The day taken off the floor cycle

A concrete frame goes up at the speed of its slowest trade, and on most residential and office frames that trade is the slab. A floor a week is the rhythm a contractor reaches when nothing is pushed: the deck is struck, cleaned and flown up, the reinforcement fixed and the concrete poured, and seven days later the slab just cast is strong enough to be struck itself. Taking a day off that cycle is worth a great deal of money on a thirty-storey frame. Taking three is what a contractor with a jump-formed core, a table-form system and a heated deck tries for.

Every essay on this frame so far has cast a floor every seven days. The strength a slab had on the day measured the concrete; the frame loaded twice over before a month shared the wet slabs out between the levels; a snug backprop took load nobody gave it once the slabs crept; a jacked backprop moved the young slab’s load onto the floor below; and a preload turned out to be impossible to take back once it was in. This essay changes only the week. The question it answers is the one the preload left: whether a faster cycle makes a jacked backprop more useful or less, and whether a second level of backprops is then the only answer.

Four days and seven, and nothing else changed #

The frame is the one every earlier essay used: slabs 250 mm thick on a 7.5 m column grid, at 20 °C throughout, one level of shores under the newest slab and one level of backprops under the slab below that. On a cycle of c days, a slab is cast, carried on shores for c days, struck and backpropped, and carried on backprops for another c days until the slab two above it is cast and its backprops come out. The slabs creep by Eurocode 2’s creep function at their equivalent age, and the props are springs.

Nothing else is changed. The slab is the same thickness, the concrete the same strength class, the props the same stiffness and the temperature the same. A real four-day cycle almost always comes with something else — heated formwork, a richer mix, an accelerator — and each of those changes the strength curve, which is the very thing this essay is about. Holding them back is deliberate: the first question is what the days alone cost, and only then what a heater could buy back.

The two lines in the figure have the same shape. A slab struck under a wet slab takes its peak at once, sheds some of it into the backprops below as it creeps, and then takes a second jump when the slab two above is cast and it becomes the slab below. On a seven-day cycle the peak load on the struck slab is 1.52 times its own weight; on a four-day cycle, 1.52; on three days, 1.51. The second jump is 1.71, 1.70 and 1.69. The loads do not depend on the cycle. They depend on how many levels of props there are and how stiff the slabs are relative to the props, and neither is changed by casting faster. Creep has slightly less time to move load between the levels on a short cycle, which is why the numbers drift down in the second decimal, and it is not enough to matter.

What depends on the cycle is the slab’s strength when the load arrives. Struck at seven days, the slab has 78 per cent of its 28-day strength, and 1.52/0.78 is the 1.95 that every earlier essay found governing. Struck at four days, it has 66 per cent, and 1.52/0.66 is 2.29. The load did not change; the denominator did. Shortening the cycle from seven days to four costs the frame a sixth of its margin, and it costs it in the youngest slab, at the moment the next one is cast.

Which slab loses most #

There are two slabs in this frame that can govern. The struck slab is one cycle old and carries the wet slab above on its shores, with its own backprops beneath it. The slab below is two cycles old and carries what its backprops pass down. On a seven-day cycle the struck slab governs, at 1.95 against the slab below’s 1.83, and a jacked backprop is useful because it moves load from the first to the second until the two are level.

On a short cycle both are younger. Whether the preload becomes more or less useful depends on which of them a short cycle hurts more, and that is not a question about loads. It is a question about the shape of the strength curve.

Concrete gains strength fastest in its first days and more slowly after that, so a day taken off a young slab costs more than a day taken off an older one. On a seven-day cycle the struck slab has 0.78 of its 28-day strength and the slab below 0.90; the first has 86 per cent of the second’s strength. On a three-day cycle they have 0.60 and 0.75, and the first has 80 per cent of the second’s. Every day off the cycle widens the gap between the two slabs that can govern.

That is the answer to the question the preload left, and it can be read straight off this figure before any frame is analysed. A preload exists to move load from the weaker slab to the stronger one. A short cycle makes the struck slab weaker relative to the slab below, so there is more to be gained by moving load between them, and the preload worth applying is larger.

The opposite intuition is natural and wrong. It says that on a short cycle the slab below is young too, so loading it with a preload is riskier than on a week, and the jack should be eased off. The slab below is indeed weaker; at three days it has 0.75 of its strength rather than 0.90. But the slab the preload relieves has lost more, and the preload is set by the balance between them, not by either one alone.

The whole frame, against the cycle #

The hero figure puts the four arrangements a contractor can choose from on one set of axes. Each line is the largest use of strength of any slab in the frame — whichever slab governs — as the cycle is shortened from seven days to three.

