The backprop that takes load nobody gave it

The arithmetic of a propped frame rests on two assumptions, and it says so: the props are rigid, and a load is shared among the slabs it reaches at the moment it arrives and never again. On those terms the result is clean. A shore passes weight down; a backprop, put in snug after the slab above it is struck, shares only what arrives later. One level of shores over one level of backprops gives every slab a peak of 1.5 times its own weight; two levels of backprops, 1.33; three, 1.25.

Both assumptions are about time and stiffness, and a concrete slab a week old is the least time-independent thing in a building. Its stiffness is still rising, it creeps under a new load faster than at any later age, and the props between it and the slab below are steel tubes a few metres long — springs, not walls. The essay that set out the arithmetic ended by asking whether a stack of soft backprops loses what the snug installation bought, and whether the worst load then moves back towards the struck slab.

That question has three parts, and they turn out to have three different answers: the props’ stiffness, which barely matters; the slabs’ creep, which matters a good deal and moves the worst load down; and the slack a backprop goes in with, which matters most and moves the worst load up.

Two springs, and how stiff each is #

Treat each slab as a spring and each level of props as another spring between them. The slab’s stiffness per square metre is the load it carries over the deflection it takes. For a flat slab on a square grid of columns the deflection is not uniform — zero at the columns, largest in the middle of each panel — but a grid of props pushes on the whole panel, so what matters is the average. For an elastic plate on a grid of point supports that average has a closed form: 0.003866·qL⁴/D, where q is the load, L the column spacing and D the plate’s bending stiffness, and the coefficient is a lattice sum, $\sum 1/(m^2 + n^2)^2$ over every pair of whole numbers except zero, divided by (2π)⁴.

For a 250 mm slab on a 7.5 m grid, in concrete with a 28-day modulus of 33 kN/mm², the slab’s own weight of 6.25 kPa deflects it by 1.71 mm on average. Its stiffness is therefore 3.66 kN per square metre for each millimetre of average deflection. The prop is a standard adjustable steel prop: an inner tube of 48 mm and an outer of 60 mm, 3.2 mm walls, about 2.9 m long, so about 36 kN/mm axially. On a 1.2 m grid that is 25 kN per square metre per millimetre.

A steel backprop is not thousands of times stiffer than the slab it links, but 6.8 times. The slab is a broad thin plate spanning 7.5 m; the prop is a short thick tube; per square metre they are within an order of magnitude of each other. The suspicion behind the question was right in kind.

It is not right in degree. Two slabs linked by a spring share a load placed on the upper one in a ratio that depends on the spring’s stiffness against theirs. Call the prop’s stiffness α in units of the slab’s, and the two slabs’ stiffnesses $E_1$ and $E_2$ as fractions of their 28-day values. The upper slab takes $E_1(E_2 + \alpha)/(E_1E_2 + \alpha(E_1 + E_2))$ of the load. With rigid props that is $E_1/(E_1 + E_2)$: the seven-day slab, at 93 per cent of its final stiffness, against the fourteen-day one at 97, takes 49 per cent. With α of 6.8 it takes 52 per cent. Halve the prop’s stiffness, for timber packers under its base plate or a bearer that crushes a little, and it takes 55.

So the struck slab carries 1.52 times its weight rather than 1.49. That is the cost of the props being springs, and it is three per cent. The curve shows why it is small: it is flat to the right of about 5, and steel props sit on the flat part. A prop half as stiff again would hardly change anything; a prop five times softer than steel would start to.

The week after the cast #

The second assumption is that loads are shared once. A snug backprop goes in after the slab above has been struck and carries nothing; the next floor’s weight then arrives and is shared; and from then until the backprop is removed, nothing is supposed to change.

Concrete creeps under sustained load, and the slab just struck has just been given a sustained load: its own weight, which a moment earlier was standing in the shores beneath it. It took that weight at seven days old, which is the age at which concrete creeps fastest, and it took it a moment before the backprop went in under it. Everything the slab does under its own weight from that moment on, it does against the backprop.

The model behind the curves superposes every load increment a slab has received, each creeping from its own day with the coefficient of the European concrete code’s creep annex at the equivalent age when it arrived. The seven-day slab under its own weight, at the humidity and size of an indoor slab, has a creep coefficient that reaches 0.43 within a day and 0.77 within the week; the fourteen-day slab below, under loads that mostly arrived a week earlier, creeps far more slowly, because creep is steepest just after a load arrives. The upper slab wants to sag further than the lower one, the backprop will not let it, and the backprop’s force rises until the two are compatible again.

