, 2.25 on two, 2.37 on three, 2.44 on four — Grundy and Kabaila’s own numbers. With the extra levels as backprops it falls as $(n+1)/n$. The props are identical; what differs is when they entered the load path.

A shore goes in under the formwork before the concrete is poured. It is in the load path when the wet weight arrives and carries all of it, and when the level below it is removed, everything it was carrying — including the weight of every slab it held up while they were wet — lands on the stack beneath. Each level of shores is a column of stored weight, and the stack’s base slab receives it. A backprop goes in snug after the slab above it has been struck and is already carrying its own weight. It carries nothing at first and only a share of what arrives later — the next wet slab’s weight, divided among the linked floors. A shore passes weight down; a backprop only shares what comes after it. The conservation argument is short: when floor $k$ is cast on one level of shores over $n-1$ levels of backprops, the $n$ linked slabs each already carry their own weight and share one more, so each carries $1 + 1/n$.

The shores do buy something, and the next figure shows what: with more levels the peak arrives on older concrete. Two levels peak at fourteen days and three at twenty-one, when the concrete has 90 and 96 per cent of its strength, against 78 per cent at seven. That is the trade a shoring scheme actually makes, and it is a weak one: 2.25 on 90 per cent strength is 2.50 of a 28-day slab’s worth, and 2.00 on 78 per cent is 2.57.

The worst floor is the third #

With shores, the first floors are not typical. The shores under floor one stand on the ground, and the ground, being rigid, takes the whole of every wet slab while the stack reaches it; the first floor therefore carries little. The third floor, with three levels, is the first slab to become the base of a full stack that no longer reaches the ground, and it collects the stored weight of the stack at once: 2.37 times its weight. After that the peaks oscillate and settle to 2.00, the steady state that every number of shore levels reaches. So the classical peak of 2.37 is a start-up effect, and a start-up effect on the floors least likely to be inspected — the ones straight out of the ground, cast while the site was still being organised.

With backprops the floors settle at once. The first floor is struck and backpropped like every other, and every floor’s history is the same.

The same load on younger concrete #

Two things decide how hard that load bites: how old the concrete is when it arrives, and how fast the concrete gains strength at the site’s temperature. The first is the floor cycle. The second is the maturity argument of the earlier essay: concrete curing at 5 °C gains strength at about half the rate it does at 20 °C, so a seven-day cycle in winter meets the load with three-and-a-half-day concrete.

The ratio is the load a 28-day slab would have to carry to be used as highly as the young one is. On a seven-day cycle in summer, with two levels of shores, it is 2.50 — above the slab’s factored design load of 2.27. On a three-day cycle, 3.01. In a cold week on a three-day cycle, 3.83. A fast programme in winter can ask a floor for one and two-thirds times its factored design load, in the concrete it has.

Backprops move every curve down, but the temperature curve moves them back. One level of shores and two of backprops on a three-day cycle at 5 °C gives 3.14, because the struck slab, which takes its own weight the moment its formwork comes out, is then less than two days old in equivalent age and has 42 per cent of its strength. Striking early is the other half of the arithmetic: backpropping lowers the peak by taking weight off the shores, and it does that by putting the slab’s own weight onto the slab the day it is struck. The striking age is then the governing check, and it is the one the earlier essay’s striking criterion exists for.

Which strength is the right denominator is not settled by this ratio. A slab’s flexural capacity is set mostly by its reinforcement and hardly depends on the concrete’s age; its punching resistance at a column goes as roughly the cube root of the concrete’s strength, and a check made on a perimeter is exactly the one that a flat slab under construction loads fails. The load is the certain part of the picture and the resistance the uncertain part, which is the wrong way round from normal design.

Why the stiffness barely matters #

The earlier essay said the stiffnesses of slabs of different ages are close enough that the load shares out nearly equally. The method here can test that directly, by weighting every share by each slab’s modulus at its age instead of splitting it equally.

It holds. At seven days the modulus is 93 per cent of its 28-day value, because stiffness follows strength only to the power 0.3, and weighting by it moves the worst floor by less than three per cent. The concrete’s age barely changes where the load goes; it changes only how well the concrete carries it. The classical method with equal shares is as good as the weighted one, and the real uncertainties are elsewhere — in the props, which are not rigid, and in the ground or foundation under the first stack, which is not either.

The one load that is certainly all there #

The comparison with the design load flatters the design load, and it is worth saying why. The imposed load a floor is designed for is a fifty-year extreme — the office packed with filing cabinets, the crowd at the party — and on any given day most of it is absent; the load that is never all there at once is the essay about how little of it a large floor ever carries. The factor on it covers the chance that the extreme is worse than assumed. The construction load is not an extreme. The weight of the wet slab above is certainly there, in full, on the day the programme says, and its only uncertainty is whether the slab was poured thicker than drawn, which it usually is by a few per cent.

