The prop that holds what the roller left

The pressure the roller leaves behind found that a vibrating roller compacting backfill against a wall locks in a lateral pressure the soil’s weight alone would never make — Ingold’s envelope, constant with depth below a critical depth of half a metre or so, and far above the active pressure at the top of the wall. The stem that bends away from the roller let the wall move. A cantilever stem leans as the fill rises, a residual locked against it relaxes as later lifts push it on, and a 3 m stem 300 mm thick ends with 87 per cent of a stiff stem’s base moment: some relief, never much, because the base never moves and the last lifts have nothing placed after them.

That essay’s stem was free at its top. It named the case where it is not. A basement wall is usually propped by its ground-floor slab, and the slab is often cast before the backfill is placed, so that the wall spans between its base and the slab and deflects most in its middle. The essay asked 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 is loaded by a residual nobody designed it for.

The first part of the answer is that a propped wall hardly sheds anything anywhere. The second part follows from the first.

A stem that spans instead of leaning #

The calculation is the cantilever’s, with one change. The stem is fixed at its base and its top is held by a prop, rigid or a spring of given stiffness. Each 250 mm lift is compacted by a 60 kN-a-metre roller and locks in its residual at every depth below it where that residual beats the pressure the stem already feels. A locked residual relaxes toward active pressure in proportion to how far the stem moves at that depth after it was locked, completely after 3 mm. After each lift the pressure and the deflection are solved together, since each depends on the other, and the prop’s force comes from the stem’s compatibility at its top.

The free cantilever’s pressure, dashed, dips below the stiff stem’s over most of its height, where it leaned away and the residual let go. The propped stem’s, solid, lies on top of the stiff stem’s everywhere. From half a metre down it carries the roller’s residual, the saw-tooth of the separate lifts still visible in it, at about four times the active pressure at mid-height. Nothing has relaxed.

A sixth of a millimetre #

The reason is in the deflections. A cantilever under a distributed pressure deflects most at its free top, and the 300 mm stem leaned 2.61 mm there. A propped stem has both ends held and deflects only in its span, as a beam fixed at one end and pinned at the other, and a beam’s deflection between two supports is a small fraction of a cantilever’s: this one bows 0.15 mm at most, a seventeenth of the cantilever’s lean.

The residual relaxes in proportion to how far the stem moves after it is locked, against the 3 mm or so the fill needs to let go fully. The cantilever’s top moved most of that and relaxed most; its base moved nothing and kept everything; the answer was 87 per cent. The propped stem’s largest movement is a twentieth of the 3 mm, and it is at mid-height. It relaxes almost nothing, anywhere.

So the earlier essay’s question has an answer it did not expect. The propped wall does not shed its pressure where the cantilever kept it, and does not keep it where the cantilever shed it. It keeps it everywhere: 99 per cent of a stiff stem’s base moment, against the cantilever’s 87.

Why a second support takes away nearly all the movement #

The ratio is no accident of the numbers. A cantilever of span $L$ under a uniform pressure $w$ deflects $wL^4/8EI$ at its tip; the same member fixed at one end and propped at the other deflects at most $wL^4/185EI$, twenty-three times less. A triangular or saw-toothed pressure changes the factors and not their size: a second support removes almost all of a member’s freedom to move. One support too many made that point about redundant supports in general — they take stiffness from the member and give it to the support. Here the stiffness that was taken is the stiffness the fill needed the stem not to have.

It also explains why the cantilever kept anything at all. The cantilever’s movement was concentrated at its top, where the fill was shallow and the residual large, and its base moved nothing. The propped stem’s movement is concentrated in its middle third, where the residual is smaller than near the top, and its top, where the residual is largest relative to the soil’s own pressure, now moves nothing either. The propped stem has lost the one place the cantilever was letting the fill go.

