Nine piles, and four times the settlement

Three problems in this collection have the same arithmetic and are never taught together. A bolt group under an eccentric shear. A section under axial force and biaxial bending. And a pile cap.

In each case a rigid plate connects a set of discrete elements to a load that does not pass through their centroid, and in each case the share each element takes is

$$P_i = \frac{N}{n} + \frac{M_x\,y_i}{\sum y^2} + \frac{M_y\,x_i}{\sum x^2}$$

— three terms, in that order, from the same three equations of statics applied to a rigid cap. The bolt that carries more than its share is that arithmetic in a connection; this is the same arithmetic in the ground, and everything that is different about it comes from what the elements are standing in rather than from how the load divides.

Which free body produced the number #

The cap, taken as a rigid body, with the applied load on top of it and the pile heads pushing up from underneath.

Three equations — vertical force and two moments — and $n$ unknowns. That is short by $n-3$, and the extra information is the assumption that the cap is rigid, so the pile heads lie on a plane. Equal stiffnesses then give a load proportional to displacement, and the plane’s three parameters are exactly the three the equations fix.

Two corollaries fall out of that immediately, and both are worth stating because both are regularly got wrong on drawings.

A pile at the centroid carries the mean load and no moment at all. It adds to $n$ and to neither $\sum y^2$ nor $\sum x^2$, so every other pile’s moment term is untouched. Somebody worried about the corner piles who adds one in the middle has bought nothing.

Piles a long way from the centroid do all the moment work. The share goes as $y_i/\sum y^2$, and $\sum y^2$ is dominated by the outer rows, so widening the cap by one row helps out of proportion to the pile it adds. Which is the same conclusion the material far from the middle reaches about a section, arrived at by the same sum.

Where the analogy stops #

Bolts do not know about each other. Piles do, through the ground between them, and the two things that produces are of quite different importance from the ones usually emphasised.

Block failure is the one everybody checks. Piles close together in clay drag the soil between them down as a single block, and the capacity of that block — perimeter shear plus base bearing over the whole plan — can be less than the sum of the individual pile capacities. It is a real mechanism and it belongs to the plan of the group rather than to any pile.

The arithmetic of when it governs is instructive. The block’s perimeter grows as $\sqrt{n}$ and the sum of the piles’ perimeters grows as $n$, so the two cross — and they cross at a number of piles, not at a spacing. A pair of piles at three diameters has no group problem at all. Sixty-four of them at two diameters do.

On the three-by-three group drawn, at five diameters, the block check gives a ratio of 4.5 — it does not come close to governing. Bring the spacing to two diameters and a five-by-five group gives 1.23; an eight-by-eight gives 1.01; a nine-pile group in soft clay with long piles and a high adhesion factor gives 0.92. The check that is always made governs in a corner of the parameter space that most groups are not in.

What the block check is actually asking is worth drawing once at the scale of a single pile, because the mechanism is the same one and the group version is only larger. A pile’s base pushes a wedge of soil down and a pair of wedges sideways and up, and the capacity is the load at which that motion becomes possible.

The effect that does govern #

Settlement. And it is not in the efficiency formula anywhere.

A single pile loaded at its head carries most of the load in shaft friction, and the soil it stresses is a sleeve a few diameters across. A group of piles at three or four diameters has stress bulbs that overlap all the way down, and the soil beneath the whole group is loaded to a depth comparable with the plan dimension of the group rather than with the diameter of a pile.

So the ground under a group is loaded much deeper, and it settles much more. Randolph and Wroth’s interaction factor — the settlement of one pile caused by another, as a fraction of its own — falls only logarithmically with spacing, so a pile is affected by every other pile in the group and not merely by its neighbours. Summing over the plan gives a settlement ratio for the group: 2.3 for four piles, 3.9 for nine, 5.7 for sixteen, 13.0 for a hundred.

That is the number a pile group is designed by, on almost every project, and it is the number that makes the design a settlement calculation with a strength check attached rather than the other way round. The beam that sits on the ground is the same conclusion for a footing: bearing capacity is a collapse load and settlement is a serviceability displacement, and the second decides the size.

Which pile settles, and what the cap does about it #

The elastic distribution above assumed a rigid cap, and the settlement argument shows why that assumption is doing more work than it looks.