So a construction peak of 2.0 is not “a little under the factored design load of 2.27”. It is a certain load of 2.0 against a factored combination that, in service, will almost never be approached, applied to concrete that has three quarters of its strength. The honest comparison is between the construction peak and the load the slab will actually see in use, around 1.2 to 1.4 times its weight in an office, and by that measure the week a floor is used as the base of a propping stack is the most heavily loaded week of its life, by a wide margin.

It is also a load nobody on the slab’s design team chose. The number of levels of props, whether they are shores or backprops, the floor cycle and the striking age are the temporary works designer’s decisions, taken to suit a programme, and the permanent works designer usually sees them as a method statement to approve rather than a load case to compute. That is the general condition of a structure that was never complete until it was finished: every stage of it is a different structure, and the stages are designed by different people. The propped frame is the concrete counterpart of the steel frame whose most dangerous day is before its bracing is in — a temporary condition with no drawing of its own.

A stack by hand #

The steady-state numbers can be checked with nothing but equilibrium. With one level of shores, the slab under the wet slab carries its own weight and the wet slab’s: $1 + 1 = 2$. With one level of shores and one of backprops, the struck slab carries its own weight, 1; the next wet slab adds 1, shared equally between the two backpropped slabs because they deflect together: $1 + \tfrac{1}{2} = 1.5$. With two of backprops, $1 + \tfrac{1}{3} = 1.33$.

With two levels of shores the start is where the 2.25 comes from, and it is worth following once. Floor 1 is cast on shores on the ground, and floor 2 on shores on floor 1, which still stands on its own shores; both floors carry nothing, and the ground-level shores carry both, 2 slab weights. At day 21 the ground-level shores come out: floors 1 and 2, linked by the shores between them, now hold their own two weights and share them equally, 1 each. Floor 3 is cast on shores on floor 2, and its weight is shared by the same two floors: 1.5 each. At day 28 the shores between floors 2 and 1 come out; they were carrying 0.5, so floor 1 falls to its own weight, 1.0, and the 0.5 lands on floors 2 and 3, now linked, a quarter each: floor 2 goes to 1.75, floor 3 to 0.25. Floor 4 is cast on floor 3 and its weight is shared by floors 2 and 3: floor 2 reaches $1.75 + 0.5 = 2.25$. That is the whole method — rigid props, equal shares and careful counting — and the 2.25 is nothing but a floor that received its partner’s weight twice before it was allowed to hand any of it on.

What the model assumes #

The props are rigid. Real props shorten under load, and a stack of props with slabs between them behaves like springs in series, the same arithmetic as a beam carried on beams: stiff props share more like rigid ones, soft props let each slab take more of its own load before the props engage. The rigid assumption moves load down the stack and is the pessimistic direction for the base slab, which is why Grundy and Kabaila’s peaks are regarded as upper bounds and refined methods find a few per cent less.

Every slab is the same. A transfer floor, a thicker plant floor or a floor with a different span changes the stiffness shares, and a stiff floor in the stack attracts load exactly as the stiffest path takes the load says it should — which is sometimes used on purpose, by backpropping through to a thick podium slab.

The props line up. A prop under a slab that is not directly over a prop on the slab below loads the lower slab in bending at a point it was not designed for, and a slab that is propped near its columns shares differently from one propped at mid-span. The method treats each floor as one spring.

Construction live load is left out, which it should not be. Workers, equipment and stored materials on the slab being cast are typically a sixth to a third of a slab’s weight, and the whole of it goes down the shores with the wet concrete; with a quarter of a slab’s weight on the wet slab, two levels of shores peak at 2.38 rather than 2.25.

What the pictures cannot show #

That the peak load is short and the damage it does is not. A slab loaded to twice its weight at seven days cracks at a lower moment than a 28-day slab would, and its cracks do not close when the load comes off; its long-term deflection starts from a cracked stiffness and arrives years later as a sag nobody can trace to a week of construction. The creep coefficient of concrete loaded at seven days is also larger than at twenty-eight, so the same construction load leaves more permanent deflection on a young slab than an old one. None of that is a strength failure, and all of it is paid for in service by a floor that is strong and unusable.

Nor can they show the day it goes wrong. The recorded collapses of concrete frames under construction are, with very few exceptions, collapses of the propping system or of a slab struck too early in cold weather — the two cases this essay’s last figures are about.

Still open: the props that are springs #

Every number here assumes rigid props. A backprop is a steel prop a few metres long at a small fraction of its capacity, and its shortening under load is comparable to the deflection of a slab spanning between columns, so the stack is really a column of springs alternating with plates. Whether a stack of soft backprops loses the advantage the snug installation bought — because a soft backprop engages late, as if it had been installed after the load arrived — or keeps it, and whether the worst floor then moves back down the stack towards the struck slab, is the question that turns this arithmetic into a specification for how stiff a backprop must be.