The stem that bends away from the roller

The pressure the roller leaves behind followed a vibrating roller up the back of a wall. Each pass of the roller pushes the soil beneath it sideways against the wall; when the roller moves on, the vertical stress it caused goes with it, but much of the lateral stress stays, locked in by friction in a soil that has been compacted around it. The residual is largest a short depth below each lift’s surface and is capped near the top by what the soil above can hold. Taken over every lift, the most any lift leaves at each depth is Ingold’s envelope: a constant pressure down most of the wall, far above the active triangle a soil’s own weight would make. For a 3 m wall compacted by a 60 kN-per-metre roller the envelope more than triples the base moment.

That essay assumed the wall stayed where it was while the fill went up, and ended by asking what happens when it does not. A cantilever stem bends a little with every lift. A residual locked in against the stem at one depth is then pushed against by a stem that keeps moving away, and a soil’s lateral stress falls fast when the wall it bears on moves away from it — that is how the active pressure arises in the first place. Does a bending stem shed its compaction pressure, and how much?

Locking in against a stem that moves #

The calculation follows the backfilling lift by lift. A 3 m stem, fixed at its base, is backfilled with soil of 20 kN/m³ and 30° in 250 mm lifts, each compacted by the 60 kN-per-metre roller of the earlier essay. When a lift is compacted, the residual it would lock in at each depth below it is the earlier essay’s, and where that residual is more than the pressure already on the stem there, it replaces it — locked against the stem where the stem then is. As later lifts push the stem further, each locked residual relaxes toward the active pressure in proportion to how much further the stem has moved at its depth since it was locked: completely, once that further movement reaches 3 mm. The stem’s deflection depends on the pressure and the pressure on the deflection, so after each lift the two are solved to agreement before the next lift goes on.

That relaxation rule is an assumption, and the most important one here: how much movement a compacted fill needs to let go of a locked-in stress is not well measured, and the last figure shows how much rides on it. It is chosen so that it can be checked against its limits. A stem that cannot bend recovers the earlier essay’s lift-by-lift build-up exactly; a fill with no roller recovers the active triangle exactly.

On a stem that cannot bend, the residuals locked in lift by lift give a base moment of 95 kN·m per metre run — a little under the 106 of Ingold’s continuous envelope, because discrete lifts leave a sawtooth of residuals rather than a flat top. On a 300 mm concrete stem the base moment is 83 kN·m, 87 per cent of the rigid stem’s. The active pressure alone would be 30. So the stem sheds something, and a modest something: an eighth of the moment, not the two thirds a designer hoping for active pressure would need.

Where the stem sheds it #

The relief is not spread evenly, and its shape is the argument. At the base it is nothing at all: the stem is fixed there, it never moves, and a residual locked at the base stays locked however much the stem above it bends. In the top lift it is small, because the top lift’s residual was locked after nearly all the stem’s movement had happened, and nothing comes after it to push the stem further. In between it peaks — 20 per cent of the rigid pressure at 0.9 m below the finished surface — at depths locked in when the fill was still low and then left to watch the stem bend away from them for the rest of the backfilling.

That answers the question the earlier essay left in the form of an either–or. A bending stem does relieve the compaction pressure locked in by earlier lifts, and the upper lifts are compacted against a stem that has already moved — and both are true, at different depths. The relief lives where the pressure was locked early and the stem moved late. Neither end of the stem offers that combination.

Watching the fill go up shows the mechanism directly. With the fill at 0.8 m the residuals near the base are as the rigid wall would have them, because the stem has barely begun to bend. By 1.5 m the stem’s top has moved, the residuals locked at mid-height have begun to relax, and the new lifts are locking theirs against a stem already leaning away. By 3 m the middle of the stem has lost a fifth of its residual and the bottom none.

