The Reese (1974) sand p-y model

The Reese (1974) sand p-y model (Reese, Cox & Koop, OTC 2080) predicts the lateral resistance of sand against a pile as a piecewise curve: an initial straight line of slope k·z, a parabolic transition, a linear segment, and an ultimate resistance pᵤ = A·pₛ, where pₛ is the lesser of a near-surface wedge failure and a deep flow-around failure. It remains the default sand criterion in COM624P, LPILE, and RSPile.

Where the model comes from

The criterion was calibrated against full-scale lateral load tests on two instrumented 24-inch steel pipe piles driven into clean, submerged fine sand at Mustang Island, Texas. Reese, Cox & Koop published the method in 1974 as OTC paper 2080, “Analysis of Laterally Loaded Piles in Sand.” Both static and cyclic loading were applied, which is why the model ships with separate static and cyclic adjustment coefficients. It was subsequently adopted by COM624/COM624P (Wang & Reese, 1993, FHWA-SA-91-048) and carried into every mainstream laterally loaded pile program since — including PileCalc, where the implementation is checked term-by-term against the published equations (see validation).

If you are new to the p-y method itself, start with what a p-y curve is; this page assumes that background and goes straight into the model.

Ultimate resistance: wedge failure vs flow-around failure

Near the surface, a laterally loaded pile pushes a passive wedge of sand up and out. At depth, the overburden suppresses the wedge and sand flows around the shaft horizontally. Reese derived a theoretical ultimate resistance for each mechanism and takes the smaller of the two at every depth. With φ the friction angle, γ′ the effective unit weight, b the pile diameter, z the depth, and the auxiliary angles α = φ/2 and β = 45° + φ/2, the near-surface wedge resistance (Eq. 15 of the paper) is:

pst = γ′·z·[ K₀·z·tanφ·sinβ / (tan(β−φ)·cosα)
          + (tanβ / tan(β−φ))·(b + z·tanβ·tanα)
          + K₀·z·tanβ·(tanφ·sinβ − tanα)
          − Kₐ·b ]

and the deep flow-around resistance (Eq. 16) is:

psd = Kₐ·b·γ′·z·(tan⁸β − 1) + K₀·b·γ′·z·tanφ·tan⁴β

with K₀ = 0.4 (at-rest coefficient) and Kₐ = tan²(45° − φ/2) (Rankine active coefficient). The theoretical ultimate is pₛ = min(pst, psd) — force per unit length of pile. The tan⁸β term makes the flow-around expression grow very fast with φ, so for typical sands the wedge governs down to many diameters of depth. In PileCalc, when a sloping ground surface is specified, the slope correction (after Reese & Van Impe) modifies only the wedge equation; the flow-around expression always keeps the level-ground Kₐ.

The A and B adjustment coefficients

The Mustang Island tests showed that the theoretical pₛ needed empirical adjustment. Reese introduced two depth-dependent coefficients, published only as charts against the depth ratio z/b: A scales the ultimate resistance (pᵤ = A·pₛ) and B scales an intermediate point on the curve (pₘ = B·pₛ). Both come in static and cyclic versions:

The static A curve starts near 2.85 at the surface and decays to a deep asymptote of 0.88; the cyclic A curve starts around 0.70, bulges slightly above 1.0 near z/b ≈ 1.25, and merges with the static curve at the same 0.88 asymptote. Because these were published as figures rather than numbers, implementations historically carried their own digitizations. PileCalc uses the exact numeric A/B tables of COM624P v2.0 itself — recovered from the official FHWA-distributed program binary and verified against every coefficient value the program's manual prints — so its lookup matches the program of record verbatim. The engine also accepts user-supplied A(z/b) and B(z/b) functions when a project standard prescribes exact values.

How the curve is constructed

Two characteristic points anchor the backbone: point M at deflection yₘ = b/60 with resistance pₘ = B·pₛ, and point U at yᵤ = 3b/80 with the ultimate pᵤ = A·pₛ. The full curve, exactly as implemented:

• Initial straight line, p = k·z·y, where k is the initial modulus of subgrade reaction (force/length³). This represents the small-strain soil stiffness growing linearly with depth.
• Parabolic segment, p = C·y^(1/n), fitted so it passes through M with the same slope as the next segment: n = pₘ/(m·yₘ) and C = pₘ/yₘ^(1/n), where m = (pᵤ − pₘ)/(yᵤ − yₘ).
• Straight segment from M to U with slope m.
• Plateau at pᵤ for y ≥ 3b/80.

The initial line intersects the parabola at a point Reese labels k; rather than solving for that intersection explicitly, PileCalc evaluates the curve as min(k·z·y, backbone) — the lower envelope reproduces the intersection automatically and is robust when the geometry degenerates (e.g. pₘ ≥ pᵤ at some depths, where the construction collapses to a straight line to the ultimate point).

