The shelter that stops at the interface

The layer a pile feels put one laterally loaded pile into two layers of ground and found that it loads its ground over a short length near the head — half the energy in the top 0.71 characteristic lengths — so that a soft layer one length thick raises the head’s deflection by 70 per cent, and that soft ground over stiff gives a larger bending moment than either ground alone, because the pile bends as a pile in soft ground would and then meets ground that holds it. The peak follows the interface down.

That essay ended with the piles as they are actually built: in groups, under a cap. A group pushed sideways shares its push between piles that interact through the ground between them. The leading row pushes into ground nobody has disturbed. The rows behind push into ground the rows in front have already pushed, and that ground gives less. Practice accounts for it with a factor for each row on the soil’s reaction — for piles three diameters apart, about 0.8 for the leading row, 0.4 for the second and 0.3 for the rest — calibrated on full-scale groups pushed in uniform ground. This essay asks what that factor means when the ground is not uniform, and finds that the calibration does not say.

Two readings of one factor #

The piles are the earlier essay’s: 20 m long, of bending stiffness $120\ \text{MN·m}^2$, in ground whose modulus grows with depth at 5.0 MN/m³ — a characteristic length of 1.89 m — under a soft layer at a tenth of that, 0.5 MN/m³, two characteristic lengths (3.8 m) thick. Nine of them stand in three rows of three under a rigid cap that does not rotate, and the cap is pushed along the rows by 300 kN a pile.

With the cap rigid, every pile’s head moves by the same amount, so each pile carries a shear in proportion to its own head stiffness, and the cap’s push divides between the rows in proportion to how stiff each row’s soil makes them. In uniform ground that is the whole calculation: the row factor multiplies the soil’s modulus, the stiffness follows, and the leading row takes the most.

In layered ground there are two ways to apply the factor and the calibration does not choose between them. Applied at every depth, as a code’s table is usually read, the rows behind are sheltered in the stiff ground below the interface as much as in the soft ground above it. Applied only in the soft layer, the rows behind are sheltered where the ground is soft and the piles move a long way through it, and bear on undisturbed stiff ground below, where the piles barely move and the leading pile has hardly disturbed anything. Neither is a measurement. The second is the physical picture the first was calibrated on — the shadow is a zone of ground the leading pile has already strained, and its size scales with how far the pile moves — but no group test separates the two, and the difference is the subject.

Under the usual reading the leading row carries between 1.24 and 1.35 times the average whatever the layering, and the last row between 0.75 and 0.83. The layering moves the shares a little, and in the direction the earlier essay would predict: a soft layer makes every pile’s head stiffness depend more on the stiff ground below, where all three rows have the same factor’s proportional effect, so the rows’ stiffnesses come a little closer together, and then part again as the soft layer grows thick enough to dominate.

Under the second reading the shares start equal — a thin soft layer shelters almost nothing — and separate only as the soft layer thickens. At two lengths the leading row carries 1.07 times the average and the last 0.96. The same table of factors, read two ways, gives the leading row a quarter more than the average or a fifteenth more. By four lengths the readings have nearly converged, because the soft layer is then the ground the piles feel and both readings shelter it.

The leading pile and the last #

Every pile is fixed against rotation at the cap, so its largest moment is at the head, hogging, and it reverses through the soft layer to a sagging peak just below the interface — the peak the earlier essay found following the interface down. The two readings separate the piles at both places, but not in the same way.

At the cap, the readings differ by how much shear each row takes. Under the usual reading the leading pile takes 1.24 times the average push and carries 1,009 kN·m at its head; under the second it takes 1.07 times and carries 859. The last pile goes the other way, from 773 to 819. Below the interface the readings differ in kind. Under the usual reading the last pile is sheltered in the stiff ground too, so its deflected shape reaches deeper and its sagging peak is lower and smaller: 289 kN·m against the leading pile’s 408. Under the second reading every pile meets the same unsheltered stiff ground below the interface, and every pile’s peak is the same within a kilonewton-metre: 358, 359, 359.

