The plane where the pile and the ground agree

A pile driven through ground that is still settling carries more than is put on it. The ground that hangs on instead of holding up drew the free body: above some depth the ground moves down past the shaft and its friction pulls the pile down; below it the pile moves down past the ground and the friction holds it up. The depth where the friction turns round is the neutral plane, and the force in the pile is greatest there — the head load plus all the drag above it. That essay found the plane from equilibrium: the load and the drag above it balanced by the shaft below and the base, with every metre of friction fully developed in one direction or the other and the base’s resistance given, 300 kN.

It ended on the other way of finding the plane, which is the way the plane is defined. The friction turns round where the pile stops moving down faster than the ground and starts moving down slower — where the two settle by the same amount. That is a statement about displacements, not forces, and a calculation built on it has to know how far the ground settles, how much movement the friction needs, and how the toe resists. It also stops needing the one number equilibrium cannot supply: the toe’s resistance, which is whatever the toe’s own settlement mobilises.

The pile in the ground, by compatibility #

Take the pile of the earlier essay: 24 m long, 600 mm across, concrete, carrying 800 kN, in ground whose friction is $\beta\gamma'z$ with $\beta = 0.3$ and $\gamma' = 9$ kN/m³. Let the ground’s top 18 m settle — 100 mm at the surface, less with depth, nothing at 18 m — as a soft layer consolidates under new fill, with the pile’s lower 6 m in firmer ground.

Make the friction depend on movement, as friction that is whatever it needs to be always does. Where the ground and the pile move together there is none; as one moves past the other the friction grows, reaching its full value $\beta\gamma'z$ once the relative movement is 5 mm, in whichever direction the movement is. Make the toe resist by settling: half of its 1,500 kN capacity at 15 mm, rising toward the whole of it as it settles further. And make the pile an elastic column, shortening under the force in it.

Then guess how far the toe settles, work out what that mobilises at the toe, and climb the pile: at each step the force grows by the friction there and the settlement grows by the pile’s shortening, and the friction there depends on how the pile’s settlement compares with the ground’s. Adjust the guess until the force at the head is the 800 kN applied. The neutral plane is not assumed anywhere. It is wherever the solution makes the pile and the ground move together.

The pile settles almost uniformly — 13.6 mm at the head, 10.6 at the toe, the difference its own shortening — while the ground’s settlement falls from 100 mm at the surface to nothing at 18 m. The two lines cross at 15.9 m, and there the force in the pile peaks at 1,407 kN. The toe, having settled 10.6 mm, is carrying 623 kN.

Equilibrium agrees, once it is told the toe #

Now feed equilibrium the toe force compatibility found. With 623 kN at the toe instead of 300, the earlier essay’s closed form puts the neutral plane at 15.91 m — the same depth — and the largest force at 1,444 kN, within 3 per cent. The small difference is a transition zone a few metres long around the plane, where the friction is turning round and is not fully developed either way, which equilibrium’s fully mobilised friction cannot represent.

So equilibrium is not wrong. It is incomplete in a precise way: it needs the toe force as an input, and the toe force is an output of the settlements. Given the 300 kN the earlier essay assumed, equilibrium puts the plane at 13.8 m and the largest force at 1,283 kN — a plane two metres too high and a force a tenth too small, from a toe force that was a guess. The 300 kN is not an unreasonable guess for a toe; it is simply not what this toe does when the ground drags it down.

Drag grows with the ground’s settlement #

Equilibrium treats the drag as all or nothing: the ground either moves past the shaft or it does not. Compatibility makes it a matter of degree, and the degree is how far the ground settles. With 2 mm of settlement at the surface the ground moves less than the pile does under its own load, the friction everywhere holds the pile up, and there is no drag at all. At 10 mm the upper shaft begins to drag, the plane appears at 10.4 m and the largest force is 887 kN. At 32 mm it is 1,219 kN at 14.4 m; at 100 mm, 1,407 at 15.9; at 316 mm, 1,520 at 16.9.

Most of it arrives early. A third of the largest settlement drawn produces more than four fifths of the largest drag, because the friction needs only a few millimetres of relative movement to develop fully and the ground’s settlement soon exceeds that over most of the upper shaft. After that, extra settlement only pushes the plane deeper, slowly, and lengthens the dragging zone by a little.

