The Reese (1997) weak rock p-y model

The Reese (1997) weak rock p-y model predicts the lateral resistance of weak rock (unconfined compressive strength roughly 0.5–5 MPa) around a pile or drilled shaft. Ultimate resistance is pᵤr = αr·qᵤ·b·(1 + 1.4·zr/b) near the rock surface, capped at 5.2·αr·qᵤ·b at depth, where αr = 1 − ⅔·(RQD/100) reduces intact strength for jointing. The curve rises through a linear segment and a fourth-root power law to that ultimate.

What counts as “weak rock”

Reese published the method in 1997 (“Analysis of Laterally Loaded Piles in Weak Rock,” ASCE Journal of Geotechnical and Geoenvironmental Engineering, building on Reese & Nyman, 1978) as an interim criterion for materials that are too strong to treat as stiff clay and too weak and fractured to treat as intact rock: claystones, siltstones, weak sandstones, weathered shales, chalk — broadly, qᵤ in the 0.5–5 MPa range. Materials much stronger than that fail differently (brittle wedge breakout controlled by joint sets rather than the smeared mass behavior assumed here), and the model should not be stretched to cover them. The method was calibrated against a small number of full-scale lateral load tests on instrumented drilled shafts, which is why Reese himself flagged it as interim; two decades on it remains the standard weak-rock criterion in COM624P-family software, LPILE, and RSPile, and it is one of the seven built-in models in PileCalc's lateral pile engine.

Ultimate resistance and the αr reduction

The model starts from the intact strength qᵤ and discounts it for the fracturing of the rock mass using the Rock Quality Designation:

αr = 1 − (2/3)·(RQD/100)

so αr runs from 1.0 for completely broken rock (RQD = 0) down to ⅓ for perfect core recovery (RQD = 100). Counterintuitive at first glance — better rock gets a bigger reduction — but the logic is that highly fractured weak rock behaves like a frictional mass whose full smeared strength participates, while intact rock fails in brittle fashion and cannot be counted on for its full qᵤ in this formulation. With b the shaft diameter and zr the depth below the top of the rock (not below the ground surface — overlying soil counts only as surcharge), the ultimate resistance per unit length is:

pur = αr·qu·b·(1 + 1.4·zr/b)      near the surface
pur = 5.2·αr·qu·b                 at depth
pur = min of the two

The two expressions cross at zr = 3b: within three diameters of the rock surface a wedge of rock can displace upward and resistance grows linearly with depth; below that, the flow-around limit of 5.2·αr·qᵤ·b governs.

Initial stiffness: Kir = kir·Eir

The initial slope of the p-y curve comes from the rock-mass modulus Eir scaled by a dimensionless depth factor:

kir = 100 + 400·zr/(3b)    for zr < 3b
kir = 500                  for zr ≥ 3b

Kir = kir·Eir              (initial slope, force/length²)

Eir is the modulus of the rock mass, ideally from pressuremeter or dilatometer tests in the socket, otherwise from intact-core modulus reduced for jointing. Because kir is in the hundreds, the initial branch is extremely stiff — deflections in the linear range are measured in hundredths of a millimetre, and the working part of the curve is almost always the power-law segment.

Constructing the curve

The backbone has three segments, with y_rm = krm·b setting the deflection scale:

• Linear segment: p = Kir·y at very small deflections.
• Power-law segment: p = (pᵤr/2)·(y/y_rm)^(1/4), taking over where it undercuts the linear branch.
• Ultimate: p = pᵤr for y ≥ 16·y_rm — the deflection at which the power law reaches pᵤr, since 16^(1/4) = 2 exactly.

As with PileCalc's other piecewise models, the implementation evaluates the linear branch as a lower envelope — p = min(Kir·y, power-law capped at pᵤr) — which reproduces the segment intersections without solving for them. The constant krm is the strain at 50% strength mobilization, analogous to ε₅₀ in clay models; Reese recommends 0.0005 down to 0.00005. It is the least-constrained parameter in the model: a smaller krm shifts the whole curve left (stiffer response at a given deflection), so when the socket response governs a design it is worth running the bounds.

