The seasons that stop reaching the wall

Retained soil is a load that depends on what carries it: the pressure on a wall is decided by how far the wall moves. Compaction adds a history to that. Each lift of fill rolled against a wall locks in a lateral stress larger than the active pressure, and the pressure the roller leaves behind relaxes only as the wall moves away from it. A basement stem propped at its top by a ground-floor slab cast before the fill does not move away, so the prop holds what the roller left — 22.6 kN a metre into the slab, and a base moment of 26.6 kN·m a metre on a 3 m stem that would carry 82.9 as a free cantilever.

Then the slab dries. The slab that shrinks away from the wall followed its shrinkage: 2.08 mm at each end and a slab that only bears on the stem has let go of it, the residual near the top relaxes as the top leans after the slab, and the base moment rises to 69.6 kN·m — 2.6 times what the stem was given. A 30 m slab at 60 per cent humidity gets there in about a year.

But a slab does not only shrink. It moves with the seasons, a few tenths of a millimetre at each end, every year for the life of the building, and a stem that has gone slack is touched again when the slab expands. The question this leaves is whether those annual pushes press the stem back into its fill and rebuild the pressure the shrinkage relieved — the ratchet the roller makes, repeated once a year for decades — or whether the fill has long since moved where a few tenths of a millimetre no longer reach it.

The seasons on the shrinkage #

The seasonal movement of a slab’s end is its half-length times its temperature swing times the coefficient of expansion — the movement nobody applied, which a free slab makes and a restrained one turns into stress. A ground-floor slab inside a heated building swings a few kelvin over a year; at ±3 K and $10^{-5}$ per kelvin, the end of a 30 m slab moves ±0.45 mm, a 20 m slab’s ±0.30 and a 10 m slab’s ±0.15. Those are the few tenths of a millimetre the question is about.

They ride on the shrinkage, which is not finished when the fill is. EN 1992-1-1’s drying shrinkage for a slab with a notional size of 500 mm at 60 per cent humidity runs on for years, and the end of a 30 m slab has shrunk back 2.2 mm by the end of the first year, 3.8 mm by the sixth and 4.4 mm by the thirtieth.

The three curves answer most of the question at a glance. The 30 m slab’s seasonal swing is real and visible, but it rides on a shrinkage that climbs through the slack line in the first year and keeps climbing. Its swing brings it back below the line once, a little over a year in, by a sixth of a millimetre, and never again: it last touches the stem 16 months after the fill is finished. The 20 m slab crosses later and more slowly, its winters dipping back below the line for three years before the shrinkage carries them clear; it last touches after 3.4 years.

The 10 m slab never reaches the slack line at all. Its whole shrinkage, 1.47 mm at thirty years, is less than the 2.08 mm it would take, so it bears on the stem in every season of the building’s life, and the seasons press it a little harder and a little less hard each year.

So the ratchet the question imagined needs a slab long enough to have let go and short enough for the seasons to bring it back, and on this stem that window closes within a few years on any slab long enough to go slack. A shrinkage that reaches several millimetres outruns a seasonal swing of a few tenths, and drying, unlike temperature, does not come back.

What the stem carries in the meantime #

Follow the stem’s base moment over thirty years, for the three slabs.

The 30 m and 20 m slabs do what the shrinkage essay said they would, with a short flicker on top: their winters bring the base moment down from the slack 69.6 kN·m for a year or three, and then never again. The stem spends its life as the cantilever with a top it does not use.

The 10 m slab is the case the shrinkage essay counted as safe, and it is not what the stem was designed for either. Its shrinkage never lets it go, but 1.5 mm is enough to let the stem’s top lean away from the fill by half the movement at which the compaction residual near the top relaxes completely. The base moment climbs from 26.6 kN·m to 54–60 over thirty years — most of the way to the slack slab’s 69.6, while the slab is still in contact. A slab that bears is not the same as a slab that props as designed: the prop’s effect is set by how much of the roller’s residual it still holds, and it holds less with every tenth of a millimetre the slab has drawn back.

On top of that drift the seasons swing the base moment by about 6 kN·m a year. That is the annual push, and on the short slab it does exactly what the question suggested: each winter the slab presses the stem’s top back into the fill and the base moment falls; each summer it lets the top lean and the moment rises. What it does not do is ratchet. The band moves up with the shrinkage, not down with the pushes.