Choosing φ, γ′, and k

The model needs three soil inputs. φ and γ′ come from lab tests or SPT correlations. The initial modulus k is taken from Reese's recommendations by relative density:

Use the effective (buoyant) unit weight below the water table. k mostly controls the small-deflection response; pᵤ controls behavior near failure. For serviceability-governed designs, put your site-investigation effort into k and φ; for strength-governed designs, φ and γ′ dominate.

A worked mini-example

Medium-dense submerged sand, φ = 35°, γ′ = 10 kN/m³, k = 16.3 MN/m³, pile diameter b = 0.5 m, depth z = 1.5 m (z/b = 3), static loading. The wedge equation gives pst ≈ 92.5 kN/m and the flow equation psd ≈ 403 kN/m, so the wedge governs: pₛ ≈ 92.5 kN/m. At z/b = 3 the static coefficients are A ≈ 1.03 and B ≈ 0.74, so pᵤ ≈ 95.7 kN/m at yᵤ = 3(500)/80 ≈ 18.8 mm, and pₘ ≈ 68.4 kN/m at yₘ = 500/60 ≈ 8.3 mm. The initial line has slope k·z = 24.45 MN/m² and intersects the parabola at y ≈ 1.7 mm (p ≈ 41 kN/m). Sampling the assembled curve: p ≈ 24 kN/m at 1 mm (initial line), 58 kN/m at 5 mm (parabola), 68 kN/m at 8.3 mm (point M), and the full 95.7 kN/m at 18.8 mm and beyond. You can reproduce this curve point-for-point in the free p-y curve generator.

Limitations — and when to prefer API sand

The model is empirical at its core: the A and B coefficients encode two test piles at one sandy site. PileCalc's tables are the exact COM624P program values, which removes the historical implementation-to-implementation digitization spread on PileCalc's side — but other codes' chart reads can still differ by a few percent, and no table changes the narrow empirical base. It assumes clean sand — no cementation, no significant fines plasticity — and, like all p-y formulations, treats the soil as independent nonlinear springs with no continuum coupling. It is not applicable to liquefied sand.

The API sand model (O'Neill & Murchison, 1983) replaces the chart coefficients and piecewise construction with a single hyperbolic-tangent curve built on closed-form C₁/C₂/C₃ coefficients — the theoretical ultimate is algebraically the same, only the curve shape differs. Prefer API sand when a code or client standard requires API RP 2A (routine offshore work), when you want a fully closed-form model with no empirical coefficient tables, or when comparing against software defaults that use it. Prefer Reese (1974) when matching COM624P/LPILE runs — PileCalc uses COM624P's own coefficient tables, so it reproduces those programs' sand curves exactly — or when the project standard names it explicitly. Running both and enveloping the results costs nothing and brackets the model-choice question entirely.

Common questions

What is the ultimate resistance in the Reese sand p-y model?

It is pᵤ = A·pₛ, where pₛ is the lesser of a near-surface passive-wedge expression and a deep flow-around expression, both functions of φ, γ′, depth, and diameter, and A is an empirical depth-dependent coefficient from the Mustang Island tests (≈2.85 at the surface decaying to 0.88 at depth for static loading). The ultimate is mobilized at a deflection of 3b/80.

What is the difference between static and cyclic Reese sand curves?

Only the A and B coefficients change. Static A starts near 2.85 at the ground surface; cyclic A starts near 0.70, reflecting degradation of near-surface resistance under repeated loading. Both converge to the same deep asymptote of 0.88 below about five diameters, so cyclic loading mainly penalizes the shallow soil that dominates lateral response.

What value of k should I use for sand?

Reese's recommendations, used by COM624P and LPILE: submerged sand — 5.4 MN/m³ (loose), 16.3 MN/m³ (medium), 33.9 MN/m³ (dense); above the water table — 6.8, 24.4, and 61 MN/m³ respectively. k is the initial modulus of subgrade reaction and controls the small-deflection stiffness of the curve through the initial line p = k·z·y.

Is Reese (1974) sand the same as API sand?

They share the same theoretical ultimate resistance — the API C₁/C₂/C₃ coefficients are the Reese wedge and flow-around expressions regrouped into closed form. They differ in curve shape: Reese uses a four-segment piecewise curve with chart-digitized A and B coefficients, while API uses a single smooth hyperbolic tangent with a simple A factor.

To see the model in action, generate Reese sand curves at any depth — and overlay the API sand equivalent — in the free p-y curve generator, or run a full pile analysis with it in the PileCalc app.