The piles that would lift the ground

A basement is a boat: the water under it pushes up with its full depth of pressure, and what holds it down is its own weight and nothing else. A 20 × 30 m substructure dug 6 m below the water table, with a 0.9 m slab, has a factor against floating of 0.53 empty — the water pushes up with 36 MN and the structure weighs 19. Building on it raises the factor, but it starts below one and has to be held down meanwhile, by about 17 MN.

The earlier essay named the ways: more weight, which is the honest answer and slow; drainage, which is a structural decision dressed as a services one; and tension piles or ground anchors, which pin the slab to the ground below it. This essay takes the third, and finds that a group of tension piles has a ceiling on what it can hold that has nothing to do with the piles.

What one pile holds #

A tension pile resists being pulled out by friction on its shaft, and like all friction it is set by what presses the surfaces together, not by how large they are. The friction at a depth is a share β of the effective vertical stress there, which in ground below the water table is the submerged unit weight times the depth — about 10 kN/m³ times $z$. So the friction grows linearly down the pile, and a pile of diameter $d$ and length $L$ holds

$$Q = \pi d \cdot \beta \gamma' \cdot \frac{L^2}{2}.$$

For a 600 mm pile 15 m long in ground with β = 0.3, that is 636 kN. Twenty-seven of them make up the 17 MN the basement needs, at about 4.7 m centres under the raft.

What a group holds #

A single pile pulled from the ground takes a thin skin of soil with it and leaves the rest. A group of piles pulled together does the same as long as the ground between them stays put. Pack them close enough and it does not: the friction on every shaft pulls the ground between the shafts upward too, and the cheapest way for the group to come out is as one block — the piles and the ground they enclose, sliding up together on the block’s outer faces.

The block’s resistance is its own submerged weight, and (more doubtfully) the friction on its outside. Taking the weight alone, the block under a 20 × 30 m raft to the piles’ 15 m depth weighs 90 MN in the water. That is a hard ceiling. However many piles are in it and however well each grips, the group cannot hold down more than the ground it would bring with it.

So the group holds the lesser of two numbers — the piles’ sum, and the block’s weight — and EN 1997-1 asks for both checks, for exactly this reason. In compression the same block check rarely governs at ordinary spacings, because a block pushed down has the whole bearing capacity of the ground beneath it to overcome; pulled up, it has only its own weight.

Where the block takes over #

On a square grid of spacing $s$ the raft has $B L_r/s^2$ piles, and their summed pull-out rises steeply as the spacing closes — as one over its square. The block’s weight does not move. They meet where

$$\frac{B L_r}{s^2}\cdot\frac{\pi d \beta \gamma' L^2}{2} = \gamma' B L_r L \quad\Longrightarrow\quad s^2 = \frac{\pi d \beta L}{2}.$$

The raft’s size cancels. So does the water, the soil’s weight and the load. What is left is the pile — its diameter, its length and the friction its ground gives — and for these piles it is 2.06 m, 3.4 diameters. Closer than that, the group lifts its block before the piles slip, and each pile added is a pile the block’s weight has already paid for.

That is the condition, not the spacing anyone would choose. The basement needs only 27 piles, and at 4.7 m centres they are far inside the range where each one counts. The block becomes the governing case when piles are packed close for some other reason — a deep basement with an enormous deficit, a raft that must also carry heavy columns between which the piles cluster, or a contractor who prefers many short piles to a few long ones.

The number past which none helps #

Counted in piles, the ceiling is a number: about 142 of these under this raft, after which the line goes flat. The block is what a designer would ask for if every pile were free, and it is 90 MN; the deficit is 17. So for this basement the ceiling is five times above what is needed and is not the governing check.

It becomes one when the deficit approaches the block. A basement twice as deep has twice the uplift and, on 15 m piles, the same block; a raft on shallower piles has a smaller block. Where the needed uplift resistance is more than the block’s weight, no arrangement of piles of that length can provide it — the answer is longer piles, which reach a bigger block, or a structure heavy enough not to need them.

Longer piles stand further apart #

The crossing grows with the pile’s length, and the reason is that the two things it balances grow at different rates. A pile’s pull-out grows as its length squared — a longer shaft, and higher friction at its foot because the effective stress there is larger — while the block’s weight grows only as the length. A longer pile is therefore worth more against a block that has grown less, and it must stand further from its neighbours before the block stops capping it: 1.30 m for piles 6 m long, 2.06 m at 15 m, 2.91 m at 30 m.