Parameters at a glance

A worked mini-example

A 0.9 m drilled shaft socketed into weak rock: qᵤ = 2.0 MPa, RQD = 50%, Eir = 200 MPa, krm = 0.0005, evaluated at zr = 1.8 m below the rock surface (zr/b = 2). The reduction is αr = 1 − ⅔·(0.5) = 0.667. The near-surface expression gives pᵤr = 0.667·2000·0.9·(1 + 1.4·2) ≈ 4,560 kN/m; the deep cap is 5.2·0.667·2000·0.9 ≈ 6,240 kN/m, so the shallow value governs. The stiffness factor is kir = 100 + 400·(1.8)/(3·0.9) ≈ 367, giving Kir ≈ 73,300 MN/m² — so the linear branch hands off to the power law at about 0.01 mm. With y_rm = 0.0005·0.9 m = 0.45 mm, the curve gives p ≈ 2,280 kN/m at 0.45 mm (half the ultimate, by construction), ≈ 3,310 kN/m at 2 mm, and the full 4,560 kN/m at 16·y_rm = 7.2 mm. Note the scale: resistances are tens of times higher than a sand or clay curve, mobilized at millimetres — this is why even short rock sockets dominate the lateral response of a shaft. You can reproduce the curve in the free p-y curve generator.

Limitations

Reese labeled the method interim for a reason: its experimental base is a handful of instrumented load tests, far thinner than the sand and clay criteria enjoy. It is not applicable to strong rock (qᵤ well above ~5 MPa) or to rock whose behavior is controlled by discrete joint sets, steeply dipping bedding, or karst — those demand wedge analyses on the actual discontinuity geometry, or the vuggy-limestone/massive-rock criteria that some programs offer. The RQD-based αr is a crude proxy for mass behavior; where RQD and observed mass quality disagree, judgment beats the formula. And the p-y springs describe lateral resistance only — axial side shear and end bearing of a rock socket follow entirely different mechanics (Horvath & Kenney-type socket friction, settlement-based mobilization), covered in our drilled shaft documentation. Deflections predicted in the linear range are small enough that structural stiffness of the shaft — including cracking, see moment–curvature — often matters more than the rock parameters.

Common questions

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

pᵤr = αr·qᵤ·b·(1 + 1.4·zr/b) near the rock surface, capped at 5.2·αr·qᵤ·b for depths beyond three diameters below the top of rock, where qᵤ is the intact compressive strength, b the diameter, zr the depth into the rock, and αr = 1 − ⅔·(RQD/100) the strength reduction for jointing.

Why does higher RQD reduce the resistance?

Because αr = 1 − ⅔·(RQD/100) discounts intact strength, not mass strength. Highly fractured weak rock deforms as a frictional mass and mobilizes its full smeared strength, while intact rock fails in brittle fashion and cannot be relied on for its full qᵤ in this formulation. At RQD = 100 the reduction bottoms out at αr = ⅓.

What is krm and what value should I use?

krm is a strain factor, analogous to ε₅₀ in clay: y_rm = krm·b sets the deflection scale of the curve, half the ultimate is mobilized at y_rm, and the full ultimate at 16·y_rm. Reese recommends 0.0005 to 0.00005; PileCalc defaults to 0.0005 (the softer end). If the socket governs your design, analyze both bounds.

Can I use this model for strong rock?

No. Above roughly qᵤ = 5 MPa, behavior is controlled by brittle fracture and discrete joint geometry rather than the smeared mass response assumed here, and the interim criterion has no calibration there. Strong-rock and massive-rock p-y criteria (e.g. Hoek–Brown-based wedge models) exist in some programs; for jointed rock, a discontinuity-based wedge analysis is more defensible.

Generate weak rock p-y curves for your socket in the free p-y curve generator, or analyze the full shaft — soil layers, rock socket, nonlinear section — in the PileCalc app.