The pile that is too short to bend

A pile has no length until the ground gives it one, and the length the ground gives it is

$$T = \left(\frac{EI}{n_h}\right)^{1/5}$$

Everything about a laterally loaded pile is decided by how many of those the pile is. That is a ratio rather than a dimension, and it separates two structures that share a name and share almost nothing else.

Four lengths, and then nothing #

The first thing the ratio decides is how much of the pile is doing anything.

A lateral check is a check on the top four T of a pile, whatever the pile is doing below. The 20 m pile above is 10.6 T long, so the bottom 12 m of it is present for its axial load and is entirely irrelevant to the lateral one — which is why lengthening a pile that deflects too much achieves nothing, and why the answer is always a larger diameter.

That is the long pile, and it is the case every published solution is for. Its defining property is that its foot has no influence: the moment, the shear and the deflection have all decayed to nothing before the end arrives, so the boundary condition down there could be anything at all.

Where the coefficients begin to drift #

Shorten the pile and the foot begins to be asked for something.

At 3.2 T the drift is seven per cent and would be lost in the uncertainty of the ground. What matters is the direction and the mechanism: the pile is beginning to rotate rather than bend, so the moment falls, the deflection rises and the reaction concentrates.

The pattern below the pile’s tip is the giveaway. In a long pile the reaction decays smoothly to zero; in a short one it is still finite at the toe, which means the ground down there is being asked to push back, and a rigid rotation has begun.

The monopile, which is a different structure #

That drift stops being academic at a scale of structure that did not exist when the coefficients were published.

An offshore wind turbine’s monopile is 8 m in diameter, 80 mm thick and 30 m into the seabed. Its flexural stiffness is enormous — 3.4 × 10¹⁸ N·mm² against the 1.2 × 10¹⁴ of the 600 mm pile — so its characteristic length is 14.67 m, and thirty metres of embedment is two of them.

The two coefficients move in opposite directions, and that is the part worth carrying. A short pile deflects far more than the long-pile solution predicts and carries far less moment, because it is rotating about a point rather than bending about a curve. A designer who takes the tabulated deflection is unsafe; one who takes the tabulated moment is wasteful; and nothing about the ground or the equation changed.

Reading the ratio before the analysis #

The useful consequence of all this is that a single division done on the back of an envelope says which of two quite different problems is in front of the designer, and it can be done before any analysis at all.

Compute $T$ from the section and a plausible modulus gradient. Divide the embedment by it. Above four, the pile is long: the tabulated coefficients apply, the toe is irrelevant, and length is not a design variable. Below two, the pile is short: it rotates, the toe carries reaction, the coefficients are wrong by tens of per cent in both directions, and embedment is one of the few things that helps. Between two and four is a transition where neither idealisation is safe and the equation has to be solved.

For ordinary building piles the answer is almost always “long”, which is why the subject feels settled. A 450 mm bored pile in medium sand has $T$ of about 1.6 m and is typically 12 to 18 m deep — eight to eleven characteristic lengths, comfortably inside the tabulated case, and a designer can spend a whole career without meeting anything else.

The exceptions are the interesting ones and they are all recent or all marine: monopiles, large-diameter caissons, drilled shafts for bridge piers, and the short stubby piles used under transmission towers where the load is mostly overturning. Each of those is a structure where the section was chosen for something other than the lateral load — buckling, driveability, a hole size — and ended up with a stiffness that put it in the other category by accident.

What the two structures fail by #

The categories differ in their strength calculation as well as their stiffness one, and the difference is not a refinement.

A long pile fails when a plastic hinge forms in the section at the depth of the peak moment. The ground below that hinge is untouched and could carry more; the pile simply runs out of section. It is therefore a structural failure with a structural remedy — a thicker wall, a higher grade — and the ground’s ultimate strength barely enters.

A short pile fails when the ground gives way over the whole of its length, rotating about a point near the toe with the soil in front pushing at its ultimate resistance above the point and behind it below. Nothing yields in the steel. It is a geotechnical failure with a geotechnical remedy — more embedment, a larger diameter — and the section’s own strength barely enters.

Two failure modes, two disciplines, and the ratio that picks between them is a fifth root. That is the sharpest reason to compute $L/T$ early: it decides not only which equations apply but which specialist owns the problem, and a design that has been checked by the wrong one of the two can be perfectly correct and about the wrong thing.

