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 offers two methods. Linear mode 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.

Nonlinear compression path

The nonlinear mode follows the load-path procedure in Ensoft LPILE User Manual 2026 §3.14.7. Constant head shear and moment (or zero head rotation) act on an initially straight elastic pile. Constant axial compression increases from zero through the declared increments, using the preceding converged deflection field as the next initial state. The physical tip is free.

All nonzero load points enter the least-squares fit. Static Reese/API sand, Matlock soft clay, and elastic layers are supported. Supply axial and moment limits valid for the elastic section over the entire compression range. Nonlinear section stiffness, P–M capacity, cyclic degradation, geometric initial curvature and post-buckling behavior are outside this mode.

A failed or reversed load step invalidates the estimate. Material-limit crossings, rotations above 0.1 rad, deflections above 10% of pile length, fewer than ten requested compression increments, amplification below two, load-fit RMS error above 5%, or fine/coarse path or estimate differences above 3% also reject. A separate load-step check compares N with 2N increments at the same fine spatial mesh: estimate and shared-point displacement changes must each be at most 1%. The returned path and fit use the refined 2N grid (at most 200 increments). These numerical acceptance gates do not certify physical stability at the extrapolated estimate.

The response includes the full path, fit error, mesh comparison and supplied material limits. Its fitted asymptote is not an exact critical load or design resistance. In the official Example 4a numeric output, the fitted asymptote is 22,606.92 kN; the manual figure's 19,592 kN is the last computed thrust after an unsuccessful next step. PileCalc does not label that failure cutoff as a critical load. The fitter is checked against the source's numerical path. With the sand modulus recovered from independently printed local spring values, all 49 source head-displacement points agree within 0.5%. The complete solver is also checked against an independently derived elastic beam solution.

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.