The top keeps moving after it is filled #

The stem’s own deflection is small. The top of a 300 mm stem moves 2.6 mm by the end of backfilling — less than the 3 mm that relaxes a residual fully — and most of that movement comes from the last few lifts, which load the cantilever furthest from its base. A depth half-way up the stem moves about a third as much as the top. A residual locked there when the fill reached it sees the stem move away perhaps a millimetre afterwards, which is enough to take a fifth of it.

A stiffer stem keeps more, and no stem sheds it all #

Thin the stem and it sheds more: a 200 mm stem, whose top moves 7.0 mm, keeps 70 per cent of the rigid stem’s base moment; a 400 mm stem, moving 1.2 mm, keeps 94. That is the expected direction. What is not expected is where it stops. The softest stem drawn, which bends nearly 30 mm at its top, still keeps 68 per cent — while the active pressure alone, which is what a wall that moved freely would see, would give 32.

The floor is the base and the last lifts. However far the stem bends, its base never moves, and the residual locked there at the very start of backfilling stays for good; and however flexible the stem, the last lift’s residual is locked against the stem’s final position, with nothing afterwards to relax it. Those two zones, and the zones near them that move little, carry the floor. A cantilever stem cannot be made flexible enough to reach active pressure, because the place where it is fixed is the place where its moment is decided.

How easily the fill lets go #

Everything above rests on the 3 mm a locked residual needs to relax, and the relief is sensitive to it. If the fill lets go after half a millimetre, the 300 mm stem keeps only 60 per cent of the rigid base moment; after one millimetre, 72; after six, 93; after twelve, 96 — nearly the rigid wall’s. The active state of a dense granular fill is reached at a movement of about a thousandth of the wall’s height, 3 mm here, which is why that value was used; but a compacted residual is not the same thing as an at-rest stress, and how far a wall must move to undo it is exactly the quantity the field data are thinnest on.

The honest reading is therefore a range. A stem’s flexibility buys somewhere between a twentieth and two fifths of the base moment back, depending on how readily the compacted fill lets go, and a designer who wants to take credit for it needs a number for the second quantity before the first can be counted.

Residuals, and the structure they are locked into #

The general shape is familiar from stresses that were there before the load: a self-equilibrating field locked in during construction, which the structure’s later deformation can relieve only where it deforms. A welded section’s residual stresses are relieved where it yields and kept where it stays elastic; a compacted fill’s residual pressure is relieved where the wall moves away from it after it was locked and kept where the wall does not. In both, the relief is a matter of sequence as much as of stiffness — what happened after the residual was locked, not merely how much the structure can move.

It also explains why compaction pressure is treated so differently for different walls. A gravity wall that slides or tilts on its base moves everywhere at once, base included, and can shed a residual over its whole height; a propped basement wall moves hardly at all and keeps nearly all of it, as every prop in an excavation finds when it is asked to hold back ground that never got the chance to let go; a cantilever stem lies between, and the figure for it here — an eighth of the moment for an ordinary stem — is a statement about a wall fixed at the one place where its moment is largest.

The order of construction is part of the load #

What the calculation finally says is that the pressure on the finished wall depends on the order in which it was loaded. Two walls with the same fill, the same roller and the same stem end with different pressures if one was backfilled before its footing’s ground had settled and the other after, or if one had its top propped and the other did not. That is the same lesson a bay cast last teaches about restraint cracking and a self-anchored bridge teaches about its own erection: a structure that acquires its stiffness and its load in stages carries the history of the stages, not just the final state.

It also means that a wall check that applies Ingold’s envelope to a finished cantilever is checking a wall that was never built: one that stood perfectly still while every lift was compacted against it. The real wall moved, a little, at every stage, and the envelope is its upper bound. For a stocky stem the bound is close; for a thin stem on a soft footing it is a quarter too high — but for no stem is it twice too high, which is the margin a designer would need to justify active pressure instead.

