Pile buckling

A pile is normally well braced by soil, but stability can matter for slender piles, micropiles, and piles with scour, water, or stick-up above grade. The buckling tool returns the first elastic bifurcation load and its signed mode for a finite, partially embedded pile. It does not infer failure from a lateral perturbation or an arbitrary amplification threshold.

Generalized eigenproblem

Cubic Hermite Euler–Bernoulli beam elements assemble depth-varying bending stiffness K_b, explicitly linear Winkler support K_s, and the unit-load geometric stiffness K_g. The smallest positive eigenvalue is the critical compressive load:

Element integration samples the actual EI and soil modulus with depth. Pile-section, ground-surface, and soil-layer interfaces are inserted as exact mesh nodes. This directly represents finite length and unsupported stick-up instead of substituting a single effective length. The implementation is checked against the ideal-column equations in NCHRP Report 343, the long-pile Winkler limit, and the partially embedded worked case published by Davisson and Robinson (1965).

The unit-load geometric matrix represents constant compression over the full pile length. An axial force that changes with depth because of shaft transfer, downdrag, or self-weight is outside this formulation and requires a separately validated stability method.

Head and tip fixity

The head and physical tip are declared independently. Do not select a fixed tip solely because the pile is long; use a restraint supported by the structural and geotechnical model.

Linear-support boundary

This endpoint accepts elastic layers whose E_py is constant or varies linearly with depth, plus API requests using a user layer with a single constant es. Nonlinear p-y models fail validation. A nonlinear curve has no unique unloaded eigenproblem until a reference equilibrium state and a tangent, initial, or secant linearization are explicitly chosen; silently using one would make P_cr load-path dependent.

A model with zero embedded support must declare enough independent end restraint to remove rigid translation and rotation. Otherwise the elastic-stiffness matrix is singular and the request fails before an eigenvalue is reported.

Inputs

• Total finite pile length from head to physical tip and the ground-surface depth below the head.
• Uniform, sectioned, or tapered pile stiffness EI; the API preserves changes with depth.
• A contiguous linear foundation profile covering the entire embedded length.
• Independent head and tip translation/rotation restraints.
• Optional mesh, eigen-residual, iteration, and mesh-comparison tolerances.

Convergence evidence

Every result includes the normalized eigenpair residual, eigenvalue change, iteration count, active degrees of freedom, and the relative P_cr difference between the requested mesh and an independent half-resolution mesh. The engine fails closed when the eigenpair or mesh does not meet its declared tolerance.

Reading the results

• Critical load P_cr is the smallest positive elastic bifurcation eigenvalue, not an allowable axial resistance.
• Mode shape contains signed lateral displacement and rotation at every node. Displacement is normalized so its largest absolute value is one; its amplitude is arbitrary.
• Convergence documents the eigen residual and mesh-refinement difference used to accept the result.