Nine piles and a third of the pull

The pile the ground pulls apart put one pile in clay that swells. The clay rises round the upper shaft and drags the pile with it, the toe lifts away from the ground beneath with nothing to hold it, and the deep part of the shaft, which stands in ground that is not moving, becomes what holds it down. Between the two the pile is in tension, a concrete column being pulled apart at 17 m depth with a house on top. The essay ended on the fact that nobody builds on one pile. Piles in swelling clay come in groups, under ground beams and strip foundations, and the clay between them is held by all of them at once.

Two opposite guesses were on the table. Clay cannot rise past one pile without rising past its neighbours, which pull it back, so perhaps a group shares its uplift and each pile carries less. Or the corner and edge piles, with clay rising freely on their outer sides and their neighbours’ restraint on only one side, might carry more than a pile alone would, as the corners of a settling group take more than their share of the load. Nine piles, and four times the settlement had already found that a group in settling ground interacts strongly through the soil between its piles, always to its disadvantage. The question is what that same interaction does to a group when the ground is rising.

Each shaft is a cylinder, and so is the clay beside it #

A shaft carrying a shear stress $\tau$ moves the soil around it. Randolph and Wroth’s picture of this is a set of concentric cylinders of soil in equilibrium with the shaft, the stress falling as $1/r$ and the movement as a logarithm of radius, so that the soil at $r$ moves by

$$w(r) = \frac{\tau d}{2G}\,\ln\frac{r_m}{r},$$

out to a radius $r_m = 2.5L(1 - \nu)$ beyond which the disturbance is taken to have died away, here 30 m. Put $r = d/2$ and that same expression is the pile’s own elastic slip against the far soil, so a shaft’s stiffness and its effect on a neighbour are the same formula at different radii. The shaft law here is that, capped at the friction $\beta\sigma'_v$.

The neighbours’ movements add. So the clay beside any one pile is not the free heave $s(z)$ but

$$s_i(z) = s(z) + \sum_{j \ne i} \frac{\tau_j(z)\,d}{2G}\,\ln\frac{r_m}{r_{ij}}.$$

A pile whose upper shaft is being dragged upward has a positive $\tau$ on it, and the reaction on the clay is downward, so its neighbours see clay that has been held back. A pile whose deep shaft is being held down has the opposite, and pushes up on the clay beside its neighbours’ deep shafts. Both signs matter, as will be seen.

The group is a square of nine, 24 m long and 0.6 m in diameter at three diameters’ spacing, each pile carrying 200 kN in clay heaving 100 mm at the surface, with the heave falling to nothing at 18 m. Friction is 0.3 of the effective overburden. The clay’s shear modulus is 5 MPa, which is a low value appropriate to a softened, swelling clay. The soil in the solution is taken as elastic and the neighbour terms as linear, while the shaft law is not. A pile alone in this picture is in tension to 403 kN at 16.5 m, where the earlier essay’s 520 had a stiffer shaft law, reaching full friction in 5 mm; the 403 is the fair comparison here because it has the same soil as the group. A pile has no length until the ground gives it one makes the same point sideways: what a shaft does is a property of the ground beside it, and the number has no meaning without the ground’s.

The force down each pile #

The answer to the first question is plain in the figure. Every pile of the group is pulled less than a pile alone. The corner, the worst of them, is at 214 kN, 53 per cent of the single pile. The edge is at 123. The pile in the middle, with neighbours on all sides, is at 12, which on a pile of this size is effectively nothing. Averaged over the nine, the group’s peak tension is 151 kN: more than three fifths of the pull has gone.

So the fear that a corner pile is exposed and pays for its neighbours’ shelter is wrong in the direction it was stated. A corner pile’s outer face is no more exposed than a single pile’s whole circumference is, and its inner side has neighbours holding clay back. Corners do carry more than the inside of the group, and the difference is large, so the group does distribute its uplift unevenly. It just distributes it unevenly below what one pile would take.

From the head to about 9 m all four lines lie on top of each other, and that is informative. Over the top of the shaft the clay is rising so far past the pile that the friction is at its limit, $\beta\sigma'_v$, on every shaft alike, group or no group. The friction there has no idea how far past it the clay is going: a shaft at its limit drags no harder for being overtaken faster. That length of the shaft has simply taken the head’s 200 kN to nothing at about 9 m, and it does the same on every pile. The depth where force and movement change sense is the same plane where the pile and the ground agree that a settling site has, with its signs reversed. The whole of the group effect lives in the lower part of the shaft, below the depth where the friction has stopped being saturated. That is where the differences between the piles begin.