By hand: the stem’s own movement #

The stem’s movement can be checked against the simplest case. Under the active triangle alone, $K_a\gamma z$ with $K_a = 1/3$, a cantilever 3 m high has a top deflection of $K_a\gamma H^5/(30EI) = (1/3) \times 20 \times 243/(30 \times 67{,}500) = 0.80$ mm. Under the compaction pressure, roughly constant at about 25 kPa over the lower two thirds and building up above, the base moment is near 83 kN·m and the top deflection near $wH^4/(8EI)$ with $w \approx 22$ kPa, $22 \times 81/(8 \times 67{,}500) = 3.3$ mm — the right size for the 2.6 mm found, since the real pressure falls off toward the top. A 300 mm stem 3 m high is stiff enough that its movement is comparable to the movement the fill needs to let go, which is why it sheds a modest share; a much more flexible one moves well past it at the top and still cannot move at all at the base.

A rule for relaxing, a fixed base and a stiff fill #

The residual relaxes linearly with movement. A compacted fill’s lateral stress probably falls faster at first and then slower; the linear rule over 3 mm is a stand-in, and the last figure shows the answer’s dependence on it.

The stem’s base is fixed, and the footing is the first thing to relax. A real stem stands on a footing that rotates a little on the ground, which is a spring, and a footing that rotates under the base moment tilts the whole stem away from the fill. Computed with the footing as a rotational spring, a stiffness of 100,000 kN·m per radian a metre — a footing that rotates three quarters of a milliradian under this moment — takes the 300 mm stem from 87 to 78 per cent of the rigid stem’s base moment, and its top to 4.6 mm; a footing half as stiff, to 72. Even a perfectly rigid stem on that footing sheds an eighth. But the pressure at the very base is still untouched: a stem rotating about its base point moves everywhere except there.

And the fill’s own stiffness is left out. The stem’s movement is set by the pressure alone; the fill behind it, being stiff, also resists the movement and holds the stem back, so the real movement is a little smaller than computed and the relief a little less.

Footings, drainage and the second summer #

They cannot show the footing. A stem whose footing tilts outward as the fill rises behind it moves at its base too; how much of the relief that adds depends on the ground under the toe, which is a separate problem.

They cannot show water. A backfill that becomes saturated adds water pressure to everything here, and a residual locked in a free-draining fill may not survive a winter of wetting and drying.

And they cannot show time. Compacted residuals relax slowly in service as the fill creeps and as traffic on the fill above vibrates it — a slow version of the same relief, needing no movement of the wall at all, and the reason long-term measurements behind old walls rarely find the full envelope. A stem designed for the residual on the day the backfill was finished carries less a year later, a relief its stiffness has nothing to do with; the day the backfill was finished, and the load that depends on what carries it on that day, is still the one it must be designed for.

Locked early, moved late #

A bending stem sheds compaction pressure only where it moves after the pressure was locked in. A 3 m stem 300 mm thick ends with 87 per cent of the rigid stem’s base moment, 83 kN·m a metre against 95.

The relief peaks a third of the way down and is zero at the base, which never moves, and small in the last lifts, which nothing comes after.

No stem sheds it all. A 200 mm stem keeps 70 per cent and the softest stem drawn 68, where active pressure would give 32.

And the relief depends on how easily the fill lets go — 60 per cent of the rigid moment at half a millimetre, 96 at twelve — which is the number a designer needs before a stem’s flexibility can be counted.

Still open: the wall that is propped before it is filled #

Every stem here is free at its top while the fill goes up. Many basement walls are not: the ground-floor slab is cast first and props the wall’s top before the backfill is placed, so the wall spans between its base and the slab and deflects most in the middle. A residual locked in near the middle when the fill reaches it is then pushed against by a wall that moves most there, while the propped top hardly moves at all — the reverse of the cantilever. Whether a propped wall sheds its compaction pressure where the cantilever kept it and keeps it where the cantilever shed it, and whether the slab that props it is loaded by a residual that nobody designed it for, is the question the order of construction puts to this one.