The prop’s force #

The same story read at the top of the stem is starker.

The 10 m slab carries 22.6 kN a metre on the day the fill is finished and 8.2 at most thirty years on — little over a third. Its winters press harder than its summers, and in the thirtieth year the force swings between 4.9 and 8.2 kN a metre. The prop was designed for the roller’s residual, and the residual has gone, mostly in the first decade, while the prop was still there to be designed against.

A prop sized to hold that residual is not wasted: it has to carry 22.6 kN a metre for the months when the residual is still there. But the slab–wall connection sees the larger force once and the smaller one every year afterwards, and the base of the stem sees the reverse, a small moment once and a larger one for ever. A design that checks the propped stem at its as-built moment and the prop at its as-built force has checked each at the end of its life that matters least.

What a push can give back #

Whether the pushes could ever ratchet depends on what the fill does when the stem is pressed back into it, and that is the part of the calculation that has to be stated rather than derived.

The fill’s residual relaxes towards the active pressure in proportion to how far the stem has moved since it was locked, completely after 3 mm. When the stem is pressed back, the relaxation is taken to reverse along the same path: the fill recovers the part of its residual the stem’s movement had taken, and no more, because nothing new is compacted against the stem. The figure shows what that gives. In the twentieth winter the stem’s thrust is 60.3 kN a metre and in the twentieth summer 58.7, both well below the 67.4 the roller left and well above the slack stem’s 56.0. The push gives back a little of what the drying took, and only that.

That is an upper bound on what a push can do if the fill behaves like the compacted fill the whole calculation has assumed — that its lateral stress is set by its compaction and its movement since. A granular fill cycled against a wall can do more: the integral-bridge literature records lateral stresses that build up over many cycles of a stiff abutment pushed into its backfill, towards a limit that grows with the movement’s amplitude relative to the wall’s height. On this stem the amplitude is 0.15 mm over 3 m — a twenty-thousandth of the height — and the cycles number thirty. Nothing here says that ratcheting is impossible; the calculation says it is not what the compaction residual does, and the size of the movement says it is far smaller than an abutment’s.

Why the residual does not come back #

The asymmetry between a push and a pull is the whole mechanism, so it is worth stating plainly. The roller’s residual was never a property of the fill alone. It was a stress locked in by compaction against a wall that was where it was at the time, and a stem that bends away from the roller loses it in proportion to how far it moves. That loss is what the shrinkage buys: every tenth of a millimetre the slab draws back lets the top of the stem follow it, and the fill near the top, where the residual was largest, unloads towards the active state.

A push reverses the movement but not the history. Pressing the stem back towards where it stood recovers the stress the fill had lost on the way out, along the same path, as an elastic body would — and then it stops, because the stress the fill had at the start was the roller’s, and the slab is not a roller. To lock in more, something would have to compact the fill again, at a pressure many times the slab’s, and a slab pressing a few tenths of a millimetre at the top of a stem does nothing of the kind. The stem’s pressure has its as-built value as a ceiling it cannot pass, and the shrinkage lowers the whole seasonal range beneath that ceiling year by year.

This is the reverse of a member built to the wrong length, where a misfit locks in a force in proportion to the stiffness that resists it. Here the misfit grows, the stiffness that resists it is the fill’s unloading, and what it locks in is a loss.

How long the seasons reach the stem #

The slab’s length decides which of the two regimes a stem lives in: touched for good, or touched for a few years and then left. The seasonal swing decides how long the few years are.

Up to about 14 m between walls every slab touches its stems for the life of the building, whatever the seasonal swing. Between 15 and 20 m the answer turns on the swing, because the shrinkage there is only just past the 2.08 mm the slab needs to let go, and a larger swing keeps bringing the slab’s winters back below it for longer: at 20 m, 2.5 years at ±1.5 K, 3.4 at ±3 and 8.3 at ±6. Past about 25 m the swing hardly matters — the shrinkage overwhelms it in the first year or two — and a ±6 K swing on a 30 m slab buys only a month more than ±3.

The transition is sharp because both movements scale with the slab’s length but only one of them grows with time. The seasonal swing is the same every year; the shrinkage keeps rising, as $(t - t_s)/(t - t_s + 0.04 h_0^{1.5})$, toward a final value twice what it was at the end of the first year. Wherever the final shrinkage is less than the slack movement plus the swing, the slab keeps touching; wherever it is more, it stops, and the only question is when.