At the cap, and in the ground #

At the cap the leading row carries the most under every reading, and the layering raises every row’s head moment by about a third over the uniform stiff ground, which is the earlier essay’s soft-layer effect multiplied through the group. What the readings change is the spread. Under the usual reading the last row’s head moment is 77 per cent of the leading row’s; under the second, 95 per cent. A designer who sizes the leading pile and repeats it across the group is covered under both. A designer who sizes each row for its own moment is not: the last row sized at 773 kN·m under the usual reading is 6 per cent short under the second.

In the ground the difference is larger. The usual reading has the leading row carrying 41 per cent more than the last; the second reading has them equal. A pile whose reinforcement is curtailed down its length — as most bored piles are, because the head moment is two or three times the moment in the ground — is curtailed at a depth set by the moment below the interface, and a last-row pile curtailed to the usual reading’s 289 kN·m is 24 per cent short of the 359 the second reading gives it, at the depth where the earlier essay found the peak and where the steel has already been reduced.

The interface does not change which row works hardest at the cap. It can erase the difference between the rows below it, which is where the shortfall would be if each row were designed for its own share.

How thick the soft layer has to be #

The moment in the ground grows with the soft layer under both readings, because a thicker soft layer lets the pile bend further before the stiff ground catches it. Under the usual reading the leading row’s peak stays about 40 per cent above the last row’s throughout. Under the second reading the two are equal for every layer up to about three lengths, and only then begin to separate — the leading row’s ahead — as the soft layer becomes deep enough to hold most of the pile’s deflected length and the difference in its shelter starts to matter below the head as well.

That is the shape of the argument. Below the interface the piles all bear on the stiff ground, and whether they bear on it equally depends entirely on whether the shelter reaches it. Above the interface every reading agrees that the rows behind are sheltered. The calibration is silent in exactly the ground that carries the peak.

What a designer does with a bracket #

A factor that can be read two ways is a factor that has to be designed for both, and the figures say what that costs. At the cap, the leading pile is governed by the usual reading and the last pile by the second: 1,009 and 819 kN·m. In the ground, the leading pile again by the usual reading, 408, and the last by the second, 359. The envelope is not either reading. It is the usual reading for the leading row and the second reading for every row behind it, and it is what a designer who sizes rows individually has to use.

Most designers do not size rows individually. The common practice is to find the worst pile and repeat it, and under that practice the bracket costs nothing: the leading pile under the usual reading governs everything, at the cap and in the ground, and every pile in the group is at least as strong as the worst reading asks. The bracket matters where practice has moved on to economy — groups whose rear rows are lighter, or shorter, because the factors said they carried less — and there it matters most in the ground below a soft layer, where the usual reading says the rear rows carry 29 per cent less than the leading row and the second reading says they carry the same.

The factors have a precedent in bolted joints. The hole made bigger so the steel would fit found that the reduction a slotted hole costs a friction joint is a ratio of test results, carried into the design table as if it were a property of the hole. A row factor is the same kind of number: a ratio of what a sheltered pile carried to what a lone pile carried, in the groups that were tested. It is reliable inside the conditions the tests covered and silent outside them, and layered ground is outside them.

The group’s stiffness #

The group efficiency — how stiff the cap is, as a share of nine piles pushed independently — is the number a frame analysis takes from the foundation, and it decides the cap’s movement under wind or earthquake. Under the usual reading it is 0.74 at two lengths; under the second, 0.90. The cap’s deflection under the same push differs by the inverse ratio, a fifth, and a superstructure whose drift limit was close would read a different answer from the same foundation depending on a choice the foundation’s calibration never made.

The direction matters for a seismic design, where a stiffer foundation is not automatically the safe assumption: it attracts more force to the piles and less to the structure’s own ductility, and a designer who wants to bound the pile forces takes the stiffer reading and a designer who wants to bound the drift takes the softer. In layered ground the two bounds are now a fifth apart rather than a single number.

