The Welch & Reese (1972) stiff clay p-y model (above the water table)

The Welch & Reese stiff clay p-y model (1972) defines the lateral resistance of stiff clay above the water table — dry or moist clay with no free water at the pile face. It keeps Matlock's ultimate resistance envelope but uses a stiffer quarter-power curve, p = 0.5·pᵤ·(y/y₅₀)^(1/4) with y₅₀ = 2.5·ε₅₀·b, reaching pᵤ at 16·y₅₀ with no post-peak loss.

Where the model comes from

Welch and Reese derived the criteria from static and cyclic lateral load tests on an instrumented reinforced-concrete drilled shaft, roughly 0.76 m (30 in) in diameter, at a Houston, Texas site in stiff clay with no free water present. The work was published as Lateral Load Behavior of Drilled Shafts (Welch & Reese, 1972, University of Texas at Austin). Two observations drove the formulation. First, dry stiff clay loads up much more stiffly than soft clay but is still ductile — the measured resistance kept climbing to large deflections with no brittle drop. Second, cyclic loading did not destroy resistance the way it does in submerged clay; it simply accumulated deflection as a function of the number of cycles. Both observations are baked directly into the equations below.

When to use it — and when not to

Use Welch & Reese for stiff clay above the water table: no free water at the pile, undrained strength typically 50 kPa and up. The distinction from the Reese, Cox & Koop (1975) model is not strength but water. Below the water table, stiff fissured clay behaves brittlely — a gap opens, free water is forced in and out of it, and resistance degrades sharply past the peak. Above the water table that mechanism is absent, and the response stays ductile. Choosing between the two stiff-clay models is therefore a question about the groundwater, not the consistency. For soft to medium submerged clay, use the Matlock (1970) criteria instead.

Ultimate resistance

The ultimate resistance envelope is the same wedge / flow-around pair Matlock proposed, with undrained strength c, effective overburden built from the unit weight γ, depth z, diameter b, and the empirical factor J (default 0.5):

pᵤ = min[ (3 + γ·z/c + J·z/b) · c · b ,  9 · c · b ]

Because c is large for stiff clay, the 9cb flow-around cap sits deep and the shallow wedge governs over most of the length that matters for lateral response. Above the water table the total unit weight applies (there is no buoyancy to subtract).

The static curve: a quarter-power backbone

p = 0.5 · pᵤ · (y/y₅₀)^(1/4),   y₅₀ = 2.5 · ε₅₀ · b
p = pᵤ  for  y ≥ 16·y₅₀

Compare the exponents: Matlock's soft clay uses (y/y₅₀)^(1/3), Welch & Reese use (y/y₅₀)^(1/4). The smaller exponent makes the curve rise faster at small deflections and flatten sooner in shape — the signature of a stiff soil — while pushing the deflection at full mobilization out to 16·y₅₀ (versus Matlock's 8·y₅₀). Since ε₅₀ for stiff clay is also several times smaller than for soft clay, y₅₀ itself is small and the curve is stiff in absolute terms too. There is no descending branch: like the soft-clay model, and unlike the submerged stiff-clay model, the static curve never sheds resistance.

Cyclic loading: deflection accumulates with N

Welch & Reese observed that cycling shifts the curve to larger deflections rather than cutting its peak. After N cycles, the deflection at a given resistance level is

y_cyclic = y_static + 9.6 · y₅₀ · (p/pᵤ)⁴ · log₁₀N

Substituting the static backbone (y_static = 16·y₅₀·(p/pᵤ)⁴ at resistance p) gives the closed form PileCalc evaluates:

p = pᵤ · ( y / y_max )^(1/4),   y_max = y₅₀ · (16 + 9.6·log₁₀N)

The cyclic curve is the static curve stretched horizontally by the factor (16 + 9.6·log₁₀N)/16 — about 1.6× at N = 100 — with pᵤ unchanged. This is the crucial contrast with both Matlock and Reese-Cox-Koop: no strength is lost, only stiffness. It is also the only one of the three clay models where the cycle count N is an explicit input.

