Reese, Cox & Koop (1975) stiff clay p-y model (below the water table)

The Reese, Cox & Koop stiff clay p-y model (1975) defines the lateral resistance of stiff clay below the water table. Ultimate resistance is p_c = min[2cb + γ′bz + 2.83cz, 11cb], the initial stiffness is a straight line of slope k·z, and — uniquely among the standard clay criteria — the curve degrades sharply past its peak to a low residual, reflecting the brittle behavior of submerged stiff fissured clay.

Where the model comes from

Reese, Cox and Koop developed the criteria from static and cyclic lateral load tests on instrumented steel pipe piles (roughly 0.6 m diameter) at a site near Manor, Texas, in heavily overconsolidated stiff fissured clay, published as Field Testing and Analysis of Laterally Loaded Piles in Stiff Clay, Offshore Technology Conference paper OTC 2312 (1975). The companion paper to their 1974 sand criteria, it completed the original COM624 family of soil models. The Manor data showed something the Matlock soft clay and Welch-Reese dry stiff clay tests had not: measured resistance that peaked and then fell as deflection grew.

Why submerged stiff clay is brittle

Stiff overconsolidated clay is typically fissured. Under lateral load a gap opens behind the pile and, below the water table, free water is drawn into it. As the pile cycles — or simply pushes on — water is jetted in and out of the gap and along the fissures, softening and eroding the clay at exactly the depths where resistance matters most. The result is a p-y curve with a genuine post-peak drop: resistance climbs steeply (stiff clay), peaks early, and degrades toward a residual that can be a small fraction of the peak. Above the water table this mechanism is absent, which is why the ductile Welch & Reese (1972) model applies there instead. The choice between the two stiff-clay models is a groundwater question, not a strength question.

Ultimate resistance

The theoretical ultimate p_c is the lesser of a shallow wedge and a deep flow-around mechanism (the paper's Eqs. 11–12), with average undrained strength c, effective unit weight γ′, depth z, and diameter b:

p_ct = 2·c·b + γ′·b·z + 2.83·c·z     (shallow wedge)
p_cd = 11·c·b                        (deep flow-around)
p_c  = min(p_ct, p_cd)

The crossover depth is z = 9cb/(γ′b + 2.83c) — much shallower than in Matlock's model, because the 2.83c·z term grows so fast. Note that p_c is a theoretical ultimate: the actual peak of the curve is scaled down by the empirical coefficient A below.

y₅₀, the A coefficient, and the k constants

Three definitions trip people up when moving from the other clay models to this one:

• y₅₀ = ε₅₀·b — without the 2.5 factor used by Matlock and Welch-Reese. ε₅₀ is the usual triaxial strain at half the maximum stress difference (typically 0.007 / 0.005 / 0.004 for c of 50–100 / 100–200 / 200–400 kPa).
• A is an empirical coefficient, published only as a chart against depth ratio z/b, that scales the peak of the curve. The static curve A_s runs from about 0.2 at the surface to a deep asymptote of 0.60 by z/b ≈ 4; the cyclic A_c runs from the same ≈0.2 origin to 0.30 by z/b ≈ 2. PileCalc ships the exact numeric A tables recovered from the official COM624P v2.0 program (static surface value 0.223) and lets you override them with exact per-layer values for design work.
• k (k_s static, k_c cyclic) is the initial modulus of subgrade reaction: the p-y curve starts as the straight line p = (k·z)·y, applied as an upper bound on the early curve. Use the value matching your load type.

The representative k values from the criteria, as reproduced in the LPILE and RSPile manuals:

The static curve, segment by segment

The static backbone is a five-part piecewise curve. With y₅₀ = ε₅₀·b and A_s from the chart:

1. initial line:  p = (k_s·z)·y                    (governs until it meets the parabola)
2. parabola:      p = 0.5·p_c·(y/y₅₀)^(1/2)
3. offset branch, y > A_s·y₅₀:
      p = 0.5·p_c·(y/y₅₀)^(1/2) − 0.055·p_c·[(y − A_s·y₅₀)/(A_s·y₅₀)]^1.25
4. straight decline, 6·A_s·y₅₀ < y ≤ 18·A_s·y₅₀:
      p = 0.5·p_c·(6·A_s)^(1/2) − 0.411·p_c − (0.0625·p_c/y₅₀)·(y − 6·A_s·y₅₀)
5. residual plateau, y > 18·A_s·y₅₀:
      p = p_c·(1.2247·√A_s − 0.411 − 0.75·A_s)

The curve peaks within the offset branch and then sheds resistance — first gradually, then along the straight decline — before flattening at the residual. For deep elements (A_s = 0.60) that residual is only about 0.09·p_c: past roughly 18·A_s·y₅₀ of deflection, submerged stiff clay retains under a tenth of its theoretical ultimate. No other model in the standard COM624P set degrades this hard under static load.

