The pressure the roller leaves behind

The load that depends on what carries it found earth pressure to be a load decided by the wall: a wall that moves away from its soil by a thousandth of its height is pressed by the active triangle, a third of the vertical stress for a sand at 30°, and one that cannot move by the at-rest triangle, half of it. Among the things the model left out it listed one plainly: “Compaction is not in it. Backfill rolled in layers is pushed against the wall by the roller, and the locked-in pressure near the top can exceed $K_0$ substantially.”

Most retaining walls are backfilled, and most backfill is compacted, because loose fill settles and a road or a floor on top of it would settle with it. So the soil behind most walls is not the soil the active triangle describes. This essay follows the roller.

A line load, and what it leaves behind #

A roller passing over the backfill is a line load on its surface, $p$ kN for every metre of its width — its static weight, and for a vibrating roller the centrifugal force of its eccentric as well. A line load on the surface of a soil produces a vertical stress that falls off with depth as $2p/(\pi z)$, the Boussinesq result for a strip, and the soil beneath it presses sideways against the wall by the active ratio of that stress.

The roller moves on, and the vertical stress it caused goes with it. The sideways stress does not all go. Soil pushed laterally by a passing load does not spring back — its grains have moved and rearranged, and the wall behind them holds them — so a large part of the horizontal stress the roller produced stays, locked in against the wall. The next lift adds another layer on top and its roller does the same thing a lift higher, and a point on the wall keeps the most that any pass ever gave it.

What limits how much is kept is the soil above the point. A lateral stress can only be held against the wall if the soil above can resist being pushed up and out by it — passive resistance, $K_p$ times the vertical stress, with $K_p = 1/K_a$. Close under the roller the vertical stress from the soil’s own weight is small, and passive failure caps what can be locked in; deeper, the roller’s influence has spread out and fallen. Between the two there is a depth at which the most is kept, and below it every point has at some stage been at that depth under some lift.

That is Ingold’s envelope, from 1979. The locked-in lateral pressure rises from the surface along the passive limit $\gamma z/K_a$ to a critical depth

$$z_c = K_a\sqrt{\frac{2p}{\pi\gamma}}$$

and is constant below it at

$$\sigma_{hm} = \sqrt{\frac{2p\gamma}{\pi}}$$

until the soil’s own active pressure $K_a\gamma z$ grows past it, at $h_c = \sigma_{hm}/(K_a\gamma)$.

For backfill of 20 kN/m³ and $\phi' = 30°$, rolled by a 60 kN/m vibrating roller — an ordinary single-drum machine — the plateau is $\sqrt{2 \times 60 \times 20/\pi} = 27.6$ kPa, reached 0.46 m below the surface and held down to 4.15 m. Behind a 3 m wall the roller’s pressure is the whole of the load below the top half-metre, and the soil’s own triangle never catches up with it. The active triangle at the base is 20 kPa; the plateau is 27.6 all the way up the wall.

The integrated effect is larger than the peak suggests, because the plateau moves the load up the wall. The thrust is 76.6 kN per metre of wall, two and a half times the active 30.0; the moment about the base is 106 kN·m, three and a half times the active 30.0, because the resultant sits at the wall’s mid-height rather than a third of the way up. It is more than the at-rest moment of 45 kN·m that an immovable wall would carry. A wall that yields cannot shed compaction pressure by yielding: the active triangle was never the load.

Why a roller outweighs three metres of soil #

The size of the effect is easier to believe once the roller is compared with the soil. At the critical depth, 0.46 m under a lift’s surface, the 60 kN/m roller adds a vertical stress of $2p/(\pi z) = 2 \times 60/(\pi \times 0.46) = 83$ kPa. That is the weight of more than four metres of the same soil. Every lift, at the moment it is rolled, is pressed as if it were buried four metres deep, and its lateral share — a third of 83 kPa, 28 kPa — is left against the wall when the roller moves on. The wall is being loaded by a series of four-metre overburdens applied one lift at a time and never removed.

That is also why the plateau is constant with depth. Each lift contributes the same peak to the soil just below it, however deep the lift is in the finished fill, so every point below the critical depth ends with the same residual. The soil’s own weight, which grows steadily with depth, catches up with that residual only at $h_c$, and for this roller that is below the base of any wall under four metres high.

