The layer a pile feels

A pile has no length until the ground gives it one found that a pile pushed sideways in ground whose stiffness grows steadily with depth has exactly one length in it, $T = (EI/n_h)^{1/5}$, and that every answer about it — the head deflection, the largest moment and where it is, the depth below which nothing happens — is a multiple of that length. The pile that is too short to bend used the same length to tell a slender pile from a stubby one. Both put the pile in one ground.

Real ground comes in layers. A river crossing puts piles through soft silt into dense sand; a reclaimed site puts them through fill into the clay the fill was placed on; an offshore foundation starts in a loose surface layer the sea has disturbed. Both essays named the case and set it aside: “layered profiles, where a single characteristic length stops existing”. This essay puts one layer over another and asks which of them the pile feels.

One pile, two grounds #

The pile is the one used before: 20 m long, a bending stiffness of $120\ \mathrm{MN\,m^2}$, its head free and pushed sideways by 150 kN. The stiff ground has a modulus growing with depth at 5 MN/m³, a medium-dense sand, which gives the pile a characteristic length of 1.89 m. The soft ground is ten times softer, 0.5 MN/m³. In the stiff ground alone the head moves 20.4 mm and the largest moment is 219 kN·m; in the soft ground alone, 81.4 mm and 346 kN·m. Ten times softer ground costs four times the deflection and only 1.6 times the moment, which is the fifth root at work: the head deflection goes as $T^3$, the moment as $T$, and $T$ as the ground’s stiffness to the power −1/5.

Which ground the pile asks for #

Before layering anything, it is worth asking where along its length a pile in uniform ground actually uses the ground. A soil reaction does work, and the energy the ground stores at each depth is its stiffness there times the square of the pile’s deflection there. Where the pile hardly moves, the ground is not being asked for anything, however stiff it is.

The weight is a hump near the top. The ground just below the surface is weak — its modulus grows from nothing — and the ground far down is not deflected, so what the pile uses is a band around one characteristic length deep. Half of everything the ground does happens in the top 0.71 characteristic lengths, 1.3 m of a 20 m pile; nine tenths in the top 1.38 lengths. A layer is felt in proportion to its share of this curve, not of the pile’s length. A soft layer 2 m thick is a tenth of the pile and most of what the pile asks of its ground.

The weight is not an invention for the purpose. It is the ground’s term in the pile’s strain energy, and averaging the ground over the pile’s own deflected shape is exactly the Rayleigh quotient that finds a load from a guessed shape, applied to the ground instead of the column.

Soft over stiff, and stiff over soft #

The two curves are the weight curve turned into deflections. A soft layer at the top matters at once and a lot: half a characteristic length of it, 0.9 m, adds a fifth to the head’s movement, and one length adds 70 per cent. Three lengths of soft ground and the stiff ground below might as well not be there. The stiff crust over soft ground works the same way in reverse, and is the more useful result: half a characteristic length of stiff ground at the surface — a compacted platform, a layer of rock fill — nearly halves the head deflection of a pile in soft ground, and one length of it brings the pile within a quarter of what the stiff ground throughout would give.

Both curves are followed closely by the average. For every thickness and both orders, the pile in uniform ground whose modulus is the layered ground weighted by the square of its own deflection moves within 8 per cent of the layered pile. The average was iterated — the deflected shape it weights by is the one the averaged ground itself produces — and it errs slightly on the low side, as an estimate built on an assumed shape tends to: the layered pile bends into a shape a uniform ground never produces, and forcing it into the uniform pile’s shape stiffens it.

The moment the average cannot find #

The head deflection of the layered pile lies between the two uniform piles’, as an average would suggest. Its moment does not. With soft ground 3.8 m deep over stiff, the pile’s largest moment is 427 kN·m — twice the stiff-ground pile’s, and a quarter more than the soft-ground pile’s. Layered ground can bend a pile harder than either of its layers could.