On one level of snug backprops the worst use is 1.95 at seven days, 2.03 at six, 2.14 at five, 2.29 at four and 2.53 at three. The steps grow as the cycle shortens: a day off a seven-day cycle costs 0.08, a day off a four-day cycle costs 0.24, three times as much. That is the shape of the strength curve again. Near a week the curve is flattening, and a day either side changes little; near three days it is still steep, and a day is a large part of the strength the slab has.

Jacking the backprops to their best preload lowers every point, and lowers the short cycles most: by 0.08 at seven days and by 0.23 at three. A second level of backprops, put in snug, lowers the line by 0.17 at seven days and by 0.22 at three, nearly a constant amount, because a second level shares the young slab’s load with one more slab at every cycle length. A second level jacked to its own best preload does both, and is lowest everywhere, at 1.67 for a week and 1.99 at three days.

Read against the dotted line, the figure says what each arrangement can afford. Two levels jacked can run at three days and be scarcely worse than one level snug at seven. One level jacked crosses the dotted line between five and six days. Two levels snug cross it at five. The order of the four arrangements does not change with the cycle; the distance between them does.

Where the struck slab and the slab below meet #

The best preload on a week was 0.07 of a slab’s weight, the point at which the struck slab’s falling use and the slab below’s rising use cross. A short cycle moves the crossing, and the figure shows both crossings on one set of axes.

Snug, at zero preload, the struck slab on a week is at 1.95 and the slab below at 1.82; on four days, at 2.29 and 2.02. The gap is 0.13 on a week and 0.27 on four days. The preload closes it. Each tenth of a slab’s weight passed down lowers the struck slab’s use and raises the slab below’s, and the two lines meet where the gap is gone. A wider gap needs more preload to close: 0.07 on a week, 0.11 on four days.

The slopes of the two lines tell the rest. The struck slab’s use falls by the preload over the struck slab’s strength, and the slab below’s rises by the preload over its own. On a short cycle both strengths are lower, so both lines are steeper; the struck slab’s is steeper by more, because its strength is the lower of the two. A preload on a four-day cycle buys more relief for the struck slab per unit of extra load on the slab below than the same preload buys on a week. That is the mechanism under the first figure’s arithmetic, read off the frame.

At the crossing on four days both slabs are at 2.12. That is lower than the 2.29 a four-day frame gives on snug backprops, and still well above the 1.87 a jacked frame gives on a week. The preload has helped more than it did on a week — 0.17 against 0.08 — but it has not bought the week back.

The preload worth turning a collar for #

The best preload against the cycle, on one level and on two, is the plainest statement of the essay’s finding.

The best preload doubles as the cycle is cut from seven days to three, and what it saves nearly triples. On two levels it is larger still, because the young slab’s load is now shared with two slabs below rather than one, so passing a preload down costs each of them less. At three days on two levels it is 0.19 of a slab’s weight, and it saves 0.31 of the frame’s use of strength — a sixth of the whole.

This is the opposite of what the slab below’s age suggests and the same as what the ratio of strengths says. A contractor shortening the cycle who eases off the jack, on the reasoning that the slab below is now young and should not be loaded, is moving the wrong way. The slab below is young, but the slab above it is younger in the way that matters, and the preload is the one tool that trades load between them.

A preload cannot be taken back, and on a short cycle that matters more. The larger preload goes into the slab below the instant the collar is turned, and the slab below is at its worst within hours. Every argument that essay made about slackening holds on four days as it did on seven, with larger numbers: slackening a backprop jacked to 0.11 on four days releases its whole force onto a slab with two-thirds of its strength.

Two levels and a jack, against a week #

The last figure asks the question a programme asks. Taking the frame on a seven-day cycle and one level of snug backprops as the standard of safety — its worst use, 1.95, is what every earlier essay on this frame accepted — how short a cycle can each arrangement run at and be no worse?

One level of backprops jacked to its best preload can run at 5.7 days. A preload alone, on a frame with nothing else changed, buys 1.3 days — not quite a day and a half, at the price of a gang with a torque wrench and a stricter sequence. A second level of backprops, put in snug, buys two days: the frame can run at five. Two levels jacked buy 3.7 days, and the frame can run at 3.3.

So the question the preload left has two answers. A faster cycle makes a preload more useful, and on a frame where a second level of backprops is impossible — a floor below that must be handed over to the fit-out, or a frame with too few props on site — a preload is the only thing that buys any days at all. But it does not buy many. A four-day cycle on one level of backprops cannot be made as safe as a week by jacking; the best a preload can do there is 2.12, and that is above the week’s 1.95. Below about five and a half days, a frame that wants a week’s safety needs the second level. With it, the preload again becomes worth turning: two levels jacked reach four days with room to spare, at 1.86.