Which free body produced the number #

The free body is the whole pile, cut free of the ground at its perimeter, with the soil replaced by a distributed reaction proportional to the local deflection.

That substitution is the entire model and it is worth being exact about what it assumes. The reaction at any depth depends on the deflection at that depth and nowhere else, so the ground is a bed of independent springs with no shear between them. The springs get stiffer with depth, linearly, at $n_h$ per unit depth — which is what makes a sand profile and is why the fifth root appears rather than the fourth root a beam on a uniform bed gets.

Vertical equilibrium of that free body is trivial and horizontal equilibrium is the whole problem: the applied load has to be balanced by the integral of the reaction, and the reaction is an unknown function of the deflection, which is an unknown function of the load. What closes it is the beam equation

$$EI\frac{d^4y}{dx^4} + n_h x\, y = 0$$

solved with two conditions at the head and two at the toe. The two conditions at the toe are what separates the long pile from the short one. For a long pile they are irrelevant because everything has decayed; for a short one they decide the answer, and the classical solutions simply do not contain them.

The fifth root, and what it makes unimportant #

Because $T$ is a fifth root, everything that goes into it is flattened.

A hundredfold range of ground is the whole of what a site investigation can distinguish, from a soft clay to a dense gravel, and it moves the answer by a factor of two and a half. The most uncertain input in the calculation is also the least influential, which is an unusually comfortable arrangement and is worth knowing before commissioning another borehole.

The same root works the other way on the section. Doubling the pile’s stiffness — a heavier wall, a larger diameter — raises $T$ by only 15 per cent, so a designer cannot buy much characteristic length either. What a larger diameter does buy is more soil contact per unit depth, which is a different term and is the reason diameter is the effective variable.

And it is the fifth root that makes the monopile’s problem arrive so suddenly. Going from a 600 mm pile to an 8 m one multiplies $EI$ by about 28,000 and $T$ by only 7.8 — but the embedment goes up by only 1.5, so the ratio $L/T$ collapses from 10.6 to 2.0. The structure changed category because its stiffness grew faster than its length, and nothing in the design process announces that.

What “short” actually does #

Three things change when a pile drops below about two characteristic lengths, and each of them inverts a habit formed on slender piles.

Length becomes useful again. On the flat part of the curve embedment buys nothing; on the steep part it buys a great deal. A monopile’s embedment is a real design variable and a bored pile’s is not.

The moment falls and the rotation rises. A rigid pile rotating about a point near its toe carries much less bending than a flexible one bending about a point near its head. That is good for the section and bad for everything the pile is holding up, because a wind turbine is governed by the rotation at its base rather than by the stress in the steel.

The toe carries load. A long pile’s toe is unloaded and a short pile’s toe pushes back hard. That reaction is applied to ground at a depth where the vertical stress is highest and the modulus is stiffest, which is why the linear $n_h x$ assumption is worst exactly where the short pile needs it most.

There is a fourth change, and it is about the group rather than the pile. A slender pile’s lateral influence dies out within four T, so two piles four T apart interact hardly at all laterally even though they interact strongly in settlement. A short pile’s influence extends over its whole embedment and beyond, so a group of them shares ground the way a group of piles shares it axially — which is a much stronger interaction than the p-multipliers written for slender piles allow for.

The other direction, where length is the problem #

It is worth setting the lateral case beside the axial one, because the two give opposite advice about the same dimension.

Downdrag makes a longer pile worse, and it makes it worse as a square. Lateral behaviour makes a longer pile no better at all past four T. Only the axial capacity rewards length, and it rewards it linearly, which means the length of a pile is decided by one of its three actions and is then simply endured by the other two.

That is worth stating because the instinct in foundation design is that deeper is safer. It is safer for end bearing, neutral for lateral load, and actively worse for drag — and a pile group makes all three worse again, because the ground under a group has already been used.

The rotation point, which the drawing shows #

There is a way of reading the deflection profiles above that makes the two categories visible without any arithmetic, and it is worth learning because it survives every refinement of the soil model.