Slender ground anchors meet the block sooner and at closer spacing. A 250 mm anchor 15 m long holds 265 kN and crosses at 1.33 m — closer than most anchor grids are laid, but not by much, which is why anchor groups are the place this check is most often the one that governs.

Between the block and the slab #

The block bounds the spacing from below. The raft bounds it from above: between the piles, the slab carries the water pressure less its own weight as a slab spanning two ways between point supports, and the further apart the piles the larger the moment. With 0.3 per cent of steel each way, a 0.9 m slab spans 14.7 m under its 37.5 kN/m² of net uplift; a 0.4 m slab only 5.1. The working window is the space between the two.

The two limits behave differently. The slab’s limit moves with the slab, and a designer controls it. The block’s limit is a property of the piles alone and does not move with anything the raft does. A thin raft on long piles has a narrow window; a thick raft on short piles a wide one.

What the ground between the piles is doing #

The block is not a figure of speech. Each pile pulls on the ground around it through a shear stress at its shaft, and that shear does not stop at the shaft’s surface: it spreads outward, falling off roughly as one over the distance, and drags up a cylinder of ground whose movement decays into the soil around it. A single pile’s cylinder is narrow, and the ground beyond it is undisturbed.

Put a second pile beside the first and their cylinders overlap. The ground between them is being pulled up from both sides, and the share of the pull it can resist by shearing against still ground falls, because there is less still ground left. At close spacing all the ground inside the group is moving up together and the only still ground is outside it — which is exactly the block mechanism, arrived at continuously rather than by switching. The two-line minimum in the figures is the simple bound of a gradual change; a real group starts to lose efficiency somewhat before the crossing and reaches the block somewhat after it.

The partial factors move the crossing #

A design does not compare the two characteristic values. It compares a factored pile resistance — the characteristic pull-out divided by a resistance factor, typically between 1.25 and 1.6 for tension piles — with a factored block weight, the weight being a stabilising action and multiplied by 0.9. Both numbers fall, but not equally: dividing the piles by 1.4 and multiplying the block by 0.9 changes their ratio by 0.79.

The crossing goes as the square root of that ratio, so the factored crossing is 0.89 of the characteristic one — 1.83 m instead of 2.06 for these piles. The factors make the block govern at closer spacing, not wider, because they are harder on the piles than on the ground. That is the right direction for the uncertainty: the block’s weight is the most reliable number in the whole calculation, and the friction on a shaft is among the least.

Where the 17 MN came from #

The deficit the piles make up is not the characteristic one either. The water’s 36 MN of uplift is a destabilising permanent action, taken at its full value; the basement’s 19 MN of weight is a stabilising one, taken at 0.9 of it — the factor that a weight that has to be known before it can be found earns when it is the thing holding a structure down. The piles must make up $36 - 0.9 \times 19 = 18.9$ MN rather than 17, and a further margin if the water table could rise above the level the uplift was computed for.

None of that changes the shape of the argument. It changes how many piles are needed — with a resistance factor of 1.4 on each, about forty-two rather than twenty-seven — and leaves the ceiling about four times above the need, at 0.9 of 90 MN. The block check matters for this basement only as the answer to a different question: what happens if somebody tries to save money with more, shorter piles.

More, shorter piles #

Shorten the piles and each holds much less, because the pull-out goes as the length squared, while the block they would lift shrinks only in proportion. The arithmetic of the factored design for this basement runs: 15 m piles, 42 of them at 3.8 m centres against a crossing of 2.06; 10 m piles, 94 at 2.5 m against 1.68; 6 m piles, 260 at 1.52 m against 1.30 — still clear, but only just.

At 4 m the line is crossed. Each pile holds 45 kN, the factored need is 585 of them, and on the raft they stand 1.01 m apart, inside the 1.06 m crossing. The group lifts its block first — and the block of 4 m of ground under the raft weighs 24 MN, 21.6 factored, barely above the 18.9 needed. A designer counting piles would have found the number that holds the basement down; the ground would have come up with them at a little more than the load, with almost no margin and a mechanism that no pile test would have shown.

Each pile alone, or the block #

Two free bodies, and the check is which is cheaper to pull out.