Built up one lift at a time #

The envelope is smooth because it assumes every point was, at some moment, exactly the critical depth below a lift’s surface. Backfill is not placed that way. It goes in lifts of a quarter of a metre or so, and a point on the wall sits at a set of discrete depths below the successive lift surfaces.

Simulated lift by lift — each roller pass at a lift’s surface pressing on every point below with $K_a\cdot 2p/(\pi d)$, capped by the passive limit of the soil above, and every point keeping its largest value — the residual is a sawtooth. Points that happened to lie a critical depth below some lift reach the plateau; points midway between get less, down to 21 kPa for quarter-metre lifts. The envelope is the sawtooth’s upper bound, which is the right thing to design for, since the lift levels are not controlled to the centimetre, and it is also a reminder of why measured compaction pressures scatter: two instruments a few centimetres apart on the same wall can read a fifth apart.

The simulation also shows the order of events. The upper part of the wall acquires its pressure late, from the last few lifts, and the lower part early, from the first; a wall propped during backfilling, or a wall whose base has not yet been restrained by its slab, meets the lower part of its load while it is least able to take it — the argument every prop has its own worst day makes for excavations, run in reverse for fill.

A heavier roller reaches deeper #

The plateau and the depth it reaches both grow as the square root of the roller’s weight. That is the design lever and its limitation at once. A pedestrian roller or a vibrating plate of 10 kN/m locks in 11 kPa and dominates the top 1.7 m; a heavy vibrating roller of 150 kN/m locks in 44 kPa and dominates more than 6.5 m. Halving the roller’s weight reduces the locked-in pressure by only three tenths. Specifications that keep heavy plant a couple of metres back from a wall and allow only light plant close to it are using exactly this lever, and it is a weaker lever than it looks.

Short walls are loaded by the roller #

The roller’s plateau is fixed by the roller and the soil, and the soil’s own triangle grows with the square of the wall’s height. So compaction pressure is a problem of short walls. A 2 m wall backfilled with the 60 kN/m roller carries nearly five times the active moment at its base; a 3 m wall three and a half; a 6 m wall twice; a 10 m wall 1.4 times. The walls built in the largest numbers — garden walls, abutment wing walls, basement walls under a single storey, culvert headwalls — are exactly the ones for which the roller is the governing load and the active triangle a distant second, and they are the walls least likely to be designed by someone who has seen a roller’s weight on a specification.

The ratio of thrusts is smaller than the ratio of moments at every height, because the plateau puts its extra load high on the wall where the lever arm to the base is longest. Stability checks that use the thrust — sliding — are less affected than those that use the moment — overturning, the stem’s bending at its base, the bearing pressure at the toe.

Five moments on one wall #

Side by side, the choices a designer makes about the soil’s pressure and the contractor makes about the plant are of the same size, and the contractor’s is the larger. The difference between active and at-rest pressure — the question of whether the wall can move, which the load that depends on what carries it spent its length on — is a factor of 1.5. The difference between the lightest and heaviest plant that might reasonably be used behind the wall is a factor of 2.4, and even the light plant nearly doubles the active moment.

The wall that cannot move #

A basement wall propped by its floors, or a culvert wall held by its roof slab, cannot move away from its soil, and its design pressure is the at-rest triangle, half the vertical stress rather than a third. That makes the soil’s own share larger and the roller’s relatively smaller, but not small. For the same 3 m of fill and the same roller, the at-rest triangle overtakes the plateau at 2.76 m rather than 4.15, so the plateau still governs three quarters of the wall’s height; the moment about the base is 106 kN·m against the at-rest 45, a factor of 2.4 rather than 3.5.

For a propped wall the moment that matters is usually the span moment between props, not a base moment, and there the plateau’s shape matters as much as its size: a uniform pressure between two supports produces a larger mid-span moment than a triangle of the same resultant, because more of it sits in the middle. A stiff, propped wall is exactly the case Ingold’s method was calibrated for — nothing relieves the locked-in stress — so it is the case in which the envelope should be believed in full.

What can be done about it #

There are three levers, and none of them is free. The first is the plant: keep the heavy roller back from the wall and compact the zone behind it with light plant, which is what most earthworks specifications do. The envelope says the saving is the square root of the weight ratio — a 15 kN/m plate against a 60 kN/m roller halves the plateau — and that a light machine still dominates the top of the wall. Heavy plant kept a metre or two back loads the wall through the soil between, at a stress that falls with distance but does not vanish, so the saving is real and partial.