Choosing ε₅₀

As in the soft-clay model, ε₅₀ is the triaxial strain at half the maximum principal stress difference, and it sets y₅₀ directly. Measured values are best; absent lab data, the commonly reproduced recommendations for stiff clay are:

Average undrained strength c (kPa)	ε₅₀
50–100	0.007
100–200	0.005
200–400	0.004

Worked example

A 0.6 m pile in stiff clay above the water table: c = 100 kPa, γ = 19 kN/m³, ε₅₀ = 0.005, J = 0.5, at depth z = 2 m:

wedge:  pᵤ = (3 + 19·2/100 + 0.5·2/0.6) · 100 · 0.6
           = (3 + 0.38 + 1.67) · 60 = 302.8 kN/m
flow:   pᵤ = 9 · 100 · 0.6 = 540 kN/m   →   pᵤ = 302.8 kN/m

y₅₀ = 2.5 · 0.005 · 0.6 = 0.0075 m = 7.5 mm
static, y = 5 mm:  p = 0.5 · 302.8 · (5/7.5)^(1/4) = 136.8 kN/m
cyclic, N = 100:   y_max = 7.5 · (16 + 9.6·2) = 264 mm
                   p = 302.8 · (5/264)^(1/4) = 112.3 kN/m

At 5 mm the soil already carries 45% of ultimate — an order of magnitude stiffer response than the soft-clay example at the same relative deflection. A hundred load cycles soften that to 37%, with the ultimate untouched.

How well does the implementation check out?

PileCalc's Welch-Reese implementation is verified term-by-term against the published equations and benchmarked against LPILE: on the standard single-layer dry stiff clay problem (0.5 m pile, c = 100 kPa, ε₅₀ = 0.005, 100 kN head load), PileCalc computes a head deflection of 0.87 mm versus LPILE's 0.85 mm, with maximum moment 99 vs 100 kN·m — within chart-reading tolerance. The full benchmark set is published at /docs/validation.

Limitations

Water table position is load-bearing. If the water table can rise into the zone of lateral response over the structure's life, the submerged (brittle) model may govern; check both.
Single-site calibration. One drilled shaft in Houston clay underlies the criteria; decades of use support them, but they remain empirical.
Cyclic form needs N. The log₁₀N accumulation was calibrated to the test program's cycle counts; extrapolating to millions of cycles (machine foundations) outruns the data.
No brittle drop. If your stiff clay is heavily fissured and wet — even perched water counts — the ductile curve can overpredict post-peak resistance.

Common questions

Welch-Reese vs Reese-Cox-Koop — which stiff clay model do I use?

Choose by groundwater, not strength. Welch & Reese (1972) applies to stiff clay above the water table and gives a ductile quarter-power curve with no strength loss. Reese, Cox & Koop (1975) applies below the water table, where free water in the pile-soil gap makes the response brittle, with sharp post-peak degradation. If the water table may rise, analyze both cases.

What makes the Welch-Reese curve "stiffer" than Matlock's?

Two things: the exponent and the parameter. The quarter-power (y/y₅₀)^(1/4) rises faster at small deflections than Matlock's cube root, and stiff clay's ε₅₀ (≈0.004–0.007) is a third or less of soft clay's (≈0.02), so y₅₀ = 2.5·ε₅₀·b is proportionally smaller. Both effects concentrate resistance at small deflections.

How does cyclic loading affect stiff clay above the water table?

It accumulates deflection without destroying resistance. Welch & Reese express this as y_cyclic = y_static + 9.6·y₅₀·(p/pᵤ)⁴·log₁₀N: each tenfold increase in cycle count adds a fixed increment of deflection, largest near ultimate resistance. The ultimate pᵤ itself is unchanged — unlike submerged clay, where cycling cuts the peak.

What pile types was the model developed for?

The calibration test was a bored pile — an instrumented drilled shaft about 0.76 m in diameter. The criteria are routinely applied to driven piles in dry stiff clay as well, since the p-y formulation depends on soil properties and diameter rather than installation method, but that extension is by practice rather than by the original test data.

To see the quarter-power shape and the log₁₀N stretch for your own soil parameters, plot the curves in the free p-y curve generator — or run the full pile with it in the lateral pile design workflow.