And the toe follows the ground. Its mobilised force climbs from 184 kN with no drag to 800 kN under the heaviest settlement drawn, because the drag pushes the pile down and the toe resists by being pushed. The toe force equilibrium needed as an input is, in fact, set by the ground’s settlement, and by every other property in the problem along with it.

The pile settles because the ground does #

The pile settles too, and the reason is the drag. With the ground settling 10 mm the head goes down 5.3 mm; at 100 mm, 13.6; at 316 mm, 20.2. At the neutral plane the pile and the ground settle by the same amount by definition, so the pile’s settlement is the ground’s settlement at the plane plus the pile’s shortening above it — and the ground at 16 m, under a surface settling 100 mm, settles about 11 mm.

That is the practical content of downdrag, and the reason its modern treatment begins with settlement rather than with force. The ground is a spring as well as a load here, and the structural capacity of a concrete pile is rarely threatened by an extra 600 kN at 16 m. What the building on it notices is that its piles settle by the ground’s settlement at their neutral planes, and that piles in different parts of a site — near the new fill, away from it — sit in ground settling by different amounts and therefore settle differently from each other, which is the settlement that matters. A pile was supposed to carry the building down past the soft ground; downdrag brings part of the soft ground’s settlement back up the pile.

Three quarters, not half #

The earlier essay found a striking rule in its closed form: every kilonewton added to the head raises the largest force in the pile by half a kilonewton, because the extra load pushes the pile down, the neutral plane rises to meet it, and the drag above the plane shrinks by the other half. It is a correct consequence of its assumptions, and one of them is that the toe’s resistance does not change.

By compatibility the largest force rises 0.77 kN for each kilonewton added. The extra load pushes the whole pile down, the toe included, and the toe answers by mobilising more of its own resistance — about half of each added kilonewton here. Written out, the closed form’s largest force is $Q/2 + Q_b/2 + aL^2/4$, so its sensitivity to the head load is $\tfrac12 + \tfrac12\,dQ_b/dQ$: a half from the plane rising, and half of whatever the toe picks up. With the toe fixed, a half; with this toe, about three quarters.

So the rule overstates how well a pile in settling ground shrugs off added load, by about half again. That matters when a designer checks a pile’s structural capacity at the neutral plane under the full factored load, and it matters more for what the rule seemed to say about it — that downdrag is a displacement the pile shares with its load rather than a force added to it. It is partly both.

Friction that has not had room to turn round #

The friction diagrams show where the two methods part. With the ground settling 200 mm, the friction is fully negative down to a few metres above the plane, turns round over about a metre, and is fully positive below — almost exactly equilibrium’s picture, which is why equilibrium gets the large-settlement case nearly right once it has the toe. With the ground settling only 10 mm, the upper shaft’s friction is partly mobilised, never reaching its full value, because the ground there has not moved far enough past the pile; the plane is at 10.4 m and the force at it is much smaller.

The movement the friction needs is the property that decides the width of that transition. Make it 2 mm and the largest force at 100 mm of ground settlement is 1,429 kN; 5 mm, 1,407; 10 mm, 1,370; 20 mm, 1,263, with the plane at 15.3 m. Stiffer interface, sharper turn, more drag — and the fully-mobilised limit that equilibrium assumes is the limit of a friction that needs no movement at all.

The pile with nothing on it #

One number from the compatibility calculation has no equivalent in equilibrium. Take the 800 kN off the head entirely.

Equilibrium with a fixed toe force still finds a plane and a drag; compatibility, with the toe mobilised by movement, finds that the pile with no load on it at all carries 726 kN at 17.4 m, sits on a toe carrying 268 kN and settles 4.2 mm. The ground is holding the pile up below the plane and dragging it down above, and the toe takes the difference. Adding the building’s 800 kN to that does not give 1,526 kN: the pile then carries 1,407, because the load pushes the plane up from 17.4 m to 15.9, sheds drag as it rises and hands part of itself to the toe: the 800 kN adds 681 to the worst force, the same sharing seen from the other end. A pile driven and left unloaded in settling ground is already carrying most of the force it will ever carry, before anything is built on it, which is why a pile load test made after the fill has been placed reports a capacity that already has the drag inside it.