What the clay does between the piles #

Beside the piles the clay is not doing what it would do alone. At 10 m free heave would have been 44 mm. Beside the inside pile the clay has risen 16 mm, beside the corner 23. Lower down the reverse holds: below 18 m free clay stands still, but beside the inside pile it has been lifted by 15 mm, the amount by which the piles themselves have been lifted. The inside pile moves 15 mm, the clay beside it moves about 15 mm from a depth of 9 m to the toe, and a pile that moves with its clay has no friction on it.

This is the mechanism. The middle of the group has become a block of clay and piles moving as one thing, and it has been lifted by the friction over the top 9 m of the shafts around the outside and inside it. The inside pile has no relative movement at any depth below 9 m, so it has no tension to develop and nothing deep to pull against, and the upper friction does nothing to it but take up its head load. The outer piles sit at the block’s boundary. Their outer faces see undisturbed clay at some distance, and they are held by their deep shafts against the clay outside the group, so the tension of the whole group ends up there.

There is a second, less welcome reading of the same line. The deep shaft is doing less because the clay around it has been lifted, which is the second sign in the sum. A group’s deep clay is pushed up by all the deep shafts in the group, and so the group’s whole movement is a rise of 15 mm, nearly twice the 8.9 mm of the pile alone. The building has less tension in its piles and more movement at its base, and a row of frames is not a foundation is a reminder of what that movement can mean when it is not uniform under the frames carried on top.

How much heave the shelter lasts for #

A single pile’s tension rises quickly with heave and then saturates, as the friction above it does, which the earlier essay showed. The group’s lines sit below that curve, and they have a different shape: they begin later, then run parallel for a time, and go on climbing long after the single pile has levelled off. Below about 80 mm the inside pile has no tension at all, and a small heave of 40 mm leaves the corner at a third of what a pile alone would carry. A shallow heave is not able to overtake the block, and it would be a mistake to read that as there being no heave to design for: the group still rises, whether or not a tension develops.

At large heave the sheltering shrinks. In absolute terms it is steady, about 136 kN off the corner at 300 mm against 189 at 100, as if a fixed quantity of movement were being taken from the heave; proportionally, 26 per cent off at 300 mm against 47 per cent at 100. That has to be the case: a neighbour’s shelter is proportional to its shaft stress, and that stress has a limit, whereas the free movement is limited only by the geology. At enough heave the piles’ shelter is a small fraction of the rise, and every pile approaches the single pile’s friction-limited tension from below.

So a site with a modest swell potential gets the largest benefit from the group. A 40 mm site has a third of the tension at a corner and none in the middle. It is at the extreme, in 300 mm of swell on a hot site where the trees have just been felled, that the single-pile number becomes the right one again, and even there the corner is at three quarters of it.

The spacing that shelters #

It is natural to ask whether a wider group gets rid of the interaction, as it does in settling ground. In the sense of the figure it does not, at any spacing a foundation would use. The soil’s movement falls with the logarithm of distance, and a logarithm is a very slow function. At 6 diameters, twice the normal spacing, the corner has only moved from 214 to 242, and the inside pile from 12 to 77. At 15 diameters, 9 m apart, a group that nobody would regard as a group, the corner is still at 78 per cent of a pile alone. Piles spaced at 3 diameters to get their bearing out of the ground are well inside this range, so the shelter comes without anyone having designed for it.

At the other end of the figure, it does not gain very much by going closer, since the shaft law’s own cylinder is stiffer than any neighbour’s. Between 2 and 3 diameters the corner moves from 208 to 214 kN. What decides the amount of shelter is the number of shafts inside the radius $r_m$ and the logarithm of that radius over the spacing, and a designer has already fixed both of those on other grounds.

More rows #

A larger group makes everything better, but mostly the inside of it. The friction doing this is the friction the ground that hangs on instead of holding up followed down a pile in settling ground, with its sign reversed. Two rows, four piles, are already down to 246 kN each, and the whole of what the corner stands to gain is largely spent by then. At six rows the corner is still at 187, while the mean has fallen from 151 to 54. All of the group’s tension is now concentrated in the 20 piles of the perimeter, and among them the corners take the most.

This is the shape a designer would want to reinforce, and since the group’s corners are its worst piles they deserve the reinforcing steel while the inside piles need almost none. It runs the same way as the distribution of head loads in a settling group, whose stiff corners attract the load, though for the opposite reason. The earlier essay gave 1,196 mm² of steel for the single pile’s tension at 17 m. A corner in this group needs a little over half of it, and the pile in the middle needs the minimum. Whether saving that steel is worth having two sorts of cage on one site is a practical matter, and the mistake a plan could easily make is to reverse it, on the idea that the middle is the most confined and so must be working hardest.

