The slab that shrinks away from the wall

The prop that holds what the roller left cast a basement’s ground-floor slab before the wall was backfilled, so that the slab propped the top of the wall while the fill was compacted against it. The wall bowed by a sixth of a millimetre and kept nearly all of the roller’s pressure; the slab carried 22.6 kN a metre, nearly four times what an active-pressure design gives it; and the stem’s base moment fell to a third of what it would have been as a free cantilever. It looked like a gain bought with a stiffer prop.

That essay left one movement out, and it is the slab’s own. A ground-floor slab is cast weeks before the backfill and dries for years afterwards. Every millimetre it shortens moves its ends away from the walls it props, toward its middle. The stem’s top was held still while the fill was placed; now the support that held it is walking away. This essay follows what the stem does as its prop leaves, and what happens if the slab is tied to the wall so that it cannot.

The same stem, with its support moving #

The stem, the fill and the roller are the earlier essays’: a 3 m reinforced concrete stem of bending stiffness $67\,500\ \mathrm{kN\,m^2}$ a metre, fixed at its base, backfilled in 250 mm lifts with a 60 kN-a-metre roller over sand with a friction angle of 30°. Each lift locks in a residual pressure where it beats the pressure the stem already feels, and a locked residual relaxes toward active pressure in proportion to how far the stem moves at that depth afterwards, completely after 3 mm. The slab is cast first and holds the stem’s top while the fill goes up.

The new step comes after the last lift. The slab’s end moves away from the fill by a given amount. A slab that only bears on the stem pushes on it only while the stem’s top would otherwise move further than the slab’s end; once the slab has shrunk past that, the two part and the slab carries nothing. A slab tied to the stem — dowelled, or cast with the wall’s starter bars running into it — pulls the stem’s top with it. In both cases the stem moves, the residual locked behind it relaxes where it moves, and the pressure and the deflection are solved together again, exactly as they were between lifts.

The hero figure is the stem at the three states that matter: propped as filled, slack, and tied after 4.5 mm. Propped, the moment changes sign down the stem, with 15 kN·m a metre of span moment at about a metre’s depth and 27 at the base. Slack, the span moment is gone and the base carries 70: the stem has become what it would have been without the slab, with a little less pressure on it because the fill has relaxed as it leaned. Tied, the base carries 120.

Two millimetres #

How much shrinkage it takes is the first number, and it is small.

The prop’s 22.6 kN a metre falls by about 10.9 kN for every millimetre the slab shrinks, and it is gone at 2.08 mm. Over the same two millimetres the base moment climbs in a straight line from 26.6 to 69.6 kN·m a metre. There is nothing gradual about the arithmetic: the stem is stiff, so a small movement of its top takes a large force to make, and a prop that has moved 2 mm has given up all of the force it held.

The 2.08 mm is the distance the stem’s top would lean if the slab were not there, under the pressure the fill exerts once the stem has leaned that far. It is less than the 2.6 mm the same stem leaned as a cantilever in the stem that bends away from the roller, and the reason is the order of events. The cantilever leaned while the lifts were still going in, so each new lift locked its residual against a stem that had already moved, and only the movement after that relaxed it. The propped stem’s residual was every bit of it locked against a stem that had not moved at all, and the whole of the release counts against it: the same fill, released at once, relaxes more and pushes less.

A bearing slab has nothing more to give after 2.08 mm, and the bearing line is flat from there. The slab goes on shrinking and the gap between it and the stem opens, but the stem’s moment is set by the fill alone. The propped wall has become a cantilever, and it was not designed as one. Its base moment is 2.6 times the propped value, and nearly six times the 12 kN·m a metre a propped cantilever under active pressure would carry — the number an ordinary design for a propped basement wall would start from.

Where the roller’s pressure goes #

The stem that moves away from the fill relieves some of the pressure on it, and it relieves it near the top first.

The roller’s residual sits highest in the fill, from half a metre down, where each lift’s compaction pressure beats the weight of the fill above it. That is also where a stem fixed at its base moves most when its top is released, so it is where the residual relaxes first. With the slab slack the stem’s top has moved 2.08 mm, and the residual half a metre down has fallen from 25 kPa to 14. With a tied slab pulling the top 4.5 mm, the top half-metre is back at active pressure and the residual below it has largely gone.