The backprop that went in snug carries 0.71 of a slab’s weight by the end of the week, against the 0.48 the arithmetic gives it, and the slab below carries 1.71 times its weight rather than 1.48. Nobody put that load into the backprop. The slab above did, by creeping.

Two checks run through the figure. What the lower slab gains is exactly what the backprop gains, to the last digit, because the lower slab has nothing else under it. And what the upper slab loses is exactly the same amount, because creep moves load and creates none. The total in the two slabs is 3.00 slab weights before and after, their own two plus the wet slab above.

The rate matters as much as the size. Four fifths of the transfer happens in the first day, which is a property of how Eurocode 2’s creep function starts — steeply, as a power of the time under load — and it means the redistribution is not a slow drift to be caught by an inspection later in the week. It has mostly happened by the time anyone could look.

The heaviest day moves down a floor #

What creep does to a single slab’s life is the clearest way to see where the worst load ends up.

Read the dashed history first. On one level of shores and one of backprops, every slab carries the peak of its life twice: once when it has just been struck and the next floor is cast on it, and once a week later, when it is the slab below and the next floor again shares onto it. In the arithmetic the two episodes are equal, near 1.5 each.

Creep makes them unequal in a definite direction. In the first, the slab is the one creeping, and it sheds load all week — from 1.52 down to 1.29. In the second, the slab is a week older and creeping slowly, while the slab above it is the young one creeping onto it, and it gains all week — from 1.48 up to 1.71. The heaviest load moves from the youngest slab to the one below it, and grows by 15 per cent on the way.

That direction matters, because the slab it moves to is older. A slab carrying 1.71 w at 21 days has 96 per cent of its 28-day strength, and its load over its strength that day is 1.77. The struck slab at the moment of the cast carries 1.52 w with 78 per cent of its strength: a ratio of 1.95. Creep raises the heaviest load in the stack and leaves the heaviest use of strength exactly where it was, on the seven-day slab at the instant the next floor arrives — because creep needs time, and that instant has none. The struck slab’s worst moment is decided before creep can touch it, and the arithmetic’s number for it survives.

The support that had no moment when it was cast is the same mechanism in a finished structure: a concrete frame whose arrangement changes after a load is applied moves that load, by creep, towards where it would have gone had the final arrangement been there from the start. Here the arrangement changes every week, and each backprop goes in a moment after a load it would have shared had it been there first. It is also the same mechanism as the deflection that arrives years late, run in a week rather than a decade: a sustained load on young concrete, and a structure that resists the creep it causes.

What the extra levels were buying #

The arithmetic says that each added level of backprops spreads the next wet slab over one more floor, so the worst load falls as (n + 1)/n: 1.5, 1.33, 1.25. That is the whole case for more than one level.

The springs alone take a little off the gain: with steel props the deeper levels share less effectively, because each spring in a chain lets the slab above it carry a little more than its share, and the soft link compounds down the stack. Creep takes most of the rest. Each newly struck slab creeps onto the stack beneath it under its own weight, and with more levels of backprops there are more young slabs in the stack each doing so. Going from one level of backprops to three takes 0.25 w off the worst slab by the arithmetic and 0.17 w once the slabs creep, and the third level buys 0.06.

Two consequences follow. A third level of backprops is close to worthless on this cycle: it takes the worst load from 1.59 to 1.53, on slabs five weeks old with all their strength. And the worst load in every scheme with backprops now sits on the oldest slab in the stack rather than the youngest, which is the slab a temporary works design would least expect to be governing. It still carries less than a one-level shore scheme puts on a one-week slab, but the margin is no longer the one the method statement shows — and the method statement is the only drawing a structure that was never complete has of this stage.

A millimetre of slack #

Everything so far assumed that snug means snug: the backprop touches both slabs and carries nothing, and any movement that closes it loads it at once. A backprop is in fact screwed up by hand until it is tight, against a timber head plate or a packer, and what tight means is a matter of feel. A gap of a few tenths of a millimetre between the prop and the slab, or in the pin and collar of the prop itself, or in the bearer crushing as it takes up, is a backprop that engages late.

A backprop that engages late is, for the load that arrives before it engages, a backprop that was not there. The struck slab carries the next floor’s weight alone until it has deflected through the slack, and only the rest is shared.