The cyclic curve

Cycling makes it worse. The cyclic curve peaks at A_c·p_c (A_c ≤ 0.30 — half the static coefficient at depth) at a characteristic deflection y_p = 4.1·A_c·y₅₀:

y ≤ 0.6·y_p:        p = A_c·p_c·[1 − |(y − 0.45·y_p)/(0.45·y_p)|^2.5]
0.6·y_p → 1.8·y_p:  p = 0.936·A_c·p_c − (0.085·p_c/y₅₀)·(y − 0.6·y_p)
y > 1.8·y_p:        p = 0.936·A_c·p_c − (0.102·p_c/y₅₀)·y_p

with the initial line p = (k_c·z)·y — using the lower cyclic k — again bounding the start. There is no cycle-count parameter: like Matlock's cyclic criteria, this is an envelope for many cycles.

Worked example

A 0.6 m pile in submerged stiff clay: c = 100 kPa, γ′ = 9 kN/m³, ε₅₀ = 0.005, k_s = 135 MN/m³, static loading, at depth z = 3 m:

wedge:  p_ct = 2·100·0.6 + 9·0.6·3 + 2.83·100·3 = 985 kN/m
flow:   p_cd = 11·100·0.6 = 660 kN/m   →   p_c = 660 kN/m

y₅₀ = 0.005 · 0.6 = 0.003 m = 3 mm
z/b = 3/0.6 = 5  →  A_s = 0.60
initial slope k_s·z = 135,000 · 3 = 405,000 kN/m²  (governs to y ≈ 0.22 mm)

at y = 5 mm (offset branch, since y > A_s·y₅₀ = 1.8 mm):
  p = 0.5·660·(5/3)^(1/2) − 0.055·660·[(5 − 1.8)/1.8]^1.25
    = 426.0 − 74.5 = 351.5 kN/m

residual (y > 18·A_s·y₅₀ = 32.4 mm):
  p = 660·(1.2247·√0.60 − 0.411 − 0.75·0.60) ≈ 58 kN/m

The same element loaded cyclically (A_c = 0.30, k_c = 55 MN/m³) would peak at just A_c·p_c = 198 kN/m near y = 1.7 mm and settle to a plateau of about 102 kN/m by 6.6 mm. PileCalc evaluates exactly these expressions at every node of the lateral pile solution, and the implementation is checked term-by-term against the RSPile theory manual's equations — see how we validate.

Limitations

• Chart-based coefficients. A_s and A_c were never published as numbers, only as figures, so implementations historically digitized them. PileCalc uses the exact tables recovered from the official COM624P v2.0 program and accepts exact user-supplied values per layer.
• Sensitivity to k. The initial slope k·z controls small-deflection response, and the recommended values span a factor of four over the strength range. Where serviceability governs, bracket k.
• Severe residuals. The static residual near 0.09·p_c at depth is calibrated to Manor's fissured clay; in intact, lightly fissured clays it may be conservative. It is not a reason to switch models — it is the model's point.
• Single-site origin. Like the other classical criteria, the model comes from one test program, extended by decades of practice.

Common questions

When do I use Reese-Cox-Koop instead of Welch-Reese?

Use Reese, Cox & Koop (1975) when stiff clay lies below the water table, where free water in the pile-soil gap makes the response brittle with sharp post-peak loss. Use Welch & Reese (1972) above the water table, where the response stays ductile. If the water table could rise into the upper pile length during the structure's life, run both and design for the worse case.

What are the A_s and A_c coefficients?

Empirical scale factors on the theoretical ultimate p_c, published as chart curves against depth ratio z/b. Both start near 0.2 at the ground surface; the static A_s rises to a deep asymptote of 0.60 by z/b ≈ 4, while the cyclic A_c reaches only 0.30 by z/b ≈ 2 — quantifying how much less resistance cycling leaves in submerged stiff clay.

What k value should I use for stiff clay below the water table?

The criteria recommend an initial modulus of subgrade reaction tied to strength and load type: static k_s ≈ 135 / 270 / 540 MN/m³ and cyclic k_c ≈ 55 / 110 / 220 MN/m³ for average undrained strengths of 50–100 / 100–200 / 200–400 kPa. The curve starts along p = (k·z)·y, so k directly controls predicted deflections at working loads.

Why is y₅₀ different in this model?

Reese, Cox & Koop define y₅₀ = ε₅₀·b, without the 2.5 factor that Matlock and Welch-Reese apply (y₅₀ = 2.5·ε₅₀·b). Using the wrong definition stretches or compresses every segment of the piecewise curve, so it is one of the most common sources of disagreement when hand-checking software output against the paper.

The fastest way to see the brittle shape — peak, decline, residual — is to plot it: the free p-y curve generator draws the static and cyclic Reese-Cox-Koop curves at any depth from your c, ε₅₀, and k, using the same validated engine as the full PileCalc lateral analysis.