Under a rigid cap #

So far each head has carried its own 200 kN. A pile cap joins them, forcing the heads to one movement and letting the load be shared as it comes. Under a small heave the pattern is the classic one for a group, with the stiff corners carrying more and the sheltered inside pile less: 233 kN against 132 at 5 mm of heave. The soil between the piles settles a little with them, and the outer piles have stiffer soil to work with.

Then it reverses, because heave reverses the sense of what is pushing. At 100 mm of heave the three heads are about equal, and at 200 mm the inside pile is carrying 300 kN and the corner 140. The block in the middle is being pushed up less than the clay around it is pushing the corners, the corners are carried up with a larger friction force, and the cap transfers its load to the inside pile to keep the heads level. An ordinary group analysis gives only the first half of this figure, and the second half is the one that loads the inside pile heavily in compression. A pile in the middle of a heaving group is not the safe one to under-design at its head, even as it is safe at its deep cross-section.

The tension below hardly notices the cap: 219, 118 and 15 kN at 100 mm against 214, 123 and 12. A head force has moved by up to a quarter of its value, but the tension is set in the deep shaft by the friction there and by the clay’s own movement, and the head reaches it only by moving the neutral depth a little.

The hand check, and one that does not work #

At the depth where all the piles part company, the head’s 200 kN has been taken off by friction. The upper shaft carries $\pi d \int \beta\gamma' z\,dz = 1.885 \times 2.7\, z^2/2$, which reaches 200 kN at $z = 8.9$ m. The figure’s crossing is at 8.9 m.

For the sheltering, the coefficient on a neighbour at three diameters is $(0.6/10{,}000) \times \ln(30/1.8) \times 1{,}000 = 0.17$ mm per kPa. The corner has eight neighbours, from 1.8 to 5.1 m away, whose logarithms add to 18.1, so all of them together are worth 1.09 mm per kPa. At 10 m the friction limit is 27 kPa: if every neighbour were at it, that would hold back 29 mm of the free 44, leaving 15. The computed rise is 23 mm, because the neighbours are not all at the limit at that depth, and the sum gives an upper bound on the shelter and the right size for it. The inside pile cannot be done by hand: its result depends on the near-cancellation of the upper and lower effects, and it needs the solution.

What the cylinders leave out #

Swelling is not an imposed movement. Clay swells because it has taken up water and its effective stress has fallen, and the amount it rises depends on what is holding it down. A shaft that holds clay back raises the stress in it and reduces the swell itself, which the additive picture leaves out. The heave is imposed here as a movement of the soil at some distance, with the neighbours’ effect laid over it, and that is a fair sketch of what the piles do to the soil’s movement and not a theory of the clay.

The modulus decides the amount. The clay’s shear modulus sets the size of every sheltering term, and of the pile alone. At 1.5 MPa the pile alone is in tension to 216 kN and the corner to 54, the edge and the middle next to nothing, since soft clay yields round a shaft rather than dragging it. At 20 MPa the pile alone is at 545 and the group at 427, 349 and 281: the mean is 69 per cent of the single pile instead of 37, and the claim that the middle is unloaded has gone. In a swelling clay, softened near its surface and fissured at depth, a single value of $G$ is a large simplification, and a real site’s figure should be taken as a range.

No clay is ever parted from the shaft. Dry-season cracks open between clay and pile near the surface, in exactly the places where a fully mobilised friction is assumed. The group effect is mostly in the deep shaft, so the difference to the distribution is modest, but the 200 kN crossing moves down.

A symmetrical square. Strip foundations under a house have one row, not a square, so that all the piles are edge piles and the shelter is that of a line of neighbours. A row of nine has much less shelter than this square, and the two-row figure is closer to it.

No coating. A slip coating over the upper shaft takes down the friction that does the lifting, and the coating that takes the resistance with it priced it for a pile in settling ground. In a group its effect would have to be read against what the shelter has already removed.

The piles stay whole. The tension here is elastic, in a pile whose steel carries the crack. A cracked pile has a much lower axial stiffness, and that would change how it moves past its neighbours; the shelter that stops at the interface found a similar limit on what a group’s reduction factors can be read to mean sideways.

Still open: the row under a wall, and a season at a time #

A square group is the sheltered case, and most houses on piles stand on single rows under ground beams. Shelter in a row comes from two sides rather than eight, the row has no inside, and the clay can flow round its ends as well. The ground also moves in seasons, up in winter and down in summer, and each year the shafts’ friction reverses over some of their depth while the ground beam, carried on the group, is asked to take the differences. Whether a row’s piles work more in the winter’s uplift than in the summer’s settling, and whether a tension cycle in a concrete pile at 17 m that has already cracked does any further harm, is what the one heave of a square group leaves for a row in a real clay climate.