The thrust falls, from 67.4 kN a metre propped to 56.0 slack and 44.6 tied, and a reading of that alone would say the wall is better off. It is not, because a moment depends on where the support is as much as on how hard the load pushes. The propped stem carried its thrust between two supports, the base and the slab, and the slab took a third of it. The slack stem carries a smaller thrust on one support, and its base takes all of it, on a lever arm that runs from the base to wherever the pressure acts. Less load on a worse structure is a larger moment. That is the trade the load that depends on what carries it described in general, and here the structure gets worse by more than the load gets smaller.

How long the slab takes #

The two millimetres arrive at the rate the slab dries, which EN 1992-1-1’s shrinkage gives for a given slab, concrete and humidity.

The slab is 250 mm thick, C30/37, cast against the ground and drying from its upper face only, so its notional size is 500 mm and it dries slowly: by EN 1992-1-1’s expression for drying shrinkage the half-time is more than a year. It also shrinks autogenously, most of that in its first weeks. The fill is finished at 28 days, and only the shrinkage after that moves the slab away from a stem that is already held against fill.

At 60 per cent humidity — a heated building in a temperate climate — the end of a 30 m slab moves 2.08 mm after a year and 4.5 mm in the end. At 50 per cent, which a well-heated building reaches in winter, the slab lets go after ten months. At 80 per cent, an unheated basement car park, it takes 2.7 years. In each case the wall was propped when it was built and inspected, and stopped being propped long after anybody was looking. A slab that dries from one face curls as well as shortening, and its edges lift; the movement here is its average shortening, which is what moves its ends, and the same shortening is what a slab that shrinks onto a finished beam pushes into the steel under it.

A design that expires #

The stem’s base moment over the life of the building puts the two calculations side by side.

The propped value, 26.6, is true on the day the fill is finished and for a few months after it. From there the base moment rises with the slab’s shrinkage, slowly at first and then through its first year as the slab dries. At a year the bearing slab lets go and the moment stays at 69.6 for the rest of the building’s life. That number — not 26.6 — is the one the stem lives with for ninety-nine of its hundred years.

The tied slab’s line is the more dangerous one, because it is the line a careful designer draws. A slab that bears on a wall can lose contact with it, so the obvious detail is to tie them: starter bars from the wall into the slab, so that the slab cannot leave. The slab leaves anyway, because shrinkage is not a load that a tie can resist; it is a change in the slab’s own length. A tie turns the slab’s shortening into a pull on the top of the stem, and the stem, a cantilever of $67\,500\ \mathrm{kN\,m^2}$, is far less stiff than the slab is in tension. The pull grows with the shrinkage, to 24.8 kN a metre at 4.5 mm, and the stem’s base moment with it: past the cantilever’s 82.9 after 1.9 years and to 120.7 at a hundred.

A propped basement wall is a structure whose support is temporary. A structure that was never complete described the frame that is only ever analysed in its finished state when its worst moment comes during construction. This wall is the same problem with the order reversed: the analysis was made in the state the wall was built in, and the state it spends its life in comes later.

The length between the walls #

The movement at each end is half the slab’s shrinkage over its whole length, so the distance between the walls decides when the prop goes.

A slab shrinks toward its middle, and each end moves by its strain times half its length. At 60 per cent humidity a 60 m slab lets go of its walls after five months, a 30 m slab after a year and a 20 m slab after 2.6 years. Below about 12 m a slab never shrinks 2.08 mm at each end, and it props its walls for good. At 80 per cent the threshold is about 20 m.

That is a short list of basements. A house’s basement, a plant room under a small building, a services trench between two walls — these are short enough to stay propped. A car park, a basement under an office block, a storage hall: their slabs are 40 to 100 m between walls, and their slabs let go of the walls within the first year or two. The basements whose walls are most likely to be designed as propped by the slab, because propping saves the most steel on a long wall, are the ones whose slabs are long enough to leave.