The base moment falls, and the moment does not go away #

Propping the stem changes its moments completely, and it is easy to misread the change as relief. The free cantilever carried 82.9 kN·m a metre at its base. The propped stem carries 26.6 there and 14.9 of the other sign in its span — a third of the base moment, because a beam with two supports carries its load with far less moment than a beam with one.

That is the arithmetic of a propped cantilever, not of any relief, and it is the same with or without compaction. Under active pressure alone the propped stem would carry 12.0 at its base and 5.4 in its span. The roller’s residual more than doubles both. A designer who compared the propped wall with the cantilever would see the base moment fall by two-thirds and conclude that propping had dealt with the compaction. A designer who compared the propped wall under compaction with the propped wall under active pressure, which is the comparison that matters, would see every moment in it more than double.

Where the pressure goes instead #

The stem’s base carries a third of what the cantilever’s did. The rest is in the prop.

As the fill goes up the prop’s force rises slowly at first, while the fill is deep in the stem’s span and pushes mostly on the base, and then steeply, as the last lifts load the stem near its top. With the fill complete the slab carries 22.6 kN a metre of wall: 3.8 times the 6.0 that active pressure would put into it, and 2.5 times the 9.0 of soil at rest — the pressure a careful designer might use for a wall that cannot move. It passes the at-rest figure when the fill is three-quarters of the way up.

The reason it is so much larger is the shape of Ingold’s envelope. The residual is constant with depth, and it is largest relative to the soil’s own pressure near the top, where the soil’s weight is small and the roller’s residual is not. Of all the parts of a propped stem, the top is the one that loads the prop most. And of all the parts of a propped stem, the top is the one that never moves, because the prop is there. The prop holds still exactly where the roller’s residual is largest, so it collects all of it.

The load that depends on what carries it made the general point that earth pressure is not a load in the ordinary sense but a response to how far the wall gives. Propping a wall before filling it is a decision that it will not give, made before the load arrives.

A softer stem does not rescue it #

On the cantilever, a softer stem relieved more, down to about two-thirds of the stiff stem’s moment. On the propped stem a softer stem relieves more too, but from a much smaller start. At the stiffness of a 300 mm stem it keeps 99 per cent. A stem with a sixty-seventh of that stiffness, thin enough to bow 6.5 mm under the fill — more than twice the movement that relaxes a residual completely — still keeps 64 per cent of the base moment and 72 per cent of the prop force, because its top and its base still do not move. There is no practical stem thin enough to let a propped fill go.

The prop force is the more stubborn of the two, because it is collected from the top of the stem, where even a soft stem moves least.

The order of casting decides who carries it #

The prop’s stiffness decides how much of the residual goes to it. A prop softer than the stem’s own top, about 7,500 kN/m per metre for this stem, is barely a prop: the stem leans almost as a cantilever, sheds a little, and carries most of the moment at its base. A prop much stiffer takes everything the top of the stem would have shed. A ground-floor slab spanning to the opposite wall of a basement has an axial stiffness of hundreds of thousands of kN/m per metre. It is a rigid prop.

So the order in which the basement is built decides where the roller’s residual ends up. Backfill first, and the stem carries it as a cantilever, at 83 kN·m a metre in its base. Cast the slab first, and the stem’s base carries 27 while the slab carries 22.6 kN a metre of horizontal thrust. Neither order makes the residual go away. The structure that was never complete found that the order of building is a load case; here it is a decision about which element the same load case lands on.

The slab propping a basement wall is usually designed for the wall’s soil pressure — often at rest, since a propped wall cannot move — and for its own floor loads. A horizontal force of 22.6 kN a metre along a 30 m wall is 680 kN pushing on the slab’s edge, to be carried across the basement to the opposite wall, which is pushing back with its own. The slab is in compression from both sides, which it can usually take; what it must not have is an opening, a movement joint or a ramp in the line of that thrust, because the residual will go wherever the slab gives it a path.

What can be changed #

If the residual cannot be shed by the wall, it has to be made smaller where it is made, or put somewhere it can be carried.