Look at where the deflection curve crosses zero. In the slender pile it crosses once, near the head, and then oscillates with rapidly decaying amplitude — the signature of a beam on an elastic foundation, and the same decaying oscillation a raft under a column shows. In the monopile it crosses once, low down, and the profile either side of the crossing is nearly straight.

A nearly straight deflection profile is a rigid rotation, and the crossing is the point it rotates about. That point is at about 0.7 of the embedment for a rigid pile in a linearly increasing modulus, and finding it is the whole of the classical short-pile analysis: take moments about it, and the two soil pressure blocks above and below give the capacity directly, with no differential equation anywhere.

So the two categories have two entirely different hand methods, and each is trivial while the transition between them is not. That is the usual shape of a problem with a dimensionless ratio in it: the limits are simple and the middle is where the computer earns its keep.

Where the model stops #

The ground is a bed of independent springs. Real soil transmits shear between depths, so a deflection at one level drags the soil beside it and the reaction is not local. That is what a continuum or a finite-element analysis adds, and it stiffens the answer.

The modulus grows linearly and without limit. At the head the linear profile gives almost no resistance, which is conservative; at depth it gives more than the soil can supply, since the ultimate lateral resistance of soil is a strength and not a stiffness. The p–y method exists to bound both ends and is what a real design uses.

Everything is linear and monotonic. The load is applied once, in one direction, and the soil is elastic. A monopile sees ten million cycles, and the accumulated rotation under cyclic load is the quantity the whole structure is actually governed by — it is not in this equation anywhere.

The pile is a beam. At a diameter-to-length ratio of 8 m to 30 m the monopile is not slender, and shear deformation and cross-section ovalisation both matter. Beam theory is being used at proportions it was never intended for.

The diameter enters only through the modulus. The model has one soil term, $n_h x$ per unit length, and the pile’s width appears in it only through whatever value of $n_h$ was chosen. That is defensible for slender piles and increasingly poor as the diameter grows, because a wide pile mobilises soil over a much larger volume and the modulus is not a property of the ground alone.

And the head condition is idealised. A free head and a fully fixed head are the two extremes; a real cap or a real transition piece is somewhere between, and the answer is sensitive to where.

What a designer should carry away #

Three things, and the first is the only one that needs remembering.

Compute $L/T$ before anything else. It costs one fifth root and it decides which of two structures is being designed, which set of published coefficients applies, which failure mode governs and whether embedment is worth anything.

Do not extrapolate a coefficient across the ratio. The tabulated values are not a soil model or a material property; they are the solution of one differential equation with one pair of boundary conditions, and the conditions are the thing that changes. A table with no $L/T$ column is a table for long piles, whether or not it says so.

And expect the ground to matter less than it feels like it should. The fifth root flattens a hundredfold range of soil into a factor of two and a half, and the same root flattens the designer’s own choice of section into nearly nothing. What is left with real leverage is the diameter, because it acts on the soil contact rather than through the root — which is why every remedy for a lateral pile problem, at every scale, turns out to be a bigger pile rather than a longer or a stiffer one.

The same competition between a member’s own stiffness and the ground’s turns up twice more in this field. A beam that sits on the ground is the horizontal version with the same characteristic length in it, and the ground that hangs on instead of holding up is the axial one, where the interface changes sign along the pile.

The ladder from here #

Later rungs on this anchor: the p–y method proper, with the curves for sand and for clay and the way they bound both ends of the linear model. Layered profiles, where a single characteristic length stops existing and the pile has one near the surface and another lower down. Group effects and the p-multipliers, which are asymmetric with the axial group behaviour in a way nothing explains simply. Cyclic degradation and accumulated rotation, which is what governs a monopile and is not a strength calculation. The pile-head connection, which is where a fixed head’s moment goes and is usually detailed as though the head were pinned. And lateral load tests, which are the only measurement in this subject and which calibrate a model rather than a soil.

The coefficients quoted throughout are Matlock and Reese’s, published in 1960 for slender driven piles in sand, and they have been used almost unchanged ever since. The monopile arrived forty years later at a diameter twenty times larger and an embedment ratio nothing in that work covers, and the offshore industry spent much of the 2010s discovering by measurement that the piles were considerably stiffer than the method said. The equation was never wrong; the piles had left the range the coefficients were tabulated over, and the ratio that says so is printed on every one of these figures.