The API sand p-y model (O'Neill & Murchison 1983)

The API sand p-y model (O'Neill & Murchison, 1983; adopted by API RP 2A) describes lateral sand resistance with a single hyperbolic-tangent curve, p = A·pᵤ·tanh(k·z·y / (A·pᵤ)), where the ultimate pᵤ = min[(C₁·z + C₂·b), C₃·b]·γ′·z and C₁, C₂, C₃ are closed-form functions of the friction angle φ. It is the standard sand criterion for offshore pile design.

From Reese (1974) to API RP 2A

The model is a deliberate simplification of the Reese, Cox & Koop (1974) sand criterion developed from the Mustang Island load tests. O'Neill and Murchison evaluated the available sand p-y formulations against a database of field tests in 1983 and proposed replacing Reese's four-segment piecewise curve — which requires two empirical coefficients published only as charts — with one smooth hyperbolic-tangent expression. The American Petroleum Institute adopted it in API RP 2A, and it has been the offshore standard ever since. The theoretical ultimate resistance is unchanged; what the simplification buys is a fully closed-form model with no empirical coefficient charts, which is why every implementation of API sand computes identical numbers. (Reese implementations historically differed by a few percent through independent chart reads; PileCalc closes that on its side by using COM624P's own numeric A/B tables.)

Ultimate resistance and the C₁, C₂, C₃ coefficients

The ultimate resistance per unit length is the lesser of a shallow (wedge) and a deep (flow-around) expression, with b the pile diameter, z the depth, and γ′ the effective unit weight:

pus = (C1·z + C2·b)·γ′·z        (shallow wedge)
pud = C3·b·γ′·z                (deep flow-around)
pu  = min(pus, pud)

C₁, C₂ and C₃ depend only on φ. API RP 2A presents them as a chart, but they are closed-form. With α = φ/2, β = 45° + φ/2, K₀ = 0.4, Kₚ = tan²(45° + φ/2), and Kₐ = tan²(45° − φ/2):

C1 = tanβ·[ Kp·tanα + K₀·( tanφ·sinβ·(1/cosα + 1) − tanα ) ]
C2 = Kp − Ka
C3 = Kp²·(Kp + K₀·tanφ) − Ka

Evaluating these for common friction angles:

A point worth knowing when you cross-check tools: these coefficients are not a new theory. They are the Reese (1974) wedge and flow-around equations regrouped algebraically — for example, C₂ = Kₚ − Kₐ is exactly Reese's tanβ/tan(β−φ) − Kₐ, and C₃ equals the tan⁸β flow-around coefficient identically (since Kₐ·Kₚ⁴ = Kₚ³). Reese sand and API sand therefore share the same theoretical pᵤ at every depth; they diverge only in how the curve rises to it.

The tanh backbone and the A factor

The full curve is a single expression:

p = A·pu·tanh( (k·z·y) / (A·pu) )

At small y, tanh(x) ≈ x, so the curve starts on the line p = k·z·y — the same initial stiffness as Reese sand, with k the initial modulus of subgrade reaction and z the depth. At large y it approaches the asymptote A·pᵤ. The factor A adjusts the ultimate for loading type:

A = 3 − 0.8·(z/b)  ≥ 0.9    (static)
A = 0.9                     (cyclic)

Static A starts at 3.0 at the surface and decreases linearly until it hits the 0.9 floor at z/b = 2.625, below which static and cyclic curves are nearly identical. This mirrors the role of Reese's chart-based A coefficient — strong near-surface enhancement under static loading, degradation to a modest deep value under cyclic loading — in one line of algebra.

Choosing k

API RP 2A gives k as a function of relative density or friction angle; the values match Reese's recommendations:

Because the tanh argument is k·z·y/(A·pᵤ), k sets both the initial slope and how quickly the curve saturates: a stiff k with a modest pᵤ reaches the asymptote at small deflections, which is exactly the behavior observed in dense sands.