The second is to design for it. For a short wall the compaction moment is the design moment, and the additional steel in a stem designed for 106 kN·m rather than 30 is modest in a wall whose thickness was set by durability and construction. The overturning and bearing checks on the base are harder to satisfy, because the resultant sits higher; a block’s stability is decided by where its resultant falls, and compaction moves it towards the toe.

The third is to let the fill push against something that gives. A compressible layer against the back of the wall — expanded polystyrene, a crushable foam, a void former — accepts the roller’s lateral push by squashing, so the locked-in stress falls to what the layer can hold. The same device protects the abutments of integral bridges from the ratchet of their backfill, and it works for the same reason: the load is an imposed deformation of the soil, and a soft layer turns it back into a small force.

The envelope, by hand #

The three numbers need a calculator and nothing else. With $p = 60$ kN/m, $\gamma = 20$ kN/m³ and $K_a = 1/3$: the plateau is $\sqrt{2 \times 60 \times 20/\pi} = \sqrt{764} = 27.6$ kPa; the critical depth is $K_a\sqrt{2p/\pi\gamma} = \tfrac13\sqrt{1.91} = 0.46$ m; and the depth to which it governs is $27.6/(\tfrac13 \times 20) = 4.15$ m. Against a 3 m wall the thrust is a triangle to 0.46 m and a rectangle below it: $\tfrac12 \times 0.46 \times 27.6 + 2.54 \times 27.6 = 6.4 + 70.1 = 76.5$ kN/m. The moment about the base adds each part’s lever arm: $6.4 \times 2.69 + 70.1 \times 1.27 = 17 + 89 = 106$ kN·m. The active triangle’s are $\tfrac12 \times \tfrac13 \times 20 \times 3^2 = 30$ kN/m acting at 1 m, 30 kN·m.

Where the method stops #

The wall is taken as able to hold what the roller pushes. Ingold’s derivation assumes the locked-in pressure is retained, which needs a wall stiff enough not to move away under it. A cantilever wall that deflects outward by a thousandth of its height as the backfill rises relieves some of the compaction pressure below it, as it would relieve at-rest pressure, and a flexible sheet-pile wall may relieve most of it. A stiff basement wall propped by its floors keeps all of it.

The soil is granular and dry. A clay backfill compacted wet locks in pressure that swells or relaxes with its moisture, and a clay that swells after placement can push harder than any roller. A backfill that is later saturated has its effective stresses changed, and its compaction pressure with them — and gains a hydrostatic pressure of its own that a basement is a boat found to be the largest load such a wall carries.

The roller is a line load at the surface, running parallel to the wall. A roller running at right angles to the wall, or working close to its face so that the wall itself confines the load’s spread, pushes differently; the envelope is a design model calibrated on measurements, not a solution of the soil’s behaviour, and its accuracy is that of the measurements — a fifth or so.

What the pictures cannot show #

That the roller’s load does not stay where the wall was designed to receive it. Compaction pressure is a residual stress, like the stress a weld leaves in a plate, and the thing that decides what happens to it is the wall’s movement over years: creep of the stem, rotation of the footing on its soil, a season of freezing in the backfill. Measurements behind real walls show compaction pressures decaying slowly in some cases and persisting unchanged in others, and nothing in the envelope says which.

Nor can they show the soil arching that a stiff wall induces in a narrow backfill. A wall retaining a thin wedge of fill between itself and a rock face or an excavation slope carries the fill’s weight partly by friction, as the pipe decides what the soil weighs found for a buried pipe and the pressure that stops growing for a silo, and its compaction pressure then sits against a soil whose vertical stress is less than its weight.

Still open: the wall that moves while it is backfilled #

The envelope treats the wall as fixed while the fill goes up and then asks whether it moved. A cantilever wall moves during backfilling, a little with every lift, and each lift’s compaction is locked in against a wall that is already leaning away from the lifts below. Whether a wall that deflects as it is backfilled carries less compaction pressure than Ingold’s envelope — because each lift’s contribution is partly relieved by the next lift’s deflection — or more, because the upper lifts are compacted against a wall that has already moved and has no further movement left to give, is the question of how a residual load and a flexible structure divide the pressure between them, and it is the one that would let a designer take credit for a stem that bends.