There is a cost on the other side of the ledger that the figure does not carry. A second level of backprops ties up a second floor below the deck, and the trades who would have moved onto that floor wait another cycle. On a four-day cycle that wait is four days rather than seven, and the frame is moving faster, so the floor is released sooner by the calendar even though it is held for one more cycle. The programme arithmetic is a contractor’s, and this frame’s arithmetic is only the half of it that concerns the concrete.

The two days the curve gives back #

A rule of thumb follows from the figures, and it is worth stating so it can be checked against a different concrete. On this frame a second level of backprops is worth two days of cycle, nearly exactly: two levels snug at five days give 1.95, the same as one level snug at seven. It is a coincidence of this concrete’s strength curve and this frame’s prop stiffness, not a law, and it will move with both. A concrete that gains strength faster in its first days — a Portland cement of a rapid-hardening class — flattens the curve near three days and makes every day off cheaper; a slag blend steepens it and makes them dearer.

The rule’s form is more durable than its number. Every arrangement of props sets the loads, and the strength curve sets what those loads cost on a given day, so the shortest safe cycle for a given arrangement is found by reading the curve at the strength the loads require. A frame’s designer who knows the peak load on the struck slab — 1.52 of its weight on one level, less on two — and the use of strength the frame must not exceed can read the shortest cycle straight off the strength curve: the cycle at which the slab has 1.52/1.95 = 0.78 of its 28-day strength. That is seven days for this concrete at 20 °C, which is why a week suits it.

Checking the four-day frame by hand #

The four-day slab’s peak use can be checked from two numbers. The load on the struck slab when the next slab is cast is 1.52 of its weight on either cycle: its own weight, plus the share of the wet slab above that its shores pass down, less what its backprops take. At four days and 20 °C the slab has 0.66 of its 28-day strength by the Eurocode 2 strength curve the first essay used, and 1.52/0.66 = 2.30, against the figure’s 2.29.

The best preload on four days can be checked from the crossing. Snug, the struck slab is at 2.29 and the slab below at 2.02, and their strengths are 0.66 and 0.80 of the 28-day strength. A preload p moves the first down by roughly p/0.66 and the second up by roughly p/0.80 — roughly, because creep takes back part of any preload over the cycle, as the jacked-backprop essay found. Setting 2.29 − 1.5p equal to 2.02 + 1.25p gives p = 0.10, against the figure’s 0.11. On seven days the same arithmetic gives 0.13/(1.28 + 1.11) = 0.05, against 0.07; creep’s share of the relief is larger on a week, because it has longer to act.

Where the four-day frame stops being this one #

The strength curve is Eurocode 2’s, at 20 °C, for one class of cement. A four-day cycle on site is rarely run on an unheated deck with a standard mix, and both the heater and the mix change the curve this essay depends on. The figures say what the days alone cost; they do not say what a heater buys back, and the model applies one temperature to every slab for its whole life, which a heated deck does not.

The creep at a day or two old is extrapolated. Eurocode 2’s creep function is calibrated on concrete loaded at a few days and older, and a three-day cycle loads concrete younger than its data. The loads at the peaks are set mostly by the stiffness of the props and slabs and only slightly by creep, so the peak uses are robust; the best preloads, which depend on how much of the preload creep takes back, are less so.

The slab’s strength is the only resistance. A use of strength above one means the slab carries more, relative to its strength that day, than its own weight does relative to its 28-day strength; whether that is safe depends on the margin between a slab’s self-weight and its design load, which is a design decision and not this frame’s. A young slab is also weaker in punching than in bending, and a punching check is made on a perimeter whose concrete contribution grows with the cube root of the strength — less steeply than the strength itself, so a young slab loses less in punching than in bending, and this essay’s comparison by strength is the conservative one for punching.

The sag a slab loaded young keeps #

They cannot show deflection, which is set by the slab’s stiffness rather than its strength and is less sensitive to age, nor the long-term sag a young slab carries for life after being loaded early — the deflection that arrives three years late is larger for a slab first loaded at three days than at seven, by the same creep function that moves the props. They cannot show the variability of a real cycle, where a pour slips a day and the next is rushed. And they cannot show the cost, which is what decides whether two days are worth a second level of backprops — only that, on this frame, two days is what a second level is worth.

Still open: the deck that is heated and the floor below that is not #

A heated deck brings the young slab to its striking strength sooner and leaves the slab below at the air’s temperature. That changes the ratio of strengths in the opposite direction to a short cycle: the struck slab becomes stronger relative to the slab below, the gap the preload closes narrows, and the best preload falls. Whether heating the deck on a four-day cycle restores the week’s balance between the two slabs — so that the preload a week needed is again the right one — or overshoots it, so that the slab below governs and a preload makes the frame worse, is the question this frame leaves. It turns on the strength a slab has at its equivalent age, with the young slab and the slab below each at its own temperature, and on the creep that belongs to each member at that temperature.