Civil, Structural & Geotechnical Engineering
Matlock (1970) Soft-Clay Lateral Pile p-y Curve Formula: N_p, y_50 & Cubic Spring
Generate nonlinear lateral soil-pile spring resistance p(y) (kN/m) in submerged soft clay using Hudson Matlock’s (1970, API RP 2GEO / LPILE) bearing factor N_p = MIN(9, 3 + gamma_prime*z/c_u + J*z/D), y_50 = 2.5*eps_50*D, and cubic parabola.
Model nonlinear soil yielding around offshore jackets, wind monopiles, and bridge shafts!
Linear Winkler springs overpredict pile stiffness at working lateral deflections! Standardized in API RP 2GEO, ISO 19901-4, and LPILE from Sabine, Texas field pile tests by Hudson Matlock (1970, OTC 1204), the Soft-Clay Static Lateral p-y Curve defines Ultimate Lateral Resistance per unit shaft length p_u = N_p * c_u * D (kN/m), where dimensionless Bearing Factor N_p transitions smoothly from shallow wedge heave (N_p = 3 at mudline z = 0) to deep full flow-around failure (N_p = 9 below transition depth z_R):
N_p = MIN(9, 3 + (gamma_prime * z) / c_u + (J * z) / D), where c_u is undrained shear strength (kPa), gamma_prime is submerged unit weight (kN/m^3), D is pile outer diameter (m), and J ~= 0.5 for soft normally consolidated clay (0.25 for stiffer clay)! Scaling lateral pile deflection y (m) by Half-Strength Reference Deflection y_50 = 2.5 * eps_50 * D (where eps_50 ~= 0.005–0.020 is axial strain at 50% peak deviator stress in UU triaxial testing) gives Matlock’s Static Cubic Parabola: p(y) = p_u * MIN(1, 0.5 * ((y / y_50)^(1/3))), reaching 0.50 * p_u at y = y_50 and full plastic plateau 1.00 * p_u at y = 8 * y_50!
| Lateral Deflection Ratio y / y_50 (D = 1.0 m, eps_50 = 0.010 -> y_50 = 0.025 m) | Deflection y (mm) | Mobilized Fraction p / p_u | Soil Reaction p (at p_u = 285 kN/m) |
|---|---|---|---|
| Small Working Load (y / y_50 = 0.125) | 3.125 mm | 0.5 * (0.125)^(1/3) = 0.250 | 71.25 kN/m |
| Half-Capacity Reference (y / y_50 = 1.00) | 25.00 mm | 0.5 * (1.00)^(1/3) = 0.500 | 142.50 kN/m |
| Full Plastic Flow Plateau (y / y_50 >= 8.00) | >= 200.0 mm | 1.000 (Capped) | 285.00 kN/m |
Matlock Ultimate Lateral Clay Resistance p_u (kN/m) at Depth z
Evaluates MIN(9, 3 + (gamma_prime * z) / c_u + (J * z) / D) * c_u * D across depth z >= 0 (B2 m), pile diameter D (B3 m), undrained shear strength c_u (B4 kPa), submerged unit weight gamma_prime (B5 kN/m^3), and empirical J (B6, e.g., 0.5).
=IF(OR(B2 < 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Depth, Diameter D, Shear Strength c_u, gamma_prime, or J", ROUND(MIN(9, 3 + (B5 * B2) / B4 + (B6 * B2) / B3) * B4 * B3, 2))Mobilized Static p-y Spring Resistance p(y) (kN/m) at Deflection y
Computes y_50 = 2.5 * eps_50 * D (with eps_50 in B7) and evaluates p_u * MIN(1, 0.5 * ((y / y_50)^(1/3))) for lateral deflection y >= 0 (B8 m).
=IF(OR(B2 < 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0, B7 <= 0, B8 < 0), "Invalid Matlock p-y Inputs", ROUND((MIN(9, 3 + (B5 * B2) / B4 + (B6 * B2) / B3) * B4 * B3) * MIN(1, 0.5 * ((B8 / (2.5 * B7 * B3)) ^ (1 / 3))), 2))Try the example
Use the original values or change them to test your understanding.