What the seasons still do in the twentieth year #

Where the slab keeps touching, the seasons keep moving the stem’s base moment, and the swing is a cyclic load on the stem’s base for as long as the building stands.

The swing is in proportion to the seasonal temperature: 6.2 kN·m at ±3 K on a 10 m slab, double at ±6. It is a small fraction of the moment, and it comes once a year, so in a lifetime it is a few dozen cycles — nothing the cycles that do not count would bother to count for fatigue. What it matters for is serviceability: a crack at the base of the stem, opened by a base moment twice what it was designed for, breathes by about a tenth of its width every year, and a crack that breathes is the one water finds.

Two movements, compared by hand #

The question whether the seasons reach a slack stem is one comparison. The slab’s end moves away by its shrinkage, $\varepsilon_{cs}(t)\,L/2$, and back by its swing, $\alpha\,\Delta T\,L/2$. The stem is touched while

$$\big(\varepsilon_{cs}(t) - \varepsilon_{cs}(t_0)\big)\,\frac{L}{2} - \alpha\,\Delta T\,\frac{L}{2} < \delta_{slack},$$

where $\delta_{slack}$ is the movement at which the bearing slab lets go, 2.08 mm on this stem. Both sides of the left scale with $L$, so the length at which the slab never lets go is $L = 2\delta_{slack}/(\Delta\varepsilon_{cs,\infty} - \alpha\Delta T)$ — about 16 m with $\Delta\varepsilon_{cs,\infty}$ near 300 millionths and $\alpha\Delta T$ of 30, against the 15 m at which the drawn slab, whose shrinkage is not quite final at thirty years, still touches in its thirtieth. The year in which a longer slab last touches is the year its shrinkage reaches $\delta_{slack}\cdot 2/L + \alpha\Delta T$.

The base moment in between needs the stem, the fill and their history together — the compaction residual, its relaxation with the stem’s movement, the stem’s bending and the prop — which is the calculation drawn. But the regime the stem lives in is set by that one comparison of two movements.

What the stem’s arithmetic leaves out #

The stem is fixed at its base on a footing that does not rotate, and the slab is a rigid bearing at its top that pushes but cannot pull. A slab tied into the wall pulls once it has shrunk past it, as the shrinkage essay found, and a tied slab’s seasonal swing becomes a cycle of pull and less pull rather than of touch and release. A footing that rotates leans the whole stem with the slab and gives the residual more movement to relax.

The shrinkage is EN 1992-1-1’s for one slab thickness and one humidity, and the slab is taken as moving freely between the walls. A slab that dries from one face curls as well as shortening, and its edges lift, which moves the bearing on the stem. And the seasonal swing is a sine of fixed amplitude; a building whose heating fails one winter, or whose basement is left unheated for a year during a refit, gives the slab one large contraction that it cannot take back, and a stem that was touched every year is then left for good.

Touched, and still not propped #

The seasons do reach the stem, on a short slab, every year. They do not rebuild its pressure, because a push on this fill recovers only the residual the stem’s own movement took, and the shrinkage that took it does not come back. On a long slab the seasons stop reaching the stem within a year or two of the fill, and the stem is the slack cantilever for the rest of its life. On a short one, the seasons reach it for good and it is propped in name: the prop that carried 22.6 kN a metre carries 8, and the base moment it was meant to hold at 26.6 has risen to between 54 and 60.

So the two movements that decide a propped stem’s life are not the seasonal one and the shrinkage but the shrinkage and the fill’s relaxation, and the seasonal one decides only how many years the transition takes. A stem propped by a slab has to be designed for the moment it will have once the slab has dried — which on any slab longer than a few metres is most of the free cantilever’s.

Still open: the wall that was backfilled after its slab had dried #

Every stem here was filled while its slab was young, so the roller’s residual was locked against a slab that then shrank away. A basement whose slab is cast months before the fill goes up — because the floor was needed for access, or the fill was late — has a slab that has already done much of its shrinking, and a prop that holds a residual locked against a slab that will shrink only a little more. Whether the delay, by letting the slab dry first, keeps the roller’s residual in the prop for the building’s life; whether the stem then carries the propped moment it was designed for; and whether the cost of the delay is a prop force that never relaxes, is what the order of slab and fill would ask of the movement that decided everything here.