The reason is visible in the shape of the curve. Through the soft layer the pile bends as a pile in soft ground bends, with its moment climbing slowly because the soft ground is resisting little. In uniform soft ground that climb would peak and turn a couple of characteristic lengths down, where the soft ground had built up enough reaction. Here, before it can, the pile meets the stiff ground, which holds it almost as a fixed support holds a cantilever — and a cantilever’s moment is largest at its root. The soft layer supplied the length and the stiff layer supplied the fixity, and neither ground alone supplies both.

Swept over thickness, the moment’s behaviour is the opposite of the head’s. The averaged ground, which followed the head deflection to 8 per cent, underestimates the moment in soft-over-stiff ground by up to 28 per cent, and at exactly the thicknesses where the moment is largest. An equivalent uniform ground gets the head right and the moment wrong, because the head is a global quantity — an integral of the whole pile’s flexibility — and the moment’s peak is made locally, at the interface, which an average does not have.

The reverse arrangement does the reverse. A stiff crust over soft ground holds the pile near the head, and the pile below bends in soft ground under a load already partly carried: its largest moment falls to 169 kN·m, less than in either ground alone. Again the average cannot see it, and here it errs on the safe side.

Where the moment goes #

The depth of the peak tells the same story. In uniform ground it is at 1.34 characteristic lengths, which is the classical result. A thin soft layer barely moves it. From about one length of soft ground the peak follows the interface down, sitting a little below it, until the soft layer is so thick that the pile has found its peak within the soft ground itself and the interface is too deep to matter.

That matters for the pile’s reinforcement or wall thickness, which is usually varied down its length. A pile designed for the moment at 1.34 characteristic lengths of the stiff ground — 2.5 m — has its strongest section in the wrong place when the soft layer is 3.8 m thick and the peak is at 4.1 m. The same applies to where a long pile is spliced: a joint placed below the uniform-ground peak, where the moment has fallen away, can find itself at the layered pile’s peak. And a reinforcement cage curtailed at three characteristic lengths of the stiff ground, 5.7 m, has only a metre and a half of full cage below a peak that the layering has moved to 4.1 m, and the moment a metre and a half below a peak made by a concentrated reaction has not fallen far.

The step in the reaction #

The mechanism is in the reaction. The pile’s deflection is continuous through the interface, and the ground’s stiffness jumps by a factor of ten, so the soil’s reaction jumps by the same factor: from 21 kN per metre just above to 200 just below. The stiff ground grabs the pile along a short band beneath the interface, and a concentrated reaction on a beam is what makes a peak in its moment. If the band’s reaction is large enough it reaches the stiff ground’s own limit — a wedge of soil pushed out ahead of the pile — and the band widens as it yields, which lowers the peak somewhat and moves it deeper. The soft layer above, meanwhile, supplies little reaction over a long length, so the moment from the head load is carried down to the interface almost undiminished — the free length of a cantilever that happens to be surrounded by soil.

This is also the case the lateral load test is worst at revealing. A test that pushes the head and measures its movement is measuring the global quantity the average gets right. The moment, and the depth it sits at, are what fail the pile, and a head-deflection test that matches the design’s prediction does not say they were right.

Two springs in parallel, and why the order matters #

The head deflection behaves as if the two layers were two springs sharing one load, each contributing in proportion to how much of the pile’s deflection it sees. That picture explains why the average works for the head: the pile is a flexible beam, a beam on a bed of springs, and its head stiffness is a sum over depth of what each depth’s ground contributes, weighted by how far the beam moves there. Change the order of the layers and the weights change with them, which is why soft over stiff and stiff over soft are not mirror images: a soft layer at the top sits where the weight is largest, and the same soft ground under a stiff crust sits where the weight has already fallen away.

The moment is not a sum of that kind. It is the result of equilibrium at one section — the head load’s lever arm less the moment of the soil reactions above the section — and the reactions above a section in soft-over-stiff ground are small until the section reaches the stiff ground. A sum over depth can be averaged; a lever arm that ends at an interface cannot.

The same profile, pulling down #

The profile that bends a pile hardest sideways is also the one most likely to be dragging it down. Soft ground over stiff is usually a soft clay or a fill over a bearing stratum, and soft clay under the weight of new fill goes on settling past the pile’s shaft, turning its friction round and hanging its weight on the pile. So the pile carries its largest axial force near the interface, where the settling ground stops and the stiff ground starts, and — from the figures here — its largest bending moment just below the same interface. Two checks that are usually made separately, by different people, are looking at the same metre of pile.

