Foundation stiffness (springs)

The foundation stiffness tool evaluates a pile at one explicit service state. It returns a signed reciprocal 2×2 lateral–rotation tangent matrix from the nonlinear p-y solver, together with vertical secant and local tangent stiffness from the t-z/q-z load–settlement response. It does not hide a load-independent spring behind a default load.

The declared service state

Supply downward compression Q_s, signed head shear V_s, and signed head momentM_s. The vertical load is used both to locate the Q–s operating point and as the beam-column axial compression in every lateral solve. Zero shear or zero moment is valid; a derivative remains defined even when the corresponding component ratio is not.

One state, two kinds of stiffness

Tangent stiffness predicts a small increment about the service state. Secant stiffness reproduces the total response from the origin to that state. The response names both definitions explicitly.

Signed tangent matrix

Five p-y runs are required: the service state, V_s ± ΔV, and M_s ± ΔM. Their centered differences form the flexibility Jacobian. Clockwise head rotation is defined asθ = -dy/dz, making moment and rotation a work-conjugate pair:

[dy, dθ]ᵀ = Fₜ [dV, dM]ᵀ · Kₜ = Fₜ⁻¹

Service-state lateral flexibility and stiffness

The two raw coupling flexibilities ∂y/∂M and ∂θ/∂V must agree within the declared 2% numerical tolerance. Only then are they averaged and the matrix inverted. The published terms K_yθ and K_θy retain their sign and are exactly reciprocal. A nonconverged perturbation, nonpositive diagonal flexibility, singular matrix, or failed reciprocity check aborts the analysis; there is no partial fallback.

Do not take absolute values

The coupling sign depends on the documented clockwise-positive rotation convention. Taking absolute values or combining independent V/y and M/θ ratios produces a different, non-work-conjugate support model.

Vertical secant and tangent

K_z,sec = Q_s / s_s · K_z,tan = dQ/ds |_s

Vertical service-state definitions

The service load must lie on the computed loading branch; extrapolation is rejected. The secant uses the interpolated settlement s_s. The tangent is the slope of the bracketing t-z/q-z load–settlement segment. The axial integration and curve sampling are run at a fine mesh and at half resolution. Settlement and secant stiffness must agree within 3%, and local tangent stiffness within 5%, or the result fails closed.

Inputs and applicability

L, D, EI, EOne physical pile

The axial and lateral objects must have identical length, section bounds, and external diameters. The present method supports a straight vertical pile with its head at ground. Tapered lateral sections are blocked because the axial schema cannot represent the same geometry unambiguously.

Qₛ, Vₛ, MₛService loads

A positive vertical service load is mandatory. Shear and moment may be signed or zero. Tangent steps must be positive and no more than 5% of the matching action. At a zero lateral action, Qₛ is the shear reference and QₛD is the moment reference.

p-y, t-z/q-zTwo explicit soil profiles

The lateral profile selects the p-y models; the axial profile supplies shaft and tip transfer. Both must start at depth zero, remain ordered and contiguous, and cover the full pile. Groundwater requires an explicit water unit weight. Cyclic/history-dependent p-y response is outside this static tangent method and is rejected.

Reading the results

Kyy, Kyθ, Kθy, Kθθ — the signed lateral–rotation tangent block. Use the four terms together when the structural program accepts coupled support stiffness.
Kz,secant — reproduces total vertical service settlement; Kz,tangent— predicts a small incremental vertical response about that load.
Apparent V/y and M/θ are diagnostics only. With simultaneous shear and moment they contain coupling and are not uncoupled spring coefficients.
Diagnostics preserve all five lateral convergence records, raw reciprocity mismatch, vertical fine/half meshes, and the three refinement differences.

Axial–lateral cross-derivatives are not included

Vertical compression participates in the lateral P-delta state, but this response does not calculate ∂V/∂s, ∂M/∂s, ∂Q/∂y, or ∂Q/∂θ. Do not present the lateral block plus Kz as a fully coupled 3×3 tangent matrix.

Validation status

Method ID method.stiffness.coupled-foundation-springs is validated: the signed lateral matrix is gated against the independent semi-infinite homogeneous Winkler solution published by Crispin and Mylonakis (2022), including reciprocity, SI/US twins, elastic load scaling, and mesh refinement. FHWA-HIF-18-046, the 2019 Manual for Refined Analysis in Bridge Design and Evaluation, supplies the public foundation-stiffness guidance; FHWA GEC 12 supplies the axial load-transfer basis.

See the API contract for exact field names and units.