If every pile had the same stiffness, a rigid cap would give them equal loads under a concentric force. They do not: the centre pile of a group is softer than a corner one, because it has neighbours on all sides pushing the ground down around it, while a corner pile has neighbours on two sides only. So under a rigid cap the corner piles attract more load than the equal share, and the centre piles less — by a factor of two or more in a large group.

The consequence is the reverse of the intuition a designer brings from the strength calculation. The corner piles are the ones that reach their capacity first, and the middle of the group is under-used. A “creep pile” design deliberately exploits this: use fewer piles than capacity demands, let them all work near their limit, and let the raft carry the rest — which turns the group into a beam on springs with the piles as the springs and the raft as the beam.

And a flexible cap does none of this. A large raft on piles is not rigid at all across its own width, so the pile heads do not lie on a plane, and the whole three-term arithmetic is a local approximation valid over a few pile spacings. The honest analysis then is a plate on a set of non-linear springs, which is a computation rather than a formula.

The general shape of that answer is a set of discrete supports of finite stiffness under a member of finite stiffness, and the share each one takes depends on the ratio of the two. It runs continuously between two limits nobody meets: a perfectly rigid cap, where every pile head lies on a plane and the three-term formula is exact, and a perfectly flexible one, where only the piles under the load take anything. Every real cap is somewhere between, and where it sits is a ratio rather than a category.

What a single pile is carrying, and where #

Before the group argument can be trusted, it is worth being clear about what one pile does, because the group effect is different for the two mechanisms and the split between them is rarely known.

A pile carries load in shaft friction along its length and in end bearing at its base, and the proportions are wildly different between pile types. A long slender bored pile in stiff clay may take 85% of its load in friction; a short driven pile onto rock takes nearly all of it in end bearing. The distinction matters here because the two produce completely different stress fields in the ground.

A friction pile sheds its load continuously down its length, so the soil it stresses is a long sleeve and most of the stress is delivered high up. An end-bearing pile delivers its whole load at one level, at the base, and stresses a bulb beneath it. Interaction between friction piles is strong, because their sleeves overlap all the way down. Interaction between end-bearing piles on rock is negligible, because rock is stiff and the bulbs are small.

So the settlement ratios above belong to friction piles. A group of piles bearing on rock settles very nearly as much as one of them — which is why a designer’s first question about a group effect ought to be what the piles are standing on, and why the same efficiency table cannot cover both.

The ground’s stiffness dominates all of it, and it does so through a fourth root. The characteristic length over which a load is felt in the soil moves as the fourth root of the stiffness ratio, so an order of magnitude of difference between one soil and another is less than a factor of two in the reach — while the pressure under the load moves by very much more. That is why the pile type and the founding stratum settle the group question before any arithmetic about spacing begins.

The number that a load test gives, and the one it does not #

Almost every significant pile group is preceded by a test on a single pile, and it is worth being precise about which of this essay’s numbers such a test settles.

It settles the capacity of one pile in that ground, which is the hardest quantity to compute and the easiest to measure. A maintained-load test to twice the working load, or a rapid test, or a bi-directional cell test, gives a load–settlement curve for one pile with all its construction defects in it, and there is no calculation that competes with it.

It does not settle the group settlement at all, and this is where the argument bites hardest. The tested pile settles a few millimetres at working load because it is stressing a sleeve a metre or two across. The group of nine, at the same load per pile, settles four times that, because it is stressing the ground to twenty metres. A designer who takes the test’s settlement as the group’s has under-predicted by a factor that grows with the number of piles — and the error is worst on the largest, most expensive groups.

That is a general property of testing a component of a structure whose parts interact, and it is worth naming: a test on one element measures the element and not the interaction, and the interaction is what the group added. The same warning applies to testing one bolt of a long joint, one anchor of a group, or one panel of a floor that will be walked on as a whole.

Uplift, which is a different pile #

A moment large enough to reverse the sign of the outer term puts a pile into tension, and a pile in tension is not the same pile.

Its base contributes nothing at all — a pile cannot pull on the soil beneath it — so the capacity is shaft friction alone, and the shaft friction available in tension is generally taken as less than in compression, because the pile’s Poisson contraction under tension reduces the normal stress on the shaft. Something like 70% is a common assumption, and it is an assumption rather than a measurement in most ground.

Two things follow. A cap designed for a large moment is often governed by its tension piles, which the compression arithmetic will not reveal. And the connection changes: a compression pile needs a cap that bears on it, and a tension pile needs reinforcement anchored into the cap and into the pile, which is a detail that has to exist before anybody notices it is needed. The failure that is in the concrete is the way that detail goes wrong.