The scale of the slack that matters is set by the slab’s deflection, and that is small. Under the next floor’s weight a seven-day slab deflects about 1.8 mm on average, so a slack of a millimetre lets it carry more than half that weight before the backprop takes any. At 0.5 mm the worst use of strength rises from 1.95 to 2.12; at 1 mm, to 2.29, which is what two levels of shores give with no backprops at all; at 2 mm the backprop never engages under the cast and the scheme is one level of shores, 2.57.

A millimetre of slack gives back half of what a backprop saves, and it gives it back where it costs most. The load returns to the struck slab, the youngest in the stack, at the instant of the cast — exactly the position that creep had left alone. That is the answer to the second half of the original question. Soft backprops do not move the worst load down towards the struck slab; it is already there. Late backprops push more onto it.

Creep then partly closes the slack, and the history figure shows how: the slab with a millimetre of slack starts its week at 1.78 w, and within a day it has crept through what was left of the gap and is handing load down again. But the peak happened at the cast, and nothing that happens afterwards reduces a peak that has already been carried.

In a cold week the whole picture lifts, because the struck slab has less of its strength, and the slack costs proportionately the same. At 5 °C the snug scheme already uses its struck slab at 2.44 times what a 28-day slab would carry at the same proportion, and half a millimetre of slack adds another 0.19 to it. Against a factored design load of 2.27 slab weights for a typical office floor, the snug scheme in winter is past it before any slack is counted; a punching check at the columns, which falls with the concrete’s strength, is where that bites.

What a specification can say #

The three parts come out as three different kinds of instruction.

The prop’s stiffness needs no specification. A standard steel prop on a normal grid is six to seven times the slab’s stiffness per square metre, and the curve is flat there. A prop with soft packing under it is half that, and still costs only a few hundredths of a slab’s weight. Nothing is gained by asking for stiffer props.

Creep needs no specification either, because nothing can be done about it, but it does need to be counted. The heaviest load in a backpropped stack is about 1.7 slab weights on the slab below the struck one, not 1.5, and it arrives two or three weeks after casting rather than one. Where the slab’s design is checked against the construction stage, that is the load case, and on a stack of two or three levels of backprops the extra levels should be credited with roughly two thirds of what the arithmetic says.

The slack needs a number, and the number is small. To keep the struck slab within a tenth of a slab’s weight of what a snug backprop gives it, the slack must be under about 0.3 mm — a fifth of the slab’s own deflection under its weight. That is not a tolerance a hand-tightened prop meets by feel. It is met by tightening the backprops before the formwork above the struck slab is fully removed, so that the slab’s own sag closes them, or by preloading them with wedges or jacks to a measured force. The second converts a backprop from a support that shares what arrives later into one that already carries some of what is there, and moves the scheme back towards shores — the trade the striking-age arithmetic makes from the other side.

What the springs and the creep leave out #

Cracking. The slab’s stiffness is the uncracked plate’s, and a cracked slab is a softer one, if not as soft as its cracked section says. A seven-day slab carrying 1.5 times its weight is very likely cracked over the columns, which makes it softer, deflect more, and close any slack sooner — and makes the props relatively stiffer. Both effects are in the direction of the rigid-prop arithmetic, but neither is in the numbers here.

A uniform plate. The props push on a slab whose deflection varies from zero at the columns to its maximum mid-panel. Using the panel’s average deflection treats every prop as carrying the same share; in practice the props near the middle of a panel carry more, which is a question of how a single prop is loaded rather than how a whole level is.

The creep function at early ages. Eurocode 2’s creep coefficient is fitted to tests loaded at days and measured over weeks and years. Its behaviour within the first day of loading — where four fifths of the transfer happens here — is an extrapolation of a power law, and the transfer could be faster or slower than drawn. The total by the end of the week depends much less on that.

Shrinkage and temperature. A slab drying from its faces shortens and curls, and a slab warmed by its own hydration then cools; both move it against props that will not move with it. Neither is a load in the sense used here, but each changes the gap a backprop sees.

Still open: whether a preloaded backprop keeps its preload #

A backprop jacked to a measured force at installation is the obvious cure for slack, and it introduces a load path that the snug arithmetic does not have: some of the struck slab’s own weight goes directly into the slab below from the moment of striking. The creep shown here then works on a different starting point. The struck slab creeps under less of its own weight, because the preload carries part of it, while the slab below creeps under more. Whether a preload of a third or a half of the slab’s weight survives the first week — or is shed back by the lower slab’s creep, so that the slab above ends up carrying what it would have carried without it — is the question that decides whether preloading is a design choice or only an installation tolerance.