And the soft layer does not stay as soft as the site investigation found it. A clay that consolidates under a load gains strength and stiffness as the water leaves it, so a pile through a soft layer under a new embankment sits in ground whose upper layer stiffens for years. The pile’s head deflection falls with it and its moment peak moves back up; the day the embankment is finished is the day the soft-over-stiff arithmetic applies in full.

Scour, which is a soft layer of nothing #

The limit of a soft upper layer is a layer of no stiffness at all, and a river provides it. Scour around a bridge pier’s piles removes the top of the ground during a flood, and the piles become cantilevers standing free for the scoured depth before they reach ground. Their head deflection then grows as the cube of the free length and their moment linearly with it — the curves here with the soft ground’s modulus taken to zero. A pile designed for the soil around it on the day it was driven is, after a flood, a pile in layered ground whose upper layer is water, and its peak moment has moved down to the new ground surface with the scour, as the peak moved with the interface above.

The layered pile, by hand #

A rough hand estimate follows from the weight curve. With a soft layer one characteristic length thick, the top 1.89 m of ground, which in the uniform pile does 73 per cent of the ground’s work, is ten times softer; the other quarter is unchanged. An averaged modulus is then about $0.73 \times 0.5 + 0.27 \times 5 = 1.7$ MN/m³ weighted by energy — a crude version of what the figures iterate — and in ground of 1.7 MN/m³ the characteristic length is $(120 \times 10^{12}/0.0017)^{1/5} = 2.34$ m. The head deflection of a free-headed pile is $2.435\,PT^3/EI = 2.435 \times 150{,}000 \times 2{,}340^3/1.2\times10^{14} = 39$ mm, against 35.3 from the layered solution: the right size, a little high because the crude weighting charges the soft layer with too much of the work. The moment such an estimate gives, $0.772\,PT = 271$ kN·m, is too low by the same margin the figures show.

A Winkler ground, linear and layered cleanly #

The ground is a bed of independent springs, each with a stiffness that grows linearly with depth in its own layer. Real soil is continuous: the stiff ground just below an interface is loaded partly through the soft ground above, and the soft ground just above is stiffened by the stiff ground below, so the step in stiffness is smoothed over a distance comparable to the pile’s diameter. The step in the reaction is correspondingly softened, and the peak moment somewhat reduced.

The springs are linear. Near the head the soft ground yields at small deflections, which a p–y curve would show as a stiffness falling with load, and the soft layer’s effect grows with load rather than staying a fixed proportion of it. The stiff ground’s reaction just below the interface is also the reaction most likely to reach its limit, and a limited reaction spreads the band that makes the moment’s peak.

The interface is horizontal and sharp, and the pile vertical. A sloping interface puts different piles of a group into different layers, which is how the piles of one pile cap come to carry very different moments under one load.

What the pictures cannot show #

That soil layers are not what a borehole log draws. A log shows a boundary at a depth, and the ground between two boreholes has boundaries at other depths, in which the characteristic length is several metres and a metre’s error in the depth of the interface moves the peak moment by the same metre. The figures show what the pile does in a known ground; the design has to be made in a ground known to within the spacing of the boreholes.

Nor can they show the pile’s own section. Every figure takes the bending stiffness to be constant, and a pile whose wall or reinforcement is reduced below the depth of the uniform-ground moment peak has less stiffness exactly where the layered ground puts its largest moment.

Still open: the group in layered ground #

A single pile in layered ground is a question about one beam and one soil profile. A group of piles under one cap shares the soil between them: the leading piles shelter the trailing ones, and the p-multipliers that account for it were calibrated in uniform ground. In layered ground the shadow each pile casts is deeper in the soft layer and shallower in the stiff, and the band of concentrated reaction beneath the interface is shared by piles whose deflections there differ. Whether the group effect in layered ground is the uniform-ground effect applied layer by layer, or whether the interface changes which pile in the group carries the largest moment, is the question that follows this one into the foundations that are actually built.