What can be done instead #

The figures point to three honest designs. The first is to design the stem as a cantilever under the compacted fill from the start, and to treat the slab’s propping as a construction convenience that reduces the moment during backfilling and is not relied on afterwards. The stem then needs the cantilever’s reinforcement, 83 kN·m a metre at the base, and the slab need not be tied.

The second is to delay the slab’s connection to the wall: cast the slab with a gap at its edges, or a compressible filler, and close it with a later pour once most of the shrinkage has happened. A slab closed after a year at 60 per cent humidity has 2.5 mm still to shrink and will let go again; one closed after three years has 1.3 mm to go and keeps 8.8 of its 22.6 kN a metre, about two-fifths. It needs temporary props for the backfilling, which is what one support too many found the cost of a prop to be: something that must be removed or stays for good.

The third is to make the prop independent of the slab’s shrinkage, with a separate strut or a slab short enough between its movement joints to stay inside 2.08 mm at its ends. A slab broken into 10 m bays with sliding joints between them props its walls only at the end bays, but it props them for good.

The movement by hand #

The prop’s stiffness against the stem’s top can be checked from the stem alone. A cantilever of $67\,500\ \mathrm{kN\,m^2}$ over 3 m deflects $H^3/3EI$ = 27/(3 × 67,500) = 0.133 mm for every kN at its top, so 22.6 kN of prop force corresponds to 3.0 mm of movement on a stem whose pressure did not change. The fill’s pressure relaxes as the stem moves, which lowers the force the stem needs to be held still, and the prop is gone at 2.08 mm rather than 3.0: the difference is the residual the stem’s own movement released.

The slack stem’s base moment can be checked from its thrust and where it acts. Its thrust is 56.0 kN a metre and its base moment 69.6, so the resultant acts 69.6/56.0 = 1.24 m above the base, at 0.41 of the height — higher than the third of active pressure, because the roller’s residual sits in the upper part of the fill. The tied stem’s base moment is the slack stem’s under its own relaxed pressure plus the pull times the height: under 44.6 kN of thrust acting about a metre up, and 24.8 kN × 3 m = 74 from the pull, about 120.

Where the wall’s arithmetic stops #

The slab is free to shrink. A ground-bearing slab drags on its sub-base as it shortens, and the friction restrains the movement at its ends a little; for a 30 m slab on a granular sub-base the restraint is a few per cent of the free movement. A suspended ground floor on piles has no such restraint at all.

The residual relaxes with the stem’s movement and nothing else. Sand under a sustained lateral stress creeps, and over years some of the roller’s residual may relax even against a stem that does not move. Nothing here credits that, and it would lower all three moments, the propped one included, without changing the order between them.

The stem’s stiffness is uncracked and constant. A stem whose base moment rises from 27 to 70 cracks at its base, softens, and leans further under the same fill. The cracked stem’s top would move further, and a tied slab’s pull would be smaller for a given shrinkage; the base moment of a slack stem is set by the fill, and does not change much.

The wall’s ends and its starter bars #

They cannot show the wall at its ends, where a return wall stiffens it and the slab’s movement is two-dimensional. They cannot show the starter bars of a tied slab yielding or the slab cracking at the wall, either of which would cap the pull — a tied slab whose bars yield at 25 kN a metre becomes a bearing slab with a bent connection. And they cannot show the water in the fill, whose pressure a basement is a boat found usually governs a deeper wall, and which neither comes nor goes with the slab.

Still open: the wall that leans back when the slab does #

The slab shrinks once; it also moves with the seasons. A ground-floor slab in a building whose heating comes on and goes off every year expands and contracts by a few tenths of a millimetre at each end, every year for the life of the building. A stem that has gone slack is touched again each winter as the slab expands, and released each summer. Whether those annual pushes, each locking a little more of the fill’s pressure as the stem is pressed back into it — the ratchet that the pressure the roller leaves behind found in each lift of compaction — build the pressure on the stem back up over decades, or whether the fill has long since relaxed to where a few tenths of a millimetre no longer reach it, is a question about a cycle the fill sees far more often than the roller.