The roller is the first lever. The residual a roller locks in grows with the square root of its line load at shallow depth, until the envelope’s limit takes over below. Halving the roller from 60 to 30 kN a metre takes the prop’s force from 22.6 to 16.2 kN a metre; a 15 kN light roller or a plate compactor takes it to 11.6. Specifications for basement backfill commonly restrict the plant within a metre or so of a wall for exactly this reason, and the calculation says that on a propped wall the restriction matters most in the top metre, against the prop, rather than near the base.

The order is the second. Backfilling before the slab is cast lets the stem lean as a cantilever and shed an eighth, at the price of a base moment three times larger; casting the slab first protects the base and loads the slab. A third choice, common where the slab must go in first for stability, is to leave a strip of the slab uncast along the wall until the backfill is complete, so that the wall leans as a cantilever while the fill goes up and is propped only afterwards. The prop then collects only what is added after it is made — groundwater, surcharge, the soil’s long-term pressure — and none of the roller’s residual.

None of these is a matter of the wall’s own design. They are decisions about plant and sequence, made on site by people who may never see the calculation. That is the common thread of every prop has its own worst day, where an excavation’s props are loaded most not when the hole is deepest but at some intermediate stage of digging: the load a prop sees is set by the order of work as much as by the finished geometry.

Residual pressure as residual stress #

There is a broader way to see the result. The roller’s residual is a lateral stress in the fill that remains after the roller has gone, held in place by the wall — a residual stress, in the same sense as the stress that was there before the load in a rolled steel section. A residual stress is in equilibrium with nothing outside the body that holds it, and it can only be released by deformation. The cantilever stem released part of it by deforming; the propped stem cannot deform enough, so the residual stays, and the wall-and-slab system carries it as a self-equilibrating set of forces between the fill, the stem and the prop.

That is also why the prop’s force does not depend on the slab’s strength or the wall’s design, only on the stiffnesses. It is the force needed to hold the fill at the deformation the roller left it at. A basement wall propped first is a basement that is a boat in more than the hydrostatic sense: its walls are held apart by a compressed slab against a pressure that the ground and the roller put into it before the building existed.

What the calculation leaves out #

The relaxation law is linear in movement. A residual relaxes in proportion to the stem’s movement after it was locked, completely after 3 mm. The earlier essay showed the cantilever’s answer depends on that figure, from 60 per cent of the rigid moment at half a millimetre to 96 at twelve. The propped stem moves so little that the figure hardly matters: at half a millimetre it still keeps 94 per cent of the base moment and 96 of the prop force.

The slab is placed and then left. A real slab shrinks as it dries, which moves the top of the stem away from the fill by a millimetre or two over months — far more than the stem’s own bow. That shrinkage happens after the fill is in, against a residual already locked, and it would relax part of the top’s residual and of the prop’s force. It is a time-dependent relief the calculation does not have, and it is the most plausible reason a propped wall’s prop force is ever less than this.

Groundwater, surcharge and the floor’s own load are not here. They add to the soil’s pressure in the usual way, and they add to a propped wall whose prop has already been loaded by compaction.

The fill is granular and the roller is a line load. Ingold’s envelope is for free-draining fill compacted by a roller of known weight; a clay fill, or a plate compactor, leaves a different residual.

Still open: the slab that shrinks away from the wall #

The one movement the calculation lacks is the slab’s own. A ground-floor slab cast before the backfill dries and shrinks for months after, and every millimetre it shortens moves the top of every wall it props away from the fill — exactly where the roller’s residual is largest and has so far relaxed least. A 30 m slab shrinking by three hundred millionths moves each wall’s top by 4.5 mm, more than enough to let the residual near the top go. Whether that shrinkage relieves the prop’s force before the basement is put into service, or whether the slab is tied into walls that hold it, so that the shrinkage is resisted and becomes a tension in the slab instead, is the question of which of two slow movements the wall’s top is waiting for.