Monopile p-y curves: what PISA was needed to fix

An offshore wind turbine on a monopile is a cantilever in the seabed, and its foundation design is governed by lateral stiffness rather than by axial capacity. The industry method for that, until recently, was the p-y approach: replace the soil with a series of independent nonlinear springs, each described by a curve relating lateral resistance $p$ per unit length to lateral displacement $y$ at that depth.

The API formulation for sand has been in the offshore recommended practice for decades and works. The question is what it was calibrated on, and whether a monopile is that.

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The methodPermalink to “The method”#

For sand, the ultimate lateral resistance is the smaller of two mechanisms. Near the surface the soil fails by lifting a passive wedge; deeper down it fails by flowing horizontally around the pile.

$$p_{us} = (C_1 z + C_2 D)\,\gamma' z, \qquad p_{ud} = C_3 D\,\gamma' z$$

The coefficients depend only on the friction angle. At $\varphi' = 35°$, with $\alpha = \varphi'/2$, $\beta = 45° + \varphi'/2$ and $K_0 = 0.4$:

$$C_1 = 2.970, \qquad C_2 = 3.419, \qquad C_3 = 53.79$$

and the curve itself is

$$p = A\,p_u \tanh\!\left(\frac{k\,z}{A\,p_u}\,y\right)$$

with $k$ the initial modulus of subgrade reaction and $A$ a loading factor, 0.9 for cyclic loading. $k$ is read from a chart in the recommended practice against friction angle, with separate curves for sand above and below the water table; the worked example below takes $k = 24400$ kN/m³, and the last section shows that nothing in the argument turns on that choice.

The transition depth does not know about the pilePermalink to “The transition depth does not know about the pile”#

Setting $p_{us} = p_{ud}$ gives the depth at which the mechanism changes:

$$z_t = \frac{(C_3 - C_2)\,D}{C_1}$$

$D$ appears once on the top and nowhere else, so $z_t/D$ depends on the friction angle alone: 16.96 at $\varphi' = 35°$, and 12.83 at the 28.5° reported for the sand at Mustang Island, where Reese and co-workers ran the tests this formulation was fitted to. The transition depth therefore scales with the diameter and with nothing else:

The test pile crosses its transition depth 7.83 m down, a third of the way along its embedded length, and so mobilises both mechanisms — which is what the formulation was fitted against. The monopile’s transition depth is 135.67 m, four and a half times its embedded length. The deep flow branch never activates anywhere on the pile; every metre of it is in the shallow wedge regime.

That is not necessarily wrong — a rigid pile plausibly does fail by wedge mechanisms over its whole length — but it means the monopile is being designed by half a formula, and the half that runs is being extrapolated a long way outside the geometry that fitted it.

The stiffness has no diameter in itPermalink to “The stiffness has no diameter in it”#

The initial slope of the p-y curve, from the derivative of the hyperbolic tangent at the origin, is exactly $k z$. At 5 m depth:

$$\left.\frac{\mathrm{d}p}{\mathrm{d}y}\right|_{y=0} = 24400 \times 5.0 = 122000\ \text{kN/m per metre}$$

for the 0.61 m test pile and for the 8 m monopile alike, though their diameters differ by a factor of 13.1 in width. A pile thirteen times wider pushes on thirteen times as much soil and is credited with the same stiffness per unit length.

The saturation behaviour follows from the same omission. At 5 m depth on the 8 m monopile, $p_u$ is 2110 kN/m from the wedge mechanism, so at a displacement of 2% of the diameter — 0.16 m, which is small for a structure whose serviceability limit is measured in milliradians of tilt — the argument of the hyperbolic tangent is 10.28 and

$$p = 0.9 \times 2110 \times \tanh(10.28) = 1899\ \text{kN/m}$$

which is 100.00% of $A p_u$ to four figures. The spring is fully mobilised. Beyond that point the model has no stiffness left to offer and the computed response is governed entirely by whatever is below.

That conclusion does not rest on the $k$ that was assumed. Repeating the calculation across the range of the chart, the curve is at 99.7636% of its ultimate at $k = 8000$ kN/m³ and indistinguishable from 100% above about 11756 kN/m³ — which is below every value the chart gives for a sand dense enough to carry a monopile. The saturation is a consequence of the diameter appearing in the displacement and not in the stiffness, and no admissible $k$ removes it.

What the springs leave outPermalink to “What the springs leave out”#

The deeper problem is structural rather than parametric. A p-y model represents the soil as springs that act only laterally and only independently. For a slender pile that is a good approximation, because the pile bends and each depth genuinely does respond largely on its own.

A monopile at $L/D = 3.75$ does not bend appreciably. It rotates about a point some way down its length, and in doing so it mobilises three reactions the spring model contains no term for: shear on the pile shaft from vertical relative movement, shear across the pile base, and a moment at the base from the same. All three resist rotation, and all three are absent.

That is what the PISA design method addressed. Rather than a single load-transfer curve, it represents the soil reaction with four components — distributed lateral load, distributed moment, base horizontal force and base moment — with the shape of each calibrated against field tests and three-dimensional finite element analysis over the range of slenderness that monopiles actually occupy. The reported consequence is a stiffer computed response and, for a given design requirement, a shorter pile.

Why this is a modelling lesson rather than a monopile lessonPermalink to “Why this is a modelling lesson rather than a monopile lesson”#

The failure mode here is general and worth naming. A method was fitted on one geometry, was formulated in a way that hid which parts of it were geometry-dependent, and was then applied to a geometry an order of magnitude away. Nothing in the formulas complains: they return numbers, the numbers are plausible, and the analysis converges.

The only defence is the one that applies to every other correlation on a site investigation: knowing what the fitting database contained, and checking whether the case in hand is inside it. When it is not — as here, where $L/D$ differs by a factor of nine — the honest options are to go back to a formulation that resolves the mechanism, which for this problem means three-dimensional finite element analysis, or to use a method calibrated for the geometry.

And the finite element route brings its own obligations. A three-dimensional model of a monopile is only as good as its constitutive model and its initial stress state, which for a laterally loaded pile in sand means the dilatancy angle and the K0 procedure are not details — they are the model.