The toe force, by hand #

The compatibility answer can be reached with nothing more than the earlier essay’s closed form and a pencil, by iterating on the one number equilibrium could not supply. Start from the assumed toe force, 300 kN. Equilibrium puts the plane at 13.8 m. At that depth the ground settles $100 \times (1 - 13.8/18) = 23.5$ mm, and since the pile settles with the ground at the plane, its toe settles that less the pile’s shortening between the plane and the toe — about 1 mm, the force averaged over the 10 m below the plane divided by $EA$ — so 22.5 mm. The toe’s hyperbola gives $1{,}500 \times 22.5/(22.5 + 15) = 900$ kN. Far more than was assumed.

Try halfway, 600 kN. Equilibrium puts the plane at 15.8 m, where the ground settles 12.4 mm; the toe settles 11.4 and mobilises 648 kN. Try 624: plane at 15.9 m, ground 11.6 mm, toe 10.6, toe force 620. It has converged in three steps, to 622 kN at 15.91 m — the same answer as the full calculation, with the toe settling the same 10.6 mm.

The iteration shows the structure of the problem better than the full solution does. Equilibrium relates the plane to the toe force; compatibility relates the toe force to the ground’s settlement at the plane. Two curves, and the answer is where they cross. Equilibrium alone has one curve and a guess.

Movement-dependent friction, a hyperbolic toe and a straight settlement profile #

The friction is rigid-plastic after a fixed movement. Real interfaces mobilise friction gradually and can soften after the peak; a softening interface would reduce the drag at large settlement, and the 5 mm used here is a typical figure rather than a measured one.

The toe’s resistance follows one hyperbola. Toe response depends on the soil under it and on how the pile was installed; a stiffer toe takes less of each added kilonewton and moves the sensitivity back toward a half, a softer one toward one.

And the ground settles in a straight line. A consolidating layer settles most near its drainage boundaries and the profile with depth is curved; the neutral plane’s depth depends on where the pile’s settlement meets that curve, and a different profile moves it.

Time, groups and the long term #

They cannot show time. Consolidation takes years — the ground arrives after the weight — and the drag grows with it; the curves here are states the pile passes through, from the first settlement to the last, rather than a single design condition.

They cannot show a pile pushed sideways. The same settling ground that drags a pile down also pushes on it sideways wherever the fill spreads, and a pile in the soft layer is a beam whose length the ground decides; the axial force found here is the compression that beam has to carry while it bends.

They cannot show a group. In a group of piles the ground between them is held up by all of them at once, and the drag on an interior pile is limited by the weight of the block of ground it shares with its neighbours.

And they cannot show a slip coating. A pile coated to shed its drag has a friction that needs more movement and reaches less, which is the same calculation with a different interface — and a coating that ends at the wrong depth puts the plane where compatibility says, not where the coating’s designer meant it.

Settlement first, force second #

The neutral plane is where the pile and the ground settle alike. For the 24 m pile carrying 800 kN in ground settling 100 mm, 15.9 m, with 1,407 kN in the pile there.

Equilibrium finds the same plane only when told the toe force, and the toe force — 623 kN here, not 300 — is set by how far the ground drags the toe down.

Drag grows with the ground’s settlement, from nothing at 2 mm to 1,520 kN at 316 mm, and the pile’s own settlement grows with it, from 5.3 mm to 20.2.

And each kilonewton on the head costs three quarters of one, not half, because the toe takes some of it — the half survives only with the toe held fixed.

Still open: the ground that rises instead #

Every settlement here is downward. Swelling ground — a clay that takes up water after the trees on it are felled, or after a basement relieves its overburden — moves up past the pile instead, and the same compatibility puts the pile in tension above a neutral plane at the depth where pile and ground rise alike. A concrete pile with little tension reinforcement is then being pulled apart at a depth set by the same arithmetic, and the toe that resisted settlement by moving down now holds the pile by moving up. Whether the heave plane sits where the settlement plane did, mirrored, or whether the toe’s different behaviour in uplift moves it, is the question the swelling site asks of this one.