A worked mini-example

Submerged sand, φ = 30°, γ′ = 10 kN/m³, k = 5,400 kN/m³, pile diameter b = 0.5 m, depth z = 2 m, static loading. From the table, C₁ = 1.91, C₂ = 2.67, C₃ = 28.7. Then pus = (1.91·2 + 2.67·0.5)·10·2 ≈ 103 kN/m and pud = 28.7·0.5·10·2 ≈ 287 kN/m, so the shallow mechanism governs: pᵤ ≈ 103 kN/m. At z/b = 4 the static A = max(3 − 3.2, 0.9) = 0.9, giving an asymptote A·pᵤ ≈ 92.8 kN/m. With k·z = 10,800 kN/m², the curve gives p ≈ 10.8 kN/m at y = 1 mm, 48.6 kN/m at 5 mm, and 92.3 kN/m at 25 mm — already 99% of the asymptote. These parameters match PileCalc's published validation case: a 0.5 m pile in this sand under a 100 kN head load deflects 7.33 mm at the head versus LPILE's 7.3 mm (see the validation page for the full benchmark set).

Why offshore practice standardized on it

Three reasons. It is reproducible — closed-form coefficients mean two implementations cannot disagree by chart-reading error, which matters when a design is checked across contractors and certifying authorities. It handles cyclic loading with a one-parameter switch, and wave loading makes essentially all offshore lateral design cyclic. And it is smooth: the tanh curve has a continuous derivative everywhere, which helps the nonlinear beam-column solution converge cleanly — piecewise curves introduce slope discontinuities that iterative solvers must negotiate.

API vs Reese sand in the mid-range

Since both models share the initial slope k·z and (in effect) the ultimate resistance, the differences live in the mid-range of deflection. The Reese construction passes through a prescribed intermediate point (B·pₛ at y = b/60) and reaches its full ultimate exactly at y = 3b/80, then stays flat. The tanh curve approaches its asymptote only asymptotically — at the deflection where Reese has fully plateaued, API sand is still a few percent below A·pᵤ. Depending on depth and the A/B coefficient values, the two curves can cross once or twice in the working range, and head-deflection predictions typically differ by a few percent between the two models. If that margin matters to your design, run both models and envelope the results.

Limitations

The model inherits the p-y method's assumptions: independent nonlinear springs, no continuum coupling between depths, calibration on driven piles of modest diameter. Recent offshore research (e.g. the PISA project for large-diameter monopiles) has shown the tanh formulation underestimates stiffness for very low slenderness ratios — for L/b below roughly 6, treat results with caution. It applies to clean sands only: no cementation, carbonate sands (notoriously crushable) excluded, and no liquefaction. For layered profiles, PileCalc applies the Georgiadis equivalent-depth construction so overburden from dissimilar layers is carried consistently into pᵤ.

Common questions

What is the API sand p-y equation?

p = A·pᵤ·tanh(k·z·y / (A·pᵤ)), where pᵤ = min[(C₁·z + C₂·b), C₃·b]·γ′·z is the ultimate resistance, C₁/C₂/C₃ are closed-form functions of φ, k is the initial modulus of subgrade reaction, and A = 3 − 0.8·z/b (≥ 0.9) for static loading or 0.9 for cyclic loading. It is the sand criterion of API RP 2A, from O'Neill & Murchison (1983).

How is API sand different from Reese (1974) sand?

The theoretical ultimate resistance is algebraically identical — C₁/C₂/C₃ are the Reese wedge and flow-around expressions regrouped. The difference is the backbone: API uses one smooth hyperbolic tangent with a simple A factor, while Reese uses a four-segment piecewise curve with empirical A and B coefficient tables (PileCalc uses COM624P's own). Predictions typically differ by only a few percent.

What are typical C₁, C₂, C₃ values?

At φ = 30°: C₁ ≈ 1.91, C₂ ≈ 2.67, C₃ ≈ 28.7. At φ = 35°: C₁ ≈ 2.97, C₂ ≈ 3.42, C₃ ≈ 53.8. At φ = 40°: C₁ ≈ 4.62, C₂ ≈ 4.38, C₃ ≈ 104. All three grow rapidly with φ — C₃ climbs from about 16 at φ = 25° to over 100 at φ = 40° — so the friction angle is by far the most consequential input.

When does the static A factor matter?

Only in the top 2.6 diameters of the profile: A = 3 − 0.8·z/b starts at 3.0 at the surface and reaches its 0.9 floor at z/b = 2.625, where it equals the cyclic value. Below that depth, static and cyclic API sand curves are the same, so cyclic degradation is purely a shallow-soil effect — which is also where lateral response is decided.

Generate API sand curves — and compare them side-by-side against Reese (1974) — in the free p-y curve generator, or run a complete laterally loaded pile analysis in the PileCalc app.