The share by hand #

The shares can be checked from single-pile stiffnesses. A fixed-head pile in ground whose modulus grows linearly with depth has a head stiffness that scales as $n_h^{3/5}$ — the earlier essays’ characteristic length goes as $n_h^{-1/5}$, and the head stiffness as $EI/T^3$ — so a factor $f$ on the whole of the ground multiplies the stiffness by $f^{3/5}$. In uniform ground the three rows’ stiffnesses are then in the ratio $0.8^{0.6} : 0.4^{0.6} : 0.3^{0.6} = 0.875 : 0.577 : 0.486$, whose mean is 0.646, and the leading row’s share is $0.875/0.646 = 1.35$ and the last row’s $0.486/0.646 = 0.75$ — exactly the figure’s values in uniform ground, and the group efficiency is the mean itself, 0.65.

In layered ground the usual reading loses the shortcut too, and the reason is worth a sentence because it explains why the shares move at all. A factor on the whole of a linearly increasing ground is the same as a change of characteristic length, $f^{-1/5}$ — the softer rows behind are simply longer piles in the same ground. With an interface at a fixed depth that self-similarity breaks: the last row, with the longest characteristic length, reaches furthest past the interface into the stiff ground, so more of its stiffness comes from ground that is ten times stiffer, and its share rises from 0.75 to 0.83 while the leading row’s falls. The rows’ stiffnesses come together because they are measured against the same fixed interface with rulers of different lengths.

The second reading in layered ground has no such shortcut, because the factor acts on only part of the ground the pile feels. But the earlier essay’s weighting gives its shape: a pile loads its ground in proportion to the square of its deflection, half of it in the top 0.71 lengths, so a soft layer two lengths thick holds most of the weight and the rows’ stiffnesses differ by a large share of what the full factor would give — yet the stiff ground below holds enough of the weight to pull them together. The figure’s 1.07 is that compromise.

The cap that shares the push #

The cap is the free body that sets the shares: cut through the nine piles just below it, and the cap’s push is balanced by nine shears, its overturning by the piles’ axial forces (which this model does not need, the cap being held from rotating), and its rotation by the nine head moments. Rigid and non-rotating, it imposes one displacement on every head, and the shears follow the stiffnesses. Each pile is then its own free body: a beam on a bed of springs whose stiffness is the soil’s modulus times the row’s factor wherever the reading applies it, loaded by its share of the shear and held from rotating at its head.

The rows interact only through the factor. The pile the ground pulls apart and the plane where the pile and the ground agree showed how much a pile’s internal forces depend on what the ground around it is doing, and a factor is the bluntest possible statement of that dependence. The essay on nine piles and four times the settlement found that a group in settling ground interacts through the soil between its piles, the ground moving under one pile because of the load on another. A factor on each pile’s own springs is a much cruder account — it lets the rows behind feel the leading row only as a softening of their own ground — and it is the account practice uses, which is why it is the one examined here.

What a factor on each pile’s springs leaves out #

The factors are the calibration’s, not the ground’s. Whether the shelter a leading pile gives reaches into stiff ground below a soft layer is exactly what no group test in uniform ground could show. The two readings are a bracket on that ignorance, not two models of which one is right.

The ground is linear. Real soil reactions soften with deflection, and the shelter itself is a non-linear effect: a leading pile that has pushed its ground to yield leaves a gap behind it that a trailing pile’s ground falls into. In the soft layer, where deflections are large, that makes the second reading more plausible; at small loads, where nothing yields, both may overstate the shelter.

The cap is rigid and does not rotate. A cap that rotates relieves the head moments and loads the piles axially; a flexible cap shares the push in proportion to its own stiffness as well as the piles’. The pile that is too short to bend is a reminder that a pile’s head condition decides most of its moment.

The rows are rows. The factors here vary only along the push; the piles across a row are taken to share equally, and the corner piles of a real group, sheltered on one side only, carry more than the middle ones.

Still open: the group that is pushed both ways #

Everything here is a push in one direction, so the leading row is always the same row. Wind and earthquake reverse the push, and on the reversal the leading row becomes the last and the last the leading. A soft layer that has been pushed to large deflections by one row on the outward stroke is ground the other row now pushes into on the return. Whether a group cycled in soft-over-stiff ground ends with every row sheltered in the soft layer — the second reading becoming true for every row at once because each has been the trailing row in turn — and whether the moment below the interface then grows in every pile rather than in one, is the question the reversing load puts to a factor that was calibrated on a single push, and it is where a pile’s length that the ground gives it meets the history of the ground.