The first is each pile alone: a cylinder of shaft with the ground around it held still, the pull at its head resisted by friction on its surface. Its resistance adds pile by pile, and so does the group’s — on the assumption that the ground between the shafts stays where it is.

The second is the block: the raft’s footprint carried down to the piles’ tips, with everything inside it — piles and soil — moving together. Its resistance is its own weight below the water table and whatever friction the outside of the block finds, and the piles inside it do not appear at all. They are part of the free body, not forces on it. That is why their number drops out, and why a group’s capacity can be independent of how many piles it has. The block is a mechanism in the sense that bearing capacity is a mechanism: a chosen way for the ground to move, whose cost is computed, and the answer is the cheapest of the ways tried.

The 600 mm group by hand #

The effective vertical stress at the foot of a 15 m pile is $10 \times 15 = 150$ kPa, and the friction there is $0.3 \times 150 = 45$ kPa; at the head it is zero. The average over the shaft is 22.5 kPa, and the shaft’s area is $\pi \times 0.6 \times 15 = 28.3\ \text{m}^2$, so $Q = 22.5 \times 28.3 = 636$ kN.

The block is $20 \times 30 \times 15 = 9{,}000\ \text{m}^3$ of ground at $10\ \text{kN/m}^3$ submerged: 90 MN.

The crossing: $s^2 = \pi \times 0.6 \times 0.3 \times 15/2 = 4.24\ \text{m}^2$, so $s = 2.06$ m. Check it the long way: at 2.06 m centres the raft has $600/4.24 = 142$ piles, and $142 \times 636 = 90$ MN.

What the model assumes #

Friction that grows with depth and nothing else. Real shaft friction in sand depends on how the pile was installed — a driven pile densifies the ground, a bored pile loosens it — and it can turn round altogether where the ground settles past the shaft; in clay it is a share of the undrained strength rather than of the effective stress. The square of the length is a property of the effective-stress model; in clay the pull-out grows only linearly with length, like the block, the length cancels as well, and the crossing becomes $s^2 = \pi d \alpha c_u/\gamma'$ — a property of the pile’s diameter and the clay alone.

A block that is the raft’s footprint. A block can be wider than the group — the soil it drags up fans out — or narrower, if the outer piles are far from the raft’s edge. EN 1997-1 leaves the shape to the designer, and the friction on the block’s outer faces, which is the doubtful part, is left out here; including it raises the block and lowers the crossing to 1.76 m.

And water that stays where it is. The uplift and the block’s submerged weight both take the water table as fixed. A water table that rises reduces the block’s effective weight and increases the uplift at the same time.

What the figures cannot show #

They cannot show cyclic loading. A tension pile under a water table that rises and falls with the seasons or the tide is loaded and unloaded every cycle, and shaft friction degrades under repeated loading. The 636 kN is a first-load capacity.

They cannot show the pile in tension as a member. A pile carrying 636 kN in tension is a reinforced concrete tie, and the steel in it must be continuous down to where the friction is developed — most of it near the foot.

And they cannot show the order of construction. The piles are needed when the basement is empty and the pumps are off; the building above eventually holds the basement down by itself and the piles carry nothing, or carry the compression of the columns above. A tension pile is usually a pile that changes sign during the life of the structure, and like a tie that spends an afternoon as a strut it has to be checked as both members it will be.

What it comes to #

One pile holds by its shaft’s friction, which grows with the square of its length: 636 kN for 600 mm piles 15 m long.

A group holds the lesser of its piles’ sum and its block’s weight. Under a 20 × 30 m raft, at most 90 MN however many piles.

The two meet at $s = \sqrt{\pi d \beta L/2}$, which contains no raft, no water and no load: 2.06 m here, 3.4 diameters.

Longer piles must stand further apart, because their pull-out grows faster than their block; and the slab spanning between them closes the window from the other side.

Still open: the water that rises while the building is built #

Every number here has the water table fixed at the level the uplift was designed for. A basement built inside a dewatered excavation sees the water come back over weeks once the pumps stop, while the building above it adds weight floor by floor — and the tension piles carry the difference only while the water is ahead of the building. Whether the piles can be fewer if the pumps are kept running until a given floor is cast, and how many floors a month of pumping is worth against how many piles, is a question about two rates rather than two weights — the rate the ground refills and the rate the building rises — and it decides whether the tension piles are a permanent structure or a temporary one.