The cap is more concrete than the piles #

Every quantity in this essay is a pile load or a ground movement. The object that produced them is the cap, it is assumed rigid throughout, and it is the largest single item in the foundation.

Size it for the group drawn. Nine 600 mm piles at 3.0 m centres — five diameters, which is where the efficiency argument would like them — give a cap $2 \times 3.0 + 0.6 + 2 \times 0.3 = 7.2$ m square. Its depth follows from the moment the outer row applies about the column face: three piles at 1,250 kN, at a lever of 2.5 m, is

$$M = 3 \times 1{,}250 \times 2.5 = 9{,}375\ \text{kNm}$$

which across 7.2 m of width and an effective depth of 1.5 m needs $A_s = M/(0.9df_{yd}) = 15{,}960$ mm², or about 2,200 mm² per metre in each direction, in both faces.

So the cap is 7.2 × 7.2 × 1.8 m — 93 m³ of concrete. The nine piles it joins, at 20 m long, are 51 m³. The cap is nearly twice the concrete of the foundation it is described as connecting, and it has not appeared in a single equation on this page.

That reframes the spacing decision entirely. Bring the piles to 1.8 m — three diameters, where the group effects the essay has been about start to matter — and the cap becomes 4.8 m square and about 1.2 m deep: 27.6 m³, a saving of seventy per cent.

So the price of the tighter group is a quarter more settlement and a corner pile working 14 per cent harder, and the block check that is always made is still nowhere near governing.

The spacing of a pile group is a concrete decision wearing a soil-mechanics argument, and that is why real groups sit at two and a half to three diameters rather than at the five the interaction factors would prefer. The soil argument wants them apart; the cap pays for every metre, in a member whose depth grows with the lever arm and whose plan grows with its square.

Where the model stops #

The piles were identical and vertical. A raked pile takes lateral load axially and changes the whole arithmetic — the cap now has horizontal equilibrium to satisfy as well, and the “plane of pile heads” assumption becomes a rigid-body movement with three components in plane.

The ground was uniform. It is not, and a pile founded on rock behaves nothing like a friction pile beside it; a group with both is a group of very different springs, and the stiff ones take almost everything. That is the stiffest path takes the load with a factor of ten in the stiffness ratio.

Lateral load was ignored. A pile group under horizontal load has a shadowing effect of its own — the front row takes far more than the back rows, because the back rows are pushing into soil the front row has already displaced — and the reduction factors for it are empirical and large.

Negative skin friction was ignored. Ground that is settling around a pile — a recent fill, a consolidating clay, a lowered water table — drags the pile down rather than holding it up, and the shaft friction over that depth reverses sign and becomes a load. It is a load that arrives with no structure on the pile at all, and it is the movement nobody applied with a shaft to act on.

And the piles were assumed to have been built. The single largest source of variability in a pile group is construction: a bored pile with a soft toe, a driven pile that met an obstruction, a continuous-flight-auger pile with a necking. Which is why pile testing exists, and why a group’s design is usually a statement about the weakest pile rather than about the average one.

The generalisation #

The idea worth carrying is that discrete elements sharing a continuum interact, and elements sharing a plate do not — so the same arithmetic gives a good answer for one and a partial answer for the other.

The three-term share is exact for a bolt group because the bolts are connected only through the plate, and the plate is the thing assumed rigid. It is a first approximation for a pile group because the piles are connected through the plate and through the soil, and the second connection is invisible in the equations. Everything that is difficult about pile groups — the efficiency, the settlement ratio, the corner piles taking more, the front row of a laterally loaded group taking more — is that second connection asserting itself.

The habit that follows is to ask, of any group of elements sharing a load: what else are they sharing? Bolts share a plate. Piles share soil. Bracing members share a floor diaphragm. Columns share a frame. Fasteners in a timber joint share a piece of timber that can split — the smallest of six failures is what happens when they do. Wherever the shared thing is a continuum rather than a rigid connector, the equal-share arithmetic is where the analysis begins and not where it ends.

There is a second habit beside it, and this essay is a clean case of it: ask which check is the one that governs, and how often. Group efficiency is computed on every pile group ever designed and decides almost none of them; settlement decides nearly all of them and is computed on rather fewer. A profession’s checklist is a record of what once went wrong rather than a ranking of what usually does, and the two drift apart quietly — which is the same observation the check that cannot see the error makes about verification.